CMSC 425: Lecture 16 Motion Planning: Basic Concepts

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1 : Lecture 16 Motion lanning: Basic Concets eading: Today s material comes from various sources, including AI Game rogramming Wisdom 2 by S. abin and lanning Algorithms by S. M. LaValle (Chats. 4 and 5). eca: reviously, we discussed navigation meshes as a techniue for lanning the motion of walking agents in games. In this and future lectures, we will delve deeer into the theory and ractice of motion lanning as it relates to game rogramming. Configuration Saces: To begin, let us consider the roblem of lanning the motion of a single agent among a collection of obstacles. Since the techniues that we will be discussing originated in the field of robotics, henceforth we will usually refer to a moving agent as a robot. The environment in which the agent oerates is called its worksace, which consists of a collection of geometric objects, called obstacles, which the robot must avoid. We will assume that the worksace is static, that is, the obstacles do not move. 1 We also assume that a comlete geometric descrition of the worksace is available to us. 2 For our uroses, a robot will be modeled by two main elements. The first element is the robot s geometric model, say with resect to its reference ose (e.g., ositioned at the origin). The second is its configuration, by which we mean a finite seuence of numeric arameters that fully secifies the osition of the robot. Combined, these two elements fully define the robot s exact shae and osition in sace. For examle, suose that the robot is a 2-dimensional olygon that can translate and rotate in the lane (shown as a triangle in Fig. 1). Its geometric reresentation might be given as a seuence of vertices, relative to its reference osition. Let us assume that this reference ose overlas the origin. We refer to the location of the origin as the robot s reference oint. The robot s configuration may be described by its translation, which we can take to be the (x, y) coordinates of its reference oint after translation and an angle θ that reresents the counterclockwise angle of rotation about its reference oint (see Fig. 1). Thus, the configuration is given by a trile (x, y, θ). We define the sace of all valid configurations to be the robot s configuration sace. For any oint = (x, y, θ) in this sace, we define () to be the corresonding lacement of the robot in the worksace. The dimension of the configuration sace is sometimes referred to as the robots number of degrees of freedom (DOF). In 3-dimensional sace, a similarly rigid object can be described by six arameters, the (x, y, z)-coordinates of the object s reference oint, and the three Euler angles (θ, φ, ψ) that 1 The assumtion of a static worksace is not really reasonable for most games, since agents move and structures may change. A common techniue for dealing with dynamic environments is to searate the static objects from the dynamic ones, lan motion with resect to the static objects, and then adjust the lan incrementally to deal with the dynamic ones. 2 The assumtion of a known worksace is reasonable in comuter games. Note that this is not the case in robotics, where the world surrounding the robot is either unknown or is known only aroximately based on the robots limited sensor measurements. Lecture 16 1 Sring 2018

2 (1, 3, 0) (4, 2, 30, 60 ) 60 (0, 0, 0) eference ose (6, 2, 45 ) 45 (0, 0, 0, 0) eference ose 30 Fig. 1: Configurations of: translating and rotating robot and a translating and rotating robot with a revolute joints. define its orientation in sace.) 3 A more comlex examle would be an articulated arm consisting of a set of links, connected to one another by a set of revolute joints. The configuration of such a robot would consist of a vector of joint angles (see Fig. 1). The geometric descrition would robably consist of a geometric reresentation of the links. Given a seuence of joint angles, the exact shae of the robot could be derived by combining this configuration information with its geometric descrition. Free Sace: Because of limitations on the robot s hysical structure and the obstacles, not every oint in configuration sace corresonds to a legal lacement of the robot. Some configurations may be illegal because: The joint angle is outside the joint s oerating range. (E.g., you can bend your knee backwards, but not forwards... ouch!) The lacement associated with this configuration intersects some obstacle (see Fig. 2). Such illegal configurations are called a forbidden configurations. Given a robot and worksace S, the set of all forbidden configurations is denoted C forb (, S), and all other lacements are called free configurations, and the set of these configurations is denoted C free (, S), or free sace. These two sets artition configuration sace into two distinct regions (see Fig. 2). C-Obstacles and aths in Configuration Sace: Motion lanning is the following roblem: Given a worksace S, a robot, and initial and final configurations s, t C free (, S), determine whether it is ossible to move the robot from one configuration by a ath (s) (t)consisting entirely of free configurations (see Fig. 3). Based on the definition of configuration sace, it is easy to see that the motion lanning roblem reduces to the roblem of determining whether there is a ath from s to t in con- 3 A uaternion might be a more reasonable reresentation of the robot s angular orientation in sace. You might rotest that the use of a uaternion will involve four arameters rather than three. But remember that the uaterions used for reresenting rotations are unit uaternions, meaning that once three of the arameters are given, the fourth one is fixed. Lecture 16 2 Sring 2018

