Reference frames in learning and control

REFERENCE FRAMES IN LEARNING AND CONTROL.. .

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Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

REFERENCE FRAMES IN LEARNING AND CONTROL

N. Kirupaharan .. W. P. Dayawansa **,1

.. Department of Mathematics and Statistics Texas Tech University . Lubbock, Tx 79409 , USA [email protected] " .. Department of Mathematics and Statistics

Texas Tech University Lubbock, Tx 79409, USA [email protected]

Abstract: This article is aimed at introducing the theory of reference frames in mathematical psychology to the control theory community and to point out its relevance in reducing the complexity of control systems design. As an example we discuss the balancing problem for a multi-segment pendulum. Copyright
Keywords: reference frames, balancing, human posture control.

methodology. This realization has been the impetus for a renewed and a vigorous call to enumerate a set of fundamental principles which will render biological systems less misterious and even more importantly, win allow human designers to design engineering systems to behave "intelligently".

1. INTRODUCTION

It is becoming increasingly clear that there are many lessons to be learnt from biological systems, for they are adept at fault tolerance, adaption, learning and extrapolation and many other capabilities which still live in the imagination of designers of engineering systems. For example human brain is capable of breaking down a complex concept, or a task, into several simpler tasks, each of which is familiar to us, thereby making it possible to solve a novel and complex problem by solving several simpler familiar problems. Examples of such occur in reading, facial recognition, learning, sports, and almost everything else we do. On the other hand engineering systems are designed to carry out certain specific tasks in a very orderly manner. Baring a few exceptions they are incapable of learning from their past experiences, and, do not rely upon a hierachical organization as a problem solving

Our aim here is to point out some relevant concepts from cognitive psychology. They have been interested in developing models of cognition and perception rather than asking how these are carried out at a cellular level. The most fundamental questions from this viewpoint are what do we mean by cognition and perception and how are these mechanisms organized in our psychy. Only a few minutes of thought is required to convince the relevance of the following basic question: How does our brain represent the outside world. Classically the cognitive system was thought to act as a dynamical system with a very high dimensional state-space representing events in the outside world as points in this state-space. This notion was challenged in 1982 in a series of landmark papers by Michael Leyton (see (?), (?). (?)). He argued that the brain represent each outside object and event as a hierarchy of machines or dynamical systems. To ellaborate. each

Research was funded through NSF Grants ECS 9707927 & 9720357.

1

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REFERENCE FRAMES IN LEARNING AND CONTROL.. .

event is associated with several simpler events which in turn are associated with even simpler events and so on. For example, each sentence can be broken down into a set of phrases which in turn are broken down into a set of "ideas" etc., and machine descriptions of these arise from rules in linguistic constructions. In this framework cognition of a new and complex object boils down to an organization of a set of links to an existing hierarchy of machines and via them constructing a new machine. Leyton argues forcefully that these structural features are present in every aspect of human cognition, perception, learning and control. Our main pOint of contention here is that if we are to understand control theory from the stand point of biological systems, then we need to be cognizant of the need to model our systems as a hierarchical structure and device control design tools to design control algorithms which can be built up from the lowest level to the highest level. Unfortunately such methodologies are scarce and the need to develop such has not received due attention. For example we have no tools comparable to wavelet analysis in communication theory and multigrid methods in numerical ·analysis. This paper is organized as follows. In section 2 we will briefly describe the theory of reference frames in cognition from the viewpoint of Michael Leyton' s work in (1), (?), (1) etc. This will be illustrated via an example dealing with the cognition of geometric figures. In section 3 we will discuss as an example, the problem of balancing a multi-segmented pendulum. The control law derived here is motivated by the reference frame theory. In section 4 we will give some concluding remarks.

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To explain this awareness classical notion would require us to mentally carry out these transformations, i.e. the cognitive system does the computations, and test and see whether the transformed parallelogram matches up with a new geometric figure. In contrast Leyton's theory says our cognitive representation of parallelograms consists of a state-space of parallelograms and transformations which act upon it. This viewpoint obviates the need for a cognitive system to carry out computations to match objects. Instead, the natural symmetries of objects under consideration are stored in together with the objects in our cognitive representation. A crucial aspect of the cognitive representation is the notion of reference frames. Eventhough the notion of references has an old history in psychology, Leyton seems to be the first person to mathematically define this notion. Each dynamical system in the outside world is associated with several levels of simpler versions of it. For example, a sentence is associated with a certain set of rules (i.e. dynamics). In linguistics phrases are at a simpLer level than sentences, and the rules they have to obey are simpler than those for sentences. Theory of reference frames states that in cognitive representation each dynamical system is associated with a set of simpler dynamical systems, which in turn are associated with even simpler ones and so forth.

