1. Actuation families: stored momentum vs momentum exchange

Practical spacecraft often combine multiple actuator types. A useful mental model is to separate what happens internally (momentum exchange) from what happens via the environment (momentum dumping).

(A) Momentum exchange (internal)

• Reaction wheels: change wheel speed → change wheel momentum
• CMGs: steer the direction of a large rotor momentum vector using gimbals

(B) Momentum dumping (external torque)

• Magnetorquers: torque from interaction with Earth’s magnetic field
• Thrusters: propellant torque (strong and fast, but consumes fuel)

(C) Passive / environmental use

• Gravity-gradient stabilization, aerodynamic torques, etc. (rarely primary for agile pointing)

Domain D.4 focuses on (A), and how (B) is used to keep (A) usable over long mission time.

2. Reaction wheel systems (precise but limited torque)

A reaction wheel is a motor-driven flywheel. Accelerating the wheel changes its angular momentum; the spacecraft responds with an equal and opposite change to conserve total angular momentum.

2.1 Angular momentum bookkeeping

Let the spacecraft body angular momentum be $J\\,\\omega$ and the total stored wheel momentum be $h_w$ (vector sum over wheels). A compact conservation statement is:

\[ H_{\text{total}} = J\,\omega + h_w. \]

In the ideal “no external torque” case:

\[ \dot{H}_{\text{total}} \approx 0 \;\Rightarrow\; J\,\dot{\omega} \approx -\dot{h}_w. \]

2.2 Wheel torque generation

For a single wheel with inertia $J_w$ and wheel speed $\Omega$, one common model is $h_w = J_w\,\Omega$ along its spin axis. Then the torque delivered to the spacecraft is:

\[ u = -\dot{h}_w = -J_w\,\dot{\Omega}. \]

2.3 Strengths and limits

• Strengths: high pointing smoothness, low jitter (good for imaging), continuous torque within limits.
• Limits: modest torque compared to CMGs, and bounded stored momentum.

\[ \lVert h_w \rVert \le h_{w,\max}. \]

Long-term disturbances push wheels toward saturation (see Section 6).

3. Control Moment Gyros (high torque via precession)

CMGs are designed for agility: rapid slews, fast retargeting, and strong disturbance rejection—especially important for agile imaging spacecraft that repeatedly rotate between targets.

3.1 The essential mechanism

A CMG stores a large rotor momentum vector $h$. Unlike wheels (which mainly change the magnitude), CMGs primarily change the direction of $h$ using gimbals. The spacecraft torque is associated with the momentum-rate:

\[ \tau = \dot{h}. \]

For a gimbaled rotor, a common geometric form is:

\[ \dot{h} = \omega_g \times h, \]

where $\omega_g$ is the gimbal-rate vector.

3.2 Why CMGs are “agile-satellite actuators”

If $\lVert h \rVert$ is large, moderate gimbal rates can generate large torque. CMGs are therefore used when missions require:

• repeated large-angle slews (roll/pitch/yaw)
• rapid settling and frequent retargeting
• high torque authority without continuous thruster use

4. Momentum envelope (what is achievable, not just what is stored)

Both wheels and CMGs have finite capacity. The set of momentum vectors the actuator set can realize is the momentum envelope. The envelope matters because it constrains not only “how much” momentum can be stored, but also “which directions” are practically reachable during maneuvers.

4.1 Reaction wheel envelope

With three orthogonal wheels and independent axis bounds, the reachable momentum set is box-like in momentum space: simple to reason about, but rigid—each axis can saturate independently.

4.2 CMG envelope depends on geometry

CMG envelopes depend on array layout (pyramid, skewed, parallel gimbals, etc.). Two consequences:

• High stored momentum does not guarantee good torque authority in every direction.
• Envelope geometry is tightly linked to singularity behavior (Section 8).

5. Disturbances and why momentum builds up

Even with “perfect” control laws, the environment injects torque. If wheels/CMGs counteract those torques, actuator momentum accumulates (drifts) over time.

