  An impedance-based control scheme is introduced for cooperative manipulators
grasping a rigid load. The position and orientation of the load are to be
maintained close to a desired trajectory, trading off tracking accuracy by low
energy consumption and maintaining stability. To this end, the augmented
dynamics of the robots, their actuators and the load is formed, and an
impedance control is adopted. A virtual control strategy is used to decouple
torque control from actuator control. An optimization problem is then
formulated using energy balance equations. The optimization finds the damping
and stiffness gains of the impedance relation such that the energy consumption
is minimized. Furthermore, L2 stability techniques are used to allow for
time-varying damping and stiffness in the desired impedance. A numerical
example is provided to demonstrate the results.
