We used an optimal control method involving covariant control equations as optimality conditions, to command the actuators of robot manipulators. These form a coupled system of second order nonlinear ordinary differential equations when associated with the robot motion equations. By solving this system, the control action required to take the robot from an initial to a final state is optimized in a prescribed time. However, the target set of equations exhibited stiffness. Therefore, an adequate solution could only be found for short trajectory durations with readily available numerical methods. We examined a time discretization procedure based on cubic and quintic Hermite finite elements which exhibited superconvergence properties for interpolation. This motivated us to develop a time integration algorithm based on Hermite's technique, where motion and control equations were perturbed to solve the optimal control problem. The optimal motion of a robotic manipulator was simulated using this algorithm. Our method was compared with a commercial differential equations solver on the basis of specific indicators. It outperformed the commercial solver by effectively solving the stiff set of equations for longer trajectory durations, with the cubic elements performing better than the quintic ones in this sense. The convergence analysis of our method confirmed that the quintic elements are more precise at the cost of increased computational burden, but converge at a lower rate than expected. Controlled motion experiments on a robotic manipulator validated our methodology. Trajectories were smoothly tracked and results exposed further methodology improvements.

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.