Riemannian Formulation of Pontryagin’s Maximum Principle for the Optimal Control of Robotic Manipulators

Abstract

In this work, we consider robotic systems for which the mass tensor is identified to be the metric in a Riemannian manifold. Cost functional invariance is achieved by constructing it with the identified metric. Optimal control evolution is revealed in the form of a covariant second-order ordinary differential equation featuring the Riemann curvature tensor that constrains the control variable. In Pontryagin’s framework of the maximum principle, the cost functional has a direct impact on the system Hamiltonian. It is regarded as the performance index, and optimal control variables are affected by this fundamental choice. In the present context of cost functional invariance, we show that the adjoint variables are the first-order representation of the second-order control variable evolution equation. It is also shown that adding supplementary invariant terms to the cost functional does not modify the basic structure of the optimal control covariant evolution equation. Numerical trials show that the proposed invariant cost functionals, as compared to their non-invariant versions, lead to lower joint power consumption and narrower joint angular amplitudes during motion. With our formulation, the differential equations solver gains stability and operates dramatically faster when compared to examples where cost functional invariance is not considered.

Simpson’s Variational Integrator for Systems with Quadratic Lagrangians

“…An improved nonlinear Simpson's variational integrator could find its application in simulating complex non-linear mechanisms. Some application examples could involve a system of synchronized pendulums [36]; the discrete optimal control of robotic systems [37]; the modal analysis of dynamical systems [38]; the motion analysis of multibody systems evolving in fluid environments [39]; or the motion prediction of sliding rods [24] and soft robots [25].…”

A finite and fast time-optimal trajectory planning based on velocity sinking method for robots along specified path

“…The first methods are indirect methods, which mainly consist of the Pontryagin Maximum Principle and Dynamic programming (Shin et al, 1985, Rojas-Quintero et al, 2022, Shin et al, 1986& Singh et al, 1987. The time optimal trajectory solved by using the Pontryagin Maximum Principle is considered a "bang-bang" trajectory type in the plane, and can be calculated by successive integration of the maximum and minimum acceleration.…”