Chapter 1], and basically, sensor positioning algorithms are based on these ... to the navigation frame; finally, integrate to obtain the position. .... When moving higher in the atmosphere, there is less air above and hence a lower ..... Implementing ..... The modified Cholesky decomposition of a matix P is easy to construct if its ...
TKT-2546 Methods for Positioning
Jussi Collin Helena Leppäkoski Martti Kirkko-Jaakkola
2010
Preface This hand-out continues the hand-out of the course MAT-45806 Mathematics for Positioning (http://math.tut.fi/courses/MAT-45806/). These courses are closely related and used to share the same hand-out in the past years. This year, however, each course has its own hand-out. Mathematics for Positioning is a highly recommended but not mandatory prerequisite for this course. The purpose is that Mathematics for Positioning presents mathematical principles and concepts, particularly statistics and optimization, lying under many positioning methods. In this course, these tools are applied to practical positioning problems. Any errata found in the text will be reported on the course website at http://www.tkt.cs. tut.fi/kurssit/2546/. We would like to thank Simo Ali-Löytty, Niilo Sirola, and Henri Pesonen especially for developing the LATEX template used for this hand-out. Furthermore, Hanna Sairo is acknowledged for her contributions in the previous versions of the hand-out, based on which the beginning of Chapter 2 has been rewritten.
Tampere, 19 Feb 2010 authors
2
Contents 1 Sensor-Assisted Positioning 1.1 Accelerometers and Gyroscopes 1.2 Odometers . . . . . . . . . . . . 1.3 Altitude Measurement . . . . . 1.4 Sensor Measurement Errors . . .
. . . .
4 4 7 8 9
2 Carrier Phase Based Satellite Positioning 2.1 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Differential Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Real-Time Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 15 18 21
3 Integrity and Reliability 3.1 Positioning Performance Metrics . . . . . . . . . . 3.2 Statistical Inference and Hypothesis Testing . . . . 3.3 Residuals . . . . . . . . . . . . . . . . . . . . . . 3.4 Receiver Autonomous Integrity Monitoring . . . . 3.5 Reliability Testing Based on Global and Local Tests
27 27 30 35 37 39
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Index
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References
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3
Chapter 1 Sensor-Assisted Positioning J USSI C OLLIN
Because of the very low transmit power of satellite signals, users frequently encounter situations where a position solution is not available. For example, underground parking lots and shopping malls are places where an accurate position solution could be quite useful but a GPS receiver does not help. In such places, e.g., a WLAN network can enable radiolocation. However, a device equipped with proper sensors can obtain information about the motion of its user without any external help. In this chapter, sensors suitable for positioning are presented. Furthermore, resolving the user positioning using sensor information is addressed, and finally, some simple sensor error models are discussed. Coordinate frame transformations were studied in the course “Mathematics for Positioning” [2, Chapter 1], and basically, sensor positioning algorithms are based on these operations: first, measure the change in position in the device body frame; then, transform the measured change to the navigation frame; finally, integrate to obtain the position. This procedure yields a route on the map, as long as the initial position is obtained from some other source. In practice, external aiding is required more frequently than once due to the accumulation of biases in integration.
1.1 Accelerometers and Gyroscopes The term inertia refers to the tendence of a body to retain its motion state when not subject to external forces. Accelerometers, and some gyroscopes, are based on this principle. To construct an accelerometer, a test mass, a spring, a housing, and an indicator of the position of the test mass with respect to the housing are needed (see Figure 1.1)∗ . Since the relationship between force and acceleration is known, the magnitude (and direction) of the external force can be computed from the strain (or compression) of the spring. ∗ There
may also be a feedback coupling to keep the test mass at a fixed position with respect to the housing.
4
Figure 1.1: Functioning principle of an accelerometer. Source: [27]
Unfortunately, the problem in such acceleration measurements is the lack of the gravitational acceleration g: an accelerometer triad measures the vector aB − gB . The comperession of the spring in Figure 1.1 can be due to either the sum of a normal force and gravitation (when the sensor is stationary) or an acceleration in the inertial frame (without gravitation). The sensor cannot distinguish between these two scenarios. Anyway, using Newtonian laws of motion requires measuring gravitational forces as well, but accelerometers are not directly capable of this! For this reason, gravitational forces are introduced in the computations using a gravitation model. On the other hand, if the Newtonian acceleration is known in the sensor body frame, the gravitation vector can be computed from the measurement equations. If the sensor triad is known to be stationary (i.e., aB = 0), it is actually measuring an upward acceleration vector, yielding an inclination measurement. Moreover, accelerometrs are capable of measuring periodic vibrations, e.g., steps, enabling indirect estimation of the covered distance. This will be studied in more detail in the pedestrian dead reckoning section of the lectures. For measuring angular rate, various possibilities exist. Mechanical gyros are also based on inertia, but unlike accelerometers, on the tendency of a moving sensor element to resist rotations. One way of measuring rotations is to mount a rotating test mass M on bearings to a housing B such that wBBM = [0 0 ωM ]T . Hang the test mass such that it can rotation in the direction uBout = [0 ± 1 0]T , but not in the direction uin = [±1 0 0]T . Then, a motion wBIB = [ωin 0 0]T will cause the mounting to rotate in the direction wBBM × wBIB which is an axis parallel to uout . By measuring rotation with respect to this axis, the angular rate with respect to the axis uin can be obtained. If there are restrictions on the size or power consumption of the sensor, a rotating mass may be impractical. Another possibile mechanical gyro is a vibrating gyro where the motion is oscillatory. Figure 1.2 shows the functioning principle of a tuning fork gyro. The upper fork is made resonate using an electric current. Now, if the sensor is rotating on its input axis, the Coriolis force causes an oscillatory motion perpendicular to the forced resonance and the input axis (see the arrows on the lower fork in Fig. 1.2). This motion can be measured (usually 5
Figure 1.2: An example of a Coriolis force based gyro. Source: [27]
capacitively), resulting in a signal modulated by the angular velocity; the angualar velocity may be obtained from the signal. Microelectromechanical (MEMS) gyros are based on this principle. For measuring rotation, moving parts are not necessarily needed because the mechanical phenomena related to rotation have an electromagnetic counterpart: the Sagnac effect. A rigorous explanation of the phenomenon would require the theory of relativity, but in the context of this course, we will simplify a little. Let us emit a laser beam and use mirrors to direct it back to its origin clockwise. Now emit another beam counterclockwise. This will result as two standing waves, and their frequency difference is to be measured. When the housing is rotating w.r.t. the I-frame on the normal of the plane of the beam route, this frequency differ-
Figure 1.3: Ring Laser Gyro. Source: [27]
6
ence changes∗ . Gyroscopes operating on this principle are called Ring Laser Gyros (RLG); Figure 1.3 is an illustration of an RLG. Optical measurements have various advantages: • High accuracy.
• Unlimited input bandwidth.
• No moving parts, hence insensitive to vibrations and more reliable than mechanical gyros. • Unaffected by linear accelerations.
On the other hand, manufacturing costs, size, and power consumption make RLGs infeasible for personal applications—at least thus far.
1.2 Odometers A fundamental problem in acceleration-based displacement measurements is double integration: even small biases will accumulate over time. For this reason, sensors measuring the rotation of the wheels of a vehicle and other odometers almost invariably perform more accurately. The largest error source of odometers is the scale factor: for instance, the radius of a wheel is not precisely known. In this case, the position error is proportional to the covered distance instead of time. As opposed to purely inertial measurements, these methods are somewhat dependent on external factors: for example, a slippery road surface may cause wheelspin. Using a Doppler radar is another means of computing the covered distance, particularly useful in heavy equipment vehicles where wheel sensors are hard to mount or maintain. Measured speed (km/h)
Doppler radar speed
3
0.5 0
True speed (km/h) -3
-3
-0.5
0.5
3
Figure 1.4: The output of a Doppler radar as a function of the velocity. The measurement is unsigned, and at low speeds, the output is truncated to 0.
∗ The
beam that moves in the direction of rotation will cover a longer distance in the I-frame. . .
7
1.3 Altitude Measurement When moving higher in the atmosphere, there is less air above and hence a lower pressure. Thus, air pressure measurements give altitude information. Unfortunately, the ambient pressure changes as a function of weather conditions (and, e.g., air conditioning when indoors). Hence, pressure altimeters must be calibrated frequently. In personal positioning applications, even a few meters of accuracy may be significant (e.g., for identifying the floor number inside a building). In these cases, another pressure sensor should be mounted at a nearby location at a known altitude. Then, the altitude difference can be accurately determined based on the pressure difference. 178 stairs 3rd floor
174
altitude (m)
5th floor
4th floor
176 3rd floor
172
stairs
170
2nd floor
168 Elevator
166
1st floor stairs
164 Stairs to outside
162 160
0
50
100
150
200
250
300
350
400
450
time (s)
Figure 1.5: Altitude solution from a MEMS air pressure sensor indoors. The solution is accurate to meters from sea level when a reference barometer is available in the same building at a known altitude.
1.3.1 Inertial Navigation Equations * (Inertial navigation is beyond the scope of this course, and is covered in the course TKT2556 Basics of Inertial Navigation. However, the equations are given here as they may be useful in other applications as well.) The equations given in this section follow the formulations of [22]. The origin of the E-frame serves as the origin of the location vector, and gL is computed separately (e.g., using a gravitation model). The derivative of the relationship between the B- and L-frame is ˙ LB = CBL (wBIB ×) − (wLIL ×)CLB . C
(1.1)
The initial condition CLB is obtained using a separate initialization algorithm (see Problem 1.8). The angular velocity term wBIB is obtained from a gyro triad, and the term wLIL is the sum of the rotation of the Earth and the rotation of the local coordinate frame due to ground velocity: wLIL = wLIE + wLEL . 8
(1.2)
The local frame L retains its z-axis perpendicular to the surface of the spherical (or ellipsoidal) Earth; thus, as the user moves on the surface of the Earth, the frame is rotating at an angular velocity of wLEL = Fc (uLZL × vL ) + ρZL uLZL .
(1.3)
The structure of the 3 × 3-matrix Fc is determined by the Earth model used (spherical or ellipsoidal). Furthermore, the term ρZL adjusts the motion of the North axis of L (Problem 1.1). uLZL is an unit vector pointing “upwards” in the L-frame, i.e., based on our definitions, uLZL = [0 0 1]T . The measured acceleration vector (denoted as aBSF = aB − gB ) is transformed to the L-frame; aLSF = CLB aBSF .
(1.4)
Finally, we need the gravitation (gP =gravitation plus the rotation of the Earth) gLP = gL − (wLIE ×)(wLIE ×)RL ,
(1.5)
the change in the velocity vector (w.r.t. the Earth, E-frame) v˙ L = aLSF + gLP − (wLEL + 2wLIE ) × vL ,
(1.6)
the rate of the horizontal position C˙ EL = CEL (wLEL ×),
(1.7)
h˙ = uLZL · vL .
(1.8)
and the altitude rate
That was all. We can now solve for position and velocity using the differential equations (1.6)– (1.8) given suitable initial conditions. Let us now group the most important terms for a better general view: • Information from sensors: aBSF and wBIB • Earth-related terms: wIE , g, Fc • What we originally wanted to compute: RE (location vector in the E-frame; CEL and altitude h together are equivalent, see Problem 1.2), vL (ground velocity), CLB (attitude of the INS device w.r.t. the L-frame).
1.4 Sensor Measurement Errors Consider a gyro triad output y as the first example. According to the course “Mathematics for Positioning” [2, Chapter 1], the ideal measurement is wBIB , and one possible error model could be y = MwBIB + b + n 9
(1.9)
where the matrix M contains scale factor and misalignment errors. Noise terms are divided to bias-type (b) and uncorrelated noise (n). Note that the error term is now v = (M − I)wBIB + b + n, i.e., scale factor and misalignment errors cause the error to be a function of the measured quantity. This renders statistical error analysis somewhat problematic. A basic triad assembly consists of three sensors whose measurement axes are orthogonal. In that case, the diagonal elements of the 3 × 3 matrix M correspond to the scale factors of the respective sensors. Misalignment is due to the fact that in reality, it is impossible to mount the sensors exactly perpendicular to each other; hence, the data of a certain axis is somewhat visible at the other sensors as well. In high-quality INS devices, misalignment is in the order of 10−3 degrees, and scale factors are a few parts per million (ppm) of the signal magnitude. Note that it is not necessary to mount the sensors orthogonally as long as the angles between them are known. It is also possible to have more than three sensors to account for failure situations; in that case, of course, it is impossible to mount all sensors orthogonally. The error term b is probably the most interesting and, in practice, the most challenging. This term refers to errors that remain (almost) constant for a longer time. The classification between b and n is by no means obvious. The extreme situation where b would always remain constant and n would be totally uncorrelated between samples does not exist in real life: if it did, the bias term could be resolved and corrected for during factory calibration. In practice, the correlation time of b is in the order of the interval between device startups (hours to months) whereas the correlation period of n is considerably shorter. Treating the error term n was already addressed in the previous paragraphs. Once again, note that using sensors for positioning almost always requires integrating their measurements, causing error correlation with time to significantly affect the positioning accuracy. Even if a short sample of data looks noisy, no definitive conclusions must be drawn: it is a severe mistake to compare gyroscope quality using standard deviation estimates based on a few minutes of measurements. Model (1.9) is also for accelerometers as such, as long as the term wIB is replaced by aBSF . Table 1.1 shows what is required of INS sensors to retain the positioning error under 0.1 or 1 nautical miles after one hour of navigation. The requirements are very strict, especially for gyros. In addition, depending on the application, there may be requirements on dynamics: for example, it may be required that a rotation of 500 degrees per second is measurable. INS devices of this quality cost around $100 000–$200 000, although the prices are decreasing constantly. Furthermore, the availability of such devices may be limited because of export restrictions: highquality INS equipment is regarded as military technology in many countries. MEMS sensors are not even close to fulfilling these requirements but, on the other hand, are easily available and cheaper. Even if they are incapable of INS as such, they can give valuable additional information to positioning algorithms [5].
10
Table 1.1: Sensor accuracy requirements when the position error may grow no more than 0.1 or 1 nautical mile (1.852 km) per hour [23].
Error source
Required accuracy 0.1 nmph
Accelerometer
Gyro
bias 5 µg scale factor 40 ppm misalignment 1/3600 ◦ bias 0.0007 ◦ /h scale factor 1 ppm misalignment 0.7/3600 ◦
1 nmph 40 µg 200 ppm 7/3600 ◦ 0.007 ◦ /h 5 ppm 3/3600 ◦
1.4.1 Error Processes The most simple error process consists of uncorrelated random variables nt with mean 0 and variance σ2n < ∞. This process is called white noise. Furthermore, suppose that these random variables obey a Gaussian distribution. Figure 1.6 shows a realization of white noise and its cumulative sum t
xt =
∑ nt .
(1.10)
t=1
with σ2n = 1. AR(1) process
Next, we will introduce some autocorrelation in the process; first, the xt = ρxt−1 + nt .
(1.11)
10 r=randn(500,1) cumsum(r)
5 0 −5 −10 −15 −20 −25 −30
0
100
200
300
400
500
Figure 1.6: A realization of white noise, and random walk obtained by integrating it.
11
4 2 0 −2 −4
0
100
200
300
400
500
0
100
200
300
400
500
150 100 50 0 −50
Figure 1.7: A realization of AR(1) and its cumulative sum.
In order to obtain a stationary process, we require that |ρ|< 1. When generating a realization σ2n of the process, x0 must be drawn from the distribution N 0, 1−ρ2 to ensure that stationarity
holds for a finite-length sequence (see e.g. [24]). Now choose ρ = 0.9 and σ2n = 1 − ρ2 to get a process with variance equal to that of the previous process. When integrated, the sequence starts to get very high values: the standard √deviation of the last random variable is around 96.5 in the integrated sequence as opposed to 500 ≈ 22.4 in the case of uncorrelated noise. These numbers will be verified as an exercise.
These noise models are fairly simple, and unfortunately, more complex processes are often encountered in reality. Consider 1/ f noise (ARFIMA(0,0.5,0) in discrete time; see e.g. [4, 29] with their interesting examples) as an example. A realization of 1/ f noise can be generated by half-integrating white noise. Since the lower triangular matrix 1 0 0 ··· 0 1 1 0 ··· 0 A = .. .. .. .. .. (1.12) . . . . . 1 1 1 ··· 1
integrates the time series vector x once when multiplied as Ax, we will compute the matrix B satisfying BB = A. The matrix square root [2, Chapter 1] is computed using the familiar command sqrtm yielding 1 0 0 ··· 0 0.5 1 0 ··· 0 0.375 0.5 1 · · · 0 . (1.13) B= . . 0.3125 0.375 0.5 . . .. .. .. .. .. . 1 . . . 12
3 2 1 0 −1 −2 −3 −4 −5 −6 −7
0
100
200
300
400
500
Figure 1.8: A realization of discrete-time 1/ f noise.
Now use B as a coefficient matrix for white noise (σ2n = 1); an example result is shown in Figure 1.8. Bias oscillation is characteristic of this noise type: a large part of its power occurs at very low frequencies. For identifying the noise process presented in this section, the Allan variance (two-sample variance) [3, 19] 1 σ2x (τ) = E(x¯2 − x¯1 )2 , 2
(1.14)
is often used. Here, the mean values are computed from consecutive blocks with length τ 1 τ−1 ∑ xt τ t=0
(1.15)
1 2τ−1 ∑ xt . τ t=τ
(1.16)
x¯1 =
x¯2 =
An estimate of Allan variance is obtained by gathering the sample in m blocks: σˆ 2x (τ, m) =
m−1 1 ∑ (x¯k+1 − x¯k )2 . 2(m − 1) k=1
Under certain conditionsfor the process xt , this yields an unbiased estimate [20].
13
(1.17)
Problems 1.1. Consider the equation wLEL = Fc (uLZL × vL ) + ρZL uLZL . What kind of a matrix is Fc when modeling the Earth as a sphere with radius R? What does the term ρZL do? Hint: Consider an aircraft flying near the North pole. If the y-axis of the L-frame always points towards North, what will happen to the vector wLEL ? 1.2. Is it possible to compute a unique position solution given the matrix CEL and altitude h only? 1.3. In INS applications, three gyros are enough for navigation, but more of them may be used to account for failure situations. Suppose there are three mutually orthogonal gyroscopes, and a fourth gyro is mounted such that its axis is not parallel to any of the axes of the other gyros. Is it now possible to detect if one of the gyros is giving grossly erroneous measurements? If yes, is it possible to identify (and exclude) the malfunctioning gyro? Does the situation change if there are five gyros such that no three measurement axes lie on the same plane? 1.4. Show that a bias in g (gravitation model error) causes a positive feedback to INS altitude error. 1.5. Suppose that one day, a MEMS gyro with analog output meets the 1 nmph INS requirements (Table 1.1, page 11). For digitalizing the signal, an A/D converter is needed. Estimate the required resolution of the converter, i.e., how many bits are required for the quantization. 1.6. Through how many integrators is the term wBIB (i.e., gyro data) fed in the INS equations? What happens to (almost) uncorrelated noise after that many integrations? How about bias-type noise? 1.7. Computer problem: Feedback couplings in INS mechanization. 1.8. Computer problem: Initializing CLB based on Earth rotation and normal force.
14
Chapter 2 Carrier Phase Based Satellite Positioning M ARTTI K IRKKO -JAAKKOLA
While tracking a satellite signal, a GNSS receiver measures both the phase of the modulated ranging code and the phase of the sinusoidal carrier wave. These two, i.e., code phase and carrier phase, are the basic GNSS measurements, and based on them, pseudoranges and the integrated Doppler observable can be constructed. In this chapter, using the carrier phase measurements for positioning is discussed. In fact, carrier measurements were not originally designed to be used for positioning at all; however, in the late 1970s, Counselman et al. [7] demonstrated their potential for high positioning accuracies. As shown in Figure 2.1, code measurements are significantly more noise than carrier measurements: code phase measurements typically contain decimeter-level noise whereas even low-cost receivers are capable of measuring the carrier phase at centimeter precision. However, utilizing carrier measurements is not straightforward because they only express the phase of the wave modulo 2π∗ – by measuring the phase, one does not know the amount of full carrier cycles which is an integer-constrained unknown for each satellite. Therefore, the use of carrier measurements in personal positioning is usually limited to estimating the change in the distance, allowing for smoothing the noise of code measurements (carrier smoothing, see e.g. [21]). Implementing carrier positioning with consumer equipment is nevertheless a popular research topic [1].
