Novel techniques for robotic systems based on filtering Lie groups and large deviations

Abstract

The dynamics profile of the robot is modelled with the Gaussian and Poisson processes as a non-desirable (noise) input along with the desired input using Lie groups and Lie algebra to incorporate various advantages of Lie groups. Further, the configuration of each link at any time is described by its rotation relative to the preceding link. Using this formulation, elegant formula for the kinetic energy of the d or n degree of freedom (DOF) system is obtained as a quadratic form in the angular velocities, with coefficients being highly nonlinear trigonometric functions of the angles. Properties of the Lie algebra generators and the Lie adjoint map are used to arrive at this expression. Further, the gravitational potential energy and the torque potential energy are expressed as nonlinear trigonometrical functions of the angles using the SO(3) group properties. The input torque comprises a nonrandom intentional torque component and a highly nonlinear tremor torque component. Both coordinate dependent torque and coordinate-free torque based Lagrange equations of motion (EOM) are formulated. Two models of nondesirable/non-intentional motion are developed. First, the non-desirable tremor torque is modelled as a stochastic differential equation (SDE) satisfying Ornstein-Uhlenbeck (OU) process with diffusion and damping coefficients. In this the tremor is treated as the disturbance. Second, the jerky torque is modelled as a unique feature with superposition of compound Poisson processes. The state vector of the robot, i.e., angular position and angular velocity vector, is thus a Markov process whose transition probability generator can be expressed in terms of the rate of the compound Poisson process that defines the jerky torque. newline newlineFurther, for the disturbance/tremor estimation, the extended Kalman filter (EKF) is applied to get a more accurate state estimation provided by the usual disturbance observer. The proposed optimal UKFOC provides state estimation and control simultaneously, omitting the system s need for