The combination of independent estimates

It is shown how best to combine two independent estimates of a variable to forma weighted mean value, this operation being central to the process of Kalmanfiltering [1,2]. The following development assumes only a knowledge by the readerof elementary statistical principles. A full mathematical derivation of the Kalmanfilter is beyond the intended scope of this book. Therefore, the reader interested in amathematical treatise on the subject is referred to the excellent text by Jazwinski [3].

The single-dimension case

Consider the situation in which two independent estimates, JCI and JC2, are provided ofa quantity x, where o\ and o\ are their respective variances. It is required to combinethe two estimates to form a weighted mean, corresponding to the 'best', or minimumvariance, estimate, x. In general, the weighted mean may be expressed as:

jc = w\x\ + W2X2 (A.I)

where w\ and W2 are the weighting factors and w\ + W2 = 1. The expected or meanvalue of x, written as E(x), is given by:

E(x) = W1E(Xi) + W2E(X2) (A.2)

The variance of a quantity x is defined to be E[{x — Zs (JC)}2]. Hence the variance of x,denoted a2, may be written as:

(A.3)

Appendix A

Kalman filtering

Since x\ and X2 are independent, {x\ — E(x\)) and (X2 — E(xi)) are uncorrelated,.CI(JCi — E(x\))(x2 ~ E(x2))} = 0. Hence a2 can be expressed as follows:

(A.4)

Writing W2 = w and w\ = 1 — w, the variance o2 may be expressed as:

(A.5)

The value of w, which minimises cr2, is obtained by differentiating the aboveequation with respect to w. Hence,

— - = -2(1 - w)ai + 2wo\ = 0dw;

which yields the optimum weighting factor as:

(A.6)

Substitution in eqns. (A.I) and (A.5) gives x and its variance a2:

(A.7)

(A.8)

By following this process, the two independent estimates, x\ and X2, have beencombined to form a weighted mean value, in which the weighting factor has beenselected to yield a mean with minimum variance, and hence, maximum probability.In a Kalman filter, one such estimate is usually provided by updating a previousbest estimate in accordance with the known equations of motion, whilst the otheris obtained from a measurement. If X2 is taken to be a measurement which is usedto improve an updated estimate x\, the above equations can be expressed in thefollowing form:

(A.9)

(A.10)

which shows how the estimate (x\) and its variance (<ri) are improved by themeasurement (X2). This derivation is now generalised to the multi-dimensional formnecessary for a full Kalman filter implementation.

The multi-dimensional case

Consider now the situation in which Xi and X2 are n -element vectors representingtwo independent estimates of an n-dimensional vector quantity x. The variances ofxi and X2 are represented by the two n x n matrices, Pi and P2, respectively.

The weighted mean of Xi and X2 may be expressed in the same form as presentedearlier for the single dimension case, viz:

x = (I - W)xi + Wx2

= X 1 - W ( X 1 - X 2 ) (A.ll)

where W is an n x n weighting matrix and I is a unit matrix. The best estimate of x,denoted x, will be provided by the above equation when W is selected to minimisethe variance ofx.

In most practical cases, the dimensions of the two estimates are not equal andone of them is often a function of the individual elements of x. For example, a setof m measurements may be provided, denoted y2, where y2 is related only to someof the elements of x. In this situation, the relationship between y2 and X2 may beexpressed as follows:

Y2 = Hx2 (A.12)

where H i s a n m x w matrix.It is, therefore, required to form an optimum estimate ofx from one estimate X1

of variance P1 and a second estimate y2(= Hx2), with variance denoted here by thesymbol R. If we now let the weighting matrix W = KH, where K is another arbitraryweighting matrix, then:

(A.13)

By definition, the variance (P) ofx is given by:

P = E{[x - E(X)M ~ E(X)?) (A.14)

Similar expressions can be written for the variances Pi and R. Substituting for xfrom eqn. (A.13) yields:

but since xi and y2 are uncorrelated, this reduces to:

(A.15)

It is now required to find the value of K which minimises P in the sense that thediagonal elements of P, the variances of x, are minimised.

