Strapdown Inertial Navigation Systems (INS) Sensors and UAVs Avionic

Strapdown Inertial Navigation Systems (INS) Sensors and UAVs Avionic
Strapdown Inertial Navigation Systems (INS) Sensors and UAVs Avionic By Dr. Hamid Bolandhemmat

contents Overview of Inertial Navigation Systems (INS) and their application General navigation equations in spaced fixed and rotating reference frames Inertial sensors Gyroscopes Underlying theory Rebalance loop Random process - review Sources of errors calibration Accelerometers Calibration Compass Air data systems Pressure Altitude Airspeed Global Positioning Systems (GPS) Underlying concept Pseudo ranges

contents Global Positioning Systems (GPS) - continue Clock drift
Trilateration to measure position A recursive approach for trilateration Other source of errors – GPS error budget Velocity measurements – Doppler effect Differential GPS technology INS-GPS integration Complementary aspects of INS and GPS Multi-sensor inertial navigation systems Introduction to Estimation and Kalman filtering Vision systems Other sensors

Inertial Navigation Systems overview)
In order to autonomously travel from one point to another, information regarding position and attitude vector of the UAV must be determined: INS Control laws Servos UAV Navigation Mechanization Equations Inertial Sensors Sensor Fusion Filter Guidance Aiding Sensors These states are also required to calculated other flight parameters such as wind vector, crab angle, etc

2D navigation w.r.t. spaced fixed frame:
Yb=E Xb . q Zb N earth D

2D navigation w.r.t. spaced fixed frame:
Yb=E Xb . q Zb N D UAV

2D navigation w.r.t. rotating frame:
q Yb=E . Xb N Zb XE ZE D

2D navigation w.r.t. rotating frame:
q Yb=E . Xb D N Zb XE ZE

3D Strapdown Inertial Navigation Systems:
3 accelerometers 3 Gyros UAV Onboard Navigation processor N D E

Inertial Sensors - Gyroscopes
Angular momentum vector

Angular velocity measurement (the application of the precession principle): The spinning wheel or rotor is put on a gimbal inside an instrument case to be held isolated in space; any changes in the angles of the gimbal then will represent changes in the orientation of the case with respect to the reference direction maintained by the rotor angular momentum

Spin axis output axis Input axis Torque generator or springs

Gyroscopes – Rebalance loop
With the springs: With the torque generator:

Gyroscopes – Rebalance loop
With the torque generator: PI controller instead of a gain to guarantee zero steady state error :

Gyroscopes – dynamics equations
Gimbal 𝑥 𝐼 𝑧 𝐼 𝑦 𝐼 case

Gyroscopes – fiber optic gyros (FOG):
Underlying principle (Sagnac effect): The transit time for two counter-propagating beams of light travelling in a fiber optic ring is not the same when the ring is spinning. CW wave CCW wave

two counter-propagating beams of light are travelling through a looped fiber optic coils (in a closed path) The transmit time difference or the phase shift between the two beams are used to compute the rotational speed Light source Light detector The time difference: The phase shift:

Advantages: High reliability and low maintenance cost (no mechanical and spinning parts) Wide dynamic range Insensitivity to acceleration, shock and vibration Digital output Instant start-up time Light source Light detector

Gyroscopes – Ring laser gyros (RLG):
Underlying principle (Sagnac effect): Frequency and wave length of two counter-propagating beams of laser travelling within an optical cavity (a triangular closed path with reflecting mirrors at each corner) are different when the cavity is spinning. Photo Diodes Helium-Neon Laser Laser beams The frequency difference resulting from the difference in “effective path lengths” can then be calculated as: Wavelength stretches when it had to go farther in the cavity.

The frequency difference will be then: Laser beams Helium-Neon Laser Photo Diodes

Lock-in effect: due to imperfections in the lasing cavity (mirrors), below an input rate thresholds, there would be no output frequency difference. Dead zone The lock-in dead zone is in the order of 100deg/hr (earth rotation rate is 15deg/hr) Mechanical dither the laser block at Hz, with peak amplitude of approximately 1 arc-sec to remove the dead zone.

