Member IEEE Claude Jauffret

Annie-Claude Perez

Denis Pillon

Annie-Claude Pérez

Observability

Keywords:

come



Abstract-Range-only target motion analysis (ROTMA) consists of estimating the trajectory of a target using a single platform collecting range-only measurements. Observability analysis is carried out when the target is in constant-velocity motion and the observer maneuvers gently (a constant turn motion or a constant acceleration motion). We compare observability in bearings-only target motion analysis (BOTMA) and ROTMA throughout the paper, together with the rank of the Fisher information matrix. In each case, we establish necessary and sufficient observability conditions and we identify the virtual (or ghost) targets giving the same measurements when the system is not observable. Index Terms-Target motion analysis, tracking, range-only, bearings-only, observability, Fisher information matrix, constant turn motion, constant acceleration motion.

I. INTRODUCTION

HIS PAPER presents the second part of the observability analysis in range-only target motion analysis (ROTMA), started in a previous paper [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF]. The target (or source) was assumed to be in constant velocity (CV) motion. The previous paper [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF] provided the following results: The Fisher information matrices (FIMs) in BOTMA and in ROTMA have the same rank when the source is in CV motion, whatever the trajectory of the observer. Secondly, when the observer is zigzagging, with constant speed on each leg, the trajectory of the source is observable in ROTMA if and only if it is unobservable in BOTMA. We proved this happens when and only when the bearings are constant. In this case, the FIM is singular, whereas in ROTMA, the target's trajectory is observable. Conversely, when the trajectory of the target is observable in BOTMA, it is unobservable in ROTMA. At this point, a prior question must be asked: are these antinomic observability conditions in BOTMA and in ROTMA maintained for other observers' maneuvers? Some answer elements are given in the present paper for two types of smooth maneuvers of the observer: a constant turn (CT) C. Jauffret and A.C. Pérez are with Université de Toulon, CNRS, IM2NP (UMR 7734), CS 60584, 83041 Toulon Cedex 9, France (email: annie-claude.perez, jauffret@univ-tln.fr) D. Pillon, is retired and works as counselor with IM2NP (email: pillon_denis@orange.fr) motion or a constant acceleration (CA) motion and a combination, CA-CV (for BOTMA only) and CT-CV. Note that generalizing this comparison is difficult because, unlike in BOTMA, general necessary and sufficient observability conditions do not exist in ROTMA: indeed, when the observer maneuvers, this problem cannot be expressed by a linear system, whereas in BOTMA, a linear system can be exhibited [START_REF] Dogançay | 3D Pseudolinear Target Motion Analysis from Angle Measurements[END_REF][START_REF] Lindgren | Position and Velocity Estimation via Bearing Observations[END_REF][START_REF] Whitcombe | Pseudo-state Measurements Applied to Recursive Nonlinear Filtering Proceedings of the 3rd Symposium on Estimation Theory and Its Application[END_REF]. In ROTMA, for each observer's kinematics, a specific study of observability must be conducted. Our strategy of analysis is the same as in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF]: we start with BOTMA, we deduce the rank of the FIM, and finally, we study the observability of the target in ROTMA (under the same conditions). We will prove that when the observer is in CT motion, the target is observable in BOTMA and in ROTMA as well. In this case, the FIMs are of full rank (no antinomy). When the observer is in CA motion, several situations can occur: the target can be observable in ROTMA and the FIM can be of deficient rank, or the target can be unobservable in ROTMA and the FIM can be of full-rank or deficient rank. When the trajectory of the target of interest is not observable, a set of ghost-targets exists, which misleads the operator when he or she tries to estimate the state vector. This problem also occurs when associating multi-tracks between several arrays [START_REF] Guelle | Inter-array Multitracks Association[END_REF]. Therefore, in BOTMA, the observer must maneuver in order to render the trajectory of the target observable. Unfortunately, some maneuvers are ineffective [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF] [START_REF] Nardone | Observability Criteria for Bearings-Only Target Motion Analysis[END_REF]. Here, we give some examples of ineffective maneuvers in ROTMA.

The paper is organized as follows: In Section II, the problem and the ad-hoc notations are given. The notion of observability is recalled and the models of the two types of trajectories of the observer are given, as well. We introduce the notion of a rendezvous route (and the associated criteria), which will play a crucial role in the analysis of observability when the observer is in CA motion. In Section III, observability in BOTMA is analyzed when the observer is in CT motion, in CA motion, and in CA motion followed by CV motion or the converse. For each kinematic, we give necessary and sufficient observability conditions. The set of ghost-targets is characterized in the scenarios where the target is unobservable. Section IV is devoted to ROTMA: we start by revisiting the situations in which the observer is in CT motion and then in CA motion. For each of them, we give necessary and sufficient observability conditions. In unobservable cases, we identify the set of ghost-targets. Illustrative examples are given. The conclusion follows.

II. GENERAL NOTION AND NOTATIONS A. Definitions and notations

A target (T) and an observer (O) move in the same plane, given a Cartesian system. The target has a CV all along the scenario, while the observer maneuvers (here, the term "maneuver" is employed when the observer is not in CV motion). The scenario starts at time 0  t and finishes at time f T t  . For the observer, the position and velocity at time t are respectively

  T )] ( ) ( [ t y t x t P O O O  and     T )] ( ) ( [ t y t x dt t dP t V O O O O    
. Both are concatenated into the vector

        T t y t x t y t x t X O O O O O   ) ( ) (  .
For the target, the notations are similar:

  T )] ( ) ( [ t y t x t P T T T  ,   T ] [ T T T T y x dt t dP V    
, and

    T T T T T T y x t y t x t X   ) ( ) (  .
In the sequel,   0 T X will be simply denoted as T X , which entirely defines the target's trajectory. Obviously,

    T T T V t P t P   0 . We will assume that       f O T T t t P t P , 0 ,    .
The motion of the target relative to the observer is given by      

            T t y t x t y t x t X t X t X OT OT OT OT O T OT   ) ( ) (    .
All the angles are clockwise-positive. Subsequently, we will use the symbol  to designate angles: for any pair of vectors U and

W ,   W U, 
is the angle defined by the couple   W U, referenced to U. When U is collinear to the northward direction, we will use W  only (for the bearing or heading).

The range and the bearing at time t are given respectively by    

t P t r OT  and     t P t OT    . So,                t t t r t P OT   cos sin ; that is,                        t y t t r t y t x t t r t x O T O T   cos sin .
Subsequently, we will use the following simplified notations:

  0 0    ,   0 0 r r  ,   0 OT r V v 
, and

  0 OT r V h  
for the initial bearing, range, relative speed, and heading. For convenience, ) 0 ( OT X will be denoted simply OT X .

Figure 1 displays a typical scenario.

B. Observability notion

We extend the previous notation to emphasize this dependence:  

      OT OT f X X X t X t T t      , , , , 0   . Otherwise,
the trajectory is said to be unobservable: at least, one vector

  T OG OG OG OG OG y x y x X   ) 0 ( ) 0 (  (defining a CV motion) different from OT X exists such that       f OT OG T t X t X t , 0 , , ,      .
Similarly, in ROTMA, the observability definition is

      OT OT f X X X t r X t r T t      , , , , 0
. Otherwise, again, at least one vector

  T OG OG OG OG OG y x y x X   ) 0 ( ) 0 (  different from OT X exists such that     OT OG X t r X t r , ,  . The vector O OG G X X X  
defines the "virtual" trajectory of a "ghost-target", denoted G. We define similarly

        T 0 0 0 G G G y x P  , T ] [ G G G y x V    ,     T )] ( ) ( [ 0 t y t x V t P t P G G G G G   
, and

    T G G G G G y x t y t x t X   ) ( ) (  , with the convention   0 G G X X  .
Note that, depending on various circumstances, the set of "ghosts" is finite or can be a family (an uncountable set).

Observability analysis has two aims: a) Give a necessary and sufficient condition to have unicity of the state vector OT X . b) When this condition is not satisfied, characterize the set of OG X . This will conduct our paper.

C. Observer kinematic models

As announced in the introduction, two models of smooth observer motion are considered in this paper: in the first, the observer travels in an arc of a circle at constant speed, and in the second, it has a constant acceleration vector.

1) CT motion

The observer turns around a fixed point

       C C C y x P at range 0  
, with a constant turn rate 0   (positive if the motion of the observer is clockwise) and an "initial angle"  relative to north, at the beginning of its motion. Its speed is constant. As a consequence, at time t (recall that the initial time is equal to 0), the location of the observer is given by

                     t t P t P C O cos sin
(see Fig. 2). In order to simplify the coming calculation, we will assume that 2) CA motion The position of the observer at any time t is

       0 0 C P .
          2 0 0 2 t V t P t P O O O
, where  

0 O V is the initial velocity and   T y x    
is the (non-zero) acceleration vector. The relative position of the target with respect to the observer is

          2 0 0 2 t V t P t P OT OT OT (1)
Without loss of generality, we will assume that 0  x  and 0  y  . Indeed, a suitable rotation of the entire scenario allows us to be in this case. 1 This assumption will make easier the following observability analysis. [START_REF] Bucy | Digital Synthesis of Nonlinear Filters[END_REF] The matrix of this rotation is

                 sin cos cos sin .

