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Biomimetic and Biologically Inspired Control 415

16.4.1 Impedance Control

Let us formulate a robot’s dynamic equation in contact task space as

::
I r ðxÞx þ C r ðx, ˙ xÞ˙ x ¼ u r f (16:18)

where I r xðÞ is the robot’s inertia matrix, c r x,˙ xð Þ˙ x is the centrifugal and Coriolis force vector,
respectively. u r is the robot’s control input vector and f is the contact force from the environment.
For simplicity, let us consider the environmental dynamics as
::
M e x þ D e ˙ x þ K e x ¼ f (16:19)

where M e , D e , and K e are the environmental inertia, viscosity, and stiffness, respectively, and is
assumed unknown. It is well known that by specifying nonlinear compensation of

1
u r ¼ f þ M I r (x)[u D r ˙ x k r x] þ c r (x,˙ x)˙ x (16:20)
r
the robot’s dynamics becomes
::
M r x þ D r ˙ x þ K r x ¼ u f (16:21)

where M r , D r , and K r are the robot’s inertia, viscosity, and stiffness, respectively, and u(t) is a new
control input of the robot that we should design late.
In impedance control, we usually control the robot’s position and use the force feedback to
adjust the robot’s mechanical impedance as seen from the environment so as to keep a compliant
contact with the environment (Hogan, 1985; Luo and Ito, 1993).
In detail, the control input is designed as

u ¼ C x ðx d xÞþ C f f

so that the robot dynamics as seen from the environment be as

M rd x þ D rd ˙ x þ K rd ðx x d Þ¼ f
x

where C x and C f are the robot’s position and force feedback controllers, respectively.
From the stability point of view, we usually require the robot to be passive with respect to the
environmental interactions. Passivity is defined as the property that the system does not flow energy
to outside. The robot’s passivity as seen from its environment or the manipulated object is very
useful for the stable and safely mechanical interaction. When applying impedance control, if the
desired position x d is constant, then the robot is passive. However, if the x d changes with respect to
time, then the robot may lose the passivity as seen from the environment.
Inorder fortherobot torealizethe passivity while performingthetime varying interactions, Li and
Horowitz(1999)proposedapassivevelocityfieldcontrol(PVFC),theyalsosuggestedtoapplyPVFC
to control a human interactive robot and smart exercise machines. Unlike the passivity based control
scheme by Slotine and Li (1991), in which they considered the passivity of a tracking error system,
PVFC remains passive of the robot with respect to the external environment by adding a virtual
flywheel to exchange the mechanical energy with the real robot. However, PVFC has the following
two main problems. Firstly, when specifying desired velocity vector field, PVFC does not consider
theuncertaintiesoftheenvironmentalgeometricconstraints.Secondly,although PVFC maintainsthe