Abstract
In this paper, we study some properties of several local planners for nonholonomic dynamical systems to achieve asymptotic global optimality through the RRT*. More specifically, we study the conditions that a local steering method must have to produce global optimal trajectories in an environment with obstacles. The main properties we analyse in the steering methods are the following: (1) Whether or not the steering method produces local optimal motion primitives (optimal letters). (2) Whether or not the steering method concatenates the local optimal primitives in such a way that the resulting concatenation is also optimal (optimal words). (3) Whether or not the steering method produces trajectories that respect the topological property. Experimentally, it is studied how those properties affect the speed of convergence to the globally optimal solution, moreover, their sufficiency and necessity is also validated, all making use of the problem of finding the time-optimal trajectories for a differential drive robot in the presence of obstacles. We also discard conditions that show not to be necessary and we give some insight on the necessary and sufficient conditions for the RRT* to asymptotically converge to optimal trajectories, which is indeed the sough research target.
The authors would like to acknowledge the financial support of Intel Corporation.
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- 1.
A system is small-space locally controllable at \(z \in X\), if for any neighbourhood \(\varOmega \) of z, there exists a neighbourhood \(A_\varOmega (x)\) whose points are all accessible by the system without departing from \(\varOmega \). The system is small-space locally controllable if it is SSLC at any \(z \in X\).
- 2.
The related cost is computed as \(t=s(t) + b\sigma (t)\), where s(t) is the rectified path length in \(\mathbb {R}^2\), the plane of robot position, and \(\sigma (t)\) is the rectified arc length in \(S^1\), the circle of robot orientations, see [1] for details.
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Becerra, I., Yervilla-Herrera, H., Murrieta-Cid, R. (2020). An Experimental Analysis on the Necessary and Sufficient Conditions for the RRT* Applied to Dynamical Systems. In: Morales, M., Tapia, L., Sánchez-Ante, G., Hutchinson, S. (eds) Algorithmic Foundations of Robotics XIII. WAFR 2018. Springer Proceedings in Advanced Robotics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-44051-0_48
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