2.1 Doppler Effect Navigation satellites are constantly moving kilometers per second with respect to their users; for example, the speed of GPS satellites is around 4 km/s. Furthermore, the user may move as well. The relative motion between the satellite and the user affects the frequency at which the signal is observed: if the receiver is moving towards the signal source, it encounters wave ∗ Carrier
phase is usually expressed in units of carrier cycles, not radians or meters
15
Code
Carrier
−424
−424
phase diff. [m] −426
−426
−428
−428
Figure 2.1: Noise levels in code and carrier measurements. The illustrations only show the differences of consecutive measurements (all scaled to units of meters) instead of the observables themselves, thus canceling bias-type errors (including integer ambiguities).
fronts more frequently than when standing still; on the other hand, when moving away from the transmitter, wave fronts are encountered less often. In both cases, the receiver observes the signal at a different frequency than the true transmit frequency. The effect of the relative motion of the source and the receiver on the received frequency is called the Doppler effect. For waves propagating at the speed of light∗, the Doppler shift is modeled by vr · u fR = fT 1 − c
(2.1)
where fR and fT are the received and transmitted frequencies, respectively; vr is the relative velocity between the user and the satellite; u is the unit vector pointing from the receiver antenna to the satellite antenna and c is the speed of light. The change in the frequency caused by the Doppler effect is called the Doppler shift fD : fD = fR − fT = −
vr · u . c
(2.2)
In order to track a signal, the receiver must naturally know its Doppler-affected frequency. Therefore, the receiver also measures the Doppler shift of each satellite signal. In the following sections we will see how we can take advantage of this.
∗ This equation does not hold, e.g., for sound waves (example:
the siren of an ambulance passing by), because in that case, the approximation v + vs ≈ v, with v denoting the propagation speed and vs the velocity of the wave source, is invalid.
16
2.1.1 Relation between Doppler Shift and Range Rate In the course “Mathematics for Positioning” [2, Example 14, p. 23], a model for the pseudorange measurement is presented. Let us now differentiate the pseudorange to satellite i with respect to time (this derivative is called the delta range): ρ˙i =
s−x d (ksi − xk + b) = (vi − vu ) · + b˙ dt ks − xk
(2.3)
where vi is the velocity vector of satellite i, vu is the velocity vector of the receiver, and b˙ is the receiver clock drift [s/s]. Derivation of (2.3) is left to the reader. s−x By comparing (2.2) and (2.3) we note that u = ks−xk and vr = vi − vu . Thus, the delta range can be obtained by means of an affine transformation from the Doppler shift:
˙ ρ˙i = −c fD + b.
(2.4)
Delta ranges are commonly used, e.g., for computing the velocity of the receiver. It is also possible, although significantly more error-prone because of the different measurement model, to compute the position using delta ranges [18].
2.1.2 Relation Between Doppler Shift and Carrier Phase The carrier phase measurement is also known as “integrated Doppler”. A new carrier measurement φi (t) is constructed by subtracting the integral of the Doppler shift over the measurement epoch (length T ) from the previous phase measurement φi (t − T ). It should be emphasized that it is subtraction, not addition, because the Doppler shift increases when the receiver approaches the satellite, i.e., when their mutual distance decreases. We naturally want the phase measurement to behave in the opposite way. Noting that in (2.1), the dot product between the relative velocity vr and the unit direction vector u is the projection of the velocity to the line of sight, i.e., the time derivative of the distance r, the Doppler shift (2.2) can be written in yet another form as fD = −
r˙ λ
(2.5)
where λ = c/ f is the wavelength of the signal. Now we can integrate this to obtain the carrier phase measurement: φi (t) = φi (t − T ) −
Z t
t−T
fD (τ)dτ = λ−1 (ri (t) − ri(t − T )) .
(2.6)
Let us now write φi (0) = ri (0) + b(0) + Ni where ri (0) is the true range between the receiver and satellite i at time t = 0, b(0) is the receiver clock bias at the same instant, and Ni is an unknown number of carrier cycles (as mentioned previously, we do not know the value of Ni by measuring 17
the phase) called the integer ambiguity. Now, by introducing the satellite clock bias bi (t) and the measurement error term εi (t) (including, e.g., satellite ephemeris errors, multipath propagation and receiver noise), the carrier phase measurement can be modeled as ri (t) (2.7) + b(t) − bi(t) + Ni + εi (t). λ This quantity is in units of carrier cycles. In reality, error sources analogous to pseudorange errors, e.g., atmospheric errors, should be introduced in the model, but they have been omitted for simplicity. φi (t) =
Particularly note that the integer ambiguity Ni does not depend on the time t: it is determined in the signal acquisition phase and remains constant as long as satellite i is tracked continuously. In practice, especially with low-cost equipment, the integer ambiguity Ni can sometimes change due to a short signal outage. This situation is called a cycle slip, and detecting them (possibly also correcting for them) is crucial in precise positioning because each slipped cycle corresponds to a range error of around 20 cm. Cycle slips can be detected, e.g., from the differences of consecutive carrier measurements using RAIM methodology (p. 37), see e.g. [14].
2.2 Differential Positioning As stated before, the measurement model (2.7) is not realistic as it lacks some significant error sources. When positioning in postprocessing mode where the results are allowed to have a latency of a couple of weeks, one can use precise ephemeris and atmosphere data provided by, e.g., the International GPS Service (IGS) [9] to correct for these biases. In real-time applications, however, this is not feasible. Another way of mitigating the errors is to take advantage of their spatial and temporal correlations: for instance, errors caused by atmospheric effects are practically equal for receivers located close to each other. This is the key idea in differential positioning where a reference receiver located “sufficiently close”∗ At its simplest form, differential positioning can function as follows. A reference receiver whose exact location is known estimates how much error is included in its measurements to move the position estimate out of the true position. Then, this estimate of the total error, known as the differential correction, is broadcast to the users via, e.g., a radio link. In the absence of Selective Availability (SA), these errors do not vary rapidly in time, and therefore, the correction is valid for a longer time, i.e., a new correction is not needed to be computed and broadcast at each measurement epoch. The estimate of the total error also includes the clock bias of the reference receiver; however, this does not matter as long as the users do not use both differential-corrected and uncorrected signals at the same time: the users just solve for the sum of the clock biases of their own and the reference receiver, which is done in the exactly same way as solving for the user clock bias only without differential corrections. ∗ “Sufficiently close”
is defined totally by the situation: For instance, for single-frequency receivers in a bad weather is can be a couple of kilometers whereas for dual-frequency receivers in a cloudless weather, even a hundred kilometers may be ok [21].
18
si
Baseline Reference User Figure 2.2: Single difference to satellite i.
If multiple reference receivers are available, total error estimates are not that useful because it would be better if each reference would estimate the contributions of each error source separately. This is how many GPS augmentations, such as the Wide Area Augmentation System (WAAS) in the U.S., work. A similar system, the European Geostationary Navigation Overlay System (EGNOS) is available in Europe (officially since October 2009). As the name suggests, these systems use geostationary satellites for broadcasting correction data to a larger area. Unfortunately, geostationary satellites are barely visible at near-polar latitudes (e.g., in Finland) because their orbits are located directly above the Equator.
2.2.1 Relative Positioning In applications where it is sufficient to only know the distance vector between two receivers, usually called the baseline, measurements made by different receivers can be directly subtracted from each other without constructing separate error estimates. In this case, however, absolute position information is lost. If the location of one of the receivers—we will call this the reference receiver—is known, we can solve for the absolute position of the other receiver (which we call the user, although in literature it is commonly referred to as the rover receiver) at the accuracy of the reference location or the baseline, depending on which one is less accurate. Using carrier measurements benefits relative positioning significantly. Since bias-type errors can be mitigated by the differential method, measurement noise becomes a major error source when using code measurements. Carrier measurements, however, are a few orders more precise. The integer ambiguities Ni , however, are problematic. If they can be solved for, the baseline can be estimated possibly at a millimeter-level precision.
2.2.2 Single Difference Between Receivers If measurements according to model (2.7) are available from the user and a reference receiver r “sufficiently close” to the user, as depicted in Figure 2.2, we can construct the difference of the
19
si
sj
Baseline Reference User Figure 2.3: Double difference to satellites i and j.
measurements made by these receivers to satellite i, called the single difference ∆r φi (t) = ∆r
ri (t) + ∆r b(t) + ∆r Ni + ∆r εi λ
(2.8)
where the operator ∆r denotes single differencing with receiver r. The satellite clock bias bi (t) is equal to both receivers and hence is canceled out from (2.8). Furthermore, atmospheric effects are almost common because of the proximity assumption, and thus their effect is mitigated and they are omitted from the model. In contrast, the receiver clock biases, integer ambiguities, measurement noise and multipath are not correlated between receivers and hence are not cancelled but redefined. Fortunately, even when differenced, these treated analogously to those in the original models, and the integer ambiguities preserve their integer nature. Random measurement noise, however, is actually √ amplified: its standard deviation is increased 2-fold, i.e., var ∆r εi = 2 var εi
(2.9)
when the measurement noises of the receivers are assumed to be independent and identically distributed. Denoting the baseline as ∆r x = x − xr where x and xr are the positions of the user and reference receiver, respectively, and computing the first-order Taylor series of (2.8) yields ∆r φi (t) ≈ λ−1
xr (t) − si(t) · ∆r x(t) + ∆r Ni + ∆r b(t) + ∆r εi (t) kxr (t) − si(t)k
(2.10)
when the position of satellite i is denoted by the vector si . Deriving this equation is left to the reader.
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2.2.3 Double Difference Two single differences corresponding to different satellites can be used to construct a double difference∗ (Figure 2.3): ∆r φi j (t) = ∆r φi (t) − ∆r φ j (t) xr (t) − s j (t) xr (t) − si(t) −1 ≈λ · ∆r x(t) + ∆r Ni j + ∆r εi j (t). − kxr (t) − si(t)k kxr (t) − s j (t)k
(2.11)
Differencing measurements cancels receiver-dependent biases, the most important of which being the clock bias ∆r b(t). The price of this operation is losing one measurement and further amplifying measurement noise (see problem 2.2). The integer ambiguity ∆r Ni j remains as an integer, but not necessarily equal to the single-differenced ambiguity. There are two alternative main principles for choosing the satellite pairs to be differenced. In the fixed reference method, one satellite is chosen as the reference!satellite and is subtracted from the others. Usually, the satellite at the highest elevation angle is chosen as the reference because its signals propagate the shortest distance through the atmosphere and are thus probably least affected by atmospheric errors. Another possibility is sequential differencing where the first double difference is computed between satellites 1 and 2, the next between satellites 2 and 3, etc. In both ways, k single differences yield k − 1 nonredundant double differences. Fixed reference is the more common way, but its drawback is that losing the reference satellite signal causes losing all resolved integer ambiguities, as opposed to the sequential differencing where losing one signal causes the loss of at most two ambiguities.
