Optimal Active Target Localisation Strategy with Range-only

Measurements

Shaoming He

a

, Hyo-Sang Shin

b

and Antonios Tsourdos

c

School of Aerospace, Transport and Manufacturing, Cranﬁeld University, College Road, Cranﬁeld, MK43 0AL, U.K.

Keywords: Target Localisation, Range-only Measurement, Optimal Manoeuvre.

Abstract:

This paper investigates the problem of one-step ahead optimal active sensing strategy to minimise estimation

errors with range-only measurements for non-manoeuvring target. The determinant of Fisher Information

Matrix (FIM) is utilized as the objective function in the proposed optimisation problem since it quantiﬁes the

volume of uncertainty ellipsoid of any efﬁcient estimator. In consideration of physical velocity and turning

rate constraints, the optimal heading angle command that maximises the cost function is derived analytically.

Simulations are conducted to validate the analytical ﬁndings.

1 INTRODUCTION

The past few years have witnessed an increasing inter-

est in the development and employment of small-scale

unmanned robots in both civil and military applica-

tions. One particular interesting mission of small-

scale unmanned robots is to track and localize targets

of interest in an automated fashion since reliable tar-

get tracking is a fundamental and key enabling tech-

nology in many practical applications, such as vehicle

navigation, situational awareness and public surveil-

lance (Atanasov et al., 2014; Salaris et al., 2017).

One main challenge for the operations of these small-

scale robots in target localisation is that they are typi-

cally constrained by limited payload, power and en-

durance. Therefore, only limited information, e.g.,

bearing-only or range-only, can be gathered due to the

limits of sensor availability.

It is well-known that the relative geometry be-

tween the observer and the target poses great effects

on the achievable localisation performance (Bishop

et al., 2010). For this reason, active sensing that ﬁnds

the optimal path or trajectory for information gain

maximisation can yield signiﬁcant beneﬁts to improv-

ing the perceptual results in target localisation (Bajcsy

et al., 2018). Through numerically maximising the

determinant of FIM over a ﬁnite horizon, optimal ob-

server trajectory for bearing measurements gathering

a

https://orcid.org/0000-0001-6432-5187

b

https://orcid.org/0000-0001-9938-0370

c

https://orcid.org/0000-0002-3966-7633

to localize a single target was proposed in Tokekar

et al. (2011). The rationale of leveraging FIM lies

in that it prescribes a lower bound of target localisa-

tion error covariance of any efﬁcient estimation algo-

rithm (Taylor, 1979). Therefore, the determinant of

FIM can be utilized as a performance metric to quan-

tify the volume of the error uncertainty ellipsoid. Ex-

cept for FIM, the trace of error covariance, which di-

rectly quantiﬁes the average estimation performance,

was also utilized to ﬁnd the proper path for target

localisation using bearing-only measurements (Logo-

thetis et al., 1997; Zhou et al., 2011). Note that most

approaches utilize numerical methods to ﬁnd the op-

timal solutions and consequently require high com-

putational burden. To alleviate the computational is-

sue, an analytical solution was proposed in (He et al.,

2019) using geometric analysis for bearing-only tar-

get localisation.

Apart from bearing-only-based target localisation,

range-only-based active sensing strategy is another

emerging low-cost solution for target localisation us-

ing small-scale robots due to the proliferation of

lightweight and low-cost LIDAR/infrared range ﬁnd-

ers. Mart

´

ıNez and Bullo (2006); Bishop et al. (2010)

analysed the optimal relative target-observer geomet-

ric, that maximises system observability, for multiple

static sensors to localize a stationary target, where the

determinant of FIM was utilized as the cost function.

A range-only-based sliding mode controller was pro-

posed in Matveev et al. (2011) to drive a wheeled mo-

bile robot to a predeﬁned distance from a manoeu-

vring target and makes the robot follow the target at

He, S., Shin, H. and Tsourdos, A.

Optimal Active Target Localisation Strategy with Range-only Measurements.

DOI: 10.5220/0007928400910099

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 91-99

ISBN: 978-989-758-380-3

Copyright

c

2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

91

this distance. A similar problem was considered in

Matveev et al. (2016), but this reference additional ad-

dressed the issue of turning rate limit pertaining to the

robot, thus providing more realistic results. As an ex-

tension of Matveev et al. (2016), a three-dimensional

circumnavigating algorithm for multiple moving tar-

gets was proposed in Matveev and Semakova (2017).

