PaperlayerView exact revision on arXiv

Source-linked AI summary

Safe and Fast Tracking on a Robot Manipulator: Robust MPC and Neural Network Control

Julian Nubert, Johannes Köhler, Vincent Berenz, Frank Allgöwer, Sebastian Trimpe

arXiv:1912.10360v2cs.ROcs.LGeess.SY

TL;DR

Modern robotic tracking must handle nonlinear, uncertain, constrained systems while providing fast feedback and safety guarantees. The paper combines robust MPC with neural-network approximation to unify planning and control, guarantee safe operation, and accelerate runtime control on a KUKA LBR4+ manipulator.

  • Problem

    Complex robotic tracking involves nonlinear, high-dimensional systems, hard safety constraints, and separate planning and control layers that can create corner cases.

  • Method

    The approach combines robust MPC for nonlinear systems with offline-trained neural-network approximation, producing a single-layer control law that directly computes commands from references and constraints.

  • Results

    Experiments on a KUKA LBR4+ demonstrate safe robust MPC tracking and neural-network control with a reported two-orders-of-magnitude speed improvement.

  • Takeaways & Limitations

    Robust MPC unifies planning and control while the neural-network approximation replaces online optimization with fast evaluation for real-robot tracking.

Abstract

from arXiv · show

Fast feedback control and safety guarantees are essential in modern robotics. We present an approach that achieves both by combining novel robust model predictive control (MPC) with function approximation via (deep) neural networks (NNs). The result is a new approach for complex tasks with nonlinear, uncertain, and constrained dynamics as are common in robotics. Specifically, we leverage recent results in MPC research to propose a new robust setpoint tracking MPC algorithm, which achieves reliable and safe tracking of a dynamic setpoint while guaranteeing stability and constraint satisfaction. The presented robust MPC scheme constitutes a one-layer approach that unifies the often separated planning and control layers, by directly computing the control command based on a reference and possibly obstacle positions. As a separate contribution, we show how the computation time of the MPC can be drastically reduced by approximating the MPC law with a NN controller. The NN is trained and validated from offline samples of the MPC, yielding statistical guarantees, and used in lieu thereof at run time. Our experiments on a state-of-the-art robot manipulator are the first to show that both the proposed robust and approximate MPC schemes scale to real-world robotic systems.

I. INTRODUCTION

The paper targets safe, fast tracking for nonlinear, uncertain, constrained robotic systems while avoiding the complexity of separate planning and control layers. It combines robust setpoint-tracking MPC with neural-network approximation and demonstrates both approaches on a real robot.

  • Motivation: Modern robotics requires controllers that handle nonlinear, high-dimensional dynamics while guaranteeing stability and hard state and input constraints.Robust safety is especially important in applications involving humans.
  • Motivation: Fast update rates are needed for robotics, but resource-limited hardware and separate planning, inverse-kinematics, and control layers complicate deployment.The paper specifically identifies millisecond-scale operation and redundant-robot inverse kinematics as challenges.
  • Approach: The proposed single-layer approach combines robust MPC for nonlinear systems with supervised neural-network approximation of the MPC law.The MPC directly computes control commands from references and constraints, while the NN addresses online optimization complexity.
  • Experiments: Experiments on a KUKA LBR4+ manipulator demonstrate the feasibility of both the robust MPC and its neural-network approximation.The paper presents these as the first experimental implementations of the proposed robust and approximate schemes on real robotic systems.
  • Contributions: The robust setpoint-tracking MPC extends prior nonlinear and robust MPC methods to a real-world robotic system with safety properties guaranteed by design.The approach combines robust constraint handling with online setpoint tracking and optimized terminal ingredients.
  • Contributions: The approximate MPC uses offline supervised learning and a less conservative validation criterion that accounts for model mismatch while providing statistical safety guarantees.The NN is intended to replace the online optimization during runtime.

III. METHODS: RMPC SETPOINT TRACKING & AMPC

The method combines a continuous stabilizing feedback law with sampled robust MPC and uses incremental stability to construct uncertainty-aware tubes around nominal predictions.

  • Robust MPC design: The control architecture combines a piece-wise constant MPC input with a continuous-time feedback law κ(x) for fast feedback.The MPC input is updated at sampling instances while κ acts continuously between them.
  • Incremental stability: The RMPC design assumes that κ provides incremental exponential stability and uses an incremental Lyapunov function to characterize deviations between trajectories.The assumptions support stability conditions and the construction of robust tubes.
  • Tube construction: The incremental Lyapunov function parameterizes a tube around the nominal trajectory, with a scalar tube size representing the uncertainty bound.The nominal prediction follows the system dynamics under the continuous feedback law.

