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U P C G
U NIVERSITAT P OLITÈCNICA
DE
C ATALUNYA
Programa de Doctorat
AUTOMATIZACIÓ AVANÇADA I R OBÒTICA
Tesi Doctoral
G RASP P LANNING
UNDER
TASK- SPECIFIC
C ONTACT C ONSTRAINTS
Carlos J. Rosales Gallegos
Directors:
Lluís Ros Giralt and Raúl Suárez Feijóo
Institut d’Organizació i Control de Sistemes Industrials
Novembre de 2012
Universitat Politècnica de Catalunya
Programa de doctorat:
Automatizació Avançada i Robòtica
Aquesta tesi ha estat realitzada a:
Institut de Robòtica i Informàtica Industrial, CSIC-UPC
Institut d’Organizació i Control de Sistemes Industrials, UPC
Dirigida per:
Lluís Ros Giralt and Raúl Suárez Feijóo
Realitzada per:
Carlos J. Rosales Gallegos
Compartida segons:
Creative Commons License Attribution-NonCommercial 3.0
.
To the memory of Ignacio Fresnedo Lourerio (1972-2011),
a good friend and Mechanical Engineer at heart.
G RASP P LANNING UNDER TASK- SPECIFIC
C ONTACT C ONSTRAINTS
Carlos J. Rosales Gallegos
Abstract
Several aspects have to be addressed before realizing the dream of a robotic handarm system with human-like capabilities, ranging from the consolidation of a proper
mechatronic design, to the development of precise, lightweight sensors and actuators,
to the efficient planning and control of the articular forces and motions required for
interaction with the environment. This thesis provides solution algorithms for a main
problem within the latter aspect, known as the grasp planning problem: Given a robotic
system formed by a multifinger hand attached to an arm, and an object to be grasped,
both with a known geometry and location in 3-space, determine how the hand-arm
system should be moved without colliding with itself or with the environment, in order
to firmly grasp the object in a suitable way.
Central to our algorithms is the explicit enforcement of a pre-speceified set of handobject contact constraints in the grasp configuration, imposed by the particular manipulation task to be performed with the object. This is a distinguishing feature from
other grasp planning algorithms given in the literature, where a means of ensuring
precise hand-object contact locations in the resulting grasp is usually not provided.
These conventional algorithms are fast, and nicely suited for planning grasps for pickan-place operations with the object, but not for planning grasps required for a specific
manipulation of the object, like those necessary for holding a pen, a pair of scissors,
or a jeweler’s screwdriver, for instance, when writing, cutting a paper, or turning a
screw, respectively. To be able to generate such highly-selective grasps, we assume that
a number of surface regions on the hand are to be placed in contact with a number of
corresponding regions on the object, and enforce the fulfilment of such constraints on
the obtained solutions from the very beginning, in addition to the usual constraints of
grasp restrainability, manipulability and collision avoidance.
The proposed algorithms can be applied to robotic hands of arbitrary structure,
possibly considering compliance in the joints and the contacts if desired, and they
can accommodate general patch-patch contact constraints, instead of more restrictive
contact types occasionally considered in the literature. It is worth noting, also, that
while common force-closure or manipulability indices are used to assess the quality
iii
iv
of grasps, no particular assumption is made on the mathematical properties of the
quality index to be used, so that any quality criterion can be accommodated in principle.
The algorithms have been tested and validated on numerous situations involving real
mechanical hands and typical objects, and find applications in classical or emerging contexts like service robotics, telemedicine, space exploration, prosthetics, manipulation in
hazardous environments, or human-robot interaction in general.
Acknowledgements
I would like to express my gratitude to my advisors Lluís Ros and Raúl Suárez for their
invaluable guidance. Also, to Josep M. Porta and Jan Rosell for their unconditional
support, and for leading the projects of such wonderful frameworks, the CUIK and
Kautham suites, in which this thesis relies on. I would like to thank Federico Thomas
who, together with Raúl, came with the idea of joining the worlds of grasping and
kinematics that resulted in this PhD research. I am thankful to Antonio Bicchi and
Marco Gabiccini for letting me accomplish a fruitful research stay in Pisa; Chapter 5 is
a joint work with them. I’d like to highlight the work of Alexander Perez, a friend, a coauthor, and coder of a large part of the Kautham suite; Montserrat Manubens, Léonard
Jaillet, Oriol Bohigas, friends and contributors to the CUIK suite; Patrick Grosch, a friend
who provided some of the 3D models used here as well as great ideas; and Leopold
Palomo, a friend who introduced me to the open source world and provided technical
support. Thus, I would like to acknowledge the work of the open source and free
software communities since most of the content of this document has been generated
using freely-available tools, such as Ubuntu, Debian, LaTeX, Inkscape, GIMP, Geomview,
Blender, among many others, freely-available 3D models, such as the SketchUp handarm model used in Figure 1.3 shared by Daniel Murray, and freely-available LaTex templates, such as the PhD thesis style used for this document shared by Adolfo Rodriguez,
all of them free as in free beer, and some as in freedom. I would like to acknowledge
the reviewing work of numerous researchers who have improved this work with their
valuable comments on publications. I’d like to thank my relatives for their support.
And last, but not least for sure, I am more than pleased to have met very nice friends
and colleagues at the IOC and IRI institutes, as well as countless friends aside who
collabarated in different ways and means to the development of this work.
Carlos J. Rosales Gallegos
Barcelona, Catalunya
November 7, 2012
This work has been partially supported by the Spanish Ministry of Education and Science through the
contracts:
DPI2007-63665 PROA: Grasping and Repositioning of Objects using Anthropomorphic Dexterous Manipulation: Analytic and Learning Approaches (Leader: R. Suárez).
DPI2007-60858 Analysis and Motion Planning of Complex Robotic Systems (Leader: L. Ros).
DPI2010-15446 MUMA: Multi-hand Systems for Complex Robotized Manipulation Tasks (Leader: R.
Suárez).
DPI2010-18449 CUIK++: An Extension of Branch-and-Prune Techniques for Motion Analysis and Synthesis of Complex Robotic Systems (Leader: L. Ros).
Also, by the “Comunitat de Treball dels Pirineus” under contract 2006ITT-10004, and by the Interdepartamental Research Center “E. Piaggio” from Pisa.
v
Contents
Abstract
iii
Acknowledgements
v
I Preliminaries
1
1 Introduction
1.1 Motivation . . . . . . . . .
1.2 Objectives and scope . . .
1.3 Outlook at the dissertation
1.4 Glossary . . . . . . . . . .
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2 Related Work
2.1 Grasp synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Approach path planning . . . . . . . . . . . . . . . . . . . . . . . . . . .
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II Grasp Synthesis
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3 Finding Grasp Configurations
3.1 The hand-object system . . . . . . . . . . . . . . . . . .
3.1.1 Structure of an anthropomorphic hand . . . . . .
3.1.2 Contact constraint specification . . . . . . . . . .
3.2 Kinematic equations . . . . . . . . . . . . . . . . . . . .
3.2.1 Link constraints . . . . . . . . . . . . . . . . . . .
3.2.2 Joint assembly constraints . . . . . . . . . . . . .
3.2.3 Joint limit constraints . . . . . . . . . . . . . . .
3.2.4 Contact constraints . . . . . . . . . . . . . . . . .
3.2.5 Final system of equations . . . . . . . . . . . . .
3.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . .
3.3.1 Equation expansion . . . . . . . . . . . . . . . .
3.3.2 Equation solving . . . . . . . . . . . . . . . . . .
3.3.3 Box shrinking . . . . . . . . . . . . . . . . . . . .
3.4 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Equations for the Schunk Anthropomorphic hand
3.4.2 Computed solutions . . . . . . . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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viii
CONTENTS
4 Optimizing Grasp Configurations
4.1 Kinematically-feasible grasps . . . . . . . . . .
4.2 Relevant grasps . . . . . . . . . . . . . . . . . .
4.3 Grasp quality optimization . . . . . . . . . . . .
4.3.1 Tracing the manifold of relevant grasps .
4.3.2 Evaluating the quality of relevant grasps
4.4 Test cases . . . . . . . . . . . . . . . . . . . . .
4.4.1 A planar hand . . . . . . . . . . . . . . .
4.4.2 The Schunk Anthropomorphic hand . .
4.5 Summary . . . . . . . . . . . . . . . . . . . . .
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5 Accounting for Compliance in the Grasp
5.1 Kinetostatic formulation of a grasp . . . . . . . . . . . .
5.1.1 Model description . . . . . . . . . . . . . . . . .
5.1.2 Characterizing kinematically-feasible grasps . . .
5.1.3 Characterizing restrained grasps . . . . . . . . .
5.1.4 System overview and dimension analysis . . . . .
5.2 Solution strategy . . . . . . . . . . . . . . . . . . . . . .
5.3 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 A simple planar hand grasping an ellipse . . . . .
5.3.2 THE First hand grasping an ellipsoid . . . . . . .
5.3.3 Experiments with THE First hand grasping a ball
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
III
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Approach Path Planning
6 Reducing the Hand Configuration Space
6.1 Finger motion coordination . . . . .
6.1.1 Experimental set-up . . . . .
6.1.2 Experimental protocol . . . .
6.1.3 Data analysis . . . . . . . . .
6.2 Wrist orientation constraint . . . . .
6.3 Summary . . . . . . . . . . . . . . .
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7 Finding Approach Paths
7.1 Sample generation . . . . . . . . . . . . . . . . . . . . . . .
7.2 Sample interconnection . . . . . . . . . . . . . . . . . . . .
7.3 The approach path planner . . . . . . . . . . . . . . . . . .
7.4 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Evaluation of the use of PMDs . . . . . . . . . . . . .
7.4.2 Evaluation of the use of wrist orientation constraints
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
IV Closing Remarks
ix
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8 Conclusions
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8.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 Future research directions . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A List of Publications
131
References
133
Figures
1.1
1.2
1.3
1.4
Commercially available robotic hands . . . . . . .
Alternative versus anthropomorphic designs . . .
The grasp planning problem . . . . . . . . . . . .
A comparison of our approach versus the GraspIt!
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2.1 Form- and force-closure grasps . . . . . . . . . . . . . . . . . . . . . . .
2.2 Related work on approach path planning . . . . . . . . . . . . . . . . . .
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3.1
3.2
3.3
3.4
3.5
3.6
Constraint-based specification of a grasp on a scalpel
The URR structure of a finger . . . . . . . . . . . . .
Elements intervening in a contact constraint . . . . .
Assembly constraints of revolute and universal joints
Polytope bounds within box Bc . . . . . . . . . . . .
Geometric parameters of the SA hand . . . . . . . . .
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4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
Two different grasps of a can for drink service . . . . . . . . . . . . .
Elements involved in the grasp optimization framework . . . . . . .
The continuation method on a two-dimensional manifold in 3D space
The process of chart construction . . . . . . . . . . . . . . . . . . . .
Three stages of the construction of an atlas over a sphere . . . . . . .
A simple planar hand with three fingers holding an object . . . . . .
The atlas, and the best and worst grasps on the planar hand example
The same atlas of Figure 4.8 colored with a different criterion . . . .
The same atlas of Figures. 4.8 and 4.9 colored with multiple criteria .
Optimization of a force-closure quality index for the SA hand . . . .
Optimization of a manipulability index for the SA hand . . . . . . . .
The best grasp of the SA hand under a serial combination of criteria .
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5.1
5.2
5.3
5.4
5.5
5.6
Different hand actuations system . . . . . . . . . . . . . . . . .
Kinetostatic model of a grasp . . . . . . . . . . . . . . . . . . .
Representation of a spatial spring modeling a compliant contact
Solutions obtained for a simple hand . . . . . . . . . . . . . . .
Solution obtained for the Schunk Anthropomorphic hand . . . .
Executing a solution on THE First hand . . . . . . . . . . . . . .
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6.1
6.2
6.3
6.4
6.5
6.6
Mapping between a mechanical hand and a sensorized glove . .
A human hand with a sensorized glove . . . . . . . . . . . . . .
Schematic representation of the experimental setup . . . . . . .
Correlation of joints in the hand configuration space . . . . . .
Configurations of the SA hand using two PMDs independently .
Configurations of the SA hand using combinations of two PMDs
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approach
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xii
FIGURES
6.7
6.8
6.9
6.10
Total variance covered when using an increasing number of PMDs. . .
The goal configuration used to define the point of interest . . . . . . .
The 4-dimensional submanifold that satisfies the orientation constraint
Rotation matrix R1 used to define the orientation constraint . . . . . .
. 97
. 99
. 100
. 100
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
A 2-dimensional space Xh modelled with two PMDs . . . . . . . . . .
Samples of the hand-arm system . . . . . . . . . . . . . . . . . . . .
Roadmap under construction . . . . . . . . . . . . . . . . . . . . . .
Qualitative comparison between the classical and proposed approach
Simulation and real execution of a solution path . . . . . . . . . . . .
Hand-arm goal configurations for some of the benchmark tasks . . .
Solution paths of two benchmark tasks . . . . . . . . . . . . . . . . .
Snapshots of the task execution using the orientation constraint . . .
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120
Tables
3.1 Representative hand designs adopted during the last decade . . . . . . .
3.2 Test cases and their computed solutions . . . . . . . . . . . . . . . . . .
3.3 Parameters of the SA hand . . . . . . . . . . . . . . . . . . . . . . . . . .
26
39
42
5.1 Notation used in the kinetostatic model of a grasp . . . . . . . . . . . . .
5.2 Kinematic and elastic parameters for the planar hand . . . . . . . . . . .
5.3 Kinematic and elastic parameters of THE First hand . . . . . . . . . . . .
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6.1 Correspondence between the glove sensors and the hand acuators . . . .
94
7.1
7.2
7.3
7.4
Parameters used in the approach path planner . . . . . . . . . . . . . . .
Quantitative comparison of the proposed and classical approaches . . . .
Comparison of the planner performance on several benchmark tasks . .
Computation time of the planner when considering and omitting the
orientation constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
116
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119
Part I
Preliminaries
1
Introduction
The human hand, the most versatile end-effector provided by Nature, can perform
such a variety of motions and subtle adjustments that mimicking its abilities
mechanically has been qualified as the Holy Grail of robotic end-effectors [9].
This chapter outlines the main motivations underlying the pursuit of such abilities,
delimits particular objectives and scope of this Ph.D. work, and summarizes the
structure of the dissertation.
1.1 Motivation
When Karel C̆apek first introduced the term in the 1920’s, robots were conceived as
human-like machines. However, the first industrial robots that appeared a few decades
later were still far from exhibiting anthropomorphic features or dexterous manipulation
abilities. These manipulators allowed more flexibility than hard automation machines
for large-scale production, and were able to execute a variety of tasks by programming
them, but they still required costly end-effector retoolings for each specific task, and
could not accurately perform precise movements of the payload, such as those that
might be required in an assembly or fine manipulation task. In the early 1980’s,
increasing needs of more flexible tools in the industry made robot designers think of
more versatile grippers, giving rise to the appearance of the first multifingered endeffectors inspired by the human hand. Since a robot hand has many degrees of freedom,
it provides a way to grasp a large class of objects with a single end-effector. Also, it is
usually smaller than the arm to which it is attached, which allows improving the overall
accuracy of the robot.
4
Introduction
Figure 1.1: Commercially available robotic hands by Shadow Robot Company [107]
(left) and Schunk GmbH & Co. KG [106] (right).
The price of using a mechanical hand is the complexity of the overall system. Truly
anthropomorphic designs involve from four to five fingers, and up to forty actuators
to provide agonist/antagonist motions (Figure 1.1), which complicates the kinematic
and dynamic analyses of the system substantially. Since a hand is in contact with the
object being manipulated, the kinematics and dynamics of a mechanical system with
contact constraints must be analysed, and the increased number of degrees of freedom
complicates the planning of a feasible grasp to perform a given task. It is for all of these
reasons that an anthropomorphic hand may not be the answer to every manipulation
problem. The use of end-effectors tailored to particular tasks can effectively cover many
needs in areas such as manufacturing and materials transport. Alternative end-effectors,
like classical grippers or underactuated hands, or innovative ones based on the jamming
of granular material, can achieve stable grasps using a simpler design (Figure 1.2,
top), whereas a similar grasp using a fully articulated hand typically requires the use
of feedback control and might, eventually, result in an overall decrease of the grasp
stability. Despite these difficulties, however, it seems clear that the vision of ubiquitous
robotic assistants, whether in home, the industry, or in space, will not be realized
without the ability to grasp and dexterously manipulate usual objects in human environments, for which fully-actuated hands seem to provide the ideal solution (Figure 1.2,
1.1 Motivation
5
bottom). Driven by the potential benefits of such hands, progress towards building
hands for practical applications is underway, and the first commercial hands, such as
the Shadow [107] and Schunk [106] hands, have recently come to light, allowing to
picture out an increasing number of applications in the future, in classical or emerging
contexts like service robotics, telemedicine, space exploration, prosthetis, manipulation
in hazardous environments, or human-robot interaction.
Figure 1.2: Alternative versus anthropomorphic designs. The underactuated hand from
the Yale GRAB lab [29] (top-left) and the Universal Robotic Gripper [12] (top-right)
perform stable grasps of different glasses using a single actuator. Thanks to their
multiple acturators, the Shadow hand [107] (bottom-left) and the Schunk hand [106]
(bottom-right) are able to perform more complex grasps, like those requiring the
screwing of various objects.
6
Introduction
1.2 Objectives and scope
Several aspects have to be addressed before realizing the dream of a robotic hand
with human-like capabilities, ranging from the consolidation of a proper mechatronic
design, to the development of precise, lightweight sensors and actuators, to the efficient
planning and control of the finger forces and motions required for interaction with the
envrionment. The aim of this thesis is to provide suitable solutions to a main problem
within the latter aspect, known as the grasp planning problem: Given a robotic system
formed by a multifinger hand attached to an arm, and an object to be grasped, both
in a known geometry and location in 3-space, determine how should the hand-arm
system be moved without colliding with itself or with the environment, in order to
firmly grasp the object in a suitable way, allowing to perform a specific manipulation
task (Figure 3.1).
The solution to the grasp planning problem will take the following hypotheses and
requirements into account. First, note that many of the objects and tools used on
everyday activities are designed to be grasped and manipulated in particular ways.
Consider, for instance, how a pen, a scalpel, or a jeweler’s screwdriver are held, to
properly write on a paper, cut a tissue or turn a screw, respectively. To perform these
tasks with a robotic hand, the grasp planner must be able to place the hand in contact
with the object at specific locations, allowing the manipulation to properly occur. In
accordance to such requirement, we will assume that a given set of contact constraints
is to be satisfied between predefined regions on the object and hand surfaces, including
both the finger and palm surfaces in the latter case. This is a distinguishing feature
from other existing algorithms in the literature, such as those embedded in the GraspIt!
suite [20, 37, 64] for example, where constraints of such kind are not explicitly enforced
(Figure 1.4). Second, while no particular assumption will be made on how the contact constraints are derived—they could be inferred from learned experience through
examples [77], automatically generated from contact synthesis algorithms [70, 88]
and known patterns of static prehension [47], or directly informed by a human, for
instance—these constraints will be assumed to be general patch-patch contact constraints between the hand and object surfaces, instead of the more restrictive contact
constraints assumed in the literature, so as to be able to accommodate the endless
possible shapes of objects that may be encountered in practice. Third, since robotic
1.2 Objectives and scope
7
Figure 1.3: Grasp planning: Given a robotic hand-arm system and an object (left),
determine how should the system be moved in order to grasp the object in a suitable
way, avoiding collision with itself or with the environment (right).
end-effectors may adopt a variety of designs, with a range of kinematic topologies, link
geometries, and actuation schemes, we will require our algorithms to be of sufficient
generality to be able to apply them to any particular instance of such designs. This will
entail adopting general numerical techniques for kinematic constraint solving in their
implementation, since they are the only ones that have proved to be powerful enough to
solve non-linear systems of equations of the kind that arise. Fourth, we will assume that
accurate geometric models of the object, the hand-arm system, and the environment
are available, so that the planning of motion paths can safely be made using them.
A divide-and-conquer strategy will be used to overcome the complexity of the grasp
planning problem, breaking it up into the following subproblems:
1. The grasp synthesis problem: Determine a grasp configuration satisfying a number of hand-object contact constraints, while guaranteeing the manipulability and
restrainability of the object at the same time.
2. The approach path planning problem: Determine a joint-space path allowing
the hand-arm system to place the hand in the previously-obtained grasp configuration, without colliding with itself or with the environment.
8
Introduction
Figure 1.4: While the GraspIt! suite [20, 37, 64] is nicely suited to generate random
grasps able to hold an object (left), it cannot produce specific grasps allowing a
particular manipulation of the object (right). This PhD work aims at solving the grasp
planning problem in the latter context.
A solution to the grasp planning problem will be obtained by devising modules
able to solve the previous two subproblems. The grasp synthesis module provides a
hand configuration in contact with the object, which can be fed to the approach path
planning module to generate a goal configuration for the overall hand-arm system,
using standard inverse kinematics techniques for the supporting arm. Once such a goal
configuration is available, the latter module can start the search for a joint-space path
connecting the current configuration of the system to the mentioned goal configuration.
1.3 Outlook at the dissertation
The dissertation is structured into four parts, which can be outlined as follows:
• Part I provides the motivation, objectives and scope of the work (Chapter 1), and
encloses a short account on the state of the art on the two topics related to this
thesis (Chapter 2).
• Part II presents the solution given to the grasp synthesis problem. Initially, an
algorithm is developed for solving a relaxed version of the problem. Namely,
1.4 Glossary
9
the computation of a configuration of the hand-object system that respects all
assembly constraints imposed by the hand joints, and all hand-object contact
constraints considered (Chapter 3). Next, a second algorithm is developed that,
departing from the grasp configuration computed by the previous algorithm, is
able to explore the manifold of hand configurations in contact with the object, to
find those that are the maximally restrained and manipulable. The technique is
general enough to cope with usual force-closure or manipulability metrics, combinations of such metrics, or any other metric that might be defined (Chapter 4).
Finally, the fact that both the joints and the hand-object contacts are actually
non-rigid is taken into account, and a way to synthesize grasp configurations
where such joints and contacts behave according to given compliance models is
introduced (Chapter 5). The technique is general enough to tackle the kinematic
and restrainability constraints at once, and only requires to be initialized with an
estimation of the solution, which can be computed using the techniques developed
in the preceding chapters.
• Part III presents the solution given to the approach path planning problem. Firstly,
a meaningful way to reduce the dimension of the hand configuration space is
proposed. The approach draws inspiration from how the human hand moves in
the free space to lessen the finger freedom by coordinating the joint movements,
and biases the wrist orientation towards the goal by imposing relational positioning constraints (Chapter 6). This reduction is later on exploited to develop an
algorithm that solves the approach path planning problem in an adaptive way,
adjusting the configuration space dimension where needed, according to the local
obstacle complexity encountered at each step (Chapter 7).
• Part IV, finally, draws the main conclusions of the thesis, summarizes its contributions, and enumerates points deserving further attention (Chapter 8).
1.4 Glossary
For the sake of clarity, we next define basic terminology used throughout the manuscript.
Hand: Following Salisbury and Craig [103], a hand is defined as a collection of serial
kinematic chains acting as fingers, connected to a common base acting as a palm,
10
Introduction
intended to provide an end-effector of similar versatility to that of the human
hand. The number of fingers must be at least three to grant sufficient stability.
The fingers must have at least two links acting as phalanges, articulated through
lower-pair joints. The number of joints must be at least three per finger, with at
least three actuators per finger. The joint distribution must allow the finger to
produce abduction/adduction and flexion/extension motions.
Task-specific contact constraint: A contact constraint that must necessarily be fulfilled in order to manipulate the object in a specific way. Such constraints are
assumed to be given, and take the form of general patch-patch contact constraints,
so that a surface patch defined on the hand is enforced to be in contact with
a corresponding patch defined on the object, with the normal vectors to the
patches aligned. The hand patches can be defined anywhere on the finger or
palm surfaces.
Configuration of the hand-object system: A description of the position and orientation of all bodies involved in the hand and the object, respecting the joint assembly
constraints of the hand only. In such a configuration, the hand may or may not be
in contact with the object.
Grasp: A particular configuration of the hand-object system in which the hand is in
contact with the object, but not necessarily satisfying any of the task-specific
contact constraints assumed.
Kinematically-feasible grasp: A grasp that satisfies all of the imposed task-specific
contact constraints.
Restrained grasp: A grasp that is able to firmly hold an object, and to control the
internal forces applied on the object. A restrained grasp is said to be force- or
