Navigation Sensors andSystemsReferenceused:Titterton, D.H., and J.L.Weston1997.Strapdowninertial navigationtechnologyPeterPeregrinusandIEELondon.Massachusetts Instituteof TechnologySubject2.017
Navigation Sensors and Systems Reference used: Titterton, D.H., and J.L. Weston 1997. Strapdown inertial navigation technology. Peter Peregrinus and IEE, London. Massachusetts Institute of Technology Subject 2.017
What is Inertial Navigation?forceNavigation:Locating oneself in any(t+St)environment, e.g., dead-reckoning.Inertial: use of Newtonian mechanics:oymBody in linear motion stays in motion unlessacted on by an external force, causing any(t)acceleration:f = d(m y)/dt > m dv/dt ( * if dm/dt = 0!)yawAmechanicalaccelerometeris effectivelyatorqueload cellRotational velocity is given by a gyroscopiceffect:spin↑ = d (J @) /dtoryaw torque = Jspin X spin_rate X pitch_rateAmechanicalrategyroiseffectivelyapitchgyroscope with a load cell.MassachusettsInstituteofTechnologySubject2.017
What is Inertial Navigation? • Navigation: Locating oneself in an force 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: v(t+Gt) v(t) m Gv f = d(m v)/dt Æ m dv/dt ( * if dm/dt = 0!) spin torque yaw – A mechanical accelerometer is effectively a load cell. – Rotational velocity is given by a gyroscopic effect: W = d (J Z) /dt or yaw torque = Jspin X spin_rate X pitch_rate – A mechanical rate gyro is effectively a pitch gyroscope with a load cell. Massachusetts Institute of Technology Subject 2.017
Svy(t+8t)@(t+St)spin@(t)@m=massV= velocity vectorpitchJ= inertia matrixE=force vectory(t))@ = rotation rate vector=torque vectorE = d/dt(my)= m &y / St *↑ = d/dt(J @)= J@ /8t *AccelerometermeasuresRategyromeasurestotalaccelerationininertialplatform-referencedframe,projected ontoangular rates:platform frame.p (roll rate)q (pitch rate)Includes, e.g.,r (yaw rate)centrifugal effect, andradius x do/dtMassachusetts InstituteofTechnologySubject2.017
Gv v(t) v(t+Gt) Accelerometer measures total acceleration in inertial frame, projected onto F = d/dt(mv) = m Gv / Gt* m = mass v = velocity vector F = force vector platform frame. Includes, e.g., centrifugal effect, and radius x dZ/dt Massachusetts Institute of Technology Subject 2.017 spin Z(t) Z(t+Gt) GZ pitch J = inertia matrix Z = rotation rate vector W = torque vector W = d/dt(J Z) = J GZ / Gt* Rate gyro measures platform-referenced angular rates: p (roll rate) q (pitch rate) r (yaw rate)
What does accelerometer give? Sum ofmeasuredaccelerationactuallinearaccelerationatsensorsensorPLUSprojection ofgravityaxis 1sensoraxis 2Suppose a 2D sensor is inclined atApparent0angle 0. Then measurements are:acceleration dueactualto gravitym, = dv,/dt + g sin 0accelerationm2 = dv2/dt + g cos 0atthesensorCase of three sensors:Suppose you integrate:m, = dv,/dt + g R(Φ,0,y)m2 = dv2/dt + g R2(Φ,0,)v is sensorreferencedvelocity,m3 = dv3/dt + g R3(Φ,0,)relatedtovelocity in a base framebyORV = RT(,0,)m = dy/dt + g R(Φ,0,)[Φ,0,y] are Euler angles; theycompletely define the attitude of thesensorMassachusettsInstituteofTechnologySubject2.017
What does accelerometer give? Sum of actual linear acceleration at sensor PLUS projection of gravity Suppose a 2D sensor is inclined at angle T. Then measurements are: m = dv1/dt + g sin T 1 m = dv2/dt + g cos T 2 Case of three sensors: m1 = dv1/dt + g R1(IT\) m = dv2/dt + g R2(IT\) 2 m = dv3/dt + g R3(IT\) 3 OR m = dv/dt + g R(IT\) Massachusetts Institute of Technology Subject 2.017 measured acceleration T acceleration sensor axis 1 sensor axis 2 Apparent acceleration due actual to gravity at the sensor Suppose you integrate: v is sensor referenced velocity, related to velocity in a base frame by v = RT(IT\)v b [IT\] are Euler angles; they completely define the attitude of the sensor
CoordinateObjective:to express avector ginvariousframesofreferenceFramesAnyframecanbetransformedtoanotherframethroughatranslationand arotationthrough three Euler angles [Φ,o,y]. One ofz,z'>twelvepossible sequences is:>g0WBaseframeis[x,y,z][x,y',z]a.Rotateaboutz bytoqiveb. Rotate about y'by 0 to give[x",y",z"]c. Rotate about x" by y to give [x",y",z""]Wy',y"dLetgbegiveninthebaseframe-thengY(given in the rotated frame) is:g" = R(,0,) gwhereRistherotationmatrix0+AXBoard example!x",x"MassachusettsInstituteofTechnologySubject2.017
Coordinate Frames z,z’ z’’ Objective: to express a vector q in various frames of reference Any frame can be transformed to another frame through a translation and a rotation through three Euler angles [IT\]. One of z’’’ twelve possible sequences is: Base frame is [x,y,z] x y y’,y’’ y’’’ I I T T \ \ q a. Rotate about z by Ito give [x’,y’,z’] b. Rotate about y’ by T 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(IT\) q where R is the rotation matrix x’ Board example! x’’,x’’’ Massachusetts Institute of Technology Subject 2.017
