Advances in Robot Kinematics

440 Pages·2000·19.395 MB·English

by  A. Gfrerrer (auth.), J. Lenarčič, M. M. Stanišić (eds.)

ADVANCES IN ROBOT KINEMATICS
ADVANCES IN
ROBOT KINEMATICS
Edited by
J. Lenarcic
Department ofA utomatics, Biocybernetics and Robotics,
J. Stefan Institute, Ljubljana,
Slovenia
and
M. M. Stanisi6
Aerospace and Mechanical Engineering,
University ofNotre Dame,
Notre Dame, Indiana,
U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-5803-2 ISBN 978-94-011-4120-8 (eBook)
DOI 10.1007/978-94-011-4120-8
Printed on acid-free paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Preface
This book presents the most recent research advances in the theory, design,
control and application of robotic systems, which are intended for a variety
of purposes such as manipulation, manufacturing, automation, surgery,
locomotion and biomechanics. The issues addressed are fundamentally
kinematic in nature, including synthesis, calibration, redundancy, force
control, dexterity, inverse and forward kinematics, kinematic singularities,
as well as over-constrained systems. Methods used include line geometry,
quaternion algebra, screw algebra, and linear algebra. These methods are
applied to both parallel and serial multi-degree-of-freedom systems. The
results should interest researchers, teachers and students, in fields of
engineering and mathematics related to robot theory, design, control and
application.
All articles in the book were reported at the seventh international
symposium on Advances in Robot Kinematics that was organised in June
2000 in the beautiful ancient Mediterranean town of Piran in Slovenia. The
preceding symposia of the series took place in Ljubljana (1988), Linz (1990),
Ferrara (1992), Ljubljana (1994), and Piran (1996), and Salzburg (1998).
These symposia were organised under the patronage of the International
Federation for the Theory of Machines and Mechanisms. In all these years,
ARK has attracted the most outstanding authors in the area and has
managed to create a perfect combination of professionalism and friendly
atmosphere. Many important and original scientific results were for the
first time reported and discussed at ARK symposia. Weare glad to observe
that, in spite of the strong worldwide competition of many international
meetings, workshops and conferences, ARK has become one of the most
recognised events in robot kinematics and continues to provide a unique
scientific impact. ARK has developed a remarkable scientific community in
the area of robot kinematics.
Each contribution in this book had been rigorously reviewed by two or
three independent reviewers and 45 articles had been recommended for
publication. We are grateful to the authors of the articles for their valuable
contributions and for their efficiency in preparing their manuscripts in
time, and to the reviewers for their timely reviews of the articles. We are
also indebted to our collaborators at the Institute Jozef Stefan in Ljubljana
and to the personnel at Kluwer Academic Publishers for their excellent
technical and editorial support.
J adran Lenarcic and Michael M. Stanisic, Editors
Table of Contents
1. Methods in Kinematics
A. Gfrerrer: Study's kinematic mapping - A tool for motion
design 7
E. Staffetti, F. Thomas: Kinestatic analysis of serial and parallel
robot manipulators using Grassmann-Cayley algebra 17
V. Milenkovic, P.R. Milenkovic: Unit quaternion and CRV:
Complementary non-singular representations of rigid-body
orientation 27
I.D. Coope, A.B. Lintott, G.R. Dunlop, M.L Vuskovic:
