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Implementation of Nonlinear Model Predictive Path-Following Control for an Industrial Robot
Timm Faulwasser, Tobias Weber, Juan Pablo Zometa, Rolf Findeisen
TL;DR
The paper studies real-time nonlinear model predictive control for industrial-robot path following under state and input constraints. It implements a controller that jointly plans motion along a Cartesian path and computes tracking inputs, with or without reference-speed assignment. Experiments on a KUKA LWR IV demonstrate real-time feasibility and path-following behavior, including errors below 1 mm in one experiment and constraint-limited speed tracking in another.
Problem
Robotic applications require accurate following of prescribed geometric paths, motivating real-time constrained path-following control with flexible speed assignment.
Method
The controller repeatedly solves a predictive optimal-control problem that jointly plans motion along the path and computes system inputs, with weights and constraints adapted to speed assignment.
Results
The KUKA LWR IV implementation is real-time feasible; path-following errors stayed below 1 mm in one experiment, while angular-velocity constraints limited reference-speed tracking in another.
Takeaways & Limitations
The proposed MPFC concept supports constrained path following with and without speed assignment in a laboratory robot implementation.
Abstract
from arXiv · showhide
Many robotic applications, such as milling, gluing, or high precision measurements, require the exact following of a pre-defined geometric path. In this paper, we investigate the real-time feasible implementation of model predictive path-following control for an industrial robot. We consider constrained output path following with and without reference speed assignment. We present results from an implementation of the proposed model predictive path-following controller on a KUKA LWR IV robot.
I. INTRODUCTION
The paper addresses path-following tasks where a system follows a geometric curve while its speed along the curve is not fixed in advance. It develops and implements constrained nonlinear model predictive path-following control on a three-joint KUKA LWR IV robot, achieving sampling periods on the order of 1 ms.
- Path following steers a system along a prescribed geometric curve without fixing the speed along that curve beforehand.
- Existing approaches often decompose path following into trajectory generation followed by trajectory tracking.
- Closed-loop path-following controllers integrate reference-motion generation and input computation while modulating speed along the path.
- Only a few laboratory implementations of predictive path-following control for mechatronic and robotic systems had been reported.
- The proposed sampled-data nonlinear model predictive controller handles input and state constraints on a three-joint KUKA LWR IV with Cartesian-space reference paths.
- A suitable numerical implementation achieves a sampling period in the order of 1 ms.
II. PREDICTIVE PATH FOLLOWING
The paper formulates nonlinear systems with constrained states and inputs as the basis for predictive path-following control.
- The considered nonlinear system has state x, input u, and output y governed by differentiable dynamics and output maps.
- State and input trajectories are required to remain within the admissible sets X and U.
A. Path-following Problems
Path-following treats the desired motion as a geometric output-space path whose timing can be selected by the controller. The paper distinguishes unconstrained timing from speed-assigned path following.
- The geometric path is P = {y ∈ R^nu | y = p(θ)}, where θ is the path parameter and p(θ) parametrizes P.
- Unlike trajectory tracking, path following does not generally prescribe when the system must reach each point on the path.
- Constrained output path following: Constrained output path following requires convergence to P, forward motion along P, and satisfaction of state and input constraints.
- Speed-assigned path following: The speed-assigned formulation additionally requires the path-parameter velocity to converge to a prescribed profile.
- Speed-assigned path following: Speed assignment is not equivalent to trajectory tracking because it does not specify a unique output reference p(θ(t)).
- Timing law: The timing law treats θ as a virtual state whose evolution is influenced by an additional bounded virtual input.
- Augmented formulation: In the augmented formulation, the output contains the path-following error e = h(x) − p(θ) and the virtual state z.
B. Model Predictive Path-following Control
Model predictive path-following control repeatedly solves a finite-horizon optimal control problem for the physical system and virtual path dynamics. The formulation directly plans motion along the path while accounting for constraints and can support both path-following variants.
- MPFC addresses constrained output path following with and without speed assignment using continuous-time sampled-data NMPC.
- At each sampling instant, MPFC repeatedly solves a finite-horizon optimal control problem over the physical input and virtual timing input.
- The optimization simultaneously plans a reference motion on P and computes system inputs to track that motion.
- The optimal physical input is applied, and the problem is solved again at the next sampling instant using updated initial conditions.
- The controller is a dynamic feedback strategy because the timing-law state is initialized from the previous predicted trajectory.
- The paper focuses on implementation rather than a detailed analysis of convergence conditions, although terminal ingredients can provide path convergence and recursive feasibility.
C. Problems with and without Speed Assignment
The paper uses a quadratic cost function and changes its weighting and constraints to handle path-following with or without speed assignment.
