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Adaptive Control Allocation for Underactuated Time-Scale Separated Non-Affine Systems

Daniel M. Cherenson, Dimitra Panagou

TL;DR

The paper addresses control of uncertain, underactuated nonlinear systems with non-affine input mappings and time-scale separation. It introduces an adaptive control architecture that combines a reduced-order model, a state predictor, and dynamic control allocation to select feasible inputs under input constraints. Stability and bounded tracking are established via singular perturbation theory and Lyapunov analysis, and the approach is validated on a VTOL quadplane across cruise, transition, and hover. This framework enables robust performance without complex mode-switching or gain scheduling in highly nonlinear, state-dependent actuation scenarios.

Abstract

Many robotic systems are underactuated, meaning not all degrees of freedom can be directly controlled due to lack of actuators, input constraints, or state-dependent actuation. This property, compounded by modeling uncertainties and disturbances, complicates the control design process for trajectory tracking. In this work, we propose an adaptive control architecture for uncertain, nonlinear, underactuated systems with input constraints. Leveraging time-scale separation, we construct a reduced-order model where fast dynamics provide virtual inputs to the slower subsystem and use dynamic control allocation to select the optimal control inputs given the non-affine dynamics. To handle uncertainty, we introduce a state predictor-based adaptive law, and through singular perturbation theory and Lyapunov analysis, we prove stability and bounded tracking of reference trajectories. The proposed method is validated on a VTOL quadplane with nonlinear, state-dependent actuation, demonstrating its utility as a unified controller across various flight regimes, including cruise, landing transition, and hover.

Adaptive Control Allocation for Underactuated Time-Scale Separated Non-Affine Systems

TL;DR

The paper addresses control of uncertain, underactuated nonlinear systems with non-affine input mappings and time-scale separation. It introduces an adaptive control architecture that combines a reduced-order model, a state predictor, and dynamic control allocation to select feasible inputs under input constraints. Stability and bounded tracking are established via singular perturbation theory and Lyapunov analysis, and the approach is validated on a VTOL quadplane across cruise, transition, and hover. This framework enables robust performance without complex mode-switching or gain scheduling in highly nonlinear, state-dependent actuation scenarios.

Abstract

Many robotic systems are underactuated, meaning not all degrees of freedom can be directly controlled due to lack of actuators, input constraints, or state-dependent actuation. This property, compounded by modeling uncertainties and disturbances, complicates the control design process for trajectory tracking. In this work, we propose an adaptive control architecture for uncertain, nonlinear, underactuated systems with input constraints. Leveraging time-scale separation, we construct a reduced-order model where fast dynamics provide virtual inputs to the slower subsystem and use dynamic control allocation to select the optimal control inputs given the non-affine dynamics. To handle uncertainty, we introduce a state predictor-based adaptive law, and through singular perturbation theory and Lyapunov analysis, we prove stability and bounded tracking of reference trajectories. The proposed method is validated on a VTOL quadplane with nonlinear, state-dependent actuation, demonstrating its utility as a unified controller across various flight regimes, including cruise, landing transition, and hover.

Paper Structure

This paper contains 11 sections, 3 theorems, 42 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Method
  5. Reduced System Reformulation
  6. State Predictor
  7. High-Level Controller
  8. Control Allocation and Parameter Adaptation
  9. Theoretical Arguments
  10. Simulation
  11. Conclusion

Key Result

Theorem 1

Let $e_x(t,\epsilon)$ be the solution to eq:slow_sp and $\bar{e}_x(t)$ be the solution to eq:reduced_slow on the interval $[t_0,\infty)$. Then there exists a positive constant $\epsilon^*$ such that for all $0 < \epsilon < \epsilon^*$, $e_x(t,\epsilon) - \bar{e}_x(t) = O(\epsilon)$ holds uniformly f

Figures (4)

  • Figure 1: Block diagram of the proposed adaptive control architecture. The gray blocks are user-specified. The blue and green blocks are detailed in \ref{['sec:method']}, and our specific contributions appear in the green blocks.
  • Figure 2: Control inputs over time during the three VTOL landing phases: cruise, transition, and hover.
  • Figure 3: Parameter estimation error over time. Initial estimates are initialized at 1.5 times the true values.
  • Figure 4: Slow state tracking error vs. time-scale separation $\epsilon$.

Theorems & Definitions (8)

  • Definition 1: Underactuated
  • Example 1: Planar Quadrotor with Pusher Propeller
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof