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Robust Convex Model Predictive Control with collision avoidance guarantees for robot manipulators

Bernhard Wullt, Johannes Köhler, Per Mattsson, Mikeal Norrlöf, Thomas B. Schön

TL;DR

A novel model predictive control solution for manipulators, where the two main components are a robust tube MPC and a corridor planning algorithm to obtain collision-free motion, which results in a convex MPC, which can solve fast, making the method practically useful.

Abstract

Industrial manipulators are normally operated in cluttered environments, making safe motion planning important. Furthermore, the presence of model-uncertainties make safe motion planning more difficult. Therefore, in practice the speed is limited in order to reduce the effect of disturbances. There is a need for control methods that can guarantee safe motions that can be executed fast. We address this need by suggesting a novel model predictive control (MPC) solution for manipulators, where our two main components are a robust tube MPC and a corridor planning algorithm to obtain collision-free motion. Our solution results in a convex MPC, which we can solve fast, making our method practically useful. We demonstrate the efficacy of our method in a simulated environment with a 6 DOF industrial robot operating in cluttered environments with uncertainties in model parameters. We outperform benchmark methods, both in terms of being able to work under higher levels of model uncertainties, while also yielding faster motion.

Robust Convex Model Predictive Control with collision avoidance guarantees for robot manipulators

TL;DR

A novel model predictive control solution for manipulators, where the two main components are a robust tube MPC and a corridor planning algorithm to obtain collision-free motion, which results in a convex MPC, which can solve fast, making the method practically useful.

Abstract

Industrial manipulators are normally operated in cluttered environments, making safe motion planning important. Furthermore, the presence of model-uncertainties make safe motion planning more difficult. Therefore, in practice the speed is limited in order to reduce the effect of disturbances. There is a need for control methods that can guarantee safe motions that can be executed fast. We address this need by suggesting a novel model predictive control (MPC) solution for manipulators, where our two main components are a robust tube MPC and a corridor planning algorithm to obtain collision-free motion. Our solution results in a convex MPC, which we can solve fast, making our method practically useful. We demonstrate the efficacy of our method in a simulated environment with a 6 DOF industrial robot operating in cluttered environments with uncertainties in model parameters. We outperform benchmark methods, both in terms of being able to work under higher levels of model uncertainties, while also yielding faster motion.

Paper Structure

This paper contains 29 sections, 4 theorems, 54 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Problem formulation
  5. Robust convex MPC for manipulators
  6. Feedback linearization and model error
  7. Auxiliary controller and tube dynamics
  8. Robust convex MPC problem
  9. Real-time MPC with convex obstacle avoidance constraints
  10. Obstacle avoidance through SCDF
  11. Parallel MPC formulation
  12. Robust corridor planning
  13. Numerical experiments
  14. Setup
  15. Convex acceleration set
  16. ...and 14 more sections

Key Result

Proposition 1

For all $\textbf{x} = (\textbf{q}, \dot{\textbf{q}}) \in \mathcal{X}$, $\textbf{a} \in \mathcal{A}$ and $\bm{\theta} \in \Theta$, we have with $\Delta_{\bm{\theta}},\Delta_{\text{disc}}$ from eq:dynamics_discretized and $a,b,c$ according to eq:me_a-eq:me_c.

Figures (3)

  • Figure 1: Illustration of our convex robust MPC. We produce collision-free motion that is robust to model errors. The closed loop trajectory (transparent red meshes) connect a given start and goal state (white and green meshes) while avoiding the obstacles (gray spheres).
  • Figure 2: All plots are illustrated in the configuration space. The yellow and purple symbolize the free and the obstacle regions, respectively. Left: From a collision-free path (blue line) a corridor is defined by the SCDF (orange region) Mid left: The corridor is discretized, resulting in a sequence of balls (black circles). The idea is to select a set of collision-free balls and a temporary goal state, such that the predicted trajectory makes progress to the goal and stays within the corridor. A feasible trajectory from the previous iteration is used as help (dotted green curve). Mid right: The first step is to compute the balls. For each state along a given feasible trajectory, a ball is allocated. An example state is highlighted (green point). The balls that contain the state are selected (black and red circles). Then the ball with the largest margin to the point is selected (red circle). Right: Illustration of how the virtual goal is computed. The ball corresponding to the last point along the trajectory is selected (red circle). Then, a virtual goal (cyan point) is computed by finding a point along the path, contained inside the ball, with the furthest progress along the path.
  • Figure 3: Left: Illustrates our problem scenario, where a 6 DOF robot is supposed to pick an object from a bin and place it on a shelf. The wrist frame position for the resulting motion is illustrated as the blue curve. The starting pick configuration is illustrated as the transparent red mesh. The goal shelving configuration is illustrated as the green transparent mesh. All obstacles are colored gray. Middle: Presents the mean time to goal for the different MPC methods. The horizontal axis represents the amount of uncertainty of mass and damping around nominal values. The vertical axis presents the time to goal. Oracle refers to a nominal MPC without any model errors, Rigid refers to a fixed sized tube, Flexible refers to our approach. A nominal MPC with model error was also tested, but it failed to reach the goal region due to infeasibility. Right: Presents the computation time for the MPC methods in the experiments. The dashed line represents the mean, the shaded region represents the min and max computation times. The black dashed line represents the real time limit of 40 [ms] for the robust methods, that is, our and the rigid approach.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof