A decoupled image space approach to visual servo control of a robotic manipulator.

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A decoupled image space approach to visual servo

control of a robotic manipulator.

R. Mahony, Tarek Hamel, François Chaumette

To cite this version:

R. Mahony, Tarek Hamel, François Chaumette. A decoupled image space approach to visual servo

control of a robotic manipulator.. IEEE Int. Conf. on Robotics and Automation, ICRA’02, 2002,

Washington DC, United States. pp.3781-3786. �inria-00352092�

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A D e c o u p l e d I m a g e Space A p p r o a c h to V i s u a l Servo C o n t r o l

of a R o b o t i c M a n i p u l a t o r

Robert Mahony

Dep. of Eng.,

Australian Nat. Uni.,

ACT, 0200 Australia.

email: mahony0ieee, org

Tarek Hamel

Cemif-SC FRE-CNRS 2494,

40 rue du Pelvoux,

91020 Evry France.

email: t h a m e l 0 i u p . u n i v - e v r y , f r

Francois Chaumette

Irisa/Inria Rennes,

Campus Universitaire de Beaulieu,

35042 Rennes, France.

email: Francois. C h a u m e t t e @ i r i s a . f r

A b s t r a c t

An image-based visual servo control is presented for a robotic manipulator. The proposed control design ad- dresses visual servo of 'eye-in-hand' t y p e systems. Us- ing a novel representation of the visual error based on a spherical representation of t a r g e t centroid information along with a measure of the rotation between the c a m e r a and the target, the control of the position and orientation is decoupled. A non-linear gain introduced into the orientation feedback kinematics prevents the t a r g e t image from leaving the visual field. Semi-global convergence of the closed loop system is proved.

1 I n t r o d u c t i o n

Visual servo algorithms have been extensively developed in the robotics field over the last ten years [7, 19]. Most visual servo control has been developed for serial-link robotic m a n i p u l a t o r s with the c a m e r a typically m o u n t e d on the end-effector [12]. Visual servo systems m a y be divided into two main classes [15]: Position-based visual

scrvo (PBVS) involves reconstruction of the t a r g e t pose with respect to the robot and results in a Cartesian too- tion planning problem. Problems with this approach in- elude the necessity of an accurate 3D model of the target, sensitivity to c a m e r a calibration errors, poor robustness of pose estimation and the t e n d e n c y for image features to leave the c a m e r a field of view during the task. Image-

based visual scrvo (IBVS) t r e a t s the problem as one of controlling features in the image plane, such t h a t moving features to a goal configuration implicitly results in the task being accomplished [7]. Feature errors are m a p p e d to a c t u a t o r inputs via the inverse of an image Jacobian matrix. Features can be derived from image features such as points, lines and circles. Issues associated with opti- real selection of features and partitioned control where some task degrees of freedom are controlled visually and others by position or force [13, 2] have been considered in depth. IBVS avoids m a n y of the robustness and calibration problems associated with PBVS, however, it has its own problems [31 . Foremost in the classical approach is

the need to determine d e p t h information for each feature point in the visual data. Various approaches have been reported including estimation via partial pose estimation [15], adaptive control [17] and estimation of the image Ja- cobian using quasi-Newton techniques [11, 18]. More re- cently, there has been considerable interest in hybrid control m e t h o d s whereby translational and rotational control are t r e a t e d separately [15, 6]. Hybrid methods, however, share with the P BVS approach a difficulty to keep all image features within the camera's field of view. Con- t e m p o r a r y work [16, 4] aims to address these issues. In this p a p e r we propose an image-based visual servo control for a robotic manipulator. T h e model considered is t h a t of an 'eye-in-hand' type configuration, where the c a m e r a is a t t a c h e d to the end effector of the robotic arm. The approach taken is based on recent work by the authors [9] in which an image based error is introduced for which the dynamics have certain passivity-like prop- erties. T h e visual error representation used is based on a spherical representation of target centroid information. Using this representation it is possible to express the position and orientation kinematics of the visual error as a partially decoupled system. An exponentially stable feedback law is proposed for stabilization of the position. T h e feedback law for the orientation kinematics is modi- fied via a simple non-linear gain to ensure t h a t the t a r g e t image never leaves the field of vision of the camera.

