Robot kinematics is the study of robot motion without considering forces. It's crucial for understanding how robots move and interact with their environment. This topic covers forward and inverse kinematics, which determine robot configurations and movements.

Kinematics involves concepts like link lengths, joint angles, and transformation matrices. These tools help describe robot structures and calculate their positions and orientations in space. Understanding kinematics is essential for planning robot movements and controlling their actions.

Forward and inverse kinematics

• Forward kinematics determines the end-effector position and orientation given the joint angles or positions, enabling the robot to understand its current configuration in space
• Inverse kinematics calculates the joint angles or positions required to achieve a desired end-effector position and orientation, allowing the robot to plan and execute movements to reach a target
• Understanding both forward and inverse kinematics is crucial for autonomous robots to perform tasks that require precise positioning and manipulation of objects in their environment

Link length and twist angle

• Link length is the distance between two adjacent joint axes along the common normal, determining the size of each robot link (upper arm, forearm)
• Twist angle is the angle between two adjacent joint axes, measured in a plane perpendicular to the common normal, defining the relative orientation of the links
  • A twist angle of 0° indicates parallel joint axes, while 90° represents perpendicular axes
• Link length and twist angle are essential parameters in the Denavit-Hartenberg (DH) convention for describing robot kinematics, enabling consistent and systematic modeling of robot structures

Link offset and joint angle

• Link offset is the distance between two adjacent common normals along the joint axis, accounting for any displacement between the links (e.g., due to the thickness of the joint)
• Joint angle is the angle between the extension of the previous link and the current link, measured in a plane normal to the joint axis, representing the rotation of each joint
  • A joint angle of 0° indicates a fully extended configuration, while positive or negative angles represent rotations in different directions
• Link offset and joint angle, along with link length and twist angle, form the four DH parameters that uniquely define the kinematic relationship between adjacent robot links

Homogeneous transformation matrices

• Homogeneous transformation matrices (HTMs) are 4x4 matrices that represent the position and orientation of a coordinate frame with respect to another coordinate frame, enabling efficient computation of robot kinematics
• HTMs combine rotation and translation information into a single matrix, simplifying the process of transforming points and vectors between different coordinate frames
• In robotics, HTMs are used to describe the relationship between the base frame, the end-effector frame, and any intermediate frames associated with the robot's links and joints

Rotation matrices

• Rotation matrices are 3x3 matrices that represent the orientation of one coordinate frame relative to another, preserving vector lengths and angles during rotations
• Common rotation matrices include:
  • Rotation about the x-axis (roll)
  • Rotation about the y-axis (pitch)
  • Rotation about the z-axis (yaw)
• Rotation matrices can be combined through matrix multiplication to represent a sequence of rotations, enabling the description of complex robot orientations

Translation vectors

• Translation vectors are 3x1 vectors that represent the position of one coordinate frame's origin relative to another frame's origin, describing the linear displacement between frames
• In homogeneous transformation matrices, the translation vector is appended to the rotation matrix as the fourth column, creating a 4x4 matrix that encapsulates both rotation and translation information
• Translation vectors allow for the compact representation of robot positions and the efficient computation of coordinate transformations in 3D space

Kinematic chains and robot configurations

• A kinematic chain is a series of links connected by joints, forming the structure of a robot manipulator or mobile robot, and determining its degrees of freedom (DOF) and range of motion
• The configuration of a robot refers to the specific arrangement of its links and joints at a given instant, fully describing the robot's pose in space
• Analyzing kinematic chains and robot configurations is essential for understanding the capabilities, limitations, and singularities of a robot system

Open vs closed kinematic chains

• Open kinematic chains have a single sequence of links connected by joints, with one end (the base) fixed and the other end (the end-effector) free to move in space (industrial robotic arms)
• Closed kinematic chains have one or more loops in the link-joint sequence, with multiple paths connecting the base to the end-effector, constraining the motion of the robot (parallel robots, delta robots)
• Open kinematic chains offer a larger workspace and simpler control, while closed kinematic chains provide higher stiffness, accuracy, and load-carrying capacity

Serial vs parallel robot configurations

• Serial robot configurations consist of a single open kinematic chain, with each joint actuated independently, resulting in a large workspace but lower accuracy and stiffness (most industrial robots)
• Parallel robot configurations have multiple kinematic chains connecting the base to the end-effector, with the actuators located on or near the base, providing high stiffness and accuracy but a smaller workspace (Stewart platform, delta robot)
• Hybrid robot configurations combine serial and parallel kinematic chains to achieve a balance between the advantages of both architectures, such as high stiffness and a large workspace (tricept robot)

