MR-MPCC: Model Predictive Contouring Control for Robotic Manipulator Tracking under Moving Reference Frames

The gist: MR-MPCC presents a Model Predictive Contouring Control (MPCC) framework for velocity-controlled robotic manipulators with 6-DoF end-effector pose regulation under MovingReference disturbances.

Introduction and Motivation

Maintaining small path-tracking errors under moving reference frames is essential for robotic manipulation tasks such as machining and surgical automation, where geometric fidelity is prioritized over strict timing adherence Conventional time-indexed controllers, including tracking Model Predictive Control (MPC), may accumulate path-relative error under abrupt reference-frame motion or timing mismatch. Model Predictive Contouring Control (MPCC) offers an alternative by reformulating tracking relative to the reference path, decomposing the deviation into a contouring component orthogonal to the path and a lag component along the path, allowing geometric path-following accuracy and path progress to be regulated separately. Extending MPCC to robotic manipulators is challenging due to nonlinear kinematics and joint-level constraints which complicate real-time task-space control.

MR-MPCC Framework

The MR-MPCC framework adopts a hierarchical control architecture consisting of an upper-level MPCC that optimizes path progress while penalizing contouring and orientation errors, and a lower layer constrained resolved motion rate control (RMRC) that maps the commanded Cartesian twist to feasible joint velocities under joint-level and safety constraints. The method is implemented in a ROS2–MuJoCo pipeline and validated on a physical UR5e platform under moving-reference disturbances.

Upper Layer: Task-Space MPCC Formulation

The upper layer task-space MPCC controller generates a desired end-effector twist together with a path progress command based on the current robot state. The optimization problem minimizes the sum of weighted contouring error, lag error, and orientation error, alongside input deviation from the reference trajectory. The objective function is formulated as min over N steps of [D ek+i squared Qc + (e l k+i) squared Ql + φ k+i squared Qo + Duk+i - Du d(s,k+i) squared Ru − qsvs,k] subject to kinematic and control constraints.

Lower Layer: Joint-Space Realization with Constraints

The lower layer realizes the task-space MPCC output by solving a constrained quadratic program using a constrained RMRC-based QP [25]. This layer computes a feasible joint-velocity command using the least-squares solution to track the task-space command while enforcing actuation and safety constraints. Singularity avoidance is incorporated by regularizing the RMRC using a configuration-dependent damping term and imposing a manipulability barrier constraint to prevent further loss of manipulability near minimum values. Joint velocities are also restricted by elementwise bounds derived from position and acceleration consistency calculations.

Experimental Validation

The framework was evaluated in a unified ROS2–MuJoCo simulation and physical UR5e experimental pipeline under moving-reference disturbances. Disturbances are applied directly to the path frame and measured using an electromagnetic tracking system. The evaluation metrics include contouring error as the primary measure of geometric path fidelity, along with lag and orientation errors. Simulation results showed that MR-MPCC achieves lower contouring error than PI control and time-indexed tracking MPC under matched disturbance, demonstrating improved path-following performance under reference-frame motion. Physical experiments confirmed that the proposed MPCC-based framework exhibits a lower median contouring error and a smaller interquartile range than PI and tracking MPC across tested paths, indicating more consistent path relative accuracy under moving reference disturbances.

Conclusion

MR-MPCC is effective for robotic manipulator path following when the desired path is defined relative to a moving reference frame because the online progress adaptation enables the controller to maintain more accurate path adherence without requiring strict synchronization with a time-indexed reference. The method demonstrates that MR-MPCC is effective for robotic manipulator path following when the desired path is defined relative to a moving reference frame. Future work will incorporate sensing uncertainty, delayed and asynchronous measurements, and reference-frame state prediction into the estimation and control framework. The method will also be extended to contact-rich manipulation by introducing force-aware objectives and constraints.

