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Proceedings of the Seventh

CISM-IFToMM Symposium

on Theory and Practice

of Robots and Manipulators

3 Edited by

A Morecki, G Bianchi and K Jaworek RoManSy 7

Proceedings of the Seventh CISM-IFToMM Symposium

on Theory and Practice of Robots and Manipulators

Edited by

A. Morecki, G. Bianchi and K. Jaworek RoManSy 7

Proceedings of the Seventh

CISM-IFToMM Symposium

on Theory and Practice

of Robots and Manipulators

Edited by

A. Morecki, G. Bianchi and K. Jaworek

Sponsored by CISM - Centre International des Sciences Mécaniques, IFToMM

International Federation for the Theory of Machines and Mechanics, in associa

tion with the IV Technical Division of the Polish Academy of Sciences

HERMES © Hermès, 1990

Editions Hermès

34, rue Eugène Flachat

75017 Paris

ISBN 2-86601-215-1 Contents

Organizing and Programme Committee 9

Editorial Note 11

Opening Lecture - M. Vukobratovic3

Part - Mobile Robots and Walking Machines 1 37

1. K. Kircanski, M. Vukobratovic, M. Kircanski, A. Timcenko

Mobile Robots - Customized symbolic models. Simulation and control 38

2. V. Kumar, K.J. Waldron

Gait analysis for walking machines for omnidirectional locomotion on

uneven terrain 46

3. A. Takanishi, Y. Egusa, M. Tochizawa, T. Takeya, I. Kato

Realization of dynamic biped walking stabilized with trunk motion ... 68

4. K. Jaworek, G. Ferrigno, G.C. Santambrogio

On the assessment of human locomotion 80

5. P. Bellamy, J.-C. Guinot

Mobile robots : Gripping ground 93

Part 2 - Control of Motion 1 105

1. A.K. Bejczy, B. Hannaford, Z. Szakaly

Multi-mode manual control in telerobotics6

2. K. Bouazza-Marouf, J. Hewit

Active control of robot arms on ramdomly moving mobile basis 119

3. O. Khatib

Inertial characteristics and dextrous dynamic coordination of macro/mi

cro-manipulator systems 131

4. P. Kiriazov, P. Marinov

On the independent dynamics of manipulator systems 140

5. F.W. Paul, L. Guvenc

Active end-effector control for robot assisted material removal8 RoManSy 7 6

Part 3 - Sensing 159

1. H. Diensing, B. Heimann, U. Schmucker

Multicomponent force/torque-sensors in robotics 160

2. R. Palm

Sensor guided robot control by fuzzy logic8

3. K. Tanie, M. Park, K. Komoriya, M. Kaneko, N. Suzuki

An imaging tactile sensor for robot hand using a light conductive plate 176

4. R.M. Cutkosky, R.D. Howe

Dynamic tactile sensing 184

5. R.P. Paul, X. Xu, X. Yun

Terminal link force and position control of a robot manipulator 192

Part 4 - Mobile Robots and Walking Machines 2 211

1. A.P. Bessonov, S.I. Goncharov, N.U. Umnov

Evaluation of the top limit velocity for walking robots2

2. A. Morecki, Z. Busko, K. Jaworek, W. Pogorzelski, T. Zielinska,

J. Fraczek, G. Malczyk

Robotic system : Elastic manipulator combined with a quadruped walking

machine 221

3. V.B. Larin

A walking machine control 235

4. S. Hirose, A. Morishima

Articulated body mobile robot 243

Part 5 - Synthesis and Design 1 25

1. M.S. Konstantinov

Mechatronics in robotics2

2. S. Sugano, H. Suzuki, O. Matsumoto, I. Kato

Design of the piano playing manipulator with four-dof-thumb 265

3. F Duditza, D. Diaconescu, I. Staretzu

Structural synthesis and systematization of grippers with redundant

mobilities 276

4. A.E. Samuel, K.H. Hunt

Design of a fast cartesian robot 288

5. J.-P. Merlet

Parallel manipulators. 301

Part 6 - Synthesis and Design 29

1. Harrison H.S. Wang, B. Roth

Position errors of manipulators 310 7 Contents

2. M. Ceccarelli, A. Vinciguerra

A design method of three revolute open chain manipulators 318

3. J. Wojnarowski, J. Swider

The research of manipulator model of dynamical flexibility with the me

thod of matrix hibridgraphs 326

4. R.W. Daniel, M. A. Irving

A compliant direct drive industrial robot arm 334

5. H. Asama, K. Yokota, H. Yoshikawa

A knowledge-based task sequence planning system for maintenance mani- 342

pulators

Part 7 - Control of Motion 2 351

1. D.M. Stokic, M.K. Vukobratovic, D. Devedjic

Export system for synthesis of dynamic control of manipulation robots 352

2. R. Fournier, P. Gravez, J. Morillon, A. Detolle, M. Dupont

Mobile robot computer aided teleoperation : A generalized master-slave

control concept application 362

3. J. Ackermann, P.C. Miiller

Compensation of Coulomb friction in the position control of elastic robots 370

4. F. Pfeiffcr

Trajectory control for elastic manipulators 378

5. H. Henrichfreise, W. Moritz, H. Siemensmeyer

Control of a light, elastic manipulation device 38

Part 8 - Mechanics 1 399

1. V. Parenti Castelli, C. Innocenti

Spatial open Kinematic chains : Singularities, regions and subregions 400

2. M. Hiller, C. Woernle

A solution of the inverse kinematic problem for robots with non-intersec-

tiong wrist axes 408

3. H. Fujimoto, H. Takita, M. Arita, K. Koganezawa, I. Kato

A/K prosthesis for ascending and descending stairs 41

Part 9 - Mechanics 2 431

1. P. Genova, R. Kazandjieva, K. Yankov

Computational algorithm for identification of inertia and dissipation para

meters of manipulating systems2

2. S. Dubowsky, I. Paul, H. West

An analytical and experimental program to develop control algorithms for

mobile manipulators 440 8 RoManSy 7

3. A.I. Korendyasev, B.L. Salamandra, L.I. Tyves

Mechanics of robots with dynamically decoupled motions 448

4. M. Riemer, J. Wauer

Equations of motion for flexible multibody systems with revolute and pris

matic joints 456

Part 10 - Application and Performance Evaluation 1 467

1. E.J. Kreuzer, W.O. Schiehlen, E. Settelmeyer

Robot controller for uncertainties in modelling and tracking8

2. H.J. Warnecke, W.C. Wanner

Application and experience with software tools for the design of industrial

robots 47

3. K. Ikuta

Application of shape memory alloy actuator for clean gripper 482

4. G.V. Krejnin

Positioning pneumatic drive in robotics 490

Part 11 - Application and Performance Evaluation 29

1. A. Rovetta, R. Sala, C. Sala, D. Bugo, F. Zecchini

Assembly with single and double arms robots by structured logic 500

2. B.D. Petriashvili, V.O. Margvelashvili, M.A. Bilashvili

Methods of maintenance of assigned parameters of multi-legged walking

robot motion 510

3. H. Iwata, K. Marukawa, K. Matsushima

Development of an instruction system for robotics and manipulators 518

4. M.T. Mason, K.Y. Goldberg, R.H. Taylor

Planning sequences of squeeze-grasps to orient and grasp polygonal objects 531

List of Participants 539 Organizing and Programme Committee

Chairmen

Prof. G. Bianchi

CISM, Piazza Garibaldi 18, 33100 Udine, Italy

Prof. A. Morecki

Warsaw University of Technology

Al. Niepodleglosci 222, 00-663 Warsaw, Poland

Members

Prof. A.P. Bessonov

Academy of Sciences of the USSR

Griboedova 4, Moscow-Centre 101000, USSR

Prof. J.-C. Guinot

Université P. et M. Curie

Lab. de Mécanique et Robotique, IMTA

4, place Jussieu-Tour 66

75230 Paris cedex 05, France

Dr. B. Heimann

Academy of Sciences of GDR

Institute of Mechanics

Reichenhainer 88, 9010 Karl Marx Stadt, GDR

Dr. J.R. Hewit

Loughborough University of Technology

Dept. of Mechanical Engineering h LE11 3TU, England

Prof. S. Hirose

Tokyo Institute of Technology

Dept. of Physical Engineering

2-12-1 Ookayama, Meguro-ku, Tokyo, Japan

Prof. M. Konstantinov

Central Laboratory for Manipulators and Robots

Darwenitza, Sofia, Bulgaria

Prof. A.I. Korendyasev

Academy of Sciences of the USSR

Griboedova 4, Moscow-Centre 101000, USSR 10 RoManSy 7

Prof. K. Matsushima

Institute of Engineering Mechanics

University of Tsukuba, Sakura, Ibaraki 300-31, Japan

Prof. W.O. Schiehlen

Universitat Stuttgart, Institut B fur Mechanik

Pfaffenwaldring 9

7000 Stuttgart 80, GFR

Prof. M. Vukobratovic

Institute Mihajlo Pupin

Volgina 15

Beograd, Yugoslavia

Prof. K.J. Waldron

Dept. of Mechanical Engineering

The Ohio State University

Colombus, Ohio 43210, USA

Scientific Secretary

DR. K. Jaworek

Warsaw University of Technology

Al. Niepodleglosci 222, 00-663 Warsaw, Poland

Secretary

Dr. A. Bertozzi

CISM, Piazza Garibaldi 18, 33100 Udine, Italy Editorial Note

This volume contains the papers accepted for the Seventh Symposium on

Theory and Practice of Robots and Manipulators "RoManSy'88" held in Udi-

ne, Italy, 12-15 September 1988.

"RoManSy'88" was attended by 71 participants from 14 countries, who

were selected experts in the field of robotics.

