GPS, Inertial Navigation and LIDAR Sensors

GPS, Inertial Navigation and LIDAR Sensors

Introduction GPS- The Global Positioning System Inertial Navigation
Accelerometers Gyroscopes LIDAR- Laser Detection and Ranging Example Systems

The Global Positioning System
Constellation of 24 satellites operated by the U.S. Department of Defense Originally intended for military applications but extended to civilian use Each satellite’s orbital period is 12 hours 6 satellites visible in each hemisphere

GPS Operating Principles
Position is determined by the travel time of a signal from four or more satellites to the receiving antenna Three satellites for X,Y,Z position, one satellite to cancel out clock biases in the receiver Image Source: NASA

Time of Signal Travel Determination
Code is a pseudorandom sequence Use correlation with receiver’s code sequence at time shift dt to determine time of signal travel

GPS Signal Formulation

Signal Charcteristics
Code and Carrier Phase Processing Code used to determine user’s gross position Carrier phase difference can be used to gain more accurate position Timing of signals must be known to within one carrier cycle

Triangulation Equations Without Error

Sources Of Error Geometric Degree of Precision (GDOP)
Selective Availability Discontinued in 5/1/2000 Atmospheric Effects Ionospheric Tropospheric Multipath Ephemeris Error (satellite position data) Satellite Clock Error Receiver Clock Error

Geometric Degree of Precision (GDOP)
Relative geometry of satellite constellation to receiver With four satellites best GDOP occurs when Three satellites just above the horizon spaced evenly around the compass One satellite directly overhead Satellite selection minimizes GDOP error

Good Geometric Degree of Precision

Bad Geometric Degree of Precision

Pseudorange Measurement
Single satellite pseudorange measurement

Error Mitigation Techniques
Carriers at L1 and L2 frequencies Ionospheric error is frequency dependent so using two frequencies helps to limit error Differential GPS Post-Process user measurements using measured error values Space Based Augmentation Systems(SBAS) Examples are U.S. Wide Area Augmentation System (WAAS), European Geostationary Navigational Overlay Service (EGNOS) SBAS provides atmospheric, ephemeris and satellite clock error correction values in real time

Differential GPS Uses a GPS receiver at a fixed, surveyed location to measure error in pseudorange signals from satellites Pseudorange error for each satellite is subtracted from mobile receiver before calculating position (typically post processed)

Differential GPS

WAAS/EGNOS Provide corrections based on user position
Assumes atmospheric error is locally correlated

Inertial Navigation Accelerometers measure linear acceleration
Gyroscopes measure angular velocity

Accelerometer Principles of Operation
Newton’s Second Law F = mA Measure force on object of known mass (proof mass) to determine acceleration

Example Accelerometers
Force Feedback Pendulous Accelerometer

Example Accelerometers
Micro electromechanical device (MEMS) solid state silicon accelerometer

Accelerometer Error Sources
Fixed Bias Non-zero acceleration measurement when zer0 acceleration integrated Scale Factor Errors Deviation of actual output from mathematical model of output (typically non-linear output) Cross-Coupling Acceleration in direction orthogonal to sensor measurement direction passed into sensor measurement (manufacturing imperfections, non-orthogonal sensor axes) Vibro-Pendulous Error Vibration in phase with pendulum displacement (Think of a child on a swing set) Clock Error Integration period incorrectly measured

Gyroscope Principles of Operation
Two primary types Mechanical Optical Measure rotation w.r.t. an inertial frame which is fixed to the stars (not fixed w.r.t. the Earth).

Mechanical Gyroscopes
A rotating mass generates angular momentum which is resistive to change or has angular inertia. Angular Inertia causes precession which is rotation of the gimbal in the inertial coordinate frame.

