4.4 Part B — Control Authority, Constraints, Uncertainty, and Verification
Part A established how individual actuators generate force, torque, momentum exchange, and impulse.
Part B now moves from the physics of an individual actuator to the capability of the complete spacecraft actuation system.
The main questions are
\[ \boxed{ \text{What force and torque can the spacecraft actually generate?} } \]
\[ \boxed{ \text{Which control directions are strong, weak, or unavailable?} } \]
\[ \boxed{ \text{What physical constraints reduce the available authority?} } \]
\[ \boxed{ \text{How do uncertainty, degradation, and failure affect actuation?} } \]
\[ \boxed{ \text{How is the complete actuator model verified?} }. \]
The progression is
\[ \boxed{\text{Control Authority}} \rightarrow \boxed{\text{Constraints}} \rightarrow \boxed{\text{Uncertainty / Failure}} \rightarrow \boxed{\text{Actuator Models}} \rightarrow \boxed{\text{Verification}}. \]
Each topic below opens a dedicated page where the mathematics, engineering interpretation, examples, simulation methods, and spacecraft applications can be developed in detail.
Explore Part B
Select a topic below to study the corresponding actuation concept in detail.
Control Authority and Feasibility
These topics describe what the complete spacecraft actuator set can physically generate and whether the controller's desired wrench is actually achievable.
4.4.47 Definition of Control Authority
Focus: Desired spacecraft wrench versus physically achievable wrench.
Meaning of control authority, actuator capability, feasible commands, actuator limits, and the fundamental relation \( \mathbf w = \mathbf B_a\mathbf u \).
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4.4.48 Force Authority
Measures: Available translational force capability.
Net spacecraft force, body-axis force components, thruster summation, spacecraft acceleration, mass dependence, and maximum translational effort.
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4.4.49 Torque Authority
Measures: Available rotational control capability.
Body-axis torque, reaction-wheel or thruster torque, spacecraft inertia, angular acceleration, rigid-body rotational response, and maximum rotational effort.
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4.4.50 Axis-Wise Control Authority
Focus: Compare control capability along individual spacecraft axes.
\(X\), \(Y\), and \(Z\) force authority, roll, pitch, and yaw torque authority, anisotropic control capability, and body-frame actuator effectiveness.
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4.4.51 Positive and Negative Direction Authority
Focus: Control authority may not be symmetric in opposite directions.
Positive and negative force limits, positive and negative torque limits, asymmetric geometry, unequal actuator strength, operational restrictions, and directional margins.
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4.4.52 Coupled Force–Torque Authority
Focus: One actuator action may generate both force and torque.
Translation–rotation coupling, off-center forces, residual torque, residual translation, six-degree-of-freedom wrench generation, and coupled actuation geometry.
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4.4.53 Actuator Effectiveness Matrix Rank
Question: How many independent control directions can the actuator system generate?
Rank of \( \mathbf B_a \), full-rank and rank-deficient systems, independent wrench directions, six-DOF actuation, and geometric loss of authority.
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4.4.54 Singular Values and Weak Control Directions
Tool: Singular Value Decomposition of the actuator effectiveness matrix.
Strong and weak actuation directions, minimum singular value, matrix conditioning, near-singular configurations, directional authority, and comparison with observability concepts.
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4.4.55 Redundant Actuation
Meaning: More than one actuator combination can generate the same wrench.
Redundancy, null-space freedom, actuator load sharing, fault tolerance, secondary objectives, actuator balancing, and the connection to control allocation.
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4.4.56 Underactuated Systems
Meaning: Not every desired control direction is instantaneously available.
Rank deficiency, insufficient independent control directions, nonlinear controllability, sequential manoeuvres, time-varying geometry, and magnetorquer examples.
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4.4.57 Overactuated Systems
Meaning: More actuator degrees of freedom than required wrench dimensions.
Multiple feasible solutions, actuator optimization, load balancing, redundancy, minimum propellant solutions, and allocation freedom.
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4.4.58 Failed-Actuator Authority
Question: What control capability remains after an actuator is lost?
Complete actuator failure, removal of effectiveness columns, degraded matrices, remaining rank, remaining force and torque authority, and post-failure mission feasibility.
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4.4.59 Degraded Control Authority
Focus: Partial loss of actuator effectiveness.
Reduced force or torque capability, degradation factors, weaker singular values, reduced redundancy margin, weakened directions, and degraded reachable sets.
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4.4.60 Reachable Force/Torque Set
Defines: The complete set of physically achievable spacecraft wrenches.
Actuator bounds, feasible wrench regions, force–torque geometry, configuration changes, failed-actuator cases, and control feasibility.
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Actuator Constraints and Nonlinearities
Real actuator hardware cannot generate arbitrary commands instantly or continuously. These sections describe the physical limitations that the controller and allocator must respect.
