  This paper presents a method to reduce the computational complexity of
including second-order dynamics sensitivity information into the Differential
Dynamic Programming (DDP) trajectory optimization algorithm. A tensor-free
approach to DDP is developed where all the necessary derivatives are computed
with the same complexity as in the iterative Linear Quadratic Regulator~(iLQR).
Compared to linearized models used in iLQR, DDP more accurately represents the
dynamics locally, but it is not often used since the second-order derivatives
of the dynamics are tensorial and expensive to compute. This work shows how to
avoid the need for computing the derivative tensor by instead leveraging
reverse-mode accumulation of derivative information to compute a key
vector-tensor product directly. We benchmark this approach for trajectory
optimization with multi-link manipulators and show that the benefits of DDP can
often be included without sacrificing evaluation time, and can be done in fewer
iterations than iLQR.
