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This book presents a unified mathematical treatment of diverse problems in the

general domain of robotics and associated fields using Clifford or geometric alge-

bra. By addressing a wide spectrum of problems in a common language, it offers

both fresh insights and new solutions that are useful to scientists and engineers

working in areas related with robotics.

It introduces non-specialists to Clifford and geometric algebra, and provides ex-

amples to help readers learn how to compute using geometric entities and geomet-

ric formulations. It also includes an in-depth study of applications of Lie group

theory, Lie algebra, spinors and versors and the algebra of incidence using the

universal geometric algebra generated by reciprocal null cones.

Featuring a detailed study of kinematics, differential kinematics and dynamics

using geometric algebra, the book also develops Euler Lagrange and Hamiltoni-

ans equations for dynamics using conformal geometric algebra, and the recursive

Newton-Euler using screw theory in the motor algebra framework. Further, it

comprehensively explores robot modeling and nonlinear controllers, and discusses

several applications in computer vision, graphics, neurocomputing, quantum com-

puting, robotics and control engineering using the geometric algebra framework.

The book also includes over 200 exercises and tips for the development of future

computer software packages for extensive calculations in geometric algebra, and a

entire section focusing on how to write the subroutines in C++, Matlab and Maple

to carry out efficient geometric computations in the geometric algebra framework.

Lastly, it shows how program code can be optimized for real-time computations.

An essential resource for applied physicists, computer scientists, AI researchers,

roboticists and mechanical and electrical engineers, the book clarifies and demon-

strates the importance of geometric computing for building autonomous systems

### Chapter 1. Geometric Algebra for Modeling in Robotic Physics

Abstract
In this chapter, we will discuss the advantages for geometric computing that geometric algebra offers for solving problems and developing algorithms in the fields of artificial intelligence, robotics, and intelligent machines acting within the perception and action cycle. We begin with a short tour of the history of mathematics to find the roots of the fundamental concepts of geometry and algebra.
Eduardo Bayro-Corrochano

### Chapter 2. Introduction to Geometric Algebra

Abstract
This chapter gives a detailed outline of geometric algebra and explains the related traditional algebras in common use by mathematicians, physicists, computer scientists, and engineers.
Eduardo Bayro-Corrochano

### Chapter 3. Lie Algebras, Lie Groups, and Algebra of Incidence

Abstract
We have learned that readers of the work of Hestenes and Sobzyk [1, Chap. 8] and a late article of Doran et al. [2] section IV may have difficulties to understand the subject and practitioners have difficulties to try the equations in certain applications. For this reason, this chapter reviews concepts and equations most of them introduced by Hestenes and Sobzyk [1, Chap. 8] and the article of Doran et al. [2, Sect. IV]. This chapter is written in a clear manner for readers interested in applications in computer science and engineering. The explained equations will be required to understand advanced applications in next chapters.
Eduardo Bayro-Corrochano

### Chapter 4. 2D, 3D, and 4D Geometric Algebras

Abstract
It is the belief that imaginary numbers appeared for the first time around 1540 when the mathematicians Tartaglia and Cardano represented real roots of a cubic equation in terms of conjugated complex numbers. A Norwegian surveyor, Caspar Wessel, was in 1798 the first one to represent complex numbers by points on a plane with its vertical axis imaginary and horizontal axis real. This diagram was later known as the Argand diagram, although the true Aragand’s achievement was an interpretation of $$i=\sqrt{({-}1)}$$ as a rotation by a right angle in the plane. Complex numbers received their name by Gauss, and their formal definition as pair of real numbers was introduced by Hamilton in 1835.
Eduardo Bayro-Corrochano

