1.1 Robot kinematics

Robot kinematics forms the backbone of understanding robot motion and control. It delves into the geometric relationships between robot components, focusing on joint types, degrees of freedom, and kinematic chains. These concepts are crucial for designing and analyzing robotic systems effectively.

Forward and inverse kinematics are key aspects of robot motion planning. Forward kinematics calculates end-effector position from joint angles, while inverse kinematics determines joint angles for a desired end-effector pose. These principles enable precise control and task execution in robotics applications.

Fundamentals of robot kinematics

• Provides foundation for understanding robot motion and control in Robotics and Bioinspired Systems
• Focuses on geometric relationships between robot components and their movement in space
• Crucial for designing, analyzing, and controlling robotic systems effectively

Types of robot joints

Top images from around the web for Types of robot joints

• 

  MS - Architectural singularities of parallel mechanisms with prismatic joints due to special ... View original

  Is this image relevant?

• 

  Frontiers | Influence of Compliant Joints in Four-Legged Robots View original

  Is this image relevant?

• 

  Frontiers | Influence of Compliant Joints in Four-Legged Robots View original

  Is this image relevant?

• 

  MS - Architectural singularities of parallel mechanisms with prismatic joints due to special ... View original

  Is this image relevant?

• 

  Frontiers | Influence of Compliant Joints in Four-Legged Robots View original

  Is this image relevant?

1 of 3

Top images from around the web for Types of robot joints

• 

  MS - Architectural singularities of parallel mechanisms with prismatic joints due to special ... View original

  Is this image relevant?

• 

  Frontiers | Influence of Compliant Joints in Four-Legged Robots View original

  Is this image relevant?

• 

  Frontiers | Influence of Compliant Joints in Four-Legged Robots View original

  Is this image relevant?

• 

  MS - Architectural singularities of parallel mechanisms with prismatic joints due to special ... View original

  Is this image relevant?

• 

  Frontiers | Influence of Compliant Joints in Four-Legged Robots View original

  Is this image relevant?

1 of 3

• Revolute joints allow rotational motion around a single axis
• Prismatic joints enable linear motion along a single axis
• Cylindrical joints combine rotational and linear motion
• Spherical joints permit rotation around three orthogonal axes
• Planar joints allow movement in a two-dimensional plane

Degrees of freedom

• Represent independent variables needed to specify robot configuration
• Determined by number and type of joints in robotic system
• Cartesian robots typically have 3 DOF (x, y, z translations)
• 6 DOF robots (PUMA, KUKA) can reach any position and orientation in 3D space
• Redundant robots have more DOF than required for task (7+ DOF)

Kinematic chains

• Series of rigid bodies connected by joints forming robot structure
• Open-chain manipulators have single path from base to end-effector
• Closed-chain mechanisms form loops in kinematic structure (parallel robots)
• Branching chains have multiple end-effectors or tool points
• Kinematic chains determine robot workspace and dexterity

Forward kinematics

• Calculates end-effector position and orientation given joint angles
• Essential for robot control, path planning, and workspace analysis
• Builds foundation for more complex kinematic and dynamic analyses

Denavit-Hartenberg parameters

• Standardized method to describe kinematic relationship between adjacent links
• Consists of four parameters ($\theta, d, a, \alpha$) for each joint
• $\theta$ represents joint angle for revolute joints or joint offset for prismatic joints
• $d$ denotes link offset along previous z-axis or joint variable for prismatic joints
• $a$ describes link length along common normal
• $\alpha$ defines link twist angle between z-axes
• Simplifies forward kinematics calculations for serial manipulators

Homogeneous transformations

• 4x4 matrices representing position and orientation of robot links
• Combine rotation and translation in single matrix operation
• General form: $T = \begin{bmatrix} R & p \\ 0 & 1 \end{bmatrix}$
  • $R$ represents 3x3 rotation matrix
  • $p$ denotes 3x1 position vector
• Used to transform coordinates between different reference frames
• Multiplication of transformation matrices yields end-effector pose

Link coordinate frames

• Assigned to each link in robotic manipulator
• Origin typically placed at joint axis or intersection of joint axes
• z-axis aligned with joint axis for revolute joints
• x-axis points along common normal between joint axes
• y-axis completes right-handed coordinate system
• Consistent frame assignment crucial for applying DH parameters