Gyroscopes – MEMS gyros:
In 2D, polar position coordinate of the object is: Double differentiating with respect to time, gives the acceleration:

Driven vibrations Coriolis forces Tuning fork Torsional vibration Size and shape of the tuning fork is designed such as the torsional vibration frequency is identical to the flexural frequency of the tuning fork. Courtesy of Ref. 3

Usually the sensing forks are coupled to a similar fork which produces the rate output signal: Courtesy of Ref. 2 Coriolis forces The piezo-electric drive tines are oscillated at precise amplitudes. In the presence of the angular velocity, the tines of the pick up fork move up and down in an out of the plane of the fork assembly. An electrical output signal is then produced by the pick up amplifier which is proportional to the input angular rate.

Random process – review
Expected Value (first moment of a random variable): It is the weighted average of a random variable: Let 𝑥 1 , 𝑥 2 ,… are instances of a random variable 𝑋. Also, the corresponding probabilities are 𝑝(𝑥 1 ), 𝑝 𝑥 2 ,…, then 𝑥 , the expected value of 𝑋, is defined by 𝑥 =𝐸 𝑋 = 𝑘 𝑥 𝑘 𝑝( 𝑥 𝑘 ) Expected value of 𝑓(𝑋), a function of 𝑋, is also defined as: 𝑓 =𝐸 𝑓(𝑋) = 𝑘 𝑓(𝑥 𝑘 )𝑝( 𝑥 𝑘 ) Variance of a random variable (the second moment) : It is an indication of how wide the random variable instances are spread around the expected value, For the random variable 𝑋 with expected value 𝑥 , variance is given by: 𝜎 2 =𝑉𝑎𝑟 𝑋 =𝐸 (𝑋− 𝑥 ) 2 =𝐸 𝑋 2 − 𝑥 2 Covariance of a random variable shows the distribution of the random variable with respect to its mean value, it resembles moment of inertial for a mass Standard deviation of a random variable is the square root of its variance, here it is 𝜎. The same concept is extended to multi-variable random variables.

Gyroscopes – error Terms
Constant bias [4]: Bias is the sensor average output when zero output is expected (no rotation). A constant bias error 𝜀 causes ∆𝜃=𝜀.𝑡 Thermo-mechanical noise: Bandwidth of The thermo-mechanical noise on the (MEMS) gyroscopes are much larger than the sampling frequency. As a result, the noise due to the thermo-mechanical fluctuations behaves similar to white noise. Assume that 𝑁 𝑖 is the 𝑖 𝑡ℎ sample of the sensor white noise sequence, then 0 𝑡 𝜀 𝑡 𝑑𝑡= ∆𝑡 𝑖=1 𝑛 𝑁 𝑖 Covariance of a random variable shows the distribution of the random variable with respect to its mean value, it resembles moment of inertial for a mass

Hence, the accumulated angle random walk error has a mean and covariance of [4]: Hence the output angle random walk has zero mean and a variance which grows proportional to square root of time:

Ref. 4

Flicker noise / Bias stability [4] Bias of the MEMS gyro changes due to flicker noise (low frequency noise with a 1/f spectrum) Bias fluctuations due to flicker noise is modelled as random walk (not accurate as the bias variance doesn’t grow with time). Bias stability parameter (1𝜎 value) is defined to show how the sensor bias changes over a specified period of time (usually 100sec – constant temperature) for example, if the bias stability (based on 100 seconds time period) is calculated to be 0.01 𝑑𝑒𝑔 ℎ𝑟 , it means that the sensor bias after 100sec, would have a mean equal to the original sensor bias and an standard deviation of 0.01 𝑑𝑒𝑔 ℎ𝑟 . If using the random walk model for the bias variations, then the variance change is expressed to be proportional to square root of time: 𝐵𝑅𝑊 𝑑𝑒𝑔 2 ℎ𝑟 = 𝐵𝑆( 𝑑𝑒𝑔 ℎ𝑟 ) 2 𝑡(ℎ)

Flicker noise / Bias stability [4] The attitude error due to the bias fluctuations (with bias random walk model), would then be a second order angle random walk model. Temperature variations: Changes in the sensor bias due to the temperature changes (could be also caused due to the electronics self heating) The effect is nonlinear usually of order 3 Either the sensors must be calibrated with temperature or the sensors unit must be temperature controlled.

