(713d) Discrete Mechanics Optimal Control (DMOC) and Model Predictive Control (MPC) Synthesis for 3D Spatial Actuation

The common problem of the optimal controller synthesis for the process and mechanical robotic system is explored in this work. Namely, the optimal actuator motion control law and actuation policy for the reaction-diffusion process is considered. The common in process control practice realization admits the transport-reaction system coupled with servo-robotic mechanical device. For example, the welding arm scans over the specified domain in attempt to achieve the desired speed of welding.

In the considered model, the system consists of the four-degree robotic arm modeled by the discrete Lagrange-d’Alembert principle and a reaction-diffusion process governed by the parabolic partial differential equation (PDE). To address the optimal operational problem for the finite dimensional mechanics coupling with infinite-dimensional representation of reaction-diffusion process in different time scale, a two-layer control scheme is proposed in this research. The regulation of the robotic arm is performed in the lower layer by the discrete mechanics optimal control (DMOC) framework [1], which preserves the exact dynamics of the actuator even after the state discretization, to minimize the energy consumption of the arm’s motion and meet the tight constraints of the velocity and position. An optimization problem is solved in real time to chose the control forces in the dual of tangent bundle (TQ*) and generate the optimal trajectory over the configuration space (Q).For the higher level controller, synthesis benefits from the linear operator theory, spectrum decomposition and eigenfunction expansion [2], such that the parabolic PDE for the reaction-diffusion process is decomposed by the standard spectral technique and converted to a set of ordinary differential equations (ODEs). Then the modal model predictive control (MMPC) [3] is employed on this reduced model to optimize the actuation rate as well as guarantee the satisfactions of both the states and inputs constraints. The coordinated law of MMPC and DMOC for this two-layer controllers is established to achieve the overall processing optimality. Finally, the simulation study is conducted for the robotic application in the welding process of the 3D object [4], which evaluates and demonstrates the effectiveness of the control framework presented in this research.

References

[1] J. Mardson and M. West, Discrete mechanics and variational integrators, Acta Numerica, vol. 10, pp.357-514, 2001.

[2] R. F, Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer, 1995. 

[3] S. Dubljevic, N. H. El-Farra, P. Mhaskar and P. D. Christofides, Predictive control of parabolic PDEs with state and control constraints, International Journal of Robust and Nonlinear Control, vol. 14, pp.133-156,2004.

[4] R. M. Murry, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation,CRC Press, 1994.