The International Journal of Robotics Research

[Pages:17]The International Journal of Robotics Research



Nonparametric Second-order Theory of Error Propagation on Motion Groups Yunfeng Wang and Gregory S. Chirikjian

The International Journal of Robotics Research 2008; 27; 1258 DOI: 10.1177/0278364908097583

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Yunfeng Wang

Department of Mechanical Engineering, The College of New Jersey, Ewing, NJ 08628, USA jwang@tcnj.edu

Gregory S. Chirikjian

Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA gregc@jhu.edu

Nonparametric Second-order Theory of Error Propagation on Motion Groups

Abstract

Error propagation on the Euclidean motion group arises in a number of areas such as in dead reckoning errors in mobile robot navigation and joint errors that accumulate from the base to the distal end of kinematic chains such as manipulators and biological macromolecules. We address error propagation in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE132, can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors that are small (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second order, whereas prior efforts only considered the first-order case. Our formulation is non-parametric in the sense that it will work for probability density functions of any form (not only Gaussians). Numerical tests demonstrate the accuracy of this second-order theory in the context of a manipulator arm and a flexible needle with bevel tip.

KEY WORDS--recursive error propagation, Euclidean group, spatial uncertainty.

1. Introduction

Error propagation on the Euclidean motion group arises in a surprising number of different areas. For example, consider a

The International Journal of Robotics Research Vol. 27, No. 11?12, November/December 2008, pp. 1258?1273 DOI: 10.1177/0278364908097583 1c SAGE Publications 2008 Los Angeles, London, New Delhi and Singapore Figure 1 appears in color online:

robotic manipulator for which each joint angle has some backlash. If we describe this backlash as a distribution of possible angles around the nominal one, how will these joint errors add up to produce pose errors at the end effector? Similar problems arise in the study of chainlike biological macromolecules that undergo thermal fluctuations in solution. See, for example, Zhou and Chirikjian (2006) and Kim and Chirikjian (2005). As another example, consider a non-holonomic mobile robot that executes an open loop trajectory. Uncertainties in pose will add up along the path, and if many trials are performed, what will the distribution of terminal poses be? Many such problems in "probabilistic robotics" can be imagined with the recent popularity of simultaneous localization and mapping (SLAM) (Thrun et al. 2005).

If the errors are small, Jacobian-based methods or firstorder error propagation theories can be used. However, what if the errors are very large? Here we address the propagation of large errors in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE132, can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors of moderate size (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second order, whereas prior efforts only considered the firstorder case. Our formulation is non-parametric in the sense that it will work for probability density functions (pdfs) of any form (not only Gaussians).

In the remainder of this section we review the literature on error propagation, and review the terminology and notation used throughout the paper. In what follows, bold lower case letters denote vectors, N and n are positive integers, G denotes either the groups SO132 or SE132, all upper case letters (Roman

1258

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Wang and Chirikjian / Nonparametric Second-order Theory of Error Propagation on Motion Groups 1259

or Greek) (except for N and G) denote matrices, lower case letters denote scalars and group elements, and a lower case letter followed by parenthesis denotes a scalar-valued function.

In Section 2, important definitions from the basic theory of Lie groups and probability and statistics are reviewed. In Section 3, several new theorems are proved. This forms the core of our paper. In Section 4, sampling is discussed and the theory is adapted for the case when a whole pdf is not available. Then numerical tests demonstrate the accuracy of this recursive second-order propagation formula relative to baseline truth generated by brute force. In Section 5 our conclusions are presented. Three appendices provide more detailed background material that is important for understanding the definitions and proofs presented in the main body of the paper. The remainder of the current section reviews the literature and basic definitions and notation used throughout the paper.