D. Rendezvous routes in CA motion

The observability criteria when the observer is in CA motion will be shown to be linked to the rendezvous (or collision) route in Sections III. C. (BOTMA) and IV. B. 2. (ROTMA).

Definition: the rendezvous route

The target and the observer are said to be on a rendezvous route (RDVR), when they are in the same place at a time c t .

Actually, this rendezvous instant is purely virtual: before 0  t and after f T , O and T were and will be free to choose their own trajectories. Note that this motion model is not the pursuit curve motion, which has not been studied in the TMA observability problem, from our knowledge. ). This equality is equivalent to 0 2

1) The two types of RDVR

                    ) 3 ( 0 0 0 ) 2 ( 0 2 1 0 0 2 OT c OT x c OT c OT y t y t x t x    If   0 0  OT y , then (
1 2          x c c t t    or 0 2 2 2      c c t t
              0 0 0 0 0 0 2 2 OT OT OT OT OT OT x y x y x y y       .
Proof:

Equation (3) implies that     0 0 OT OT c y y t   
. Substituting into (2), we get

            0 0 0 2 1 0 0 0 0 2 2    x OT OT OT OT OT OT y y x y y x     . We end up with               0 0 0 0 0 0 2 2 OT OT OT OT OT OT x y x y x y y       QED.
Note that, an unique RDV instant exists for an RDVD-II. In the two following propositions, we give a criterion based on the measurements (range or bearing), which allows us to know which type of RDVR we are on.

2) Criterion of RDVR-I

From the remark following Proposition 2, when the target and the observer are on an RDVR-I, the bearings are constant. In this case, the trajectory of the target is not observable in BOTMA. In ROTMA, we will see that this type of route must be considered in the observability analysis (see Proposition 15). The following proposition gives us a criterion of RDVR-I based only on range. In practice, when the measurements are corrupted by additive noises, it can help to construct a statistical test to decide if the two mobiles are on an RDVR-I.

Proposition 4: Criterion on range of RDVR-I

The target and the observer are on RDVR-I if and only if a scalar r exists such that, for

  f T t , 0  , either r t r t t r x     0 2 2 1 ) (  , or r t r t t r x     0 2 2 1 ) (  and x r r  0 2 2    .
Proof: First, assume that the target and the observer are on an RDVR-I.

From

(1), we have . Note that they can be the same.

      x x OT OT OT t t t x t x t x                 2 2 2 1 2 0 0  ,
If     2 1 , , 0 t t T f  , then     t x t r OT   . If 1 t T f  , or 0 2  t then     t x t r OT 
. So, we get the result.

Conversely, suppose now that either

r t r t t r x     0 2 2 1 ) (  , or r t r t t r x     0 2 2 1 ) (  for   f T t , 0  and x r r  0 2 2   
. Let us prove that the observer and the target are on an RDVR. To avoid repetition, we will only consider the case

r t r t t r x     0 2 2 1 ) (  and x r r  0 2 2    .
We have to solve the following equation:

    2 2 0 2 2 2 1 2 1 0 0             x OT OT t r t r t V t P   .
After developing the left and right expressions and equating the coefficients of each power of t, we end up with a system of four equations:

  2 0 2 0 r P OT  (4)     r r V P OT T OT  0 0 0  (5)     x T OT OT r r P V  0 2 2 0 0      (6)   x T OT r V     0 (7)
In terms of components, they are equivalent to

    2 0 2 2 0 0 r y x OT OT   (8)         r r y y x x OT OT OT OT    0 0 0 0 0   (9)       x x OT OT OT r r x y x   0 2 2 2 0 0 0        (10)   x x OT r x      0 (11)
From [START_REF] Lindgren | Position and Velocity Estimation via Bearing Observations[END_REF], we get

  r x OT    0
. Substituting into ( 9) and ( 10), we obtain

        r r x y y OT OT OT   0 0 0 0    (12)       x OT OT r x y  0 2 0 0    (13)
Now, from (8), we have

          0 0 2 0 0 0 r x r x y OT OT OT     (14)
Taking the square of each side of ( 12), after replacing

  0 2 OT y  and   0 2 OT y
with their respective expressions given by ( 13) and ( 14), we get

            2 2 0 0 2 0 0 0 0 r r x r x r x OT x OT OT       
. [START_REF] Sathyan | Multiple Hypothesis Tracking with Multiframe Assignment Using Range and Range-rate Measurements[END_REF].

Let us examine two cases:

Case 1 :

  0 0 r x OT  Equation (15) is equivalent to              x OT r r x  2 0 0  . Substituting into (14), we finally get   x x OT r r r y   2 2 0 2 2 0              . Consequently, 0 2 2 0   x r r   ; that is, 0 2 2 0   r r x   .
But, by assumption, we have 0 2

2 0   r r x   . Hence 0 2 2 0   r r x   . (16)
It follows that

  0 0  OT y ; hence   2 0 2 0 r x OT  . We conclude that   0 0 r x OT  
. Reporting this value into [START_REF] Nardone | Observability Criteria for Bearings-Only Target Motion Analysis[END_REF], we get 0  r  . From [START_REF] Whitcombe | Pseudo-state Measurements Applied to Recursive Nonlinear Filtering Proceedings of the 3rd Symposium on Estimation Theory and Its Application[END_REF], we have 0 0  r , which is incompatible with the assumption

  0  t P OT for   f T t , 0 
. This must be discarded.

Case 2 :

  0 0 r x OT  From ( 13) and ( 14 QED.

3) Criterion of RDVR-II

Similarly to ROTMA, BOTMA observability conditions (see Proposition 9) need to have a criterion of RDVR-II.

Proposition 5: Criterion on bearing of RDVR-II

O and T are on RDVR-II if and only if .

  t t 1 0 tan      , with

Proof:

We have to prove the equivalence

                0 0 0 0 0 0 2 tan 2 1 0 OT OT OT OT OT OT x y x y x y y t t             We have           0 0 2 1 0 0 tan 2 OT OT x OT OT y t y t x t x t         . If   t t 1 0 tan      then             t y t y t x t x OT OT x OT OT 1 0 2 0 0 2 1 0 0           .
As a consequence,

    0 0 OT OT c y y t    is a root of     x OT OT t x t x  2 2 1 0 0    . We get the equality               0 0 0 0 0 0 2 2 OT OT OT OT OT OT x y x y x y y       . Conversely, if               0 0 0 0 0 0 2 2 OT OT OT OT OT OT x y x y x y y       , then we readily verify that     0 0 OT OT c y y t    is a root of     x OT OT t x t x  2 2 1 0 0   
. Hence, there are two real numbers 0 

and 1  such that            t y t y t x t x OT OT x OT OT 1 0 2 0 0 2 1 0 0           .
QED.

To resume, the two mobiles are on an RDVR-II   0 and , tan

1 1 0         t t , the two mobiles are on an RDVR-I   0 tan     t .
It is well known that when the observer is itself in CV motion and the bearings are constant, the observer will meet eventually the target (the collision instant can equal to infinity if the two routes are parallel. To avoid this collision hazard well known by sailors, the observer must maneuver. However, if it accelerates and if the bearing tangent is linear, it may still remain on a rendezvous route.

III. OBSERVABILITY CRITERIA IN BOTMA

Let us recall that in BOTMA, the target trajectory is observable if and only if the FIM is nonsingular. We give hereafter two general results, necessary to prove observability when the trajectory of the target is a combination of CA and CV motions (see proposition 11).

A. Two general results

Proposition 6: Observability equivalence between two observers in BOTMA

Let there be two observers measuring the same bearings. If the target is observable from one, it will be observable from the other (or equivalently, if it is unobservable from one, it will be unobservable from the other).

Proof:

We recall that BOTMA has a linear version, whatever the trajectory of the ownship. Indeed, the noise-free measurement

equation                       t y y t y t x x t x t O T T O T T   0 0 tan 1  can be transformed into the linear equation                     N O O T t
             or, in short,   Z X T   A , where the k-th line of    A is           k k k k k k t t t t t t     sin cos sin cos  
, and the k-th [START_REF] Dogançay | 3D Pseudolinear Target Motion Analysis from Angle Measurements[END_REF][START_REF] Lindgren | Position and Velocity Estimation via Bearing Observations[END_REF][START_REF] Whitcombe | Pseudo-state Measurements Applied to Recursive Nonlinear Filtering Proceedings of the 3rd Symposium on Estimation Theory and Its Application[END_REF].

element of Z is         k k O k k O t t y t t x   sin cos  ; see
The observability is hence brought by the set

      N t t   , , 1  , which means that if two observers collect the same set       N t t   , , 1 
, if the target is observable from one, it will be observable from the second one. In this case, the state vector is computed by

      Z X T 1     A A T .
QED.