form-closed, depending on whether such ability relies on frictional contacts or
not, respectively. Provided that the number of contacts is at least the minimum
required to hold the object, a restrained grasp is obtained when all forces applied
on the object are in equilibrium, and in agreement with the frictional model
assumed, if any. Various metrics can be defined to assess how far is a restrained
grasp from loosing this property.
1.4 Glossary
11
Manipulable grasp: A grasp that is able to perform controllable object motions along
arbitrary directions of its pose space. Again, various metrics can be defined to
assess the quality of a manipulable grasp.
Arm: A serial kinematic chain acting as a carrying device for a robotic hand. The
arm must be able to move the hand throughout large portions of its pose space,
that is, both in position and orientation. Six-revolute arms will be considered for
concreteness, but the presented techniques can be applied to general serial arms.
Approach Path: A path defined in the joint space of the hand-arm system, connecting
two points corresponding to initial and final configurations of the system. The
final configuration is assumed to be one in which the hand is directly in contact
with the object, forming a grasp, or very close to realizing such contact, forming
a so-called pre-grasp.
2
Related Work
This chapter provides a short account of previous work related to the problems
of grasp synthesis and approach path planning dealt with in this Thesis. Along
the way, the reader is pointed to related sections of the manuscript where such
techniques play an important role, or where an enhancement or alternative to
them is given.
2.1 Grasp synthesis
The earliest studies on grasping can be traced back to the 19th century, when Reuleux
defined the notions of form- and force-closure grasp [86] in the context fixturing and
machine design. A restrained grasp is one that is able to counteract external disturbances applied on the object, and it is force- or form- closed, depending on whether such
ability relies on frictional contacts or not, respectively (Figure 2.1). In the frictionless
case, Reuleaux [86] and Somov [110] showed that a minimum of four (resp., seven)
contact points were necessary to achieve closure of 2D (resp., 3D) objects. About a
century later, with the arrival of robot hands, these results were revisited [28, 66],
and Markenscoff et al. [62] proved that, under very general conditions, four and seven
contact points were also sufficient to respectively generate 2D and 3D form-closed
grasps [62]. They also showed that when Coulomb friction is taken into account, three
fingers are sufficient in the 2D case, and four in the 3D case. Although these results did
not provide constructive procedures for synthesizing proper contact points, they laid the
14
Related Work
Figure 2.1: Whereas a force-closed grasp relies on friction (left), a form-closed grasp
relies on shape to restrain the object (right). [Images courtesy of the Human Grasping
Database [31].]
foundations for finding them. From that point, the external wrench applied to the object
was related to the forces applied at the contacts through the so-called grasp matrix, and
qualitative tests were provided to determine whether a given set of wrenches restrained
an object [102], which naturally led to quantitative tests [33, 89, 116]. By using these
tests, numerous approaches have been developed for synthesizing contact points in
2D [22, 58, 70, 73] and 3D [10, 56, 77, 78, 87] objects.
In parallel, another issue considered has been the determination of the contact forces
and joint torques required to optimally balance a given external force applied on the
object [4, 53, 124]. Since the contact forces are related to the joint torques through
the hand Jacobian, by adjusting such torques it is possible to achieve a set of contact
forces that lie inside the friction cones of the contact points. Assuming that the overall
configuration of the hand-object system is known, both the grasp matrix and the hand
Jacobian are known, and by linearizing the cone constraints, it is possible to rapidly
determine which joint torques allow to grasp the object with maximum stability, by
resorting to Linear Programming techniques [18, 49]. Solutions that avoid the latter
linearization have been given as well, but at the expense of using more costly nonLinear Programming techniques [13, 16, 122, 125]. The connections of the problem to
the force analysis of walking robots [72, 120], multi-arm manipulation systems [54], or
tensegrity frameworks [46], have also been investigated.
2.1 Grasp synthesis
15
All of the works just mentioned provide useful insights into key aspects of grasp
planning, but the resulting algorithms are of limited applicability in practice, because
the hand configuration in contact with the object is either neglected, despite being
fundamental to the force-closure concept adapted to grasping [5], or assumed to be
given, but never synthesized in any case. Recently, a more comprehensive approach to
the problem has been attempted, which emphasizes the computation of hand configurations in contact with the object from the very beginning, in addition to searching
for an optimal grasp [20]. The main difficulty in this case is that the set of hand
configurations in contact with the object is a complex manifold, implicitly defined by a
system of non-linear equations that expresses all joint-assembly and contact constraints
involved in the hand-object system. To avoid this complexity, Ciocarlie and Allen [20]
initially relax the contact constraints, resulting in a search space that coincides with
the configuration space of the hand. Typically, this space is of a high dimension, but
meaningful simplifications derived from the use of hand postural synergies [104] can
be introduced to narrow the search to a subspace explorable in reasonable times, using
simulated annealing. The hand configurations obtained, however, are not exactly in
contact with the object and, thus, they must be evaluated with pre-grasp quality indices,
which involves purely geometric features only. Unfortunately, a grasp qualified as good
using these indices does not always result in a high-quality grasp once the contact with
the object is finally enforced using local techniques, or by simply closing the fingers
until contact is achieved. The final hand-object contacts, moreover, are not necessarily
suitable for a specific task and, hence, the technique is nicely suited to generate random
grasps able to hold an object, but not those grasps allowing a particular manipulation
of the object.
To generate a specific grasp for a given task, both the joint-assembly constraints of
the hand and the contact constraints imposed by the task need to be enforced simultaneously. The problem to be solved to this end can be stated as follows. Given a
number of regions on the surface of the hand, and a number of corresponding regions
on the surface of the object, determine how the hand should be configured relative to
the object, so that each hand region establishes contact with its corresponding object
region. This problem has mostly been addressed with local search methods to date.
Examples of such methods include those proposed by Borst et al. [11], who cast the
problem into one of unconstrained optimization, where the constraints are enforced
using penalty terms in the objective function, Gorce and Rezzoug [38], who rely on a
16
Related Work
neural network to learn the finger inverse kinematics, and on reinforcement learning to
optimize the pose of the hand, and Rosell et al. [95], who propose an iterative method
to compute joint displacements that maximally reduce the distance from the fingertips
to the desired contact points. Such methods are usually fast and return solutions in
many cases, but their convergence is not always guaranteed, even if a solution exists.
Some of the methods, moreover, require a sufficiently-good initial estimation of the
solution [11, 95], which might not always be available.
A way around such limitations will be worked out in Chapter 3, by proposing a new
algorithm of guaranteed convergence; i.e., one that always provides a hand configuration satisfying the required contact constraints whenever one exists. This algorithm does
not require an initial estimation of the solution and can, in fact, solve a superclass of the
configuration problems dealt in [11, 38, 95], because all contact constraints considered
in such works can be seen as particular cases of more general ones tractable herein.
Regardless of the adopted method to compute a grasp, it is worth noting that the
grasp is usually not optimized in terms of any quality criterion, so that a final optimzation process is needed to obtain a suitable high-quality grasp. Implementing such
a process is not trivial though, since trying to optimize the grasp in a generate-andtest fashion is computatinally too expensive, and local optimization methods like [75]
are likely to get trapped into local optima because, except in simpler cases [57, 111],
grasp quality indices present local extrema. A grasp optimization procedure that avoids
such drawbacks will be proposed in Chapter 4, based on tracing a set of relevant grasps
exhaustively using a higher-dimension continuation technique. A rigid-grasp model is
assumed in such a procedure, but a way to take grasp compliance into account will be
given in Chapter 5.
2.2 Approach path planning
When planning the motions of an articulated system, the common practice is to compute some representation of the collision-free regions of the configuration space of the
system, using e.g. cell-decompositions, potential field methods, or roadmap techiques,
and then search for a valid path between the start and goal configurations using this
representation. The application of such techniques to the approach path planning
problem is not straightforward however, because of the high number of joints involved
in an anthropomorphic hand-arm system, and the fact that the goal configuration is
2.2 Approach path planning
17
either very close to the obstacle region, or directly on its border. A main trend since
the early attacks to the problem, thus, has been to simplify the problem in some way or
another, either by using simpler hands or restricted actuation schemes, or by assuming
favorable uncluttered environments. Nevertheless, the increase of computational power
over the years, and the appearance of succesful probabilistic roadmap methods, has
allowed to progressively drop some of these simplifications, gradually increasing the
range of solvable problem instances (Figure 2.2).
One of the earliest works on the problem was that by Lozano-Perez et al. [61] on the
Handey system, which involved a classical one-degree-of-freedom gripper mounted on
a six-revolute arm, aimed at performing pick-and-place operations for assembly tasks.
In this system, the planning of an approach path to the object was subdivided into
two-stages. First, using a cell-decomposition method, a path involving gross motions of
the arm was computed during a global stage in order to bring the gripper in close
proximity to the object. Then, a simple path taking the system to the final grasp
was generated during a local stage, using a potential field method. Although several
simplifying assumptions were made in both stages —including planar-faced objects,
motions involving only three joints, and a relatively uncluttered environment around
the object— the Handey system was taking many aspects of the problem into consideration, and many subsequent works are still reminiscent of the two-stage strategy
adopted. The work in [61] was extended by Pollard [76] later on, replacing the gripper
by the more complex, three-fingered Salisbury hand. The planning of the path was done
under similar assumptions, using cell-decomposition and potential field methods for the
global and local stages, but more involved grasp synthesis methods were introduced to
account for the increased complexity and reconfiguration capacity of the newer hand.
The grasp configuration was synthesized by choosing proper contact points for the
fingertips, together with a feasible pose for the wrist heuristically, and an attempt to
connect this configuration to an intermediate configuration at a given distance from the
object was then made using a straight-line path possibly deformed locally, so that the
latter configuration could be easily reached by the global planner.
The local problem of computing a motion path from a pre-grasp configuration to one
in contact with the object was considered by Kragic et al. [50]. In this work, the pregrasp configuration is provided by a human guiding the hand wrist, and the GraspIt!
system is used to evaluate whether a closing of the fingers can yield a sufficiently good
grasp of the object. The objects are again plane-faced, and, strictly speaking, there is
18
Related Work
no planning of the approach path to the object, since the pre-grasp configuration is
informed, and the closing of the fingers is deterministic. However, Morales et al. [67]
extended this approach to account for more complex objects and hands, alleviating the
need of informing manually the system with pre-grasp configurations. In their case,
the pre-grasp configuration is automatically selected from a database of previouslycomputed pre-grasps, and the final approach path to the object, taken as well from the
database, follows a straight-line pattern similar to the one used by Pollard [76], but
allowing the rotation of the wrist about the path if necessary. In simulation, the planner
iteratively translates the hand along the path, and possibly rotates it, checking along the
way whether the resulting hand configuration can properly reach the object by closing
the fingers.
Aiming to deal with more cluttered environments, Berenson et al. [2] propose to
rank all possible approach paths a priori before trying them, using a score function that
quantifies the promise of the path to yield a high-quality, collision-free grasp in the
end. The paths considered are identical to those tried by Morales et al. [67] but, once
one of them is found to be valid, the system is able to plan a motion from the start
configuration of the hand-arm system, to the pre-grasp configuration determined, using
bi-directional Rapidly-exploring Random Trees (RRTs). The hand configuration is kept
constant during such planning in order to grow the trees in six-dimensional joint spaces
only. Berenson and Srinivasa [3] present a refinement of the technique in which all
degrees of freedom of the hand are allowed to move in a coordinated way during the
exploration of the final approach path, adding one additional degree of freedom that
increases the chances of succeeding along the path.
A slight twist to these approaches has been recently introduced by Vahrenkamp
et al. [118], in an attempt to integrate the grasp synthesis and approach path planning
phases to the largest possible extent. In their work, a global planner iteratively grows
an RRT from the initial configuration of the hand-arm system, selecting one of its
nodes randomly from time to time for exploration towards the object. This exploration,
which is similar to that in previous approaches, consists in trying a linear path until a
threshold position around the object is reached, then closing the fingers until contact is
established, and finally evaluating the grasp using typical grasp quality metrics.
Despite the significant advances to the problem, three principal weaknesses can be
identified in all previous works. First, note that although all methods may perform well
in their particular contexts of application, there is little or no control on whether the
2.2 Approach path planning
19
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.2: Grasp planners of increasing complexity proposed over the years. (a) The
Handey system by Lozano-Perez et al. [61], which assumes a classical gripper. (b) The
extension considered by Pollard [76], which changes the gripper by the Salisbury hand
and considers possibly-deformed straight-line approach paths. (c) Morales et al. [67]
introduces rotations about such paths. (d) Berenson et al. [2] rank the paths a priori
before trying them, according to a score function considering grasp quality and potential
collisions. (e) Berenson and Srinivasa [3] add finger-coordinated motions to increase
the chances of finding a collision-free path. (f) Vahrenkamp et al. [118] integrate grasp
synthesis and approach path planning under a randomized approach.
20
Related Work
final grasp configuration will be suitable to a particular manipulation task involving
the object. In most grasp planning systems the hand is simply closed from a pre-grasp
configuration, so that it is difficult to predict in which precise object regions will the
fingers finally land. Second, the overall grasp planner is usually a combination of global
and local planners whose domain of application is heuristically defined with rule-ofthumb conditions that seem artificial to the problem. Third, the hand-arm system is
usually set to move under simplified motion paths only, which restricts the range of
possible movements, and hence the ability to find proper solutions in complex situations
with cluttered environments.
An alternative planner overcomming such limitations is proposed in Chapters 6
and 7, based on the idea of introducing principal motion directions and wrist orientation
constraints. As it will be shown, such a planner is able to drive the hand-arm system
to a pre-grasp or a final kinematically-feasible grasp by increasing or decreasing the
mobility of the system as needed, according to the local complexity of the obstacle
region encountered.
Part II
Grasp Synthesis
3
Finding Grasp Configurations
This chapter presents a method to compute a kinematically-feasible grasp, i.e.,
one that satisfies a given collection of hand-object contact constraints. In contrast
to previous algorithms given for the same purpose, the one presented here allows
specifying such contact constraints between free-form surface patches on the hand
and object surfaces, and always returns a solution whenever one exists. The
method is based on formulating the problem as a system of polynomial equations,
and then exploiting the special form of the equations to isolate the solutions,
using a numerical technique based on linear relaxations. Although explained
for anthropomorphic hands, the approach is general, in the sense that it can
be applied to any grasping mechanism with lower-pair joints. The approach can
also accommodate as many hand-object contacts as required, involving any of the
hand links, including the palm.
Given a number of regions on the surface of the hand, and a number of corresponding regions on the surface of the object, our goal is to determine how should the
hand be configured relative to the object so that each hand region establishes contact
on its corresponding object region (Figure 3.1). To see the constraints involved into
such a problem, we first describe the kinematic structure of common anthropomorphic
hands, and how the hand-object contact constraints are assumed to be mathematically
specified. We then show that, reflecting such constraints, the problem can be posed as
one of solving a system of polynomial equations, for which a complete method able to
find a solution, if one exists, is proposed. We finally illustrate the performance of this
method on the particular case of the Schunk Anthropomorphic hand (SA) on various
tasks requiring an object to be grasped in a special way. The method will be useful in
Chapter 4 to synthesize an initial grasp from which to start an optimization process,
aimed at obtaining high-quality, restrained and manipulable grasps.
24
Finding Grasp Configurations
h1
h2
h3
h4
o4
o2
o3
o1
Figure 3.1: A typical grasp for a scalpel can be specified by requiring the contact of
regions h1 , . . . , h4 of the hand, with regions o1 , . . . , o4 on the object (left). The problem
is to determine how should the hand be configured relative to the object, in order to
bring the hand regions into contact with their corresponding object regions (right).
3.1 The hand-object system
3.1.1 Structure of an anthropomorphic hand
Although each anthropomorphic hand follows a particular design, all hands are in
general made up of a palm and several fingers, one of them acting as the thumb.
Usually, all fingers are aligned with each other and with the palm, except the thumb,
which is mounted asymmetrically so that it can push against the other fingers. Each
finger is composed of several phalanges, usually articulated through revolute (R) or
universal (U) joints, whose freedom may be actuated, underactuated, or coupled to
those of other joints. Mechanical limitations usually exist, that constrain these joints to
take values within prescribed ranges.
Many finger designs follow an URR structure or slight variations of it (Figure 3.2).
This structure closely mimics that of the human finger [69]. It mounts a universal joint
at the finger base, to model the metacarpophalangeal joint, and two additional revolute
joints, to model the proximal and distal interphalangeal joints. The axis of the U joint
that is fixed to the palm is responsible for abduction/adduction movements, and the
3.1 The hand-object system
25
remaining axes, which are usually
parallel, are responsible for flexion/extension movements of the
finger. The thumb structure is more
diverse and controversial [36, 119].
R
Designs are found where the thumb
adopts the same structure as that
of the remaining fingers, which
R
facilitates the construction of the
hand, like in the SA hand, for
instance.
Other designs either
decrease or increase the mobility of
U
the thumb, by removing or adding
joints with respect to the basic URR
design. In all cases, however, the tip
of the thumb is allowed to face all
other fingertips, so as to be able to
grasp and manipulate objects under
Figure 3.2: The URR structure of a finger.
stable prehensions.
A summary of representative hand designs adopted during the last decade is provided in Table 3.1. In order to reduce the number of actuators, note that many hands
have coupled degrees of freedom. The coupling of two joints A and B is indicated as
⁀ in Table 3.1, meaning that a rotation about an axis of A produces a proportional
AB
⁀ coupling only the parallel axes are coupled.
rotation about an axis of B. On a UR
Such couplings can be implemented in software or in hardware, with their respective
advantages and disadvantages. In the first case, the idea is to reduce the computations
in the planning/control stages without losing dexterity, since the coupling can easily be
switched off if further dexterity is required. Implementing the couplings in hardware
has the benefit pf reducing the number of actuators but at the cost of compromising
dexterity [34, 82]. A natural way to implement these couplings in any of the two cases
is through the use of postural synergies [104]. Such synergies are obtained by studying
human grasps quantitatively, and picking up the most useful coupling patterns for a
given set of grasps of common objects. It is worth noting that this is a different concept
from real underactuation [9].
26
Finding Grasp Configurations
#Actuated
Hand
DIST hand [1998]
Finger designs
d.o.f.
Little
16
-
Robonaut hand [1999]
12
LMS hand [2001]
16
Ultralight Anthropom. hand [2001]
10
GIFU II hand [2002]
16
Ring
Middle
Index
Thumb
URR
⁀
URR
⁀ RR
⁀
RR
URR
-
⁀ RR
⁀
U
⁀
RRR
⁀
URR
Shadow Robot hand [2003]
18
⁀
RURR
DLR II hand [2004]
13
-
UBH 3 hand [2004]
20
MA-I hand [2005]
16
-
SA hand [2012]
13
-
Twendy-One hand [2009]
13
-
⁀
UR
URR
⁀
URR
⁀
URR
UUR
⁀
RURR
URR
URR
⁀
URR
⁀
URR
⁀
RURR
RUR
Table 3.1: Representative hand designs adopted during the last decade.
3.1.2 Contact constraint specification
The contact constraints are assumed to be given as a collection of pairs (hc , oc ), for
c = 1, . . . , b contacts, where hc and oc are two-dimensional regions on the hand and
object surfaces, respectively. The constraint (hc , oc ) is meant to require the contact of hc
and oc at some point, with the normals to hc and oc aligned at such point, to avoid the
interpenetration of the regions (Figure 3.3).
By convention, hc and oc are assumed to be given as polynomial patches. That is, it
is assumed that a polynomial function of the form
p = p(u, v),
(3.1)
is given for each region, providing the parametric coordinates p = (px , py , pz ) of a
point P in the region, in terms of some scalar parameters u and v, bound to lie within the
interval [0, 1]. To properly align the normals of hc and oc , the parameterization p(u, v) is
supposed to be non-degenerate, in the sense that, if pu and pv are the partial derivatives
of p(u, v) with respect to u and v, then the normal vector to the patch, defined as
n = pu × pv ,
(3.2)
3.1 The hand-object system
27
never vanishes for (u, v) ∈ [0, 1] × [0, 1].
For ease of explanation, p(u, v) will adopt the form of a standard Bézier patch of
some given degree M × N ,
p(u, v) =
M X
N
X
bi,j · Bi,M (u) · Bj,N (v),
(3.3)
i=0 j=0
where bi,j denote the Bézier control points of the patch, and Bi,j (x) =
j
i
xi (1 − x)j−i
is the ith Bernstein polynomial of degree j. Note that any polynomial paramaterization
p(u, v) can be converted into such form, by using an appropriate change of basis [30].
Ln
Oc
oc
n̂c
m̂c
Hc
hc
Lk
Figure 3.3: Elements intervening in a contact constraint (hc , oc ). The constraint is
satisfied when points Oc ∈ oc and Hc ∈ hc coincide, with the normals on such points
aligned.
28
Finding Grasp Configurations
3.2 Kinematic equations
A kinematically-feasible grasp can be formulated as a number of constraints that the
poses of the hand and object links must fulfill. This section formulates such constraints
mathematically, following the methodology proposed by Porta et al. [80]. Once gathered together, the constraints form a system of polynomial equations characterizing all
possible solutions of the problem. The special structure of this system will be beneficial
to solve the problem numerically, as it will be shown in Section 3.3.
3.2.1 Link constraints
It will be convenient to label the hand and object links as L0 , L1 , . . . , Ln , where L0 is the
palm link, L1 , . . . , Ln−1 are the various phalange links, and Ln is the object link. The
joints of the hand will also be labelled for reference, as J1 , . . . , Jm .
Each link Ll , l = 0, . . . , n, will be furnished with a local reference frame Fl , and we
will let the reference frame of the palm link, F0 , to act as the absolute frame. Moreover,
each frame will have an associated vector basis, and we will write vFl to refer to the
coordinates of vector v, written in the basis of Fl . Vectors with no superscript will either
be expressed in the basis of the absolute frame, or in no particular frame, depending on
the context.
With the previous notation, a configuration of the hand-object system will be an
assignment of a pose (rl , Rl ) to each link Ll , l = 1, . . . , n, where rl ∈ R3 is the position
of the origin of Fl with respect to F0 , and Rl ∈ SE(3) is a 3 × 3 rotation matrix giving
the orientation of Fl relative to F0 . The elements of the rotation matrices are not
independent, because if Rl has the form (ĉl , d̂l , êl ), then it must be
kĉl k2 = 1,
(3.4)
kd̂l k2 = 1,
(3.5)
ĉl · d̂l = 0,
(3.6)
ĉl × d̂l = êl ,
(3.7)
for l = 1, . . . , n, in order for Rl to represent a valid rotation. Note that the joints, the
contacts, and the mechanical limits impose additional constraints on the link poses.
These constraints are next formulated explicitly.
3.2 Kinematic equations
29
3.2.2 Joint assembly constraints
Since most hand designs only resort to revolute or universal joints (Table 3.1), we focus
on formulating the constraints imposed by such joints, but other joint types would be
formulated in a similar way [80].
In terms of spatial constraints, the assembly of two links Lj and Lk , through a
revolute joint Ji , is equivalent to imposing the coincidence of two points, Pi and Qi ,
and the alignment of two unit vectors, ûi and v̂i , respectively fixed to Lj and Lk
(Figure 3.4a). These two points and vectors are chosen on the axis of the joint, and they
(a)
(b)
Pi
Pi , Qi
Qi
Lj
Lj
Lk
Lk
ûi
v̂i
(c)
ûi , v̂i
ûi
90o
(d)
ûi
v̂i
Lj
v̂i
Pi
Lj
Pi , Qi
Qi
Lk
Lk
Figure 3.4: (a,b): The assembly of two links through a revolute joint is specified by
imposing the coincidence of two points and the alignment of two vectors. (c,d): The
assembly through a universal joint is specified by imposing the coincidence of two points
and the orthogonality of two vectors. [Figure adapted from [80].]
30
Finding Grasp Configurations
coalesce into a single point and vector when the two links get assembled (Figure 3.4b).
The coincidence and alignment conditions can be written, respectively, as
F
(3.8)
F
(3.9)
k
rj + Rj pi j = rk + Rk qF
i ,
Rj ûi j = Rk v̂iFk ,
F
k
refer to the position vectors of Pi and Qi in frames Fj and Fk ,
where pi j and qF
i
respectively. The valid poses of the two links, hence, are those that fulfill Eqs. (3.8)
and (3.9) simultaneously.
Similarly, if Ji is a universal joint, the valid poses of Lj and Lk are those that fulfill
F
k
