Continuous-time successive convexification for constrained trajectory optimization

Abstract

We present continuous-time successive convexification (ct- scvx ), a real-time-capable solution method for constrained trajectory optimization, with continuous-time constraint satisfaction and guaranteed convergence. The proposed solution framework only relies on first-order information, and it combines several key methods to solve a large class of nonlinear optimal control problems: (i) exterior penalty-based reformulation of the path constraints; (ii) generalized time-dilation; (iii) multiple-shooting discretization; (iv) ${\ell }_1$-exact penalization of the nonconvex constraints; and (v) the prox-linear method, a sequential convex programming (SCP) algorithm for convex-composite minimization. The proposed reformulation of the path constraints enables continuous-time constraint satisfaction even on sparse temporal discretization grids and obviates the need for mesh-refinement heuristics. Through the prox-linear method, we guarantee that: (i) ct-scvx converges to stationary points of the penalized problem; (ii) the converged stationary points that are feasible for the discretized and control-parameterized optimal control problem are also Karush–Kuhn–Tucker (KKT) points. Furthermore, we specialize this property to global minimizers of convex optimal control problems and obtain stronger convergence results by exploiting convexity. In addition to theoretical analysis, we demonstrate the effectiveness and real-time performance of ct-scvx by means of numerical examples from real-world optimal control applications: dynamic obstacle avoidance, and 3-degree-of-freedom (3-DoF) and 6-DoF autonomous rocket landing.