The tracking performance of mobile robot is often affected by uncertainties from the deviation of initial conditions, external disturbances and varying loads, etc. An Udwadia-Kalaba based adaptive robust control is proposed for the trajectory tracking of an omnidirectional mobile robot in the presence of uncertainties. The proposed control includes nominal control part based on Udwadia-Kalaba theory and adaptive robust control part. The desired trajectory is considered as a virtual servo constraint applied to the robot system and converted into the second order standard form. So that the analytical form of constraint force could be obtained via Udwadia-Kalaba Fundamental Equation (UKFE). The system will precisely obey the given constraint (i.e., the desired trajectory) under the obtained constraint force in ideal cases. No auxiliary variables are required and it is effective whether the constraints are holonomic or nonholonomic. The designed adaptive law is in leakage type and the adaptive parameters are adjusted according to the performance of the system in order to compensate for the effect caused by uncertainty in the system. No extra information of uncertainty is needed except for the existence of uncertainty bound. Comparing with PID control, it can be found that the proposed control has better performance and can realize higher precision trajectory tracking control.

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The clearances, which universally exist in the joint parts of mechanical systems, are a main cause of various operation deteriorating problems, including noises generating, material wearing, even the early fatigue of members. In this paper, a new method to determine the clearance magnitude in the beam support is proposed with the description of a numerical example. The growing clearance will increase the vibration period of realistic system, and the accurateness of measurement is relevant to the frequency resolution of the instrument. Through the time series of the midpoint response of a simply supported beam, the period of the first modal free vibration can be obtained using the FFT Algorithm. Only two responses from different initial conditions are needed to identify the magnitude of the clearance. It is estimated the accurateness could be below 15%. The investigation also suggests this free vibration period approach may be adapted to handle the energy losses situations with the damping ratio considered in the formulas.

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