Modeling and Control of Geared Servomechanisms

T.K. Shing; Advisors, Prof. L.W. Tsai and Prof. P.S. Krishnaprasad


A major problem associated with the use of gear trains for power transmission is backlash. Backlash between meshing gear teeth can cause impact, reduce system stability, generate noise and undesired vibrations. Uncertainty caused by backlash can also decrease the repeatability and accuracy of geared servomechanisms. Most recent studies on backlash have concentrated on models of the instantaneous impact phenomena of a simple gear pair of such complexity that the models are not suitable for the purpose of control. Another major problem associated with geared servomechanisms is friction. Typical errors caused by friction in geared servomechanisms are steady-state error and tracking error. Even though friction models have been widely studied, most of the work on the effects of friction in geared servomechanisms is primarily concerned with friction in journal bearings which, at low speed, may have only a minor role in comparison with the meshing friction between gear teeth. Furthermore, the resulting models are not robust enough to accurately predict the performance of a servomechanism in all situations. Thus, application engineers have to adopt specialized friction model for each mechanism to obtain satisfactory friction compensation for a desired task. Also, the control methods implemented on geared servomechanisms usually are conventional feedback laws, such as PD controllers, which do not consider the effects of backlash. Therefore inaccuracy and tracking errors cannot be avoided in such systems. Control engineers have pursued various strategies to overcome such problems but still with some drawbacks. Therefore our goals are to establish a simplified dynamic model for real time control and find a good control strategy to achieve high precision.


First, a new dynamic model which accounts for backlash effects is proposed for the dynamics of spur gear systems. This dynamic model is mainly developed for the purpose of real time control. The complicated variation of the meshing stiffness as a function of contact point along the line of action is studied. Then the mean value is used as the stiffness constant in the improved model. To further include both backlash and friction effects, another new model is proposed for the dynamics of a spur gear system. The model estimates average friction torque and uses it to replace the instantaneous friction torque to simplify the dynamical equations of motion. The average friction model will reduce the original thirteen cases of operation into three cases. As for the control strategy, a new open-loop optimization-based control strategy is developed here. This new controller is expected to achieve high precision. But with load disturbance from the environment, a state feedback compensation for small corrective actions becomes necessary. Therefore, a systematic method to find such a state feedback law is proposed as follows. First, a simplified linear model is used for estimating feedback gains through H design due to the complexity of the real system. Later, the obtained data is used as starting points and surface plots around each such point are made to help obtain the "best" feedback gains.


Results and Future Work From the simulation results, the proposed models are judged to be more realistic for real time control of electromechanical systems to reduce gear noise and to achieve high precision. The proposed control strategy also works well in simulation. Therefore, our future work will primarily concentrate on experimental verification. A design of a suitable experiment is under way.