Part IV
Advanced Topics
Introduction to Part IV
In this last part of the textbook we present some advanced issues on robot
control. We deal with topics such as control without velocity measurements
and control under model uncertainty. We recommend this part of the text for
a second course on robot dynamics and control or for a course on robot control
at the first year of graduate level. We assume that the student is familiar with
the notion of functional spaces, i.e. the spaces L
2
and L
∞
. If not, we strongly
recommend the student to read first Appendix A, which presents additional
mathematical baggage necessary to study these last chapters:
• P“D” control with gravity compensation and P“D” control with desired
gravity compensation;
• Introduction to adaptive robot control;
• PD control with adaptive gravity compensation;
• PD control with adaptive compensation.
13
P“D” Control with Gravity Compensation and
P“D” Control with Desired Gravity
Compensation
Robot manipulators are equipped with sensors for the measurement of joint
positions and velocities, q and
˙
q respectively. Physically, position sensors may
be from simple variable resistances such as potentiometers to very precise
optical encoders. On the other hand, the measurement of velocity may be
realized through tachometers, or in most cases, by numerical approximation

d
dt
, the com-
ponents of ϑ ∈ IR
n
are given by
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ϑ
1
ϑ
2
.
.
.
ϑ
n
⎤
⎥
⎥
⎥
⎥

.
.
.
.
.
.
.
.
.
.
.
00···
b
n
p
p + a
n
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢

b
i
p
p + a
i

q
where a
i
and b
i
are strictly positive real constants but otherwise arbitrary, for
i =1, 2, ···,n.
A state-space representation of Equation (13.1) is
˙
x = −Ax −ABq
ϑ = x + Bq
where x ∈ IR
n
represents the state vector of the filters, A = diag{a
i
} and
B = diag{b
i
}.
In this chapter we study the proposed modification for the following con-
trollers:
• PD control with gravity compensation and
• PD control with desired gravity compensation.
Obviously, the derivative part of both control laws is no longer proportional

q.
13.1 P“D” Control with Gravity Compensation 293
A possible modification to the PD control law with gravity compensation
consists in replacing the derivative part (D), which is proportional to the
derivative of the position error, i.e. to the velocity error
˙
˜
q =
˙
q
d
−
˙
q,bya
term proportional to
˙
q
d
− ϑ
where ϑ ∈ IR
n
is, as said above, the result of filtering the position q by means
of a dynamic system of first-order and of zero relative degree.
Specifically, the P“D” control law with gravity compensation is written as
τ = K
p
˜
q + K
v
[

Σ
Σ
Σ ROBOT
g(q)
K
v
K
p
B
ϑ
˙
q
d
q
d
τ
q
(pI +A)
−1
AB
Figure 13.1. Block-diagram: P“D” control with gravity compensation
Define ξ = x + Bq
d
. The equation that describes the behavior in closed
loop may be obtained by combining Equations (III.1) and (13.2)–(13.3), which
may be written in terms of the state vector

ξ
T
˜

⎢
⎢
⎣
−Aξ + AB
˜
q + B
˙
q
d
˙
˜
q
¨
q
d
− M(q)
−1
[K
p
˜
q + K
v
[
˙
q
d
− ξ + B
˜
q] −C(q,
˙

as
τ = K
p
˜
q − K
v
diag

b
i
p
p + a
i

q + g(q),
which is close to the PD with gravity compensation control law (7.1), when
the desired position q
d
is constant. Indeed the only difference is replacement
of the velocity
˙
q by
diag

b
i
p
p + a
i


as
d
dt
⎡
⎢
⎢
⎢
⎣
ξ
˜
q
˙
q
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
−Aξ + AB
˜
q
−
˙
q

is constant, constitutes an autonomous
differential equation. Moreover, the origin

ξ
T
˜
q
T
˙
q
T

T
= 0 ∈ IR
3n
is the
unique equilibrium of this equation.
With the aim of studying the stability of the origin, we consider the Lya-
punov function candidate
V (ξ,
˜
q,
˙
q)=K(q,
˙
q)+
1
2
˜
q

the robot. Notice that the diagonal matrix K
v
B
−1
is positive definite. Con-
sequently, the function V (ξ,
˜
q,
˙
q) is globally positive definite.
The total time derivative of the Lyapunov function candidate yields
˙
V (ξ,
˜
q,
˙
q)=
˙
q
T
M(q)
¨
q +
1
2
˙
q
T
˙
M(q)

