🛰️Spacecraft Attitude Control Unit 4 – Spacecraft Attitude Dynamics

Key Concepts and Terminology

• Spacecraft attitude refers to the orientation of a spacecraft with respect to a reference frame, typically an inertial reference frame
• Attitude dynamics describes the motion and behavior of a spacecraft's attitude over time, considering various forces and torques acting on the spacecraft
• Rigid body dynamics assumes the spacecraft is a rigid body, meaning its shape and size remain constant, and the distance between any two points on the spacecraft does not change
• External torques are forces that cause the spacecraft to rotate about its center of mass, altering its attitude (gravity gradient, solar radiation pressure, magnetic torques)
• Disturbances are unwanted external forces or torques that perturb the spacecraft's attitude, requiring compensation from the attitude control system (atmospheric drag, thruster misalignments, fuel slosh)
• Attitude representation methods include Euler angles, quaternions, and direction cosine matrices (DCM), each with their own advantages and disadvantages
• Attitude sensors measure the spacecraft's orientation in space (star trackers, sun sensors, magnetometers, gyroscopes)
• Attitude actuators apply torques to control and adjust the spacecraft's attitude (reaction wheels, control moment gyroscopes, thrusters, magnetic torquers)

Spacecraft Attitude Representation

• Euler angles represent spacecraft attitude using three angles: roll (rotation about X-axis), pitch (rotation about Y-axis), and yaw (rotation about Z-axis)
  • Euler angles are intuitive and easy to visualize but suffer from singularities and gimbal lock
• Quaternions represent attitude using a four-element vector, consisting of a scalar component and a three-element vector component
  • Quaternions are singularity-free and computationally efficient but less intuitive than Euler angles
• Direction cosine matrices (DCM) represent attitude as a 3x3 matrix that transforms vectors from the body frame to the reference frame
  • DCMs are singularity-free and provide a direct mapping between frames but require more storage and computation than quaternions
• Rotation matrices describe the orientation of the spacecraft body frame relative to the reference frame, enabling the transformation of vectors between frames
• Attitude kinematics describes the relationship between the spacecraft's angular velocity and the rate of change of its attitude representation
• Attitude dynamics relates the external torques acting on the spacecraft to its angular acceleration, angular velocity, and attitude
• Coordinate frames used in attitude representation include the inertial frame (fixed with respect to stars), body frame (fixed to the spacecraft), and orbital frame (aligned with the spacecraft's orbit)

Rigid Body Dynamics

• Newton's laws of motion form the basis for understanding spacecraft attitude dynamics, relating forces and torques to linear and angular accelerations
• Moment of inertia tensor is a 3x3 matrix that describes the distribution of mass in a rigid body and its resistance to rotational acceleration
  • Principal axes are the eigenvectors of the moment of inertia tensor, representing the axes about which the spacecraft rotates without inducing torques on the other axes
• Euler's rotational equations of motion describe the relationship between the external torques acting on a spacecraft and its angular acceleration, angular velocity, and moment of inertia tensor
• Angular momentum is a vector quantity that represents the product of the spacecraft's moment of inertia tensor and its angular velocity
  • Conservation of angular momentum states that the total angular momentum of a system remains constant in the absence of external torques
• Torque-free motion occurs when no external torques are acting on the spacecraft, resulting in a constant angular momentum vector and a predictable attitude evolution
• Nutation is the wobbling motion of a spacecraft's rotation axis about its angular momentum vector, caused by misalignment between the principal axes and the angular velocity vector
• Precession is the slow rotation of a spacecraft's angular momentum vector about a fixed axis, often caused by external torques such as gravity gradient or solar radiation pressure
• Energy dissipation mechanisms, such as fuel slosh or flexible appendages, can dampen the spacecraft's rotational motion and affect its attitude dynamics