2.3 Real-Time Kinematics Real-Time Kinematic (RTK) positioning is based on the mixed integer programming model obtained by omitting the noise term from the double difference model (2.11). Then, the unknowns to be solved for are the baseline ∆r x and the double-differenced integer ambiguities ∆r Ni j . Unfortunately, the integer constraint renders the problem hard: Even linear integer programming is a difficult problem for which general solution algorithms are not known, let alone nonlinear cases. The most obvious solution method, i.e., finding a solution without the integer constraint and rounding it, does not generally work. Integer ambiguity resolution algoritms have been extensively researched and developed. Already in the 1980s, the Ambiguity Function Method (AFM) [6] was published, and in the early 1990s, Fast Ambiguity Resolution Approach (FARA) [10] and Least-Squares Ambiguity Search Technique (LSAST) [11] became known. The probably most popular method, i.e., Least-Squares Ambiguity Decorrelation Adjustment (LAMBDA) [26], was developed in the mid-1990s in the Netherlands, and is briefly presented in Section 2.3.2. literature, differencing between satellites is sometimes denoted by the operator ∇, and thus double differencing is denoted by the operator pair ∇∆.
∗ In
21
Although rounding does not usually yield a good solution estimate, the first step in RTK positioning is usually constructing the nonconstrained float solution, which can be done by means of filtering (in the dynamics model, the ambiguities are assumed to remain constant, but cycle slips must be excluded). Integer programming methods are usually based on exhaustive searching, and the search space is chosen to lie around the float estimate. Unfortunately, the volume of the search space increases rapidly: with 7 double differences available and a tolerance of ±5 cycles (approximately ±1 m), the search space will contain 117 candidate vectors, and searching through them is obviously computationally burdensome. Moreover, when computing the float solution, the reliability of the solution is usually monitored, typically including cycle slip detection (and usually correction because the slips are known to be multiples of the wavelength). If the integer ambiguity resolution succeeds, the final estimate called the fixed solution can be computed. To do this, the resolved ambiguities are subtracted from the respective double differences, allowing for accurate estimation of the baseline.
2.3.1 Advantages of Multifrequency Receivers In absolute pseudorange positioning, multifrequency receivers mostly serve for estimating the ionospheric delay as ionosphere is dispersive and affects different frequencies at different powers. In RTK, however, the ionospheric error is not as significant, particularly at short baselines, and at longer baselines, it can be estimated by means of filtering (where dual-frequency measurements naturally are helpful). Nevertheless, multifrequency receivers do have special use in RTK because measurements of the range from the user to the satellite made at multiple frequencies can be combined to construct pseudo-observables where the wavelength, and thus the spacing of the integer ambiguity candidates, is different from the original measurements. Suppose we have doubledifferenced carrier phase measurements made at two different frequencies with corresponding wavelengths are λ1 and λ2 , respectively. Let us now construct a linear combination of these measurements (omitting atmospheric and clock errors): ri ri ri n1 + (n1 Ni1 + n2 Ni2 ) + εei + Ni1 + εi1 + n2 + Ni2 + εi2 = (2.12) λ1 λ2 λe f f where n1 , n2 ∈ R. The effective wavelength of the new observable is given by λe f f =
λ1 λ2 . n1 λ 2 + n2 λ 1
(2.13)
Now we can construct, e.g., the widelane combination of GPS observables with n1 = 1 and n2 = −1, obtaining an effective wavelength of around 90 cm—notably longer than the GPS L1 and L2 wavelengths (approx. 19 cm and 24 cm); thus, the corresponding search space is much more sparse than the original space. When constructing these linear combinations, the ambiguities preserve their integer nature if the coefficients n1 , n2 ∈ Z. The drawback of widelaning is that the standard deviation of the noise term εei also changes as a function of n1 and n2 , and for the widelane combination it is amplified by a factor of 5.7 [21]. In 22
contrast, for example, the narrowlane combination (n1 = n2 = 1) has an effective wavelength of about 11 cm but a lower noise variance than in the original observables. Thus, the narrowlane combination allows for more precise positioning than the original measurements, but resolving the integer ambiguities is more difficult. It is also possible to totally eliminate the ionospheric advance by choosing the coefficients n1 and n2 suitably (see problem 2.4).
2.3.2 LAMBDA method The LAMBDA method is based on the idea that the rounding approach would work if the integer ambiguities were not so highly correlated∗ . The input of LAMBDA consists of a float e and its covariance matrix P, both of estimate vector of the double-differenced ambiguities N which are obtained, e.g., by means of filtering. In the heart of the method is a linear transformation Z that decorrelates the ambiguities as well as possible, i.e., the covariance matrix ZPZT of e is almost diagonal. Then, the search space can be drastically the transformed ambiguities ZN shrunk, and, in principle, if the ambiguities could be totally decorrelated, the optimum would be obtained by rounding. The objective function of the optimization is chosen to be the difference of the float and integer estimates weighted by the inverse of the covariance P (Integer Least Squares; vector a denotes the integer estimate) T −1 e e e a−N . ka − NkP−1 = a − N P
(2.14)
The LAMBDA method consists of multiple steps, and its progress is outlined in Algorithm 1. Algorithm 1 Outline of the LAMBDA method. e ∈ Rn is a real-valued estimate of the integer ambiguities and P = cov N. e Precondition: N 1. Compute the modified Cholesky decomposition P = LDLT .
2. Construct the LAMBDA transformation Z based on the matrices L and D. e 3. Find the two best vectors a1 and a2 according to argmina∈Zn ka − ZNk . (ZPZT )−1 4. Return Z−1 a1 and Z−1 a2 .
Definition 1. An invertible matrix Z ∈ Zn×n is a LAMBDA transformation if Z−1 ∈ Zn×n . If Z is a LAMBDA transformation, then in the image Z(S) of the set S ⊆ Rn×n , all integer vectors of S are integer vectors, and conversely, the inverse images of all integer vectors of Z(S) are integer vectors in S. This means that the integer ambiguities can be transformed to a decorrelated space using Z: in the decorrelated space the optimum is easier to find, and one can be sure that the obtained candidate also maps to an integer vector of the original space. ∗ Double
differences are highly correlated because of the way they are constructed.
23
The decorrelating transformation is constructed as a sequence of two different types of LAMBDA transformations. The first type is the Gauss transformation, expressed in 2×2 dimension as 1 0 (2.15) a 1 where a ∈ Z. Essentially, the Gauss transform adds the (k + 1):th column of the original matrix multiplied by a to the kth. The other necessary transformation type is the column permutation of the form 0 1 . (2.16) 1 0 Efficient implementations of LAMBDA [8] apply these two transformation types on the modified Cholesky decomposition of the covariance matrix of the ambiguities. The modified Cholesky decomposition of a symmetric positive definite matrix P is of the form P = LDLT
(2.17)
where L is unit lower triangular (all diagonal elements equal to 1) and D is diagonal. Based on this decomposition, the decorrelating transformation Z and the corresponding decomposition of the decorrelated covariance matrix can be efficiently computed. The modified Cholesky decomposition of a matix P is easy to construct if its conventional Cholesky decomposition P = Ln Ln T , computed in Matlab by chol(P), is available. However, computing the modified decomposition directly is more efficient than using the conventional decomposition as an intermediate step because no square roots need to be computed for the modified decomposition. The LAMBDA transformation is usually computed using two main functions. The purpose of one of them is to diagonalize, and the other reorders the ambiguities such that the variances of the decorrelated ambiguities would be approximately equal. Algorithm 2 describes the diagonalization process; the reordering algorithm is not addressed in the scope of this course because of its complexity, but a detailed description is available in literature [8, algorithm SRC1]. Algorithm 2 modifies the unit lower triangular matrix L such that all subdiagonal elements of of the transformed matrix are in the interval (−0.5, 0.5] while retaining the diagonal entries as ones, and returns the modified L and the corresponding transformation Z. According to intuition, the more diagonal L, the more diagonal the product LDLT as the product of diagonal matrices is known to be diagonal. Once the transformation matrix Z has been constructed, the actual integer programming is carried out in the transformed space. This can be done, e.g., by means of a normal depth-first e The optimization method should return a predefined amount of search around the image of N. the “best” candidate vectors (according to the criterion (2.14)). Usually, the two best candidates are required because there must be a way to ensure that the best solution candidate is reliable. Therefore, the rounding method is not the most useful as it only finds one candidate (which is not necessarily even the best). Finally, the candidate vectors are transformed back to the original space by multiplying with Z−1 and returned. 24
Algorithm 2 “Diagonalization” of a unit lower triangular matrix [8, algorithm ZTRAN] Require: L ∈ Rn×n unit lower triangular Z := I for i := n − 1, n − 2, . . ., 1 do for j := i + 1, i + 2, . . ., n do µ := ROUND (l j,i ) # Rounding to nearest integer l j...n,i := l j...n,i − µl j...n, j z1...n,i := z1...n,i − µz1...n, j end for end for return (Z, L)
2.3.3 Solution Validation Once some estimate of the integer ambiguities has been obtained using some optimization method, one should decide if the estimate can be relied on. This is called solution validation, and, unfortunately, no direct validation method is known. Usually, validation is based on the residuals (2.14) of the two best candidates: if the residual of the second-best candidate is, say, at least two times as large as that of the best, the best candidate is accepted. However, this test is totally heuristic and has no theoretical justification. Nevertheless, more sophisticated solution methods have been developed; see e.g. [28]. It is also possible to compute the success probability of integer ambiguity resolution. Then, if the probability is not high enough, ambiguity resolution may be decided not even to be attempted.
