JPRS ID: 10172 USSR REPORT CYBERNETICS, COMPUTERS AND AUTOMATION TECHNOLOGY

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• Document Creation Date: November 1, 2016
• Sequence Number: 26
• Content Type: REPORTS

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JPRS L/ 10172
: g December 1981
~ USSR Re ort
p
- CYBERt~ETICS, ~OMPUTERS AND
AUTOMATION TECHNOLOGY ~
CFOUO 25/81 ~
FB~S FOREIGN ~ROADC/~S~ INFORMATION SERVlCE
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JPRS L/10172
9 December 1981
USSR REPORT ~
CYBERNETICS, COMPUTERS AND AUTOMATION TECHNOLOGY
(FOUO 25/81)
CONTENTS
HARDWARE .
Ministry of the Electronics Industry Produces Electronic
Games 1
Processing of Information Regresented in Matrix Form on
Computer With Rearrangeable Structure..........~ 'L
Some Evaluations of Efficiency of Parallel Crnuputations.......... 7
Functional Data Processors ...................................o... 19
Cylindrical Magnetic Domains in Computer Hardware Elements....... 24
Microprocessor, Microcomputer Applications in Instrument
Making~ Research 28 .
Bit-Slice Microprocessors 31 �
Sixteen-Bit Microprocessor Parameters 33
SOFTWARE
Handbook on Standard Simulation Programa 34
Microprocessor Process Control Computer Complex for Overhead
Thrust Conveyera 38
Extension of Algorithmic Language FORTRAN IV 40
Abstracts From the Journal 'ALGORITHMS A.~1D PROGRAMS', May 1981... 47
More Abstracts From the Journal 'ALGORITI~SS AND PROGRAMS',
- May 1981......e 49
~ - a- [III - USSR - 21C S&T FOUO]
~f?D /lCL'i!'T A i i TCL' AATT
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Additional Abstracts From the Journal 'ALGORITf~iS AND PROGRAMS',
May 1981 54
A~sembler Language Programming on Unif ied System of Computers.... 57
WALKING ROBOTS
Determining Energy Expenditures for Six-Legged Moving
Apparatus When It Is in Motion 68
Mechanice and Motion Control of Robots With Artificial
Intelligence Components 74
PUBLICATIONS ~
Methods for Automating Geophysical Research 111 ,
Abstracts From the Journal 'TECHNICAL CYBERNETI:CS',
,~uiy-August i9si iis
Table of Contents From the Journal 'ENGINEERING CYBERNETICS'..... 121
Systems and Methods for Automating Scientific Research........... 123
Principles of Constructing Common Memory Module in Multimachine
Systems Based on Microcomputers With Cammon Line 126
- b
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MIrTISTRY OF THE ELECTRONICS INDUSTRY PRODUCES ELEGTROr1IC GAMES
Moscow SOVETSKAYA ROSSIYA in Russian 18 Sep 81 p 4 ~
~Article by T. Nazaxenko~
[Text~ "Any electronic game~ including those which we are developing~ is a dis-
tinctive sma.ll computer in the home~" the conversa.tion was begun by G. S. Klisskiy,
chief of the laboratory of the US5R Ministry of theElectronics.Industry. "Here axe
our latest two novelties--the "Eksi-video" and the "Videosport." They both work as
' attachments to a television set.
"Very likely many have seen such games on store counters, or pla.yed them. Remember
how a sports field burns on the screen, how the electranic "blip" darts acro~s the
glowing screen. Each such game, strictly speaking~ represents a set of games:
football, hockey, tennis, ba,sketball..."
"I have heard more than once the opinion," continues Genadiy Semenovich, "tha,t
electronic games give nothing to the player. In pla.ying with an unreal puck you
do not develop your muscles. You develop your reactions instea,d. And not w~orse
than in other sportsmen. ,
"'Eksi-video~'for e~.mple~ was created in such a way that the complexity of the
game can be constantly increased, as if you axe pla.ying wit~ a partner who is a1-
ways growing stronger. The ga.me a?so develops the intellect, for the winning situ-
ation must be rapidly ca.lculated and the position on the field weighed. He who
thinks that the computer plays mur~otonously is m~staken.
