1、Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Mechatronics - Foundations and Applications Position Measurement in Inertial Systems,JASS 2006, St.Petersburg Christian Wimmer,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technica
2、l University of Munich,Content,Motivation Basic principles of position measurement Sensor technology Improvement: Kalman filtering,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Motivation,Johnnie: A biped walking machineOrientation Stabilization Na
3、vigation,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Motivation,Automotive Applications:Drive dynamics Analysis Analysis of test route topology Driver assistance systems,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical
4、 University of Munich,Motivation,Aeronautics and Space Industry:Autopilot systems Helicopters Airplane Space Shuttle,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Motivation,Military Applications:ICBM, CM Drones (UAV) Torpedoes Jets,Lecture: Positi
5、on Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Motivation,Maritime Systems:Helicopter Platforms Naval Navigation Submarines,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Motivation,Industrial robotic Systems:Main
6、tenance Production,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Principles,Measurement by inertia and integration:Acceleration Velocity Position,Newtons 2. Axiom:F = m x a BASIC PRINCIPLE OF DYNAMICS,Measurement system with 3 sensitive axes3
7、 Accelerometers 3 Gyroscope,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Principles,Gimballed Platform Technology:3 accelerometers 3 gyroscopes cardanic Platform,ISOLATED FROM ROTATIONAL MOTION TORQUE MOTORS TO MAINTAINE DIRECTION ROLL, PITC
8、H AND YAW DEDUCED FROM RELATIVE GIMBAL POSITION GEOMETRIC SYSTEM,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Principles,Strapdown Technology:Body fixed 3 Accelerometers 3 Gyroscopes,Lecture: Position Measurement in Inertial Systems Christia
9、n Wimmer Technical University of Munich,Basic Principles,Strapdown Technology:The measurement principle,SENSORS FASTENED DIRECTLY ON THE VEHICLE BODY FIXED COORDINATE SYSTEM ANALYTIC SYSTEM,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Princi
10、ples,Reference Frames:i-frame e-frame n-frame b-frameAlso normed: WGS 84,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Vehicles acceleration in inertial axes (1.Newton):Problem: All quantities are obtained in vehicles frame (local) Euler Derivative
11、s!,Basic Principles,Interlude: relative kinematics,Differentiation:,trans,cor,rot,cent,Inertial system: i,Moving system: e P = CoM,O,P,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Frame Mechanisation I: i-FrameVehicles velocity (ground speed) and
12、Coriolis Equation:abbreviated:Differentiation: Applying Coriolis Equation (earths turn rate is constant):subscipt: with respect to; superscript: denotes the axis set; slash: resolved in axis set,Basic Principles,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University
13、of Munich,Frame Mechanisation II: i-FrameNewtons 2nd axiom:abbreviated:Recombination: i-frame axes: Substitution:subscipt: with respect to; superscript: denotes the axis set; slash: resolved in axis set,Basic Principles,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical Uni
14、versity of Munich,Basic Principles,Frame Mechanisation III: Implementation,BODY MOUNTED GYROSCOPES,ATTITUDE COMPUTER,RESOLUTION OF SPECIFIC FORCE MEASUREMENTS,BODY MOUNTED ACCELEROMETERS,NAVIGATION COMPUTER,CORIOLIS CORRECTION,GRAVITY COMPUTER,INITIAL ESTIMATES OF VELOVITY AND POSITION,INITIAL ESTIM
15、ATES OF ATTITUDE,POSITION INFORMATION,POSITION AND VELOVITYESTIMATES,POSSIBILITY FOR KALMAN FILTER INSTALLATION,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Principles,Strapdown Attitude Representation:Direction cosine matrixQuaternionsEuler
16、 angles,No singularities, perfect for internal computations,singularities, good physical appreciation,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Principles,Strapdown Attitude Representation: Direction Cosine Matrix,For Instance:,Simple Der
17、ivative:,Axis projection:,With skew symmetric matrix:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Principles,Strapdown Attitude Representation: Quaternions,Idea: Transformation is single rotation about one axis,Components of angle Vector, d
