# Learning Dynamic-Objective Policies from a Class of Optimal Trajectories

## TL;DR

This paper introduces a new method combining trajectory optimisation, homotopy continuation, and imitation learning to efficiently synthesize dynamic-objective state-feedback controllers capable of switching objectives online.

## Contribution

It presents a novel approach to generate and train policies for multiple objectives using numerical continuation and imitation learning, enabling online objective switching.

## Key findings

- Policies produce near-optimal trajectories in simulations.
- Method effectively learns families of optimal controllers.
- Demonstrated on inverted pendulum and spacecraft transfer.

## Abstract

Optimal state-feedback controllers, capable of changing between different objective functions, are advantageous to systems in which unexpected situations may arise. However, synthesising such controllers, even for a single objective, is a demanding process. In this paper, we present a novel and straightforward approach to synthesising these policies through a combination of trajectory optimisation, homotopy continuation, and imitation learning. We use numerical continuation to efficiently generate optimal demonstrations across several objectives and boundary conditions, and use these to train our policies. Additionally, we demonstrate the ability of our policies to effectively learn families of optimal state-feedback controllers, which can be used to change objective functions online. We illustrate this approach across two trajectory optimisation problems, an inverted pendulum swingup and a spacecraft orbit transfer, and show that the synthesised policies, when evaluated in simulation, produce trajectories that are near-optimal. These results indicate the benefit of trajectory optimisation and homotopy continuation to the synthesis of controllers in dynamic-objective contexts.