"But these games are not the limit of development of electronic.games. We axe now
developing new variations. These games do not need a television set~ they work by
themselves and in a,ddition themselves assign a play program and logic. For exam-
ple~ el~ctronic chess, which is pla.yed with the power of a third-rank chess-player.
And besides~ chess, checkers, sea batile... All these games will be eombined in a
single electronic system... The style, variations and ~echnology of game-type com-
puters are pra.ctica,lly unlimited."
2174
CSO: 1863/13
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rux urr,~~wL u~~, uNLY
. ~
~
PROCESSING OF INFORMATION REPRESENTED IN MATRIX FORM ON COMPUTER WITH
REARRANGEABLE STRUCTURE '
Tbilisi TEKHNICHESKAYA KIBERNETIKA in Russi;in No (225), 1980 (manuscript
- received 17 Dec 79) pp 29-33
[Article by K. N. Kamkamidze and A. S. Papandopulo]
[Text] Automated systems for management and control of various industries, organi~
~ zations and power networks are currently being implemented.
Lar~e arrays af information circulate in a power system which includes a large
number of transducers located at power retransmission stations, substations,
GRES, TES, GES [state regionai, thermal and hydroelectric power stations],~and ~
enterprises-power consumers. Processing of these arrays of information under the
conditions of on-line control of the power.syste~n requires develoument of new high-
speed algorithms and programs that support issuance of control commands in real
time~ Representation of the information in ma.trix form is convenient for a number
of on-line calculations. Let us examine a method for.accelerating calculations
with information in matrix form when a fourth-generation computer ~,ith rearrange-
J able structures (PS) with paralleling of computations is uaed to control the power
s~stem conditions. In solving power~problems, the problem of inversion of real'`and
complex matrices of a large order often arises. As a rule, tterative methods are
used to solve a system of linear equations of a large order in contemporary third-
generation computers. Since it is difficult to 'i~ver~ matrices of a large order
(greater than or.equal to 60) in one pro~edure, a number of artificial methods ~
((diakoptika), various types of equivalence) that allow solving a problem piecemeal
- has been developed in power engineering. It is necessary to ncte that, as a rule,
iterative methods are inadequate for paralleling. But direct (precise) methods to
invert , matri.ces of a large order require much machine time which is not feasi~le
- due to technoiogical reasons. This is why it makes sense to find new algorithmic
~ solutions that would make it easy to pargllel the computation process applicable
~ to the new machines. These algorithmic solutions must preserve or increase the
speed obtainable with iterative methods. To this end, we have developed a series
of algorithms that allow r.epresenting real and complex matrices in the form of
hypercomplex matrices similar to them. To p~rform operations on hypercomplex
matrices, it is necessary to have a program that performa matrix hype~complex oper-
ations. As in conventional matrix algebra, the operation of matr3x multiplication
~ of a hypercomplex matrix is central. The cited algortthm for multiplying
hypercomplex matrices has been compiled in such a way that the operation of compu-
ting each hypercomplex number is divided into four independent operations t:~at
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compute tae real and imaginary parts of the hypercomplex numbers. To realize this
method, the desired hypercomplex matrices are represented as simple matrices with
a fourfold greater number of columns, where 6oth the real and imaginary parts of
the element of the hypercomplex matrix are represented as an independent element of
= a simple matrix. Computation is performed in a loop; computed at once in each loop
are the four elements that are the real and imaginary parts of an element of the
hyFercomplex matrix. Power engineering problems that make use of this algorittm
are solved in real time and require fast methods of solution. In view of the high
orders of the matrices being calculated, the existing program for multiplication of
hypercomplex matrices, written in FORTRAN [2], requires a large amount of machine
time (an order of 1000 requires 30 minutes on a YeS 1033 computer). In this case,
one would expect a s~gnificant effect from using a computer with parallel execut~on
of operations. The technique of the algorithm fully satisf ies paralleling of the
process of execution at the level of branches. Independent execution of the real
and imaginary parts of an element of the hypercomplex matrix can be split into four
branches; the four computational operations of the ioop are perfo~ed in parallel
simult;ineously in the different branches (fig. 1). The capability of paralleling
at the level of branches is supported by modules for processing without calls to
the operating system, consequently, without additional loss of titne spent by the
operating system. Used as the program are compound statements for processing
scalar operands and compound statements for processing vector operands; buffering
of intermediate results is provided for, and system throughput is enhanced by re-
ducing storage requests. The homogeneity of program instructions iznposes no
special requirements on the dynamic allocation of computing resources.
Shown in fig. 2 is the flowchart where the initial matrices are not input, but are
generated in the program itself for any order. The very program for computation of
the multiplication of the hypercomplex matrices begins at once from the paralleling
at the levels of the branches. The process of computation of a matrix occurs fully
independently and in parallel in each branch to the very end. The results of the
computation of each branch are the f inal result of a specif ic part of the matrix.
The main branch waits for the end of the other three branches only before the print-
out itself is made on the terminal so that no time is spent on waiting for the
intermediate results in the execution of the branches. Each branch consists of
nested loops: branches main, II and III of three loops, and branch I of two loops.
Encountered in the main branch is the statement for processing of the vector oper-
ands, being realized sequentially instruction after instruction on the elements of
the vector, multiple addition and multiplication. It performs the calculation .,t
the real parts of the elements of the hypercomplex matrices. The first block per-
forms the basic operation of execution and the following blocks are trr~nsitions
between the loops and perform fetching of the elements of the matrices to be com-
puted precisely in this branch. Branches I, II and III compute the imaginary parts
of the elements of the hypercomplex matrices.
- Since a real matrix is transfoYmed into a hypercomplex and a hypercomplex intosparse
matrices with fixed positioning of zeros without a single division or multipli-
cation, the main time in the method is apent on inversion of the fours.parselq filled
matrices, which is also not great because of their zero structure. The parallel
structure of the algorithm can be tracked by the steps c~f the algoritYmt for inver-
sion of a hypercomplex matrix (the formula was derived especially for this method)
~l~ �
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ruK ~rr~~~aL w~ uNr.,v
S . . .
~i> - ' - - - -
a B c d~
cz> - - -
,q e ~ ~
- (3) - - - - -
F~ Y,~ X Z
~ ~
(4)-----_- -
_ f
j . ?f~ x ~
(5) - - - - - - -
s-' .
Fig. 1. Flowchart of parallel computations of inversion of a symmetrical real
matrix by using the BPU [multiplication program]
Key:
1. Derivation of hypercomplex matrix
2. Derivation of hypercomplex sparsely filled matrix by using the BPU
_ 3. Block inversion of sparsely filled hypercomplex matri~c
4. Reduction of inverted hypercomplex matrix by using the BPU
5. Reduction of inverted real matrix
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Key:
. HOy0A0 Start
1. Branch I
2. Branch II
Pe39pea,oe- gesE~ve
3 . Branch III ~aNUc ~saw-
n memory
- 4. Computation
operation eMepaK~~ ~Generation of �
5. Fetch ~~~~A'~ matrices A, B ~
addresses of No
next element Main branch ~~t ~=6~, Ho++o~ ler~i,
6. Fetch ?~.~eT6a ~1e~ hbyo~ - (3)
addresses of ~ D and a finite set of points . We must determine whether Q falls within con-
vex hull /p
'ihe al~;orithm5 solving this problem are written on the basis of the statement which we
will now write.
T.~ t ~pny represenl ~li~~ cc~nvex hull of set ~p and the boundary of cirele O
I.et - a poinL- on thc: plane iiot lying within Q . From this point we may draw two
~a?igencs to,$' (see Figure 3). Let ~~k represent the arc of circumference lying be-
tween these tangents . We will assume hat ifxE ,~~j~ ~ x; if Xe Q 1~ ,~~X, c~-
empty set.
~ c c or, v' IP ft-~ L' D(/~)
Statement . p ,
That is, circle lies within the convex hull of set ~p when and only when the ares
p~P> from all points P of set /~J encor~pass the entire circumference
The triith of this statement follows dire.ctly from the theorem of the mutual location
of convex hull.s, which is proved below.
The stability check algorithm sequentially constructs ares D(P~~ for all projections
of foctho~ds onto the horizontal plane. If as it being constructed arc D~p)
intersects ar~~~he two combine into a single arc. The process of checking stability
i5 comrlete when either the ares constructed encompass the entire circumference or
Ll~erc~ nre no more L'u~~thulcl5.