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Problems 2.1. Show that Equation (2.3) holds. 2.2. Show that the standard deviation of the noise in double-differenced measurements is two times as large as that of the original undifferenced measurements (cf. (2.9) for single differences), if all the original measurements are independent and identically distributed. Assume a Gaussian distribution for noise. Hint: [2, Theorem 4, p. 10] 2.3. Derive Equation (2.10). Assume that the unit vector from the user location to satellite i is equal to that from the reference location to the same satellite (as the baseline ∆r x(t) is orders of magnitude shorter than the distance to the satellite). 2.4. According to the Appleton–Hartree model, the ionospheric advance expressed in units of carrier cycles is directly proportional to the wavelength. It is known that for the GPS L1 L1 and L2 frequencies, ffL2 = 77 60 . Construct a linear combination of measurements made at these frequencies such that the ionosphere term is canceled while preserving the ambiguities as integers. What is the effective wavelength of the combination? 2.5. Show that a matrix Z ∈ Zn×n is a LAMBDA transformation if and only if det Z = ± 1. Hint: Cramer’s rule. 2.6. Prove that the weighted norm (2.14) computed in the original space is equal to that computed in the transformed space if the weighting is done using the covariance matrix of the respective space, i.e., e −1 = kZa − ZNk e ka − Nk . P (ZPZT )−1
2.7. Computer problem: Write a Matlab function that computes the modified Cholesky decomposition (2.17) of a symmetric positive matrix. You may call the function chol. 2.8. Computer problem: Let the float estimate and its covariance be 3.06 −1.35 −0.25 −3.37 106.68 −1.35 4.15 −0.42 3.11 . e = 5.93 N P = −0.25 −0.42 0.70 −9.56 0.30 −3.37 3.11 0.30 5.14 −83.24
a) Compute the value of the target function (2.14) for the integer estimate obtained directly by rounding.
b) Compute the corresponding value for the estimate obtained using the LAMBDA method when the transformation Z is constructed directly using Algorithm2 and the optimization method is rounding in the transformed space (we only want one candidate now).
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Chapter 3 Integrity and Reliability H ELENA L EPPÄKOSKI
From a user’s point of view, positioning cannot be considered reliable if it possible that, because of a system or device failure, positioning either cannot be carried out at all or yields significantly erroneous results despite successful execution of the algorithms. This chapter presents methods used in satellite positioning for detecting gross position errors and excluding erroneous measurements from position computations. Reliability is related to many criteria used to describe positioning performance; these are addressed below. Then, basics of statistical inference and hypothesis testing are reviewed. Finally, fault detection and exclusion methods for static positioning methods based on these concepts are presented.
3.1 Positioning Performance Metrics In this section, positioning performance is considered in the context of satellite positioning. Some of the methods are, however, applicable on other positioning methods as well. The topic is covered in more detail in [25].
3.1.1 Accuracy Positioning accuracy describes how close to the true position the estimates computed from the measurements are located: the closer to the true position the estimates lie, the more accurate they are. Accuracy is often estimated using the root mean square error (RMSE). Another common way of quantifying accuracy is to present the statistical distribution of positioning errors, which can be done by means of a probability density function or a cumulative density function. A 27
simplification of these is the circular error probable (CERP) which describes the radius of the circle centered at the true location containing a certain percentage of the (horizontal) position estimates; for example, 95 % CERP 56 m means that at least 95 % of position estimates have at most 56 meters of horizontal position error. The three-dimensional equivalent of CERP is the spherical error probable (SERP) where the radius of a sphere is given instead of the radius of a circle∗ Besides measurement signal quality, positioning accuracy depends on how possible measurement errors affect the error in the position estimate. This is determined by the satellite geometry and is quantified by the dilution of precision (DOP).
3.1.2 Integrity The integrity of a positioning system refers to its ability of giving timely warnings when the system should not be used for positioning. The purpose of integrity information is to warn the user about situations where the system outputs a positioning signal that is ostensibly normal but actually so faulty that using it corrupts the resulting position estimate significantly. GPS satellites do broadcast integrity information to the users, but unfortunately, this information is only available after a delay: The terrestrial control segment must first detect and identify the satellite signal error and upload the information to the satellites for broadcasting to users. Using the faulty signal during this delay may have disastrous results in, e.g., aviation applications. A critical situation may occur in terrestrial vehicle positioning as well if, e.g., an ambulance is navigating to its destination using GPS but loses its way after using a contaminated satellite signal, arriving at the destination too late. A similar situation occurs if the source of an emergency call is located using GPS, and the position is computed using a defective satellite. Because of the inherent latency of GPS integrity information, methods of investigating the integrity of the signals inside the receiver have been developed. These methods are called Receiver Autonomous Integrity Monitoring (RAIM) and are based on evaluating the consistency of the positioning signals. RAIM can be used if redundant measurements are available, i.e., if the system of positioning equations is overdetermined. A lack of signal consistency can be due to an error in a signal or in the measurement model. Originally, RAIM was developed for civil aviation purposes. The idea was to eliminate gross measurement errors caused by, e.g., a satellite clock failure, and measurement model errors caused by inaccurate knowledge of the position of a satellite (ephemeris error). Thus, the purpose of RAIM was to detect system errors caused by the space or control segment. Such errors are rare: their frequency is estimated to be one error per 18 to 24 months. Because of this, a fundamental assumption in many RAIM schemes is that no more than one error may occur at a time. In practice, however, the positioning signal may be corrupt because of other reasons than those caused by the space and control segments. Signal reflections and multipath can cause ∗ In
literature, CERP and SERP are sometimes abbreviated CEP and SEP, respectively.
28
measurement errors significantly larger than normal measurement noise. With high sensitivity GPS (HSGPS) equipment, highly attenuated signals can be tracked at the cost of a strong amplification of noise. In the case of a poor signal-to-noise ratio, it is also possible that the receiver locks itself to the wrong satellite, causing the possibly correct signal to be associated with a wrong equation, yielding in a result similar to the case of a gross ephemeris error. This kind of errors are typical in personal positioning. Since they affect the positioning estimate in the same way as space or control segment errors do, they can be detected and excluded using the same methodology as in conventional RAIM. Designing RAIM algorithms for personal positioning has some additional challenges: these errors occur considerably more often than space or control segment errors, and thus, it is more likely that multiple errors occur simultaneously. In addition to in-receiver integrity checking, external GPS monitoring station networks are available. These networks also broadcast information on GPS integrity, but unfortunately, this information comes with a latency similar to that of the GPS navigation message. Moreover, monitoring networks cannot detect errors caused by the surroundings of the user.
3.1.3 Reliability In general, reliability can be defined in many ways: • Applicability of a device or system to its purpose as a function of time
• The ability of a device or system to perform the required operation under given conditions is maintained during a given period of time • Probability that an operational unit carries out the required operation for a given period of time under given conditions • A device or system meets its performance specifications • Robustness of a device or system
In the context of satellite navigation, reliability may refer to two things: the reliability of operation of the system or the statistical reliability of the positioning results. In the former case, reliability is associated with the reliability of operation of devices and components, i.e., the probability of their correct operation, where reliability is quantified using, e.g., the failure probabilities of the components as a function of time. In contrast, statistical reliability refers to the consistency of the measurement sequence. In satellite positioning, measurements made simultaneously from multiple sources are available. This allows for examining their mutual consistency if the positioning problem is overdetermined, i.e., if more measurements than unknowns are available. In the context of this course, positioning reliability is considered from the point of view of the statistical reliability of the positioning results. In the analysis of geodetic measurement networks, reliability refers to the capability of the estimation to detect gross errors [15, 16]. This definition is analogous to the integrity monitored by RAIM algorithms. Such a definition of reliability is applicable to analyzing the performance of personal positioning [17]. 29
The geometry of the positioning problem affects not only the accuracy of positioning but also the efficiency of reliability and integrity monitoring based on consistency tests. The traditional DOP value used for estimating positioning accuracy is not in all cases sufficient for describing the applicability of the problem to consistency evaluation..
3.1.4 Availability The availability of positioning is affected by many factors. In satellite positioning, the first requirement for availability is that the receiver must be able to receive and track a sufficient number of satellite signals. This does not only depend on the satellite constellation but on the surroundings as well—the weak satellite signals cannot penetrate thick concrete structures. Although positioning availability may be excellent on the roof of a skyscraper, it may be poor inside the building, on the street level, in tunnels, or in underground facilities. When assessing positioning availability, there may be additional requirements on positioning quality (i.e., accuracy and reliability) which usually decrease the quantity describing availability (probability or time percentage of availability). For instance, we may need to estimate positioning availability when it is required that all satellites used for positioning must be at an elevation of at least 5◦ and they must constitute a geometry with PDOP no larger than 6. It is also possible to possible to consider the availability of reliability or integrity information separately—to get this kind of information, more satellites are needed than for positioning only, and additional constraints are imposed on the geometry. If a position solution is rejected because it is suspected to be erroneous based on integrity or reliability evaluations and there is no possibility to exclude the error, or if sufficient reliability is not attained because of a deficient number of satellites or a bad geometry, the availability of positioning decreases. On the other hand, if integrity and reliability evaluations are omitted and all computed position solutions are accepted, the mean positioning accuracy is usually degraded, particularly in personal positioning.
3.2 Statistical Inference and Hypothesis Testing This chapter uses the frequentistic interpretation of probability as opposed to the Bayesian interpretation [2, p. 34]. Many algorithms for positioning integrity and reliability monitoring are based on statistical inference and hypothesis testing. The methods presented in this chapter are reviewed in more detail in [12, 15, 16, 17, 30]. The purpose of statistical inference is to find out the state of affairs based on evidence. A typical question would be, e.g., “Is this industrial process producing lowerquality products than before, or can the bad quality of this sample be only a coincidence?”. When considering a set of measurements, we often deal with the question if an anomalous measurement is erroneous or can the observed anomaly be a coincidence, given the normal variance of the observable.