Equations of Precession
Angular Momentum vector H Torque vector T Torque is proportional to Angular Rate omega cross H plus A change in angular momentum

Problems with Mechanical Gyroscopes
Large spinning masses have long start up times Output dependent on environmental conditions (acceleration, vibration, sock, temperature ) Mechanical wear degrades gyro performance Gimbal Lock

Gimbal Lock Occurs in two or more degree of freedom (DOF) gyros
Planes of two gimbals align and once in alignment will never come out of alignment until separated manually Reduces DOF of gyroscope by one Alleviated by putting mechanical limiters on travel of gimbals or using 1DOF gyroscopes in combination

Gimbal Lock

Optical Gyroscope Measure difference in travel time of light traveling in opposite directions around a circular path

Types Ring Laser Gyroscope Fiber Optic

Ring Laser Gyro Change in traveled distance results in different frequency in opposing beams Red shift for longer path Blue shift for shorter path For laser operation peaks must reinforce each other leading to frequency change.

Lock In and Dithering Lasers tend to resist having two different frequencies at low angular rates Analogous to mutual oscillation in electronic oscillators Dithering or adding some small random angular accelerations minimizes time gyro is in locked in state reducing error

Fiber Optic Gyroscope Measure phase difference of light traveling through fiber optic path around axis of rotation

Example Complete GPS/INS System
Applanix POS LV-V4 Used in Urbanscape Project Also includes wheel rate sensor

Pulse LIDAR Measures time of flight of a light pulse from an emitter to an object and back to determine position. Sensitive to atmospheric effects such as dust and aerosols

Conceptual Drawing

The Math d = Distance from emitter/receiver to target
C = speed of light (299,792,458 m/s in a vacuum) Δt = time of flight

Determining Time of Flight

From Depth to 3D Use angle of reflecting mirror to determine ray direction Measurement is 3D relative to LIDAR sensor frame of reference Transform into world frame using GPS/INS system or known fixed location

Error Sources Aerosols and Dust
Scatter Laser reducing signal strength of Laser reaching target Laser reflected to receiver off of dust introduces noise Minimally sensitive to temperature variation (changes path length inside of receiver and clock oscillator rate) Error in measurement of rotating mirror angle Specular Surfaces Clock Error

Example Pulse LIDAR Characteristics
Sample specification from SICK

Doppler LIDAR Uses a continuous beam to measure speed differential of target and emitter/receiver Measure frequency change of reflected light Blue shift- target and LIDAR device moving closer together Red shift- target and LIDAR device moving apart

Application of Doppler LIDAR
Speed Traps

Combined Sensor Systems

Inertial Navigation Advantages
instantaneous output of position and velocity completely self contained all weather global operation very accurate azimuth and vertical vector measurement error characteristics are known and can be modeled quite well works well in hybrid systems

Inertial Navigation Disadvantages
Position/velocity information degrade with time (1-2NM/hour). Equipment is expensive ($250,000/system) - older systems had relatively high failure rates and were expensive to maintain newer systems are much more reliable but still expensive to repair Initial alignment is necessary - not much of a disadvantage for commercial airline operations (12-20 minutes)

Inertial Navigation – Basic Principle
If we can measure the acceleration of a vehicle we can integrate the acceleration to get velocity integrate the velocity to get position Then, assuming that we know the initial position and velocity we can determine the position of the vehicle at ant time t.

Inertial Navigation – The Fly in the Ointment
The main problem is that the accelerometer can not tell the difference between vehicle acceleration and gravity We therefore have to find a way of separating the effect of gravity and the effect of acceleration

Inertial Navigation – The Fly in the Ointment
This problem is solved in one of two ways Keep the accelerometers horizontal so that they do not sense the gravity vector This is the STABLE PLATFORM MECHANIZATION Somehow keep track of the angle between the accelrometer axis and the gravity vector and subtract out the gravity component This is the STRAPDOWN MECHANIZATION

Inertial Navigation – STABLE PLATFORM
The original inertial navigation systems (INS) were implemented using the STABLE PLATFORM mechanization but all new systems use the STRAPDOWN system We shall consider the stable platform first because it is the easier to understand

Inertial Navigation – STABLE PLATFORM
There are three main problems to be solved: The accelerator platform has to be mechanically isolated from the rotation of the aircraft The aircraft travels over a spherical surface and thus the direction of the gravity vector changes with position The earth rotates on its axis and thus the direction of the gravity vector changes with time

