Tài liệu Advances in Robot KinematicsMechanisms and Motion Edited byJADRAN pdf

ADVANCES IN ROBOT KINEMATICS

Advances in Robot Kinematics

Edited by

I

Jo ef Stefan Institute

Ljubljana, Slovenia

and

B. ROTH

Stanford University

California, U.S.A.

Mechanisms and Motion

ý ý

ž

JADRAN LENAR

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-4940-4 (HB)

ISBN-13 978-1-4020-4940-8 (HB)

ISBN-10 1-4020-4941-2 (e-book)

ISBN-13 978-1-4020-4941-5 (e-book)

Published by Springer,

P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

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All Rights Reserved

© 2006 Springer

No part of this work may be reproduced, stored in a retrieval system, or transmitted

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Printed in the Netherlands.

Preface

This is the tenth book in the series of Advances in Robot Kinematics.

Two were produced as workshop proceedings, Springer published one

book in 1991 and since 1994 Kluwer published a book every two years

without interruptions. These books deal with the theory and practice

of robot kinematics and treat the motion of robots, in particular robot

manipulators, without regard to how this motion is produced or con￾trolled. Each book of Advances in Robot Kinematics reports the most

recent research projects and presents many new discoveries.

The issues addressed in this book are fundamentally kinematic in

nature, including synthesis, calibration, redundancy, force control, dex￾terity, inverse and forward kinematics, kinematic singularities, as well as

over-constrained systems. Methods used include line geometry, quater￾nion algebra, screw algebra, and linear algebra. These methods are ap￾plied to both parallel and serial multi-degree-of-freedom systems. The

en

application.

All the contributions had been rigorously reviewed by independent

reviewers and fifty three articles had been recommended for publica￾tion. They were introduced in seven chapters. The authors discussed

their results at the tenth international symposium on Advances in Robot

Kinematics which was held in June 2006 in Ljubljana, Slovenia. The

symposium was organized by Jozef Stefan Institute, Ljubljana, under

the patronage of IFToMM - International Federation for the Promotion

of Mechanism and Machine Science.

We are grateful to the authors for their contributions and for their

efficiency in preparing the manuscripts, and to the reviewers for their

timely reviews and recommendations. We are also indebted to the per￾sonnel at Springer for their excellent technical and editorial support.

Jadran Lenarˇciˇc and Bernard Roth, editors

results should interest researchers, teachers and students, in fields of

gineering and mathematics related to robot theory, design, control and

Contents

Methods in Kinematics

J. Andrade-Cetto, F. Thomas

Wire-based tracking using mutual information 3

G. Nawratil

The control number as index for Stewart Gough platforms 15

C. Innocenti, D. Paganelli

Determining the 3×3 rotation matrices that satisfy three linear

equations in the direction cosines 23

P.M. Larochelle

A polar decomposition based displacement metric for a finite

region of SE(n) 33

J.-P. Merlet, P. Donelan

On the regularity of the inverse Jacobian of parallel robots 41

P. Fanghella, C. Galletti, E. Giannotti

Parallel robots that change their group of motion 49

A.P. Murray, B.M. Korte, J.P. Schmiedeler

Approximating planar, morphing curves with rigid-body linkages 57

M. Zoppi, D. Zlatanov, R. Molfino

On the velocity analysis of non-parallel closed chain mechanisms 65

Properties of Mechanisms

H. Bamberger, M. Shoham, A. Wolf

Kinematics of micro planar parallel robot comprising large joint

clearances 75

H.K. Jung, C.D. Crane III, R.G. Roberts

Stiffness mapping of planar compliant parallel mechanisms in a

serial arrangement 85

Y. Wang, G.S. Chirikjian

Large kinematic error propagation in revolute manipulators 95

A. Pott, M. Hiller

A framework for the analysis, synthesis and optimization

of parallel kinematic machines 103

Z. Luo, J.S. Dai

Searching for undiscovered planar straight-line linkages 113

X. Kong, C.M. Gosselin

Type synthesis of three-DOF up-equivalent parallel

manipulators using a virtual-chain approach 123

A. De Santis, P. Pierro, B. Siciliano

The multiple virtual end-effectors approach for human-robot

interaction 133

Humanoids and Biomedicine

J. Babiˇc, D. Omrˇcen, J. Lenarˇciˇc

Balance and control of human inspired jumping robot 147

J. Park, F.C. Park

A convex optimization algorithm for stabilizing whole-body

motions of humanoid robots 157

R. Di Gregorio, V. Parenti-Castelli

Parallel mechanisms for knee orthoses with selective recovery

action 167

S. Ambike, J.P. Schmiedeler

Modeling time invariance in human arm motion coordination 177

M. Veber, T. Bajd, M. Munih

Assessment of finger joint angles and calibration of instrumental

glove 185

R. Konietschke, G. Hirzinger, Y. Yan

All singularities of the 9-DOF DLR medical robot setup for

minimally invasive applications 193

G. Liu, R.J. Milgram, A. Dhanik, J.C. Latombe

On the inverse kinematics of a fragment of protein backbone 201

V. De Sapio, J. Warren, O. Khatib

Predicting reaching postures using a kinematically constrained

shoulder model 209

viii Contents

Analysis of Mechanisms

D. Chablat, P. Wenger, I.A. Bonev

Self motions of special 3-RPR planar parallel robot 221

A. Degani, A. Wolf

Graphical singularity analysis of 3-DOF planar parallel

manipulators 229

C. Bier, A. Campos, J. Hesselbach

Direct singularity closeness indexes for the hexa parallel robot 239

A. Karger

Stewart-Gough platforms with simple singularity surface 247

A. Kecskem´ethy, M. T¨andl

A robust model for 3D tracking in object-oriented multibody

systems based on singularity-free Frenet framing 255

P. Ben-Horin, M. Shoham

Singularity of a class of Gough-Stewart platforms with three

concurrent joints 265

T.K. Tanev

Singularity analysis of a 4-DOF parallel manipulator using

geometric algebra 275

R. Daniel, R. Dunlop

A geometrical interpretation of 3-3 mechanism singularities 285

Workspace and Performance

J.A. Carretero, G.T. Pond

Quantitative dexterous workspace comparisons 297

E. Ottaviano, M. Husty, M. Ceccarelli

Level-set method for workspace analysis of serial manipulators 307

M. Gouttefarde, J P. Merlet, D. Daney

Determination of the wrench-closure workspace of 6-DOF

parallel cable-driven mechanisms 315

G. Gogu

Fully-isotropic hexapods 323

P. Last, J. Hesselbach

A new calibration stategy for a class of parallel mechanisms 331

M. Krefft, J. Hesselbach

The dynamic optimization of PKM 339

.

