Here deals with a controller design for an acceleration-controlled system. Let us consider a system that has a sensors for position $x$ and force $f$, and the transfer function of an acceleration controller

whose states get an initial value of 0.

A transfer function from an input to output is a double integral system. Putting position feedback-controller $C_{\rm p}$ on this system, a transfer function gets

In general, position control is performed using feedforward control, because a trajectry is given in advance. A feeddback controller should be designed in compliance with the inner-model principle.

A force controller makes an action force gets a desired value, i.e., a reaction force gets that. Let us consider an environment that returns a reaction force depending on a strain caused by pushing:

Installing a force feedback-controller $C_{\rm f}$ enables a reaction control track a reference:

A force controller works as it reduces a virtual inertia of a system and suppresses an ineartial force in a contact motion.

An application requires both position-based task and force-based task. A position/force hybrid controller is composed of two controllers, and generate a reference as

Pure position control and force control can be available by disabling either.

Besides that, enabling both controllers and setting force command to 0 realizes compliant position control.

This index indicates a virtual impedance arround an equilibrium point. It is called a control stiffness. Adjusting a control stiffness is important in terms of controlling energy exchange between connecting systems, because a work is a result of that phenomenon. Such a concept can be seen in vibration-suppression control that requires the impedance matching between a drive-side (input) and load-side (output).

A contact motion sometimes excites a vibration of an environment. To realize a stable contact, a system should dissipate a residual energy circulating in an environment. This can be achieved by adding a viscosity to a system and indirectly providing a dissipation pass of energy to an environment in a contact. Let us consider an environment that returns a reaction force depending on a strain caused by pushing

and install a controller

The responses are calculated as

Here, a control stiffness is set to

It should be noted that velocity feedback certainly suppresses a vibration but degrades a back-drivability.