1.1. Literature Review

The Lie-group-theoretic notation and terminology which has now become standard vocabulary in the robotics community is presented by Murray et al. (1994) and Selig (1996). Chirikjian and Kyatkin (2001) formulated many problems in robot kinematics and motion planning as the convolution of functions on the Euclidean group. The representation and estimation of spatial uncertainty has also received attention in the robotics and vision literature. Two classic works in this area are Smith and Cheeseman (1986) and Su and Lee (1992). Recent work on error propagation describes the concatenation of Gaussian random variables on groups and applies this formalism to mobile robot navigation (Smith et al. 2003). In all three of these works, errors are assumed to be small enough that covariances can be propagated by the formula (Wang and Chirikjian 2006a,b)

3122 3 Ad1g241231 AdT1g2412 5 324

(1)

where Ad is the adjoint operator for SE132 (see the appendix for a review of terminology). This equation essentially says that given two "noisy" frames of reference g14 g2 6 SE132, each of which is a Gaussian random variable with 6 7 6 covariance matrices1 31 and 32, respectively, the covariance of g1 8 g2 will be 3122. This approximation is very good when errors are very small. We extend this linearized approximation

to the quadratic terms in the expansion of the matrix exponential parameterization of SE132. The origin of (1) will become clear for the special case of small errors in our more general

nonparametric derivation.

1. Exactly what is meant by a covariance for a Lie group is quanti1ed later in the paper.

1.2. Review of Rigid-body Motions

The Euclidean motion group, SE132, is the semi-direct product

of 13 with the special orthogonal group, SO132. We represent

elements of SE132 using 4 7 4 homogeneous transformation

matrices

13

Rt

g32

44

0T 1

and identify the group law with matrix multiplication. The in-

verse of any group element is written as

1 RT

g41 3 2 0T

3 4RTt

45 1

For small translational (rotational) displacements from the identity along (about) the ith coordinate axis, the homogeneous transforms representing infinitesimal motions look like

gi 162 39 exp16 E

i 2 I4 5 6 E

i

where I4 is the 4 7 4 identity matrix and

1

3

1

3

00 0 0

0 010

E

1

3

5555555200

0 1

41 0

0066666664

E

2 3 55555552401

0 0

0 0

0066666664

00 0 0

0 000

1

3

0 41 0 0

E

3

3

5555555210

0 0

0 0

0066666664

0 0 00

1

3

0001

E

4 3 5555555200

0 0

0 0

0066666664

0000

1

3

0000

E

5

3

5555555200

0 0

0 0

1066666664

0000

1

3

0000

E

6 3 5555555200

0 0

0 0

0166666664 5

0000

These are related to the basis elements

Ei for so132 (the

Lie algebra corresponding to the rotation group, SO132) as

1

3

E

i 3 2Ei

0 4

0T 0

when i 3 14 24 3. Each E

i has a corresponding natural unit basis vector ei 6 16. For example, e1 3 [14 04 04 04 04 0]T, e2 3 [04 14 04 04 04 0]T, etc.

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1260 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / November/December 2008

Large motions are also obtained by exponentiating these matrices. For example,

1

3

cos t 4 sin t 0 0

exp1t E

32

3

55555552si0n t

cos t 0

0 1

0066666664

and

0

0 01

1

3

1000

exp1t E

62

3

5555555200

1 0

0 1

0t 66666664 5

0001

More generally, it can be shown that every element in the neighborhood of the identity of a matrix Lie group G can be described with the exponential parameterization

7

9

8n

g 3 g1x14 x24 5 5 5 4 xn2 3 exp

xi E

i

(2)

i 31

where n is the dimension of the group. For SO132 and SE132, n 3 3 and 6, respectively, and the exponential parameterization extends over the whole group.

One defines the "vee" operator, , such that

13

x1

7 8n

i 31

xi

9 E

i

3

55555555552xx5552366666666664

xn

The vector, x 6 1n, can be obtained from g 6 G from the

formula

x 3 1log g25

(3)

For SO132 and SE132 this is defined except on a set of measure zero, which for all intents and purposes in the probability and statics problems that we consider means that the exponential and logarithm maps are "effectively" bijective. See the appendix for details.

When integrating a function over SO132 or SE132, a weight 71x2 is defined such that

f 1g2 dg 3 f 1g1x2271x2 dx5

G

1n

The exact form of the weighting function is

71x2 3 det Jr1x2 where

Jr1x2

3

 g41

8g 8 x1

4

4

 g41

8g 8 xn

5

(4)

This is derived for SO132 and SE132 in the Appendix B and C, respectively. The weighting function is even in the sense that 71x2 3 714x2.