Contents ix

-

J.A. Snyman

On non-assembly in the optimal synthesis of serial manipulators

performing prescribed tasks 349

Design of Mechanisms

W.A. Khan, S. Caro, D. Pasini, J. Angeles

Complexity analysis for the conceptual design of robotic

architecture 359

D.V. Lee, S.A. Velinsky

Robust three-dimensional non-contacting angular motion sensor 369

K. Brunnthaler, H.-P. Schr¨ocker, M. Husty

Synthesis of spherical four-bar mechanisms using spherical

kinematic mapping 377

R. Vertechy, V. Parenti-Castelli

Synthesis of 2-DOF spherical fully parallel mechanisms 385

G.S. Soh, J.M. McCarthy

Constraint synthesis for planar n-R robots 395

T. Bruckmann, A. Pott, M. Hiller

Calculating force distributions for redundantly actuated

tendon 403

P. Boning, S. Dubowsky

A study of minimal sensor topologies for space robots 413

M. Callegari, M.-C. Palpacelli

Kinematics and optimization of the translating 3-CCR/3-RCC

parallel mechanisms 423

Motion Synthesis and Mobility

C.-C. Lee, J.M. Herv´e

Pseudo-planar motion generators 435

S. Krut, F. Pierrot, O. Company

On PKM with articulated travelling-plate and large tilting angles 445

C.R. Diez-Mart´ınez, J.M. Rico, J.J. Cervantes-S´anchez,

J. Gallardo

Mobility and connectivity in multiloop linkages 455

K. Tcho´n, J. Jakubiak

Jacobian inverse kinematics algorithms with variable steplength

for mobile manipulators 465

x

-based Stewart platforms

Contents

J. Zamora-Esquivel, E. Bayro-Corrochano

Kinematics and grasping using conformal geometric algebra 473

R. Subramanian, K. Kazerounian

Application of kinematics tools in the study of internal

mobility of protein molecules 481

O. Altuzarra, C. Pinto, V. Petuya, A. Hernandez

Motion pattern singularity in lower mobility parallel

manipulators 489

Author Index 497

Contents xi

Methods in Kinematics

J. Andrade-Cetto, F. Thomas

Wire-based tracking using mutual information

G. Nawratil

C. Innocenti, D. Paganelli

Determining the 3×3 rotation matrices that satisfy three

linear equations in the direction cosines

P.M. Larochelle

A polar decomposition based displacement metric for a finite

region of SE(n)

J.-P. Merlet, P. Donelan

On the regularity of the inverse Jacobian of parallel robots

P. Fanghella, C. Galletti, E. Giannotti

Parallel robots that change their group of motion

A.P. Murray, B.M. Korte, J.P. Schmiedeler

Approximating planar, morphing curves with rigid-body

linkages

M. Zoppi, D. Zlatanov, R. Molfino

On the velocity analysis of non-parallel closed chain

mechanisms

3

15

23

33

41

49

57

65

The control number as index for Stewart Gough platforms

WIRE-BASED TRACKING USING

MUTUAL INFORMATION

Juan Andrade-Cetto

Computer Vision Center, UAB

Edifici O, Campus UAB, 08193 Bellaterra, Spain

[email protected]

Federico Thomas

Institut de Rob`otica i Inform`atica Industrial, CSIC-UPC

Llorens Artigas 4-6, 08028 Barcelona, Spain

[email protected]

Abstract

ing devices. They consist of a fixed base and a platform, attached to

the moving object, connected by six wires whose tension is maintained

along the tracked trajectory. One important shortcoming of this kind

of devices is that they are forced to operate in reduced workspaces so

as to avoid singular configurations. Singularities can be eliminated by

adding more wires but this causes more wire interferences, and a higher

force exerted on the moving object by the measuring device itself. This

paper shows how, by introducing a rotating base, the number of wires

can be reduced to three, and singularities can be avoided by using an

active sensing strategy. This also permits reducing wire interference

problems and the pulling force exerted by the device. The proposed

sensing strategy minimizes the uncertainty in the location of the plat￾form. Candidate motions of the rotating base are compared selected

automatically based on mutual information scores.

Keywords:

1. Introduction

Tracking devices, also called 6-degree-of-freedom (6-DOF) devices, are

used for estimating the position and orientation of moving objects. Cur￾rent tracking devices are based on electromagnetic, acoustic, mechani￾cal, or optical technology. Tracking devices can be classified according

to their characteristics, such as accuracy, resolution, cost, measurement

range, portability, and calibration requirements. Laser tracking systems

exhibit good accuracy, which can be less than 1µm if the system is well

calibrated. Unfortunately, this kind of systems are very expensive, their

3

J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 3–14.

© 2006 Springer. Printed in the Netherlands.

Wire-based tracking devices are an affordable alternative to costly track￾Tracking devices, Kalman filter, active sensing, mutual information,

parallel manipulators

calibration procedure is time-consuming, and they are sensitive to the

environment. Vision systems can reach an accuracy of 0.1mm. They are

low-cost portable devices but their calibration procedure can be compli￾cated. Wire-based systems can reach an accuracy of 0.1mm, they are

also low cost portable devices but capable of measuring large displace￾ments. Moreover, they exhibit a good compromise among accuracy,

measurement range, cost and operability.