The Symposium Programme included :

Opening and Closing Sessions attended by CISM and IFToMM officials, g Lecture given by Professor M. Vukobratovic,

Working Sessions (mobile robots and walking machines, control of mo

tion, sensing, synthesis and design, mechanics, application and perfor

mance evaluations),

Round table discussion,

Film and video session.

The papers in this book are in the same sequence as the sessions mentio

ned above.

All linguistic and terminology correctivus have been kept to a minimum.

The Proceedings of the previous Six Symposia are available in the final form.

Thes of the "RoManSy'73" (Sept. 5-8, 1983, Udine, Italy) may be

obtained from Springer-Verlag, Vienna, or CISM, Udine, Italy. Those of the

"RoManSy'76" (Sept. 14-17, 1976, Jadwisin, Poland) and "RoManSy'78" (Sept.

12-15, 1978, Udine, Italy) may be purchased from Elsevier (Amsterdam, Hol

land) od PWN - Polish Scientific Publishers (Warsaw, Poland). Proceedings

of the "RoManSy'81" (Sept. 8-12, 1981, Zaborow, Poland) may be obtained

from PWN (Warsaw, Poland). The Proceedings of the "RoManSy'84" (June

26-29, 1984, Udine, Italy) and "RoManSy'86" (Sept. 9-12, 1986, Cracow, Po

land) may be obtained from Editions Hermes (Paris), Kogan Page (London)

and MIT Press (Cambridge, Massachusetts, USA).

The next Symposium "RoManSy'90" will be held in Cracow, Poland, in July

1990.

The Editors express treir thanks to Mr. S. Ménascé for his excellent coo

peration in the publishing of the Proceedings.

A. Morecki, G. Bianchi and K. Jaworek Opening Lecture Customized Software for Dynamic Analysis

and Control Synthesis of Robotic Manipulators

Miomir Vukobratovic

Mihajlo Pupin Institute, Beograd, Yugoslavia

ABSTRACT - To reduce computational requirements of modelling

algorithms for real-time applications, it is necessary to cus

tomize the algorithms for specific robot design. Various methods

appeared in this field in the last several years constituting so

called computational robot dynamics. Some of these methods are

based on Lagrangian formulation while the other utilize recursi

ve Newton-Euler formulation. These methods are essentially sin

gle-step methods. It means that the output customized equations

are obtained after one step from the general equations by elimi

nating unnecessary algebraic operations. In this paper we shall

focus our attantion on a new class of methods: multi-step opti

mizing customization methods. These methods are based on polinomial

representation of model variables and optimization ofl

computations. The program support for design and synthesis of control

for robots is also presented in the paper. The program

includes several software packages which enable precise simulation

of flexible manufacturing cells, robot design, robot modelling and

synthesis of control for robots. The packages are compatable, so

the user may easily combine them to design and synthesize con

trol of robots which are appropriate for particular flexible manu

facturing cell.

1 . INTRODUCTION

An adequate study of robotic mechanisms starts with the development

of computer oriented methods for forming the mathematical models

of kinematics, notable dynamics of spatial active mechanisms. By

means of this the barriers in complete insight of the dynamic

robot models significance were jumped, aimed at the synthesis of

dynamic control laws with various classes of dynamic manipulation

tasks.

The raising rate of the development of robotic mechanism mathema

tical models went the way from the numerical-iterative computer 15 Opening Lecture

methods, over the numeric-symbolic ones and, finally, to the

forming of mathematica1 model s in a symbolic form. While the first

version, even the second, are predominantly research versions

of the computer procedures in forming mathematical models of

robotic mechanisms, the third one, a symbolic model form, is a

typical implementation version.

Mathematical models of robot kinematics and dynamics are not the goal,

but just the tools f the synthesis of dynamic control of o r

these mechanisms. Thus, the customized software for the synthe

sis of dynamic control laws of various complexity is a signifi

cant step towards the realization of control system, the per

formances of which satisfy the ever more complex tasks of indus

trial manipulation, as well as the manipulation being performed

in some unconventional working environments.

The further and even more complex role of robots should be

sought in the scope of flexible production cells, and broader,

of intelligent technological systems. Hence the customized soft

ware should be expanded towards the simulation and control of

flexible modules, by means of which a specific, new and more

extensive promotion of the robotic technology is carried out.

In the course of the several last years, there has been published

a great number of papers, rendering an important contribution to

the development of computer methods for the forming of mathema

tical models of robotic mechanisms, as well as the computer syn

thesis of non-adaptive and adaptive dynamic control of robotic

systems of various complexity and deducation.

The author of this paper had to mention, for understandable

reasons, only some, according to his opinion, characteristic

results dedicated to the newest results directly or indirectly

connected with customized software for mathematical modelling

and the synthesis of control laws of manipulation robots.

At the same time the author has taken the liberty to represent

in more details the results of customized software, developed in 16 RoManSy 7

the course of the last years in the Laboratory for Robotics and

Flexible Automation of the Mihajlo Pupin Institute.

2 . AN OVERVIEW OF GENERAL MODELLING METHODS

The modelling methods may be classified with respect to the laws

of mechanics on the basis of which motion equations are formed.

One may distinguish methods based on Lagrange, Newton-Euler,

Appel and other formalisms for dynamic modelling of interconnec

ted multibody systems. The other criterion is whether the method

yields the solution to both the direct and the inverse problem

of dynamics or only to the inverse problem itself (computatation of dri

ving forces given desired motion of manipulator). The number of

multiplications/additions is yet another criterion of comparing

the methods. This criterion is certainly the most important one

from the point of view of their real-time applicability.

2.1 . Lagrangian formalism

The first result in the class of Lagrange methods, which are

used to solve either the direct or the inverse problem of dyna

mics was reported by Uicker, [1]. Since the method referred

mostly to certain classes of closed loop mechanisms, Kahn [2]

elaborated an algorithm for modelling open kinematic chains.

The method was modified and reformulated by Woo and Freunden-

stein [3] through introducing screw calculus. Improved versions

of this method in the sense of a reduced number of numerical

operations were developed by Mahil [4] and Renaud [5]. A proper

ty common to these algorithms is that they do not employ recursive

relations but closed form expressions for the elements of model

matrices. However, the number of multiplications/additions in

4

these methods depends on n (n - number of joints) and is very

large for microprocessor implementation.

Recursive Lagrangian formalism developed by Vukobratovid and Potko-

njak [6] improves the computational efficiency several times com

paring to Kahn - Uicker's method. Though it is still difficult Opening Lecture 17

for real-time applications since it requires more than two

thousand multiplications for a standard 6-joint manipulator.

The referred to methods that yield the solution to both the direct

and the inverse dynamic problem, are obviously not suitable for

real-time applications. Thus, Waters and Hollerbach developed

algorithms for solving only the inverse problem of dynamics on

the basis of Uicker-Kahn's method, and reduced the number of

2

operations to n [7], or even n. Here the inertia matrix is not

explicitely calculated. Nevertheless the Newton-Euler formalism

yields even less operation then the mentioned methods.

2.2. Newton-Euler formalism

The application of Newton-Euler dynamic equations to modelling

of interconnected rigid bodies may be found in papers by Kane

[8]. Huston adapted Kane's results for robotic systems [9).

These algorithms solve both the direct and the inverse problem

of dynamics, using closed form expressions for the elements of

dynamic model matrices. A more efficient computer method was

elaborated in [10, 11] by introducing recursive relations into

the algorithm and reduced the number of operations to n^. Walker

and Orin [12] reorganized the computations and reduced the num

ber of operations to n . A special case of Newton - Euler method

that yields the solution to the inverse problem of dynamics was

treated by Luh, Walker and Paul [13]. By expressing the equations

of motion of each link with respect to local coordinate frames

they obtained a much more efficient method comparing to the previ

ous methods and reduced the number of multiplications to 150n-48,

i.e. 852 multiplications for a 6 joint manipulator. It appeared

that the customized algorithms offered much more efficient models

without losing the accuracy of computation. 18 RoManSy 7

3. SINGLE-STEP CUSTOMIZED ALGORITHMS

The common property of the methods described in Sec. 2 is that

they do not depend on any specific manipulator design. They employ

general laws of kinematics and dynamics of joint-interconnected

rigid bodies. The kinematic and dynamic model of a specific ma

nipulator may comprise a considerably less number of numerical ope

rations. For real - time applications it is suitable to customize

the algorithm fora specific manipulator dcsign.This is first re

cognized by Aldon [14], Vukobratovic" and Kirdanski [15, 16], and

Renaud [17]. The method [15, 16] belongs to the class of multi-

-step optimizing methods and will be described in more details

in the next section. The methods given in [14] and [17] are not

computer-oriented while the methods referred to in [15, 16j are

fully automatic from the standpoint of computer generation of

model source code in a high level computer language (FORTRAN'77).

As surveyed in [18] several papers dealing with computer symbolic

model generation appeared in 1984 (for example [19]). Most of

them were based on inefficient Lagrange formalism. From 1985 to

1988 many papers based on Newton - Euler formalism appeared

[2 0 - 26]. These methods belong to the class of single-step

algorithms, because the generated models strictly follow the ge

neral models that represent the input of the algorithm. The un

necessary arithmetic operations as multiplication by 1 and addi

tion by 0 are simply eliminated from the source code. These me

thods do not take into account the data-dependency graph and do not

optimize the symbolic expressions. The comparison between seve

ral single-step customized algorithms is given in Figure 1.

The number of F.P. operations is still high (about 500) for

implementation within low-cost industrial robot controllers.

