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MR-MPCC: Model Predictive Contouring Control for Robotic Manipulator Tracking under Moving Reference Frames
Chonghao Cheng, Tiancheng Li, Jonathan Woolfrey, Liang Zhao, Shoudong Huang
TL;DR
Time-indexed tracking can accumulate path-relative error under timing mismatch or moving-reference disturbances. MR-MPCC combines path-progress optimization with constrained joint-velocity control for 6-DoF manipulators. It achieves lower contouring errors than PI control and time-indexed tracking MPC under moving-frame disturbances, with progress adaptation trading temporary lag for average contouring accuracy.
Problem
Time-indexed controllers rely on fixed reference timing, while timing mismatch and reference-frame motion make accurate path-relative tracking a challenge for manipulators.
Source: §I · p. 1Method
MR-MPCC formulates full 6-DoF manipulator tracking in a path-attached frame, optimizes path progress, and uses constrained RMRC to produce feasible joint velocities.
Source: §VI · p. 7Results
MR-MPCC achieves lower contouring RMSE and MAE than PI control and time-indexed tracking MPC under translation-only, rotation-only, and combined reference-frame disturbances.
Source: §VI · p. 7Takeaways & Limitations
Online progress adaptation supports more accurate path adherence without strict synchronization to a time-indexed reference, while transient progress reduction can temporarily accumulate lag.
Source: §VI · p. 7
Abstract
from arXiv · showhide
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 Moving-Reference 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
The introduction motivates geometric path fidelity under disturbances and timing mismatch, then presents MR-MPCC as a constrained, hierarchical approach evaluated against PI control and time-indexed MPC.
I. INTRODUCTION
- Robotic manipulation often prioritizes geometric fidelity over strict timing, motivating control that minimizes spatial path deviation under disturbances and modeling uncertainty.The paper cites industrial assembly, surgical automation, and machining as examples.Source: §I · p. 1
- Time-indexed MPC relies on fixed reference timing, while abrupt reference-frame motion or timing mismatch can accumulate geometric error and impair simultaneous position and orientation regulation.Manipulator MPCC also faces nonlinear kinematics and joint constraints, and stable joint-level control alone may not ensure small path-relative errors during abrupt frame motion.Source: §I · p. 1
- MR-MPCC combines task-space MPCC with constrained RMRC in a ROS2–MuJoCo pipeline and validates the framework on a physical UR5e.The paper also compares contouring, lag, and rotational tracking errors against PI control and time-indexed tracking MPC.Source: §I · pp. 1–2
- MR-MPCC achieves lower contouring error than PI control and time-indexed tracking MPC in simulation and physical experiments under matched paths and disturbance.The framework regulates full 6-DoF end-effector pose and combines task-space MPCC with constrained RMRC for feasible joint-velocity commands.Source: §I · pp. 1–2
A. Manipulator Control and Disturbance Rejection
- Feedback compensation and repetition-based refinement strengthen robustness, but do not explicitly optimize future motion under constraints.This motivates predictive control formulations that incorporate constraints and anticipate disturbance effects.Source: A. Manipulator Control and Disturbance… · p. 2
B. MPC for Manipulators
This material reviews constrained MPC and MPCC, then establishes the path representation and time-indexed tracking MPC baseline used in the paper.
B. MPC for Manipulators
- MPC can encode multivariable dynamics, actuation limits, and safety constraints, while robust and learning-augmented variants address uncertainty and model fidelity.Examples include tube MPC designs and learning-augmented MPC.Source: B. MPC for Manipulators · p. 2
- Time-indexed MPC can preserve temporal alignment while accumulating spatial deviation under execution delays or moving-reference disturbances; MPCC instead penalizes path-relative contouring and lag errors.This decouples path fidelity from time parameterization.Source: B. MPC for Manipulators · p. 2
C. Evolution of MPCC
- MPCC has been extended from contouring systems to multi-axis platforms and robotics, including 3D path errors and constraint-aware formulations.Prior work also used online path-progress optimization for adaptive speed regulation.Source: C. Evolution of MPCC · p. 2
- Manipulator MPCC remains relatively limited; prior work addressed safe feasible following, whereas this paper focuses on accurate tracking under external reference-frame disturbances.The preliminaries then derive contouring and lag errors and an MPCC problem that jointly optimizes control input and path progress.Source: C. Evolution of MPCC · p. 2, §III · p. 2
A. Reference Path
- The reference path is represented by Cartesian waypoints interpolated into a smooth curve, with a predefined timing law specifying its time-parameterized reference.The path parameter lies in [0,1].Source: A. Reference Path · p. 2
B. Time-Parameterized Tracking MPC
- Time-indexed MPC computes a finite-horizon control sequence using a nonlinear discrete-time model and weighted matrices, with a predefined timing law for the reference.The provided passage describes the optimization setup but does not give its full formulation.Source: B. Time-Parameterized Tracking MPC · p. 2, A. Reference Path · p. 2
C. Model Predictive Contouring Control
MPCC represents position deviation through path-relative contouring and lag components, then optimizes control and virtual path progress over a finite horizon.
