Passivity-based Visual Force Feedback Control for Eye-to-Hand Systems

329

18
x

Passivity-based Visual Force Feedback
Control for Eye-to-Hand Systems
Hiroyuki Kawai 1, Toshiyuki Murao 2 and Masayuki Fujita 3

1 Kanazawa Institute of Technology, Ishikawa

Japan

2 Advanced Institute of Industrial Technology, Tokyo

Japan

3 Tokyo Institute of Technology, TokyoJapan

1. Introduction
Robotics and intelligent machines need sensory information to behave autonomously indynamical environments. Visual information is particularly suited to recognize unknown
surroundings. Vision based control of robotic systems involves the fusion of robot
kinematics, dynamics, and computer vision to control the motion of the robot in an efficientmanner. The combination of mechanical control with visual information, so-called visualfeedback control or visual servoing, is important when we consider a mechanical system
working under dynamical environments (Chaumette & Hutchinson, 2008).
For the theoretically problem of three dimensional(3D) visual servo control based on the
robot control theory, (Kelly et al., 2000) considered a simple image-based controller under
the assumption that the objects' depths are known. (Chen et al., 2007) addressed the field-ofview problem for 3D dynamic visual feedback system using an image-space navigation
function. In our previous works, we discussed the dynamic visual feedback control for 3D
target tracking based on passivity (Fujita et al., 2007) (Murao et al., 2008). On the other hand,applications of visual feedback system are also increasing in many fields. For example,
recent applications of visual feedback system include the autonomous injection of biologicalcells (Yu & Nelson, 2001), laparoscopic surgery (Omote et al., 1999) and others. Althoughvisual information is necessary in order to recognize environments, only visual information
is not enough to complete tasks in these applications. For example, not only visual
information but also force information are needed to inject DNA to biological cells. Hence,
integrating visual feedback control with force control is important for the modern robot.
(Xiao et al., 2000) developed sensor fusion scheme for controlling an end-effector to follow
an unknown trajectory on a contact surface. (Baeten et al., 2003) addressed a hybrid control
structure for the eye-in-hand vision and force control. Although many practical methods arereported with experimental results, rigorous results have hardly been obtained in terms ofthe nonlinear control aspects. For this problem, (Dean-Leon et al., 2006) has combined
image-based visual feedback control with force control and discussed the stability of the330

Robot Manipulators, New Achievements

nonlinear system. The authors have proposed passivity based visual force feedback control
law for force control with target tracking (Kawai et al., 2007). Although these control laws
guarantee Lyapunov stability and are effective for the visual force feedback system, they are
restricted to planar manipulators.
This chapter deals with 3D visual force feedback control for eye-to-hand systems as depicted
in Fig. 1. In our proposed method, we can control not only the position but also theorientation of the robot hand with a contact force in the visual force feedback system. The
main contribution of this chapter is to show that the 3D visual force feedback system has thepassivity which allows us to prove stability in the sense of Lyapunov. Both the passivity ofthe manipulator dynamics and the passivity of the visual feedback system are preserved inthe 3D visual force feedback system. Finally simulation results are shown to verify thestability of the proposed method.

Fig. 1. Visual force feedback system with an eye-to-hand configuration

2. Visual Feedback System
This section mainly reviews our previous works (Fujita et al., 2007) (Murao et al., 2008) via
the passivity based visual feedback control. Throughout this chapter, we use the notationˆe ab  33 to represent the change of the principle axes of a frame  b relative to a frame

 a .  ab  3 specifies the direction of rotation and  ab   is the angle of rotation. For

simplicity we use ˆ ab to denote ˆab ab . The notation '  ' (wedge) is the skew-symmetric
operator such that ˆ     for the vector cross-product  and any vector   3 . The

notation '  ' (vee) denotes the inverse operator to '  ', i.e., so( 3)   3 . Recall that a skewsymmetric matrix corresponds to an axis of rotation (via the mapping a  aˆ ). We use the4  4 matrix
 eˆ ab p ab g ab  
(1)
 0
1 

