Passivity-based Iterative Learning Control for Visual Feedback System
Toshiyuki Murao, Hiroyuki Kawai and Masayuki Fujita

Abstract— This paper investigates iterative learning control
based on passivity for three-dimensional (3-D) visual feedback
systems. Firstly, a brief summary of a visual motion observeris given. Next, a pose control error system for iterative learning control that has an output strictly passivity property isconstructed. Then, iterative learning control for 3-D visualfeedback systems is proposed. The transient response of the
proposed control law should be improved because of therepeatability. Convergence analysis of the closed-loop system
is discussed based on passivity. Finally, simulation results areshown in order to confirm the proposed method.

I. I NTRODUCTION
Visual feedback control is now a very flexible and useful
method in robot control [1]. Recently, Lippiello et al. [2]presented position-based visual servoing for a hybrid eyein-hand/eye-to-hand multicamera system by using an extended Kalman filter and a multiarm robotic cell. Gans
and Hutchinson [3] proposed hybrid switched-system control
which incorporates both an image-based visual servoing
controller and a position-based one. Hu et al. [4] proposed
quaternion-based robust visual servo control in presenceof camera calibration error. Swensen et al. [5] presented
featureless kernel-based visual servoing and discussed a
domain of attraction of the method through experiments.
Allibert et al. [6] reported comparison results between two
image prediction models for an image-based visual servoing
scheme based on nonlinear model predictive control. The
authors have been proposed passivity-based visual feedback
control for three-dimensional (3-D) target tracking in a seriesof papers [7]–[10].
On the other hand, iterative learning control has become
an attractive control method for improving the transient
response and tracking performance of uncertain dynamic systems that operate repetitively [11]. Examples include not only
robotics, chemical processes and biomedical applications,but also surgical assistants recently [12]. Nowadays, there
are several papers related to the iterative learning control
for visual feedback systems. Mansard et al. [13] proposeda jacobian learning method for coarsely calibrated visualfeedback systems. In particular, iterative learning control approaches for visual feedback systems which address stability
analysis were investigated in a number of papers [14]–[16].
In [14] and [15], a iterative learning scheme for nonlinear
This work was not supported by any organization
T. Murao is with Master Program of Innovation for Design and Engineering Advanced Institute of Industrial Technology, Tokyo 140-0011, Japanmurao-toshiyuki@aiit.ac.jp
H. Kawai is with Department of Robotics, Kanazawa Institute of Technology, Ishikawa 921-8501, Japan
M. Fujita is with Department of Mechanical and Control Engineering,Tokyo Institute of Technology, Tokyo 152-8550, Japan

Camera5c
gcoTargetObjectFig. 1.

gwc

5o

gwo

5w
World Frame

Eye-in-hand visual feedback system.

visual feedback control with initial resetting error and with an
unknown homography matrix were presented, respectively.
Jiang et al. [16] reported indirect iterative learning control
using neural networks without a vision camera model. Although good iterative learning control methods are reported
in those papers, each control approach has the assumptionthat a depth along optical axis can be measured by a robot
position sensor [14], a robot motion plane is parallel to an
image plane [15] and selected feature points are restricted toonly three points, in other words, redundant image features(over four feature points) cannot be applied [16].
In this paper, we propose iterative learning control basedon passivity for 3-D visual feedback systems with an eyein-hand configuration as shown in Fig. 1. Firstly, a briefsummary of a visual motion observer in order to estimate
relative rigid body motion is given. Secondly, a pose control
error system for iterative learning control that has an output
strictly passivity property is constructed. Next, we propose
iterative learning control to ensure that an actual relative
rigid body motion from a camera to a target object tracks
the desired one. Our control law is based on the Arimototype iterative learning controller [17] that is much efficientfor motion control of robot systems. Convergence analysisof the closed-loop system is discussed based on passivity.
Compared with our previous work [7], the transient response
can be improved through repetition to track desired trajectories. Finally, control performance of the proposed control
scheme is evaluated through simulation results. In particular,
the simulation under a realistic system model is performed
from the more practical point of view.
II. V ISUAL M OTION O BSERVER
This section mainly reviews our previous work [7] via
passivity-based visual feedback control for eye-in-hand systems.