In a GPS-denied environment when only range mea-

surement is available, Cao (2015) suggested a circum-

navigating algorithm to improve active perceptual re-

sults. Although these range-only-based circumnav-

igating algorithms show good performance in both

simulations and experiments, they fail to maximise or

minimise a meaningful performance measure, which

is of paramount importance in active sensing. Us-

ing simpliﬁed numerical search, Yang et al. (2014)

proposed an optimal sensor coordination strategy for

active target localisation with range-only measure-

ments. Although numerical optimisation methods

provide exact solution in active sensing, these algo-

rithms might not be suitable for low-cost robots due

to limited computational power. For this reason, it is

meaningful to derive analytically optimal solutions to

improve the estimation performance.

This paper aims to to develop an analytically opti-

mal active sensing strategy in consideration of phys-

ical constraints for non-manoeuvring target localisa-

tion with range-only measurements. This work is an

extension of our previous results (He et al., 2019)

to range-only scenarios. Similar to previous stud-

ies, the determinant of FIM is leveraged as the cost

function in the optimisation problem. As one of the

main contributions, this paper derives a closed-form

solution that maximises the cost function. Further-

more, physical constraints such as minimum turning

rate and velocity limits are also considered in deriv-

ing the optimal solution to support practical appli-

cations. The resultant analytical solution, given as

heading angle input command, is simple to be imple-

mented in practice. Theoretical observability analysis

is also performed to support the proposed localisation

algorithm.

2 PROBLEM FORMULATION

This section provides some necessary preliminaries

of vehicle kinematics model and range-only measure-

ment model to facilitate the analysis in the following

sections. The problem formulation of this paper is

also stated.

O

,max

o s

V T

,

r

t k

k

X

Y

,

r

o k

Observer

Target

r

k

max

2

Figure 1: Geometric relationship between the observer and

the target at time step k in an inertial coordinate.

2.1 Vehicle Kinematics

This work assumes that the observer is equipped with

a high-performance low-level control system that pro-

vides velocity tracking, heading and altitude hold

functions. This study aims to design guidance in-

put to this low-level controller for target localisation

and only concerns the two-dimensional (2D) motions.

The vehicle’s kinematics in a 2D environment is given

by

˙x

o

= V

o

cosγ

o

˙y

o

= V

o

sinγ

o

(1)

where (x

o

,y

o

) stands for the observer’s position in an

inertial coordinate. γ

o

is the observer’s heading an-

gle and V

o

∈(0,V

o,max

] denotes the observer’s velocity

with V

o,max

being the maximum permissible velocity.

Note that the observer’s velocity V

o

is assumed to be

larger than the target’s velocity.

In practice, the observer heading change between

two consecutive time steps is constrained due to phys-

ical turning rate limitation as

γ

o,k

−γ

o,k−1

≤ γ

max

∆

= ω

max

T

s

(2)

where γ

o,k

represents the heading angle at time step

k, ω

max

the maximum permissible turning rate of the

observer, and T

s

the sampling time.

2.2 Measurement Model

Figure 1 shows the geometric relationship between

the observer and the target at time step k, where the

observer is represented by a magenta pentagram and

the red circle denotes the target. The red vector stands

for the observer heading direction at previous time

step. X −O −Y is the inertial coordinate. The no-

tations r

o,k

= [x

o,k

,y

o,k

]

T

and r

t,k

= [x

t,k

,y

t,k

]

T

repre-

sent the position vectors of the observer and the tar-

get at time step k, respectively. r

k

= r

o,k

−r

t,k

de-

notes the relative position vector between the observer

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

92

and the target. η

k

denotes the bearing angle at cur-

rent time step, which represents the direction of the

relative position vector. With heading constraint (2),

the maximum region the observer can travel at current

time step is described by the blue sector with radius

V

o,max

T

s

and angle 2γ

max

, as shown in Fig. 1.