It is shown in Reference 1 that this is achieved when

K = PiHT [HPiH1 + R]" 1 (A. 16)

Under such conditions, the best estimate of x is given by:

X = X 1 - K [ H x 1 - Y 2 ] (A.17)

and its variance is given by:

P = P1-KHP1 (A. 18)

where K takes the value given by eqn. A. 16. The weighting process, defined byeqns. (A.16)-(A.18), is implemented in a Kalman filter as discussed in the followingsection.

The Kalman filter

In this section, we begin by considering the application of Kalman filtering to linearsystems, before moving on to show the extensions which are necessary to overcomeany system non-linearity. It is noted that the algorithms presented below are applicableto systems which are linear and time varying, not simply constrained to constantparameter systems. The explicit dependence of some of the parameters with time hasbeen omitted to aid the clarity of the development.

Linear systems

The dynamical behaviour of a linear system may be represented by a set of first orderdifferential equations of the form

— = Fx + Gu + Dw (A. 19)at

where the elements of the n -vector, x(t), are called the states of the system, u(f) is ap-vector of deterministic inputs and w(t) is the system noise. F is an n x n matrix,known as the system matrix and G is an n x p system input matrix. F, G and Dare constant or time-varying matrices. The noise w(f) has zero mean and is normallydistributed (Gaussian), with a power spectral density of Q.

Let us now assume that there are m measurements of the system, which are alinear combination of the states, but are corrupted with noise. This can be expressedin terms of the system states by the following equation:

y = Hx + n (A.20)

Here, the m-vector, y(t), is called the measurement vector and H is an m x nmeasurement matrix. n(t) represents the measurement noise, which also has zeromean and is normally distributed, with power spectral density R.

Figure A. 1 Block diagram of a Kalman filter

The Kalman filter for the system described here seeks to provide the best estimatesof the states, x, using:

• the measurements, y;• a model of the system provided by the matrices F, G, H, and D;• knowledge of the system and measurement statistics given in the matrices

Q and R.

The deterministic or measurable inputs are processed by both the system and themodel of the system, as shown in the block diagram representation in Figure A.I.

The measurements of the true system are compared with predictions of thosemeasurements, derived from the latest best estimates of the states provided by thesystem model. The difference between the true and predicted measurements are fedback through a weighting matrix, the Kalman gain matrix, to correct the estimatedstates of the model.

The Kalman gains are selected to provide best estimates of the states in a least-squares sense. It can be shown that this is equivalent to the best estimate in themaximum likelihood sense in the linear, Gaussian noise system described above. Itshould be noted that, it is because there is a feedback of a noisy signal in the Kalmanfilter that the system must be linear and the noise Gaussian, or normally, distributed.It is only because the distribution of the sum of two normally distributed signals isitself normally distributed, and also because a normally distributed signal remainsso after passing through a linear system, that a least-squares or maximum likelihoodestimation procedure can be applied repeatedly.

Whilst it is usual for the system to be described mathematically in the continuousdifferential equation form given above, the measurements are in practice provided atdiscrete intervals of time. To cope with this, and to provide a computationally efficientfiltering algorithm, it is customary to express the continuous equations in the form of

Random inputs

Known inputs, U Linear systemMeasurements

Kalmangains

Linear model Predictions

difference equations as shown below:

x*+i = <&*** + r*u* + Ajt w* (A.21)

with measurements:

y*+i = Hjt+i Xit+i + njt+i (A.22)

where x^ is the state at time tk, u& is the input at time tk, w^ is the system noise attime tk, n*+i is the measurement noise at time tk+i, ^k is the state transition matrixfrom time tk to time f^+i, H^+i is the measurement matrix calculated at time ^ + 1 ,and Fk and Ak are appropriate input matrices.

The noise is zero mean, but now discrete, and will be characterised by thecovariance matrices Q^ and R^, respectively.

These equations are used to formulate a recursive filtering algorithm. In sucha formulation, it is necessary to consider two distinct sets of equations. The first setis concerned with the prediction of the state of the system based on the previousbest estimate, whilst the second involves the updating of the predicted best estimateby combining the prediction with a new measurement.

