机械设计英文资料13

Journal of the Franklin Institute 344(2007)1021–1038

Velocity ?eld control of robot manipulators by using

only position measurements

Javier Moreno-Valenzuela ?

Centro de Investigacio

′n y Desarrollo de Tecnolog?′a Digital,CITEDI-IPN,Ave.del Parque 1310,Mesa de Otay,Tijuana,BC 22510,Mexico

Received 3September 2005;accepted 17May 2007

Abstract

In the velocity ?eld control approach the robot motions are speci?ed through a vectorial function that assigns the desired velocity to each point of the con?guration space.In other words,a velocity ?eld de?nes the robot desired velocity in the operational space as a function of its current position.In this paper is introduced a new algorithm to solve the velocity ?eld control formulation in the robot operational space.The proposed approach assumes only joint position measurements and is based on a hierarchical structure that results of using the kinematic control concept and a joint velocity controller.To estimate the joint velocity,nonlinear ?ltering of the joint position is used.r 2007The Franklin Institute.Published by Elsevier Ltd.All rights reserved.

Keywords:Velocity ?eld control;Operational space;Position measurements;Lyapunov function;Stability

1.Introduction

The passive velocity ?eld control introduced in [1,2],attempts to be an alternative to motion control.In this control philosophy,the task to be accomplished by the robot is coded by means of a smooth desired velocity vector ?eld de?ned in the operational con?guration space G and denoted as a map

v ey T:

G !T G y !v ey T;

https://www.360docs.net/doc/777259095.html,/locate/jfranklin

?Tel.:+526646231344;fax:+526646231388.

E-mail address:moreno@citedi.mx

where T y G is the tangent space of G at the speci?c con?guration y ,and T G ?S

y 2G T y G denotes the tangent bundle of G [3].

A velocity ?eld de?nes the desired end-effector velocity at every point of the robot operational con?guration space.Fig.1illustrates the speci?cation of motion by means of a velocity ?eld.This ?gure depicts a velocity ?eld de?ned in the three dimensional Cartesian space of a robot arm which assigns a desired velocity vector (arrow)to each point in the operational space.

The velocity ?eld control is reformulated in this paper without regard of closed-loop passivity requirements [4]:

The velocity field control objective in operational space is established as the design of the torques input s so that

lim t !1

?v ey et TTà_y et T ?0,(1)

where the difference between the desired velocity ?eld v ey Tand the manipulator end-effector

velocity _y

de?nes the velocity ?eld error .Instead of requiring the arm end-effector tip to be at speci?c location at each instant time such as it is imposed in trajectory tracking control formulation,in the velocity ?eld control approach the arm tip has to match with the ?ow lines of the desired velocity ?eld,as it can be seen in Fig.1.This way of formulating the robot motion is specially important when coordination and synchronization is more important than the time following of a desired trajectory.Velocity ?eld control is well adapted to contour following tasks such as cutting,milling and deburring [5].

The problem of velocity ?eld control in operational space was explored in [4],showing the practical feasibility of two controllers through experimental results in a two degrees-of-freedom direct drive robot.The problem of achieving velocity ?eld control without knowing the robot model has been addressed [6].There,a proportional-integral controller was analyzed.More recently,the problem of friction compensation when controlling the robot motion along the ?ow lines produced by a desired velocity ?eld has been discussed in [7].

Practical implementation of control algorithms on robot arms requires velocity measurements,which can be contaminated with noise,especially if tachometers are being used as sensors.This situation together with the discretization of the controller limit the values of the controller gains.As practical solution in the implementation of a controller for mechanical systems,high resolution encoders and hardware that allows high sampling

Fig.1.Desired velocity ?eld in Cartesian space.

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frequency can be used to approach the velocity measurements via numerical differentia-tion.However,without high quality hardware one generally?nds that numerical differentiation does not work well,especially as the sampling interval decreases,due to the encoder measurement noise[8].

In order to avoid contaminated velocity measurements and simple numerical differentiation of the joint position to estimate the velocity,an approach consists in using the Lyapunov theory to design a controller/?lter guaranteeing the robot task execution,no mattering if an estimate of the velocity and acceleration can be possible with the obtained design.This approach increments the robustness with respect to measurement noise and model uncertainties.

The hierarchical control is based on two loops of feedback:A kinematic control loop for joint velocity resolution and an asymptotically stable joint velocity loop.This approach has been exploited in[9]to generate a class of trajectory tracking controllers.

This paper introduces a velocity?eld controller based on the hierarchical structure of[9] and[10],i.e.,a two loops of feedback controller.By using the ideas in[11–13],the proposed solution follows the concept of replacing the joint velocities by the?ltering of the joint positions and the desired velocity?eld via a stable?rst order?https://www.360docs.net/doc/777259095.html,ing the Lyapunov theory framework,we present a rigorous stability analysis for the proposed algorithm.

The remaining of this paper is organized as follows.Section2deals with the robot dynamics and kinematics.In Section3,the proposed controller is introduced as well as the discussion on its stability analysis.In Section4,some remarks on the proposed controller and the stability analysis are presented.Section5is devoted to show the simulation results.

Finally,some concluding remarks are drawn in Section6. Throughout the following notation will be adopted.k x k?

????????

x T x

stands for the norm of

vector x2R n.k BexTk?

???????????????????????????????

l M f BexTT BexTg

stands for the induced norm of a matrix BexT2

R m?n for all x2R n.l min f AexTg and l Max f AexTg denote the minimum and maximum eigenvalues of a symmetric positive de?nite matrix AexT2R n?n for all x2R n,respectively.

2.Robot dynamics and kinematics

The dynamics in joint space of a serial-chain n-link robot manipulator considering the presence of friction at the robot joints can be written as[14–16]

MeqT€qtCeq;_qT_qtgeqTtF v_q?s,(2) where q is the n?1vector of joint displacements,_q is the n?1vector of joint velocities,s is the n?1vector of applied torque inputs,MeqTis the n?n symmetric positive de?nite manipulator inertia matrix,Ceq;_qT_q is the n?1vector of centripetal and Coriolis torques, geqTis the n?1vector of gravitational torques,and F v is a n?n diagonal positive de?nite matrix which contains the viscous friction coef?cients of each joint.

Based on the requirement that matrix Ceq;_qTis expressed in terms of the Christoffel symbols,the following properties are satis?ed.

Property1.The matrix Ceq;_qTand the time derivative of the inertia matrix_MeqTsatisfy

x T?1

2_MeqTàCeq;_qT x?08x;q;_q2R n,(3) J.Moreno-Valenzuela/Journal of the Franklin Institute344(2007)1021–10381023

eq T?C eq ;_q TtC T eq ;_q T8q ;_q 2R n .(4)

Property 2.For all x ;y ;z 2R n we have that matrix C ex ;y Tsatis?es

C ex ;y Tz ?C ex ;z Ty ,(5)k C ex ;y Tz k p k C k y k k z k ,(6)

where k C is strictly positive constant.

Denoting h eq T:R n !R m the robot direct kinematics,then the position and orientation y 2R m of the end-effector is given by

y ?h eq T.

(7)

The time derivative of the direct kinematic model (7)yields the differential kinematic model

?d d t h eq T?q h

q q

?J eq T_q ,(8)

where J eq Tis the so-called analytical Jacobian matrix [17].The robot Jacobian describes a

map from velocities in joint space to velocities in operational space.The Jacobian right pseudoinverse [17],is given by

J eq Ty ?J eq TT ?J eq TJ eq TT à1,

assuming that J eq TJ eq TT is nonsingular.

The analytical Jacobian J eq Tis assumed of full-rank (rank ?m )and bounded by k J 40,i.e.,

k J eq Tk p k J

8q 2R n .

(9)

At the same time,it is also assumed that

k J eq Ty k p k y J

8q 2R n ,

(10)

with k y J 40.

The following assumptions on the Jacobian right pseudoinverse time derivative will be useful in the analysis presented in this paper.

Assumption 1.The map expressed by the time derivative of the Jacobian right pseudoinverse satis?es the following relation:

eq ;x ty Ty ?_J eq ;x Ty t_J eq ;y Ty 8x ;y 2R n .