rj + Rj pi j = rk + Rk qF
i ,
F
Rj ûi j · Rk v̂iFk = 0,
(3.10)
(3.11)
where Eqs. (3.10) and (3.11) impose the coincidence of two points Pi and Qi , and the
orthogonality of two unit vectors ûi and v̂i , respectively fixed on Lj and Lk . The points
are located on the center of the universal joint, on Fj and Fk . The vectors are aligned
F
with the axes of the joint on such frames (Figures 3.4c and 3.4d). Since vectors pi j ,
F
Fk
j
k
are known a priori, the only unknowns in Eqs. (3.8)-(3.11) are the
qF
i , ûi , and v̂i
poses of the two links (rj , Rj ) and (rk , Rk ).
3.2.3 Joint limit constraints
For a revolute joint Ji incident to links Lj and Lk , the relative angle between Lj and Lk ,
denoted φi , is the angle between two unit vectors âi and b̂i orthogonal to the axis of Ji ,
fixed in Lj and Lk , respectively. Usually, due to the existence of mechanical limits, φi
can only take values within a prescribed interval which, using a proper location for âi
and b̂i , can always be written in the form [−αi , αi ], with αi ∈ [0, π]. In our formulation,
these limits can be taken into account by constraining the cosine of φi . For this, we
define a new variable ci = cos(φi ), and observe that the constraint φi ∈ [−αi , αi ] is
equivalent to the constraint ci ∈ [cos αi , 1]. Then we note that
ci = âi · b̂i ,
(3.12)
3.2 Kinematic equations
31
where
F
âi = Rj âi j ,
(3.13)
k
b̂i = Rk b̂F
i .
(3.14)
Thus, to constrain φi to the range [−αi , αi ] it is only necessary to add Eqs. (3.12)-(3.14)
to the system to be solved, taking into account that ci can only take values in the range
[cos αi , 1]. Joint limits for a universal joint can be imposed in a similar way.
3.2.4 Contact constraints
Let us suppose that in the required grasp some hand link Lk is required to be in contact
with the object link Ln , where the contact has to be established between given regions
hc and oc defined on Lk and Ln , respectively (Figure 3.3). Let Hc ∈ hc and Oc ∈ oc be
Fn
k
relative to Fk and Fn ,
two points on such regions, with position vectors hF
c and oc
respectively, and let m̂c and n̂c denote unit normal vectors to the link surface at such
points. Then, the poses of Lk and Ln that bring the two regions in contact through Hc
and Oc are those that fulfill
Fn
k
rk + Rk hF
c = rn + R n o c ,
Fn
k
Rk m̂F
c = −Rn n̂c ,
(3.15)
(3.16)
where Eq. (3.15) imposes the coincidence of Hc and Oc , and Eq. (3.16) establishes the
alignment of m̂c and n̂c .
All vectors and matrices in Eq. (3.15) are unknowns. However, since Hc and Oc are
bound to lie on hc and oc , the additional constraints
Fk
k
hF
c = hc (uc , vc ),
(3.17)
Fn
n
oF
c = oc (sc , tc ),
(3.18)
k
must be taken into account to properly formulate the contact, where hF
c (uc , vc ) and
n
oF
c (sc , tc ) are parametric descriptions of regions hc and oc , given in the form of Eq. (3.3).
Fn
k
Note that the Bézier control points of the patches hF
c (uc , vc ) and oc (sc , tc ) must be
given in frames Fk and Fn , respectively.
32
Finding Grasp Configurations
Fn
k
Analogously, the unit vectors m̂F
c and n̂c in Eq. (3.16) must also be related to the
patch parameters. This relationship can be established by taking into account that, for
a parametric patch p(u, v) of the form of Eq. (3.3), the normal vector n(u, v) defined by
Eq. (3.2) can be written as
n(u, v) =
−1
2M
−1 2N
X
X
i=0
b′i,j · Bi,2M −1 (u) · Bj,2N −1 (v),
(3.19)
j=0
so that it can be thought of as a new Bézier patch, but now of degree (2M −1)×(2N −1).
Explicit formulas for computing the control points b′i,j in this expression, in terms of the
Fn
k
can
control points bi,j of p(u, v), are given by Yamaguchi [123]. Thus, m̂F
c and n̂c
Fn
k
be related to the patch parameters by defining two unnormalized vectors mF
c and nc ,
k and n̂Fn through the
and their norms µc and νc , placed in correspondence with m̂F
c
c
constraints
k 2
µ2c = kmF
c k ,
(3.20)
n 2
νc2 = knF
c k ,
(3.21)
Fk
k
mF
c = µc m̂c ,
(3.22)
Fn
n
nF
c = νc n̂c ,
(3.23)
and setting the additional constraints
Fk
k
mF
c = mc (uc , vc ),
(3.24)
Fn
n
nF
c = nc (sc , tc ),
(3.25)
whose right-hand sides follow the form of Eq. (3.19).
3.2.5 Final system of equations
Summarizing, the final system of equations defining the possible grasp configurations
will be formed by Eqs. (3.4)-(3.7) for each link, Eqs. (3.8) and (3.9) for each revolute
joint, Eqs. (3.10) and (3.11) for each universal joint, equations of the form of (3.12)(3.14) for each joint limit constraint, and Eqs. (3.15)-(3.18) and (3.20)-(3.25) for each
contact constraint. Note that the variables involved in this system are:
3.3 Numerical Solution
33
• The pose variables (rl , Rl ) corresponding to links Ll , l = 1, . . . , n.
• The variables âi , b̂i , and ci corresponding to the joint limit constraints on all joints
Ji , i = 1, . . . , m.
Fk
Fn
Fn
k
• The contact point coordinates hF
c and oc , associated normal vectors mc , nc ,
Fn
k
m̂F
c , n̂c , vector norms µc and νc , and parameters uc , vc , sc , and tc , corresponding
to all contact constraints (hc , oc ), c = 1, . . . , b.
It is worth mentioning that the rl variables of this system can actually be eliminated
through a process explained in detail in [80]. The elimination is based on the fact that,
for a loop of links pairwise constrained by joint or contact constraints, Eqs. (3.8), (3.10),
and (3.15) occurring along the loop can be substituted by an equivalent “loop-closure”
equation which is their sum, which does not contain any of the rl variables. This process
simplifies the system, and can always be invoked if desired, but the numerical method
that follows is equally applicable to both the original and the simplified system.
3.3 Numerical Solution
Let ne and nv be, respectively, the number equations and variables of the final system
described in Section 3.2.5. This system can be compactly written as
Φ(q) = 0,
(3.26)
where q = (q1 , . . . , qnv ) refers to a vector encompassing all of its variables, and Φ is a
polynomial function from Rnv to Rne describing its equations. We now wish to develop
a method to find a point q satysfying Eq. 3.26, whenever one exists.
Several strategies have been given to solve systems of equations of this kind. For example, algebraic-geometric methods, including those based on resultants and Gröbner
bases, use variable elimination to reduce the initial system to a univariate polynomial,
so that the roots of such polynomial, once backsubstituted into other equations, yield
all solutions of the original system [23]. Continuation methods, in contrast, begin with
an initial system whose solutions are known, and then transorm it gradually to the
system whose solutions are sought, tracking all solution paths along the way [109].
34
Finding Grasp Configurations
In a third approach, branch-and-prune methods use approximate bounds of the solution set to rule out portions of the search space that contain no solution, reducing
the initial domain as needed until a fine-enough approximation of the solution is obtained [17, 27, 63, 80, 85].
While algebraic-geometric and continuation methods are general, they have a number of limitations in practice. On the one hand, algebraic-geometric methods usually
explode in complexity, may introduce extraneous roots, and can only be applied to
relatively simple systems of equations. On the other hand, continuation methods should
be implemented in multi-precision arithmetic to avoid numerical instabilities, leading
to important memory requirements and, like elimination methods, they must compute
all possible roots, even the complex ones, which are physically meaningless in our case,
thus slowing the process substantially on systems with a small fraction of real roots.
Branch-and-prune methods are also general, and present a number of advantages that
make them preferable in our case: (1) Contrarily to many elimination methods, they do
not require intuition-guided symbolic reductions, (2) they directly isolate the real roots,
(3) they can be made numerically robust without resorting to extra-precision airthmetic,
and (4) some of them can tackle under- and over-constrained problems without needing
modifications. These are the main reasons that motivate the approach we adopt next,
which belongs to this latter category. The approach, which is based on the one proposed
in [80], entails expanding the equations to a standard form (Subsection 3.3.1) and
then using a branch-and-prune method exploiting this form to isolate the solutions
(Subsections 3.3.2 and 3.3.3).
3.3.1 Equation expansion
We distinguish two groups of equations in the final system Φ(q) = 0. A first group
encompassing Eqs. (3.17), (3.18), (3.24), and (3.25), whose polynomials follow the
Bézier form of Eqs. (3.3) and (3.19), and a second group encompassing the remaining
equations, whose polynomials only contain monomials of the form qi , qi2 and qi qj . Note
that all equations of the second group can be easily converted into linear form by
introducing the changes of variables
pi = qi2
(3.27)
bk = qi q j
(3.28)
3.3 Numerical Solution
35
for all qi2 and qi qj monomials occuring in them. After such changes, we obtain a new
system of the form
Λ(x) = 0
Ψ(x) = 0
)
,
(3.29)
where x is an nx -dimensional vector encompassing all of the original qi variables, and
the newly-introduced pi and bk ones. Here, Λ(x) = 0 represents a collection of linear
equations in x, and Ψ(x) = 0 represents a collection of equations, each of which can
only adopt one of these three forms:
xk = x2i ,
(3.30)
xk = xi xj ,
(3.31)
xk = f (xi , xj ).
(3.32)
While the first two forms correspond to the changes of variables in Eqs. (3.27) and (3.28),
the latter form corresponds to the scalar components of Eqs. (3.17), (3.18), (3.24),
and (3.25), so that f (xi , xj ) refers to a Bernstein-form polynomial of degrees di and dj
in xi and xj , respectively.
3.3.2 Equation solving
It can be seen that, under the used formulation, each variable xi of x can only take
values within a prescribed interval [80], so that the Cartesian product of all such
intervals defines an initial nx -dimensional box B ⊂ Rnx which bounds all solutions of
Eqs. (3.29). The algorithm to isolate such solutions recursively applies two operations
on B: box shrinking and box splitting.
Using box shrinking, portions of B containing no solution are eliminated by narrowing some of its defining intervals. This process is repeated until either 1) the box
is reduced to an empty set, in which case it contains no solution, or 2) the box is
“sufficiently” small, in which case it is considered a solution box, or 3) the box cannot be
“significantly” reduced, in which case it is bisected into two sub-boxes via box splitting
(which simply bisects the box through its largest interval). To converge to all solutions,
the whole process is recursively applied to the new sub-boxes, until one ends up with a
collection of solution boxes, whose side lengths are below a given threshold σ.
36
Finding Grasp Configurations
As it turns out, this algorithm explores a binary tree of boxes, whose internal nodes
correspond to boxes that have been split at some time, and whose leaves are either
solution or empty boxes. By properly implementing the bookkeeping of boxes awaiting
to be processed, this tree can be explored either in depth- or breadth-first order, the
choice of order depending on whether one wishes to isolate just one solution, or the
entire solution set.
Note that the algorithm is complete, in the sense that the solution boxes it returns
include all solution points of Eqs. (3.29). Thus, the algorithm will always succeed in
isolating a solution, whenever one exists, provided that a small-enough value of the σ
parameter is used. Detailed properties of the algorithm, together with examples of its
output, are given by Porta et al. [79, 80].
3.3.3 Box shrinking
We next see how a given sub-box Bc ⊆ B can be reduced, discarding portions of the box
that contain no solution. Observe that the solutions of Eqs. (3.29) lying within Bc ⊆ B
must lie on the linear variety defined by Λ(x) = 0. Thus, in principle, we might shrink
Bc to the smallest possible box bounding this variety inside Bc . The lower and upper
limits of the shrunk box along dimension xi , i = 1, . . . , nx , would respectively be found
by solving the two linear programs
LP1: Minimize xi , subject to: Λ(x) = 0, x ∈ Bc ,
LP2: Maximize xi , subject to: Λ(x) = 0, x ∈ Bc .
The sub-box Bc may be further reduced, however, because the solutions must also
satisfy the equations Ψ(x) = 0. These equations can be taken into account by noting
that for each equation it is possible to define a convex polytope that bounds the equation
solutions within Bc . Thus, to better delimit the solutions of the system, Bc can be safely
reduced to the smallest possible box enclosing the intersection of Λ(x) = 0 and the
polytopes of all equations in Ψ(x) = 0. This reduction can be implemented by representing the individual polytopes with linear inequalities, and adding such inequalities to
the constraint set of the linear programs LP1 and LP2. We next see how such polytopes
can be derived, for each one of Eqs. (3.30)-(3.32). The notation [li , ui ] will refer to the
interval of Bc relative to xi .
3.3 Numerical Solution
37
xk
(a)
xk
A2
B1
B4
li
A1
A3
li
lj
B2
uj
xj
ui
xi
ui
B3
(b)
xi
xk
(c)
C11
C10
C02
C21
C20
C01
C12
C22
li
lj
uj
xj
ui
xi
Figure 3.5: Polytope bounds within Bc . (a) The points on xk = x2i are bound by the
triangle A1 A2 A3 . (b) The points on xk = xi xj are bound by the tetrahedron B1 B2 B3 B4 .
(c) The points on xk = f (xi , xk ) are bound by the convex hull of the points Cpq . In this
example, f (xi , xk ) is a Bernstein-form polynomial of degree two in xi and xj , so that the
control points Cpq form a grid of size 3 × 3.
To derive a polytope for xk = x2i , note that the portion of the parabola xk = x2i lying
within Bc is bounded by the triangle A1 A2 A3 in the xi -xk plane, where A1 and A2 are the
points where the parabola intercepts the lines xi = li and xi = ui , and A3 is the point
where the tangent lines at A1 and A2 meet (Figure 3.5a). Thus, the polytope of xk = x2i
is defined by the triangle A1 A2 A3 , which can be represented by three inequalities that
correspond to the three edges of this triangle.
38
Finding Grasp Configurations
To derive a polytope for xk = xi xj , we realize that the portion of the surface xk = xi xj
included in Bc is bound by a tetrahedron B1 B2 B3 B4 in the xi -xj -xk subspace, whose vertices Bi are obtained by lifting the four corners of the rectangle [li , ui ] × [lj , uj ] vertically
to the surface xk = xi xj (Figure 3.5b). Thus, the polytope of xk = xi xj is defined by
the tetrahedron B1 B2 B3 B4 , and can be represented by four inequalities, corresponding
to the four faces of this tetrahedron. Finally, to derive a polytope for xk = f (xi , xj ),
we resort to the subdivision and convex-hull properties of Bernstein polynomials [30].
Using the subdivision property, on the one hand, f (xi , xj ) is written in the form
f (xi , xj ) =
dj
di X
X
bp,q · Bp,di (xi ) · Bq,dj (xj ),
p=0 q=0
where the scalars bp,q are the so-called control points of f (xi , xj ) relative to the interval [li , ui ] × [lj , uj ]. Using the convex-hull property, on the other hand, we know that
the surface xk = f (xi , xj ) must be contained inside the convex-hull of the 3D points Cpq
with coordinates
cpq
p
q
= li + (ui − li ), lj + (uj − lj ), bp,q ,
di
dj
(3.33)
for p = 0, . . . , di , and q = 0, . . . , dj (Figure 3.5c). This convex hull defines a polytope for
equation xk = f (xi , xj ), which can be encoded as a set of inequalities by resorting to an
algorithm for convex-hull computations [1].
3.4 Test cases
The presented method has been implemented in C, extending the libraries of the CUIK
suite [80]. This section illustrates the performance of the method under this implementation, on various test cases where an object needs to be grasped in a particular way, in
order to fulfill a given task.
The tests involve the solution of various instances of the problem defined on a
scalpel, a teapot, and a guitar, where each problem involves a number of regions to be
placed in contact, imposed by the specific requirements of the task to be accomplished
with the object (Table 3.2). In all test cases, the SA hand has been used to grasp the
objects (Figure 3.6), but the presented methodology is equally applicable to any other
hand. While the area of all contact regions defined on the hand is approximately 40%
3.4 Test cases
39
2 (slightly constrained)
3 (moderately constrained)
4 (highly constrained)
S CALPEL tasks
Task requirements
Upholding
It requires picking the scalpel
up using the index finger and
the thumb.
Incision
It requires a pencil-like grasp
using two fingers, the thumb,
and the palm.
nv , ne , d
CPU time [s]
219, 209, 17
106
Handling
It requires handling the
scalpel delicately using the
middle finger, the thumb, and
the palm.
243, 235, 16
255
Number of contacts
331, 324, 18
418
Computed solution
T EAPOT tasks
Task requirements
Lid lifting
The lid must be pulled up
through its knob using the
index finger and the thumb.
nv , ne , d
CPU time [s]
219, 209, 17
114
Service
The hand is required to hold
the teapot by its handle, placing the thumb on top, while
the index and middle fingers
embrace it.
288, 278, 19
262
Tunning
It requires the hand to grasp a
given key using the index finger and the thumb to tune the
tension of the corresponding
string.
219, 209, 17
68
Playing
The fingertips must be at
specified strings and frets to
perform a chord, while the
thumb contacts the guitar
neck.
307, 298, 19
229
Transportation
The palm contacts the bottom
of the teapot, while the fingers
enclose it so that it does not
slide out.
312, 305, 17
375
Computed solution
GUITAR tasks
Task requirements
nv , ne , d
CPU time [s]
Holding
It requires a whole-hand grasp
on a specific region where the
guitar can not be damaged
while being transported.
331, 324, 18
664
Computed solution
Table 3.2: Test cases and their computed solutions.
40
Finding Grasp Configurations
of the fingertip area (the dark patches on the upper limbs in Figure 3.6b), the area of
the contact regions on the object varies from experiment to experiment, from 2% of the
fingertip area on the teapot knob (“lid lifting” experiment), to 9000% of such area on
the guitar neck (“playing” experiment).
We next explain how the equations of the hand can be set up, and later discuss the
algorithm’s performance on the mentioned tests.
3.4.1 Equations for the Schunk Anthropomorphic hand
The Schunk Anthropomorphic hand is composed of four identical fingers that follow the
anthropomorphic structure illustrated in Figure 3.2. Three of these fingers are directly
mounted on the palm, and act as ring, middle, and index fingers. The fourth finger is
mounted on an intermediate link articulated with the palm through a revolute joint,
which allows this finger to act as a thumb (Figure 3.6). The hand has a total of fourteen
links (one palm and thirteen phalanges) and thirteen joints (nine revolute joints and
four universal joints).
To set up the equations, the links of the hand are labelled as L0 , . . . , L13 , as shown
in Figure 3.6, and the joints as J1 , . . . , J13 , letting Ji be the joint between Li−1 and Li
(for clarity, joint labels are not shown in Figure 3.6). Twenty-six points and unit vectors
are then defined, that provide the positions and orientations of all rotation axes of the
hand relative to the involved links. The points correspond to the centers of the universal
joints and to the midpoints of the revolute joints. The vectors correspond to unit vectors
aligned with the rotation axes of the joints. These points and vectors are displayed in
Figure 3.6 and their coordinates are given in Table 3.3, in milimeters. All reference
l
frames Fl are located with their origin in Ql , so that qF
l = (0, 0, 0), for l = 0, . . . , 13.
The orientations of such frames can be deduced easily from the coordinates provided in
Table 3.3. Taking into account these definitions, Eqs. (3.4)-(3.11) can readily be written
for all links and joints involved.
To write down the equations of Section 3.2.3, the mechanical limits of the SA hand
must be considered. Regarding the universal joints, the rotations about their ûi and
v̂i axes are limited to the ranges [−15o , 15o ] and [−4o , 75o ], respectively. Regarding the
revolute joints, all of them can only rotate in the range [4o , 75o ], except for the revolute
joint at the base of the thumb, which is restricted to the range [0o , 90o ]. The reference
configuration corresponding to all rotation angles at zero is shown in Figure 3.6b.
3.4 Test cases
41
middle
ring
index
L6
L3
v̂3
Q3
û3
P3
Q9
û6
û2
L1
Q1
L8 û9
v̂5
Q5
P5
Q8
û5
L4
v̂1
Q4
P4
P1
v̂4
P7
(a)
û13
û8
v̂12
û12
Q10
û11
P11
Q13
P13
L12
Q12
P12
L11
v̂11Q11
û7
û10
v̂13
v̂7
Q7
P10
palm L
0
L7
L13
v̂8
P8
û4
û1
thumb
v̂9
P9
L5
L2 v̂2
Q2
P2
L9
v̂6
Q6
P6
R
R
R
U
R
L10
U
v̂10
Q0
(b)
R
Figure 3.6: Geometric parameters (a) and reference configuration (b) of the Schunk
Anthropomorphic Hand. The various joint types are indicated in (b).
Finally, it must be taken into account that not all joints of the SA hand are independently actuated. The two distal joints of each finger are coupled, so that when one of
such joints is actuated, a rotation of the same angle about the other is produced. In the
adopted formulation, the coupling of two rotation angles is simply imposed by equating
the sine and cosine of such angles.
3.4.2 Computed solutions
A system of equations has to be solved for each task of Table 3.2, encompassing the
equations of the SA hand, together with the contact equations that impose the specific
requirements of the task. It must be noted that Eqs. (3.4)-(3.11) relative to fingers not in
contact with the object can actually be removed from this system, because such fingers
42
Finding Grasp Configurations
Ring
Joint
type
R
R
U
Middle
Index
Thumb
Par.
Value
Par.
Value
Par.
Value
Par.
Value
2
pF
3
(30, 0, 0)
4
pF
6
(30, 0, 0)
7
pF
9
(30, 0, 0)
12
pF
13
(30, 0, 0)
2
ûF
3
F3
v̂3
(0, 1, 0)
(0, 1, 0)
(0, 1, 0)
12
ûF
13
F13
v̂13
(0, 1, 0)
(0, 1, 0)
8
ûF
9
F9
v̂9
(0, 1, 0)
(0, 1, 0)
5
ûF
6
F6
v̂6
1
pF
2
(67.80, 0, 0)
7
pF
8
(67.80, 0, 0)
11
pF
12
(67.80, 0, 0)
(0, 1, 0)
7
ûF
8
v̂8F8
(0, 1, 0)
11
ûF
12
F12
v̂12
(0, 1, 0)
(67.80, 0, 0)
4
pF
5
1
ûF
2
v̂2F2
(0, 1, 0)
(0, 1, 0)
4
ûF
5
v̂5F5
(0, 1, 0)
0
pF
1
(−4.30, −40.16, 145.43)
0
pF
4
0
(−4.30, 0, 145.43) pF
7
10
(−4.30, 40.16, 145.43) pF
11
(1, 0, 0)
0
ûF
4
F4
v̂4
0
ûF
7
F7
v̂7
10
ûF
11
F11
v̂11
0
ûF
1
F1
v̂1
(0, 1, 0)
(1, 0, 0)
(0, 1, 0)
(0, 1, 0)
(1, 0, 0)
(0, 1, 0)
0
pF
10
0
ûF
10
F10
v̂10
R
(0, 1, 0)
(0, 1, 0)
(97, 6, −87)
(cos 55o , 0, sin 55o )
(0, 1, 0)
(−3, 27.10, 0)
(0, 0, −1)
(1, 0, 0)
Table 3.3: Parameters of the Schunk Anthropomorphic hand.
do not intervene in any kinematic loop, and hence impose no loop-closure constraint on
the overall system.
Table 3.2 provides the size of the equation system Eq. (3.26) to be solved in each
case, in terms of the number of variables, nv , and equations, ne , it involves, and the
dimension of its solution space, d, predicted as the number of variables minus the
number of non-redundant equations. Note in this regard that Eq. (3.9) introduces
equations that are redundant in terms of predicting such dimension, because ûi and
v̂i are unit vectors, and it is sufficient to establish two out of the three components
of Eq. (3.9) to determine the alignment of Lk relative to Lj . The third component of
Eq. (3.9), however, is needed to remove a sign ambiguity in such alignment. Since
a similar redundancy is introduced by Eq. (3.16), there will be as many redundant
equations as the number of joints and contacts involved in the problem at hand.
As it can be observed from Table 3.2, typical problems yield solution spaces of rather
high dimension. To avoid the curse of dimensionality as much as possible, and converge
to one solution rapidly, the proposed algorithm must be set to explore in depth-first
order (Subection 3.3.2). Running the algorithm in this order, we have obtained the
hand-object configurations depicted, in the CPU times indicated in each case. All times
reported correspond to a parallelized version of the algorithm, executed on a grid of 8
3.5 Summary
43
DELL Poweredge computers, equipped with two Intel Quadcore Xeon E5310 processors
and a 4Gb RAM each one, using a threshold of σ = 0.1. Note that the cost of computing a
solution increases with the number of contact constraints to be satisfied. This is because
the size of the linear programs to be solved during box shrinking is proportional to the
number of polytope inequalities introduced by such constraints, which increases the
cost of each iteration of the algorithm (Subsection 3.3.3).
3.5 Summary
This chapter has presented a new method to compute kinematically-feasible grasps.
When compared to other approaches for the same problem, the proposed method is
always guaranteed to converge to a solution whenever one exists. Additional features
of the method are the ability to deal with general region-to-region contact constraints,
as opposed to point-to-region [11, 95] or point-to-point [38] constraints, and the possibility to define the regions as general Bézier patches, to better adapt to the shape
of real surfaces on the hand and object considered. The proposed algorithm is based
on formulating the problem as a system of polynomial equations of special form, and
then exploiting such form to solve the equations, using an extended version of a recent
method based on linear relaxations [80]. It must be noted that, whereas the algorithm
in [80] can deal with lower-pair mechanisms of general structure, it cannot be directly
applied to mechanical hands, because it is unable to cope with general contact constraints between free-form surfaces. Here, we have extended that algorithm to be able
to specify such constraints between Bézier patches defined anywhere on the object or
on the hand, and to solve the corresponding equations.
4
Optimizing Grasp Configurations
This chapter provides a method for optimizing the quality of a robotic grasp,
subject to satisfying a number of hand-object contact constraints of the kind
assumed in the preceding chapter. Due to the multi-modal nature of typical grasp
quality measures, approaches that resort to local optimization methods are likely
to get trapped into local extrema on such problem. An additional difficulty of the
problem is the fact that the set of feasible grasps is a highly-dimensional manifold,
implicitly defined by a system of non-linear equations. The proposed procedure
finds a way around these issues by focusing the exploration on a relevant subset
of grasps of lower dimension, and tracing this subset exhaustively using a higherdimensional continuation technique. Using this technique, a detailed atlas of the
subset is obtained, on which the highest-quality grasp according to any desired
criterion, or a combination of criteria, can be readily identified.
Regardless of the method used to generate an initial configuration of the hand
in contact with the object, the returned grasp may not be optimal in terms of any
quality criterion, so that an optimization process is necessary to generate a suitable
high-quality grasp satisfying the specified contacts (Figure 4.1). The grasp optimization
procedure proposed here entails characterizing the manifold of kinematically-feasible
grasps as a system of equations to be fulfilled (Section 4.1), then extending this system
with meaningful equations to reduce the dimension of its solution set (Section 4.2),
and finally performing an exhaustive search over a point grid discretizing such set
at a desired resolution, to determine the highest-quality grasp attainable on the grid
(Section 4.3). Test cases are provided that validate the approach on simple and complex
robotic hands grasping objects with different geometries, under typical force-closure
and manipulability indices (Section 4.4).
46
Optimizing Grasp Configurations
Figure 4.1: Two grasps of a can for drink service. While both grasps are force-closed and
manipulable, the grasp on the left is preferable. The fingers are almost fully extended
in the grasp on the right, limiting the possibility to move the can in one direction.
4.1 Kinematically-feasible grasps
Recall that, a kinematically-feasible grasp is a configuration of the hand-object system
in which a number of regions of the hand hi , for i = 1, . . . , b, are in contact with
corresponding regions oi on the object. The regions and their pairings are pre-specified,
and the contact between hi and oi is assumed to be established with a point Hi ∈ hi
coinciding with another point Oi ∈ oi , keeping aligned the surface normals at such
points, m̂i and n̂i , to avoid local hand-object inter-penetrations. We further assume
that the hand joints are independently actuated or mechanically coupled, but do not
consider the case of adaptive underactuated hands [9].
Independently of the particular formulation adopted, a grasp configuration can be
represented by a vector x = (x⊤h , x⊤o , x⊤c )⊤ ∈ Rn of generalized coordinates, where xh
and xo encompass the configuration variables of the hand and the object, respectively,
and xc encompasses contact-related variables. Without loss of generality, we will assume that the absolute reference frame is attached to the palm of the hand, so that its