--- Page 1 ---

MR-MPCC: Model Predictive Contouring Control for Robotic Manipulator Tracking under Moving Reference Frames Chonghao Cheng1, Tiancheng Li1, Jonathan Woolfrey1, Liang Zhao2, and Shoudong Huang1 Abstract— Maintaining small path-tracking errors under moving reference frames is essential for robotic manipulation tasks with tight execution tolerances, such as machining and surgical automation. Conventional time-indexed controllers, including tracking Model Predictive Control (MPC), rely on a fixed temporal correspondence with the reference trajectory and may therefore accumulate path-relative error under abrupt reference-frame motion or timing mismatch. This paper presents MR-MPCC, a Model Predictive Contouring Control (MPCC) framework for velocity-controlled robotic manipulators with 6-DoF end-effector pose regulation under MovingReference disturbances. A hierarchical control architecture is adopted, in which an upper-level MPCC optimizes path progress while penalizing contouring and orientation errors, and a constrained resolved motion rate control (RMRC) layer maps the commanded Cartesian twist to feasible joint velocities under joint-level and safety constraints. MR-MPCC is implemented in a ROS2–MuJoCo pipeline and validated on a UR5e platform under moving-reference disturbances, where the path frame is perturbed and tracked by electromagnetic sensors. Compared with PI control and time-indexed tracking MPC under matched disturbance, MR-MPCC achieves lower geometric path-following error and smaller deviations in the moving path frame, demonstrating improved path-following performance under reference-frame motion. I. INTRODUCTION Accurate end-effector motion control is fundamental to robotic manipulation in industrial assembly, surgical automation, and machining [1,2]. In such tasks, performance is usually determined by geometric fidelity to a prescribed path rather than by strict adherence to a preassigned timing law [3]. The control objective is therefore to minimize spatial deviation from the path in the presence of disturbances and modeling uncertainty. Conventional controllers such as proportional–integral–derivative (PID) control provide intuitive and computationally efficient solutions [4], but do not explicitly predict future system evolution or systematically handle state and input constraints. Model Predictive Control (MPC) addresses these issues by optimizing future control actions over a finite horizon [5,6]. However, standard MPC formulations are time-indexed and rely on a fixed time parameterization of the reference [7]. Under disturbances that induce timing mismatch or abrupt motion of the reference frame, timeindexed tracking can accumulate geometric error, leading to degraded robustness, especially when simultaneous position and orientation regulation is required. Model Predictive Contouring Control (MPCC) offers an alternative by reformulating tracking relative to the reference path [8]. It decomposes the tracking deviation into a contouring component orthogonal to the path and a lag component along the path, allowing geometric path-following accuracy and path progress to be regulated separately. Despite these advantages, its application to robotic manipulators remains in its infancy, with only a small number of recent works beginning to explore this domain [9]. Extending MPCC to robotic manipulators is challenging because nonlinear kinematics, and joint-level constraints complicate real-time task-space control [10]. The challenge increases when the reference frame moves abruptly, since stable joint-level control does not necessarily ensure small path-relative errors. In applications such as surgical milling and high-accuracy assembly, these errors can affect task quality and safety. This motivates direct regulation of pathrelative error under reference motion and joint constraints. This paper presents MR-MPCC, a task-space MPCC framework for velocity-controlled manipulators with full 6DoF end-effector pose regulation. The method combines MPCC with constrained RMRC to generate feasible jointvelocity commands in real time. It is implemented in a ROS2–MuJoCo pipeline (Fig. 1) and validated on a physical UR5e, where reference-frame motion is measured using electromagnetic tracking and the controller operates at 500 Hz. Under matched paths and disturbance, MR-MPCC achieves lower contouring error than PI control and time-indexed tracking MPC under matched disturbance, demonstrating improved path-following performance under reference-frame motion. II. RELATED WORK A. Manipulator Control and Disturbance Rejection Robotic manipulators are subject to modeling uncertainties and external disturbances that can degrade tracking accuracy and closed-loop stability. Classical controllers such as PID and feedback linearization provide reliable baseline performance, but disturbance attenuation often depends on careful tuning and model fidelity [4]. Robust control techniques explicitly shape uncertainty margins to improve disturbance rejection, including slidingmode control [11] and H∞ designs [12]. In practice, these methods can trade tracking smoothness or responsiveness for robustness, and they do not directly address the timing-sensitivity induced by a time-parameterized reference. Iterative learning control leverages task repetitiveness to reduce tracking errors across repeated executions, which has been widely adopted in machining and contouring applications [13–15]. However, its effectiveness diminishes when disturbances are non-repetitive or when the task conditions change rapidly between trials. Overall, these approaches strengthen robustness via feedback compensation or repetition-based error refinement, but they do not explicitly optimize future motion under constraints. This motivates predictive control formulations that can incorporate constraints and anticipate disturbance effects. B.

1. Bold task-space MPCC formulation for full 6-DoF path following under moving reference frames: The system can regulate geometric path-following accuracy and path progress to be regulated separately by decomposing tracking deviation into a contouring component orthogonal to the path and a lag component along the path.

2. Unified simulation and physical evaluation pipeline: The improved AI system can perform robust validation across environments by using a unified simulation and physical evaluation pipeline for reference-frame disturbances, allowing comparison of control performance under both simulated and real-world moving-reference conditions.

3. Hierarchical control architecture implementation: The system will utilize a two-layer hierarchy where the upper layer optimizes path progress while penalizing contouring and orientation errors and the lower layer uses constrained resolved motion rate control (RMRC) to map the commanded Cartesian twist to feasible joint velocities.

4. Moving-reference disturbance compensation: The controller can actively suppress undesirable behavior during transients by using its mechanism where MR-MPCC suppresses the progress rate vs during large deviations, which reduces transient deviation in contouring error at the cost of increased lag error.

5. Constraint handling via RMRC: The system ensures operational safety and feasibility by employing a constrained resolved motion rate control (RMRC) layer that enforces joint-level constraints, including singularity avoidance at both the objective and constraint levels and Self-collision avoidance.