Optimal controller applied to robotic systems using covariant control equations

Rojas-Quintero, Juan Antonio; Rojas-Estrada, J.A.; Villalobos-Chin, Jorge; Santibáñez, Víctor; Bugarin, Eusebio

doi:10.1080/00207179.2020.1865570

Cited by 7 publications

(16 citation statements)

References 35 publications

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“…We have followed the optimal control methodology presented in [25]. Therefore, this section summarizes the main ideas of the cited methodology.…”

Section: Optimal Control Methodologymentioning

confidence: 99%

“…However such a cost function is not invariant under a change of coordinates. Instead, as proposed in [25], we select the invariant cost function γ = 1 2 u i u i = 1 2 M ij u i u j so that the performance index becomes…”

Section: B Optimal Controlmentioning

confidence: 99%

“…The reader is referred to [25], where the complete analysis of how equations (7) lead to (8), is conducted. These control equations are nonlinear second order ODE that restrict the torques during motion.…”

Section: B Optimal Controlmentioning

confidence: 99%

“…The left hand side of these equations involve: the second covariant time derivative of the covariant torque tensor components u i in the first term (see equation ( 2)); the Riemann-Christoffel curvature tensor in the second term; the second covariant derivative of potential V in the third term; and contravariant torque tensor components u l in the third term (see equation (5)). Definitions for these objects can be found in [25], [29] and have the following expressions:…”

Section: B Optimal Controlmentioning

confidence: 99%

“…Due to the FE nature of our method, it is not excluded that our algorithm can be adapted to solve PDE that arise in DP. We will focus in the ODE case of the optimal control methodology described in [25] which involves covariant control equations. This methodology is similar to PMP, however, the resulting set of nonlinear ODE are of the second order instead, but also exhibit stiffness.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Control of Robotic Systems Using Finite Elements for Time Integration of Covariant Control Equations

2021

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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.

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“…We have followed the optimal control methodology presented in [25]. Therefore, this section summarizes the main ideas of the cited methodology.…”

Section: Optimal Control Methodologymentioning

confidence: 99%

Section: B Optimal Controlmentioning

confidence: 99%