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Extra resources for Algorithmic Foundation of Robotics VII: Selected Contributions of the Seventh International Workshop on the Algorithmic Foundations of Robotics
We then pick a point x ∈ FW , as described above, and compute qx so that at q = (qx , qθ ), the robot’s feature point f lies at x and the robot has orientation qθ . For an articulated robot, the conﬁguration q speciﬁes its joints parameters q 1 , q 2 , . .. Suppose that the feature point f lies in the th link of the robot. To sample a conﬁguration, we again pick a point x ∈ FW and then ﬁnd the joint parameters q1 , q2 , . . , q that place f at x by solving the robot’s inverse kinematics (IK) equations.
2 K-Order Deformation Deﬁnition 1. A K-order deformation is a particular homotopic deformation such that each curve transforming a point of τ into a point of τ is an angle line of K segments. a c b τ τ τ τ τ τ Fig. 2. (a) General homotopic deformation. (b) ﬁrst order deformation: the deformation surface is a ruled surface. (c) Second order deformation: the deformation surface is obtained by concatenating two ruled surfaces. Therefore, a ﬁrst-order deformation surface describes a ruled surface and a Korder deformation is obtained by concatenation of K ruled surfaces.
This process starts with the shortest path found and stops when a visible deformation is possible (then the conﬁguration is rejected) or when all the possible paths have been tested (then the conﬁguration and the edges n1 −qv and n2 −qv are inserted). TestRedundancy(τ, n1 , n2 , G) 1 τ ← BestPath(n1 , n2 , G) 2 While τ = ∅ 3 If VisibDeformation(τ, τ ) = T rue 4 Return T rue 5 End If 6 τ ← BestPath(n1 , n2 , G) 7 End While 8 Return False Fig. 13. Visibility deformation test between a path τ and a roadmap path The VisibDeformation function (line 3 of algorithm 13) tests whether two paths τ and τ can be visibility deformed one into the other.
Algorithmic Foundation of Robotics VII: Selected Contributions of the Seventh International Workshop on the Algorithmic Foundations of Robotics by Erion Plaku, Lydia E. Kavraki (auth.), Srinivas Akella, Nancy M. Amato, Wesley H. Huang, Bud Mishra (eds.)