Actuator Models and Control Authority

4.4 Actuator Models and Control Authority

Spacecraft guidance, navigation, and control ultimately require the vehicle to physically change its motion.

Guidance may determine where the spacecraft should go, navigation may estimate where the spacecraft currently is, and a controller may calculate the force or torque required to reduce an error.

However, none of these calculations by themselves move the spacecraft.

Physical motion is produced by actuators.

Depending on the spacecraft and mission, actuators may include reaction wheels, control-moment gyroscopes, magnetorquers, chemical thrusters, cold-gas thrusters, electric propulsion systems, gimballed engines, control surfaces, or other force- and torque-producing devices.

Controller Output Is Not Automatically Physical Effort

One of the most important distinctions in practical GNC is

\[ \boxed{ \text{Controller Output} \neq \text{Physical Force/Torque Automatically} }. \]

The controller generates a desired control effort. The actuator system must then convert that command into realizable physical force, torque, impulse, or momentum exchange.

Conceptually,

\[ \boxed{\text{State Estimate}} \rightarrow \boxed{\text{Guidance / Control}} \rightarrow \boxed{\text{Desired Wrench}} \rightarrow \boxed{\text{Actuator Commands}} \rightarrow \boxed{\text{Realized Force / Torque}} \rightarrow \boxed{\text{Spacecraft Motion}}. \]

Actuator modelling therefore forms the physical bridge between control mathematics and spacecraft dynamics.

What Is an Actuator Model?

An actuator model describes how a commanded input produces a physical effect on the spacecraft.

A simple ideal actuator might appear to obey

\[ u \rightarrow F \]

or

\[ u \rightarrow \tau. \]

A real actuator is normally more complicated.

A general command-to-actuation chain may be represented as

\[ \boxed{u_{\mathrm{cmd}}} \rightarrow \boxed{\text{Command Interface}} \rightarrow \boxed{\text{Actuator Dynamics}} \rightarrow \boxed{\text{Constraints / Nonlinearities}} \rightarrow \boxed{\text{Realized Effort}}. \]

The realized physical effort may differ from the requested effort because of effects such as

• saturation,
• minimum command levels,
• minimum impulse bit,
• minimum on-time,
• finite rise and fall time,
• valve delay,
• friction,
• deadband,
• hysteresis,
• backlash,
• rate limits,
• quantization,
• finite bandwidth,
• thermal constraints,
• power constraints,
• mounting and alignment errors,
• calibration errors,
• aging and degradation,
• partial or complete actuator failures.

Consequently,

\[ \boxed{ \mathbf u_{\mathrm{commanded}} \neq \mathbf u_{\mathrm{realized}} } \]

in general.

Understanding this distinction is essential when moving from ideal control equations to realistic spacecraft simulation and flight implementation.

Force, Torque, Impulse, and Momentum

Different actuators influence spacecraft motion through different physical quantities.

Force

A force changes translational momentum:

\[ \boxed{ \mathbf F = \frac{d\mathbf p}{dt} }. \]

For a spacecraft of mass \(m\),

\[ \boxed{ m\mathbf a = \mathbf F }. \]

Torque

A torque changes angular momentum:

\[ \boxed{ \boldsymbol{\tau} = \frac{d\mathbf H}{dt} }. \]

A force acting at position \(\mathbf r\) relative to the spacecraft center of mass produces the moment

\[ \boxed{ \boldsymbol{\tau} = \mathbf r \times \mathbf F }. \]

Impulse

For finite-duration force application, the linear impulse is

\[ \boxed{ \mathbf J = \int_{t_0}^{t_f} \mathbf F(t)\,dt }. \]

If spacecraft mass remains approximately constant over the firing,

\[ \boxed{ \Delta\mathbf v \approx \frac{\mathbf J}{m} }. \]

Angular Impulse

Similarly,

\[ \boxed{ \Delta\mathbf H = \int_{t_0}^{t_f} \boldsymbol{\tau}(t)\,dt }. \]

These relationships connect actuator behaviour directly to spacecraft translational and rotational dynamics.

Actuator Coordinate Frames and Geometry

Actuator capability depends not only on actuator magnitude, but also on where the actuator is mounted and in which direction it acts.

For actuator \(i\), define

\[ \mathbf r_i \]

as its position relative to the spacecraft center of mass, and

\[ \hat{\mathbf d}_i \]

as its actuation direction expressed in a common spacecraft frame.

For a thruster,

\[ \boxed{ \mathbf F_i = F_i\hat{\mathbf d}_i }. \]

The corresponding torque is

\[ \boxed{ \boldsymbol{\tau}_i = \mathbf r_i \times \mathbf F_i }. \]

Therefore two identical thrusters can produce different spacecraft effects because of differences in

• mounting position,
• direction,
• cant angle,
• lever arm,
• reference frame,
• spacecraft center-of-mass location.

Actuator-Frame to Body-Frame Transformation

If an actuator direction is defined in the local actuator frame \(A_i\), it may be transformed into the spacecraft body frame \(B\) according to

\[ \boxed{ \hat{\mathbf d}_i^{B} = \mathbf C_{A_i}^{B} \hat{\mathbf d}_i^{A_i} }. \]

Coordinate-frame consistency is therefore as important for actuator modelling as it is for sensor modelling.

Actuator Effectiveness Matrix

Individual actuator contributions can be collected into a compact mathematical representation.

Define the generalized spacecraft wrench

\[ \boxed{ \mathbf w = \begin{bmatrix} \mathbf F\\ \boldsymbol{\tau} \end{bmatrix} }. \]

Equivalently,

\[ \mathbf w = \begin{bmatrix} F_x\\ F_y\\ F_z\\ \tau_x\\ \tau_y\\ \tau_z \end{bmatrix}. \]

For a collection of actuators,

\[ \boxed{ \mathbf w = \mathbf B_a\mathbf u }. \]

Here,

• \(\mathbf u\) is the actuator-command or actuator-effort vector,
• \(\mathbf B_a\) is the actuator effectiveness matrix,
• \(\mathbf w\) is the resulting generalized spacecraft force and torque.

This relationship connects

\[ \boxed{\text{Actuator Geometry}} + \boxed{\text{Actuator Strength}} \rightarrow \boxed{\text{Spacecraft Force / Torque Capability}}. \]

For a thruster system,

\[ \boxed{ \begin{bmatrix} \mathbf F\\ \boldsymbol{\tau} \end{bmatrix} = \mathbf B_T\mathbf u_T }. \]

The effectiveness-matrix concept later becomes the mathematical foundation for control authority analysis and control allocation.

Different Actuators Produce Control Differently

Spacecraft actuators do not all generate control effort using the same physical mechanism.

Reaction Wheels

Reaction wheels exchange angular momentum internally with the spacecraft.

Approximately,

\[ \boxed{ \boldsymbol{\tau}_{body} = -\dot{\mathbf H}_w }. \]

Reaction wheels provide accurate and continuously variable torque, but they possess finite momentum-storage capability.

Saturation occurs as

\[ |H_w| \rightarrow H_{w,\max}. \]

Momentum unloading or momentum dumping may then be required.

Control-Moment Gyroscopes

Control-moment gyroscopes generate spacecraft torque by changing the direction of stored angular momentum.

CMGs can provide substantial torque, but their achievable control depends strongly on gimbal geometry and may be reduced near kinematic singularities.

Magnetorquers

Magnetorquers generate a commanded magnetic dipole moment \(\mathbf m\).

Interaction with the local geomagnetic field produces torque:

\[ \boxed{ \boldsymbol{\tau} = \mathbf m \times \mathbf B }. \]

Because this torque is perpendicular to the instantaneous geomagnetic field vector, magnetorquer control authority is direction-dependent.

Thrusters

Thrusters generate external forces and can therefore affect both spacecraft translation and rotation.

For thruster \(i\),

\[ \boxed{ \mathbf F_i = F_i\hat{\mathbf d}_i } \]

and

\[ \boxed{ \boldsymbol{\tau}_i = \mathbf r_i \times F_i\hat{\mathbf d}_i }. \]

Thruster modelling also requires consideration of

• specific impulse,
• propellant consumption,
• minimum impulse bit,
• minimum on-time,
• valve dynamics,
• rise and fall time,
• thrust magnitude uncertainty,
• misalignment,
• timing uncertainty,
• plume constraints,
• simultaneous firing restrictions.

Actuator physics therefore directly determines what control actions are physically possible.

