Globally-Attractive Logarithmic Geometric Control of a Quadrotor for Aggressive Trajectory Tracking

Johnson, Jacob C.; Beard, Randal W.

doi:10.1109/lcsys.2022.3141066

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…The trace function [13] has been introduced and applied in [2], [4], [5], [7], [11], [14] to indicate the configuration error of rotational motion. However, this error function may lead to slow error convergence [3], [15] when the rotational error is large. To solve the above problem, the logarithmic error has been applied [3], [15], [16].…”

Section: Introductionmentioning

confidence: 99%

“…However, this error function may lead to slow error convergence [3], [15] when the rotational error is large. To solve the above problem, the logarithmic error has been applied [3], [15], [16]. However, the [3], [16] does not prove the stability property.…”

Section: Introductionmentioning

confidence: 99%

“…However, the [3], [16] does not prove the stability property. The proof in [15] is specific to SO(3) and requires the left Jacobian of SO (3), thus making it less general for systems on Lie groups.…”

Section: Introductionmentioning

confidence: 99%

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Lie Algebraic Cost Function Design for Control on Lie Groups

Teng¹,

Clark²,

Bloch³

et al. 2022

Preprint

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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 π 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).

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