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The Planar Motion with Two Bounded Controls - the Acceleration and the Derivative of the Curvature

Elena Degtiariova-Kostova 1
1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We study the minimum time problem to go from one given point on the plane to another with given initial and final tangent angles, curvatures and absolute values of speed, the paths joining these given points being $C^1$ and along them the derivative of the curvature and acceleration remaining bounded by two constants $B$ and $A$ respectively (we denote by $u_1$ (by $u_2$) the control of acceleration (of the derivative of the curvature respectively)). After the application of the Maximum Principal of Pontryagin and after the study of all possible forms of concatenation of arcs of curves of any extremal path we obtain the following results: 1) any general optimal path is a $C^1$-jonction of line segments in one and the same direction $\varphi $ ($u_2\equiv 0$; $\varphi \in [0,2\pi]$ and it is defined by the initial and final conditions) and of arcs of curves with linear curvature ($u_2\equiv \pm B$); 2) along any general optimal path the point moves with piecewise-linear absolute value of the speed ($u_1\equiv \pm A$); 3) any optimal path contains at most one line segment; 4) if for some optimal path the point moves along the line segment in the direction $\varphi +\pi $ $(\bmod (2\pi ))$, then this optimal path contains an infinite number of concatenated arcs of curves with linear curvature ($u_2\equiv \pm B$) which accumulate towards each endpoint of the line segment.
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Elena Degtiariova-Kostova. The Planar Motion with Two Bounded Controls - the Acceleration and the Derivative of the Curvature. RR-3444, INRIA. 1998. ⟨inria-00073246⟩

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