  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).