A. Vision Camera Model

B. Visual Motion Observer

Visual feedback systems with an eye-in-hand configuration
use three coordinate frames which consist of a world frameΣw , a camera frame Σc , and a object frame Σo as in Fig. 1.
Let pco ∈ R3 and eξ̂θco ∈ SO(3) be a position vector and arotation matrix from the camera frame Σc to the object frame
Σo . Then, a relative rigid body motion from Σc to Σo canbe represented by gco = (pco , eξ̂θco ) ∈ SE(3) 1 . Similarly,
gwc = (pwc , eξ̂θwc ) and gwo = (pwo , eξ̂θwo ) denote rigid
body motions from the world frame Σw to the camera frame
Σc and from the world frame Σw to the object frame Σo ,respectively.
Position-based visual feedback control needs, in general,
to derive the relative rigid body motion from image data. The
relative rigid body motion from Σc to Σo can be led by using
the composition rule for rigid body transformations [18] asfollows:

In this subsection, we consider a nonlinear observer (wecall the visual motion observer) in order to estimate therelative rigid body motion gco from the image feature f .
Using the body velocity of the relative rigid body motionˆbgco (2), we choose estimates ḡco = (p̄co , eξ̄θ̄co ) and V̄coof
the relative rigid body motion and velocity, respectively as

−1
gwo .
gco = gwc

(1)

The relative rigid body motion involves a velocity of eachrigid body. We define a body velocity of the camera relativeb
T
T T
to the world frame Σw as Vwc= [vwcωwc] , where vwc andωwc represent a velocity of an origin and an angular velocity
from Σw to Σc , respectively [18]. Differentiating Eq. (1) withrespect to time, the body velocity of the relative rigid body
motion gco can be written as follows (See [7]):
b
b
b
−1 V
Vco
= −Ad(gco) wc + Vwo ,(2)where Ad(gab ) is the adjoint transformation associated withb
gab and Vwo
is the body velocity of the target object relativeto Σw .
In this visual feedback system, we use a pinhole camera
model with a perspective projection. Here, we considerm(≥3) feature points on the rigid target object in thispaper. Let λ be a focal length, poi ∈ R3 and pci ∈ R3
be position vectors of the target object’s i-th feature point
relative to Σo and Σc , respectively. Using a transformationof the coordinates, we have pci = gco poi , where pci andpoi should be regarded, with a slight abuse of notation, as[pTci 1]T and [pToi 1]T . We assume that multiple point features
on a known object poi are given. The perspective projectionof the i-th feature point onto the image plane gives us theimage plane coordinate fi := [fxi fyi ]T ∈ R2 as
λ
fi =
zci



xci
yci


,

(3)

where pci = [xci yci zci ]T . It is straightforward to extend thismodel to m image points by simply stacking the vectors of
T T
the image plane coordinate, i.e., f := [f1T · · · fm
] ∈ R2m .
The image feature f only depends on the relative rigid body
motion gco .
1 The notation of the homogeneous transform is denoted in Appendix.

b
b
−1 V
V̄co
= −Ad(ḡco) wc + ue .

(4)

The new input ue is to be determined in order to drive thebestimated values ḡco and V̄coto their actual values.
In order to establish an estimation error system, we define
an estimation error between the estimated value ḡco and theactual relative rigid body motion gco as−1
gee := ḡco
gco .

(5)

ˆ

Note that pco = p̄co and eξ̂θco = eξ̄θ̄co if and only if gee = I4 ,i.e., pee = 0 and eξ̂θee = I3 . We next define an error vector of
the rotation matrix eξ̂θei as rei := sk(eξ̂θei )∨ where sk(eξ̂θei )denotes 12 (eξ̂θei − e−ξ̂θei ). Using this notation, the vector of
the estimation error is given by
TT
ee := pTee ree.
(6)Note that ee = 0 iff pee = 0 and eξ̂θee = I3 . Suppose anattitude estimation error θee is small enough that we can leteξ̂θee  I + sk(eξ̂θee ). Therefore, using a first-order Taylor
expansion approximation, the estimation error vector ee canbe obtained from the image feature f and the estimated valueof the relative rigid body motion ḡco (i.e., the measurement
and the estimate) as follows:
ee = J † (ḡco )(f − f¯),

(7)

where f¯ is an estimated value of the image feature andJ(ḡco ) is an image Jacobian-like matrix [7]. In the same way
as Eq. (2), the estimation error system can be represented bybb−1 ue + V
Vee
= −Ad(gee
wo .
)