2 I m a g e B a s e d E r r o r s

In this section an image based error m e a s u r e m e n t for a visual servo task is introduced.

Let Z = {Ex, Ey, Ez} denote a right-hand inertial frame. Let the vector ~ = ( x , y , z ) denote the position of the focal centre of the c a m e r a in the inertial frame. Let A - {E~, E~, E~} be a (right-hand) b o d y fixed frame for the camera. The orientation of the c a m e r a is given by the r o t a t i o n / ~ : A + Z.

Let P represent a s t a t i o n a r y point t a r g e t visible to the

Proceedings of the 2002 IEEE International Conference on Robotics & Automation

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c a m e r a expressed in t h e c a m e r a frame. T h e i m a g e point observed by t h e c a m e r a is d e n o t e d p a n d is o b t a i n e d by rescaling onto t h e i m a g e surface 8 of t h e c a m e r a (cf. Fig- ure 1)

1

p

--

r(p) P.

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Following t h e a p p r o a c h i n t r o d u c e d in [10] we consider a c a m e r a w i t h a spherical i m a g e plane. T h u s 1 r ( P ) - P .

Let V C ,,4 d e n o t e t h e linear velocity a n d gt C ,,4 d e n o t e t h e a n g u l a r velocity of t h e c a m e r a b o t h e x p r e s s e d in t h e c a m e r a frame. T h e d y n a m i c s of an i m a g e point for a spherical c a m e r a of i m a g e surface r a d i u s u n i t y are (see E 9, 8]) 7I'p

(p)v,

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w h e r e :rp - ( I 3 - p p r ) is t h e p r o j e c t i o n :rp" R 3 + T p S 2, t h e t a n g e n t space of t h e s p h e r e S 2 at t h e point p ¢ S 2. 1 ', /~ = - ~ x p - -r-Xp v ,: S r i r t O P

Figure 1: Image dynamics for spherical camera image ge- ometry.

For a given t a r g e t t h e centroid is defined to be n

q " - c a 3 ( 3 ) i = 1

w h e r e t h e t a r g e t consists of n points {P~}. Using cen- t r o i d i n f o r m a t i o n is an old t e c h n i q u e in visual servo control [1, 14, 211. A m o n g t h e a d v a n t a g e s of using t h e t a r g e t centroid as a visual f e a t u r e one has t h a t it is not necessary to m a t c h observed i m a g e points directly to desired f e a t u r e s a n d t h a t t h e calculation of a i m a g e c e n t r o i d is a highly r o b u s t p r o c e d u r e . T h e s e p r o p e r t i e s are e x p e c t e d to ensure s t r o n g p r a c t i c a l r o b u s t n e s s of t h e p r o p o s e d control design. N o t e t h a t t h e centroid defined above contains m o r e i n f o r m a t i o n t h a n a classical definition. This m a y be seen from t h e fact t h a t t h e n o r m of q is r e l a t e d to t h e av- erage d e p t h of t h e t a r g e t points.

1 T h e n o t a t i o n x[ r e p r e s e n t s t h e n o r m of a n y v e c t o r x C R n

( ~ 1 - ~ ) .

Consider t h e case of a t a r g e t c o m p r i s i n g a finite n u m b e r of point t a r g e t s . Recalling Eq. 2 it m a y be verified t h a t

× q - 0v,

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w h e r e

/ ~ 1 7rp~

O - .