Gyroscopes – error Terms – Allan variance
Allan variance method [Ref. 4]: It is a time domain technique to characterize the noise Is a function of “averaging period” For an averaging period 𝑇, the Allan variance is calculated as: Create 𝑛 bins of data out of a long sequence of the sensor readings where each bin contains data with the length of the averaging period 𝑇 (at least 9 bins are required). Take average of the data for each bin 𝑎(𝑇) 1 , 𝑎(𝑇) 2 ,…, 𝑎(𝑇) 𝑛 , where 𝑛 is total number of bins. Calculate the Allan variance by: Also, Allan deviation (equivalent to standard deviation) would be the square root of Allan variance: 𝐴𝑉𝐴𝑅 𝑇 = 1 2(𝑛−1) 𝑖=1 𝑛 ( 𝑎 𝑇 𝑖+1 − 𝑎 𝑇 𝑖 ) 2 𝐴𝐷 𝑇 = 𝐴𝑉𝐴𝑅(𝑇)

Gyroscopes – error Terms – Allan variance
Allan variance method is plotted as a function of the averaging period T on a log-log scale Different random process usually appear in different region of T: T Courtesy of Ref. 5

Gyroscopes – calibration
Calibration of the sensor suite is accomplished on temperature- controlled precise turn-tables (1 arcsec tilt accuracy) to determine the sensors bias, scale factor, misalignment factors, g-sensitivity factors, etc: For gyroscopes: For accelerometer: Kalman Filter or Least Square method to determine the unknown calibration parameters.

Accelerometers - Concept
Accelerometers measure specific force 𝑓= 𝑎−𝑔 = 1 𝑚 ( 𝐹 𝐴𝑒𝑟𝑜 + 𝐹 𝑇ℎ𝑟𝑢𝑠𝑡 ) g + − Displacement pick-off Signal proportional to specific force 𝑓 Proof mass Acceleration with respect to inertial space case 𝑎

Accelerometers – rebalance loop
Accelerometers measure specific force 𝑓= 𝑎−𝑔 = 1 𝑚 ( 𝐹 𝐴𝑒𝑟𝑜 + 𝐹 𝑇ℎ𝑟𝑢𝑠𝑡 ) Input axis 𝜃 Proof mass 𝑎 Angle pick-off controller Torque generator 𝑦 𝑧 𝑥

Accelerometers – rebalance loop
Euler’s law for the proof mass pendulous in the instrument case: 𝐼 𝜃 + 𝜑 +𝑐 𝜃 = 𝐹 𝑦 𝑏+ 𝐹 𝑧 𝜃𝑏− 𝑇 𝑇𝑔 1 𝐼 𝑠 2 +𝑐𝑠 𝐾(𝑠) Σ 𝑚𝑏 𝑘 𝑇𝑔 𝐹 𝑦 + − 𝑖 𝜃 𝑓 𝑧 𝜑 𝐼

Accelerometers – MEMS sensors
Spring and mass from Silicon Change in the displacement causes an output voltage due to the change in capacitance Courtesy of Ref. 6

COMPASS Compass is used to determine the UAV heading with respect to the magnetic north, i.e. 𝜓 𝑚 True North Magnetic North Local magnetic field 𝜓 𝑚 𝜓 𝛿 𝛿 is the declination angle: the angle between the true north and the magnetic north The declination angle varies from one location to another on the surface of the Earth ( function of longitude and latitude) [7]. The declination angle, at each longitude and latitude, can be obtained from WMM (World Magnetic Model). The earth magnetic field is very similar to that of a dipole. The magnetic field lines running normal to the earth’s surface at the poles and “parallel to the earth at the equators. The heading angle of the UAV would then be: 𝜓= 𝜓 𝑚 +𝛿

COMPASS A 3D compass measures components of the Earth’s local magnetic field intensity in the body frame (Earth’s magnetic field is 3 dimensional), N Magnetic North 𝛿 To calculate 𝜓 𝑚 , components of the magnetic field strength should be expressed in a local tangential frame (parallel to the local horizon) using the transformation matrix 𝐶 𝑏 𝑡 (𝜃,𝜑): 𝜓 𝑚 𝑦 𝑏 𝜓 𝑚 𝑚 𝑥 𝑏 𝑚 𝑡 = 𝐶 𝑏 𝑡 (𝜃,𝜑) 𝑚 𝑏 Where 𝑚 𝑏 is the compass measurements (in nano-Tesla) in the body frame: 𝑚 𝑧 𝑏 The earth magnetic field is very similar to that of a dipole. The magnetic field lines running normal to the earth’s surface at the poles and “parallel to the earth at the equators. 𝑚 𝑏 = 𝑚 𝑥 𝑏 𝑚 𝑦 𝑏 𝑚 𝑧 𝑏 𝑇 𝐶 𝑏 𝑡 (𝜃,𝜑) is the transformation matrix from the body frame to the tangential frame

