FAST ESTIMATION FOR RANGE IDENTIFICATION IN THE

PRESENCE OF UNKNOWN MOTION PARAMETERS

Lili Ma, Chengyu Cao, Naira Hovakimyan, Craig Woolsey

Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24061-0203

Warren E. Dixon

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250

Keywords:

Fast adaptive estimator, Range identiﬁcation.

Abstract:

A fast adaptive estimator is applied to the problem of range identiﬁcation in the presence of unknown mo-

tion parameters. Assuming a rigid-body motion with unknown constant rotational parameters but known

translational parameters, extraction of the unknown rotational parameters is achieved by recursive least square

method. Simulations demonstrate the superior performance of fast estimation in comparison to identiﬁer based

observers.

1 INTRODUCTION

A variety of 3D motion estimation algorithms have

been developed since 1970’s, inspired by such dis-

parate applications as robot navigation, medical imag-

ing, and video conferencing. Even though motion es-

timation from imagery is not a new topic, continual

improvements in digital imaging, computer process-

ing capabilities, and nonlinear estimation theory have

helped to keep the topic current. Assuming that the

motion of the moving object follows certain structure,

which can have parametric uncertainties, extended

Kalman ﬁlter (EKF) has been used to estimate the

states and parameters of the nonlinear system asso-

ciated with the moving object dynamics. Application

of EKF assumes linearization about the estimated tra-

jectory. However, for the motion estimation from im-

agery the geometric structure of the perspective sys-

tem can be lost during the linearization

(Ghosh et al.,

1994; Dixon et al., 2003). Refs. (Jankovic and Ghosh,

1995; Chen and Kano, 2002; Dixon et al., 2003; Kara-

giannis and Astolﬁ, 2005; Ma et al., 2005) have con-

sidered nonlinear observers for perspective dynamic

systems (PDS) arising in visual tracking problems. In

general, a PDS is a linear system, whose output is

observed up to a homogeneous line(Chen and Kano,

002). This class of nonlinear observers is referred to

as perspective nonlinear observers.

Perspective nonlinear observers (Jankovic and

Ghosh, 1995; Chen and Kano, 2002; Dixon et al.,

2003; Karagiannis and Astolﬁ, 2005; Ma et al.,

2005) are used quite often for determining the un-

nown states (i.e., the 3D Euclidean coordinates) of

a moving object with known motion parameters. For

example, an identiﬁer-based observer was proposed

in(Jankovic and Ghosh, 1995) to estimate a station-

ary point’s 3D position using a moving camera. An-

other discontinuous observer, motivated by sliding

mode and adaptive methods, is developed in(Chen

and Kano, 2002) that renders the state observation

error uniformly ultimately bounded. A state esti-

mation algorithm with a single homogeneous obser-

vation (i.e., a single image coordinate) is presented

in(Ma et al., 2005). A reduced-order nonlinear ob-

server is described in(Karagiannis and Astolﬁ, 2005)

to provide asymptotic range estimation. All these re-

sults are based on the assumption that the object is

following a known motion dynamics in the 3D space.

In this paper, we discuss a situation when some

of the motion parameters, more speciﬁcally, the rota-

tional parameters, are unknown constants. The objec-

tive is to achieve fast state estimation and parameter

convergence.

One model for the relative motion of a point in

the camera’s ﬁeld of view is the following linear

system

(Jankovic and Ghosh, 1995; Chen and Kano,

002; Dixon et al., 2003; Karagiannis and Astolﬁ,

157

Ma L., Cao C., Hovakimyan N., Woolsey C. and E. Dixon W. (2007).

FAST ESTIMATION FOR RANGE IDENTIFICATION IN THE PRESENCE OF UNKNOWN MOTION PARAMETERS.