Numerically stable methods for converting rotation matrices
to Euler parameters 35
M. Keler: Geometry of homokinematic spatial Cardan shafts by
dual methods 43
C. Bombin, L. Ros, F. Thomas: A concise Bezier clipping
technique for solving inverse kinematics problems 53
S. Besnard, W. Khalil, G. Garcia: Geometric calibration of robots
using multiple plane constraints 61
2. Kinematic Synthesis
J. Angeles, D. Chablat: On isotropic sets of points in the plane.
Application to the design of robot architectures 73
Y. Pang, V. Krovi: Fourier methods for synthesis of coupled
serial chain mechanisms 83
A. Perez, J. M. McCarthy: Dimensional synthesis of spatial RR
ro~~ ~
P. Larochelle: Approximate motion synthesis via parametric
constraint manifold fitting 103
J.M. Rico, B. Ravani: Designing linkages with symmetric
motions: The spherical case 111
A. Murray, M. Ranchak: Kinematic synthesis of planar
platforms with RPR, PRR, and RRR chains 119
3. Force Analysis
J.P. Schmiedeler, K.J. Waldron: Impact analysis as a design tool
for the legs of mobile robots 129
K. Iagnemma, S. Dubowsky: Vehicle wheel-ground contact angle
estimation: with application to mobile robot traction control 137
2
S.R. Lucas, C.R. Tischler, A.E. Samuel: The Melbourne hand 147
K. Harada, M. Kaneko, T. Tsuji: Active force closure for multiple
objects 155
G. Duchemin, E. Dombre, F. Pierrot, E. Degoulange: SCALPP: A
6-dof robot with a non-spherical wrist for surgical
applications 165
D. Ruspini, O. Khatib: A framework for multi-contact multi-body
dynamic simulation and haptic display 175
F. Caccavale, G. Ruggiero, B. Siciliano, L. Villani: On the
dynamics of a class of parallel robots 187
4. Kinematic Redundancy
R. Schaufler, C.H. Fedrowitz, R. Kammiiller: A simplified
criterion for the repeatability of redundant manipulators 199
G. Schreiber, G. Hirzinger: An intuitive interface for nullspace
teaching of redundant robots 209
S.B. Nokleby, R.P. Podhorodeski: Methods for resolving velocity
degeneracies of joint-redundant manipulators 217
M.W. Hannan, I.D. Walker: Novel kinematics for continuum
n7
ro~~
Y. Zhang, J. Duffy, C. Crane: The optimum quality index for a
spatial redundant 4-8 in-parallel manipulator 239
G. Antonelli, S. Chiaverini: Fuzzy inverse kinematics for
underwater vehicle-manipulator systems 249
5. Parallel Mechanisms and Workspace Analysis
G.8. Chirikjian: Symmetries in workspace densities of discretely
actuated manipulators 259
M.J.D. Hayes, M.L. Husty: Workspace characterization of
planar three-legged platforms with holonomic higher pairs 267
R. Verhoeven, M. Hiller: Estimating the controllable workspace
of tendon-based Stewart platforms 277
J.A. Snyman, A.M. Hay: The chord method for the determination
of non-convex workspaces of planar parallel platforms 285
J. Kim, F.C. Park: Elasto-kinematic design tools for parallel
mechanisms 295
P. Wenger, D. Chablat: Kinematic analysis of a new parallel
machine tool: the orthoglide 305
J-P. Merlet, M-W. Perng, D. Daney: Optimal trajectory planning
of a 5·axis machine-tool based on a 6-axis parallel
manipulator 315
3
6. Analysis and Application of Parallel Mechanisms
J. Lenarcic, M.M. Stanisic, V. Parenti-Castelli: A 4-dof parallel
mechanism simulating the movement of the human
sternum-clavicle-scapula complex 325
V. Parenti-Castelli, R. Di Gregorio: Parallel mechanisms
applied to the human knee passive motion simulation 333
A. Kecskemethy, C. Lange, G. Grabner: A geometric model for
cylinder-cylinder impact with application to vertebrae
motion simulation 345
M. Husty, A. Karger: Architecture singular parallel
manipulators and their self-motions 355
K. Wohlhart: Architectural shakiness or architectural mobility of
platforms 365
O. Roschel: Mobius mechanisms 375
7. Parallel Mechanisms and Screw Algebra
J.C.F. Shum, P.J. Zsombor-Murray: Direct kinematics of the
double-triangular manipulator: An exercise in geometric
thinking 385
M. Karouia, J.M. Herve: A three-dof tripod for generating
spherical rotation 395
G. Olea, N. Plitea, K. Takamasu: Kinematical analysis and
simulation of a new parallel mechanism for robotics'
application 403
M. Ceccarelli: Early studies in screw theory 411
LA. Parkin: On deriving infinitesimal twists and velocity screws
from finite displacement screws 423
A. Frisoli, D. Checcacci, F. Salsedo, M. Bergamasco: Synthesis
by screw algebra of translating in-parallel actuated
mechanisms 433
Author Index 441
1. Methods in Kinematics
A. Gfrerrer: Study's kinematic mapping - A tool for motion design
E. Staffetti, F. Thomas: Kinestatic analysis of serial and parallel robot
manipulators using Grassmann-Cayley algebra
V. Milenkovic, P.R. Milenkovic: Unit quaternion and CRY: Complementary
non-singular representations of rigid-body
orientation
l.D. Coope, A.B. Lintott, G.R. Dunlop, M.l. Vuskovic: Numerically stable
methods for converting rotation matrices to Euler parameters
M. KeIer: Geometry of homo kinematic spatial Cardan shafts by dual
methods
C. Bomhin, L. Ros, F. Thomas: A concise Bezier clipping technique for
solving inverse kinematics problems
S. Besnard, W. Khalil, G. Garcia: Geometric calibration of robots using
multiple plane constraints
5
STUDY'S KINEMATIC MAPPING-A TOOL
FOR MOTION DESIGN
A.GFRERRER
University of Technology, Institute for Geometry
A-80lO Graz, Austria.
email: [email protected].
