Problem statement

Given a C-space Q, planning a trajectory is finding a curve

where [0,1] is the parameterization interval and s is the arc-length parameter of C. If xstart and xgoal are the start and the goal configurations respectively, then C(0) = xstart and C(1) =

An optimal trajectory is a curve C that minimizes a set of internal and external constraints (time, fuel consumption or danger for instance). It is assumed in this chapter that the complete set of constraints is described in a cost function t:

2.3 Metric space

In this chapter the metric space Q we refer to is the usual C-space equipped with the metric p defined as:

where C is a trajectory between two configurations x1 and x2, and t is the cost function. This metric can be seen as the "cost-to-go" for a specific robot to reach x2 from x1. At a configuration x, t(x) can be interpreted as the cost of one step from x to its neighbours. If a C-obstacle in some region S is impenetrable, then t(S) will be infinite. The function t is supposed to be strictly positive for an obvious physical reason: t(x) = 0 would mean that free transportation from some configuration x is possible.

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This first volume will guide you through the basics of Photoshop. Well start at the beginning and slowly be working our way through to the more advanced stuff but dont worry its all aimed at the total newbie.

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