In Proceedings of the Four th International Conference on Informatics in Control, Automation and Robotics, pages 157-164

DOI: 10.5220/0001642001570164

 SciTePress

2005):





X(t)

Y(t)

Z(t)









0 w

−w

0 w

−w









X(t)

Y(t)

Z(t)













(1)

where the matrix [w

] presents the rotational dynam-

ics, the vector [b

] corresponds to the translational mo-

tion, while [X, Y, Z]

⊤

are the coordinates of the point

in the camera frame. From the 2D image plane, the

homogeneous output observations are given by

(t) = X(t)/Z(t), x

(t) = Y(t)/Z(t). (2)

These equations might model either a stationary

point’s 3D position as observed from a moving cam-

era (assuming that the moving camera’s velocities can

be measured

(Jankovic and Ghosh, 1995)) or a mov-

ing point’s 3D position as observed from a stationary

camera(Tsai and Huang, 1981). In general, w

can be

time-dependent, but in this paper we limit the discus-

sion to constant w

’s.

Let

x(t) = [x

(t), x

(t)]

⊤

= [X(t)/Z(t), Y(t)/Z(t), 1/Z(t)]

⊤

(3)

The system (1) with output observations (3) is equiv-

alent to the system













˙x

(t)

˙x

(t)





−b





+ w

−w

+ w



˙x

(t) = (w

+ w

−b

(4)

with the output

y(t) = [x

(t),x

(t)]

⊤

. (5)

Estimation of x

(t) from the measurements

(t),x

(t)) constitutes the range identiﬁcation

problem. Refs.

(Jankovic and Ghosh, 1995; Chen

and Kano, 2002; Dixon et al., 2003; Karagiannis and

Astolﬁ, 2005; Ma et al., 2005) have solved this prob-

lem assuming that the motion parameters w

and b

(1) are known (where i ∈ {1,2, 3}). Here, we assume

that the parameters w

are unknown. The objective,

then, is to estimate x

(t) as well as the unknown

parameters w

. This problem can be formulated in a

way such that an existing i

dentiﬁer-based observer

(IBO), described in

(Jankovic and Ghosh, 1995),

can be applied, such that under certain assumptions,

the approach provides exponential convergence of

both the range and the parameter estimates. A more

general case of the problem is discussed in

(Ma et al.,

2007)

, where the rotational matrix is represented by a

3×3 matrix instead of the skew-symmetric matrix as

in (1).

In this paper, a recently-developed novel adaptive

estimator is applied for the estimation of x

(t) along

with the unknown parameters w

. A numerical com-

parison of the performance of this adaptive estimator

with the IBO observer is provided.

The paper is organized as follows. Range identi-

ﬁcation in the presence of unknown parameters via

the IBO is presented in Sec. 2. A brief review of

the fast estimator is given in Sec. 3. In Sec. 4, fast

estimation for the range identiﬁcation problem with

unknown motion parameters is presented. Section 5

presents the simulation results. Section 6 extends the

analysis to general afﬁne motion. Finally, section 7

concludes the paper.

2 RANGE IDENTIFICATION IN

THE PRESENCE OF UNKNOWN

PARAMETERS VIA IBO

Consider the state estimation problem for the perspec-

tive dynamic system (7), where the motion parameters

(for i = 1,2, 3) are assumed to be unknown con-

stants. Let θ be a vector of these unknown constants

deﬁned as

θ = [w

, w

]

⊤

. (6)

The system (4) can be rewritten as



˙x

(t)

˙x

(t)



= w

⊤

)





, (7a)



˙x

(t)









+ w

−b

{z }

)

3×1







, (7b)

with

⊤

) =



−b

1+ x

−b

−x

1+ x



(8)

which ﬁts into the form of the general nonlinear sys-

tem to which IBO might be applicable, by regarding

= [x

]

⊤

, x

= [x

,θ

⊤

]

⊤

, and φ(x

,u) = 0 (please

refer to

(Jankovic and Ghosh, 1995) for details of the

IBO).

