Pergamon 09670661(94)000557
ControlEn&.Practice,Vol.2, No. 6, pp. 10311038,1994 CopyrightO 1994ElsevierSdenceLtd Printed in Great Britain.All rightsreserved 09670661/94$7.00+ 0.00
AUTONOMOUS GRASP CONTROL OF LINK MECHANISM BY VIBRATING POTENTIAL METHOD H. Yokoi* and Y. Kakazu**
*HumanEnvironmentSystem Department,National Institute of Bioscience and Humantechnology, M J.TJ., 114 Higashi, Tsukuba, lbara/d 305, Japan **Faculty of Engineering, Hokkaido University, N13WS, Sapporo060, Japan
A b s t r a c t . Living c r e a t u r e s differ from machines in t h a t they have no prefixed objective function. These basic processing elements (creatures or cells) are influenced by their objective functions and change their properties based upon this influence. This recursive structure is the basis of living creatures' autonomous behavior. This paper aims a t the development of a new control method based on a decentralized m a n a g e m e n t approach to describing a mathematical model of autonomous machines, called the vibrating potential method (VPM). In the VPM, the system consists of a group of units, and the objective functions of the e l e m e n t a r y units are combined so t h a t they become the objective function of the system. A field is employed as the medium for information transmission; its application to the problem of grasping freeform objects is discussed. K e y W o r d s ; autonomous grasp c o n t r o l / v i b r a t i n g potential m e t h o d / w a v e m o t i o n / i n t e r a c t i o n among link element/ information processing.
and output information for an entire system, a prerequisite condition is that the system be given an objective function (which is subject to feedback) for the output of the system. That is, construction of the system is based on the existence of the (human or designerselected) objectoriented behaviors, and is thus a topdown approach. However, the objectoriented behaviors of living creatures consist of a summation of the objectives of individual cells, so the purpose cannot be set by humans. In other words, an objective function is determined for each unit of a system, and the objective function of the entire system has to be adaptable, depending on the system size, the system form and environment, and various other factors. The current machine control methods lack these qualities, but the authors of this paper believe that any proposed method should include them.
I. INTRODUCTION The purpose of this paper is to develop a mathematical representation of a control method for autonomous machines based on decentralized management. One of the engineering applications of the model is an approach to the autonomous grasping problem. Conventional research on the grasping problem has been concerned with the field of robotics, specifically: the problem of selecting a grasping point by utilizing the compliance of the top of an endeffector, the problem of reconstructing the object to be grasped using information from a touch sensor on the endeffector, and the problem of controlling multilink movable robots encircling an object to be grasped. This research has assumed that the link touches the object to be grasped, and thus has not been concerned with how to approach the object to be grasped. Because the mechanism is specified, the system has difficulty allowing the replacement, addition, or subtraction of elements in the mechanism, and consequently the units depend on a description of the procedure, and have difficulty expressing unity and enlargement.
When animals try to catch their food, their behaviors are those of grasping, trapping, repressing, adhering, and binding. For grasping behavior, they pass through the following three processes: Finding the object to be grasped, approaching the object, grasping the object. Animals higher on the evolutionary scale realize grasping behavior through observing the patterns of objects using their eyes and ears, analyzing patterns and deciding their own behavior accordingly, and then passing appropriate infomation through the nervous system to the muscles. Lower animals also use a similar process to decide their own behavior, and even if the differences between a nerve organ, a sense organ, and a muscle organ are not clear, they still work cooperatively. Thus, the process of observing and information processing supports the animals' decision making. Using the analogy between living things and machines, one can find a desirable control method that permits autonomous and decentralized management of machines, and also
It is well known that living creatures have many excellent qualities, such as the ability to adapt to changes in the environment. A living creature viewed as an autonomous and distributed system can serve as a useful model for engineering. The Cybernetic approach (by Winner) has shown the possibilities of living creatures' capabilities being replicated by machines and computers and algorithms, and also their effectiveness. Currently many researchers are trying to find ways to describe living creatures in terms of mechanical systems. In cybernetics, because the objective of study is clarifying the relationship between input information 1031
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H. Yokoi and Y. Kakazu
permits the enlargement of a system through the unified representation of machines. The mathematical expression of just such a method has been developed, and named the Vibrating Potential Method (VPM). In this study, the VPM is treated as an analysis of the movements of groups of units in a multiple Vibrating Potential Field (VPF). The VPM's fundamental mathematical expression has sufficient unity to describe the movement of machines as units. Furthermore, the system is "autonomous" in that the system can achieve a goal by itself through the interaction of units, and the goal finding rule is described as a single 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 its solution. A concrete example of this model is applied to the problem of controlling a link mechanism grasping freeform objects.