(11)

Assumption 2.The time derivative of the Jacobian right pseudoinverse satis?es the following bound:

d d t J eq Ty

?k _J eq ;_q Ty k p k J 1k _q k .(12)From a practical point of view,Assumptions 1and 2resemble properties more than assumptions.

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Some properties of the tangent hyperbolic function will be used.The tangent hyperbolic function is de?ned as

tanh ex T?

e x àe àx

e te ,

and it can be arranged in a vector in the following way:

tanh ez T??tanh ez 1T;...;tanh ez n T T

and the following properties are accomplished by tanh ez T(a)The Euclidean norm of tanh ez Tsatis?es

k tanh ez Tk p k z k 8z 2R n ;

???

n p 8z 2R n :(

(b)The time derivative of tanh ez Tis given by

d t

tanh ez T?sech 2ez T_z

,where sech 2ez T?diag f sech 2ez 1T;...;sech 2ez n Tg and

sech ex T?

2e x te àx ?

cosh ex T

.(c)The maximum eigenvalue of the matrix sech 2ez Tis one for all z 2R n ,i.e.,

l Max f sech 2ez Tg ?1

8z 2R n .

3.A hierarchical approach without velocity measurements 3.1.Velocity ?eld kinematic control

Two important assumptions in the results presented in this paper are k v ey Tk p v M 1

8y 2R m

(13)

and

q v ey Tq y

p v M 2

8y 2R m .

(14)

Because the analytical robot Jacobian J eq Tis assumed full-rank,and inspired from the resolved motion rate control philosophy [18],we propose the following control law to generate the desired joint velocity x d

x d eq T?J eq Ty v ey eq TT.

(15)

Note that we have used the notation y eq Tbecause the posture y 2R m is a function of the joint position q 2R n .Inspired in [19],Eq.(15)is called velocity ?eld kinematic control.

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Let us de?ne the joint velocity error as

?x d eq Tà_q .(16)

In this way,from the differential kinematic (8)and the de?nition of the joint velocity error

(16)we have

?v ey TàJ eq T~x .(17)

On the other hand,we can interpret the joint velocity error (16)as the dynamics of the joint position q 2R n ,i.e.,

?x d eq Tà~x .(18)

Real-time implementation of the kinematic control law (15)requires a joint velocity

controller.Then,as soon as the joint velocity error ~x

vanishes,the operational space robot velocity _y

will match the desired velocity ?eld v ey T.3.2.Joint velocity control

In practice,it is often to ?nd that velocity measurements are corrupted by noise.This

situation limits the performance of the robot motions and makes dif?cult the tuning of the control algorithm.In order to overcome these situations,our key idea is to use a joint velocity controller together with a ?rst order ?lter to carry out an estimation of the joint velocity.The proposed joint velocity controller is written as follows:

s ?M eq T_x %d tC eq ;x d Tx d tg eq TtK v tanh e_n TtF v x d ,

e19T_n

?x d à!,e20T

where K v is an n ?n diagonal positive–de?nite matrix,!2R n is the output of a ?rst order ?lter,the desired joint velocity x d is given by (15),and desired joint acceleration x %

d is

_x %d ?_J eq ;x d Ty v ey TtJ eq Ty q v ey Tq y

v ey T.Fig.2shows a general block diagram of the proposed scheme of velocity ?eld control

which uses only position measurements as feedback information.It will be useful to de?ne the function

g eq ;~x T?_x d à_x %d

?à_J

eq ;~x Ty v ey TtJ eq Ty q v ey Tq y

J eq T~x ,e21T

which is obtained by using Assumption (11)and Eq.(17).Moreover,it is worthy to

observe that g eq ;~x

Tin (21)satis?es the following bound:k g eq ;~x

Tk p k _J eq ;~x Ty v ey Tk tk J eq Ty q v ey Tq y

J eq T~x k p ?k J 1v M 1tk y J v M 2k J k ~x

k ?k k ~x

k ,e22T

where Assumptions (9)–(11),(13)and (14)were used.

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The signal!involved in the joint velocity controller(19)–(20)is obtained from the following?rst order?lter

_x?tanhex dà!Tàx d,e23T!?x dtA xtA q,e24Twhere A?diag f a1;...;a n g is positive de?nite.