Inverse kinematics

• Determines joint angles required to achieve desired end-effector pose
• Critical for robot motion planning and control in task space
• Generally more complex than forward kinematics due to multiple solutions

Analytical vs numerical methods

• Analytical methods provide closed-form solutions for specific robot geometries
• Closed-form solutions exist for 6 DOF manipulators with specific joint configurations
• Numerical methods (Newton-Raphson, gradient descent) used for complex kinematics
• Iterative techniques approximate solution through successive refinement
• Trade-offs between computation speed and accuracy for different approaches

Singularities and redundancy

• Singularities occur when manipulator loses one or more degrees of freedom
• Types include boundary singularities (workspace limits) and internal singularities
• Redundancy arises when robot has more DOF than required for task
• Redundant manipulators offer increased dexterity and obstacle avoidance
• Null space motion allows joint movement without affecting end-effector pose

Workspace analysis

• Determines reachable positions and orientations of robot end-effector
• Dexterous workspace includes positions with all orientations achievable
• Reachable workspace encompasses all attainable end-effector positions
• Workspace shape affected by joint limits, link lengths, and kinematic structure
• Visualization techniques (3D plots, cross-sections) aid in workspace evaluation

Differential kinematics

• Relates joint velocities to end-effector velocities and angular velocities
• Essential for real-time robot control and trajectory generation
• Provides framework for analyzing robot dynamics and force control

Jacobian matrix

• Linear transformation mapping joint velocities to end-effector velocities
• Composed of partial derivatives of end-effector position w.r.t. joint variables
• General form: $v = J(\theta)\dot{\theta}$
  • $v$ represents end-effector velocity vector
  • $J(\theta)$ denotes Jacobian matrix
  • $\dot{\theta}$ represents joint velocity vector
• Dimensions depend on number of DOF and task space dimensions
• Inverse Jacobian used for velocity-based control schemes

Velocity and acceleration analysis

• Joint velocities determined through inverse kinematics of end-effector velocity
• Acceleration analysis involves time derivative of Jacobian ($\dot{J}$)
• End-effector acceleration: $a = J(\theta)\ddot{\theta} + \dot{J}(\theta)\dot{\theta}$
• Crucial for dynamic control and trajectory optimization
• Enables smooth motion planning and execution in robotic systems

Singularity configurations

• Occur when Jacobian matrix loses full rank (determinant becomes zero)
• Result in infinite joint velocities for finite end-effector velocities
• Types include boundary singularities, internal singularities, and algorithmic singularities
• Strategies for handling singularities damped least squares, workspace restriction
• Singularity analysis important for robot design and control system development

Trajectory planning

• Generates time-dependent path for robot to follow during task execution
• Balances factors like smoothness, accuracy, and execution time
• Critical for efficient and safe robot operation in various applications

Joint space vs task space

• Joint space planning works directly with robot joint angles
• Simplifies collision avoidance and respects joint limits
• Task space planning operates in Cartesian coordinates of end-effector
• Allows intuitive specification of desired end-effector motion
• Hybrid approaches combine advantages of both spaces for complex tasks

Interpolation methods

• Linear interpolation provides simplest path between waypoints
• Polynomial interpolation (cubic, quintic) ensures smooth velocity profiles
• Spline interpolation connects multiple waypoints with continuous derivatives
• Bezier curves offer intuitive control over path shape
• Trapezoidal velocity profiles balance smooth acceleration and constant velocity

Time-optimal trajectories

• Minimize execution time while respecting kinematic and dynamic constraints
• Bang-bang control achieves time-optimality for point-to-point motions
• S-curve profiles provide smoother motion with bounded jerk
• Convex optimization techniques used for complex multi-axis systems
• Trade-offs between execution time, energy consumption, and wear on robot components

Kinematic calibration

• Improves robot accuracy by identifying and compensating for kinematic errors
• Essential for high-precision tasks in manufacturing and medical robotics
• Iterative process involving measurement, modeling, and parameter estimation

Error sources and modeling

• Manufacturing tolerances lead to deviations from nominal kinematic parameters
• Joint offset errors affect zero position of robot joints
• Link length errors impact overall robot geometry
• Non-geometric errors (joint compliance, backlash) also contribute to inaccuracies
• Kinematic error models incorporate these factors into forward kinematics equations

