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Variational estimation of a mechanical system is based on the application of variational principles from mechanics to state estimation of the system evolving on its configuration manifold. If the configuration manifold is a Lie group, then the underlying group structure can be used to design nonlinearly stable observers for estimation of configuration and velocity states from measurements. Measured quantities are on a vector space on which the Lie group acts smoothly. We formulate the design of variational observers on a general finite-dimensional Lie group, followed by the design and experimental evaluation of a variational observer for rigid body motions on the Lie group SE(3). 

We consider estimation of motion on spheres as a variational problem. The concept of variational estimation for mechanical systems is based on application of variational principles from mechanics, to state estimation of mechanical systems evolving on configuration manifolds. If the configuration manifold is a symmetric space, then the overlying connected Lie group of which it is a quotient space, can be used to design nonlinearly stable observers for estimation of configuration and velocity states from measurements. If the configuration manifold is a sphere, then it can be globally represented by an unit vector. We illustrate the design of variational observers for mechanical systems evolving on spheres, through its application to estimation of pointing directions (reduced attitude) on the regular sphere S^2. 

This article presents an extended state observer for a vehicle modeled as a rigid body in three-dimensional translational and rotational motions. The extended state observer is applicable to a multi-rotor aerial vehicle with a fixed plane of rotors, modeled as an under-actuated system on the state-space TSE(3), the tangent bundle of the six-dimensional Lie group SE(3). This state-space representation globally represents rigid body motions without singularities. The extended state observer is designed to estimate the resultant external disturbance force and disturbance torque acting on the vehicle. It guarantees stable convergence of disturbance estimation errors in finite time when the disturbances are constant, and finite time convergence to a bounded neighborhood of zero errors for time-varying disturbances. This extended state observer design is based on a Hölder-continuous fast finite time stable differentiator that is similar to the super-twisting algorithm, to obtain fast convergence. Numerical simulations are conducted to validate the proposed extended state observer. The proposed extended state observer is compared with other existing research to show its advantages. A set of experimental results implementing disturbance rejection control using feedback of disturbance estimates from this extended state observer is also presented.