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In this paper, motion of a micro-robot with two perpendicular vibrational actuators is studied. In this work, separation of the micro-robot body from the substrate of motion is modeled. If the applied vertical force to the micro-robot body is greater than its weight, body of the micro-robot jumps apart from the substrate and then returns to it. This motion will continue intermittently until it is damped. In this condition, the motion mechanism of "stick-slip" is not valid, and a hybrid motion according to "stick-slip-jump" mode is governed. By increasing the applied vertical force, the motion velocity of the micro-robot becomes disordered and unrepeatable. By numerical solving of the equation of motion and investigating the dynamic response, we find the micro-robot motion is chaotic, related to the restitution factor of the motion substrate and the ratio of vertical force to micro-robot weight.

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A novel partial differential equation observer is proposed to be used in boundary attitude and vibration control of flexible satellites. Solar panels’ vibrations and attitude dynamics form a coupled partial differential equation–ordinary differential equation system which is controlled directly without discretization. Few feedback signals from boundaries are required which are estimated via a partial differential equation observer. Consequently, just satellite attitude and angular velocity should be measured and still the control system benefits information from continuous part vibrations. The closed-loop system is proved to be asymptotically stable. Simulations with a finite element technique illustrate good performance of this observer-based boundary controller.