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MPC-based motion planning for non-holonomic systems in non-convex domains

Matthias Lorenzen, Teodoro Alamo, Martina Mammarella, Fabrizio Dabbene

TL;DR

The paper tackles guaranteed convergence for MPC-based motion planning of non-holonomic mobile robots in non-convex environments with obstacles. It introduces an MPC scheme that replaces traditional terminal constraints with an artificial steady-state tied to a reference path, solving for $(\mathbf{x}_k,\mathbf{u}_k, s_k)$ to minimize $J_N$ while enforcing collision avoidance and path-consistent terminal conditions. Under verifiable assumptions, including a local controllability-like property near steady states, the closed-loop system is recursively feasible and $x_k$ converges to the target $x_T = g_p(1)$. Simulations on a differential-drive platform demonstrate obstacle avoidance and the ability to obtain shorter, smoother trajectories as the horizon increases, outperforming simple path-tracking approaches in presence of non-convex constraints.

Abstract

Motivated by the application of using model predictive control (MPC) for motion planning of autonomous mobile robots, a form of output tracking MPC for non-holonomic systems and with non-convex constraints is studied. Although the advantages of using MPC for motion planning have been demonstrated in several papers, in most of the available fundamental literature on output tracking MPC it is assumed, often implicitly, that the model is holonomic and generally the state or output constraints must be convex. Thus, in application-oriented publications, empirical results dominate and the topic of proving completeness, in particular under which assumptions the target is always reached, has received comparatively little attention. To address this gap, we present a novel MPC formulation that guarantees convergence to the desired target under realistic assumptions, which can be verified in relevant real-world scenarios.

MPC-based motion planning for non-holonomic systems in non-convex domains

TL;DR

The paper tackles guaranteed convergence for MPC-based motion planning of non-holonomic mobile robots in non-convex environments with obstacles. It introduces an MPC scheme that replaces traditional terminal constraints with an artificial steady-state tied to a reference path, solving for to minimize while enforcing collision avoidance and path-consistent terminal conditions. Under verifiable assumptions, including a local controllability-like property near steady states, the closed-loop system is recursively feasible and converges to the target . Simulations on a differential-drive platform demonstrate obstacle avoidance and the ability to obtain shorter, smoother trajectories as the horizon increases, outperforming simple path-tracking approaches in presence of non-convex constraints.

Abstract

Motivated by the application of using model predictive control (MPC) for motion planning of autonomous mobile robots, a form of output tracking MPC for non-holonomic systems and with non-convex constraints is studied. Although the advantages of using MPC for motion planning have been demonstrated in several papers, in most of the available fundamental literature on output tracking MPC it is assumed, often implicitly, that the model is holonomic and generally the state or output constraints must be convex. Thus, in application-oriented publications, empirical results dominate and the topic of proving completeness, in particular under which assumptions the target is always reached, has received comparatively little attention. To address this gap, we present a novel MPC formulation that guarantees convergence to the desired target under realistic assumptions, which can be verified in relevant real-world scenarios.
Paper Structure (8 sections, 3 theorems, 24 equations, 3 figures)

This paper contains 8 sections, 3 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Main Results
  4. MPC Algorithm
  5. Illustrative Example
  6. Obstacle Avoidance
  7. Target reaching vs. path following
  8. Conclusions and Future Work

Key Result

Theorem 1

Suppose that for a given initial state $x_0$ and path $p$, Assumptions ass:problemSetup, ass:cost, and ass:controllability hold, and the system is controlled by the proposed MPC control law $\kappa_{MPC}$. If Problem eq:mpcOpt is initially feasible, then it remains feasible for all $k>0$, the system

Figures (3)

  • Figure 1: With the proposed implementation, the robot does move around the obstacle and reaches the target pose. In contrast, with an MPC implementation following Limon2018_NMPCForTracking and ignoring the assumption of a convex constraint set, the robot gets stuck in a local minimum in front of the target.
  • Figure 2: With a longer horizon in the optimal control problem, the robot can "smooth out" the reference path and finds a shorter route with a lower cost. The reference path is straight, $90^\circ$ turn, and straight again.
  • Figure 3: The result of a global path planner using an RRT$^*$ algorithm after 1000 iterations : It is generally characterized by many corners if a fast result is necessary, hence exact path tracking is often not desirable.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof
  • proof