Passivity-based Visual Force Feedback Control for Eye-to-Hand Systems

331ˆ

as the homogeneous representation of g ab  ( pab , e ab )  SE( 3) describing the configurationof a frame b relative to a frame  a . The adjoint transformation associated with g ab isdenoted by Ad( g ab ) (Murray et al., 1994).
2.1 Basic Representation for Visual Feedback System
Visual feedback systems with an eye-to-hand configuration typically use four coordinate
frames which consist of a world frame  w , a target object frame  o , a camera frame  cˆand a hand (end-effector) frame  h as in Fig. 1. Then, g wh  ( pwh , e wh )  SE( 3) ,ˆˆg wc  ( pwc , e wc )  SE( 3) and g wo  ( pwo , e wo )  SE( 3) denote the rigid body motion from w to  h , from  w to c and from  w to  o , respectively. Similarly, the relative rigid

body motion from c to  h , from c to  o and from  h to  o can be represented byˆˆˆgch  ( pch , e ch )  SE( 3) , gco  ( pco , e co )  SE( 3) and gho  ( pho , e ho )  SE( 3) , respectively,

as shown in Fig. 1. The objective of the visual feedback control is to bring the actual relativeˆrigid body motion gho to a given reference gd  ( pd , e d ) which is constant in this chapter.

In other words, our goal is to determine the motion of hand by using the visual information.
The relative rigid body motion from c to  o can be led by using the composition rule forrigid body transformations ((Murray et al., 1994), Chap. 2, pp. 37, eq. (2.24)) as follows:1
gco  g wc
g wo

ref 1(2)The relative rigid body motion involves the velocity of each rigid body. To this aid, let us
consider the velocity of a rigid body as described in (Murray et al., 1994). We define thebTbody velocity of the camera relative to the world frame  w as Vwc
 vTwc  wc

 , where v
Twc

and  wc represent the velocity of the origin and the angular velocity from  w to  c ,respectively ((Murray et al., 1994), Chap. 2, eq. (2.55)).
Differentiating (2) with respect to time, the body velocity of the relative rigid body motion
gco can be written as follows (See (Fujita et al., 2007)):bb
Vcob  Ad( g 1 )Vwc
 Vwo
co

ref 2(3)bwhere Vwois the body velocity of the target object relative to  w . In the case of the eye-tob
 0 , the model of the relative rigid body motion gco can be
hand configuration, i.e. Vwc

rewritten asb
Vcob  Vwo

ref 3(4)332

Robot Manipulators, New Achievements

Roughly speaking, if both the camera and the target object move, then the relative rigidbandbody motion gco will be derived from the difference between the camera velocity Vwcb. Hence, the model of the relative rigid body motion from cthe target object velocity Vwobto  o equals the target object velocity Vwo.

2.2 Estimation Error System
The relative rigid body motion gco can not be immediately obtained in the visual feedback

system, because the target object velocity is unknown and furthermore can not be measured
directly. Hence, we consider the estimation problem of the relative rigid body motion gco .

The visual feedback control task requires information of the relative rigid body motion gco .
Since the measurable information is only the image information f ( gco ) in the visualfeedback system, we consider a visual motion observer in order to estimate the relative rigid
body motion gco from the image information f ( gco ) .

Firstly, using the basic representation (4), we choose estimates gco and Vcob of the relative

rigid body motion and velocity, respectively as

Vcob  ue .

ref 4(5)The new input ue is to be determined in order to drive the estimated values gco and Vcob totheir actual values.
In order to establish the estimation error system, we define the estimation error between theestimated value gco and the actual relative rigid body motion gco as
1g ee  gco
gco .

ref 5(6)ˆUsing the notation eR ( e ) , the vector of the estimation error is defined asˆˆ

ee : [ pTee eTR ( e ee )]T . Note that ee  0 iff pee  0 and eR ( e ee )  I 3 . Therefore, if the vector of
the estimation error is equal to zero, then the estimated relative rigid body motion gco
equals the actual relative rigid body motion gco .
Supposeˆ eeetheattitudeˆ ee I  sk( eestimation

error  ee

is

small

enough

that

we

can

let

) . Therefore, using a first-order Taylor expansion approximation, theestimation error vector ee can be obtained from image information f ( gco ) and theestimated value of the relative rigid body motion gco as follows (Fujita et al., 2007) :ee  J  ( gco )( f  f ),(7)

Passivity-based Visual Force Feedback Control for Eye-to-Hand Systems

333

where f is the estimated value of image information. In the same way as the basic
representation (4), the estimation error system can be represented by¶(9pt) \label{eqn:V_ee}bVeeb  Ad( g 1 )ue  Vwo.

ref 6(8)ee¶(9pt)It should be noted that if the vector of the estimation error is equal to zero, then theestimated relative rigid body motion gco equals the actual one gco .

