We propose a formulation of the finite horizon optimal control problem (FHOCP) based on inverse dynamics for general open-chain rigid-body systems, which reduces the computational cost from the conventional formulation based on forward dynamics. We regard the generalized acceleration as a decision variable and inverse dynamics as an equality constraint. To treat under-actuated systems with inverse dynamics that are well defined only to fully actuated systems, that is, to consider passive joints in this FHOCP, we add an equality constraint to zero the corresponding generalized torques. We include the contact forces in the decision variables of this FHOCP and treat the contact constraints using Baumgarte's stabilization method for numerical stability. We derive the optimality conditions and formulate the two-point boundary-value problem that can be efficiently solved using the recursive Newton–Euler algorithm (RNEA) and the partial derivatives of RNEA. We conducted three numerical experiments on model predictive control based on the proposed formulation to demonstrate its effectiveness. The first experiment involved simulating a swing-up control of a four-link arm with a passive joint and showed that the proposed formulation is effective for under-actuated systems. The second one involved comparing the proposed formulation with the conventional forward-dynamics-based formulation with various numbers of joints and showed that the proposed formulation reduces computational cost regardless of the number of joints. The third experiment involved simulating a whole-body control of a quadruped robot, a floating-base system having four contacts with the ground, and showed that the proposed formulation is applicable even for floating-base systems with contacts.

中文翻译：

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基于逆动力学的刚体系统有限时域最优控制问题的公式化

我们提出了一种基于逆动力学的有限水平最优控制问题 (FHOCP) 的公式，用于一般开链刚体系统，这降低了基于前向动力学的传统公式的计算成本。我们将广义加速度视为决策变量，将逆动力学视为等式约束。为了处理具有仅对完全驱动系统明确定义的逆动力学的欠驱动系统，即考虑该 FHOCP 中的被动关节，我们添加了一个等式约束来将相应的广义扭矩归零。我们将接触力包括在此 FHOCP 的决策变量中，并使用 Baumgarte 的数值稳定性稳定方法处理接触约束。我们推导了最优性条件并制定了可以使用递归牛顿-欧拉算法 (RNEA) 和 RNEA 的偏导数有效解决的两点边值问题。我们基于所提出的公式对模型预测控制进行了三个数值实验，以证明其有效性。第一个实验涉及模拟带有被动关节的四连杆臂的摆动控制，并表明所提出的公式对于欠驱动系统是有效的。第二个涉及将所提出的公式与具有各种关节数量的传统基于前向动力学的公式进行比较，并表明无论关节数量如何，所提出的公式都降低了计算成本。第三个实验涉及模拟四足机器人的全身控制，