🛰️Spacecraft Attitude Control Unit 2 – Spacecraft Attitude: Coordinates & Representations

Key Concepts

• Spacecraft attitude refers to the orientation of a spacecraft with respect to a reference frame
• Attitude determination involves measuring the spacecraft's orientation using sensors (sun sensors, star trackers, magnetometers)
• Attitude control involves adjusting the spacecraft's orientation using actuators (reaction wheels, thrusters, magnetic torquers)
• Reference frames are coordinate systems used to describe the spacecraft's attitude (Earth-Centered Inertial (ECI), Earth-Centered Earth-Fixed (ECEF), body-fixed)
• Attitude representations mathematically describe the spacecraft's orientation using various methods (Euler angles, quaternions, rotation matrices)
• Coordinate transformations allow conversion between different reference frames and attitude representations
• Spacecraft missions require precise attitude control for tasks (pointing antennas, solar panels, cameras)

Coordinate Systems

• Earth-Centered Inertial (ECI) frame has its origin at the Earth's center, with axes fixed relative to celestial objects
  • X-axis points towards the vernal equinox
  • Z-axis is aligned with the Earth's rotation axis
  • Y-axis completes the right-handed orthogonal system
• Earth-Centered Earth-Fixed (ECEF) frame also has its origin at the Earth's center but rotates with the Earth
  • X-axis intersects the prime meridian and equator
  • Z-axis aligns with the Earth's rotation axis
  • Y-axis completes the right-handed orthogonal system
• Body-fixed frame is attached to the spacecraft, with axes defined by the spacecraft's geometry
  • Origin is typically the spacecraft's center of mass
  • X-axis often points along the spacecraft's main axis
  • Y and Z axes are orthogonal to the X-axis
• Orbital frame is centered on the spacecraft, with axes defined by the spacecraft's orbit
  • X-axis points in the direction of the spacecraft's velocity
  • Z-axis points towards the Earth's center
  • Y-axis completes the right-handed orthogonal system
• Sensor frames are coordinate systems specific to individual attitude sensors (star trackers, sun sensors)

Attitude Representations

• Euler angles represent spacecraft orientation using three sequential rotations about the coordinate axes
  • Rotations are typically performed in a specific order (yaw, pitch, roll)
  • Euler angles are intuitive but suffer from singularities (gimbal lock)
• Quaternions are four-dimensional vectors that represent spacecraft orientation without singularities
  • Consist of a scalar part and a vector part
  • Provide a compact and computationally efficient attitude representation
  • Quaternion multiplication allows for attitude composition and transformation
• Rotation matrices are 3x3 matrices that describe the rotation between two coordinate frames
  • Orthogonal matrices with determinant equal to 1
  • Can be derived from Euler angles or quaternions
  • Matrix multiplication allows for attitude composition and transformation
• Axis-angle representation describes a rotation about a single axis by a specified angle
  • Axis is represented as a unit vector
  • Angle is measured in radians
  • Can be converted to and from quaternions and rotation matrices

Quaternions

• Quaternions are an extension of complex numbers, consisting of a scalar part and a vector part
  • Scalar part represents the magnitude of rotation
  • Vector part represents the axis of rotation
• Unit quaternions, also known as versors, are used to represent rotations in 3D space
  • Have a magnitude of 1
  • Can be normalized by dividing each component by the quaternion's magnitude
• Quaternion multiplication is non-commutative and allows for the composition of rotations
  • The product of two quaternions represents the combined rotation
  • Order of multiplication matters, as rotations are applied from right to left
• Quaternions can be interpolated using methods like spherical linear interpolation (SLERP)
  • Allows for smooth transitions between orientations
  • Useful for attitude control and animation applications
• Advantages of quaternions include avoiding singularities, compact representation, and efficient computation
  • Quaternions do not suffer from gimbal lock, unlike Euler angles
  • Require only four elements, compared to nine for rotation matrices