To apply the IBO observer, we need the following

assumption for the system in (7):

Assumption 2.1

1. Let x(t) =



(t),x

(t),θ

⊤



⊤

be bounded

kx(t)k< M, M > 0 for every t ≥ 0. Let Ω = {x ∈

: kx(t)k < M}. Further, for some ﬁxed con-

stant γ > 1, let Ω

= {x ∈ R

: kx(t)k< γM}.

2. The function w

) and its ﬁrst time deriva-

tive are piecewise smooth and uniformly bounded.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

158

Suppose that there exist positive constants L

such that

⊤

)k < L



⊤

)



< L

. (9)

Further, there do not exist constants κ

(for i =

1,2, 3,4) with

∑

i=1

6= 0 such that

(τ)+ κ

(τ) = 0, (10)

for all τ ∈ [t,t + µ], where µ > 0 is a sufﬁciently

small constant, and v

(τ) denotes the i

column

in w

in (8).

It is straightforward to verify that, under Assump-

tion 2.1, the system in (7) veriﬁes the assumptions re-

quired for the application of IBO. Estimation of x

(t),

along with the unknown motion parameters θ, can be

obtained via direct application of the IBO, as given

below.

Letting e

= ˆx

−x

, e

= ˆx

−x

, e

= ˆx

−x

, the

following observer can be designed for the system (7)













ˆx



= GA





+ w

⊤

)



ˆx





ˆx



= −G







, ˆx

, ˆw

)

3×1



x(t

) = M

x(t

−

)

x(t

−

(11)

where x denotes [x

,θ

⊤

]

⊤

θ denotes the esti-

mation of θ, and the sequence of t

is deﬁned as

= min {t : t > t

i−1

andk

x(t)k≥ γM}, t

= 0, (12)

for some ﬁxed constant γ > 1. The closed-loop error

dynamics can be derived from (7) and (11) as













˙e



= GA





+ w

⊤

)





˙e

= −G







, ˆx

, ˆw

) −g

)

3×1



(13)

where

θ =

θ−θ and

θ =

θ, since θ is assumed to be

a constant vector. The main claim is that there ex-

ists a positive constant G

, such that the estimation

errors [e

⊤

]

⊤

converge to zero exponentially

if the constant G in (11) is chosen to be larger than

(Jankovic and Ghosh, 1995).

3 FAST ESTIMATOR

Range identiﬁcation in the presence of unknown mo-

tion parameters is further pursued using a recently-

developed fast adaptive estimator. The adaptive es-

timator enables estimation of the unknown time-

varying parameters in the system dynamics via fast

adaptation (large adaptive gain) and a low-pass ﬁlter.

If the time-varying unknown signal is linearly param-

eterized in unknown constant parameters, the adap-

tive estimator can be further augmented by a recur-

sive least-square algorithm (RLS) to estimate the un-

known constant parameters asymptotically

(Cao and

Hovakimyan, 2007)

In the following, main results of the the adaptive

estimator are given for the purpose of completeness.

More details are presented in

(Cao and Hovakimyan,

2007)

3.1 Preliminaries

Some basic deﬁnitions from linear system theory are

given in this section.

Deﬁnition 3.1 For a signal ξ(t), t ≥ 0, ξ ∈ R

, its

∞

norm is deﬁned as

kξk

∞

= max

i=1,...,n



sup

τ≥0

|ξ

(τ)|



, (14)

where ξ

is the i

component of ξ.

Deﬁnition 3.2 The

gain of a stable proper single–

input single–output system H(s) is deﬁned as:

kHk

∞

|h(t)|dt, (15)

where h(t) is the impulse response of H(s).

Deﬁnition 3.3 For a stable proper m input n output

system H(s) its

gain is deﬁned as

kHk

= max

i=1,...,n

∑

j=1

, (16)

where H

(s) is the i

row j

column element of H(s).

