Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993
AUTONOMOUS GRASP CONTROL OF LINK MECHANISM BY VIBRATING POTENTIAL METHOD H. Yokol and Y. Kakazu Faculty of Engineering, Department of Precision Engineering, Holclcaido University, Sapporo 060 , Japan
Abs tract. This paper aims at developing a new control method based on the decentralized management approache to describing a mathematical model of the autonomic machiens. In the field of engineering, there were formerly two categories of the control method . One was based on a central control , the other was on parallel di stribution . However, these former models are based on the connection between processing units. Thus each processing unit must be controled by a host unit. Both of these units have to be programmed by a sequence of procedures, and all the units are hardly able to express unification and enlargement. However, in Nature there is an important method that utilizes the propagation of information, for example light waves, sonic waves and smell; and it is known that information transmission among all life units, not just animal, depends on this method . Furthermore, the information processing of a life unit is a selforganizing process that can be achieved by utilizing a field to exchange information. By this method, there are many cases where high performance information processing can be attained without supervision. I le re, in order to develop a more generalized theory for the control of autonomic machines, the Vibrating Potential Method (VPM) is introduced. The VPM is described by placing each unit's potential functions on its own harmonic wave as axes. Each unit receives potential energy from the Vibrating Potential Field (VPF) by convolution and decides its own strategy of motion independently. This paper proposes a new control method using field and discusses its application to the grasping problem of free form objects.
Key Words. Automatic control; information theory; parallel distributed processing; vibrati on me as urement; vibrating potential method; wave motion
1. INTRODUCTION
2. OBJECTIVE STATUS SPACE
In recent years, the autonomous control of machines has drawn much attention in the engineering field. Here a n "autonomous" machine means a machine that has the ability to construct a frame of unified information interchange and a strategy for deciding a behavioral goal for itself. In this paper, idealized autonomic machines and their features are introduced and discussed. Specifically, how to develop a theory that can describe the behavior of autonomic machines in future FA systems. It is well known that I ife has many excellent qualities, for instance, the ability to adapt to changes in the environment. Life as an autonomic and distributed system can serve as a useful model for engineering. However, the current control method of machines does'nt possess any similar qualities, and therefore any proposed method must include them. From this view point, a mathematical expression for this control method has already been developed, and that is the Vibrating Potential Method (VPM). In this study we treat the VPM as an analysis of the movements of groups of units in a multiple Vibrating Potential Field (VPF). The VPM has sufficient unity of fundamental mathematical expression to describe the movement of machines as units. Furthermore, "autonomic" means that the system can achieve a goal by itself through the interaction of units, and the goal finding rule is described as one objective function. By using this method and setting the potential functions of each unit, each unit searches for a stable place. This paper approaches problem solving by regarding the initial set of the VPF as a problem and the stable arrangement of the units as it's solution. A concrete example is shown of this model applied to the problem of the control of a link mechanism to grasp free fonn objects.
The control method of machines proposed here , is basically a PDP system but there are differences concerning information transmission and processing methods. We discuss an objective status space in this model, the required status space being as follows: a) No connection among processing units: The information processing model doesn 't need the precondition that communication among units mu st be dependant on a connection among units. b) Unity expression of all units: We assume that all units can be described by a mathematical expression. c) Autonomous movement of units: All the units mutually cooperate in approaching the solution of an objective problem by the law of cause and effect determined in the field of information processing and using the local information surroundings. A status space that satisfies the above requirements is shown in Fig.I.
1. A field of information processing: The relation in a processing unit between input and output is mediated by a field. This field is categorized into two classes : Onc is the whole field (Vibrating Potential Field) that includes all the phy s ical matter of a unit, and the other is the individual field of each unit. 2. A mechanism of information transmission: The whole information in a field is expressed by the simultaneous summation of multiple information. Therefore, the mechanism of infonnation transmission aims at processing the whole information as a selfcontained unit. To achieve this purpose, we try using a wave motion as our mechanism. Therefore, the transmission of information can be processed as a unit through the mutual interference of wave motion. 859
the link mechanism. The friction problem means how to estimate the amount of interaction energy among the surfaces of objects . Furthermore, in order to decide its own behavior, each unit has to observe its own situation in the environment.
