Sub-Riemannian Problem on Lie Group of Motions of Pseudo Euclidean Plane.
The sub-Riemannian problem on group of motions of pseudo Euclidean plane is considered. From engineering perspective, this is the optimal control problem of a unicycle moving on a hyperbolic plane (plane with constant negative curvature). The dynamical system comprises real analytic left invariant vector field with 2-dimensional linear control vector – the control variables being the translational and the angular velocity of the unicycle. The sub-Riemannian optimal control problem seeks to determine optimal control input and the corresponding optimal trajectory between the given initial and terminal states such that the sub-Riemannian length on this trajectory is minimized.
At the onset we prove the controllability of the control distribution. We define the left invariant Hamiltonian for the system under consideration and apply the Pontryagin’s Maximum Principle (PMP). We prove that the extremal trajectories in the abnormal case are not strictly abnormal and the corresponding control vector is a constant i.e., identically zero. We then consider the normal Hamiltonian system. Through suitable coordinate transformation, we prove that the vertical subsystem is a double covering of a mathematical pendulum. This fact allows us to introduce Jacobi elliptic functions for integration of the nonlinear state equations that would otherwise be analytically intractable. Using specific elliptic coordinates, we calculate the extremal trajectories parametrized by Jacobi elliptic functions corresponding to various energy levels of the pendulum describing the vertical subsystem. We gain further insight into the nature of extremal trajectories through simulation and qualitative analysis.
As PMP gives only first order necessary optimality conditions, the extremal trajectories resulting from the integration of normal Hamiltonian system are candidate optimal only. Hence, second order optimality conditions are applied to eliminate the trajectories that cease to be optimal at certain point. It is known that a normal extremal trajectory ceases to be optimal either because it meets another candidate optimal trajectory at a point called Maxwell point where both have equal sub-Riemannian length, or because there exists a point called a conjugate point where a family of extremal trajectories has an envelope. We find surfaces in the state space M containing all Maxwell points and obtain a description of the Maxwell points in terms of roots of function forming these surfaces. Once the Maxwell sets are calculated, an upper bound on the cut time (the time at which an extremal trajectory loses global optimality) is obtained.
As argued, an extremal trajectory can lose optimality due to the existence of conjugate points. Conjugate points are the critical points of the exponential mapping and are found as roots of the Jacobian of the exponential mapping. The time at which the first conjugate point exists is called the first conjugate time. We conclude that the first conjugate point occurs later than the first Maxwell point and therefore the upper bound on cut time is given by the first Maxwell time. In the end, we present 3-dimensional plots of some important objects in sub-Riemannian problem on Lie group SH(2).