Commanded Wrench and Realized Wrench

Suppose the spacecraft controller requests the generalized wrench

\[ \mathbf w_c = \begin{bmatrix} \mathbf F_c\\ \boldsymbol{\tau}_c \end{bmatrix}. \]

The actuator system attempts to generate this wrench through

\[ \mathbf w_a = \mathbf B_a\mathbf u. \]

Ideally,

\[ \mathbf w_a = \mathbf w_c. \]

In practice,

\[ \boxed{ \mathbf w_a \neq \mathbf w_c } \]

may occur because the requested wrench exceeds actuator capability or violates physical or operational constraints.

A more realistic actuation chain is therefore

\[ \boxed{\mathbf w_c} \rightarrow \boxed{\mathbf u_c} \rightarrow \boxed{\text{Actuator Dynamics}} \rightarrow \boxed{\text{Constraints}} \rightarrow \boxed{\mathbf u_a} \rightarrow \boxed{\mathbf w_a}. \]

This distinction becomes particularly important during precision pointing, rendezvous and proximity operations, docking, landing, formation flying, and other control-sensitive spacecraft manoeuvres.

What Is Control Authority?

Having an actuator onboard does not mean that the spacecraft can generate an arbitrary force or torque.

Control authority describes the set and magnitude of control actions that the installed actuator system can physically produce.

Consider

\[ \mathbf w = \mathbf B_a\mathbf u \]

with actuator limits

\[ \mathbf u_{\min} \leq \mathbf u \leq \mathbf u_{\max}. \]

The corresponding achievable wrench set may be represented as

\[ \boxed{ \mathcal W = \left\{ \mathbf B_a\mathbf u \;\middle|\; \mathbf u_{\min} \leq \mathbf u \leq \mathbf u_{\max} \right\} }. \]

This represents the spacecraft's reachable force/torque set.

The central engineering question is therefore not only

\[ \boxed{ \text{What force or torque do we want?} } \]

but also

\[ \boxed{ \text{Can the installed actuator system physically generate it?} }. \]

Axis-Wise and Directional Control Authority

A spacecraft does not necessarily possess identical control capability in every direction.

For translational control,

\[ F_{x,\max} \neq F_{y,\max} \neq F_{z,\max}. \]

Similarly, rotational authority may satisfy

\[ \tau_{x,\max} \neq \tau_{y,\max} \neq \tau_{z,\max}. \]

Even positive and negative control authority along the same axis may differ:

\[ \boxed{ F_{x,\max}^{+} \neq \left|F_{x,\max}^{-}\right| }. \]

Directional asymmetry may result from

• asymmetric actuator placement,
• different actuator sizes,
• plume restrictions,
• center-of-mass offsets,
• failed actuators,
• mission-specific firing restrictions.

Control authority must therefore be evaluated in both magnitude and direction.

Coupled Force–Torque Authority

A spacecraft actuator may generate multiple physical effects simultaneously.

For example, an off-center thruster intended to provide translation also produces torque:

\[ \mathbf F_i \quad \Rightarrow \quad \boldsymbol{\tau}_i = \mathbf r_i \times \mathbf F_i. \]

Likewise, a combination of thrusters chosen to generate a torque may introduce unwanted translational force.

Therefore,

\[ \boxed{ \text{Translation and Rotation May Be Coupled} }. \]

Actuation analysis should therefore consider the complete six-dimensional spacecraft wrench:

\[ \boxed{ \mathbf w = \begin{bmatrix} F_x & F_y & F_z & \tau_x & \tau_y & \tau_z \end{bmatrix}^{T} }. \]

This coupling becomes one of the central problems later in control allocation.

Rank, Singular Values, and Weak Control Directions

The actuator effectiveness matrix contains important information about the capability of the complete actuation system.

For

\[ \mathbf w = \mathbf B_a\mathbf u, \]

the quantity

\[ \operatorname{rank}(\mathbf B_a) \]

indicates the number of independent wrench directions that can be generated locally by the actuator configuration.

If the rank is insufficient, some desired force or torque directions cannot be generated independently.

Singular Value Decomposition

Rank alone does not describe how strong the available control directions are.

The singular value decomposition is

\[ \boxed{ \mathbf B_a = \mathbf U \boldsymbol{\Sigma} \mathbf V^{T} }. \]

The singular values contained in \(\boldsymbol{\Sigma}\) provide information about actuation strength in different directions.

Small singular values indicate weakly actuated directions.

Connection with Observability

This creates a useful conceptual symmetry between estimation and actuation.

On the estimation side,

\[ \boxed{ \text{Weak Measurement Sensitivity} \rightarrow \text{Weak Observability / Estimability} }. \]

On the actuation side,

\[ \boxed{ \text{Weak Actuator Effectiveness} \rightarrow \text{Weak Control Authority} }. \]

Sensor geometry helps determine what the spacecraft can know, while actuator geometry helps determine what the spacecraft can do.

