Preparation of Papers for AIAA Technical Conferences

ZipMode: Implementation of a Receiver Localization System using ASTRA Satellites Markus Gross* and Guy Harles† and Georges Krier‡ SES ASTRA SA, Château de Betzdorf, 6815 Betzdorf, Luxembourg Thomas Nicolay§ Rohde & Schwarz GmbH Co. KG, Mühldorfstrasse. 15, 81671 Munich, Germany

The authors present the implementation of a new localization system for satellite receivers. The system uses two geostationary collocated satellites to measure the range difference to a stationary DVB-S receiver. Algorithms and hardware implementation of a prototype system are introduced. Measurements and field tests are performed to prove the feasibility of the system.

I. ∆ρ ρ λi c c[n] di h0 n ui Sj,x

r b r dx r vi r x r x0 λ C H J

A. = = = = = = = = = =

Nomenclature

Scalars range difference pseudo range ith eigenvalue of the covariance matrix C speed of light Counter value for reception of packet n Distance from Satellite i to the user Height over reference geoid Index of DVB-S ranging packets Distance from uplink i station to Satellite i X component of the jth Satellite position

B. Vectors = Measurement residuals = Variation of the assumed user position x0 = ith eigenvector of the covariance matrix C = User position = Assumed user position C. = = = =

Matrices Variance matrix Covariance Matrix Visibility Matrix Transformation Matrix from Cartesian to Spherical coordinates

*

Computer and Communication Engineer, Ground Systems and Solutions Senior Earth Station Engineer, Ground Systems and Solutions ‡ Senior Flight Dynamics Engineer, Flight Dynamics Section § Head of Development, Radiocommunications Division †

1 American Institute of Aeronautics and Astronautics

II. Introduction he development of the Digital Video Broadcasting1 (DVB) standard in the 90s provided not only better viewing quality but also paved the way for new techniques like high definition TV2 (HDTV) and interactive Television3 (iTV). In particular iTV, which enables the spectator to contribute to the distributed TV content, delivers added value services for end-users. Location based services (LBS), like automatic language selection or regional newsletters, enhances iTV applications. For commercial integration of location based services in iTV the need for a cost efficient localization system which is easily embeddable in satellite receivers has triggered the development of a Geostationary Colocated Satellite Positioning (GCSP) system4. Such a system uses geostationary TV satellites in colocated orbits to determine the position of DVB-S (Digital Video Broadcast – Satellite) satellite receivers. Range differences from the satellites to the user are measured by the satellite receiver. In the framework of the SATMODE5 project of the European Space Agency (ESA), SES ASTRA has developed, implemented, and tested a GCSP system called ZipMode. Purpose of the ZipMode system is to perform georegioning and offer a position equivalent to a zip code. This paper presents the algorithms, the implementation, and the results of a prototype implementation.

T

III. Positioning using colocated satellites The GCSP proposed in Ref. 4 is using colocated geostationary satellites6,7 for position estimation of a stationary DVB-S receiver. Precise satellite orbits and at least three independent measurements are required for threedimensional localization. Roundtrip delay measurements are performed by inserting dedicated ranging packets into the MPEG (Motion Picture Expert Group) transport stream (TS) and measuring the transmission- and reception time. As the receiver clock and the transmitter clock are not time synchronized, a clock-offset toff between both stations occurs. Due to the clock offset the delay measurement results in a pseudo range ρi+c*toff. By computing the range difference ∆ρ, the offset can be eliminated4.

∆ρ [n] = ρ 2 [n] + c ⋅ t off − (ρ1 [n] + c ⋅ t off ) = ρ 2 [n] − ρ1 [n]

(1)

To get a sufficient number of independent measurement equations the system takes advantage of the relative motion of the satellites. If the satellite receiver is assumed stationary, only two colocated satellites8 are needed to get a sufficient number of equations. Range difference equations measured at different times are independent as the satellites constellation is changing over time. So multiple measurements can be performed with the same satellite pair to localize the receiver. The special geometry of colocated geostationary satellites has an important impact on the accuracy of the system. The Dilution of Precision9 (DOP) can be used as metric for the quality of the satellite constellation. It is depending on the satellite constellation, the user position and the number of measurements taken used for position estimation. Table 1 presents the DOP for the satellites ASTRA1G and ASTRA1H using 60 range measurements for position estimation. The ZipMode system is assuming that the receiver is on the surface of the earth. Without this height assumption the Vertical Dilution of Precision (VDOP) of the system is extremely bad and in consequence the Position Dilution of Precision (PDOP) is also bad. Using the height assumption, the Horizontal Dilution of Precision (HDOP) is the most restricting factor for the accuracy. Without the height assumption positioning is impossible.