1.3. The Baker?Campbell?Hausdorff Formula

Given any two elements of a Lie algebra, X and Y , the Lie

bracket is defined as [X4 Y ] 3 XY 4 Y X . An important re-

lationship called the Baker?Campbell?Hausdorff (BCH) for-

mula exists between the Lie bracket and matrix exponential

(see Baker (1904), Campbell (1897) and Hausdorff (1906)).

Namely, the logarithm of the product of two Lie group ele-

ments written as exponentials of Lie algebra elements can be

expressed as

Z 1X4 Y 2 3 log1eX eY 2

where

Z1X4Y2 3 X 5 Y

5 1[X4 Y] 2

5 1 1[X4 [X4 Y ]] 5 [Y4 [Y4 X]]2 12

5 1 1[Y4 [X4 [Y4 X ]]] 5 [X4 [Y4 [Y4 X]]]2 48

5 5

(5)

This expression is verified by expanding eX and eY in Taylor

series of the form in (36), and then substituting the result into (37) with g 3 eX eY . If the operation is applied (see the appendix for a review), (5) can be written as

z 3 x 5 y 5 1 ad1X 2y 5 1 1ad1X 2ad1X 2y

2

12

5 ad1Y 2ad1Y 2x2 5 1 1ad1Y 2ad1X2ad1Y 2x 48

5 ad1X2ad1Y 2ad1Y 2x2 5

1.4. Probability and Statistics in 1n: Multivariate Analysis

In 1n, a pdf is defined by the conditions

f 1x2 0 x 6 1n and

f 1x2 dx 3 1

1n

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Wang and Chirikjian / Nonparametric Second-order Theory of Error Propagation on Motion Groups 1261

where dx 3 d x1 d x2 d xn is the usual Lebesgue integration measure. The mean of a pdf, f 1x2, is defined as

1 3 x f 1x2 dx or

1x 4 12 f 1x2 dx 3 05 (6)

1n

1n

Note that 1 minimizes the cost function

c1x2 3 x 4 y2 f 1y2 dy

(7)

1n

where

v

3

v

v

is

the

2-norm

in

1n

.

The covariance of the same pdf about the mean is defined

as

3 3 1x 4 121x 4 12T f 1x2 dx5

(8)

1n

It follows that

C 3 xxT f 1x2 dx 3 3 5 11T4

(9)

1n

where C is the covariance about the origin and 3 is the co-

variance about the mean.

Pdfs are often used to describe distributions of errors. If

these errors are concatenated, they "add" by convolution:

1 f1 2 f221x2 3 f1122 f21x 4 22 d25

(10)

1n

The mean and covariance of convolved distributions are found

as

1122 3 11 5 12 and 3122 3 31 5 325

(11)

In other words, these quantities can be propagated without explicitly performing the convolution computation, or even knowing the full pdfs. This is independent of the parametric form of the pdf. Often one does not have access to the full pdf, but only samples from a process with an underlying pdf. In this case, the unbiased sample mean and covariance are defined as (Anderson 2005)

11N 2

3

1 8N

N

xi

and

i 31

31N2

3

N

1 4

1

8N 1xi

i 31

4

11 N 2 21xi

4

11 N 2 2T 5

The reason for division by N 4 1 rather than N is explained in the literature on multivariate analysis, such as Anderson (2005). As the sample size becomes large, the difference between N and N 4 1 becomes negligible and these sampled quantities converge to those corresponding to the underlying pdf.

Our main purpose in this paper is to develop equations analogous to (11) to describe the propagation of error on the motion group SE132. In the process, we also do so for the rotation group SO132.

It is often convenient to use the Gaussian (or normal) distribution to model errors in 1n. This parametric distribution is completely defined by its mean and covariance. We have no need to assume that densities are Gaussian. Our results are non-parametric, and therefore more general.

2. Definitions and Properties of Mean and Covariance on SE132

In this section we provide definitions of the mean and covari-

ance of Lie-group-valued functions and illustrate some of their

properties. We note in passing that a pdf that is a symmetric function, 91g2 3 91g412, always satisfies the condition

1log g291g2 dg 3 04

(12)

G

for G 3 SO132 or G 3 SE132. This is easy to see if we let 901x2 3 91eX 2. Then 901x2 3 9014x2. This is an even function in the exponential coordinates, and so the odd function

x901x2 integrates to zero over a symmetric domain of integration in the space of exponential parameters that maps to G.

See the appendix for a discussion of integration measures. In

our case this domain is the ball of radius

(for SO132), or the Cartesian product of this ball with 13. Both of which are sym-

metric. Hence, the integral in (12) vanishes. More generally, if

91g2 then

is for

asyni 3m1 mnietordicd,function

on

G

3

SO132

or

G

3

SE132

n

[1log g2 ei ]ni 91g2 dg 3 05

(13)

G i31

This is because the integrand is an odd function of the compo-

nents of x. For example,

1log g22k5191g2 dg 3 0n5

G

Here 0n is the n-by-n zero matrix with n 3 3 or n 3 65

Definition 1. If a unique value 6 G exists for which

[log141 8 g2] f 1g2 dg 3 04

(14)

G

will be called the mean of a pdf f 1g2 on G, which is a straightforward extension of (6). Furthermore, the covariance

about the mean will be computed as

3 3 log141 8 g2[log141 8 g2]T f 1g2 dg5 (15)

G

Note that while in the case of Euclidean space (6) and minimization of (7) both give the same value of the mean, the minimization of a functional of the form

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1262 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / November/December 2008

c1h2 3 [log1h41 8 g2]2 f 1g2 dg

G

does not generally return a value hmin that is equal to . However, in the special case when f 1g2 is unimodal and very concentrated, hmin .

The equality (12) can be thought of as a statement of when the mean is at the identity. If 91g2 has mean at the identity, then f 1g2 3 91a41 8 g2 has mean at a. We use 91g2 to denote pdfs with mean at the identity, and f 1g2 to denote pdfs that can have the mean at some other group element.

Theorem 1. If f 1g2 has mean and covariance 3, then to second order

m 3 [I 5 F1132]1

(16)

where the following shorthand is used:

m 3 1log1g22 f 1g2 dg and 1 3 1log1224 (17)

G

and the matrix-valued function F1132 is defined as

[log1 8 g2]91g2 dg

G

3

x

5

86

i4 j31

i

j

1 12

[ E

i

4

[E

j

4

X

]]

5

1 48

1[

E

i

4

[X4

[E

j 4

X ]]]

5

[X4 [E

i 4

[E

j 4

X ]]]2

5

5

The first expression in the statement of the theorem results from the definition of the adjoint and from keeping the 1rst

two terms in the above expansion. Likewise,

[log1

8

g22]

[log1

8

g22]

T

9

1g2

d

g

G

3

x 5 y 5 1 ad1X2y 5

G

2

7

x

5

y

5

1

ad 1 X 2y

5

T

91g2

dg5

2

Expanding out the product and eliminating terms linear in y

results in the second statement of the theorem.

1

F1 13 2

3

1 12

86  i j ad1E

i 2ad1E

j 2

i4 j31

(18)

and

C 3 1log1g2211log1g222T f 1g2 dg

G

3 3 5 1log 211log 22T

5

1

3

a

d

T1log

2

5

a d 1log

23

5

(19)

2

Here C is the covariance about the identity, which is defined

in an analogy with the concept of covariance about the origin in the context of probability and statistics in 1n.

Proof. Let f 1g2 3 9141 8 g2 where 91g2 has mean at the identity. Then

1log g2 f 1g2 dg 3 1log g29141 8 g2 dg

G

G

3 1log1 8 g2291g2 dg5

G

Expanding using the BCH formula (5) with 3 exp X and g 3 exp Y , and using the linearity of the Lie bracket, we find that since 91g2 is a pdf with mean at the identity,

3. Propagation of the Mean and Covariance of pdfs on SE132

Let 14 2 6 SE132 be two precise reference frames. Then 1 8 2 is the frame resulting from stacking one relative to the other. Now suppose that each has some uncertainty. Let

hi and

k j be two sets of frames of reference that are distributed

around the identity. Let the first have N1 elements, and the second have N2. How will the covariance of the set of N1 N2 frames

1182241818hi 828k j (which are also distributed around the identity) look?