For high performance robots (with high torque direct drive actu

ators) computing time for dynamic compensations (model) and re

gulators must be below 1 ms. Thus, the model itself must be com

puted in about 500 MS. During this time about 500 F.P. operati

ons must be carried out, so, the speed of the microprocessor

must be about 1 Mflop's. Since this computing power is too high

for todays technology of low-cost microcomputers, the develop

ment of new multi-step optimization algorithms as well as paral

lel computing methods is necessary. Opening Lecture 1'

METHOD S MULTIPLICATIONS ADDITIONS

150n-48 Luh's method (numeric) 131n-48

852 Stanford arm 738

Burdick's method (symbolic)

Puma 560 401 254

Horak's method (hybrid)

Stanford arm 361 256

Kircanski, Vukobratovic,

Timcenko's method

Stanford arm 196 167

Neuman's method (symbolic)

192 Stanford arm 154

1Khalil , Kleinfingers

method (symbolic, min.

inert , parameters)

Stanford arm 142 99

General robot 101n-129 90n-118

n = 6 477 422

Fig. 1. Comparison between several single-step

customized algorithms

4 . MULTISTEP OPTIMIZING CUSTOMIZED ALGORITHMS

Multistep optimizing algorithms are much more sophisticated com

paring to the methods described in the previous section. Single-

-step methods represent actually the first step of the multistep

optimizing customized algorithms. The output of the first step

is the source program of the robot model customized for a speci

fic manipulator design. This source code is the input for the

second step. In the second step we introduce the optimization RoManSy 7 20

procès to minimize the number of numerical operations. The out

put of the second step is the source code of the optimized model.

Given the same parameters and joint coordinates, velocities and

accelerations, the output of the generated programs is exactly

the same after the first, second and other steps. Only the num

ber of operations becomes less. Theoretical minimum is still not

known for a specific manipulator design, but the already existing

methods show that the number of operation is at least 2 times less

comparing to single step customization.

As pointed out in the previous section the first result in this

field appeared in 1983 (Kircanski, Vukobratovic') [15] . In this paper

as well as in the elaborated versions of this method £16-20] the

basic idea was to represent the expressions of the model in a clo

sed polynomial form (arguments were cosines and sines of the joint

coordinates) and to find out the algorithm for the computation of

the obtained polynomials with a minimal number of multiplications.

In this method the parameters are treated numerically, while the

joint coordinates are treated symbolically. A generalization of

this method where the parameters and the joint coordinates, velo

cities and accelerations are treated symbolically, is described

in £27] . Let us describe the essence of this method.

The output program source of the first step represents a FORTRAN

subroutine for computing driving torques in joints if the joint

coordinates are given, velocities and accelerations. The matrix

form of the model is

T

P = Hq + q Cq + g (q) (1)

where q represents n vector of the joint coordinates,H(q) - nxn sy-

metric, positive definite inertia matrix, the term q Cq repre-

. T

sents n vector which i-th element is of the form q C^q where

is n«n matrix due to Coriolis and centrifugal effects: 21 Opening Lecture

• T

q c.q

• T

C C•T • * i î

q Cq = (2)

• T

q C.q _

The term g(q) is an n vector due to the gravitational forces.

The general form of any expression in the output source program

after the first step is of the form

z = [-] x_1 [* y_1] ± • • • ± x_n [* y_1] (3)

where x_i and y_i are symbolic arguments, and [* y_i] denotes

that the operation * y_i is optional. The symbolic arguments

are initialized as symbolic constants, joint coordinates, joint

velocities, joint accelerations, and sines or cosines of joint

coordinates. Thereafter, the recursive relations of the form

/} / starting with these initial values follow the Newton-Euler

formalism. Unnecessary operations are avoided.

In the second step the output program with the expressions of

the type (3) is further optimized. The recursive relations are

first converted to closed-form expressions. In [28] we proved

that any variable in the model can be represented in the unified

manner as

T C s q q q

1 1 1 4 1z = I c. n (cos q ) (sin q.) q. q q (4)

1 11 i = 1 i =

where c is the coefficient that depends on manipulator parame(

ters (it can represent a symbolic expression involving kinema

tic and dynamic parameters of links) , e ^ is the constant expo

nent that can be 0, 1 or 2. Here v describes the corresponding

argument type.

The polynomials are stored up as structures consisting of symbolic

coefficients and numeric exponents. More details concerning the use 22 RoManSy 7

of the polynomials (3) are given in £29-3lJ . The set of poly

nomials (3) is further optimized with respect to the number of

multiplications. The common terms (monomials) are identified

and replaced by auxiliary variables. Thus, some new recursive rela

tions betweeny variables are identified> and, the

output source code after the second step is the same as the out

put code of the first step (Fig. 2). The difference is only in

the number of operations. It is illustrated in Fig. 3. Three ty

pical industrial robots are considered (Puma, Scara and Stand-

ford arm). The difference between the number of operations after

the first and the second step shows the effectiveness of the

proposed optimisation. An example of the generated program code

for the Puma robot with 3 joints is given in Fig. 4.

HEADER

TRIGONOMETRICAL

EXPRESSIONS

ALGEBRAIC

EXPRESSIONS

EN D

Fig. 2. Model source program organizations

after the second step

I step outpu t II step outpu t

MANIPULATOR n * +/- +/-

PUMA 4 13 5 1 08 78 45

SCARA 4 AC 33 14 14

4 STANFORD 1 ul 80 61 44

Fig. 3. Number of multiplications (*) and additions/subtractions

(+/-) after the first and the second step of the propo

sed method 23 Opening Lecture

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U2n--uu222-uu233 UMS22- U221 RF33 - A3IBFF332

U233--UJ211 -LU222 UMS23 - U23HBMS2I BrH331 - SPHt >H<31 »RF31

UU3I1- 0M33H0M331 &FB21 - ftri2iUu221 AW032 - SPfC-OIC32'RF32»BMSW32

UU312 - 0tt331 I0K332 BFC21 - MFB21 *UMS21 Bt«533- SPfG.BHC33-RF33-flM5V33 UU313- 0M33U0M323 BFG22 - BM2IU0222 BMG21 - C3iBff<331-S3IBM032

UU322 - 0M33210M332 AFC22- BFB22-UF1S22 AHC22 - S3iflt«331-C3iMf«332

UU3S3 - 0K332i0rO23 frfn23- «M2IUP23 ««C2I - (U-T02I-BK21 .RF21

UU333 - OM323IOK323 WC23- BFB23-UMS23

«««22- BM022-BfC22.RF22.fcMSUe2

U31 1--UJ322-UL033

,UMS31 - U31 1 IBMS31 i«e23- fH«333-HHC23-RF23-MftS<.23

U333--UJ31 1 -UU322 UMS32- U321IUMS3I

Fdiei2- S2iBt«C21-C2i«#«22

Er C I F UMS33- U33IIOMS3I ftfffll2- Bf-Tf212*Bf<12

FlfH3f- MM3nX>33l P1--AW112

••a MOM Frequency t*FC31 - BFB31 «UMS3I P2- OMC23

0M3211--002IQD 1 i.FF.32- BM3H0332

P3- FMG33

orosii- d'omw -saioooi HFC32- f(FB32»urTS32 Et*

0fC222--S2i0rO2 l 1 <2i000 t MFB33 - BM3iUC'333

Fig. 4. Computer-generate d program of Puma robot with 3

joints and 3 links

5. SOFTWARE FOR CONTROL SYNTHESIS OF ROBOTIC MANIPULATORS

The major application of roboti c system today an in near future

seems to lie in robotic flexible manufacturing system (FMS),

The selection and/or design of roboti c systems for particular

flexible manufacturing cell (FKC) is very complex and tedious

job requiring the designer' s experience and a great deal of

iterative search. In orde r to improve solving of this problem,

both a computer aided design of robotic FMC and a computer-aided

synthesis of the controller for robots have to be applied. 24 RoManSy 7

In Mihajlo Pupin Institute several software packages have been

developed for design, modelling and control synthesis of robotic

systems. Also the program package for simulation of robotic FMC

has been developed [32, 33]. The software packages for robot de

sign include selection of link parameters and selection of the

actuators. The packages for generating of robot kinematic and

dynamic models are also developed. The package for control syn

thesis enables selection of control law and computation of con

trol parameters. Combining all these packages we develop a pro

gram support for selection and design of robots for robotic FMC.

As already stated above, our software support consists of seve

ral packages which are compatible to each other.

The basic software package is for simulation of FMC enables simu

lation of FMC of various structures consisting of up to two ro

bots, several NC machines, conveyors or buffers. This package in

cludes simulation of complete hierarchical control of FMC (if the

user selects this option). The highest control level is simulated.

At the first phase of the package development the highest control

level is implemented by Petri nets [34], but the new language for

programing of FMC is also developed. The package includes a geometri

cal representation of allthe subsystems in FMC, i.e. it includes a geo

metrical modelling of robots and other subsystems in FMC. A 3-D

graphical representation of the FMC scene is provided. Each sub

system might be modelled by regular geometrical bodies [35].

Using this mode the user may : a) select location of robots and

other subsystems in the working space b) test geometrical struc

ture of the robot (whether the robot can reach all the desired points

in the working space within the cell or not), c) test algorithms

for path planning at the strategical control level of the robot

controller (i.e. the user can test whether the selected robot

could move between the obstacles in the particular FMC) Q36]. It

should be noted that this package includes modelling of robot in

such a way that each link of the robot may be represented by a

set of regular geometrical bodies. 25 Opening Lecture

A special attention in developing this package has been paid

to simulation of robots. The package includes various types of

models of robots: complete dynamic models, kinematic models and

models of robots in the form of finite automate[;;7] . By this

package the designer of FMC may test the behaviour of the robot

system within the particular FMC which the user considers.