Wire-based tracking devices consist of a fixed base and a platform

connected by six wires whose tension is maintained, while the platform is

moved, by pulleys and spiral springs on the base, where a set of encoders

give the length of the wires. They can be modelled as 6-DOF parallel

manipulators because wires can be seen as extensible legs connecting

the platform and the base by means of spherical and universal joints,

respectively.

Dimension deviations due to fabrication tolerances, wire-length un￾certainties, or wire slackness, may result in unacceptable performance of

rors can be eliminated by calibration. Some techniques for specific errors

have already been proposed in the literature. For example, a method

for compensating the cable guide outlet shape of wire encoders is de￾tailed in Geng and Haynes, 1994, and a method for compensating the

deflections caused by wire self-weights is described in Jeong et al., 1999.

In this paper, we will only consider wire-length errors which cannot be

compensated because of their random nature.

Another indirect source of error is the force exerted by the measuring

device itself. Indeed, all commercial wire encoders are designed to keep

a large string tension. This is necessary to ensure that the inertia of the

mechanism does not result in a wire going slack during a rapid motion.

If a low wire force is used, it would reduce the maximum speed of the

object to be tracked without the wires going slack. On the contrary, if a

high wire force is used, the trajectory of the object to be tracked could

be altered by the measuring device. Hence, a trade-off between accuracy

and speed arises.

The minimum number of points on a moving object to be tracked for

pose measurements is three. Moreover, the maximum number of wires

attached to a point is also three, otherwise the lengths of the wires will

not be independent. This leads to only two possible configurations for

the attachments on the moving object. The 3-2-1 configuration was pro￾posed in Geng and Haynes, 1994. The kinematics of this configuration

was studied, for example, in Nanua and Waldron, 1990 and Hunt and

Primrose, 1993. Its direct kinematics can be solved in closed-form by

using three consecutive trilateration operations yielding 8 solutions, as

a wire-based tracking device. In general, the effects of all systematic er￾4 J. Andrade-Cetto and F. Thomas

(a) (b) (c)

p platform

base

(d)

x

z

a1

a2

a3

ρ1

a¯

yA

θA

xA

Figure 1. The main two configurations used for wire-based tracking devices: (a) the

“3-2-1”, (b) the “2-2-2”, and (c) the proposed tracking device, with (d) the rotating

in Thomas et al., 2005. The 2-2-2 configuration was first proposed in

Jeong et al., 1999 for a wire-based tracking device. The kinematics of

this configuration was studied, for example, in Griffis and Duffy, 1989,

Nanua et al., 1990, and Parenti-Castelli and Innocenti, 1990 where it

was shown that its forward kinematics has 16 solutions. In other words,

there are up to 16 poses for the moving object compatible with a given

set of wire lengths. These configurations can only be obtained by a nu￾merical method. The two configurations above were compared, in terms

of their sensitivity to wire-length errors, in Geng and Haynes, 1994. The

conclusion was that they have similar properties.

This paper is organized as follows. Section 2 contains the mathemat￾ical model of our proposed 3-wire-based sensing device, while Section 3

derives the filtering strategy for tracking its pose. Given that this device

has a moving part, Section 4 develops an information theoretic metric

for choosing the best actions for controlling it. A strategy to prevent

possible wire crossings is contemplated in Section 5. Section 6 is de￾voted to a set of examples demonstrating the viability of the proposed

approach. Finally, concluding remarks are presented in Section 7.

2.

In order to reduce cable interferences, singularities, and wire tension

problems we choose to reduce the number of cables from six to three, and

to have the base rotate on its center. Provided the tracked object mo￾tion is sufficiently slow, two measurements at different base orientations

would be equivalent to a 2-2-2 configuration.

More elegantly, and to let the tracked object move at a faster speed,

measurements can be integrated sequentially through a partially observ￾able estimation framework. That is, a Kalman filter.

Wire-based Tracking Using Mutual Information 5

base.