### Chapter 5. Kinematics of the 2D and 3D Spaces

Abstract
This chapter presents the geometric algebra framework for dealing with 3D kinematics. The reader will see the usefulness of this mathematical approach for applications in computer vision and kinematics. We start with an introduction to 4D geometric algebra for 3D kinematics. Then we reformulate, using 3D and 4D geometric algebras, the classic model for the 3D motion of vectors. Finally, we compare both models, that is, the one using 3D Euclidean geometric algebra and our model, which uses 4D motor algebra.
Eduardo Bayro-Corrochano

### Chapter 6. Conformal Geometric Algebra

Abstract
The geometric algebra of a 3D Euclidean space $$G_{3,0,0}$$ has a point basis and the motor algebra $$G_{3,0,1}$$ a line basis. In the latter geometric algebra, the lines expressed in terms of Plücker coordinates can be used to represent points and planes as well.
Eduardo Bayro-Corrochano

### Chapter 7. The Geometric Algebras , , ,

Abstract
The geometric algebra of a 3D Euclidean space $$G_{3,0,0}$$ has a point basis and the motor algebra $$G_{3,0,1}^+$$ a line basis. In the latter, the lines expressed are expressed in terms of Plücker coordinates and the points and planes in terms of bivectors.
Eduardo Bayro-Corrochano

### Chapter 8. Programming Issues

Abstract
In this chapter, we will discuss the programming issues to compute in the geometric algebra framework. We will explain the technicalities for the programming which you have to take into account to generate a sound source code. At the end, we will discuss the use of specialized hardware as FPGA and Nvidia CUDA to improve the efficiency of the code processing for applications in real time.
Eduardo Bayro-Corrochano

### Chapter 9. Rigid Motion Interpolation

Abstract
This chapter presents the motor interpolation, and it is based on our previous works [1, 2]. We will use this technique when we interpolate geometric objects like points, lines, planes, circles, and spheres.
Eduardo Bayro-Corrochano

### Chapter 10. Robot Kinematics

Abstract
This chapter presents the formulation of robot manipulator kinematics within the geometric algebra framework. In this algebraic system, the 3D Euclidean motion of points, lines, and planes can be advantageously represented using the algebra of motors. The computational complexity of direct and indirect kinematics and other problems concerning robot manipulators are dependent on the robot’s degrees of freedom as well on its geometric characteristics. Our approach makes possible a direct algebraic formulation of the concrete problem in such a way that it reflects the underlying geometric structure. This is achieved by describing parts of the problem based on motor representations of points, lines, planes, circles, and spheres where necessary.
Eduardo Bayro-Corrochano

### Chapter 11. Robot Dynamics

Abstract
The study of the kinematics and dynamics of robot mechanisms has employed different frameworks, such as vector calculus, quaternion algebra, or linear algebra; the last is used most often. However, in these frameworks, handling the kinematics and dynamics involving only points and lines is very complicated. In previous chapter, the motor algebra was used to treat the kinematics of robot manipulators using the points, lines, and planes. We also used the conformal geometric algebra which includes for the representation also circles and spheres. The use of additional geometric entities helps even more to reduce the representation and computational difficulties.
Eduardo Bayro-Corrochano

### Chapter 12. Control of Robot Manipulators

Abstract
In this chapter, we present the localized control policy which is possible due to the computing of the local dynamic model at each robot joint computed using the recursive Newton–Euler algorithm. We compute the local Hamiltonians at each joint and derive their localized controllers as well. In the experimental part, we compare the performance of PD, Bang–Bang, and sliding mode controllers.
Eduardo Bayro-Corrochano

### Chapter 13. Robot Neurocontrol

Abstract
Biological creatures are able to perform complex tasks, due to the capacity of the brain to store information and to adapt its neuro connections as necessary, and this is known as synaptic plasticity [1]. The neuroplasticity was investigated and later used in Artificial Neural Networks (ANN), where these ANN were called the third generation of neural networks [2]. The main advantage of the third generation of neural networks, or Spiking Neural Networks (SNN), is the ability to mimic the biological behavior, where this characteristic can be used in a variety of applications.
Eduardo Bayro-Corrochano