Table 1: Contribution of the height assumption

VDOP

HDOP

PDOP

Without height assumption

4664 km

5.3 km

4664.3 km

With height assumption

0.1 km

5.7 km

5.7 km


IV. ZipMode Infrastructure The ZipMode system (Fig.1) consists of the ground segment, the space segment and the user segment. The ground segment comprises all equipment required for operating the ZipMode system. A ground station generates the ranging packets. These are dedicated MPEG packets with which the transmission time is measured. The ranging packets are transmitted via two DVB-S links to the satellites. A precise orbital determination system10 tracks the position of the two satellites and offers orbit data to the position estimation. The position estimation is located at the ground segment and computes the receiver position based on the range difference measurements performed by the receiver. Performing the position computation at the ground segment reduces the necessary processing power at the receivers and enables a low-cost implementation. The space segment comprises the two satellites ASTRA1G and ASTRA1H. These space crafts are colocated in one station keeping box, so reception of both with one common satellite dish is possible. The satellites are broadcasting the signals of the ground segment to the user segment. The user segment consist of a multitude of ZipMode enabled receivers.. These receivers have additional GCSP hardware to measure the reception time of incoming ranging packets. Based on the reception time the range difference to the satellites is computed. The receiver uses the return channel provided by the iTV application to send the measurement data to the ground segment.

Figure 1: ZipMode System overview Figure 2 represents the implementation of the time-stamping at the ground segment. Ranging packets (RP) are generated each second by a Test Data Generator (TDG). The ranging packets are connected via an ASI (Asynchronous Serial Interface) link to a multiplexer. The multiplexer distributes the RP and additional MPEG payload on two different Transport Streams (TS). DVB-S modulation of both signals is performed (Parameters: 22Msymb/s, CR: 5/6). The DVB-S signal is converted to the uplink frequency and transmitted to the satellites. The transmitted RF signal is coupled out after up-conversion and deployed to a Dual Ranging Receiver11 (DRR). The DRR consists of two Ranging Receivers (RR). Each RR measures the transmission time of the ranging packets by sampling the incoming signal and performing correlations. Table 2 presents the accuracy of the RR. The mean error of the system is constant and can be eliminated by a calibration process. Precise time stamping is possible by small standard deviations of 0.14 meters. Table 2: Accuracy of the DRR Range [m] Mean error 15.78 0.142 1σ – raw data 3 American Institute of Aeronautics and Astronautics

Figure 2: Uplink Time-stamping system at the Ground Segment

As the uplink time is measured after the DVB-S modulation, the stochastic delays of the multiplexer (MUX) and modulator (MOD) have no impact on the measurement result. The uplink timestamp, measured by the RR, is send to the Ranging Information Inserter (RII). This device modifies the payload of the ranging packets generated by the TDG. It inserts the uplink time of the previous packet into the payload of the actual packet. So uplink time is transmitted by the ranging packets to the user segment. In addition the RII can be used to transmit the satellite orbits or other commands to the receiver. The method using DVB payload signals for delay measurements occupies only a small data bandwidth (188 bytes/s) and has the advantage that no additional interfering signals are transmitted by the transponder.

Figure 3: ZipMode enabled receiver 4 American Institute of Aeronautics and Astronautics

Figure 3 represents the setup of a ZipMode enabled receiver. The receiver uses a standard satellite dish with Low Noise Block (LNB) for reception of the colocated satellites. The receiver itself consist of a tuner and a DVB-S demodulator which provides the TS and the packet start indicator (PAC). Within the GCSP hardware a free running clock is realized by a 32 bit counter. The counter is driven by a local 27 MHz low-cost crystal oscillator. A PID (Packet Identifier) filter monitors the incoming TS for ranging packets. If a trigger occurs the local clock is latched and the payload of the ranging packet is stored in a FIFO (First In - First Out). A microcontroller (µC) communicates with a PC by RS232 and controls the receiver via I2C bus. If a ranging packet is received, the µC provides the transponder number, the measured packet timestamp and the payload of the ranging packet to the PC. The PC performs data collection, range difference computation and position estimation. The ZipMode system uses the existing receiver hardware to measure the reception time and avoids additional tuner and correlation structures at the receiver. The implementation of the ZipMode prototype is represented in figure 4. The prototype consists of a power supply, a receiver board and the GCSP hardware board. A combined tuner and demodulator, the Fujitsu MB86A15, is used as DVB-S receiver. The GCSP hardware consists of a FIFO, a microcontroller and a CPLD (Complex Programmable Logic Device). The CPLD is a configurable logic device which implements the counter, the PID filter, and the latch. Figure 4: Prototype implementation of a ZipMode receiver But since this low-cost implementation performs reception time measurements after the DVB-S demodulation, the delays within the demodulator have a significant influence on the accuracy. The delays within the receiver are shown in Figure 5. The distribution of the delay shows clearly five equidistant separated levels of delay with a width of 20 ns for each level. The maximum delay difference is approximately 130 ns peak-peak which is equivalent to a range error of 39 meters. As the system is sensitive to noise due to the unfavorable satellite geometry, these stochastic delays reduce the accuracy of the system significantly. This degradation of accuracy is compensated by taking a multitude of range difference measurements.

a) b) Figure 5: a) Delays within the DVB-S receiver b) Distribution of the delays within the DVB-S receiver 5 American Institute of Aeronautics and Astronautics

V. Algorithm implementation For the localization of a satellite receiver three different algorithms must be performed. The algorithm for the measurement of the range difference is executed every 20 seconds. In between the measurements a second algorithm estimates the frequency of the local oscillator to stabilize the receiver clock. If sufficient range difference values are collected, a third algorithm is performs a position estimation based on the measurements and computes an error ellipsoid. A. Range difference measurement Range difference measurements are performed by subtracting pseudo range measurements from different satellites. Normally satellite receivers are equipped with a single tuner. As the receiver can receive only one satellite at a time simultaneous measurement of both pseudo ranges is not possible. So ZipMode measures the pseudo range of both satellites consecutively. The micro controller (µC) measures the reception time of the actual satellite. Then it changes the transponder to the second satellite and measures the reception time of the second satellite. The transmission time of both satellites is transmitted in the payload of the ranging packets to the receiver. The range difference ∆ρ can be computed knowing the transmission time tTX, the frequency of the crystal oscillator fosc, and the counter c[n] value at the reception time:

⎛ (c[n] − c[n − 1]) ⎞ ∆ρ − c ⋅ (t Re c [n] + t Re c [n − 1]) = c ⋅ ⎜⎜ − t TX 2 [n] + t TX 1 [n − 1]⎟⎟ f osc [n] ⎝ ⎠

(2)

Equation (2) shows that range difference measurement is perturbed by the stochastic delays tRec during the DVB-S demodulation. According to figure 4 the maximum value for tRec is ± 65ns.

B. Frequency estimation For range difference computation in Eq. (2) the frequency fosc of the local oscillator (LO) is required. As the pseudo range measurements are performed consecutively, the time between both measurements is approximately 1 second. The drift of the receiver clock during this period causes a range difference error. A frequency offset of 1Hz results in a range difference error of 11 meters. The frequency of the applied low-cost oscillator must be computed continuously as the frequency varies over time. Consecutive packets received from the same satellite are used for frequency estimation. The receiver measures the number of clock cycles between two packets. The known transmission time tTX1 is used as time reference. Reformulation of Eq. (2) for receptions of packets from the same satellite gives:

⎛ (c[n] − c[n − 1]) ⎞ ∆ρ ' ⎟⎟ = ⎜⎜ − (t Re c [n] + t Re c [n − 1]) + t TX 1 [n ] − tTX 1 [n − 1] c f osc [n ] ⎠ ⎝

(3)

Were ∆ρ‘ represents the change of the pseudo range ρ1 between the reception of both packets.

∆ρ ' = ρ1 [n] − ρ1 [n − 1]

(4)

Equation (3) can be used to estimate the frequency fosc of the local oscillator. The frequency measurements are disturbed by the stochastic delays tRec of the DVB-S demodulator. To reduce these perturbations a low pass filter is applied on the frequency measurements. In the actual implementation an averaging filter with a span of 60 tabs is used. The averaging filter was chosen to minimize the required processing power at the receiver. Figure 6 presents the distribution of the frequency error of the measured frequency. The error without averaging is maximum 3.5 Hz; with averaging the frequency accuracy is better than 1.5 Hz.