The user may choose one of the above mentioned models of the

robot, depending on which aspect of the robot performance and

on which level of the robot controller he wants to test. For

example, if he wants just to check the "kinematic capabilities"

of the robot to reach certain desired positions in the particu

lar FMC and to move in a desired way between the obstacles, the

user may choose a kinematic model of the robot. The simulation

of FMC by such a model of robot is much less time consuming than

by a dynamic model. However, if the user wants to get answer how

fast and how precise robot can a perform certain movements (requi

red by the highest control level of FMC), the user must use a

complete dynamic model of the robot for his simulation.

This package obviously requires both kinematic and dynamic models

of these robotic systems. The packages for automatic computer gene

rating of programs which compute kinematic and dynamic models of

the arbitrary type and structure of robots have been described

above.

The package for simulation of robot FMC requires the models of

robot actuators, too. The package for selection of robot actua

tors has been developed [38] enabling the robot designer to

choose either D.D. electro-motors or hydraulic actuators in each

joint of the considered robotic system. This package analyzes the

behaviour of the robot with various types of actuators, helping

the user to select the most appropriate actuators. In selection

of actuators the package uses the symbolic model of the robot

generated as explained above. This package will be extended to in

clude a selection of pneumatic actuators, direct drive actuators,

A.C. electro-motors etc. 26 RoManSy 7

The simulation of robots within the FMC includes simulation of

all control levels of the robotic controller, namely te robot

controllers usually includes three basic control levels £39] :

strategical, tactical and executive control levels. The strate

gical control level consists of various algorithms for path

planning, which might be either general (independent of the par

o rticular robot structure and characteristic) £i-G] » they might

be developed just for a particular robot structure. In the for

mer case, the algorithm for path planning might be selected from

the base of the algorithms which also have been developed £(l] .

In the latter case the particular algorithms for path planning

must be supplied by the user.

The tactical control level represents an inverse kinematic model.

If there exists any analytical solution of the so-called inverse

kinematic problem for a particular robot structure under consider

ation, it may be taken from the base of inverse kinematic models

which have been built, too. Otherwise, a numerical solution of

inverse kinematic problem must be adopted. This numerical solution

of inversec problem is applicable for any possible struc

ture of robot, but it requires significantly more computation time

than analytical solutions of inverse kinematic problem. The nume

rical solution uses the kinematic model of the robot in a symbo

lic form, generated by the computer as described above.

In case the user decides to simulate a complete dynamic model of

the robot in FMC, the simulation of the executive control level

of the robot controller will be included, too. Thel law

at. the executive control level may be synthetized using a soft

ware package for computer-aided control synthesis of manipulation

1robots [^2, *3j. Using this package the user may select one of

the available control laws: non-dynamic control law (i.e. simple

local servos around each joint of the robot), dynamic control

laws, and/or adaptive control laws. Out of dynamic control laws

the user may choose one of these: global control in the form of

force feedback or by on-line computation of (approximative) dyna

mic model of robot £l2, 43J, controller synthesized by so-called

inverse dynamic approach £^1 , robust controller £^'3l , etc. Opening Lecture 27

Various forms of adaptive control laws are also available in the

package £^oJ . The package helps the user select the simplest

control law which could satisfy certain requirements regarding

the robot speed and precision in the particular robotic FMC and

for particular tasks for which the robot is intended. The pack-

cage also computes feedback gains for the selected control law.

In doing this the package analyses the stability of the robotic

system with the selected control law. The flow-chart of thio

package is presented in Fig. 5. The stability analysis is per

formed on the base of a complete dynamic model of the robot in

symbolic form and the models of selected actuators. The package

can generate the program for microprocessor implementation of the

chosen control law which can be used directly in simulation of

robot performance in particular FMC. It should be noted that the

program for implementation of control at the executive control

level may include computation of (approximative) dynamic model

of robot (if dynamic control law has been selected). It that case

the symbolicc model is used, too, due to its high effici

ency regarding the processor time consumption. We are developing

an expert system for selection of control law according to the

user's requirements, which should help user make the final deci

sion about the most appropriate control law [^J •

In this way a complete robot control may be synthesized by the

computer and checked by the simulator of robot FMC. Each of the

above listed packages may be used separately, but due to their

compatibility they can be easily combined in order to select

and/or design robot and its controller for particular robotic

FMC. 28 RoManSy 7

Settin g data on mechanism structure:

- number of d.o.f.

- joint axes

- linkss etc.

Settin g data on nominal trajectory:

- internal coordinates, or

- external (task-oriented) coordinate

- velocity profile

- movement duration, etc.

- constraints and obstacles

Computation of nominal kinematics:

- trajectories of joint coordinates

Settin g data on mechanism dynamic

parameters:

- masses

- moments of inertia

- friction coefficients

Settin g of actuators data:

- type of actuators: d.c . motors, a.c.

motors, d.d. drive, hydraulic or

pneumatic drives

- data from data sheets

Ô Opening Lecture 29

Q

Stage of nominal

Calculation of nominal regimes

driving torques

Option ^

choic e of nominal

^programmed control^

Synthesi s of nominal Synthesi s of nominal

control on centralized control on decoupled

model subsystem models

Settin g of

control task

requi rement

Stab i l ity Tolerances

degree

Computation of

tolerance s in joint

coordinate s

Selectio n of control structure

- local feedback loops

Synthesi s of local control

- optimal regulator

- pole placement

- Bode's method etc.

Analysi s of linearized model of robot

(computation of eigen-values) RoManSy 7 30

0

N O

Stable

?

YES

Analysis of stability of

nonlinear model

sfa oba> NO NO Stabl e

control

YES

Selectio n of global control structure

Selectio n

of global control

^ type

On-line computation ot Force feedback

i coupling

y— Selection^v

of components to

"— compute -

Inerti a Gravi ty

terms term Opening Lecture 31

Selectio n of global gains

Stabilit y analysis of nonlinear model

NO

Stabl e

\YES

Selectio n of sampling period

Synthesi s of control in time

- discrete domain

Stabilit y analysis in time

- discrete domain

NO

^Stabl e

?

YES

Determination of control equipment

- number of uP

Simulation

- analysis of various control law

-s of energy consumption

Fig. 5. Flow-chart of package 32 RoManSy 7

6. CONCLUSION

The final goal of simulation and microprocessor implementation

of robotic mechanisms, machine tools and integrated technologi

cal equipment is the integration of software containing geometri

cal modelling of the industrial scene, the simulation of the kine

matics of motion mechanisms in the scope of the technological

unit, simulation of the dynamics of technical subsystems, sen

sors, of a hierarchical control the highest level of which must

contain a specific control language supported by an expert system

for a broader class of industrial tasks and robotized flexible

modules. According to this goal it is necessary to realize a

distributed control hardware suitable for the software imple

mentation so conccpted.

In scope of the software of flexible technological modules it

is also important to dispose with the simulation models of coope

rative work of robotic systems, whereby the robotic units re

present one to the other moving obstacles, the circling of

which and collision avoidance complicate the software, but at

the same time raise the efficiency of the production system.

Apart from the cooperative work of robots in scope of the fle

xible production system simulation complex, it is necessary to

introduce also mobile robotized units, which significantly con

tribute to the flexibility of such systems.

Hence, the age of conventional robotics is behind us. In the

scope of such robotics there are left behind both the application

and the development of research in the field of composite mater

ials application in the construction of robots, in the applica

tion of new types of actuators, in the development and

tion of new sensors, in the synthesis of new robotic mechanisms,

etc. However, the basic reason for a new growth of robotic tech

nology application lies in its integration into flexible produc

tion systems for which the simulation procedures, the user's

software and its microcomputer implementation must be intensi

vely developed. Opening Lecture 33

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[12] Walker M.W., Orin D.E., Efficient Dynamic Computer Simulation

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[14] Aldon M.J., Liégeois A., "Generation et Programmation Automa

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[16] Vukobratovid M., Kirdanski N., "Computer Assisted Generation

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candea Mathematicae, Int. J. of Applying Mathematics and

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[17] Renaud N., "An Efficient Iterative Analytical Procedure for

Obtaining a Robot Manipulator Dynamic Model", Proc. 1st Int.

Symp. Robotics Research, Bretton Woods, New Hampshire, USA,

1983.

[18] Neuman P. Ch., Muray J.J., "Computational Robot Dynamics:

Foundations and Applications", J. Robotic Systems, Vol. 2,

No. 4, pp. 425-452, 1985.

[19] Cesareo G., Nicolo F., Nicosia S., "DYMIR: A Code for Genera

ting Dynamic Model of Robots", Proc. 1st IEEE Conf. on Ro

botics, Atlanta, GA, pp. 115-120, 1984.

[20] Khosla P.K., Neuman CP. , "Computational Requirements of Cus

tomized Newton-Euler Algorithms", J. Robotic Systems, Vol. 2,

No. 3, pp. 309-327, 1985.

[21] Izaguirre A., Paul R., "Automatic Generation of the Dynamic

Equations of the Robot Manipulators Using a LISP Program:,

Proc. IEEE Int. Conf. on Robotics and Automation, San Fran

cisco, pp. 220-227, 1986.

[22] Burdick J., "An Algorithm for Generation of Efficient Manipu

lator Dynamic Equations", Proc. IEEE Int. Conf. on Robotics

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[23] Khalil W. , Kleinfinger J.F., Gautier M., "Reducing the Compu

tational Burden of the Dynamic Models of Robots", Proc. IEEE

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-532, 1986.

[24] Khalil W., Kleinfinger J.F., "Minimum Operations and Minimum

Parameters of the Dynamic Models of Three Structure Robots",

IEEE J. Robotics and Automation, Vol. RA-3, No. 6, pp. 517-

-526, 1987.

