Rigid body motion estimation based on the Lagrange-d'Alembert principle

Abstract

Stable estimation of rigid body pose and velocities from noisy measurements, without any knowledge of the dynamics model, is treated using the Lagrange-d'Alembert principle from variational mechanics. With body-fixed optical and inertial sensor measurements, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in velocity estimation error and the sum of two artificial potential functions; one obtained from a generalization of Wahba's function for attitude estimation and another which is quadratic in the position estimate error. An additional dissipation term that is linear in the velocity estimation error is introduced, and the Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation. This estimation scheme is discretized using discrete variational mechanics. The presented pose estimator requires optical measurements of at least three inertially fixed landmarks or beacons in order to estimate instantaneous pose. The discrete estimation scheme can also estimate velocities from such optical measurements. In the presence of bounded measurement noise in the vector measurements, numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states. 88003 USA. {mi,sashi}@nmsu.edu 2 A. K. Sanyal is with the

Finite time stable attitude estimation of rigid bodies with unknown dynamics

“…Thus the estimated value of the attitude converges to actual attitude and angular velocity values in finite time from almost all initial estimates except from a set of zero measure. Similarly, angular velocity measurements could also have errors, that is, Ω m will be as in Equation (22). Due to the almost global finite time stable nature of the estimation scheme, the convergence from almost any bounded initial condition to the desired equilibrium configuration (Q, ) = (I, 0) in finite time is guaranteed if there are no measurement errors as shown in Theorem 1.…”

“…The set SO(3) is a matrix manifold and forms a Lie group under the operation of matrix multiplication. Attitude estimation schemes based on the Lagrange-d'Alembert principle of variational mechanics was first proposed in [20] and subsequently developed in [21,22]. The shortcomings such as unwinding and singularities can be avoided by making use of the unique global representation of attitude given by SO (3), to design the estimation scheme.…”

“…Complimentary filters proposed in are one of the most commonly used set of filter‐like algorithms for attitude estimation. Attitude estimation schemes based on the Lagrange‐d'Alembert principle of variational mechanics was first proposed in and subsequently developed in .…”

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

“…Hence, the derivative of the Morse-Lyapunov function is negative semi-definite. Note that the error dynamics for the pose estimate error h is given by (11), while the error dynamics for the velocities estimate error ϕ is given by (46). Note that D(t), as a function of time, is piecewise continuous and uniformly bounded.…”

Symmetry Reduction in Optimal Control of Multiagent Systems on Lie Groups

“…Methods for trajectory tracking and estimation algorithms for pose and attitude of mechanical systems evolving on Lie groups are commonly employed for improving accuracy on simulations, as well as to avoid singularities by working with coordinate-free expressions in the associated Lie algebra of the Lie group to describe behaviors in multi-agent systems [13] (i.e., a set of equations depending on an arbitrary choice of the basis for the Lie algebra). More recently, this framework has been used for cooperative transportation [19].…”