3.2 Problem Formulation

Consider the following system dynamics:

˙x(t) = A

x(t) + ω(t), x(0) = x

, (17)

where x ∈ R

is the system state vector (measurable),

ω(t) ∈ R

is a vector of unknown time-varying sig-

nals or parameters, and A

is a known n×n Hurwitz

matrix. Let

ω(t) ∈ Ω, (18)

where Ω is a known compact set. The signal ω(t) is

further assumed to be continuously differentiable with

uniformly bounded derivative

ω(t)k≤ d

< ∞, ∀ t ≥ 0, (19)

where d

can be arbitrarily large. The estimation ob-

jective is to design an adaptive estimator that provides

fast estimation of ω(t).

FAST ESTIMATION FOR RANGE IDENTIFICATION IN THE PRESENCE OF UNKNOWN MOTION

PARAMETERS

159

3.3 Fast Adaptive Estimator

The adaptive estimator consists of the state predictor,

the adaptive law and a low-pass ﬁlter, which extracts

the estimation information.

State Predictor: We consider the following state

predictor:

ˆx(t) = A

ˆx(t) +

ω(t), ˆx(0) = x

, (20)

which has the same structure as the system in (17).

The only difference is that the unknown parameters

ω(t) are replaced by their adaptive estimates

ω(t) that

are governed by the following adaptation laws.

Adaptive Laws: Adaptive estimates are given by:

ω(t) = Γ

Proj(

ω(t),−P˜x(t)),

ω(0) =

, (21)

where ˜x(t) = ˆx(t)−x(t) is the error signal between the

state of the system and the state predictor, Γ

∈ R

the adaptation rate, chosen sufﬁciently large, and P is

the solution of the algebraic equation A

⊤

P + PA

−Q, Q > 0.

Estimation: The estimation of the unknown sig-

nal is generated by:

(s) = C(s)

ω(s), (22)

where C(s) is a diagonal matrix with its i

diagonal

element C

(s) being a strictly proper stable transfer

function with low-pass gain C

(0) = 1. One simple

choice is

(s) =

s+ θ

. (23)

3.4 Convergence Results

The fast adaptive estimator in Sec. 3.3 ensures that

(t) estimates the unknown signal ω(t) with the ﬁnal

precision:

k1−C(s)k

kωk

∞

√

, (24)

where k·k

denotes the

gain of the system.

To quantify this performance bound between

(t) and ω(t), an intermediate signal ω

(t) is intro-

duced as:

(s) = C(s)ω(s). (25)

The following theorem gives the performance

bound between ω

(t) and ω

(t). Details of the proof

can be found in

(Cao and Hovakimyan, 2007).

Theorem 3.1 For the system in (17) and the fast

adaptive estimator in (20), (21) and (22), we have

kω

−ω

∞

≤

√

, (26)

where

= kC(s)H

−1

(s)k

min

(P)

, (27a)

H(s) = (sI −A

)

−1

, (27b)

= max

ω∈Ω

4kωk

+ 2

max

(P)

min

(Q)



max

ω∈Ω

kωk



, (27c)

and k·k

∞

denotes the

∞

norm of the signal.

Corollary 3.1 For the system in (17) and the fast

adaptive estimator in (20), (21) and (22), we have

lim

→∞

(ω

(t) −ω

(t)) = 0, ∀ t ≥ 0. (28)

We further characterize the performance bound

between ω

(t) and ω(t). For simplicity, we use a ﬁrst

order C(s) as in (23). It follows from (25) that

(t) = −θ

(t) + θ

ω(t), ω

(0) = 0. (29)

We note that ω

(t) can be decomposed into two com-

ponents:

(t) = ω

(t) + ω

(t), (30)

where ω

(t) and ω

(t) are deﬁned via:

(t) = −θ

(t)+ θ

ω(t), ω

(0) = ω(0),(31a)

(t) = −θ

(t), ω

(0) = −ω(0). (31b)

It follows from (31a) that

kω

−ωk

∞

= k1−C(s)k

kωk

∞

. (32)

Since

lim

→∞

k1−C(s)k

= 0, (33)

the norm kω

− ωk

∞

can be rendered arbitrarily

small by increasing the bandwidth of C(s). Further,

(t) decays to zero exponentially and the settling

time is inverse proportional to the bandwidth of C(s).