Let 9i 1g2 be a unimodal pdf with mean at the identity and which has a preponderance of its mass concentrated in a unit

ball around the identity (where distance from the identity is measured as 1log g2). Then 9i 1i41 8 g2 will be a distribution with the same shape centered at i . In general, the convolution of two pdfs is defined as

1 f1 2 f221g2 3 f11h2 f21h41 8 g2dh4

G

and, in particular, if we make the change of variables k 3 41 18 h, then

91141 1 8 g2 2 92142 1 8 g2

3

911k292142 1 8 k41 8 41 1 8 g2 dk5

G

Making the change of variables g 3 1 8 2 8 q, where q is a relatively small displacement measured from the identity, the

above can be written as

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Wang and Chirikjian / Nonparametric Second-order Theory of Error Propagation on Motion Groups 1263

912211 8 2 8 q2 3 911k292142 1 8 k41 8 2 8 q2 dk5 (20)

G

The essence of this paper is the efficient approximation of

covariances about the mean of 9122 in (20) when the covariances about the means of 91 and 92 are known. In cases when 122 3 1 8 2, the problem reduces to the efficient approximation of

3122 3

log1q 2 [log1q 2 ]T 9 1 1k 29 2

GG

7 142 1 8 k41 8 2 8 q2 dk dq

3

log142 1 8 k 8 2 8 q2

GG

7 [log142 1 8 k 8 2 8 q2]T911k2921q2 dk dq5 (21)

Lemma 1. The convolution of pdfs with mean at the identity

results (to second order) in a pdf with mean at the identity. Furthermore, if 91 2 92 3 92 2 91 and 9i 1g2 3 9i 1g412, then this result becomes exact.

where

F1A4 B2 3 1C1A4 B2 4

5 1 A B 5 1 A B2T 5 B A 5 1B A2T 4 12

A 3 Ad142 1231 AdT142 124 B 3 324

and C1 A4 B2 and A are computed as follows:

1

A 3 2

A11 4 tr1 A112I3

A12 5 AT12 4 2tr1 A122I3

3

03

44

A11 4 tr1 A112I3

where A is divided into 3-by-3 blocks A11, A12, A21, A225 We define B in the same way with B replacing A every-

where in the expression. The blocks of C are computed as

C11 3 4D114114 C12 3 41D214112T 4 D11412 3 C214

Proof.

We have

1log g2191 2 9221g2 dg

G

3

1log g2911h2921h41 8 g2 dh dg

GG

3

log1h 8 k2911h2921k2 dh dk5

GG

To second order, all terms in the BCH expansion of log1h 8 k2

are linear in either log h or log k (or both), and therefore at

least one of the above integrals integrates to zero.

If 91 2 92 3 92 2 91 and 9i 1g2 3 9i 1g412 then it is easy to show that 191 2 9221g2 3 191 2 9221g412, which automat-

ically means that the function 191 2 9221g2 has mean at the

identity due to (12).

1

Theorem 2. If fi 1g2 is a pdf on SE(3) that has mean i and covariance 3i for i 3 14 2, then to second order, the mean and

covariance of 1 f1 2 f221g2 are, respectively,

122 3 1 8 2

(22)

and

3122 3 A 5 B 5 F1 A4 B24

(23)

C22 3 4D22411 4 D21421 4 1D214122T 4 D114224

where Di j4kl 3 D1 Ai j 4 Bkl 2, and the matrix-valued function D1A4 B2 is defined relative to the entries in the 3 7 3 blocks A and B as

d11 3 4a3 3b2 2 5 a3 1b3 2 5 a2 3b2 3 4 a2 2b3 34 d12 3 a3 3b2 1 4 a3 2b3 1 4 a1 3b2 3 5 a2 1b3 34 d13 3 4a2 3b2 1 5 a2 2b3 1 5 a1 3b2 2 4 a1 2b3 24 d21 3 a3 3b1 2 4 a3 1b3 2 4 a2 1b1 3 5 a2 1b3 34 d22 3 4a3 3b1 1 5 a3 1b3 1 5 a1 3b1 3 4 a1 1b3 34 d23 3 a2 3b1 1 4 a2 1b3 1 4 a1 3b1 2 5 a1 1b3 24 d31 3 4a3 2b1 2 5 a3 1b2 2 5 a2 2b1 3 4 a2 1b2 34 d32 3 a3 2b1 1 4 a3 1b2 1 4 a1 2b1 3 5 a1 1b2 34 d33 3 4a2 2b1 1 5 a2 1b2 1 5 a1 2b1 2 4 a1 1b2 25