Jacobian matrix

• The Jacobian matrix is a matrix-valued function that relates the joint velocities of a robot to the linear and angular velocities of the end-effector, enabling the computation of robot motion in Cartesian space
• The Jacobian matrix is essential for robot motion planning, control, and analysis, as it provides a linear approximation of the robot's kinematic behavior around a given configuration
• The dimensions of the Jacobian matrix depend on the number of degrees of freedom (DOF) of the robot, with a 6xn matrix for a robot with n joints and a 6-DOF end-effector (3 for position, 3 for orientation)

Velocity kinematics

• Velocity kinematics is the study of the relationship between joint velocities and end-effector velocities, using the Jacobian matrix to map joint space velocities to Cartesian space velocities
• The forward velocity kinematics problem involves computing the end-effector velocity given the joint velocities, using the equation: $\dot{x} = J(q) \dot{q}$, where $\dot{x}$ is the end-effector velocity, $J(q)$ is the Jacobian matrix, and $\dot{q}$ is the joint velocity vector
• The inverse velocity kinematics problem involves determining the joint velocities required to achieve a desired end-effector velocity, using the equation: $\dot{q} = J(q)^{-1} \dot{x}$, where $J(q)^{-1}$ is the inverse of the Jacobian matrix

Singularities and manipulability

• Singularities are robot configurations in which the Jacobian matrix becomes rank-deficient, resulting in a loss of degrees of freedom and the inability to move the end-effector in certain directions (e.g., fully extended or folded arm)
• Manipulability is a measure of a robot's ability to move and exert forces in different directions at a given configuration, quantifying the ease of motion and force transmission
  • The manipulability ellipsoid, derived from the Jacobian matrix, visualizes the robot's motion and force capabilities in Cartesian space
• Analyzing singularities and manipulability is crucial for designing robot workspaces, avoiding singular configurations during motion planning, and optimizing robot performance

Trajectory planning

• Trajectory planning is the process of generating a feasible path for a robot to follow, considering the robot's kinematics, dynamics, and any constraints imposed by the environment or the task
• Trajectory planning involves specifying the desired positions, velocities, and accelerations of the robot's joints or end-effector over time, ensuring smooth and efficient motion
• Common trajectory planning techniques include:
  • Point-to-point motion: Moving the robot from an initial configuration to a goal configuration without specifying the intermediate path
  • Interpolation: Generating a smooth path between waypoints using polynomial functions (e.g., cubic splines) or other interpolation methods
  • Optimization-based planning: Formulating the trajectory planning problem as an optimization problem, considering criteria such as minimum time, energy, or jerk

Joint space vs Cartesian space

• Joint space trajectory planning involves specifying the desired positions, velocities, and accelerations for each joint of the robot over time, resulting in a set of joint trajectories that the robot follows
  • Joint space planning is simpler and more computationally efficient, as it directly controls the robot's actuators
• Cartesian space trajectory planning involves specifying the desired positions, velocities, and accelerations of the end-effector in Cartesian space, which are then mapped to joint trajectories using inverse kinematics
  • Cartesian space planning is more intuitive and task-oriented, as it directly controls the end-effector motion, but it requires solving the inverse kinematics problem and may lead to singularities
• The choice between joint space and Cartesian space trajectory planning depends on the specific application, the robot's architecture, and the available computational resources

Trapezoidal velocity profile

• A trapezoidal velocity profile is a common trajectory planning technique that generates a smooth motion profile with linear acceleration, constant velocity, and linear deceleration segments
• The trapezoidal velocity profile consists of three phases:
  1. Acceleration phase: The robot accelerates from rest to a maximum velocity with a constant acceleration
  2. Constant velocity phase: The robot maintains the maximum velocity for a specified duration
  3. Deceleration phase: The robot decelerates from the maximum velocity to rest with a constant deceleration
• Trapezoidal velocity profiles provide a simple and efficient way to generate smooth trajectories, minimizing jerk and ensuring continuous velocity and acceleration profiles
• The duration of each phase and the maximum velocity can be adjusted based on the robot's capabilities, the task requirements, and any velocity or acceleration constraints

Redundant manipulators

• Redundant manipulators are robots with more degrees of freedom (DOF) than necessary to perform a given task, providing increased flexibility and dexterity in motion planning and execution
• Kinematic redundancy occurs when the number of joints (n) is greater than the number of DOF required for the task (m), resulting in an infinite number of joint configurations for a given end-effector pose
• Redundant manipulators offer several advantages:
  • Obstacle avoidance: Redundant DOF allow the robot to navigate around obstacles while maintaining the desired end-effector pose
  • Singularity avoidance: Redundancy enables the robot to move away from singular configurations, improving manipulability and robustness
  • Task optimization: Redundant DOF can be exploited to optimize secondary criteria, such as minimizing joint torques, maximizing manipulability, or minimizing energy consumption

Kinematic redundancy

• Kinematic redundancy refers to the excess DOF in a robot manipulator, allowing for multiple joint configurations that result in the same end-effector pose
• The degree of redundancy (DOR) is the difference between the number of joints (n) and the number of task-space DOF (m): DOR = n - m
• Kinematic redundancy introduces challenges in robot control and motion planning, as the inverse kinematics problem has an infinite number of solutions, requiring additional criteria or constraints to select a unique joint configuration