2) Tube:

The tube construction bounds deviations caused by model mismatch and propagates that bound through continuous- and discrete-time dynamics. The resulting robust tracking formulation avoids requiring an inverse-kinematics mapping for the desired output.

  • Tube definition: The predicted tube is defined by an incremental Lyapunov sublevel set around the nominal state, with scalar size sτ|t controlling its radius.The tube is used to represent uncertainty around the nominal trajectory.
  • Tube dynamics: Model mismatch is over-approximated by a disturbance bound, which drives the tube dynamics together with the contraction rate.The continuous-time tube evolves according to ˙st = −ρcst + wc.
  • Tube dynamics: The discrete-time tube size is derived from the sampling time, contraction rate, and discrete disturbance bound, and it bounds uncertainty around the nominal trajectory.The contraction rate determines tube growth while the tube size bounds model-induced deviations.
  • Setpoint formulation: Standard setpoint MPC would require a state-and-input target obtained from the desired output, often through inverse kinematics.For redundant manipulators, this mapping may be unknown or difficult to compute.
  • Setpoint formulation: The proposed robust setpoint-tracking optimization avoids requiring the mappings from output references to target states and inputs.It combines the robust MPC formulation with output setpoint tracking.
  • Optimization: The optimization solves for an artificial steady state and applies the first element of the optimal input sequence as the MPC control law.Predictions use the nominal dynamics, while constraints are tightened using the tube size.

1) Objective Function:

The objective introduces an artificial steady state whose output tracks the desired reference, allowing robust MPC to stabilize a safely reachable setpoint.

  • Objective function: The setpoint-tracking formulation introduces artificial steady-state variables so the steady-state output can track the desired output reference.The objective steers the system toward the artificial steady state while penalizing output-reference mismatch.
  • Stability result: Theorem 1 establishes exponential stability of the optimal safely reachable steady state in the robust setting.This extends earlier setpoint-tracking stability results to the robust formulation.

2) New Terminal Ingredients:

The approach redesigns terminal ingredients for nonlinear robust setpoint tracking by optimizing terminal-set size online, supporting recursive feasibility and constraint satisfaction.

  • New terminal ingredients: Online optimization of the terminal-set size α makes the design less conservative for varying setpoints.The terminal set is parameterized with the incremental Lyapunov function Vδ, yielding bounds on α for robust recursive feasibility.
  • New terminal ingredients: The terminal constraints and controller provide sufficient conditions for robust recursive feasibility.These conditions apply for fixed artificial steady state and terminal-set parameters.
  • New terminal ingredients: The terminal set remains recursively feasible, while tightened state and input constraints hold inside the terminal region.Both properties are stated for fixed xs, vs, and α.
  • Algorithm: The design combines offline preparation of Lyapunov and terminal-cost functions with online MPC optimization and control application.Offline steps determine κ, Vδ, disturbance quantities, and Vf; online steps solve the MPC problem and apply the resulting law with κ.

4) Closed-Loop Properties:

The closed-loop scheme guarantees recursive feasibility and constraint satisfaction under its assumptions, with practical exponential convergence to a safely reachable steady state; the assumptions are restrictive for redundant manipulators.

  • Safety and convergence: The safely reachable steady-state output set is defined by the system’s steady-state constraint functions and disturbance margin.The optimal safely reachable setpoint is selected from this set.
  • Safety and convergence: If the initial MPC problem is feasible, the closed loop remains recursively feasible and satisfies the posed constraints.This establishes safe operation under Assumption 1.
  • Safety and convergence: Under Assumption 2 and a constant reference, the optimal safely reachable setpoint is practically exponentially stable and the output converges practically exponentially to its reachable steady state.The theorem links convergence to the safe-reference-set assumptions.
  • Safety and convergence: With model mismatch bounded, the closed loop achieves practical exponential stability around the optimal steady state.The bound accounts for the effect of model mismatch.
  • Assumptions and scope: Convexity of the safe output-reference set and uniqueness of steady-state mappings are strong assumptions that fail for the considered redundant 7-DOF manipulator.Despite this, the paper states that safety properties are unaffected and experiments typically converge to some steady state.

C. Approximate MPC

Approximate MPC replaces the online robust MPC law with a learned neural-network controller while adding robustness and statistical validation to preserve safety and stability guarantees probabilistically.

