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Reading Assignment 41-59 - Coggle Diagram

- - - - Moving frame denoted as Fₙ,ₒ,ₐ
      - n-axis: normal (x-axis)
      - o-axis: orientation (y-axis)
      - a-axis: approach (z-axis)
  - - - Both location of the origin and direction of axes must be described
      - Expressed as three directional vectors and one location vector
      - Vector w = 1 for origin location
      - Vectors w = 0 for directional unit vectors
      - Scale factor ensures correct location and orientation
      - Example: 2.3 (Frame description at specific units)
  - - - Rigid body: represented by a frame fixed to it
      - Position and orientation of the object always known
      - Described by:
      - Three vectors for orientation (9 values)
      - One vector for position (3 values)
      - Total of 12 values constrained by six equations to reduce to six degrees of freedom (DOF)
  - - - Constraint Equations:
      - Unit vectors n, o, and a must be perpendicular
      - Each unit vector must have a magnitude of 1
      - Six constraint equations ensure correct representation
      - Use dot product for equations or cross product to ensure right-hand rule
  - - - Prefer square matrices (easier to multiply and invert)
      - 4x4 matrix: combines orientation and position (with scale factors)
      - 3x3 matrix: for orientation only (no scale factors)
  - - - Transformation: a movement in space
      - Pure translation
      - Pure rotation
      - Combination of translations and/or rotations
      - Transformations follow the same mathematical representation as a frame
  - - - Assume frame is at the origin and parallel to the reference frame
      - Rotation matrix for x-axis, y-axis, or z-axis:
      - cos(θ) and sin(θ) used for rotational components
- - - - Transformations can be made relative to a moving frame instead of the fixed reference frame
        
        Example: Rotation of 90 degrees about the n-axis of the moving frame instead of the x-axis of the reference frame
        
        Post-multiplication is used when calculating changes in coordinates relative to a moving frame
    - - It is possible to combine transformations relative to both the fixed reference frame and the moving frame
        Pre- and post-multiplication is used to apply the correct transformations to the frame of interest
    - - Inverse matrices are required for many robotic tasks, such as determining joint angles or the position of an end-effector
        Example: Robot moving towards a part to drill a hole requires knowledge of multiple transformations (frames for the universe, robot base, end effector, and part)
      - Unitary matrices: Rotation matrices are unitary, meaning their inverse is their transpose
        Homogeneous matrices: Inverting homogeneous matrices involves transposing the rotation portion and negating the dot products of the position vector with the frame axes
    - - Forward kinematics: Calculates the position and orientation of the robot's hand when the configuration (link lengths and joint angles) is known
        
        Inverse kinematics: Determines the joint angles and link lengths necessary to place the robot’s hand at a desired location and orientation
        
        Example: The importance of inverse kinematics for robot control systems
        
        Degrees of freedom (DOF): The hand frame of the robot needs six pieces of information to be fully defined in space (3 for position and 3 for orientation)