Euler Angles

• Euler angles represent spacecraft orientation as a sequence of three rotations about the coordinate axes
  • Rotations are typically denoted as yaw (ψ), pitch (θ), and roll (φ)
  • The order of rotations is important and must be specified (e.g., ZYX, XYZ)
• Yaw (ψ) represents rotation about the Z-axis
  • Positive yaw is counterclockwise when viewed from the positive Z-axis
• Pitch (θ) represents rotation about the Y-axis
  • Positive pitch is counterclockwise when viewed from the positive Y-axis
• Roll (φ) represents rotation about the X-axis
  • Positive roll is counterclockwise when viewed from the positive X-axis
• Euler angles are intuitive and easy to visualize but have limitations
  • Suffer from singularities at certain orientations (gimbal lock)
  • Not unique, as different sequences of rotations can lead to the same orientation
• Conversion between Euler angles and other attitude representations (quaternions, rotation matrices) is possible
  • Involves trigonometric functions and matrix operations
  • Care must be taken to handle singularities and order of rotations

Rotation Matrices

• Rotation matrices are 3x3 matrices that describe the rotation between two coordinate frames
  • Orthogonal matrices with determinant equal to 1
  • Columns represent the unit vectors of one frame expressed in the other frame
• Rotation matrices can be derived from Euler angles using a sequence of trigonometric functions
  • Each rotation (yaw, pitch, roll) is represented by a separate matrix
  • The overall rotation matrix is the product of the individual rotation matrices
• Rotation matrices can also be derived from quaternions using a set of algebraic equations
  • Involves the components of the quaternion
  • Provides a direct relationship between quaternions and rotation matrices
• Matrix multiplication allows for the composition of rotations
  • The product of two rotation matrices represents the combined rotation
  • Order of multiplication matters, as rotations are applied from right to left
• Rotation matrices are used for coordinate transformations between different frames
  • Multiply a vector in one frame by the appropriate rotation matrix to express it in another frame
• Advantages of rotation matrices include their direct geometric interpretation and ease of matrix operations
  • Disadvantages include the larger storage requirement (nine elements) and potential numerical instability

Attitude Transformations

• Attitude transformations involve converting between different attitude representations and coordinate frames
• Euler angle to rotation matrix transformation:
  • Construct individual rotation matrices for yaw, pitch, and roll using trigonometric functions
  • Multiply the rotation matrices in the appropriate order to obtain the overall rotation matrix
• Quaternion to rotation matrix transformation:
  • Use a set of algebraic equations involving the quaternion components
  • The resulting matrix represents the same rotation as the quaternion
• Rotation matrix to Euler angle transformation:
  • Extract the Euler angles from the rotation matrix using trigonometric functions
  • The order of rotations and quadrant of the angles must be considered
• Quaternion to Euler angle transformation:
  • Convert the quaternion to a rotation matrix, then extract the Euler angles
  • Alternatively, use a set of trigonometric equations directly relating quaternion components to Euler angles
• Coordinate frame transformations:
  • Multiply a vector in one frame by the appropriate rotation matrix to express it in another frame
  • The rotation matrix represents the relative orientation between the two frames
• Composition of rotations:
  • Multiply rotation matrices or quaternions to combine multiple rotations
  • The order of multiplication matters, as rotations are applied from right to left

Practical Applications

• Attitude determination:
  • Spacecraft use sensors (sun sensors, star trackers, magnetometers) to measure their orientation
  • Sensor measurements are processed using attitude estimation algorithms (Kalman filters, QUEST)
  • The estimated attitude is used for navigation, control, and data analysis
• Attitude control:
  • Spacecraft use actuators (reaction wheels, thrusters, magnetic torquers) to adjust their orientation
  • Control algorithms (PID, LQR) generate actuator commands based on the desired and estimated attitude
  • Precise attitude control is crucial for pointing antennas, solar panels, and cameras
• Orbit determination and control:
  • Attitude information is combined with orbital measurements (GPS, ranging) to estimate the spacecraft's trajectory
  • Orbit control maneuvers (thruster firings) are planned based on the desired and estimated orbit
  • Accurate attitude knowledge is necessary for executing orbit maneuvers
• Earth observation and remote sensing:
  • Spacecraft cameras and instruments require precise pointing to capture high-quality images and data
  • Attitude control ensures the spacecraft is oriented correctly during data acquisition
  • Attitude information is used to geo-reference and process the collected data
• Astronomical observations:
  • Space telescopes and observatories require extremely precise and stable pointing
  • Attitude control systems maintain the desired orientation for long exposure times
  • Attitude knowledge is used to calibrate and process the astronomical data