˜
q and M(q)
¨
q, and
canceling out some terms we obtain
˙
V (ξ,
˜
q,
˙
q)=−[ξ −B
˜
q]
T
K
v
B
−1
A [ξ −B
˜
q]
= −
⎡
⎣
ξ
˜
q
˙
q
⎤

˙
q
T

1
2
˙
M(q) −C(q,
˙
q)

˙
q =0,
which follows from Property 4.2.
Clearly, the time derivative
˙
V (ξ,
˜
q,
˙
q) of the Lyapunov function candidate
is globally negative semidefinite. Therefore, invoking Theorem 2.3, we con-
clude that the origin of the closed-loop Equation (13.4) is stable and that all
solutions are bounded.
Since the closed-loop Equation (13.4) is autonomous, La Salle’s Theorem
2.7 may be used in a straightforward way to analyze the global asymptotic
stability of the origin (cf. Problem 3 at the end of the chapter). Neverthe-
less, we present below, an alternative analysis that also allows one to show
global asymptotic stability of the origin of the state-space corresponding to
the closed-loop Equation, (13.4). This alternative method of proof, which is

→ 0 ∈ IR
3n
as t →∞(for all
initial conditions), the origin is a globally asymptotically stable equilibrium.
It is precisely this property that we show next.
In the development that follows we use additional properties of the dy-
namic model of robot manipulators. Specifically, assume that q,
˙
q ∈ L
n
∞
.
Then,
296 P“D” Control
• M(q)
−1
,
d
dt
M(q) ∈ L
n×n
∞
• C(q,
˙
q)
˙
q ∈ L
n
∞
.

˙
q
•
1
2
˜
q
T
K
p
˜
q
•
1
2
[ξ − B
˜
q]
T
K
v
B
−1
[ξ − B
˜
q] .
Since the time derivative
˙
V (ξ,
˜

q ∈ L
n
∞
and also that
C(q
d
−
˜
q,
˙
q)
˙
q is bounded for all q
d
,
˜
q,
˙
q ∈ L
n
∞
, it follows that the time deriva-
tive of the state vector is also bounded, i.e.
˙
ξ,
˙
˜
q,
¨
q ∈ L


d
dt
M(q)

M(q)
−1
[K
p
˜
q − K
v
[ξ − B
˜
q] −C(q,
˙
q)
˙
q]
+ M(q)
−1

K
p
˙
˜
q − K
v

˙

M(q)

M(q)
−1
.
In (13.7) and (13.8) we have already concluded that ξ,
˜
q,
˙
q,
˙
ξ,
˙
˜
q,
¨
q ∈ L
n
∞
then, from the properties stated at the beginning of this analysis, we obtain
¨
ξ,
¨
˜
q, q
(3)
∈ L
n
∞
, (13.10)

q. Using (13.13), (13.7), (13.9)
and (13.11), we get from this lemma
lim
t→∞

˙
ξ(t) −B
˙
˜
q(t)

= 0 .
Consequently, using the closed-loop Equation (13.4) we get
lim
t→∞
−A[ξ(t) −B
˜
q(t)] + B
˙
q = 0 .
From this expression and (13.13) we obtain
lim
t→∞
˙
q(t)=0 ∈ IR
n
. (13.14)
Now, we show that lim
t→∞
˜

n
. (13.15)
298 P“D” Control
The last part of the proof, that is, the proof of lim
t→∞
ξ(t)=0 follows
trivially from (13.13) and (13.15). Therefore, the origin is a globally attractive
equilibrium point.
This completes the proof of global asymptotic stability of the origin of the
closed-loop Equation (13.4).
We present next an example with the purpose of illustrating the perfor-
mance of the Pelican robot under P“D” control with gravity compensation.
As for all other examples on the Pelican robot, the results that we present are
from laboratory experimentation.
Example 13.1. Consider the Pelican robot studied in Chapter 5, and
depicted in Figure 5.2. The components of the vector of gravitational
torques g(q) are given by
g
1
(q)=(m
1
l
c1
+ m
2
l
1
)g sin(q
1
)+m

