Actuator Fundamentals, Physics, and Models

4.4.1 Actuator Role in the GNC Loop

Focus: Controller output, actuator commands, realized physical effort, and the connection between control laws and spacecraft motion.

Commanded versus realized effort, desired spacecraft wrench, actuator interfaces, and the actuator as the physical output stage of the GNC loop.

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4.4.2 Force, Torque, Impulse and Momentum Fundamentals

Focus: Physical quantities through which actuators influence spacecraft motion.

Linear momentum, angular momentum, force, torque, linear impulse, angular impulse, momentum exchange, and their connection to translational and rotational dynamics.

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4.4.3 Actuator Coordinate Frames and Geometry

Focus: Where the actuator is mounted and in which direction it acts.

Spacecraft body frames, actuator frames, mounting positions, direction vectors, lever arms, direction-cosine matrices, center-of-mass dependence, and frame transformations.

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4.4.4 Actuator Effectiveness Matrix

Focus: Mapping individual actuator effort into spacecraft force and torque.

Generalized wrench, actuator command vectors, effectiveness columns, configuration geometry, and the fundamental relationship \( \mathbf w = \mathbf B_a \mathbf u \).

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4.4.5 Reaction Wheel Fundamentals

Produces: Spacecraft body torque through internal angular-momentum exchange.

Wheel construction, rotor acceleration, conservation of angular momentum, attitude-control applications, and basic wheel operation.

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4.4.6 Reaction Wheel Torque and Momentum

Key relation: \( \boldsymbol{\tau}_{body} = -\dot{\mathbf H}_w \)

Wheel angular momentum, spin-axis representation, wheel acceleration, momentum exchange, and spacecraft reaction torque.

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4.4.7 Wheel Speed Dynamics

State: Wheel rotational speed \( \Omega_w \)

Motor torque, wheel inertia, acceleration and deceleration, friction torque, wheel-speed evolution, and dynamic response.

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4.4.8 Momentum Storage and Saturation

Limit: \( |H_w| \leq H_{w,\max} \)

Maximum wheel speed, stored momentum, disturbance accumulation, saturation, loss of torque capability, and momentum-storage sizing.

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4.4.9 Wheel Friction and Non-Idealities

Includes: Friction, imbalance, vibration, ripple, and deadband.

Bearing friction, viscous friction, Coulomb friction, motor torque ripple, imbalance disturbances, vibration, and non-ideal wheel behaviour.

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4.4.10 Reaction Wheel Assemblies

Architecture: Multiple wheels for three-axis spacecraft control.

Three-wheel orthogonal layouts, four-wheel pyramids, redundant clusters, wheel-axis matrices, and assembly-level torque generation.

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4.4.11 Wheel Geometry and Torque Authority

Focus: How wheel orientation determines spacecraft torque capability.

Wheel-axis geometry, three-axis torque authority, redundant configurations, body-axis capability, and failed-wheel geometry.

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4.4.12 Momentum Dumping Requirements

Purpose: Remove accumulated reaction-wheel momentum.

Environmental disturbance torque, wheel desaturation, momentum unloading, external torque generation, magnetorquer dumping, and thruster-assisted dumping.

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4.4.13 Control Moment Gyroscope Fundamentals

Principle: Generate torque by changing the direction of stored angular momentum.

Spinning rotors, gimbals, stored momentum vectors, gyroscopic reaction torque, CMG architecture, and comparison with reaction wheels.

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4.4.14 CMG Torque Generation

Key relation: \( \boldsymbol{\tau} = -\dot{\mathbf H} \)

Gimbal angular velocity, rotating momentum vectors, gyroscopic torque, reaction torque, and CMG torque magnitude.

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4.4.15 CMG Steering Geometry

Mapping: Desired spacecraft torque into CMG gimbal-rate commands.

Gimbal angles, gimbal rates, CMG clusters, configuration Jacobians, steering matrices, and geometric torque mapping.

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4.4.16 CMG Singularities

Problem: Loss or weakening of torque directions because of CMG geometry.

