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LQ-OCP: Energy-Optimal Control for LQ Problems

Logan E. Beaver

TL;DR

The paper tackles energy-efficient trajectory planning for linear-quadratic systems with linear constraints by deriving an analytic motion primitive generator via Pontryagin's minimum principle in Brunovský-normal-form coordinates. It yields a complete set of dynamical motion primitives, including unconstrained and constrained arcs, with explicit junction and optimality conditions that enable exact, closed-form solutions for each primitive. The approach allows offline precomputation of primitives, enabling real-time generation of energy-optimal trajectories and delivering substantial speedups and energy savings compared to an energy-focused LQR baseline in a submersible-in-cave scenario. This work advances constrained optimal control by providing a principled method to select constraint activations and connect optimal trajectory segments, with potential extensions to broader problem classes and multi-agent settings.

Abstract

This article presents a method to automatically generate energy-optimal trajectories for systems with linear dynamics, linear constraints, and a quadratic cost functional (LQ systems). First, using recent advancements in optimal control, we derive the optimal motion primitive generator for LQ systems--this yields linear differential equations that describe all dynamical motion primitives that the optimal system follows. We also derive the optimality conditions where the system switches between motion primitives--a system of equations that are bilinear in the unknown junction time. Finally, we demonstrate the performance of our approach on an energy-minimizing submersible robot with state and control constraints. We compare our approach to an energy-optimizing Linear Quadratic Regulator (LQR), where we learn the optimal weights of the LQR cost function to minimize energy consumption while ensuring convergence and constraint satisfaction. Our approach converges to the optimal solution 6,400% faster than the LQR weight optimization, and that our solution is 350% more energy efficient. Finally, we disturb the initial state of the submersible to show that our approach still finds energy-efficient solutions faster than LQR when the unconstrained solution is infeasible.

LQ-OCP: Energy-Optimal Control for LQ Problems

TL;DR

The paper tackles energy-efficient trajectory planning for linear-quadratic systems with linear constraints by deriving an analytic motion primitive generator via Pontryagin's minimum principle in Brunovský-normal-form coordinates. It yields a complete set of dynamical motion primitives, including unconstrained and constrained arcs, with explicit junction and optimality conditions that enable exact, closed-form solutions for each primitive. The approach allows offline precomputation of primitives, enabling real-time generation of energy-optimal trajectories and delivering substantial speedups and energy savings compared to an energy-focused LQR baseline in a submersible-in-cave scenario. This work advances constrained optimal control by providing a principled method to select constraint activations and connect optimal trajectory segments, with potential extensions to broader problem classes and multi-agent settings.

Abstract

This article presents a method to automatically generate energy-optimal trajectories for systems with linear dynamics, linear constraints, and a quadratic cost functional (LQ systems). First, using recent advancements in optimal control, we derive the optimal motion primitive generator for LQ systems--this yields linear differential equations that describe all dynamical motion primitives that the optimal system follows. We also derive the optimality conditions where the system switches between motion primitives--a system of equations that are bilinear in the unknown junction time. Finally, we demonstrate the performance of our approach on an energy-minimizing submersible robot with state and control constraints. We compare our approach to an energy-optimizing Linear Quadratic Regulator (LQR), where we learn the optimal weights of the LQR cost function to minimize energy consumption while ensuring convergence and constraint satisfaction. Our approach converges to the optimal solution 6,400% faster than the LQR weight optimization, and that our solution is 350% more energy efficient. Finally, we disturb the initial state of the submersible to show that our approach still finds energy-efficient solutions faster than LQR when the unconstrained solution is infeasible.
Paper Structure (13 sections, 38 equations, 4 figures, 3 tables)

This paper contains 13 sections, 38 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Optimal Control Problem
  3. Dynamical Motion Primitives
  4. Constraint Junctions
  5. Tangency Conditions
  6. Optimality Conditions
  7. Summary
  8. Finding a Constraint Sequence
  9. Simulation Results
  10. Dynamical Motion Primitives
  11. Unconstrained Solution
  12. Constrained Solution
  13. Conclusions and Outlook

Figures (4)

  • Figure 1: A submersible robot at point $(x, y)$ with the (net) buoyancy force and forward thrust shown; horizontal lines correspond to the minimum and maximum height.
  • Figure 2: Resulting submerisble trajectories for the unconstrained case.
  • Figure 3: Resulting trajectories for the constrained case.
  • Figure 4: Change in buoyant force ($a_y$) for the optimal control and LQR solutions; the left vertical dashed lines demarcates the junction for the $p_y = 0$ case, and the right vertical dashed line demarcates the junction for $p_y = h$.