robot positioning system based on a rotating receiver

7/31/2019 Robot Positioning System Based on a Rotating Receiver

1/5

ANALYSIS OF A ROBOT POSITIONING SYSTEM BASED ON A ROTATINGRECEIVER, BEACONS, AND CODED SIGNALS

V. Pierlot and M. Van Droogenbroeck

INTELSIG Laboratory, Montefiore Institute, University of Lige, Belgium

ABSTRACT

Positioning is a fundamental issue in mobile robot applications thatcan be achieved in multiple ways. Among these methods, trian-gulation with active beacons is widely used, robust, accurate, andflexible. In this paper, we analyze the performance of an originalsystem, introduced in one of our previous papers, that comprises arotating receiver and beacons that send an On-Off Keying modu-lated infrared signal. The probability density functions of the mea-sured angles are established and discussed. In particular, it is shownthat the proposed estimator is a non biased estimator of the beaconangular position. We also evaluate the theoretical results by means

of both a simulator and measurements.

1. INTRODUCTION

In a previous paper [3], we have presented an original system forrobot positioning used during the Eurobot contest since 2008. Thesystem comprises a turning turret that measures angles based on sig-nals sent by several infrared beacons (see Figure 1 for an illustrationof the Eurobot setup). These angles, denoted i, are combined, dur-ing a triangulation calculus, to compute the current position of amobile robot in a 2D plane [1, 4]. In order to identify a beaconand to increase the robustness against noise, each beacon sends aunique binary sequence encoded as an On-Off Keying (OOK) am-plitude modulated signal.

In this paper, we evaluate the error made on measured angles re-sulting from the coding of signals sent by the beacons. As explained

in [3], the modulation of the 455 [kHz] carrier beacon signal makesit possible to identify beacons but, as a drawback, introduces errorsthat occur when no signal is sent, that is during an OFF period ofthe sequence. In the following, we address the issues raised by theuse of an OOK modulation mechanism. An angle estimator is pro-posed and we characterize its performance by means of a simulatorand experiments.

The paper is organized as follows. In Section 2, we first brieflypresent the principle of the original positioning system describedin [3]. Section 3 discusses the origin of the errors. In Section 4,we propose several estimators and compute the probability densityfunctions of the measured angles. Then we present and discuss theexperimental results in Section 5, and conclude the paper in Sec-tion 6.

2. ANGLE MEASUREMENT PRINCIPLE

The system described in [3] is an angle measurement sensor basedon fixed infrared beacons, a rotating turret turning at a constantspeed , and an infrared receiver (TSOP7000). Each beacon con-tinuously emits an infrared (IR) signal in all directions in a horizon-tal 2D plane. Let us denote by the current angular position of thereceiver. As the turret turns at a constant speed, the angular position is directly proportional to time

(t) = t. (1)

As a result, we can either talk about time or angular position in-differently. In fact, the processing unit of the positioning systemmeasures times corresponding to angles thanks to eq. (1). Althoughthe system measures times, we prefer to think in terms of angles,because it is more intuitive for the developments.

1

3

2 B2

B1

B3

R

Figure 1: Left-hand side drawing: triangulation setup in the 2Dplane. The circle and the small squares represent the mobile robot

(R) and the beacons (Bi) respectively. i are the angle measure-ments for each Bi, relatively to the robot reference orientation .Right-hand side image: playing field of the EUROBOT contest 2009.

The receiver has a limited field of view and, consequently,the amount of infrared power collected at the receiver, denoted byPIR(), depends on the angle. Moreover, this power depends onthe power emitted by the beacon and the distance between the bea-con and the receiver. The exact shape of PIR () depends on thehardware used (receiver, optical components and geometry of tur-ret) and, unfortunately, the information available at the receiver out-put is too sparse to derive a precise curve. Indeed, we have onlyaccess to the demodulated signal and no information about power isavailable. Therefore, we make some basic assumptions on PIR ().