1. Field of Information Processing Vibrating Potential Field 2. Mechanism of Information Transmission Unit Potential Unit Wave ~
4. DYNAMICS OF THE AUTONOMIC UNITS
3. Information Processing Uniti.~~~0~"":"" Unit_____ L . . : 4. Obiective
4.1. Description of the Prllhlcm Solving System
Problem~ ' ~ .
B~undary Condition ~
. = Ki~E netlc ~rgy
E
M.ulJ.ipkJ[ibmling Potential Field Fig. I. Objective space
3. Autonomic Infonnation Processing Unit: Every unit can move autonomously in a style individual to each problem. Not to spoil the capability for spatial interaction, we set an individual field for each unit. Each unit can then interact with the whole field through this individual field. The form of propagated information is defined in three parts : Unit Potential, Unit Wave and Unit Information . These are discussed in detail in chapterS.
The objective problem so lving sys tem is basically POP system. Such system should be constructed of several processing units , and each processing unit should be independent and uniform . The automaton theory fits well for this system wh e re each processing unit is described by automata. This paper describes the units as an automaton in preparation for the problem . The problem that the system deals with is given by a human operator. The automaton theory is famous as the basic model for considering a logical and distributed system, and it aims at des cri bin g the h a m i I ton i a n a s a successive approximation. Therefore, it may be a useful frame for the group problem . The objective automaton is eq.(l). A
4. The Objective problem: The objective problem in this model includes both a physical interaction problem and an information processing problem to be tackled simultaneously. This kind of group problem is characterised by restriction conditions, potential function forms and entropy forms. Furthermore, autonomic problem solving is achieved by the unprogrammed information transmiss ion of units and the searching for a local stable state for each unit. Hereafter, we will call the information style of each unit a "potential function" , and th e individual field for each unit a "unit coordinates axis".
=( S, I, 6,11 )
(1)
where, S is a finite set of state value, 2' is a finite alphabet of input. {) is the status transi tion function, and J/ is a set of the fields . A problem solving system can be constructed by this automaton which solves a problem through information processing as follows : First, the problem and prerequisite conditions are translated to a form of entropy or potential function by a tran s lator. This translated problem or prerequi si te conditions arc built up on the vibrating potential field (VI'F). On receiving information from this VI'F, the units interact with each other and they minimize the increase of entropy and potential within the VI'F. Finally, the units' group produces a solution that is an addition of the units' positions and waves.
3. AUTONOMOUS GRASP PROBLEM
4.2. Objectin function of unit
The objective problem is how to control a link mechanism to grasp an object. This problem consists of the following three parts: the power transmission problem of link units , the friction problem among o b j e c t s t hat h a v e f re e for m s h ape san d the recognition problem of the environment.
To control the units of the system by using the local information, an objective function for the decentralized control is necessary. The most suitable objective function of the decentrali zed system should be leading the units to an orderly configuration. In this section, the objective function of each unit is set by using the concept of life's mechanics in the statistical mechanics field. Total heat capacity ~ is eq.(2). From the definition of thermodynamics, entropy S is eq.(3). Where T(I') is a value determined by the relative value of P. And Q(t) is the internal heat capacity of a units' group and expresses the relative energy head of all control values for a unit.
Units have to catch the object .
set) = E(t) + Q(t) + I'(t)
(2)
dS=~
(3)
l(l'(t»)
Fig.2. An outline of the Autonomous Grasp Problem
Where, E(l) is an inrtowing heat capacity into a group of units. pet) is the outl1owing heat capacity through a group of units . Now, by u sing life's mechanics, the system conserves it s own order by
The power transmission problem is what amount of torque should be transmitted between link units . Each link unit should cooperate and decide its transmission torque depending on the total length of 860
minimizing the gain of entropy S. Then we try to rewrite eq(3) to eq(5).
axes: One is that the VPF can maintain different kinds of interactions by using this expression, and the other is that each unit can achieve interaction through an easy operation.
°
~ (5) dt dt lip (t» Eq.(5) becomes the objective function of a unit.
liS. = dl Eet) + Q(!) + P(!))