The prediction process

The best estimate of the state at time tk is denoted here by Xk/k- Since the systemnoise, w&, has zero mean, the best prediction of the state at time ^ + 1 is given by:

x*+i/* = ®k*k/k (A.23)

whilst the expected value of the covariance at time tk+1 predicted at time tk, is given by:

P*+i/* = < № / * * * + AitQjfc Aj (A.24)

The measurement update

On arrival of a new measurement y^+i, at time tk+\, it is compared with the predictionof that measurement derived from the system model. The measurement is then usedto update the prediction to generate a best estimate, following the procedure outlinedin the previous section. Hence, the best estimate of the state at time ^ + 1 is given by:

x*+i/*+i = XjH-i/jfc - Kjfc+i[H*+iXjk+i/jfc - y*+i] (A.25)

and its covariance by:

P*+l/iH-l = P*+l/* - Kfc+iHfc+iPfc+i/£ (A.26)

where the Kalman gain matrix is given by:

K*+i = P*+i/*Hj+1 [H*+iPik+iAHj+1 + R*+!]-1 (A.27)

where HT denotes the transpose of the measurement matrix, H.The system states may therefore be updated each time a measurement is received

by implementing eqns. (A.25)-(A.27).

Non-linear systems - the extended Kalman filter

So far, we have considered only linear dynamical systems with zero mean, Gaussiannoise type disturbances. For such systems, the Kalman filter is optimal in the least-squares sense or maximum likelihood sense. If the system is not linear or if the noiseis not Gaussian, the Kalman filter is no longer optimal. In these cases, the only wayto regain the 'optimality' of the filtering is to design an algorithm specifically forthe system under consideration. This, however, is not usually feasible in practice,as the filter would be of infinite dimension. Therefore, it is normal to accept thatthe performance will be sub-optimal, and use the Kalman filter in such a way as tomake the performance as close to optimal as possible. This may involve, for instance,predicting the system and its covariance matrix over relatively short time intervalsduring which the conditions for linearity hold.

Consider a continuous non-linear dynamical system described by the equations:

(A.28)

with discrete measurements given by:

y = h(x,r)x + v (A.29)

For simplicity, the explicit dependence on time will be dropped, that is, f(x, t) willbe written as f(x). This system may be approximated by linearising it about a nominalset of states commonly referred to as a nominal trajectory.

This approximation will only be valid for short periods of time after which thesystem will have to be re-linearised. The linearisation technique most often used isthe truncation of the Taylor series. Thus, for the function f(x), the Taylor series abouta nominal trajectory x is given by:

(A.30)

and similarly for other non-linear functions. Defining the nominal trajectory as:

^ = f (x)x + g(x)u (A.31)

aty = h(x)x (A.32)

and subtracting from the original equations, we obtain differential equationsgoverning the deviations from the nominal trajectory, viz:

(A.33)

(A.34)

If we define:

(A.35)

then the discrete Kalman filter equations may be used as given in the previous section.At each measurement interval, the following steps must now be taken:

1. Linearise the equations about the nominal trajectory. The nominal trajectory isusually taken to be the latest estimate of the states.

2. Calculate the transition matrix and other matrices of the discrete equivalent tothe linearised system.

3. Integrate the state prediction equations. The actual estimated states may be usedhere, as there is no difference between this and integrating the differential equa-tions for the nominal trajectory and the deviations from it separately and thenadding the result.

4. Implement the Kalman filter equations. This will provide best estimates ofthe deviations from the nominal. The corrections, given by the Kalman gainsmultiplied by the measurement differences, can again be added directly to thepredicted state estimates.

5. Continue to the next time interval, that is, return to step 1.

There are situations in which the measurement update rate may be relatively low,in these cases the prediction stage of the filter will need to be run with a smaller timeinterval. If this is not done, the non-linearities begin to dominate the deviations of thestate estimates from the nominal.

Although this form of the Kalman filter, the so-called extended Kalman filter,is used for systems which are explicitly non-linear, it is also used when there is arequirement to identify certain unknown parameters in the system. In this situation,the unknown parameters are defined as states of the system and the state vector isaugmented to include them.