Proposition 7: Observability equivalence for time-reversed bearings

Let there be two observers #1 and #2. Observer #2 measures the same bearings as observer #1, but in the inverse temporal

order, that is         k N k t t t   1 2   , where     k i t

 is the bearing measured at time k t by observer #i. If the target detected by

observer #1 is observable, then the target detected by observer #2 will be, and the converse.

Proof: Let us define the matrix

    i  A whose k-th line is                   k i k k i k k i k i t t t t t t     sin cos sin cos   . Let us prove that             2 1 Rank Rank   A A  . We note that the k-th line of     2  A is                                     k N k k N k k N k N k k k k k k t t t t t t t t t t t t t t t t          1 1 1 1 2 2 2 2 sin cos sin cos sin cos sin cos        
We construct a third matrix denoted A ~ by permutation of the lines of

    2  A , that is, the first line of A ~ is the N-th line of     2  A , the second line of     2  A is the (N -1)-th of     2  A ,
and so on. In other words, we flip the matrix in the up/down direction. Obviously,

        2 Rank Rank  A A 
. We note that the first two columns of A ~ are the first two columns of

    1  A .
The third (resp. fourth) column of

A ~ is N t multiplied by the first (resp., second) column of N t , minus the third (resp., fourth) column of     1  A . Hence,         1 Rank Rank  A A  . Consequently,             1 2 Rank Rank   A A  . We readily deduce that                     2 2 1 1 Rank Rank     A A A A T T  .
QED.

Note that these properties cannot be extended for any measurements such as frequency measurements because the Doppler effect is not time-reversible.

B. Observer in CT motion

The CT motion was defined in Section II.C.1. We propose the following result when the noise-free bearings are continuously available during   f T , 0 .

Proposition 8: Observability in BOTMA for CT motion If the observer is traveling along an arc of a circle, then any target moving with a constant velocity is observable in BOTMA.

Proof: Suppose that a ghost (G) moving in CV motion is detected in the same bearings as the target for any

  f T t , 0  . The equality       t P t P t OG OT      is equivalent to       t P t k t P OT OG  for certain   0  t k               f O T O G T t t P t P t k t P t P , 0 ,                   t P t k t P t k t P O T G 1                                        t t t k V t P t k V t P T T G G cos sin 1 0 0   1   t k .
In other words, no such ghost exists: the trajectory of the target is hence observable.

QED.

Note that if the observer's trajectory contains at least one arc of a circle, then the trajectory of any target having a constant velocity is observable in BOTMA.

C. Observer in CA motion

  t t 1 0 tan      .
If

0 1  
, the target and the observer are on a RDVR-II and the ghosts are on a RDVR-II with the observer. Their trajectories are defined by

        0 0 T G X X
, where  is a scalar and  is a vector in the null space of the matrix

                              1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 OT x OT OT OT OT OT y y x y x y   



. The values of 0  and 1  are given in Proposition 5.

If

0 1  
, the trajectories of the ghosts are defined by

    Ξ   0 0 T G X X , with   T 0 0 2 1       Ξ , for any 1   and 2   such that 0 2 1     .
Proof of the two previous propositions:

We have to solve the equation 

   OT X t X t , ,    . Since the implication         OT OT X t X t X t X t , tan , tan , ,        holds, we concentrate our effort on the equation     OT X t X t , tan , tan    . We define the components of X by   T y x y x   .
Two cases must be studied:

Case (1):

  0 0  OT y or   0 0  OT y  .       OT X t X t , tan , tan           0 0 2 0 0 2 2 2 OT OT x OT OT x y t y t x t x y t y t x t x              , for any t in   f T , 0 .             y t y t x t x y t y t x t x x OT OT OT OT x                             2 0 0 0 0 2 2 2 ,   f T t , 0   .
After reordering the terms of this equation, we get

                                   0 0 2 0
           
We end up with the system

) (  B X M with                                1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 OT x OT OT OT OT OT y y x y x y     M and                              0 0 2 0 0 0 0 0 OT OT x OT OT OT OT y y y x y x B      We note that OT X is a solution of ) ( .
The vector OT X is the unique solution of ) ( if and only if

  0 det  M
, in other words, the trajectory of the target is observable. Conversely, the trajectory of the target is not observable if and only if

  0 det  M . Consequently, the discussion is about   M det . We readily get               x OT OT OT OT OT OT y y x y y x  0 2 1 0 0 0 0 0 det 2 2       M . If   0 det  M
, we have to discriminate the subcase where

  0 0  OT y from the subcase   0 0  OT y . If   0 0  OT y , then   0 0  OT x . The case   0 0  OT x means that
the target and the observer are co-localized at the initial time. This does not satisfy the assumptions given in II A. This case is hence discarded.

If   0 0  OT y , then               0 0 0 0 0 0 2 2 OT OT OT OT OT OT x y x y x y y       ;
that is, the observer and the target are on an RDVR-II (cf. Proposition 3). Since the acceleration x  is not equal to zero,   0 OT y  cannot be equal to zero. As the consequence, the first two columns of M are not collinear. We remark also that the fourth column of the matrix M cannot be simultaneously collinear with another one, so the rank of M is equal to 3. The set of solutions of ) ( is the set of the vectors defined by

    OT OG X X
, where  is a scalar and

  T 0 3 2 1     
is a nonzero vector in the null space of M . We verify that

  t  has a special form:     t t 1 0 1 tan       , with     0 0 0 OT OT y x   and           0 0 0 0 0 2 1 OT OT OT OT OT y x y x y      . Note that         t y t y t x t x OG OG x OG OG 1 0 2 0 0 2 0 0           is equivalent to             0 0 2 0 0 1 0 2 OG OG x OG OG y t y t t x t x           .
Hence, at time

        0 0 0 0 OT OG OG OG c y y y y t       (
  0 0  OT y and   0 0  OT y  ; that is,     0 , 0 OT OT V P and
 are collinear. Note that O and T are not necessarily on an RDVR-I.

Then     2 , ,       OT X t X t 0    y t y  ; that is, 0   y y  .
The bearing rate is zero and the set of solutions is the line of sight of the target: any ghost traveling in this line (the X-axis) is detected in the same (constant) bearing

       2
 as the target of interest. The target's trajectory is not observable.

QED.

D. Observer in CV and then CA motion and the converse

Observability must be studied only when the target is not observable during the CA motion of the observer, that is, when O and T are on an RDVR of type II or the bearings are constant. To do this, we need the notion of an "angleequivalent non-maneuvering observer" [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF] and [START_REF] Nardone | Observability Criteria for Bearings-Only Target Motion Analysis[END_REF] when O is in CA motion; the existence of a virtual observer (E) in CV motion continuously collecting the same bearings as the observer will be proved in the sequel.

In the rest of the paragraph, we will assume that

  t t 1 0 tan     
(during the CA motion only).

              T T 0 0 0 0 0 O O E E E y x y x P   and   T E E E y x V    , respectively.
If the bearings are constant, then     0

0 0   O O y y  . The virtual observer (in CV motion) such that     0 0 O E P P  and   0 O E V V  collects the same bearings as O.
If O and T are on a rendezvous route of type II, suppose that a virtual observer E in CV motion exists. The equality

                  ET OT ET OT y t y x t x t   0 0 tan 1  implies that     t y t y x t x ET OT ET OT 1 0 0 0         . Consequently 0  ET y 
, and

  1 0   OT ET y x  . Since           0 0 0 0 0 2 1 OT OT OT OT OT y x y x y      (see Proposition 10), we get         0 0 0 0 OT OT OT OT ET y x y x x     
. We end up with

                      T OT OT OT O E y y x y x V    0 0 0 0
. Note that this virtual observer is unique.

QED.

Hereafter, we propose an example of a scenario where the observer has a higher order dynamic than the target and the target's trajectory still remains unobservable. We have chosen a scenario with an RDVR-II (hence satisfying Proposition 3):

      ) m/s ( ] 7 6 [ and , ) m ( ] 4000 3000 [ 0 , ) m/s ( ] 2 10 [ 0 , ] 0 0 [ 0 T T T T       T T O O V P V P with 0416 . 0   x  m/s 2 .
The duration is 6 minutes. Figure 6 depicts the maneuvering observer together with the target (thick lines) and four ghosts (thin lines). Moreover, the trajectory of the virtual observer E is plotted. Four lines of sight are given.

    1 0 T P P O E  and   0 O E V V 
(see lemma 1). We deduce that the bearings during   1 , 0 T are also constant, that is, the respective trajectories of the virtual observer and of the target are on the same line. The target is non observable.

If O and T are on an RDVR-II, following Lemma 1, the observer collects the same bearings as the non-maneuvering observer E. Hence, the observer collects the bearings acquired by an observer whose trajectory is composed of two legs: the first one is defined by

  0 O V
, and the second one by E V . Note that E V cannot be equal to

  0 O V , otherwise   0 O T y y    and consequently 0  x 
(see Proposition 3). In short, the observer acquires the same bearings as the ones collected by a leg-by-leg maneuvering observer. Following [START_REF] Le Cadre | Discrete-Time Observability and Estimability Analysis for Bearings-Only Target Motion Analysis[END_REF], if the bearings are not constant, the target's trajectory is observable from the equivalent observer E. Proposition 6 completes the proof.