} = diag{30} [Nm/rad] ,
K
v
= diag{k
v
} = diag{7, 3} [Nm s/rad] ,
A = diag{a
i
} = diag{30, 70} [1/s] ,
B = diag{b
i
} = diag{30, 70} [1/s] .
The components of the control input τ are given by
τ
1
= k
p
˜q
1
− k
v
ϑ
1
+ g
1
(q)
τ
2
= k
p

2
q
2
ϑ
1
= x
1
+ b
1
q
1
ϑ
2
= x
2
+ b
2
q
2
.
The initial conditions corresponding to the positions, velocities and
states of the filters, are chosen as
q
1
(0) = 0,q
2
(0) = 0
˙q
1
(0) = 0, ˙q

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....

.. ...

state is
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ξ(0)
˜
q(0)
˙
q(0)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢

0.3141
0.1047
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Figure 13.2 presents the experimental results and shows that the
components of the position error
˜
q(t) tend asymptotically to a small
nonzero constant. Although we expected that the error would tend
to zero, the experimental behavior is mainly due to the presence of
unmodeled friction at the joints. ♦
In a real implementation of a controller on an ordinary personal computer
(as is the case of Example 13.1) typically the joint position q is sampled
periodically by optical encoders and this is used to compute the joint velocity
˙
q. Indeed, if we denote by h the sampling period, the joint velocity at the
instant kh is obtained as
˙
q(kh)=
q(kh) −q(kh −h)
h

compensation, studied in Chapter 7, and whose characteristic is that it does
not require the velocity term
˙
q in its control law. The original references on
this controller are cited at the end of the chapter.
This controller, that we call here P“D” control with desired gravity com-
pensation, is described by
τ = K
p
˜
q + K
v
[
˙
q
d
− ϑ]+g(q
d
) (13.16)
˙
x = −Ax −ABq
ϑ = x + Bq (13.17)
where K
p
,K
v
∈ IR
n×n
are diagonal positive definite matrices, A = diag{a
i

p
K
v
q
d
˙
q
d
Figure 13.3. Block-diagram: P“D” control with desired gravity compensation
13.2 P“D” Control with Desired Gravity Compensation 301
Comparing P“D” control with gravity compensation given by (13.2)–(13.3)
with P“D” control with desired gravity compensation (13.16)–(13.17), we im-
mediately notice the replacement of the term g(q)bythefeedforward term
g(q
d
).
The analysis of the control system in closed loop is similar to that from
Section 13.1. The most noticeable difference is in the Lyapunov function con-
sidered for the proof of stability. Given the relative importance of the controller
(13.16)–(13.17), we present next its complete study.
Define ξ = x + Bq
d
. The equation that describes the behavior in closed
loop is obtained by combining Equations (III.1) and (13.16)–(13.17), which
may be expressed in terms of the state vector

ξ
T
˜
q

⎣
−Aξ + AB
˜
q + B
˙
q
d
˙
˜
q
¨
q
d
−M(q)
−1
[K
p
˜
q+K
v
[
˙
q
d
−ξ+B
˜
q]+g(q
d
)−C(q,
˙

p
˜
q − K
v
diag

b
i
p
p + a
i

q + g(q
d
),
which is very close to PD with desired gravity compensation control law (8.1)
when the desired position q
d
is constant. The only difference is the substitution
of the velocity term
˙
q by
ϑ = diag

b
i
p
p + a
i



ξ
T
˜
q
T
˙
q
T

T
as
d
dt
⎡
⎢
⎢
⎢
⎣
ξ
˜
q
˙
q
⎤
⎥
⎥
⎥
⎦
=

q−g(q
d
−
˜
q)]
⎤
⎥
⎥
⎥
⎦
(13.18)
which, since q
d
is constant, is an autonomous differential equation. Since
the matrix K
p
has been picked so that λ
min
{K
p
} >k
g
, then the origin