Tu reduce thE calculation required, the coordinates of the ends of the ares ~~~i~ are
mea5ured in nonlinear angular units. Let us explain this in greater detail. Let us
l.ocate the origin of the sy5tem of Cartesian coordinates 0 XZ X~ in the center 0 of
circlc . Now let the projection of a foothold have coordinates J~~ P~, (see Fig-
i,re 4). Arc ~~jo) l.ies between points A and B. Point A lies on segment (jR~ with
coordinates
= F, u; t P, 8.
iAz ~ F,u - PI B (2~
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- ~
Point B lies on segment with coordinates
~G)
~3~=P=cc-P,B
/32 = p2 cC + F1 6 ~3~
S
~ Figure 3 Here Q and $ are calculated in accordance with
formulas
X2 P 2� z ' B= _,�'+Pi_t 21
e p, +P2 ~ pz~'+~2
6
~'a x Formulas (2, 3) describe the rotation of vector OP to
_ ~ ~ angles POA and POB with a stretch factor of�~/~P~ ,~Ps/
Let us observe that vectors OA and OB have a unit norm.
Fihure 4 After computing the coordinates of points A and B we
compute nonlinear angles ~ and y~ ~ . If ~ = ~ L>-
XZ the unit vector on the plane, the nonlinear angle yy
corresponding to it will be calculated according to the
Y-~�casd ~P-Sn-c formula
~ ~ L ~ f ~ ~t ~'i 0
{ X, = 1'~fi~~, ~ D, ~L> O
`P-rs~npc V-3�cosd
Z ~L /~a~~ol~j
I'igure 5 3~!- i~ >Di !~z ~ ~
If we let D~ represent the angle between vector 0~ and
- y _ axis OX (see Figure 5) , the change in will be
- as shown in Figure 6.
3 I
The value of is computed according to the following
2 ~ ~ formulas:
I
~ i ~ ~ ~ty d. Wh~rJ ~~�'c- ~,~3 ~ , we add 4 to ~p~ This
operation is equivalent to increasing the corresponding linear angle by 2jJ. With
this calculation S/~ ~ is always equal to or less than f/~~ .
When we transfer foothold projections P~ onto the hori.zontal plane we obtain the
segments rSO~i ~ t~B'~ which, in case they intersect, are replaced by a single
~ common segment. If the length of a given segment has exceeded 4, this means that
ares D~P~~ have encompassed the entire circumference, that is, that there is in
fact stat c stability. But if the footholds are used up before the length of a given
segmenl: exceeds 4, then these footholds will not provide static stability of motion
at our given point ,~5' in time.
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Let us now prove the theorem upon which is based the,algorithm for determining the
static stability of. walking-machine motion. Because of the great simplicity of the
proof it will be shown for the general case.
Let us first introduce some designations and definitions.
Let/R Euclidean n-dimensional space and /P a set within it. CDl1V will '
represent convex hull J/~ ; will designate the frontier of set //d and .t~P> its
interior. If x,y~./iQ" ,~X~y] will represent the seg~ent connecting X and y.
~ will represent an empty set.
n ~X~ will be referred to as a su ort
- Let Q be a set in ~/p Linear functional fy pP
functional with respect to Q at point ~re if
yq..r� t.? fq i~G~) < i'% ftGJ ;
If a support functional exists, then ~E ~(Q~ . We will refer to point `~E as
an ordinary point if a support functional exists at that point and is unique.
Let (,~C f/~n be a given closed set. For any point XE //Q� let us calculate the set
Dlx1 ={~E/"!G~)/[4,,r](1 I(CJ1=~}
This is a set of points such that segtnent ~qi~X] has no other points of inter-
sectiun witti except points r~q)~
Theorem. Let be a convex compact set in fjQM , each point r(lJ' an ordinary
point and /P a compact set in/~N ; then .
