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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

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Content 1.Motivation 2.Basic principles of position measurement 3.Sensor technology 4.Improvement: Kalman filtering

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Motivation Johnnie: A biped walking machine Orientation Stabilization Navigation

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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

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Motivation Aeronautics and Space Industry: Autopilot systems Helicopters Airplane Space Shuttle

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Motivation Military Applications: ICBM, CM Drones (UAV) Torpedoes Jets

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Motivation Maritime Systems: Helicopter Platforms Naval Navigation Submarines

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Motivation Industrial robotic Systems: Maintenance Production

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Basic Principles Measurement by inertia and integration: Acceleration Velocity Position Newton‘s 2. Axiom: F = m x a BASIC PRINCIPLE OF DYNAMICS Measurement system with 3 sensitive axes 3 Accelerometers 3 Gyroscope

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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, PITCH AND YAW DEDUCED FROM RELATIVE GIMBAL POSITION GEOMETRIC SYSTEM

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Basic Principles Strapdown Technology: Body fixed 3 Accelerometers 3 Gyroscopes

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Basic Principles Strapdown Technology: The measurement principle SENSORS FASTENED DIRECTLY ON THE VEHICLE BODY FIXED COORDINATE SYSTEM ANALYTIC SYSTEM

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Basic Principles Reference Frames: i-frame e-frame n-frame b-frame Also normed: WGS 84

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Vehicle‘s acceleration in inertial axes (1.Newton): Problem: All quantities are obtained in vehicle’s frame (local) Euler Derivatives! Basic Principles Interlude: relative kinematics Differentiation: transcorrot cent Inertial system: i Moving system: e P = CoM O P

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Frame Mechanisation I: i-Frame Vehicle‘s velocity (ground speed) and Coriolis Equation: abbreviated: Differentiation: Applying Coriolis Equation (earth‘s turn rate is constant): subscipt: with respect to; superscript: denotes the axis set; slash: resolved in axis set Basic Principles

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Frame Mechanisation II: i-Frame Newton’s 2nd axiom: abbreviated: Recombination: i-frame axes: Substitution: subscipt: with respect to; superscript: denotes the axis set; slash: resolved in axis set Basic Principles

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University 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 ESTIMATES OF ATTITUDE POSITION INFORMATION POSITION AND VELOVITY ESTIMATES POSSIBILITY FOR KALMAN FILTER INSTALLATION

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Basic Principles Strapdown Attitude Representation: Direction cosine matrix Quaternions Euler angles No singularities, perfect for internal computations singularities, good physical appreciation

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Basic Principles Strapdown Attitude Representation: Direction Cosine Matrix For Instance: Simple Derivative:Axis projection: With skew symmetric matrix:

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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, defined with respect to reference frame Magnitude of rotation: Operations analogous to 2 Parameter Complex number

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Basic Principles Strapdown Attitude Representation: Euler Angles Rotation about reference z axis through angle Rotation about new y axis through angle Rotation about new z axis through angle Singularity: Gimbal angle pick-off!

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Accelerometers Physical principles: Potentiometric LVDT (linear voltage differential transformer) Piezoelectric Newton’s 2nd axiom: gravitational part: Compensation

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Accelerometers Potentiometric + -

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Accelerometers LVDT (linear voltage differential transformer) Uses Induction

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Accelerometers Piezoelectric

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Accelerometers Servo principle (Force Feedback) Intern closed loop feedback Better linearity Null seeking instead of displacement measurement 1 - seismic mass 2 - position sensing device 3 - servo mechanism 4 - damper 5 - case

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Gyroscopes Vibratory Gyroscopes Optical Gyroscopes Historical definition:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Gyroscopes: Vibratory Gyroscopes Coriolis principle: 1. axis velocity caused by harmonic oscillation (piezoelectric) 2. axis rotation 3. axis acceleration measurement Problems: High noise Temperature drifts Translational acceleration vibration

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Gyroscopes: Vibratory Gyroscopes

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Sensor Technology Gyroscopes: Optical Gyroscopes Sagnac Effect: Super Luminiszenz Diode Beam splitter Fiber optic cable coil Effective path length difference LASER INTERFERENCE DETECTOR MODULATOR Beam splitter Beam splitter