At time step k, the observer only has access to the

relative range provided by a range ﬁnder. Therefore,

the sensor measurement z

k

is given by

z

k

=

k

r

k

k

+ v

k

=

q

x

2

k

+ y

2

k

+ v

k

(3)

where v

k

denotes the sensor measurement noise,

which is assumed to be Gaussian white as v

k

∼

N

0,σ

2

r

with σ

r

being the standard deviation of the

measurement noise.

2.3 Problem Formulation

The objective of this paper is to analytically ﬁnd an

one-step ahead optimal sensing strategy that min-

imises non-manoeuvring target localisation errors

with range-only measurements. For this optimisation

problem, we utilise the well-known FIM to formulate

the cost function since the inverse of FIM prescribes a

lower bound of the estimation error covariance of any

efﬁcient estimator. Assume that the relative position

vector r

k

is known for ﬁnding the optimal observer

manoeuvre vector at time instant k. Then, the one-

step ahead position-related FIM for range-only local-

isation is given by

FIM =

1

σ

2

r

k+1

∑

i=k

∂

k

r

i

k

∂x

t,i

2

∂

k

r

i

k

∂x

t,i

∂

k

r

i

k

∂y

t,i

∂

k

r

i

k

∂x

t,i

∂

k

r

i

k

∂y

t,i

∂

k

r

i

k

∂y

t,i

2

=

1

σ

2

r

k+1

∑

i=k

"

cos

2

η

i

sin(2η

i

)

2

sin(2η

i

)

2

sin

2

η

i

#

(4)

For computational simplicity, we consider the deter-

minant of FIM, also known as D-optimality criterion,

as the cost function in our problem. The determinant

of FIM quantiﬁes the volume of the estimation error

uncertainty ellipsoid and can be readily obtained from

Eq. (4) as

det(FIM) =

1

σ

4

r

sin

2

(η

k+1

−η

k

) (5)

Denote v

o,k

= [V

o,k

T

s

cosγ

o,k

.,V

o,k

T

s

sinγ

o,k

]

T

as the

observer manoeuvre vector at time step k and σ =

η

k+1

−η

k

. This paper formulates the following con-

strained discrete-time optimisation problem, denoted

as CDO

1

: ﬁnd the observer manoeuvre vector at time

step k, v

o,k

, which maximises the following objective

function J

J = sin

2

σ (6)

subject to

γ

o,k

−γ

o,k−1

≤ ω

max

T

s

0 < V

o,k

≤V

o,max

(7)

The aim of this paper is to derive the closed-form

solution of CDO

1

.

3 DERIVATION OF OPTIMAL

SENSING STRATEGY FOR

ACTIVE TARGET

LOCALISATION

This section will propose an analytical optimal ma-

noeuvre that maximises the cost function J for target

localisation with range-only measurement. We ﬁrst

derive the optimal solution without heading constraint

and then extend to the case that the robot has limited

turning rate to change its heading angle.

3.1 Optimal Solution without Heading

Constraints

Excluding the heading constraint, CDO

1

reduces to

CDO

2

: ﬁnd the observer manoeuvre vector at time

step k, v

o,k

, which maximises the following objective

function J

J = sin

2

σ (8)

subject to

0 < V

o,k

≤V

o,max

(9)

Change in the vehicle’s velocity over a short interval,

like over T

s

, is usually negligible. Hence, for sim-

plicity, it is assumed that the observer moves with

constant speed and constant direction between two

consecutive time steps. Let v

t,k

=

v

tx,k

,v

ty,k

T

∆

=

[V

t,k

T

s

cosγ

t,k

,V

t,k

T

s

sinγ

t,k

]

T

be the target manoeuvre

vector at time step k. Assume that the target position

and velocity vector at current time step can be ob-

tained from Kalman ﬁlter, v

t,k

is known in trajectory

optimisation. Note that this assumption will be vali-

dated by a detailed observability analysis provided in

the next section. Figure 2 shows the geometric rela-

tionship between the observer and the target within

two consecutive time steps, where δr

k

represents the

relative manoeuvre at time step k and

¯

r

k

= r

k

−v

t,k

is an auxiliary vector utilised in the analysis. Since

v

t,k

is ﬁxed,

¯

r

k

is known in trajectory optimisation.

The analytical solution of CDO

2

can be obtained us-

ing Lemmas 1 through 2.

Optimal Active Target Localisation Strategy with Range-only Measurements

93

r

k

k

1

r

k

,

v

o k

r

k

,

v

t k

Figure 2: Geometric illustration for moving target locali-

sation without turning rate limit in the relative frame. The

blue circle determines the maximum permissible region that

the observer can travel at current time step.

Lemma 1. Given the observer velocity V

o,k

, the can-

didate optimal heading angle at time step k without

any constraint is given by

γ

∗,1

o,k

= arcsin

−

V

o,k

T

s

k

¯

r

k

k

−ϑ (10)

γ

∗,2

o,k

= π −arcsin

−

V

o,k

T

s

k

¯

r

k

k

−ϑ (11)

with sin ϑ = b/

√

a

2

+ b

2

and cos ϑ = a/

√

a

2

+ b

2

,

where a = y

k

−v

ty,k

, b = x

k

−v

tx,k

.

Proof. For moving target, the relative manoeuvre

vector at time step k can be obtained as

δr

k

= v

o,k

−v

t,k

=

V

o,k

T

s

cosγ

o,k

−v

tx,k

,V

o,k

T

s

sinγ

o,k

−v

ty,k

T

(12)

Then, the relative position vector at time step k + 1 is

given by

r

k+1

= r

k

+ δr

k

= [x

k

+V

o,k

T

s

cosγ

o,k

−v

tx,k

,y

k

+V

o,k

T

s

sinγ

o,k

−v

ty,k

]

T

(13)

From Fig. 2, the separation angle σ between two

consecutive relative position vectors can be obtained

as

σ = arccos

r

T

k

·r

k+1

k

r

k

kk

r

k+1

k

(14)

Substituting Eq. (14) into Eq. (6) gives

J = 1 −cos

2

σ =

(x

k

y

k+1

−x

k+1

y

k

)

2

x

2

k

+ y

2

k

x

2

k+1

+ y

2

k+1

(15)

Since x

k

and y

k

are assumed to be known, max-

imising J is equivalent to maximising

¯

J =

(x

k

y

k+1

−x

k+1

y

k

)

2

x

2

k+1

+ y

2

k+1

(16)

Taking the partial derivative of

¯

J with respect to γ

o,k

and substituting Eq. (13) into it yields

∂

¯

J

∂γ

o,k

=

2V

o,k

T

s

x

k

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

−y

k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

)

h

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

)

2

+

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

2

i

2

×

n

(x

k

cosγ

o,k

+ y

k

sinγ

o,k

)

h

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

)

2

+

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

2

i

+

x

k

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

−y

k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

)

×

sinγ

o,k

(x

k

−v

tx,k

) −cos γ

o,k

y

k

−v

ty,k

(17)

Solving ∂

¯

J/∂γ

o,k

= 0 gives

x

k

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

−y

k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

) = 0

(18)

(x

k

−v

tx,k

)cosγ

o,k

+

y

k

−v

ty,k

sinγ

o,k

+V

o,k

T

s

×

x

2

k

+ y

2

k

−x

k

v

tx,k

−y

k

v

ty,k

+ x

k

V

o,k

T

s

cosγ

o,k

+y

k

V

o,k

T

s

sinγ

o,k

) = 0

(19)

Note that, if condition (18) is satisﬁed, e.g., the

observer heading results in the fact that the relative

manoeuvre vector δr

k

has either the same or the oppo-

site direction with r

k

, we have r

k+1

= λr

k

, λ ∈R. This

will minimises the cost function

¯

J as

¯

J = 0. Therefore,

this should be excluded from the solution candidates.

Together with the fact that the cost function

¯

J is con-

tinuous, the candidate optimal heading solutions can

be obtained from condition (19). Further simplifying

(19) yields

r

T

k+1

·v

o,k

r

T

k

·r

k+1

= 0 (20)

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

94

r

k

F

O

1

1

r

k

,

v

t k

*,1

,

v

o k

2

2

1

r

k

*,2

,

v

o k

1

r

k

Figure 3: Geometric illustration of the candidate optimal

heading solutions. AF is parallel to r

k

.

