2.1 Spatial descriptions and transformations

Spatial descriptions and transformations are crucial in robotics, allowing precise tracking of position and orientation in 3D space. Understanding coordinates, reference frames, and degrees of freedom forms the foundation for manipulating objects and navigating environments.

Homogeneous transformations simplify calculations by combining position and orientation into single matrices. This enables efficient representation of complex spatial relationships, chaining of transformations, and inverse operations, essential for robotic motion planning and control.

Spatial Descriptions and Transformations

Position and orientation in 3D

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• Position
  • Defined by x, y, and z coordinates in 3D space pinpoints exact location
  • Represents point's location relative to reference frame enables precise tracking
• Orientation
  • Describes object's rotation in 3D space defines spatial alignment
  • Represented by angles or rotation matrices captures complex rotations
• Reference frames
  • Coordinate systems describe position and orientation establish spatial context
  • Origin and axes define frame provide basis for measurements
• Degrees of freedom (DOF)
  • Six DOF in 3D space: three for position (x, y, z), three for orientation (roll, pitch, yaw)
• Euler angles
  • Roll (rotation around x-axis), pitch (y-axis), yaw (z-axis) describe orientation
  • Sequence-dependent representation leads to potential ambiguity
• Quaternions
  • Four-dimensional representation of orientation (x, y, z, w) encodes rotation
  • Avoid gimbal lock issues ensures smooth rotations

Homogeneous transformations for spatial relationships

• Homogeneous coordinates
  • Add fourth component to 3D vectors (x, y, z, w) unifies representation
  • Allows representation of position and orientation in single matrix simplifies calculations
• Transformation matrices
  • 4x4 matrices combining rotation and translation encode complete spatial information
  • Upper-left 3x3 submatrix represents rotation defines orientation
  • Right-most column represents translation specifies position
• Frame relationships
  • Describe position and orientation of one frame relative to another enables spatial reasoning
• Chain of transformations
  • Multiply matrices to represent multiple transformations combines sequential movements
• Inverse transformations
  • Represent opposite spatial relationship reverses transformation
  • Calculated by inverting transformation matrix undoes spatial change

Translations and rotations with matrices

• Translation matrices
  • Represent linear displacement along x, y, and z axes shifts position
  • Identity matrix with translation values in fourth column encodes movement
• Rotation matrices
  • Represent angular displacement around x, y, and z axes changes orientation
  • 3x3 matrices for basic rotations define elemental rotations
• Basic rotation matrices
  • Rotation around x-axis: $R_x(\theta)$ = [[1, 0, 0], [0, cos(θ), -sin(θ)], [0, sin(θ), cos(θ)]]
  • Rotation around y-axis: $R_y(\theta)$ = [[cos(θ), 0, sin(θ)], [0, 1, 0], [-sin(θ), 0, cos(θ)]]
  • Rotation around z-axis: $R_z(\theta)$ = [[cos(θ), -sin(θ), 0], [sin(θ), cos(θ), 0], [0, 0, 1]]
• Combining rotations
  • Multiply rotation matrices in desired order produces composite rotation
  • Order matters: rotations are not commutative affects final orientation
• Translation followed by rotation
  • Multiply rotation matrix by translation matrix preserves correct order
• Rotation followed by translation
  • Add translation vector to rotated coordinates maintains proper sequence

Composition of spatial transformations

• Composition of transformations
  • Multiply transformation matrices in desired order combines multiple movements
  • Represent multiple sequential transformations as single matrix simplifies calculations
• Decomposition of transformations
  • Extract individual rotations and translations from complex matrix reverses composition
  • Use methods like Euler angle extraction or quaternion conversion recovers component movements
• Homogeneous transformation matrix structure
  • $T = [R | t]$, where R is 3x3 rotation matrix and t is translation vector encodes complete transformation
• Extracting rotation
  • Use upper-left 3x3 submatrix of transformation matrix isolates orientation change
• Extracting translation
  • Use first three elements of fourth column of transformation matrix retrieves position shift
• Change of basis
  • Transform coordinates from one reference frame to another enables perspective shifts
  • Use inverse transformations to change coordinate representations converts between frames
• Screw theory
  • Represent rigid body motions as combination of rotation and translation along axis unifies motion description