Global positioning of robot manipulators with mixed revolute and prismatic joints

Abstract

The existing controllers for robot manipulators with uncertain gravitational force can globally stabilize only robot manipulators with revolute joints. The main obstacles to the global stabilization of robot manipulators with mixed revolute and prismatic joints are unboundedness of the inertia matrix and the Jacobian of the gravity vector. In this note, a class of globally stable controllers for robot manipulators with mixed revolute and prismatic joints is proposed. The global asymptotic stabilization is achieved by adding a nonlinear proportional and derivative term to the linear proportional-integral-derivative (PID) controller. By using Lyapunov's direct method, the explicit conditions on the controller parameters to ensure global asymptotic stability are obtained.

Full text

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Global Positioning of Robot Manipulators with Mixed Revolute and Prismatic Joints
Josip Kasac, Branko Novakovic, Dubravko Majetic and Danko Brezak
Faculty of Mechanical Engineering and Naval Architecture
University of Zagreb, I. Lucica 5, HR-10000 Zagreb, Croatia
e-mail: {josip.kasac, branko.novakovic, dubravko.majetic, danko.brezak}@fsb.hr
Phone: +385-1-6168-{375, 354, 348, 357}
Fax: +385-1-6168-351
Corresponding author: Josip Kasac
Abstract—The existing controllers for robot manipulators with
uncertain gravitational force can globally stabilize only robot
manipulators with revolute joints. The main obstacles to the
global stabilization of robot manipulators with mixed revolute
and prismatic joints are unboundedness of the inertia matrix and
the Jacobian of the gravity vector. In this paper a class of globally
stable controllers for robot manipulators with mixed revolute and
prismatic joints is proposed. The global asymptotic stabilization
is achieved by adding a nonlinear proportional and derivative
term to the linear PID controller. By using Lyapunov’s direct
method, the explicit conditions on the controller parameters to
ensure global asymptotic stability are obtained.
Index Terms—Position control, manipulators, robot dynamics,
stability.
I. INTRODUCTION
It is well known that a PD plus gravity compensation
controller can globally asymptotically stabilize a rigid-joints
manipulator with both revolute and prismatic joints [1]. This
approach has drawbacks since the gravitational torque vector,
which depends on some usually uncertain parameters, is
assumed to be known accurately. To overcome the parametric
uncertainties on the gravitational torque vector, an adaptive
version of PD controller has been introduced in [2], guaran-
teeing global asymptotic stability. The main drawback of this
approach is that the gravity regressor matrix has to be known.
On the other hand, most industrial robots are controlled by
linear PID controllers which do not require any component
of robot dynamics into its control law. A simple linear and
decoupled PID feedback controller with appropriate control
gains achieves the desired position without any steady-state
error. This is the main reason why PID controllers are still
used in industrial robots. The local asymptotic stability of the
linear PID controller in a closed loop with robot manipulator is
proved in [3],[4] and semiglobal asymptotic stability is shown
in [5], [6]. By looking at the proof, it can be seen that the cubic
term in the derivative of the Lyapunov’s function hampers the
global asymptotic stability. This is the reason to believe that
linear PID control is inadequate to cope with highly nonlinear
systems like robot manipulators, since the design of the linear
PID control law is based solely on local arguments.
The first nonlinear PID controller which ensures global
asymptotic stability (GAS) is proposed in [7]. In this work,
This work was supported by the National Scientific Foundation of Republic
of Croatia under Grant No. 0120025, ”Application of Artificial Intelligence
in Robotics and Manufacturing Systems”.
which was inspired by the results of [2], it is proven that
global convergence is still preserved if the regressor matrix is
replaced by the constant matrix. Since the regressor matrix is
constant, the control law can be interpreted as a nonlinear PID
controller which achieves GAS by normalized nonlinearity in
the integrator term of the control law. The second approach to
achieving GAS is the scheme of Arimoto [8] that uses a satu-
ration function in the integrator to render the system globally
asymptotically stable, just as the normalization did in [7]. A
unified approach to both above mentioned controllers, which
have a linear derivative term, linear or saturated proportional
term, and a class of nonlinear integral action is given in [9].
