Rigid body attitude estimation based on the Lagrange–d’Alembert principle

Estimation of rigid body attitude and angular velocity without any knowledge of the attitude dynamics model is treated using the Lagrange–d’Alembert principle from variational mechanics. It is shown that Wahba’s cost function for attitude determination from two or more non-collinear vector measurements can be generalized and represented as a Morse function of the attitude estimation error on the Lie group of rigid body rotations. With body-fixed sensor measurements of direction vectors and angular velocity, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential obtained from Wahba’s function. An additional dissipation term that depends on the angular velocity estimation error is introduced, and the Lagrange–d’Alembert principle is applied to the Lagrangian with this dissipation. A Lyapunov analysis shows that the state estimation scheme so obtained provides stable asymptotic convergence of state estimates to actual states in the absence of measurement noise, with an almost global domain of attraction. These estimation schemes are discretized for computer implementation using discrete variational mechanics. A first order Lie group variational integrator is obtained as a discrete-time implementation. In the presence of bounded measurement noise, numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states.