Almost global finite-time stable observer for rigid body attitude dynamics

Abstract

A state observer is proposed for rigid body at- titude motion with a given attitude dynamics model. This observer is designed on the state space of rigid body attitude motion, which is the tangent bundle of the Lie group of rigid body rotations in three dimensions, SO(3), and therefore avoids instability due to the unwinding phenomenon seen with unit quaternion-based attitude observers. In the absence of mea- surement noise and disturbance torques, the observer designed leads to almost global finite-time stable convergence of attitude motion state estimates to the actual states for a rigid body whose inertia is known. Almost global finite-time stability of this observer is shown using a Morse function as part of a Lyapunov analysis; this Morse function has been previously used for almost global asymptotic stabilization of rigid body attitude motion. Numerical simulation results confirm the analytically obtained stability properties of this attitude state observer. Numerical results also show that state estimate errors are bounded in the presence of bounded measurement noise and bounded disturbance torque. I. INTRODUCTION This paper presents a nonlinear observer for attitude and angular velocity states of a rigid body. The attitude dynamics of the rigid body is assumed to be given by Euler's equation with a known external torque, known body inertia, and a bounded but unknown disturbance torque. The observer de- sign presented here exhibits almost global finite-time stable convergence of state estimates to actual states in the absence of measurement noise and disturbance torque. This gives it fast convergence as well as robustness to bounded distur- bance torques and bounded measurement noise. Since most unmanned and manned vehicles can be accurately modeled as rigid bodies, the attitude dynamics of such vehicles when operated in uncertain or poorly known environments, can be subject to unknown but bounded disturbance torques. Therefore, fast convergence of state estimates and robustness of the observer to such unknown disturbances is essential for feedback control of such vehicles. The attitude is described directly on the Lie group of rigid body orientations in this observer design, without using local coordinates or unit quaternions for attitude representation. The attitude and angular velocity observer designed here assumes that attitude and angular velocity have been deter- mined from available measurements. Attitude is determined from direction or angle measurements while angular velocity measurements are usually obtained directly through rate gyros. The problem of attitude determination from a set of three or more vector measurements is commonly set up as