2.2 Euler angles and rotation matrices

Euler angles and rotation matrices are key tools for describing spacecraft orientation in 3D space. They use three rotations around fixed axes to represent attitude, with roll, pitch, and yaw defining specific movements.

While intuitive, Euler angles can face issues like gimbal lock. Rotation matrices offer an alternative, using 3x3 orthogonal matrices to represent orientation. Both methods have pros and cons in spacecraft attitude control.

Euler Angles and Rotation Sequences

Fundamental Concepts of Euler Angles

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• Euler angles represent orientation of rigid body in 3D space using three rotations around fixed axes
• Roll defines rotation around x-axis, measures tilt of object left or right
• Pitch describes rotation around y-axis, indicates nose-up or nose-down attitude
• Yaw specifies rotation around z-axis, determines direction object is facing
• Rotation sequence establishes order in which Euler angle rotations are applied
• 3-2-1 rotation sequence commonly used in aerospace applications
  • Applies yaw rotation first, followed by pitch, then roll
  • Aligns with typical aircraft maneuvers

Practical Applications and Limitations

• Euler angles widely used in aviation, robotics, and computer graphics
• Provide intuitive representation of object orientation
• Gimbal lock occurs when two rotation axes align, causing loss of one degree of freedom
  • Happens in 3-2-1 sequence when pitch approaches 90 degrees
  • Results in inability to distinguish between roll and yaw rotations
• Alternative representations (quaternions) used to avoid gimbal lock in certain applications

Rotation Matrices

Properties and Operations of Rotation Matrices

• Rotation matrix represents orientation of coordinate frame relative to reference frame
• 3x3 orthogonal matrix for 3D rotations
• Orthogonal matrix properties ensure columns and rows are mutually perpendicular unit vectors
• Determinant of rotation matrix always equals 1 or -1
  • Positive determinant indicates proper rotation
  • Negative determinant suggests improper rotation (reflection)
• Inverse of rotation matrix equals its transpose due to orthogonality
  • Simplifies computational operations in attitude determination algorithms

Composition and Decomposition of Rotations

• Multiple rotations combined by multiplying rotation matrices
• Order of multiplication matters due to non-commutativity of matrix multiplication
• Rotation matrix can be decomposed into individual Euler angle rotations
• Extraction of Euler angles from rotation matrix involves trigonometric operations
  • Can lead to numerical instabilities near singularities

Euler Angle Limitations

Singularities and Alternative Representations

• Euler angle singularities occur when gimbal lock happens
• Singularities cause loss of one degree of freedom in attitude representation
• Small angle approximation used for simplified calculations in certain scenarios
  • Assumes sin(θ) ≈ θ and cos(θ) ≈ 1 for small angles
  • Valid for angles typically less than 0.1 radians or about 5.7 degrees
• Quaternions provide alternative attitude representation
  • Four-dimensional representation avoids gimbal lock
  • Computationally efficient for attitude propagation and control algorithms

Practical Considerations in Spacecraft Attitude Control

• Euler angles remain useful for human interpretation and visualization
• Quaternions preferred in many spacecraft attitude determination and control systems
• Conversion between Euler angles and quaternions necessary for complete attitude representation
• Choice of representation depends on specific mission requirements and computational constraints