References

1 BARHAM, P.M., and HUMPHRIES, D.E.: 'Derivation of the Kalman filteringequations from elementary statistical principles', NATO Agardograph AG139,'Theory and application of Kalman filters', 1970

2 GELB, A. (Ed.): 'Applied optimal estimation' (Massachusetts Institute ofTechnology, Cambridge, MA, 1974)

3 JAZWINSKI, A.H.: 'Stochastic processes and filtering theory' (Academic Press,1970)

The accuracy of an inertial navigation system is often expressed as a positionuncertainty after a given period of navigation, or on reaching a given destination.Alternatively, it is expressed in terms of the rate at which the navigation error buildsup with time, in nautical miles per hour for instance. The actual form of expressionused for this overall accuracy figure depends upon the application. For example, for aninter-planetary missions, the accuracy refers to the desired point of closest approachto the 'destination' planet. For navigation in the vicinity of the Earth, it is usually theerrors in two dimensions which are of most interest, the along-track and cross-trackposition errors over the surface of the Earth. These errors are often combined to yielda single number which expresses navigation accuracy after a given navigation time,the circular error probable (CEP) or circular probable error (CPE) as it is sometimescalled. Essentially, this defines a circular area within which the navigation systemestimates its true position to be, with a certain probability. The 50 percent CEP isa frequently quoted figure. When the probability value is not stated, it usually meansa 50 percent value should be assumed.

Consider now the composition of this navigation performance figure. In practice,navigation errors propagate owing to a large number of error sources which includealignment errors, a variety of inertial sensor errors and errors attributable to computa-tional inaccuracy. In general, each errors may be regarded as comprising a repeatableor predictable component and a random or unpredictable component. The formercategory produces predictable effects which can be compensated for, if so desired,that is, electrical signals or software corrections can be applied which should offset theeffect of such errors. The remaining errors, which arise as a result of random effectswithin the system and incomplete compensation of systematic errors, will give riseto navigation inaccuracies.

For the assessment of system performance, random errors are treated statisticallyto derive a mathematical description of each term and to allow the various contributorsto the overall system errors to be combined in what is known as the system errorbudget. It is common practice to assume that the random errors within the components

Appendix B

Inertial navigation system error budgets

Figure B.I Gaussian distribution

of an inertial navigation system follow a Gaussian or normal distribution where theprobability density function, denoted px(x), is expressed mathematically as:

A Gaussian distribution is depicted in Figure B. 1.Based on this assumption, the statistical analysis techniques applicable for normal

distributions may be conveniently used for the analysis of inertial systems.If measurements are made of a bias (xi) on the measurement provided by

a gyroscope or an accelerometer, then over a large number of sensor samples, itis assumed that the error distribution approximates to this curve. It is common prac-tice to quote a lcr error or standard deviation for each sensor error. The standarddeviation of a normal distribution may be expressed in terms of the mean error (xm)and the number of samples (n) as follows:

The Ia value represents 68.26 percent of the area under the curve. Hence, if a Iagyroscope bias is quoted as being l°/h, it is expected that over a large number ofgyroscopes of that type, 68.26 percent would have a bias of between ±l°/h of themean value. Alternatively, it may be inferred there is a 68.26 percent probability ofthe bias uncertainty on a given gyroscope being within ±l°/h. The 2a and 3a biasvalues which are obtained by multiplying the Ia value by 2 and 3, respectively, mayalso be defined for the sensor, corresponding to 95.44 and 99.73 percent of the areaunder the normal distribution curve. Hence, for this example, there is a 95.44 per centprobability of the gyroscope bias being within ±2°/h and a 99.73 percent probabilityof the bias being within ±3°/h.

In an inertial navigation system, a number of sensors and components are requiredto operate together, each producing random errors. It is usual to assume that each errorcomponent is unrelated to each of the other error components, that is, the error com-ponents are said to be independent. In combining the effect of many error sources,simply summing the individual contributions to the error budget arithmetically wouldgive a very pessimistic prediction of system performance. Where a number of inde-pendent sources need to be combined, a more correct prediction of overall systemperformance is obtained by summing the individual Xo errors quadratically, i.e. bytaking the root sum square (RSS) error. Hence, if o\, 02,..., on represent a numberof independent errors which combine to produce an overall error, then the total oroverall error is obtained using:

where <TRSS is the Xo RSS error. In an inertial navigation system, an RSS positionerror can be calculated for both the along-track or down-range position error (ox) andthe cross-track position error (cry). Assuming all error contributions to both along-and cross-track errors to be Gaussian, then the behaviour of each is also describedby a Gaussian distribution curve. When ox and oy are combined, a probability ellipse isused to describe the behaviour of the total error. For the special case where Ox = oy,the probability ellipse reduces to a probability circle. In mathematical terms, the radialprobability distribution is given by:

where 0 is the standard deviation of the error in x and v. This is known asthe Rayleigh probability density function (Figure B.2). The probability of r lying

Figure B. 2 Rayleigh distribution

between O and R, Pr (</?), is given by:

Pr (< R) = 0.5 defines the 50 per cent probability circle which occurs when R/a =1.17741. The radius of this circle is the 50 percent CEP referred to earlier.

Hence,

50 percent CEP= \.Ill Ala

The reader is referred to any standard text on probability theory, Reference 1 forinstance, for a more detailed discussion of the topics outlined in this appendix.

Reference

1 LATHI, B.P.: 'An introduction to random signals and communication theory'(International Textbook Company, 1968)

Practical inertial navigation systems may take a variety of forms. These formsgenerally fall into one of two basic categories,

• stable platform systems;• strapdown systems.

Although the two types of system are very different, both physically and computa-tionally, it must be stressed that the underlying principles and functionality of the twotypes of system are identical.

Stable platform systemsInertial sensors mounted on stabilised platformInertial sensors isolated from rotational motion of vehicle

Strapdown systemsInertial sensors rigidly attached to vehicleInertial sensors subjected to vehicle turn rates

Stable platform systems

The original applications of inertial navigation technology used stable platform tech-niques. At that time, neither sensors with the necessary dynamic range nor sufficientlypowerful computers were available to allow strapdown systems to be produced. At thecore of such a system is a structure, called the platform, on which inertial sensorsare mounted. This platform is 'isolated' from the rotational motion of the vehicleusing a number of gimbals arranged to provide at least three degrees of rotationalfreedom and so minimise the angular coupling between the vehicle and the platform.Generally, three gimbals are used.

Figure C. 1 shows a schematic representation of a three gimbal stabilised platformsystem. The platform containing the inertial sensors is supported by three gimbals.

Appendix C

Inertial system configurations

The gimbals are mechanical frames, each of which is free to rotate about a single-axiswhich is nominally perpendicular to the free axis of its neighbouring gimbal(s). Torquemotors are used to rotate the gimbals with respect to one another and angular pick-offsprovide a measure of their relative orientation. The system is configured so that thethree gimbal pick-off angles correspond to the roll, pitch and yaw orientations of thehost vehicle with respect to the platform/reference frame. Sometimes, it is necessaryto add a fourth gimbal; this is required for very agile vehicle applications to allowthe platform to remain isolated from the vehicle irrespective of its orientation. Forexample, a vertical launch guided weapon test vehicle was produced some yearsago which had a four-axis gimbal system to avoid 'gimbal lock' during the dynamicturn-over manoeuvre.

The platform configuration (Figure C l ) illustrated here minimises the amountof computing needed to implement the navigation function of providing position,velocity and attitude of the host vehicle with respect to the designated navigationreference frame. Since the platform, and hence the accelerometer triad, is held inalignment with the reference frame, typically coincident with the local geographicframe (north, east, down), it is simply required to sum the accelerometer outputswith the gravity terms and to integrate the navigation equations to obtain estimates ofvelocity and position in the reference frame. Any rotational motion of the platformwould be detected by a gyroscope, the output of which would be fed back via a torquemotor to rotate the appropriate gimbal (and hence the platform) in the opposite sense,so maintaining its initial (and fixed) orientation in space. For systems required tonavigate around the Earth, it is required to torque the platform at Earth's rate plus anyrate caused by the velocity of the system with respect to the Earth (transport rate) toallow it to remain aligned with the local level frame.

x accelerometery accelerometerz accelerometer

Resolver

Torque motor

Torque motor Outer gimbal

Torque motor

Inner gimbal

Resolver

x gyroygyroz gyro

Stable platform

Figure C. 1 Three gimbal platform IN system

Resolver

Figure C. 2 Stable platform IN system

It should be noted that platform systems are still in common use, particularlyfor ships and submarines where accurate navigation is required (unaided) over longperiods of time.