[25] Kirdanski. M. , Vukobratovid M., Kircanski N., Timcenko A.,

"A New Program Package for the Generation of Efficient Mani

pulator Kinematic and Dynamic Equations in Symbolic Form",

Proc. Ill Int. Conf. on Advanced Robotics - "Towards third

generation robotics", 1987, (to appear in Robotica, 1988).

[26] Horak, D.T. "A Fast Computational Scheme for Dynamic Control

of Manipulators", Proc. of 1984. American Control Conference,

San Francisco, CA (1984).

[27] Timcenko A., Kirdanski N., "Implementation Version of Dynamic

Robot Models in Symbolic Form", Proc. IV Yug. - Soviet Symp.

on Appl. Robotics and Flex. Automation, Novi Sad, 1988.

[28] Vukobratovic M., Kircanski N., Real-Time Dynamics of Manipu

lation Robots, Series: Scientific Foundations of Robotics,

Springer-Verlag, 1984. Opening Lecture 35

[29] Kirdanski N., Vukobratovid M., Kircanski M., Timcenko 0.,

"An Approach to Development of Real-Time Robot Models",

Proc. VI IFToMM Symp. on Theory and Practice on Robots and

Manipulators, Crakov, 1986.

[30] Kirdanski N., Vukobratovid M., Kircanski M., "General-Purpose

Software System for Computer-Aided Generation of Real-Time

Robot Dynamic Models", Proc. VII World IFToMM Congress, Se-

villa, 1987.

[31] Kircanski N., Vukobratovid M., "Computer Method for Develo

ping of High-Frequency Dynamic Robot Models on Mono - and

Multiprocessor Controllers", Problems of Mashine Engineering

and Automatization, Vol. 16, Int. Center for Scien. and Tech

nical Informations, Moskva - Budapest, 1987.

[32] Vukobratovic M., Stokid D., "Concept of Program Support for

Design and Control Synthesis of Robotic FM Systems", The

Fourth Yugoslav-Soviet Symp. on Applied Robotics and Flexible

Manufacturing, Novi Sad, Yugoslavia, 1988.

[33] Stokid D., Vukobratovid M., Lekovid Dj. Jockovid M., Hristic

D., Timcenko 0., Djuric N., Rodid A., Ognjanovid Z., "Program

Package for Simulation and Development of FM Cells", The

Fourth Yugoslav-Soviet Symposium on Applied Robotics and

Flexible Manufacturing, Novi Sad, Yugoslavia, 1988.

[3 4] Jockovid M., Ognjanovid Z., "An Example of Application of

Petri Net in Control System of Flexible Manufacturing Cell",

XXXII Yugoslav Conference for ETAN, Sarajevo, 1988.

[35] Timcenko 0., Rodid A., "Importance of Solid Modelling in

Flexible Manufacturing System Simulation", XXXII Yugoslav

Conference for ETAN, Sarajevo, Yugoslavia, 1988.

[36] Timcenko 0., "Contribution to Path Planning of Manipulation

Robots in the Working Space with Obstacles", The First Yugo

slav Symposium on Applied Robotics and Flexible Manufacturing,

Bled, Yugoslavia, 1987.

[37] Lekovid Dj. Stokid D., "Software Package for Simulation and

Design of Flexible Manufacturing Systems", XXXII Yugoslav

Conference for ETAN, Sarajevo, Yugoslavia 1988.

[38] Vukobratovid M., Potkonjak V. , Applied Dynamics and CAD of

Manipulation Robots, Series: Scientific Fundamentals of

Robotics Vol. 6, Monograph, Springer-Veralg, Berlin, 1985.

[39] Popov P.E., Verescagin A.F., Zenkevic L.S., Manipulation

Robots: Dynamics and Algorithms. Series: Scientific Fundamen

tals of Robotics, "Nauka", Moskva, 1987.

[40] Kirdanski M., Timcenko 0., "Manipulator Path Planning in 3D

Space in the Presence of Obstacles", The Fourth Yugoslav-

-Soviet Symposium on Applied Robotics and Flexible Manufac

turing, Novi Sad, Yugoslavia, 1988. 36 RoManSy 7

[4 1] Devedzid V., Stokid D., "Expert System on Strategical Control

Level of Manipulation Robots", The Fourth Yugoslav-Soviet

Symposium on Applied Robotics and Flexible Manufacturing,

Novi Sad,. Yugoslavia, 1 988.

[42] Vukobratovid M. , Stokid D., "A Procedure for Interactive

Dynamic Control Synthesis of Manipulators", Trans, on Systems,

Man, and Cybernetics, Sept./Oct. issue, 1982.

[43] Vukobratovid M. , Stokid D., Control of Manipulation Robots:

Theory and Application, Series: Scientific Fundamentals of

Robotics, Vol. 2, Monograph, Springer-Verlag, Berlin, 1982.

[44] Paul R.C., Modelling, Trajectory Calculation and Servoing of

A Computer Controlled Arm, A.I. Memo 177, Stanford Artificial

Intelligence Laboratory Stanford University, September, 1972.

[45] Asada H., Slotine E.J.J., Robot Analysis and Control, John

Wiley and Sons, New York, 1986.

[46] Vukobratovid M. , Stokid D., Kircanski N., Non-Adaptive and

Adaptive Control of Manipulation Robots, Series: Scientific

Fundamentals of Robotics, Vol. 5., Monograph, Springer-Veralg,

Berlin, 1985.

[47] Stokid D., Vukobratovid M., Devedzid v., "Expert System for

Synthesis of Dynamic Control of Manipulation Robots", CISM-

-IFToMM Conference on Robots and Manipulators, "RO-MAN-SY 88",

Udine, Italy, 1988. Part 1

Mobile Robots and Walking Machines 1 Mobile Robots - Customized Symbolic Models,

Simulation and Control

N. Kircanski, M. Vukobratovic, M. Kircanski, A. Timcenko

Mihajlo Pupin Institute, Beograd, Yugoslavia

ABSTRACT: In this paper the problem of path planning, MODEL 1 I NG AND

control of mobile robots consisting of a mechanical arm mounted on

a computer controlled vehicle with a damping system is concerned.

An off-line, heuristic method FOR A TRAJECTORY GENERATION FOR THE

mobile robot as a redundant system is suggested. It provides FOR A

considerably smoother motion of the platform with respect to the

motion of the arm. The method is based on the filtration of the

user—specifled end-effector trajectory.

A virtual manipulator concept is introduced in order to model the

arm, the platform and Its damping system dynamically. This concept

provides completely the same modelling technique for the mobile

robot as for standard robots. Since the entire system with 12 or

more degrees of freedom is extremely complicated USED THE ALREADY t W E

DEVEL OPED METHOD FOR A computer generation of customized dynamic

models which are numerically much more efficient than other types

of models. Finally, we introduced a control system that compensate

for nonlinear system dynamics. An example with a 12 degree of

freedom mobile robot is presented.

1. INTRODUCTION

Autonomous land vehicles, autonomous vehicles for the disabled,

computer-control 1 ed mobile platforms, industrial mobile robots anci

other types of controllable mobile systems need to be described lr.

a uni fi ed manner from the KI nematic AND DYNAM I C poi nt of vi ew.

Although, the mobile systems may differ in many technological

detai Is it is ver y useful to f I nd out the methods that can be

applied.to anyone IN these systems. The purpose of this paper is tc

propose a new unified methodology applicable to many different

mobile robots.

A variety of design philosophies for mobile robot systems HAS BEEN

discussed in literature [1-123. A VISUAL NAVIGATION SYSTEM FOR ;

autonomous land vehicles and road-following problem is discussed lr.

[1,21. An autonomous vehicle design to transit unknown terrain is

DESCRIBED IN £3^ • AN AUTONOMOUS WHEEL CHAIR EQUIPPED WITH AN ULTRASONIC

rangefinder and a digital camera is described in [4]. The

navi gati on al gor 1 th m concern! ng the use of a contr asti ng 1 i ne or.

the pathway floor in a factory environment is proposed in CS]. The Mobile Robots and Walking Machines 39

navigation system for an intelligent mobile platform equipped with

a rotati ng ultrasonic range sensor 1 s descr1 bed in C 6- .

Path-planning for mobile platforms in 2D environment is discussed ir.

several papers [7-93. Kinematic modelling and control of specially

designed mobile platforms is considered in [10, 11]. Dynamic

equations for a circular platform with 3 wheels are presented ir,

[l2] . We ca n se e that different research topics have been concerned with

mobil e robots: sensory-data integration, navigation, path planning, kinematics,

dynamic s and control . Most of the experimenta l systems represented

mobil e platform designed to transit 20 terrain. But a very serious

practica l problem arises here, namely how the motion of the plat

for m and the manipulato r degrees of freedom (d.o.f) are to be co

ordinated , given the robot, the mobil e platform and the robot en

vironment . This problem is considered in this paper.

After analyzing the variety of particular mobile platforms we have

decided to use a general 6 d.o.f. representation describing a

platform active drive system together with its damping system. An n

1 i nk manipulator wi th both el asti c and stiff Jolnts is attached at

any point on the top of the platform. A virtual manipulator concept

is Introduced to build up a kinematic and dynamic model of the

overall mobile system Csee Sec. 25. Thelr exactly

simulates the motion of an arbitrary mobile robot.

Th e inverse kinematics of a virtua l manipulator represent a specy-

fi c problem due to the redundancy of the system and the constraints

upo n the velocit y and acceleratio n of the platform . While the 'fast'

motio n is associate d with the manipulator , the 'slow' motion recom

mende d for the platform . The acceleration s and jerk s during the mo

tio n of a platform excite resonant flexible modes originating from

dampe r elasticity. For these reasons we shall focus on developing

specia l inverse kinematics for mobile robots under assumption that

th e platform should move as smoothly as possible.