Redundant, Underactuated, and Overactuated Systems

The number and geometry of installed actuators determine the structure of the actuation problem.

Underactuated Systems

An underactuated system cannot independently command every desired control degree of freedom.

Fully Actuated Systems

A fully actuated system possesses sufficient independent actuation for the control space required by the mission.

Overactuated Systems

An overactuated system contains more actuator degrees of freedom than are strictly required to generate the desired wrench.

Redundancy may provide

• alternative actuator combinations,
• reduced individual actuator loading,
• fault tolerance,
• improved directional authority,
• flexibility in satisfying plume constraints,
• flexibility in satisfying power or thermal constraints.

However, redundancy creates another important problem:

\[ \boxed{ \text{Desired Wrench} \rightarrow \text{Which Combination of Actuators?} }. \]

This problem is addressed later through control allocation and actuator commanding.

Actuator Constraints and Nonlinearities

The ideal relation

\[ \mathbf w = \mathbf B_a\mathbf u \]

is only the starting point for practical actuator modelling.

Real systems may be constrained by

\[ \boxed{ \mathbf u_{\min} \leq \mathbf u \leq \mathbf u_{\max} }. \]

Additional limitations may include

• actuator saturation,
• deadband,
• rate limits,
• quantization,
• minimum commandable effort,
• minimum impulse bit,
• hysteresis,
• backlash,
• latency,
• command-to-actuation delay,
• finite actuator bandwidth,
• thermal limits,
• duty-cycle limits,
• power limits,
• propellant limits.

Thruster Example

For a pulsed thruster,

\[ t_{\mathrm{on}} < t_{\min} \]

may correspond to a command that cannot be reliably executed.

Reaction-Wheel Example

For a reaction wheel,

\[ |H_w| \leq H_{w,\max} \]

limits the momentum that can be stored.

Control feasibility therefore depends not only on actuator geometry, but also on the dynamic and operational limits of the hardware.

Actuator Uncertainty

Even when an actuator command is physically feasible, the resulting force or torque is not known perfectly.

Magnitude Uncertainty

A simple thrust-magnitude error model may be written as

\[ \boxed{ F_{\mathrm{actual}} = F_{\mathrm{cmd}} \left(1+\epsilon_F\right) }. \]

Direction Uncertainty

Mounting and alignment errors may cause

\[ \boxed{ \hat{\mathbf d}_{\mathrm{actual}} \neq \hat{\mathbf d}_{\mathrm{nominal}} }. \]

Timing Uncertainty

A firing or actuator response may occur at

\[ \boxed{ t_{\mathrm{actual}} = t_{\mathrm{commanded}} + \Delta t }. \]

Practical uncertainty sources may include

• magnitude uncertainty,
• direction or misalignment uncertainty,
• timing uncertainty,
• scale-factor error,
• bias,
• actuator-to-actuator variation,
• aging and degradation.

These uncertainties can propagate into spacecraft position, velocity, attitude, and angular-rate errors.

Actuator Failure and Degraded Control Authority

Actuator failure does more than remove one hardware component.

It can change the geometry and achievable wrench set of the complete spacecraft.

Nominally,

\[ \mathbf w = \mathbf B_a\mathbf u. \]

Following an actuator failure, the effective system becomes

\[ \boxed{ \mathbf w = \mathbf B_{a,\mathrm{degraded}} \mathbf u }. \]

The degraded actuator matrix may produce

• reduced maximum force or torque,
• weaker control directions,
• reduced redundancy,
• changed reachable wrench sets,
• reduced matrix rank.

Fault analysis therefore asks

\[ \boxed{ \text{Can the spacecraft still achieve the required control objectives after an actuator failure?} }. \]

Actuator Models in Simulation

Different levels of actuator-model fidelity are useful during different stages of GNC development.

Static Algebraic Model

A simple model may use

\[ \boxed{ \mathbf w = \mathbf B_a\mathbf u }. \]

This representation is useful for initial controller and allocator development.