A functional block diagram representation of a stable platform system is shown inFigure C.2. This diagram shows the functional blocks which combine to form a stableplatform ES[S. Attention is drawn to the inertial reference unit comprising the inertialsensors mounted on the platform and supported by a set of mechanical gimbals withtheir respective torque motors and angular pick-offs. This may be compared with theequivalent strapdown system configuration in Figure CA A photograph of a stableplatform is given in Figure C.3.

Strapdown systems

In strapdown systems, the inertial sensors are fastened directly (or via anti-vibrationmounts) to the vehicle and hence are not isolated from its angular motion. Conse-quently, the gyroscopes, as well as the accelerometers, experience the full dynamicmotion of the host vehicle. The signals produced by the inertial sensors are resolvedmathematically in a computer prior to the calculation of navigation information. Thisuse of a computer to establish and resolve the inertial data reduces the mechan-ical complexity of the inertial navigation system, thus frequently reducing thecost and size of the system and increasing its reliability. It is stressed that thisreduction in mechanical complexity is achieved at the expense of computationalcomplexity.

Specific forcemeasurementsin reference frame

Position

Velocity

Earth rate +transport rateestimates

Attitude

Heading

Platform inertial reference unit

Figure C. 4 Strapdown IN system

Figure C.4 shows the main functional blocks of a strapdown system for compari-son with the stable platform configuration in Figure C.2. The mechanical complexityof the platform system is seen to be replaced by additional computing tasks in thestrapdown inertial reference unit, viz. the attitude computation and specific forceresolution tasks. Attitude and heading are now computed rather than being provideddirectly by electrical pick-off devices. However, outside of the respective inertial

Figure C. 3 Mar con i stable platform

Specific forcemeasurementsin reference frame

|jj|||||||j||B|||f|||||||j|||S||l||

DCmatrix

Body ratemeasurements

Position

Velocity

Attitude

Headingl l j j§|down inertial reference unit

reference units, the navigation computations are the same, that is, both systems arerequired to solve identical equations.

To summarise, the underlying principles of the stable platform and strapdownsystems are common to both and they perform identical functions. The only differencebetween the two systems, at the functional level discussed here, is the replacementof the mechanical platform with what may be described as an 'analytic platform'.A strapdown system replaces mechanical complexity with computational complexity.

Whilst this book aims primarily to provide an appreciation of the principles ofstrapdown navigation and the related areas of technology, it should be rememberedthat the stable platform system constitutes a precise mechanical analogue of the strap-down system. It is often helpful to refer to the stable platform system to gain a clearerphysical insight into many of the processes that take place within the strapdown(analytic) system.

There are currently two satellite-based navigation systems deployed, the Americancontrolled system known as GPS or Navstar and the Russian system GLONASS.There are a number of similarities between the two systems, such as the constellationsconsist of up to 24 satellites, although the configuration within each constellation issomewhat different. There are several other similarities between the two systems, butthe characteristics of the two systems are sufficiently different to affect their operationand world coverage.

The similarities and complementary nature of the differences in coverage leadto the possibility of using the systems together in an integrated system to enhancenavigation performance.

Comparison of systems

Both systems are space-based approaches to provide the user with precision position,velocity and time data for accurate navigation. The aim of each system is to provide:

• navigation data throughout the day and night;• navigation data during all types of weather conditions;• navigation data anywhere over the world's surface, or close to it.

GPS

This system has a space segment with 21 active satellites and three active spares in six12-h orbits that are at an altitude of 20 180 km; the orbits have an inclination of 55°to one another. The message broadcast continuously from each satellite contains dataabout its precise position and clock accuracy, and less precise information about itsrelative position with regard to the other satellites in the constellation, that is, thealmanac.

Appendix D

Comparison of GPS and GLONASSsatellite navigation systems

This system has a control segment, consisting of monitor stations and a mastercontrol station to provide precise measurement of each satellite's orbital parameters.This information is sent to each satellite for re-transmission to the users.