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T h e visual servo t a s k considered is t h a t of p o s i t i o n i n g a c a m e r a relative to a s t a t i o n a r y (in t h e inertial frame) target. In a d d i t i o n to visual i n f o r m a t i o n a d d i t i o n a l inertial i n f o r m a t i o n is explicitly used in t h e error formulation. In p a r t i c u l a r , it is a s s u m e d t h a t t h e o r i e n t a t i o n of t h e c a m e r a w i t h r e s p e c t to t h e t a r g e t is known a priori in

t h e inertial frame. For example, one m a y wish to position t h e c a m e r a above t h e t a r g e t or to t h e n o r t h of t h e t a r g e t . N o t e t h a t t h e centroid of an i m a g e contains no i n f o r m a t i o n relative to t h e o r i e n t a t i o n t h a t can be used to position t h e c a m e r a . If p u r e l y centroid i n f o r m a t i o n is used t h e n a d d i t i o n a l i m a g e or inertial i n f o r m a t i o n m u s t be i n c o r p o r a t e d into t h e error to fully specify t h e c a m e r a o r i e n t a t i o n .

Formally, let b C Z d e n o t e t h e desired inertial direction for t h e visual error. T h e n o r m of b encodes t h e effective d e p t h i n f o r m a t i o n for t h e desired limit point. Define

b* " - R T b C ,,4

to be t h e desired t a r g e t vector expressed in t h e c a m e r a fixed frame. Since b* C ,,4 it inherits d y n a m i c s from t h e m o t i o n of t h e c a m e r a

b* - - g t x b*.

T h e i m a g e b a s e d error considered is t h e difference between t h e m e a s u r e d centroid a n d t h e t a r g e t vector expressed in t h e c a m e r a f r a m e

6 . - q - ( 6 )

T h e i m a g e error k i n e m a t i c s are

d -

× 6 - O v

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To r e g u l a t e t h e full pose of t h e c a m e r a , it is n e c e s s a r y to consider an a d d i t i o n a l error criterion for t h e orientation. N o t e t h a t any r o t a t i o n of t h e c a m e r a t h a t alters t h e observed i m a g e centroid m a y be controlled using centroid i n f o r m a t i o n . However, t h e i m a g e centroid is invariant under r o t a t i o n a r o u n d t h e axis of t h e centroid. Thus, t h e best t h a t can be achieved using t h e centroid f e a t u r e is to orient t h e c a m e r a such t h a t t h e i m a g e centroid lies in a desired direction. To control t h e r e m a i n i n g degrees of f r e e d o m one m u s t use a d d i t i o n a l information; either from t h e i m a g e or a prior inertial direction (linearly independent from b (see section 3.1).

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In general it is desired t h a t t h e image centroid is close to t h e focal axis of the camera• This is particularly impor- t a n t if t h e r e is a risk t h a t the image m a y leave t h e active field of view of the c a m e r a during the servo manoeuvre• For the sake of simplicity we choose to stabilise the ori- e n t a t i o n in order t h a t the image centroid converges to t h e focal axis of the camera• Choose the c a m e r a frame A such t h a t t h e focal axis of t h e c a m e r a is co-linear with t h e t h i r d c o o r d i n a t e axis. Let

{el, e2, e3}

denote t h e unit vectors of t h e c o o r d i n a t e frame A. T h e visual error used for the c a m e r a orientation is the angle 0 between the centroid and t h e focal axis

qT¢ 3

c o s 0 - -- q0Te3 a n d s i n 0 - q0 x e3 , (8)

Iql

where the normalized centroid is defined to be q0 := q/lql.

In practice, it is most convenient to work directly with the cosine of 0. Define

a " - cos 0 - q0 Te 3 .

Thus, t h e goal of the servo a l g o r i t h m is to drive 5 ~ 0 a s y m p t o t i c a l l y and stabilize a ~ 1.

3 K i n e m a t i c C o n t r o l D e s i g n

In this section a L y a p u n o v control design is given for the kinematic control of t h e visual error based on the visual errors i n t r o d u c e d in Section 2.

T h e kinematics of an image based visual servo s y s t e m are t h e first order d y n a m i c s of the visual features q and a. T h e derivative of a is

d eT3(t qeT3q

& - ~ c o s 0 - q q2

= - a ~ ( q o × ~ ) - 7~ (9)

Note t h a t angular velocity ft is t h r e e dimensional, however, only the c o m p o n e n t in t h e direction q0 x e3, t h a t r o t a t e s the focal axis in the direction of t h e observed centroid, contributes to the first order dynamics of a. Thus, the kinematics considered are given by E q n ' s 7 and 9. T h e inputs to these equations are V a n d ft.