An alternative approach to global asymptotic stabilization
of robot manipulator is ”delayed PID” (PIdD) [5]. PIdD can
be understood as a simple PD controller to which an integral
action is added after some transient of time. The idea of this
approach consists of ”patching” a global and a local controller.
The first drives the solutions to an arbitrarily small domain,
while the second yields local asymptotic stability.
None of the proposed controllers takes into account the
effects of actuators saturation. Recently, several saturated
PID controllers for robot manipulators with uncertain grav-
itational force have been reported: a semiglobal saturated
linear PID controller [10] and two global saturated nonlinear
PID controllers [11], [12]. Also, progress is made in the
adaptive set point control of robot manipulators. In [13] an
adaptive set point controller with uncertain gravity regressor
is proposed and global asymptotic convergence is proved. In
[14] a semiglobal adaptive controller with amplitude-limited
torque and uncertainty in the kinematic and dynamic models
is developed.
All of the mentioned nonlinear PID controllers can globally
or semiglobally stabilize robot manipulators with revolute
joints only. In this paper an approach to GAS of robot manip-
ulators with mixed revolute and prismatic joints is presented.
In this approach GAS is achieved by adding a nonlinear
proportional and derivative term to the linear PID controller.
Explicit conditions on controller parameters which guarantee
GAS are given.
Throughout the paper we use the notation: ?x? for the
Euclidean norm of the vector x ∈ Rn, λM{A} and λm{A}
for the maximal and minimal eigenvalues, respectively, of the
symmetric positive definite matrix A, and I for the identity
matrix of the appropriate dimension.
This paper is organized as follows. Robot dynamics and its

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main properties are presented in Section II. The main results
are presented in Section III, where a class of PID controllers
with nonlinear proportional and derivative gains is introduced
and conditions for global asymptotic stability are established.
The simulation results are presented in Section IV. Finally, the
concluding remarks are emphasized in Section V.
II. ROBOT DYNAMICS
The model of n-link rigid-body robotic manipulator, in the
absence of friction and disturbances, is represented by
M(q)¨ q + C(q, ˙ q)˙ q + g(q) = u,
(1)
where q is the n × 1 vector of robot joint coordinates, ˙ q is
the n × 1 vector of joint velocities, u is the n × 1 vector of
applied joint torques and forces, M(q) is n×n inertia matrix,
C(q, ˙ q)˙ q is the n×1 vector of centrifugal and Coriolis torques
and g(q) is the n×1 vector of gravitational torques and forces,
obtained as the gradient of the robot potential energy U(q)
g(q) =∂U(q)
∂q
.
(2)
We assume that the matrix C(q, ˙ q) is defined using the
Christoffel symbols.
The following properties of the robot dynamics with mixed
revolute and prismatic joints are important for stability analysis
(see e.g. [6], [15], [16], [17]).
Property 1. The matrix˙M(q)−2C(q, ˙ q) is skew-symmetric,
i.e.,
zT(˙M(q) − 2C(q, ˙ q))z = 0,
Property 2. The inertia matrix M(q) is a positive definite
symmetric matrix which satisfies
∀z ∈ Rn.
(3)
a1?z?2≤ zTM(q)z ≤ (a2+ c2?q? + d2?q?2)?z?2,
for all z,q ∈ Rn, where a1,a2> 0, c2,d2≥ 0.
Property 3. The Coriolis and centrifugal terms C(q, ˙ q)˙ q
satisfy
?C(q, ˙ q)˙ q? ≤ (c1+ d1?q?)?˙ q?2,
for all q, ˙ q ∈ Rn, where c1,d1≥ 0.