3.3.Closed-loop system derivation

Differentiating(24)with respect to time,and substituting(23)in the resulting expression,we can write

!?_x dtA tanhex dà!TàA x dtA_q.(25) Using the de?nitions of the joint velocity error~x in(16)and the signal_n in(20),it is possible to write(25)as

d t

_n?àA tanhe_nTtA~x.(26) Eq.(26)resumes the dynamics of the velocity observer(23)–(24).

On the other hand,substituting Eq.(19)in the robot Eq.(2),adding and subtracting Ceq;_qTx d,and using Assumption(5),we obtain

MeqT_~xt?Ceq;_qTtCeq;x dT ~xtK v tanhe_nTtF v~xàMeqTgeq;~xT?0.(27) Eqs.(18),(26)and(27)represent the closed-loop dynamics,that in state variables is given by

S1:d

d t

q?x deqTà~x,e28T

S2:d

d t

àA tanhe_nTtA~x

àMeqTà1feq;_n;~xTtgeq;~xT

,e29T

Fig.2.Block diagram of the velocity?eld controller which uses only position measurements.

J.Moreno-Valenzuela/Journal of the Franklin Institute344(2007)1021–10381027

where

f eq ;_n

;~x T??C eq ;x d eq Tà~x TtC eq ;x d eq TT ~x tK v tanh e_n TtF v ~x ,x d eq T?J eq Ty y eq Tand g eq ;~x

Tgiven in (21).3.4.Stability analysis

In order to consider systems S 1and S 2,as a cascade [20,21],we must prove that the

overall system (28)–(29)is complete that is,that the solutions can be continued for all t X 0and do not blow up in ?nite time.This allows us to consider S 2as a time-varying system,that is,we regard the robot-joint velocity controller system as a time-varying system dependent on the joint position q et T.

https://www.360docs.net/doc/777259095.html,pleteness of the closed-loop system

To prove that the system (28)–(29)is complete consider the function

W eq ;_n ;~x T?1~x T M eq T~x t1_n T K v A à1_n t1q T q ,

which is a positive de?nite and radially unbounded function.By virtue of Property 1in (3),the time derivative of W eq ;_n

;~x Talong of the closed-loop system trajectories (28)–(29)is given by

eq ;_n ;~x T?à~x T C eq ;x d eq TT~x à~x T K v tanh e_n Tà~x

T F v ~x T t~x T M eq Tg eq ;~x Tà_n

T K v tanh e_n Tt_n K v ~x tq T x d eq Tàq T ~x .It is possible to show that for all q ;_n

;~x 2R n the function _W eq ;_n ;~x Tattains the inequality

eq ;_n ;~x Tp k 1W eq ;_n ;~x Ttk 2,with k 1and k 2large enough strictly positive constants.This can be done by invoking the

Property 2in (5)and (6),properties of the tangent hyperbolic function,(10)and (13),and inequality (22).

It follows using the comparison equations method to show that

W et Tp e k 1t W e0Ttk 2

k 1?e k 1t à1 ,

that is,W et Tis bounded for all t bounded and since W et Tis positive de?nite we obtain that

the solutions ?q et TT _n

et TT ~x et TT T exist and can be continued for all t X 0.3.4.2.Proof of GES

We have proven that the solutions of the closed-loop system S 1–S 2can be continued for all t X 0thus the subsystem (29)can be interpret as a nonlinear and nonautonomous system,being the state space origin its unique equilibrium point.In order to show the asymptotic stability of the subsystem S 2in Eq.(29),the following Lyapunov function

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candidate is proposed:

Vet;~x;_nT?1

~x T MeqT~xt

X n

i?1

k vi aà1

lnej coshe_x iTjTàa tanhe_nTT MeqT~x,(30)

where a is a strictly positive constant.With the aim of showing that Vet;~x;_nTis globally positive de?nite,a lower bound can be computed on Vet;~x;_nTas follows:

Vet;~x;_nTX 1

k tanhe_nTk

k~x k

k tanhe_nTk

k~x k

with

l min f K v Aà1gàal Max f MeqTg àal Max f MeqTg l min f MeqTg "#

and the fact that

X n i?1k vi aà1

lnej coshe_x iTjTX

tanhenTT K v Aà1tanhenT

was used.A suf?cient condition to guarantee that Vet;~x;_nTis positive de?nite is that P is positive de?nite,which is satis?ed with

a o

????????????????????????????????????????

