Stability of Haptic Displays

Haptic displays can be considered to be devices which generate mechanical impedances. ‘Impedance’ here is defined as a dynamic relationship between velocity and force. The behavior of the haptic display depends on the virtual environment being rendered. For instance, if the desired behavior is that of a point mass, the haptic display must exert forces proportional to acceleration. Similarly, if the desired behavior is that of a spring, the haptic display must exert forces proportional to displacement [Colgate and Brown 94]. Passivity has proved to be a useful tool for studying both the stability and performance of haptic displays. A one-port system is passive if the integral of the power extracted over time does not exceed the initial energy stored in the system. For a translational mechanical system, power is the product of force (f) and velocity (ẋ), with the sign convention that power is positive when energy flows into the system. Typically, the initial energy is defined to be zero, resulting in the following inequality: