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Chapter 4 the Configuration Space Chapter 4 The Con¯guration Space Steven M. LaValle University of Illinois Copyright 2004 110 S. M. LaValle: Planning Algorithms between 0 and 1, except 0 and 1. However, for either endpoint, an in¯nite sequence may be de¯ned that converges to it. For example, the sequence 1=2, 1=4, : : :, 1=2i, converges to 0 as i tends to in¯nity. This means that we can get within any small, positive distance from 0 or 1, but we cannot stand exactly on the boundary of the interval. For a closed interval, such as [0; 1], these boundary points are included. The notion of an open set lies at the heart of topology. The open set de¯nition Chapter 4 that will appear here is a substantial generalization of the concept of an open interval. The concept will apply to a very general collection of subsets of some larger space. It is general enough to easily include any kind of con¯guration space The Con¯guration Space that may be encountered in planning. A set X is called a topological space if there is a collection of subsets of X called open sets such that the following axioms hold: Chapter 3 only covered how to model and transform a collection of bodies; how- 1. The union of a countable number of open sets is an open set. ever, for the purposes of planning it is important to de¯ne a whole state space. 2. The intersection of a ¯nite number of open set is an open set. The state space for motion planning is a set of possible transformations that could be applied to the robot. This will be referred to as the con¯guration space, based 3. Both X and are open sets. on the seminal work of Lozano-P¶erez [17, 15, 16], who introduced this notion in the ; context of planning. The motion planning literature was further uni¯ed around Note that in the ¯rst axiom, the union of an in¯nite number of open sets may be this concept by Latombe's book [14]. Once the con¯guration space is clearly taken, and the result must remain an open set. This will not necessarily be true understood, many motion planning problems that appear di®erent in terms of for closed sets. geometry and kinematics can be solved by the same planning algorithms. This For the special case of X = R, the open sets include open intervals, as expected. level of abstraction is therefore very important. Note that many sets that are not intervals are being included because taking This chapter provides important foundational material that will be very useful unions and intersections of open intervals generates many other open sets. For in Chapters 5 to 8 and other places where planning over continuous state spaces example, the set occurs. Many of concepts introduced in this chapter come directly from mathe- 1 1 2 ; ; (4.1) matics, particularly from topology. Therefore, Section 4.1 gives a basic overview 3i 3i i=1 of topological concepts. Section 4.2 uses the concepts from Chapter 3 to de¯ne the [ µ ¶ con¯guration. After reading this, you should be able to precisely characterize the which is an in¯nite union of intervals, is open. con¯guration space and understand its structure. In Section 4.3, obstacles in the world are transformed into obstacles in the con¯guration space, but it is important Closed sets Open sets appear directly in the de¯nition of a topological space. to understand that this transformation may not be explicitly constructed. The It next seems that closed sets are needed. Suppose X is a topological space. A implicit representation of the state space is a recurring theme throughout plan- subset C X is de¯ned to be a closed set if and only if X C is an open set. Thus, ½ n ning. Section 4.4 covers the important case of kinematic chains that have loops, the complement of any open set is closed, and vice versa. Any closed interval, such which was mentioned in Section 3.4. This case is so di±cult that even the space as [0; 1] is a closed set because its complement ( ; 0) (1; ) is an open set. ¡1 [ 1 of transformations usually cannot explicitly characterized (i.e., parameterized). For another example, (0; 1) is an open set; therefore, R (0; 1) = ( ; 0] [1; ) is a closed set. The use of \(" may seem wrong in then last expression,¡1 [but 1\[" cannot be used because and do not belong to R. Thus, the use of \(" is 4.1 Basic Topological Concepts just a notational quirk. ¡1 1 Here is a question to ponder: are all subsets of X either closed or open? 4.1.1 Topological Spaces Although it appears that open sets and closed sets are opposites in some sense, the answer is NO. For X = R, the interval [0; 2¼) is neither open nor closed (the Recall from basic mathematics, the concepts of open and closed intervals in the interval [2¼; ) is closed, and ( ; 0) is open). Note that for any topological set of real numbers R. An open interval, such as (0; 1) includes all real numbers space, X and1 are both open and¡1closed! ; 109 4.1. BASIC TOPOLOGICAL CONCEPTS 111 112 S. M. LaValle: Planning Algorithms of the name \limit point" comes from the fact that such a point might be the limit U of an in¯nite sequence of points. For example, 0 is the limit point of the sequence generated by 1=2i for each i N, the natural numbers. There are several convenien2 t consequences of the de¯nitions. A closed set, C, contains the limit point of any sequence that is a subset of C. This implies that A1 it contains all of its boundary points. The closure, cl, always results in a closed set because it adds all of the boundary points to the set. On the other hand, an x 2 open set contains none of its boundary points. These interpretations will come in x1 handy when considering obstacles in the con¯guration space for motion planning. x3 A2 Some examples The de¯nition of a topological space is so general that an incredible variety of topological spaces can be constructed. Figure 4.1: An illustration of the boundary de¯nition. Suppose X = R2, and U Example 4.1.1 (X = Rn) We should expect that Rn for any integer n is a topo- is a subset as shown. Three kinds of points appear: 1) x1 is a boundary point, 2) logical space. This requires characterizing the open sets. An open ball, B(x; ½) is x2 is an interior point, and 3) x3 is an exterior point. Both x1 and x2 are limit the set of points in the interior of a sphere of radius ½, centered at x. Thus, points. n B(x; ½) = x0 R x0 x < ½ ; (4.2) f 2 j k ¡ k g Special points From the de¯nitions and examples so far, it should seem that in which denotes the Euclidean norm (or magnitude) of x. The open balls are k¢k n points on the \edge" or \border" of a set are important. There are several terms open sets in R . Furthermore, all other open sets can be expressed as a countable 1 that capture where points are relative to the border. Let X be a topological space, union of open balls. For the case of R, note that this degenerates to representing and let U be any subset of X, and let x be any point in X. The following terms all open sets as a union of intervals, which we have done so far. n capture the position of point x relative to U (see Figure 4.1): Even though it is possible to express open sets of R as unions of balls, we prefer to use other representations, with the understanding that one could revert If there exists an open set, O, such that x O and O U, then x is called to open balls if necessary. The primitives of Section 3.1 can be used to generate ² an interior point of U. The set of all interior2points in Uis called the interior many interesting open and closed sets. For example, any algebraic primitive ex- of U, and is denoted by int(U). pressed in the form H = x Rn f(x) 0 , in which x Rn, produces a closed set. Taking ¯nite unionsfand2 intersectionsj · ofg these primitiv2 es will produce more If there exists an open set, O, such that x O and O X U, then x is ² 2 n closed sets. Therefore, all of the models from Sections 3.1.1 and 3.1.2 produce called an exterior point with respect to U. an obstacle region, , that is a closed set. As mentioned in Section 3.1.2, sets constructed only fromO primitives that use the < relation are open. ¥ If x is neither an interior point nor an exterior point, then it is called a ² boundary point of U. The set of all boundary points in X is called the boundary of U, and is denoted by @U. Example 4.1.2 (Subspace topology) A new topological space can easily be constructed from a subset of a topological space. This will be very useful in the All points in x X must be one of the three above; however, another term coming sections. Let X be a topological space, and let Y X be a subset.
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