  Purpose: This paper presents an impedance control method with mixed
$H_2/H_\infty$ synthesis and relaxed passivity for a cable-driven series
elastic actuator to be applied for physical human-robot interaction.
  Design/methodology/approach: To shape the system's impedance to match a
desired dynamic model, the impedance control problem was reformulated into an
impedance matching structure. The desired competing performance requirements as
well as constraints from the physical system can be characterized with
weighting functions for respective signals. Considering the frequency
properties of human movements, the passivity constraint for stable human-robot
interaction, which is required on the entire frequency spectrum and may bring
conservative solutions, has been relaxed in such a way that it only restrains
the low frequency band. Thus, impedance control became a mixed $H_2/H_\infty$
synthesis problem, and a dynamic output feedback controller can be obtained.
  Findings: The proposed impedance control strategy has been tested for various
desired impedance with both simulation and experiments on the cable-driven
series elastic actuator platform. The actual interaction torque tracked well
the desired torque within the desired norm bounds, and the control input was
regulated below the motor velocity limit. The closed loop system can guarantee
relaxed passivity at low frequency. Both simulation and experimental results
have validated the feasibility and efficacy of the proposed method.
  Originality/value: This impedance control strategy with mixed $H_2/H_\infty$
synthesis and relaxed passivity provides a novel, effective and less
conservative method for physical human-robot interaction control.