Figure 2 shows a possible curve for PIR () (top curve). The shape ofthis curve has not much importance in this study but is supposed toincrease from a minimum to a maximum and then to decrease fromthis maximum to the minimum. In the following theoretical devel-opments, we make three important assumptions about the curve andthe detection process itself:

1. The maximum occurs at an angle which is the angular positionof the beacon, denoted b. In other words, we have, for anyangle

PIR() PIR (b). (2)

2. The curve is also supposed to be symmetric around the maxi-mum since the turret and all optical components are symmetric.Therefore, we consider that

PIR (b) = PIR (b +). (3)

3. Finally, we suppose that the receiver reacts to 0 1 (raising)and 1 0 (falling) transitions at the same infrared power thresh-old Pth , respectively at an angle r and f

PIR(r) = PIR (f) = Pth . (4)

From eq. (3) and (4), we can see that br = fb and, conse-quently, that

b =r+f

2. (5)

If the beacons are assumed to send a non modulated IR signal(that is, a pure 455 [kHz] sine wave), the measurement of the an-gular position of a given beacon works as follows: while the turret

19th European Signal Processing Conference (EUSIPCO 2011) Barcelona, Spain, August 29 - September 2, 2011

EURASIP, 2011 - ISSN 2076-1465 1766


2/5

E3

Pth

bPIR()

R0

R1

R2

R3

E1

E2

E4

R4

fr

0

Figure 2: The upper curve PIR () is the infrared power collected atthe receiver while the turret is turning. Ei are examples of emittedsignals from the beacons. Ri are the corresponding received sig-nals at the receiver output. R0 is a special case corresponding toa non modulated infrared carrier (no blank periods). The black ar-rows represent the measured values respectively for r to the left(first Rising edge) and for f to the right (last Falling edge). Theencircled arrows emphasize errors committed on r or f.

is turning, the receiver begins to see the IR signal from that bea-con when the threshold Pth is crossed upwards (0 1 transition).The receiver will continue to receive that signal until Pth is crosseddownwards (1 0 transition). The receiver output is depicted as

R0 in Figure 2. The estimator used for r is the angular position ofthe first rising edge at the receiver output; the estimator is denotedby r hereafter. The estimator used for f is the angular positionof the last falling edge at the receiver output; it is denoted by f.These estimators are represented by the black arrows in Figure 2.The estimator b of the beacon angular position b derives fromeq. (5)

b =r+f

2, (6)

where r,f and b are random variables.Until now, we have considered that the IR signal is not mod-

ulated. But the situation is different because the IR receiver getsan On-Off Keying amplitude modulation of a 455 [kHz] carrier fre-quency, otherwise it would not be possible to distinguish betweenbeacons, and this introduces some errors.

3. SOURCE OF THE ERRORS

Even if we would use a pure 455 [kHz] carrier, noise or other irrel-evant IR signals could produce erroneous transitions at the receiveroutput, and consequently erroneous r or f. In order to identifythe different beacons and to be robust against noise, each beaconsends its own coded signal. These coded signals solve issues men-tioned previously but introduce a new error, as detailed hereafter.

The receiver gets an OOK amplitude modulated signal. Itmeans that the information can be represented only by the presenceof the carrier (denoted by a 1 or ON period) or the absence of thecarrier (denoted by a 0 or OFF period). The coding of the beaconsignals will inevitably introduce 0s in the emitted sequences. Letus now examine the effects of the code on the detection time. If abeacon emits a 1 while it enters into the receiver field of view, thereis no error onr, meaning that r = r. However, if a beacon emitsa 0 while it enters the receiver field of view, there is an error on rbecause no signal produces a 0 1 transition at the receiver output.In fact the transition occurs later (r r), at the next 1. The sameconsideration applies to f, except that the 1 0 transition couldoccur sooner (f f).

Such situations are illustrated in Figure 2. We first representthe output of the receiver for a non modulated carrier, R0. In thatcase there are no errors on the transition times because the beaconconstantly emits 1s. The four other cases represent the output ofthe receiver for four different situations using an arbitrary code. Thefirst case (R1) does not generate errors because Pth is reached twicewhile the beacon emits a 1. The second case (R2) generates anerror on r only because Pth is reached in a 0 period. The thirdcase (R3) generates an error on f only and the fourth case (R4)generates an error on both r and f. From Figure 2, one can see

that the receiver output Ri for an emitted signal Ei is the logicalAND between Ei and R0. Now suppose that the OFF periods of asequence have the same duration, denoted by 0 (this is the case bydesign). The worst case for r occurs when an OFF period starts atan angle = r, delaying the next transition at an angle r = r+0. The same reasoning applies to f when an OFF period beginsat an angle = f0. In both cases, the maximum absolute erroron r or f is equal to 0. These are the worst cases but there aremany combinations of these two errors. The next Section evaluatesthe probability density functions of the estimated angles.

