Impedance Control

Impedance Control

Consider a mechanical operation where we push against a spring that has constant stiffness.

Here, the value of the force completely determines the displacement; similarly the value of the displacement completely determines the force. It follows that, in this example, we are unable to control the force and the displacement independently at the same time.

Also, it is not possible, in this example, to apply a command force that has an arbitrarily specified relationship with displacement.

In other words, stiffness control is not possible.

Now suppose that we push against a complex dynamic system, not a simple spring element.

In this case, we should be able to command a pushing force in response to the displacement of the dynamic system so that the ratio of force to displacement varies in a desired manner.

This is a stiffness control (or compliance control) action. Dynamic stiffness is defined as the ratio (output force)/(input displacement), expressed in the frequency domain .

Mechanical impedance is defined as the ratio (output force)/ (input velocity), in the frequency domain.

 Note that stiffness and impedance both relate force and motion variables in a mechanical system.

The objective of impedance control is to make the impedance function equal to some specified function (without separately controlling or independently constraining the associated force variable and velocity variable).

Force control and motion control can be considered extreme cases of impedance control (and stiffness control). Specifically, since the objective of force control is to keep the force variable from deviating from a desired level, in the presence of independent variations of the associated motion variable (an input), force control can be considered zero-impedance control if velocity is chosen as the motion variable (or zero-stiffness control if displacement is chosen as the motion variable). Similarly, displacement control can be considered infinite stiffness control and velocity control can be considered infinite-impedance control. Impedance control has to be accomplished through active means, generally by generating forces as specified functions of associated motions.

Impedance control is particularly useful in mechanical manipulation against physical constraints, which is the case in assembly and machining tasks. In particular, very high impedance is naturally present in the direction of a motion constraint, and very low impedance is naturally present in the direction of a free motion.

Problems that arise by using motion control in applications where small motion errors would create large forces can be avoided if stiffness control or impedance control is used.

Furthermore, the stability of the overall system can be guaranteed and the robustness of the system improved by properly bounding the values of impedance parameters.

Impedance control is useful as well in tasks of fine/flexible manipulation, for example, in the processing of flexible and inhomogeneous natural material such as meat. In this case, the mechanical impedance of the task interface (i.e., in the region where the mechanical processor interacts with the processed object) will provide valuable characteristics of the process, which can be used in fine control of the processing task.

 Since impedance relates an input velocity to an output force, it is a transfer function.

 The concepts of impedance control can be applied to situations where the input is not a velocity and the output is not a force.

Still, the term “impedance control” is used, even though the corresponding transfer function is, strictly speaking, not impedance.