🛰️Spacecraft Attitude Control Unit 3 – Spacecraft Attitude Kinematics

Key Concepts and Definitions

• Spacecraft attitude refers to the orientation of a spacecraft with respect to a reference frame
• Kinematics is the study of motion without considering the forces causing the motion
• Reference frames provide a basis for describing the position and orientation of objects
  • Inertial reference frames are non-accelerating and non-rotating (Earth-Centered Inertial (ECI) frame)
  • Body-fixed reference frames are attached to the spacecraft and rotate with it
• Euler angles (roll, pitch, yaw) describe the orientation of a body-fixed frame relative to a reference frame
• Quaternions are four-dimensional vectors used to represent rotations and orientations
  • Avoid singularities associated with Euler angles
• Direction cosine matrix (DCM) is a 3x3 matrix that transforms vectors between reference frames

Coordinate Systems and Reference Frames

• Coordinate systems are used to describe the position and orientation of objects in space
• Cartesian coordinate systems consist of three orthogonal axes (x, y, z)
  • Right-handed coordinate systems follow the right-hand rule for axis orientation
• Spherical coordinate systems use radius, azimuth, and elevation to describe points in space
• Perifocal coordinate system is centered on the focus of an orbit and aligned with the orbital plane
• Local-Vertical-Local-Horizontal (LVLH) frame is centered on the spacecraft and rotates with the orbital motion
  • x-axis points along the velocity vector, z-axis points towards the center of the Earth
• Spacecraft body-fixed frame is aligned with the principal axes of the spacecraft
  • Allows for convenient description of spacecraft dynamics and control

Attitude Representations

• Euler angles represent rotations about three axes (roll, pitch, yaw)
  • Rotations are performed in a specific order (e.g., 3-1-3, 3-2-1)
  • Susceptible to gimbal lock when two axes align
• Quaternions are four-dimensional vectors that describe rotations
  • Consist of a scalar part and a vector part ($q = [q_0, q_1, q_2, q_3]$)
  • Unit quaternions have a magnitude of 1 and represent pure rotations
  • Avoid gimbal lock and provide a continuous representation of attitude
• Direction cosine matrix (DCM) is a 3x3 orthogonal matrix that transforms vectors between frames
  • Columns of the DCM represent the axes of one frame expressed in another frame
• Axis-angle representation describes a rotation by a unit vector and an angle
  • Rotation axis is the eigenvector of the rotation matrix with an eigenvalue of 1

Kinematics Equations

• Kinematics equations describe the motion of a spacecraft without considering forces
• Euler's equation relates angular velocity to the rate of change of Euler angles
  • $\dot{\boldsymbol{\theta}} = \mathbf{R}(\boldsymbol{\theta}) \boldsymbol{\omega}$, where $\boldsymbol{\theta}$ is the Euler angle vector, $\boldsymbol{\omega}$ is the angular velocity vector, and $\mathbf{R}$ is a matrix function of $\boldsymbol{\theta}$
• Quaternion kinematics equation describes the rate of change of a quaternion
  • $\dot{\mathbf{q}} = \frac{1}{2} \mathbf{q} \otimes \boldsymbol{\omega}$, where $\mathbf{q}$ is the quaternion, $\boldsymbol{\omega}$ is the angular velocity vector, and $\otimes$ is the quaternion multiplication operator
• DCM kinematics equation relates the rate of change of the DCM to angular velocity
  • $\dot{\mathbf{C}} = -[\boldsymbol{\omega} \times] \mathbf{C}$, where $\mathbf{C}$ is the DCM, $\boldsymbol{\omega}$ is the angular velocity vector, and $[\boldsymbol{\omega} \times]$ is the skew-symmetric matrix of $\boldsymbol{\omega}$