Since the optimal manoeuvre satisﬁes the condition

r

T

k

·r

k+1

6= 0, the ﬁnal solution that locally maximises

¯

J is given by r

T

k+1

·v

o,k

= 0. This implies that the opti-

mal observer manoeuvre vector that maximises sin

2

σ

is perpendicular to next step relative position vector

r

k+1

. Using the deﬁnitions of r

k+1

and v

o,k

, we have

asinγ

o,k

+ bcosγ

o,k

= −V

o,k

T

s

(21)

Solving Eq. (21) for γ

o,k

gives Eqs. (10) and (11),

which completes the proof.

Remark 1. Given the robot velocity V

o,k

, Lemma 1

revels that there exist two locally optimal heading so-

lutions and these two solutions are symmetric with re-

spect to

¯

r

k

. The geometric illustration of the candidate

optimal solutions is presented in Fig. 3, where −v

t,k

is

assumed to be located on the right hand side of r

k

. It is

easy to verify that these two candidate heading direc-

tions provide the same level of optimality if

¯

r

k

= ρr

k

with ρ ∈ R, e.g., stationary target scenario or target

moves with either the same or the opposite direction

as r

k

. Except for these particular scenarios, there ex-

ists only one global optimal solution since σ

1

6= σ

2

.

Remark 2. Deﬁne γ

m

o,k

as the heading solution that

r

k+1

= λr

k

, λ ∈ R. Then, the proof of Lemma 1

indicates that, given the robot velocity V

o,k

, the cost

function monotonically increases when the heading

rotates from γ

m

o,k

to either γ

∗,1

o,k

or γ

∗,2

o,k

since there ex-

ists no solution of ∂

¯

J/∂γ

o,k

= 0 that locates between

γ

m

o,k

and γ

∗

o,k

. This property is useful in deriving the

optimal solution of the original optimisation problem.

The following lemma analyses the effect of ob-

server’s velocity on the cost function.

Lemma 2. If the heading angle satisﬁes condition

(10) or (11), the cost function J monotonically in-

creases with respect to the observer’s velocity V

o,k

.

Proof. Evaluate the partial derivative of

¯

J with re-

spect to V

o,k

and substitute r

T

k+1

·v

o,k

= 0 into it gives

∂

¯

J

∂V

o,k

= −2T

s

cosγ

o,k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

) + sin γ

o,k

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

)

2

+

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

4

×

x

k

v

ty,k

−y

k

v

tx,k

+ T

s

V

o,k

y

k

cosγ

o,k

−T

s

V

o,k

x

k

sinγ

o,k

2

+ 2T

s

(y

k

cosγ

o,k

−x

k

sinγ

o,k

)

x

k

v

ty,k

−y

k

v

tx,k

+ T

s

V

o,k

y

k

cosγ

o,k

−T

s

V

o,k

x

k

sinγ

o,k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

)

2

+

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

2

=2T

s

(y

k

cosγ

o,k

−x

k

sinγ

o,k

)

x

k

v

ty,k

−y

k

v

tx,k

+ T

s

V

o,k

y

k

cosγ

o,k

−T

s

V

o,k

x

k

sinγ

o,k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

)

2

+

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

2

(22)

If v

∗

o,k

is located on the right hand side of r

k

(refer

to Fig. 3), we have

y

k

cosγ

o,k

−x

k

sinγ

o,k

> 0 (23)

and r

k+1

is located on the right hand side of r

k

, e.g.,

x

k

y

k

−v

ty,k

+V

o,k

T

s

sinγ

o,k

−y

k

(x

k

−v

tx,k

+V

o,k

T

s

cosγ

o,k

) < 0

(24)

Substituting Eqs. (23) and (24) into Eq. (22) yields

∂

¯

J/∂V

o,k

> 0. Similarly, one can easily verify that

∂

¯

J/∂V

o,k

> 0 when v

∗

o,k

is located on the left hand side

of r

k

. This implies that the cost function

¯

J monoton-

ically increases with the increase of observer’s veloc-

ity V

o,k

if the heading angle satisﬁes condition (10) or

(11), which completes the proof.

By using the results of Lemmas 1 and 2, the op-

timal observer manoeuvre, without any heading con-

straints, that maximises cost function J is obtained in

Optimal Active Target Localisation Strategy with Range-only Measurements

95

Theorem 1.