(8)

Then, we have the following lemma relating the input ue
to the vector form of the estimation error ee .
bLemma 1 ([7]): If Vwo= 0, then the following inequality
holds for the estimation error system (8).
 T
uTe (−ee )dt ≥ −βe ,(9)0where βe is a positive scalar.
Based on the above passivity property of the estimation
error system, we consider the following control law:ue = −Ke (−ee ),(10)

where Ke := diag{ke1 , · · · , ke6 } is the positive gain matrixof x, y and z axes of the translation and the rotation for theestimation error.
b= 0, then the equilibrium point
Theorem 1 ([7]): If Vwo
ee = 0 for the closed-loop system (8) and (10) is asymptotic
stable.

b
Vwo

Next, we consider that the input of the vision camerab
has an ideal one ui which can achieve to bring the
Vwc
actual relative rigid body motion gco to the reference one
gcd . Here, we assume that the initial relative rigid body
motion and the body velocity from the camera to the target
object are equal to the desired ones, i.e., gco (0) = gcd (0) andbbbb(0) = Vcd(0). Substituting gco and Vco
with gcd and Vcd
Vcorespectively in Eq. (2), the ideal input ui can be represented
as follows:
 b

b
.
(15)
− Vwo
ui = −Ad(gcd ) Vcd

Vision Camera
Relative Rigid gco Perspective f
Body Motion
Projection

b
Vwc

Ke

ee
ue
gco

ee

Image
Jacobian-like
Matrixfe +Vision Camera Model
Relative Rigid gco Perspective f
Projection
Body Motion
Model
Model
Visual Motion Observer

Fig. 2.

Block diagram of visual motion observer.

It should be noted that if the vector of the estimation error
is equal to zero, then the estimated relative rigid body motionḡco equals the actual one gco . Fig. 2 shows a block diagramof the visual motion observer. The estimation error vector isconfigured by available information (i.e., the measurement
and the estimate) though it is defined by unavailable one.
Thanks to the proposed visual motion observer, the unmeasurable motion gco can be exploited as the part of control
law. Our proposed visual motion observer is composed justas Luenberger observer for linear systems.
III. I TERATIVE L EARNING C ONTROL FOR
PASSIVITY- BASED V ISUAL F EEDBACK S YSTEM
Iterative learning control is an approach to improving thetransient response performance of the system that operates
repetitively [11]. In this section, an iterative learning control
law for the visual feedback system is proposed based on apassivity property.
A. Pose Control Error System for Iterative Learning Control
Let us consider the dual of the estimation error system,
which we call the pose control error system, in order toachieve the control objective. The control objective is tobring the actual relative rigid body motions gco to a givenreference one gcd . First, we define the pose control error as
follows:−1
gec = gcd
gco ,
(11)
which represents the error between the relative rigid body
motion gco and the reference one gcd . It should be remarked
that gco can be calculated by using the estimated relative
rigid body motion ḡco and the estimation error vector ee =T
[pTee ree] equivalently as follows:
gco = ḡco gee
sin−1 ree ξθee =ree ,ree 

(12)(13)

although gco cannot be measured directly (see [8] for moredetails). Using the notation rei = sk(eξ̂θei )∨ , the vector of
the pose control error is defined as

T
T
.
(14)
ec := pTec rec

It should be remarked that the ideal input ui cannot actually
be implemented for the visual feedback system since thebis unknown.
body velocity of the target object Vwo
We now propose the camera velocity for the visual feedback system asb= Ad(gcd ) Kc ec + ul ,(16)
Vwc
where Kc := diag{kc1 , · · · , kc6 } is the positive gain matrixof x, y and z axes of the translation and the rotation forthe pose control error. The new input ul is to be determinedlater by an iterative learning control scheme. Substituting
Eq. (16) into Eq. (2), the body velocity of the relative rigid
body motion gco can be obtained as follows:


b
b
−1
= −Ad(gco(17)
Vco) Ad(gcd ) Kc ec + ul + Vwo .
It should be noted that ul → ui from Eq. (17) when gco →gcd , i.e., ec → 0.
Here, we define the ideal input error as follows:b− u i ) − K c ec
uei = Ad(g−1 ) (Vwc
cd
= Ad(g−1 ) (ul − ui ).
cd

(18)

It is obvious that the ideal input error uei can be regarded as
the exact error between the actual input for the vision cameraband the ideal one ui in the case of ec = 0. Differentiating
Vwc
Eq. (11) with respect to time, the pose control error system
for the iterative learning control can be represented as
b
b
b
−1 V
Vec
= −Ad(gec) cd + Vcob−1 (uei + Kc ec ) +−1= −Ad(gec
I − Ad(gec)) Vwo .(19)Moreover, we consider the following output for the input
ul using the error vector of the pose control error:νl = −ec .