Singular configurations, Jacobian rank loss, excessive gimbal-rate demand, weak directions, steering-law sensitivity, and singularity avoidance.

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4.4.17 CMG Control Authority

Depends on: Stored momentum, gimbal rate, geometry, and current gimbal angles.

Configuration-dependent torque capability, gimbal-rate limits, singularity proximity, CMG cluster geometry, and time-varying authority.

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4.4.18 Magnetorquer Fundamentals

Produces: Magnetic dipole moment.

Torque rods, air-core coils, PCB coils, detumbling, coarse attitude control, momentum unloading, and magnetic actuation architecture.

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4.4.19 Magnetic Dipole Generation

Key relation: \( \mathbf m = NIA\hat{\mathbf n} \)

Coil current, number of turns, coil area, dipole direction, three-axis magnetic actuation, and commandable magnetic moment.

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4.4.20 Magnetic Torque Equation

Key relation: \( \boldsymbol{\tau} = \mathbf m \times \mathbf B \)

Dipole-field geometry, torque magnitude, cross-product interpretation, maximum torque, and body-frame magnetic torque.

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4.4.21 Geomagnetic Field Dependence

Focus: Magnetorquer authority varies with the local Earth magnetic field.

Orbital position, altitude, inclination, magnetic-field magnitude and direction, and orbit-dependent torque capability.

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4.4.22 Magnetorquer Authority and Limitations

Fundamental limitation: Torque cannot be generated parallel to the instantaneous geomagnetic-field vector.

Direction-dependent torque, instantaneous underactuation, dipole limits, magnetic-field dependence, and time-varying control capability.

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4.4.23 Momentum Dumping with Magnetorquers

Application: Reaction-wheel desaturation without propellant consumption.

Magnetic unloading torque, wheel momentum management, dipole commands, wheel-speed reduction, and momentum-dumping strategies.

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4.4.24 Thruster Fundamentals

Produces: External spacecraft force and, when offset from the center of mass, torque.

Propellant acceleration, reaction force, thrust generation, nozzle direction, firing commands, and spacecraft propulsion architecture.

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4.4.25 Thrust Force Model

Key relation: \( \mathbf F_i = F_i\hat{\mathbf d}_i \)

Thrust magnitude, thrust direction, force components, time-dependent thrust, and total force from multiple thrusters.

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4.4.26 Thruster Torque Model

Key relation: \( \boldsymbol{\tau}_i = \mathbf r_i \times \mathbf F_i \)

Lever arms, center-of-mass location, thruster torque contribution, net spacecraft torque, and force–torque coupling.

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4.4.27 Thruster Position and Direction Vectors

Geometry: \( \mathbf r_i \) and \( \hat{\mathbf d}_i \)

Thruster mounting coordinates, unit vectors, body-frame transformations, geometry tables, lever arms, and individual wrench columns.

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4.4.28 Thruster Cant Angles

Focus: Intentional angular orientation of the thrust vector.

Cant geometry, thrust decomposition, three-dimensional direction, translational authority, rotational authority, cross-axis effects, and geometric trade-offs.

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4.4.29 Specific Impulse and Propellant Consumption

Key relation: \( I_{sp} = F/(\dot m g_0) \)

Mass-flow rate, propulsion efficiency, finite burns, propellant depletion, spacecraft mass change, and mission-level consumption.

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4.4.30 Minimum Impulse Bit

Defines: The smallest repeatable impulse a pulsed thruster can deliver.

Minimum pulse duration, impulse integration, minimum achievable \(\Delta v\), fine translation, rendezvous, docking, and precision-control limitations.

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4.4.31 Minimum On-Time and Valve Dynamics

Focus: Real ON/OFF thruster response.

Minimum pulse duration, valve opening, valve closing, electrical delay, fluid transients, pulse repeatability, and realistic thruster-command execution.

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4.4.32 Rise Time and Fall Time

Focus: Transient thrust rather than ideal rectangular pulses.

Startup behaviour, steady thrust, shutdown behaviour, thrust-time profiles, pulse integration, and delivered-impulse error.