Define a storage function S

1 6~ (lo)

S - - ~

Taking t h e time derivative of S a n d s u b s t i t u t i n g for Eq. 7, yields

= -6TO, V

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Note t h a t Eq. 11 is i n d e p e n d e n t of the angular velocity ft.

T h e m a t r i x Q > 0 is not exactly known, however, it is known to be positive definite• Thus, a choice V = 5 is sufficient to stabilise S1. T h e kinematic control chosen for Eq. 11 is

V = k65 (12)

L e m m a 3.1 Consider the system defined by Eq. 7 and

let k6 > 0 be a positive constant. A s s u m e that the image remains in the camera field of view for all time. Then, the closed loop system Eq. 7 along with control Eq. 12 exponentially stabilises the visual error 5.

P r o o f : Recalling the derivative of S a n d s u b s t i t u t i n g the control input V by its expression, it yields

= --k66TQ6

Since Q is a positive definite matrix, classical L y p u n o v t h e o r y g u a r a n t e e s t h a t 5 converges exponentially to zero.

To control the kinematics of t h e visual feature a t h e storage function considered is

(~ - ~o~O) ~ - ~ (~ -

~)~

T ' - 2

E

"

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Following a classical L y a p u n o v control design t h e n a t u r a l choice of ft is = - k ~ (qo × e3). (14) Thus,

~ ~~oO, V)

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- (1 - ~ ) a~(qo × ~ ) +

- - k ~ ( 1 + ~ ) ( 1 - ~) ~ + q

L e m m a 3.2 Consider the rotational dynamics defined

by Eq. 9 with the control law Eq. 1~. Let V be an ezpo- nentially stable signal

IvI ~ B e -~t

for positive constants B, ~ > O. A s s u m e that ]q[ > c > 0 is bounded away f r o m zero and that

]]QI]

< A < oc is bounded. Then a ~ 1 exponentially. A s a consequence 0 ~ 0 exponentially.

An u n f o r t u n a t e possibility with t h e above control design is t h a t t h e t a r g e t image m a y leave field of view of t h e c a m e r a during the evolution of t h e closed loop system• To avoid such a situation it is necessary to modify the ori- e n t a t i o n control design• In particular, if the closed loop system evolves such t h a t the image is t e n d i n g to leave t h e field of view, high gain d y n a m i c s are i n t r o d u c e d in the ori- e n t a t i o n kinematics to keep t h e t a r g e t in t h e image plane•

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For a given initial condition it is possible to choose a gain ks > 0 sufficiently large such t h a t the exponential stability of t h e o r i e n t a t i o n dynamics d o m i n a t e s t h e c a m e r a a t t i t u d e kinematics and the c a m e r a points towards the t a r g e t centroid regardless of the linear dynamics of the camera. This a p p r o a c h is undesirable as it leads to high gain input over the full d y n a m i c s of the s y s t e m a n d re- duces system r o b u s t n e s s as a consequence. In this p a p e r a non-linear gain (linked to a barrier function potential) is used to limit t h e divergence of t h e image centroid from t h e focal axis of t h e camera.

Let the m a x i m u m allowable value for 0 be d e n o t e d 0 n > 0. T h e constraint 0 < 0,~ limits the possible orientation of the c a m e r a to a positive cone a r o u n d t h e centroid. Set a,~ = cos(0,~). T h e a p p r o a c h t a k e n is to choose the control to be

ko~

• - - ( q o ×

T h e barrier t e r m 1 / ( a - a ~ ) e n s u r e s t h a t the control

input for the orientation dynamics d o m i n a t e s any finite p e r t u r b a t i o n due to the linear velocity V if the orientation approaches the limit. Recalling the definition of the storage function T, one has

- _ ( 1 - + q

L e m m a 3 . 3 Consider the rotational dynamics defined by Eq. 9. Let the control law Eq. 17. Let V be an exponentially stable signal

IvI _<

for positive constants B, ~ > O. Let 0,~ > 0(0) (a(O) > a,~). A s s u m e that

Iql

> ~ > 0 is bounded away f r o m zero and that llQIl< A < oc is bounded. Then a > a,~ f o r all time and a ---+ 1 exponentially.