Property 4. There exist such positive constants kg1and kg2
that the Jacobian of the gravity vector satisfies
????
depends linearly on translational coordinates, while rotational
coordinates appear only in trigonometric form with period 2π.
Property 5. (See the Appendix A) For any constant vector
of desired joint positions qd∈ Rn, the functions
G(˜ q) =1
2
D(˜ q) =¯kg?˜ q?2+ kg2?˜ q?3+ ˜ qT(g(q) − g(qd)),
are globally positive definite with respect to ˜ q ∈ Rn, where
˜ q = q − qddenotes the joint position error vector, and
¯kg= kg1+ kg2?qd?.
(4)
(5)
∂g(q)
∂q
????≤ kg1+ kg2?q?,
∀ q ∈ Rn.
(6)
The property (6) follows from the fact that the potential energy
¯kg?˜ q?2+1
3kg2?˜ q?3+ U(q) − U(qd) − ˜ qTg(qd),(7)
(8)
(9)
The assumptions (4)-(6) are valid for all practically used
robot manipulators. If the robot has no prismatic joints, then
d1,d2,c2,kg2= 0, and the inertia matrix and the Jacobian of
the gravity vector are uniformly bounded. In that case we get
well known properties of the robot manipulators with revolute
joints only, where a1= λm{M}, a2= λM{M}, c1= kcand
kg1= kg(see e.g. [18], [19], [20]).
From a practical point of view, the prismatic joints of robot
manipulators are always limited by physical limits and the
uniform bounds exist in (4)-(6) within the robot workspace.
However, as pointed in [16], with the uniform bounds within
the robot workspace the entire stability analysis becomes local,
and it is necessary to know the domain of attraction. It is often
very difficult to explicitly characterize the domain of attraction
that could be much smaller then the robot workspace. So, a
global result is always more useful and exact characterization
of upper bounds in (4)-(6) for all q ∈ Rnis crucial in
establishing global stability.
III. A CLASS OF NONLINEAR PID CONTROLLERS
A. Main Result
Proposition 1: Consider the robot dynamics (1) in closed
loop with nonlinear PID controller
u = −KP˜ q − KD˙ q − KIν − ΨP(˜ q)˜ q − ΨD(˜ q)˙ q, (10)
˙ ν = ˜ q,
(11)
where KP, KD and KI are n × n constant positive definite
symmetric matrices, ΨP(˜ q) and ΨD(˜ q) are n × n positive
definite diagonal matrix functions which can be written in the
following form
ΨP(˜ q) = k(1)
ΨD(˜ q) = k(1)
P?˜ q?I,
D?˜ q?I + k(2)
Dare positive constants.
(12)
D?˜ q?2I,
(13)
where k(1)
If the following conditions are satisfied
P, k(1)
Dand k(2)
k(1)
P> kg2,
k1=λm{KP} −¯kg> 0,
k1
λM{KI}>max
where¯kgis defined by (9) and
¯ m = a2+ c2?qd? + d2?qd?2,
¯ m1= c2+ 2d2?qd?,
¯kc= c1+ d1?qd?,
then the equilibrium [˜ qT
globally asymptotically stable. ?
(14)
(15)
?
¯ m
λm{KD},
¯ m1+¯kc
k(1)
D
,d1+ d2
k(2)
D
?
, (16)
(17)
(18)
(19)
˙ qT(ν + K−1
Ig(qd))T]T= 0 is
B. Proof of Main Result
The stability analysis is based on Lyapunov’s direct method,
and can be divided in four parts. First, error equations for
the closed-loop system (1), (10), and (11) are determined.
Second, the Lyapunov function (LF) candidate is proposed.
Then, a global stability criterion on system parameters is
established. Finally, the LaSalle invariance principle is invoked
to guarantee the asymptotic stability.

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1) Error Equations: The stationary state of the system (1),
(10), (11) is ˜ q = 0, ˙ q = 0, ν = ν∗, and ν∗satisfies g(qd) =
−KIν∗.