pose variables do not intervene in x.
The variables in x are subject to a number of constraints. A first set of equations,
H(xh ) = 0,
(4.1)
enforces xh ∈ Rh to be a valid hand configuration, i.e., one respecting the assembly
4.1 Kinematically-feasible grasps
47
constraints imposed by the joints, usually revolute or universal, on the various bodies
they connect, that is, the palm and the several finger phalanges. Note that Eq. (4.1) is
not necessary if the xh coordinates are independent, as it happens for instance when
choosing joint angles to represent a configuration [26]. In our case, however, we
resort to the dependent coordinates defined in the previous chapter because they are
equations of a simple structure, which has proved to be beneficial in the context of
grasp synthesis (Chapter 3), and for the application of continuation techniques [121].
In particular, this formulation encodes the six degrees of freedom of a body with twelve
variables, providing the position vector and the rotation matrix of a reference frame
attached to the body. Therefore, in addition to the joint assembly equations, Eq. (4.1)
includes constraints to enforce the 12-tuple of each body to be a member of SE(3).
Similarly, the spatial pose of the object is encoded by twelve variables, so that a second
set of equations,
L(xo ) = 0,
(4.2)
constrains xo ∈ R12 to define a member of SE(3). Finally, a third set of equations
formulates the contact constraints between the hand and the object. As in Chapter 3,
we assume that each contact region hi on the hand is specified as a parametrized patch,
i.e., as a smooth function of the form
hi = hi (ui , vi , xh ),
(4.3)
providing the absolute coordinates of a point hi = (xi , yi , zi ) in the patch, in terms of two
bounded scalar parameters, ui and vi , and of the hand configuration xh . Analogously,
the normal to any point in this patch is assumed to be given as a function
m̂i = mi (ui , vi , xh ).
(4.4)
In general, Eqs. (4.3) and (4.4) define two-dimensional regions described, for example,
by Bézier patches [30], but they can be replaced by single-parameter curves or fixed
points, if desired. The points and normals for the contact patches on the object can be
defined in a similar way
oi = oi (si , ti , xo ),
(4.5)
n̂i = ni (si , ti , xo ),
(4.6)
48
Optimizing Grasp Configurations
in terms of bounded parameters si and ti , and the object pose xo . With the previous
definitions, the contact of hi with oi can be expressed by setting
hi − oi = 0,
(4.7)
m̂i + n̂i = 0.
(4.8)
Thus, the vector xc encompases the vectors hi , m̂i , oi and n̂i , and the patch parameters
ui , vi , si and ti intervening in Eqs. (4.3) and (4.8), for i = 1, . . . , b.
Each variable in x = (x⊤h , x⊤o , x⊤c )⊤ can only take values within a given range. For
instance, the variables defining the orientation matrices in xh and xo take values within
[−1, 1]. Also, the size of the hand provides interval bounds on the translation variables,
and, without loss of generality, the patch parameters of the contact regions can be
normalized to the [0, 1] range [30]. Thus, the Cartesian product of such ranges defines
a rectangular domain D ⊂ Rn where the search for an optimal feasible grasp is to be
confined.
In sum, the set of feasible grasps F encompasses the points x ∈ D satisfying the
system
F(x) = 0
(4.9)
formed by Eqs. (4.1) and (4.2), and Eqs. (4.3) to (4.8) for all contacts i = 1, . . . , b.
The formulation in Chapter 3 guarantees that F(x) is differentiable and, to keep
the presentation of the method as simple as possible, we assume that its Jacobian is
full rank for all x ∈ F, which is the generic situation according to Sard’s theorem of
Analysis. Thus, F can be assumed to be a smooth manifold of dimension t = n − f ,
where f is the number of scalar equations in Eq. (4.9). Now, using Eq. (4.9), the
method in Chapter 3 can be applied to determine an initial kinematically-feasible grasp
x1 ∈ F from which to start the optimization process. We emphasize, however, that other
grasp synthesis techniques could be used to compute x1 [11, 21] since the optimization
approach introduced in this chapter makes no assumption on how this configuration is
obtained.
4.2 Relevant grasps
49
4.2 Relevant grasps
Although obtaining points on F is feasible, the dimension of F is very large for a
realistic hand and contact model, which hinders the exhaustive exploration of this space
independently of the methodology adopted. In the context of grasping, however, studies
on the human behavior suggest that humans do not use all degrees of freedom of the
hand independently, but in a coordinated way [104]. Following this idea, anthropomorphic hands are usually controlled using postural hand synergies [6, 104], where fewer
degrees of freedom are used to account for the overall hand configuration space. By
taking postural synergies into consideration, the search of a good grasp can be narrowed
to a subset of relevant grasps R ⊂ F of lower dimension, thus speeding up the overall
optimization process.
In this work, we compute the postural synergies via linear dimension-reduction
techniques on a predetermined set of hand configurations Xh = {x1h , . . . , xzh }, where
each xih is a value of xh satisfying Eq. (4.1). Let h be the number of components in xh ,
x̄h the average of the configurations in Xh , and T an h × z matrix whose i-th column is
xih − x̄h . The principal component analysis of Xh can be performed by diagonalizing the
covariance of T,
T T⊤ = E S2 E⊤ .
The h × h orthonormal matrix E gives the directions of variance of the data, and the
diagonal matrix S2 is the variance in each one of these directions, sorted in decreasing
magnitude. The linear variety through x̄h generated by the first p columns of E defines
the set E of the p postural synergies. The vectors in E have null components along the
columns of Es , the matrix formed by the last s = h − p columns of E, so that E is the
solution set of
E⊤s (xh − x̄h ) = 0.
(4.10)
This equation, together with Eq. (4.9), defines the system
R(x) = 0,
(4.11)
characterizing the non-linear set R = F ∩ E of relevant grasps. In the generic case, the
Jacobian of R(x) will also be full rank for all x ∈ R, so that, like F, R can also be as-
50
Optimizing Grasp Configurations
xc , xo
F
E′
x1
R
x̄h
x1,h
xh
Figure 4.2: Schematic representation of all the elements involved in the optimization
framework presented in this chapter. See the text for details.
sumed to be a smooth manifold with no bifurcations, which allows adopting a simplified
continuation strategy below, with no provisions for branch switching operations [41].
Since t is the dimension of F, and assuming k is the desired dimension for R, then
we must set
s=t−k
(4.12)
in Eq. (4.10). For efficiency reasons, k must be small in the adopted continuation
method (typically below 5), and s must be set accordingly. However, s must be smaller
than h, which limits the amount of dimension reduction obtained by the use of postural
synergies. As we will see, however, this is not an issue in practice, since the amount of
dimension reduction to be introduced is moderate in all cases, due to the presence of
4.3 Grasp quality optimization
51
the contact constraints. Actually, these constraints allow using a significantly smaller s
than that used in existing approaches relying on postural synergies.
Nonetheless, the introduction of postural synergies might lead to an empty set R
if the contact points are not reachable by the hand when constrained to be in E. To
guarantee that R is not empty, this set is redefined as R = F ∩ E ′ hereafter, where E ′ is
the solution set of
E⊤s (xh − x1,h ) = 0,
(4.13)
with x1,h being the subvector of the initial grasp x1 containing the values of xh . In
general, the difference of using Eq. (4.13) instead of Eq. (4.10) is minor, since x1,h
is usually close to the original set of principal hand motions, but this approximation
ensures that the hand can always conform to the object surface because R will at
least include the initial feasible grasp x1 . Figure 4.2 summarizes the different elements
involved in the approach. F is the set of feasible grasps defined in the space of xh , xo ,
and xc . In this representation, the configurations in Xh are shown as black dots, the
white dot is their average, x̄h , and the original set of p principal hand motions is shown
as a dashed line in the xh plane. The latter set is approximated by a line through x1,h ,
shown dotted in the figure, which, when extended to the whole space, generates the
linear variety E ′ . Finally, the set R = F ∩ E ′ is the space where the grasp optimization is
to be performed. Note that R is one-dimensional in this schematic representation, but
it is a k-dimensional set in general.
Typically, previous methods that use principal hand motions explore E. Using local
methods, then, they try to modify points on such space to yield configurations in contact
with the object. However, the final configurations may not necessarily lie on the set F of
grasps satisfying the contact constraints required for the task. In contrast, our method
directly operates on the set R, which is fully included in F.
4.3 Grasp quality optimization
The search for an optimal grasp is performed by computing an atlas of the k-dimensional
manifold R just defined, including the relevant grasps. Such an atlas provides a collection of charts, where each chart parametrizes a portion of R, and this allows enumerating a representative collection of grasps in R, on which any quality index can be
evaluated to detect the optimal one.
52
Optimizing Grasp Configurations
Tx i R
Tx i R
xi
xi
uij
xij
R
R
xj
xj
Tx j R
Figure 4.3: The higher-dimensional continuation method applied to a two-dimensional
manifold R in 3D space. A point xj ∈ R can be obtained by orthogonally projecting
a point xij corresponding to a vector uij ∈ Txi R (top). If a new chart is defined at xj ,
it must be properly coordinated with the chart at xi so that their projections smoothly
cover the manifold (bottom).
4.3.1 Tracing the manifold of relevant grasps
Formally, a chart Ci is a local map from a parameter domain Pi ⊂ Rk to an open neighborhood around a given point xi ∈ R, initially x1 . The higher-dimensional continuation
method proposed in [40] defines the map for chart Ci using Φi , an n × k orthonormal
basis of Txi R, the k-dimensional tangent space of R at xi . The map is constructed by
first selecting a k-dimensional vector of parameters uij ∈ Txi R (Figure 4.3, top), and
then using this vector to generate a point xij ∈ Rn in the neighborhood of xi using
xij = xi + Φi uij .
(4.14)
Then, the point xj ∈ R that corresponds to the orthogonal projection of xij on R is
computed, by solving the system
R(xj ) = 0
Φ⊤ (xj − xij ) = 0
using a Newton-Raphson method initialized at xij .
)
,
(4.15)
4.3 Grasp quality optimization
53
Each point on the manifold is the potential center of a new chart (Figure 4.3), and a
method introduced by Henderson [40] can be used to select the chart centers, in order
to ensure a good coverage of the manifold. Using his approach, the domain Pi of a chart
Ci is initialized as a k-dimensional hypercube enclosing a ball Bi of radius r, where Pi
and Bi are both defined in Txi R as illustrated in Figure 4.4 (top). A vertex of Pi exterior
to Bi , with position vector v, is then employed to generate a point xij , using Eq. (4.14)
with
uij =
α
v,
kvk
(4.16)
where α is initialized to the radius r. If the projection of xij to R does not converge, or
if the new chart Cj at xj is too far or too different from Ci , i.e., if
kxj − xij k > ǫ,
(4.17)
kΦ⊤i Φj k < 1 − ǫ,
(4.18)
or
for a given threshold ǫ, then the new chart is discarded and a new attempt of chart
Pi
Pi
Bi
r
Bi
xi
xi
uij
v
Bji
Cji
Figure 4.4: The process of chart construction. The domain Pi of chart Ci is initialized as
a box in Txi R circumscribing a ball of radius r centered in xi (top). Pi is refined using
a ball Bji that approximates Cji , the projection on Ci of the part of the manifold covered
by Cj (bottom).
54
Optimizing Grasp Configurations
Figure 4.5: Three stages of the construction of an atlas over a sphere. Red and
blue polygons represent charts under expansion and charts whose domain is already
bounded, respectively.
generation is performed with a smaller α. This procedure adapts the size of the area
covered by each chart to the local curvature of the manifold. When Cj is valid, it is used
to refine Pi from the intersection between Bi and Cji , the projection on Txi R of the part
of the manifold covered by Cj . This projection is approximated by a ball Bji in Txi R, as
shown in Figure 4.4 (bottom). The intersection of Bi and Bji defines a new face of Pi
that eliminates some of its vertices (in particular the one given by v) and generates new
ones. Similarly, the polytope Pj associated with Cj is cropped using the projection of Ci
into Cj . When all vertices of the domain Pi are included in Bi , the domain is said to
be bounded. Moreover, charts whose center is out of the domain D are also considered
bounded. If R has the manifold structure everywhere and r is properly set [40], then
when all charts are bounded, the atlas fully covers the connected component of R
containing the initial point x1 (Figure 4.5).
Algorithm 4.1 summarizes the atlas computation procedure. The algorithm receives
as inputs the set F of equations implicitly defining F, the set Xh of representative hand
configurations, the initial grasp x1 , the dimension t of F, the desired dimension k for
R, and the parameters r and ǫ used to construct the atlas. As output, the algorithm
returns an atlas A of the component of R reachable from x1 . The algorithm determines
the postural synergies as described in Section 4.2 (line 1). The required number of
constraints relative to such motions is then computed as a function of t and k (line 2)
in order to obtain the set R of equations defining R (line 3). Then, A is initialized with
a chart centered in x1 (line 4), and the construction of A proceeds while any of the
4.3 Grasp quality optimization
55
Algorithm 4.1: Computation of the atlas of R
13
ATLAS C OMPUTATION(F, Xh , x1 , t, k, r, ǫ)
input : The set F of equations , the set Xh of representative
hand configurations, the initial grasp x1 , the
dimension t of F, the desired dimension k for R, and
the parameters r and ǫ used to build the atlas.
output: An atlas A of R.
E ← P OSTURAL S YNERGIES(Xh )
s←t−k
R ← F ∪ {Es (xh − x1,h )}
A ← {G ENERATE C HART(R, x1 , r)}
while not B OUNDED(A) do
Ci ←N OT B OUNDED C HART(A)
α←r
v ←E XPANDIBLE V ERTEX(Ci )
repeat
Cj ←N EW C HART(R, Ci , α, v, r)
α ← α · 0.9
until not S IMILAR C HARTS(Ci , Cj , ǫ)
A ← A ∪ {Cj }
14
return A
1
2
3
4
5
6
7
8
9
10
11
12
charts can be extended (lines 5 to 13). The extension of a chart Ci starts by selecting a
vertex of Pi not included in Bi (line 8). This vertex is used to generate a new chart Cj
(line 10) using Eqs. (4.14) and (4.15) to determine its center. If the difference between
the new chart and the previous one is too large according to Eqs. (4.17) and (4.18),
chart generation is attempted closer to xi , i.e., with a smaller α (line 11). Otherwise,
the new chart is added to A, coordinating the chart with those already included in A
(line 13). The computational cost of this algorithm is exponential in k and, therefore,
it is only practical to compute an atlas on manifolds of moderate dimension, as it is the
case of the manifold R herein considered.
4.3.2 Evaluating the quality of relevant grasps
Once the atlas is computed, we can identify the optimal grasp over R by considering
an ordered set Q of quality indices. The indices in Q can be taken into account either
in series or in parallel [113]. In the first case, only grasps with a minimum value for a
given index are evaluated with the subsequent indices. In the second case, all indices are
56
Optimizing Grasp Configurations
Algorithm 4.2: The grasp optimization process.
12
G RASP O PTIMIZATION(A, d,Q, l, w)
input : The atlas A, the number d of points to consider on
each chart, the set Q of quality measures to be
optimized, and the vectors l and w of lower bounds
and weights for the quality measures, respectively.
output: The optimal grasp xg .
g ← −∞
xg ← ∅
forall the C ∈ A do
U ← P OINTS O N C HART(C, d)
forall the u ∈ U do
x ← C HART T O M ANIFOLD(C, u)
q ←Q(x)
if q l then
q ← wT q
if q ≥ g then
xg ← x
g←q
13
return xg
1
2
3
4
5
6
7
8
9
10
11
evaluated simultaneously and combined to produce a single measure, typically using a
weighted sum. Algorithm 4.2 allows these two ways of considering the quality indices.
The algorithm iterates over all charts in the atlas A (lines 3-12). For each chart C
in A, it generates a set U of d points on the tangent space associated with this chart
(line 4). These points can be either computed on a regular grid or sampled randomly
but, in any case, they all must lie inside the domain of C. The number d of points to
generate depends on the resolution at which the optimal grasp is required, and on the
smoothness of the quality indices considered; i.e., the sharper the variations the denser
the set of points. For each one of the points in U, we obtain the corresponding point
on R (line 6) using Eqs. (4.14) and (4.15). Then, for each point on R we evaluate the
quality indices in Q (line 7). If the obtained values for the indices are all above the
required thresholds in an element-wise comparison (line 8), we combine the indices
(line 9), and then check whether the combined value is larger than that of the best
grasp found up to the moment (lines 10-12). By setting the appropriate thresholds
in l and using w = (0, . . . , 0, 1)⊤ we will obtain a serial evaluation of the indices in
which the last index in Q will be optimized. A parallel evaluation can be obtained
4.4 Test cases
57
by using l = (−∞, . . . , −∞)⊤ and setting the desired values in w. Mixed evaluation
schemes can be obtained too, by adequately setting l and w. By iterating over all charts
and points, the optimal grasp over the computed points of R is finally identified and
returned (line 13). In an extreme case, the algorithm could return no grasp if the quality
indices for all of the considered grasps are below the given thresholds in l. Note that if
an optimal grasp is required with a finer precision, we only need to use Algorithm 4.2
with a larger value of d, and there is no need to recompute the atlas from scratch. In
a multi-resolution optimization context, this refinement can be focused into the most
promising areas previously identified. In any case, the overall cost of this algorithm
is bilinear with the number of charts in the atlas, and with the number d of points
considered for each chart.
4.4 Test cases
For clarity, we first exemplify the application of the method on a simple hand, and then
summarize the results for the Schunk Anthropomorphic hand. All results correspond to
an implementation in C of Algorithm 4.1, and in Matlab of Algorithm 4.2, running on
an Intel Core 2 Duo processor at 3 GHz.
4.4.1 A planar hand
Figure 4.6 shows a planar hand with three fingers and two phalanges per finger holding
an object composed of circles, where the OXY absolute reference frame is attached to
the base of one of the fingers. The length and absolute orientation of the j-th phalanx of
the i-th finger are given by the parameter li,j and the unit vector v̂i,j ∈ R2 , respectively.
Since the lengths are fixed, the configuration of the hand can be encoded in a simplified
form in this case, by the vector xh = (v̂⊤1,1 , . . . , v̂⊤3,2 )⊤ subject to the constraints
kv̂i,j k2 = 1,
(4.19)
for all phalanges. Thus, Eq. (4.1) is the system formed by Eqs. (4.19). Since this system
contains 6 equations in 12 variables, its solution set is of dimension 6, which agrees with
the number of degrees of freedom of the hand. Note that xh only includes parameters
for the orientation of the hand links, since their position vectors can be computed from
the orientations and the constant length parameters [80].
58
Optimizing Grasp Configurations
li,2
li,1
Y′
v̂i,2
li,3
ci
v̂i,1
X′
O′
to
ai
Y
O
X
Figure 4.6: A simple planar hand with three fingers holding an object. The parameters
are indicated for one finger only, but apply to the three fingers.
A local reference frame O′ X ′ Y ′ is attached to the object, whose pose in the absolute
frame is given by xo = (to , v̂o ), where to = (xo , yo )⊤ is the position vector of O′ and
v̂o = (so , co )⊤ is a unit vector aligned with the X ′ axis. Then, Eq. (4.2) becomes
kv̂o k2 = 1.
(4.20)
In this example, the contact regions in each fingertip reduce to a point and the explicit
expression of Eq. (4.3) is
hi = ai + li,1 v̂i,1 + li,2 v̂i,2 ,
(4.21)
where ai is the position vector of the palm anchor point of finger i in the absolute frame,
and Eq. (4.4) providing the associated normal is
m̂i = v̂i,2 .
(4.22)
4.4 Test cases
59
Moreover, the contact regions on the object are arcs of circumference given by a single
parameter. Thus, for each one of such arcs, Eq. (4.5) boils down to the following
expression
oi = to +
"
co −so
so
co
#
( ci + li,3 ŵ(ui )),
(4.23)
where ci is the center of the circumference in local coordinates of the object, li,3 is its
radius, and
ŵ(ui ) =
"
cos ui
sin ui
#
,
(4.24)
with ui ∈ [ai , bi ], is the angular range defining the arc for contact patch i. Similarly,
Eq. (4.6) reduces to
n̂i =
"
co −so
so
co
#
ŵ(ui ),
(4.25)
in this case. Thus, Eq. (4.9) encompasses Eqs. (4.19) to (4.25), defining a set F of
feasible grasps of dimension t = 3. The proposed optimization procedure can be
directly applied to problems of this dimension. However, to complete the example
and to facilitate the visualization of the results, it is better to reduce the dimension to
obtain a set of relevant grasps R of dimension k = 2. Thus, according to Eq. (4.12),
s = 1 linear constraints given by Eq. (4.13) must be added to Eq. (4.9), in order to
get Eq. (4.11). In this example, the set Xh used to reduce the dimensionality contains
randomly generated hand configurations.
Figure 4.7 shows the results obtained when applying the proposed method to this
example. Two complementary views of the computed atlas are depicted (left), together
with the best and worst grasps found (center and right, respectively). The atlas was
obtained using Algorithm 4.1 with r = 0.125 and ǫ = 0.4 in about 0.1 seconds. It
contains a total of 750 charts, whose polytopes Pi form the shown hexagonal-like mesh.
To optimize the grasp, the atlas was evaluated using Algorithm 4.2 under the forceclosure quality index reported in [83] normalized to the range [0, 1], obtaining the
results shown in Figure 4.7, where green and red charts respectively correspond to
configurations with a large and low value of the index. Thus, Algorithm 4.2 was
called with I containing only this index in this case, and setting l = 0 and w = 1.
The contacts were modeled as frictional point contacts with a friction coefficient of
60
Optimizing Grasp Configurations
atlas
best grasp
worst grasp
Figure 4.7: Two views of the atlas of the set R of relevant grasps on the planar hand
example, together with the best and worst grasp configurations obtained, according to
the force-closure quality index in [83]. The views have been obtained by projecting the
atlas on three of the problem variables. Green and red points in the views correspond
to grasps with large and low values of the index, respectively.
µ = 1, resulting in the shown friction cones of 45◦ . Since in this case the quality index
is smooth at the resolution of the computed atlas, we set d = 1 in Algorithm 4.2,
which corresponds to evaluating one point of each chart only, e.g. its center point.
Algorithm 4.2 evaluated the whole atlas in about 10 seconds in this situation. Note that
if the kinematic constraints of the hand were neglected, the optimal force-closed grasp
would have the contact normals equi-distributed in the plane, i.e., with angles of 120◦
between them [65]. However, this configuration is not reachable in our case, because
it does not satisfy the joint-assembly and contact constraints of the hand-object system
considered. The example, thus, emphasizes the relevance of taking into account both
the kinematic and contact constraints when optimizing a given grasp, as proposed in
this chapter.
To illustrate a case where the quality index exhibits multiple local optima over R,
the same atlas was evaluated according to the normalized inverse condition number
of the manipulability ellipsoid proposed in [7], which is actually the ratio between the
smallest and largest axis lengths of this ellipsoid. Algorithm 4.2 took 1.5 seconds in
4.4 Test cases
atlas
61
best grasp
worst grasp
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
-0.02
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
-0.1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Figure 4.8: The same atlas of Figure 4.7, but now colored according to the
manipulability index defined in [7]. Green, red points correspond to grasps with a large
and low value of this index, respectively. Black corresponds to near-singular grasps.
Note that, this measure yields three local optimal grasps.
this case, and produced the results shown in Figure 4.8, where the atlas views coincide
with those in Figure 4.7, but are now colored according to the new index. In the figure,
green and red correspond to grasps with large and low values of this index, and black
corresponds to grasps where the index is below 10−3 . In such near-singular grasps, the
manipulability ellipsoid flattens in at least one direction, meaning that the hand can
hardly move the object along that direction while maintaining the required contacts.
Figure 4.8 shows 2D projections of this ellipsoid revealing this fact, for the best and
worst grasp configurations found. One of the fingers is fully extended in the worst
configuration, which is in agreement with the inverse singularity condition of the 3-RRR
parallel manipulator equivalent to this grasp [? ]. Note from the figure that this quality
index would pose difficulties to local optimization methods, since the index yields three
local optimal grasps.
Finally note that, in a precision manipulation task, both the force-closure and the
manipulability criteria may need to be taken into account. However, as it can be
appreciated when comparing Figures 4.7 and 4.8, these two criteria are conflicting in
this example. The global optimum in Figures 4.8, for example, corresponds to a point
62
Optimizing Grasp Configurations
atlas (serial combination)
best grasp
Figure 4.9: The same views of the atlas of Figures. 4.7 and 4.8, but now evaluated
under a serial approach that combines the force-closure and manipulability indices.
The regions with a force-closure index above 0.2 are indicated in blue in the atlas.
These regions are then evaluated according to the same manipulability index used in
Figure 4.8 to select the best configuration on them.
with a low value of the force closure index in Figures 4.7. We use a serial evaluation
approach, and optimize the manipulability index only for those grasps with a minimum
value of the force-closure index. Figures 4.9 shows the result of such strategy, obtained
by applying Algorithm 4.2 with I= (I1 , I2 )⊤ , where I1 and I2 are the force-closure index
and the inverse condition number of the manipulability ellipsoid, respectively, and using
l = (0.2, 0)⊤ and w = (0, 1)⊤ . These thresholds are set so that about half of the charts are
discarded using the force-closure index. Clearly, the optimal grasp would be different if
the indices were considered in the reverse order or with different thresholds.
4.4.2 The Schunk Anthropomorphic hand
To validate the approach on a complex grasping device, we have applied it to the Schunk
Anthropomorphic hand. Assuming that all joints are independently actuated, this hand
has 13 degrees of freedom and, formulated according to Chapter 3, Eq. (4.1) is a hand
system of 88 equations in 101 variables. The tests are performed using three objects:
4.4 Test cases
63
(1) a can, (2) a jeweler’s screwdriver, and (3) a Marquina oil bottle. In all cases, the
grasps are to be performed using three fingers, and the complexity of the examples is
determined by the dimension and the distribution of the contact patches.
In the case of the can, the contact patches at the fingertips are punctual. Moreover,
the contact patch on the object for the thumb is reduced to a line along the can to avoid
repeated solutions caused by the axial symmetry of the can, but the two other fingers are
allowed to contact an identical cylindrical patch defined all over the surface of the can.
Despite involving only one contact patch, the large extension of this patch makes this
test case a hard one, because a large atlas will have to be computed. In the case of the
screwdriver, a point on the index fingertip is set to be in contact with the flat head (to
be able to apply forces that ensure a proper contact of the screwdriver with the screw),
a point on the thumb is limited to move on a line along the screwdriver’s body (to avoid
symmetric solutions), and a curve on one side of the last phalanx of the third finger is
constraint to contact the screwdriver’s body at any point. Thus, this example illustrates
the applicability of the method under different contact models. Finally, to properly
