A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.

In this paper, we propose a motion control system for a low-cost differential drive mobile robot. The robotic platform is equipped with two driven wheels powered by Beckhoff motors, instrumented with incremental encoders. The control system is designed and implemented using Beckhoff's TwinCAT 3 automation software, running on an industrial PC. The system is tested and experimentally tuned to achieve optimal performance. The method allows addressing both odometry motion accuracy and motion correction in order to obtain minimum trajectory errors. Test results on linear and angular robot trajectories show errors below 0.02 and 0.03%, respectively, after tuning of the motion parameters. The proposed approach can be expanded, tweaked and applied to other differential drive TwinCAT 3 based robotic solutions. This will contribute to expanding mobile robot applications to a variety of fields, such as industrial automation, logistics, warehouse management, health care, ocean and space exploration and a variety of other industrial and non-industrial activities.