Tracking Overview and Mathematics Christoph Krautz Tracking Motivation

Tracking Overview and Mathematics Christoph Krautz

Tracking Motivation Technologies Mathematics Content n Motivation n Technologies – Advantages and Disadvantages – – – – Common Problems and Errors Acoustic Tracking Mechanical Tracking Inertial Tracking Magnetic Tracking Optical Tracking Inside-out versus Outside-in n Mathematics – Transformations in the 2 D-space – Transformations in the 3 D-space n Discussion Christoph Krautz 2

Tracking Motivation Technologies Mathematics Motivation What is tracking? n The repeated localization of the position and orientation (pose) of one or several real physical objects Why is tracking needed in AR? n Integration of virtual objects into real world (images) Christoph Krautz 3

Tracking Motivation Technologies Mathematics Content n Motivation n Technologies – Advantages and Disadvantages – – – – Common Problems and Errors Acoustic Tracking Mechanical Tracking Inertial Tracking Magnetic Tracking Optical Tracking Inside-out versus Outside-in n Mathematics – Transformations in the 2 D-space – Transformations in the 3 D-space n Discussion Christoph Krautz 4

Tracking Motivation Technologies Mathematics Common Problems and Errors n High update required (usually in real-time systems) n Dynamic tracker error, e. g. sensor‘s motion n Distortion due to environmental influences, e. g. noise n Long-term variations – Cause readings to change from one day to the next day Christoph Krautz 5

Tracking Motivation Technologies Mathematics Acoustic Tracking n The Geometry From [1] – 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. n Ultrasonic – 40 [k. Hz] typical Christoph Krautz (Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch) 6

Tracking Motivation Technologies Mathematics Acoustic Tracking - Methods n 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 n Phase Coherence – Measures phase difference between transmitted and received sound waves – Relative to previous measurement • still absolute!! Christoph Krautz (Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch) 7

Tracking Motivation Technologies Mathematics Acoustic Tracking – Discussion n Advantages – Small and lightweight (miniaturization of transmitters and receivers) – Only sensitive to influences by noise in the ultrasonic range n 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 Motivation Technologies Mathematics Mechanical Tracking n Ground-based or Body-based n Used primarily for motion capture n Provide angle and range measurements – Gears – Bend sensors n Elegant addition of force feedback Christoph Krautz (Slide taken from SIGGRAPH 2001 Course From [1] 11 – Slides by Allen, Bishop, Welch) From [1] 9

Tracking Motivation Technologies Mathematics Mechanical Tracking – Discussion n Advantages – Good accuracy – High update rate – No suffering from environmental linked errors n Disadvantages – Small working volume due to mechanical linkage with the reference Christoph Krautz 10

Tracking Motivation Technologies Mathematics Inertial Tracking n Inertia – Rigidity in space n Newton’s Second Law of Motion – F = ma – M = I (linear) (rotational) n Accelerometers and Gyroscopes – Provide derivative measurements Christoph Krautz 11

Tracking Motivation Technologies Mathematics Inertial Tracking - Accelerometers n Measure force exerted on a mass since we cannot measure acceleration directly. n Proof-mass and damped spring – Displacement proportional to acceleration From [1] n Potentiometric and Piezoelectric Transducers Christoph Krautz (Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch) 12

Tracking Motivation Technologies Mathematics Inertial Tracking - Gyroscopes n Conservation of angular momentum n 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 Christoph Krautz 13

Tracking Motivation Technologies Mathematics Inertial Tracking - Gyroscopes From [1] Christoph Krautz 14

Tracking Motivation Technologies Mathematics Inertial Tracking - Gyroscopes Christoph Krautz From [1] 15

Tracking Motivation Technologies Mathematics Inertial Tracking - Gyroscopes Christoph Krautz 16

Tracking Motivation Technologies Mathematics Inertial Tracking - Gyroscopes Christoph Krautz 17

Tracking Motivation Technologies Mathematics Inertial Tracking – Discussion n Advantages – Lightweight – No physical limits on the working volume n 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 Motivation Technologies Mathematics Magnetic Tracking n 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 6 D pose n AC at low frequency n DC-pulses Christoph Krautz (Parts of the slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch) 19

