Minimal time splines on the sphere

Abstract

An interesting problem in geometric control theory arises from robotics and space science: find a smooth curve, controlled by a bounded acceleration, connecting in minimum time two prescribed tangent vectors of a Riemannian manifold Q. The state equation is $$ \nabla _{\dot{\gamma }} \dot{\gamma } = u \in TQ, |u| \le A$$ . Applying Pontryagin’s principle one gets a Hamiltonian system in $$T^*(TQ)$$ . We consider this problem in $$S^2(r)$$ . Seemingly, it has not been addressed before. Via the SO(3) symmetry, we reduce the four degrees of freedom system to the five variables $$(a,v, M_1,M_2,M_3),$$ where v is the scalar velocity, conjugated to a costate variable a and $$(M_1,M_2,M_3)$$ are costate variables that satisfy $$\{ M_i, M_j\} = \epsilon _{ijk} M_k$$ . We derive the reduced equations and find special analytical solutions, that are organizing centers for the dynamics. Reconstruction of the curve $$\gamma (t)$$ is achieved by a time dependent linear system of ODEs for the orthogonal matrix R whose first column is the unit tangent vector of the curve and whose last column is the unit normal vector to the sphere.