Inertial Navigation – Aircraft Axes Definition
The three axes of the aircraft are: The roll axis which is roughly parallel to the line joining the nose and the tail Positive angle: right wing down The pitch axis which is roughly parallel to the line joining the wingtips Positive angle: nose up The yaw axis is vertical Positive angle: nose to the right

Inertial Navigation – Aircraft Axes Definition
ROLL PITCH YAW

Inertial Navigation – Platform Isolation
The platform is isolated from the aircraft rotation by means of a gimbal system The platform is connected to the first (inner) gimbal by two pivots along the vertical (yaw) axis. This isolates it in the yaw axis The inner gimbal is the connected to the second gimbal by means of two pivots along the roll axis. This isolates the platform in the roll axis. The second gimbal is connected to the INU (Inertial Navigation Unit) chassis by means of two pivots along the pitch axis. This isolates it in the pitch axis.

Inertial Navigation – Platform Isolation
Now the platform can be completely isolated from the aircraft rotations

Inertial Navigation – Gyroscopes
To keep the platform level we must be able to: Sense platform rotation and Correct for it To do this we mount gyroscopes on the stable platform and install small motors at each of the gimbal pivots. The gyroscopes sense platform rotation in any of the three axes and then send a correction signal to the pivot motors which then rotates the relevant gimbal to maintain the platform at the correct attitude

Inertial Navigation – Alignment
Before the INS can navigate it must do two things: Orient the platform perpendicular to the gravity vector Determine the direction of True North Also it must be given: Initial Position: Input by the Pilot (or navigation computer) Velocity: This is always zero for commercial systems

Inertial Navigation – Orientation
In the alignment mode the INU uses the accelerometers to send commands to the pivot motors to orient the platform so that the output of the accelerometers is zero. Note that the earth (and therefore the INU) is rotating so that it will be necessary to rotate the platform in order to keep it level.

Inertial Navigation – Gyrocompassing
The rotation of the platform to keep it level is used to determine the direction of True North relative to the platform heading.

The platform is being rotated around the X and Y axes at measured rates: RX=ΩcosΦcosα RY=ΩcosΦsinα Since Ω is known ( º/hour) we have two equations in two unknowns and can calculate Φ (Latitude) and α (platform heading)

Inertial Navigation – Navigation
Once the INU has been aligned it can be put into NAVIGATE mode . In navigate mode, the outputs of the accelerometers are used to determine the vehicle’s position and the gyroscopes are used to keep the platform level. This involves compensating for the earth’s rotation compensating for travel over the earth’s (somewhat) spherical surface

Inertial Navigation – Schuler Oscillation
To compensate for the travel over the surface of the earth the platform must be rotated by an amount d/R where d is the distance travelled and R is the radius of curvature of the earth R s θ

This leads to a phenomenon know as Schuler oscillation At the end of the alignment procedure the accelerometers are almost never perfectly level.

Assume for now that the aircraft remains at rest The measured acceleration causes the INU to compute a velocity and hence a change in position. This in turn causes the gyros to rotate the platform

Assume for now that the aircraft remains at rest The measured acceleration causes the INU to think that it is moving an it computes a velocity and hence a change in position. This in turn causes the gyros to rotate the platform

The direction of the rotation tends to level the accelerometer but when it is level, the computer has built up a considerable speed and thus overshoots. (this is like pulling a pendulum off centre and letting it go)

Characteristics of the oscillation: a=-gsinθ or –gθ for small angles θ = s/R where R is the radius of curvature differentiating twice

This is a second order differential equation whose solution is: θ = θ0cos(ωt) where θ0 is the initial tilt angle and The period of this oscillation is 84 minutes

Inertial Navigation – Accelerometers
Requirements: high dynamic range (10-4 g to 10g) low cross coupling good linearity little or no asymmetry Exacting requirements dictate the use of Force-Rebalance type of devices

Types: Pendulum floating flexure pivot Vibrating String or Beam MEMS (micro electromechanical systems)