dispense oil with the Marquina oil bottle, a point on the index finger must contact the
top of the bottle along a curve, and points on the two other fingers must touch patches
in the middle and bottom sections of the bottle, respectively. Therefore, the fingers
contact the object on three disjoint patches with different sizes and orientations, which
represents a general situation for the proposed approach.
After the contact constraints are imposed, Eq. (4.9) involves 128 equations in 136
variables (and hence t = 8) in the first example, 131 equations in 141 variables (and
hence t = 10) in the second case, and 133 equations in 142 variables (and hence t = 9)
in the third example. Thus, in order to obtain a set R of dimension k = 2, s must be set
to 6, 8, and 7, respectively. By removing at most 8 postural synergies out of 13, we still
retain more than 99% of the motion capability of the hand. In this case, the synergies
are computed from a database of human hand configurations.The atlases for the three
examples were obtained in 170, 180, and 18 seconds using Algorithm 4.1 with r = 1 and
ǫ = 0.5, and they include 4900, 4800, and 400 charts, respectively. These times agree
with the fact that the stronger the contact constraints, the less the number of charts in
the atlas, and thus, the faster the generation of such atlas.
Figure 4.10 shows the results of applying Algorithm 4.2 with d = 1 on the atlases
obtained on the three examples, using the force-closure quality index with the same
friction coefficient µ = 1 used in the planar hand example. In this case, the spatial
64
Optimizing Grasp Configurations
atlas
best grasp
worst grasp
Figure 4.10: Optimization of the force-closure quality index described in [83] for the
Schunk anthropomorphic hand grasping a can (top), a jewelry screwdriver (middle),
and an oil drizzler (bottom), assuming punctual contacts with a friction coefficient
of µ = 1. Two views of the atlas obtained for each object are shown, where green
and red points correspond to configurations with large and small values of the index,
respectively, and black points correspond to non-force-closed grasps.
4.4 Test cases
65
atlas
best grasp
worst grasp
Figure 4.11: Optimization of the inverse of the condition number of the manipulability
ellipsoid defined in [7], for the Schunk anthropomorphic hand grasping the same
objects as in Figure 4.10. Two views of the atlas obtained for each object are shown,
where green, red, and black points respectively correspond to grasps with a large, low,
and null value of this index.
66
Optimizing Grasp Configurations
atlas (serial combination)
best grasp
Figure 4.12: The best grasp found when optimizing the manipulability criterion on
those grasps that exhibit a minimum value of the force-closure index (the blue regions
indicated in the atlas).
friction cones are linearly approximated using 8 generators. The optimization for the
three examples took 115, 120, and 9 seconds, respectively. These times basically scale
with the number of charts in the atlas.
Figure 4.11 shows the results of applying again Algorithm 4.2 on the same three
atlases, but now considering the inverse condition number of the manipulability ellipsoid, as done in Figure 4.9. The evaluation took 17, 16, and 1.5 seconds, respectively. In
this case the execution of Algorithm 4.2 is faster because here we only have to solve a
relatively small eigenvalue problem for each tested grasp.
In the oil bottle test case, we also optimized the manipulability criterion with k = 3,
simply by considering one additional principal hand motion. In this case, the atlas
includes 12800 charts and the optimization takes 1050 seconds. However, the optimal
grasp returned is almost the same as that obtained with k = 2, which confirms that,
in this example, 6 postural synergies are enough to capture most of the mobility of the
hand.
4.5 Summary
67
Finally, we use the screwdriver example to illustrate the results of optimizing the
grasp under a criterion combining two quality indices. In this case, we have computed
the grasp that maximizes the manipulability criterion, subject to exhibit a normalized
force-closure quality index equal to or larger than 0.675 (i.e., using l = (0.675, 0)⊤
and w = (0, 1)⊤ in Algorithm 4.2), obtaining the grasp shown in Figure 4.12. Note how
removing the lower quality force-closed grasps results in two separate regions where
to optimize the manipulability. Despite one of the regions yields better grasps than the
other, a local optimizer might get stuck in the bad region, if not initialized in the region
including the optimum grasp.
4.5 Summary
This chapter has proposed a procedure for optimizing the quality of a robotic grasp
subject to satisfying the specified contact constraints established between the hand
and the object. The procedure enforces the satisfaction of all kinematic and contact
constraints of the hand-object system during the whole optimization process, and it is
global, in the sense that it explores the whole set of relevant grasps attainable from a
given point, determining the optimal grasp at the desired resolution, without getting
trapped into local extrema. The method is also general, meaning that it can be applied
to any hand structure, and to any desired set of quality indices. The efficiency of the
method critically depends on the dimension of the traced manifold. In the case of
grasps, however, postural synergies allow reducing the dimension of such manifold
considerably. Actually, the proposed method can keep a large number of postural
synergies (up to 7 out of 13 for the Schunk hand), while previous methods [20] use
a smaller number of them (typically 2). This is because the method proposed here
integrates the contact constraints a priori, which already introduces a large dimension
reduction.
5
Accounting for Compliance in the Grasp
This chapter proposes a solution to the grasp synthesis problem in the case
where compliance is present in both the hand and the object. In the two
previous chapters, a grasp was obtained under the assumption that the contacting
bodies and the actuation were rigid. However, there are cases in which small
deformations occur when applying the required forces through the contacts, and
the hand joints should accommodate for such deformations to keep the object
firmly held. To tackle this issue, we present a kinetostatic formulation that models
the compliance at the joints and the contacts by introducing springs. This provides
a proper framework to find a kinematically feasible and restrained grasp by
considering simultaneously the necessary constraints. As a consequence, a solution
of the proposed system of equations results in a set of hand configurations that
allow to execute the grasp using a position controller.
Several hands are designed with soft elements, with the purpose of increasing the
performance of the grasp under different types of situations, such as uncertainty in
the geometric models and the finger positioning, and the lack of precise contact force
measurements (Figure 5.1). Even though the rigid body approach is good enough in
many practical applications, considering the finger and contact compliances introduced
by the various soft elements would yield a more accurate grasping model. A similar
problem is found as well on the synthesis of stable configurations of tensegrities [46,
55], since having a structure supported by beams and cables results in a considerably
elastic structure. The compliance in a grasp has been determined to be a function
of servo compliance, structural compliance, changes in the grasp geometry, coupling
among the different joints and fingers, and different contact types and changes in
70
Accounting for Compliance in the Grasp
the contact locations, e.g. due to rolling or sliding [25]. Recent works account for
these compliances to some extent [19, 44], but they either miss some of the necessary
constraints, limit the approach to planar grasps, or use a reduced number of fingers.
This chapter proposes a more general solution to the synthesis of compliant grasps
using a kinetostatic formulation inspired by the notion of soft synergies [6, 34, 81,
82]. The approach effectively introduces an elastic model of the hand whereby the
physical hand is attracted towards a reference hand through a set of virtual springs at
the joints, while being repelled by the object through springs at the contacts, which
are also supposed to be compliant. A system of equations is proposed to account for
the behaviour of such model, which implicitly enforces the necessary constraints to find
a kinematically-feasible and restrained grasp in the presence of compliant elements.
As a consequence, any solution to this system allows to execute the grasp using a
position controller, because it provides the actuated-joint setpoints that keep the object
held after the deformation of the hand-object system. The dimension of the problem
is considerably large, so that the use of a local search method becomes inevitable
to compute a solution. However, the method can be properly initialized using an
estimation of the solution computed with the methods in Chapters3 and 4, for instance,
in combination with joint torque determination methods [49].
5.1 Kinetostatic formulation of a grasp
The formulation involves the specification of three hand configurations, as shown in
Figure 5.2. The outer hand configuration (black) accounts for the kinematic-feasibility
constraints, i.e. whether the fingertips can actually reach the object surface given the
kinematic structure of the hand. The inner hand configuration (blue) produces joint
torques that attract the outer and yield an interaction with the object. Such interaction
generates reaction forces that repel the hand. Thus, the middle hand configuration
(green) is the actual grasp where the joint torques and contact forces are balanced
accounting for restrainability.
5.1.1 Model description
For simplicity, in this chapter we assume precision grasps, i.e. that only one contact per
finger at the fingertip is used. Precision grasps are most often used for dextrous manip-
5.1 Kinetostatic formulation of a grasp
71
Figure 5.1: THE First hand [8] (right), developed at the Interdepartamental Research
Center “E. Piaggio”, is cable-driven, which is considerably more compliant than a geardriven system, such as that of the Schunk Anthropomorphic hand [106] (left), based on
the DLR II hand.
ulation [24]. While the soft synergy model can easily account for inner-hand contacts
and power grasps, the algorithmic complexity grows with the number of contacts to
place and with the number of kinematic constraints.
A robotic hand is usually composed of several articulated fingers attached to a
palm. The hand palm is positioned and oriented with respect to the world using the
matrix TH ∈ SE(3). The hand is composed of n fingers, each of them articulated
P
through mi revolute joints, for i = 1, . . . , n, which sum up to m = ni=1 mi hand joints.
The rotation angle of the j-th joint at the i-th finger is the joint value qij ∈ S, where
S denotes the angular nature of values. The phalanges are positioned and oriented
with respect to the world using the homogeneous matrix Tij , which depends on the
joint values, qij , for j = 1, . . . , mi and i = 1, . . . , n. Thus, by collecting all joint values
in the vector q = (q11 , . . . qij . . . , qnmn ), a configuration of the hand is represented by the
pair (q, TH ) ∈ Sm × SE(3).
There are n given contact points on the hand, specifically at the fingertips. A
reference frame, positioned and oriented using the matrix Xc ∈ SE(3), is placed at
the contact point xc ∈ R3 , for c = 1, . . . , n. The outward normal vector with respect to
72
Accounting for Compliance in the Grasp
TH
t
q11
r
q11
κq11
r
qi1
t
qi1
g
q11
κqi1
S1
g
qi1
Sc
t
q1j
r
q1j
κq1j
r
qij
g
q1j
κqij
Xt1
Xg1
t
qij
g
qij
K1
Xgc
TO
Kc
Xtc
Figure 5.2: Kinetostatic model using springs at the joints and at the contact. The
grasp is characterized by three hand configurations accounting for the reachability and
restrainability of a grasp.
the fingertip surface at the contact point is denoted as n̂c . Both the contact point and
the contact normal depend on the hand configuration (q, TH ). The number of contact
locations, n, is assumed to be, at least, the minimum required to restrain the object
according to the chosen contact model (e.g. n = 7 for non-frictional contacts spatial
grasps (Section 2.1).
The object position and orientation with respect to the world is defined using the
matrix TO ∈ SE(3). Without loss of generality, the object frame represented by TO
is considered to be the absolute coordinate frame. On the object surface, there is
a given contact region, Sc , for each contact point, xc , on the hand, defined by the
5.1 Kinetostatic formulation of a grasp
Symbol
TH
TO
m
n
qij
κqij
qt
qg
qr
δq
Xgc
Xtc
δXc
Kc
hc
Sc
γc
ac
bc
sc
m̂c
J
G
Kq
K
73
Definition
Hand reference frame
Object reference frame
Number of joints
Number of fingers and contacts
Value of the j-th joint at the i-th finger
Stiffness of torsional spring of the j-th joint at the i-th finger
Touching joint configuration
Grasping joint configuration
Reference joint configuration
Joint displacement from qg to qr
Contact reference frame, with origin at the point xgc
Contact reference frame, with origin at the point xtc
The displacement from Xtc to Xgc
Stiffness of the spatial spring at the c-th contact point
Number of linear springs used at the c-th contact point
Contact region on the object corresponding to the c-th contact point
Parameter that defines a point on Sc
Dimensionality of region Sc (point, curve, surface)
Orientation freedom at the c-th contact (normal alignment or not)
Contact point on the region Sc defined by γ c
Normal vector at the point sc defined by γ c
Hand Jacobian evaluated at the grasping configuration
Grasp matrix evaluated at the grasping configuration
Hand stiffness matrix
Contact stiffness matrix
Table 5.1: Notation used in the kinetostatic model of a grasp.
parametrization sc (γ c ), where γ c ∈ Rac , with ac = 0, 1, 2, the vector of parameters
defining a point, a curve or a surface, respectively. The parametric inward normal
vector with respect to the object at the point sc , is denoted as m̂c (γ c ).
The contact forces and joint torques are modeled using a spatial spring at the contacts and a torsional spring at the joints with known stiffness constants Kc and κqij ,
respectively. The rest position for the contact springs is defined by the touching configuration of the hand, (qt , TH ), that makes the desired contact point, xtc , reach the corresponding region, Sc , on the object. The reference configuration of the hand, (qr , TH ),
pulls the fingers inside the object surface loading the springs at the joints. The grasping
74
Accounting for Compliance in the Grasp
configuration, (qg , TH ), floats between those two, modifying the lenght of the joint
springs when the contact frame is moved to Xgc , loading the contact springs and pushing
the fingers out of the object back to Xtc . The list of symbols is summarized in Table 5.1.
5.1.2 Characterizing kinematically-feasible grasps
A grasp is kinematically-feasible when the touching configuration of the hand, (qt , TH )
makes the fingertips contact properly on the corresponding regions on the object,
i.e. xtc ∈ Sc . Thus, the contact constraint is written as
kxtc − sc (γ c )k2 = 0.
(5.1)
In order to avoid interpenetrations of the fingertips into the object, the vectors
normal to the fingertip surface, n̂tc , and to the object surface, m̂c at the contacting
points are aligned by requiring
n̂tc · m̂c (γ c ) = 1.
(5.2)
And, the position vector rij of matrices Tij are forced to be outside the object by
requiring
m̂Tc (sc − rij ) > 0,
(5.3)
that is, a positive projection of the vector going from rij to sc onto the diretion normal
to the object at the contact point.
Finally, the joint values of real robotic hands are subject to mechanical limitations.
Hence, the touching configuration must fulfill the preceding constraints while the joint
values stay within the valid range
ql ≤ qt ≤ qu ,
(5.4)
with ql and qu being the vectors of minimum and maximum values, respectively.
5.1.3 Characterizing restrained grasps
A grasp is restrained when the object motion due to external perturbations is prevented
by applying valid contact forces according to the contact model and to the grasping
configuration (object equilibrium), and such contact forces are balanced by applying
joint torques according to the reference configuration (hand equilibiurm) [19]. It is
5.1 Kinetostatic formulation of a grasp
75
worth noting that, the restrainability condition, together with the assumption that n is
the minimum number of contacts required to restrain the object according to the contact
model, yields a force-closure grasp as defined in [5].
Object equilibrium
Each contact force is modeled using a spatial springs conformed of hc linear springs
connecting the frames Xtc and Xgc at the contact (see Figure 5.3). Thus, we express the
effect of these springs acting on the object, i.e. the c-th contact force, as the sum of all
spring forces as wc = [ p̂1,c . . . p̂hc ,c ]λc , where p̂k,c ∈ R6 represents the supporting
line of the k-th spring passing through the contact point xgc , and λc = [ λ1,c . . . λhc ,c ]T
collects the force magnitudes of the springs obtained as λk,c = −κk,c dk,c , where dk,c
is the spring elongation and κk,c the stiffness constant of the k-th spring. Thus, the
magnitude of the contact force can be written as λc = −K̃c dc using the diagonal matrix K̃c = diag(κ1 , . . . , κhc ) and dc = [ d1,c . . . dhc ,c ]T . Then, introducing the matrix
Pc = [ p̂1,c . . . p̂hc ,c ], we express the c-th contact force as
wc = −Pc K̃c dc .
(5.5)
The displacement that goes from Xtc to Xgc can be parametrized using six independent
variables, known as the exponential coordinates [68], expressed as
Xgc = e(δXc ) Xtc ,
(5.6)
where e(δXc ) is the exponential map representing the relative finite rigid body displacement, δXc ∈ se(3), between Xtc and Xgc . The spring elongations are obtained by
projecting the displacement onto the supporting lines of the springs as
dc = PTc δXc .
(5.7)
Substituting Eq. (5.7) in Eq. (5.5) yields the expression of the contact force as a function
of the touching and grasping configuration,
wc = −Kc δXc ,
where Kc = Pc K̃c PTc models the compliance of the c-th contact.
(5.8)
76
Accounting for Compliance in the Grasp
Xtc
κ2,c
κ1,c
κ3,c
Xgc
p̂3,c
p̂2,c
p̂1,c
Figure 5.3: Representation of the spatial spring placed at the contact locations,
using hc = 3 linear springs. The triangles are drawn for clarity, however, the vertices are
coincident with the contact point, making p1,c , p2,c and p3,c , pass through the origin of
the frame placed at the contact point at the touching (black) and the grasping (green)
configuration, represented by Xtc and Xgc , respectively. The rest lenght of the springs
correspond to the configuration where Xgc = Xtc .
Since we are assuming that n is the minimum number of required contact points, we
P
need to enforce the equilibrium of all contact forces, i.e. nc=1 wc = 0. Thus, considere-
ing the matrix G = [ P1 . . . Pn ], the block diagonal matrix K = blkdiag(K̃1 , . . . , K̃n ),
the block diagonal matrix P = blkdiag(P1 , . . . , Pn ), and collecting all contact displacements in δX = [ δXT1 . . . δXTn ]T , the object equilibrium can be expressed as
GKPT δX = 0.
(5.9)
Additionally, the contact forces must comply with the contact model. Typical contact
models in grasping include: the point contact without friction (PC), point contact with
5.1 Kinetostatic formulation of a grasp
77
friction (PCWF), and contact with a soft finger (SF) [83]. They can be implemented
by considering springs only along the constrained directions, with hc = 1, 3, 4, depending whether we use a PC, PCWF, or SF, respectively. Without loss of generality,
the supporting lines of the springs are chosen such that p̂1,c indicates a translation
along the inward normal direction, p2,c and p̂3,c indicates translations along the tangent
directions, and p̂4,c indicates a rotation about the normal direction, with respect to the
object. Thus, the linear and torsional friction coefficients, µc and νc , define an additional
constraint on the vector wc = [ w1,c . . . w6,c ], which must belong to the generalized
friction cone
Cc = {wc ∈ R6 |kwc k∆ ≤ w1,c },
q
q
2
2
2
2
where kwc k∆ can take the form 0, µ1c w2,c
+ w3,c
or µ1c w2,c
+ w3,c
+
(5.10)
1
|w4,c |
νc
depending
on whether we use the PC, PCWF, or SF model, respectively, as proposed in [83].
Hand equilibrium
Each joint torque is modeled using a torsional spring connecting the grasping and the
reference configuration at the joints. The resultant force due to the spring elongation is
written as
wij = ẑij τij ,
(5.11)
where ẑij ∈ R6 represents the supporting line that coincides with the joint rotation axis
at the grasping configuration, and τij is the torque magnitude obtained as τij = κij dij ,
where dij is the spring elongation. The joint displacement of the i-th finger is expressed
as δqi = qri − qgi . Thus, introducing the matrix Zi = [ ẑTi,1 . . . ẑTi,mi ]T , the joint torque
magnitudes that result of applying a force wc at the c th contact point is
Kqi δqi = ZTi wc ,
(5.12)
where Kqi = diag(κqi1 , . . . , κqimi ), for c = 1, . . . , n and i = c in turns, models the compliance of the i-th joint.
Finally, we must ensure that the contact forces are compensated by joint torques.
Since the fingers are independent, the force applied at the c-th should be compensated
by torques at the i-th finger, with i = c. Thus, introducing the block diagonal matrix
J = blkdiag(Z1 , . . . , Zn ) and the vector w = [ w1T . . . wnT ]T = PKPT δX, the hand
78
Accounting for Compliance in the Grasp
equilibrium can be expressed as
Kq δq = JT PKPT δX,
(5.13)
where Kq and δq consider all joints, ordered accordingly to the corresponding row
of JT .
In addition, the grasping configuration must be attainable by the actual hand, hence
the joint value limitations are again applicable here as
ql ≤ qg ≤ qu .
(5.14)
Finally, the joint torques are subject to mechanical limitations as well, written as
|Kq δq| ≤ τ u ,
(5.15)
with τ u being the vector of maximum torques that the joint motors can exert, and the
absolute value and the inequality must be read component-wise. Even when qr is not
subject to the mechanical limitations, the fingers must push against the object, enforcing
a minimum torque τ l by including
|Kq δq| ≥ τ l .
(5.16)
5.1.4 System overview and dimension analysis
A grasp configuration y = (qr , qg , qt , Th , γ) is kinematically feasible and restrained when
it fulfills Eqs. (5.1), (5.2), (5.9), and (5.13) collected in
Meq (y) = 0,
(5.17)
and (5.3), (5.10), (5.15), and (5.16), transformed in less-than-equal inequalities and
collected in
Mineq (y) ≤ 0,
(5.18)
while staying within the valid ranges defined by (5.4) and (5.14), for c = 1, . . . , n
contacts and j = 1, . . . , mi joints with i = c.
The number of variables, in this case, correspond with the internal degrees of free-
5.2 Solution strategy
dom nv = 3m +
Pn
c=1
79
ac +
Pn
c=1 bc ,
where m is the total sum of joints per hand configu-
ration, ac defining the whether the contact region is a surface, a curve or a point, and bc
being 1 or 3 depending whether the normal vector at the contact points are aligned or
not. The number of algebraic constraints is ne = D(n − 1) + m + D, i.e. the constraint
due to the (n − 1) independent loops in D-space, m equations due to the hand equilibrium constraint, D equations from the object equilibrium constraint, with D being 3 or 6
whether it is a planar or a spatial grasp, respectively. Assuming ac = a and bc = b, for c =
1, . . . , n, the dimension of the solution space is then ns = nv − ne = 2m + (a + b − D)n.
In general, this number is relatively high.
5.2 Solution strategy
There may be multiple solutions to the system given by Eqs. (5.17) and (5.18) due to
high nonlinearities in the constraints, even for 0-dimensional solution spaces. Thus,
we propose the minimization of the potential energy of the springs at the joints as the
selection criterion, which is expressed by the objective function
1
Ψ(y) = δqT Kq δq.
2
(5.19)
It is worth noting that, when the constraints are met, the substitution of Eq. (5.13) in
Eq. (5.19), yields
1
Ψ′ (y) = δqT (JT PKPT δX).
2
(5.20)
and, additionally, δX ≈ J(qg − qt ) for small joint displacements. Thus, introducing the
block diagonal matrix K′ = PKPT , Eq. (5.20) becomes
1
Ψ′′ (y) = (qr − qg )T JT K′ J(qg − qt ).
2
(5.21)
By comparing Eqs. (5.21) and (5.19), the criterion can be rewritten as
Ψ′′′ (y) = kKq − JT K′ Jk,
(5.22)
i.e. when the constraints are met and joint displacements are small, the criterion selects
the configuration y in which the contact stiffness seen from the joints equals the joint
stiffness.
80
Accounting for Compliance in the Grasp
The problem can be casted as: Given a hand with n articulated fingers to grasp
an object, with a kinematic configuration defined by the pair (q, Th ), a contact on the
fingertip Xc , its corresponding contact region on the object surface Sc , m joint spring
stiffnesses κij and nhc contact spring stiffnesses κk,c , and friction coefficients µc and νc ,
find a configuration y that minimizes the objective function expressed by Eq. (5.22)
subject to the constraints in Eqs. (5.17-5.18), (5.4), and (5.14). This non-linear optimization problem is in the form required by the MATLAB routine fmincon. We
select the SQP algorithm due to its ability to work out of the solution manifold using a
feasibility reformulation. This slows down the process, however it is desired when the
method is starting and the configurations are far from satisfying the constraints [71].
5.3 Test cases
The method is illustrated here with three experiments. The first one consists of a simple
planar hand grasping an ellipse, the second one consists of a complex robotic hand
grasping an ellipsoid, and the third one is a an informed instance of the problem with a
real execution. The details of each case are shown next.
5.3.1 A simple planar hand grasping an ellipse
In this example, we use a simple planar hand with n = 2 fingers, and mi = 2 joints per
finger, for a total of m = 4 joints. The object to be grasped is an ellipse, and the contact
regions cover competely the ellipse boundary (Figure 5.4). The normal vectors are not
aligned at the contact points, hence bc = 1 (one extra internal degree of freedom due
to free orientation of contacting link). The dimension of the solution space in this case
is ns = 6. The kinematic structure, spring constants and friction coefficients needed to
write the grasp synthesis problem as stated in Section 5.2 are shown in Table 5.2. In
this case, the initial guesses are randomly generated, but biased towards the center of
the search range for each variable. The results using the proposed method from two
different initial guesses are shown in Figure 5.4. The relation between the touching,
grasping and reference configurations can be appreciated, for instance, in the left figure,
where most of the work in the right finger is done by the outer joint, not the case for
the left finger, as expected from their respective touching configurations and the need
of pushing against each other. For a different touching configuration and the same
5.3 Test cases
81
2
0.5
0
1.5
-0.5
1
-1
0.5
-1.5
0
-2
-0.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 5.4: Two solutions obtained for a simple hand satisfying all constraints. The
color code corresponds to that of Figure 5.2, black for the outer hand configuration,
blue for the inner hand configuration, and green for the middle hand configuration.
pushing requirement, which is the second solution found, the grasping and reference
configurations are different. Note, in this case, the little work made by the joints in the
left finger, where most of the contact force is absorbed by the structure, that traduces
into high Cartesian stiffness, requiring less load in the joint spring. Though the hand is
symmetric and their parameters are symmetric, the solutions are not symmetrical due
to the randomness for their initial guesses.
5.3.2 THE First hand grasping an ellipsoid
The second example employs a complex hand, in this case, using the paremeters of teh
Schunk Anthorpomorphic shown in Figure 5.1(right). The grasp uses n = 3 fingers
(three out of the four available), with m1 = m2 = 3 and m3 = 4 joints, for a total
of m = 10. The object is an ellipsoid, and the contact regions completely cover its
surface (Figure 5.5). The normal vectors must be aligned, hence bc = 1. The dimension
of the solution space in this case is ns = 11. The kinematic structure, spring constants
and friction coefficients needed to write the grasp synthesis problem as stated in Section 5.2 are shown in Table 5.2. In this case, the initial guess was set by introducing the
constraints sequentially, such that it was as close as possible from the solution manifold.
The result using the proposed method is shown in Figure 5.5.
82
Accounting for Compliance in the Grasp
a) Kinematic structure and limit values
Finger base reference frame
Phalanx length [cm]
Joint limits [deg]