COMPASS The transformation matrix, 𝐶 𝑏 𝑡 𝜃,𝜑 , is given by:
𝐶 𝑏 𝑡 𝜃,𝜑 = 𝑐𝑜𝑠𝜃 0 𝑠𝑖𝑛𝜃 −𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜑 −𝑠𝑖𝑛𝜑 0 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠𝜑 Where 𝜃 is the pitch attitude of the UAV and 𝜑 is its bank attitude. The magnetic heading can then be calculated by: 𝜓 𝑚 =−𝑎𝑡𝑎𝑛2( 𝑚 𝑦 𝑡 , 𝑚 𝑥 𝑡 ) Where 𝑎𝑡𝑎𝑛2 is the four quadrant inverse tangent function, resulting the heading in −𝜋,+𝜋 range (should be usually converted to the 0,+2𝜋 range for heading) The earth magnetic field is very similar to that of a dipole. The magnetic field lines running normal to the earth’s surface at the poles and “parallel to the earth at the equators. Compass measurements are sensitive to other electromagnetic fields which could be generated by UAV motors and servos or any other ferrous materials. Also, the measurements could be corrupted due to interferences from electric power lines, buildings, cars, etc.

Air Data Systems – Airspeed
The pitot-static sensors measure the dynamic pressure which is proportional to the airspeed square with half of the air density 𝜌 : 𝑝 𝑑 = 1 2 𝜌 𝑣 𝑎 2 Dynamic pressure is the difference between the total pressure and the static pressure: 𝑝 𝑑 = 𝑝 𝑡 − 𝑝 𝑠 Where 𝑝 𝑡 is the total pressure (stagnation pressure) and 𝑝 𝑠 is the static pressure (ambient pressure), The difference is directly measured by the pressure transducer inside the pitot-static tube (should be calibrated for pressure). The earth magnetic field is very similar to that of a dipole. The magnetic field lines running normal to the earth’s surface at the poles and “parallel to the earth at the equators.

Air Data Systems – Airspeed
Where 𝑝 𝑡 is the total pressure (stagnation pressure) and 𝑝 𝑠 is the static pressure (ambient pressure), The difference is directly measured by the pressure transducer inside the pitot-static tube (should be calibrated for pressure) [7]. 𝑝 𝑠 𝑝 𝑡 Diaphragm – differential pressure sensor 𝑣 𝑎 Static pressure 𝑝 𝑠 Total pressure 𝑝 𝑡 Pitot-static tube The earth magnetic field is very similar to that of a dipole. The magnetic field lines running normal to the earth’s surface at the poles and “parallel to the earth at the equators.

Global Positioning System (GPS) - Overview
Consists of a 24 satellites constellation orbiting the earth at a period of approximately 12hrs The orbital distance from the Earth is approximately 21000km (20180km) The orbital radius is approximately 26568km With no obstruction, at any point on the earth, there should be 4-6 satellites available to observe:

Global Positioning System (GPS) - Overview
The satellites information are submitted through two different codes: Coarse Acquisition (CA) and Precision (P). Carrier frequencies are 𝐿 1 = 𝑀𝐻𝑧 and 𝐿 2 =1227.6𝑀𝐻𝑧 CA is modulated only at 𝐿 1 frequency, P is carried with the both frequencies Everything is about measuring the travel time of the submitted radio waves to the receiver, Because of the synchronization error between the satellite clocks (atomic) and the receiver clocks, the distances are calculated with error also known as pseudo ranges. Trilateration of 4 pseudo ranges can then yield location of the receiver (longitude, latitude an height above WGS84) as well as the receiver clock bias. Low bandwidth measurements compare to the inertial sensors, usually between 1−4𝐻𝑧

Global Positioning System (GPS)
Pseudo ranges and time measurement (1 𝜇𝑠𝑒𝑐 at speed of light = 300m error): 𝑠 3 𝑠 2 𝑠 1 𝑠 4 𝑅 3 𝑅 2 𝑅 1 𝑅 4

