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Overview and Mathematics

Motivation

Technologies

Mathematics

Content

- 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
- Transformations in the 3D-space

- Discussion

Motivation

Technologies

Mathematics

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)

Motivation

Technologies

Mathematics

Content

- 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
- Transformations in the 3D-space

- Discussion

Motivation

Technologies

Mathematics

Common Problems and Errors

- 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

Tracking

Motivation

Technologies

Mathematics

Acoustic Tracking

- 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

(Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

Motivation

Technologies

Mathematics

Acoustic Tracking - Methods

- 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!!

(Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

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

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

From [1]

Tracking

Motivation

Technologies

Mathematics

Mechanical Tracking

- Ground-based or Body-based
- Used primarily for motion capture
- Provide angle and range measurements
- Gears
- Bend sensors

- Elegant addition of force feedback

(Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

Motivation

Technologies

Mathematics

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

Motivation

Technologies

Mathematics

Inertial Tracking

- Inertia
- Rigidity in space

- Newton’s Second Law of Motion
- F = ma (linear)
- M = I (rotational)

- Accelerometers and Gyroscopes
- Provide derivative measurements

Tracking

Motivation

Technologies

Mathematics

Inertial Tracking - Accelerometers

- Measure force exerted on a mass since we cannot measure acceleration directly.
- Proof-mass and damped spring
- Displacement proportional to acceleration

- Potentiometric and Piezoelectric Transducers

(Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

Motivation

Technologies

Mathematics

Inertial Tracking - Gyroscopes

- 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

Motivation

Technologies

Mathematics

Inertial Tracking – Discussion

- Advantages
- Lightweight
- No physical limits on the working volume

- Disadvantages
- Error accumulation due to integration (numerical)
- Periodic recalibration
- Hybrid systems typical

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

- Error accumulation due to integration (numerical)

Motivation

Technologies

Mathematics

Magnetic Tracking

- 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)

- Varies with
- Three measurements apiece (three receiver coils)
- Nine-element measurement for 6D pose

- Induced current in the receiver coils is measured

- Each transmitter coil activated serially
- AC at low frequency
- DC-pulses

(Parts of the slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

Motivation

Technologies

Mathematics

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

Tracking

Motivation

Technologies

Mathematics

Optical Tracking

- Provides angle measurements
- One 2D pointdefines a ray
- Two 2D pointsdefine a pointfor 3D position
- Additional pointsrequired fororientation

- Speed of Light
- 2.998 * 108 [m/s]

(Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

Tracking

Motivation

Technologies

Mathematics

Optical Tracking – Active Targets

- Typical detectors
- Lateral Effect PhotoDiodes (LEPDs)
- Quad Cells

- Active targets
- LEDs

Tracking

Motivation

Technologies

Mathematics

Optical Tracking – Passive Targets

- Typical detectors
- Video and CCD cameras
- Computer vision techniques

- Video and CCD cameras
- Passive targets
- Reflective materials, high contrast patterns

Motivation

Technologies

Mathematics

Optical Tracking – Discussion

- Advantages
- Good update rate (due to the speed of light)
- Well suited for real-time systems

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

Motivation

Technologies

Mathematics

Content

- 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
- Transformations in the 3D-space

- Discussion

Tracking

Motivation

Technologies

Mathematics

Position and Orientation (Pose)

- Representation
- x, y, z (position) and , , (orientation)
- with respect to a given reference coordinate system

2

1

X

2

3

1

Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space

- Translation

Y

2

1

X

X

2

3

1

Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space

- Rotation

Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space

- Scale and Rotation can be combined by multiplication of their matrices
- Translation cannot be combined with them by multiplication
- Introduction of Homogeneous Coordinates

Motivation

Technologies

Mathematics

Transformations in the 3D-space

- e.g. Rotation through about the z axis

Motivation

Technologies

Mathematics

Transformations in the 3D-space

- Rotation-Sequences
- Concatenation of several rotations
- Can be performed by using
- Rotation matrices (matrix multiplication)
- Euler-angles
- Quaternions

Motivation

Technologies

Mathematics

Transformations in the 3D-space

- 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, , )

- Three angles , and

Motivation

Technologies

Mathematics

Transformations in the 3D-space

- Quaternions

- Unit Quaternions

- A unit quaternionrepresents a rotation about the axisthrough the angle

Motivation

Technologies

Mathematics

Transformations in the 3D-space

- Multiplication-operator for quaternions:

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

Motivation

Technologies

Mathematics

Transformations in the 3D-space

- 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

Motivation

Technologies

Mathematics

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

Motivation

Technologies

Mathematics

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

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