Tracking Overview and Mathematics

Tracking

Overview and Mathematics

Tracking

Motivation

Content

Technologies Mathematics

 Motivation

 Technologies – Advantages and Disadvantages

– Common Problems and Errors

– Acoustic Tracking

– Mechanical Tracking

– Inertial Tracking

– Magnetic Tracking

– Optical Tracking

– Inside-out versus Outside-in

 Mathematics

– Transformations in the 2D-space


 Discussion

Christoph Krautz 2

Tracking

Motivation

Motivation


What is tracking?

 The repeated localization of the position and orientation

(pose) of one or several real physical objects

Why is tracking needed in AR?

 Integration of virtual objects into real world (images)

Christoph Krautz 3

Tracking

Motivation

Content


 Motivation

 Technologies – Advantages and Disadvantages








 Mathematics



 Discussion

Christoph Krautz 4

Tracking

Motivation Technologies

Common Problems and Errors

Mathematics

 High update rate required (usually in real-time systems)

 Dynamic tracker error, e.g. sensor‘s motion

 Distortion due to environmental influences, e.g. noise

 Long-term variations

– Cause readings to change from one day to the next day

Christoph Krautz 5

Tracking


Acoustic Tracking

Mathematics

From [1]

 The Geometry

– The intersection of two spheres is a circle.

– The intersection of three spheres is two points.

• One of the two points can easily be eliminated.

 Ultrasonic

– 40 [kHz] typical

Christoph Krautz

(Slide taken from SIGGRAPH 2001 Course

11 – Slides by Allen, Bishop, Welch)

6

Tracking


Acoustic Tracking - Methods

Mathematics

 Time of Flight

– Measures the time required for a sonic pulse to travel from a transmitter to a receiver.

– d [m] = v [m/s] * t [s], v = speed of sound

– Absolute range measurement

 Phase Coherence

– Measures phase difference between transmitted and received sound waves

– Relative to previous measurement

• still absolute!!



Christoph Krautz 7

Tracking

Motivation Technologies Mathematics

Acoustic Tracking – Discussion

 Advantages

– Small and lightweight (miniaturization of transmitters and receivers)

– Only sensitive to influences by noise in the ultrasonic range

 Disadvantages

– Speed of Sound (~331 [m/s] in air at 0°C)

• Varies with temperature, pressure and humidity

•  Slow  Low update rate

Christoph Krautz 8

Tracking


Mechanical Tracking

Mathematics

 Ground-based or Body-based

 Used primarily for motion capture

 Provide angle and range measurements

– Gears

– Bend sensors

 Elegant addition of force feedback

Christoph Krautz

From [1]



From [1]

9

Tracking


Mechanical Tracking – Discussion

 Advantages

– Good accuracy

– High update rate

– No suffering from environmental linked errors

 Disadvantages

– Small working volume due to mechanical linkage with the reference

Christoph Krautz 10

Tracking


Inertial Tracking

Mathematics

 Inertia

– Rigidity in space

 Newton’s Second Law of Motion

– F = ma

– M = I 

(linear)

(rotational)

 Accelerometers and Gyroscopes

– Provide derivative measurements

Christoph Krautz 11

Tracking


Inertial Tracking - Accelerometers

 Measure force exerted on a mass since we cannot measure acceleration directly.

 Proof-mass and damped spring

– Displacement proportional to acceleration

From [1]

 Potentiometric and Piezoelectric Transducers



Christoph Krautz 12

Tracking


Inertial Tracking - Gyroscopes

Mathematics

 Conservation of angular momentum

 Precession

– If torque is exerted on a spinning mass, its axis of rotation will precess at right angles to both itself and the axis of the exerted torque

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Tracking



Mathematics

Christoph Krautz

From [1]

14

Tracking



Mathematics

Christoph Krautz From [1] 15

Tracking



Mathematics

Christoph Krautz 16

Tracking



Mathematics

Christoph Krautz 17

Tracking


Inertial Tracking – Discussion

Mathematics

 Advantages

– Lightweight

– No physical limits on the working volume

 Disadvantages

– Error accumulation due to integration (numerical)

• Periodic recalibration

– Hybrid systems typical

– Drift in the axis of rotation of a gyroscope due to the remaining friction between the axis of the wheel and the bearings

Christoph Krautz 18

Tracking


Magnetic Tracking

Mathematics

 Three mutually-orthogonal coils

– Each transmitter coil activated serially

• Induced current in the receiver coils is measured

– Varies with

» the distance (cubically) from the transmitter and

» their orientation relative to the transmitter (cosine of the angle between the axis and the local magnetic field direction)

• Three measurements apiece (three receiver coils)

• Nine-element measurement for 6D pose

 AC at low frequency

 DC-pulses

Christoph Krautz

(Parts of the slide taken from SIGGRAPH

2001 Course 11 – Slides by Allen, Bishop,

Welch)

19

Tracking


Magnetic Tracking – Discussion

 Advantages

– Small

– Good update rate

 Disadvantages

– Small working volume

– Ferromagnetic interference

– Eddy currents induced in conducting materials

 Distortions

 Inaccurate pose estimates

– Use of DC transmitters overcomes that problem

– Sensitive to electromagnetic noise

Christoph Krautz 20

Tracking


Optical Tracking

Mathematics

 Provides angle measurements

– One 2D point defines a ray

– Two 2D points define a point for 3D position

– Additional points required for orientation

 Speed of Light

– 2.998 * 10 8 [m/s]



Christoph Krautz

From [1]