External Torques and Disturbances

• Gravity gradient torque arises from the variation in gravitational force across the spacecraft's body, causing a torque that aligns the spacecraft's minimum moment of inertia axis with the local vertical
• Solar radiation pressure torque is caused by the uneven reflection and absorption of solar photons on the spacecraft's surfaces, creating a net torque that depends on the spacecraft's geometry and surface properties
• Magnetic torques result from the interaction between the spacecraft's residual magnetic dipole and the Earth's magnetic field, inducing torques that vary with the spacecraft's position and orientation
• Atmospheric drag torque is caused by the uneven distribution of atmospheric particles impacting the spacecraft's surfaces, creating a torque that is more significant in low Earth orbits
• Thruster misalignments and uncertainties can introduce unwanted torques during thruster firings, requiring careful calibration and compensation
• Fuel slosh occurs when liquid propellant moves within its tanks, shifting the spacecraft's center of mass and creating disturbance torques
• Flexible appendages, such as solar arrays or antennas, can vibrate and create disturbance torques that couple with the spacecraft's rigid body dynamics
• Outgassing from the spacecraft's materials can create small but persistent disturbance torques, particularly in the early stages of the mission

Attitude Sensors and Actuators

• Star trackers are optical sensors that measure the positions of stars in the spacecraft's field of view, providing high-accuracy attitude determination
  • Star catalogs are used to identify stars and calculate the spacecraft's orientation based on their relative positions
• Sun sensors detect the direction of the Sun relative to the spacecraft, providing coarse attitude information during sunlit portions of the orbit
  • Coarse sun sensors have a wide field of view and provide basic attitude information, while fine sun sensors offer higher accuracy over a narrower field of view
• Magnetometers measure the direction and strength of the Earth's magnetic field, enabling attitude determination and control using magnetic torquers
• Gyroscopes measure the spacecraft's angular velocity, providing high-frequency attitude information and short-term attitude propagation
  • Rate gyros measure angular velocity directly, while rate-integrating gyros measure angular displacement over time
• Reaction wheels are momentum exchange devices that apply torques by accelerating or decelerating a flywheel, allowing for precise attitude control
  • Reaction wheel assemblies typically consist of three or more wheels mounted in orthogonal axes to provide three-axis control
• Control moment gyroscopes (CMGs) are momentum exchange devices that apply torques by changing the orientation of a constantly spinning rotor, offering high torque capability and agility
• Thrusters generate forces by expelling propellant, providing attitude control and momentum management capabilities
  • Thruster configurations, such as couples and pyramids, are designed to provide efficient and redundant attitude control
• Magnetic torquers, also known as magnetorquers, are electromagnetic coils that interact with the Earth's magnetic field to generate control torques, often used for momentum management and coarse attitude control

Attitude Determination Algorithms

• Deterministic methods use sensor measurements directly to calculate the spacecraft's attitude, without considering measurement noise or uncertainties
  • TRIAD algorithm determines attitude using two vector measurements, typically from sun sensors and magnetometers
  • Q-Method calculates attitude by minimizing the error between measured and reference vectors, using an eigenvalue-eigenvector approach
• Stochastic methods incorporate statistical models of sensor noise and uncertainties to estimate the spacecraft's attitude and associated uncertainties
  • Kalman filters recursively estimate the spacecraft's attitude by combining sensor measurements with a dynamic model of the spacecraft's motion
    • Extended Kalman filters (EKF) linearize the nonlinear attitude dynamics and measurement models, enabling the use of the standard Kalman filter equations
    • Unscented Kalman filters (UKF) use a deterministic sampling approach to capture the mean and covariance of the attitude state, avoiding the need for linearization
  • Particle filters represent the attitude state using a set of weighted samples, allowing for the estimation of non-Gaussian and multimodal distributions
• Batch least-squares methods estimate the spacecraft's attitude by minimizing the error between measured and predicted sensor observations over a batch of data
  • Wahba's problem seeks to find the rotation matrix that minimizes the weighted sum of squared errors between measured and reference vectors
  • Singular value decomposition (SVD) and quaternion estimator (QUEST) are common algorithms for solving Wahba's problem
• Sensor fusion techniques combine measurements from multiple sensors to improve attitude determination accuracy and robustness
  • Complementary filters blend the high-frequency information from gyroscopes with the low-frequency information from star trackers, sun sensors, or magnetometers
  • Optimal sensor selection chooses the most informative and reliable sensor measurements based on their expected quality and the spacecraft's attitude uncertainty