l min f K v g l min f MeqTg

?????????????????

l Max f A g

l Max f MeqTg

.(31)

The time derivative of Vet;~x;_nTalong of the closed-loop system trajectories(29)is given by

_Vet;~x;_nT?à~x T Ceq;x

T~xà~x T F v~xt~x T MeqTgeq;~xT

àtanhe_nTT K v tanhe_nTta tanhe_nTCeq;x dT~x

ta tanhe_nTT K v tanhe_nTta tanhe_nTT F v~x

àa tanhe_nTT MeqTgeq;~xTàa tanhe_nTT Ceq;_qTT~x

ta~x T MeqTsech2e_nTA tanhe_nTàa~x T MeqTsech2e_nTA~x,

where Property1in Eqs.(3)–(4),and Property2in Eq.(5)were used.By virtue of the same Properties(Eqs.(3)–(6)),and inequality(22),bounds on each term of the Lyapunov function time derivative_Vet;~x;_nTcan be computed:

à~x T Ceq;x dT~x p k C k x d k M k~x k2,

à~x T F v~x T pàl min f F v gk~x k2,

~x T MeqT?_x dà_x%

p l Max f MeqTg k k~x k2,

àtanhe_nTK v_n pàl m f K v gk tanhe_nTk2,

a tanhe_nTT Ceq;x dT~x p a k C k x d k M k tanhe_nTkk~x k,

a tanhe_nTT K v tanhe_nTp al Max f K v gk tanhe_nTk2,

J.Moreno-Valenzuela/Journal of the Franklin Institute344(2007)1021–10381029

a tanh e_n

TT F v ~x p al Max f F v gk tanh e_n Tkk ~x k ,àa tanh e_n TT M eq T?_x d à_x %d p a k l Max f M eq Tgk tanh e_n Tkk ~x

k ,àa tanh e_n

TT C eq ;_q TT ~x p a k C ???

n p k ~x k 2ta k C k x d k M k tanh e_n Tkk ~x k ,a ~x

T M eq Tsech 2e_n TA tanh e_n Tp al Max f M eq Tg l Max f A gk tanh e_n Tkk ~x k ,àa ~x

T M eq Tsech 2e_n TA ~x p àal min f M eq Tg l min f sech 2e_n Tg l min f A gk ~x k 2.An upper bound on _V et ;~x ;_n Tcan be obtained by using the previous inequalities:_V et ;~x ;_n Tp àk _n k k ~x k "#T Q k _n k

k ~x k "#àf k ~x

k 2,where the entries of the matrix Q are

Q 11?l min f K v g àal Max f K v g ,

Q 12?à12a ?g 1tl Max f A g l Max f M eq Tg ,Q 21?à12a ?g 1tl Max f A g l Max f M eq Tg ,Q 22?l min f F v g ,and

f ?a ?l min f M eq T

g l min f sec

h 2e_n Tg l min f A g àk C ???

n p àg 2

,with

g 1?2k C k x d k M tl Max f F v g tk l Max f M eq Tg ,e32Tg 2?k C k x k M tk l Max f M eq Tg .

e33T

Suf?cient conditions for the matrix Q to be positive de?nite are a o

l min f K v g

2l Max f K v g

e34Ta o

???????????????????????????????????l min f K v g l min f F v g

p g 1tl Max f A g l Max f M eq Tg

,e35Twith g 1expressed in Eq.(32).Let us de?ne the set

D ?f _n

2R n :k _n k p d g .(36)

Then,for all _n

2D and l min f A g 4

k C ???n

p l min f M eq Tg sech ed T,(37)

we have that the inequality

a 4

k C k x d k M tk l Max f M eq Tg

l min f M eq Tg l min f A g sech ed Tàk C ???

p (38)guarantees f X 0.