Distribution of frequency error (Lab setup) 12

before averaging after averaging

Probability P[%]

10

8

6

4

2

0 -4

-3

-2

-1

0

1

2

3

4

Frequency error [Hz]

Figure 6: Distribution of the frequency error Without averaging (blue line); with averaging (red line) As the satellites motion relative to the observer is slow, the change of the pseudo range is also small (∆ρ‘ < 3 m). So further simplifications can be applied on Eq. (3). Neglecting ∆ρ‘ the measured frequency only depends on the measured transmission time, counter values, and the stochastic delays of the demodulator.

(c[n] − c[n − 1]) = t [n] − t [n − 1] + t [n] − t [n − 1] TX 1 TX 1 Re c Re c f osc [n]

(5)

(c[n] − c[n − 1]) t TX 1 [n ] − t TX 1 [n − 1] + t Re c [n] − t Re c [n − 1]

(6)

f osc [n] =

This simplification results in a frequency error which is analyzed in Figure 7. The absolute frequency error is smaller than 0.25 Hz; the relative frequency error is smaller than 2*10-8. The simplified frequency estimation algorithm is implemented, as the frequency accuracy is sufficient.

Figure 7: Frequency error due to the simplified frequency estimation algorithm 7 American Institute of Aeronautics and Astronautics

C. Positioning estimation The position estimation algorithm computes the receiver position, using an iterative least square fit. Calculations are based on the known uplink positions TX1, TX 2, the positions of the satellites S1,, S 2 and a set of n range differences ∆ρi (Fig. 8).

Figure 8: Measurement setup We can describe a set of n range difference measurements according to Eq. (1) with:

v r v r ∆ρi = S2i − x + u2 − S1i − x − u1

i = 1...n

(7)

Equation 7 describes a set of nonlinear equations representing the range difference measurements. We propose an iterative least square algorithm for solving this nonlinear problem. The algorithm performs a linearization of the r r equations by a first order Taylor approximation around a guessed user position x 0 and refines the position of x 0 during each iteration step.

r r r x = x0 + d x

(8)

The linearization of Eq. (7) is (see Appendix):

r r ri r S2i − xr0 r ri r S1i − xr0 r ∆ρ = S2 − x0 + ri r ⋅ dx + u2 − S1 − x0 − ri r ⋅ dx − u1 S2 − x0 S1 − x0 i

(9)

r

We can rearrange this equation to separate d x :

ri r ⎞ ⎛ Sr i − xr ⎜ 2 0 − S1 − x0 ⎟ ⋅ drx = ∆ρ i − Sr i − xr − u + Sr i − xr + u ri r ⎟ 2 0 2 1 0 1 ⎜⎜ Sr i − xr 144444 42444444 3 S − x0 ⎟ 2 0 1 ⎝144424443⎠ bi

(

)

Hi


(10)

To enhance the accuracy of the system, especially in vertical direction, we assume the receiver on the surface of the earth. This height information can be represented by:

r x = h + r⊕ (θ )

(11)

h is the assumed height above the reference ellipsoid and r⊕ (θ ) is a function of the latitude θ that gives the distance from the center of the earth to the surface of the reference geoid at the assumed point x, It equals the equatorial radius R⊕ for zero latitude and (1 − f )R⊕ for a latitude of 90°. The WGS8412 (World Geodetic System 1984) model is used as geoid: R⊕ = 6378137 m , flattening factor f = 1 298.257 . Equation (11) can be linearised to:

r r r x0 r ∂r⊕ ( x ) r x0 + r ⋅ d x = h + r⊕ (θ 0 ) + r ⋅ dx x0 ∂x

(12)

r

Separating dx , we get the linear equation (when neglecting the last term in Eq. (12)):

r r x0 r r ⋅ dx = h + r⊕ (θ 0 ) − x0 144244 3 x0 { b0

(13)

H0

We can combine the range difference Eq. (10) and the height Eq. (13) in one linear system of equations. As range difference measurements and height information have different variances relative weights are applied on each equation.

⎛ H 0 σ 02 ⎞ ⎜ ⎟ ⎜ H1 σ 12 ⎟ r 2 ⎜ M ⎟ ⋅ dx = b0 σ 0 ⎜ ⎟ ⎜H σ 2 ⎟ ⎝ n n⎠

(

b1 σ 12 L bn σ n2

)

(14)

The standard deviations of the range difference σ1… σn are defined by the user. For the used receiver the typical value is 20 m. The assumed standard deviation σ0 of the height is 100 m. We can write above in a compact matrix equation by combining standard deviations in a separate variance matrix λ = Diag(σ12… σn2).