Theorem 1. The optimal solution of CDO

2

is given

by v

∗

o,k

=

h

V

o,max

T

s

cosγ

∗

o,k

,V

o,max

T

s

sinγ

∗

o,k

i

T

where

γ

∗

o,k

= max

γ

o,k

∈

n

γ

∗,1

o,k

,γ

∗,2

o,k

o

J (γ

o,k

) with

γ

∗,1

o,k

= arcsin

−

V

o,max

T

s

k

¯

r

k

k

−ϑ (25)

γ

∗,2

o,k

= π −arcsin

−

V

o,max

T

s

k

¯

r

k

k

−ϑ (26)

Proof. From Lemmas 1 and 2, the proof of Theorem

1 is straightforward.

Corollary 1. Under the proposed algorithm without

heading constraint, the relative range at next time

step is given by

k

r

k+1

k

=

q

k

¯

r

k

k

2

−(V

o,max

T

s

)

2

(27)

Proof. By deﬁnition of the relative range, we have

k

r

k+1

k

2

=

x

k

+V

o,max

T

s

cosγ

∗

o,k

−v

tx,k

2

+

y

k

+V

o,max

T

s

sinγ

∗

o,k

−v

ty,k

2

= (x

k

−v

tx,k

)

2

+

y

k

−v

ty,k

2

+ (V

o,max

T

s

)

2

+2V

o,max

T

s

h

(x

k

−v

tx,k

)cosγ

∗

o,k

+

y

k

−v

ty,k

sinγ

∗

o,k

i

(28)

Substituting Eq. (21) using V

o,max

into Eq. (28) leads

to the proof of Eq. (27).

Remark 3. From Corollary 1, one can note that the

relative range at next time step is inﬂuenced by the tar-

get moving speed and its direction. However, for lo-

calizing stationary target, the relative range is mono-

tonically decreasing. This is given by the following

corollary.

Corollary 2. Under the proposed algorithm without

heading constraint, the relative range monotonically

decreases with respect to time as

d

k

r

k

k

dt

= −

V

2

o,max

T

s

2

k

r

k

k

(29)

Proof. For stationary target, it is clear that

k

r

k

k

=

k

¯

r

k

k

. Then, Eq. (27) becomes

k

r

k+1

k

=

q

k

r

k

k

2

−(V

o,max

T

s

)

2

(30)

By deﬁnition, the rate of the relative range is deter-

mined by

d

k

r

k

k

dt

=

k

r

k+1

k

−

k

r

k

k

T

s

(31)

Substituting Eq. (31) into Eq. (30) and using V

o,max

yields

d

k

r

k

k

dt

=

q

k

r

k

k

2

−

(

V

o,max

T

s

)

2

−

k

r

k

k

T

s

=

k

r

k

k

s

1−

(

V

o,max

T

s

)

2

k

r

k

k

2

−

k

r

k

k

T

s

≈

k

r

k

k

1−

1

2

(

V

o,max

T

s

)

2

k

r

k

k

2

−

k

r

k

k

T

s

= −

V

2

o,max

T

s

2

k

r

k

k

(32)

3.2 Optimal Solution with Heading

Constraints

As discussed in Remark 1, if the target is stationary

or moves with either the same or the opposite di-

rection as r

k

, the candidate heading solutions, given

by Lemma 1, provide the same level of optimality.

This might generate multiple candidate trajectories

and zigzag heading change in practice. Therefore, it

is necessary to choose only one heading solution that

satisﬁes the practical heading change constraint when

implementing Theorem 1 in real applications. For this

reason, this subsection will propose the optimal solu-

tion of the constrained problem, e.g., CDO

1

.

Let γ

∗

o,k−1

be the optimal heading angle at

the previous time step. Without loss of gen-

erality, assume that γ

∗,1

o,k

is closer to γ

∗

o,k−1

than

γ

∗,2

o,k

, e.g.,

γ

∗,1

o,k

−γ

∗

o,k−1

≤

γ

∗,2

o,k

−γ

∗

o,k−1

. Denote

Ξ as the permissible heading angle set, e.g., Ξ

∆

=

n

γ

o,k

γ

∗

o,k−1

−γ

max

≤ γ

o,k

≤ γ

∗

o,k−1

+ γ

max

o

, then, the

solution of CDO

1

is determined as follows.