P r o o f : First it is necessary to show t h a t T remains b o u n d e d . Indeed, from Eq. 13 it follows t h a t T < 2. I n t r o d u c i n g b o u n d s on q a n d Q, the derivative of T m a y be b o u n d e d by the following expression

k ~ ( l + a) 4 A B e ~t

T < -

- - -

Taking t h e limit of t h e above b o u n d on T when a ~ a,~, it yields

T < 0 This proves t h a t a > a,~ for all time.

T h e exponentially decreasing p e r t u r b a t i o n does not de- stroy t h e classical L y a p u n o v convergence result t h a t forces 5 b ~ 0 exponentially a n d g u a r a n t e e s t h a t a ~ 1

exponentially. •

3 . 1 S t a b i l i s a t i o n o f r e m a i n i n g o r i e n t a t i o n w i t h r e f e r e n c e t o v i s u a l d a t a .

In this subsection an auxiliary control is p r o p o s e d t h a t will regulate the r e m a i n i n g degree of freedom in the atti- t u d e of t h e c a m e r a using an additional error criteria. To control t h e r e m a i n i n g o r i e n t a t i o n of a c a m e r a via an error in image space it is necessary to consider error criteria t h a t d e p e n d s on a n o t h e r vector direction fixed in t h e c a m e r a frame. A key observation for t h e p r o p o s e d control design is t h a t such an error is chosen and minimized af- ter the regulation of the visual errors 5 a n d 0 respectively. Thus, t h e best t h a t can be achieved is to orient the camera within the set 5, 0 = 0 and the auxiliary error must be chosen with this in mind. T h e control objective considered is to align el as closely as possible with new visual 'feature' vector c o m p u t e d from image m e a s u r e m e n t s

i=n

- Z ( i s )

i=1

where a~ are a set of real constants. It is not required t h a t a~ > 0 and differences between observed points m a y be used to g e n e r a t e t h e visual feature q~ vector.

Define cr to be t h e error between t h e visual feature di-

qa

rection q~ (q~ -- iqal) and the c a m e r a frame direction C1.

cr - q~ - el (19)

Taking t h e derivative of or, yields

1

~ - - - - Q X cr + - - 7 c a Q a V - e2 Q3 + e3 Q2 (20)

qa qo

Let U be the storage function for the error criterion or.

1 2

U - 2 c r

Deriving U and recalling t h e derivative expression of or, yields

_ _o_T ( 1

-~ff£-iTCq~QaV - e3Q 2 + e2 Q3

k

q

Choose t h e control for gt 3 to be

~'-~3 __ kcre T O- ( 2 1 )

T h e derivative of U becomes now,

0 -- -o-T~,-yV @ O-Te3 ~2 -- ]~crGTe2eTG 1

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L e m m a 3.4 Consider the remaining rotational dynam- ics defined by Eq. 19 with the control law Eq. 21. Let V and f~2 be exponentially stable signals

Q2 V < B e - m

for positive constants B, ~ > O. A s s u m e that qa > c > 0 is bounded away f r o m zero and that Qa < A < cx~ is bounded. Then ~ ~ 0 exponentially.