Trilateration to find the absolute position (longitude and latitude) of the receiver by using the pseudo-ranges (4 satellites are required) 𝑠 3 𝑠 2 𝑠 1 𝑠 4 𝑅 2 𝑅 3 𝑅 1 𝑅 4 𝑋 𝑖 𝑋 𝑢 ( 𝑥 𝑢 , 𝑦 𝑢 , 𝑧 𝑢 ) 𝑥 𝐼 𝑧 𝐼 𝑦 𝐼

Global Positioning System (GPS) - Trilateration
Trilateration to find the absolute position (longitude and latitude) of the receiver by using the pseudo-ranges (4 satellites are required) Courtesy of Ref. 7

GPS – Pseudo Random Code
The pseudo random codes (they just look like random signals) received from each satellite is synchronized with the identical code generated in the receiver to measure the travel time (Code-Phase based) 1 1 1 1 1 1 1 1 1 1 … Receiver code ∆ 𝑡 1 … 𝑠 1 sat code ∆ 𝑡 2 … 𝑠 2 sat code ∆ 𝑡 3 … 𝑠 3 sat code ∆ 𝑡 4 … 𝑠 4 sat code The receivers clock is not as accurate as the satellites atomic clock (based on oscillation of a particular atom). The ∆ 𝑡 𝑖 ′ 𝑠 are corrupted with the receiver clock bias B

Because of the imperfection in the travel time measurements only the pseudo ranges 𝑅 𝑖𝑝 can be calculated: 𝑅 𝑖𝑝 =𝑐∆ 𝑡 𝑖 And relation between the pseudo range and the actual ranges is: 𝑅 𝑖𝑝 = 𝑅 𝑖 +𝑏 where 𝑏=𝐵𝑐 1𝜇𝑠𝑒𝑐 error in travel time measurement would be equivalent to approximately 300m range error. 𝑠 2 ( 𝑥 2 , 𝑦 2 ,𝑧 2 ) 𝑠 3 ( 𝑥 3 , 𝑦 3 ,𝑧 3 ) 𝑠 4 ( 𝑥 4 , 𝑦 4 ,𝑧 4 ) 𝑅 1 𝑅 2 𝑅 3 𝑅 4 𝑠 1 ( 𝑥 1 , 𝑦 1 ,𝑧 1 ) ( 𝑥 𝑢 , 𝑦 𝑢 ,𝑧 𝑢 )

GPS –Trilateration Considering the receiver clock bias, there are 4 unknown, so 4 satellite data are required to obtain the absolute position of the receiver( 𝑥 𝑢 , 𝑦 𝑢 ,𝑧 𝑢 ): 𝑠 2 ( 𝑥 2 , 𝑦 2 ,𝑧 2 ) 𝑠 3 ( 𝑥 3 , 𝑦 3 ,𝑧 3 ) 𝑠 4 ( 𝑥 4 , 𝑦 4 ,𝑧 4 ) 𝑅 1 𝑅 2 𝑅 3 𝑅 4 𝑠 1 ( 𝑥 1 , 𝑦 1 ,𝑧 1 ) ( 𝑥 𝑢 , 𝑦 𝑢 ,𝑧 𝑢 ) The concept of Trilateration from 3D geometry can be used to develop 4 nonlinear equations: 𝑅 𝑖 =[( 𝑥 𝑢 − 𝑥 𝑖 ) 2 +( 𝑦 𝑢 − 𝑦 𝑖 ) 2 +( 𝑧 𝑢 − 𝑧 𝑖 ) 2 ] = 𝑅 𝑖𝑝 −𝑏 , 𝑖=1,2,3,4 Where 𝑅 𝑖 are actual ranges, 𝑅 𝑖𝑝 are the pseudo ranges, and 𝐵 is the receiver clock bias 𝑅 𝑖𝑝 = 𝑅 𝑖 +𝑏 where 𝑏=𝐵𝑐

GPS – A Recursive solution for trilateration
Define vector 𝑧 as following: 𝑧 ≜ 𝑅 𝑢 𝑏 Where 𝑅 𝑢 is the receiver position vector, then 𝑦 = 𝜌 𝑧 + 𝑣 𝜌 is the vector of pseudo ranges 𝜌 = 𝑅 1𝑝 𝑅 2𝑝 𝑅 3𝑝 𝑅 4𝑝 𝑇 𝑅 𝑖𝑝 =[( 𝑥 𝑢 − 𝑥 𝑖 ) 2 +( 𝑥 𝑢 − 𝑥 𝑖 ) 2 +( 𝑥 𝑢 − 𝑥 𝑖 ) 2 ] 𝑏 And 𝑦 is the vector of scaled time measurements, i.e., 𝑦 =[ ⋯ 𝑐∆ 𝑡 𝑖 ⋯ ] 𝑇 , 𝑖=1,2,3,4 Finally, 𝑣 is the zero-mean white noise sequence corrupting the receiver time measurements