Rategyrosarepure-theygiveexactlythe sensor-referenced rates→cancombination of three accelerometers and three rate gyros provide navigation?Accelerometers contain g projected through the attitude.Gyrosgive only angular rate;an integral will drift overtime!Consideronerategyroandtwoaccelerometers:mg1 = do/dtma1 = dv,/dt + g sin 0ma2 = dvz/dt + g cos 0One procedure for an attitude package (if accelerations are small compared to go):dv/dt ~ 0ma2dv,/dt ~ 0ma1g cos()g sin()mg1eestimateintegratekMassachusettsInstituteofTechnologySubject2.017
Rate gyros are pure – they give exactly the sensor-referenced rates Æ can combination of three accelerometers and three rate gyros provide navigation? 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: m = dTdt g1 ma1 = dv1/dt + g sin T ma2 = dv2/dt + g cos T One procedure for an attitude package (if accelerations are small compared to gT : integrate T estimate g sin() g cos() dv2/dt ~ 0 dv1 m /dt ~ 0 a1 ma2 mg1 k + _ _ + _ + Massachusetts Institute of Technology Subject 2.017
L:Some Gyro Corrections:lattitude@e: earth rotation vector;Rotation of the earth:magnitudeis0.0042deg/sR:QE COS LEarthradius.6400kmV:platformvelocityCurvature of the earth:V/Ry(t+St)ovOECoriolis acceleration:QEXYy(t)(viewfromaboveNorthPole)Some Accelerometer Corrections:Centripetal acceleration due to Earth rotation:WE? /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 JMassachusettsInstituteofTechnologySubject2.017
Some Gyro Corrections: L: lattitude Rotation of the earth: ZE cos L ZE: earth rotation vector; magnitude is 0.0042 deg/s R: Earth radius, 6400km Curvature of the earth: v: platform velocity v / R Coriolis acceleration: ZE x v ZE v(t+Gt) Gv v(t) (view from above North Pole) Some Accelerometer Corrections: Centripetal acceleration due to Earth rotation: ZE 2 / R cos L Variation of gravity field with lat./long.: e.g., g(z=0) = 9.780318 * [1 + 0.00530 sin2 L – 0.000006 sin2 2L ] Massachusetts Institute of Technology Subject 2.017
Accelerometermodel of Earth'smeasurementsgravitational fieldGyromeasurementsTransform toResolveglobal frameIntegrate,Correctionsfromattitudeintegrategravity, Earth rotationIntegrateand curvature, andangularCorioliseffectsratesvelocity,The General CasepositionMassachusetts Instituteof TechnologySubject2.017
model of Earth’s gravitational field Integrate Resolve Corrections from gravity, Earth rotation and curvature, and Integrate, integrate Transform to global frame attitude angular rates velocity, position Accelerometer measurements Gyro measurements The General Case Coriolis effects Massachusetts Institute of Technology Subject 2.017
Gyroscope TypesMechanical:0.05-20degrees perhourdrift.Vibration (e.g., tuning fork): 360 -3600 degreesCheap and small!perhour.Optical (ring laser):0.001-10degrees perhour.Optical (fiber optic) : 0.5 - 50 degrees per hour.ImageremovedforcopyrightreasonsAccelerometer TypesHoneywell HG1700 IMU.·Displaced spring.Pendulous mass: 0.1-10 mg bias.SiliconMEMS:<25mgSmall,canbecheapCrossbow IMU700:Honeywell HG1700:LittonLM100INS:20 deg/hr fiber optic (3)1 deg/hr ring-laser (3),0.003degree/hrringlaser9 mg silicon (3)1 mg silicon (3)0.025 mg siliconMassachusetts InstituteofTechnologySubject2.017
Gyroscope 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 Image removed for copyright reasons. Honeywell HG1700 IMU. • Displaced spring • Pendulous mass: 0.1-10 mg bias • Silicon MEMS: < 25 mg Small, can be cheap Honeywell 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 laser 0.025 mg silicon Massachusetts Institute of Technology Subject 2.017
What is achievable with INs?The Litton LN100 alone achieves ~1mile/hr drift; dependsstrongly on errors in initialization.INTEGRATEDNAVIGATIONSYSTEMaugmentstheinertialsystemwith complementary sources-i.e.,anabsolutemeasurement: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 dipangle, relative to a mapTerrain/scenematching,relativetoanimagedatabaseEtc.MassachusettsInstituteofTechnologySubject2.017
What is achievable with INS? The Litton LN100 alone achieves ~1mile/hr drift; depends strongly on errors in initialization. INTEGRATED NAVIGATION SYSTEM augments the inertial 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 map Terrain/scene matching, relative to an image database Etc. system with complementary sources – i.e., an absolute Massachusetts Institute of Technology Subject 2.017