(Vibrating Potential Field) that includes all the ph.ysical matter of a unit; or the field for an individual unit.
2. OBJECTIVE STATUS SPACE
iv, 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 characterized by restriction conditions, potential function forms and entropy forms. Furthermore, autonomous problem solving is achieved through the unprogrammed information transmission of units and each unit's search for a local stable state.
The proposed control method for machines is basically a parallel distributed processing system but with differences in the information transmission and processing methods. An objective status space in this model is discussed, the required status space being as follows: a) No connections among processing units: The information processing model does not need the precondition that communication among units depend on a connection among units. b) Unified expression of all units: It is assumed that all the objective units can be described by a mathematical expression.
ii. A mechanism for information transmission. All the information in a field is expressed by the simultaneous summation of multiple information. Therefore, the information transmission mechanism aims at processing all the information as a selfcontained unit. To achieve this purpose, wave motion is used as the mechanism. Therefore, the transmission of information can be processed as a unit through the mutual interference of wave motion. iii. Autonomous Information Processing Unit. Each unit can move autonomously in a style unique to each problem. In order to preserve the capability for spatial interaction, an individual field is set for each unit. Each unit can then interact with the entire 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 section 4.
Hereafter, the information style for each unitwill be referred to as a "potential function", and the individual field for each unit a "unit coordinates axis".
c) Autonomous movement of units: All the units 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. A status space that satisfies the above requirements is shown in Fig. 1.
q. Fieldof Information
l Processing / VibratingPotentialField.~.~1~¢ i~ ,,. Mechanismof Information"l~ / . .TransmissionI._.l.; ~ unit Potentiat~"~/'...:.'..J / UnitWave.._ ~ ~ Z . . ¢ ' ~
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Fig. 1. Objective space i. A field of information processing. The relation in a processing unit between input and output is mediated by a field. This field may be categorized into two classes: the whole field
Fig.2. An outline of the Autonomous Grasp Problem In this paper, the objective problem is how to find and grasp objects, and it consists of three processes. Those are (i) the process of finding the object to be grasped, (ii) the process of approaching the object, and (iii) the grasping process itself. In the engineering field, conventional research realizes the finding process by using visual feature detection methods. The estimation of approach path and grasping torque is derived from the approach process, and the grasping process is based on the position and shape of the object. Here, the vision image consists of the pattern of the amplitude of the light waves that are propagated from the objects. Namely, it is assumed that all objects give out
Autonomous Grasp Control of Link Mechanism information about their own features (location, shape, motion etc.) through wave motion. Thus, the problem is regarded as one of processing information received in the form of wave motion in the VPF. In order for each unit to decide its own behavior, it must observe its own situation in the environment. 3. DYNAMICS OF THE AUTONOMOUS UNITS Animals which live in groups (and also cells in bodies) are able to coordinate their offensive and defensive manuevers despite lack of connections. They use some sort of spatial information transmission and are able to adjust their behavior according to information received. Humans use forms of dispersing informatiotvwave propagation, for examplemwhich provides both the media of information and multiple memories. For the criticism of a system's behavior, the kinematics of living creatures which minimize increasing entropy rates must be effective. Assuming that input information is virtual energy for continuous energy changing, and that the information is used to adjust the energy consumption rate, the relation is set between information and energy after the real energy input. Furthermore, in order to deal with the interaction of group of unit in a field, the powerful tool of quantum mechanics is employed. The meanfield approximation is effective for this purpose. Because this paper requires many fields, superposition of fields with orthogonal coordinates is used.
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(5) •
•
where E(t) is the unit group's capacity for infiowing heat. P(t) is the unit group's capacity for outflowing heat. Now, using the mechanics of living creatures, the system conserves order by minimizing the gain of entropy S. Then Eq(2) is rewritten to Eq(6). dS = d{ E(t) + Q(t) + P(t)} >_0 dt dt T(P(t)) (6) dS To minimize the value of dt, Eq(7) becomes the objective function of a unit.
Mini TdE(t)+ dQ(t)~ }, Maxl T}dP(t)
(7)
Then the objective function for the problem is given by Eq.(7).