Special attention is paid to the dynamic model of the overall

system. High nonlinearities and coupling between degrees of freedom

as well as vibrations of the platform are described by extremely

complicated dynamic equations. Thus, we employed virtual

manipulator representation and applied computer program for

automatic generation of customized symbolic models. The obtained

models are nearly optimal in numeric sense. Finally, we introduced

position, velocity and force-feedback control loops and simulated

complete dynamic behaviour of a 12 d.o.f. mobile robot during the

execution of a prescribed task.

2. Virtual manipulator concept

A mobile platform consists of a drive unit with several actuators,

driving wheels and a damping system preventing transmission of

Jerks from the wheels to the upper level of the platform. For the

sake of simplicity we shall consider the 20 case when the platform

moves in xy plain of a fixed Oxyz frame. The torques that are to be

produced about wheel shafts can be reduced to Fx, Fy forces along x

and y axis and the torque Mz about the z' axis of the local

coordinate frame of the platform CFig. lb}. Thus , the rotation anc

the translation of them can be modelled by 3 d.o.f. RoManSy 7 40

manipulator with two sliding and a revolute Joint Cfirst 3 Joints —

in Fig. lb}. The damping system is described by 3 additionals

to describe oscillations along z axis Csliding Joint} and about x'

Croll} and y' Cpitch} axes CFig. 13. The maximal number of virtual.

Joints describing drive unit Is 6 for the platform moving along 3D

surfaces. Since the damping system can generally be extended to C

virtual joints, we can see that up to 12 joints of virtual manipulate

are necessary for describing the kinematic and dynamic behaviour of E

arbitrary platform. Notice that the first 3 Joints in Fig. 1 are

powered by a virtual drive mechanism, while the next 3 Jointse

passive, non-controllable Joints consisting of virtual spring -

damper mechanisms. ^

manipulator

vehicle

damping system

vehicle

drive system

Fig. 1. Real system a3 and its virtual manipulator b3

One of the motivations for developing the virtual manipulator-

representation a s the extensibility of the model's complexity. Now it is w

quite simple to add several Joints and links of a desired robot

mounted on the platform to the platforms Cthe last 0 Joints ir.

Fig. 13.

3. Trajectory generation

The trajectory generation problem may be formulated in different

ways. Here we will consider the problem of finding the time history

of Joint positions, platform position and platform orientation,

given the trajectory of manipulator end - effector in the fixec 41 Mobile Robots and Walking Machines

reference frame. We will consider the platforms where the platform

orientation coincides with the direction of the velocity vector

C the most f requent l y used pi atf orm s i n practice}. We wil 1 not

consider the problem of path planning in the presence of obstacles,

but resolving the redundancy of the system. Obviously the passlve

d.o.f. corresponding to the platform damping system are not

included in trajectory generation.

In this paper we have adopted a heuristic, off-line method for the

trajectory generation. It is based on the filtration of the end -

effector trajectory in order to obtain a slow varying platform

path. The algorithm consists of the following steps which are

illustrated in Fig. 2.

Fig. 2. x, y projections of manipulator end - effector tip

Ccurve 13, gripper base C2.3D. manipulator base

C4,5D and center of platform C6D trajectories

At the first step the user - specified trajectory x , y , z . y •

& , p of the end - effector Ccurve no. 1 at Fig. 2D is used tc

evaluate the corresponding fast varyingy of the base of

the end - effector x , y^, z^CtD - curve no. 2. Then, curves x CtD fa fc

and yCtD are filtrated by a low-frequency filter and scaled in fc

time in order to eliminate time delay introduced by the filter.

Thus the trajectories x^CtD and y CtD are obtained Ccurve no. 3D. fa

The position of the manipulator base x CtD, y CtDe no. 4D is

obtal ned heur isticall y in such a way that it is for a f i xec

distance y, far away from x , y and belongs to the line orthogona-

f DC

to the tangent to the curve x , y^. The parameter y is selected tc fe f

be about a half of the maximal reach of the manipulator. In

case the radius of the curve x , y^ is less than y the fe f

obtained trajectory x , y will have loops CFig. 3D. This is not

acceptable for the platform path. So we filtrate the trajectory x^,

y CtD and obtain smooth trajectory of the manipulator base x^,

y CtD - curve no 5. Since the center of the platform 0' and the

posi ti on of the mani pulator base O do not necessari1 y coi nci de

CFig. laD, the position of the platform xCtD, yCtD in the base

frame Ccurve no. 6D does not coincide with the position of the 42 RoManSy 7

manipulator base (curve no. 5) , either. The orientation of the platform

V^CO is evaluated as the angle between x axis and the tangent to

the curve x^. v^CO. Once the position and orientation of the

platform are determined, the manipulator Joint angles are uniquely

defined.

4. Dynamic model, simulation and control

The virtual manipulator representation is very useful to avoid the

derivation of complicated dynamic equations for each particular

mobile robot. We have to find the transformations between the

real drive system of the platform ande virtual one. After that

it is easy to form the model because we can apply any modelling

method developed for open - chain linkages [13]. The dynamic model

of the virtual manipulator is of the forrr.

P - H q • hCq, q 3 C13

Twhere q = C q^ ... q ] is the vector of Joint coordinates, H is nxr. r

positive-definite inertia matrix, hCq, q) Is n - vector due tc

Coriolis, centrifugal and gravitational forces, and P is n - vector

of driving torques/forces in Joints.

The virtual Joints corresponding to the actuators driving the

platform as well as robot Joints are powered. For the sake of

simplicity we shall accept that the actuators are DC motors modelée

as

u=aP+bq . C23

where u. is the input voltage, P is the produced torque in Joint

i, q is Joint coordinate, and a. and b are parameters. The

inductive effects are neglected. The virtual Joints representing

dampings are modeled by

c Cq - q.3 - v. q. = P. C3D

v. m i v ». I

where qs a constant equilibrium position of the spring in

Joint i, c^ is the spring constant and v. is the damping friction

coefficient. From C23 and3 it follows that the vector P =

[P . . . P ] can be expressed as

P = Au - Bq + C<j + Cq C 43

where A, B and are diagonal constant matrices. Here u = lu ...

u 0...0 u . . . u 1 Is the input vector. The number of

r n n • n. i n

p p d

powered virtual Joints of the platform is denoted by n^. and the

number of virtual Joints corresponding to damping effects by r.^.

T The q vector has the form q= [ 0... O q ... q ... O...0]

m

mm j j

wher e I - n + 1, and J = n + n . 1

Combining C13 and C43 we obtain the model of the entire system Mobile Robots and Walking Machines 43

Hq + hCq, q> + Bq + Cq = Au + Cq^ C53

1

Since H is a positive - definite n x n matrix and H always

exists, we can express q as a function of q and q. Thus, both

direct and inverse dynamics is provided. It is interesting to note

that the system C53 can be unstable even when u - 0 if the elastic

modes are not damped. In reality viscous frictions are always

combined with elastic effects and the oscillations are damped.

The control system for both the platform and the manipulator

consists of feed-forward term u and feedback term u

FF Fl

au u + u C CVJ

FF Fl

Various types of control laws are applicable. We shall adopt the

following algorithm

u = k . q

FFv Vi \ CTJ

u = kq - q ) + k Cq - q.D + k P

FBI Pi v. v. W i i Fl .

for controllable Joints 1 = 1 n , n+n. + l n. Thus

p p c

we 1ntroduced posi ti on and velocity feedback. as wel 1 as force -

feedback. This control 1 aw would make the overal1 system decoupiec

and 11 near if no uncontrol1able Joi nts exi st. Unfortunately, the

mobile r obot 1n our case sti11 exhlbi ts nonl1 near osci11ator y

effects due to the el asti c effects i n the damp! ng system of the

piatfor m. Fur ther i mpr ovement s of the control1er C 65 , C 75 can be

achieved by introducing new sensors measuring the platform

osci 11 atl ons. The robot mounted on the pi atf or m can, to some

extent, compensate for the oscillations of its bottom plane. The

proposed control scheme and the simulation module is shown in Fig.

3. The real drive system and the real sensory equipment Cshaft

encoder s and tacho - gêner ator s at the wheel s etc.5 as wel1 as the

algebraic transformations between the real and the virtual drive

system are not introduced in Fig. 3 for the sake of simplicity.

task path i nverse

pianning description kinematics

mobi le

control

robot

1 aw

dynamics

measuring

system

Fig. 3. Control system for a mobile robot 44 RoManSy 7

5. EXAMPLE

The proposed algorithm for a trajectory generation, modelling and

contro l synthesis was examined on the 12 d.o. f. robot shown in Fig.

1. In order to test the method of the trajectory generation the

end-effecto r trajectory was chosen to have abrupt changes in the

directio n of the motion Ccurve no. 1. in Fig. 2. S . As we can see

from this figure the obtained manipulator base path is very smooth.

The corresponding time histories of the Joint coordinates of the

virtua l robot are shown in Fig. 4.

Fig. 4. Time histories of the Joint coordinates

o f the virtual robot

The control law shown in Fig. 3. results in the tracking errors of

th e first 3 Joints of the robot CJoints 7. 8, and S of the virtual

manipulator s shown in Fig. 5. Also the behaviour of the first

non-controllabl e Joint of the damping system Cfourth Joint of the

virtua l manipulators is shown in Fig. 6. It can be easily seen that

tha t Joint induces the damped oscillation s into the system.

time [MS]

Fig. 5. Tracking errors of the first 3 robot Joints 45 Mobile Robots and Walking Machines

tine [MSI

Fig. 6. Trajectory of the 4. Joint of the

virtua l manipulator

REFERENCES ;

[ 1 ] Waxman A. et al. , "A Visual Navigation System for Autonomous

Land Vehicles". IEEE J. Rob. & Automat., Vol. RA-3, No. £,

1987 , pp. 124 - 141.