q ~~o�~ ~ v A rP~ ~ r~c~)
v~ ~a
Proof. 1. Let Q~CO?7v/P , Let us assume that the theorem is false. Then
3~ F rrc~~ v afli'), dPC cvr?11~ ~4)
Let be the point of maximum,f on
f( 4 t)= rrrax f! 2')
g ' E c.~
_ then 1. E /~~Q~
2. conv /P
3. is a suppor.t functional with respect to Q at point ~f,'~
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We prc~ve that ~'~'EpE/P D~/?~ and arrive thereby at a contradiction. Let us now assume
the opposite case, that 3pxE /P such that ~i'~~~'~/f L~Q~= Q. According to the Han-

~ Banach tlieorem, there exists a functional j~ dividing ~~~'P~~ and Q , that is,
a line:~r functional suct~ thatb~X E~Q~ PJ and dq'c-I~wl,~~cx) , but /1 is a support
functional to Q at point Q* ; hence, by virtue of the simplicity of points !~(Q) /~~,f,'
but then from (4)f~~J'~~>a>,{(~x~ , which cannot be.
Thus, 2*~V DCP) , w}iich completes the proof of the theorem.
p E /P
Plotting the foothulds. For the sake of simplicity in notation, let us disregard the
index I1 in designr*.in~ current and subsequent states; and we will assume that before
beginning to plot 4 new seqtience of �ootholds we know the current initial stateryo ,
it.s existence time ~S�~, Se ~ and the corresponding footholds with their existence
tin~es. We al.so know the two subsequent machine states $1 and
Before we begin to compute the footholds for state we calculate S~ , where $
represents the moment at which the machine should move into state $j~ If the transition
from ~o to Y~1 is accomplished only with the raising of certain legs, then$~ is calcu-
latel as the minimum time S during which the supporting legs of state provide
state motion stability with margin >D and which satisfies the condition s6 ~~~t-~',
- tiere ~ represents the value of the time of the simultaneous existence of states
~p and $y , which is required for redistribution of forces in the supporting legs
durinF, tl~e transition from to
if during the transiti~n from state ~o to~i some legs pass from the transfer phase
into the support phase, then we calculate S~ in accordance with one of the following
three formulas:
s; = s~ - s cs>
lss+sE)_~' (6)
S r = 3 s~ �f 3~e -s
If during state `~t only the rear legs (L = 3, 4) pass into the support phase, we
employ formula (5); if the middle legs = 2, 5) pass into the support phase as well
we employ formula (6); but if even one of the front legs ( 1, 6) enters the sup-
port phase, we calculateS~ in accordance with formula (7) . This rule is heuristic.
It has been adopted because relationships (5-7) are satisfieci with a"galloping" mo-
tion, which t?:is proved most suitable in negotiating cylindrical obstacles [2].
Let us note tliat frum furmulas (5-7) and from the fact that the existence segment of
state R 0 is at least ~ in length it follows that
SQ < Si < 5e - ~
The set of points on the supporting surface on which leg G may rest at moment S
represents the total foothold area /P~ ~S~ of leg ~[3] . Fur point P to lie within/P'~S~
_ it is necessary and sufficient that certain conditions be satisfied:
1, point/~ must a].reacly l~ave been measured, that is, that the height of the supporting
surface at this poiut be known;
- 2. point P n?ust be abl.e to be used as a supporting surface, that is, the slope of the
supporting surface at this point is sufficiently small;
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3. the foot of leg G may be placed at point P at moment S within the, limits of the
kinematic and geometric constraints upon the degree of freedom of this root;
4. if the foot of leg ~ is placed at point/~ at moment leg t will not touch or
intersect any other points on the supporting surface except point p
5. if the foot of leg~. is placed at point P at moment $ leg [ will not touch or
intersect tY~e supporting legs of. state c~t , for which footholds have already been com-
puted and spatial positions fixed.
If E~~>=~S,~Ss~ is ttie existence segment of foothold ~ of leg L, then
P ~ (1 IP; l s)
ScF[P)
Let L represent the set of numbers of legs for which it is necessary to compute
Eoutholds in state ~ 1
L4{L/ %e_p~ %1=1~ ~~t,.1. �-,6}
Let u5 designate these points P'~~ E G . At the initial moment SB of the existence
of state ~1 footholds p~ must already exist; therefore
P~ F rP~ ~ s ~ F c
'The algorithm employeci to locate footholds Pi selects these points in footholds areas
The fewer the points in these areas, the more rapidly the algorithm can acco~-
plish its task. To accelerate the calculations involved a special algorithm is employed
to form the f.amily of sets ,~C
i
c 2~ G... G~N= IP~ ( s~) i c- L.