5. AN APPROACH TO AUTONOMOUS GRASP CONTROL
4.3. Mathematical Model of Mutual Action (Vibrating Potential Method)
5.1. Potential Function of the Unit
In this section, a mathematical expression of the Vibrating Potential Method (VPM) is proposed. It is constructed by VPF and the law of movement. Any objects that concern us are set as a unit, the problem and restricted conditions are translated into potential functions and the VPF is described by those potential functions . This method deals with a case where an interaction force acts upon the cent er of gravity and the direction of a unit, and can achieve a stable state as a solution through interaction. In concrete terms, the VPF is constructed by the potential functions (hi(r,n) and Wi(r,n») that unit puts on each harmonic wave axis (unit coordinates axis "PC 4» or X( 4> »). H(r,n) = ~ Ihi(r,n)li'(4» + wi(r,n)x(4»1
The information exchanging in this model is represented as an interaction among the potential functions of the unit in the VPF . The potential function works as an information that propagates to other units and informs those other units as to the status of the objective unit. The potential functions of a unit are defined as follows: Each potential function is defined by the distance between two units. There are two types of potential function of a unit. One of the properties of a unit is represented by the unit potential function hi(r,n) as in eq(lO). By this potential function, each unit can express basic behavior such as the closing together and repulsion of other units. Each potential function is defined by the distance between two units.
(7)
I
(l)"B.(r,n),q. h1(r, n )_ ~ ~ 1
Where, H(r,n) is the environment function for a field. hi(r,n) is the potential function of a unit. Wi(r,n) is the wave function that unit puts on a field. r is the position vector. n is the direction vector. v is the velocity vector. Q) is the angular velocity vector. From this VPF and Lagrange's equation of motion, momentum p and angular momentum q ca n be derived by using a convolution as follows:
.·0
Another property for each unit is the capability to propagate a unit wave from the unit boundary as in eq(ll). Each unit gets an amplitude of the unit wave in the VPF by the value ki(r,n). wi(r, n) = B.(r, n) (wlrirl)expUwolr. r!  Kot)
ii~.,P H(r, n)'IjJ(4))d4> dz
pet) =M '       :     
iir iif,~ H(r, n)'IjJ(4)) d$ dz
q(t) =I '      iin
~
ki(r, n)= ko +
(8)
°
t(4)) = 0, ~ I=a, b)
0, (t(4» ='IjJ(4)), x(4)))
~I;ftf H(ri, ni)  X(4)) d4>1
(11) (12)
Here, each unit gets an amplitude of the potential in the VPF by the value ki(r,n). ko is the basic value of ki(r,n). W is the dumping factor. wo is the space frequency. !Co is the wave number. ak, nand kl are the constant. Bk(r,n) is an effective coefficient that determines a unit's figure. Now, to confirm the function of Bk(r,n), we represent an interaction angle e as the inner product between the direction vector of a unit and the position vectorr on a unit coordinates axis as in eq(13).

Where, dz is the curvilinear integral at ohii(r)] = [ or r  z , and z is named as the control point. M is the inertia mass. I is the tensor of inertia. Unit communication is mediated by the VPF H in eq(7). However, since the description of H is only a summation of the unit information, and the objects that we are dealing with have many properties, the information from it alone would be confused. Therefore, to avoid this confusion, we prepared the unit coordinate axes to be at a right angle to each other. Each unit coordinates axis is described as a wave equation. Boundary conditions should be different for each property to avoid confusion.
¥ d~~~) + E.t(4)) =
(10)
a.+(rir)·IIt
n.{r. or)
co~ = In .Iir.  ~
(13)
Table!. Function of control point and unit shapes Name Function of Control Point Unit Shape Rectangle Rectangle ;t=ifany Square; T.=n.1
(8)
Lemniscate
(9)
Lemniscate ;k=2 TriIeaves ;k=3 Pentaleavesjk5
Where, '\jJ( 4» and xC 4» are unit coordinates axis. ~ is Interaction energy. !C(a) is Parameter which determines the spatial distance among unit coordinates axes. a, b : Initial coefficient.
For example, when the unit shape is elliptical, the effective coefficient Bk(r,n) satisfies eq(14) at every control point z.
