LMI-based impedance control synthesis: a case study on fixed-stiffness actuators (FSAs)

Impedance control is frequency-dependent in the twofold sense: (1) the impedance matching error can only be attenuated arbitrarily small over a low-frequency range due to the water bed effect (Freudenberg et al., 2003), and (2) the impedance matching error always diverges to infinity as the frequency tends to infinity since no causal controller can track arbitrarily fast signals. To sum up, we can only shape the robot impedance over some low-frequency range which may be very small due to the limitation resulting from the robot's inherent high-frequency characteristics. Unfortunately, this fact was underestimated in the existing impedance control schemes. In this paper, a frequency-aware linear impedance control synthesis technique is developed for FSAs based on H∞ control theory, which is also suitable for the general robot system in the condition that the robot dynamics is linearised around an equilibrium configuration or compensated with its inverse dynamics model to form an exact-linearised system a priori. With the powerful tool, we study the impedance shaping technique and address the two long-standing issues: (1) Under what conditions, an impedance-controlled FSA can exceed its physical stiffness? (2) Can a stiff FSA render a wider stiffness range than its softer counterparts?