State 21 footholds are sought first in sets ~l ~LE L ; if the search is unsuccessful
the Eoothold-search algorithm looks for them in sets e G etc. In the last step
of the calculation footholds are sought in the full foothold regions
~~=IP< ~~ `~2~+0)>0 (2.15)
In this case we exclude neither the separation of point B from the surface nor the
preservation of its immobility. The problem of what occurs after impact can be solved
by introducing the law of impact fri.ction or by studying the interaction of leg OB
witli tiie surface with examination in great detail of the "microstructure" of the sur-
face at i~s point oC contact with leg OB.
Thus, if for the three-member mechanism dealt with in section 1 only cases (2.1?.) and
(~.14) can occur, for the three-member mechanism examined in this section cases (2.13)
, and (2.15) can occur as well.
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The qualitative nature of the regions in Figure 5 remains unchanged with char~ges in /u~
Let us note that when S~~P= 3~- inequality (2.10) takes the form svt2et ~ , that is,
it does not depend upon ~t . A similar situation occurs for the three-member mechanism
- in Section 1 as well.
Conclusion. This article has thus investigated the impact problem for two three-member
mechanisms: one with fixed and one with articulated legs (rods). The problem was fully
solved for the first mechanism; we can progress toward solution of the problem for the
second mechanism by introducing the law of friction with impact i~teraction or by de-
tailed examination of the structure of the support surface. It should be pointed out
that, witli small leg aperture angles (small angles a), the supporting leg of both
mechanisms always reUounds from the surface.
BIBLIOGRAPHY
1. Velershteyn, R. A. and Formal'skiy, A. M., "The Locomotion of an Anthropomorphic
Mechanism (with Legs) with Pulse Effects. The Single-Support Phase. I,1� IZV. AN
SSSR, MTT, 1979, Nu 5.
' 'L. Velershteyn, R. A. and Formal'skiy, A. M., "The Locomotion of an Anthropomorphic
Mechanism (with Legs) with Pulse Eff.ects. The Double-Support Phase. II," IZV.
AN SSSR, MTT, 1980, No 1.
- 3. Beletskiy, V. V., "The Dynamics of Biped Locomotion. I, II," IZV. AN SSSR. MTT,
1975, Nos 3, 4.
4. Larin, V. B., "Upravleniye shagayushchimi agparaCami" [Walking Machine Control],
Kiev, Naukova Dumka, L980.
5. Gulliday, C. L. and Hemami, H., "An Approach to Analyzing Biped Locomotion Dynamics
and Designing Robot Locomotion Controls," IEEE TRANS. ON AUTOMAT. CONTROL, 1977,
vol 22, No 6.
6. Lur'ye, A. I., "Analiticheskaya mekhanika" [Analytical Mechanics], Moscow, Fizmat-
giz, 1961.
7. Appel', P. "Teoreticheskaya mekhanika" [Theoretical Mechanics], Moscow, Fizmatgiz,
1960.
8. Panovko, Ya. G., "Vvedeniye v teoriyu mekhanicheskogo udara" [Introduction to the
Theory of Mechanical Impactj, Moscow, Nauka, 1977.
3. The Spatial Prohlem of Optimizing the Dynamics of Biped Locomotion over an Irregular
Surface
- A. G. Orlav (Institu*_e of. Applied Mathematics imeni M. V. Keldysh of the USSR Academy
of Sciences, Moscow)
Annotation. This work is a study of the spatial problem of optimizing the dynamics of
_ biped locomotion during movement over an irregular surface at the level of a model de-
- scribing the primary integral characteristics of the system. Biped locomotian is for-
malized as a ~omplex, discrete-continuous process. The problem is solved with an algo-
rithm for improving complex gradient processes.
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Introduction. At the stage of the mathematical model.ing of biped locomotion processes,
which precedes the creaCion of biped walking machines, there arises the problem of se-
lecting specific modes of movement with the desired characteristics. These modes must
correspond to judicious requirements imp~sed by a possible machine, by its design, its
sources of energy etc. These would include, first of all, requirements for minimiza-
tion of energy expenclitures and a smoothness of motion insuring reliable operation of
the spatial orientation system.
This selection is in many instances made on the basis of intuitive considerations or
- of limited criteria within the framework of specific models of locomotion. Meanwhile,
however, important movement characteristics by and large frequently depend to only a
slight degree upon the specificity of the model.
- The process of biped locomotion is fairly difficult to model mathematically. Its
unique characteristics would include the following [1]: the presence of a number of
support phases for each step, the dependence of solutions upon the continuity of the
path of movement and the discrete-continuous nature of the set of controlling forces-
support r.eactions. Relationships of this nature mean that movement in different time
segments will be described by different systems of differential equations with param-
eters varying discretely in time.
'The present work presents a fairly general scheme and a solution of the problem of
- optimizing ttie dynamics of biped locomotion at the level of basic integral character-
_ istics of the system: the position and velocity of the center of mass and the total
kinetic ;nomentum of the machine indirectly taking into account the other components of
the system in the form of finite limitations. It is a study of the spatial problem of
the movement of a walking machine three steps forward, to include single- and double-
support phases, fi:om a given initial position to a final position. The machine moves
over ari irregular surface modeled with the uae of a set of programs developed at the
Institute of Applied Mathematics imeni M. V. Keldysh of the USSR Academy of Sciences [2].
A sheaf of functionals reflecting the requirements for smoothness of motion, energy
and maintenance of the prof;ram for location of center of mass.
1. The problem formulated. Machine motion will be studied within a�ixed ri~ht-handed
system of coordinates (Figure 1) and occurs as a movement of the machine's center of
mass and a change in its total kinetic momentum under the effect of support reactions
produced on the given supporting surface.
x We will assume here that there is no momentum in the foot
� ~ ancl that the points at which the support reactions are ap-
P plied at each step are fixed.
r " = The dynamics of this machine may be described by the fol-
~ , Y lowing system of conventional differential vector equations:
x �~r~'o
d~,/d~'=vo ; dYo~d'i =(R~tRZ+P)/M; (1)
d ~,/d~'=(zt-z,)xR,+(z2-z,)xR~ .
Figure 1
Here 2'o is the radius vector of the center of mass of the system; ~C~� Zt are the
radius vectors of ttie support points; Ri,RQ the support reactions; I~ total ki-
netic momentum and 1`~ the mass of the machine. 20,~/0,~ are treated as phase
variables; Rt~Qp as continuous controls and 2t,~2 as discrete controls.
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Reactions may be produced only in a region of possible supports, which is to be under-
stood as the intersection of the supporting surface with a sphere with its center at 7~
(the set of support points permitted by the design of the mechanism with the given
center of mass)
~ c~~~o~=~z : ~~z~-a} ~{z~ ~Z-zo~~g~ . c2~
� The set of possible reaction~ for each leg is determined by the location of the center
of mass and the support point.
]C~z~~Z:~7o~= {0~ Z~,r2~ ~~To)~ K1z~or:~~ Z1+ZzE (~(70~, (3)
where ~z~~~=~ is a given cone .
_ The c~~iter of mass of the system cannot be nearer than a certain minimum distance from
the supporting surface
(zo,a) 3 0. c4>
We ar.e given the sets of initial and end conditions
zo~,~. V ~~~~~H KE rN~K . (5)
,
where ~H~S-K represent the beginning and end times of the process respectively.
As a criterion we take the sheaf of functionals ,
~K 3
. I ~N ~ d
A~ dc , c6)
.
where a(L is the weight coefficient; A~ = Rz +Ry +(RZ-P)2+~�~~.~ ~ Rx,Ry,RZ is the sum
of tt~e projections of the support reactions and
A2-1eh~~ F13=~4~,~~ ~~=~o~T'-hhP~'~'~~ 0~=~~~~-I,.tiP~~~~ h~~ ~nP
a given prugram movenient.
Component A1 takes i:~to account the linear and angular accelerations of the machine and
in a certain sense conveys the idea of smooth motion as even motion with low angular
_ accelerations; the term (i(L)2 minimizes the moment of the forces of reaction and, as
may be anticipated, the inCernal moments as well. Components A2 and A3 take into
account the square of the deviation of the phase variables from programmed movement.