AutonomicMachine
Source of Wave
3. I. Ol~jective function of unit Controlling the units of a system by using local information requires an objective function for decentralized control. An ideal objective function of a decentralized system leads the units to an orderly configuration. In this section, the objective function for each unit is set by using the concept of the mechanics of living creatures from the field of statistical mechanics. Total heat capacity ~ is Eq.(1). From the definition of thermodynamics, entropy S is Eq.(2), where T(P) 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, where M,I,K,L and A indicate the coefficients of inertia mass. ~(0 = E(0 + Q(t) + P(t)
dS
(1)
d~(t) 'I~P(t))
(2)
According to the objective problem, E(t), Q(t) and P(t) are set using the information of the VPF H(r, n) and the quantities of each unit as in Fig.3. E(t) =
Q(t) =
2
Ai•
H(r, n) X(¢?) d~)
(3)
(,w.vo xri), (,wavoXni)' Ki
+
Ji
(4)
~
(t)
E(t) (
VPM H(r.n)
]
Fig.3. Inflowing and outflowing heat capacity in the VPF 3.2. Mathematical Model of Mutual Action (Vibrating Potential Method) In this section a mathematical expression of the Vibrating Potential Method (VPM) is proposed. The model is constructed using a vibrating potential field (VPF) and the law of movement. Any objects of concern are set as a unit, and the problem and restrictions are translated into potential functions which are used to describe the VPF. This method deals with a case where an interaction force acts upon the center 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 of the potential functions (hi(r,n), M~j(r, n) and wi(r,n)) that a unit puts on each harmonic wave axis (unit coordinates axis ~F(~)or X(O)). These equations ((8)(13)) describe the movement of the units.
1034
H. Yokoi and Y. Kakazu
H(r,n) = ~ . { hi
equation. Each property's boundary conditions should be different in order to avoid confusion.
(r,n)~(¢) + wi (r,n)z(¢)
I
+ ~jfi~j(r,n)(pi(Cp)(pj(O)} (8) where H(r,n) : the environment function for a field. hi(r,n) : the potential function of a unit. ~j(r, n) : the elastic potential function of a unit. wi(r,n): the wave function that a unit puts on a field. r : the position vector: r(t) = (rx, ry, rz). n : the direction vector: n(t) = (nx, ny, nz). v : the velocity vector: v(0 = P(t). w : the angular velocity vector: ago =/z(t). As long as it does not affect the parity of the units, any method may be selected. To attain mutual cooperation in the movement of units, Lagrange's equation of motion has been selected as the informationprocessing method. This method is used to deal with a case where an interaction force acts upon the center of gravity and the direction of a unit. Lagrangean L is defined as Eq(9) by using the VPF H as a potential. L = (2Lv(t)"p(t)+ ½ al(t)" q(t))" q/(¢)  H(r,n) (9) Lagrange's equations of motion are represented in Eq(10). dt/~]
~~ = 0,
=
Vz,
(10) From this VPF and Lagrange's equation of motion, momentum p and angular momentum q can be derived by using a convolution as follows:
r(a) ~((~)2 + F_,~(¢) = O, (~(¢) = V(¢), z(¢) ) 2 de 2 (12) 4(¢) = O, (~¢~a, b)
(13) where ~(t)), X(O): Unit coordinates axis. E¢ : Interaction energy among units. r(a) : Parameter which determines the spatial distance among unit coordinates axes. a, b : Initial coefficient. The change rule of interaction axes is caused by coefficients a and b in Eq(13). The unit coordinates axis conforms to wave Eq(12). There are two benefits to this type of expression of interaction 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.
4. AN APPROACH TO AUTONOMOUS GRASP CONTROL 4.1. Potential Function of the Unit Information exchange in this model is represented as an interaction among the potential functions of the units in the VPF. The potential functions work as information that propagates to other units and informs them of the status of the objective unit. The potential functions of a unit are defined by the distance between two units. There are two types of potential function for a unit. One of the properties of a unit is represented by the unit potential function hi(r,n) as in Eq(14). Using this potential function, each unit expresses basic behavior such as the approaching and repelling of other units.
0 ~ . , ~ H(r, n)(V(¢) + ~i((~))d0 dz /~(0 = M . q(t) = I .
hi(r, n)= ~r 0 ~ . . , ~ H(r, n)(~l/(q))+ 9i(*))d* dz
~n
(11)
3 (1)"B,(r, n).q, n=0 a,+(ri_r) 2"h,
(14)
In order to bond the units, the elastic potential fi~j(r, n) is set as a joint among two link units.
where
Z
[Ohij(r)__...~ = 0 : the curvilinear integral at [ Or Jr = z : the control point.
