A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon

By Sanjay P. Bhat, Dennis S. Bernstein. 2000

875 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e496de1f9192c03a961cf3574d5eba9b039a5a33

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Control and Systems Engineering

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Attitude Tracking and Disturbance Rejection of Rigid Spacecraft by Adaptive Control (2009, background): “In fact, references [2], [4], and [5] have shown that, because of topological obstructions, it is impossible to achieve a general global attitude tracking of a rigid body using a single coordinate representation like the quaternions by a static smooth controller.”
Robust Adaptive Attitude Tracking on ${\rm SO}(3)$ With an Application to a Quadrotor UAV (2013, background): “These three undesired equilibria cannot be avoided for any continuous attitude control system because of a topological property of SO(3) [4], [9].”
Rigid-Body Attitude Control (2011, background): “This property is related to the fact that every continuous time-invariant closed-loop vector field on SO 13 2 and S(2) has more than one closed-loop equilibrium, and hence, the desired equilibrium cannot be globally attractive [9]; see “The Impossibility of Global Attitude Stabilization Using Contin”
Geometric tracking control of a quadrotor UAV on SE(3) (2010, background): “Without these considerations, a quaternion-based controller can exhibit an unwinding phenomenon, where the controller unnecessarily rotates the attitude through large angles [15].”
Quaternion-Based Hybrid Control for Robust Global Attitude Tracking (2011, neutral): “In particular, [4] discusses how this control law leads to unwinding.”
Almost global finite‐time stabilization of rigid body attitude dynamics using rotation matrices (2016, background): “Unwinding is a type of instability of attitude control systems using continuous quaternion feedback that arises when the ambiguity in the two-to-one mapping from the space of unit quaternions (the unit hypersphere S(3) R(4)) to SO(3) is not accounted for in the control law [2, 7].”
Nonlinear robust tracking control of a quadrotor UAV on SE(3) (2011, neutral): “Therefore, depending on the particular choice of control inputs, a quaternion-based control system may become discontinuous when applied to actual attitude dynamics [7], and it may also exhibit unwinding behavior, where the controller rotates a rigid body through unnecessarily large angles [8].”
Globally exponentially stable cascade observers for attitude estimation (2012, background): “Finally, the proposed solution does not exhibit any of the aforementioned drawbacks common to attitude estimation solutions such as singularities, unwinding phenomena, slow convergence near unstable equilibrium points or topological limitations for achieving global asymptotic stabilization on SO(3),”
Inertia-Free Spacecraft Attitude Tracking with Disturbance Rejection and Almost Global Stabilization (2009, neutral): “This unwinding phenomenon is discussed in [4].”
Unit Quaternion-Based Output Feedback for the Attitude Tracking Problem (2008, background): “For more details about the topological obstruction to continuous global stabilization of rotational motion, the reader is referred to [2].”
High-Performance Spacecraft Adaptive Attitude-Tracking Control Through Attracting-Manifold Design (2008, background): “It is pertinent here to observe the fact that the set of special group of rotation matrices that describe body orientation in three dimensions SO 3 is not a contractible space, and hence quaternion-based formulations do not permit globally continuous stabilizing controllers [1,2].”
Geometric Adaptive Tracking Control of a Quadrotor Unmanned Aerial Vehicle on SE(3) for Agile Maneuvers (2014, background): “This ambiguity should be carefully resolved in quaternion-based attitude control systems, otherwise they may exhibit unwinding, where a rigid body unnecessarily rotates through a large angle even if the initial attitude error is small [27].”
Finite‐time angular velocity observers for rigid‐body attitude tracking with bounded inputs (2017, background): “As a result, the unwinding phenomenon can occur, yielding an unnecessary full rotation even for small initial attitude errors, as discussed in [17].”
Attitude Synchronization of a Group of Spacecraft Without Velocity Measurements (2009, background): “Almost global asymptotic stability results are obtained in the sense that the closed loop system has several equilibria, that represent the same physical configuration, but only one of them is an attractor [21].”
Satellite Attitude Control by Quaternion-Based Backstepping (2005, background): “A discussion concerning this point can be found in [20], where it was proved that no continuous control law can globally stabilize a system on SO(3).”