They have been introduced to maintain the linear movement of the center of mass of the
machine at a given lieight with zero lateral and transverse components of kinetic momen-
tum.
Our problem consists in finding a mode of movement satisfying the conditions we have
enumerated and minimizing the given criterion.
_ 2. Algorithm of solution. To solve the problem we have formulaeed we employed an ap-
paratus for optimizing complex processes based upon the superpositaon of V. F. Krotov's
suf.ficient optimum conditions for discrete and continuous processes [3].
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We formalize the lucomotion as a complex process as follows. We introduce a set of
discrete stages T corresponding to double the number of machine steps and from it
single out the subset of odd [T]-- T*. Locomotion consists of the selection of discrete
controls (points of support placement) at the even-numb:::red stages (physical time in
this instance does not change) and of movement by virtt~: of the differential equstions
(1) at successive uneven-numbered stages. In these differential equations the discrete
controls selected above play the role of parameters. The result is a discrete-continu-
ous process in which the group ~f permissible processes serves as control for set T*.
The f.ormalization we en~ployeci permitted us to use for solution of our problem an algo-
rithm for improving complex processes [4] based upon the general sufficient conditions
for complex processes. The algorithm is a variation of the "shuttle" methc;ds analogous
for continuous systems to the Eneyev and Chernous'ko-Kxylov methods.
3. Results of numerical calc:~lations. Within the framework of our model and the formu-
lation of our problem, we have studj.ed the spatial problem of biped locomotion over an
irregular surface with single-support and double-support phases at each of three steps.
Tlie maclline moves over a given irregular surface {Figure 2) in the direction of axis
Z
AL _ _ - - ~
Y
X A~-.A~ Figure 2
s
~ 5
~ ~ ? ~ X ~
_ ~ '~C ~t tt +l k K ~'d ~ ~ V
~ v , Y
Figure 3
The machine is assumed to be anthropomorphic and has the following characteristics: it
has a mass of 70 kg; the initial and final height of its center of mass above the base
surface is 0.8 m, the initial length of a step is 0.7 m, the maximum length of a single
step U.9 m and the initial width of its track 0.1 m. The machine moves by regular loco-
= motion at an initial velocity of Vy = 1.4 m/^ec, VX VZ = 0; the initial values of
kinetic momentum Lx = 20 Nm/sec, Ly = LZ = 0. Minimizing the indicated criterion and
satisfying the boundary conditions '~n aYf;~ ~ L;n t L~;q , we are to move it in
ttiree steps from an initial position ~ xp . p~4g y, ~.?�5 r, Po = I to a final
position X,~ ~ C~~l3~r,~~}. ~ ~,35 ~ f= 11n � As a base surface for computing devia-
tion of the height of the center of mass we employ a surface 1 m equidistant from co-
ordinate plane NXY. Total time required by the process is given. Each step takes
U.S sec; the rati~ of the duration of the double-support phase to that of a single
step is 0.2.
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We solve our probl~m Uy employing the improvement algorithm referre3 to above in combi-
nation with an al.gorithm for modeling external environment [1]. The environmental mod-
el is realized in the form of an algarithm providing a discrete, three-level descrip-
tion of the geometry of the supporting surface. The surface environment is described
in terms of a piecewise-translational, piecewise-linear, periodic function of the hori-
zontal coordinates.
The initial ~.pproximation with respect to the continuous controls (broken line - reac-
tions R, R, R) has been taken from [S].
X y Z
- Figures 3-5 show some of the results computer calculations. The broken line represents
the initial approximation.
~ R=a~ Figure 4 shows the behavior of the
functional of the iteration problem
~ r in the process cf improvement. Fig-
I+~ ~ ure 3 shows the supporting surface
~ I�hw ' . ~ ~I
cut through by the plane in which the
~ center of mass is moving. Points 1, 2,
3, 4 represent the initial support posi-
arw ~ tior~, points 1*, 2*, 3*, and 4* the
~ final support position. Figure 5 shows
; Rr~~~ projections of the support reactions.
~
- ~ _ Let us now note the special character-
- g istics of spatial locomotion over an
~~r'~l?e 7 T irregular surface. Representation of
~ ~