3 Free Worksace Forbidden Free Configuration Sace (Loosely interreted) Forbidden C forb (, S) C free (, S) Fig. 2: Worksace showing free and forbidden configurations and a ossible configuration sace. figuration sace (as oosed to the robot s worksace) that lies entirely within the robot s free configuration subsace (see Fig. 3). Thus, we have reduced the task of lanning the motion of a robot in its worksace to the roblem of finding a ath for a single oint through free configuration sace. Work sace Configuration sace Fig. 3: Motion lanning: worksace with obstacles and configuration sace and C-obstacles. Configuration Obstacles and Minkowski Sums: Since high-dimensional configuration saces are difficult to visualize, let s consider the simle case of translating a convex olygonal robot in the lane amidst a collection of olygonal obstacles. In this cased both the worksace and configuration sace are two-dimensional. We claim that, for each obstacle in the worksace, there is a corresonding configuration obstacle (or C-obstacle) that corresonds to it in the sense that if () does not intersect the obstacle in the worksace, then does not intersect the corresonding C-obstacle. For simlicity, let us assume that the reference oint for our robot is at the origin. Let () denote the translate of the robot so that its reference oint lies at oint. Given a olygonal obstacle, the corresonding C-obstacle is formally defined to the set of lacements of Lecture 16 3 Sring 2018

4 that intersect, that is C( ) = { : () }. One way to visualize C( ) is to imagine scraing along the boundary of and seeing the region traced out by s reference oint (see Fig. 4). C( ) + Q Q Fig. 4: Minkowski sum of two olygons. Given and, how do we comute the configuration obstacle C( )? To do this, we first introduce the notion of a Minkowski sum. Let us think of oints in the lane as vectors. Given any two sets and Q in the lane, define their Minkowski sum to be the set of all airwise sums of oints taken from each set (see Fig. 4), that is, Q = { + :, Q}. Also, define S = { : S}. (In in the lane S is just the 360 rotation of S about the origin, but this does not hold in higher dimensions.) We introduce the shorthand notation to denote {}. Observe that the translate of by vector is () =. The relevance of Minkowski sums to C-obstacles is given in the following claim. Claim: Given a translating robot and an obstacle, C( ) = ( ) (see Fig. 5). roof: Observe that C( ) iff () intersects, which is true iff there exist r and such that = r+ (see Fig. 5), which is true iff there exist r and such that = + ( r) (see Fig. 5), which is euivalent to saying that ( ). Therefore, C( ) iff ( ), which means that C( ) = ( ), as desired. It is an easy matter to comute in linear time (by simly negating all of its vertices) the roblem of comuting the C-obstacle C( ) reduces to the roblem of comuting a Minkowski sum of two convex olygons. We ll show next that this can be done in O(m + n) time, where m is the number of vertices in and n is the number of vertices in. Note that the above roof made no use of the convexity of or. It works for any shaes and in any dimension. However, comutation of the Minkowski sums is most efficient for convex olygons. We will not resent the algorithm formally here, but here is an intuitive exlanation. First, comute the vectors associated with the edges of each olygon and merge them into a single list, sorted by angular order. Then link them together end-to-end (see Fig. 6). (It is not immediately obvious that this works, but it can be roved to be correct.) Lecture 16 4 Sring 2018

5 C( ) ( ) r r Fig. 5: Configuration obstacles and Minkowski sums. u 1 u 2 u u 1 3 v 1 u 2 v 4 v 3 + u 3 Q v 4 v 3 v 1 v 2 Q v2 Fig. 6: Comuting the Minkowski sum of two convex olygons. C-Obstacles for otating obots: When rotation is involved, this scraing rocess must consider not only translation, but all rotations that cause the robot s boundary to touch the obstacle s boundary. (One way to visualize this is to fix the value of θ, rotate the robot by this angle, and then comute the translational C-obstacle with the robot rotated at this angle. Then, stack the resulting C-obstacles on to of one another, as θ varies through one comlete revolution. The resulting twisted column is the C-obstacle in 3-dimensional sace.) Note that because the configuration sace encodes not only translation, but the joint angles as well. Thus, a ath in configuration sace generally characterizes both the translation and the individual joint rotations. (This is insanely hard to illustrate, so I hoe you can visualize this on your own!) When dealing with olyhedral robots and olyhedral obstacles models under translation, the C-obstacles are all olyhedra as well. However, when revolute joints are involved, the boundaries of the C-obstacles are curved surfaces, which reuire more effort to rocess than simly olyhedral models. Comlex configuration saces are not tyically used in games, due to the comlexity of rocessing them. Game designers often resort to more ad hoc tricks to avoid this comlexity, and the exense of accuracy. Lecture 16 5 Sring 2018