Calibration techniques

• Open-loop methods use external measurement systems (laser trackers, CMMs)
• Closed-loop techniques rely on robot's own sensors and known reference objects
• Self-calibration approaches exploit robot redundancy for parameter estimation
• Optimization algorithms (least squares, maximum likelihood) fit error models to data
• Machine learning techniques (neural networks, Gaussian processes) for complex error patterns

Performance evaluation

• Absolute accuracy measures deviation from commanded position in global frame
• Repeatability quantifies variation in reaching same position multiple times
• Resolution defines smallest controllable increment of robot motion
• ISO 9283 standard provides guidelines for robot performance testing
• Evaluation metrics guide robot selection and assess calibration effectiveness

Advanced kinematic structures

• Explores non-traditional robot designs for specialized applications
• Offers unique capabilities and challenges in kinematics and control
• Pushes boundaries of robot performance and versatility

Parallel manipulators

• Multiple kinematic chains connect base to end-effector platform
• Provide high stiffness, accuracy, and speed for certain tasks
• Stewart platform widely used for flight simulators and precision positioning
• Delta robot excels in high-speed pick-and-place operations
• Closed-form inverse kinematics often simpler than serial manipulators

Redundant manipulators

• Possess more degrees of freedom than required for task execution
• Enable obstacle avoidance, singularity avoidance, and joint limit avoidance
• Null space control allows secondary objectives without affecting primary task
• Challenges include resolving redundancy and handling increased complexity
• Examples include 7-DOF arms (KUKA LBR iiwa) and hyper-redundant snake-like robots

Soft robots

• Constructed from compliant materials inspired by biological systems
• Offer inherent safety and adaptability in human-robot interaction
• Continuum kinematics models deformation of flexible structures
• Challenges include modeling material properties and achieving precise control
• Applications in minimally invasive surgery, search and rescue, and adaptive gripping

Kinematics in mobile robotics

• Extends kinematic principles to robots capable of locomotion
• Crucial for navigation, localization, and path planning in diverse environments
• Integrates concepts from classical mechanics and differential geometry

Wheeled robot kinematics

• Describes motion constraints imposed by wheel-ground contact
• Holonomic robots (omnidirectional) can move in any direction instantaneously
• Non-holonomic robots (car-like) have restricted motion due to no-slip condition
• Kinematic models include differential drive, Ackermann steering, and omnidirectional
• Odometry uses wheel encoders to estimate robot pose over time

Legged robot kinematics

• Models articulated limb structures for walking, running, and climbing
• Gait analysis studies periodic patterns of leg movements
• Static stability maintains center of gravity within support polygon
• Dynamic stability allows temporary instability for more efficient locomotion
• Examples include bipedal humanoids, quadrupedal robots, and hexapods

Aerial robot kinematics

• Describes 6-DOF motion of flying robots in three-dimensional space
• Quadrotor kinematics relate rotor speeds to translational and rotational motion
• Fixed-wing aircraft kinematics incorporate aerodynamic principles
• Euler angles or quaternions represent orientation in 3D space
• Challenges include underactuation and coupling between degrees of freedom

Kinematics simulation tools

• Enable virtual prototyping and testing of robotic systems
• Accelerate development process and reduce hardware costs
• Provide platform for algorithm development and performance evaluation

Software packages

• ROS (Robot Operating System) offers extensive libraries for kinematics and control
• MATLAB Robotics Toolbox provides functions for kinematic analysis and simulation
• V-REP (CoppeliaSim) enables detailed robot modeling and physics-based simulation
• Gazebo integrates with ROS for realistic robot simulations
• OpenRAVE focuses on motion planning and kinematic analysis for manipulators

Virtual robot models

• URDF (Unified Robot Description Format) standardizes robot kinematic descriptions
• CAD models provide accurate geometric representations of robot components
• Physics engines (ODE, Bullet) simulate dynamics and interactions with environment
• Sensor models emulate real-world sensor data for algorithm testing
• Libraries of pre-built robot models available for common industrial and research platforms

Visualization techniques

• 3D rendering engines create realistic visual representations of robots
• Interactive GUIs allow real-time manipulation of robot joints and parameters
• Augmented reality overlays kinematic data on physical robot systems
• Motion trails and vector fields visualize robot trajectories and workspaces
• Heat maps and color coding highlight kinematic performance metrics