18、efined with respect to reference frame,Magnitude of rotation:,Operations analogous to 2 Parameter Complex number,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Basic Principles,Strapdown Attitude Representation: Euler Angles,Rotation about reference
19、 z axis through angle Rotation about new y axis through angle Rotation about new z axis through angle,Singularity:,Gimbal angle pick-off!,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,AccelerometersPhysical principles:Potentiometr
20、ic LVDT (linear voltage differential transformer) Piezoelectric,Newtons 2nd axiom:,gravitational part: Compensation,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,AccelerometersPotentiometric,+,-,Lecture: Position Measurement in In
21、ertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,AccelerometersLVDT (linear voltage differential transformer)Uses Induction,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,AccelerometersPiezoelectric,
22、Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,AccelerometersServo principle (Force Feedback)Intern closed loop feedback Better linearity Null seeking instead of displacement measurement,1 - seismic mass 2 - position sensing device
23、 3 - servo mechanism 4 - damper 5 - case,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,GyroscopesVibratory Gyroscopes Optical Gyroscopes,Historical definition:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Tec
24、hnical University of Munich,Sensor Technology,Gyroscopes: Vibratory GyroscopesCoriolis principle: 1. axis velocity caused by harmonic oscillation (piezoelectric) 2. axis rotation 3. axis acceleration measurementProblems: High noise Temperature drifts Translational acceleration vibration,Lecture: Pos
25、ition Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,Gyroscopes: Vibratory Gyroscopes,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Sensor Technology,Gyroscopes: Optical GyroscopesSagnac Effect: Su
26、per Luminiszenz Diode Beam splitter Fiber optic cable coil Effective path length difference,LASER,INTERFERENCE DETECTOR,MODULATOR,Beam splitter,Beam splitter,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,The Kalman Filter A stochastic
27、 filter methodMotivation:Uncertainty of measurement System noise Bounding gyroscopes drift (e.g. analytic systems) Higher accuracy,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,The Kalman Filter what is it?Definition: Optimal recursiv
28、e data processing algorithm. Optimal, can be any criteria that makes sense.Combining information: Knowledge of the system and measurement device dynamics Statistical description of the systems noise, measurement errors and uncertainty in the dynamic models Any available information about the initial
29、 conditions of the variables of interest,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,The Kalman Filter Modelization of noiseDeviation: Bias: Offset in measurement provided by a sensor, caused by imperfections Noise: disturbing value
30、 of large unspecific frequency rangeAssumption in Modelization: White Noise: Noise with constant amplitude (spectral density) on frequency domain (infinite energy); zero meanGaussian (normally) distributed: probability density function,Lecture: Position Measurement in Inertial Systems Christian Wimm
31、er Technical University of Munich,Kalman Filter,Basic Idea:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Combination of independent estimates: stochastic Basics (1-D)Mean value:Variance:Estimates: Mean of 2 Estimates (with weighting
32、factors):,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Combination of independent estimates: stochastic Basics (1-D)Weighted mean:Variance of weighted mean:Not correlated: Variance of weighted mean:,Quantiles are independent!,Lecture
33、: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Combination of independent estimates: stochastic Basics (1-D)Weighting factors:Substitution:Optimization (Differentiation): Optimum weight:,Lecture: Position Measurement in Inertial Systems Chris
34、tian Wimmer Technical University of Munich,Kalman Filter,Combination of independent estimates: stochastic Basics (1-D)Mean value:Variance:Multidimensional case:Covariance matrix:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Interlude