ξ
T
˜
q
T
˙

v
B
−1
(ξ − B
˜
q) (13.19)
where
K(q
d
−
˜
q,
˙
q)=
1
2
˙
q
T
M(q
d
−
˜
q)
˙
q
f(
˜
q)=U(q
d

, we have from Lemma 8.1 that f(
˜
q)isa
(globally) positive definite function of
˜
q. Consequently, the function V (ξ,
˜
q,
˙
q)
is also globally positive definite.
The time derivative of the Lyapunov function candidate yields
˙
V (ξ,
˜
q,
˙
q)=
˙
q
T
M(q)
¨
q +
1
2
˙
q
T
˙

K
v
B
−1

˙
ξ − B
˙
˜
q

.
Using the closed-loop Equation (13.18) to solve for
˙
ξ,
˙
˜
q and M(q)
¨
q, and
canceling out some terms, we obtain
˙
V (ξ,
˜
q,
˙
q)=−(ξ −B
˜
q)
T

v
A 0
000
⎤
⎦
⎡
⎣
ξ
˜
q
˙
q
⎤
⎦
(13.20)
where we used (cf. Property 4.2)
13.2 P“D” Control with Desired Gravity Compensation 303
˙
q
T

1
2
˙
M(q) −C(q,
˙
q)

˙
q =0.


ξ(t)
T
˜
q(t)
T
˙
q(t)
T

T
→ 0 ∈ IR
3n
when t →∞(for all initial conditions),
i.e. the equilibrium is globally attractive, then the origin is a globally asymp-
totically stable equilibrium. It is precisely this property that we show next.
In the development below we invoke further properties of the dynamic
model of robot manipulators. Specifically, assuming that q,
˙
q ∈ L
n
∞
we have
• M(q)
−1
,
d
dt
M(q) ∈ L
n×n

The Lyapunov function V (ξ,
˜
q,
˙
q) given in (13.19) is positive definite and
is composed of the sum of the following three non-negative terms
•
1
2
˙
q
T
M(q)
˙
q
•U(q
d
−
˜
q) −U(q
d
)+g(q
d
)
T
˜
q +
1
2
˜

˙
q) is bounded along
trajectories. Therefore, the three non-negative listed terms above are also
bounded along trajectories. Since, moreover, the potential energy U(q)of
robots having only revolute joints is always bounded in its absolute value, it
follows that
˙
q,
˜
q, ξ, ξ − B
˜
q ∈ L
n
∞
. (13.21)
Incorporating this information in the closed-loop Equation (13.18), and
knowing that M(q
d
−
˜
q)
−1
and g(q
d
−
˜
q) are bounded for all q
d
,
˜

q ∈ L
n
∞
, (13.22)
and therefore, it is also true that

˙
ξ − B
˙
˜
q

∈ L
n
∞
. (13.23)
Using again the closed-loop Equation (13.18), we can compute the second
time derivative of the variables state to obtain
¨
ξ = −A
˙
ξ + AB
˙
˜
q
¨
˜
q = −
¨
q

p
˙
˜
q − K
v

˙
ξ − B
˙
˜
q

−
d
dt
(C(q,
˙
q)
˙
q)+
d
dt
g(q)

where q
(3)
denotes the third time derivative of the joint position q and we
used
d
dt

¨
ξ,
¨
˜
q, q
(3)
∈ L
n
∞
, (13.24)
and therefore also
¨
ξ − B
¨
˜
q ∈ L
n
∞
. (13.25)
On the other hand, from the time derivative
˙
V (ξ,
˜
q,
˙
q), expressed in
(13.20), we get
ξ − B
˜
q ∈ L

q(t)] + B
˙
q = 0 .
From this last expression and since we showed in (13.27) that lim
t→∞
ξ(t)−
B
˜
q(t)=0 it finally follows that
lim
t→∞
˙
q(t)=0 . (13.28)
We show next that lim
t→∞
˜
q(t)=0 ∈ IR
n
. Using again Lemma A.6 with
(13.28), (13.21), (13.22) and (13.24) we have
lim
t→∞
¨
q(t)=0 .
Taking this into account in the closed-loop Equation (13.18) as well as
(13.27) and (13.28), we get
lim
t→∞
M(q
d