Inequalities (31),(34)and (35)can be satis?ed for small enough a ,while (38)is satis?ed if the product l min f A g sech 2ed Tis suf?ciently large.Summarizing,using proper values for

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the observer gain A and the control gain K v we can assure that the inequality

k C k x d k M tk l Max f M eq Tg

l min f M eq Tg l min f A g sech ed Tàk C ???

p o a o min

????????????????????????????????????????l min f K v g l min f M eq Tg p l Max f M eq Tg l Max f A g ;l min f K v g 2l min f K v g ;????????????????????????????????????l min f K v g l Max f F v g

p g 1tl Max f A g l Max f M eq Tg

()

e39T

holds for all ~x

2R n and _n 2D .A simple tuning procedure to select the gains A and K v that satisfy the condition (39)is shown in Appendix A.Roughly speaking,the tuning rule consists in using large enough gains A and K v .

To prove asymptotic stability we invoke similar arguments to the ones used in [22–24].

The Lyapunov function V et ;~x

;_n Tis globally positive de?nite and radially unbounded,and its time derivative _V

et ;~x ;_n Tis negative de?nite into the set R A ?f ~x

2R n g ?f _n 2D g ,(40)

where D is described in (36).The domain of attraction contains the largest level set O c of

V et ;~x

;_n Tin R A ,where O c ?_n ~x "#2R A :V et ;~x ;_n Tp c ()

,and c is a strictly positive constant.Due to the fact that V et ;~x

;_n Tis globally positive de?nite and radially unbounded,all level sets O c are contained in R A ,which means that for

any initial condition ?_n

e0TT ~x e0TT T 2R A it is always possible to ?nd a constant c such that ?_n

e0TT ~x e0TT T 2O c .Therefore,the set R A &R 2n in (40)is forward invariant and all the conditions to prove that the state space origin of the closed-loop system (29)is

exponentially stable are satis?ed [25].As result,it is possible to claim that ?_n

et TT ~x et TT T !0as t !1for all initial condition ?_n

e0TT ~x e0TT T starting at the domain of attraction R A in (40).The latter together with the assumption that the robot Jacobian is full-rank and bounded allow the conclusion from Eq.(17)that

lim t !1

?_y

et Tàv ey et TT ?lim t !1

J eq et TT~x et T?0,which is equivalent to

lim t !1

?_q

et Tàx d eq et T ?lim t !1

~x et T?0.4.Discussions

Let us remark that the kinematic control was originally proposed to be used in the joint

trajectory tracking control framework and in applications for obstacle avoidance of redundant manipulators.Roughly speaking,the kinematic control problem can be stated as follows:For a given trajectory in the operation space,let us say y d et T,?nd a joint space trajectory q d et Tsuch that h eq d et TT?y d et T.To satisfy this requiring,the

kinematic control approach considers that the system input is the joint velocity _q

and the differential kinematic model (8)is considered as the robot model [15,17,19].

The real-time implementation of kinematic control requires that the robot at hand is equipped with a joint velocity control system guaranteeing in practice asymptotic

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velocity tracking,which corresponds to the controller in Eqs.(19)–(20).This agrees with the actual situation of many industrial robots in that the control of each electro-mechanical axis is carried out using inner joint velocity loops in addition to outer position loops [26,27].

In practice,the assumption that J eq Tis full-rank for all q 2R n cannot be satis?ed.Therefore,the controller and the desired velocity ?eld must be constrained to a singularity-free region of the operational space where the Jacobian J eq Tis full-rank.When computation of the inverse or pseudoinverse of the Jacobian J eq Ty is required,a damped least-squares inverse can be adopted to improve the robustness in the neighborhood of the kinematic singularities [28].

The stability result guarantees asymptotic vanishing of the velocity ?eld error

for any singularity-free initial con?guration y e0T2R m ,joint velocity error ~x

e0T2R n and _n

e0T2D .Notwithstanding,the assumption of viscous friction at the robot joints allows to invoke the reasoning introduced in [29]to prove the global asymptotic stability of the closed-loop system (29)for a suf?ciently small range of desired joint velocity x d .

The attraction domain can be arbitrarily expanded by choosing a large constant d ,which enlarges the set D and in consequence the region of attraction R A .However,increasing of the constant d implies that a large enough numerical value of l min f A g must be speci?ed to satisfy the conditions (37)and (39).In order to achieve this,a tuning procedure is proposed in Appendix A.

