  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.