¶(14pt)2.3 Control Error System
In this subsection, let us consider the dual of the estimation error system, which we call thecontrol error system, in order to establish the visual feedback system. We assume that g wc

and g wh can be obtained accurately by a prior calibration procedure, then the estimated1
gco where gco is the estimated value which discussed
value of gho is calculated as gho  gch

in the previous subsection. Here, we define the control error between the actual relative
rigid body motion gho and desired one gd as
g ec  gd1 gho .

ref 7(9)¶(9pt)It should be noted that gho can not be measured directory. Similar to the definition of ee ,ˆthe vector of the control error is defined as ec : [ pTec eTR ( e ec )]T .
Here we have to consider the way of deriving g ec (9), because gho can not be measured
directory. Using g ee , the control error can be transformed as¶(9pt) \label{eqn:g_ec_kai}

1
1
g ec  gd1 gho  gd1 gho
gho gho  gd1 gho
g ee .

ref 8(10)¶(9pt)In Equation (10), gd and gho are available information. While the estimation error vector
ee can be obtained as Equation(7), the estimation error

matrix g ee cannot be directly obtained, because g ee is defined using non-measurable value
gco as Equation (6). Therefore, we consider the way of deriving g ee from ee .
ˆBecause of the definition of the estimation error vector ee , i.e., ee  [ pTee eTR ( e ee )]T , theposition estimation error pee can be derived directly from ee . Under the condition2  ee 2,  ee can be derived as follows (Murao et al., 2008):
ˆ ee sin 1 eR ( e ee )ˆ eeeR ( e)Hence, g ee can be derived from ee through  ee .

ˆeR ( e ee )ref 9(11)334

Robot Manipulators, New Achievements

The reference of the relative rigid body motion gd is constant in this chapter, i.e., g d  0 ,b. Thus, the control error system can be represented ashence, Vecb  Vho¶(9pt) \label{eqn:V_ec}bb
Vecb  Ad( g 1 )Ad( g 1 )Vwh
 Vwo
.
ec

d

ref 10(12)¶(9pt)
This is dual to the estimation error system.

¶(14pt)2.4 Passivity of Visual Feedback System
Combining (8) and (12), we construct the visual feedback system as follows:¶(9pt) \label{eqn:VFS }0Vecb  Ad( g ec1 )I  b uce   Vwo b
0Ad1( g ee ) I 
Vee  

ref 11(13)b T T T) ue ] denotes the control input. For the design of the visual
where uce : [(Ad( g 1 )Vwhdbfeedback system, it is assumed that the hand velocity Vwh
can be directly chosen. Let us

define the error vector of the visual feedback system as e : [ ecT eTe ]T which consists of thecontrol error vector ec and the estimation error vector ee .

Next, we show an important relation between the input and the output of the visualfeedback system.
b
Lemma 1 (Fujita et al., 2007) : If Vwo
 0 , then the visual feedback system (13) satisfies¶(9pt) \label{eqn:PaVFS }T T

0 uce ce dt   ce , T  0

ref 12(14)where  ce is defined as  ce :  e and ce is a positive scalar.

Using the following positive definite function, we can prove Lemma 1.
¶(9pt) \label{eqn:V_ce }Vce  E( g ec )  E( g ee )ref 13(15)

ˆ
ˆ
ˆ
1
1
2
pab   ( e ab ) and  ( e ab ) : tr( I  e ab ) is the error function of the22
rotation matrix (see, e.g., (Bullo & Lewis, 2004)).
The block diagram of the passivity of the visual feedback system is shown in Fig. 2. Let us
take uce as the input and  ce as its output in Fig. 2. Thus, Lemma 1 implies that the visual

where E( g ab ) :feedback system (13) is passive from the input uce to the output  ce as in the definition in(Schaft, 2000).

Passivity-based Visual Force Feedback Control for Eye-to-Hand Systems

335

Fig. 2. Block diagram of the visual feedback system¶(14pt)3. Dynamic Visual Force Feedback Control¶(6pt)A force is important to complete tasks which need a contact with environments. This section
considers the visual force feedback control with a contact force (we call the dynamic visual
force feedback control) based on the passivity which is a main contribution in this chapter.

¶(14pt)3.1 Dynamic Visual Force Feedback System
The dynamics of n-link rigid robot manipulators with the end-effector constraint can be
written as follows (Liu et al., 1999)¶(9pt) \label{eqn:ManDyn}
M(q )q  C (q , q )q  g(q )    J (q )
ref 14(16)where q , q and q are the joint angle, velocity and acceleration, respectively,  is the vectorof the input torque. M (q )  nn is the manipulator inertia matrix, C (q , q )  nn is the
Coriolis matrix and g(q )  n is the gravity vector.    is the contact force, J (q )  n is
the normalized Jacobian of the kinematic constraint  (q )  0   and defined as follows.
¶(9pt) \label{eqn:Jvarphi}T  (q ) nJT (q )q  0 , J ( q )  
 

q



ref 15(17)¶(9pt)
Equation (16) possesses several important properties which will be used in the sequel. The
manipulator dynamics (16) is passive from  to q , that isT  qdt   where  is a
0

T

m

m

 (q )  2C (q , q ) is skew-symmetric by defining C (q , q ) using thepositive scalar. Moreover, M
Christoffel symbols.
Now, we propose the control law for the manipulator as¶(9pt) \label{eqn:tau}  M(q )qr  C ( q , q )q r  g( q )  J bT (q )AdT( g 1 )ee  J d  us  J uF ref 16(18)dwhere¶(9pt) \label{eqn:dqr}¶(9pt)q r  Q ( q )q d  J Fe .

ref 17(19)336Robot Manipulators, New Achievements is positive constant and¶(9pt)

ttFe : (  d )d  e d  
0

ref 18(20)0where e :   d is the force error. Then, the following relation holds with respect to about

the force errorFe  e .

r

ef 19(21)¶(9pt)
Because we consider the single point contact in this chapter, the projection matrix Q (q ) canbe simply defined as¶(9pt) \label{eqn:Qvarphi}
Q (q )  I  J (q ) JT (q )re f 20(22)which arises on the tangent space at the contact surface  (q )  0 (Liu et al., 1999).
b
On the other hand, the body velocity of the hand Vwh
is given by¶(9pt) \label{eqn:J_b}b
Vwh J b (q )q  J b (q )Q q

ref 21(23)where J b (q ) is the body manipulator Jacobian (Murray et al., 1994). Moreover, we define the
T T
desired body velocity of the hand uh  [ vTuh uh] which will be obtained from the visualfeedback system. Then, uh can be represented as uh  J b (q )Q q d by using the projection

matrix Q (q ) .
We define the error vector with respect to the joint velocity of the manipulator dynamics as¶(9pt) \label{eqn:s}
s : q  q r  n

ref 22(24)JT s  Fe .

ref 23(25)¶(9pt)
Here, we know that the following relation holds (Liu et al., 1999)¶(9pt) \label{eqn:Js}¶(9pt)
Using (13)(16) and (18), the visual force feedback system with manipulator dynamics (wecall the dynamic visual force feedback system) can be derived as follows:¶(9pt) \label{eqn:FixedDVFFS}00 s   M 1  Cs  J bT AdT 1 ec  J e   M  1 M 1 J0 ( gd )  F  
 
0
0


0
 0
 u  0  w 24(26)e e 
 
0Vecb  
I 
0  Ad( g ec1 ) Ad( g 1 ) J bs
  0 b  ho Ad( g 1 )   00 I 
0
Vee  
0
ee 

 
where
¶(9pt)

Passivity-based Visual Force Feedback Control for Eye-to-Hand Systems

337

us


s


F 
u
F
e
 w : V b ,x :   u : 
wo
Ad 1 uh  ec  ( gd )  ue ee are defined as the state, the input and the disturbance of the dynamic visual force feedback
system, respectively. Here, we formulate the manipulator control problem as follows:
Control problem : For the dynamic visual force feedback system with the eye-to-handconfiguration described by (26), design a control input u such that¶(9pt)
lim s  0 , lim Fe  0 , lim ec  0 and lim ee  0.