4.4.61 Actuator Saturation
Limit: Maximum and minimum achievable actuator output.
Saturation functions, output clipping, unreachable commands, reduced tracking performance, actuator margin, stability implications, and integrator windup.
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4.4.62 Deadband
Meaning: Small commands may produce no physical actuator response.
Command thresholds, valve deadband, motor deadband, fine-control limitations, limit cycles, and nonlinear response near zero command.
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4.4.63 Rate Limits
Limit: Maximum rate at which actuator output can change.
Reaction-wheel acceleration limits, CMG gimbal-rate limits, control-surface rates, engine-gimbal rates, and transient response restrictions.
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4.4.64 Quantization
Focus: Continuous desired commands represented by finite digital resolution.
Command resolution, digital levels, quantization step size, quantization error, minimum command increment, and precision-control consequences.
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4.4.65 Minimum Commandable Effort
Defines: The smallest reliable nonzero actuator output.
Minimum wheel torque, minimum gimbal motion, command thresholds, actuator resolution, precision limits, and fine-control behaviour.
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4.4.66 Minimum Impulse Bit
Defines: Smallest repeatable impulse available from a pulsed thruster.
Minimum pulse width, minimum momentum change, minimum achievable \(\Delta v\), quantized translation, RPO, docking, and precision manoeuvring limits.
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4.4.67 Hysteresis
Meaning: Actuator output depends partly on previous command history.
Magnetic hysteresis, mechanical hysteresis, friction-related memory effects, nonlinear command paths, and history-dependent actuator response.
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4.4.68 Backlash
Meaning: Mechanical free motion before commanded motion is transmitted.
Gear backlash, direction reversal, lost motion, gimbal mechanisms, control-surface actuation, and backlash-induced tracking error.
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4.4.69 Latency
Meaning: Time between issuing a command and beginning physical response.
Software delay, communication delay, electronic-driver delay, valve latency, motor response, and mechanical response time.
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4.4.70 Command-to-Actuation Delay
Model: \(u_a(t)=u_c(t-\tau_d)\)
Explicit time delay, delayed commands, Laplace-domain delay representation, phase lag, closed-loop stability, and high-bandwidth control implications.
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4.4.71 Actuator Bandwidth
Measures: How rapidly the actuator can follow changing commands.
Time constants, first-order response, frequency response, actuator cutoff behaviour, controller bandwidth, and dynamic tracking limits.
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4.4.72 Thermal Constraints
Limit: Actuator temperature and allowable thermal operating range.
Heat generation, thermal states, continuous versus transient operation, cooling, maximum temperature, and temperature-dependent command restriction.
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4.4.73 Duty-Cycle Constraints
Limit: Maximum allowable actuator operation over a specified time interval.
ON-time fraction, thermal duty cycle, propulsion duty cycle, repeated firing restrictions, and time-dependent actuator availability.
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4.4.74 Power Constraints
Resource: Spacecraft electrical power available for actuation.
Individual actuator power demand, simultaneous operation, spacecraft power budgets, high-power propulsion, and power-constrained actuator selection.
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4.4.75 Propellant Constraints
Resource: Finite onboard propellant.
Mass-flow rate, remaining \(\Delta v\), propellant depletion, spacecraft mass change, center-of-mass shift, mission lifetime, and propellant-aware actuation.
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Actuator Uncertainty and Failure
These topics describe how the real actuator differs from its nominal mathematical model and how actuation capability changes with degradation or failure.
4.4.76 Magnitude Uncertainty
Error: Actual actuator magnitude differs from commanded magnitude.
Thrust variation, torque variation, multiplicative uncertainty, manufacturing tolerance, calibration uncertainty, and realized-wrench error.
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4.4.77 Direction/Misalignment Uncertainty
Error: Actual actuator direction differs from nominal geometry.
Mounting error, direction-vector uncertainty, small-angle rotations, cross-axis force, parasitic torque, and alignment tolerance.
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4.4.78 Timing Uncertainty
Error: Actual actuation timing differs from commanded timing.
Activation uncertainty, shutdown uncertainty, command jitter, valve timing, synchronization errors, pulse-width variation, and impulse error.
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4.4.79 Scale-Factor Error
Error: Command-to-output gain differs from the nominal value.
Wheel-torque scaling, thrust scale factor, magnetic-dipole scale factor, calibration gain error, and systematic actuator gain mismatch.
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4.4.80 Bias
Error: Constant or slowly varying actuator output offset.
Residual torque, magnetic offset, gimbal zero error, leakage, persistent control disturbance, and additive-bias modelling.
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4.4.81 Actuator-to-Actuator Variation
Focus: Nominally identical actuators are not physically identical.
Variation in maximum output, delay, bandwidth, alignment, scale factor, temperature response, and manufacturing tolerance.
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4.4.82 Aging and Degradation
Focus: Actuator performance changes over mission life.