Kinematics of the Proposed Sensor

Consider the 3-wire parallel device in Figure 1(c). It is assumed that

Let the pose of our tracking device be defined as the 14-dimensional array

x =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

p

θ

v

ω

θA

ωA

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

, (1)

where p = (x, y, z)

is the position of the origin of a coordinate frame

fixed to the platform, θ = (ψ, θ, φ)

is the orientation of such coordinate

frame expressed as yaw, pitch and roll angles, v = (vx, vy, vz)

and

ω = (ωx, ωy, ωz)

are the translational and rotational velocities of p,

respectively; and θA and ωA are the orientation and angular velocity of

the rotating base.

Assume that the attaching points on the base ai, i = 1, 2, 3, are

distributed on a circle of radius ¯a as shown in Figure 1(d). Then, the

coordinates of ai can be expressed in terms of the platform rotation

angle θA as

⎡

⎣

axi

ayi

azi

⎤

⎦ =

⎡

⎣

a¯ cos(ρi + θA)

a¯ sin(ρi + θA)

0

⎤

⎦ . (2)

Moreover, let ei be the unit norm vector specifying the direction from

ai to the corresponding attaching point bi in the platform; and let li

be the length of the i-th wire, i = 1, 2, 3. The value of bi is expressed

in platform local coordinates, where R is the rotation matrix describing

the absolute orientation of the platform. Then, the position of the wire

attaching points in the platform, in global coordinates, are

b

i = ai + liei = p + Rbi . (3)

3. State Estimation

We adopt a smooth unconstrained constant-velocity motion model, its

pose altered only by zero-mean, normally distributed accelerations and

staying the same on average. The Gaussian acceleration assumption

means that large impulsive changes of direction are unlikely. In such

model the prediction of the position and orientation of the platform at

time t plus a time interval τ is given by



p(t + τ )

θ(t + τ )



=



p(t) + v(t)τ + δa(t)τ 2/2

θ(t) + ω(t)τ + δα(t)τ 2/2



, (4)

6 J. Andrade-Cetto and F. Thomas

the platform configuration is free to move in any direction in IR3×SO(3).

with δa and δα zero mean white Gaussian translational and angular

acceleration noises. Moreover, the adopted model for the translational

and angular velocities of the platform is given by



v(t + τ )

ω(t + τ )



=



v(t) + δa(t)τ

ω(t) + δα(t)τ



. (5)

By the same token, the adopted models for the orientation and angular

velocity of the base are



θA(t + τ )

ωA(t + τ )



=



θA(t) + ωA(t)τ + (αA(t) + δαA(t))τ 2/2

ωA(t)+(αA(t) + δαA(t))τ



, (6)

in which the control signal modifying the base orientation is the accel￾eration impulse αA.

Since in practice, the measured wire lengths, li, i = 1, 2, 3, will be

corrupted by additive Gaussian noise, δzi, we have that

zi(t) = li(t) + δzi(t) =
p(t) + R(t)bi − ai(t)
+ δzi(t) . (7)

Lastly, the orientation of the moving base is measured by means of

an encoder. Its model is simply

z4(t) = θA(t) + δz4(t) . (8)

Eqs. 4 and 5 constitute our motion prediction model f(x, αA, δx).

Now, an Extended Kalman Filter can be used to propagate the platform

pose and velocity estimates, as well as the base orientation estimates,

and then, to refine these estimates through wire length measurements.

To this end, δx ∼ N(0, Q), δz ∼ N(0, R), and our plant Jacobians with

respect to the state F = ∂f/∂x, and to the noise G = ∂f/∂δx become

F =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

I τ I

I τ I

I

I

1 τ

1

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

and G =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

τ 2I

2 τ 2I

2

τ I

τ I τ 2

2

τ

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

. (9)

The measurement Jacobians H = ∂h/∂x are simply

Hi(t) =

ei(t) bi × ei(t) 0 0 ∂hi

∂θA

0

, (10)

with

ei(t) = p(t) + R(t)bi − ai(t)

p(t) + R(t)bi − ai(t)
. (11)

Wire-based Tracking Using Mutual Information 7

Eqs. 7 and 8 complete our measurement prediction model h(x, δz).