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Sketch of P r o o f Similar to the case encountered in L e m m a 3.2 the Lyapunov function U need not be mono- tonically decreasing and stabilizing part of the right-hand of Eq. 22 is not negative definite. Nevertheless, using the fact t h a t b o t h V and gt 2 are exponentially stable (cf. L e m m a s 3.1 and 3.2) then a result proving convergence of eTcr to zero is possible. L e m m a ' s 3.2 and 3.4 provide a complete characterisation of the convergence of the orientation of the camera. This can be seen by confirm- ing t h a t L e m m a 3.2 guarantees t h a t the e3 axis of the c a m e r a frame is aligned in the direction of the centroid q. T h e c a m e r a m a y still be r o t a t e d around the e3 axis without changing the convergence result obtained, however, L e m m a 3.4 ensures convergence of q~ into the plane given by span{el, e3}. Since e3 is g u a r a n t e e d to lie in the direction of the centroid q t h e n e l must lie in the unique direction

e l C I F~ ~ •

/k

4 E x p e r i m e n t a l results

In this section experimental results are presented. Our aim is to validate the proposed image-based approach, showing, in real conditions, the convergence and robustness.

T h e control law has been tested on a six d.o.f. Cartisian robot A F M A (at Irisa/Inria-Rennes). T h e c a m e r a was m o u n t e d on the end-effector of the robot. In this experiment a basic stabilization task is considered. T h e target is the four black m a r k s on the vertices of a pla- nar square. Each vertex is of 0.04m of length. T h e signals available are the normalized co-ordinates of the center of the four marks observed by the camera, de- noted { ( / Z l , / ) 1 ) , ( / Z 2 , / ) 2 ) , ( / Z 3 , / ) 3 ) , ( / Z 4 , / ) 4 ) } . These coordinates are transformed, in the computer, into normalized spherical coordinates (x~, y~, z~).

T h e desired t a r g e t vector b* is chosen such t h a t the camera set point is located at 0.2m above the square. Figure 2 shows the initial and goal image a p p e a r a n c e of the target. T h e end-effector of the robot is moved according to the control laws Eqn's 12, 17 and 21.

In this experiment, the barrier t e r m is obtained from the initial condition; it is chosen to be ( c ~ = 0.g cos0(0)). T h e following control gains are used, ks = k~ = 0.07. T h e control gain /ca is chosen to be a constant m a t r i x /c6 = 0.07Q *-1. This choice is used to avoid the con- ditioning problem of the m a t r i x Q and to decouple the dynamics of the components of 5. We have proved t h a t Q Q , - 1 is positive definite matrix, however, due to space limitations the proof is o m i t t e d and m a y be obtained on application to the authors. T h e constants a~ used

i!

iii~::.

!.

F i g u r e 2: Initial image and desired appearance of the target.

in the definition of the variable cr have been fixed to [1, 1, - 1, - 1]. T h e algorithm performance is satisfactory : the pixel trajectories of the image points (cf. Figures 4 and 3) show a good convergence of the difference between the e x t r a c t e d and desired features. The convergence and stability of the positioning process is shown in Figure 5.

/ . . . f / / / i . - / i / ... i i i i i i i 0 'tO.dat' ,~i dc, t 't2.dat' 't3 d~t'

F i g u r e 3: Image point trajectories

'er::;," :ia*: .sing i 0.15 'error.dat' using 3 .:,,.,:,: ,:,,:.::: ,,:.::;: 01 i ". . ... . .... g • 'error.c:at' using . . . i( f -o.2 -0.25 -O.S , , , , , , , , 0

F i g u r e 4: Error on the coordinates of the image features.

5 C o n c l u d i n g r e m a r k s

In this p a p e r a novel representation of the visual error, based on a spherical representation of t a r g e t centroid information, along with a Lyapunov control design is used to decouple the position and orientation servo control design for an image based visual servo system. In contrast to other recent work [5] the proposed servo control design does not need separate d e p t h of target estimates. Moreover, the framework used leads to a simple barrier

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0.6 0.4 ,-, 0.2 o - -o.2 -o.4 -o.6 -o.8 -1 -1.2 -1.4 o , / 'del[a.da'_ r' t~sir~g 1 'clelta dat' using 3

F i g u r e 5: Exponential evolution of the error 5.

function technique that is proposed to guard against the observed target leaving the field of view of the camera without otherwise effecting the performance of the algorithm.

6 A c k n o w l e d g m e n t s

The authors wish to thank Peter Corke for numerous helpful discussions.

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