Proposition 7 allows us to determine when the observer is first in CA motion and then in CV motion.

QED.

IV. OBSERVABILITY CRITERIA IN ROTMA We proved in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF] that the target's trajectory is not observable in ROTMA when the observer is in CV motion. Actually, in this case, the ROTMA problem has a linear version, similarly to BOTMA. Indeed, even if the range does not depend linearly on the vector X, the square of the range can be expressed in linear form, relative to another vector denoted Z:

          t V t P V t P t r tV P t r OT OT T OT OT OT OT        2 2 2 2 0 2 0 0 We define                             3 2 1 2 2 0 0 Z Z Z V P V P Z OT OT T OT OT . So,      , 3 , 2 , 1 , 1 2 2   k Z t t t r k k k
. This is another proof of the non-observability of the target's trajectory in ROTMA, since this trajectory depends (mathematically speaking) on a three-dimensional vector, whereas the trajectory is defined by X. The vector Z is nevertheless observable: it defined the set of solutions given in Section IV-B of [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF], which is an uncountable set (due to this linearity).

In the two next subsections, we will consider two types of kinematics of the observer: CT motion and CA motion.

A. Observer in CT motion

Proposition 12: Observability in ROTMA for CT motion

If the observer is traveling along an arc of a circle, then any target moving with a constant velocity is observable by range measurements only.

Proof:

Again, the proof is made in continuous time as in BOTMA. Suppose now that another target G moving with a constant velocity, say G V , is at the same range as the target of interest T. The square of the range at any time

t is                           2 2 2 2 2 t y t y t x t x t y t y t x t x t r O G O G O T O T                                  . cos 0 sin 0 cos 0 sin 0 2 2 2 2 t t y t y t x t x t y t y t x t x G G G G T T T T                                 or, equivalently,                                                         . , cos sin 2 cos 0 sin 0 2 0 0 2 0 0 cos sin 2 cos 0 sin 0 2 0 0 2 0 0 2 2 2 2
G G G G G G G G G G G G T T T T T T T T T T T T                                                                
This implies the following five equalities: 17) and ( 18) are equivalent to

                                                                             ) 18 ( . cos sin cos sin ) 17 ( cos 0 sin 0 cos 0 sin 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2                 t y t x t y t x t y t x t y t x y y x x y y x x y x y x y x y x t T T G G T T G G T T T T G G G G T T G G T T G G             Equations (
          t P P t t T G            0 0 0 cos sin T     and       t V V t t T G            0 cos sin T     . Since                 t t cos sin spans the whole two-dimensional space,     0 0 T G P P  and T G V V  . QED.
Obviously, if the observer's trajectory contains at least one arc of a circle, then the trajectory of any target having a constant velocity is observable in ROTMA.

B. Observer in CA motion

We proved in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF] that, when the observer has a leg-by-leg motion, the trajectories of the ghosts are obtained by orthogonal transformations of the trajectory of the target of interest. The matrices of these transformations are shown to be constant (they are independent of time). They are denoted H .

Here, the matrices of these transformations can depend on time. This makes the analysis much more complex. In the following subsection, two examples are given: one with a constant H and another with a non-constant matrix denoted   t H .

1) Examples of constant and non-constant orthogonal transformation matrices

First of all, we consider the case where   t P OT is not collinear with  . Let us prove that a ghost-target exists. Indeed, consider the time-dependent vector defined by

                2
    0 0 OT OG P P H  ,     0 0 OT OG V V H 
, and    H . This last equality implies that H is the orthogonal matrix of the symmetry around the line spanned by the vector  . This constant orthogonal matrix will be denoted S in the sequel.

         0 0 2 u P OT (for a certain value 0  u ) and          0 0 0 OT V
. Then a ghost exists and its relative trajectory is defined by

         0 0 2 u P OG and          u V OG 2 0 0 . We have           2 0 0 2 t V
               2 2 2 2 2 2 2 2 1 t u t u t u t u t u t H .
The second example leads us to think that analyzing observability must not be reduced to seeking constant orthogonal matrices, conversely to the case of a leg-by-leg trajectory of the observer [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF].

2) Necessary and sufficient observability condition

The following analysis will be conducted for the relative motion with respect to the observer's trajectory. The question is to identify, if they exist, vectors

  T y x y x X    such that   t t P t y t y x t x OT             , 2 2   ( 19
)
Such a vector will be called a solution. The unknowns are the initial relative ghost-target position     T y x P OG  0 and its initial relative velocity

    T y x V OG    0 .
From the above subsection, we know that the set of solutions contains, of course, the vector defining the trajectory of the target, that is,

          T 0 0 0 0 OT OT OT OT OT y x y x X    and the vector             T 0 0 0 0 OT OT OT OT OT y x y x X      S I 2
, where the symbol  denotes the Kronecker product and 2 I is the 2 2 identity matrix. For convenience, the vector   OT X S I 2  is denoted S OG X . It defines the trajectory of a ghost.

We will show below that most of the time, other ghost-targets exist and the vectors defining their trajectories are linked to OT X by a time-dependent orthogonal matrix

  t H I 2  .
Before starting our analysis, we give a fundamental result in the following lemma: it establishes the equivalence between observability in discrete time and observability in continuous time. Therefore, we will conduct our analysis in continuous time, but the results will be valid for discrete time.

Lemma 2

If four different times

  4 , 3 , 2 , 1 , , 0  n T in f n  , satisfy       n OT n OG n OG P V P        2 0 0 2 (20) then for any   f T in t , 0         f OT OG OG T t t P t tV P , 0 , 2 0 0 2       . (21)
Proof: First of all, note that eq. ( 21) is equivalent to

      2 2 2 2 0 0 t P t tV P OT OG OG     (22)
Each term of this equality is a polynomial function of degree 4:

    4 2 3 2 2 2 4 2 2 0 0 t t d t c t b a t tV P G G G G OG OG          , with             ) 23 ( 0 ) 23 ( 0 0 ) 23 ( 0 0 ) 23 ( 0 2 2 2 2 2 2 d x V d c x y x P V c b y y x x V P b a y x P a x OG G x OG OG G OG OG G OG G                           T T T   4 2 3 2 2 4 2 t t d t c t b a t P T T T T OT       , with               ) 24 ( sin 0 ) 24 ( sin 0 0 ) 24 ( cos 0 0 ) 24 ( 0 0 0 2 2 0 0 2 0 2 d h v V d c r v P V c b h v r V P b a r P a r x r OT T x r OT OT T r r OT OT T OT T                      T T T Eq. ( 20
), which is equivalent to       4 , 3 , 2 , 1 , 2 0 0 2 2 2      n P V P n OT n OG n OG   
gives us four instants where these two polynomial functions take the same values. Because they have a common coefficient (the 4 th degree coefficient which is 4 2  ), these two polynomial functions are equal. QED.

Eq. ( 23) and (24) yield the following lemma on which the search of ghost-target is based:

Lemma 3

The set of solutions of ( 19) is defined by the following equations:

  ) 28 ( sin ) 27 ( sin cos ) 26 ( cos ) 25 ( 0 0 2 2 2 2 0 0 2 0 2 2 r x r x x r r x r r h v x r h v x y x h v r y y x x r y x                      Note that eq (28) implies   0 OT x x    .
The next proposition characterizes the set of ghost-targets. the set of solutions among the set of four-dimensional vectors whose first component is a root of a certain quadratic equation given below, and the third component is

  0 OT x  .

Proposition 14

The set of targets at the same range as the target of interest is composed by the target whose trajectory is defined by

  OT OG X X S I 2 S  
and by those whose trajectories are

defined by   T y x y x   , such that a) x is equal to   0 OT x or satisfies   0 2 sin 2 0 0 2 0 2 2      r h v r x v x r x r x r    b) and   0 OT x x    .
Due to its length, the proof (based on Lemma 3) is given in Appendix A.

The two other components of OG X , that is, y and y  , remain to be identified using Eqs. ( 25), (26), and (27).

For convenience, we now define  

2 0 0 2 r S r x x x Q       with x r v   2  and   r h S 2 sin 0    . Note that  is negative.
By convention, the heading

r h is zeroed when 0  r v .
Now, we are able to give a necessary and sufficient observability condition.

Proposition 15: Observability condition in ROTMA for a CA motion

Assuming that the observer is traveling with a constant acceleration vector, the trajectory of the target is observable from at least four range measurements acquired at different times if and only if O and T are on an RDVR-I.