Redundancy resolution techniques

• Redundancy resolution techniques are methods for determining a unique joint configuration for a redundant manipulator, given a desired end-effector pose and additional optimization criteria
• Common redundancy resolution techniques include:
  • Pseudoinverse-based methods: Using the Moore-Penrose pseudoinverse of the Jacobian matrix to find the least-squares solution for the joint velocities, minimizing the Euclidean norm of the joint velocity vector
  • Null-space projection methods: Exploiting the null space of the Jacobian matrix to optimize secondary criteria, such as obstacle avoidance or joint limit avoidance, while maintaining the desired end-effector motion
  • Optimization-based methods: Formulating the redundancy resolution problem as a constrained optimization problem, considering multiple criteria and constraints, and solving it using numerical optimization techniques (e.g., quadratic programming, nonlinear programming)
• The choice of redundancy resolution technique depends on the specific application, the robot's architecture, and the desired optimization criteria, balancing computational efficiency, robustness, and task performance

Dynamics in robot kinematics

• Robot dynamics is the study of the relationship between the forces and torques acting on a robot and the resulting motion, considering the robot's mass, inertia, and other dynamic properties
• Incorporating dynamics into robot kinematics enables more accurate and realistic modeling of robot behavior, accounting for the effects of gravity, friction, and external forces on the robot's motion
• Two main approaches for formulating robot dynamics are the Lagrangian formulation and the Newton-Euler formulation, which differ in their mathematical representation and computational efficiency

Lagrangian formulation

• The Lagrangian formulation is an energy-based approach that derives the equations of motion using the Lagrangian, which is the difference between the kinetic and potential energy of the robot system
• The Lagrangian equations of motion are obtained by applying the Euler-Lagrange equation to the Lagrangian, resulting in a set of second-order differential equations that describe the robot's dynamics
• The Lagrangian formulation provides a systematic and intuitive way to derive the equations of motion, making it suitable for analyzing the dynamics of complex robot systems with multiple degrees of freedom
• However, the Lagrangian formulation can be computationally expensive, as it involves the computation of the kinetic and potential energy terms and their derivatives

Newton-Euler formulation

• The Newton-Euler formulation is a force-based approach that derives the equations of motion by applying Newton's second law and Euler's equations of motion to each link of the robot
• The Newton-Euler equations describe the linear and angular motion of each link, considering the forces and torques acting on the link, as well as the coupling effects between adjacent links
• The Newton-Euler formulation is computationally efficient, as it involves recursive calculations that propagate the forces and torques from the base to the end-effector and back, avoiding the need for explicit computation of energy terms
• However, the Newton-Euler formulation can be more complex to implement and less intuitive than the Lagrangian formulation, as it requires careful bookkeeping of the forces, torques, and motion variables for each link

Kinematics of mobile robots

• Mobile robot kinematics describes the motion of wheeled, legged, or tracked robots in their environment, considering the constraints imposed by the robot's locomotion mechanism and the terrain
• Unlike manipulators, mobile robots have a non-fixed base and must navigate in a 2D or 3D space while maintaining stability and avoiding obstacles
• The kinematics of mobile robots involves analyzing the relationship between the robot's wheel or leg velocities and the resulting motion of the robot in the environment, as well as the effects of slip, skid, and other non-ideal conditions

Holonomic vs non-holonomic constraints

• Holonomic constraints are constraints on the robot's configuration that can be expressed as algebraic equations involving only the robot's position variables and time, allowing the robot to move in any direction without reorienting its wheels (omnidirectional robots)
• Non-holonomic constraints are velocity-dependent constraints that cannot be integrated to obtain position constraints, limiting the robot's instantaneous motion and requiring the robot to reorient its wheels to change direction (car-like robots, differential drive robots)
• Holonomic robots have greater mobility and can easily perform tasks that require precise positioning, while non-holonomic robots are simpler to design and control but have more limited motion capabilities

Ackermann steering geometry

• Ackermann steering geometry is a wheel configuration commonly used in car-like robots, where the front wheels are steered independently to follow a curved path while the rear wheels remain fixed
• In Ackermann steering, the inner front wheel turns more sharply than the outer front wheel during a turn, ensuring that all wheels rotate around a common center point and minimizing slip and tire wear
• The Ackermann steering angle $\delta$ is given by: $\cot{\delta_{o}} - \cot{\delta_{i}} = \frac{w}{l}$, where $\delta_{o}$ and $\delta_{i}$ are the outer and inner wheel steering angles, $w$ is the track width, and $l$ is the wheelbase
• Ackermann steering provides a simple and effective way to control the motion of car-like robots, but it assumes a planar surface and low-speed motion, and may not be suitable for rough terrain or high-speed applications