### Chapter 14. Robot Control and Tracking

Abstract
This chapter presents the computing of the dynamic model, the generation of trajectories using quadratic programming with geometric constraints, and nonlinear control for robot manipulators using the geometric algebra framework.
Eduardo Bayro-Corrochano

### Chapter 15. Rigid Motion Estimation Using Line Observations

Abstract
This chapter is dedicated to the estimation of 3D Euclidean transformation using motor algebra. Two illustrations of estimation procedures are given: the first uses a batch approach for the estimation of the unknown 3D transformation between the coordinate reference systems of a robot neck, or arm, and of a digital camera. This problem is called the hand–eye problem and it is solved using a motion-of-lines model.
Eduardo Bayro-Corrochano

### Chapter 16. Tracker Endoscope Calibration and Body-Sensor Calibration

Abstract
In general, when the sensors are mounted on a robot arm, one can use the hand–eye calibration algorithm to calibrate them. In this chapter, we present the calibration of an endoscopic camera with respect to a tracking system and the case of a mobile robot for that one has to calibrate the robot’s sensors with respect to the robot’s global coordinate system.
Eduardo Bayro-Corrochano

### Chapter 17. Tracking, Grasping, and Object Manipulation

Abstract
In this chapter, we utilize the conformal geometric algebra for the development of concepts and computer algorithms in the domain of robot vision. We present an interesting application of fussy logic and conformal geometric algebra for grasping using the Barret Hand. We present real-time algorithms for a real scenario of perception, approach, and action that handles a number of real grasping and manipulation tasks.
Eduardo Bayro-Corrochano

### Chapter 18. 3D Maps, Navigation, and Relocalization

Abstract
This chapter presents applications of body–eye calibration algorithms using motors of the conformal geometric algebra. A scan-matching algorithm, based on such algorithm, aligns the scans by representing the scan points as lines. We show then a path-following procedure that also uses the conformal geometric algebra techniques to estimate the geometric error for a control law. The chapter extends the applications of body-sensor calibration for the case of stereo vision and laser scanner. Such sensors are used for building 3D maps and tackling the relocalization problem. For the relocalization, we resort to an approach based in the Hough transform, where the desired position is searched in the line Hough space.
Eduardo Bayro-Corrochano

Abstract
Unmanned autonomous vehicles, especially multi-copters, are becoming nowadays ubiquitous. Its popularity is due to its relative maneuverability in civil field for performing a wide range of applications, for example, monitoring roads or areas at risk, remote surveillance, inspection of power lines, etc. However, some of these applications require more specific control tasks, even more, robust controllers because external dynamics or disturbances, in general, affect the flight.
Eduardo Bayro-Corrochano

### Chapter 20. Modeling and Registration of Medical Data

Abstract
In medical image analysis, the availability of 3D models is of great interest to physicians because it allows them to have a better understanding of the situation, and such models are relatively easy to build. However, sometimes and in special situations (such as surgical procedures), some structures (such as the brain or tumors) suffer a (nonrigid) transformation and the initial model must be corrected to reflect the actual shape of the object.
Eduardo Bayro-Corrochano

### Chapter 21. Geometric Computing for Minimal Invasive Surgery

Abstract
In this chapter, we show the treatment of a variety of tasks of medical robotics handled using a powerful, non-redundant coefficient geometric language. This chapter is based on our previous works [1, 2]. You will see how we can treat the representation and modeling using geometric primitives like points, lines, and spheres. The screw and motors are used for interpolation, grasping, holding, object manipulation, and surgical maneuvering. We use geometric algebra algorithms in three scenarios: the virtual world for surgical planning, the haptic interface to command the robot arms, and the visually guided robot arms system for operation of ultrasound scanning and surgery. Note that in this work, we do not present a complete system for computer-aided surgery, here we illustrate the application of geometric algebra algorithms for some relevant tasks in minimal invasive surgery.
Eduardo Bayro-Corrochano