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Mechatronics - Foundations and Applications Position Measurement in Inertial Systems. JASS 2006, St.Petersburg Christian Wimmer. Motivation Basic principles of position measurement Sensor technology Improvement: Kalman filtering. Content. Motivation. Johnnie: A biped walking machine

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mechatronics foundations and applications position measurement in inertial systems

Mechatronics - Foundations and ApplicationsPosition Measurement in Inertial Systems

JASS 2006, St.Petersburg

Christian Wimmer

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

content
Motivation

Basic principles of position measurement

Sensor technology

Improvement: Kalman filtering

Content

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

motivation
Motivation
  • Johnnie: A biped walking machine
  • Orientation
  • Stabilization
  • Navigation

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

motivation4
Motivation
  • Automotive Applications:
  • Drive dynamics Analysis
  • Analysis of test route topology
  • Driver assistance systems

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

motivation5
Motivation
  • Aeronautics and Space Industry:
  • Autopilot systems
  • Helicopters
  • Airplane
  • Space Shuttle

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

motivation6
Motivation
  • Military Applications:
  • ICBM, CM
  • Drones (UAV)
  • Torpedoes
  • Jets

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

motivation7
Motivation
  • Maritime Systems:
  • Helicopter Platforms
  • Naval Navigation
  • Submarines

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

motivation8
Motivation
  • Industrial robotic Systems:
  • Maintenance
  • Production

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles
Basic Principles

Measurement by inertia and integration:

Acceleration

Velocity

Position

  • Measurement system with 3 sensitive axes
  • 3 Accelerometers
  • 3 Gyroscope

Newton‘s 2. Axiom:

F = m x a

BASIC PRINCIPLE OF DYNAMICS

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles10
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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles11
Basic Principles
  • Strapdown Technology:
  • Body fixed
  • 3 Accelerometers
  • 3 Gyroscopes

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles12
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 principles13
Basic Principles
  • Reference Frames:
  • i-frame
  • e-frame
  • n-frame
  • b-frame

Also normed: WGS 84

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles14
Basic Principles

Interlude: relative kinematics

Moving system: e

P = CoM

Vehicle‘s acceleration in inertial axes (1.Newton):

Problem: All quantities are obtained in vehicle’s frame (local)

Euler Derivatives!

P

O

Differentiation:

Inertial system: i

cent

trans

rot

cor

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles15
Basic Principles

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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles16
Basic Principles

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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles17
Basic Principles

Frame Mechanisation III: Implementation

POSITION

INFORMATION

GRAVITY

COMPUTER

CORIOLIS

CORRECTION

RESOLUTION

OF

SPECIFIC FORCE

MEASUREMENTS

BODY

MOUNTED

ACCELEROMETERS

NAVIGATION

COMPUTER

POSITION AND

VELOVITY

ESTIMATES

POSSIBILITY FOR KALMAN FILTER INSTALLATION

BODY

MOUNTED

GYROSCOPES

ATTITUDE

COMPUTER

INITIAL ESTIMATES OF

ATTITUDE

INITIAL ESTIMATES OF

VELOVITY AND POSITION

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles18
Basic Principles
  • Strapdown Attitude Representation:
  • Direction cosine matrix
  • Quaternions
  • Euler angles

No singularities, perfect for internal

computations

singularities, good physical appreciation

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles19
Basic Principles

Strapdown Attitude Representation: Direction Cosine Matrix

Axis projection:

Simple Derivative:

For Instance:

With skew symmetric matrix:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles20
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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

basic principles21
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

Gimbal angle pick-off!

Singularity:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology
Sensor Technology
  • Accelerometers
  • Physical principles:
  • Potentiometric
  • LVDT (linear voltage differential transformer)
  • Piezoelectric

Newton’s 2nd axiom:

gravitational part: Compensation

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology23
Sensor Technology

Accelerometers

Potentiometric

-

+

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology24
Sensor Technology
  • Accelerometers
  • LVDT (linear voltage differential transformer)
  • Uses Induction

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology25
Sensor Technology

Accelerometers

Piezoelectric

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology26
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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology27
Sensor Technology
  • Gyroscopes
  • Vibratory Gyroscopes
  • Optical Gyroscopes

Historical definition:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology28
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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology29
Sensor Technology