The user segment comprises the equipment required to track the satellites that arevisible to the receiver to enable navigation data to be determined.

The GPS satellites use two carrier frequencies in the L band, known asLi (1575.42 MHz) and L2 (1227.6 MHz). Each of these signals is modulated byeither or both of the precise positioning service (PPS or P) signal (10.23 MHz) andthe standard positioning service (SPS), also called the coarse/acquisition (C/A) signal(1.023MHz). The binary signals are created by a P-code or C/A-code, which ismodulo 2 added to 50 bps data. The P-code and C/A-code are added to Li in phasequadrature. Note that only the P-code is present on the L2 signal.

The P-code is a pseudo-random sequence with a period of one week. This contrastswith the C/A-code, which has a period of 1 ms and has a Gold code. A receiver dupli-cates either or both of these codes and the transmission time is derived from the offsetthat has to be generated at that position to synchronise the locally generated code withthe one emitted from a satellite. The C/A-code is a 1 MHz 1.023 bit sequence froma set of Gold codes, it repeats at 1 ms intervals, enabling a receiver to search rapidlythrough the received signal to obtain lock. The P(r) code is a 10 MHz signal. Receivertechnology to acquire the P(v) code is only just becoming available, even so it is expen-sive and power hungry. The C/A code was designed to enable rapid acquisition of timeto allow the position of the P(v) code to be established. However, the P-code providesan accuracy of about a factor of 2 or 3 better than the use of the C/A-code alone.

The American Department of Defence controls the precision of the navigation dataavailable through a technique known as selective availability (SA), where the satellitesignals are corrupted. Authorised users are provided with the algorithms required torecover the original satellite signals. The reduction in accuracy caused by SA to theorder of 100 m has prompted a number of techniques for recovering some of this lostprecision, examples include relative and differential GPS.

The principle of differential GPS techniques is to take advantage of the fact thata substantial component of the navigation error in a GPS measurement arises fromslowly changing biases. Moreover, these biases are correlated in both distance andtime between an array of receivers. Therefore if two receivers, or more, are operat-ing simultaneously at different locations and the position of one of them is known,then corrections can be generated in real time to the measurements from the posi-tional knowledge of one receiver and applied to the other receiver measurements.Clearly, a data link is required between each receiver to enable this technique tooperate.

GLONASS

The space segment has 24 satellites in this constellation at an altitude of 19 100 km,corresponding to an 11 h 15 min period. They are arranged in three orbital planes with arelative inclination of 64.8°. These satellites also broadcast their own precise positionas well as less accurate data on the positions of the other satellites in the constellation.

The transmitted data are in the form of Earth-centred, Earth-fixed coordinates andextrapolation terms, rather than the orbital parameters used by the GPS.

The ground-based control segment is similar to GPS and undertakes a similarfunction. Similarly, the user segment involves the use of a receiver, or set of receivers,to track the satellites.

Each of the satellites in the constellation uses two carrier frequencies in the L band,however, there are different frequencies for each satellite. The Li emission rangesfrom 1602.5625 to 1615.5MHz in jumps of 0.5625 MHz, whilst L2 ranges from1246.4375 to 1256.5 MHz in jumps of 0.4375 MHz. Each of these signals is modulatedin a similar fashion to the GPS signals: the P-code modulation is 5.11 MHz whereasthe C/A-code is 0.511 MHz. The binary signals are created in an identical fashion.However, the period of the P-code pseudo-random sequence is one second and thecorresponding C/A-code period is 1 ms, and added to the Li signal in quadrature,again with the P-code present in the L2 signal.

A single GLONASS code is used for all of the satellites in the constellation,whereas each of the GPS satellites has a unique code. The navigation function isidentical to that described above for GPS. At this stage there is no selective availabilityapplied to corrupt the fidelity of the transmitted satellite data, and the accuracy issuperior to the standard positioning service of GPS.

The table below compares the characteristics of these two satellite navigationsystems.

