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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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JASS 2006, St.Petersburg
Christian Wimmer
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Basic principles of position measurement
Sensor technology
Improvement: Kalman filtering
ContentLecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Measurement by inertia and integration:
Acceleration
Velocity
Position
Newton‘s 2. Axiom:
F = m x a
BASIC PRINCIPLE OF DYNAMICS
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
SENSORS FASTENED DIRECTLY ON THE VEHICLE
BODY FIXED COORDINATE SYSTEM
ANALYTIC SYSTEM
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Also normed: WGS 84
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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
Frame Mechanisation I: iFrame
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
Frame Mechanisation II: iFrame
Newton’s 2nd axiom:
abbreviated:
Recombination: iframe 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
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
No singularities, perfect for internal
computations
singularities, good physical appreciation
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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
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
Strapdown Attitude Representation: Euler Angles
Gimbal angle pickoff!
Singularity:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Newton’s 2nd axiom:
gravitational part: Compensation
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Accelerometers
Potentiometric

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Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Accelerometers
Piezoelectric
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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
Historical definition:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Gyroscopes: Vibratory Gyroscopes
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
INTERFERENCE
DETECTOR
Beam
splitter
LASER
MODULATOR
Beam
splitter
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Motivation:
Uncertainty of measurement
System noise
Bounding gyroscope’s drift (e.g. analytic systems)
Higher accuracy
Kalman FilterLecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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 FilterLecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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 FilterLecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Basic Idea:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Combination of independent estimates: stochastic Basics (1D)
Mean value:
Variance:
Estimates:
Mean of 2 Estimates
(with weighting factors):
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Combination of independent estimates: stochastic Basics (1D)
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
Combination of independent estimates: stochastic Basics (1D)
Weighting factors:
Substitution:
Optimization
(Differentiation):
Optimum weight:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Combination of independent estimates: stochastic Basics (1D)
Mean value:
Variance:
Multidimensional case:
Covariance matrix:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Interlude: the covariance matrix
1D: Variance – 2nd central moment
ND: Covariance – diagonal elements are variances, offdiagonal 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
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
Combination of independent estimates: (nD)
Mean value:
measurement:
Mean value:
Covariance
with:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Combination of independent estimates: (nD)
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
Combination of independent estimates: (nD)
Mean value:
Variance:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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
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
Linear Systems – the Kalman Filter:
Discrete State Model:
Sensor Model:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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
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
The Kalman Filter: Kalman Gain
Kalman Gain:
State Prediction Covariance
Innovation Covariance
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
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
NonLinear 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
NonLinear Systems – the extended Kalman Filter:
Prediction:
Correction:
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich
Example: Aiding the missile
MISSILE WITH ONBOARD 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
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
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
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
Example: Aiding the missile
HMatrix (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
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
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
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich