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By Joachim Stolze, Dieter Suter

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Let us represent a configuration of A by (x, y, (J), where x and yare the coordinates, in the frame Fw, of the midpoint R between the two rear wheels and (J E [0, 27r) is the angle between the x-axis of Fw and the main axis of A. At any instant during a motion, assuming no slipping, 28 Chapter 1: Introduction and Overview r 1 1 y 1 1 1 1 1 1 1 1 1 , / 1 1 1 1 \ ,~ "'I , 1 , 1 'en 0", Figure 12. This car-like robot A is modeled as a rectangle. Any configuration of A is represented by three parameters (x, y, B) E R2 X [0,2'11"), with modulo 27r arithmetic on B.

2 Slicing in Configuration Space Another way to reduce the dimension of configuration space is to consider a cross-section of dimension m - r through C and to plan a free path in the space defined by this cross-section. This corresponds to forbidding the motion of A along r axes. Examples: - Let A be a free-flying three-dimensional object. In order to plan a free path from the initial configuration qinit = (1init,0init) to the goal configuration qgoa/ = (Tgoa/,0 goa /), one may consider the intermediate configuration qinter = (Tgoa/, 0init) at which A has the same orientation 0init as at the initial configuration and the same position Tgoa/ as at the goal configuration.

The first extension (and possibly the second) requires time to be explicitly considered in the motion plans. The second and third extensions yield configuration spaces of arbitrarily large dimensions. 3 Moving Obstacle. In the presence of moving obstacles, the motion planning problem can no longer be solved by merely constructing a geometric path. A continuous function of time specifying the robot's configuration at each instant must be generated instead. This can be done by adding one dimension representing time to the robot's configuration space.

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