1 0 1


T11 =  0 1 0 ,
0 0 1


1 0 −1


T21 =  0 1 0 
0 0 1
All phalanges are of length 1

ql =
qu =
Torque limits [Ncm]
Contact point xc , ∀c in local [cm]
Regions Sc , ∀c [cm]
τ min
h
0 −90 0 −90
h
90 45 90 45
i⊤
,
i⊤
= 1[1], τ max = 10[1],
where [1] ∈ Rm is a vector containing ones
" #
0
x=
1
"
#
0.8 cos(γc )
s=
,
0.7 sin(γc )
with γc ∈ [0, 2π]
b) Coefficients
Joint stiffness [N/rad]
Kq = blkdiag(1, 1, 1, 1)
Friction (PCWF) µc , ∀c
µ = 0.5
Contact stiffness K̃c , ∀c [N/cm]
K̃ = blkdiag(1, 1)
Table 5.2: Kinematic and elastic parameters for the planar hand.
5.3.3 Experiments with THE First hand grasping a ball
For this example, the five-fingered hand developed at the Pisa laboratory was used,
namely THE First hand [8]. In this case, the hand was mounted on a fixed base,
therefore, the reachability constraint was calculated in a decoupled way considering
the workspace of the hand. The object was placed such that the fingers could reach
it, and the constraint was verified. The restrainability constraint was then met using
the solution strategy presented in Section 5.2. The results from the implementation are
shown in Figure 5.6. The hand first make contact fulfilling the reachability constraint
5.3 Test cases
83
-100
100
-50
50
0
100
0
50
0
-50
-100
50
-150
Figure 5.5: A solution obtained for the Schunk Anthropomorphic hand satisfying all the
constraints performing the grasp with three fingers (thumb, index and middle). The
color code corresponds to that of Figure 5.2, black for the outer hand configuration,
blue for the inner hand configuration, and green for the middle hand configuration.
Figure 5.6: A solution obtained and implemented on THE First Hand [8]. The object is
placed at the computed position (left). The fingertips make contact at the object surface
satisfying the rechability constraint (middle). The fingers perform the squeezing action
to firmly hold the object (right).
84
Accounting for Compliance in the Grasp
a) Kinematic structure and limit values
Finger base reference frame
DH parameters (aj , αj , dj )i
[cm,rad,cm]
Joint limits [deg]
Torque limits [Ncm]
Contact point xc , ∀c in local [cm]
Regions Sc , ∀c [cm]


0 −3

0 27.1 


T11 = 
,

1
0 
0
1


0
0
1 −4.3
 0.035 −0.99 0 40.165 


T21 = 
,
 0.99 0.035 0 145.43 
0
0
0
1


0 0 1 −4.3

 0 −1 0
0


T31 = 

 1 0 0 145.43 
0 0 0 1

0
0
0
 0
−π/2 0 


,

 67.8
0
0 
30
0
0
for j = 1, 2, 3, 4, for i = 1, 2, 3, and qi3 = qi4
h
i⊤
ql = 0 −15 −4 4 −15 −4 4 −15 −4 4 ,
h
i⊤
qu = 90 15 75 75 15 75 75 15 75 75
1
0
0
0
0
1
0
0
τ min = 10[1], τ max = 1000[1],
where [1] ∈ Rm is a vector containing ones
h
i⊤
x = 0 29.5 0


60 cos(γ1,c ) sin(γ2,c )