Motor wear, bearing wear, valve degradation, thermal cycling, radiation effects, material fatigue, reduced output, and time-dependent effectiveness.
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4.4.83 Failure Modes
Includes: Complete, partial, stuck, delayed, and intermittent faults.
Loss of output, reduced effectiveness, stuck-on, stuck-off, fixed-output faults, excessive delay, intermittent response, incorrect magnitude, and incorrect direction.
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4.4.84 Fault Models
Purpose: Represent actuator failures mathematically in simulation.
Loss-of-effectiveness models, complete failure, stuck actuator, additive bias faults, combined fault models, and simulation-ready failure representations.
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Actuator Modelling and Verification
These sections describe how actuator behaviour is represented mathematically and progressively verified from analytical models through software and hardware testing.
4.4.85 Continuous-Time Actuator Models
Representation: Differential-equation actuator dynamics.
Internal actuator states, wheel speed, gimbal states, motor current, valve dynamics, thermal states, and continuous-time input-output models.
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4.4.86 Discrete-Time Actuator Models
Representation: Sampled actuator dynamics for digital implementation.
Discrete state equations, update rate, zero-order hold, sample time, quantization, synchronization, timing, and flight-software implementation.
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4.4.87 Transfer-Function Models
Representation: Linear input-output actuator dynamics.
First-order and second-order models, poles, damping ratio, natural frequency, bandwidth, phase response, and frequency-domain control analysis.
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4.4.88 State-Space Actuator Models
Representation: Actuator dynamics through states, inputs, and outputs.
State vectors, system matrices, actuator commands, actuator outputs, dynamic coupling, and integration with spacecraft state-space models.
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4.4.89 Nonlinear Actuator Models
Includes: Actuator behaviour that cannot be represented by linear superposition.
Saturation, deadband, friction, hysteresis, backlash, minimum impulse bit, binary valve states, and time-varying effectiveness.
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4.4.90 Simulink Actuator Modelling
Environment: Build and integrate actuator models in MATLAB/Simulink.
Command conditioning, delay, actuator dynamics, saturation, valve logic, MIB, rate limiters, misalignment, faults, and reusable actuator subsystems.
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4.4.91 Actuator Command Interfaces
Purpose: Convert controller or allocator output into hardware commands.
Wheel torque commands, magnetic dipoles, thruster ON/OFF commands, pulse width, throttle, gimbal angles, units, sign conventions, frames, timing, and validity flags.
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4.4.92 SIL Actuator Verification
Level: Software-in-the-Loop.
Control software, software actuator models, command generation, limits, timing, frame conventions, closed-loop performance, and fault-handling verification.
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4.4.93 HIL Actuator Verification
Level: Hardware-in-the-Loop.
Real flight computers, actuator electronics, motor drivers, valve drivers, hardware emulators, real-time spacecraft dynamics, and hardware/software integration.
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4.4.94 Monte Carlo Actuator Uncertainty
Purpose: Quantify robustness to uncertain actuator behaviour.
Random sampling of magnitude, direction, timing, bias, scale factor, degradation, repeated closed-loop simulations, and statistical performance metrics.
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4.4.95 Actuator Fault Injection
Purpose: Deliberately test actuator failure and degraded-mode behaviour.
Complete failure, reduced effectiveness, stuck actuators, incorrect magnitude, direction errors, delay faults, fault detection, isolation, reconfiguration, and survivability testing.
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4.4.96 Complete Actuation Architecture
Integration: From desired spacecraft motion to realized physical actuation.
Guidance, control laws, desired wrench, control allocation, actuator commands, command interfaces, actuator dynamics, constraints, uncertainty, realized wrench, and spacecraft motion.
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From Control Authority to Control Allocation
Section 4.4 determines what the installed actuator system can physically produce.
Once a desired spacecraft wrench has been shown to be feasible, the next question is
\[ \boxed{ \text{Which individual actuators should generate that wrench?} }. \]
The transition is
\[ \boxed{\text{Desired Wrench}} \rightarrow \boxed{\text{Control Allocation}} \rightarrow \boxed{\text{Individual Actuator Commands}} \rightarrow \boxed{\text{Realized Wrench}}. \]
Detailed allocation methods such as pseudoinverse allocation, weighted least squares, constrained optimization, thruster selection, pulse-width modulation, and fault-aware allocation belong in the later section on Control Allocation and Actuator Commanding.
Return to Part A — Actuator Fundamentals, Physics, and Models
Covers: Sections 4.4.1–4.4.46
Return to actuator fundamentals, reaction wheels, control-moment gyroscopes, magnetorquers, thrusters, propulsion systems, control surfaces, and thrust-vector-control systems.
Return to Part A →
Return to 4.4 Actuator Models and Control Authority
Return to the main Section 4.4 introduction, actuator architecture, and Part A / Part B overview.
Return to Section 4.4 →