30
For statistical inference, the question is formulated as a hypothesis, i.e., as an assumption on the state of affairs. These assumptions (which can be either true or false) are called statistical hypotheses. Usually, a hypothesis deals with one parameter of the probability distribution of the phenomenon in question. The purpose of hypothesis testing is not to prove the hypothesis to be true or false. Instead, the goal is to show that the hypothesis is not plausible because it leads to a negligible probability, i.e., the observations differ significantly from measurements that would be likely if the hypothesis was true. Example 1. We want to find out if a coin used for tossing up is fair, i.e., does the coin come down heads as frequently as tails. To investigate this, we formulate the hypothesis “the coin is fair”: p = 0.5 where p is the probability of the coin coming down heads. Now we toss the coin 20 times, and 16 of the tosses result in heads. In this case we are likely to reject the hypothesis “the coin is fair” because if it was true, a result of “16 out of 20 times heads” is not credible as it differs considerably from what we expect based on the hypothesis. The hypothesis to be questioned is chosen such that it represents status quo, i.e., a neutral situation requiring no actions to be taken. For this reason, it is usually called the null hypothesis H0 . When considering the state of the production process, the null hypothesis would be “no change has occurred in product quality” (no need to identify and repair a failure); in measurement data analysis, the null hypothesis would be “all measurements are healthy” (no need to identify and exclude the contaminated observation); in the coin toss example, we would have “the coin is fair” as the null hypothesis (no need to replace the coin). Hypothesis testing can be described as a five-phase process: 1. Formulate a practical problem as a hypothesis. Let us start with the alternative hypothesis H1 . It should cover the situations to be diagnosed such that a positive test result means that we have to take some action to remedy the situation. In contrast, the null hypothesis is chosen to represent status quo, and together with the alternative hypothesis, it should cover all possible situations. For example: • H1 : “The coin is biased”, p 6= 0.5 (the coin must be replaced) H0 : “The coin is fair”, p = 0.5; p is the probability of the coin coming down heads. • H1 : “The altered process B improves the quality of the product compared with the original process A”, µB > µA H0 : “The altered process B does not improve the quality of the product compared to the original process A”, µB ≤ µA ; µA and µB are expected values of a quantity describing the quality of the product (with higher values corresponding to better quality). 2. Choose a test statistic T that is a function of the observations. A good test statistic has the following properties: (a) It behaves in a different way when H0 is true than when H1 is true. 31
(b) Its probability distribution can be computed assuming H0 is true. 3. Choose the critical region. If the value of the test statistic T lies in the critical region, H0 is rejected in favor of H1 ; otherwise, H0 is not rejected. Note that no decision of accepting H0 is made—if there is no evidence supporting H1, we only decide not to reject H0 . When choosing the critical region, the task is to decide which values of the test statistic strongly imply that H1 holds. For example, the following serve as critical regions: Right-sided H1 : T > Tcr . If the test statistic exceeds the (right-side) critical value, H0 is rejected. Left-sided H1 : T < Tcr . If the test statistic is smaller than the (left-side) critical value, H0 is rejected. Two-sided H1 : T < Tcr1 tai T > Tcr2 . If the test statistic is either greater than the rightside critical value or smaller than the left-side critical value, H0 is rejected. 4. Decide the size of the critical region. The starting point of statistical inference is that the decision made on rejecting or not rejecting H0 may be correct or wrong. The inference may lead to a situation where H0 is rejected although in reality it is true; this is called a type I error. It may also happen that H0 is not rejected even though it actually is false; such a situation is called a type II error. When determining the size of the critical region, it must be decided how big a risk of making a wrong decision is acceptable. The probability of a type I error is called the significance level (or risk level) of the test and is denoted by α. Frequently used significant levels include 5 %, 1 %, and 0.1 %. Our confidence that the decision of not rejecting the null hypothesis is correct is reflected by the confidence level 1 − α which corresponds to the probability of the correct decision “H0 is not rejected when H0 holds”. The probability of a type II error is denoted by β. The power of test 1 −β is the probability that a real change of the status quo situation represented by H0 is detected because of a change in the test statistic. Table 3.1 summarizes the four possible ways of making a (right or wrong) decision in hypothesis testing. Table 3.1: Testing a null hypothesis.
Reality
Decision H0 not rejected
H0 true
H0 rejected
correct decision type I error probability 1 − α probability α = confidence level = significance level
H0 false type II error probability β
correct decision probability 1 − β = power of test
32
5. See if the computed value of T lies in the critical region, and if it does, in which part of it. If T lies in the critical region but near the boundary, we deduce that there is some evidence suggesting that H0 should be rejected. In contrast, if T lies deep in the critical region (far from the boundary), we have strong evidence for rejecting H0. Example 2 (α, β and the magnitude of the detectable difference, [30]). We shall illustrate hypothesis testing with an example where the geometry of the problem is quite simple compared to typical positioning problems. Consider the weights of male students. The null hypothesis is that in a certain university, the mean weight of male students is 68 kilograms. Let us test the hypotheses H0 : µ = 68 kg and H1 : µ 6= 68 kg; here, the alternative hypothesis covers both µ < 68 kg and µ > 68 kg. We take the sample average X¯ as our test statistic. If it falls close to the assumed value of 68 kg, we have evidence supporting H0 ; on the other hand, if the sample average is significantly smaller or greater than 68 kg, the sample is inconsistent with H0 and supports the alternative hypothesis H1 . We choose (somewhat arbitrarily) the critical region to consist of the intervals X¯ < 67 kg and X¯ > 69 kg; the region of not rejecting H0 is 67 kg ≤ X¯ ≤ 69 kg. These regions are visualized in Figure 3.1. H0 rejected µ 6= 68 kg
H0 not rejected µ = 68 kg
H0 rejected µ 6= 68 kg test statistic
Figure 3.1: The critical region (on the sides).
Let us now compute the probabilities of type I and II errors using the decision criterion of Figure 3.1. Suppose that the standard deviation of weight in the population is σ = 3.6 kg. The sample average used as the test statistic is computed from a randomly chosen set of n = 36 samples. It is known, by the central limit theorem, that the distribution of X¯ tends to a √ Gaussian distribution with standard deviation σX¯ → σ/ n = 3.6 kg/6 = 0.6 kg. probability density
α/2
α/2 67
µ 68
69
test statistic
Figure 3.2: Critical region when testing the hypothesis µ = 68 kg against the hypothesis µ 6= 68 kg.
The significance level α of the test, i.e., the probability of a type I error (where H0 is rejected although it is true) is equal to the total area of the shaded regions in Fig. 3.2. Now, the proba2 ¯ ¯ bility that X lies in the critical region when X ∼ N 68 kg, (0.6 kg) is ¯ = 2 P (X¯ < 67 kg) = 0.0956. α = P (X¯ < 67 kg) + P (69 kg < X) 33
According to the result, 9.6 % of all sets of 36 samples would cause rejecting the hypothesis µ = 68 kg although the hypothesis would hold. In order to decrease the probability α, either the sample size n should be increased (thus decreasing the sample variance) or the non-rejecting region should be expanded. Suppose the sample size is increased to n = 64. Then, σX¯ = 3.6/8 = 0.45 and α = 2 P (X¯ < 67 kg) = 0.0263. Decreasing the probability of a type I error α does not yet guarantee a good test—we also want the probability of a type II error β to be sufficiently small. To investigate the probability β, we evaluate it for fixed values of µ that fulfill the alternative hypothesis µ 6= 68 kg. We want to ensure, e.g., that H0 is rejected with a high probability if the true expectation value µ ≤ 66 kg or µ ≥ 70 kg. In that case, β can be computed for the alternative hypotheses µ = 66 kg or µ = 70 kg. Because of the symmetry, it is sufficient to examine the probability of not rejecting the null hypothesis µ = 68 kg when the alternative hypothesis H1 : µ = 70 kg holds. In that case, a type II error occurs if the sample mean falls to the interval [67 kg, 69 kg] when H1 is true. According to the left plot of Figure 3.3, β = P (67 kg ≤ X¯ ≤ 69 kg when µ = 70 kg) . probability density
H
H1 β
β 67 68 69 70
67
68
69
test statistic
68.5
Figure 3.3: Probability of a type II error when testing µ = 68 kg against the alternative hypotheses µ = 70 kg (left) and µ = 68.5 kg (right).
Still suppose that n = 64. Then, according to the alternative hypothesis, X¯ ∼ N 70 kg, (0.45 kg2 , and
β = P (67 kg ≤ X¯ ≤ 69 kg) = P (X¯ ≤ 69 kg) − P (X¯ ≤ 67 kg) = 0.0131 − 0.0000 = 0.0131.
If the true µ equals 66 kg, then β equals the same 0.0131. For other possible values µ < 66 kg or µ > 70 kg, the probability of a type II error β is even smaller; thus, the probability of not rejecting H0 is at most 0.0131 if H0 holds. The probability β will increase if the true µ is not equal but tending to the value assumed by the null hypothesis. On the right hand side of Fig. 3.3 2 ¯ there is an example where the true mean µ = 68.5 kg. Then, X ∼ N 68.5 kg, (0.45 kg) , and β = P (67 kg ≤ X¯ ≤ 69 kg) = 0.8667 − 0.0004 = 0.8663.
In conclusion, the probabilities of type I and II errors α and β depend on each other, and β depends on the distance between the true mean and the value presumed by the null hypothesis. 34
3.3 Residuals In this section, the properties of residuals are reviewed. Moreover, some notation different from what has previously been used in this hand-out and [2] is introduced. It is assumed that the observables and unknowns are related by a linear measurement model of the form y = Hx + ε
(3.1)
where the n × 1 vector y contains the measurements, ε contains measurement errors, the m × 1 vector x is composed by the states (i.e., the parameters to be estimated), and the measurement matrix H describes the linear relation between the measurements and states. In the context of positioning, the matrix H is commonly referred to as the geometry matrix. In practical positioning applications, however, the underlying model is usually nonlinear; in such cases, H is obtained by linearizing the system of nonlinear measurement equations, x is the change of state obtained in the last iteration cycle, and y is the difference between observed and computed measurements. Suppose the position is resolved using the weighted least squares (WLS) method. If the weights are chosen as the inverse of the covariance matrix of the measurements, the WLS method yields the best linear unbiased estimate (BLUE). The covariance of measurement errors is assumed known: V (y) = V (εε) = Σ with Σ > 0. Then, the WLS solution is given by xˆ = HT Σ−1 H
−1
HT Σ−1 y = Ky.
(3.2)
Furthermore, we assume that the measurement covariance matrix is diagonal, i.e., the measurements are mutually uncorrelated. The residuals v can be written as a function of the measurements: v = Hˆx − y = HK y − y = (HK − I) y = −Ry
(3.3)
where the matrix R is called the redundancy matrix. It is left to the reader to show that the redundancy matrix is idempotent, i.e., R2 = R. The covariance of the residuals can be expressed using the geometry matrix and the covariance of the measurements (see Problem 3.2): Cv = V (v) = (HK − I)V (y) (HK − I)T = Σ − H HT Σ−1 H
−1
HT .
(3.4)
Furthermore, the redundancy matrix can be expressed using the covariance matrices of the residuals and the measurements (Problem 3.2): R = Cv Σ−1 .