Tracking Motivation Technologies Mathematics Magnetic Tracking – Discussion n Advantages – Small – Good update rate n 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 Motivation Technologies Mathematics Optical Tracking n Provides angle measurements – One 2 D point defines a ray – Two 2 D points define a point for 3 D position – Additional points required for orientation n Speed of Light – 2. 998 * 108 [m/s] Christoph Krautz From [1] (Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch) 21

Tracking Motivation Technologies Mathematics Optical Tracking – Active Targets n Typical detectors – Lateral Effect Photo. Diodes (LEPDs) – Quad Cells n Active targets – LEDs From [1] Christoph Krautz 22

Tracking Motivation Technologies Mathematics Optical Tracking – Passive Targets n Typical detectors – Video and CCD cameras • Computer vision techniques n Passive targets – Reflective materials, high contrast patterns From [1] Christoph Krautz 23

Tracking Motivation Technologies Mathematics Optical Tracking – Passive Targets From [A. R. T. Gmb. H] Christoph Krautz 24

Tracking Motivation Technologies Mathematics Optical Tracking – Discussion n Advantages – Good update rate (due to the speed of light) • Well suited for real-time systems n 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 Motivation Technologies Mathematics Inside-out versus Outside-in n Inside-out From [3] Christoph Krautz 26

Tracking Motivation Technologies Mathematics Inside-out versus Outside-in n Outside-in From [3] Christoph Krautz 27

Tracking Motivation Technologies Mathematics Content n Motivation n Technologies – Advantages and Disadvantages – – – – Common Problems and Errors Acoustic Tracking Mechanical Tracking Inertial Tracking Magnetic Tracking Optical Tracking Inside-out versus Outside-in n Mathematics – Transformations in the 2 D-space – Transformations in the 3 D-space n Discussion Christoph Krautz 28

Tracking Motivation Technologies Mathematics Position and Orientation (Pose) n Representation – x, y, z (position) and , , (orientation) – with respect to a given reference coordinate system From [1] Christoph Krautz 29

Tracking Motivation Technologies Mathematics Transformations in the 2 D-space n Translation Y 2 1 1 Christoph Krautz 2 3 X 30

Tracking Motivation Technologies Mathematics Transformations in the 2 D-space n Scale Y 2 1 1 Christoph Krautz 2 3 X 31

Tracking Motivation Technologies Mathematics Transformations in the 2 D-space n Rotation Y Y X 2 1 1 Christoph Krautz 2 3 X 32

Tracking Motivation Technologies Mathematics Transformations in the 2 D-space n Scale and Rotation can be combined by multiplication of their matrices n Translation cannot be combined with them by multiplication n Introduction of Homogeneous Coordinates From [1] Christoph Krautz 33

Tracking Motivation Technologies Mathematics Transformations in the 2 D-space Christoph Krautz 34

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n Translation Christoph Krautz 35

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n Scale Christoph Krautz 36

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n Rotation Christoph Krautz 37

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n e. g. Rotation through about the z axis Christoph Krautz 38

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n Rotation-Sequences – Concatenation of several rotations – Can be performed by using • Rotation matrices (matrix multiplication) • Euler-angles • Quaternions Christoph Krautz 39

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n 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 Motivation Technologies Mathematics Transformations in the 3 D-space n Quaternions n Unit Quaternions n A unit quaternion represents a rotation about the axis through the angle Christoph Krautz 41

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n Multiplication-operator for quaternions: n The result is a rotation p composed by the rotations q and r. Christoph Krautz 42

Tracking Motivation Technologies Mathematics Transformations in the 3 D-space n 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 Technologies Mathematics Conclusion n Each tracking technology has advantages and disadvantages n Multi-Sensor-Fusion for minimizing the measurement errors n Transformations in the 3 D-space have to be handled with care Christoph Krautz 44

Tracking Motivation Technologies Mathematics Thank you for your attention! Any questions? Christoph Krautz 45

Tracking 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 Christoph Krautz 46