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We propose a geometric attitude controller and construct a framework to automatically tune the gains and study the system's stability.
Image showing contraction of two nearby trajectories from "W. Lohmiller and J. Slotine. On contraction analysis for nonlinear systems. Automatica, 34(6), 1998."
A surface plot showing the optimal minimum guaranteed exponential convergence rate for each controller gain pair (k_d, k_v). The highest point (in yellow) indicates the best controller gains. Any gain pair with a nonzero β will produce an exponentially stable controller.
In this work, we propose a new analysis of a simple geometric attitude controller, showing that it is locally exponentially stable and almost globally asymptotically stable; the exponential convergence region is much larger than existing non-hybrid geometric controllers (and covers almost the entire rotation space). The controller’s stability is proved using contraction analysis combined with optimization. The key in this combination is that the contraction metric is a linear matrix inequality with a special structure stemming from the configuration manifold SO(3).
As an additional contribution, we propose a general framework to automatically select controller gains by optimizing bounds on the system’s convergence rate; while this principle is quite general, its application is particularly straightforward with our contraction-based analysis. We demonstrate our results through simulations.
This work was presented at and published in the 2019 IEEE 58th Conference on Decision and Control (CDC).
Advantages:
-Simple static feedback controller, therefore easy implementation
-Gain selection is straightforward
-Almost global exponential stability except for the single unstable antipodal attitude to the desired attitude
-Tuning parameters have geometric meaning making them intuitive and simple to tune
-Controller respects the geometry of the configuration space therefore avoids singularities (e.g. gimbal lock) and ambiguities (e.g. quaternion representation)
Simulation result showing the distance error from the current attitude to the desired attitude over time. The system starting near the maximum attitude away (π) from the desired attitude convergences exponentially.
Simulation result showing that the chosen bound on the initial angular speed (2) used for computation is conservative as the system can tolerate a much faster initial speed error of 17 and still converge exponentially.
Plot showing that the maximum eigenvalue of the contraction matrix is always nonpositive throughout the trajectory proving that the system is converging exponentially.
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