If a new variable z = ν −ν∗is introduced, then the system
(1), (10), (11) becomes
M(q)¨ q + C(q, ˙ q)˙ q + g(q) − g(qd) = u,
u = −KP˜ q − KD˙ q − KIz − ΨP(˜ q)˜ q − ΨD(˜ q)˙ q,
˙ z = ˜ q,
(20)
(21)
(22)
whose origin [˜ qT
equilibrium.
2) Lyapunov Function Candidate: We will now construct
a Lyapunov function for the system (20)-(22). First, an output
variable y = ˙ q + α˜ q with some α > 0 is introduced, and
the inner product between (20) and y is made, resulting in
a nonlinear differential form which can be separated in the
following way
dV (˜ q, ˙ q,z)
dt
where V (˜ q, ˙ q,z) is the Lyapunov function candidate (see the
Appendix B).
For easier determination of conditions for positive definite-
ness of function V and W, the following decompositions
are made: V (˜ q, ˙ q,z) = V1(˜ q, ˙ q) + V2(˜ q,z) and W(˜ q, ˙ q) =
W1(˜ q, ˙ q) + W2(˜ q), where
V1(˜ q, ˙ q) =1
+1
3αk(1)
V2(˜ q,z) =1
+1
3k(1)
and
˙ qT
zT]T
= 0 ∈ R3nis the unique
= −W(˜ q, ˙ q),
(23)
2˙ qTM(q)˙ q + α˜ qTM(q)˙ q +1
D?˜ q?3+1
2αzTKIz + ˜ qTKIz +1
2α˜ qTKD˜ q +
4αk(2)
D?˜ q?4,
(24)
2˜ qTKP˜ q +
P?˜ q?3+ U(q) − U(qd) − ˜ qTg(qd),
(25)
W1(˜ q, ˙ q) = −α˙ qTM(q)˙ q + ˙ qT(KD+ ΨD(˜ q))˙ q +
+ α˜ qT(˙M(q) − C(q, ˙ q))˙ q,
W2(˜ q) = ˜ qT(αKP− KI)˜ q +
+ αk(1)
(26)
P?˜ q?3+ α˜ qT(g(q) − g(qd)).
(27)
In this way, the problem of determining conditions for posi-
tive definiteness of function V (˜ q, ˙ q,z), which contains three
variables, is transformed into two simpler problems of deter-
mination of conditions for positive definiteness of functions
V1(˜ q, ˙ q) and V2(˜ q,z), which contain only two variables. The
second advantage of the above mentioned decomposition of
functions V and W is the elimination of unspecified constant
α from the final stability conditions.
3) Stability criterion determination: The following step is
determination of conditions for positive definiteness of the
function V and positive semi-definiteness of the function W.
First, we consider function V1, which can be rearranged in
the following form
V1=1
+1
3αk(1)
2(˙ q + α˜ q)TM(q)(˙ q + α˜ q) −1
2α˜ qTKD˜ q +1
2α2˜ qTM(q)˜ q +
4αk(2)
D?˜ q?3+1
D?˜ q?4,
(28)
and using property (4) we get
V1≥1
−1
+ a1?˙ q + α˜ q?2≥ 0.
Using the triangle inequality ?q? ≤ ?˜ q?+?qd?, and rearrang-
ing the previous expression we get
2α(λm{KD} + k(1)
2α2(a2+ c2?q? + d2?q?2)?˜ q?2+
D?˜ q? + k(2)
D?˜ q?2)?˜ q?2−
(29)
V1≥1
+1
2α(λm{KD} − α¯ m)?˜ q?2+1
2α(k(2)
where ¯ m and ¯ m1are defined by (17) and (18), respectively.
The function V1is positive definite if the following conditions
are satisfied
2α(k(1)
D− α¯ m1)?˜ q?3+
D− αd2)?˜ q?4+ a1?˙ q + α˜ q?2,
(30)
λm{KD}
¯ m
> α,
k(1)
D
¯ m1
> α,
k(2)
D
d2
> α.
