to be compu- : u is a vector in !Rm= !Rn-k_ Recipro- be rewritten into k m in- of the form Fi ( s, s) of the form Gi(q, q, ij) that will be useful later in the paper: to time, and u E The set Nonholonomic car-like robot. Consider a car A mod- S and are the robot's state space and control space, x, y) be the posi- state at time t and a control tion of the midpoint R between A's rear wheels and of !Rn and !Rm. both nonholonomic and the same form as Equation (1). This refor- to defining the state of A to be of the form Fi(q,q) = = ... ,k, and choosing the vector (v, ¢), where q the and v and the car's and

doi:10.1201/9781439864135-36

Chapter

u is a vector in !Rm= !Rn-k_ Recipro- be rewritten into k m in- of the form Fi ( s, s) of the form Gi(q, q, ij) that will be useful later in the paper: to time, and u E The set Nonholonomic car-like robot. Consider a car A mod- S and are the robot's state space and control space, x, y) be the posi- state at time t and a control tion of the midpoint R between A's rear wheels and of !Rn and !Rm. both nonholonomic and the same form as Equation (1). This refor- to defining the state of A to be of the form Fi(q,q) = = ... ,k, and choosing the vector (v, ¢), where q the and v and the car's and

ABSTRACT

Point-mass robot For a point-mass robot A moving in a plane, we typically want to control the force applied to A. So, we define A ’s state to be s = (x ,y ,x ,y ) , where (x, y) is A ’s position. The equations of mo tion are:

where m and (ux,uy) denote A ’s mass and the force applied to it. Bounds on the magnitudes of (x, y) and (ux ,uy) define S and Ω as subsets of R4 and R2, re spectively. Planning query Let S T denote the statextime space S x [0, +oo). Obstacles in the robot’s workspace are mapped into this space as forbidden regions. The free space T C S T is the set of all collision-free points (s, t). A trajectory s : [a, 6] -» S is admissible if, for all t G [a, 6], s(t) is an admissible state and (s(t), t) is collisionfree.