gps, inertial navigation and lidar sensors. introduction gps- the global positioning system inertial...
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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 Each satellite’s orbital period is 12 hours12 hours
6 satellites visible in each 6 satellites visible in each hemispherehemisphere
GPS Operating Principles
• Position is determined by the travel time of a signal from four or more satellites to the receiving antenna
Image Source: NASA
Three satellites for X,Y,Z Three satellites for X,Y,Z position, one satellite to position, one satellite to cancel out clock biases in the cancel out clock biases in the receiverreceiver
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
Horizon
Receiver
Bad Geometric Degree of Precision
Horizon
Receiver
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
ProofMass (m)
Direction of Acceleration w.r.t. Inertial Space
Displacement Pickup
Case
a
Spring
Example Accelerometers
• Force Feedback Pendulous Accelerometer
Hinge
Pendulous Arm
Restoring Coil
Permanent Magnet
Case
Excitation Coil
Pick-Off
Sensitive Input Axis
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 Torque is proportional to • Angular Rate omega cross H plusAngular Rate omega cross H plus• A change in angular momentumA change in angular momentum
δH = Change in angular momentum
SPIN AXIS (At time t = t + δt)
SPIN AXIS (at time t)
DISC
Precession (rate ω)H
H
O
A
B
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
Y
X
Ω
Beam Splitter Position at time t = t + δt
Beam Splitter Position at time t = t
Light Input
Light Output
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 Ω
Coupling Lens
Beam Splitter
Light Source Detector
Fiber Optic Coil
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
Photo Detector
Laser SourceHalf Silvered
Mirror
Rotating Mirror
Rotation
Sensor Case
Target
Sensor Window
Laser Beam
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
t
Calculate Cross-Correlation of Measurement and Generated Signal
Pulse generated by emitter
Pulse detected at receiver
time
Sig
nal
Ma
gnitu
de
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 NavigationAdvantages
•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 NavigationDisadvantages
•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
1. Keep the accelerometers horizontal so that they do not sense the gravity vector
This is the STABLE PLATFORM MECHANIZATION
2. 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:
1. The accelerator platform has to be mechanically isolated from the rotation of the aircraft
2. The aircraft travels over a spherical surface and thus the direction of the gravity vector changes with position
3. 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:
1. The roll axis which is roughly parallel to the line joining the nose and the tail
Positive angle: right wing down
2. The pitch axis which is roughly parallel to the line joining the wingtips
Positive angle: nose up
3. The yaw axis is vertical
Positive angle: nose to the right
Inertial Navigation – Aircraft Axes Definition
ROLL
PITCHY
AW
Inertial Navigation – Platform IsolationThe 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 IsolationNow 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.
Inertial Navigation – Gyrocompassing
Inertial Navigation – Gyrocompassing
The platform is being rotated around the X and Y axes at measured rates:
RX=ΩcosΦcosα
RY=ΩcosΦsinα
Since Ω is known (15.05107 º/hour) we have two equations in two unknowns and can calculate
Φ (Latitude) and α (platform heading)
Inertial Navigation – Gyrocompassing
The platform is being rotated around the X and Y axes at measured rates:
RX=ΩcosΦcosα
RY=ΩcosΦsinα
Since Ω is known (15.05107 º/hour) we have two equations in two unknowns and can calculate
Φ (Latitude) and α (platform heading)
Inertial Navigation – NavigationOnce 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
1. compensating for the earth’s rotation
2. compensating for travel over the earth’s (somewhat) spherical surface
Inertial Navigation – Schuler OscillationTo 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
sR
θ
Inertial Navigation – Schuler OscillationThis leads to a phenomenon know as Schuler oscillation
At the end of the alignment procedure the accelerometers are almost never perfectly level.
Inertial Navigation – Schuler Oscillation
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
Inertial Navigation – Schuler Oscillation
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
Inertial Navigation – Schuler Oscillation
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)
Inertial Navigation – Schuler Oscillation
Characteristics of the oscillation:
a=-gsinθ or –gθ for small angles
θ = s/R where R is the radius of curvature
R
g
dt
d
R
a
dt
sd
Rdt
d
2
2
2
2
2
2 1differentiating twice
Inertial Navigation – Schuler Oscillation
This is a second order differential equation whose solution is:
θ = θ0cos(ωt)
where θ0 is the initial tilt angle and
R
g
R
g
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
Inertial Navigation – Accelerometers
Types:
•Pendulum
•floating
•flexure pivot
•Vibrating String or Beam
• MEMS (micro electromechanical systems)
Inertial Navigation – Accelerometers
Floated Pendulum
Inertial Navigation – Accelerometers
Flexure Pivot Pendulum
Inertial Navigation – Accelerometers
Vibrating Beam
Inertial Navigation – Accelerometers
MEMS
Inertial Navigation – Gyroscopes
Three main types:
Spinning Mass
Ring Laser
MEMS
Inertial Navigation – Gyroscopes
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
Inertial Navigation – Gyroscopes
Construction:
Inertial Navigation – Gyroscopes
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
Inertial Navigation – Gyroscopes
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
Inertial Navigation – Gyroscopes
Ring Laser Gyro: (RLG) in service since 1986
General Principle:
Inertial Navigation – Gyroscopes
Ring Laser Gyro: (RLG) in service since 1986
General Principle:
Inertial Navigation – Gyroscopes
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”)
Inertial Navigation – Gyroscopes
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
Inertial Navigation – Gyroscopes
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
Inertial Navigation – Gyroscopes
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
Inertial Navigation – GyroscopesMEMS 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
Inertial Navigation – Gyroscopes
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 SystemsThe 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
Inertial Navigation – Strapdown Systems
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.
Inertial Navigation – Strapdown Systems
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
Inertial Navigation – Strapdown Systems
Example:
If the outputs of the accelerometers are:
ax = 0.085773
ay = 0.085773
az = 9.805265
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)
Inertial Navigation – Strapdown Systems
222ZYX aaag
Example:
Thus g = 9.806 m/s2
Θ = sin-1(0.085773/ 9.806 ) = 1º
Φ = sin-1(0.085773/(9.806 x 1) = 1º
Inertial Navigation – Strapdown Systems
Inertial Navigation – Strapdown Systems
Note that during alignment the RLGs on the x and y axes give a direct readout of the two platform rates required for gyrocompassing
Inertial Navigation – Strapdown Systems
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