¶(14pt)

t 

t 

t 

t 

3.2 Passivity of Dynamic Visual Force Feedback System
Before constructing the dynamic visual force feedback control law, we derive an importantlemma.
Lemma 2 : If w = 0, then the dynamic visual force feedback system (26) satisfies¶(9pt) \label{eqn:PaDVFFS }T u  dt    , T  0

ref 25(27)T0where  : Nx , N : diagI n ,   ,  I 6 ,  I 6  and  is a positive scalar.

Proof: Consider the following positive definite function¶(9pt) \label{eqn:V}
1
1
V  sT Ms  Fe2  E( g ec )  E( g ee ).
22¶(9pt)
Differentiating (28) with respect to time yields¶(9pt)1 s  F F
V  sT Ms  sT M
e e
2
ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ref 26(28)ˆˆ pTec e ec e  ec p ec  eTR ( e ec )e ec ec  pTee e ee e  ee p ee  eTR ( e ee )e ee ee0M 0  s 0  
000  Fe  1 T  sT 
 V b   2 s Ms.
0
0 0 Ad ˆec
(e
)

 ec
Ad ˆee  V b 
0 0
0
(e
)   ee ref 27(29)¶(9pt)
ˆObserving that the skew-symmetry of the matrices p̂ec and p̂ee , i.e., pTec pˆ ec e  d udˆ  pTec ( e  d ud ) pec  0 , pTee pˆ eeue   pTeeˆ ue pee  0 , the above equation along the

trajectories of the system (26) can be transformed into¶(9pt) \label{eqn:dot_V2}

338

Robot Manipulators, New Achievements1sV  sT Cs  sT sTb AdT( g 1 ) ec  sT J e  Fe e  ecT Ad( g 1 ) J b s  sT M
d
ho
2
0
0
M 0
  M 1 M 1 J

0
0




0
0
0
0
0
 0
  1 sT M
T 0 
 s.
s 
 0
 2
0
0
Ad
0 0 Ad ˆec

1
0
g
(
)
e
(
)


ec

Ad ˆ ee   0- Ad( g 1 ) 
0
0 0
0
0
(e
) 
ee 

1
  2C )s  eT ( J T s  F )  xT N T u  xT N T u.
ref 28(30) sT ( Me 2¶(9pt)
Integrating (30) from 0 to T, we obtain¶(9pt) \label{eqn:dot_V }T u  dtV  V(T )  V (0)  V(0)  T

ref (31)0where  is a positive scalar that only depends on the initial states of s, Fe , ec and ee .

The block diagram of the passivity of the 3D dynamic visual force feedback system is shownin Fig. 3.
Remark 1 : The visual feedback system (13) satisfies the passivity property as described in(14). It is well known that the manipulator dynamics (16) also has the passivity. In Lemma 2,
the inequality (27) says that the dynamic visual force feedback system (26) is passive from the
input u  [uTs uTF ( Ad( g 1 )uh )T uTe ]T to the output   [sT  FeT  ecT  eTe ]T as shown in Fig. 3.
dFig. 3. Block diagram of the 3D dynamic visual force feedback system¶(14pt)3.3 Passivity-based Dynamic Visual Force Feedback Control
We now propose the following control input for the interconnected system:¶(9pt) \label{eqn:V}u  K  KNx
K :  diagK s , k F , K c , K e  

ref 29(32)n  13where K s : diagks 1 , , ksn  , kF , Kc : diagkc 1 , , kc 6  and K e : diagke 1 , , ke 6  denotethe positive gain matrices.
Theorem 1 : If w = 0, then the equilibrium point x = 0 for the closed-loop system (26) and(32)is asymptotic stable.

Passivity-based Visual Force Feedback Control for Eye-to-Hand Systems

339

Proof: In the proof of Lemma 2, we have already derived that the time derivative of V
along the trajectory of the system (26) is formulated as (30). Using the control input (32),(30) can be transformed into¶(9pt) \label{eqn:dot_V2_k}
ref 30(33)V   xT N T KNx.