Increasing the bandwidth of C(s) implies that ω

(t)

decays to zero quickly.

From (26) and (32), when the transients of C(s)

due to the initial condition −ω(0) die out, ω

(t) es-

timates ω(t) with the ﬁnal precision given in (24).

It is obvious that both the ﬁnal estimation precision

and the transient time can be arbitrarily reduced by

increasing the bandwidth of C(s), which leads to

smaller

gain for k1−C(s)k

. However, the large

bandwidth of C(s) leads to further increase of γ

which requires large Γ

to keep the term

√

small.

We note that larger Γ

implies faster computation and

requires smaller integration step.

3.5 Extraction of Unknown Parameters

If the time-varying signal ω(t) can be linearly param-

eterized in unknown constant parameters and known

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

160

nonlinear functions, extraction of the unknown pa-

rameters can be achieved by recursive least-square

(RLS) algorithm under certain persistent excitation

type of condition. The RLS algorithm is reviewed be-

low.

Consider a linear scalar regression model denoted

as:

= θ

⊤

+ e

, (34)

where

θ = [θ

,θ

,··· ,θ

]

⊤

(35)

is the n×1 vector of the plant parameters, and

= [φ

k,1

,φ

k,2

,··· ,φ

k,n

]

⊤

(36)

is the n ×1 regressor vector at time instant k, while

is a zero-mean discrete white noise sequence with

variance σ

. When the observation of (ω

,φ

) has

been obtained for k = 1, ··· ,N (with N > n), the RLS

estimate for θ, denoted by

θ, can be obtained in the

following discrete form

(Verhaegen, 1989):

k−1

λ+ φ

⊤

k−1

+ L

(ω

−φ

⊤

k−1

−

k−1

⊤

k−1

λ+ φ

⊤

k−1

(37)

where P

= pI

p×p

and λ ∈ (0,1]. Coefﬁcients p and λ

are design gains and need to be chosen appropriately.

When φ

is persistently exciting during the observa-

tion period, RLS algorithm ensures the convergence

θ to θ. The convergence rate of RLS can be in-

creased by choosing large λ.

The PE condition of the regressor vector is deﬁned

(Verhaegen, 1989):

Deﬁnition 3.4 The regressor vector φ

is persistently

exciting over the observation interval k

≤ k ≤ k

with an exponentially forgetting factor λ ≤ 1, if the

following condition is fulﬁlled:

αI ≤

∑

k=k

⊤

−k

≤ βI (38)

for some positive α > 0 and β > 0.

4 FAST ESTIMATION FOR

RANGE IDENTIFICATION IN

THE PRESENCE OF

UNKNOWN PARAMETERS

Denote

η(t) =



(t)



, (39)

and write equation (7a) as



˙x

(t)

˙x

(t)



= w

⊤

)



(t)



= η(t). (40)

From equations (6), (8), and (40), we have



−b

1+ x

−b

−x

1+ x









(t)







= η(t).

(41)

Multiplying the ﬁrst equation in (41) by T

= b

−

(t) and subtracting the second equation from it

pre-multiplying it by T

= b

−b

(t), we arrive at:



+ T

(1+ x

) −T

−T

(1+ x

)



{z }

⊤

(t)









{z}

θ(t)

= [T

−T

(42)

Recursive least squares method can be used to extract

’s according to (37), with ω replaced by T

−

. Once w

(for i = 1, 2,3) are available, equation

(41) takes the form:



−b







−



1+ x

−x

1+ x











(43)

where x

(t) can be extracted using pseudo-inverse.