Proof: Firstly, suppose that the trajectory of the target is observable. From Proposition 13,  

      0 0 , 0 , 0    OT x OT x OT y x x    



, and

  0 0  OT y  . Hence   x OT x   0 2   ,     0 sin sin 0 0 0 0 0 OT x r r S r          , and consequently           0 0 0 0 2 2 2 2 OT x OT OT x OT x x x x x x x Q         .
Formally, the roots of

  0  x Q are   0 OT x x  and     x OT OT x x x  0 0 2    
. Since the trajectory of the target is observable, either

      x OT OT OT x x x  0 0 0 2     , in other words,   0 OT x is a double root (this corresponds to 2 2     ), or     x OT OT x x  0 0 2    is an unacceptable physical solution; that is     2 0 2 2 0 0 r x x x OT OT           . Because   0 2 2 0 OT x r  , this inequality is equivalent to   2 2  
. The target and the observer are on an RDVR-I from Proposition 2.

Conversely, suppose that the target and the observer are on an RDVR-I; that is

    0 OT P and      0 OT V , and   2 2   . We have             T T 0 0 0 0 0 0 x x OT OT OT OT y x y x        .
Under this assumption, the scalars

  2 2 2 2 2 , , x T x T x T c b a            
. So, the set of equations ( 25), (26), and (27) becomes

  ) 31 ( ) 30 ( ) 29 ( 2 2 2 2 2 2 2 2 2 2 x x x x x x x y y y x y x                        which is equivalent to     ) 34 ( ) 33 ( ) 32 ( 2 2 2 2 2 x x x x x x y x y y x y                 
Taking the square of (33), we obtain

  2 2 2 2 2 x y y x x        .
Then using (32) and (34), we get

     2 2 2 2 2 2 x x x x x x x x              , which can be factorized into       2 2 2 x x x x x x x              (35). If x x    , then (35) is equivalent to   x x x     2    , from which we derive   x x    2   
. Substituting into (32), we have

    2 2 2 2 2 x y        ; that is,   2
  0  t r , for   f T t , 0  . (ii) 0   . Then 0 2 2     , but by assumption, 0 2 2     . Hence 0 2 2     . We have 0  y
. Now, from (33), we get

x x   
, which is impossible.

From this discussion, we conclude that the case

QED.

Recall that the FIM in BOTMA and the FIM in ROTMA are of the same rank. As a consequence, when O and T are on an RDVR-I, the FIM is singular (since in BOTMA, the target trajectory is unobservable). This is a nontrivial example of a mismatch between the singularity of the FIM and the observability (in ROTMA). A similar example was given in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF], when the observer is zigzagging.

3) Unobservable case: construction of the set of ghosts

When the observability condition given in Proposition14 is not satisfied, identifying the set of ghosts is interesting. We propose to solve this problem by using the same approach as previously, by exploiting the roots  

2 1 , x x of   x Q .
To do this, we need a first result.

We start with the following proposition:

Proposition 16: Existence condition of only one ghost in ROTMA for a CA motion

Assume that the observer is in CA motion. One unique ghost exists if and only if O and T are on an RDVR-II. Moreover, its trajectory is defined by S OG X .

Proof: First, assume that O and T are on an RDVR-II; that is,

    0 2 1 0 0 2    x c OT c OT t x t x   and     0 0 0   OT c OT y t y  .
Any ghost is also on an RDVR-II with the observer. Consequently, we have

0 2 1 2    x c c t x t x   and 0   y t y c  . We deduce that     0 0 OT c OT c x t x x t x      . Since   0 OT x x    , we conclude that   0 OT x x 
, and

  0 OT y y   and   0 OT y y     .
We have one unique ghost given by

          S OG OT OT OT OT X y x y x X     T 0 0 0 0   .
Conversely, assume now that one unique ghost exists. We recall that its trajectory is defined by

  T y x y x X    and that   0 OT x x    .
First of all, let us prove that

  0 OT x x  . Indeed, if   0 OT x x  , then 0   y y 
(otherwise, another ghost exists whose trajectory is defined by

  T y x y x X      
, which is in contradiction with the unicity of the ghost). At this point, we necessarily have

    T 0 0 0 OT x x X   . Suppose that   0 0  OT y or   0 0  OT y  . Then, the vector           T 0 0 0 0 OT OT OT OT y x y x    
defines the trajectory of a ghost. Since the ghost is unique, we have

             T T 0 0 0 0 0 0 0 OT OT OT OT OT y x y x x x X        , which is a contradiction. Hence     0 0 0   OT OT y y  . To summarize, we have obtained       T 0 0 0 0 OT OT OT x x X   , and     T 0 0 0 OT x x X   . Consequently,   0 OT x x  ,
which is in contradiction with the assumption. Hence,

  0 OT x x  . Therefore,     0 0  OT x Q . Let us develop the expression of     0 OT x Q :         2 0 0 2 0 0 0 r S r x x x Q OT OT OT       , with x r v   2  and     . cos sin cos 2 2 sin 2 cos sin 2 sin 0 2 2 0 0 r r r r r h h h h h S                   r r r r x r x r OT OT h h h h v r v r y x Q cos sin cos 2 2 sin 2 cos sin sin 0 0 0 2 2 0 2 0 2 0 0 2                r r x r r r x r OT h h v r h h v r y cos sin cos 2 2 sin 2 cos 1 sin 0 0 2 0 2 2 2 0 0 2             r r x r x r r OT h h v r v h r y cos sin cos 2 2 cos sin 2 0 0 2 0 2 2 0 0 2                     0 0 0 2 0 0 2 0 2 2 OT OT x OT OT x OT OT y x y y x y                            0 0 0 0 0 0 2 0 0 2 OT OT OT OT OT OT x OT y x y x y y x Q         .
According to Proposition 3, the target and the observer are on an RDVR-II.

QED.

Here, the singularity of the FIM and the unobservability of the target's trajectory are consistent (see Proposition 9).

The following lemma provides us information about the two solutions of the quadratic equation   0

 x Q
, introduced in Proposition 14. This piece of information will help us to identify all the ghost-targets. We recall that

  2 0 0 2 r S r x x x Q       , with x r v   2  and   r h S 2 sin 0    .

Lemma 4

The equation

  0  x Q has two real roots defined by 2 1      x and 2 2      x , with     2 2 0 2 0 1 4 2 S r S r       . Moreover, 2 0 1 0 x r x r     .
Proof:

We readily verify that the discriminant of

  0  x Q is     2 2 0 2 0 1 4 2 S r S r      
, which is a positive quantity.

Hence the two roots are real and are given by 2

1      x and 2 2      x . Note that 2 1 x x  . We note that     1 0 0   S r r Q  , and     0 0    S r r Q  . Hence,   0 0   r Q and   0 0  r Q . We deduce that 2 0 1 0 x r x r    
and, as a consequence, that 2

x is strictly positive.

QED.

Lemma 4 will conduct our study according to the following table:

Table I: Values of   2 1 , x x 1 x 2 x Case 1 0 1 r x   0 2 r x  Case 2 0 1 0 r x r    0 2 r x  Case 3 0 1 r x  0 2 r x  Case 4 0 1 r x   0 2 r x  Case 5 0 1 0 r x r    0 2 r x  Case 6 0 1 r x  0 2 r x  Recall that     2 1 x x and   0 0 2 1 r S r x x    . We
deduce that Case 1 is the only case where 0  r v .

Case 1:

0 1 r x   and 0 2 r x  The sum of the roots is zero, that is 0     0   r v ; as a consequence,   T 0 0 cos sin 0 0 0 0   r r X OT  . From (25), for 1 x x  or 2 x x  , we get 0  y . For 1 x x  , from (27), then 0 0 0 2 sin   x x r r y     , or equivalently   0 0 2 sin 1      x r y  . Hence   0 0 sin 1       x r y 
. We get two ghosts, given by

                   0 0 0 sin 1 0 0   x r r X . If 2 0     , then these two solutions merge into one:   OT X r X    T 0 0 0 0 . For 2 x x  , from (27), then   0 0 2 sin 1     x r y  .
But the right member of this equation is negative or null (when 2 0    ), whereas the left member is positive or null. We get one solution if 2 Case 2:

0    :   OT X r X   T 0 0 0 0 . In conclusion, in Case 1, if 2 0     ,
0 1 0 r x r    and 0 2 r x  In this case,   0 0 0 1 r S r r x    and     0 1 r x . Then necessarily, 1   S ; that is, ) 2 (mod 2 2 0      r h
. Note that in this case,

0 sin  r h , then                            r x OT h r x r r X 2 0 0 0 sin 2 0 2         , with 1    ; else                             r x OT h r x r r X 2 0 0 0 sin 2 0 2         . For 2 x x  , from (27) we get   0 0 2 0 0 0 2 2 2 sin 1 sin             x r x x r r v r r v y x   , which is equivalent to      . sin 1 2 cos 1 2 sin 1 cos 0 0 2 0 0 2 2 2            x r r x r r r h v r h v y  We have 0 sin 2 cos    r h . So, we get              . cos 2 sin 1 2 2 1 sin 1 sin 1 2 sin 1 sin 1 2 2 0 0 0 0 0 0 0 0 0 2 2 r x x x x r h r r r r v y                               



The square 2 y  being positive or null and the right hand side term being negative or null, we have

2 0    . That yields the vector                0 0 0 0 OT x r X 
(which is equal to the vector defining the trajectory of the target).