[2 ] Chavez R. . Meystel A. , "Structur e of Intelligence for ar.

Autonomous Vehicle p . IEEE Int. Conf. Rob. , Atlanta. GA.

> roc

1984 , pp. 584 - 591 .

[3 ] Harmon S. , "The Ground Surveillance Robot CGSRD: An Autonomous

Vehicle s Designed to Transit Unknown Terrain",IEEE J. Rob. 8

Autom. Vol. RA-3, No. 3, 1987, pp. 266 - 27S.

[4 ] Madarasz R. , Heiny L. . Cromp R. . Mazur N. , "The Design of ar.

Autonomous Vehicle for the Disabled", No. 3, 1986, pp. 117 -125.

C5] Drake Ki , Mc Vey E. , Inigo R. , "Experimental Position anc

Rangin g Results for a Mobile Robot". IEEE J. Rob. & Autom. ,

Vol . RA - 3. No. 1. 1987, pp. 31 -41.

[61 Crowley J. . "Navigation for an Intelligent Mobile Robot". IEEE

J. Rob. & Autom. . Vol. RA - 1 , No. 1, 1985, pp. 31 -41.

[7 ] Kambhampati S. , Davis L. , "Multiresolution Path Planning for

Mobil e Robots". IEEE J. Rob. & Autom., Vol. RA-2. No. 3. 198e,

pp. 135 - 145.

[8 ] Ichikawa Y. , Ozaki N. , "Autonomous Mobile Robot", J. Rob. Syst. ,

2C1D, 1985, pp. 135 - 144.

[9 ] Palma - Villalon E. , Dauchez P. , "World Representation and Path

Plannin g for a Mobile Robot". Robotica, Vol.6, 1988, pp.35 - 4C.

£10 ) Daniel D. , Krogh H. , Friedman M. , "Kinematics and Open-Loop

Contro l of an Ilonator-Based Mobile Platform", Proc. 1985 IEEE

Conf. Rob. & Autom. , St. Luis, 198S, pp. 346 - 35i.

[11 ] Muir P. , Neuman Ch. , " Kinematic Modelling for Feedback Control

o f an Omnidirectional Wheeled Mobile Robot", Proc. 1987 IEEE

Conf. Rob. & Autom. , Raileigh. 1987, pp. 1772 - 1776.

[12 ] Goldenberg A. , Jaluba C. , Wlercienskl J. , Fenton R. , Benhabit

B. , Kinematics and Dynamics Modelling of a Class of Mobile

Robots , Techn. Report, Rob. & Autom. Lab. , Univ. Toronto, 1986.

[13 ] Kirdanski M. . Vukobratovic M. , Kirdanski N. . Timcenko A.. "A

New Program Package for the Generation of Efficient Manipulator

Kinemati c and Dynamic Equations in Symbolic Form", to appear

i n Robotica in 1988. Gait Analysis for Walking Machines for Omnidirectional

Locomotion on Uneven Terrain

Vijay Kumar*, Kenneth J. Waldron**

* Department of Mechanical Engineering and Applied Mechanics,

University of Pennsylvania, 220 S. 33 rd Street, Philadelphia, PA 19104

**t ofl Engineering, The Ohio State University,

206 W. 18th Avenue Columbus, OH 43210

0. ABSTRACT

A statically stable walking machine with a bilaterally symmetric plan is considered here. Gaits for

locomotion on level terrain in a preferred direction have been studied extensively. However, much less is

known for walking on uneven terrain with a continuously changing heading and direction of motion. Two

salient contributions to the existing work on walking gaits arc described in this paper. A crab angle, which

is defined as the angle between the direction of motion and the longitudinal axis of the vehicle, is introduced

to model the omnidirectional abilities of a legged locomotion system. In addition, the effects of rough

terrain and incrtial loading have been taken into account. The definitions prevalent in the literature on gaits

have been appropriately adapted to account for a nonzero crab angle, and the non-orthogonality of the net

external force (including incrtial forces) on the vehicle and the plane of support In particular, the wave gait

which has been heralded as optimal in terms of stability, is modified to improve the stability.

Two measures of static stability, the sialic stability margin and the longitudinal stability margin, have

been used as performance indices in the past These two measures of stability have been used to show that

a nonzero crab angle is not very detrimental to static stability, and in fact, a small crab angle may actually

improve the stability of the vehicle. The modified wave gait is optimized for stability according to the

direction of motion, incrtial loading, and the gradient of the terrain. The optimal parameters may be derived

analytically, if the longitudinal stability margin is used as a criterion for optimization. Mobile Robots and Walking Machines 47

1. INTRODUCTION

1.1 General

A mathematical model for walking gaits for statically balanced robots is presented here. The ability of a

walking machine to walk on rough terrain and the omnidirectionality of legged locomotion are two of the

greatest advantages of such systems. However, most of the existing literature on gaits considers gaits on a

smooth surface with the direction of motion along the longitudinal axis of symmetry for the vehicle. This

paper allows for omnidirectionality and terrain roughness, and in addition, analyzes the effects of incrtial

loading on walking. Two measures of static stability described in the literature have been suitably adapted

for this analysis. Optimization of walking gaits based on these measures is discussed.

\2. A Critical Review of Literature

A vast variety of examples of legged locomotion can be found in nature and many quantitative studies

of animal gaits have been undertaken with a view to discovering strategies for control and coordination in

locomotion [1, 3-9. 18-19. 22-23, 32-33]. The earliest studies of gaits date back to 1872 when Muybridge

used successive photographs to study the locomotion of animals [19] and later human locomotion [18].

However, Hildcbrand [8] was probably the first researcher to analyze gaits quantitatively. Since then, many

reports have been published on this subject [1, 3-7, 9, 32-33], motivated in some cases by an interest in

legged robots [22-23].

Wilson's report [32] on insect gaits is particularly informative. He developed a model for insect gaits

by proposing a few simple rules. A wave of protraction runs from posterior to anterior, and contralateral (on

opposite sides of the body) legs of the same segment alternate in phase. The protraction time is constant

and the retraction time is varied to control the frequency. Further, the intervals between the steps of the hind

and middle ipsilateral (on the same side of the body) legs, and those between the middle and the fore

ipsilateral legs arc constant Though these rules are not observed in all species [32-33], most of these rules

have been validated qualitatively with a variety of insects when walking on smooth horizontal surfaces. The

classical alternating tripod gait exhibited by fast moving locusts and cockroaches is a good example [5].

More recently, a cinematic analysis of locusts undertaken to study locomotion characteristics on uneven

terrain has been described [23]. Studies by Cruse [3-4] on stick insects have shown that the posterior legs

are placed close to the support sites of the immediately anterior ipsilateral legs. This follow-the-leader

(FTL) behavior may be a general strategy for walking on uneven terrain, and has also been observed in

domestic (Nubian) goals [22].

Tomovic [29] was probably the first to study gaits for walking machines. Pioneering work was done

by McGhee and his co-workers [13-13]. The italic stability margin was identified as a suitable measure of 48 RoManSy 7

static stability and a crawl gait was shown to be optimal for quadrupeds. Bessonov and Umnov [2] used

numerical expérimentation for hcxapods 10 demonstrate that the optimal gaits are regular and symmetric and

these gaits were described as "wavy" gaits. This agreed with Wilson's observation of a mctachronal pattern

of steps in insect walking.

More recently. Song [26] has carried out a detailed analysis and classification of gaits, where the wave

gait was identified as the optimal gait for hcxapods. Other gaits were found suitable for unstructured

environments, such as the F-TL gait [27] and the free gait [12]. A more extensive treatment of statically

stable gaits can be found in References [26-28].

2. THEORETICAL BACKGROUND, DEFINITIONS

The legs on the left side of a six-legged machine are numbered 1, 3, and 5, and those on the right side

are designated by even numbers, 2, 4, and 6, in accordance with previous literature [13]. It is natural to

assume a bilateral symmetry, and the pitch between the legs, P (as shown in Figure 1), is a constant. In

addition to the pitch, the width, W, or the width to pitch ratio, y, arc used to describe the geometry.

Definition 1 [29]: A leg is a sequential finite-state machine with two output states, 1 and 0. The state 0

represents the support phase of the leg or the state of being in contact with the ground (retraction). The state

1 represents the transfer phase of the leg, in which the leg is above the contacting surface (protraction).

Definition 2 [26]: A gait is symmetric if the motion of contralateral legs on the same segment is exactly

half a cycle out of phase.

Definition 3 : A gait is periodic if the motions of each of the legs are periodic. In other words, the time

fractions of the support and transfer phases during successive cycles are constant for all legs.

Dcfimion 4 [13]: The cycle time, T, defined for periodic gaits, is the time duration of a complete cycle of

motion of a leg, and hence, the time duration of a locomotion cycle.

Definition 5 [13]: The duty factor, [5, is the fraction of time (in one cycle) that a leg spends in the support

phase. It is assumed in this paper, that all legs have die same duty factor, since there is no reason to expect

one leg to be favored over another. Such a gait is said to be regular.

Definition 6 [8]: The stride length of a gait, X, is the distance through which the center of gravity of the

system translates through one locomotion cycle.

Definition 7 [27]: The stroke, R, is the distance through which the foot translates during a locomotion

cycle. The stroke to pitch ratio, tj, is an important dimensionless quantity.

