Applying Dynamics/Cost Parameter Continuation to the Optimal Guidance of Variable‐Speed Unicycle

ABSTRACT The problem of a variable‐speed unicycle guidance to the stationary target is considered. The vehicle should be guided to the origin while minimizing the energy loss due to the induced drag. The problem is formulated as a nonlinear optimal control problem with a known velocity profile and, consequently, a known drag coefficient profile. Since no analytical solution is available, a numerical parameter continuation procedure is employed. The parameter continuation algorithm comprises the parameterization of both the system dynamics and the cost by a single parameter. The system dynamics equation is parameterized in such a way that if the parameter is equal to zero, the dynamics equation is that of a quadratic‐kinematic approximation of the original system, whereas if the parameter is equal to one, it is the original nonlinear one. In the cost parametrization, the values of zero and one correspond to the cost with constant and variable speed, respectively. A numerical algorithm based on Davidenko's equation is derived. By an extensive simulation, it is shown that the proposed method converges from a wider set of initial and terminal conditions than the classical parameter continuation method and a state‐of‐the‐art alternative solver. This improvement is due to both the dynamics/cost parameterization and choosing an initial guess from the quadratic‐kinematic approximation.