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Modern Robot Kinematics

Serial Parallel, and Mobile Systems

gareth thomas

Robots don’t move by magic. Behind every manipulator, every mobile platform, every aerial or underwater vehicle is a precise mathematical engine: kinematics. If you don’t understand it, you’re guessing. And guessing doesn’t build robots that work.

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About

About

About the Book

💡 Note: This book is written for serious engineers and researchers who want rigorous, high-level technical insight. It’s dense, detailed, and assumes prior familiarity with advanced concepts.
If you value precision, depth, and professional-grade explanations—you’ll find this book indispensable.


Robots don’t move by magic. Behind every manipulator, every mobile platform, every aerial or underwater vehicle is a precise mathematical engine: kinematics. If you don’t understand it, you’re guessing. And guessing doesn’t build robots that work.

This book is not beginner fluff. It’s a rigorous, hard-hitting manual for engineers, researchers, and serious robotics professionals who need to master the geometry of motion—fast, clean, and without shortcuts.

Inside you’ll discover:

  • Serial Chains Demystified – From SCARA arms to 6-DOF manipulators, get the full breakdown of forward and inverse kinematics, workspace limits, and singularities.
  • Parallel Robots Unlocked – Stewart platforms, Delta robots, cable-driven systems… and the non-linear math that makes them tick.
  • Mobile Bases and Locomotion – Differential drive, omniwheels, tracked vehicles, and legged systems from bipeds to quadrupeds. Understand how real robots move across real terrain.
  • Velocity Kinematics and Jacobians – Learn how to bridge joint space and task space, detect singularities before they break your system, and design redundancy resolution strategies that work.
  • Beyond the Basics – Underactuated drones, underwater vehicles, aerial manipulators, and collaborative robots are dissected with the same mathematical rigor.

Every chapter is built on formal derivations, clean mathematical structure, and real-world examples—from industrial arms like the PUMA 560 to cutting-edge quadrupeds and UAVs.

If you’re serious about robotics, this book will sharpen your edge, expand your toolkit, and keep you ahead of the engineers who are still fumbling with guesswork.

No filler. No shortcuts. Just the mathematics of motion, laid bare.

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Author

About the Author

gareth thomas

Gareth Morgan Thomas is a qualified expert with extensive expertise across multiple STEM fields. Holding six university diplomas in electronics, software development, web development, and project management, along with qualifications in computer networking, CAD, diesel engineering, well drilling, and welding, he has built a robust foundation of technical knowledge.

Educated in Auckland, New Zealand, Gareth Morgan Thomas also spent three years serving in the New Zealand Army, where he honed his discipline and problem-solving skills. With years of technical training, Gareth Morgan Thomas is now dedicated to sharing his deep understanding of science, technology, engineering, and mathematics through a series of specialized books aimed at both beginners and advanced learners.

Contents

Table of Contents

Chapter 1. Fundamentals and Rigid Body Kinematics

Section 1. Mathematical Foundations

  • Vector spaces and coordinate systems
  • Homogeneous coordinates and transformations
  • Rotation matrices and Euler angles

Section 2. Spatial Transformations

  • Translation and rotation operators
  • Composition of transformations
  • Inverse transformations

Section 3. Simple Positioning Systems

  • Cartesian gantry robots
  • Cylindrical coordinate robots
  • Spherical coordinate robots

Section 4. Quaternions and Alternative Representations

  • Quaternion algebra
  • Axis-angle representation
  • Exponential coordinates

Chapter 2. Single-Chain Serial Kinematics

Section 1. Denavit-Hartenberg Convention

  • DH parameter assignment
  • Link coordinate frames
  • Transformation matrices

Section 2. Planar Serial Chains

  • 2-DOF RR manipulator
  • 3-DOF planar arm (RRP configuration)
  • SCARA robot kinematics

Section 3. Spatial Serial Chains

  • PUMA 560 robot (6-DOF revolute)
  • Stanford arm (5-DOF with prismatic joint)
  • UR5/UR10 collaborative robots

Section 4. Forward Kinematics Algorithm

  • Matrix multiplication approach
  • Recursive formulation
  • Computational considerations

Chapter 3. Inverse Kinematics and Workspace Analysis

Section 1. Geometric Solutions

  • 2-DOF planar arm solution
  • Spherical wrist decoupling
  • PUMA 560 geometric solution

Section 2. Algebraic Methods

  • Polynomial formulation
  • Elimination theory
  • Groebner basis methods

Section 3. Numerical Methods

  • Newton-Raphson iteration
  • Jacobian pseudoinverse
  • Damped least squares

Section 4. Workspace Analysis

  • Reachable workspace
  • Dexterous workspace
  • SCARA robot workspace example

Section 5. Multiple Solutions and Singularities

  • Solution multiplicity
  • Kinematic singularities
  • Singularity avoidance strategies

Chapter 4. Velocity Kinematics and Jacobians

Section 1. Differential Kinematics

  • Velocity propagation
  • Angular velocity representation
  • Linear and angular velocity relations

Section 2. Jacobian Matrix Derivation

  • Geometric Jacobian
  • Analytical Jacobian
  • Relationship between Jacobians

Section 3. Jacobian Properties

  • Manipulability ellipsoid
  • Condition number
  • Force transformation

Section 4. Industrial Robot Examples

  • KUKA KR5 arc Jacobian analysis
  • ABB IRB140 velocity limits
  • Fanuc LR Mate singularity analysis

