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.
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