Modern Robotics: Mechanics, Planning, and Control

Chapter 1: Preview

This introductory chapter establishes the scope of the book, defining a robot as a mechanism composed of rigid links connected by joints. It distinguishes between open-chain mechanisms, such as a typical robot arm, and closed-chain mechanisms, like a Stewart platform. The chapter provides a high-level overview of the key topics to be covered, setting the stage for a cohesive exploration of configuration space, rigid-body motions, kinematics, dynamics, trajectory generation, motion planning, control, and manipulation.

Chapter 2: Configuration Space

This chapter introduces the foundational concepts of Configuration Space (C-space) and Degrees of Freedom (DOF). The book explains that a robot’s configuration is a complete specification of the position of every point on the robot.

• Degrees of Freedom (DOF): The book defines DOF as the minimum number of coordinates required to represent a robot’s configuration, establishing that a rigid body has 3 DOF in a 2D plane and 6 DOF in 3D space. Grübler’s Formula is presented as a key tool for calculating the DOF of complex mechanisms.
• C-space and Constraints: The chapter explores the topology of the C-space and distinguishes between holonomic constraints (which reduce the C-space dimension, e.g., a closed loop) and nonholonomic constraints (which restrict velocity but not the reachable C-space, e.g., a rolling wheel). It also differentiates between the robot’s C-space, its task space (where the task is defined), and its workspace (what the end-effector can reach).

Chapter 3: Rigid-Body Motions

This chapter provides the mathematical toolkit for describing rigid-body motion, which is central to the book’s geometric approach.

• Mathematical Formalism: The book uses the special orthogonal group SO(3) (3×3 rotation matrices) and the special Euclidean group SE(3) (4×4 homogeneous transformation matrices) to represent orientation and full rigid-body configurations, respectively.
• Twists and Wrenches: A key contribution of the book’s methodology is the use of twists (6D vectors representing spatial velocity) and wrenches (6D vectors representing spatial forces). This unified framework simplifies the analysis of kinematics and dynamics.
• Exponential Coordinates: The chapter introduces the concept of a screw motion (a simultaneous rotation and translation about an axis). It demonstrates that any rigid-body displacement can be achieved via a screw motion, which can be represented by exponential coordinates (Sθ). This provides a singularity-free representation of motion.

Chapter 4: Forward Kinematics

This chapter focuses on calculating an end-effector’s configuration from its joint angles. The book champions the Product of Exponentials (PoE) formula as its primary method.

• PoE Formula: This formula models each joint’s motion as a screw motion. The final end-effector configuration, T(θ), is derived by composing these individual screw motions. The book presents both the space form (where screw axes are defined in a fixed base frame) and the body form (where screw axes are defined in the end-effector frame).
• Advantages over D-H: The PoE formula is presented as a more intuitive and robust alternative to the traditional Denavit-Hartenberg (D-H) convention, as it eliminates the need for complex and often arbitrary link-frame assignments.

Chapter 5: Velocity Kinematics and Statics

This chapter introduces the Jacobian, a fundamental tool that relates joint velocities to the end-effector’s twist.

• The Jacobian: The book defines the Jacobian, J(θ), as the matrix that maps joint velocities (θ̇) to the end-effector twist (V) via the equation V = J(θ)θ̇.
• Space and Body Jacobians: Consistent with the PoE formulation, the chapter derives both the Space Jacobian (Jₛ) and the Body Jacobian (Jb). A key insight is that the columns of the Jacobian are the screw axes of the joints, expressed in the appropriate reference frame.
• Applications: The Jacobian is shown to be essential for singularity analysis (identifying configurations where the robot loses mobility), statics (relating joint torques to end-effector wrenches via τ = JᵀF), and for deriving the manipulability ellipsoid, a tool for visualizing the end-effector’s ease of motion.

Chapter 6: Inverse Kinematics

This chapter tackles the inverse kinematics (IK) problem: finding the joint angles that achieve a desired end-effector configuration. The book explains that, unlike forward kinematics, IK can have multiple, one, or no solutions.

• Analytical Solutions: For common robot architectures like the PUMA-type arm, the book demonstrates how to derive closed-form analytical solutions by decoupling the problem into inverse position and inverse orientation components.
• Numerical Solutions: For general open chains, the book presents an iterative numerical method based on the Newton-Raphson algorithm. This method uses the Jacobian pseudoinverse (J†) to iteratively update the joint angles to converge on a solution.

Chapter 7: Kinematics of Closed Chains

The kinematic analysis is extended to closed-chain mechanisms, which present unique challenges due to their loop-closure constraints and the presence of passive joints. The book analyzes the forward and inverse kinematics of platforms like the Stewart-Gough platform and introduces the constraint Jacobian for velocity and static analysis. It also provides a classification of the more complex singularities found in closed chains.

Chapter 8: Dynamics of Open Chains

This chapter addresses the relationship between the forces/torques applied to a robot and its resulting motion.

• Formulations: The book details two primary methods for deriving the equations of motion (τ = M(θ)θ̈ + h(θ, θ̇)): the energy-based Lagrangian formulation and the more computationally efficient Newton-Euler recursive formulation.
• Newton-Euler Algorithm: This algorithm involves a forward pass to calculate link velocities and accelerations, and a backward pass to compute the joint forces and torques. This method is presented as being particularly well-suited for computer implementation.

Chapter 9: Trajectory Generation

This chapter focuses on creating smooth and executable paths for the robot to follow. It distinguishes between a path (the geometric curve) and a trajectory (a path with a time scaling). The book provides methods for generating point-to-point trajectories using polynomial splines and trapezoidal motion profiles. It also presents an algorithm for time-optimal time scaling, which finds the fastest possible motion along a given path subject to the robot’s full dynamics and actuator limits.

Chapter 10: Motion Planning

This chapter addresses the problem of finding a collision-free path.

• C-space Obstacles: The book explains how obstacles in the workspace are transformed into C-obstacles in the configuration space, turning the planning problem into a search for a path for a point in the free C-space.
• Planning Algorithms: It provides a comprehensive overview of major planning methods, including grid-based methods (like A* search), sampling-based methods (like PRMs and RRTs), which are highlighted for their effectiveness in high-DOF systems, and virtual potential fields.

Chapter 11: Robot Control

This chapter covers the design of feedback controllers to ensure accurate trajectory execution.

• Key Controllers: The book details the implementation of PID control and, more significantly, computed torque control. This latter method uses a model of the robot’s dynamics to cancel out nonlinearities and create a simple, linear error dynamic.
• Advanced Control: The chapter also covers force control, hybrid motion-force control (for tasks involving contact with the environment), and impedance control (which regulates the dynamic relationship between the end-effector’s motion and external forces).

Chapter 12: Grasping and Manipulation

This chapter extends the analysis to the interaction between the robot and objects in its environment. It covers contact kinematics (classifying contacts as rolling, sliding, or breaking), contact forces (using the Coulomb friction model), and the concepts of form closure (kinematically immobilizing an object) and force closure (resisting any external wrench using frictional contacts).

Chapter 13: Wheeled Mobile Robots

The final chapter applies the book’s principles to wheeled mobile robots. It distinguishes between omnidirectional and nonholonomic robots. For nonholonomic robots, it introduces the concept of controllability and uses Lie brackets to demonstrate that these systems can reach any configuration despite their velocity constraints. The chapter also covers specific motion planning and control techniques for these platforms.