Floated Pendulum

Flexure Pivot Pendulum

Vibrating Beam

MEMS

Three main types: Spinning Mass Ring Laser MEMS

Spinning Mass: Rigidity in Space: A spinning mass has a tendency to maintain its orientation in INERTIAL space Its rigidity (or resistance to change) depends on its moment of inertia and its angular velocity about the spin axis (INU gyros spin at around 25,000 RPM) Precession; If a torque τ is applied perpendicular to the spinning mass it will respond by rotating around an axis 90 degrees to the applied torque. I.e. ω× τ

Construction:

Spinning Mass Gyros: Disadvantages: sensitive to shock during installation and handling (Pivots can be damaged) requires several minutes to get up to speed and temperature expensive

Ring Laser Gyro: (RLG) in service since 1986 Advantages over spinning mass gyros: more rugged inherently digital output large dynamic range good linearity short warm up time

Ring Laser Gyro: (RLG) in service since 1986 General Principle:

Ring Laser Gyro Problems: Lock-in at low rotation rates due to weak coupling between the two resonant systems (coupling due to mirror backscatter) Analagous to static friction (stiction) in mechanical systems Causes a dead zone Alleviated by “dithering” the gyro at a few hundred Hz Random loss of pulses at the output ( causes “drift”)

Fibre Optic Gyro Similar concept to RLG except that amplification is not usesd Two strands of optical fibre are wound in opposite directions on a coil form Laser light is sent from a single source down both fibres The outputs of the two fibres are combined at a photodiode Rotation of the coil around its axis causes the two paths to have different lengths and the output of the photodiode provides a light dark pattern. Each cycle indicates an increment of angular rotation

Fibre Optic Gyro Has the advantage of being rugged and relatively cheap Sensitivity increases with length of fibre Unfortunately, the longer the fibre, the lower the output signal. Used on low performance systems

MEMS Gyro All gyros to date have been quite large in fact the sensitivity of spinning mass gyros and RLGs are a direct function of their size. Efforts are being made to apply MEMS technology to gyros as well as to accelerometers

MEMS Gyro The MEMS gyro uses the Coriolis Effect In a rotating system (such as the earth) moving objects appear to deflected perpendicular to their direction of travel. The effect is a function of the velocity if the object and the rate of rotation

MEMS Gyro In a MEMS gyro the times of a tuning fork are the moving object MEMS gyros exhibit high drift rates and thus are not suitable for commercial aviation use They are used in conjunction with GPS in “coupled” systems which use the best characteristics of each

Inertial Navigation – Strapdown Systems
The main problem for an INS is to separate the vehicle acceleration from the effect of gravity on the accelerometers In the stable platform, this is done by maintaining the accelerometers perpedicular to the gravity vector which allows us to ignore the effect of gravity Another approach is to keep track of the gravity vector and subtract its effect from the outputs of the accelerometers This is an analytical or computational implementation

As the name implies, the accelerometers are fixed or “strapped down” to the chassis of the INU and hence to the aircraft. Since the gravity vector is three dimensional, three accelerometers are required to keep track of it. In addition, three RLGs are mounted with their axes aligned with the x,y, and z axes (roll, pitch and yaw) of the aircraft respectively.

Alignment: During the alignment procedure, the INS measures the direction of the gravity vector. Notice that the outputs of the accelerometers are proportional to the Direction Cosines of the gravity vector

Example: If the outputs of the accelerometers are: ax = ay = az = What are the roll and pitch angles?

Example: If the roll and pitch angles are Φ and Θ respectively aX = gsin Θ Note: aY = gsin Φcos Θ aZ = gcos Φcos Θ Therefore: Θ=sin-1(aX/g) and Φ= sin-1(aY/gcos Θ)

Example: Thus g = m/s2 Θ = sin-1( / ) = 1º Φ = sin-1( /( x 1) = 1º

Note that during alignment the RLGs on the x and y axes give a direct readout of the two platform rates required for gyrocompassing

Note: The sensitivity of Ring Laser Gyro is: N=4A/λL Where: N is the number of fringes per radian A is the area enclosed by the path L is the Length of the path λ is the wave length of the light Note that the larger the area, the more sensitive the gyro

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