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 Sources of errors calibration Accelerometers Calibration Compass Air data systems Airspeed Pressure Altitude Global Positioning Systems (GPS) Underlying concept Pseudo ranges Clock drift

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      

Inertial Sensors - Gyroscopes 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  

Inertial Sensors - Gyroscopes         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

Gyroscopes – fiber optic gyros (FOG): 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:        

Gyroscopes – fiber optic gyros (FOG): 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.    

Gyroscopes – Ring laser gyros (RLG): The frequency difference will be then:   Laser beams   Helium-Neon Laser Photo Diodes

Gyroscopes – Ring laser gyros (RLG): 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 400-500Hz, 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:    

Gyroscopes – MEMS gyros:     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

Gyroscopes – MEMS gyros: 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.

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  

Gyroscopes – error Terms 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:    

Gyroscopes – error Terms       Ref. 4  

Gyroscopes – error Terms 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)

Gyroscopes – error Terms 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

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 962-1997 (R2003) Standard Specification Format Guide and Test Procedure fro Single-Axis Interferometric Fiber Optic Gyros, Annex C. IEEE, 2003. www.ett.bme.hu/memsedu