リー代数に基づくリー群上の制御のためのコスト関数設計
Lie Algebraic Cost Function Design for Control on Lie Groups
リー群上の制御目標をリー代数で設計し、左不変計量を用いることで大域的な指数収束を実現する枠組みを提案。軌道最適化でもiLQRの収束を大幅に高速化できることを示した。
著者: Sangli Teng, William Clark, Anthony Bloch, Ram Vasudevan, Maani Ghaffari
分類: math.OC, cs.RO, cs.SY, eess.SY
原文アブストラクト
This paper presents a control framework on Lie groups by designing the control objective in its Lie algebra. Control on Lie groups is challenging due to its nonlinear nature and difficulties in system parameterization. Existing methods to design the control objective on a Lie group and then derive the gradient for controller design are non-trivial and can result in slow convergence in tracking control. We show that with a proper left-invariant metric, setting the gradient of the cost function as the tracking error in the Lie algebra leads to a quadratic Lyapunov function that enables globally exponential convergence. In the PD control case, we show that our controller can maintain an exponential convergence rate even when the initial error is approaching $\pi$ in SO(3). We also show the merit of this proposed framework in trajectory optimization. The proposed cost function enables the iterative Linear Quadratic Regulator (iLQR) to converge much faster than the Differential Dynamic Programming (DDP) with a well-adopted cost function when the initial trajectory is poorly initialized on SO(3).