This completes the proof of global attractivity of the origin and, since we
have already shown that the origin is Lyapunov stable, of global asymptotic
stability of the origin of the closed-loop Equation (13.18).
We present next an example that demonstrates the performance that may
be achieved with P“D” control with gravity compensation in particular, on
the Pelican robot.
Example 13.2. Consider the Pelican robot presented in Chapter 5 and
depicted in Figure 5.2. The components of the vector of gravitational
torques g(q) are given by
g
1
(q)=(m
1
l
c1
+ m
2
l
1
)g sin(q
1
)+m
2
l
c2
g sin(q
1
+ q
2
)

j





= n((m
1
l
c1
+ m
2
l
1
)g + m
2
l
c2
g)
=23.94

kg m
2
/s
2

.
Consider the P“D” control with desired gravity compensation for
this robot in position control. Let the design matrices K
p

1
= k
p
˜q
1
− k
v
ϑ
1
+ g
1
(q
d
)
τ
2
= k
p
˜q
2
− k
v
ϑ
2
+ g
2
(q
d
)
˙x

1
ϑ
2
= x
2
+ b
2
q
2
.
The initial conditions corresponding to the positions, velocities and
states of the filters, are chosen as
q
1
(0) = 0,q
2
(0) = 0
˙q
1
(0) = 0, ˙q
2
(0) = 0
x
1
(0) = 0,x
2
(0) = 0 .
The desired joint positions are
q
d1

⎢
⎢
⎢
⎢
⎣
b
1
π/10
b
2
π/30
π/10
π/30
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢

˜q
2
t [s]
.
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.

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.......

.
T
˙
q
T

T
,isa
globally asymptotically stable equilibrium . Consequently, for any initial
condition q(0),
˙
q(0) ∈ IR
n
,wehavelim
t→∞
˜
q(t)=0 ∈ IR
n
.
Consider the P“D” control with desired gravity compensation for n-DOF
robots. Assume that the desired position q
d
is constant.
• If the matrices K
p
, K
v
, A and B of the controller P“D” with desired
gravity compensation are taken diagonal positive definite, and such that
λ
min

Bibliography
Studies of motion control for robot manipulators without the requirement of
velocity measurements, started at the beginning of the 1990s. Some of the
early related references are the following:
• Nicosia S., Tomei P., 1990, “Robot control by using joint position mea-
surements”, IEEE Transactions on Automatic Control, Vol. 35, No. 9,
September.
• Berghuis H., L¨ohnberg P., Nijmeijer H., 1991, “Tracking control of robots
using only position measurements”, Proceedings of IEEE Conference on
Decision and Control, Brighton, England, December, pp. 1049–1050.
• Canudas C., Fixot N., 1991, “Robot control via estimated state feedback”,
IEEE Transactions on Automatic Control, Vol. 36, No. 12, December.
• Canudas C., Fixot N.,
˚
Astr¨om K. J., 1992, “Trajectory tracking in robot
manipulators via nonlinear estimated state feedback”, IEEE Transactions
on Robotics and Automation, Vol. 8, No. 1, February.
• Ailon A., Ortega R., 1993, “An observer-based set-point controller for robot
manipulators with flexible joints”, Systems and Control Letters, Vol. 21,
October, pp. 329–335.
The motion control problem for a time-varying trajectory q
d
(t) without ve-
locity measurements, with a rigorous proof of global asymptotic stability of
the origin of the closed-loop system, was first solved for one-degree-of-freedom
robots (including a term that is quadratic in the velocities) in
• Lor´ıa A., 1996, “Global tracking control of one degree of freedom Euler-
Lagrange systems without velocity measurements”, European Journal of
Control, Vol. 2, No. 2, June.
This result was extended to the case of n-DOF robots in

˜
q
ϑ = x + B
˜
q
where K
p
,K
v
∈ IR
n×n
are diagonal positive definite matrices, A =
diag{a
i
} and B = diag{b
i
} with a
i
,b
i
real strictly positive numbers.
Assume that the desired joint position q
d
∈ IR
n
is constant.
a) Obtain the closed-loop equation expressed in terms of the state vector

x
T

c) Show that the origin of the closed-loop equation is a stable equilibrium
point.
Hint: Use the following Lyapunov function candidate
2
:
V (x,
˜
q,
˙
q)=
1
2
˙
q
T
M(q)
˙
q +
1
2
˜
q
T
K
p
˜
q
+
1
2

