gps, inertial navigation and lidar sensors. introduction gps- the global positioning system inertial...

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.


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


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

R

g

dt

d

R

a

dt

sd

Rdt

d

2

2

2

2

2

2 1differentiating twice


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


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

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


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.


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 = 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)


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



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

gps, inertial navigation and lidar sensors. introduction gps- the global positioning system inertial...

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