(20)

In the case of the ideal input ui that can drive gco to gcd , itis obvious that its output is equal to zero, i.e. νi = 0. Similar
to the ideal input error uei , the following ideal output error
between the actual output νl and the ideal one νi is obtainedas(21)νei = νl − νi = νl .
Next, we show an important relation between the input
uei and the output νei of the pose control error system forthe iterative learning control.
b
Lemma 2: If Vwo= 0, then the pose control error system(19) satisfies
 T
 T
T
uTei νei dt ≥ −β +νei
Kc νei dt, ∀T > 0 (22)00where β is a positive scalar.
Proof: Consider the following positive definite function
V =1pec 2 + φ(eξ̂θec ),2V̇ = pTec eξ̂θec e−ξ̂θec ṗec + eTR (eξ̂θec )eξ̂θec ωec−1 (uei + Kc ec )= eTc Ad(eξ̂θec ) −Ad(gec)(24)

From the property of “∧” (wedge), we have pTec p̂ec ωuc =T
T T−pTec ω̂uc pec = 0, where uei + Kc ec = [vuc
ωuc
] . Using
this fact, we can obtainV̇= −eTc (uei + Kc ec )T
= uTei νei − νei
Kc νei .

(25)

Integrating Eq. (25) from 0 to T yields
 T
 T
T
T
uei νei dt = V (T ) − V (0) +νei
Kc νei dt
0
0
 T
T
νei
Kc νei≥ −V (0) +0 TT:= −β +
νei
Kc νei ,(26)0where β is the positive scalar which only depends on theinitial state of gec = (pec , eξ̂θec ).
Lemma 2 implies that the pose control error system (19)is output strictly passive from the input uei to the output
νei as in the definition in [20]. It should be noted that thepose control error system (19) is different from that proposedin [7], in both points that the proposed camera velocity has
already been integrated into it and that the input of the posecontrol error system is the error between the actual camera
velocity and the ideal one. One of the main contributions of
this paper is that we construct the pose control error system
for the iterative learning control which has an output strictly
passivity.
B. Iterative Learning Control for Passivity-based VisualFeedback System
In this subsection, we present an iterative learning control
law for the visual feedback system. The problem of iterative
learning control is, in general, to find a recursive form of ak
= F (ukl , νei) in trial number k thatlearning control law uk+1leventually realizes the convergence νei → 0 as k → ∞ [17].
In this paper, we tackle the same problem as mentionedin [17] for the visual feedback system. We now propose thelearning control update law for the visual feedback system
as follows:= ukl − Ad(gcd ) Kl νlk ,
uk+1
l

k
= ukei − Kl νei
.
uk+1
ei

(23)

where φ(eξ̂θec ) := 12 tr(I − eξ̂θec ) is an error function of therotation matrix (see e.g. [19]). The positive definiteness ofthe function V results from the property of the error function
φ. Evaluating the time derivative of V along the trajectories
of Eq. (19) gives us= −eTc Ad(−pec ) (uei + Kc ec ) .

iterative learning. From Eqs. (18), (21) and (27), it can beeasily derived the following relationship(27)

where Kl := diag{kl1 , · · · , kl6 } is the positive gain matrixof x, y and z axes of the translation and the rotation for the(28)

Suppose that the target object is static, the following
theorem concerning the convergence of the iterative learning
control for the visual feedback system holds.
b= 0 and 0 < Kl < 2Kc ,Theorem 2: Suppose that Vwo
then the iterative learning control law (27) for the posecontrol error system (19) guarantees the convergence of
ekc = 0 in the sense of a L2 [0, T ] norm.
Proof: For Eq. (28), multiplying both sides by the
positive definite symmetric matrix Kl−1 can be transformed
intok= Kl−1 ukei − νei.
Kl−1 uk+1
ei(29)