s =  40 sin(γ1,c ) sin(γ2,c ) ,
20 cos(γ2, c)
with γ1,c ∈ [0, 2π] and γ2,c ∈ [0, π]
b) Coefficients
Joint stiffness [N/rad]
Kq = 100[blkdiag(4, 3, 2, 1, 3, 2, 1, 3, 2, 1)]
Friction (PCWF) µc , ∀c
µ = 0.25
Contact stiffness K̃c , ∀c [N/cm]
K̃ = blkdiag(5, 5, 5)
Table 5.3: Kinematic and elastic parameters of THE First hand.
5.4 Summary
85
(Figure 5.6, middle) and then perform the squeezing action fulfilling the restrainability
constraint (Figure 5.6, right), completing the grasp action.
5.4 Summary
The proposed approach tackles simultaneously the contact constraint, the object and the
hand equilibrium issues that a grasp must satisfy in order to be reachable and restrained.
This is obtained by introducing torsional springs modeling the joint compliance, and
a set of linear springs for the contact interaction. This leads to a solution where all
the variables ultimately employs configuration values, and therefore, the hand can be
commanded to grasp the object using only a position controller.
The results show practicable solutions provided by the proposed method for illustrative examples, suggesting how to: (i) reach the specified regions on object with the
fingertips, (ii) apply the forces in the directions allowed by the contact model within
the friction constraints, and (iii) compensate such forces using the hand joints, i.e. the
hand performs a reachable and restrained grasp of the object with the minimum effort
of the joints.
Part III
Approach Path Planning
6
Reducing the Hand Configuration Space
A hand configuration is determined by all the joint values and the position and
orientation of the wrist. Thus, the dimension of the hand configuration space is
very high, typically over ten. A menaningful reduction is of great importance in
order to apply motion planning techniques that work better in lower dimensions.
Concerning the finger joints, the main consideration that supports the reduction
in the finger configuration spaces is that the human hand has several joint
movements that are not completely independent, that is, there are some joints
that can be coordinated using some relation. A typical example is given by the
last two joints of each finger, which, in general, can not be moved independently
in the free space. In the same line, some other correlations can be found when the
human hand motions are carefully analyzed. These correlations can be mapped
to a mechanical hand in order to mimic human-like motions while exploring the
relevant workspace. Concerning the wrist, the main consideration that supports
the reduction is that the hand usually faces the object during the approach phase.
Therefore, relational positioning techniques can be used to introduce proper
constriaints in the wrist configuration space. This chapter provides the insights
of both finger motion coordination and wrist orientation constraint.
6.1 Finger motion coordination
The main consideration that supports the reduction in the finger configuration spaces is
that the human hand has several joint movements that are not completely independent,
that is, there are some joints that can be coordinated using some relation. A typical
90
Reducing the Hand Configuration Space
example is given by the last two joints of each finger, which, in general, can not be
moved independently in the free space. In the same line, some other correlations can
be found when the human hand motions are carefully analyzed. These correlations can
be mapped to a mechanical hand in order to mimic human-like motions while exploring
the relevant workspace.
A relevant previous work in this line [104] uses an initial set of grasping configurations to find a bidimensional grasp subspace, i.e. a set of hand configurations used to
grasp different objects. This subspace, already used in Chapter 4, has also been used
in other works to look for grasping configurations [20, 117] as well. Dimensionality
reduction techniques have also been used to synthesize human-like motion in graphic
applications [101].
Unlike the works specifically oriented to synthesize a grasp, we use here an initial set of unconstrained general hand configurations in order to model the real hand
workspace, that is, the self-collision free region of the hand configuration space Xh .
The procedure consists of acquaring a number of samples of human hand postures
using a sensorized glove, and then, mapping those samples to the mechanical hand
configuration space, Xh . The samples in Xh are analyzed using a Principal Component
Analysis (PCA) to find the direction with largest dispersion, which is iteratively repeated
considering orthogonal directions until a new base of Xh is generated. Then, by selecting
the first p vectors of this base and properly choosing a bounding box Bh aligned with
these vectors and centered at the mean value of the original set of points, a good
bounded approximation of the hand workspace. This approximation is the subspace E,
defined by the Principal Motion Directions (PMDs), which represent ways to coordinate
the fingers.
6.1.1 Experimental set-up
The experimental set-up involves: an anthropomorphic mechanical hand, a sensorized
glove, and a hand/robot simulator connected with the real elements. The main relevant
details about these elements are:
Mechanical hand: The hand is the Schunk Anthropomorphic hand (SA) [106], shown
in Figure 6.1 (left) , which was built upon the DLR hand model [14]. It has three
fingers with four joints plus the thumb with five joints, in all of them the distal and
middle flexion joints are mechanically coupled, thus there are a total of 17 joints
6.1 Finger motion coordination
91
m
q
i
l
h
12
9
j
;11 ;10
8
7
u
n
k o r
s
v
t
e d
6
5
4
c
0
3
g
f
p
b
a
2 1
Figure 6.1: Mapping between a mechanical hand and a sensorized glove. The numbers
indicate an independent joint on the SA hand (left). The letters indicate a sensor on the
CyeberGlove (right)
with only 13 independent degrees of freedom. The extra degree of freedom of the
thumb is called the “thumb base joint”, numbered with “0” in Figure 6.1 (left),
and moves the whole thumb with respect to the palm. This is a particular joint
because it has an independent controller and cannot be moved in a synchronized
way with the other hand joints. This enforces to do some particular adjustments
in the motion planner to allow real implementation in practice, as will be shown
in Chapter 7.
R
Sensorized glove: The glove is the CyberGlove
, shown in Figure 6.1 (right). It is a
fully instrumented glove that provides 22 joint-angle measurements using resistive
bendsensing technology. It includes three flexion/extension sensors per finger,
four abduction/adduction sensors between the fingers, a palm-arc sensor, and two
sensors to measure the flexion and the abduction of the wrist.
Hand and robot simulator: The simulator is used to visualize the movements of the
mechanical hand associated with the movements of the human operator hand,
which are captured with the sensorized glove, as shown in Figure 6.2. The simulator detects collision between the involved elements. For more details, see [84].
92
Reducing the Hand Configuration Space
Figure 6.2: A human hand with a sensorized glove connected to the mechanical hand
simulator.
The schema of the whole experimental set-up is illustrated in Figure 6.3, including the
type of connection between the different elements.
ethernet
serial
fcu
planner
cyberglove
sim
isa
serial
card
task
Figure 6.3: Schematic representation of the experimental setup.
sah
6.1 Finger motion coordination
93
6.1.2 Experimental protocol
The postures of a human operator hand are captured using the sensorized glove. The
operator freely moves his/her hand in an unconstrained way, i.e. without performing
any specific task, trying to cover the whole hand workspace. There is no guarantee that
the operator actually covers the whole workspace, but in this way it is expected that
he/she performs the most natural and evident hand movements, thus the most natural
and evident hand postures are captured. The operator can have a continuous visual
feedback of the mechanical hand postures associated with his/her hand postures by
means of the hand simulator, as illustrated in Figure 6.2.
The mapping of the data obtained from the glove sensors to the joints of the SA hand
is done considering the following issues (see Figure 6.1):
• The palm of the mechanical hand is rigid, therefore the palm arc sensor v and the
wrist flexion and abduction sensors b and a are ignored.
• The mechanical hand lacks the little finger, therefore the sensors u, t, s and r are
ignored.
• The mechanical hand has a one-to-one coupling between the medium and distal
phalanx of each finger, therefore the distal phalanx sensors i, m, and q are ignored.
• The abduction is measured in a relative way in the glove, i.e. sensors j and n
give the relative angle between the index and the middle fingers and between the
middle and the ring fingers, respectively. Therefore, the mapping uses the middle
finger as reference, i.e. the joint #7 is fixed to zero, and sensors j and n are
directly associated to joints #4 and #10, respectively.
• The use of sensor c to control joint #1 produces a more natural motion of the
SA hand than using sensor d, because sensor d measures the relative abduction
between the thumb and the index. Therefore sensor c is used for both joints #0
and #1.
Then, only 11 values from the 22 sensors available in the glove are used in practice to
command the joints of the SA hand. The complete mapping is shown in Table 6.1. Note
that this mapping makes the motions of the SA hand to be defined with 11 independent
parameters despite it has 13 actuated joints.
94
Reducing the Hand Configuration Space
Cyberglove Sensor
→
SA Hand Joint
Id.
Name
Id.
Name
c
c
e
f
j
g
h
k
l
n
o
p
thumb roll
thumb roll
thumb inner
thumb outer
index abduction
index inner
index middle
medium abduction
medium inner
medium medium
ring abduction
ring inner
ring medium
0
1
2
3
4
5
6
7
8
9
10
11
12
thumb base
finger base (thumb)
proximal phalanx (thumb)
medium phalanx (thumb)
finger base (index)
proximal phalanx (index)
medium phalanx (index)
finger base (medium)
proximal phalanx (medium)
medium phalanx (medium)
finger base (ring)
proximal phalanx (ring)
medium phalanx (ring)
Table 6.1: Correspondence between the CyberGlove sensors and the joints of the SA
hand shown in Figure 6.1.
6.1.3 Data analysis
Dimensionality reduction of a feature set is a common preprocessing step used for
pattern recognition and classification applications as well as in compression schemes.
Principal Component Analysis (PCA) is often used in these fields to reduce multidimensional data sets to lower dimensions for their treatment [45], and it is also used as a
tool in exploratory data analysis as well as for making predictive models. Basically, PCA
involves the computation of the eigenvalue decomposition of a data covariance matrix
or the singular value decomposition of a data matrix, usually after mean centering the
data for each attribute. The larger the eigenvalues or the singular values the larger
the dispersion of the data along the corresponding eigenvector direction. This analysis
allows the identification of the directions of the space where the samples have larger
dispersion.
In this work, PCA is used to reduce the hand configuration space, Xh , of the mechanical hand to a more tractable space, E, of smaller dimension, using for that purpose the
data obtained from the hand postures of a human operator mapped to the mechanical
hand, as described above. The dimension reduction is done based on the correlation
6.1 Finger motion coordination
95
(a) Positive correlation between proximal
phalanxes (joints #8 and #11).
(b) Negative correlation between the index and the ring abductions/adductions
(joints #4 and #10).
(c) Positive correlation between two
medium phalanxes (joints #6 and #9).
(d) Positive correlation between a medium
phalanx and an abduction/adduction
movement (joints #8 and #10).
(e) No correlation between the thumb
base and the medium phalanx of the index
(joints #0 and #5).
(f) No correlation between the thumb base
and the medium phalanx of the ring finger
(joints #0 and #12).
Figure 6.4: Correlation of joints in the hand configuration space.
96
Reducing the Hand Configuration Space
PMD1
PMD2
Figure 6.5: Configurations of the SA hand when it is moved along the first two PMDs.
PMD2
PMD1
Figure 6.6: Configurations of the SA hand when it is moved along a combination of the
first two PMDs.
that there exists between some joints of the mechanical hand when it follows the
hand postures of the human operator. For instance, for a set of 13,500 hand postures
captured with the sensorized glove, Figure 6.4 shows different examples of the obtained
correlations between some particular pairs of joints.
From the captured data it can be seen that the position of joint #0 of the mechanical
hand is rather independent of the other hand joints, of course, with the exception of
6.1 Finger motion coordination
97
joint #1 that is completely equivalent due to the selected mapping; two examples are
given in Figures 6.4(e) and 6.4(f). This, together with the fact that joint #0 of the SA
hand cannot be coordinately controlled with the other joints, motivates the selection
of joint #0 to form part of a base of E. The remaining directions of the base of E are
obtained applying PCA to the samples of the mechanical hand, which are still defined
by 11 independent parameters. The PCA returns a new base of the configuration space
Xh , with the base vectors ordered according to the dispersion of the samples along each
vector direction. The first vector indicates the direction of maximal dispersion of the
samples. The directions indicated by these vectors in Xh are called Principal Motion
Directions (PMDs). In order to illustrate the variation of the hand configuration along
the PMDs, Figure 6.5 shows the hand postures along the two first PMDs, and Figure 6.6
the postures resulting from their linear combination.
In this experimental dataset, the first PFM represents the 42.19% of the total variance, the first two components the 77.12%, and the first three components the 84.71%.
The total accumulated variance as a function of the number of selected first PMDs is
shown in Figure 6.7. Following these results, the use of four PMDs is considered enough
100
90
Total variance (%)
80
70
60
50
40
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of PMDs
Figure 6.7: Total variance covered when using an increasing number of PMDs.
98
Reducing the Hand Configuration Space
to represent, together with the thumb base, the subspace E, which, therefore would be
of dimension 5. The inclusion of the thumb base to define one of the dimensions of E
is a particularity related with the SA hand and it does not reduce the generality of the
approach. Note also that, any desired coordinated motion can be incorporated to define
E, such as the task motion direction proposed in [114], which joins the current and goal
configuration with a linear segment in the configuration space.
6.2 Wrist orientation constraint
The main consideration that supports the reduction is that the hand usually faces the
object during the approach phase. Therefore, relational positioning techniques can be
used to introduce proper constraints in the wrist configuration space. The dimension
reduction for the wrist motions can be achieved, thus, by considering virtual constraints.
In constrained-based motion planning problems, the robot is forced to move in a submanifold with lower dimension than the original space. If sampling-based methods
are used, the planning of collision-free movements of a constrained object does not
require the sampling of the whole configuration space, but only the regions of the
configuration space where the robot is allowed to move: its configuration submanifold.
These submanifolds can be described in terms of geometric constraint sets by explicitly
stating the relations that must hold between two or more geometric entities. Geometric
constraint solvers introduce constraints between the relative position of rigid bodies
using, for instance, distance and angles bewteen planes, lines and points [90], which
then are used to find the map between constraint sets and configuration submanifolds.
The orientation constraint is to be imposed to the reference frame at the wrist, F.
The frame should be oriented such that the vector û always points towards a point
of interest on the object, P , which is defined using the wrist orientation at the goal
hand configuration, as shown in Figure 6.8. This orientation constraint provides more
realistic motions during the approach path, as the wrist is always facing the object
during the approach phase and, typically, reduces post-processes such as prunning unnecessary branches in the path. The submanifold of SE(3) that satisfies this orientation
constraint is 4-dimensional, i.e. the motion of the palm is constrained to four degrees
of freedom, three of translation defining the position, O(x, y, z), of the origin of F, and
one defining the rotation of F about û with an angle α, as illustrated in Figure 6.9.
6.2 Wrist orientation constraint
99
ŵ
v̂
xh
F
û
P
Figure 6.8: The position vector of the hand wrist at a desired configuration, xh , is used
to define point P , which is the point of interest for the constraint.
The mathematical expression describing such constraint can be stated as follows. With
support in Figure 6.10, given a position, O, of the origin of F, to satisfy the orientation
constraint, the frame F must be rotated using the matrix
R1 = Rot(β, n̂),
(6.1)
where β is the angle between û and r̂, the latter being the unitary vector pointing from
O to P , and n̂ is the unitary vector normal to the plane defined by û and r̂, obtained as
n̂ =
û × r̂
.
||û × r̂||
(6.2)
In addition, the frame F is free to rotate any given angle α about û using the matrix
R2 = Rot(α, û).
(6.3)
100
Reducing the Hand Configuration Space
ŵ
v̂
O(x, y, z)
û
α
P
Figure 6.9: Parameters x, y, z and α of the 4-dimensional submanifold that satisfies the
orientation constraint.
R1 = Rot(β, n̂)
n̂
v̂
v̂
O
O
û
ŵ
β
ŵ
r̂
P
û
P
Figure 6.10: Rotation matrix R1 used to define the orientation constraint.
6.3 Summary
101
Finally, the homogeneous transformation that defines the position and orientation
of F satisfying the orientation constraint is computed as

x



 RR y 
 2 1

T=
.


