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Robust Maneuver Planning With Scalable Prediction Horizons: A Move Blocking Approach

Philipp Schitz, Johann C. Dauer, Paolo Mercorelli

TL;DR

This paper tackles the challenge of running model predictive control for long-horizon maneuvers on hardware with limited computational resources. It introduces a shrinking-horizon SHMPC that combines time-varying move blocking with a tube-MPC framework to cap the number of decision inputs and uses an optimization-based inner-approximation to drastically reduce constraint complexity. The key contributions are (i) a constructive method for generating blocking matrices that preserve recursive feasibility via truncation and interval splits, (ii) a reformulation that enables efficient constraint reduction through interval transition matrices and inner-approximations guided by a generalized Farkas lemma, and (iii) a numerical example showing an order-of-magnitude speedup with only modest increases in trajectory cost. The approach has practical impact by enabling robust, long-horizon MPC implementations on onboard platforms, demonstrated through a 300-step helicopter-landing scenario.

Abstract

Implementation of Model Predictive Control (MPC) on hardware with limited computational resources remains a challenge. Especially for long-distance maneuvers that require small sampling times, the necessary horizon lengths prevent its application on onboard computers. In this paper, we propose a computationally efficient tubebased shrinking horizon MPC that is scalable to long prediction horizons. Using move blocking, we ensure that a given number of decision inputs is efficiently used throughout the maneuver. Next, a method to substantially reduce the number of constraints is introduced. The approach is demonstrated with a helicopter landing on an inclined platform using a prediction horizon of 300 steps. The constraint reduction decreases the computation time by an order of magnitude with a slight increase in trajectory cost.

Robust Maneuver Planning With Scalable Prediction Horizons: A Move Blocking Approach

TL;DR

This paper tackles the challenge of running model predictive control for long-horizon maneuvers on hardware with limited computational resources. It introduces a shrinking-horizon SHMPC that combines time-varying move blocking with a tube-MPC framework to cap the number of decision inputs and uses an optimization-based inner-approximation to drastically reduce constraint complexity. The key contributions are (i) a constructive method for generating blocking matrices that preserve recursive feasibility via truncation and interval splits, (ii) a reformulation that enables efficient constraint reduction through interval transition matrices and inner-approximations guided by a generalized Farkas lemma, and (iii) a numerical example showing an order-of-magnitude speedup with only modest increases in trajectory cost. The approach has practical impact by enabling robust, long-horizon MPC implementations on onboard platforms, demonstrated through a 300-step helicopter-landing scenario.

Abstract

Implementation of Model Predictive Control (MPC) on hardware with limited computational resources remains a challenge. Especially for long-distance maneuvers that require small sampling times, the necessary horizon lengths prevent its application on onboard computers. In this paper, we propose a computationally efficient tubebased shrinking horizon MPC that is scalable to long prediction horizons. Using move blocking, we ensure that a given number of decision inputs is efficiently used throughout the maneuver. Next, a method to substantially reduce the number of constraints is introduced. The approach is demonstrated with a helicopter landing on an inclined platform using a prediction horizon of 300 steps. The constraint reduction decreases the computation time by an order of magnitude with a slight increase in trajectory cost.
Paper Structure (15 sections, 5 theorems, 28 equations, 4 figures, 1 table)

This paper contains 15 sections, 5 theorems, 28 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Contribution and Structure of the Paper
  3. Notation
  4. Problem Statement and Preliminaries
  5. Tube MPC
  6. Move blocking
  7. Efficient Move Blocking for SHMPC
  8. Optimization Problem Formulation
  9. Recursive Feasibility via Truncation
  10. Recursively Feasible Interval Splits
  11. Constraint Reduction
  12. Transition Matrix Formulation
  13. Constraint Set Approximation
  14. Numerical Example
  15. Conclusion and Outlook

Key Result

Proposition 1

Let $\mathcal{Z}$ be a RPI set for system eq:system. If $x_0 \in z_0 \oplus \mathcal{Z}$ and we choose $u_k = v_k - K(x_k - z_k)$, then $x_{k} \in z_{k} \oplus \mathcal{Z}$ for all $w_k \in \mathcal{W}$ and $k \in \mathbb{N}$.

Figures (4)

  • Figure 1: An example of \ref{['eq:Psi']} with $s = (1,3,1)$. The blocking vector $s'$ corresponds to the blocking matrix $RMG$. Since $j=1$, $\sigma_1$ is empty.
  • Figure 2: Left: An example of \ref{['eq:F_tilde_k_explicit']} with $s_i = 3$. Red circles denote initial states. Right: $\tilde{\mathcal{F}}$ (blue) and $\tilde{\mathcal{F}}_t$ (dashed) of a randomly generated set from Example \ref{['example:innerApprox_computation']} for $s_i = 30$.
  • Figure 3: Ratios of volumes $V$ and number of halfspaces $q$ between $\tilde{\mathcal{F}}$ and $\tilde{\mathcal{F}}_t^*$. Subscripts $r$, $min$ and $0$ correspond to the approximation, the minimal and the non-reduced representation of $\tilde{\mathcal{F}}$, respectively.
  • Figure 4: Closed-loop simulation results of a helicopter landing. For MPC-a, the attitude and center of mass are plotted every second with the initial condition marked in red. The blue region denotes the projection of the target set. MPC-0 and MPC-min lie on top of each other.

Theorems & Definitions (9)

  • Definition 1: Robust positively invariant (RPI) set
  • Proposition 1: Proposition 1 in mayneRobustModelPredictive2005
  • Definition 2: Blocking matrix
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Lemma 1: Theorem 1 of sadraddiniLinearEncodingsPolytope2019
  • Remark 1
  • Example 1