Proof. The approximation in (22) follows directly from Lemma 1. Next, let X 3 log142 1 8 k 8 22 3 42 1 K 2 where k 3 exp K , and let Y 3 log q. Using the BCH formula (5) to evaluate the log terms in the definition of covariance, and retaining all even terms to second order (since first-order terms will integrate to zero), we obtain

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1264 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / November/December 2008

1

Z

1

X4

Y

22[1

Z

1

X

4

Y

22]Teven

3

xxT

5

yyT

5

1[X4

Y

]

[X4

Y

]T

4

5

1

y

[

X4

[

X4

Y

]]

T

5

1

x

[Y4 [Y4

X ]]T

12

12

5 1 [X4 [X4 Y ]]yT 5 1 [Y4 [Y4 X ]]xT5

(24)

12

12

Each of these terms can be expanded using the adjoint concept. For example,

[

X

4

Y

]

[

X

4

Y

]T

3

ad1X 2yyTadT1X 2

and

[X4 [X4 Y ]]yT 3 ad1X 2ad1X2yyT5

(25)

In our formulation, X 3 42 1 K 2 (where k 3 eK and so

42 1 x3

A8 dk18 42122k3, theexnp142 1 K 22

3

eX ).

Defining

the

vector

A3

xxT911k2 dk

G

3 Ad142 12 1log k2[1log k2]T911k2 dk AdT142 12

G

3 Ad142 1231 AdT142 12

and since q 3 eY ,

B3

yyT921q2 dq

G

3

1log q2[1log q2]T921q2 dq 3 325

G

The following complicated looking integral (which is noth-

ing more than (21) written in exponential coordinates)

3122 3

1

Z

1

X

4

Y

22

[1

Z

1

X

4

Y

22]T

q6G k6G

7 911k2921q2 dk dq

3

q6G

k6G

1Z1X4

Y

22 [1 Z

1X4

Y

22]Teven

7 911k2921q2 dk dq

can be simplified. This is because

xi x j 3 eiT Ad142 121log k2[1log k2]T AdT142 12e j

and yk yl 3 eTk 1log q2[1log q2]Tel4

and

since

all

terms

in

1Z1X4

Y

22[1Z 1X4

Y

22]Teven

can

be

expressed as weighted sums of such products, it follows that

after integration we obtain

3122 3 A 5 B 5 F1 A4 B25

(26)

For the SE132 case

1

3

R Ad1g2 3 2

034 6 1676 and

TR R

1

3

ad1X2

3

2

034 6 1676

V

where T 3 t, V 3 v and

3 3. Then (25) becomes

[

X

4

Y

]

[

X

4

Y

]T

1

31 3

1

3

3

2

x

03

4

3 2

y

4

[3Ty

4

vTy

]

24 x

4Vx 4

Vx

x

vy

03 4 x

1

31

31

3

3

2

x

03

4

3 2

y

3Ty

3

y

vTy 4

4 2

x

4Vx 4

Vx

x

vy 3Ty vy vTy

03 4 x

and

[X4 [X4 Y ]]yT

1

31

31 3

3

2

x

03

4

2

x

03

4

3 2

y

4

[3Ty

4

vTy

]

Vx

x

Vx

x

vy

1

31

31

3

3

2

x

03

4

2

x

03

4

3 2

y

3Ty

3yvTy 4 5

Vx

x

Vx

x

vy3Ty vy vTy

If we divide the 6 7 6 symmetric matrices A 31 AdT142 12 and B 3 32 into 3 7 3 blocks as

3

Ad142 12

7

9

7

9

A 3 A11 A12 AT12 A22

and B 3 B11 B12 4 B1T2 B22

then using the specific form of ad1X2 and integrating over q

we obtain

1

[X4 [X4 Y ]]yT921q2 dq 3 2

2x

G

Vx

x 5

x Vx

3

03 4 B

2 x

and

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