Using the fast adaptive estimator described in

Sec. 3, estimation of η(t), denoted by η

(t), can be

obtained via the following steps:

• State Estimator:



ˆx



= A



˜x



η(t),



˜x





ˆx

−x

ˆx

−x



. (44)

• Adaptive Law (use large Γ

η(t) = −Γ

⊤



˜x



⊤

. (45)

• Extraction:

(s) = C(s)

η(s), C(s) =

s+C

. (46)

According to Corollary 3.1, the ﬁnal estimation pre-

cision η

(t) −η(t) and the transient time to achieve

this can be arbitrarily reduced by increasing the band-

width of C(s). Increasing the bandwidth of C(s) re-

quires larger Γ

The ﬂow chart of state and parameter estimation

of a rigid motion using the fast adaptive estimator is

illustrated in Fig. 1. In the ﬁrst step of estimating η(t),

both the estimation precision and transient time can

be arbitrarily reduced by increasing the bandwidth of

C(s) and using larger Γ

. In the second step of extract-

ing ˆw

’s from η

(t) using the recursive least square

FAST ESTIMATION FOR RANGE IDENTIFICATION IN THE PRESENCE OF UNKNOWN MOTION

PARAMETERS

161

method, fast speed can be achieved by properly tun-

ing the RLS gains. Estimation of x

(t), denoted by

ˆx

(t), can be obtained from η

(t) and ˆw

’s via pseudo-

inverse. Since the fast adaptive estimator assumes

minimization of the

gain of 1 −C(s) for perfor-

mance improvement, it is referred to as

adaptive

estimator.

)(t

)(

)(sC

Adaptive Estimator

RLS

)(

Pseudo-inverse

Figure 1: Flow chart of L

adaptive estimator.

5 SIMULATION RESULTS

State estimation of [x

(t),θ

⊤

]

⊤

using the IBO ob-

server (11) and the fast adaptive estimator (44) (46)

are implemented in Matlab, where the motion dynam-

ics are selected to be





X(t)

Y(t)

Z(t)









0 −4 −0.8

4 0 −0.6

0.8 0.6 0









X(t)

Y(t)

Z(t)









3πsin(2πt)

3πsin(2πt +π/4)





) = (1,1.5,2.5), x

= (X

,1/Z

(47)

First, we present simulation results in the ideal case

with no measurement noise. The parameters for the

IBO observer and the fast adaptive estimator are cho-

sen to be:

• IBO (referring to (11)):

G = 10, ( ˆx

(0), ˆw

(0)) = (0,0,0,0).

• Fast adaptive estimator (referring to (37), (45),

(46)):

p = 100, λ = 0.99999, A

= −I

(

(0),

(0)) = (0,0), Γ

= 2×10

, C = 200.

In both cases, we set ( ˆx

(0), ˆx

(0)) =

(0),x

(0)), M = 30, A = I

, P = −1/2 × I

where I

denotes the 2×2 identity matrix.

Estimation of w

(for i = 1,2, 3) with the use of the

IBO and the fast adaptive estimator is shown in Fig-

ures 2 and 3, respectively. Figure 4 shows the zoomed

version of Figure 3 for the steady state error. State es-

timation error of x

is plotted in Figure 5 for compar-

ison of both methods.

From Figures 2 and 3, it can be observed that the

fast adaptive estimator achieves faster estimation of

the motion parameters. The same is true for x

Simulation results are also presented in Figs. 6∼9

when the output is noise-corrupted with uniform

bound ±10

−2

. The simulation parameters are the

same as above. In this case, when extracting ˆx

(t), the

output from the pseudo-inverse is further processed

0 5 10 15

Estimation of w

− w

0 5 10 15

−2

−1

Estimation of w

− w

0 5 10 15

−2

−1

Estimation of w

− w

Figure 2: Estimation of motion parameters using IBO

(without measurement noise).

0 5 10 15

100

Estimator of w

− w

0 5 10 15

−40

−20

Estimator of w

− w

0 5 10 15

−10

Estimator of w

− w

Figure 3: Estimation of motion parameters using fast adap-

tive estimator (without measurement noise).

using a low-pass ﬁlter

s+30

to give the ﬁnal state esti-

mation. We observe that corresponding plots with or

without measurement noise are very similar.