In conclusion, in case 2, if 2 0    , then we have three ghosts; else we have two ghosts (in case 2, the target is not observable).

Case 3:

0 2 1 r x x   In this case, 0   or equivalently 1 2  S and S r 0 2   . Because  is negative, we get 1   S (hence S 2    ).
First of all, note that 2 2

1 0         r h S and x r r v r   0 0 2 2      .
From (25), we have 0

 y . Reporting 0 r x  into (27) we get   0 0 2 0 2 2 sin 0    x r x OT r v r y x         x x OT r r r x v y    0 0 0 2 2 2 sin            x OT r y   0 0 2 sin 1 0                         0 sin 1 sin 1 sin 1 sin 1 2 sin 1 2 cos 1 2 sin 1 cos sin 1 cos 0 0 0 0 0 0 0 0 0 0 0 2 0 0 2 2                                                r r r r h r h r h v x x r x r x x r r . Consequently, 0  y  .
We deduce a solution given by

    T 0 0 0 0 OT x r X   .
We verify that equations (25), ( 26), (27), and (28) are satisfied.

In conclusion, in Case 3, the target is observable if

and only if 2 0    ( 2     r h ). Otherwise, two
                                ,
                      r x OT h
                          

Table II: Summarized results of observability in ROTMA

Necessary condition to have ghosts 0 (Observability case: RDVR-I)

1 2 3 Case 1 2 0     never 2 0    2 0     Case 2 never never 2 0    2 0    Case 3 2 0    never 2 0  
t : If r t r t r k x k k     0 2 2 1  , or r t r t r k x k k     0 2 2 1  and x r r  0 2 2   
, then the trajectory of the target is observable and we compute the state vector by minimizing the criterion

       K k k k r X t r 1 2 , w.r.t. X .
If not, we compute one of the vectors X (it is not unique since the trajectory of the target is not observable) that minimizes

       K k k k r X t r 1 2 ,
, then we choose this vector as the state vector of the target of interest (of course in reality, this choice can be wrong); this choice provides us with the corresponding polynomial function  

x Q and its two roots. Finally, we compute the other solutions X by exploiting the above analysis (we only have to identify in which case we are). Of course, we are unable at the end to separate the wheat from the chaff, that is, to identify the state vector of the true target among the ghost-targets.

4) Examples

In this paragraph, we give one example for each case. In the following scenarios, the observer has the same trajectory: at the very beginning, it starts from   T

b) Examples of unobservable cases

The following example corresponds to case 5 with a RDVR-II. II (See Fig. 10). Note that if the role of T and G1 (for example) are inverted (G1 becomes the target and T becomes a ghost), then this scenario corresponds to case 3:

actually the polynomial function   x Q
is not the same (see eq. (A6) in the proof of Proposition 14). The bearings of this new target are not constant, but two ghosts only exist. V. CONCLUSION In the present paper, the results of the observability analysis in ROTMA started in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF] for a target in constant velocity motion and an observer moving leg by leg, have been extended to a smooth observer's maneuver (constant turn motion and constant acceleration motion).

When the observer is in CT motion (see for example [START_REF] Bucy | Digital Synthesis of Nonlinear Filters[END_REF] and [START_REF] Farina | Target Tracking with Bearings-Only Measurements[END_REF] in BOTMA and [START_REF] Ristic | Target Motion Analysis Using Range-Only Measurements: Algorithms, Performance and Application to ISAR Data[END_REF] and [START_REF] Sathyan | Multiple Hypothesis Tracking with Multiframe Assignment Using Range and Range-rate Measurements[END_REF] in ROTMA), observability is guaranteed in BOTMA and in ROTMA as well. If a part of the displacement of the observer is in an arc of a circle, this conclusion remains valid. When the observer is in CA motion, observability in BOTMA is guaranteed if the observer and the target are not in an RDVR and the bearings are not constant. In ROTMA, if they are in an RDVR and the bearings are constant, the trajectory of the target is observable. This proves that even if the observer kinematic is of an order greater than the kinematic of the target, observability is not guaranteed. This is not in contradiction with [START_REF] Fogel | Nth-order Dynamics Target Observability from Angle Measurements[END_REF], whose authors established a necessary (but non-sufficient) observability condition in this case for BOTMA. When the target is not observable, in ROTMA, the set of ghost-targets is finite; we give the way to construct them from noise-free measurements and we end up with three ghost-targets at most. In BOTMA, the set of ghost-targets is uncountable. Measurement-based criteria that allow it to be known whether or not the target is observable are also given. To summarize, unlike what we established in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF] (the observer trajectory was composed of several legs), that is, that the target is bearings-only observable if it is range-only unobservable and the converse, for the two types of smooth maneuvers considered here, this duality is not verified. All our results are summarized in the following Table: We extended in BOTMA our analysis when the observer's trajectory is composed of a CA motion followed by a CV motion (and inversely): arguing fundamental properties, we proved that if the bearing rate is non-null, the target is observable.

Despite the rank equality of the FIMs in ROTMA and BOTMA (see [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF]), the statuses of observability in ROTMA and in BOTMA are not necessarily the same. Even if this result is surprising, it is consistent with the theory (see for example [START_REF] Jauffret | Observability and Fisher Information Matrix in Nonlinear Regression[END_REF]): the existence of a linear form of the BOTMA problem explains: (i) the equivalence between the regularity of the FIM and observability and (ii) the unaccountability of the set of ghost-targets in non-observable situations. Conversely, in ROTMA, in the case of non-observability, the FIM can be singular or not and the set of ghost-targets is finite. This is a proof that the problem of ROTMA cannot be expressed under a linear form (otherwise the set of ghost-targets would be a linear subspace and hence uncountable). Obviously, we do not claim to have achieved a complete study of observability in ROTMA; for example, the cases when the observer's trajectory is composed of a CA motion followed by a CV motion, or when the motion of the observer is polynomial of order greater than two, or when the observer does not maneuver whereas the target does (which we studied in BOTMA in [START_REF] Clavard | Target Motion Analysis of a Source in a Constant Turn from a Nonmaneuvering Observer[END_REF] and [START_REF] Jauffret | Bearings-Only Maneuvering Target Motion Analysis from a Nonmaneuvering Platform[END_REF]), and so on must be investigated.

Our study can help as a basis.

The next step (and paper) concerning ROTMA is the estimation, when the measurements are polluted by an additive noise. The observability analysis presented here will allow us to anticipate several difficulties: (i) thanks to the given criteria, we will be able to construct a statistical test to decide whether the observer and the target are on a rendezvous route or not. (ii) Moreover, for a given solution, we will construct the set of ghost-targets in the case of a lack of observability. (iii) Finally, an initialization of any numerical routine (necessary to compute the maximum likelihood estimate of the state vector), based on this analysis, can be proposed.

APPENDIX

A. Proof of Proposition 14

We exploit here the four equations of Lemma 3. In order to render the proof lighter, we drop the subscript T in T a , T b and T c defined in (24): ). Let us denote the three solutions as

  2 , 1 , 0  i x i
(in some cases, only one or two roots exist).

For convenience, the root   0 OT x will be denoted 0

x .

To compute the two other roots, we first develop (A3): Using Lemma L1, we conclude that 0  D .

          0 cst 0 2 0 0 2 2 2 3          x OT OT OT x a x b x c x x x x          0 cst 0 2 2 3       x OT x a x b x c x x    . ( A4
QED.

Fig. 1 .

 1 Fig. 1. Typical scenario of TMA.

Figure 2 .

 2 Figure 2. Typical scenario when the observer is traveling in an arc of a circle.

Proposition 1 :

 1 General properties of RDVR If O and T are on an RDVR, then-Either

  two types of RDVR The RDVRs are called rendezvous routes of type I (RDVR-I), when noncollinear as well, the RDVRs are called rendezvous routes of type II (RDVR-II).Note that for the RDVR-II, . Fig.3and Fig.4depict an example of RDVR-I, and of RDVD-II, respectively. We can notice that the bearings are constant for the RDVR-I whereas they change in time for the RDVR-II. We will see why and how subsequently.

Figure 3 .

 3 Figure 3. Example of RDVR-I.

Figure 4 .

 4 Figure 4. Example of RDVR-II. The converse of Proposition 1 is given in the following two propositions. Proposition 2: Condition of RDVR-I Assume that   0 OT P ,

Fig. 5 ,

 5 an example of such a situation is given: The target starts at   T

Figure 5 .

 5 Figure 5. Example of RDVR-I where the target and the observer have two RDVs.

  the observer are on an RDVR-I, from Proposition 2.

Figure 6 .

 6 Figure 6. Non-observable trajectory in BOTMA, the target, some ghosts, and the bearing-equivalent-nonmaneuvering observer.

  defines the trajectory of a ghosttarget. We can conclude by the following proposition: with  , then the target is not observable. Let us present an example of this situation, depicted in Fig.7.At the beginning of the scenario, the observer starts at with a speed of 4 m/s and a heading of 0°. The duration of the simulation is 45 min. Note that with an ad-hoc rotation, we can be again in the case where 0  y  .