z


0 0 0 1
(6.4)
6.3 Summary
This chapter has presented a suitable way to reduce the dimension of the hand configuration space Xh , such that it can be used in the hand motion planning with a
reasonable computational cost and human-likeliness during the approach path phase.
The reduction of the finger configuration space is done by capturing the human hand
workspace by means of a sensorized glove and then mapping the configurations to
the mechanical hand workspace, where the most relevant finger motion coordinations
can be identified using PCA. The wrist configuration is also constrained to belong to a
configuration subspace of lower dimension. In this case, using a sphere-like constraint,
the hand moves always facing the object, which is usually the case for humans too.
Both reductions are meant to increase the efficiency when planning and, additionally,
improve the quality of the randomly generated samples, as it will be shown in Chapter 7.
It is worth noting that, even when the approach have been developed for the SA hand,
the procedures are general enough to considere any other particular hand.
7
Finding Approach Paths
Planning the motion of a hand-arm system is a hard problem due to the high
dimensionality of the combined system. This chapter deals with the problem of
motion planning of such systems avoiding collisions and trying to mimic real
human hand motions. The approach uses the results from the previous chapter to
propose a solution with a compromise between motion optimality and planning
complexity (time). The proposal is a PRM-based motion planner which is able to
determine the motions of a hand-arm system towards any desired configuration,
for instance, an optimal grasp from Part II, in other words, it provides a solution to
the approach path planning problem for grasps satisfying the task-specific contact
constraints.
In this chapter, we present a new approach for the motion planning of an anthropomorphic hand assembled on a robot arm. It is of particular interest for the
hand-arm movement close to the goal configuration in a cluttered environment, that
is, when the existence of a linear path for the arm considering a bounding volume
for the hand is unlikely to be found. Therefore the movements must be planned
in a meaningful space defined using the hand-arm degrees of freedom. Efficiency
is sought by sampling hand configurations from lower dimensional subspaces, whose
dimension is increased according to the difficulty of finding a collision-free approach
path. Simultaneously, arm configurations are sampled using two strategies. The first
one follows the classical approach, that is using a linear segment connecting the initial
and the final arm configurations, but enhancing the path by allowing samples around it,
as if a box were swept along it, that is, the resulting motion is a quasi-linear path inside
104
Finding Approach Paths
a volume. The second one uses the wrist orientation constraints proposed in Chapter 6.
In both cases, the configuration space of the arm is reduced.
Let X = Xh × Xa be the configuration space of a hand-arm system, where Xh and Xa
are the configuration spaces of the hand and of the manipulator arm, respectively. Then,
the dimension of X is equal to the number of degrees of freedom of the hand plus the
number of degrees of freedom of the arm. The problem to be solved is the following:
given an initial hand-arm configuration xi ∈ X and a final desired one xf ∈ X , find a
collision free path in X from xi to xf , i.e. a collision free path for the hand-arm system.
The dimension of the search space, X , is relatively large, and therefore conventional
solutions require high computational times. In this context, the proposed approach is
based on a reduction of the search space dimension proposed in Chapter 6, which is
done by using a representative subspace E of the hand configuration space Xh using
Principal Motion Directions (PMDs), and then looking for continuous valid paths in the
reduced subspace R = E × Xa . Of course, there may be solutions in X not included in
R, thus, the selection of a proper R (i.e. a proper E) is a relevant step in the proposed
approach. On the other hand, if a solution is found in R, for sure it is valid in X .
The following sections detail the sampling and interconnecting issues and the proposed
general planning algorithm.
7.1 Sample generation
The basic features of the procedure to sample hand-arm configurations are listed below, and then the sampling algorithms are formally presented. The features are the
following:
• A random sampling source is considered.
• Hand configurations are sampled from an axis-aligned box Bh in the subspace E
of dimension h, having side lenght 2λi , where λi is a number proportional to the
deviation of the data set in the ith PMD (Figure 7.1). Let:
- E = (ê1 , ê2 , . . . , êh )⊤ be a matrix with a base of E as columns,
- ǫ = (ǫ1 , . . . , ǫh ) with ǫi ∈ [−λi , λi ] be a sample obtained inside Bh ,
- x̄h be the mean value of the data set used for the PCA analysis.
7.1 Sample generation
105
x2,h
ê2
λê11
−λ1
xh = E⊤ ǫ + x̄h
x̄h
0
x1,h
Figure 7.1: A 2-dimensional example of a space Xh conformed by the two joint values,
x1,h and x2,h , modelled with two PMDs, ê1 and ê2 , obtained from the input dataset shown
as gray points. The subspace E is 1-dimensional and defined by E⊤ = (ê1 ). Samples are
obtained from the sampling box Bh that in this case is the segment [−λ1 , λ1 ] shown as
bigger red dots on the ê1 -axis.
Then, the joint values xh of the hand are obtained as:
xh = E⊤ ǫ + x̄h .
(7.1)
In the present work, the dimension of E is not a fixed parameter but a parameter,
h, that is iteratively increased by the planning algorithm, as required by the task.
Correspondingly, the number of columns of E⊤ is iteratively increased.
• The arm configurations are obtained selecting two strategies.
- L INEAR: A sampling region for the arm configurations is defined around the
segment sa that connects the initial and the final arm configurations, xia and
xfa . This region is defined as the union of hypercubes, Ba (xda ), of side 2ρb
centered at evenly spaced points xda ∈ sa separated a distance d ≤ ρa .
- S PHERE: A sampling region is defined as the union of hypercubes in the
Cartesian space, particularly, in the subset where the end-effector satisfies
the wrist orientation constraint (Chapter 6), that is a cube of side 2ρx for
the position and a segment of length 2ρα for the orientation centered at the
106
Finding Approach Paths
end-effector pose corresponding to the configuration xda ∈ sa . Therefore, it
relies on a known, and preferable complete, inverse kinematic procedure for
the arm to determine its configuration sub-space.
• To obtain a collision-free hand-arm configuration, an arm configuration is sampled
from the corresponding hypercube and a hand configuration sampled from Bh is
associated to it, until a non-collision hand-arm configuration is found. This is
done trying up to na arm configurations and for each of them up to nh hand
configurations, using each time an increasing number of PMDs.
Algorithms 7.1 and 7.2 detail, respectively, the sampling procedures for the arm and
the hand. They are called from the main algorithm (Section 7.3) that set the values nh
and na as fixed input parameters, and ρa as an input parameter that takes increasing
values. The following functions are used in Algorithms 7.1 and 7.2:
R AND(v, [a, b]): Returns a vector of dimension v whose components have random values
in the range [a, b].
S ELF C OLLISION(x): Takes as a parameter either a hand-arm configuration or an arm
configuration. The function returns true if x makes the corresponding system to
be in self-collision, or false otherwise.
C OLLISION(x): Returns true if the input configuration x makes the hand-arm system to
be in collision with the environment, or false otherwise.
D IM(S): Returns the dimension of the space S.
M AP(ǫ): Returns the configuration xh ∈ Xh corresponding to ǫ ∈ E, as computed by
Eq. (7.1).
G ET P OSITION(xa ): Returns the position of the end-effector with coordinates (x, y, z),
using forward kinematics.
G ET O RIENTATION(xa , xa f ): Returns the rotation angle α of the wrist frame about the
segment connecting the wrist positions at the configurations xa and xa f .
S ET W RIST C ONSTRAINT(x, y, z, α, xa f ): Returns the wrist frame represented in the homogenous matrix T that satisfies the orientation constraints.
7.1 Sample generation
107
Rn
RD IM(E )
Bh
≤ ρa
2ρa
R6
xia
xfa
Ba (xia )
Ba (xda1 )
Ba (xda2 )
Ba (xfa )
Figure 7.2: Samples of the hand-arm system as a composition of arm and hand
configurations.
I NV K IN(T): Returns the arm configuration xa where the wrist satisfies the orientation
constraint. In our case, the arm has a closed-form solution, which is complete and
very fast. If there is no solution to the inverse kinematic problem, the function
returns the N O IKS OLUTION exception.
N ULL: In the case that no collision-free sample could be obtained using the sampling
procedures, the algorithms return the N ULL exception.
108
Finding Approach Paths
Algorithm 7.1: Obtention of an arm configuration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
S AMPLE A RM(xa i , xa f , d, ρb , ρx , ρα , mode, n)
input : The initial, final and valid arm configuration, the distance d
within the range [0, 1], the half-sizes ρb , ρx and ρα of the
sampling hypercubes for position and orientation, respectively,
the strategy mode to be used, and the maximum number n of
trials.
output: An arm configuration xa free of self-collisions, if found, or
N ULL otherwise.
i←0
while i < n do
switch mode do
case L INEAR
xa ←xia +d(xfa −xia )+R AND(D IM(Xa ), [−ρb , ρb ])
case S PHERE
xa d ← xia + d(xfa − xia )
(xd , y d , z d ) ← G ET P OSITION(xa d )
(x, y, z) ← (xd , y d , z d )+ R AND(3, [−ρx , ρx ])
αd ← G ET O RIENTATION(xa d )
α ← αd + R AND(1, [−ρα , ρα ])
T ← S ET W RIST C ONSTRAINT(x, y, z, α, xa f )
try xa ←I NV K IN(T)
catch N O IKS OLUTION
return N ULL
17
if FALSE=S ELF C OLLISION(xa ) then
return xa
18
i←i+1
19
return N ULL
16
7.2 Sample interconnection
As explained in Chapter 6, the hand is not able to change the configuration along any
direction in X , because the thumb-base joint has not velocity control. Therefore, the
motion of the hand is performed by interleaving the motion of the thumb-base joint
with the coordinated motion of all of the others joints of the hand, in a Manhattan-like
manner. Then, the motion of the whole hand-arm system between two neighboring
configurations is performed by dividing the arm motion in two parts and coordinating
them with the two interleaved hand movements [99], one with the arm and th hand
7.2 Sample interconnection
109
Algorithm 7.2: Obtention of a hand-arm configuration
1
2
3
4
5
6
7
8
9
10
11
12
13
S AMPLE H AND(xa , n)
input : The configuration xa of the arm, the dimension e
of E, and the maximum number n of trials.
output: A hand-arm configuration x free of collisions if
found, or N ULL otherwise.
e←1
while e < h do
i←0
while i < n do
ǫ←R AND(e, [0, 1])
xh ←M AP(ǫ)
x← (xa ,xh )
if FALSE=S ELF C OLLISION(x) then
if FALSE=C OLLISION(x) then
return x
i←i+1
e←e+1
return N ULL
joints but the thumb base joint, and the other with the arm and the thumb base joint
only. Therefore, the interconnection must be carried out as follows.
Let the thumb range associated to a given collision-free configuration x be the set of
values that the thumb-base joint can sweep without producing any collision, provided
that no other joint is changed. Then, two configurations are considered as neighboring
configurations if the distance between them is below a given threshold and their thumb
ranges intersect.
Then, in order to build a roadmap, neighboring samples are interconnected. The
main features of the interconnection procedure are the following:
• The maximum number of neighboring samples is limited to the closest K samples,
being K a given value.
• All the samples generated within the hypercube centered at xia , Ba (xia ), are forced
to have xi as a neighboring configuration, provided that their thumb ranges intersect and irrespective of whether xia belongs to the closest K neighbors or not. The
same is done for the final configuration xfa .
110
Finding Approach Paths
Algorithm 7.3: Connect a sample to the roadmap
1
2
3
4
5
6
7
8
C ONNECT S AMPLE(G, x)
input : The roadmap G and the configuration x
output: The updated roadmap G
N ←F IND N EIGHBORS(x)
forall the xn ∈ N do
try
L ←M-L OCAL P LAN(xn , x)
A DD E DGE(L, G)
catch N O E DGE
A DD N ODE(x, G)
return G
• The local planner searches for a collision-free path between two neighboring configurations by uncoupling the motion of the thumb-base joint from the coordinated
motion of all of the others joints of the hand, in a Manhattan-like manner, and
dividing the motion of the arm in two parts and coordinating them with the two
uncoupled phases of the hand [99].
Algorithm 7.3 shows the procedure that performs the connection of a sample to the
roadmap. The following functions are used in this algorithm:
F IND N EIGHBORS(x): Finds the K-nearest neighbors of x from all the nodes of the
roadmap G. The neighboring threshold is set equal to the distance between xi
and xf .
M-L OCAL P LAN(xi , xj ): Returns the collision-free path L connecting xi and xj satisfying
that the thumb-base motions are interleaved with those of the other hand joints
using with Manhattan-like maneuvers. If it can not connect the two nodes using a
collision-free path it returns the N O E DGE exception.
A DD N ODE(x, G): Adds node x to graph G. It automatically creates a new component
and updates the graph.
A DD E DGE(L, G): Adds edge L = [xi , xj ] to graph G. It automatically adds the nonexisting node, merges the components they belong to if required, and updates the
connected components of the graph.
7.3 The approach path planner
111
7.3 The approach path planner
The main algorithm is a probabilistic roadmap planner that samples and interconnects
the configurations as detailed in the two previous sections. It is an easy-to-tune adaptive
algorithm whose principal features are:
• The dimension of the hand search space E is iteratively increased when no collisionfree hand-arm configurations is found for a given arm configuration in Xa .
• The volume of the arm search space is iteratively increased each time the attempt
to connect the initial and the final configurations fails, i.e. if no solution is found
by sampling all the hypercubes Ba (Figure 7.2), the size of hypercubes is increased,
and a new iteration of the algorithm is launched.
• The main algorithm keeps track of the connected components that contain xi
and xf in order to explore only a subset of the hypercubes Ba (pi ) that define the
sampling region for the arm configurations. This is done as follows. Let:
– di be the maximum distance from xia to the center of a hypercube that has
generated a sample that belongs to the same connected component than xia ,
– df be the minimum distance from xia to the center of a hypercube that has
generated a sample that belongs to the same connected component than xfa .
Then, only those hypercubes centered at points located at a distance d ∈ [di , df ]
from xia are likely to generate samples that aid to interconnect the connected
component of xi with that of xf , as illustrated in Figure 7.3 (take into account that
the distance from xia to xfa is considered unitary and that the hypercubes Ba are
swept following the Van der Corput sequence as explained in Section 7.1).
• There are no critical parameters to be tuned.
Algorithm 7.4 formally details the planning procedure that returns a path P connecting xi and xf . The following functions are used:
A DJUST S IZE(k, ρ): Scales the vector ρ of parameters proportional to a given constant k.
VAN D ER C ORPUT(xi , xj , ρ, D): Computes a set of points evenly spaced along the segment defined by xi and xj normalized to the interval D, such that the number
112
Finding Approach Paths
Algorithm 7.4: Algorithm that plans the motion of the hand-arm
system towards a final desired configuration.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
A PPROACH PATH(xf , xi , ρb , ρx , ρα , na , nh , mode, N )
input : The final and initial configurations, xf and xi , the initial half-sizes
ρb , ρx and ρα of the sampling hypercubes, the maximum numbers
na and nh of trials to sample arm configurations per hypercube and
to sample hand configurations per arm configuration, respectively,
the strategy mode for the arm path planning, and the maximum
number N of trials for the roadmap generation.
output: The sequence of edges P ∈ G forming the path that connects xi
and xf
P←∅
G←∅
A DD N ODE(xi , G)
A DD N ODE(xf , G)
c←2
k←0
repeat
k ←k+1
[ρb , ρx , ρα ] ←A DJUST S IZE(k, [ρb , ρx , ρα ])
D ←VAN D ER C ORPUT(xia , xfa , ρb , [0, 1])
foreach d ∈ D do
try
xda ← S AMPLE A RM(xia , xfa , d, ρb , ρx , ρα , mode, na )
xd ←S AMPLE H AND(xda , nh )
G ←C ONNECT S AMPLE(xd , G)
P ←F IND PATH(G, xi , xf )
return P
catch N OT I N S AME C OMPONENT
c←c+1
U PDATE S EQUENCE(D)
next d
catch N ULL
c←c+1
next d
until c < N
return P
of points in the set is a power of two and the distance between two points in
the configuration space is below the given threshold ρ. In our case, the interval
7.3 The approach path planner
113
s1
s2
Rn
xf
s0
xi
Ba (xda1 )
Ba (xda2 )
Ba (xda0 )
xfa
xia
R6
di
df
Figure 7.3: Roadmap under construction composed of two connected components
after having sampled three configurations. Further exploration of the sampling
region for the arm configurations is constrained to the hypercubes Ba located at a
distance d ∈ [di , df ] = [0.5, 1.0] from xia .
is initially set to [0, 1]. The elements in the set are finally re-ordered such that
they follow the Van der Corput sequence [52], for which the points would be
d = 0, 1, 0.5, 0.25, 0.75, 0.125, . . . .
U PDATE S EQUENCE(D): Updates the distances di and df , hence the interval D = [di , df ]
used to defined the set D. It automatically discards the points d ∈ D that fall
outside the interval.
F IND PATH(G, xi , xf ): First, verifies whether the nodes xi and xf belong to same connected component or not. If they belong to the same component, it returns a
path P ∈ G connecting nodes xi and xf using the A∗ algorithm and a smoothing
procedure. If they do not, it returns the N OT I N S AME C OMPONENT exception.
114
Finding Approach Paths
7.4 Test cases
All the cases presented here were studied using the robot simulation toolkit for motion
planning and teleoperation guiding, which has served to generate and validate the paths
before executing them on the physical devices. For the simulator development, three
guidelines were considered [74]: ability to run on different platforms, code accessability
and software modularity. The first two allow the use of cross-platform and open-source
tools. Regarding the software modularity, the project was conceived to be librarybased, thus, different libraries have been developed such as a Geometric library for the
treatment of the bodies and their kinematic relation, a Sampling library with different
sampling strategies, a Motion Planning library, a Device library for the communication
with different devices, and, finally, the GUI library that implements the user interface
and library management.
For the validation, the hand is mounted on the industrial arm Säubli TX90, which
has 6 revolute joints. The validation of the proposed approach has been carried out
both in a virtual environment with simulated elements, as well as in a real scene with
the actual hand-arm system. In order to evaluate the use of the PMDs, the planner
will be casted using the L INEAR strategy, and the results are compared with the case
where no PMDs are used (Subsection 7.4.1), and secondly, using the S PHERE strategy
using the PMDs, and the results are compared with the case where no strategy is used
for the sampling of arm configurations, that is, the samples are obtained over the full
configuration space (Subsection 7.4.2).
7.4.1 Evaluation of the use of PMDs
As a benchmark, the task of grasping a can on a table was selected. The result of the
proposed planner using the L INEAR strategy is compared with the case where no PMDs
are used, i.e. samples of the hand configuration are obtained from the whole hand
configuration space.
The quantitative results of using PMDs in motion planning are summarized and compared in Table 7.2. These results were obtained using a desktop computer equipped with
a 3.00GHz Intel Core2 CPU, running Windows operating system and initial parameters
of the main algorithm shown in Table 7.1. It is noticeable the decrease in the time
required to find a solution when using PMDs (more than 50 times faster), due basically
7.4 Test cases
115
Figure 7.4: Qualitative comparison between the approach that considers the full hand
configuration space (top) and the proposed approach that reduces the hand workspace
using the PMDs (bottom). The use of PMDs resulted in a path composed of a smooth
sequence of human-like postures.
Figure 7.5: Simulation of a solution path and real execution in the actual hand-arm
system.
to the fact that using PMDs collision-free samples are more easily found. Also, the
number of samples required resulted very small (less than 1% of the samples required
when no PMDs were used). The qualitative results are even more interesting (see
Figure 7.4). Even though a smoothing procedure is always applied, the solutions found
when sampling the whole hand configuration space contain awkward hand postures,
ressembling movements of humans with coordination problems. On the contrary, the
proposed method provides a sequence of well-coordinated human-like postures, which
requires less smoothing and prunning post-processing.
As it was previously mentioned, a solution path was successfully implemented on
the actual hand-arm system, the screenshots are exposed in Figure 7.5, where both
116
Finding Approach Paths
Parameters
ρa
K
N
na
nh
Value
0.001
10
1000 (10,000)
10 (20)
10 (20)
Table 7.1: Parameters used in the approach path planner. The values shown for the
adaptive parameters are the initial one; the values in parenthesis are used when no
PMDs are considered.
Comparison factors
Proposed
Classical
Time to find a solution [s]
Smoothing time [s]
16.83
5.08
915.28
15.20
Final neighboring threshold (ρb )
Maximum number of samples required (c)
Proportion of samples generated by using:
thumb-base + 2 PMDs
thumb-base + 3 PMDs
thumb-base + 4 PMDs
thumb-base + 5 PMDs
0.0013
698.82
0.0631
7274.78
22.4%
26.5%
16.5%
34.7%
–
–
–
–
Total nodes in the PRM
Total nodes in the solution path
22.55
7.27
270.44
3.22
Table 7.2: Comparison in averaged values between the proposed approach in which the
hand workspace has been reduced using PMDs and the classical approach in which the
workspace is the full configuration space.
the virtual and the real path are shown in their equivalent points on the path. The
implementation on the real hand-arm system makes the usefulness of the proposed
algorithms visible, along with the particular specific hardware constraints (the noncustomizability of the thumb-base joint controller of the SA hand) have been correctly
tackled with an appropriate local-planner.
Performance study
Assuming a given grasp or pre-grasp configuration, the proposed approach looks for
the final approaching motion, where the collisions are more likely to occur with the
hand rather than with the arm (i.e. collision-free solution paths will require finger
motions and only slight arm deviations from the straight motion). With this in mind,
the planner has been evaluated on several problems, four of them shown in Figure 7.6,
7.4 Test cases
117
with different degrees of difficulty. In comparison with the task used for the evaluation,
the tasks used for the performance shown Figure 7.6 have: (1) a narrower passage,
(2) a goal configuration closer to the obstacles, (3) a rectilinear path to the goal more
obstructed by the presence of the T-shaped object and of the shelf itself, (4) a more
cluttered environment with a longer narrow passage (this task is similar to that used
in [2]). The solution paths required motions of the finger joints, maintaining the
robot configurations close to the rectilinear path, and resulted in smooth sequences of
human-like configurations (Figure 7.7). The algorithm was run in a computer with a I5
processor with 4 cores and 4 Gb of RAM, under Windows 7 64-bit. The testing procedure
was parallelized using the MPI library [39] in order to use all cores. The quantitative
results are shown in Table 7.3. Note that the fourth task required the generation of
2
1
3
4
Figure 7.6: Goal configurations of the hand-arm system for some of the tasks used to
test the planner: 1) Cans on a desk; 2) Can in a box; 3) Cans in a shelf; 4) T-shape
object in a complex scene.
118
Finding Approach Paths
Figure 7.7: Some screen shots of the solution paths of tasks (2) and (3) illustrated in
Figure 7.6.
1
2
3
4
Time to find a solution [s]
Smoothing time [s]
21.06
0.124
7.75
0.022
71.64
0.821
24.09
0.142
Number of samples
Nodes in the PRM
Nodes in solution path
1188.7
53.6
3.9
437.7
24.6
3.4
2093.8
131.7
4.2
2205.6
81.9
4.9
Problem
Table 7.3: Comparison of the performance of the planner used to solve the problems
illustrated in Figure 7.6. These values are the means from 1000 runs.
more samples than the third task, because the environment is more cluttered and many
samples resulted in collision, but it could be solved with a PRM composed of less nodes
because the narrow passage was more aligned with the direction connecting xini and
xgoal . Therefore, the time to find a solution was larger in the third task because the
validation of the PRM edges is time-consuming.
Discussion
The value of ρb determines how far the arm path can be from the rectilinear segment
in Xa that connects xia and xfa . During the final approaching motion to grasp an object,
the potential collisions are likely to occur with the hand, not with the arm. The value of
ρb also determines the number of samples considered for each iteration of the general
loop, i.e. the number of hypercubes Ba considered, although the neighboring threshold
is an independent value and configurations sampled from non-contiguous hypercubes
can be connected in the roadmap. The value of ρb is iteratively increased, and the initial
7.4 Test cases
119
chosen value is not a critical issue. It has to be neither too small (since then its increase
could be too slow and too many samples could be required), nor to large (since then
the search space could be too large and also too many samples could be required).
Values between 0.001 and 0.05 gave good results for different tasks (the parameter is
non-dimensional since each direction of the configuration space is normalized in the
range [0, 1]).
The proposed approach searches the hand postures using as few PMDs as possible,
which results in smoother motions all along the solution path. Moreover, the use of
PMDs results in a better computational efficiency because the percentage of collisionfree samples is much higher than when sampling directly the finger joints [97].
The values nh and ns allow several trials in the difficult parts of the path, increasing
the posisblity to find a collision-free hand-arm configuration. These values are by
no means critical, since the successive iterations of the main loop also permit the
resampling of the difficult areas.
The distance threshold used to consider two configurations as neighboring samples
is set equal to the distance between the initial and the final configuration, hence, it is
not a user-defined parameter but calculated from the input of the problem.
7.4.2 Evaluation of the use of wrist orientation constraints
As an example to illustrate the S PHERE strategy, the SA hand is required to move, among
obstacles, from an unconstrained configuration to a preshape configuration to grasp a
can. Figure 7.8 (top) shows the snapshots of the solution path found when the task has
been programmed with an orientation constraint and using the PMDs as well. Snapshots
(1) and (5) show the initial and the final configurations, respectively. Snapshots (2),(3),
and (4) are intermediate configurations.
The same task has been programmed without using the orientation constraint with
Constraints
Success
# Nodes
Sampling
Time (s)
Total
Time (s)
With
Without
100%
62%
90 ± 71
37 ± 9
8.9 ± 6.7
169 ± 85
85 ± 68
198 ± 80
Table 7.4: Computation time of the planner when considering and omitting the
orientation constraint.
120
Finding Approach Paths
(1)
(2)
(3)
(4)
(5)
Figure 7.8: Snapshots of the results: (Top) task simulation using the orientation
constraint; (Middle) task execution using the orientation constraint; (Bottom) task
simulation without using the orientation constraint.
the same initial and final configurations. The result is shown in Figure 7.8 (bottom).
Table 7.4 shows the results of the comparison in terms of computational efficiency
measured as computational time (running on a [email protected]) used by the sampling procedures. The sampling time is much shorter when considering constraints because in
this case samples are more often collision-free and reachable. Using virtual constraints
further benefits in both senses as shown in Table 7.4 and in the snapshots of Figure 7.8.
The task has been executed in the real robotic system, as shown in Figure 7.8 (middle),
where it is shown how the arm movements are more realistic, in the sense that it moves
directly and smoothly towards the object.
7.5 Summary
This chapter presented a motion planner for a hand-arm robotic system. The proposal
pursues efficiency and human-likeliness in the hand postures while finding the approach
path that connects the system from any initial configuration to any desired configuration, including those grasps obtained in the Part II. Planning is done with a probabilistic
roadmap planner, and the dimensionality reduction in the hand search space poposed
7.5 Summary
121
in the previous chapter results in lower computational time. The dimension of the hand
search space and the volume of the arm search space are iteratively increased as much
as it is required by the difficulty of the task. The arm configurations are sampled using
two strategies. With the first one, the sampling is performed on a box swept around the
rectilinear segment connecting the initial and final arm configurations. With the second
strategy, that uses the orientation constraints proposed in the previous chapter, the
dimension of the search space is reduced to 4, allowing a wider workspace exploration
while keeping the path biased toward the goal. The proposed method has no critical
parameters to be tuned such as thresholds for transitions configurations.
The approach has been implemented for the SA hand, which requires a particular
local planner to decouple the motion of the thumb-base joint from the other hand joints,
but is a general procedure for any other mechanical hand. The validity of the approach
has been demonstrated both in simulation and in real experiments.
Part IV
Closing Remarks
8
Conclusions
As in any research, the present work opens more questions than it settles,
suggesting new avenues for future investigation. This chapter summarizes the
main contributions of the thesis, and highlights several points deserving further
attention.
8.1 Summary of contributions
The main contribution of the thesis is a general procedure for planning grasps subject
to specific contact constraints. The fact that such constraints allow to define particular
grasps for specific tasks is the main motivation behind our work, but the enforcement of
such constraints also provides a means of ensuring selective grasps suitable to arbitrary
purposes, such as grasps in which the fingers should avoid the touching of fragile areas,
or object regions that might be harmful to the robot.
The problem has been approached by subdividing it into the grasp synthesis and
approach path planning subproblems, for which the following solution methods have
been proposed:
1. A complete method to compute kinematically-feasible grasps (Chapter 3). The
method is based on a previous formulation for general multi-body systems articulated through lower pair joints [80], extended to include additional constraints for
modelling general patch-patch contact constraints. The formulation is suitable to
apply a linear relaxation technique to compute a solution, which allows to find a
126
Conclusions
kinematically-feasible grasp whenever one exists, or to prove grasp non-existence
otherwise. As opposed to other methods in the literature, the proposed method
does not need to be fed with an initial estimation of the final grasp and can cope
with general region-to-region contact constraints.
2. A grasp optimization method that circumvents the local minima problem
(Chapter 4). The optimization process starts with a kinematically-feasible grasp—
such as one computed by the previous method— and then focuses the exploration
on a relevant subset of feasible grasps. This subset is traced exhaustively by
using recently-developed techniques of higher-dimensional continuation [42]. A
detailed atlas of the subset is obtained as a result, on which the highest-quality
grasp according to any desired criterion, or a combination of criteria, can be
readily identified.
3. A method to account for compliant joints and contacts (Chapter 5). The approach is inspired by the notion of soft synergies [6, 82]. It effectively introduces
an elastic model of the hand whereby the physical hand is attracted towards a
reference hand configuration through a set of virtual springs representing the compliance of the musculoskeletal system, while at the same time being repelled by
the compliant surface of the object. This model allows to tackle all kinematic and
restrainablity constraints of the grasp simultaneously, and the computed solution
provides a set of hand configurations that can be used to execute the grasp using
a position controller. The method is advantageous in comparison to methods of
a similar spirit, which either neglect some of the mentioned constraints, limit the
approach to planar grasps, or use a reduced number of fingers.
4. A technique to reduce the dimension of the hand configuration space in a
meaningful way (Chapter 6). The reduction comes from finger coordination and
wrist orientation constraints. As for the former constraints, they are inspired by
the concept of hand postural synergies, with a difference on how the data points
to be processed are generated. These points are typically obtained from static
object prehensions, so that the results of a Principal Component Analysis provide
hand postural synergies. In our case, the points are obtained by moving the
hand in the free space instead, obtaining principal motion directions in the hand
configuration space that allow the coordination of the joint movements. As for the
8.2 Future research directions
127
latter constraints, two approaches were explored. The first approach is a linear
path reduction similar to the one usually performed in previous works [3, 61, 67,
118], but sweeping a box along the path and sampling in it in our case, which
allows the arm to deviate slightly from the adopted path. The second approach
introduces a way to constrain the wrist orientation, which biases the path towards
the object while keeping the palm facing to a given point in the object. In both
cases, the reduction will allow an efficient search with a tradeoff between the
mobility of the system and the probability of finding a collision-free path in a
complex scene.
5. An adaptive planner to compute approach paths to the object (Chapter 7).
The planner provides methods to generate and interconnect samples in a reduced
configuration space induced by the previously-mentioned finger-coordination and
wrist-orientation constraints. The main feature of such planner is its ability to
adapt to the complexity of the scene. If the environment is simple, only a few
principal motion directions are used. As the environment gets more complex, and
finding a path is harder, additional principal motion directions are considered,
allowing to increase the overall mobility of the system, and hence the probability
of finding a collision-free path.
Appendix A summarizes the publications through which these methods have been disseminated so far.
8.2 Future research directions
The following points deserving further attention have been identified:
1. Automatic generation of the contact constraints. A priori knowledge of the
contact constraints is assumed throughout the work. As exposed in Section 1.2,
there exist algorithms that might be helpful to determine them, but further work
needs to be done on automating the process of deciding which object regions
should be placed in contact with which specific hand regions. While some heuristic
methods have been proposed for the case in which such regions reduce to isolated
points [32, 122], algorithms able to cope with general free-form regions are still
128
Conclusions
to be developed. Furthermore, determining a suitable set of contact constraints
from a higher-level specification of the manipulation task is an open issue as well.
2. Generation of postural synergies of the hand in contact with the object. In
Chapter 4, hand postural synergies were introduced to reduce the dimension of
the hand configuration space, specifically introducing constraints in xh . But postural synergies are generally obtained using the joint values, so that the amount of
dimension reduction is limited by the number of joints. In order to obtain further
suitable reductions, an interesting approach would be to study how the contacts
on the hand may vary with respect to the hand configuration. For instance, when
we have the hand almost closed, we typically use the furthest part of the fingertip
right under the nail. Oppositely, when the hand is widely open, we tend to use the
middle part of the fingerprint. Thus, a generalized dimension reduction procedure
could be envisaged based on introducing constraints in both xh and xc .
3. Optimization of the grasp stability from a control standpoint. Whereas the
optimization framework proposed in Chapter 4 can in principle tackle any kind
of grasp quality metric, the stability of a grasp requires a more detailed study. To
achieve stability, the positive definiteness of the so-called grasp impedance matrix
must be ensured, which involves a non-linear programming problem for a known
configuration [108], thus making the problem more complex.
4. Optimization on all connected components of the manifold of kinematicallyfeasible grasps. The presented optimization framework operates in the connected
component of the manifold of relevant grasps that contains an initial kinematicallyfeasible grasp. In most robotic hands this is not an issue since, due to joint range
limitations, such manifold only contains one connected component. However,
this might not be the case in general. Ways to obtain one starting point in each
connected component of the manifold of relevant kinematically-feasible grasps
would extend this work to more general settings.
5. Study of virtual constraints for the approach path to the object. In Chapter 6,
a virtual constraint was proposed to reduce the wrist configuration space. This
constraint enforces the hand to face a center point on the object, which seems
appropriate for a wide range of cases. However, other relational positioning
constraints might be used in principle, such as enforcing the hand to face the
8.2 Future research directions
129
axis of a cylinder for grabbing long tubes, or to face a plane for particular pickand-place operations.
6. Consideration of time-varying environments. The presented grasp planning
method considers that the environment is static with objects and obstacles not
changing over time. Even thought this is a typical scenario in every-day activities,
it would be desirable to have a grasp planning method, with the ability to consider time-varying environments, including moving objects in them. In this case,
switching to Rapidly-exploring Random Tree techniques [51] seems appropriate,
since they are faster than the Probabilistic Roadmap technique used here, which
is an asset for real-time applications.
7. Optimization of the approach path to the object. The path determined by the
proposed method accounts for collision avoidance constraints, but it would be
advisable to include the treatment of objective functions so as to optimize the
approach path according to a desired criterion.
8. Consideration of uncertainty in the models and their locations. The present
work assumes that the geometric models and locations of the hand-arm system,
the object, and the environment are accurately known. Ways to introduce uncertainties in the values defining these models and locations are necessary to make
the grasp planning process more robust.
9. Unification of the grasp synthesis and the approach path planning into a
one-step method. A recent work presents an approach to unify these two subproblems [118]. However, as discussed in Chapter 2, this approach is not targeted
to find grasps that allow manipulation tasks requiring the object to be contacted
at specific locations. Even though the modules developed in Parts II and III are
able to solve the grasp planning problem, a higher-level planner needs to be
implemented to iteratively call such modules whenever a solution is not found
in a first attempt.
A
List of Publications
The results of the present work have been disseminated through the following publications:
Journal papers
• Rosales, C., L. Ros, J. M. Porta, and R. Suárez (2011). Synthesizing grasp configurations with specified contact regions. The International Journal of Robotics
Research 30(4), 431–443 [93].
• Rosell, J., R. Suárez, C. Rosales, and A. Pérez (2011). Autonomous motion planning of a hand-arm robotic system based on captured human-like hand postures.
Autonomous Robots 31(1), 87–102 [100].
(under review)
• Rosales, C., J. M. Porta and L. Ros (2012). Grasp Optimization under Taskdependent Contact Constraints (submitted to the IEEE Transactions on Robotics).
Conference papers
• Rosales, C., R. Suarez, M. Gabiccini, and A. Bicchi (2012). On the synthesis of
feasible and prehensile robotic grasps. In Proceedings of the 2012 IEEE International Conference on Robotics and Automation, pp. 550–556 [94].
• Rosales, C., J. Porta, and L. Ros (2011). Global optimization of robotic grasps. In
Proceedings of Robotics: Science and Systems VII [91].
• Rosell, J., R. Suárez, A. Pérez, and C. Rosales (2011). Including virtual constraints in motion planning for anthropomorphic hands. In Proceedings of the 2011
IEEE International Symposium on Assembly and Manufacturing, pp. 1–6 [96].
• Sun, S.-C., C. Rosales, and R. Suárez (2010). Study of coordinated motions of the
human hand for robotic applications. In Proceedings of the 2010 IEEE International
Conference on Information and Automation, pp. 776–781 [115].
132
List of Publications
• Suárez, R., J. Rosell, A. Perez, and C. Rosales (2009). Efficient search of obstaclefree paths for anthropomorphic hands. In Proceedings of the 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1773–1778 [114].
• Rosell, J., R. Suárez, C. Rosales, J. A. García, and A. Pérez (2009). Motion
planning for high DOF anthropomorphic hands. In Proceedings of the 2009 IEEE
International Conference on Robotics and Automation, pp. 4025–4030 [98].
• Rosales, C., J. M. Porta, R. Suárez, and L. Ros (2008). Finding all valid hand
configurations for a given precision grasp. In Proceedings of the 2008 IEEE International Conference on Robotics and Automation, pp. 1634–1640 [92].
Technical reports
• Rosell, J., R. Suárez, C. Rosales, and A. Pérez (2009). A practical motion-planning
approach for the Schunk Anthropomorphic Hand. Technical Report IOC-DT-P-2009,
IOC-UPC.
• Rosell, J., R. Suárez, C. Rosales, J. A. García and A. Pérez (2008). Efficient motion
planning for high DOF hands using principal motion directions. Technical Report
IOC-DT-P-2008-13, IOC-UPC
• Rosales, C., J. M. Porta, R. Suárez, and L. Ros (2008). Solving hand inverse
kinematics for precision grasps. Technical Report IOC-DT-P-2007-09, IOC-UPC.
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