# 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.

**0**Bookmarks

**·**

**245**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we design an image-based robust feedback controller to compensate parametric uncertainties. The proposed controller using dynamic compensation scheme leads to improvement in transient response while achieving asymptotic regulation in the presence of parametric uncertainties. The ultimately uniform boundedness of the closed-loop system is proved by the Lyapunov method. The performance of the proposed control system is demonstrated by experiments of a 5 link SAMSUNG FARA industrial robot manipulator with two degree of freedom.Proceedings of the American Control Conference 01/2009; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we provide an answer to the longstanding question of designing global asymptotically stable proportional-integral-derivative (PID) regulators for uncertain robotic manipulators. Our main contribution is to establish the global asymptotic stability of the controlled system with the commonly-used PID controller by using Lyapunov's direct method and LaSalle's invariance principle. Furthermore, an improved nonlinear proportional-integral plus derivative (NPI-D) controller is proposed to fast the transient. Simulations performed on a planar two degrees-of-freedom (DOF) robot manipulator demonstrate the improved performance of the proposed NPI-D controller over the commonly-used PID controller.01/2007; - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper addresses the global output feedback regulation problem for robot manipulators in the presence of actuator constraints, and a very simple bounded proportional-derivative (PD) plus desired gravity compensation scheme is formulated as solution to this problem. The global asymptotic stability of the closed-loop system is proved with Lyapunovpsilas direct method. Simulations performed on a two-degrees-of-freedom (2-DOF) manipulator demonstrate the effectiveness of the proposed approach.01/2008;

Page 1

1

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

Page 2

2

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.

Page 3

3

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)

Page 4

4

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.

Page 5

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

REFERENCES

[1] M. Takegaki and S. Arimoto, “A new feedback method for dynamic

control of manipulators,” ASME J. Dyn. Syst. Meas. Contr., vol. 103,

pp. 119–125, 1981.

[2] P. Tomei, “Adaptive PD control for robot manipulators,” IEEE Trans.

Robot. Autom., vol. 7, no. 4, pp. 565–570, 1991.

[3] S. Arimoto, T. Naniwa, and H. Suzuki, “Asymptotic stability and robust-

ness of PID local feedback for position control of robot manipulators,”

in Proc. Int. Conf. on Automation, Robotics, and Computer Vision,

Singapore, September 1990, pp. 382–386.

[4] R. Kelly, “A tuning procedure for stable PID control of robot manipu-

lators,” Robotica, vol. 13, pp. 141–148, 1995.

[5] A. Loria, E. Lefeber, and H. Nijmeijer, “Global asymptotic stability of

robot manipulators with linear PID and PI2D control,” Stability and

Control: Theory and Applications, vol. 3, no. 2, pp. 138–149, 2000.

[6] A. A. Pervozvanski and L. B. Freidovich, “Robust stabilization of robotic

manipulators by PID controllers,” Dynamics and Control, vol. 9, no. 3,

pp. 203–222, 1999.

[7] R. Kelly, “Comments on: Adaptive PD control of robot manipulators,”

IEEE Trans. Robot. Autom., vol. 9, no. 1, pp. 117–119, 1993.

[8] S. Arimoto, “A class of quasi-natural potentials and hyper-stable PID

servo-loops for nonlinear robotic systems,” Trans. Soc. Instrument Contr.

Engg., vol. 30, no. 9, pp. 1005–1012, 1994.

[9] R. Kelly, “Global positioning of robot manipulators via PD control plus a

class of nonlinear integral actions,” IEEE Trans. Autom. Control, vol. 43,

no. 7, pp. 934–938, 1998.

[10] J. Alvarez, R. Kelly, and I. Cervantes, “Semiglobal stability of saturated

linear PID control for robot manipulators,” Automatica, vol. 39, pp. 989–

995, 2003.

[11] R. Gorez, “Globally stable PID-like control of mechanical systems,”

Syst. Control Lett., vol. 38, pp. 61–72, 1999.

[12] M. J. Luis, V. Santibanez, and M. V. Hernandez, “Saturated nonlinear

PID global regulator for robot manipulators: Passivity based analysis,”

in Proc. 16th IFAC World Congress, Prague, Czech Republic, 4-8 July

2005.

[13] H. Yazarel, C. C. Cheah, and H. C. Liaw, “Adaptive SP-D control of

a robotic manipulator in the presence of modeling error in a gravity

regressor matrix: theory and experiment,” IEEE Trans. Robot. Autom.,

vol. 18, no. 3, pp. 373–379, 2002.

[14] W. Dixon, “Adaptive regulation of amplitude limited robot manipulators

with uncertain kinematics and dynamics,” in Proc. Amer. Control Conf.,

Boston, MA, 4-6 June 2004, pp. 3839–3844.

[15] S. P. Hodgson and D. P. Stoten, “Robustness of the minimal control

synthesis algorithm to non-linear plant with regard to the position control

of manipulators,” Int. J. Control, vol. 72, no. 14, pp. 1288–1298, 1999.

[16] R. Gunawardana and F. Ghorbel, “On the boundedness of the Hessian

of the potential energy of robot manipulators,” J. Robotic Syst., vol. 16,

no. 11, pp. 613–625, 1999.

[17] F. Ghorbel and R. Gunawardana, “A uniform bound for the Jacobian of

the gravitational force vector for a class of robot manipulators,” ASME

J. Dyn. Syst. Meas. Contr., vol. 119, pp. 110–114, 1997.

[18] R. Kelly, V. Santibanez, and A. Loria, Control of Robot Manipulators

in Joint Space.London: Springer-Verlag, 2005.

[19] R. Ortega, A. Loria, P. Nicklasson, and H. Sira-Ramirez, Passivity-

based control of Euler-Lagrange Systems: Mechanical, Electrical and

Electromechanical Applications.

[20] S. Arimoto, Control Theory of Nonlinear Mechanical Systems: A

Passivity-Based and Circuit-Theoretic Apprach.

Press, 1997.

[21] J. Kasac, B. Novakovic, D. Majetic, and D. Brezak, “Performance tuning

for a new class of globally stable controllers for robot manipulators,”

in Proc. 16th IFAC World Congress, Prague, Czech Republic, 4-8 July

2005.

[22] F. Ghorbel and R. Gunawardana, “On the uniform boundedness of the

derivative of the gravitational force of serial link robot manipulators,”

in Proc. ASME International Mechanical Engineering Congress and

Exposition, San Francisco, California, November 12-17 1995, pp. 22–28.

London: Springer-Verlag, 1998.

Oxford University

#### View other sources

#### Hide other sources

- Available from Josip Kasac · May 15, 2014
- Available from fsb.hr