Angular Velocity and Momentum

• Angular velocity is the rate of change of angular position
  • Expressed as a vector $\boldsymbol{\omega} = [\omega_x, \omega_y, \omega_z]^T$
• Angular momentum is the product of moment of inertia and angular velocity
  • $\mathbf{H} = \mathbf{I} \boldsymbol{\omega}$, where $\mathbf{H}$ is the angular momentum vector, $\mathbf{I}$ is the moment of inertia tensor, and $\boldsymbol{\omega}$ is the angular velocity vector
• Moment of inertia tensor describes the mass distribution of a spacecraft
  • Diagonal elements ($I_{xx}, I_{yy}, I_{zz}$) represent moments of inertia about principal axes
  • Off-diagonal elements represent products of inertia and are zero for principal axes
• Conservation of angular momentum states that the total angular momentum of a system remains constant in the absence of external torques
  • $\dot{\mathbf{H}} = \boldsymbol{\tau}$, where $\boldsymbol{\tau}$ is the external torque vector

Attitude Transformations

• Attitude transformations convert vectors and attitudes between different reference frames
• Rotation matrices (DCMs) transform vectors between frames
  • $\mathbf{v}_B = \mathbf{C}_{A}^{B} \mathbf{v}_A$, where $\mathbf{v}_A$ is a vector in frame A, $\mathbf{v}_B$ is the same vector in frame B, and $\mathbf{C}_{A}^{B}$ is the DCM from frame A to frame B
• Quaternion rotations transform vectors using quaternion multiplication
  • $\mathbf{q}_{\mathbf{v}_B} = \mathbf{q}_{A}^{B} \otimes \mathbf{q}_{\mathbf{v}_A} \otimes (\mathbf{q}_{A}^{B})^{-1}$, where $\mathbf{q}_{\mathbf{v}_A}$ and $\mathbf{q}_{\mathbf{v}_B}$ are quaternions representing the vector in frames A and B, and $\mathbf{q}_{A}^{B}$ is the quaternion rotation from frame A to frame B
• Euler angle rotations transform attitudes using a sequence of rotations about three axes
  • $\mathbf{C}_{A}^{B} = \mathbf{R}_3(\psi) \mathbf{R}_2(\theta) \mathbf{R}_1(\phi)$, where $\phi$, $\theta$, and $\psi$ are the Euler angles, and $\mathbf{R}_1$, $\mathbf{R}_2$, and $\mathbf{R}_3$ are the corresponding rotation matrices

Practical Applications

• Attitude determination involves estimating the spacecraft's orientation using sensor measurements
  • Sun sensors, star trackers, and magnetometers provide absolute attitude measurements
  • Gyroscopes measure angular velocity for relative attitude determination
• Attitude control maintains the desired spacecraft orientation using actuators
  • Reaction wheels, control moment gyroscopes, and thrusters are common actuators
• Pointing requirements dictate the allowable attitude error for a given mission
  • Earth observation satellites require precise pointing for imaging
  • Communication satellites must maintain antenna pointing for reliable data transmission
• Attitude maneuvers change the spacecraft's orientation to achieve a desired attitude
  • Slew maneuvers rotate the spacecraft to a new pointing direction
  • Spin stabilization maintains a fixed angular momentum vector for stability

Common Challenges and Solutions

• Singularities in Euler angle representations can cause attitude estimation and control issues
  • Quaternions or modified Euler angle sequences (e.g., 3-2-1) can mitigate singularities
• Sensor noise and biases can degrade attitude determination accuracy
  • Kalman filters and other estimation techniques can fuse sensor data and provide optimal estimates
• Actuator saturation and dynamics can limit the performance of attitude control systems
  • Control allocation algorithms can optimize actuator usage and prevent saturation
  • Feedforward control can compensate for known disturbances and improve performance
• Flexible modes and fuel slosh can couple with attitude dynamics and cause instability
  • Structural filters and damping techniques can attenuate flexible mode responses
  • Slosh baffles and active damping can mitigate fuel slosh effects
• Environmental disturbances (e.g., solar radiation pressure, magnetic torques) can perturb the spacecraft's attitude
  • Disturbance models and estimation can be used to compensate for these effects in the control system design