Figure 7 .

 7 Figure 7. ROTMA: The observer in constant acceleration motion, the target, and a ghost. Conversely, if

  incompatible with the fact that

  we end up with three ghosts, whose respective trajectories are defined by

  allows us to construct two ghosts: if

  As in case 2, we exploit the sum and the product of the roots:

...

  In conclusion, in Case 4, the target is observable if and only This value also satisfies Eq. (27) (see Appendix A2). Consequently, we obtain the following two solutions: In conclusion, in Case 5, the target is never observable. At most, three ghosts exist, given by

  This case includes the scenarios where T and O are on an RDVR-II. ), (26), (27), and (28) are verified.In conclusion, in Case 6, the target is observable if

  k

  The observer's trajectory is hence the same as in Section III. C. (as shown in Fig.6).In the next figures, the trajectories of the observer and target are shown by thick lines, whereas those of the ghosts are shown by thin lines. The capital letters ("O" for observer, "T" for target, and "G" for ghost) designate the moving objects. a) Example of observable case (case 1)The target starts at   T s). The observer and the target are on an RDVR-I,

Figure 8 .

 8 Figure 8. Observable case in ROTMA, RDVR-I.

Figure 9 .

 9 Figure 9. Unobservable case in ROTMA, RDVR-II. The next example corresponds to case 2. The target starts at   T 0 4000 (m) with a velocity of   T 2 25

Figure 10 .

 10 Figure 10. Unobservable case in ROTMA, with constant bearings and no rendezvous route.The last example offers a scenario for which three ghosts exist. It corresponds to case 5.The target starts at   T 3464 2000 (m) with a velocity of

Figure 11 .

 11 Figure 11. Unobservable case in ROTMA, general case (no rendezvous route and non-constant bearings).

  cubic equation (A3) has at most three real roots (one of them is

5 :

 5 Compatibility of Eqs. (26) and (27)We start this proof with two useful lemmas.

  .

	Hence, O and T are on an RDVR if and only if the equation
	t	2		2  t		2 		0	has one or two real roots (one of them is c t ),
	that is, if and only if the discriminant			  2 2 	is positive.
	QED.
	Obviously, the assumption "	OT P	  0	,	OT V	  0	and  are
	collinear " is not sufficient to characterize an RDVR-I, but
	implies that the bearings are constant. Conversely, if the
	bearings are constant, the target and the observer are not
	necessarily on an RDVR-I.
	Let us note that in this case, the bearings are piecewise equal
	to		2   , and Proposition 2 remains valid up to a rotation, that
	is, for   constant  t 	up to c t . Note also that two rendezvous
	instants may exist (depending on  ). In

Proposition 9: Observability criterion in BOTMA for CA motion

  Assume that the observer is in CA motion. The target's trajectory is observable in BOTMA if and only if O and T are not on an RDVR and the bearings are not constant.

Proposition 10: Set of ghosts in BOTMA for an RDVR-II

Assume that the observer is in CA motion.

The target's trajectory is unobservable if and only if

Proposition 11: Observability criterion in BOTMA for CA- CV and CV-CA motions

	Proof:
	We have only to consider the case where the target's trajectory
	is not observable during the maneuvering phase.
	The first leg starts at time	t		0	and ends at time	1 T t  . At this
	time, the CA motion starts and finishes at time	f t  . T
	If the bearings are constant during this phase, then

Assume that the observer is successively in CV motion and in CA motion (or in CA motion and in CV motion). The target's trajectory is observable in BOTMA if and only if the bearing rate is non-null (that is, the bearings are not constant).

  

									never
	Case 4	0   	2 				never	never	0   	2 
	Case 5	never				RDVR-II	never	Not RDVR-II
	Case 6	0  	2 				never	never	0  	2 
	The content of Table II is consistent with Propositions 14, 15
	and 16.
	At this point, we can propose the following algorithm to
	construct (if existing), all the ghosts from a set of noise-free
	measurements 	r 1	r , 2	,	, 	K r	 , with	k r the range at time

  The observer and the target are on a rendezvous route of type II. We can see in Fig.9. that one ghost exists, and it is in a rendezvous route with the observer (seeProposition 16).

	tan		  t		0023 . 0 	t		75 . 0	thanks to Proposition 5.
									The target starts at 	3000	4000	 T	(m) with a velocity of
									 	6 	7	 T	(m/s).	We	can	readily	verify	that

Table III : Synthetic results about observability

 III

	Observer's kinematics	BOTMA	ROTMA
	CT	Yes	Yes
	CA: RDVD-I	No	Yes
	CA: RDVD-II	No	No
	CA: constant bearings	No	No
	CA: others	Yes	No

Recherche Opérationnelle (CIRO) agency of the Ministère de la Défense. Since 1980 he has been employed by Thomson Compagny (now Thales Compagny), France. From 1980 to 1981 he worked on sonar array processing. From 1982 to 1986, he was consultant at the groupe d'études et de recherche en detection sous-marine (GERDSM) of the Ministère de la Défense, for which he worked on sea trial analysis and TMA. From 1987 to 1989, he worked on tracking and track clustering, and in 1990 he was appointed group leader of signal and data processing laboratory at Thomson Sintra company, Sophia Antipolis, France. From 1992 to 1999, he was consultant at the DCN sonar laboratory. He worked on new sonar arrays and various submarine combat systems. Since 2000, he has been sonar expert at Thales Underwater Systems (Sophia-Antipolis). His current researches are about passive sonar performances estimation, submarine and torpedo array processing, and long term prospective studies for submarine warfare. He is now retired and keeps working on his favourite topics (sonar, tracking, TMA …).


Marion Rolland
email: marion.rolland@chu-rennes.fr

MD Christèle Dubourg

Auriane Cospain

MD Catherine Droitcourt

PhD Laurent Pasquier

PhD ; Auriane Cospain

(Asp52Asn) PIGL pathogenic variant in association with the previously reported p.(Leu167Pro) variant: A case report

Keywords: CHIME syndrome, PIGL, neuroectodermal disorder, case report

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Coloboma, congenital heart disease, ichthyosiform dermatosis, mental retardation and ear anomalies (CHIME) syndrome is a very rare autosomal recessive neuroectodermal disorder that was initially described by Zunich and Kaye in 1983 [START_REF] Zunich | New syndrome of congenital ichthyosis with neurologic abnormalities[END_REF] . This disease entity is related to pathogenic variants in the gene coding for the phosphatidylinositol glycan anchor biosynthesis class L (PIGL) protein (OMIM #280000), an enzyme located at the membrane of the endoplasmic reticulum. PIGL catalyzes the second step in glycosylphosphatidylinositol (GPI) biosynthesis, i.e. the de-N-acetylation of Nacetylglucosaminyl-phosphatidylinositol. [START_REF] Nakamura | Expression cloning of PIG-L, a candidate Nacetylglucosaminylphosphatidylinositol deacetylase[END_REF] . GPI has a critical role in the anchoring of many membrane proteins, some of which are involved in neurogenesis [START_REF] Altassan | Hyperphosphatasia with mental retardation syndrome, expanded phenotype of PIGL related disorders[END_REF] .

To date, only eight cases of CHIME syndrome in people with confirmed PIGL variants have been reported [START_REF] Ng | Mutations in the glycosylphosphatidylinositol gene PIGL cause CHIME syndrome[END_REF][START_REF] Knight Johnson | Alu-mediated deletion of PIGL in a patient with CHIME syndrome[END_REF][START_REF] Ceroni | Large deletion in PIGL: A common mutational mechanism in CHIME syndrome?[END_REF] . Given the rarity of this syndrome, data on its clinical features and clinical spectrum are scarce. All six of the cases reported by Ng et al. [START_REF] Ng | Mutations in the glycosylphosphatidylinositol gene PIGL cause CHIME syndrome[END_REF] carried a compound heterozygous variant c.500T>C, p.(Leu167Pro). A second pathogenic variant was identified in 5 of the 6 patients 4 . In the seventh case of PIGL-related CHIME syndrome (reported by Knight Johnson et al. [START_REF] Knight Johnson | Alu-mediated deletion of PIGL in a patient with CHIME syndrome[END_REF] ), the c.500T>C, p.(Leu167Pro) variant was again detected on the first allele but the second allele carried an intragenic deletion of PIGL's exons 4-6. In the eighth case (reported by Ceroni et al. [START_REF] Ceroni | Large deletion in PIGL: A common mutational mechanism in CHIME syndrome?[END_REF] ), the c.500T>C, p.(Leu167Pro) variant on the first allele was combined with a 802 base-pair deletion (involving the 5' untranslated region and the first 50 amino acids of PIGL) on the other allele.

Here, we report on a new case of genetically confirmed PIGL-related CHIME syndrome with an uncommonly mild phenotype and a novel PIGL variant.

CASE REPORT

In November 2014, a 2-year-old boy was referred for the assessment of orange-colored skin lesions.