(3.5)
3.3.1 Quadratic Form of Residuals The quadratic form of residuals vT Σ−1 v is a common test statistic in positioning reliability analysis. We assume that measurement errors in (3.1) are normally distributed: ε ∼ N (µµ, Σ). 35
The quadratic form can be written using the measurements: vT Σ−1 v = (−Ry)T Σ−1 (−Ry) = ε T RT Σ−1 Rεε . Let us write A = RT Σ−1 R. To be able to use the result [2, Theorem 6, p. 11], we must ensure the idempotency of the matrix AΣ. Using (3.5), we may write A = RT Σ−1 R = Cv Σ−1
T
Σ−1 Cv Σ−1 = Σ−1 Cv Σ−1 Cv Σ−1 = Σ−1 RR = Σ−1 R,
(3.6)
because Σ−1 and Cv are symmetric matrices and R is idempotent. This gives AΣ and AΣAΣ: AΣ = Σ−1 RΣ AΣAΣ = Σ−1 RΣΣ−1 RΣ = Σ−1 RRΣ = Σ−1 RΣ = AΣ. Hence, RT Σ−1 RΣ is idempotent and vT Σ−1 v ∼ χ2 (n − m, λ) where rank(RT Σ−1 R) = n − m (Problem 3.3) is the number of degrees of freedom, n is the number of measurement equations, and m is the number of states to be estimated. If measurement errors are assumed to have zero mean, i.e., µ = 0, the noncentrality parameter λ equals zero and the quadratic form follows a central χ2 distribution. On the other hand, if we assume that not all measurement errors have zero mean, i.e., µ 6= 0, the noncentrality parameter becomes λ = µ T Aµµ = µ T RT Σ−1 Rµµ and the quadratic form is noncentrally χ2 distributed. Consider a case ε ∼ N (µµ, Σ) where the measurement yi is biased, i.e., its error has a nonzero mean while other measurement errors have zero mean: E (εi ) =µi 6= 0 E ε j =µ j = 0 ∀ j 6= i with j = 1 . . . n.
Denote the diagonal elements of Σ as [Σ](i,i) = σ2i . Now, the value of the noncentrality parameter λ can be computed based on (3.5) and (3.6) utilizing the diagonality of Σ: λ = µ T Aµµ = µ2i [A](i,i) = µ2i Σ−1 R (i,i) µ2 = µ2i Σ−1 Cv Σ−1 (i,i) = i4 [Cv ](i,i) . σi
(3.7)
3.3.2 Standardized Residuals Standardized residuals wi are obtained by dividing residuals by the square roots of their respective variances: wi = q
vi
,
[Cv ](i,i)
i = 1 . . . n.
(3.8)
Consider again a case ε ∼ N (µµ, Σ) with yi being biased while other measurement errors have zero mean: µi 6= 0, µ j = 0 ∀ j 6= i where j = 1 . . . n. 36
By (3.3), the expectation value of the residual vector is E v j = E [−Ry] j = − [R]( j,i) µi ,
and the expectation of the standardized residuals is q q E w j = E v j / [Cv ]( j, j) = −µi [R]( j,i) / [Cv ]( j, j) q = −µi [Cv ]( j,i) Σ−1 (i,i) / [Cv ]( j, j) q 2 = −µi /σi [Cv ]( j,i) / [Cv ]( j, j) .
The expected value of the standardized residual of measurement i becomes q E (wi ) = −µi /σ2i [Cv ](i,i) .
(3.9)
3.4 Receiver Autonomous Integrity Monitoring RAIM consists of two steps. First, the algorithm needs to find out if the measurements or measurement models are erroneous or not; this is called fault detection. In the second phase, the algorithms searches for a combination of measurement that does not contain erroneous equations. This step is carried out only if an error is detected in the first phase. There are two approaches to carrying out the second step of RAIM. The first alternative is to identify the faulty equation and exclude it from computations; this method is called Fault Detection and Identification (FDI). The other approach is to construct such a combination of the available measurements that seems to be error-free, i.e., does not trigger a fault alarm in the first step of RAIM; this approach is known as Fault Detection and Exclusion (FDE). Fault exclusion does not require pinpointing the faulty measurement: Suppose that measurement 3 is faulty. Now, if we exclude measurements 3, 4, and 5, we have eliminated the error but not identified it precisely. As FDI and FDE have the same goal (i.e., excluding the faulty equation), the whole process is often referred to as FDE. If satellite positioning is only used as a complementary means of navigation, sole fault detection may be sufficient. In such a scenario, the system will use another method of navigation, e.g., INS, if RAIM detects a fault in the satellite navigation equations. In contrast, if satellite positioning is the only available navigation method, excluding the error is obviouly preferable to beíng totally left without a position solution. Using RAIM methods poses a requirement of redundancy. While traditional satellite positioning requires a minimum of four measurements for a position solution, RAIM needs at least five measurements for fault detection and six or more for FDE. Many virtually different RAIM methods exist: e.g., the parity space method, range comparison, and the least squares residual method. In GPS literature, these have been shown to be mutually equivalent; hence, in this section we will concentrate on the approach based on least squares residuals. 37
3.4.1 Fault Detection We take (3.1) as our measurement model and Σ = σ2 I as the measurement Then, the −1covariance. T T weight matrix in the WLS solution (3.2) is identity, yielding K = H H H and a redundancy −1 T T matrix of R = I − H H H H . Our observable for fault detection is the sum of the squares of the residuals, commonly known as SSE (Sum of Squares of Errors): SSE = vT v. In the case of zero-mean measurement errors, SSE ∼ χ2 (n − 4) where n is the number of available measurements. However, if the measurements are not zero-mean, SSE follows a noncentral distribution: SSE ∼ χ2 (n − 4, λ).
The test statistic is chosen to be T=
r
SSE = n−4
s
vT v n−4
where the purpose of taking the square root and dividing by the number of degrees of freedom is to reduce dependency on satellite geometry. As the critical value we take s σ2 × χ21−p f a (n − 4) Tcr = , n−4 where χ21−p f a (n − 4) is the inverse cumulative χ2 density function with the probability 1 − p f a as its argument and the number of degrees of freedom in the parentheses. The probability p f a is the false alarm rate. In typical RAIM implementations, p f a is given a very small value; e.g., in civil aviation with GPS being the primary means of navigation, p f a = 0.333 · 10−6 [13].
3.4.2 Effect of Geometry The effect of satellite geometry on the error detection capability of RAIM is investigated by comparing the impacts of a measurement error on the position estimate and on the residual. If an error affects the residual only slightly but has a significant effect on the position solution, RAIM cannot protect the user from large position estimate errors very well. In such cases, the user must be at least warned—a bad geometry does not necessarily imply measurement faults, but if a fault occurs, it is unlikely to be noticed. In the following, the concept of Approximate Radial Error Protected (ARP) is presented as an example. The purpose of ARP is to detect geometries disadvantageous for RAIM; other methods for ensuring a geometry sufficient for RAIM have been developed as well. ARP gives a coarse approximation of the maximum position error that RAIM may fail to detect, and is computed by ARP = Smax Tcr
38
where Smax is the maximum of the slopes Si describing the relation between the test statistic and a bias in the ith measurement: v u u [K]2(1,i) + [K]2(2,i) (n − 4) t . Si = [R](i,i)
Here, we assume that the geometry is expressed in the ENU frame; then, the top two rows of the matrix K correspond to the horizontal components of the position estimate. Now, to screen out bad geometries, ARP is compared to a predefined threshold value ARPcr : if ARP < ARPcr , RAIM is considered unavailable. Otherwise, when ARP ≥ ARPcr , RAIM is just unable to detect position errors smaller than the ARP threshold.
3.4.3 Fault Identification/Exclusion If a fault is detected and n ≥ 6 measurements are available, we construct n measurement subsets such that one of the original satellites is missing from each subset. Then, the ARP value is computed for each subset; if all of them are acceptable, the corresponding test statistics are evaluated. If there is exactly one biased measurement present, it should be missing from one of the subsets, and thus, the test statistic value of that subset should be less than the critical value as opposed to other subsets: they contain the fault and their test statistic values should exceed the critical value. If the subset testing method finds only one subset whose test statistic does not trigger a fault detection alarm, the measurement missing from that subset is regarded as faulty and the position solution corresponding to that subset is accepted. In contrast, if multiple subsets have a test statistic value less than the critical value or not all subsets pass the ARP test, the error cannot be reliably identified. Then, the obtained position estimate cannot be improved and is flagged as unreliable.
3.5 Reliability Testing Based on Global and Local Tests The reliability testing method presented in this section originates from reliability analysis of geodetic measurement networks [16] where the concepts are somewhat different from those of RAIM. Nevertheless, both are based on similar principles: statistical reliability testing is used to investigate if the measurements agree with assumptions. First, the entire set of measurements is examined in a global test to detect outliers. Then, if an error was detected, local tests are conducted in order to identify the erroneous measurements. The method is theoretically based on the case of a single outlier but has been extended to practical applications with multiple simultaneous errors. For examples and further reading, see [17].
39
The null hypothesis is “no biased measurements”, under which all measurement errors have zero mean. The alternative hypothesis is “at least one measurement is biased”, i.e., the measurement errors are not zero-mean: H0 : H1 :
ε ∼ N (0, Σ) ε ∼ N (µµ, Σ) where µ 6= 0.
3.5.1 Global Test The purpose of the global test is to find out if the set of measurements contains faults or not. The quadratic form of the residuals serves as the test statistic: T = vT Σ−1 v. According to Section 3.3.1, T follows a central χ2 distribution if H0 holds, and a noncentral χ2 distribution if H1 is true. The number of degrees of freedom n − m is determined by the number of available measurements n and the number of unknown states m. In contrast, choosing the noncentrality parameter λ is a part of designing the test. As a random variable following a central χ2 distribution gets smaller values than a noncentrally χ2 distributed variable with as many of degrees of freedom, we choose a right-sided critical region. Thus, our hypotheses to be tested are H0 : H1 :
T ≤ Tcr
T > Tcr .
For determining the critical value, we decide the significance level α of the test, i.e., an acceptable probability of a type I error. The critical value Tcr is chosen such that T ∼ χ2 (n − p) implies P (T > Tcr ) = α, i.e., Tcr = χ21−α (n − m) where χ2p (·) denotes the inverse cumulative χ2 density function with p as its argument and the degrees of freedom in parentheses. Conducting the global test does not require choosing the probability β or the noncentrality parameter λ. Nevertheless, it is good to be familiar with the dependency between them. Furthermore, using these values, the parameters of the global test can be related to those of the local test. Based on Example 2 (p. 33) it was stated that the probabilities of type I and II errors are interdependent, and the magnitude of β is affected by the difference between the null hypothesis assumption and reality. This is utilized when choosing the value of λ: first, we decide an acceptable probability of wrong decision β; then, we find λ such that the cumulative density function of χ2 (n − m, λ) evaluates to β at the same value of T as where the cumulative density function of χ2 (n − m) evaluates to 1 − α, i.e., χ21−α (n − m) = χ2β (n − m, λ). 40
probability density 0.25
n−m=3 α = 0.11 β = 0.20
0.2
λ = 6.03
0.15
Tkr
0.1
β 0.05
0 −5
α
0
5
10
15
20
25
test statistic
Figure 3.4: Relation between the parameters α, β, λ, and Tcr .
Figure 3.4 visualizes the choice of the parameters in a case with 3 degrees of freedom. The probabilities α = 0.11 and β = 0.20 have been chosen such that the regions corresponding to them are clearly visible in the figure. However, in practice, these probabilities—particularly α— are chosen to be much smaller, e.g., α = 0.001; commonly used values of β are, e.g., 0.2, 0.1, and 0.05.
3.5.2 Local Test If the global test results in T > Tcr , we deduce that at least one biased measurement is present. The local test attempts to identify observations that should be suspected of blunders. According to H0 , the distribution of the measurement errors ε is N (0, Σ), and conseT quently, the residuals v follow a N 0, RΣR distribution. As the standardized residuals are, according to (3.8), obtained by scaling each residual by the reciprocal of its standard deviation, we have wi ∼ N (0, 1). The hypotheses to be tested are reduced to concern the standard normal distribution by taking the standardized residual as the test statistic. As opposed to the global test examining the entire residual vector at once, each residual is tested separately in the local test. The hypotheses concerning the residual wi are H0 : H1 :
wi ∼ N (0, 1)
wi ∼ N (δ, 1) where δ 6= 0.
The corresponding regions of the test statistic are H0,i : H1,i :
− wcr ≤ wi ≤ wcr
wi < −wcr or wi > wcr . 41
Since the distribution is symmetric, we may reduce these two-sided tests to right-sided by taking the absolute value of the standardized residual as the test statistic of the local test: |wi | ≤ wcr
H0,i :
|wi | > wcr .
H1,i :
In the local test, the alternative hypothesis states that the mean δ of the distribution is nonzero, whereas in the global test, the noncentrality parameter λ is assumed to be nonzero. These parameters are interdependent; thus, the quantities δ and λ associated with the alternative hypotheses can be determined using each other. As in both tests the probability β depends on the deviation of the parameter of the null hypothesis (λ = 0 and δ = 0) from reality and vice versa, it is natural to use the same probability β for both δ and λ. Both in the global and local test, the alternative hypothesis is based on the assumption that measurement errors are normally distributed but have a nonzero mean. Suppose all measurement errors are normally distributed with zero mean except for the ith measurement that is biased:E (εi ) = µi . Let us now compare how the bias in measurement i affects the parameter λ and the expectation value of the ith standardized residual E (wi ). Equations (3.7) and (3.9) show √ 2 that λ = E (wi ) , yielding |E (wi )| = λ. Since we are now considering the absolute value of wi in the local test, we are only interested in those distributions of the alternative hypothesis where the mean is positive. Then, √ δ = |E (wi )| = λ. 0.4
probability density
0.35
δ δ = 2.45
0.3
w cr = 1.61
a 0.25
α0 = 0.107
b
0.2 0.15
β
0.1
0 −6
α0/2
α0/2
0.05
−4
−2
0
2
4
6
8
test statistic
Figure 3.5: Relation between the parameters α0 , β, δ, and wcr .
Two of the parameters α0 , β, δ, and wcr need to be fixed for determining the rest. Figure 3.5 shows the following relations: a = n1−α0 /2 (0, 1) = wcr b = δ − nβ (δ, 1) = n1−β (δ, 1) − δ = n1−β (0, 1) + δ − δ δ = a + b = n1−α0 /2 (0, 1) + n1−β (0, 1) 42
where n p (·) is the inverse cumulative normal distribution function with p as its argument and the parameters of the distribution in parentheses.
3.5.3 Internal and External Reliability The internal reliability of a positioning problem is quantified by the minimum detectable bias (MDB; also referred to as marginally detectable bias), i.e., the smallest measurement error that can be detected by means of statistical testing. In contrast, external reliability expresses the effect of the minimum detectable bias on the position estimate. The expectance value δ corresponds to the smallest bias in a standardized residual that can be detected at probability 1 − β with a confidence level of 1 − α. The absolute value of MDB for measurement i is given by δ [Σ](i,i) . mi = q [Cv ](i,i)
(3.10)
External reliability describes the impact of the MDB on the position estimate. Suppose measurement i is biased by mi and other measurements are unbiased, i.e., the measurement error vector equals ∆yi = [0 . . . 0 mi 0 . . . 0]T . We can now compute the external reliability: ei = K∆yi = [K](1:n,i) mi = HΣ−1 HT HT Σ−1 (1:n,i) mi . (3.11)
The purpose of external reliability is to describe the magnitude of the errors from which statistical testing can protect—errors smaller than the external reliability are not detected. When considering a typical positioning problem with the first three components of the state vector referring to position coordinates, external reliability can be used for computing the radius of the sphere containing all position estimates biased by undetected measurement errors: q rPPLi = e2i1 + e2i2 + e2i3 . External reliability is also known as the position protection level (PPL). If the positioning computations have been done in the ENU frame or the external reliability ei is transformed to that frame, we may compute the horizontal protection level (HPL) q rHPLi = e2i1 + e2i2 , describing the radius of the circle containing all horizontal position biases caused by undetected measurement errors.
The protection levels rPPLi and rHPLi are different for each measurement; hence, a (somewhat pessimistic) estimate of the total protection level can be obtained by computing the protection level for each measurement and choosing the maximum: rPPLi = maxi rPPLi , and correspondingly for HPL. In this context, a larger protection level means that larger errors may pass through the testing undetected. 43
As reliability testing fails at detecting the presence of smallest possible biases, they cannot be identified either. On the other hand, a gross measurement error may have a significant impact on the position estimate the iterative (weighted) least squares solution method converges to. In such cases, the linearization point may be very far from the true location, inducing significant errors in the geometry matrix as well. Such errors tend to increase all residuals, which usually can be seen in the test statistic of the global test. In contrast, pinpointing the true outlier can be difficult because the geometry matrix used in the analysis is faulty.
3.5.4 Fault Detection and Exclusion In practice, the FDE process is iterative. Detecting the presence of measurement errors is attempted by means of global testing. If errors are detected, a local test is coducted for each standardized residual wi . As more than one error may occur at a time, multiple standardized residuals not passing the local test may be encountered. A residual vi with a large absolute value may be due to an error in some other measurement than yi . The redundancy matrix R that maps measurement errors to residuals sometimes contains offdiagonal elements satisfying [R]( j,i) > [R](i,i) or [R](i, j) > [R](i,i) .
In the first case, an error in the ith observation affects the residual j more strongly than its respective residual. In the second case, residual i is more affected by measurement j than its corresponding measurement i. Both scenarios render fault identification either hard or impossible— the possibility of such a problem may be found out beforehand by examining the elements of the matrix R.
Because of the reasons mentioned above, a test statistic i not passing the local test should not be regarded as a reason to label the corresponding measurement yi as faulty; instead, it only suggests that yi suspicious. Then, the global test is repeated without the suspected measurement(s). If it still indicates the presence of errors, the testing continues. Finally, once (or if) a measurement combination that passes the global test is found, it should be tried if any of the excluded measurements could be included again without the global test indicating errors. This is due to the possibility that a gross outlier that has been later excluded may have strongly affected residuals whose corresponding measurements have been excluded first.
44
Problems 3.1.
(a) Show that the redundancy matrix R is idempotent (i.e., R2 = R). (b) What kind of a weight matrix Σ−1 is needed for the redundancy matrix to be symmetric?
3.2.
(a) Show that for the covariance of residuals, Cv = Σ − H HT Σ−1 H
(b) Show that R = Cv Σ−1
(3.5).
3.3. Show that rank(RT Σ−1 R) = n − m. Hint: show that (a) x ∈ N (RT Σ−1 R) if and only if x ∈ N (R)
(b) x ∈ N (R) if and only if x ∈ R (H) (c) dim (R (H)) = m
45
−1
HT
(3.4).
Index accelerometer, 4 accuracy, 27 AFM, 21 Allan variance, 13 availability, 30
FDE, 37 FDI, 37 fixed reference, 21 fixed solution, 22 float solution, 22
baseline, 19, 20
geometry matrix, 35 gyroscope mechanical, 5 ring laser, 7 tuning fork, 5
carrier phase, 18 carrier phase based positioning, 15 CERP, 28 Cholesky decomposition, 24 modified, 24 clock bias, 17, 21 satellite, 18 clock drift, 17 confidence level, 32 Coriolis force, 5 critical region, 32 cycle slip, 18, 22 delta range, 17 difference double, 21 single, 19 differential correction, 18 Doppler effect, 16 integrated, 17 shift, 16 ephemeris error, 28 error ephemeris, 18, 28 satellite clock, 18, 28 type I, 32 type II, 32 FARA, 21
hypothesis alternative, 31 null, 31 statistical, 31 testing, 31 inertia, 4 inertial navigation, 8 integer ambiguity, 18, 21 integer least squares, 23 integrity, 28 LAMBDA, 21, 23 transformation, 23 LSAST, 21 measurement carrier, 15 measurement errors, 9 measurement model carrier phase, 18 MEMS, 6 misalignment, 10 narrowlane, 23 parts per million, 10 46
power of test, 32 protection level horizontal, 43 position, 43 pseudo-observable, 22 RAIM, 28, 37 real-time kinematics, 21 redundancy matrix, 35 reference fixed, 21 receiver, 18, 19 reliability, 22, 29 external, 43 internal, 43 residual, 25, 35 standardized, 36 rover receiver, 19 RTK, 21 Sagnac effect, 6 scale factor, 10 sequential differencing, 21 SERP, 28 significance level, 32 status quo, 31 type I error, 32 type II error, 32 unit lower triangular matrix, 24 weighted norm, 23 widelane, 22 WLS, 35
47
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