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