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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. 

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). 

This work analyzes and develops some fundamental results for attitude consensus control of a network of rigid-body vehicles, considered a multi-agent rigid body system (MARBS). The system is analyzed using a full rigid body dynamics model on TSO(3) for each vehicle (agent) in the network. Therefore, the state space of the system is TSO(3)^N, where N is the number of vehicles. Attitude synchronization control laws for each vehicle to reach a consensus attitude with zero angular velocity for a particular type of network are obtained, using a Morse-Lyapunov function. Some fundamental results on equilibria of the network under these attitude consensus control laws are obtained. We show that unlike cooperative control of multi-agent systems with highly simplified dynamics models for agents, like point particles or unicycles where the state space of the dynamics is modeled as a vector space, there are multiple equilibrium solutions possible for attitude consensus control laws for a MARBS with dynamics on TSO(3)^N. Further, the number of equilibria depends on the network graph topology. This is followed by numerical simulation results for two different network graphs, which show this network control framework to be effective in obtaining attitude consensus.