Essentially,the stability result suggests that by using a suf?ciently large value of the observer gain A ,the control objective is satis?ed.At the same time,this implies that we can select K v so as to avoid actuator saturation and to maintain small ampli?cation of the discretization noise,while the closed-loop system stability is ensured by using high observer gain A .

5.Simulation results

We have carried out numerical simulations using the model of the experimental robot system described in [16].See Fig.3for a scheme of the two degrees-of-freedom robot.The origin of the Cartesian frame is attached at the axis of rotation of the ?rst joint while y 1and y 2denote the horizontal and vertical axes,respectively.With this convention,the direct kinematics is given by

h eq T?

l 1sin eq 1Ttl 2sin eq 1tq 2T

àl 2cos eq 1Tàl 2cos eq 1tq 2T"#,which leads to the following Jacobian matrix:

J eq T?

l 1cos eq 1Ttl 2cos eq 1tq 2Tl 2cos eq 1tq 2T

l 2sin eq 1Ttl 2sin eq 1tq 2Tl 2sin eq 1tq 2T

"#.For the robot model l 1?l 2?0:26(m).For self-contents of this paper,the dynamics of the robot arm is described in Appendix B.

The simulation has considered a velocity ?eld that encodes the task of tracing a circle in the operational space.This means that the ?ow lines of the velocity ?eld converges to

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a circular contour.The proposed velocity ?eld v ey Tto generate the ?ow lines convergent to the circle in the y 1–y 2plane is shown in Fig.4.The desired speed at the circle is v 0?0:65(m/s).

The mathematical expression of the desired velocity ?eld v ey Tis given by

v ey T?àk ey Tf ey T2?y 1ày c 1 2?y 2ày c 2 "#

tc ey T

à2?y 2ày c 2

2?y 1àx c 1

"#,

Fig.3.Scheme of the two degrees-of-freedom robot manipulator.

0.1

0.2

0.3

0.4

0.5

-0.25

-0.2-0.15-0.1-0.0500.050.1

0.150.2

0.25y 1 coordinate

y 2 c o o r d i n a t e

Fig.4.Desired velocity ?eld used in experiments.

J.Moreno-Valenzuela /Journal of the Franklin Institute 344(2007)1021–1038

1033

where feyT??y1ày c1 2t?y2ày c2 2àr20with y c1?0:26(m),y c2?0:0(m),and r0?0:085(m).Denoting r feyTthe gradient of feyT,the functions keyTand ceyTare de?ned as

keyT?

j feyTj kr feyTkt

-0.100.10.20.30.4 -0.25

-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

0.2

0.25

y1 [m]

[

]

Fig.5.Robot path in the operational space.

00.10.20.30.40.5

-0.05

0.05

00.10.20.30.40.5

0.1

-0.05

0.05

Time [sec]

(

)

[

]

(

)

[

]

Fig.6.Velocity?eld error in operational space.

J.Moreno-Valenzuela/Journal of the Franklin Institute344(2007)1021–1038

1034

ceyT?v0expàC j feyTj kr feyTk

where k0?0:65(m/s),v0?0:65(m/s),C?50emà2T,and ?0:00075em3T.

For the velocity?eld controller(19)–(20),the control gains were

K v?diag f5:0;5:0ge1=sT,

and in the nonlinear?lter(23)–(24)the gain

A?diag f1000;1000ge1=sT

was used.The results are given in Figs.5and6.They show the Cartesian robot path and the fast convergence of the velocity?eld error in operational space veyTà_y.

6.Summary

This paper dealt with the velocity?eld control of robot arms.The main contribution has been the introduction of a controller which uses only position measurements.This solution consists in a hierarchical control structure based on a kinematic controller for the resolution of the desired joint velocity and an asymptotically stable joint velocity controller.The stability analysis showed the convergence of the operational space velocity _yetTto the desired velocity?eld veyetTT.Simulations were done to assess the practical feasibility of the proposed control law.

Acknowledgment

This work was partially supported by CONACyT Grant52174and SIP-IPN Grant 20070202.