Abstract. Via Study's kinematic mapping (5 the 6-parametric Lie group
SE(3) of direct Euclidean displacements can be identified with a certain
hyperquadric M6 in 7-dimensional real projective space. The mapping has
nice geometric properties; fot instance one parametric rotation groups are
represented by straight lines on M6, coordinate transformations in Eu
clidean 3-space are represented by special automorphisms of M6. With the
help of (5 Euclidean kinematics can be considered as a point-geometry in
the sense of Felix Klein's Erlangen program.
We give an application in the field of motion design: The problem of
constructing a motion interpolating a sequence of given positions can be
solved by constructing an appropriate curve interpolating the corresponding
points on Study's quadric.
1. Introduction
In the following we consider the Euclidean 3-space lEa as affine part of the
real projective 3-space ]ID3. A point X in ]ID3 is described by its homoge
neous coordinates (Xo, Xl, X2, X3) t. If X is a proper point (xo =1= 0) then the
inhomogeneous coordinates of X are (i;, i;, ~) t .
A direct displacement a in lEa can be described by
(:~~ ~l ~12 ~13).
a: x* = x (1)
fin20 fin 2 I fin22 fin23
fin30 fin 3 I fin32 fin33
where for the matrix M := (finijkjE{I,2,3} the conditions
finoo =1= 0, M· Mt = fin50· E, det M = fin30 (2)
have to be fulfilled. Here x and x* denote the homogeneous coordinate vec
tors of a point X and its image X* = a(X), respectively and E denotes the
7
J. LeTlilrcic and M.M. StaniSic (eds.), Advances in Robot Kinematics, 7-16.
© 2000 Kluwer Academic Publishers.
8
3 x 3-unit matrix. The set of all Euclidean displacements is a 6-parametric
Lie group which is usually denoted by SE(3). This group can be embedded
into a 7-dimensional projective space JID7 via Study's kinematic mapping:
\5 : {SE(3) ---+ JID7 t } (3)
a ---+ A ... a = (ao, ... ,a7)
where the components of the homogeneous coordinate vector a representing
the point A in JID7 are determined by the relations 1:
=
aO : ai : a2 : a3
+ + +
moo mll m22 m33 : m23 - m32 : m31 - mI3 : ml2 - m2I
m23 - m32 : moo + mll - m22 - m33 : m12 + m2I : m31 + ml3 (4)
+ + +
m31 - ml3 : ml2 m2I : moo - mll m22 - m33 : m23 m32
+ + +
ml2 - m2I : m31 ml3 : m23 m32 : moo - mll - m22 m33,
+ +
2· mOO' a4 +aI . mlO a2 . m20 a3 . m30,
+
2· mOO' a5 -aD' mlO a3 . m20 ~ a2 . m30,
(5)
+
2 . mOO' a6 -a3 . mlO - ao . m20 al . m30,
2· mOO' a7 +a2 . mlO - al . m20 - ao . m30·
The homogeneous coordinates of a point A, computed via eqs. (4), (5),
satisfy
(6)
which shows us that the points A lie on a hyperquadric M6 (Study's
quadric), given by this equation.
Conversely, for any point A ... a = (ao, ... ,a7)t on M6 with
(ao,al,a2,a3)t i- (O,O,O,O)t there is exactly one preimage a E SE(3): The
corresponding matrix-entries mij have to be computed by
2 + 2 + 2 + 2
moo ao al a2 a3'
mlO = 2· (a2 . a7 - a3 . a6 - ao . a5 + al . a4),
2 + 2 2 2
mll ao ai - a2 - a3,
+
ml2 2· (al . a2 aD· a3),
ml3 2· (a3 . al - ao . a2),
+
m20 2· (a3 . a5 - al . a7 - ao . a6 a2 . a4),
m21 2· (al . a2 - ao . a3), (7)
2 2+ 22
m22 ao - al a2 - a3,
+
m23 2· (a2 . a3 ao . al),
+
m30 2· (al . a6 - a2 . a5 - ao . a7 a3 . a4), .
+
m31 2· (a3 . al ao . a2),
m32 2· (a2 . a3 - ao . al),
2 2 2 + 2
m33 ao - al - a2 a3·
1 Compare with Study (1903), pp. 174-177. The parameters ao, ai, a2, a3 are the Euler
parameters of the rotational part of the displacement.