Position-insensitive estimation of mass from limit cycle of velocity feedback relay system with Spring

Mass measurement using relay feedback of velocity and restoring force compensation is investigated for determining the mass of an object under weightless conditions. In the measurement system, the velocity of the object is fed back through a relay with hysteresis and the force acting on the object is switched from a positive value to a negative value when the velocity reaches a positive threshold and vice versa. As a result, a limit cycle is induced in the measurement system and the mass is estimated based on the period of the limit cycle. In addition, restoring force compensation with a spring is introduced to avoid the drift of the trajectory. This compensation makes the static equilibrium state unique. However, the trajectory still drifts slightly. It causes some error in measurement when a simple formula of estimating mass is applied. To eliminate such an error, a new formula is derived to estimate the mass independently of the position of the trajectory that is determined by the switching positions in the relay actions. When the switching positions deflect from the origin at which the spring is in the natural length, the trajectory is not at the center and becomes asymmetric. It is analytically shown that the period of the limit cycle is minimum when the switching positions are at the origin. It indicates that mass is overestimated with the simple estimation formula when the trajectory is not at the center. The validity of the modified formula and the analytical results are confirmed experimentally.