Inner products of both sides of Eqs. (28) and (29) can beobtained−1 k+1T(uk+1ei ) Kl uei
k Tkk= (ukei )T Kl−1 ukei + (νei
) Kl νei− 2(ukei )T νei. (30)

Integrating of both sides Eq. (30) from 0 to T yields
 T−1 k+1T(uk+1ei ) Kl uei dt
0
 T
 T
k T
k
(ukei )T Kl−1 ukei dt +(νei
) Kl νei
dt
=
0 
0
T
k
(ukei )T νei
dt.
−2

(31)

0

This equality can be represented in terms of a L2 norm as
follows:
 T
2
k 2
k 2
k
uk+1

=
u

+
ν

−
2
(ukei )T νei
dt,
−1
−1
ei K
ei Kl
ei
K
l

0

l

where  · {·} is defined as Tukei 2K −1 :=(ukei )T Kl−1 ukei dt.
l

(32)

(33)

0

In the proof of Lemma 2, we have already shown thefollowing relationship:
 T
k
k 2
(ukei )T νei
dt = V k+1 − V k + νei
 Kc ,(34)
0

where we have regarded V (T ) and V (0) as V k+1 and V k ,respectively. From Eqs. (32) and (34), it can be easily shown
that
2
k+1
k 2
= ukei 2K −1 + 2V k − νei
(2Kc −Kl ) .
uk+1
ei K −1 + 2V
l
l
(35)

From Eq. (35), the fact that the function {ukei 2K −1 +2V k } isla monotonically non-increasing function and bounded below,
and the assumption 0 < Kl < 2Kc , it is clear that itconverges to a non-negative value when k → ∞. Then, thek 2(2Kc −Kl ) tends to zero as k → ∞, i.e.,
output error νeikνl converges to zero in the sense of a L2 [0, T ] norm. Since

gcd
gco+

gec

g to eec-0.5

Pose Computation

MemoryAd(gcd)

Kc

uk+1
+
l
Ad(gcd)
+ uk

Kl

nlk

l

Iterative Learning Controller
Fig. 3.

Block diagram of iterative learning control.

gcd
f

b
Vwc

Visual gco
Motion ee
Observer

Iterative
Learning
Controller

-0.15
0.5

Vision f
CameraIterative Leaning Controller with
Visual Motion Observer
Fig. 4. Block diagram of iterative learning control with visual motion
observer.

νlk = −ekc , it can be concluded that the pose control error
ec in trial number k converges to zero.
Theorem 2 guarantees the convergence of the relative rigid
body motion gco in trial number k to the desired one gcd
using the property that the pose control error system (19)is output strictly passive. From Theorem 2, it can be also
shown that the input error ukei 2K −1 tends to zero as k → ∞.
l

Since ekc = 0 as k → ∞ in Eq. (18), the body velocity of thebconverges to the ideal input ui in the sense
vision camera Vwc
of a L2 [0, T ] norm. Fig. 3 shows a block diagram of theiterative learning controller. It should be noted that the inputbdoes not need the ideal input ui infor the vision camera Vwc
practice as shown in Fig. 3, though it is assumed its existencebtheoretically.
using the unknown target object velocity Vwo
Fig. 4 shows a block diagram of the closed-loop system
which consists of the vision camera and the passivity-based
iterative learning controller with visual motion observer.
Since the iterative learning control approach is generally to
improve the transient response and tracking performance forthe repeatability, the control performance of the proposed
control law for the 3-D visual feedback system should be
improved compared to the previous one in [7]. This is one of
the main advantages of this work. In addition, it is a fact thatthe image Jacobian-like matrix J(·) of the pinhole camera
model is exactly the same form as that of the panoramic
camera model [9]. Therefore, the proposed iterative learning
control approach can be applied to visual feedback systems
with a panoramic camera using the image Jacobian-like
matrix J(·) in [9]. This allows us to extend the technologicalapplication area.
IV. S IMULATION R ESULTS
In this section, we present simulation results for theiterative learning control with a static target object. The
desired relative rigid body motion gcd is generated by using

0.5

1

1.5

0

0.5

1

1.5

0

0.5

1

1.5

0
-1

b
Vwo
b
Vwc

0

pecy [m]