Gyroscopes: Vibratory Gyroscopes

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

sensor technology30
Sensor Technology
  • Gyroscopes: Optical Gyroscopes
  • Sagnac Effect:
  • Super Luminiszenz Diode
  • Beam splitter
  • Fiber optic cable coil
  • Effective path length difference

INTERFERENCE

DETECTOR

Beam

splitter

LASER

MODULATOR

Beam

splitter

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

Kalman Filter

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter32
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

Kalman Filter

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter33
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

Kalman Filter

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter34
Kalman Filter

Basic Idea:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter35
Kalman Filter

Combination of independent estimates: stochastic Basics (1-D)

Mean value:

Variance:

Estimates:

Mean of 2 Estimates

(with weighting factors):

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter36
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: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter37
Kalman Filter

Combination of independent estimates: stochastic Basics (1-D)

Weighting factors:

Substitution:

Optimization

(Differentiation):

Optimum weight:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter38
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 filter39
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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter40
Kalman Filter

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:

Diagonal structure: since white gaussian noise

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter41
Kalman Filter

Combination of independent estimates: (n-D)

Mean value:

measurement:

Mean value:

Covariance

with:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter42
Kalman Filter

Combination of independent estimates: (n-D)

Covariance:

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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter43
Kalman Filter

Combination of independent estimates: (n-D)

Mean value:

Variance:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter44
Kalman Filter

Interlude: time continuous system to discrete system

Continuous solution:

Substitution:

Conclusion:

Sampling time:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter45
Kalman Filter

The Kalman Filter: Iteration Principle

CALCULATION OFKALMAN GAIN (WEIGHTING OF MEASUREMENT AND PREDICTION)

PREDICTION OF ERROR COVARIANCE BETWEEN TWO ITERATIONS

PREDICTION OF STATES (SOLUTION) BETWEEN TWO ITERATIONS

DETERMINATION OF NEW SOLUTION (ESTIMATION)

CORRECTION OF THE STOCHASTIC MODELLS TO NEW QUALITY VALUE OF SOLUTION

PREDICTION

NEXT ITERATION

CORRECTION

INITIAL ESTIMATION OF STATES AND QUALITY OF STATE

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter46
Kalman Filter

Linear Systems – the Kalman Filter:

Discrete State Model:

Sensor Model:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter47
Kalman Filter

Linear Systems – the Kalman Filter: 1. Step Prediction

Prediction:

State Prediction Covariance:

Observation Prediction:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter48
Kalman Filter

Linear Systems – the Kalman Filter: 2. Step Correction

Corrected state estimate:

Corrected State Covariance:

Innovation Covariance:

Innovation:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter49
Kalman Filter

The Kalman Filter: Kalman Gain

Kalman Gain:

State Prediction Covariance

Innovation Covariance

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter50
Kalman Filter

The Kalman Filter: System Model

+

+

-

+

Memory

+

+

For linear systems: System matrices are timeinvariant

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter51
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

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter52
Kalman Filter

Non-Linear Systems – the extended Kalman Filter:

Prediction:

Correction:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter53
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)

Measurement

Noise

Missile Motion

True Position

MISSILE

SURFACE SENSORS

Estimated INS Error

Measurement Innovations

+

_

KALMAN GAINS

INS Indicated Position

INS

MEASUREMENT MODEL

Estimated Range, Elevation and Bearing

System Noise

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter54
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!

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter55
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:

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter56
Kalman Filter

Example: Aiding the missile

Estimates of the radar measurements, z, obtained from the inertial navigation system:

Innovation: (Measurement Equation)

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter57
Kalman Filter

Example: Aiding the missile

H-Matrix (Jacobian):

Best Estimate of the errors after update:

Covariance Prediction:

Initial setup: diagonal structure

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter58
Kalman Filter

Example: Aiding the missile

Filter update:

Estimates of error:

Covariance update:

(R measurement noise, diagonal structure)

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

kalman filter59
Kalman Filter

Example: Aiding the missile

Velocity and Position Correction:

Attitude Correction:

(direction cosine matrix)

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

slide60
thank you for your attention

Lecture: Position Measurement in Inertial Systems

Christian Wimmer

Technical University of Munich

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