M
: the inertia mass: M = (mx, my, mz). l=
I
:the tensor of inertia:
 Ixy Iy  Iyz  Izx  Izy Iz
Unit communication is mediated by the VPF H(r, n) in Eq(7). However, because the description of H(r, n) is only a summation of the unit information, and the objects dealt with have many properties, the i n f o r m a t i o n from it alone would be confused.Therefore, to avoid this confusion, the unit coordinate axes are set at right angles to each other. Each unit coordinates axis is described as a wave
//t'ij(r, n )= Kij (r ri +ni D)
(15)
Another property for each unit is the capability to propagate a unit wave from the unit boundary as in Eq(16). Each unit gets an amplitude of the unit wave in the VPF by the value ki(r,n). wi(r, n) = B,(r, n) (W Irirl) exp~/.tx~01rirl  rot) (16) ki(r, n) = k0 + ~  I
H(ri, ni) • X(~) d~l
(17)
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, w0 is the space frequency, ko is the wave number, ak, n and kl are the constant~Bk(r,n) is an effective coefficient that determines a unit's figure.
Autonomous Grasp Control of Link Mechanism
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potential function with a freeform shape can be shown as in Tablel. In this paper four types of unit are used: square, rectangle, lemniscate, and ellipse.
.
4.2. Simulations of wave motion • (BI
Frame {AI
Fig.4. The coefficient B of an elliptical unit Now, to confirm the function of Bk(r,n), an interaction angle O represents the inner product between the direction vector of a unit and the position vector r on a unit coordinates axis as in Eq(18). COSO =
n,.(r,  r) In,l'lr,  ~
(18)
For example, when the unit shape is elliptical, the effective coefficient Bk(r,n) satisfies Eq(19) at every control point z. 3 2kBldr, k= 1
n)~r r*)2k'i
Using the hypothesis that the behavior selection depends on wave motion corresponding to the changing status of each machine's surroundings, the objective space must be able to propagate 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 the purpose well. An example is shown in Fig.5, where the location of the wave source is the center of gravity of a circular unit. Because the velocity of the wave propagation depends on the properties of the elastic net, the simulation shows a case where the spatial frequency is changed according to the velocity of a circular unit. Each rectangular unit observes the status of the circular unit through the wave motion. .
=0, r=z
(Xk + (r r*) 2k'l
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D
Z
~ / c o s 2 0 + sin20 a2
(20)
: : : Spatial frequency is changed according : : : t O me velocity of circular unit
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Function of Control Point
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Unit Shape Ellipse ;a=if any Sphere ;a=l
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ID cos19, 2 '~+nx < 19 < "~+n/t z=
Rectangle;t=ifany Square ; t=p
D s i n 1 9 ,
"¢+n~< 19 < '~+(n+l)n , (n0r172f') Lemniscate
z = D.sin2 (k 19)
Asteroid
Lemniscate;k=2 Td.loaved;k3 Pentaleaved ;k5 Triangle ;k=3 Asteroid ;k=4
When the unit shape is rectangular, the effective coefficient Bk(r,n) satisfies Eq(19) at every control point z. The control point z can be described as Eq(21), where r* is the position vector of a unit, n* is the direction vector, D is the length of the shorter radius, and O is the interaction angle. ; cosO, ~+nx _
4.3 The behavior of a erouo of units in the wave field
sinO ,x+nx _
This section shows a computer simulation of the behavior of a group of units in the wave field. The number of units is 14 and the frequency of the wave is c0=7. Each amplitude of unit potential function qi is adjusted by Eq.(22).
~(t) _<0 /) qi
(22)
The units are initially located in the neighborhood of the unit wave as shown in Fig.6 a). Unit oscillation is caused by observing the potential function of the other units in the VPF, and a group of units converge to a stable configuration. The result after 1500 cycles of calculation is shown Fig.6 b). The closest packed hexagonal structure is a stable configuration where there are no restrictions on the unit potential function. The motion of each unit is part of the motion of the unit group, and the unit group approaches the center of the wave motion. After
1036
H. Yokoi and Y. Kakazu
3000 cycles of calculation, the group's center of gravity overlaps the center of the wave motion as shown in Fig.6 c).
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Parameter of Unit Wave Position : (90.0, 50.0)
Frequency : m = 7.0 D a m p i n g rate : W=0.95
(c)
Parameter of Unit Potential function cp = 250.0 ql = 2 5 0 . 0
= 100.0 W = 0.0
Fig.6. Mutual action of units in the wave field This paper uses this group of units as an autonomous machine for the grasping problem. The property of approaching the center of a wave motion is applied to the finding and grasping process of autonomous machines. In order to construct an autonomous machine as a link element, the effective coefficient Bk(n,r) of the units is set as rectangular. Fig.7 shows the approaching and collision process between a rectangular unit and a threeleaved lemniscate unit.
