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This book presents the theory and applications of an advanced mathematical language called geometric algebra that greatly helps to express the ideas and concepts, and to develop algorithms in the broad domain of robot physics.
In the history of science, without essentialmathematical concepts, theorieswould have not been developed at all. We can observe that in various periods of the history of mathematics and physics, certain stagnation occurred from time to time, thanks to new mathematical developments, astonishing progress took place. In addition, we see that the knowledge became unavoidably fragmented as researchers attempted to combine different mathematical systems. Each mathematical system brings about some parts of geometry however, together, these systems constitute a highly redundant system due to an unnecessary multiplicity of representations for geometric concepts. The author expects that due to his persistent efforts to bring to the community geometric algebra for applications as a meta-language for geometric reasoning, in the near future tremendous progress in robotics should take place.
What is geometric algebra? Why are its applications so promising? Why should researchers, practitioners, and students make the effort to understand geometric algebra and use it? We want to answer all these questions and convince the reader that becoming acquainted with geometric algebra for applications is a worthy undertaking.
The history of geometric algebra is unusual and quite surprising. In the 1870s, William Kingdon Clifford introduced his geometric algebra, building on the earlier works of Sir William Rowan Hamilton and Hermann Gunther Grassmann. In Clifford’s work, we perceive that he intended to describe the geometric properties of vectors, planes, and higher-dimensional objects. Most physicists encounter the algebra in the guise of Pauli and Dirac matrix algebras of quantum theory. Many roboticists or computer graphic engineers use quaternions for 3D rotation estimation and interpolation, as a pointwise approach is too difficult for them to formulate homogeneous transformations of high-order geometric entities. They resort often to tensor calculus for multivariable calculus. Since robotics and engineering make use of the developments of mathematical physics, many beliefs are automatically inherited for instance, some physicists come away from a study of Dirac theory with the view that Clifford’s algebra is inherently quantum-mechanical. The goal of this book is to eliminate these kinds of beliefs by giving a clear introduction of geometric algebra and showing this new and promising mathematical framework to multivectors and geometric multiplication in higher dimensions. In this new geometric language, most of the standard matter taught to roboticists and computer science engineers can be advantageously reformulated without redundancies and in a highly condensed fashion. Geometric algebra allows us to generalize and transfer concepts and techniques to a wide range of domains with little extra conceptual work. Leibniz dreamed of a geometric calculus system that deals directly with geometric objects rather than with sequences of numbers. It is clear that by increasing the dimension of the geometric space and the generalization of the transformation group, the invariance of the operations with respect to a reference frame will be more and more difficult. Leibniz’s invariance dream is fulfilled for the nD classical geometries using the coordinate-free framework of geometric algebra.
The aim of this book is precise and well planned. It is not merely an exposé of mathematical theory rather, the author introduces the theory and new techniques of geometric algebra by showing their applications in diverse domains ranging from neural computing and robotics to medical image processing.
Part I Fundamentals of Geometric Algebra
Introduction to Geometric Algebra.
Geometric Algebra for Modeling in Robot Physics
Part II Euclidean, Pseudo-Euclidean, Lie and Incidence Algebras, and Conformal Geometries
2D, 3D, and 4D Geometric Algebras
Kinematics of the 2D and 3D Spaces
Lie Algebras and the Algebra of Incidence Using the Null Cone and Affine Plane
Conformal Geometric Algebra
Programming Issues
Part III Geometric Computing for Image Processing, Computer Vision, and Neurocomputing
Clifford–Fourier and Wavelet Transforms
Geometric Algebra of Computer Vision
Geometric Neuralcomputing
Part IV Geometric Computing of Robot Kinematics and Dynamics
Kinematics
Dynamics
Part V Applications I: Image Processing, Computer Vision, and Neurocomputing
Applications of Lie Filters, and Quaternion Fourier andWavelet Transforms
Invariants Theory in Computer Vision and Omnidirectional Vision
Registration of 3D Points Using GA and Tensor Voting
Applications in Neuralcomputing
Neural Computing for 2D Contour and 3D Surface Reconstruction
Part VI Applications II: Robotics and Medical Robotics
Rigid Motion Estimation Using Line Observations
Tracker Endoscope Calibration and Body-Sensors’ Calibration
Tracking, Grasping, and Object Manipulation
3D Maps, Navigation, and Relocalization
Modeling and Registration of Medical Data
Part VII Appendix
Clifford Algebras and Related Algebras
Notation
Useful Formulas for Geometric Algebra