C. Model Predictive Contouring Control
- MPCC defines the closest-point parameter on the reference path and uses the associated position error as the basis for path-relative error measures.The path parameterization supports regulating geometric deviation rather than a fixed time correspondence.Source: C. Model Predictive Contouring Control · p. 3
- The position error at the closest point on the path decomposes into a scalar lag component along the local tangent and a contouring component orthogonal to it.The contouring error is the residual after taking the tangential projection.Source: C. Model Predictive Contouring Control · p. 3
- MPCC replaces predefined timing with optimized virtual velocity and augments tracking MPC with path-progress dynamics and contouring and lag error weights.The formulation regulates geometric path deviation while optimizing progression along the path.Source: C. Model Predictive Contouring Control · p. 3
IV. HIERARCHICAL MPCC-BASED FRAMEWORK FOR MANIPULATORS
The framework uses a two-layer hierarchy and formulates tracking in a path-attached frame whose pose can move with progress-triggered disturbances.
IV. HIERARCHICAL MPCC-BASED FRAMEWORK FOR MANIPULATORS
- An upper task-space MPCC generates an end-effector twist and path-progress command, while a lower constrained quadratic program maps the command to feasible joint velocities under kinematic and safety constraints.This is the framework's hierarchical control structure.Source: §IV · p. 3
A. Problem Formulation
- Moving-reference disturbances enter through the time-varying pose of the path-attached frame relative to the world frame and are triggered by path progress.The control objective regulates end-effector position and orientation toward the moving reference while selecting progression and satisfying actuator and safety constraints.Source: A. Problem Formulation · p. 3
B. Upper Layer: Task-Space MPCC Formulation
The upper-layer MPCC predicts end-effector motion in the world frame while evaluating path-relative errors in a path-attached frame. It jointly optimizes control and path progress over a finite horizon, penalizing contouring, lag, orientation, and input-deviation terms.
B. Upper Layer: Task-Space MPCC Formulation
- The predicted state contains world-frame position and orientation plus path progress, while the estimated path-frame pose is held constant over the prediction horizon.Geometric errors are evaluated in the path-attached frame.Source: B. Upper Layer: Task-Space MPCC… · p. 4
- The controller re-expresses predicted poses and twists in the path frame to compute position and orientation errors relative to the reference path.The reference path is defined in that frame with a unit tangent derived from its position derivative; the orientation error is multiplicative.Source: B. Upper Layer: Task-Space MPCC… · p. 4
- Over a horizon of N steps, MPCC optimizes control inputs and path progress while weighting contouring, lag, orientation, input deviation, and progress.Only the first control pair is applied at each step.Source: B. Upper Layer: Task-Space MPCC… · p. 4
C. Lower Layer: Joint-Space Realization with Constraints
The lower layer maps the MPCC end-effector twist to feasible joint velocities through constrained resolved motion rate control. Its quadratic program combines task tracking with singularity, actuator, and safety handling.
C. Lower Layer: Joint-Space Realization with Constraints
- A constrained RMRC quadratic program computes joint velocities to track the MPCC end-effector twist while enforcing actuation and safety constraints.The differential kinematics relates joint velocities to the commanded twist.Source: C. Lower Layer: Joint-Space… · p. 4
- Because the least-squares RMRC solution can be ill-conditioned near singular configurations, the method adds singularity avoidance at both objective and constraint levels.Configuration-dependent damping increases as manipulability approaches a threshold, and a barrier constraint enforces non-decreasing manipulability at the boundary while relaxing away from it.Source: C. Lower Layer: Joint-Space… · p. 4
- Joint velocities are bounded using position and acceleration consistency, with effective limits computed elementwise.Source: C. Lower Layer: Joint-Space… · p. 4
- For link-pair distances below a safety threshold, linearized self-collision constraints impose velocity dampers that can be stacked across pairs.These constraints are combined with the other kinematic and safety constraints in the final RMRC quadratic program.Source: C. Lower Layer: Joint-Space… · pp. 4–5
- The resulting constrained RMRC layer maps the MPCC twist to feasible joint velocities with joint-level singularity and collision avoidance.Source: C. Lower Layer: Joint-Space… · p. 5
V. EXPERIMENT
The experiment section evaluates the proposed MPCC-based framework through controlled simulation comparisons and physical UR5e tests under moving-reference disturbances.
