Navigation Sensors and SystemsCoordinate FramesDead-ReckoningWhat is Inertial Navigation?Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Gyroscope TypesWhat is achievable with INS?Two Ranging Systems for PositioningInterpreting Latitude/Longitude2. Acoustic RangingSlide Number 15Slide Number 16Massachusetts Institute of Technology Subject 2.017Navigation Sensors and SystemsA reference used:Titterton, D.H., and J.L. Weston 1997. Strapdown inertial navigation technology. Peter Peregrinus and IEE, London.Massachusetts Institute of Technology Subject 2.017Coordinate Framesxyz,z’x’y’,y’’x’’,x’’’z’’y’’’z’’’Objective: to express a vector q in various frames of referenceAny frame can be transformed to another frame through a translation and a rotation through three Euler angles []. One of twelve possible sequences is:Base frame is [x,y,z]a. Rotate about z by to give [x’,y’,z’]b. Rotate about y’ by to give [x’’,y’’,z’’]c. Rotate about x’’ by to give [x’’’,y’’’,z’’’]Let q be given in the base frame – then q’’’ (given in the rotated frame) is:q’’’ = R() qwhere R is the rotation matrixqBoard example!Massachusetts Institute of Technology Subject 2.017Dead-Reckoning• If you have nothing but compass and an estimate of speed:• U = speed• = heading• dX/dt = U cos • dY/dt = U sin • RELATIVE ONLYtUtXYX(t),Y(t)X(t+t),Y(t+t)Massachusetts Institute of Technology Subject 2.017What is Inertial Navigation?• Navigation: Locating oneself in an environment, e.g., dead-reckoning.• Inertial: use of Newtonian mechanics:– Body in linear motion stays in motion unless acted on by an external force, causing an acceleration: f = d(m v)/dt m dv/dt ( * if dm/dt = 0!)– A mechanical accelerometer is effectively a load cell.– Rotational velocity is given by a gyroscopic effect: = d (J ) /dt oryaw torque = Jspin X spin_rate X pitch_rate– A mechanical rate gyro is effectively a gyroscope with a load cell.forcev(t+t)v(t)spinpitchyawtorquemvMassachusetts Institute of Technology Subject 2.017spinpitch(t+t)(t)v(t)v(t+t)vAccelerometer measures total acceleration in the inertial frame, projected onto sensor frame.Includes, e.g., centrifugal effect, and radius x d/dt, etc.Rate gyro measures platform-referenced angular rates: p (roll rate)q (pitch rate)r (yaw rate) = d/dt(J ) = J / t*F = d/dt(mv)= m v / t* m = massv = velocity vectorF = force vectorJ = inertia matrix = rotation rate vector = torque vectorMassachusetts Institute of Technology Subject 2.017What does accelerometer give? Sum of actual linear acceleration at sensor PLUS projection of gravitySuppose a 2D sensor is inclined at angle . Then measurements are:m1 = dv1 /dt + g sin m2 = dv2 /dt + g cos Case of three sensors: m1 = dv1 /dt + g R13 ()m2 = dv2 /dt + g R23 ()m3 = dv3 /dt + g R33 ()ORm = dv/dt + g R*,3 ()sensor axis 1sensor axis 2actual acceleration at the sensorApparent acceleration due to gravitymeasured accelerationSuppose = 0 , you know the Euler angles, and you can correct for gravity; then integrate directly: v is sensor-referenced velocity, related to velocity in an inertial frame byvi = RT()v[] are Euler angles; they completely define the attitude of the sensorMassachusetts Institute of Technology Subject 2.017Rate gyros are pure – they give exactly the sensor-referenced rates Can a combination of three accelerometers and three rate gyros provide attitude?Accelerometers contain g projected through the attitude. Gyros give only angular rate; an integral will drift over time!Consider one rate gyro and two accelerometers:mg1 = ddtma1 = dv1 /dt + g sin ma2 = dv2 /dt + g cos One procedure for an attitude package (if accelerations are small compared to g:integrate estimateg cos()g sin()dv2 /dt ~ 0dv1 /dt ~ 0ma1ma2mg1k+++_+_Massachusetts Institute of Technology Subject 2.017Some Gyro Corrections:Rotation of the earth:E cos LCurvature of the earth:v / RCoriolis acceleration:E x vSome Accelerometer Corrections:Centripetal acceleration due to Earth rotation:E2 / R cos LVariation of gravity field with lat./long.: e.g.,g(z=0) = 9.780318 * [1 + 0.00530 sin2 L – 0.000006 sin2 2L ]L: lattitudeE : earth rotation vector;magnitude is 0.0042 deg/sR: Earth radius, 6400kmv: platform velocityEv(t)v(t+t)v(view from above North Pole)Massachusetts Institute of Technology Subject 2.017model of Earth’s gravitational fieldIntegrateResolveCorrections from gravity, Earth rotation and curvature, and Coriolis effectsIntegrate, integrateTransform to global frameattitudeangular ratesvelocity, positionAccelerometer measurementsGyro measurementsThe General CaseMassachusetts Institute of Technology Subject 2.017Gyroscope Types• Mechanical: 0.05-20 degrees per hour drift.• Vibration (e.g., tuning fork) : 360 - 3600 degrees per hour. Cheap and small!• Optical (ring laser): 0.001-10 degrees per hour.• Optical (fiber optic) : 0.5 – 50 degrees per hour.Accelerometer Types• Displaced spring • Pendulous mass: 0.1-10 mg bias• Silicon MEMS: < 25 mg Small, can be cheapHoneywell HG1700: 1 deg/hr ring-laser (3), 1 mg silicon (3)Crossbow IMU700:20 deg/hr fiber optic (3),9 mg silicon (3)Litton LM100 INS:0.003 degree/hr ring laser0.025 mg siliconMassachusetts Institute of Technology Subject 2.017What is achievable with INS?The Litton LM100 alone achieves ~1mile/hr drift; depends strongly on errors in initialization.INTEGRATED NAVIGATION SYSTEM augments the inertial system with complementary sources – i.e., an absolute measurement:GPS hits (in air only)Radio beacon (aircraft)Celestial navigation (clear air only)Doppler radar (air) or Doppler acoustics (seabed)Altitude (air) or depth (water)Range using lasers (air) or acoustics (underwater)Magnetic field dip angle, relative to a mapTerrain/scene matching, relative to an image databaseEtc.Massachusetts Institute of Technology Subject 2.017Two Ranging Systems for Positioning1. GPS: Global Positioning Satellite• Speed of EM signals is 3x108 m/s in free space, covering about 30cm in 1ns a GPS system
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