Adaptive Complexity Model Predictive Control

TL;DR

This work introduces a formulation of model predictive control (MPC), which adaptively reasons about the complexity of the model while maintaining feasibility and stability guarantees, and finds that this adaptive method enables more agile motion and expands the range of executable tasks compared with fixed-complexity implementations.

Abstract

This work introduces a formulation of model predictive control (MPC) which adaptively reasons about the complexity of the model based on the task while maintaining feasibility and stability guarantees. Existing MPC implementations often handle computational complexity by shortening prediction horizons or simplifying models, both of which can result in instability. Inspired by related approaches in behavioral economics, motion planning, and biomechanics, our method solves MPC problems with a simple model for dynamics and constraints over regions of the horizon where such a model is feasible and a complex model where it is not. The approach leverages an interleaving of planning and execution to iteratively identify these regions, which can be safely simplified if they satisfy an exact template/anchor relationship. We show that this method does not compromise the stability and feasibility properties of the system, and measure performance in simulation experiments on a quadrupedal robot executing agile behaviors over terrains of interest. We find that this adaptive method enables more agile motion and expands the range of executable tasks compared to fixed-complexity implementations.

Figures (7)

• Figure 1: Adaptive complexity model predictive control selectively simplifies the model to promote efficiency without sacrificing stability. For example, during a legged leaping task joint information may be required during takeoff and landing but can be omitted elsewhere without affecting the behavior.
• Figure 2: Elements in the horizon $i$ are in the set $S_k$ if they are feasible and stay on the manifold $\mathcal{Z}^s$ (illustrated as a blue dashed 1D curve). The adaptive system allows $z^l_{i}$ to leave this submanifold while remaining in the manifold $\mathcal{Z}^c$ (the surrounding white 2D space). Elements in the set $S_k$ are denoted in blue, elements not in this set are denoted in red. In this example, $S_k = \{1, 2, 3, 7, 8, 9\}$.
• Figure 3: Adaptive complexity MPC retains recursive feasibility and admissibility by updating the simplicity set and solution at time $k+1$ with the corresponding terms from time $k$ along with the new state and control determined from the terminal policy $u_{T}$ which is applied with the terminal set $\mathcal{X}^c_{T}$. Letters at the bottom indicate the condition of \ref{['eq:ac:admissibility_conditions']} that element satisfies or violates. The complex manifold $\mathcal{X}^c$ is the 2D space while $\mathcal{X}^s$ is the embedded 1D submanifold.
• Figure 4: The complex model includes body and foot states which together define joint states, while the simple model only consists of body states.
• Figure 5: Data for Acceleration experiment. The computational efficiency of the Simple configuration permits aggressive commands despite model reductions. Both Adaptive and Mixed outperform Complex because they retain some efficiency.

Theorems & Definitions (23)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5