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter The Kalman Filter – A stochastic filter method Motivation: Uncertainty of measurement System noise Bounding gyroscope’s drift (e.g. analytic systems) Higher accuracy

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter The Kalman Filter – what is it? Definition: Optimal recursive 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 conditions of the variables of interest

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter The Kalman Filter – Modelization of noise Deviation: Bias: Offset in measurement provided by a sensor, caused by imperfections Noise: disturbing value of large unspecific frequency range Assumption in Modelization: White Noise: Noise with constant amplitude (spectral density) on frequency domain (infinite energy); zero mean Gaussian (normally) distributed: probability density function

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Basic Idea:

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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 factors):

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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!

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Lecture: 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:

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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: Multidimensional case: Covariance matrix:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Interlude: the covariance matrix 1-D: Variance – 2nd central moment N-D: Covariance – diagonal elements are variances, off-diagonal elements encode the correlations Covariance of a vector: n x n matrix, which can be modal transformed, such that are only diagonal elements with decoupled error contribution; Symmetric and quadratic

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Interlude: the covariance matrix applied to equations Equation structure: x, y are gaussian distributed, c is constant: Covariance of z: Linear difference equation: Covariance: with: Kalman Filter Diagonal structure: since white gaussian noise

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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:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Combination of independent estimates: (n-D) Covariance: Minimisation of Variance matrix‘s Diagonal elements (Kalman Gain): For further information please also read: P.S. Maybeck: ‘Stochastic Models, Estimation and Control Volume 1’, Academic Press, New York San Francisco London

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Combination of independent estimates: (n-D) Mean value: Variance:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Interlude: time continuous system to discrete system Continuous solution: Substitution: Conclusion: Sampling time:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter The Kalman Filter: Iteration Principle 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 STOCHASTIC MODELLS TO NEW QUALITY VALUE OF SOLUTION PREDICTION CORRECTION NEXT ITERATION

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Linear Systems – the Kalman Filter: Discrete State Model: Sensor Model:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Linear Systems – the Kalman Filter: 1. Step Prediction Prediction: State Prediction Covariance: Observation Prediction:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Linear Systems – the Kalman Filter: 2. Step Correction Corrected state estimate: Corrected State Covariance: Innovation Covariance: Innovation:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter The Kalman Filter: Kalman Gain Kalman Gain: State Prediction Covariance Innovation Covariance

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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

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Lecture: Position Measurement in Inertial 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 state Necessary for Kalman Gain and Covariance Calculation

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Non-Linear Systems – the extended Kalman Filter: Prediction: Correction:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Example: Aiding the missile MISSILE WITH ON-BOARD INERTIAL NAVIGATION SYSTEM (REPLACING THE PHYSICAL PROCESS MODEL; 1 ESTIMATE) AND NAVIGATION AID (GROUND TRACKER MEASUREMENT; 2 ESTIMATE) MISSILESURFACE SENSORS KALMAN GAINS INSMEASUREMENT MODEL Missile Motion Measurement Noise True Position Measurement Innovations Estimated INS Error System Noise INS Indicated Position Estimated Range, Elevation and Bearing +_+_

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Example: Aiding the missile Nine State Kalman Filter: 3 attitude, 3 velocity, 3 position errors Bounding Gyroscope’s 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!

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Example: Aiding the missile Measurement Equations with respect to radar, providing measurements in polar coordinates, i.e. Range (R), elevation ( ) and bearing ( ). Expressed in Cartesian coordinates (x,y,z): Radar Measurements:

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Example: Aiding the missile Estimates of the radar measurements, z, obtained from the inertial navigation system: Innovation: (Measurement Equation)

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Example: Aiding the missile H-Matrix (Jacobian): Best Estimate of the errors after update: Covariance Prediction: Initial setup: diagonal structure

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Example: Aiding the missile Filter update: Estimates of error: Covariance update: (R measurement noise, diagonal structure)

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich Kalman Filter Example: Aiding the missile Velocity and Position Correction: Attitude Correction: (direction cosine matrix)

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Lecture: Position Measurement in Inertial Systems Christian Wimmer Technical University of Munich thank you for your attention

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