21

Tracking


Optical Tracking – Active Targets

 Typical detectors

– Lateral Effect PhotoDiodes (LEPDs)

– Quad Cells

 Active targets

– LEDs

Christoph Krautz

From [1]

22

Tracking


Optical Tracking – Passive Targets

 Typical detectors

– Video and CCD cameras

• Computer vision techniques

 Passive targets

– Reflective materials, high contrast patterns

Christoph Krautz

From [1]

23

Tracking


Optical Tracking – Passive Targets

Christoph Krautz

From [A.R.T. GmbH]

24

Tracking


Optical Tracking – Discussion

Mathematics

 Advantages

– Good update rate (due to the speed of light)

• Well suited for real-time systems

 Disadvantages

– Accuracy tends to worsen with increased distance

– Sensitive to optical noise and spurious light

• Can be minimized by using infrared light

– Ambiguity of surface and occlusion

Christoph Krautz 25

Tracking

 Inside-out


Inside-out versus Outside-in

Mathematics

Christoph Krautz

From [3]

26

Tracking

 Outside-in


Inside-out versus Outside-in

Mathematics

Christoph Krautz

From [3]

27

Tracking

Motivation

Content


 Motivation

 Technologies – Advantages and Disadvantages








 Mathematics



 Discussion

Christoph Krautz 28

Tracking


Position and Orientation (Pose)

 Representation

– x, y, z (position) and  ,  ,  (orientation)

– with respect to a given reference coordinate system

From [1]

Christoph Krautz 29

Tracking


Transformations in the 2D-space

 Translation

P



( x , y )



P '



( x

 a , y

 b )

Y

2

1

1 2 3 X

Christoph Krautz 30

Tracking



 Scale

P '



SP

S







 s

1

0

0 s

2







P



( x , y )



P '



( s

1 x , s

2 y )

Y

2

1

1 2 3 X

Christoph Krautz 31

Tracking



 Rotation

P



( x , y )



P '



( x ' , y ' ) where x '

 x cos

  y sin

 and y '

 x sin

  y cos



P '



RP

R





 cos

 sin



 sin cos



 



Y

2

1



1 2 3 X

Christoph Krautz 32

Tracking



 Scale and Rotation can be combined by multiplication of their matrices

 Translation cannot be combined with them by multiplication

 Introduction of Homogeneous Coordinates

( x , y )



( x , y , 1 )

Christoph Krautz

From [1]

33

Tracking



P '



TP

T









1

0

0

0

1

0 a b

1





P '



SP

S s

1

 



0

0

0 s

2

0

0

0

1





P '



RP

R cos



 

 sin

0



 sin cos





0

0

0

1





Christoph Krautz 34

Tracking



 Translation

P '



TP

1

T







0

0

0

0

1

0

0

0

0

1

0 a b c





1

Christoph Krautz 35

Tracking



 Scale

P '



SP

S s

1







0

0

0

0 s

2

0

0

0

0 s

3

0

0

0

0





1

Christoph Krautz 36

Tracking



 Rotation

P '



RP r

11

R





 r

21 r

31

0 r

12 r

22 r

32

0 r

13 r

23 r

33

0

0

0

0





1

Christoph Krautz 37

Tracking



 e.g. Rotation through  about the z axis

P '



RP cos



R







 sin

0



0 sin

 cos



0

0

0

0

1

0

0

0

0





1

Christoph Krautz 38

Tracking



 Rotation-Sequences

– Concatenation of several rotations

– Can be performed by using

• Rotation matrices (matrix multiplication)

• Euler-angles

• Quaternions

Christoph Krautz 39

Tracking



 Euler-angles

– Three angles  ,  and 

• Each represents a rotation about one of the coordinate axes (X, Y and Z).

– Gimbal Lock

– Ambiguities

• R(  , 0, 0) = R(0,  ,  )

Christoph Krautz 40

Tracking



 Quaternions q :

 s

 ix

 jy

 kz



( s ,

 v ), v





( x , y , z )

 Unit Quaternions s

2  x

2  y

2  z

2 

1

 A unit quaternion through the angle

( s , v

 represents a rotation about the axis

2

) arccos

 

 v

Christoph Krautz 41

Tracking



 Multiplication-operator for quaternions: p

 qr



( q r r r

 q v r v

, q r r v

 r r q v

 q v

 r v

)

 The result is a rotation p composed by the rotations q and r.

Christoph Krautz 42

Tracking



 Advantages of quaternions:

– No gimbal lock

– Unique representation of a rotation

– Interpolation can be properly carried out

(spherical interpolation on the 4-sphere; Shoemake,

1985)

– Rotation-sequences can be easily performed

Christoph Krautz 43

Tracking

Motivation

Conclusion


 Each tracking technology has advantages and disadvantages

 Multi-Sensor-Fusion for minimizing the measurement errors

 Transformations in the 3D-space have to be handled with care

Christoph Krautz 44

Tracking


Christoph Krautz

Thank you for your attention!

Any questions?

45

Tracking


References:

[1] G. Bishop, G. Welch and B. D. Allen, „Tracking: Beyond 15 Minutes of Thought”,

SIGGRAPH 2001 Course Notes, University of North Carolina at Chapel Hill

[2] G. Bishop, G. Welch and B. D. Allen, „Tracking: Beyond 15 Minutes of Thought”,

SIGGRAPH 2001 Course Slides, University of North Carolina at Chapel Hill

[3] Ribo, Miguel, “State of the Art Report on Optical Tracking”, 2001

Christoph Krautz 46

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