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3.2 Decoupled PD-control with
Feedforward
To negate the major drawback of the decoupled PD-
control an additional feedforward term is inserted into
control law (4), yielding
Q
m
= K
P
(q
d
− q) + K
D
(
˙
q
d
−
˙
q) + u
FF
. (5)
Effectively exploiting the system knowledge, ob-
tained through the dynamic modeling and identifi-
cation, the necessary reference motor torques which
guide the tool center point along a desired trajectory,
are given with the equations of motion (1) and are so
intuitively suitable for a feedforward control
u
FF
= M(q
d
)
¨
q
d
+ G(q
d
,
˙
q
d
)
˙
q
d
+ Q(q
d
,
˙
q
d
). (6)
Obviously, if the reference trajectory is two times
continuously differentiable with respect to time, the
resulting feedforward torques will show a continu-
ous progression. This set of feedforward control is
also called exact feedforward linearization in litera-
ture, see (Hagenmeyer and Delaleau, 2003).
With this choice the PD-controller’s task is re-
duced to compensate modeling inaccuracies, param-
eter variations and external or unknown disturbances.
Proving asymptotic stability, however, still leads
to further challenges, because inserting the feedfor-
ward term and evaluating the equations of motion
transforms the dynamics to a nonlinear time-variant
system. The interested reader may find more infor-
mation in (Kugi, 2008; Hagenmeyer and Delaleau,
2003).
3.3 Computed Torque
Introducing the computed torque control law
Q
m
= M(q)
¨
v+ G(q,
˙
q)
˙
q+ Q(q,
˙
q) (7)
and inserting it into the equations of motion (1) yields
the system
¨
q = v (8)
of n double integrators, resulting in a solely linear in-
terrelation between the new system input v and the
joint angles q. A possible choice for the new system
input v is
v =
¨
q
d
+ K
D
(
˙
q
d
−
˙
q) + K
P
(q
d
− q) (9)
with the positive definite diagonal matrices K
P
and
K
D
as stabilizing gains.
By defining the joint tracking error e = q
d
−q and
inserting Equation (9) in (8) the system dynamics of
the tracking error e for the closed loop system
¨
q
d
−
¨
q+ K
D
(
˙
q
d
−
˙
q) + K
P
(q
d
− q) = 0
¨
e+ K
D
˙
e+ K
P
e = 0 (10)
is found. Equation (10) instantly allows to shape the
error dynamics by choosing the gains K
P
and K
D
, for
example by pole placement.
Please note, in literature the computed torque
method is also referred to as inverse dynamics,
feedback linearization or flatness based control, as
(Isidori, 1985; Slotine and Li, 1990; Fliess et al.,
1995) describe.
3.4 Extended Linearization
Similar to Subsection 3.3, the nonlinearities in the
equations of motion are eliminated with the control
law
Q
m
= M(q)(
¨
v+ K
0
q+ K
1
˙
q)
+ G(q,
˙
q)
˙
q+ Q(q,
˙
q). (11)
Inserting the control law (11) into the equations of
motion (1) yields
¨
q = v− K
0
q− K
1
˙
q (12)
or equivalent in state space representation
d
dt
q
˙
q
=
0 E
−K
0
−K
1
q
˙
q
+
0,
E
v,
(13)
which describes the dynamics of the closed loop sys-
tem. Apparently, this MIMO-system is controllable
with the whole repertory of tools available from linear
system theory, for example linear quadratic regulators
or pole placement, well summarized in (Chen, 1998).
Certainly, before designing a linear controller for
system (13), the matrices K
0
and K
1
need to be cho-
sen. It is suggested to pick them in a manner so
that the closed loop system (13) resembles its phys-
ical counterpart. One suggestion is that the eigenval-
ues of the linearized equations of motion may be used
as guideline for picking the eigenvalues of the closed
loop system. However, due to stability issues positive
eigenvalues in the closed loop system (13) need to be
avoided.
Asymptotic stability of Equation (13) is derived
with linear system theory.
3.5 Friction Feedback Issues
One common problem with feedback linearization
methods occurs when friction models are part of the
inverse dynamics. This is usually the case for any me-
chanical or robotic application. If the classical model
for friction
Q
fric
= d
v
˙q+ d
c
signum ( ˙q) (14)
is part of the feedback loop, then the noisy measure-
ment of the velocity signal ˙q will be amplified and fed
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