This paper reports a novel result: with proper robot models on matrix Lie groups, one can formulate the kinodynamic motion planning problem for rigid body systems as \emph{exact} polynomial optimization problems that can be relaxed as semidefinite programming (SDP). Due to the nonlinear rigid body dynamics, the motion planning problem for rigid body systems is nonconvex. Existing global optimization-based methods do not properly deal with the configuration space of the 3D rigid body; thus, they do not scale well to long-horizon planning problems. We use Lie groups as the configuration space in our formulation and apply the variational integrator to formulate the forced rigid body systems as quadratic polynomials. Then we leverage Lasserre's hierarchy to obtain the globally optimal solution via SDP. By constructing the motion planning problem in a sparse manner, the results show that the proposed algorithm has \emph{linear} complexity with respect to the planning horizon. This paper demonstrates the proposed method can provide rank-one optimal solutions at relaxation order two for most of the testing cases of 1) 3D drone landing using the full dynamics model and 2) inverse kinematics for serial manipulators.

翻译：本文报道了一个新颖结果：通过矩阵李群上合适的机器人模型，可将刚体系统的动力学运动规划问题表述为可松弛为半定规划（SDP）的精确多项式优化问题。由于刚体动力学非线性，刚体系统的运动规划问题是非凸的。现有基于全局优化的方法未能恰当处理三维刚体的构型空间，因此难以扩展至长时域规划问题。我们在公式中采用李群作为构型空间，并应用变分积分器将受迫刚体系统表示为二次多项式。随后利用Lasserre层级结构通过半定规划获得全局最优解。通过稀疏化构造运动规划问题，结果表明所提算法在规划时域上具有线性复杂度。本文证明：在1）完整动力学模型的三维无人机着陆，2）串联机械臂逆运动学这两个测试案例中，所提方法能在松弛阶次为二时提供秩一最优解。