Robotics and Autonomous Systems 9 (1992) 1 - 3 I Elsevier

Guest editorial

Towards formal theories of autonomous goal-directed behavior

Leo Dorst

After many years of focussing on various aspects of the pure motion planning problem (well documented in Latombe's book [3]), the time has come for the path planning community to widen the scope of the planning of goal-directed behavior, and to include the sensory actions. This will connect the work from the motion planning area to the theoretical work in sens- ing of, for instance, [2].

This special issue brings together some ap- proaches that have been taken in the formal study of goal-directed behavior for autonomous systems. The papers reflect the incompleteness and tentativeness of the results sofar, but also make plausible'that more definite results seem to be within reach. As such, they should be inspiring to many readers. An exciting aspect of the papers is that, though they treat vastly different aspects of autonomous behavior, they are written in a common language: geometry.

Fig. 1 is helpful in visualizing the locations and relationships of the different contributions within the

Leo Dorst received his B.Sc., M.Sc. (in '82) and Ph.D. (in '86) from Delft University of Technology in The Netherlands. His research was in the field of image analysis theory. From '86 to '92 he was a Senior Member of Research Staff at Philips Laborato- ries, where he worked on general path planning algorithms for autonomous system. His method of wave-propa- gation based path planning has led to several patents. Since April '92 he is an associate professor at the Univer-

sity of Amsterdam, The Netherlands. Current research inter- est are: the translation and discretization of the continuous structures of the world into discrete (algebraic) mathematical structures, and the (geometrical) unification of sensory plan- ning and motion planning in the context of the task to be performed.

total field of autonomous behavior. Let us divide the field according to four aspects: motion, sensing, knowledge, and task, represented as the four corners of a tetrahedron. - T h e paper 'Task encoding: Toward a scientific

paradigm for robot planning and control' by Koditschek focusses on the motion tasks within an autonomous system. Coming from the geometric representation of motion in physics and control theory, the author investigates how motion tasks could be encoded in an extension of that formal- ism. Physical motion is not goal-directed, and the standard means to make it so by control theory involve the application of external forces. This was the original inspiration for the potential field tech- niques of path planning (see [3]). Navigation func- tions are an extension of this idea with the intention to prevent the occurrence of local minima. Koditschek's paper investigates the coding of robot tasks into spaces in which such navigation func- tions can be used as a paradigm for task plans which leads to directly executable task representa- tions. In Fig. 1, I have placed his paper in the 'motion' corner, extending towards the task. Sens- ing is not really an issue; Koditschek assumes a world that is fully known.

- The paper 'Constructive recognizability for task-di- rected robot programming' by Donald and Jen- nings deals with the sensory aspects of robot tasks. The authors take the point of view that traditional descriptions of world geometry are insufficient for autonomous behavior; at the very least these de- scriptions need to be extended with a framework that enables error detection and recovery (such as

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2 Guest Editorial

TASK STRUCTURE

EDGE

MO

SENSING

Fig. 1. The relationship of the papers in this special issue, with respect to the aspects of motion, sensing, knowledge, and task. Darker areas have reached a higher degree of formality.

the uncertain geometry of [2]). Specializing this to sensing, they investigate how to arrive at a proper 'calculus of the observable'. Such a calculus should enable us to translate a high-level description of a task and its goals automatically into appropriate low-level physical sensory and motive actions. The authors show how the collection of sensory data on the wodd can be represented as a bundle over the world, and that the collection of such bundles has a lattice structure. The most detailed recognizable wodd isthe upper bound of this lattice; the connec- tion of such bundles for different configurations gives a ~ graph which is the basis for 'knowledge navigation'. The underlying spatial part of the rep- resentation is very reminiscent of the geometrical representation of motion in Koditschek's paper. In Fig. 1, the paper is represented in the 'sensing' corner, extending equally towards the motive, knowledge, and task aspects.

The authors investigate how concepts such as a 'wall' are retrievable as ~ sections in the bundle structure. Their results in this area still are qualitative rather than quarO, ative but they give an interesting ~ representation of issues that B e ~ i n , in his paper, treats more algebraically (see below).

- The paper 'The geometry ofknowledgeacquisition by sensing and motion' by Dorst and Wu sets up a geometrioal knowledge space, in which the change of knowledge about the state of the world as caused by the execution of motion or the collection of observations becomes representable as a path. This can be seen as an alternative repreaentation to Donald and Jennings', now also incorporating motion. The mostly linear nature of the knowledge updating equations suggest an implementation in a standard linear algebra package, and use of the linear algebraic techniques of e'~gerwec~m analysis to decompose a problem into subproblems. In Fig. I, I have represented it in the ~ c ~ m m r , towards sensing, motion and task.

- The paper 'Reformu/a~ path ~ by task.preserving ~ ' by ~ i n em- phasizes the structure of actions, and b i ~ s to

monitor the ~ ~ of the ~ ! ~ ~ - ite s ~ ~ to an ~ ~ t h e ~ o b ' lem; h ~ r that ~ ~ ~ p t l l m ~ the original task. The theory is i ~ I ~ a sim~i-

Guest Editorial 3

fled problem involving a discrete approximation of a non-holonomic robot in the presence of obsta- cles. Though at first sight the techniques that are used are very algebraic (involving such constructs as Green's relations), they are actually historically closely related to quantitative geometrical tech- niques and representations as studied in, for in- stance, homology theory. As a consequence, the theory should be fairly easily connectable to the work by Koditschek, Donald and Jennings, and Dorst and Wu - which motivates its location in Fig. 1. The abstract algebraic representations also con- nect naturally to task representations used in theo- retical Artifical Intelligence. The paper ' The submetric formalism for task repre- sentation' by Kent and Rosar presents a slightly unusual geometric formalism for task representa- tion which can incorporate various constraints: laws of physics, simultaneous motion, arbitrary 'rules of the game'. The representation for pure motion is closely related to the standard tangent-space rep- resentation also described by Koditschek, but the inclusion of the other constraints provides a chal- lenge to the more classical representations that seem better met by the submetric formalism. The paper is offered here as a reminder that the exist- ing language of geometry may need extension to deal sensibly with issues arising in the study of goal-directed behavior. In Fig. 1., the paper is represented by a region in the motion-sensing-task plane; the submetric formalism does not deal with the acquisition of knowledge.

The papers in this issue represent the current state of

the art in the development of a practical formal theory of autonomous goal-directed systems. More qualita- tive results on the structure of such systems have been reached long ago by Arbib and Manes [1], who used the formulations of category theory to prove deep theorems on, for instance, the formal duality of reachability and observability. Category theory pro- vides extreme clarity of formulation, but at present at the expense of definiteness. I believe that a structural connection of the geometric approaches presented in this issue, inspired by the qualitative abstractions of category theory, may eventually lead to a practical theory of goal-directed autonomous systems that is as quantitative as it could possibly be on each of its levels of description, and in which these levels are connected by well understood abstraction and spe- cialization operators.

Though the exact representations of the concepts may differ from paper to paper, the authors all appear to be directed towards that same goal. If efforts such as these can be brought together, we may hope to have a paradigm of goal-directed behavior before the end of this millenium.

References

[1] M. Arbib and E. Manes. Machines in a category: An expository introduction, SIAM Reciew 16 (2) (1974).

[2] H.F. Durrant-Whyte, buegration, Coordination and ('on- trol of Multi-Sensor Robot Systems (Kluwer, Dordrecht, 1988).

[3] J.-C. Latombe, Robot Motion Plam#ng (Kluwer, I)or- drecht, 1991).