List of Figures -- List of Tables -- 1 Introduction -- 1.1 Background and Motivation -- 1.2 The State of the -- 1.3 Objective and Scope of the Study -- 1.4 Book Organization -- References -- 2 Magnetic Field Modeling -- 2.1 Introduction -- 2.2 Configuration of Rotor Poles -- 2.3 Magnetic Scalar Potential -- 2.3.1 Relations Between H and B for Three Regions -- 2.3.2 Laplace’s Equations for Three Regions -- 2.3.3 General Solution of Laplace’s Equation -- 2.4 Spherical Harmonic Expansion of M0r -- 2.5 Boundary Conditions -- 2.5.1 Boundary Condition A or Far Field Boundary Condition (BIrjr!¥ = 0, BIq jr!¥ = 0 and BIf jr!¥ = 0) -- 2.5.2 Boundary Condition B (BIrjr=Rr = BIIrjr=Rr ) -- 2.5.3 Boundary Condition C (HIf jr=Rr = HIIf jr=Rr and  HIq jr=Rr = HIIq jr=Rr ) -- 2.5.4 Finite Boundary Condition D at r = 0 (BIIIrjr=0 6= ¥, BIIIq jr=0 6= ¥ and BIIIf jr=0 6= ¥) -- 2.5.5 Boundary Condition E (BIIrjr=Rb = BIIIrjr=Rb ) -- 2.5.6 Boundary Condition F (HIIf jr=Rb = HIIIf jr=Rb and HIIq jr=Rb = HIIIq jr=Rb ) -- 2.5.7 Solution of Coefficients x mnI and kmnI -- 2.6 Solutions of Scalar Potential and Flux Density -- 2.7 Simplification of Magnetic Field Model -- 2.8 Summary -- References -- 3 Torque Modeling -- 3.1 Introduction -- 3.2 Formulation of Actuator Torque -- 3.2.1 Torque Generating Component of Flux Density -- 3.2.2 Torque Model for a Single Coil -- 3.2.3 Torque Model for Complete Set of Coils -- 3.2.4 Orientation Dependance of Torque Model -- 3.3 Solution of Inverse Electromagnetics -- 3.3.1 Nonsingularity of the Workspace -- 3.3.2 Minimum Right-inverse Solution of Electromagnetics -- 3.4 Summary -- References -- 4 Prototype Development -- 4.1 Introduction -- 4.1.1 Prototype of PM Spherical Actuator -- 4.1.2 Equations for Actuator Design -- 4.2 Rotor Pole Design -- 4.2.1 Longitudinal Angle a versus a -- 4.2.2 Latitudinal Angle b versus c -- 4.2.3 Rotor Radius Rr versus d4 -- 4.2.4 Rotor Core Radius Rb versus d4 -- 4.2.5 Relative Permeability mr versus d4 -- 4.2.6 Result of PM Pole Design -- 4.3 Coil Pole Design -- 4.3.1 Geometric Parameters of Coil -- 4.3.2 Increase Number of Winding Turns -- 4.3.3 Material of Coil Frame -- 4.4 Stator -- 4.5 Spherical Bearing -- 4.6 Summary -- References -- 5 Experimental Investigation -- 5.1 Measurement of PM Rotor Magnetic Field -- 5.1.1 Flux Density Measurement Apparatus -- 5.1.2 Flux Density Data Processing -- 5.1.3 Visualization and Analysis of Experimental Result -- 5.2 Measurement of Actuator Torque Output -- 5.2.1 Experiment on Torque Generated by a Single Coil -- 5.2.2 Experiment on Torque Generated by Multiple Coils -- 5.3 Summary -- References -- 6 Three Degree-of-freedom Optical Orientation Measurement -- 6.1 Introduction -- 6.2 Operating Principle -- 6.3 Algorithm for Computing Rotation Angles -- 6.3.1 Definition of Coordinate Systems -- 6.3.2 Calculation of Tilting Angles -- 6.3.3 Calculation of Spinning Angle -- 6.4 Experimental Measurement -- 6.4.1 Experimental Measurement on Apparatus 1 -- 6.4.2 Experimental Measurement on Apparatus 2 -- 6.5 Conclusion -- References -- 7 Conclusions -- 7.1 Accomplishments and Contributions -- 7.2 Recommendation for Future Research -- References -- Index.

A spherical actuator is a novel electric device that can achieve 2/3-DOF rotational motions in a single joint with electric power input. It has advantages such as compact structure, low mass/moment of inertia, fast response and non-singularities within the workspace. It has promising applications in robotics, automobile, manufacturing, medicine and aerospace industry. This is the first monograph that introduces the research on spherical actuators systematically.  It broadens the scope of actuators from conventional single-axis to multi-axis, which will help both beginners and researchers to enhance their knowledge on electromagnetic actuators. Generic analytic modeling methods for magnetic field and torque output are developed, which can be applied to the development of other electromagnetic actuators. A parametric design methodology that allows fast analysis and design of spherical actuators for various applications is proposed. A novel non-contact high-precision 3-DOF spherical motion sensing methodology is developed and evaluated with experiments, which shows that it can achieve one order of magnitude higher precision than conventional methods. The technologies of nondimensionalization and normalization are introduced into magnetic field analysis the first time, and a benchmark database is established for the reference of other researches on spherical actuators.