Section 5. Redundancy Resolution

  • Null space projection
  • Optimization-based methods
  • 7-DOF redundant manipulators

Chapter 5. Constrained Systems - Parallel Robots

Section 1. Parallel Mechanism Fundamentals

  • Closed kinematic chains
  • Constraint equations
  • Degrees of freedom analysis

Section 2. Planar Parallel Mechanisms

  • 3-RRR planar parallel robot
  • 5-bar linkage mechanism
  • 3-RPR parallel manipulator

Section 3. Spatial Parallel Mechanisms

  • Stewart platform (Gough-Stewart)
  • Delta robot (Clavel's design)
  • Hexaglide machine tool

Section 4. Forward Kinematics of Parallel Robots

  • Nonlinear equation systems
  • Stewart platform forward kinematics
  • Delta robot closed-form solution

Section 5. Parallel Robot Jacobians

  • Constraint Jacobian
  • Velocity analysis
  • Singularity analysis

Section 6. Cable-Driven Parallel Robots

  • Cable constraints and tensions
  • Wrench-feasible workspace
  • SkyCam and NIST RoboCrane examples

Chapter 6. Mobile Bases - Adding Translation

Section 1. Wheeled Robot Fundamentals

  • Wheel constraints
  • Holonomic vs nonholonomic systems
  • Instantaneous center of rotation

Section 2. Nonholonomic Wheeled Robots

  • Differential drive (Pioneer, TurtleBot)
  • Car-like robots (Ackermann steering)
  • Bicycle model kinematics

Section 3. Holonomic Wheeled Robots

  • Mecanum wheel platforms (KUKA youBot)
  • Swedish wheel (omni-wheel) robots
  • Four-wheel holonomic designs

Section 4. Tracked Vehicles

  • Skid-steer kinematics
  • Slip modeling
  • PackBot and military robot examples

Section 5. Motion Planning Considerations

  • Configuration space
  • Nonholonomic path planning
  • Dubins curves and Reeds-Shepp paths

Chapter 7. Articulated Mobility - Legged Systems

Section 1. Legged Locomotion Fundamentals

  • Static vs dynamic stability
  • Support polygon concept
  • Center of mass considerations

Section 2. Bipedal Robot Kinematics

  • Human-like leg mechanisms
  • ASIMO and Atlas robot kinematics
  • Zero moment point (ZMP)

Section 3. Quadrupedal Robot Systems

  • Trotting and bounding gaits
  • Boston Dynamics Spot kinematics
  • ANYmal quadruped analysis

Section 4. Multi-Legged Systems

  • Hexapod insects and robots
  • Wave gaits and tripod gaits
  • RHex and bio-inspired designs

Section 5. Contact Kinematics

  • Foot placement planning
  • Ground contact models
  • Terrain adaptation

Chapter 8. Flexible and Continuum Systems

Section 1. Continuum Robot Fundamentals

  • Infinite degrees of freedom
  • Backbone curve parameterization
  • Constant curvature assumption

Section 2. Snake-Like Robots

  • Hyper-redundant manipulators
  • Serpentine locomotion
  • OmniTread and ACM-R series

Section 3. Soft Robot Kinematics

  • Pneumatic soft actuators
  • Soft Robotics Toolkit designs
  • Elephant trunk inspired robots

Section 4. Beam Theory Applications

  • Euler-Bernoulli beam model
  • Cosserat rod theory
  • Finite element approaches

Section 5. Tendon-Driven Systems

  • Cable routing kinematics
  • Shadow Hand and dexterous grippers
  • Continuum surgical robots

Chapter 9. Underactuated and Flying Systems

Section 1. Underactuated System Theory

  • Degrees of freedom vs actuators
  • Dynamic coupling effects
  • Controllability analysis

Section 2. Quadrotor Kinematics

  • 6-DOF motion with 4 actuators
  • Euler angle singularities
  • DJI Phantom and research platforms

Section 3. Fixed-Wing Aircraft

  • Aircraft coordinate systems
  • Kinematic equations of motion
  • Dubins airplane model

Section 4. Helicopter Dynamics

  • Main rotor and tail rotor coupling
  • Cyclic and collective controls
  • Single and multi-rotor configurations

Section 5. Underwater Vehicles

  • 6-DOF underwater kinematics
  • REMUS and Bluefin AUV examples
  • Buoyancy and hydrodynamic effects

Chapter 10. Integrated Systems

Section 1. Mobile Manipulator Fundamentals

  • Kinematic coupling
  • Workspace expansion
  • Redundancy resolution

Section 2. Ground-Based Mobile Manipulators

  • PR2 and TurtleBot with arms
  • KUKA youBot system
  • Clearpath Ridgeback platforms

Section 3. Aerial Manipulators

  • Quadrotor with robotic arm
  • Dynamic coupling challenges
  • DJI Matrice with gripper

Section 4. Climbing and Wall-Climbing Robots

  • Contact switching kinematics
  • Gecko-inspired designs
  • Magnetic wheel climbing systems

Section 5. Reconfigurable Systems

  • Modular robot kinematics
  • Self-assembling systems
  • M-Blocks and metamorphic robots

Section 6. Human-Robot Collaboration

  • Shared workspace kinematics
  • Safety considerations
  • Cobotic system design

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