Appendix A.Tuning procedure

This Appendix concerns the selection of the gains K v and A used by the controller (19)–(20)and(23)–(24).From the analysis presented in Section3.4,it should be noticed that condition(39)synthesizes the way in that the matrices A and K v can be selected. Namely,the matrix A is involved in the denominator of both sides of the condition(39). The proposed tuning procedure departs from

A? 2

I n?n and K v?k v I n?n.(41) We summarize the three possibilities of condition(39),which were obtained after some direct but tedious algebraic manipulations:

By using the control gains(41)it is possible to rewrite the inequality

k C k x d k Mtk l Max f MeqTg l min f MeqTg l min f A g sech2edTàk C ???n p o

????????????????????????????????????????

l min f K v g l min f MeqTg

l Max f MeqTg l Max f A g

J.Moreno-Valenzuela/Journal of the Franklin Institute344(2007)1021–10381035

as a quadratic condition on the gain a

2a l min f M eq Tg sech 2ed T|????????????????{z????????????????}

C 1

à a l Max f M eq Tg?k C k x d k M tk l Max f M eq Tg

??????????????????????k v l min f F v g

p |???????????????????????????????????{z???????????????????????????????????}C 2

àk C ???n p |??{z??}C 3

40,

which is always satis?ed with

a 4C 2t??????????????????????????C 22t4C 1C 3q 2C 1

(42)

Secondly,after substituting the gains (41)into the inequality

k C k x d k M tk l Max f M eq Tg l min f M eq Tg l min f A g sech 2ed Tàk C ???

n p o l min f K v g

2l min f K v g we obtain the simple condition

42?k C k x d k M tk l Max f M eq Tg tk C ???

n p l min f M eq Tg sech 2ed T

.(43)

Finally,through the inequality

k C k x d k M tk l Max f M eq Tg

l min f M eq Tg l min f A g sech ed Tàk C ???

n p o ????????????????????????????????????l min f K v g l Max f F v g p g 1tl Max f A g l Max f M eq Tg

we obtain the tuning rule

2a 4l min f M eq Tg sech 2ed Tk C k x d k M tk l Max f M eq Tg

??????????????????????k v l min f F v g p àl Max f M eq Tg à1?

k C ???n p ??????????????????????

k v l min f F v g

p k C k x d k M tk l Max f M eq Tg àg 1 e44T

which is subject to

?????k v p 4l Max f M eq Tg?k C k x d k M tk l Max f M eq Tg l min f M eq Tg sech ed T?????????????????

l min f F v g

p .(45)

The above written inequalities suggest to set the control gain k v such that the inequality (45)is satis?ed and,therefore,to guarantee the existence of the constant a

involved in the Lyapunov function V et ;~x

;_n Tin (30)is enough to select a large value of observer gain a such that (42),(43)and (44)are attained.In addition,the condition (37)becomes

2a 4

k C ???

p l min f M eq Tg sech 2ed T

(46)

and it must also be satis?ed.

It is noteworthy that the attraction domain can be expanded arbitrarily by choosing a large constant d and a large enough to satisfy (42),(43),(44)and (46).

J.Moreno-Valenzuela /Journal of the Franklin Institute 344(2007)1021–1038

1036

Appendix B.Robot model

We present below the entries of the robot dynamics described in[16].The elements M ijeqTei;j?1;2Tof the inertia matrix are:

M11eqT?0:3353t0:02436coseq2T,

M12eqT?0:01267t0:01218coseq2T,

M21eqT?0:01267t0:01218coseq2T,

M22eqT?0:01267.

The elements C ijeq;_qTei;j?1;2Tof the centrifugal and Coriolis matrix are C11eq;_qT?à0:01218sineq2T_q2,

C12eq;_qT?à0:01218sineq2T_q1à0:01218sineq2T_q2,

C21eq;_qT?0:01218sineq2T_q1,

C22eq;_qT?0:0.

The entries of the gravitational torque vector geqTare given by:

g1eqT?g?1:1731sineq1Tt0:04685sineq1tq2T ,

g1eqT?g?0:04685sineq1tq2T ,

where g?9:81em=s2T.

The parameters of the viscous friction of the arm are f v

1?0:2741(Nm s/rad)and

f v

?0:1713(Nm s/rad).

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