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4.4.33 Thrust Magnitude Error

Error: Actual thrust differs from nominal or commanded thrust.

Scale-factor error, additive error, chamber pressure, temperature, manufacturing variation, calibration, and propulsion uncertainty.

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4.4.34 Thruster Misalignment

Error: Actual thrust direction differs from the nominal direction.

Small-angle mounting errors, direction-vector perturbation, cross-axis forces, parasitic torque, installation tolerances, and Monte Carlo modelling.

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4.4.35 Valve Delay and Timing Uncertainty

Error: Commanded and actual firing times do not perfectly coincide.

Valve opening delay, valve closing delay, ignition uncertainty, shutdown uncertainty, pulse-width error, timing jitter, and resulting impulse error.

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4.4.36 Plume and Firing Constraints

Focus: Thrusters may be physically available but operationally prohibited.

Plume impingement, contamination, sensor exclusion zones, solar-array interaction, docking safety, thermal restrictions, and mission-phase firing constraints.

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4.4.37 Simultaneous Thruster Firings

Focus: Combining several thrusters to generate a desired net wrench.

Active thruster sets, force summation, torque summation, paired firings, translation with torque cancellation, pure torque, and combined manoeuvres.

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4.4.38 Translational Control Authority

Measures: Available spacecraft force and acceleration capability.

Net force, body-axis acceleration, spacecraft mass dependence, positive and negative translation, and available translational control effort.

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4.4.39 Rotational Control Authority

Measures: Available spacecraft torque and angular acceleration capability.

Net torque, spacecraft inertia, lever arms, angular acceleration, body-axis rotational authority, and thruster firing geometry.

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4.4.40 Thruster Configuration Matrices

Mapping: Individual thruster commands into the complete spacecraft wrench.

Six-degree-of-freedom configuration matrices, force rows, torque rows, actuator columns, thruster geometry, wrench mapping, and preparation for control allocation.

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4.4.41 Cold-Gas Thrusters

Principle: Generate thrust by expanding stored pressurized gas through a nozzle.

Cold-gas architecture, pressure regulation, nozzle flow, thrust, specific impulse, minimum impulse capability, simplicity, and precision-control applications.

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4.4.42 Chemical Thrusters

Principle: Generate thrust through chemical energy release.

Monopropellant and bipropellant concepts, chamber pressure, mass flow, specific impulse, thrust level, pulsed operation, and spacecraft manoeuvring applications.

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4.4.43 Electric Propulsion Actuators

Characteristics: Low thrust and high specific impulse.

Electrically accelerated propellant, ion and Hall-effect concepts, throttle levels, power dependence, long-duration thrust, orbit raising, and deep-space applications.

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4.4.44 Gimballed Thrusters

Control variable: Thrust-vector direction.

Gimbal angles, thrust-vector rotation, force direction, torque generation, gimbal-rate dynamics, steering limits, and propulsion-vector control.

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4.4.45 Control Surfaces for Atmospheric Vehicles

Produces: Aerodynamic forces and moments.

Ailerons, elevators, rudders, elevons, canards, dynamic pressure, surface deflection, aerodynamic derivatives, and atmosphere-dependent authority.

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4.4.46 Thrust-Vector-Control Systems

Application: Attitude control by redirecting the propulsion force vector.

Engine or nozzle vectoring, gimbal geometry, force-vector rotation, generated moment, launch vehicles, powered descent, gimbal limits, and TVC dynamics.

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Part B — Control Authority, Constraints, Uncertainty, and Verification

Covers: Sections 4.4.47–4.4.96

Effectiveness-matrix rank, singular values, weak control directions, redundancy, reachable wrench sets, actuator limits, nonlinearities, uncertainty, failures, continuous and discrete modelling, SIL/HIL verification, Monte Carlo analysis, and fault injection.

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Return to 4.4 Actuator Models and Control Authority

Return to the main Section 4.4 introduction, complete actuation architecture, and Part A / Part B overview.

Return to Section 4.4 →