It is assumed that only the vector of positions θ of motors axes, but not
the velocities vector
˙
θ, is measured. We require that q(t) → q
d
, where q
d
is constant.
A variant of the P“D” control with desired gravity compensation is
3
τ = K
p
˜
θ − K
v
ϑ + g(q
d
)
˙
x = −Ax −ABθ
ϑ = x + Bθ
where
˜
θ = q
d
− θ + K
−1
g(q
d
)

d
dt
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ξ
˜
q
˜
θ
˙
q
˙
θ
⎤
⎥
⎥
⎥
⎥
⎥

θ
M(q)
−1

−K(
˜
θ −
˜
q)+g(q
d
) −C(q,
˙
q)
˙
q − g(q)

J
−1

K
p
˜
θ − K
v
(ξ − B
˜
θ)+K(
˜
θ −
˜

{K
p
} >k
g
and λ
min
{K} >k
g
, then the origin is a
stable equilibrium point.
Hint: Use the following Lyapunov function and La Salle’s Theorem
2.7.
3
This controller was proposed and analyzed in Kelly R., Ortega R., Ailon A.,
Loria A., 1994, “Global regulation of flexible joint robots using approximate dif-
ferentiation”, IEEE Transactions on Automatic Control, Vol. 39, No. 6, June, pp.
1222–1224.
Problems 311
V (
˜
q,
˜
θ,
˙
q,
˙
θ)=V
1
(
˜

B
−1

ξ − B
˜
θ

where
V
1
(
˜
q,
˜
θ,
˙
q,
˙
θ)=
1
2
˙
q
T
M(q)
˙
q +
1
2
˙

2
(
˜
q)=U(q
d
−
˜
q) −U(q
d
)+
˜
q
T
g(q
d
)
and verify that
˙
V (
˜
q,
˜
θ,
˙
q,
˙
θ)=−

ξ − B
˜

Obviously, while these considerations allow one to prove certain stability
and convergence properties for the controllers studied in previous chapters,
they must be taken with care. In robot control practice, either of these assump-
tions or both, may not hold. For instance, we may be neglecting considerable
joint elasticity, friction or, even if we think we know accurately the masses
and inertias of the robot, we cannot estimate the mass of the objects carried
by the end-effector, which depend on the task accomplished.
Two general techniques in control theory and practice deal with these
phenomena, respectively: robust control and adaptive control. Roughly, the
first aims at controlling, with a small error, a class of robot manipulators
with the same robust controller. That is, given a robot manipulator model,
one designs a control law which achieves the motion control objective, with a
small error, for the given model but to which is added a known nonlinearity.
Adaptive control is a design approach tailored for high performance ap-
plications in control systems with uncertainty in the parameters. That is,
uncertainty in the dynamic system is assumed to be characterized by a set
of unknown constant parameters. However, the design of adaptive controllers
requires the precise knowledge of the structure of the system being controlled.
314 14 Introduction to Adaptive Robot Control
Certainly one may consider other variants such as adaptive control for
systems with time-varying parameters, or robust adaptive control for systems
with structural and parameter uncertainty.
In this and the following chapters we concentrate specifically on adaptive
control of robot manipulators with constant parameters and for which we
assume that we have no structural uncertainties. In this chapter we present
an introduction to adaptive control of manipulators. In subsequent chapters
we describe and analyze two adaptive controllers for robots. They correspond
to the adaptive versions of
• PD control with adaptive desired gravity compensation,
• PD control with adaptive compensation.

To emphasize the dependence of the dynamic model on the dynamic pa-
rameters, from now on we write the dynamic model (14.1) explicitly as a
function of the vector of unknown dynamic parameters, θ, that is
2
,
1
Under the ideal conditions of rigid links, no elasticity at joints, no friction and
having actuators with negligible dynamics.
2
In this textbook we have used to denote the joint positions of the motor shafts
for models of robots with elastic joints. With an abuse of notation, in this and the