¶(9pt)
This completes the proof.
Theorem 1 shows the stability via Lyapunov method for the dynamic visual force feedback
system. It is interesting to note that stability analysis is based on the passivity as describedin (27). Our proposed method is valid for the 3D dynamic visual force feedback system,
while previous works (Dean-Leon et al., 2006) (Kawai et al., 2007) consider the 2D dynamic
visual force feedback control. Hence, we can control not only the position but also theorientation of the robot hand with a contact force in the visual force feedback system.

¶(14pt)4. Simulation Results¶(6pt)The simulation results on 3DOF planar manipulator as depicted in Fig. 4 are shown in orderto understand our proposed method simply, though it is valid for 3D dynamic visual force
feedback systems.

Fig. 4. Coordinate frames for dynamic visual force feedback system with three degree of
freedom manipulator
We present results for the stability analysis with a static target object. The simulation iscarried out with the conditions pwo  0.47 0.05 0T[m],  wo  0 0 0T [rad],340

Robot Manipulators, New Achievements

pwc  0.47 0.05 1T [m],  wc  0 0 0T [rad]. The lengths of the three links of themanipulator are l1  0.2 [m], l2  0.2 [m] and l3  0.1 [m], respectively. The initial angles ofthe manipulator is set as q1 (0 )   / 2 [rad], q 3 (0 )   / 2 [rad] and q 2 (0 )   / 2 [rad]. In
other words, the initial relative rigid body motion is pwo  0.15  0.27 0 T [m],ho  0 0   / 2 T [rad]. The desired force d and the desired relative rigid bodyˆmotion gd  ( pd , e d ) are given by d  5 [N], pd  0.03 0 1T [m] and  d  0 0 0 T[rad] in this simulation. The initial errors of force and vision are calculated as e  5 [N],pec  0.12  0.27 0T [m],  ec  0 0   / 2 T [rad], pee  0.53 0.95 0T [m] and ee  0 0  / 4 T [rad], respectively. The controller parameters for Equation (32) were
empirically selected as K s  diag10,10,10 , kF  25 ,   1 , Kc  diag40,40,20,20,20,40 andK e  50 I 6 . The simulation results are shown in Figs. 5-7.

Figs. 5-7 illustrate the control error ec , the estimatoin error ee , and the contact force  ,respectively. In Figs. 5 and 6, we focus on the errors of the translations of x and y and therotation of z, because the errors of the translation of z and the rotations of x and y are zeros
ideally on the 3DOF planar manipulator. The control error ec and the estimation error eetended to zero, thus we can confirm that the relative rigid body motion gho coincided withthe desired one gd by using image information. In Fig. 7, the pulse siglnal means thecontact transition at around 0.2 [s]. The contact force  tended to 5 [N], i.e., converged tothe desired one d . From these figures, the asymptotic stability can be also confirmed.

Fig. 5. The control error ec which defined the error between the relative rigid body motionof the robot hand gho and desired one gd . Initial control errors are pecx  0.12 [m],pecy  0.27 [m] and  ecz   / 2 [rad], respectively.

Passivity-based Visual Force Feedback Control for Eye-to-Hand Systems

341

Fig. 6. The estimation error ee which defined the error between the estimated value gco andthe actual relative rigid body motion gco . Initial estimation errors are peex  0.53 [m],peey  0.95 [m] and  eez   / 4 [rad], respectively.

Fig. 7. The contact force trajectory ¶(14pt)5. Conclusions¶(6pt)

This chapter considers 3D visual force feedback control for eye-to-hand systems. In our
approach, we can control not only the position but also the orientation of the robot handwith a contact force by using visual information. The proposed method can be regarded as
an extension of the hybrid position/force control to the hybrid vision/force control. The
main contribution of this chapter is to show that the visual force feedback system has thepassivity which allows us to prove stability in the sense of Lyapunov. Both the passivity ofthe manipulator dynamics and the passivity of the visual feedback system are preserved in342

Robot Manipulators, New Achievements

the visual force feedback system. Finally simulation results are shown to verify the stabilityof the proposed method.

¶(14pt)

6. References¶(6pt)

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