Pseudo range (travel time) measurement equations are written as: 𝑦 = 𝜌 𝑧 + 𝑣 Suppose that 𝑧 0 is an initial guess for 𝑧 . Linearize the above equation around 𝑧 0 : 𝑦 ≈ 𝜌 𝑧 𝐻 𝑧 − 𝑧 𝑣 Where 𝐻 is the Jacobian matrix of 𝜌 evaluated at 𝑧 = 𝑧 0 : 𝐻= 𝜕 𝜌 𝜕 𝑧 𝑧 = 𝑧 0 And, the pseudo-range vector 𝜌 𝑧 is given by: 𝜌 𝑧 0 = 𝑅 1𝑝 0 ( 𝑧 0 ) 𝑅 2𝑝 0 ( 𝑧 0 ) 𝑅 3𝑝 0 ( 𝑧 0 ) 𝑅 4𝑝 0 ( 𝑧 0 ) 𝑇 Where the initial guess for the pseudo range to the 𝑖 𝑡ℎ satellite is given by: 𝑅 𝑖𝑝 0 ( 𝑧 0 )=[( 𝑥 𝑢 0 − 𝑥 𝑖 ) 2 +( 𝑦 𝑢 0 − 𝑦 𝑖 ) 2 +( 𝑧 𝑢 0 − 𝑧 𝑖 ) 2 ] 𝑏 0

Linearized equations of pseudo range measurements around the initial guess 𝑧 0 : 𝑦 ≈ 𝜌 𝑧 𝐻 𝑧 − 𝑧 𝑣 Where 𝐻 is the Jacobian matrix: 𝐻= 𝜕 𝜌 𝜕 𝑧 𝑧 = 𝑧 0 The next guess for 𝑧 can be given by the Maximum Likelihood (ML) estimate: 𝑧 1 = 𝑧 0 +( 𝐻 𝑇 𝑅 −1 𝐻 ) −1 𝐻 𝑇 𝑅 −1 [ 𝑦 − 𝜌 𝑧 0 ] Where 𝑅 is the “scaled” time measurement noise covariance matrix ( 𝑣 𝑖 ’s have unit of length): 𝑅=𝐸 𝑣 . 𝑣 𝑇 = 𝜎 𝑣 2 𝐼 In steady state, when 𝜌 𝑧 𝑘 → 𝑦 , the ( 𝑧 𝑘 − 𝑧 𝑘−1 ) difference becomes negligible. In the other word: 𝑧 𝑘−1 → 𝑧 𝑘 .

GPS - A Recursive solution for trilateration
Then the receiver position estimation error characteristic is given by: 𝑒 = 𝑧 − 𝑧 ≈( 𝐻 𝑇 𝑅 −1 𝐻 ) −1 𝐻 𝑇 𝑅 −1 𝑣 And covariance of error will then be: 𝐸[ 𝑒 . 𝑒 𝑇 ]≈ 𝜎 𝑣 2 ( 𝐻 𝑇 𝐻 ) −1 Note that the error vector 𝑒 has defined in the earth-fixed coordinate system. To get the position errors in the NED frame, the transformation matrix from the earth frame to the NED frame, 𝐶 𝐸 𝑁 , is used 𝐸 ≜𝑆 𝑒 Where 𝑆 is a 4×4 matrix given by (𝑆 is orthogonal): 𝑆= 𝐶 𝐸 𝑁 The covariance in position error in the NED frame can then be obtained from:

GPS – Dilution of Precision (DOP)
The covariance in position error in the NED frame can then be obtained from: 𝐸 𝐸 . 𝐸 𝑇 =𝑆𝐸[ 𝑒 . 𝑒 𝑇 ] 𝑆 𝑇 Replacing for 𝐸[ 𝑒 . 𝑒 𝑇 ] from 𝐸[ 𝑒 . 𝑒 𝑇 ]≈ 𝜎 𝑣 2 ( 𝐻 𝑇 𝐻 ) −1 yields: 𝐸 𝐸 . 𝐸 𝑇 = 𝜎 𝑣 2 𝑆( 𝐻 𝑇 𝐻 ) −1 𝑆 −1 = 𝜎 𝑣 2 (𝑆 𝐻 𝑇 𝐻 𝑆 −1 ) −1 = 𝜎 𝑣 2 (𝐻 𝑡 𝑇 𝐻 𝑡 ) −1 Where 𝐻 𝑡 ≜𝐻 𝑆 𝑇 , now lets show the (𝐻 𝑡 𝑇 𝐻 𝑡 ) −1 diagonal elements by 𝑘 𝑖 ’s: (𝐻 𝑡 𝑇 𝐻 𝑡 ) −1 ≜ 𝑘 𝑘 𝑘 𝑘 4 Dilution Of Precision (DOP) coefficients are then defined to characterize how the range measurement errors are propagated into the receiver position errors in the North, East and altitude directions (NED frame).

The covariance in position error in the NED frame can then be obtained from: 𝐸 𝐸 . 𝐸 𝑇 = 𝜎 𝑣 𝑘 𝑘 𝑘 𝑘 4 Variance of the receiver position error in the horizontal NE plane is then given by 𝜎 𝑣 2 𝑘 1 + 𝑘 2 , and Horizontal Dilution Of Precision (HDOP) is defined as: 𝐻𝐷𝑂𝑃≜ 2 𝑘 1 + 𝑘 2 Variance of the receiver altitude error is given by 𝜎 𝑣 2 𝑘 3 , and Vertical Dilution Of Precision (VDOP) is defined as: V𝐷𝑂𝑃≜ 2 𝑘 3 Variance of the receiver position error is given by 𝜎 𝑣 2 𝑘 1 + 𝑘 2 +𝑘 3 , and Position Dilution Of Precision (PDOP) is defined as: 𝑃𝐷𝑂𝑃≜ 2 𝑘 1 + 𝑘 2 + 𝑘 3

The covariance in position error in the NED frame can then be obtained from: 𝐸 𝐸 . 𝐸 𝑇 = 𝜎 𝑣 𝑘 𝑘 𝑘 𝑘 4 Variance of the receiver time error is given by 𝜎 𝑣 2 𝑘 4 , and Time Dilution Of Precision (TDOP) is defined as: T𝐷𝑂𝑃≜ 2 𝑘 4 And finally Geometric Dilution Of Precision (GDOP) is defined as: 𝐺𝐷𝑂𝑃≜ 2 𝑘 1 + 𝑘 2 + 𝑘 3 + 𝑘 4 Receivers lock on a set of satellites with smaller PDOP.

GPS – Position Dilution of Precision (PDOP)
PDOP can be used by receivers as a measure to lock on a particular set of satellites. Distribution of the satellites affects the PDOP and accuracy of the position estimate: PDOP is smaller as satellites are widely spread causing a smaller area of uncertainty

GPS – Position Dilution of Precision (PDOP)
PDOP and distribution of the satellites: PDOP is larger as as satellites are close to each other causing a larger area of uncertainty

GPS – other source of errors
Main source of errors are Orbit errors (Ephemeris errors), Ionosphere errors and the multi-path (reflections): Atmospheric delays Charged particle Satellite Ephemeris error: Satellites are injected into very high orbits and therefore are relatively free from perturbing effects of the earth’s upper atmosphere Ionosphere error (Atmospheric errors): Radio waves slow down slightly from the speed of light in vacuo due to the “charged particles” in the ionosphere and water vapour in troposphere. Ionosphere buildings Troposphere Multi-path

GPS – other source of errors
Typical error budget in meter (1𝜎) : Satellite clocks m Orbit errors m Ionosphere m Troposphere m Receiver noise m Multi-path m

GPS – velocity measurements
Two approach to measure velocity from position measurements: 1.A) Filtering the position measurements (coarse estimate), 𝑠 𝜏𝑠+1 𝑣(𝑡) 𝑥(𝑡) Where 𝜏 is variable depending on the signal to noise ratio (SNR) 1.B) Kalman filtering can be used to provide estimates of the receiver velocity with better accuracy 𝑥 = 𝑣 (𝑡) 𝑣 = 𝑤 (𝑡) Are the state equations, 𝑤 (𝑡) is a white noise, the measurement equation is given by 𝑦 𝑡 = 𝑥 + 𝑛 Where 𝑛 is “scaled” measurement noise associated with the receiver clock

