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Table of Contents
Intro; Preface; How to Use This Book; Acknowledgements; Abbreviations; Contents; 1 Introduction; 1.1 Recording Movement and Orientation; 1.2 Conventions and Basics; 1.2.1 Notation; 1.2.2 Coordinate Systems; 1.3 Software Packages; 1.3.1 Python Package scikit-kinematics; 1.3.2 Matlab 3-D Kinematics Toolbox; 1.3.3 Source Code for Python and Matlab; 1.4 Warm-Up Exercises; 2 Measurement Techniques; 2.1 Marker-Based Measurements; 2.1.1 Image Formation; 2.2 Sensor-Based Measurements; 2.2.1 Overview; 2.2.2 Linear Accelerometers; 2.2.3 Gyroscopes; 2.2.4 Ultrasound Sensors-Trilateration
2.2.5 Magnetic Field Sensors3 Rotation Matrices; 3.1 Introduction; 3.2 Rotations in a Plane; 3.2.1 Rotation in Cartesian Coordinates; 3.2.2 Rotation in Polar Coordinates; 3.2.3 Application: Orienting an Object in a Plane; 3.3 Rotations About Coordinate Axes in 3-D; 3.3.1 3-D Rotations About Coordinate Axes; 3.3.2 Rotations of Objects Versus Rotations of Coordinate Systems; 3.4 Combined Rotations; 3.4.1 3-D Orientation with Sequential Rotations; 3.4.2 Gimbal Lock; 3.5 Homogeneous Coordinates; 3.5.1 Definition; 3.6 Applications; 3.6.1 Two DOF-Targeting an Object in 3-D
3.6.2 Two DOF-Projection onto a Flat Surface3.6.3 Three DOF-3-D Orientation Measurements with Search Coils; 3.6.4 Nested or Cascaded 3-D Rotation Sequences; 3.6.5 Camera Images; 3.7 Exercises; 4 Quaternions and Gibbs Vectors; 4.1 Representing Rotations by Vectors; 4.2 Axis-Angle Euler Vectors; 4.3 Quaternions; 4.3.1 Background; 4.3.2 Quaternion Properties; 4.3.3 Interpretation of Quaternions; 4.3.4 Unit Quaternions; 4.4 Gibbs Vectors; 4.4.1 Properties of Gibbs Vectors; 4.4.2 Cascaded Rotations with Gibbs Vectors; 4.4.3 Gibbs Vectors and Their Relation to Quaternions; 4.5 Applications
4.5.1 Targeting an Object in 3-D: Quaternion Approach4.5.2 Orientation of 3-D Acceleration Sensor; 4.5.3 Calculating Orientation of a Camera on a Moving Object; 4.5.4 Object-Oriented Implementation of Quaternions; 5 Velocities in 3-D Space; 5.1 Equations of Motion; 5.2 Linear Velocity; 5.3 Angular Velocity; 5.3.1 Calculating Angular Velocity from Orientation; 5.3.2 Calculating Orientation from Angular Velocity; 6 Analysis of 3-D Movement Recordings; 6.1 Position and Orientation from Optical Sensors; 6.1.1 Recording 3-D Markers; 6.1.2 Orientation in Space; 6.1.3 Position in Space
6.1.4 Velocity and Acceleration6.1.5 Transformation from Camera- to Space-Coordinates; 6.1.6 Position; 6.2 Position and Orientation from Inertial Sensors; 6.2.1 Orientation in Space; 6.2.2 Position in Space; 6.3 Applications: Gait Analysis; 6.4 Exercises; 7 Multi-sensor Integration; 7.1 Working with Uncertain Data; 7.1.1 Uncertain Data in One Dimension; 7.1.2 Uncertain Data in Multiple Dimensions; 7.2 Kalman Filter; 7.2.1 Idea Behind Kalman Filters; 7.2.2 State Predictions; 7.2.3 Measurements and Kalman Equations; 7.2.4 Kalman Filters with Quaternions; 7.3 Complementary Filters; 7.3.1 Gradient Descent Approach
2.2.5 Magnetic Field Sensors3 Rotation Matrices; 3.1 Introduction; 3.2 Rotations in a Plane; 3.2.1 Rotation in Cartesian Coordinates; 3.2.2 Rotation in Polar Coordinates; 3.2.3 Application: Orienting an Object in a Plane; 3.3 Rotations About Coordinate Axes in 3-D; 3.3.1 3-D Rotations About Coordinate Axes; 3.3.2 Rotations of Objects Versus Rotations of Coordinate Systems; 3.4 Combined Rotations; 3.4.1 3-D Orientation with Sequential Rotations; 3.4.2 Gimbal Lock; 3.5 Homogeneous Coordinates; 3.5.1 Definition; 3.6 Applications; 3.6.1 Two DOF-Targeting an Object in 3-D
3.6.2 Two DOF-Projection onto a Flat Surface3.6.3 Three DOF-3-D Orientation Measurements with Search Coils; 3.6.4 Nested or Cascaded 3-D Rotation Sequences; 3.6.5 Camera Images; 3.7 Exercises; 4 Quaternions and Gibbs Vectors; 4.1 Representing Rotations by Vectors; 4.2 Axis-Angle Euler Vectors; 4.3 Quaternions; 4.3.1 Background; 4.3.2 Quaternion Properties; 4.3.3 Interpretation of Quaternions; 4.3.4 Unit Quaternions; 4.4 Gibbs Vectors; 4.4.1 Properties of Gibbs Vectors; 4.4.2 Cascaded Rotations with Gibbs Vectors; 4.4.3 Gibbs Vectors and Their Relation to Quaternions; 4.5 Applications
4.5.1 Targeting an Object in 3-D: Quaternion Approach4.5.2 Orientation of 3-D Acceleration Sensor; 4.5.3 Calculating Orientation of a Camera on a Moving Object; 4.5.4 Object-Oriented Implementation of Quaternions; 5 Velocities in 3-D Space; 5.1 Equations of Motion; 5.2 Linear Velocity; 5.3 Angular Velocity; 5.3.1 Calculating Angular Velocity from Orientation; 5.3.2 Calculating Orientation from Angular Velocity; 6 Analysis of 3-D Movement Recordings; 6.1 Position and Orientation from Optical Sensors; 6.1.1 Recording 3-D Markers; 6.1.2 Orientation in Space; 6.1.3 Position in Space
6.1.4 Velocity and Acceleration6.1.5 Transformation from Camera- to Space-Coordinates; 6.1.6 Position; 6.2 Position and Orientation from Inertial Sensors; 6.2.1 Orientation in Space; 6.2.2 Position in Space; 6.3 Applications: Gait Analysis; 6.4 Exercises; 7 Multi-sensor Integration; 7.1 Working with Uncertain Data; 7.1.1 Uncertain Data in One Dimension; 7.1.2 Uncertain Data in Multiple Dimensions; 7.2 Kalman Filter; 7.2.1 Idea Behind Kalman Filters; 7.2.2 State Predictions; 7.2.3 Measurements and Kalman Equations; 7.2.4 Kalman Filters with Quaternions; 7.3 Complementary Filters; 7.3.1 Gradient Descent Approach