リー群上の凸幾何学的運動計画：モーメント緩和によるアプローチ
Convex Geometric Motion Planning on Lie Groups via Moment Relaxation
リー群上で剛体システムの運動計画を正確な多項式最適化問題として定式化し、半正定値計画緩和で大域的最適解を求める手法を提案。ドローン着陸やマニピュレータの逆運動学に適用し、計画ホライズンに対して線形の計算量を実現した。
著者: Sangli Teng, Ashkan Jasour, Ram Vasudevan, Maani Ghaffari
分類: cs.RO, cs.SY, eess.SY
原文アブストラクト
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.