The change rule of interaction axes is caused by coefficients a and b in eq(9). The unit coordinates axis is in conformity to wave eq(8). There are two benefits to this type of expression of interaction
l2kl =, r=z (14) k I ak + (r ·r·, Where, r is the position vector of a unit, no is the
~ 2k'B"r, n){r .r·fkl
£J
861
°
direction vector, a is radial ratio, and D is the length of the shorter radius. e is the interaction angle.
c)
z = r=~D~==;;== A
/
cos28+~
(15)
~2
V
By using this effective coefficient Bk(n,r), a potential function with a free fonn shape can be shown as in Table1. In this paper, four types of unit, square, rectangle, lemniscate and ellipse are used. 5.2. Simulations of wave motion Since we set the hypothesis that the decision of behavior depends on the wave motion that corresponds to the changing status of each machine's surroundings, the objective space must be able to propagate the wave motion. There are two requirements for this wave motion: One is that the wave source can be set anywhere, and the other is that the change can be simulated in real time in any oscillation mode. To fulfill these requirements, the approximation of an elastic net suits our purpose well. Thus, wave propagation in the objective space can be simulated by approximation of a elastic net.
..
. . . . . .
..
Fig.4 Computer Simulation of Autonomoc Grasp Control of Link Mechanizm
5.3 Simulation of autonomous grasp control The following computer simulation shows the result of the grasp problem using a link mechanism. Fig 4(a),(b),(c) shows the grasping process. In these figures, the shape of the link unit is rectangular, and the object to be grasped has a trileaved lemniscate shape. In order to know the location, the wave propagates from the latter. Each unit observes this wave motion in its surroundings, and approaches it's target. Finally, figure (c) shows how a link mechanism can find and enclose the grasped object.
. .
'C i rCl1lar Un; t (SaJrce of Wave) .
. ·Co 11· i 5
i:O~ : . : \
..
. ..
6. CONCLUSIONS
~
This paper approaches the autonomous grasp control of a link mechanism by proposing a Vibrating Potential Method (VPM). Each link unit can decide its own behavior autonomously by the VPM. Since the control of each unit is independent, the VPM presents an autonomous and decentralized control. From the computer simulations, the properties of the link unit are shown as follows: 1. Proposed method can simulate the interaction of a link unit that's shapes have been enclosed in the Riemannian Space. 2. The link mechanism approaches the object to be grasped by observing the wave mode of its surroundings. 3. The behavior of the link unit depends on the continuity in the rate of change of the wave motion. 4. This method is independent of the total number of links and their shape.
. •
.(
'\\\\\ .. ..... ........ . :.
':.' : .. :: Uave. /'Iot·i'on . . : : .
: Spa lal frequency ;'5: changed' accord i ng: :
tb.the: velobity of .circtHar: Lirlit: : :
Fig.3. Computer simulation of wave motion
a)
For the engineering application, by setting the potential functions, this method can be applied to the autonomous grasp control of a link mechanism . These examples show some of the potential of this method for engineering problem solving.
REFERENCES Baraquand, J., Langlois, B . and Latombe, J.c. (1992). Numerical Potential Field Techniques for Robot Path Planning, IEEE TransactionsOn Systems, Man, and Cybernetics, Vo\. 22, No.2. Fetter, A.L. and Walecka, J.D . (1971). Quantum theory of manyparticle systems, McGrawHiII, Inc. Giralt, G ., Chatila, R. and Vaisset, M. An Integrated Navigation and Motion Control System for Autonomous Multisensory Mobile Robots, IEEE Computer Society Press, AUTONOMOUS MODIL ROBOTS. Koide, S. (1969). Quantum mechanics (I), Syoukabo. Harken, H. (1977). Synergetics, SpringerVerlag., Inc. KaffeJ, M. and Dube, Y. (1992). An Autonomous Mobile Robot, Proceedings of Singapore International Conference on Intelligent Control and Instrumentation. Yokoi, H . and Kakazu, Y.(1992). Approach to the Traveling Salesman Problem by a Bionic Model, Heuristics, The Journal of Knowledge Engineering, lAKE, pp. 13  27.
b)
862