Abstract
Many mobile robots such as wheeled robots, boats, or plane are described by nonholonomic differential equations. As a consequence, they have to satisfy some differential constraints such as having a radius of curvature for their trajectory lower than a known value. For this type of robots, it is difficult to prove some properties such as the avoidance of collisions with some moving obstacles. This is even more difficult when the initial condition is not known exactly or when some uncertainties occur. This paper proposes a method to compute an enclosure (a tube) for the trajectory of the robot in situations where a guaranteed interval integration cannot provide any acceptable enclosures. All properties that are satisfied by the tube (such as the non-collision) will also be satisfied by the actual trajectory of the robot.
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Notes
- 1.
A trajectory \(\mathbf {x}\), which is a function from \(\mathbb {R}\) to \(\mathbb {R}^{n}\), can be denoted equivalently \(\mathbf {x}\left( t\right) \) or \(\mathbf {x}\left( \cdot \right) \). When no ambiguity may exist, i.e., when t is already used in the same paragraph, we shall often prefer \(\mathbf {x}\left( t\right) \), for simplicity.
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Jaulin, L. et al. (2016). Computing Capture Tubes. In: Nehmeier, M., Wolff von Gudenberg, J., Tucker, W. (eds) Scientific Computing, Computer Arithmetic, and Validated Numerics. SCAN 2015. Lecture Notes in Computer Science(), vol 9553. Springer, Cham. https://doi.org/10.1007/978-3-319-31769-4_17
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