1. JASS 2006, St.Petersburg Christian Wimmer
Mechatronics - Foundations and Applications Position Measurement in Inertial Systems
JASS 2006, St.Petersburg
Christian Wimmer
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich

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

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

4. 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

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

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

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

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

9. 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

10. 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

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

12. 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

13. 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

14. 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
trans
rot
cor
cent
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

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

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

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

26. 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

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

28. 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

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

30. 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

31. 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
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich

32. 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
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich

33. 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
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich

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

35. 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

36. Kalman Filter
Combination of independent estimates: stochastic Basics (1-D)
Weighted mean:
Variance of weighted
mean:
Not correlated:
Quantiles are independent!
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich

37. 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

38. 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

39. Kalman Filter Interlude: the covariance matrix Covariance of a vector:
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

40. 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

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

42. 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
Lecture: Position Measurement in Inertial Systems
Christian Wimmer
Technical University of Munich

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

44. 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

45. Kalman Filter The Kalman Filter: Iteration Principle PREDICTION
PREDICTION OF ERROR COVARIANCE BETWEEN TWO ITERATIONS
CALCULATION OFKALMAN GAIN (WEIGHTING OF MEASUREMENT AND PREDICTION)
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

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

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

48. 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

49. 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

50. 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

51. 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

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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

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