GPS – velocity measurements- Doppler effect
Receiver velocity can be estimated by using the Doppler effect, Frequency of signal received by the receiver is shifted from the original transmission frequency, 𝑓 𝑡 , from the satellite 𝑓 𝑟𝑖 = 𝑓 𝑡 − 𝑅 𝑖 𝑐 𝑓 𝑡 The phase difference 𝜑 due to the frequency shift is measured in the receiver between time 𝑡 𝑗 to 𝑡 𝑗+1 : 𝑐 𝜑 𝑖 (𝑗) 𝑓 𝑡 ( 𝑡 𝑗+1 − 𝑡 𝑗 ) = 𝑓 𝑏 𝑓 𝑡 𝑐+ 𝑅 𝑖 Where 𝑓 𝑏 is the frequency bias, also 𝑓 𝑡 is known, so the left hand side of the above measurement can be all considered as the measurement 𝑚 𝑖 =𝑓+ 𝑅 𝑖 +𝑤 𝑚 𝑖 ≜ 𝑐 𝜑 𝑖 (𝑗) 𝑓 𝑡 ( 𝑡 𝑗+1 − 𝑡 𝑗 ) 𝑓= 𝑓 𝑏 𝑓 𝑡 𝑐 , and

The range between the receiver and the 𝑖th satellite is given by: 𝑅 𝑖 2 =( 𝑋 𝑢 − 𝑋 𝑖 ) 𝑇 ( 𝑋 𝑢 − 𝑋 𝑖 ) Therefore time derivative of range from the 𝑖th satellite to the receiver is given by: 𝑅 𝑖 =− 1 𝑅 𝑖 ( 𝑋 𝑢 − 𝑋 𝑖 ) 𝑇 ( 𝑋 𝑢 − 𝑋 𝑖 ) Replacing in the phase difference measurements yields: 𝑚 𝑖 =𝑓− 1 𝑅 𝑖 ( 𝑋 𝑢 − 𝑋 𝑖 ) 𝑇 𝑋 𝑢 − 𝑋 𝑖 +𝑤 Define new measurements 𝑞 𝑖 given the fact that satellite positions , 𝑋 𝑖 ’s are available in the Ephemeris data: 𝑞 𝑖 ≜ 𝑚 𝑖 + 1 𝑅 𝑖 ( 𝑋 𝑢 − 𝑋 𝑖 ) 𝑇 . 𝑋 𝑖 = 1 𝑅 𝑖 ( 𝑋 𝑢 − 𝑋 𝑖 ) 𝑇 . 𝑋 𝑢 +𝑓+𝑤

Define new measurements 𝑞 𝑖 given the fact that satellite positions , 𝑋 𝑖 ’s are available in the Ephemeris data: 𝑞 𝑖 ≜ 𝑚 𝑖 + 1 𝑅 𝑖 ( 𝑋 𝑢 − 𝑋 𝑖 ) 𝑇 . 𝑋 𝑖 = 1 𝑅 𝑖 ( 𝑋 𝑢 − 𝑋 𝑖 ) 𝑇 . 𝑋 𝑢 +𝑓+𝑤 Again, there are 4 unknowns: the receiver velocity vector 𝑋 𝑢 (3 unknowns) and 𝑓. At least 4 measurements would be required. Note that the equations are linear.

References 1. A. Lawrence, Modern Inertial Technology, Springer, 1998
2. R.P.G. Collinson, Introduction to Avionic Systems, third edition, Springer. 3. D.H. Titterton and J.L. Weston, Strapdown Inertial Navigation Technology, Peter Peregrinus, Ltd., 1997. O.J. Woodman, An Introduction to inertial navigation. Technical Report, University of Cambridge, 2007. IEEE Std (R2003) Standard Specification Format Guide and Test Procedure fro Single-Axis Interferometric Fiber Optic Gyros, Annex C. IEEE, 2003. R. W. Beard, T. W. Mclain, Small Unmanned Aircrafts, Theory and Practice, Princton University Press, 2012. GPS tutorial of National Geodetic Survey – National Oceanic and Atmospheric Administration

Strapdown Inertial Navigation Systems (INS) Sensors and UAVs Avionic

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