  This letter presents a method to reduce the computational demands of
including second-order dynamics sensitivity information into the Differential
Dynamic Programming (DDP) trajectory optimization algorithm. An 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 also show how the structure of the dynamics can be used to further
accelerate these computations in rigid-body systems. Benchmarks of this
approach for trajectory optimization with multi-link manipulators show that the
benefits of DDP can often be included without sacrificing evaluation time, and
can be done in fewer iterations than iLQR.
