Nonlinear control of a space robot for fast tracking pose trajectories

The optimized post trajectory operates under an open-loop control system. To enhance the accuracy of pose trajectory tracking, we implement a Model Predictive Controller (MPC) on the fiber bundle, referred to as geometric MPC. Conventional MPC utilizes a receding horizon approach, adjusting itself based on measured motion to improve alignment with desired trajectories. Additionally, we incorporate a Zero Order Holder to adopt and reject control signals, as illustrated in the accompanying diagram.

While this controller is robust and suitable for complex tasks, it necessitates nonlinear optimization at each step, resulting in significant computational demands. In contrast, the geometric controller addresses problems within Lie algebra, providing rapid and precise computation. Specifically, the computational complexity for conventional MPC is proportional to the cube of (3n + 20), whereas for geometric MPC, it is proportional to the cube of (2n + 16). For a space robot with six joints, this translates to computational costs of 38 cubed and 28 cubed, respectively, yielding a 2.5 times speed improvement for geometric MPC.

To further enhance control performance, we integrate momentum shaping with geometric MPC. This enhancement aims to impose momentum constraints within a safe or zero bound during the prediction horizon. As depicted in the figure, we sample momentum (\Pi^0) at time (t_k) and predict momentum (\Pi^j) at future moments (t_k + j), where (j) ranges from 1 to (N). The predicted momentum can fall into three scenarios, each requiring specific constraints for shaping. In the first and third cases, represented by the blue regions on the left and right of the figure, we ensure that absolute momentum remains within a defined safety bound (capital delta I) or diminishes to a smaller bound (small delta I). In the second case, we employ piecewise shaping to transition absolute momentum from a high value to a lower bound, as illustrated in the middle section of the figure.