(λ

−1

)

r r −1 H ⋅ dx = λ ⋅ b

(15)

r

Equation 15 is overdetermined for n > 2 and is solved for dx using the pseudo-inverse H- according to Ref. 9:

(

)

r r −1 −1 −1 T T dx = H λ H ⋅ H λ ⋅ b 144424443 H

(16)

−

r With Eq. 8 we can update the assumed user position x 0 . The next iteration step is performed with the new guessed user position. The iteration process will be interrupted if the change between two iteration steps is sufficiently small.


D. Error ellipsoid: An error ellipsoid can give information about the reliability of position estimation. The error ellipsoid is computed based on the covariant matrix:

(

r r T −1 cov(dx , dx ) = H λ H

)

−1

⎛ var( x) cov( x, y ) cov( x, z ) ⎞ ⎜ ⎟ = ⎜ cov( y, x) var( y ) cov( y, x) ⎟ ⎜ cov( z , x) cov( z , y ) var( z ) ⎟⎠ ⎝

(17)

As the computations are performed in a Cartesian earth-fixed frame, the variances are also given in the Cartesian coordinates. The interpretation in Cartesian coordinates is difficult, while a representation of the error ellipsoid on a geographic map is more intuitive for the user. Therefore a coordinate transformation to spherical coordinates is performed.

⎛ρ⎞ r ⎜ ⎟ ds = ⎜ φ ⎟ ⎜θ ⎟ ⎝ ⎠ ( ρ radius,

(18)

φ

longitude, θ latitude)

The transformation from Cartesian to spherical coordinates is:

r r r ∂s r ds = J ⋅ dx = r ⋅ dx ∂x

(19)

With:

⎛ x ⎜ ⎜ ρ ⎜ xz J =⎜ 2 ⎜ ρ x2 + y2 ⎜ −y ⎜ 2 2 ⎝ x +y

y

ρ yz

ρ

⎞ ⎟ ⎟ ρ 2 2 ⎟ − (x + y ) ⎟ ρ 2 x2 + y2 ⎟ ⎟ 0 ⎟ ⎠ z

x2 + y2 x 2 x + y2

2

(20)

The covariance matrix in spherical coordinates is:

r r r r cov(ds , ds ) = J cov(dx , dx )J T

(21)

The height information contributed by the range difference measurements is negligible small, such that the height is decoupled from the latitude and longitude.

0 0 ⎛ var( ρ ) ⎞ ⎟ r r ⎜ cov(ds , ds ) = ⎜ 0 cov(φ , φ ) cov(φ , θ )⎟ ⎜ 0 cov(θ , φ ) cov(θ , θ )⎟⎠ ⎝


(22)

The two dimensional error ellipsoid is given by the covariance matrix C of latitude and longitude:

⎛ cov(φ , φ ) cov(φ ,θ )⎞ ⎟⎟ C = ⎜⎜ ⎝ cov(θ , φ ) cov(θ ,θ )⎠

(23)

The ellipsoid can be described by the major- and the minor axes σ1 and σ2. Orientation and length of the axes is given by the corresponding eigenvalues λi and eigenvectors vi of the covariance matrix C:

r r C ⋅ v i = λ i ⋅ vi

i = 1,2

(24)

The axes of the error ellipsoid are given by:

r vi σ i = λi ⋅ r vi

(25)

The resulting ellipsoid is centered at the origin and must be translated to the user position. The error ellipsoid gives a prediction of the error probability depending on the satellite constellations, the user position and the measurement noise. We assume that the variance of the white Gaussian mean-free noise is well known. Then the error ellipsoid represents the 1 sigma probability line.

VI. Result To prove the receiver localization concept, initial measurements have been performed. The results in figure 9 show that positioning is possible. The distance between the exact (blue circle) and the estimated position (red plus) is 1.2 km. The error predicted by the error ellipsoid gives a good guess of the real error.

