Linguistic Geometry for autonomous navigation

To discover the inner properties of human expert heuristics, which were successful in a certain class of complex control systems, we develop a formal theory, the Linguistic Geometry. This paper reports two examples of application of Linguistic Geometry to autonomous navigation of aerospace vehicles that demonstrate dramatic search reduction. The problem of aerospace combat management requires new artificial intelligence techniques of timeconstrained decision-making under dynamically changing information patterns. There are many realworld problems where human expert skills in reasoning about complex systems are incomparably higher than the level of modern computing systems. At the same time there are even more areas where advances are required but human problem-solving skills can not be directly applied. Unfortunately, problems of planning and automatic control of autonomous aerospace agents such as aircrafts, space vehicles, stations and robots with cooperative and opposing interests are the problems of this type. Reasoning about such complex systems should be done automatically, in a timely manner, and often in a real time. However, there are no highly-skilled human experts in these fields ready to substitute for robots (on a virtual model) or transfer their knowledge to them. There is no grand-master in robot control, although, of course, the knowledge of existing experts in this field should not be neglected it is even more valuable. It is very important to study human expert reasoning about similar complex systems in the areas where the results are successful, in order to discover the keys to success, and then apply and adopt these keys to the new, as yet, unsolved problems. There have been many attempts to find the optimal operation for real-world complex systems. One of the basic ideas is to decrease the dimension of the real-world problem following the approach of a human expert in a certain field, by breaking the problem into smaller subproblems (Simon, 1980; Albus, 199 1 ; Mesarovich et al., 1970; Botvinnik, 1984). These ideas have been implemented for many problems with varying degrees of success. Implementations based on the formal theories of linear and nonlinear planning (Fikes and Nilsson, 1971; McCarthy, 1980; Nilsson, 1980; Sacerdoti, 1975; Stefic, 1981; Chapman, 1987; McAllester and Rosenblitt, 1991) meet hard efficiency problems. An efficient planner requires an intensive use of heuristic knowledge. On the other hand, a pure heuristic implementation is unique. There is no general constructive approach for such implementations. In the 1960's a formal syntactic approach to the investigation of properties of natural language resulted in the fast development of a theory of formal languages by (Chomsky, 1963; Ginsburg, 1966), and others (Knuth, 1968; Rozenkrantz, 1969). This development provided an interesting opportunity for dissemination of this approach to different areas. In particular, there came an idea of analogous linguistic representation of images. This idea was successfully developed into syntactic methods of pattern recognition by (Fu, 1982; Narasimhan, 1966), and (Pavlidis, 1977), and picture description languages by (Shaw, 1969; Feder, 197 1 ; Rosenfeld, 1979). The power of a linguistic approach might be explained, in particular, by the recursive nature and expressiveness of language generating rules, i.e., formal grammars. Searching for the adequate mathematical tools formalizing human heuristics of dynamic hierarchy, we have transformed the idea of linguistic representation of complex real-world and artificial images into the idea of similar representation of complex hierarchical systems (Stilman, 1985, 1993a). This approach was called a Linguistic Geometry. The appropriate languages possess more sophisticated attributes than languages usually used for pattern description. They describe mathematically all of the essential syntactic and semantic features of the system and search, and are generated by certain "controlled" grammars (Stilman, 1993a). The origin of such languages can be traced back to the origin of SNOBOL-4 programming language and the research on programmed attribute grammars and languages by (Knuth, 1968; Rozenkrantz, 1969; Volchenkov, 1979). A mathematical environment (a "glue") for the formal implementation of this approach was developed following the theories of formal problem solving and planning by (Fikes and Nilsson, 1971), (Nilsson, 1980), (Sacerdoti, 1975), (McCarthy and Hayes, 1969), and others, based on first order predicate calculus. In this paper we investigate heuristics extracted in the form of hierarchical networks of planning paths of autonomous agents. Employing Linguistic Geometry tools the dynamic hierarchy of networks is represented as a hierarchy of formal attribute languages. The main ideas of this methodology are shown on the pilot examples of solution of 2D and 3D optimization problems for the autonomous robotic vehicles in aerospace environment. These examples include the actual generation of the hierarchy of languages, with some details of trajectory generation. They demonstrate the dramatic reduction of search in comparison with conventional search algorithms. Copyright O American Institute of Aeronautics and Astronautics, Inc., 1995. All rights reserved. American Institute of Aeronautics and Astronautics 1. COMPLEX SYSTEM A Complex System is the following eight-tuple: , where X={xi) is a finite set of points; P={pi) is a finite set of elements; P is a union of two non-intersecting subsets P i and P2; Rp(x, y) is a set of binary relations of reachability in X (x and y are from X, p from P); ON(p)=x, where ON is a partial function of placement from P into X; v is a function on P with positive integer values; it describes the values of elements. The Complex System searches the state space, which should have initial and target states; Si and St are the descriptions of the initial and target states in the language of the first order predicate calculus, which matche: 4th each relation a certain Well-Formed Formula (WFF). Thus, each state from Si or St is described by a certain set of WFF of the form { O N ( p j ) = x k } ; T R is a set of operators, TRANSITION(p, x, y), of transition of the System from one state to another one. These operators describe the transition in terms of two lists of WFF (to be removed and added to the description of the state), and of WFF of applicability of the transition. Here, Remove list: ON(p)=x, ON(q)=y; Add list: ON(p)=y; Applicability list: (ON(p)=x)"R,(x, y ), r where p belongs to P i and q belongs to P2 or vice versa. The transitions are carried out with participation of elements p from P I and P2. According to definition of the set P, the elements of the System are divided into two subsets P1 and P2. They might be considered as units moving along the reachable points. Element p can move from point x to point y if these points are reachable, i.e., Rp(x, y) holds. The current location of each element is described by the equation ON(p)=x. Thus, the description of each state of the System {ON(pj)=xk) is the set of descriptions of the locations of the elements. The operator TRANSITION(p, x, y) describes the change of the state of the System caused by the move of the element p from point x to point y. The element q from point y must be withdrawn (eliminated) if p and q belong to the different subsets P i and P2. The problem of the optimal operation of the System is considered as a search for the optimal sequence of transitions leading from one of the initial states of Si to a target state S of St. It is easy to show formally that robotic system can be considered as the Complex System (see below). Many different technical and human society systems (including military battlefield systems, systems of economic competition, positional games) which can be represented as twin-sets of movable units (of two or more opposing sides) and their locations, thus, can be considered as Complex Systems. With such a problem statement for the search of the optimal sequence of transitions leading to the target state, we could use formal methods like those in the problem-solving system STRIPS (Fikes, Nilsson, 1971) nonlinear planner NOAH (Sacerdoti, 1975), or in subsequent planning systems. However, the search would have to be made in a space of a huge dimension (for nontrivial examples). Thus, in practice no solution would be obtained. We devote ourselves to the search for an approximate solution of a reformulated problem. 2. MEASUREMENT O F DISTANCES To create and study a hierarchy of dynamic subsystems we have to investigate geometrical properties of the Complex System. A map of the set X relative to the point x and element p for the Complex System is the mapping: MAP,,p: X -> Z+, (where x is from X, p is from P), which is constructed as follows. We consider a family of reachability areas from point x, i.e., a finite set of the following nonempty subsets of X { M ~ , , ~ )