+
+

0.05
0

xq ecz [rad]

b
Vwc

0

time [s]Fig. 5. Pose control error ec : dotted; with k = 1: dashed; with k = 5:
solid; with k = 10.
0.6
perx [m]

gee++

0
-0.2
0.2
0

0

0.5

1

1.5

0

0.5

1

1.5

0

0.5

1

1.5

pery [m]e to g-0.6
0.5

xq erz [rad]ee0.5pecx [m]

gco

0
-1

time [s]Fig. 6. Pose control error er : dotted; with k = 1: dashed; with k = 10:
solid; with k = 15.

the image space navigation function-based path planner [10].
The control objective is that the vision camera tracks thestatic target object to keep all target feature points (fourpoints) inside the camera field of view. In other words, it isbring the actual relative rigid body motion gco to the given
time-varying reference one gcd , i.e., it can be achieved tomake the pose control error zero.
The simulation is carried out with the initial conditionpco = [0.2 0.2 − 1.35]T m, ξθco = [0 0 0]T rad. The final
desired relative rigid body motion is pcdf = [−0.23 − 0.3 −1.35]T m, ξθcdf = [0 0 π/2]T rad. The gains were selected
as Ke = 5I, Kc = 5I and Kl = I satisfy 0 < Kl < 2Kc .
The simulation result is presented in Fig. 5. Fig. 5 shows
the pose control error ec . The dotted, dashed and solid lines
denote the errors applying the update control law in trial
number k = 1, k = 5 and k = 10, respectively. We focus on
the errors of the translations of x and y and the rotation of z.
In Fig. 5, the asymptotic stability can be confirmed by steadystate performance. Moreover, the rise time applying theiterative learning control is shorter with increasing repetition.
Compared to [7] that is equivalent to the iterative learning
control approach in the case of k = 1, it can be easily verified
that the control performance is improved.
Next, we consider a more realistic situation which has

TABLE I
N ORM OF THE POSE CONTROL ERROR er AT 1.5 S .
Trial Number
k=1
k = 10k = 15

Norm
0.04820.02450.0096the influence of the friction for the vision camera motion.
In particular, if the elements of the camera body velocity
input is calculated as |vwci | < 0.1 and |ωwci | < 0.1, wherei = x, y, z, then we regards it as zero, i.e., the friction makes
the vision camera immovable in the case of |vwci | < 0.1 and|ωwci | < 0.1. Fig. 6 shows the actual control error er , whichis the error vector between the current relative rigid body
motion gco and the final desired one gcdf , instead of the timevarying desired one gcd . The norm of the pose control error
er at 1.5s is also shown in Table I. From Fig. 6 and Table I,
the slight error remains at the steady state because of thefriction force. However, it can be confirmed that it decreases
with increasing repetition. This simulation results indicatesthat the steady state error in trial number k − 1 influences
directly generation of the control input in the next trial. Since
the control input increases, the steady state error is reduced,
consequently. This is the great merit of the iterative learning
control for the more realistic visual feedback system.
V. C ONCLUSIONSThis paper proposes iterative learning control based on
passivity for 3-D visual feedback systems. The main contribution of this paper is to show that the iterative learning
control law which can improve the transient response andtracking performance for the repeatability is designed for thevisual feedback system. Convergence analysis of the closedloop system is discussed based on passivity. Simulation
results are presented to verify the control performance ofthe proposed control scheme and the previous one [7].
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A PPENDIX
In this paper, we use the notation eξ̂θab ∈ R3×3 torepresent the change of the principle axes of a frame Σb
relative to a frame Σa . ξab ∈ R3 specifies the direction ofrotation and θab ∈ R is the angle of rotation. For simplicity
ˆ ab to denote ξ̂ab θab . The notation ‘∧’ (wedge) is
we use ξθthe skew-symmetric operator such that ξ̂θab = ξab × θabfor the vector cross-product × and any vector θ ∈ R3 .
The notation ‘∨’ (vee) denotes the inverse operator to ‘∧’,i.e., so(3) → R3 . Recall that a skew-symmetric matrix
corresponds to an axis of rotation (via the mapping a → â).
We use the 4 × 4 matrixeξ̂θab pab
gab =
(36)
0
1
as the homogeneous representation of gab = (pab , eξ̂θab ) ∈SE(3) describing the configuration of a frame Σb relative toa frame Σa . The adjoint transformation associated with gab ,written Ad(gab ) , is given byAd(gab ) =eξ̂θab0p̂ab eξ̂θabeξ̂θab.

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