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(d)
Fig.8. Computer Simulation of Autonomous Grasp Control of Link Mechanism (the object to be grasped has a circular shape)
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4.4 Computer ~imulation of autonomous grasp control The following computer simulation shows the result of the grasping problem using a link mechanism. The concrete parameters of the link unit are W=0; the object to be grasped is W=I. Fig.8(a) indicates the initial set of units, and (b),(c),(d) shows the grasping process. In these figures, the link unit is rectangular, and the object to be grasped is circular.
(e)
Fig.9. Computer Simulation of Autonomous Grasp Control of Link Mechanism (the object to be grasped has a trileaved iemniscate shape)
AutonomousGraspConu'olof Link Mechanism Fig.9(a),(b),(c),(d),(e) also shows the grasping process. In these figures, the shape of the link unit is rectangular, and the object to be grasped has a threeleaved lemniscate shape. Waves propagate from the latter, revealing its location. Each unit observes this wave motion in its surroundings, and approaches its target. The final result, of course, is that the link mechanism grasps the object. Thus these computer simulations show how the link mechanism finds and encloses the object to be grasped.
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1037
and (d) show the grasping process. In these figures, the link unit is rectangular, and the object to be grasped is square. (a) shows the initial arrangement of the link mechanism and the object. Waves propagate from the object's center of gravity. (b) and (c) show an approach process, and (d) shows that the link units can reach and fix the object. However, the final configuration of link units does not enclose the object. The reason for the configuration is that the waves propagate from the center of the object, and the object has a large offset; consequently the object became an obstacle for the link unit. For the present problem, wave propagation from the edge of the object is used. Fig.11 shows the final configuration, and the link units succesfuUy enclose the square object.
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Fig.12(a),(b),(c),(d) shows the grasping process. In these figures, the shape of the link unit is rectangular, and the object to be grasped has a threeleaved lemniscate shape. As in the previous simulation, waves propagate from the latter, revealing its location. Each unit observes this wave motion in its surroundings, and approaches its target. Finally, figure (d) shows how the link mechanism finds and encloses the object. The next example (Fig.13) shows that an object with a more complex shape (hexleaved lemniscate). The link unit failed to enclose it.
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Fig.10. Computer Simulation of Autonomous Grasp Control of Link Mechanism (Square object) The following computer simulation shows the result of the grasping problem using a link mechanism. The concrete parameter of the link unit is W=0; the parameter of the object is W=I. Fig.10(a),(b),(c),
This paper approaches the autonomous grasp control of a link mechanism by proposing a Vibrating Potential Method (VPM). Each link unit can, using the VPM, decide its own behavior autonomously. Since the control of each unit is independent, the VPM allows autonomous and decentralized control. From the computer simulations, the properties of the link unit are shown as follows: 1. The proposed method can simulate the interaction of a link unit whose shape has 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 of the wave motion's rate of change. 4. This method is independent of the total number of links and their shape.
1038
H. Yokoi and Y. Kakazu
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Giralt, G., Chatila, R., and Vaisset, M. (1992): An Integrated Navigation and Motion Control System for Autonomous Multisensory Mobile Robots, IEEE Computer Society Press, Autonomous Mobil Robots. Hirose,S., Morishima,A., and Nagai,K. (1989): Articulated Body Mobile Robot, JRSJ Vol.7 No. 1, pp.5661. Kaffel, M., and Dube, Y. (1992): An Autonomous Mobile Robot, Proc. of Singapore International Conference on Intelligent Control and Instrumentation, pp.790804. Kaneko, M., and Tanie,K. (1990): Contact Point Detection between Multifingered Robot Hand and Unknown Object Using SelfPosture Changeability, JRSJ Vol.8 No. 6, pp.5661. Krogh, B.H., and Feng, D.: Dynamic Generation of Subgoals for Autonomous Mobile Robots Using Local Feedback Information, IEEE Computer Society Press, Autonomous Mobil Robots. Okada,T., Miyakoshi,T., and Takano,E. (1990): A Method of Determining Support Forces at Multiple Points for Stable Grasping, JRSJ Voi.8 No. 5, pp.505512. Yokoi, H., and Kakazu, Y. (1991): A Study on Behavior of Autonomous Machines in the Vibrating Potential Field  An autonomous construction of a structure by machine elements Proc. of '91 Annual Conference ROBOMEC, vol.A, pp.183. (in Japanese).