35、: the covariance matrix1-D: Variance 2nd central moment N-D: Covariance diagonal elements are variances, off-diagonal elements encode the correlationsCovariance of a vector:n x n matrix, which can be modal transformed, such that are only diagonal elements with decoupled error contribution; Symmetric
36、 and quadratic,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Interlude: the covariance matrix applied to equationsEquation structure:x, y are gaussian distributed, c is constant:Covariance of z: Linear difference equation:Covariance:with:,Kalman Fi
37、lter,Diagonal structure: since white gaussian noise,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Combination of independent estimates: (n-D)Mean value:measurement:Mean value:Covariance with:,Lecture: Position Measurement in Inertial
38、Systems Christian Wimmer Technical University of Munich,Kalman Filter,Combination of independent estimates: (n-D)Covariance:Covariance:Minimisation of Variance matrixs Diagonal elements (Kalman Gain):,For further information please also read: P.S. Maybeck: Stochastic Models, Estimation and Control V
39、olume 1, Academic Press, New York San Francisco London,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Combination of independent estimates: (n-D)Mean value:Variance:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Te
40、chnical University of Munich,Kalman Filter,Interlude: time continuous system to discrete systemContinuous solution:Substitution:Conclusion:Sampling time:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,The Kalman Filter: Iteration Princ
41、iple,INITIAL ESTIMATION OF STATES AND QUALITY OF STATE,PREDICTION OF STATES (SOLUTION) BETWEEN TWO ITERATIONS,PREDICTION OF ERROR COVARIANCE BETWEEN TWO ITERATIONS,CALCULATION OFKALMAN GAIN (WEIGHTING OF MEASUREMENT AND PREDICTION),DETERMINATION OF NEW SOLUTION (ESTIMATION),CORRECTION OF THE STOCHAS
42、TIC MODELLS TO NEW QUALITY VALUE OF SOLUTION,PREDICTION,CORRECTION,NEXT ITERATION,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Linear Systems the Kalman Filter: Discrete State Model:Sensor Model:,Lecture: Position Measurement in Iner
43、tial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Linear Systems the Kalman Filter: 1. Step PredictionPrediction:State Prediction Covariance:Observation Prediction:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filte
44、r,Linear Systems the Kalman Filter: 2. Step CorrectionCorrected state estimate:Corrected State Covariance:Innovation Covariance:Innovation:,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,The Kalman Filter: Kalman GainKalman Gain:,State
45、 Prediction CovarianceInnovation Covariance,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,The Kalman Filter: System Model,Memory,+,-,+,+,+,+,For linear systems: System matrices are timeinvariant,Lecture: Position Measurement in Inerti
46、al Systems Christian Wimmer Technical University of Munich,Kalman Filter,Non-Linear Systems the extended Kalman Filter:Nonlinear dynamics equation: Nonlinear observation equation:Solution strategy: Linearize Problem around predicted state: (Taylor Series tuncation),Error Deviation from Prediction st
47、ate Necessary for Kalman Gain and Covariance Calculation,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Non-Linear Systems the extended Kalman Filter:Prediction:Correction:,Lecture: Position Measurement in Inertial Systems Christian Wi
48、mmer Technical University of Munich,Kalman Filter,Example: Aiding the missileMISSILE WITH ON-BOARD INERTIAL NAVIGATION SYSTEM (REPLACING THE PHYSICAL PROCESS MODEL; 1 ESTIMATE) AND NAVIGATION AID (GROUND TRACKER MEASUREMENT; 2 ESTIMATE),MISSILE,SURFACE SENSORS,KALMAN GAINS,INS,MEASUREMENT MODEL,Miss
49、ile Motion,Measurement Noise,True Position,Measurement Innovations,Estimated INS Error,System Noise,INS Indicated Position,Estimated Range, Elevation and Bearing,+_,Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich,Kalman Filter,Example: Aiding the mis
50、sileNine State Kalman Filter: 3 attitude, 3 velocity, 3 position errors Bounding Gyroscopes and accelerometers drifts by long term signal of surface sensor on launch platform (complementary error characteristics)Extended Kalman Filter: Attention: All Matrices are vector derivatives! Linearisation around trajectory)Error Model: (truncated Taylor series)Discrete Representation: (System Equation)Attention: All Matrices are vector derivatives matrices!,