4. ERROR ANALYSIS

In order to determine the underlying statistics of the errors, we haveto analyze the influence of the code on the detection times. In thissection, we determine the probability density functions (PDFs) ofseveral random variables: the angle

rwhen the beacon enters the

field of view of the receiver, the angle f when the beacon leavesthe field of view of the receiver, and the angular position of thebeacon b, according to eq. (6).

4.1 Notations

In the following, we use these notations:

p0, p1: the probability to get a 0 or a 1 respectively at theIR power threshold (rising or falling edge), that is the frequencyof 0 (or 1) symbols in the code. As usual, we have p0 + p1 = 1.

T0: the duration of a blank period (0 symbol) in a code. Theonly requirement in this study is that the blank periods of a codemust all have the same duration.

0: the angle corresponding to the blank period T0

0 = T0. (7)

Note that, in our design, we have: p0 = 1/6, T0 = 30.8 [s], =10 [2/s], and 0 = 0.111 [degree].

The Uniform PDF, used later, is defined as

U(a,b) (x) =

1

ba if ax b,

0 otherwise(8)

whose variance is(ba)2

12 . The symmetric Triangular PDF is

T(a,b) (x) =

2

ba 2|2xab|

(ba)2if a x b,

0 otherwise(9)

1767


3/5

ff()

fb ()

p1

fr() p12

p0

0

2p1p00

b

2p00

r+0r fb +02

f0b02

Figure 3: Probability density functions ofr (left), f (right) and b (center) in the case of independentr and f.

where |x| denotes the absolute value of x and whose variance is(ba)2

24 . The mean of these two PDFs is given bya+b

2 . Note thatwith these notations, and ifba = d c, we have [2, page 137]

U(a,b) (x)U(c,d) (x) = T(a+c,b+d) (x) , (10)

where denotes the convolution product.

4.2 Probability density function ofr

Errors on r originate if a beacon is emitting a 0 symbol whileentering the receivers field of view. Assuming time stationarity andas there is no synchronization between the beacons and the receiver,the probability to determine the correct angle, that is to have r =r, while the beacon enters the field of view of the receiver is givenby p1. On the other hand, when the beacon emits a 0, the value ofr is uniformly distributed between r and r + 0. Therefore, ifwe define (x) as the DIRAC delta function, then the PDF ofr isgiven by

fr () = p1(r) + p0 U(r,r+0) () (11)

for [, ]. The mean and variance ofr are

r = r+ p002, (12)

2r

= p0203p20

204. (13)

4.3 Probability density function off

Because the configuration is symmetric when the source leaves thefield of view of the receiver, a similar result yields for f

ff () = p1

f

+ p0 U(f0,f) () , (14)

for [, ]. The mean and variance off are

f = fp002, (15)

2f

= p0203p20

204. (16)

4.4 Characteristics of the r and f estimators

The PDFs ofr and f are drawn in Figure 3. The expectations

ofr and f have a bias given by p002 respectively (see eq. (12)

and (15)). The bias is proportional to the blank period 0 and theproportion of 0s in a code p0. The variances ofr and f areequal.

4.5 Characteristics of the estimatorb

The aim of the system being to estimate the beacon angular positionb, we are now interested in finding the mean and variance of b.Generally the mean and variance of a random variable are calculatedwith the help of the PDF. In the case ofb, it is not necessary sincethe estimator is a function ofr and f (eq. (6)), whose PDFs areknown. Let us first consider the mean of

b

b =E{r}+E

f

2

=

r+ p0

02

+fp0

02

2

,

=r+f

2= b. (17)

As can be seen, the mean ofb is unbiased, despite that both theentering angle r and leaving angle f estimators are biased. This

justifies the construction of a symmetric receiver and the use of thatestimator.

Let us now derive the variance ofb

2b

= var

r+f

2

=

varr+f

4

. (18)

Ifr and f are uncorrelated, we have [2, page 155]

2b

=var{r}+ var

f

4

=2r

2=

2f

2, (19)

since 2r

= 2f

. This could also have been derived from the PDF

ofb, that is given by, in the case of independent r and f

fb () = p21(b)

+ 2p1p0 U(b02,b+

02

)()

+ p20 T(b02,b+

02

)() . (20)

This result is obtained by convolving the PDFs ofr and f [2,page 136], using eq. (10) and rescaling the result by using these

properties [2]: 1) if Y = X, then fY (y) =1||

fX

y

, and 2)

(x) = 1||

(x). This density is also depicted in Figure 3 (cen-

ter). However, the non correlation or independence ofr and fare questionable in our case as explained below.