(31)
Further, we consider function V2, which can be rearranged in
the following form
?√αz +
+1
3k(1)
−1
Applying property (7) we get
?
+ G(˜ q) ≥ 0,
which is positive definite if the conditions
V2=1
2
1
√α˜ q
?T
KI
?√αz +
1
√α˜ q
?
+
P?˜ q?3+1
α˜ qTKI˜ q.
2˜ qTKP˜ q + U(q) − U(qd) − ˜ qTg(qd) −
(32)
V2≥1
2
k1−1
αλM{KI}
?
?˜ q?2+1
3
?
k(1)
P− kg2
?
?˜ q?3+
(33)
α >λM{KI}
k1
,
(34)
and (14) are satisfied, where k1is defined by (15). Comparing
(34) with (31), the following conditions for positive definite-
ness are obtained
k1λm{KD} > λM{KI}¯ m,
k1k(1)
D
> λM{KI}¯ m1,
k1k(2)
D
> λM{KI}d2,
(35)
(36)
(37)
including condition (14). Note that, in the above-stated condi-
tions, the unspecified positive constant α is eliminated.
The following step is the condition which ensures that the
time derivative of LF is a negative semi-definite function, i.e.
W ≥ 0. First, we consider function W1. Applying properties
(4) and (5) we get
W1≥ (λm{KD} + k(1)
− α(a2+ c2?q? + d2?q?2)?˙ q?2−
− α(c1+ d1?q?)?˜ q??˙ q?2≥ 0.
D?˜ q? + k(2)
D?˜ q?2)?˙ q?2−
(38)

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Using triangle inequality ?q? ≤ ?˜ q? + ?qd?, and rearranging
the previous expression we get
W1≥ [λm{KD} − α¯ m]?˙ q?2+
+ [k(1)
+ [k(2)
where¯kc is defined by (19). The function W1 is positive
definite if the following conditions are satisfied
k(1)
D
(¯ m1+¯kc)> α,
Further, we consider function W2. Using property (8) we
get
D− α(¯ m1+¯kc)]?˜ q??˙ q?2+
D− α(d1+ d2)]?˜ q?2?˙ q?2,
(39)
λm{KD}
¯ m
> α,
k(2)
D
(d1+ d2)> α. (40)
W2≥ (αk1− λM{KI})?˜ q?2+ α
+ αD(˜ q) ≥ 0,
which is positive definite if the conditions (34) and (14) are
satisfied.
Comparing (34) with (40) the following conditions for
positive definiteness are obtained
k1λm{KD} > λM{KI}¯ m,
k1k(1)
D
k1k(2)
D
> λM{KI}(d1+ d2),
including condition (14)-(15). Also, in the above-stated con-
ditions, the unspecified positive constant α is eliminated.
Notice that the conditions (35)-(37) are trivially implied by the
conditions (42)-(44). So, the conditions (42)-(44), including
(14)-(15), are the final stability conditions which guaranty
global stability. Finally, by invoking the LaSalle invariance
principle, we conclude asymptotic stability.
The conditions (42)-(44) can be represented in a more
compact form by the expression (16).
Remark 1. From the stability conditions it follows that the
choice of the controller gains depends on the norm of the
desired position ?qd?. In other words, larger gains should be
chosen if the desired position qdis larger. This property of the
stability conditions is consequence of the fact that the inertia
matrix and the Jacobian of the gravity vector are not uniformly
bounded. The larger values of the robot position q increase
the upper bounds of the mentioned matrices, implying also
larger values of the controller gains in the stability conditions.
Replacing ?qd? from expressions (9), (17)-(19) with the norm
of the maximal desired position of robot manipulators ?qd?M,
we get the stability conditions which are valid for any qdin
the robot workspace.
