  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.
