Smooth Computation without Input Delay: Robust Tube-Based Model Predictive Control for Robot Manipulator Planning

TL;DR

This work tackles computation-induced delay in model predictive control (MPC) for robot manipulator planning under disturbances. It introduces a robust tube-based smooth MPC that linearizes the nonlinear dynamics around the current nominal state, bounding the linearization error with $\eta_2$ and combining it with the disturbance bound $\eta = \eta_1 + \eta_2$. A key idea is to predict the next real-state region via the ellipsoidal set $\mathbb{X}_\\omega(x^*(t_k),m)$ and to solve an OCP ahead to obtain the control sequence $\mathbf{v}^*(t_k)$, applying the first $m$ steps to the plant. Empirical results on both numerical simulations and a real UR5 manipulator show substantial improvements in response speed (up to about $90\%$) and robustness compared with conventional time-triggered MPC, enabling real-time robust planning on resource-constrained platforms.

Abstract

Model Predictive Control (MPC) has exhibited remarkable capabilities in optimizing objectives and meeting constraints. However, the substantial computational burden associated with solving the Optimal Control Problem (OCP) at each triggering instant introduces significant delays between state sampling and control application. These delays limit the practicality of MPC in resource-constrained systems when engaging in complex tasks. The intuition to address this issue in this paper is that by predicting the successor state, the controller can solve the OCP one time step ahead of time thus avoiding the delay of the next action. To this end, we compute deviations between real and nominal system states, predicting forthcoming real states as initial conditions for the imminent OCP solution. Anticipatory computation stores optimal control based on current nominal states, thus mitigating the delay effects. Additionally, we establish an upper bound for linearization error, effectively linearizing the nonlinear system, reducing OCP complexity, and enhancing response speed. We provide empirical validation through two numerical simulations and corresponding real-world robot tasks, demonstrating significant performance improvements and augmented response speed (up to $90\%$) resulting from the seamless integration of our proposed approach compared to conventional time-triggered MPC strategies.

Figures (6)

• Figure 1: In this figure, we compare the implementation of the conventional MPC with our designed approach. Compared with the conventional control fashion, our approach predicts the next real states $x(t_{k+1})$ by the estimated predictive disturbed state set $\mathbb{X}_{\omega}(x^*(t_k),m)$ to obtain the next optimal control input $\textbf{u}^*(t_{k+1})$. Thus, when the next real states $x(t_{k+1})$ come, we can directly use $\textbf{u}^*(t_{k+1})$ and compute $\textbf{u}^*(t_{k+2})$ to achieve smoothness. Compared with the conventional MPC, our approach improves the response speed and keeps the optimal control performance.
• Figure 2: (a) The UR5 manipulator in the Rviz simulator along with its geometries. (b) The manipulator moves within the plane. (c) The lateral view of the UR5 manipulator in the real environment.
• Figure 3: The comparison of three control methods over the Position Tracking task, which contain the state trajectory, position error, cost function value and control input.
• Figure 4: The comparison of three control methods over the Trajectory Tracking task, which contain the state trajectory, position error, cost function value and control input.
• Figure 5: The trajectories of the end-effector in position tracking task with different control strategies and video frames of experimental scenario.