Remark 2. The control law (10)-(13) can also be applied
to robot manipulators with all revolute joints. In that case,
parameters kg2, d1, c2, d2vanish and the following simplified
control law
?
k(1)
P− kg2
?
?˜ q?3+
(41)
(42)
(43)
> λM{KI}(¯ m1+¯kc),
(44)
u = −KP˜ q − KD˙ q + k(1)
provides global asymptotic stability [21], if the following
condition is satisfied
D?˜ q?˙ q − KI
?t
0
˜ q(τ)dτ,
(45)
λm{KP} − kg
λM{KI}
> max
?
λM{M}
λm{KD},
kc
k(1)
D
?
.
(46)
???
0x
0y
1x
1
cl
1q
2 q
1
m
2
m
1y
g
g
Fig. 1.The robot manipulator with revolute and prismatic joints.
So, the control law (45) provides an alternative approach to
the set-point control of robot manipulators with all revolute
joints without using saturation functions.
IV. SIMULATION EXAMPLE
We consider a 2-DOF manipulator with revolute and pris-
matic joints [22], [16], as shown in Fig. 1. The matrices M(q),
C(˙ q,q) and vector g(q) in (1) are given by
?m1l2
?
?m1lc1g cosq1+ m2gq2cosq1
where q = [q1q2]T, q1is the revolute joint variable and q2is
the prismatic joint variable.
Inserting (47), (48) and (49) in expressions (4), (5) and (6)
and using definitions and properties of matrix norms, we get
c1= 0, d1=√2m2, c2= 0, d2= m2,
a2= max{m1l2
kg1= max{2m1lc1g,2m2g}, kg2= 2m2g.
The numerical values of the model parameters are: m1= 1 kg,
m2= 1 kg, lc1= 0.8 m, I1+ I2= 0.1 kgm2.
Initial conditions are q1(0) = ˙ q1(0) = ˙ q2(0) = 0, and
q2(0) = 1 m. The final desired positions are q1d= π/2 rad
and q2d= 2 m.
The controller gains are chosen in accordance with stability
conditions (14)-(16) as
KP= diag{200,200}, KD= diag{20,20},
KI= diag{300,300}, k(1)
Fig. 2. shows the position errors ˜ q1and ˜ q2obtained from
simulation. It is obvious from the figure that, after a transient
due to error in initial condition, the position errors tend
asymptotically to zero.
M(q) =
c1+ m2q2
2+ I1+ I2
0
0
m2
?
?
,
(47)
C(˙ q,q) =
m2˙ q2q2
−m2˙ q1q2
m2˙ q1q2
0
,
(48)
g(q) =
m2g sinq1
?
,
(49)
c1+ I1+ I2,m2},
P
= 20, k(1)
D= 20, k(2)
D= 10.

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5
01234
−2
−1.5
−1
−0.5
0
0.5
Time (sec)
q1 − q1d (rad)
q2 − q2d (m)
Fig. 2.The position errors.
V. CONCLUSIONS
In this paper a class of globally stable controllers for robot
manipulators with mixed revolute and prismatic joints has
been presented. The stability criterion in terms of Lyapunov’s
direct method is proposed to guarantee the global asymptotic
stability. The stability criterion provides explicit conditions on
the controller gains in terms of a few parameters extracted
from the robot dynamics.
Of course, some open problems remain to be solved and
are currently under research. The first one is the evaluation
of the robustness of the proposed controller in the face of
measurement noise and joint flexibility. The second problem
is the extension of the proposed controller with actuator
constraints and velocity estimation through the filtered position
measurement, following ideas reported in [11], [5]. Some
preliminary results guarantee only semiglobal asymptotic sta-
bility.
APPENDIX A
This Appendix presents a proof of Property 5 following
ideas reported in [2] and [9].