Condition 1: γ

∗,1

o,k

∈Ξ and γ

∗,2

o,k

∈Ξ. Under this con-

dition, the optimal heading angle is obviously given

by γ

∗

o,k

= max

γ

o,k

∈

n

γ

∗,1

o,k

,γ

∗,2

o,k

o

J (γ

o,k

). If these two candidate

heading solutions provide the same level of optimal-

ity, the heading direction, that is closer to previous

time step heading, is determined as the current head-

ing direction to generate a consistent and unique so-

lution, thus avoiding large heading change. With this

in mind, the optimal heading angle is given by

γ

∗

o,k

=

γ

∗,1

o,k

, if J

γ

∗,1

o,k

= J

γ

∗,2

o,k

max

γ

o,k

∈

n

γ

∗,1

o,k

,γ

∗,2

o,k

o

J (γ

o,k

), otherwise

(33)

Condition 2: γ

∗,1

o,k

∈ Ξ and γ

∗,2

o,k

/∈ Ξ. Since the

cost function monotonically increases when the head-

ing rotates from γ

m

o,k

to either γ

∗,1

o,k

or γ

∗,2

o,k

, the optimal

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

96

heading solution is given by

γ

∗

o,k

= max

γ

o,k

∈

n

γ

∗,1

o,k

,

¯

γ

∗,1

o,k

,

¯

γ

∗,2

o,k

o

J (γ

o,k

) (34)

where

¯

γ

∗,1

o,k

= γ

∗

o,k−1

−γ

max

and

¯

γ

∗,2

o,k

= γ

∗

o,k−1

+ γ

max

.

Condition 3: γ

∗,1

o,k

/∈ Ξ and γ

∗,2

o,k

/∈ Ξ. Under this

condition, the optimal heading solution is obviously

located at the boundary of Ξ as

γ

∗

o,k

= max

γ

o,k

∈

n

¯

γ

∗,1

o,k

,

¯

γ

∗,2

o,k

o

J (γ

o,k

) (35)

Remark 4. Note that when γ

∗,2

o,k

is closer to γ

∗

o,k−1

than

γ

∗,1

o,k

, i.e.,

γ

∗,1

o,k

−γ

∗

o,k−1

>

γ

∗,2

o,k

−γ

∗

o,k−1

, similar re-

sults can also be obtained.

Remark 5. It is worthy pointing out that the ob-

server, under the proposed algorithm, will never col-

lide with the target since the minimum achievable rel-

ative range is limited by the turning rate constraint.

By summarizing the aforementioned three condi-

tions, the proposed active sensing strategy for target

localisation using range-only measurements is sum-

marized in Algorithm 1.

Algorithm 1 : Optimal Active Target localisation Strategy

with Range-Only Measurement.

Input: Estimated target position vector r

t,k

, estimated

target manoeuvre vector v

t,k

, previous observer ma-

noeuvre vector v

o,k−1

, maximum allowable heading

angle change γ

max

Output: Optimal observer heading angle γ

∗

o,k

1: Calculate the candidate optimal heading angles

γ

∗,1

o,k

and γ

∗,2

o,k

using Eqs. (25) and (26)

2: if

γ

∗,1

o,k

−γ

∗

o,k−1

≤

γ

∗,2

o,k

−γ

∗

o,k−1

then

3:

¯

γ

∗

o,k

= γ

∗,1

o,k

4: else

5:

¯

γ

∗

o,k

= γ

∗,2

o,k

6: end if

7: if

γ

∗,1

o,k

−γ

∗

o,k−1

≤ γ

max

and

γ

∗,2

o,k

−γ

∗

o,k−1

≤ γ

max

then

8: γ

∗

o,k

is given by Eq. (33)

9: else if

¯

γ

∗

o,k

−γ

∗

o,k−1

≤ γ

max

then

10: γ

∗

o,k

is given by Eq. (34)

11: else

12: γ

∗

o,k

is given by Eq. (35)

13: end if

4 OBSERVABILITY ANALYSIS

The baseline assumption that we utilised in deriving

the optimal heading solution is that current target po-

sition and velocity vector are available to the observer.

This information, however, is extracted from an esti-

mator in practical applications. For this reason, it is

necessary to analyse system observability since r

k

is

estimable only when target is observable. The results

are presented in the following proposition.

