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RoManSy 7
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
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
Prof. G. Bianchi
CISM, Piazza Garibaldi 18, 33100 Udine, Italy
Prof. A. Morecki
Warsaw University of Technology
Al. Niepodleglosci 222, 00-663 Warsaw, Poland
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
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
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.
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
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.
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
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
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
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'
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
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
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.
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
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
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
- 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
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
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
Analysis of stability of
nonlinear model
sfa oba> NO NO Stabl e
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
Stabl e
Selectio n of sampling period
Synthesi s of control in time
- discrete domain
Stabilit y analysis in time
- discrete domain
^Stabl e
Determination of control equipment
- number of uP
- analysis of various control law
-s of energy consumption
Fig. 5. Flow-chart of package 32 RoManSy 7
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
[1] Uicker J.J., On Dynamic Analysis of Spatial Linkages Using 4«4
Matrices, Ph.D. dissertation. Northwestern University, Evan-
stone, 1965.
[2] Kahn M.E., The Near Minimum Time Control of Open Loop Articu­
lated Kinematic Chains, Ph.D. Thesis, Stanford University,
MEMO AIM. 106, 1969.
[3] Woo L.S., Freudenstein F., Dynamic Analysis of Mechanisms
Using Screw Coordinates , ASME J. of Engineering for Industry,
1971 .
[4] Mahil S.S., "On the Application of Lagrange's Method to the
Description of Dynamic Systems", IEEE Trans. SMC, Vol. SMC-12,
No. 6, pp. 877-890, 1982.
[5] Renaud M., "Contribution a 1-Etude de la Modélisation et de la
Commande des Systèmes Mécaniques Articules", These de Doc. -
Ing., Toulouse, 1975.
[6] Vukobratovié M., Potkonjak V., "Contribution to Automatic For­
ming of Active Chain Models via Lagrangian Form", ASME J. Appl.
Mech., No. 1, 1979.
[7] Hollerbach M., "A Recursive Lagrangian Formulation of Manipu­
lator Dynamics and a Comparative Study of Dynamics Formulation
Complexity", IEEE Trans, on SMC, Vol. 10, No 11, pp. 730-736,
{?>] Kane T.R., Wang CF., "On the Derivation of Equations of
Motion", J. Soc. for Ind. and Appl. Math., Vol. 13, pp. 487-
-492, 1965.
[9] Huston R.L., Passerello C.E., Harlow M.W., "Dynamics of Multi
- Rigid - Body Systems", ASME J. Appl. Mechanics, Vol. 45, pp.
889-894, 1978.
[10] Stepanenko Y., Dynamics of Spatial Mechanisms, (in Russian),
Mathematical Institute, Beograd, 1974.
[11] Vukobratovic M., Stepanenko Y., "Mathematical Models of Gene­
ral Anthropomorphic Systems", Math. Biosc, Vol. 17, pp. 191-
-242, 1973.
[12] Walker M.W., Orin D.E., Efficient Dynamic Computer Simulation
of Robotic Mechanisms, JACC, Charlotteville, 1981.
[13] Luh J.Y.S., Walker M.W., Paul R.P.C., "On-Line Computational
Scheme for Mechanical Manipulators", ASME J. Dynamic Systems,
Meas. and Control, Vol. 102, No. 2, pp. 69-76, 1980.
[14] Aldon M.J., Liégeois A., "Generation et Programmation Automa­
tiques des Equations de Lagrange des Robots et Manipulators",
Rapport de Recherche, INRIA, 1984. 34 RoManSy 7
t 15] Kirdanski N. , Vukobratovic" M. , "Computer-Aided Procedure of
Forming of Dynamic Motion Equations of Arbitrary Manipulation
Robots", Proc. VI IFToMM Congress, New Delhi, 1983.
[16] Vukobratovid M., Kirdanski N., "Computer Assisted Generation
of Robot Dynamic Models in an Analytical Form", Acta Appli-
candea Mathematicae, Int. J. of Applying Mathematics and
Mathematical Applications, Vol. 3, pp. 49-70, 1985.
[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,
[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
and Automation, San Francisco, pp. 212-218, 1986.
[23] Khalil W. , Kleinfinger J.F., Gautier M., "Reducing the Compu­
tational Burden of the Dynamic Models of Robots", Proc. IEEE
Int. Conf. on Robotics and Automation, San Francisco, pp. 525-
-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.
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
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. ^
damping system
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
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
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]
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
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
au u + u C CVJ
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
1 aw
Fig. 3. Control system for a mobile robot 44 RoManSy 7
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
[ 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
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
1.1 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].
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
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.
(•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. 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
4.1 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
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.
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).

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