V. EXPERIMENT
- Simulation provides a controlled comparison under multiple disturbance profiles.Source: §V · p. 5
- Physical experiments on a UR5e manipulator assess feasibility under moving-reference disturbances.Source: §V · p. 5
- The section evaluates the proposed MPCC-based framework using both simulation and physical evaluation.Source: §V · p. 5
1) Simulation Setup:
The evaluation compares controllers in matched simulation trials, applies several moving path-frame disturbance configurations, and measures path-relative errors in the board-attached frame.
1) Simulation Setup:
- Controllers share the UR5e model, update rate, kinematic model, and joint-level constraints, with results averaged across five matched trials using identical disturbance realizations.The controllers are evaluated in MuJoCo under joint-velocity control.Source: 1) Simulation Setup · p. 5
2) Disturbance Models:
- Disturbances perturb the path-attached frame and include translation-only, rotation-only, and combined cases triggered at the same nominal progress.Combined-case transients use an impulse-driven ARMA time series, with temporal structure varied by order and coefficients; Fig. 3 shows the profiles and their translation and rotation components.Source: 2) Disturbance Models · p. 5, Figure 3 · p. 5
3) Evaluation Metrics:
- Metrics are evaluated in the board-attached frame, with contouring error as the primary geometric-fidelity measure alongside lag and orientation errors.RMSE, MAE, and MAX are computed per trial and averaged over five matched trials.Source: 3) Evaluation Metrics · p. 5
4) Simulation Results:
In simulation, MR-MPCC improves average geometric path fidelity under moving-reference disturbances, while transient progress adaptation trades lag for contouring accuracy. The accompanying physical-setup material describes a board-defined path frame and its sensing arrangement.
4) Simulation Results:
- Across RT configurations, MR-MPCC achieves the lowest contouring RMSE and MAE, while maximum contouring errors remain similar at about 50–53 mm.The peak deviation is attributed largely to the disturbance transient rather than steady tracking performance.Source: 4) Simulation Results · pp. 5–6
- After disturbances, MPCC suppresses progress and drives joint velocities toward their limits, improving average contouring accuracy at the cost of temporary lag.The response prioritizes geometric path recovery over temporal synchronization.Source: Figure 4 · p. 5, 4) Simulation Results · p. 6
1) Experimental Setup:
- Physical validation uses a 200 mm path attached to a movable board, with EM sensors tracking board and tool poses for online path-relative pose estimation.The board defines the path frame, and the ring tool serves as a physical tolerance gauge.Source: 1) Experimental Setup · p. 6, Figure 5 · p. 7
2) Disturbance Protocol:
- The physical setup fixes the reference path to a movable board defining FD, with two EM sensors measuring board and tool poses for control and evaluation.Fig. 6 shows the trajectories used in the physical experiment.Source: Figure 5 · p. 7, Figure 6 · p. 7
3) Evaluation Procedure:
Physical results show lower and more consistent path-relative contouring error for MR-MPCC than for PI and tracking MPC. The paper frames this performance within a path-frame evaluation procedure and a hierarchical controller, while identifying sensing and contact-rich extensions as future work.
3) Evaluation Procedure:
- Evaluation computes contouring error in path frame FD from estimated end-effector poses; ring contact indicates lateral deviation beyond the 16 mm clearance.The ring tool acts as a physical tolerance gauge.Source: 3) Evaluation Procedure · p. 7
4) Physical Validation Results:
- Across tested physical paths, MR-MPCC has lower median contouring error and a smaller interquartile range than PI and tracking MPC; representative trajectories also stay closer to the reference.Fig. 7 aggregates the distributions, while Fig. 8 shows a representative reconstructed trajectory after disturbance onset.Source: 4) Physical Validation Results · p. 7, Figure 7 · p. 7, Figure 8 · p. 8
VI. CONCLUSION
- MR-MPCC optimizes progress in a path-attached frame, while a constrained RMRC-based QP maps Cartesian twists to feasible joint velocities under joint and safety constraints.The unified evaluation uses ROS2–MuJoCo simulation and a physical UR5e pipeline with electromagnetic tracking.Source: §VI · p. 7
- Future work targets sensing uncertainty, delayed or asynchronous measurements, reference-frame state prediction, and force-aware objectives and constraints for contact-rich manipulation.The paper concludes that MR-MPCC is effective when the desired path is defined relative to a moving reference frame.Source: §VI · p. 7