GPS

24 satellites in six planes with a~12 h period and 55° inclination

Spread spectrum system

Code division multiplexing

C/A-codes and P-codes with selectiveavailability

Broadcast satellite orbital parametersupdated every hour

WGS-84 Earth model(E-C, E-F reference frame1)

GPS time synchronised with UTC(USNO)

GLONASS

24 satellites in three planes with a~ 11 h 15 min period and ~65°inclination

Spread spectrum system

Frequency division multiplexing

C/A-codes and P-codes but noselective availability

Broadcast satellite E-C, E-F position,velocity and acceleration updatedevery j hour

Soviet geocentric co-ordinate systemSGS-90 or PZ-90 Earth model(E-C, E-F reference frame)

GLONASS time synchronised withUTC (Moscow)

1 Earth Centred, Earth Fixed reference frame.

The differences in the orbital planes of these two constellations of these systems leadto differences in coverage throughout the world. GPS provides very good coverageat mid latitudes, whereas GLONASS provides good coverage at higher latitudes.

Schemes have been devised to enable users to combine information derived fromthese two systems. The advantages are:

• faster acquisition in the cold-start mode, as more satellites are potentiallyavailable;

• better coverage in obstructed environments, as again a larger number of satellitesshould be available;

• improved accuracy;• enhanced system integrity as the user is not dependent on a single navigation

system and its continued availability;• integrated operation.

There is a timing synchronisation and definition requirement for dual GPS andGLONASS operation, owing to the different clock time bases used by the two systems.In terms of a GPS-only receiver, or a GLONASS-only receiver, all measurementsinclude a receiver-clock error with respect to either GPS or GLONASS defined time.This error is common to all measurements from a given constellation, therefore itwill only affect the time estimate and will not have an impact on the position andvelocity estimates. However, in the case of dual satellite navigation system opera-tion using both systems, some measurements will include a GPS-to-receiver clockerror, whilst others will include GLONASS-to-receiver clock error. The relation-ship between the GPS and GLONASS time references, therefore, has to be knownor derived for acceptable position and velocity estimates to be formulated duringintegrated operations.

This factor is normally accounted for in terms of the 'two receiver' clock offsets.The state vectors in the Kalman filter could be increased by two to five, to accountfor the additional unknown parameters. An alternative approach to correcting for thisproblem is to correlate all of the measurements to the GPS (or GLONASS) timebase, which requires knowledge of the relative receiver to clock off-sets prior to themeasurements being taken.

GPS time is related to the universal time (UT) standard, but the GLONASSsystem is related to Moscow time. Analysis has shown that the differences canreach several microseconds, hence this is extremely significant for both positionand velocity measurements made by mixing measurements from the two satellite-based systems. The other important issue in dual-mode operation is the differencebetween the two reference systems used by GPS and GLONASS. As noted above,GPS uses the WGS-84 ellipsoid representation of the Earth, whereas GLONASS usesSGS-90 (also known as PZ-90). Again, this difference between the two systems canbe resolved by using a suitable transformation matrix, which allows the data to bereferred to a common reference.

The overall accuracy that can be obtained from the use of data transmitted by thetwo systems, depends on:

• the combined effect of the improved position dilution of precision over thatoperating for either system, coupled with the difference in accuracy possiblefrom either system alone;

• the residual error between the two time-base references.

The combined use of the two systems offers a significant increase in systemintegrity as well as improved accuracy. A major issue is the determination of anyrogue data that would reduce the accuracy of the navigational estimates. A highlyintegrated system architecture may be used to make the best use of available data fromthe two systems. Individual satellite tracking channels can be arbitrarily assigned toany satellite available in either constellation.

The issues concerning the advantages and disadvantages of combined operation,compared with single system use, are given in the table below.

Advantages

Better accuracy owing to

• better coverage at all latitudes;• better geometry;• GLONASS performance not impaired

by selective availability.

Enhanced integrity owing to a highernumber of satellites in the combinedconstellations.

Enhanced availability owing to a greaterselection of satellites in-view at anyinstant.

Independent of US DoD control

Disadvantages

A more complex receiver and morecomplex processing.

GLONASS future availability andreliability?

GLONASS not under 'Western'control.

Clearly, a number of the disadvantages will be resolved with the introduction ofthe Galileo system. Moreover, this has been designed to be compatible with the GPS.