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We introduce a time-varying reference trajectory in our local controller which allows the controller to achieve global stability. The framework is improved to use less conservative bounds and a more efficient method of searching the gain space.
The Nelder-Mead algorithm is used to efficiently search the gain space for locally optimal gains near some pre-determined initial gains. (Above is a visualization of the Nelder-Mead algorithm over the Rosenbrock banana function taken from wikipedia.org)
The Schur complement lemma can be used to transform nonlinear quadratic constraints with optimization parameters m_{i} to equivalent Linear Matrix Inequalities (LMI) which are convex and solvable.
A simplified representation of the product space TSO(3) x SO(3). The rigid body dynamics evolve on the TSO(3) manifold which consist of the attitude and angular velocity (R, ω) (shown on left). The reference trajectory evolves on the SO(3) manifold with independent dynamics going from R_{ref} to the desired attitude R_{d} (shown on right).
In this work, we present a simple geometric attitude controller that is globally, exponentially stable. To overcome the topological restriction, the controller is designed to follow a reference trajectory that in turn converges to the desired equilibrium (making it discontinuous in the initial conditions, but continuous in time). The system and reference dynamics are studied as a single augmented system that can be analyzed and tuned simultaneously. The controller’s stability is proved using contraction analysis (on the manifold), and the bounds on the convergence rate can be found via a semi-definite program with linear matrix inequalities. Additionally, our approach allows the use of the Nelder-Mead algorithm to automatically select controller gains and reference trajectory parameters by optimizing the aforementioned bounds. The resulting controller is verified through simulations.
This work will be presented at the 2020 IEEE 59th Conference on Decision and Control (CDC) from December 14-18, 2020 and published in 2021.
Advantages:
-Simple static feedback controller, therefore easy implementation
-Gain selection is straightforward
-Global exponential stability
-Tuning parameters have geometric meaning making them intuitive and simple to tune
-Framework generalizes well to configuration spaces with a product structure such as the n-dimensional torus (e.g. robotic arms and manipulators) and SE(3) (robots with 3-dimensional position and orientation)
-Controller respects the geometry of the configuration space therefore avoids singularities (e.g. gimbal lock) and ambiguities (e.g. quaternion representation)
The distance error over time from the current attitude to the desired attitude is shown above. The system starting at the maximum distance (π) away from the desired attitude convergences exponentially.
The angular velocity error as a function of time is shown above. Initially the system increased speed to follow the reference trajectory, but eventually slows down when the system reaches the desired attitude.
The control torque in 3-dimensions is shown. The controls are smooth making them ideal for real motors. In other popular hybrid approaches, the control may exhibit sudden jumps or drastic shift in torque requirements when switching between local controllers making them problematic for actuators.
The maximum eigenvalue of the contraction matrix (used to prove exponential stability) along the system's trajectory is always nonpositive, thus the system is exponentially converging.
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