- Cost function: The quadratic cost function uses diagonal Q and R matrices, with Q positive semi-definite and R positive definite.The weights determine whether path convergence, endpoint stopping, or reference-speed matching is prioritized.
- Without speed assignment: For path-following without speed assignment, the formulation penalizes z1 − θ1 and uses wθ > 0 with wθdot = 0.The corresponding domain is Z = [θ0, θ1] × [0, ∞) × R^(r−1).
- With speed assignment: For path-following with speed assignment, the formulation penalizes z2 − θdotref and uses wθdot > 0 with wθ = 0.The corresponding domain is Z = [θ0, ∞) × [0, ∞) × R^(r−1).
A. Robot Model and MPFC Design
The robot model represents three actuated joints with constrained dynamics, uses Cartesian tip position as the output, and parameterizes paths with continuously differentiable polynomial splines.
- Robot model: The robot dynamics satisfy B(q)q̈ + C(q, q̇)q̇ + τF(q̇) + g(q) = τ.The model includes inertia, centrifugal and Coriolis effects, friction, gravity, and joint torques.
- MPFC design: The dynamics are rewritten in implicit state-space form with x1 = q, x2 = q̇, and u = τ.The resulting representation uses E = diag(I, B(x1)).
- MPFC design: The controller uses Cartesian tip position y = hca(x1) as its output because path-following tasks are formulated in the robot’s operational space.An integrator chain of length two is added to obtain the augmented system description.
- Path representation: The path parametrization p(θ) is represented by polynomial splines whose coefficients ensure continuous differentiability.The construction uses path segments with polynomial order and coefficients determined across the partition.
B. Interfacing and Real-Time Feasible Optimization
The implementation combines a 1 kHz robot interface with simplified real-time optimization, state estimation from encoder measurements, and a sampling period of 1 ms.
- Interfacing: The Fast Research Interface supports sampling rates up to 1 kHz and superposes externally computed MPFC torques in joint-specific impedance mode.The external computer communicates with the robot through Ethernet.
- Optimization: The implementation approximates Coulomb friction with an arctan and omits gravity from the OCP because the robot provides gravity compensation.The OCP is solved online using ACADO direct single shooting and one SQP iteration per controller step.
- Optimization: The prediction horizon is T = 100 ms and the MPFC sampling period is δ = 1 ms.Piecewise-constant inputs use 10 equidistant intervals of 10 ms, with one SQP iteration at each sampling time.
- State estimation: Joint angular velocities are reconstructed from finite differences and low-pass filtering of measured joint angles.The controller’s internal state z = (θ, θdot)T does not need to be estimated.
- Real-time performance: 0.48 ms is the maximum OCP-solve time, while total latency remains below 0.92 ms, enabling a 1 kHz experimental sampling rate.The mean and median total latency are both 0.42 ms.
IV. EXPERIMENTAL RESULTS
Experiments on clover and Hello paths demonstrate real-time path following with the proposed controller. The controller rapidly reduces path deviations, adapts speed to curvature and constraints, and recovers after external disturbances.
- Experimental setup: The experiments draw a three-leaved clover and the word Hello using the MPFC scheme, with essentially identical parameters across experiments.Only wθ, ẇθ, and box constraints differ between the experiments, without recompiling the generated code.
- Clover Path: Three clover turns are executed with speed assignment, producing periodic inputs and angular velocities along the closed curve.The path parameter speed varies periodically with curvature, while the controller accelerates on straight sections and slows at sharp corners.
- Clover Path: Angular-velocity constraints prevent the path-parameter velocity from tracking its reference value of 250 s−1 when the fourth-joint bound is reached.The observed speed behavior is attributed to the robot’s angular-velocity constraint.
- Clover Path: After the MPFC scheme is activated at t = 2.5 s, all three path-following errors decrease rapidly and remain below 1 mm.The controller compensates for the initial deviation and follows the clover path accurately in operational space.
- Hello Path: External forces during five intervals cause large Hello-path deviations, but the robot returns toward the reference after each disturbance ends.The disturbance can stop the robot or move it away from the path; the zoomed experiment shows recovery after release.
V. SUMMARY AND CONCLUSIONS
The paper designs and implements nonlinear model predictive control for constrained robotic path following. Results on a KUKA LWR IV indicate real-time feasibility and promising control performance for paths with or without speed assignment.
- Summary and Conclusions: The paper designs and implements continuous-time nonlinear model predictive control schemes for constrained path following by robotic manipulators.The schemes address constrained output path following both with and without speed assignment.
- Summary and Conclusions: An augmented system description directly handles Cartesian-space paths together with input and state constraints.The implementation demonstrates the approach on a KUKA LWR IV robot and supports both considered path-following formulations.