To compute eq. (19) and (20), we have assumed that r andf were uncorrelated or independent. But, of course, the codes aredeterministic and not random. In fact, depending on the code andthe field of view fr, it is not certain that an error is possible onr and f simultaneously (because the durations between blankperiods are fixed and known). Depending on the receiver field ofview, the rotating speed and the code, four situations are possible:(1) no error is encountered, (2) an error occurs for r only, (3) anerror occurs for f only, and (4) an error occurs for both angles.

1768


4/5

These remarks show that the r and f variables are linked,and that the nature of the relationship depends on the field of viewand the coding scheme. To establish this relationship, we shouldanalyze, in full details, the four previous cases in function of thefield of view and the different codes. However, the mean ofbis always given by eq. (17), and despites the relationship betweenr and f, b remains unbiased. To the contrary, the variance ofb is no longer given by eq. (19) when r and f are correlated.

Fortunately, it is possible to derive an upper bound for 2b

. We can

expand eq. (18) in [2, page 155]

2b

=2r

+2f

+ 2C{r,f}

4, (21)

where C{r,f} is the covariance ofr andf. Since, the squareof the covariance is upper bounded [2, page 153]

C2{r,f} 2r2f

, (22)

and that 2r

= 2f

, we can give the upper bound of2b

by using

eq. (21) and (22), resulting in

2b 2

r= p0

203p20

204. (23)

5. EXPERIMENTAL RESULTS

The expression of the upper bound of2b

and the exact values for

the variances ofr andf (see eq. (13) and (16)) leads to the sameconclusions that p0 should be kept as low as possible and 0 as shortas possible to minimize the effects of the OOK modulation. In or-der to validate this result, we have performed some simulations andmeasurements. In a practical situation, we have to consider the nat-ural (noise) variance of the system inherent to the complete system,even for a non modulated carrier (perfect case with no blank pe-riods). This noise originates from the quartz jitter, rotation jitter,etc, and, to a larger extend, from the receiver jitter at the 0 1 and1 0 transitions. The variance ofb computed theoretically canbe seen as the power of additional noise induced by the OOK mod-ulation. From a theoretical point of view, it is correct to considerthat both noise are independent and therefore that the total varianceis the sum of the natural noise and the noise induced by the codes.

5.1 Measurements for different codes

In our tests, we measured the variance of the beacon angular po-sition estimator b for different codes and fields of view. Thecodes used are the code 0 (non modulated carrier) to compute thenatural variance of the system and three variations of code 5 withincreasing blank durations (111110111110, 11111001111100,1111100011111000). The three last codes have a zero symbolprobability p0 equal to respectively: 1/6, 2/7, and 3/8, and a blankangle 0 equal to respectively 19.25, 38.5, and 57.75 angle units

(one angle unit represents 36062400 = 0.00577 [degree]). In the follow-

ing, these codes are referred to, respectively, as C0, C5a, C5b, andC5c (details about codes may be found in [3]).

5.2 Modifying the angle of view

The field of view can be modified by changing the distance be-tween the beacon and the receiver or by changing the beacon emit-ted power. For practical reasons, we choose, in our experiments,to modify the emitted power, for a fixed working distance, but wewill plot the measures with respect to the field of view because thevalue of the emitted power has no particular relevance in this study;the receiver has only access to the field of view via the demodulatedsignal and it is not capable to know if the distance or the powerhave been modified. For each code, fifty different emitted powervalues were chosen ranging from about 3 [mW] to 150 [mW] to getan approximately linear increase of the field of view (these power

C0C5aC5bC5c

1000

1050

1100

1150

1200

1040 1060 1080 1100 1120 1160 1180

Meanfieldofview(

0.