Proposition 1. Under the proposed algorithm, the

target is always observable.

Proof. The necessary and sufﬁcient condition to

guarantee system observability for localising non-

manoeuvring target with range-only measurement is

given by (Song, 1999)

r(t) 6=

a

11

+ a

12

∆t

a

21

+ a

22

∆t

(36)

for some t ∈ (t

0

,t

f

] with t

0

being the initial time,

where ∆t = t −t

0

. The coefﬁcients a

i j

in Eq. (36)

are arbitrary constants but not all zero.

If the target is unobservable, if follows from con-

dition (36) that there exists a line such that the relative

position always locates on the line with equal length

between two consecutive time steps, e.g., δr

i

= δr

j

,

∀i 6= j. This means that observer manoeuvre that

changes either the magnitude of the relative velocity

or the direction of the relative velocity is required for

range-only target localisation scenario. From previ-

ous analysis, we know that the proposed algorithm

leverages the maximum observer velocity and gen-

erates a unique solution that forces the relative po-

sition vector rotate either clockwise or anti-clockwise

around the target. Therefore, the direction of the rela-

tive velocity changes at every time instant, which im-

plies that there exists some t ∈ (t

k

,t

f

] such that con-

dition (36) is satisﬁed ∀t

k

. This means that target is

always observable under the proposed approach.

5 SIMULATION STUDIES

In this section, estimator-in-the-loop simulations are

performed to validate the proposed optimal sens-

ing strategy. The initial position of the robot is

(20m,50m) with initial heading 0

◦

. The maximum

permissible velocity of the robot is set as V

o,max

=

20m/s. The turning rate of the robot is constrained

by ω

max

= 1rad/s and the sampling time is set as

T

s

= 0.1s. The target initially locates at (0m, 0m) and

moves in a straight line with constant velocity V

t

=

5.83m/s as well as constant heading γ

t

= 0.54rad.

Optimal Active Target Localisation Strategy with Range-only Measurements

97

-50 0 50 100 150 200 250 300

X (m)

-50

0

50

100

150

200

Y (m)

Target

Numerical Optimization

Proposed Approach

Figure 4: Moving trajectories.

0 5 10 15 20 25 30 35 40 45 50

Time (s)

0

2000

4000

6000

8000

10000

12000

14000

Determinant of FIM

Numerical Optimization

Proposed Approach

Figure 5: Determinant of FIM.

To implement the proposed algorithm, the required

information on target position and velocity are ex-

tracted from the well-known extended Kalman ﬁlter

in conjunction with a constant velocity model. For

validation, the numerical optimal solution of CDO

1

obtained from Particle Swarm optimisation algorithm

is also presented.

Figure 4 presents the moving trajectories of the

target and the observer. From this ﬁgure, one can

note that the proposed algorithm forces the relative

position vector rotate clockwise in the considered sce-

nario for target localisation. The determinant of FIM

is shown in Fig. 5, demonstrating that det(FIM) in-

creases monotonically with respect to time. Since

det(FIM) quantiﬁes the volume of the estimation er-

ror uncertainty ellipsoid, target localisation perfor-

mance can be improved using the proposed active

sensing algorithm. Figure 6 provides the history of

observer’s heading rate, which reveals that the phys-

ical turning rate constraint is satisﬁed. Furthermore,

from the obtained results, we can clearly observe that

the proposed analytical solution coincides with the

numerical optimal solutions, demonstrating the effec-

tiveness of the proposed algorithm. Future work in-

cludes extending the proposed approach to heteroge-

neous sensors.

0 5 10 15 20 25 30 35 40 45 50

Time (s)

-1.5

-1

-0.5

0

0.5

1

1.5

Heading Rate (rad/s)

Turning Rate Limit

Numerical Optimization

Proposed Approach

Figure 6: Turning rate.

6 CONCLUSIONS

The problem of active target localisation using range-

only measurements is studied in this paper. By lever-

aging the determinant of one-step ahead FIM as the

cost function and heading angle command as the con-

trol input, the discrete-time optimal heading is de-

rived analytically to minimise target localisation error.

Both velocity and turning rate limits are considered in

the proposed optimisation approach. Numerical sim-

ulations are performed to validate the analytical ﬁnds.

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