Position and Orientation of a Rigid Body

Position Representation

Arbitrary Point on Rigid Body

• Use an arbitrary point on the rigid body to represent its position.
• Position vector represented as ( \vec{OP} \).

Coordinates for Position

• 2D Position: 2 coordinates needed (2 degrees of freedom).
• 3D Position: 3 coordinates needed (3 degrees of freedom).

Forms of Representation

Cartesian Coordinates

• Reference frame with coordinates (x, y, z).

Cylindrical Coordinates

• Projection on XY plane, defining ( \theta \).
• ( \rho \): Length of projection vector.
• (z): Height.
• Transformations:
  • Cylindrical to Cartesian.
  • Cartesian to Cylindrical.

Spherical Coordinates

• (R): Radius (always positive).
• Angles ( \theta \) (0 to π) and ( \phi \) (0 to 2π).
• Transformations:
  • Spherical to Cartesian.
  • Cartesian to Spherical.
  • Spherical to Cylindrical.
  • Cylindrical to Spherical.
• Note: ( \theta \) is different between spherical and cylindrical coordinates.
  • Spherical: ( \theta_s \).
  • Cylindrical: ( \theta_c \).

Orientation Representation

Relation Between Frames

• Assign a reference frame to the body.
• Represent the relationship between the body frame and the reference frame.
• Frame relations define the orientation.

Coordinates for Orientation

• 2D Orientation: 1 coordinate (1 degree of freedom).
• 3D Orientation: 3 coordinates (3 degrees of freedom).
• General Formula: In n dimensions, number of coordinates: ( \frac{n(n-1)}{2} \).

Complexity of Orientation

• Orientation is less straightforward than position due to topology.
• Analogous to representing points on a sphere (latitude and longitude issues at poles).
• In 3D space, the complexity increases.

Rotation Representation

• Rotation Matrix: Implicit representation, introduces constraints.
• Exponential Coordinates: To be covered later.
• Axis-Angle Representation: Minimal (3 parameters).
• Euler Angles: Minimal (3 parameters).
• Quaternions: 4 parameters; not minimal but useful.

Minimal Representations

• Axis-Angle, Euler Angles: 3 parameters for 3 degrees of freedom.
• Quaternions: Implicit representation with 1 extra constraint.