To keep things simple in this section Let us (as Leyton does) keep our discussion restricted to a very simple class of dynamical systems; those of group actions. Thus we have an objectrepresented'as: a state-space which is a manifold M

2. LEYTON'S VIEW OF THE THEORY OF REFERENCE FRAMES. In a series of landmark papers (1), (?), (1) Michael Leyton questioned the classical notion of a human cognitive system perceived of as a computer delegated with the task of carrying out computations about SO(rounding objects. His proposal was to replace this notion by that of cognitive representation of the outside world as consisting of dynamical systems. He defined the cognition of a stimulus set as the description of the latter as a dynamical system. (In his terminology a dynamical system is the same as a discrete time control system for us. We will continue to use the term dynamical system soa SllOt to confuse it with some control tasks to be discussed later). Perhaps the simplest example to illustrate this point is to consider a geometric object, e.g. a parallelogram. Classical viewpoint would say that our cognitive system would form an image of a parallelogram. Let us think a bit harder about how we think. of a parallelogram. We are very aware (almost unconciously) that a parallelogram will remain a parallelogram under rotations, translations and certain other deformations.

an input-space which is a Lie group G its dynamics :

G x M

->

M,

a left action of the group G on M. Let us represent this object as (G,M). A reference frame for the object would be another dynamical system (R,N) where N is a submanifold of M and H is a subgroup of G. To illustrate this let us consider the cognitive representation of parallelogram objects. The description below is a bit more complex than Leyton's description, but is similar in spirit. The state-space in our cognitive representation is an abstract space M. However we can describe the manifold structure of M as follows. Take a parallelogram and identify it as a point in ~2 x ~2 X 1R2 by choosing a vertex and two vectors representing the sides meeting at the vertex. Any nearly point can be identified with a nearly point in 1R2 x 1R2 X ~2 . This now sets up local coordinates on the manifold M. Input group of the parallelograms representation is G£(2,~) x ?R2 which acts by "translating" a parallelogram and "defonning" it in an obvious way in

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accordance with its action on ~2. Thus we have the cognitive representation of parallelograms

To motivate the discussion of reference frames let us ask ourselves the following question. If we were asked to draw a parallelogram which of the parallelograms in figure 1 would be closest to ours?

~0 (a)

(b)

j '---r:o:T------/"-----' (c)

Fig. 1. Parallelogram Objects It is our contention that chances are very high that it would be (c). Thus, eventhough we know parallelograms in general we have a "preference" to those that "lie flat". Such parallelograms are cognitively represented as a dynamical system,

where N is a submanifold of M, and G = H X 1R2, where H is the group of "shears" on 1R2. Thus parallelograms that lie "flat" fonn a reference frame to an parallograms in our cognitive representation. In fact this "referencing" can be carried out further to get rectangle and square objects with their associated dynamics. In general one can think of a cognitive representation of an object as a representation of it as a control system and a reference frame for the cognitive representation as a control system derived from the original one by restricting the input and the state-space appropriately.

3. BALANCING OF A MULl-LINK PENDULUM Here we ask: What can reference frame theory offer in the form of new insight into a standard control problem.As an example to illustrate a possible use of the idea of reference frames in contol theory let us consider the problem of balancing an inverted multisegmented pendulum with multiple control inputs. We take the pendulum to consist of n segments and the number of control torques is taken to be n. Here we disregard friction. Let us assume that the system consists of k subsystems and denote by Si. the ith subsystem. We group control torques acting at join~. in Si' together and denote it by Si' Subsystems S'-1 and Si are joined at joint Po, and PI is assumed to be fixed. This is schematically represented in Figure 2. We wish to remark here that the ideas outlined here can be extended to study the human posture control problem. In this case 8 1 would consist of the two legs. S2 would be the lower torso, 8 3 would be the upper torso and 8 4 would be the head.

Fig. 2. Multi-link Inverted Pendulum Our objective here is to study the balancing problem for the multi segment pendulum from the view point of the theory of reference frames. Thus we wish to consider all subsystems below 8 i as a reference frame to the collection of subsystems above Si. More precisely, we aim at the following. Let us first consider the top most subsystem Sk. Suppose we were able to find a control law (for Tk ) that would balance it in the presence of motion of subsystems below it. We wish to employ it to derive a balancing law (for Tk-l ) for the subsystem Sk-l with minimal modifications of its dynamics. This in turn would be used to derive a balncing law for 'T"';-2 etc, until we end up with balancing control laws Tl, 72, ... Tic for the entire system. The usefulness of a strategy of this type is quite abvious. We stress that at no point in our contol design we will have to deal with the entire systm, thus reducing the complexity quite conSiderably. In spite of tremendous advances made over the past few years in computer technoloy, still we aren't close to being able to control a biped or a many degrees of freedom robot arm with precision. It is our hope that control design with reference frames in mind will shed some new light into these hard problems. Getting back to the control problem at hand, let us start from subsystem S/<. Let us represent its state vector by

Z/c

=

[ ~:]

where Ok are the angles each link

make to the vertical direction and velocities of the links.

Wk

are the angular

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Let us note that the equation of motion for S k depends on the velocity and acceleration of joint Plc. Thus we have

ih =

linearized. In order to model any computational delay in the feedback loop we take velocity and acceleration

measurements of delay.

Xi

with a 0.2 second transmission

Wk

Wk = fk(Zk, Xk, Xk)

+

Gk(Zk, Xk, X,\;)Tk'

(1)

Now, by the assumption that there are as many control torques as links it follows that G Ic is invertible, and therefore, we can find a control law, Tic = O'.k ( Z k, :h , X k ), which would stabilize 8 k at its vertical equilibrium configuration.

Figure 3 ilustrate simulation results. Note that the control torque for the upper link is much smaller in comparision with the lower one, as it oUght to be in any realistic situation. This is in view of the fact that the control law only ask for the uper link to "take care" of its own dynamics.

Now let us fix ab and consider the stabilization problem for 8 k and 8 1e- 1 together. The crucial point of the whole idea presented here from the viewpoint of control theory is that the nature of Tic renders the dynamics of §k U 31e-1 block upper triangular. Thus, the stabilization problem reduces to stabilizing §k U §K-1 while asswning that Sk is constrained to be vertical at all times. Let us represent the state vector of SK-l

1.,--_ _

~-

_

___,

OD

by

ZIe-1

=

[Ok-I] where BIcWk-l

1

ur-------,

-"7.,,:----;'"

.'o):----;--,~.

.""'-

and Wk-l and the

,.

angular positions and angular velocities of its link. The new contol problem is of dimension 2d k - 1 , instead of 2(dK - 1 + die) as in the convensional sense. This explains the reduction of compexity offered by this method.

-30

-~.-~-,~.-~~"

-'''''::-.-~-,~.--:-:-~

III'M(_)

Now, let us suppose that the equation of motion of Sk-h while keeping Sk vertically constrained, is

lItT'-.(M«>II,

- - -... Fig. 3. BalanCing a Two-link pendulum

. Ok-l =Wk-l

Wk-l

4. CONCLUDING REMARKS

= h(Zk-l, Xk-l' Xk-t) + Gk(Zk-t, :Ek-l, xk-dTk-l.

Our aim in the paper was to point out that thinking along the lines of the theory of reference frames bring some new insight into how one may approach & for S k, a control law can be found which will stabilize«?))oftheform,rk -1 = O!k-l(Zk-l,h-l,Xk-l). the problem of reducing the complexity of controlling Continuing itteraively, at the ith step we have a multi-link mechanisms. Admittedly, it doesn't produce any new algorithms, but it indicates new guidedi dimensional stabilization problem, instead of a lines for cOming up with such algorithms. It is clear 2(a"j=idj) problem in the convensional sense. that this line of thinking will only produce a crude The control laws, aI, ... ,O'.k will now provide a set solution to a control problem comparable to the unof stabilizing control laws for the system. steady walk of an infant, and undoubtedly much more sophisticated processing takes place after generating an initial solution. Unfortunately, the theory of ref3.1 Simulation results erence frames doesn't answer the question of how one may go about understanding this post processing process. What natural structures are there to facilitate To illustrate this approach we include below some such processing, i.e. go down the ladder of reference simulation results for a two segment pendulum with frames rather than going up as in here, is a challenging controls at both joints. Each link is considered as research problem. a seperate subsystem. Simulations were carried out using the Simulink package. Here, S 1 consists of a link of two meters in length with a mass of one Kg concentrated at its center, and 8 2 is a link one meter in length with a cencentrated mass of one Kg at its end. Control laws employed cancel out all Xi and Xi. terms and make the subsystems feedback

5. REFERENCES [1]

M. Leyton, "Principles of Information Structure Common to Six Levels of the Human Cognitive Systems," Information Sciences, Vol. 38, 1-120, 1986.

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[2]

M. Leyton, "A Theory of Information Structures: 1. General Principles;' J. Mathematical

[3J

M. Leyton, "A Theory of Information Structures: n. Perceptual Organization," J. MathelIUltical Psychology, Vol. 30, 257-305, 1986. T. R. Kane and D. A. Levinson, Dynamics: Theory and Applications, McGraw-Hill, 1985.

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Psychology, Vol. 30, 103-160, 1986.

[4]

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