Dynamic Actuator Model

A more complete actuator model may contain internal states:

\[ \boxed{ \dot{\mathbf x}_a = f_a(\mathbf x_a,\mathbf u_c,t) } \]

with actuator output

\[ \boxed{ \mathbf y_a = h_a(\mathbf x_a,\mathbf u_c,t) }. \]

Higher-Fidelity Models

Higher-fidelity models may include

• motor dynamics,
• wheel friction,
• gimbal dynamics,
• valve dynamics,
• pressure variation,
• thrust transients,
• minimum impulse bit,
• saturation,
• thermal effects,
• command delays.

Actuator models may therefore be represented using continuous-time, discrete-time, transfer-function, state-space, or nonlinear simulation models depending on the required fidelity.

Actuator Modelling and Verification

An actuator model should be verified progressively as the GNC system moves from mathematical analysis toward hardware implementation.

A typical development sequence is

\[ \boxed{\text{Analytical Model}} \]

\[ \downarrow \]

\[ \boxed{\text{Numerical / Simulation Model}} \]

\[ \downarrow \]

\[ \boxed{\text{Software-in-the-Loop}} \]

\[ \downarrow \]

\[ \boxed{\text{Hardware-in-the-Loop}} \]

\[ \downarrow \]

\[ \boxed{\text{Integrated Vehicle Testing}}. \]

Monte Carlo Verification

Monte Carlo analysis can vary uncertain actuator parameters such as

\[ F, \qquad \hat{\mathbf d}, \qquad t_{\mathrm{delay}}, \qquad t_{\mathrm{rise}}, \qquad J_{\min}, \qquad H_{w,\max} \]

to determine how actuator uncertainty influences closed-loop spacecraft performance.

Fault Injection

Actuator fault injection can be used to simulate

• actuator loss,
• reduced effectiveness,
• stuck actuators,
• delayed response,
• incorrect magnitude,
• incorrect direction,
• intermittent failure.

This allows degraded control authority and fault tolerance to be evaluated before flight.

Complete Actuation Architecture

The concepts introduced above can be combined into a general spacecraft actuation architecture.

Complete Signal Flow

\[ \boxed{\text{Desired Spacecraft Motion}} \]

\[ \downarrow \]

\[ \boxed{\text{Controller}} \]

\[ \downarrow \]

\[ \boxed{\text{Desired Force / Torque}} \]

\[ \downarrow \]

\[ \boxed{\text{Control Allocation}} \]

\[ \downarrow \]

\[ \boxed{\text{Actuator Commands}} \]

\[ \downarrow \]

\[ \boxed{\text{Actuator Dynamics}} \]

\[ + \]

\[ \boxed{\text{Constraints}} + \boxed{\text{Uncertainty}} \]

\[ \downarrow \]

\[ \boxed{\text{Realized Force / Torque / Momentum / Impulse}} \]

\[ \downarrow \]

\[ \boxed{\text{Spacecraft Dynamics}} \]

\[ \downarrow \]

\[ \boxed{\text{Changed Spacecraft State}}. \]

This architecture provides the connection between actuator physics, control authority, control algorithms, control allocation, and spacecraft dynamics.

Connection Between Sections 4.1, 4.2, 4.3, and 4.4

The first four sections form a progression from sensing the physical spacecraft to estimating its state and finally acting on the physical spacecraft.

Section 4.1 — Sensor Models

Section 4.1 asks:

How does the physical spacecraft state become a sensor measurement, and what imperfections are introduced?

\[ \boxed{\text{True Physical State}} \rightarrow \boxed{\text{Sensor Model}} \rightarrow \boxed{\text{Measurement}}. \]

Section 4.2 — Sensor Fusion

Section 4.2 asks:

How are multiple imperfect measurements combined with spacecraft models to obtain a navigation solution?

\[ \boxed{\text{Measurements}} + \boxed{\text{Spacecraft Model}} \rightarrow \boxed{\text{Sensor Fusion}} \rightarrow \boxed{\hat{\mathbf x},\mathbf P}. \]

Section 4.3 — Estimation Algorithms

Section 4.3 asks:

How are unknown states and parameters mathematically reconstructed from observations?

\[ \boxed{\text{Observations}} + \boxed{\text{Dynamic / Measurement Models}} + \boxed{\text{Uncertainty}} \rightarrow \boxed{\text{Estimation Algorithm}} \rightarrow \boxed{\hat{\mathbf x},\hat{\boldsymbol{\theta}}}. \]

Section 4.4 — Actuator Models and Control Authority

Section 4.4 asks:

How does a commanded control action become a physically realizable spacecraft force, torque, momentum exchange, or impulse, and what actions can the installed actuator system actually produce?

\[ \boxed{\text{Desired Control Effort}} \rightarrow \boxed{\text{Actuator System}} \rightarrow \boxed{\text{Realized Physical Effort}}. \]

Information Side and Action Side of GNC

Sections 4.1 to 4.4 establish two complementary directions of the GNC loop.

The information side is

\[ \boxed{\text{Physical World}} \rightarrow \boxed{\text{Sensors}} \rightarrow \boxed{\text{Estimation}}. \]

The action side is

\[ \boxed{\text{Control Decision}} \rightarrow \boxed{\text{Actuators}} \rightarrow \boxed{\text{Physical World}}. \]

Together they form the physical and computational foundation of the closed-loop spacecraft GNC architecture.

Explore Actuator Models and Control Authority

Actuator modelling spans two closely related layers: the physical behaviour of individual actuator technologies and the control capability of the complete spacecraft actuator system.

Section 4.4 is therefore divided into two major parts.

Part A — Topic Overview

Actuator Fundamentals

4.4.1–4.4.4

Actuator role in the GNC loop; force, torque, impulse and momentum; coordinate frames and geometry; actuator effectiveness matrices.

Reaction Wheels

4.4.5–4.4.12

Reaction-wheel fundamentals, torque and momentum exchange, wheel-speed dynamics, momentum storage, saturation, friction, assemblies, geometry, torque authority, and momentum dumping requirements.

Control-Moment Gyroscopes

4.4.13–4.4.17

CMG fundamentals, torque generation, steering geometry, singularities, and control authority.

Magnetorquers

4.4.18–4.4.23

Magnetic dipole generation, magnetic torque, geomagnetic-field dependence, authority limitations, and momentum dumping.

Thrusters

4.4.24–4.4.40

Thruster force and torque, position and direction vectors, cant angles, specific impulse, propellant use, minimum impulse bit, minimum on-time, valve dynamics, rise and fall time, thrust magnitude error, misalignment, timing uncertainty, plume constraints, simultaneous firings, translational authority, rotational authority, and thruster configuration matrices.

Other Actuator Technologies

4.4.41–4.4.46

Cold-gas thrusters, chemical thrusters, electric propulsion actuators, gimballed thrusters, atmospheric control surfaces, and thrust-vector-control systems.

Part B — Topic Overview

Control Authority and Feasibility

4.4.47–4.4.60

Definition of control authority, force authority, torque authority, axis-wise authority, positive and negative direction authority, coupled force–torque authority, effectiveness-matrix rank, singular values, weak control directions, redundant actuation, underactuated and overactuated systems, failed-actuator authority, degraded authority, and reachable wrench sets.

Actuator Constraints and Nonlinearities

4.4.61–4.4.75

Saturation, deadband, rate limits, quantization, minimum commandable effort, minimum impulse bit, hysteresis, backlash, latency, command-to-actuation delay, bandwidth, thermal constraints, duty-cycle constraints, power constraints, and propellant constraints.

Actuator Uncertainty and Failure

4.4.76–4.4.84

Magnitude uncertainty, direction and misalignment uncertainty, timing uncertainty, scale-factor error, bias, actuator-to-actuator variation, aging, degradation, failure modes, and actuator fault models.

Actuator Modelling and Verification

4.4.85–4.4.96

Continuous-time models, discrete-time models, transfer-function models, state-space models, nonlinear models, Simulink actuator modelling, actuator command interfaces, software-in-the-loop verification, hardware-in-the-loop verification, Monte Carlo actuator uncertainty, actuator fault injection, and complete actuation architecture.

Learning Progression Through Section 4.4

The complete section follows the progression

\[ \boxed{\text{What the Actuator Is}} \]

\[ \downarrow \]

\[ \boxed{\text{How It Produces Force / Torque}} \]

\[ \downarrow \]

\[ \boxed{\text{How Actuator Geometry Combines}} \]

\[ \downarrow \]

\[ \boxed{\text{What Total Control Authority Is Available}} \]

\[ \downarrow \]

\[ \boxed{\text{What Constraints Reduce That Authority}} \]

\[ \downarrow \]

\[ \boxed{\text{How Uncertainty and Failure Affect It}} \]

\[ \downarrow \]

\[ \boxed{\text{How the Complete Actuation Model Is Verified}}. \]

This provides the foundation required before developing the later sections on control algorithms, control allocation, actuator commanding, and complete closed-loop GNC implementation.