Definition 8 [27]: The leg phase, is the time fraction of a cycle time period by which the instant of

contact of the leg lags behind a reference time instant. This reference time is selected to be the instant when

leg 1 contacts the ground. Mobile Robots and Walking Machines 4 9

Definition 9 [13]: An event of a gait is the placing or lifting of any of the feet during locomotion. For a ri

le ggcd animal or machine, there are 2n events in an event sequence or locomotion cycle.

Definition 10 [ 14]: A support pattern is the minimum area convex point set in the support plane obtained

from the contact points of the feet on the ground at a given time.

Definition 11 [14]: The stability margin (SM) for a support pattern is equal to the shortest distance from

the vertical projection of the center of gravity of the machine to any point on the boundary of the support

pattern.

Definition 12 [14]: The longitudinal stability margine (LSM) for the support pattern is the shortest distance

along the direction of projection from the center of gravity of the machine to an exgc of its support pattern.

Definition 13 [26]: The local phase, is the fraction of the locomotion cycle elapsed since the placing

event of Jhe leg. In other words, $ l » 0 when the leg is placed, and $ l = P when the leg is lifted.

Definition 14 [2]: A wave gait is a regular, symmetric gait with the difference in leg phases between any

pair of adjacent ipsilateral legs being equal In other words, if mcd, represents the modula operator,

Pl = P2 = P3 = ...»p6:*2 = 1/2. *i - (4>i-l + 1/2) mod. l,i = 2,4,6.

and

(•i+2 - 4>i)mai l = - mod. 1.1 £i.js4 .

Definition IS [8]: A gait diagram is a graphical tool used to describe gaits. The duration of the support

phase or retraction is recorded by a line on a time scale, and the transfer phase or protraction is represented

by the gap between the lines (An example may be seen in Figure 3).

3. METHODOLOGY OF GAIT ANALYSIS

3. 1 New Definitions and Ideas

3.1.1 General

Some of the definitions have to be modified to allow for an uneven supporting surface and motion in

directions other than the longitudinal direction. Several new definitions and concepts arc also described here.

It should be noted that the wave gait (Definition 14) has been defined differently in the past [26-28].

The definition presented here is based on the definition proposed by Bcssonov and Umnov [2]. A perfect

wave gait is a wave gait in which 4n = f5, and consequently,

•5 = 2Pmodl.4>4 = (p+ 1/2).

and so on.

3.1.2 Omnidirectionality 50 RoManSy 7

A crab gait is defined as a gait in which the direction of progression of the center of gravity of the

machine need not coincide with the longitudinal axis of the body. The angle that the direction of

progression forms with the longitudinal axis of the body is called the crab angle, a.

A crab wave gait is a crab gait which is regular and periodic and which has the same phase difference

between any pair of ipsilateral, adjacent legs. Notice that this model is a modified wave gait which agrees

with Wilson's observations on insect walking [32]. The sequence of events in a crab wave gait is

characterized by three parameters:

(1) The duty factor, fJ

(2) The longitudinal phase difference, y, between any pair of ipsilateral legs

(3) The lateral phase difference, 0, between any pair of contralateral legs

In the special case in which the crab angle is equal to zero, 8 is equal to 1/2 in accordance with Definition

14, as shown later.

3.1.2 Rough Terrain and Inertia! Loading

The definition of a support pattern and support plane in Definition 10 is modified. The support plane,

denoted by IT, is defined as a best fit plane obtained by doing a linear regression on the points of support at

any given stale. The load wrench , a , is the wrench which is the resultant of die weight of the vehicle

body, and the incrtial forces and moments. The legs are assumed to be massless in comparison to the

weight of the vehicle body. The desired support plane, <J, is a plane passing through the ccntroid of the

support points, perpendicular to the axis of the load wrench, denoted by $, in Figure 2. (It is desirable that

fl be coincident with O, for reasons which arc described in [11] but are outside the scope of the present

discussion.) The support pattern is described by Definition 10, but now, the projections of the support

points along S on O, must be used. Further, the projection of the velocity of the point of intersection C of

$ with O" on a, Y_£, is used as an effective vehicle velocity. The crab angle, ct, in Figure 2, is the angle

between the projection of the longitudinal axis of the vehicle on O", and die effective vehicle velocity. The

definitions for stability margins (11-12) are modified to describe quasi-static stability. The new definitions

of SM and LSM arc based on the point C and the support pattern obtained by projection in Figure 2. and

they reduce to die existing definuons, if incrtial loading were absent, and the terrain were smooth. The

longitudinal stability margin for a periodic gait is the minimum LSM over an entire gait cycle. The LSM

is positive if and only if the SM is positive. The SM is a more conservative estimate of the quasi-static

stability than the LSM (see Figure 3).

3.2 Modeling the Finite Stale Logic

Gaits for hcxapods can be described as sequence of 12 events, namely, the lifting and placing events for

each of the six legs. For a crab wave gait, these events occur at cycle times, and the critical time instants Mobile Robots and Walking Machines S I

corresponding to instants immediately after a lifting event or just before a placing event are given by Table

1. The corresponding events for the wave gait would require 6 to be equal to 1/2, and for a perfect wave gait,

in addition, y must equal p. It is assumed here, that the angular velocity of the body is zero and that the

linear velocity of the body, Y, and also the velocity Xc. arc constant over a cycle. Equivalcntly, the point C

can be assumed to be stationary with the support plane moving at a velocity equal to -Yx.

In Figure 4, the distance dj decreases and the distance d2 increases as the support pattern translates. The

event of placing a leg has the effect of either increasing dj and/or dj or of leaving them unchanged.

Similarly, the event of lifting a leg has the effect of decreasing di and/or or of leaving them.

The LSM of a support pattern, by definition, is the minimum of di and dj. The quantity di has to be the

smallest just before one of the six events of placing a leg and di, the smallest, just after one of the six

lifting events. As the LSM has to be equal to either di or di, it is at a minimum at one of the twelve

instants described by Table 1.

Therefore, the following theorem may be stated:

Theorem 1

The longitudinal stability margin of the support pattern in a regular, periodic gait is at a minimum

just before any one of the six placing events, or just after any one of the six lifting events.

Although the result of this theorem may be intuitively obvious to some readers, the authors arc not

aware of any formal recognition of this simple conclusion. The computation of the LSM can be based on

the calculation of di over six instants and dj. over another six instants (enumerated in Table 1). Further,

analytical expressions may be derived for di and d2, which simplifies calculations considerably.

The analysis of support patterns is further simplified if certain symmetries can be detected. The

following two theorems arc stated here without proof.

Theorem 2 1261

For a regular gait with the same phase difference between any pair of ipsilateral adjacent legs, any

support pattern has a mirror symmetric pattern reflected about the frontal plane through the point C.

If the two time instants corresponding to the support patterns arc tj and t2, then

(t; + 12 - 2y) mod. 1 = P

Both the theorems (1 and 2) also apply to wave gaits, since wave gaits only have the added restriction

that 6 must be 1/2. In addition, the support patterns with 8 equal to 1/2 are symmetric about the sagittal

plane [27]. This leads to the following theorem [27]:

Theorem 3:

The LSM of a wave gait is at a minimum just before any one of the placing events in a half

locomotion cycle. 52 RoManSy 7

4. CRAB GAIT FOR OMNIDIRECTIONAL LOCOMOTION

4.1 Symmetry of the Crab Gait and Stability

In omnidirectional motion, the crab angle, a, must be taken into account. To facilitate analytical

treatment, it is assumed here, that the support plane is perpendicular to the weight, and that there arc no

incrtial forces. In other words, the effects of a nonzero crab angle, and that of a rough terrain with incrtial

loading, on the gait arc treated independently. In the X'-Y' system in Figure 4, with the assumption that the

incrtial forces are absent and that the load wrench is perpendicular to Ihe support plane, the projections of

the feet on a at a time, t, arc given by (F is the mod 1 function):

xy » P Cos a + W/2 Sin a + R/2 - (R/P) t; yi' « W/2 Cos a - P Sin a

xi = P Cos a - W/2 Sin a + R/2 - (R/P) F(t - 9); y£ = -W/2 Cos a - P Sin a

x ' = W/2 Sin a + R/2 - (R/P)F(t-y); yi « W/2Cosa 3

X4 « -W/2 Sin a + R/2 - (R/p) F(t - 9 - y); « -W/2 Cos a

X5- , -P Cos a + W/2 Sin a + R/2 - (R/p) F(t - 2y); ys' = W/2 Cos a + P Sin a

x ' = -PCosa - W/2 Sin a + R/2 - (R/p) F(t - 2 y - 9); y' = -W/2 Cos a + P Sin a (1) 6 6

The temporal relationship between the 12 events is shown in the gait diagram in Figure 5. If two

instants ti and 12 arc considered so that

(ti +12) mod. i = (e + 2<y + p) mod i

then

xi(ti) = -xg (tz) ; yi(ti) _ -yi (12)

xidi) = -xi (12) ; yiOi) - -yi Ci)

xi(n) = -xi (t2) ; yidi) = -yi (12) This also applies to die special case in which a equals zero.

Theorem 4 :

In a regular periodic crab gait, in which the phase difference between any pair of ipsilateral adjacent legs

is equal to a constant, y. any support pattern has another symmetric pattern reflected about the origin

and the lime instants, ti and t2, arc given by:

(ti +12) mod. 1 = (9 + 2 y + P) mod 1

where 8 is the phase difference between any pair of contralateral legs.

Corollary:

The LSM of a regular crab gait in which the phase difference between any pair of ipsilateral adjacent

legs is equal to a constant, may be computed as the minimum distance, dj. over the time instants just

before each of the six placing events. These time instants are 0", 9', >f, [(9 + w)" mod. 1 ], [(2w)- mod

l],and[(9 + 2v)-mfld 1).