00577[degree

])

Mean field of view ofC0 ( 0.00577 [degree])1140

21.66

11.003.21

21.84

11.373.52

30.82

19.7611.82

C5cC5b

C5a

Bias Th Simul Exp

Experimental Simulated

Figure 4: Mean values of the field of view at the receiver: ex-perimental values (left-hand side) and simulated values (right-handside). The table provides, respectively, the theoretical, simulated,

and experimental fields of view biases for C5a, C5b and C5c.

values/fields of view are practical values encountered in our setup).For each code and power value, 1000 angle measurements are takento compute the variance ofb. The field of view w is defined asfr. An estimator of the field of view is given by

w =fr. (24)

The mean ofw is equal to

w = Ef

E{r} ,

= fp002 r+ p0

02 ,

= wp00. (25)

As can be seen, the mean ofw has a bias given by p00. There-fore, the fields of view used to plot the data are the means of thefields of view measured for C0, for each power value (since the biasis null for C0 as p0 = 0). The fields of view are shown in Figure4 (left-hand side). The curves are linear as expected but the biasesobserved in the fields of view are larger than the theoretical biases(the values are given in the table of Figure 4). However they in-crease with p0 and 0 as predicted, and the increments between theexperimental biases are consistent with the theory.

5.3 Simulator

We developed a simulator to evaluate our theory. The three param-eters considered by the simulator are the coding scheme (symbols

and durations), the turret period, and the field of view. The code andthe turret period are known precisely in our experiments. However,to match the reality, the simulator fields of view and natural vari-ance were extracted from the experimental measurements of C0.Simulated and experimental results are thus fully comparable. Thesimulated fields of view and variances are presented respectively inFigure 4 (right-hand side) and Figure 5 (left-hand side). Simulationsconfirm our theoretical results as bounds on variances correspond topredicted bounds and the fields of view have a bias almost equal tovalues predicted by eq. (25) (the numerical values are given in thetable included in Figure 4).

5.4 Measurements ofb

The variances of the measurements are shown in Figure 5 (right-hand side). The upper bound for each code is computed after eq.

1769


5/5

100

200

300

400

500

600

700

1040 1060 1080 1100 1120 1140 1160 1180

Variance(0.00

00333[degree2])

Mean field of view ofC0 ( 0.00577 [degree])

C0C5aC5bC5c

upper bound for C5aupper bound for C5bupper bound for C5c

Simulated

100

200

300

400

500

600

700

1040 1060 1080 1100 1120 1140 1160 1180

Variance(0.00

00333[degree2])

Mean field of view ofC0 ( 0.00577 [degree])

C0C5aC5bC5c

upper bound for C5aupper bound for C5bupper bound for C5c

Experimental

Figure 5: Variance values of the beacon angular position: simulated values (left-hand side) and experimental values (right-hand side).

(13) and added to the natural variance (derived from code C0) foreach power value. From Figure 5, one can see that the variancesare close but not always smaller than the corresponding predictedbounds for each power values, especially for C5a.

The slight discrepancy between the experimental results andtheoretical bound means that one or more hypotheses about the sys-tem or the variance calculus are incorrect. But which ones? Oursimulator provides values for the variances and biases of the fieldof view that matches our theory. This tends to confirm our majorassumptions for the simulator (and the theory), which are: (1) thenatural variance is independent of the variance added by a code and(2) the natural variance is the same whatever the code used. How-ever, a detailed analysis of the receiver hardware shows the presenceof an Automatic Gain Control (AGC) loop between the input andthe demodulator. Typically, the gain is set to a high value when nosignal is present for a long time, resulting in a very noisy first

transition (r in our case). This gain then decreases over time,resulting in sharper transitions (especially the last one, f in ourcase). This characteristic is clearly identifiable from the varianceofr and f for a non modulated signal (C0). Indeed, for C0, thevariance ofr stretches from about 250 to 500 whereas the varianceoff stretches from about 7 to 20. It appears that the gain valuedepends on the past values of the received signal and the durationof blank periods, and this produces a non constant natural varianceover time. If the natural variance evolves over time, we have to con-sider this effect to tighten the agreement between theoretical andpractical results. But it is not trivial to consider this effect becauseit relates to the hardware used. Finally, note that despite that evolu-tions of bothr and f variance are quasi linear, the curve ofb isnot linear with respect to the field of view. This confirms that thereexists a dependency between r and f (this effect is most visiblefor the C5c curve).

6. CONCLUSIONS

This paper analyzes the errors on the measured angles of a position-ing system described in [3]; one of the main issues is that it usesan On-Off Keying modulation mechanism. We propose a statisti-cal estimator for the angular localization of a beacon and show thatthis estimator is unbiased and that its variance is theoretically upper

bounded by p0203 p

20204 . This variance represents the power noise

due to the OOK modulation; it increases with the blank angle0 and

with the proportion of zero symbols in a code p0 (if p0

robot positioning system based on a rotating receiver

Documents