Notice first that G(0) = 0. The gradient of G(˜ q) with respect
to ˜ q is given by
F(˜ q) =∂G(˜ q)
∂˜ q
where we used (2). The gradient F(˜ q) vanishes at ˜ q = 0, thus
G(˜ q) has a critical point at ˜ q = 0. The Hessian matrix of G(˜ q)
with respect to ˜ q is given by
H(˜ q) =¯kgI + kg2?˜ q?I + kg2˜ q ˜ qT
The Hessian is a positive definite matrix because of
zTH(˜ q)z = (¯kg+ kg2?˜ q?)?z?2+ kg2(zT˜ q)2
?
=¯kg˜ q + kg2?˜ q?˜ q + g(˜ q + qd) − g(qd), (50)
?˜ q?+∂g(˜ q + qd)
∂[˜ q + qd].
(51)
?˜ q?
?
+ zT∂g(q)
∂q
z
≥
kg1+ kg2?q? −
????
∂g(q)
∂q
????
?z?2≥ 0,
(52)
for all z,q ∈ Rn, where we used triangle inequality ?q? ≤
?˜ q? + ?qd?, and property (6).
Therefore, function G(˜ q) is a globally strictly convex func-
tion vanishing at the unique global minimum ˜ q = 0. This
implies that G(˜ q) is a globally positive definite function which
holds for any constant qd. Further, positive definiteness of
the Hessian H(˜ q) implies that the gradient F(˜ q) is a strictly
increasing vector function with respect to ˜ q, i.e. ˜ qTF(˜ q) =
D(˜ q) ≥ 0.
APPENDIX B
This Appendix presents the construction of the Lyapunov
function V = V1+ V2defined by (24) and (25). Taking inner
product between (20) and output variable y = ˙ q + α˜ q we get
the following nonlinear differential form
˙ qTM(q)¨ q + ˙ qTC(q, ˙ q)˙ q + ˙ qT(g(q) − g(qd)) +
+˙ qTKP˜ q + ˙ qTKD˙ q + ˙ qTKIz +
+k(1)
+αk(1)
+α˜ qTM(q)¨ q + α˜ qTC(q, ˙ q)˙ q + α˜ qT(g(q) − g(qd)) +
+α˜ qTKP˜ q + α˜ qTKD˙ q + α˜ qTKIz = 0.
P?˜ q?˙ qT˜ q + k(1)
P?˜ q?3+ αk(1)
D?˜ q??˙ q?2+ k(2)
D?˜ q?˜ qT˙ q + αk(2)
D?˜ q?2?˙ q?2+
D?˜ q?2˜ qT˙ q +
(53)
Some terms in the differential form (53) can be represented
in the following way
˜ qTM(q)¨ q =
d
dt
d
dt
d
dt
d
dt
d
dt
d
dt
?˜ qTM(q)˙ q?− ˙ qTM(q)˙ q − ˜ qT ˙M(q)˙ q,(54)
2˙ qTM(q)˙ q
?1
?˜ qTKIz?− ˜ qTKI˜ q,
2zTKIz
?
where we used ˙ z = ˜ q in (57) and (58). Further, using definition
(2) we get the following identity
˙ qTM(q)¨ q =
?1
?
,
−1
2˙ qT ˙M(q)˙ q,
(55)
˙ qTKj˜ q =
2˜ qTKj˜ q
?
j = P,D,
(56)
˙ qTKIz =
(57)
˜ qTKIz =
?1
?
,
(58)
?˜ q?k˜ qT˙ q =
1
k + 2?˜ q?k+2
?
,k = 1,2,
(59)
˙ qT(g(q) − g(qd)) =d
dt
?U(q) − U(qd) − ˜ qTg(qd)?. (60)
By inserting expressions (54)-(60) into (53) and separating
terms in the form of the time derivatives (first terms on the
right sides of equations (54)-(60)) on the left side of the
equality and the rest of the terms on the right side, we get
the decomposition (23).
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers
for their helpful comments and suggestions.

Page 6

6
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