6 FURTHER EXTENSION

In this paper, rigid-body motion is considered that

contains only three rotational parameters (w

)

as given in (1). For general afﬁne motion described





X(t)

Y(t)

Z(t)

















X(t)

Y(t)

Z(t)













, (48)

the rotational matrix contains nine parameters. As-

suming that the [a

] (for i, j = 1, 2,3) are unknown

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

162

2 4 6 8 10 12 14

−0.5

0.5

Estimator of w

− w

2 4 6 8 10 12 14

−0.5

0.5

Estimator of w

− w

2 4 6 8 10 12 14

−0.5

0.5

Estimator of w

− w

Figure 4: Enlarged view of Fig. 3 (without measurement

noise).

0 5 10 15

−5

−4

−3

−2

−1

Figure 5: Comparison of state estimation errors (without

measurement noise).

constants, the method described in Sec. 4 cannot lead

to extraction of the nine unknown parameters in a

straightforward way.

The system (48) with output observations (3) is

equivalent to the system













˙x

(t)

˙x

(t)





−b





+ (a

−a

+ a





−a

− a

−a

− a



˙x

(t) = −(a

+ a

−b

(49)

with the output (5). The above system can also be

rewritten in the form of (7a), where θ and w

⊤

)

take the forms

θ = [a

, a

]

⊤

(50)

0 5 10 15

Estimation of w

− w

0 5 10 15

−2

−1

Estimation of w

− w

0 5 10 15

−2

−1

Estimation of w

− w

Figure 6: Estimation of motion parameters using IBO (with

measurement noise).

0 5 10 15

100

150

Estimator of w

− w

0 5 10 15

−30

−20

−10

Estimator of w

− w

0 5 10 15

Estimator of w

− w

Figure 7: Estimation of motion parameters using fast adap-

tive estimator (with measurement noise).

and

⊤

) =



−b

1 0 0 0

−b

0 0 0 x

−x



(51)

respectively. Following the logic in Sec. 4, we can

write the following system of algebraic equations

⊤

)



··· a



⊤





(52)

with the w

⊤

) given in (51). Again, multiplying

the ﬁrst equation in (52) by T

= b

−b

and sub-

tracting the second equation from it pre-multiplying it

FAST ESTIMATION FOR RANGE IDENTIFICATION IN THE PRESENCE OF UNKNOWN MOTION

PARAMETERS

163

2 4 6 8 10 12 14

−0.5

0.5

Estimator of w

− w

2 4 6 8 10 12 14

−0.5

0.5

Estimator of w

− w

2 4 6 8 10 12 14

−0.5

0.5

Estimator of w

− w

Figure 8: Enlarged view of Fig. 7 (with measurement

noise).

0 5 10 15

−1.5

−1

−0.5

0.5

estimation error of x

IBO

fast estimator

Figure 9: Comparison of state estimation errors (with mea-

surement noise).

by T

= b

−b

, we arrive at:

,1), T

,1), (b

−b

)(x

,1)]

{z }

afﬁne

(t)













= [T

−T

(53)

The nine columns in φ

afﬁne

(t) in (53) are linearly de-

pendent. It is obvious that the 7

, and 9

columns

can be presented as linear combinations of the ﬁrst

six columns. For example, column

can be written

as column

= column

−column

. Thus, extraction

of the nine unknown parameters cannot be performed

by the recursive least square method since it violates

the PE condition in (38). Further research will ex-

plore the use of adaptive observers for general afﬁne

motion identiﬁcation.

7 CONCLUSION

A recently developed fast adaptive estimator is ap-

plied to the range identiﬁcation problem of a rigid

motion in the presence of unknown motion parame-

ters. Fast convergence speed is achieved compared to

existing nonlinear perspective observers.

ACKNOWLEDGEMENTS

This work was sponsored in part by ONR Grant

#N00014-06-1-0801 and AFOSR MURI subcontract

F49620-03-1-0401.

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