Figure 9: Position estimation Geographic map of the estimated position (red plus) with error ellipsoid and exact position (blue circle) 11 American Institute of Aeronautics and Astronautics

In addition extended field test have been performed with the proposed implementation. Multiple measurements have been collected at more than 20 different sites (Fig. 10) distributed over the whole satellite footprint. The duration of each measurement varies between 6 and 48 hours. Range differences have been taken every 20 seconds. The distribution of the position error in figure 11 proves that positioning within the complete footprint of the satellites is possible. The field test shows that the Circular Error Probability13 (CEP50) of the system is 1.5 km. The average error is 2.8 km.

Figure 10: Sites of the extended field tests

Figure 11: Distribution of the position error


VII. Discussion Existing Global Navigation Satellite Systems (GNSS) can offer accurate and instantaneous localization. Hence GNSS systems are designed for outdoor localization, indoor positioning is difficult. The low power level of the GPS signals and multi path are the main problems for indoor localization. Therefore the localization of satellite receivers, which are normally are placed indoors, with normal GPS receivers is not possible14. In consequence highly sensitive GPS systems must be used or outdoor antennas must be installed by the user. As the costs of these systems are too high for commercial applications, these systems might be not feasible for such applications. Instead the proposed ZipMode system can be implemented cheaply and needs only small changes in the receivers. The installation process needs only the exchange of the existing receiver by a ZipMode enabled one. As the system uses the existing satellite antenna no additional changes at the user site are needed. The operator of the system can use the existing satellite infrastructure and only a DRR time stamping instance must be installed. As the system uses DVB-S payload signals for positioning, no additional interfering signals are transmitted and only a negligibly small data bandwidth is occupied by the ranging packets. The ZipMode localization service is available in the complete overlap of both satellite footprints and is totally independent from other systems. The accuracy of the system is 1.5 km (CEP 50). This is fully sufficient for the targeted applications like regional TV broadcast, automatic language selection or local advertisement.

Appendix A. Linearization of a range difference equation: The non-linear range difference equation in Eq. (7) can be linearised using a first order Taylor approximation around the point x0:

v r r v r ∆ρ = f (x) = S2 − x + u2 − S1 − x − u1

(26)

The Taylor approximation of f(x) is:

()

r r r r ∂f d r f (x ) = f (x0 ) + ∂d

r ⋅ dx

(27)

r r d = x0

We can compute the derivative:

(

) (

) (

r r r r v v v v r S d u S d u S d ∂ S − d ∂ − ∂ − + − − − 2 2 1 1 2 1 ∂f (d ) r r = r r − = ∂d ∂d ∂d ∂d

)

(28)

With:

(

)

r r r v v v r v ∂⎛⎜ ( S j , x − d x ) 2 + ( S j , y − d y ) 2 + ( S j , z − d z ) 2 ⎞⎟ ∂ Sj −d ⎠ j = 1,2 r r = ⎝ ∂d ∂d r v ⎛ 2 S j,x − d x ⎞ ⎜ v r ⎟ 1/ 2 = r 2 r 2 r 2 ⎜ 2 Sv j , y − dr y ⎟ = v v v ( S j , x − d x ) + ( S j , y − d y ) + ( S j , z − d z ) ⎜⎜ 2 S − d ⎟⎟ j,z z ⎠ ⎝

( ( (

) (Sv ) v ) S


j j

r −d r −d

)

(29)

Inserting Eq. (29) in Eq. (28) we get:

r r r v v ∂f (d ) S 2 − d S1 − d r = v r − v r ∂d S 2 − d S1 − d

(30)

In addition the range difference for the initial point x0 is:

v r v r r f ( x0 ) = S 2 − x 0 + u 2 − S1 − x0 − u1

(31)

Inserting Eq. (27) and Eq. (30) in (31) we get:

⎛ v r v r r r ⎜ ∆ρ = f ( x0 + dx ) ≈ S 2 − x 0 + u 2 − S1 − x0 − u1 + ⎜ ⎜ ⎝

r v v r S2 − d S1 − d r − v r v S2 − d S1 − d

⎞ ⎟ r ⎟⎟ ⋅ dx ⎠

(32)

Rearranging the equation we get:

⎛ ⎜ ⎜⎜ ⎝

r v v r S2 − d S1 − d r − v r v S2 − d S1 − d

⎞ v r v r ⎟ r ⋅ = ∆ − − + − d x ρ S x u S 2 0 2 1 − x 0 − u1 ⎟⎟ ⎠

(

)

(33)

Acknowledgments This work was supported by SES ASTRA and LIASIT (Luxembourg International Advanced Studies in Information Technologies). The authors also want to thank Edward Cardew for his assistance with the hardware implementation.

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