Attitude Control Strategies

• Spin stabilization maintains the spacecraft's attitude by rotating the entire spacecraft about a single axis, using the gyroscopic effect to resist disturbances
  • Dual-spin spacecraft have a despun platform for instruments and a spinning section for attitude stability
• Three-axis stabilization controls the spacecraft's attitude about all three axes independently, allowing for more precise pointing and maneuverability
  • Reaction wheel control uses reaction wheels to apply torques and maintain the desired attitude, with thrusters or magnetic torquers used for momentum management
  • Control moment gyroscope (CMG) control uses CMGs to generate large control torques, enabling rapid slewing and agile maneuvers
• Momentum bias control maintains a constant angular momentum vector using a single reaction wheel or CMG, providing passive stability about two axes and requiring active control about the third axis
• Gravity gradient stabilization uses the gravity gradient torque to passively align the spacecraft's minimum moment of inertia axis with the local vertical, reducing the need for active control
• Magnetic control uses magnetic torquers to generate control torques by interacting with the Earth's magnetic field, often used for momentum management and coarse attitude control
• Thruster-based control uses thrusters to apply forces and torques directly, providing attitude control and momentum management capabilities
  • Pulse-width modulation (PWM) and pulse-width pulse-frequency (PWPF) modulation techniques are used to generate precise and efficient thruster commands
• Hybrid control strategies combine multiple control methods to leverage their respective strengths and compensate for their weaknesses
  • Momentum wheel and magnetic torquer hybrid control uses momentum wheels for fine attitude control and magnetic torquers for momentum management
  • Thruster and reaction wheel hybrid control uses thrusters for coarse attitude control and momentum management, while reaction wheels provide fine pointing

Applications and Case Studies

• Earth observation satellites require precise attitude control to maintain pointing accuracy and stability for high-resolution imaging and remote sensing
  • WorldView-3, a commercial Earth observation satellite, uses a three-axis stabilized platform with reaction wheels and star trackers for sub-meter resolution imaging
• Communication satellites often employ spin stabilization or momentum bias control to maintain antenna pointing and minimize attitude disturbances
  • Intelsat 35e, a geostationary communication satellite, uses a combination of spin stabilization and electric propulsion for attitude control and station-keeping
• Interplanetary spacecraft face unique attitude control challenges due to long communication delays, limited power, and varying environmental conditions
  • Voyager 1 and 2, launched in 1977, use a combination of spin stabilization, thrusters, and star trackers to maintain attitude control during their ongoing journey beyond the solar system
• Space telescopes require extremely precise and stable attitude control to enable high-resolution imaging and accurate pointing
  • James Webb Space Telescope (JWST) uses a fine guidance sensor, reaction wheels, and fine sun sensors to achieve nanoradian-level pointing accuracy for infrared observations
• Small satellites and CubeSats often rely on passive attitude control methods and low-cost sensors and actuators due to size, weight, and power constraints
  • ASTERIA, a 6U CubeSat, demonstrated arcsecond-level pointing accuracy using a miniaturized star tracker, reaction wheels, and closed-loop control algorithms
• Formation flying missions require coordinated attitude control among multiple spacecraft to maintain precise relative positions and orientations
  • Proba-3, an ESA mission, will demonstrate autonomous formation flying and attitude control for a large-scale coronagraph using GPS, optical sensors, and electric propulsion
• Rendezvous and docking operations require accurate and responsive attitude control to ensure safe and successful proximity operations
  • SpaceX Crew Dragon uses a combination of thrusters, Draco engines, and a vision-based navigation system for autonomous rendezvous and docking with the International Space Station
• Attitude control system design and analysis involve trade-offs between performance, complexity, cost, and reliability
  • Simulations, hardware-in-the-loop testing, and on-orbit calibration are essential for validating and optimizing attitude control system performance