# Trajectory Optimization on Manifolds: A Theoretically-Guaranteed Embedded Sequential Convex Programming Approach

## TL;DR

This paper extends Sequential Convex Programming (SCP) to manifold-constrained trajectory optimization problems by leveraging geometric embeddings, providing theoretical guarantees and demonstrating practical effectiveness.

## Contribution

It introduces a novel SCP algorithm for manifold problems using geometric embeddings, bridging the gap between Euclidean and manifold trajectory optimization.

## Key findings

- The proposed method achieves theoretical guarantees similar to Euclidean SCP.
- Numerical experiments demonstrate the practical effectiveness of the manifold SCP approach.
- The approach enables trajectory optimization with manifold constraints like loop closure.

## Abstract

Sequential Convex Programming (SCP) has recently gained popularity as a tool for trajectory optimization due to its sound theoretical properties and practical performance. Yet, most SCP-based methods for trajectory optimization are restricted to Euclidean settings, which precludes their application to problem instances where one must reason about manifold-type constraints (that is, constraints, such as loop closure, which restrict the motion of a system to a subset of the ambient space). The aim of this paper is to fill this gap by extending SCP-based trajectory optimization methods to a manifold setting. The key insight is to leverage geometric embeddings to lift a manifold-constrained trajectory optimization problem into an equivalent problem defined over a space enjoying a Euclidean structure. This insight allows one to extend existing SCP methods to a manifold setting in a fairly natural way. In particular, we present a SCP algorithm for manifold problems with refined theoretical guarantees that resemble those derived for the Euclidean setting, and demonstrate its practical performance via numerical experiments.