  • Approximate MPC: The approximate controller compensates for neural-network inaccuracies by adding robustness within the RMPC design.Without such modifications, approximate MPC generally does not satisfy constraints.
  • Guarantees: If the approximation satisfies the stated mismatch condition, AMPC inherits the RMPC properties from Theorem 1.The condition is required for every feasible state under the designed disturbance bound.
  • Limitations: The statistical guarantee relies on a deterministic disturbed-system map and cannot directly verify infinite-length trajectories in practice.The paper therefore validates trajectories operationally rather than checking the infinite-horizon definition exactly.
  • Statistical validation: The validation procedure evaluates full trajectories generated from i.i.d. initial conditions and setpoints.The indicator detects whether any trajectory step exceeds the allowed discrepancy between approximate and ideal MPC.

3) Algorithm:

The AMPC procedure designs robust MPC offline, learns an explicit approximation, validates it, and repeats learning if validation fails; the study applies the methods to a KUKA LBR4+ manipulator.

  • AMPC algorithm: The AMPC workflow selects disturbance margins, designs RMPC, learns an approximation to πMPC, and validates the learned controller.Learning is repeated when validation fails.
  • Experimental system: The proposed methods are demonstrated on a KUKA LBR4+ robotic manipulator.The manipulator is modeled using nonlinear rigid-body dynamics.
  • Experimental system: The output constraints visualize differentiable quadratic over-approximations of non-differentiable obstacles.This representation supports obstacle-related constraints in the control design.

1) System Formulation:

The manipulator tracks task-space references using a nonlinear kinematic model with joint, velocity, input, obstacle-avoidance, and self-collision constraints. The formulation uses conservative static-obstacle approximations and characterizes disturbances empirically.

  • 1) System Formulation:: The controller uses inverse-dynamics feedback linearization so joint accelerations serve as the direct control input for task-space end-effector tracking.
  • 1) System Formulation:: The task-space output is computed through the robot’s forward kinematics.
  • 2) Constraints:: Joint angles, velocities, and inputs are constrained by polytopic bounds, including joint rotation limits and a velocity limit of 2.3 rad/s.
  • 2) Constraints:: Differentiable nonlinear output constraints approximate Cartesian obstacles and prevent self-collision.
  • 2) Constraints:: The obstacle formulation is a conservative over-approximation for static obstacles; exact and moving-obstacle reformulations are identified as alternatives.
  • 2) Constraints:: At a 2.5 Hz sampling rate, observed disturbance is plotted against applied acceleration, with proportionality-like behavior reported.

B. Robust MPC Design

The robust MPC accounts for model mismatch, computes safety-oriented control for constrained tracking, and is approximated offline by a neural network for faster execution. Experiments show safe reference tracking, while validation remains incomplete for the learned controller.

  • B. Robust MPC Design: The RMPC uses a tube-based disturbance description, incremental Lyapunov function, pre-stabilizing feedback, and online optimization implemented in real-time C++.
  • B. Robust MPC Design: The model mismatch is bounded empirically by ||d_w,d||∞ ≤ 0.06 and conservatively set to ||d_w,d||∞ ≤ 0.1 for robust design.
  • C. Experimental Results RMPC: The RMPC tracks user-specified end-effector references while maintaining safe distance from an obstructing obstacle.
  • C. Experimental Results RMPC: The RMPC and AMPC are compared under moving, unreachable, and stepped references over the intervals [0,11] s, [11,15] s, and [15,19] s.
  • C. Experimental Results RMPC: During changing setpoints, RMPC has larger tracking error because computational delay limits responsiveness, whereas AMPC uses faster sampling to apply more aggressive input.
  • D. AMPC Design: Validation satisfies the stated criterion for approximately 90% of sampled points, which is insufficient for high-probability guarantees over full trajectories.

E. Experimental Results AMPC

The approximate MPC replaces online optimization with neural-network evaluation to accelerate robot control. It achieves a 200-fold evaluation-time reduction, but the experiment includes occasional input-constraint violations and lacks complete a priori validation.

  • E. Experimental Results AMPC: Evaluation time falls by a factor of 200 to 1 ms, reducing the AMPC response interval to [1,40] ms instead of [400,800] ms for RMPC.
  • E. Experimental Results AMPC: The AMPC produces more aggressive inputs but sometimes violates input constraints because of high pre-stabilization gain and measurement noise.
  • E. Experimental Results AMPC: The results demonstrate significant performance for nonlinear MPC with 11 parameters, beyond the typical scale of standard explicit MPC approaches.
  • E. Experimental Results AMPC: The combined robust-MPC and neural-network approach achieves safe and fast tracking control on complex robotic systems.
  • E. Experimental Results AMPC: The experiments provide real-robot demonstrations of robust MPC and NN-based approximate control, while improved procedures are still needed for a priori statistical guarantees.
Loading 1912.10360v2…