He was the second son of healthy, non-consanguineous parents. Both parents were tall (height: 187 cm for the father and 176 cm for the mother) but had a normal body weight (73 and 58 kg, respectively) and a normal head circumference (56.5 and 54 cm, respectively). The family's medical history was otherwise unremarkable, and the boy's older brother was healthy, with normal growth parameters (height and weight).

The patient was being followed by the genetics department after the antenatal diagnosis of nephromegaly, hydramnios, and macrosomia. It is noteworthy that the antenatal karyotype was normal. The boy was born full-term, with a birth weight of 4550 g (99 th percentile), length of 52 cm (97 th percentile), and head circumference of 38 cm (99 th percentile). A clinical examination soon after birth revealed hepatomegaly and macroglossia. The antenatal diagnosis of nephromegaly was confirmed postnatally using ultrasound. Delayed psychomotor development and febrile seizures were diagnosed at the age of 3 months. Although language acquisition and communication skills were normal at 15 months of age, the delay in motor development was more persistent. He started to walk at the age of 22 months and was able to run at 3.5 years. Throughout his childhood, he suffered from frequent gastrointestinal, upper respiratory and pulmonary viral infections. Screening for primary immunodeficiency (a complete blood count and immunophenotyping of circulating lymphocytes) was normal. Beckwith-Wiedemann syndrome was initially suspected but was not confirmed by molecular analysis (i.e. no abnormal methylation patterns in the 11p15 region in blood and skin samples, and normal array CGH findings). Furthermore, the ocular, ear, nose and throat (ENT), and cardiac abnormalities usually observed in Beckwith-Wiedemann syndrome were not present. Liver function tests were normal and in particular, alkaline phosphatase (ALP) value was measured at 311 IU/L, within the normal range (142 to 335). At age 7, exome sequencing revealed two heterozygous, missense variants in PIGL in a compound heterozygous state (NM_004278.4). The paternal allele carried the missense variant c.500T>C, p.(Leu167Pro), already classified in the ClinVar database as "pathogenic" [START_REF] Ng | Mutations in the glycosylphosphatidylinositol gene PIGL cause CHIME syndrome[END_REF][START_REF] Knight Johnson | Alu-mediated deletion of PIGL in a patient with CHIME syndrome[END_REF][START_REF] Ceroni | Large deletion in PIGL: A common mutational mechanism in CHIME syndrome?[END_REF] (class 5 according to the American College of Medical Genetics (ACMG) classification [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: A joint consensus recommendation of the American College of Medical Genetics and Genomics and the Association for Molecular Pathology[END_REF] ). The maternal allele carried the missense variant c.154G>A, p.(Asp52Asn), which had not previously been described in the GnomAD and ClinVar databases. This variant affects a residue that is highly conserved across species (from yeast up to man) and is predicted to have a deleterious effect on protein function by various in silico prediction tools (Align GVGD, SIFT, Mutation Taster, and PolyPhen, accessed through Alamut Visual). Hence, this variant was classified as "likely pathogenic" (ACMG class 4 [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: A joint consensus recommendation of the American College of Medical Genetics and Genomics and the Association for Molecular Pathology[END_REF] ). We submitted both variants to ClinVar; the accession numbers are respectively SCV001963616 for the variant c.154G>A, p.(Asp52Asn) and SCV001963617 for the variant c.500T>C, p.(Leu167Pro). The previously reported PIGL variants and the newly described variant are summarized in Figure 2.

On examination at the age of 9, the boy's growth had slowed down: his weight was 32.4 kg (+1 SD), height was 140 cm (+1.5 SD) and head circumference was 57 cm (+3 SD). The aforementioned skin abnormalities were stable, and in particular, hair growth remained very slow (Figure 4). Mild perifollicular hyperkeratosis was observed under dermoscopy of his scalp, and yellow crusts and mild erythema were noted. His neurodevelopmental status and social skills were normal.

DISCUSSION

In our case, the genotype/phenotype correlation partially matched a diagnosis of CHIME syndrome.

Along with the core features of this syndrome, other clinical signs have been described: epileptic seizures, facial dysmorphism (including a prominent forehead, hypertelorism, and epicanthal folds), widely spaced teeth, and skin, hair and nail anomalies (such as sparse, thin hair, thick palms and soles, and migratory ichthyosis) [START_REF] Knight Johnson | Alu-mediated deletion of PIGL in a patient with CHIME syndrome[END_REF][START_REF] Ceroni | Large deletion in PIGL: A common mutational mechanism in CHIME syndrome?[END_REF][START_REF] Shashi | Neuroectodermal (CHIME) syndrome: An additional case with long term follow up of all reported cases[END_REF][START_REF] Tinschert | Zunich neuroectodermal syndrome: Migratory ichthyosiform dermatosis, colobomas, and other abnormalities[END_REF][START_REF] Schnur | Acute lymphoblastic leukemia in a child with the CHIME neuroectodermal dysplasia syndrome[END_REF] . Our patient's clinical features , including early-onset epilepsy, hair, tooth and nail abnormalities, and ichthyosiform dermatitis were consistent with a diagnosis of CHIME. Persistent, excessive growth has also been reported in two cases of CHIME syndrome [START_REF] Ceroni | Large deletion in PIGL: A common mutational mechanism in CHIME syndrome?[END_REF] . The clinical and molecular features of the eight previously reported PIGL-related CHIME syndrome are summarized in the supplementary data (Table S1).

Our patient exhibited a mild phenotype. In particular, no ophthalmological, echocardiographic, or significant ENT anomalies were found. Furthermore, the boy's psychomotor and neurological impairments receded with age. It is noteworthy that in a cell-based functional assay, the c.500T>C, p.(Leu167Pro) variant was associated with a moderately but significantly lower level of GPI anchor markers, relative to other GPI deficiencies [START_REF] Ng | Mutations in the glycosylphosphatidylinositol gene PIGL cause CHIME syndrome[END_REF] . Interestingly, this variant is the common denominator of the eight previously reported cases and has never been reported in a homozygous state. Given that the c.500T>C, p.(Leu167Pro) variant appears to have a small effect per se, the severity of the phenotype might be determined by the variant on the other allele. This might explain the phenotype in the present case: assuming that the effect of the p.(Asp52Asn) variant has a small impact, the combined effect of the two variants might result in a moderate clinical expression of the CHIME syndrome.

Consequently, our patient's phenotype provides new data on the clinical spectrum of CHIME syndrome. It is noteworthy that PIGL gene variations are also associated with Mabry syndrome (characterized by hypotonia, seizures, intellectual disability and increased blood levels of tissue nonspecific alkaline phosphatase), which is phenotypically distinct from CHIME syndrome [START_REF] Altassan | Hyperphosphatasia with mental retardation syndrome, expanded phenotype of PIGL related disorders[END_REF][11][START_REF] Pagnamenta | Analysis of exome data for 4293 trios suggests GPI-anchor biogenesis defects are a rare cause of developmental disorders[END_REF][START_REF] Mogami | Early infancy-onset stimulation-induced myoclonic seizures in three siblings with inherited glycosylphosphatidylinositol (GPI) anchor deficiency[END_REF] .

Our case illustrates phenotypic/genotypic variations in CHIME syndrome. Exome sequencing was particularly useful for diagnosis, widening the clinical spectrum of CHIME syndrome and enabling accurate genetic counseling [START_REF] Altassan | Hyperphosphatasia with mental retardation syndrome, expanded phenotype of PIGL related disorders[END_REF]11 . S1: Clinical and molecular features of patients with PIGL-related CHIME syndrome

FIGURE LEGENDS

  A dermatological examination revealed ichthyosiform changes, with xerosis and eczema on the cheeks and keratotic, follicular papules on the knees.Examination of the skin at age 5 years revealed koilonychia, internal epicanthal folds, slowly growing, sparse, thin eyebrows and reduced hair density on the temporoparietal scalp (Figures 1.A to 1.C). Examination of the mouth showed gingival hypertrophy and conical, irregularly spaced teeth. These clinical features were strongly consistent with a diagnosis of ectodermal dysplasia. The patient's neurological impairments became less severe with age, and the delay in psychomotor development eventually disappeared. Antiepileptic medication was discontinued at the age of 6, with no clinical relapse. Growth parameters exceeded normal values: at 6 years of age, the boy weighed 26.5 kg (+2 standard deviations [SD]), was 126 cm tall (+2 SD), and had a head circumference of 56 cm (+3 SD).

Figure 1 :

 1 Figure 1: Cutaneous features characteristic of CHIME syndrome: 1.A: Thin eyebrows and internal epicanthal folds at age 9 years 1.B: Slowly growing, sparse, thin hair, with reduced density on the temporoparietal scalp with an area of hypotrichosis at the junction of the vertex and the upper parietal scalp at age 9 years 1.C: Conical, irregularly spaced teeth at age 5 years

Figure 2 :

 2 Figure 2: Reported PIGL gene variants. The green boxes correspond to the seven exons. The figure was created with Proteinpaint (https://pecan.stjude.cloud/proteinpaint)

This research did not receive any specific funding from agencies or organizations in the public, commercial, or not-for-profit sectors.