Common sources:

• gravity-gradient torque (attitude-dependent)
• solar radiation pressure (geometry-dependent)
• aerodynamic torque (LEO)
• magnetic effects (residual dipole interacting with Earth field)

6. Saturation (the “actuator fills up” failure mode)

6.1 Reaction wheel saturation

When $\lVert h_w \rVert$ approaches its limit, the wheel cannot provide further torque in the required direction without violating speed bounds. Practically:

• control margin shrinks
• authority becomes axis-dependent
• pointing quality degrades (or fails) unless unloading is performed

6.2 CMG “limits” vs CMG singularity

CMGs also face hardware constraints:

• gimbal rate limits
• gimbal angle limits (mechanical stops)
• distribution/geometry limits (not all momentum directions are equally reachable)

7. Momentum unloading (dumping) — returning actuators to a “healthy zone”

Unloading applies an external torque so internal actuators can be driven back toward mid-range operating conditions.

7.1 Magnetorquer unloading

Magnetorquers create a magnetic dipole $m$ and interact with Earth’s field $B$:

\[ \tau = m \times B. \]

• torque is always perpendicular to $B$ (direction is constrained)
• unloading is slower than wheel/CMG control
• works best in LEO and is commonly used as a routine “management mode”

7.2 Thruster unloading

Thrusters provide strong torque quickly but consume propellant. Typical use cases:

• rapid recovery of control margin
• magnetic unloading is insufficient (orbit/geometry/power constraints)
• mission constraints demand immediate detumble or retarget capability

8. CMG singularities (the geometry-driven control bottleneck)

CMGs map gimbal rates into a momentum-rate (and therefore torque) through a configuration-dependent mapping. A common abstract form is:

\[ \dot{h} = A(\delta)\,\dot{\delta}, \]

• $\delta$: vector of gimbal angles
• $\dot{\delta}$: gimbal rate commands
• $A(\delta)$: Jacobian-like mapping dependent on geometry and configuration

A singularity occurs when this mapping loses rank:

\[ \operatorname{rank}(A(\delta)) < 3. \]

Engineering meaning

• some torque directions become unreachable
• required gimbal rates can blow up near the singular set
• tracking becomes unreliable even if the attitude controller is “correct”

9. Steering logic (how we turn a desired torque into gimbal motion)

The attitude controller outputs a desired control torque $u$. CMG steering converts that command into gimbal rates $\dot{\delta}$ while respecting hardware constraints.

9.1 Baseline: pseudoinverse steering

\[ \dot{\delta} = A^{\dagger}u. \]

In many implementations, $A^{\dagger}$ is formed using a generalized inverse such as $A^T(AA^T)^{-1}$ when well-conditioned, with modifications near singularity.

9.2 Singularity-robust steering (regularization)

To avoid numerical blow-up near singular configurations, a damped/regularized inverse is common:

\[ A^* = A^T(AA^T + \lambda I)^{-1}, \]

where $\lambda > 0$ increases as singularity is approached.

9.3 Null-space motion (do two jobs at once)

With redundancy, you can add a component that does not change the commanded momentum-rate:

\[ \dot{\delta} = A^*u + \gamma n, \]

• $n$ is chosen from the null space of $A$ (so it does not affect $\dot{h}$ from the commanded $u$)
• $\gamma$ sets how strongly you bias configuration shaping

9.4 Real actuator constraints

Practical steering must obey bounds such as:

1: gimbal angle bounds 2: gimbal rate bounds 3: torque / slew-rate constraints from attitude controller outputs

\[ |\delta_i| \le \delta_{\max}, \qquad |\dot{\delta}_i| \le \dot{\delta}_{\max}. \]

That is why flight steering implementations include saturation, rate limiting, and configuration bias terms (penalties for approaching stops, damping near singularity, etc.).

10. Slew-rate/torque limiting in the attitude controller (why limiters show up)

High-agility commands must still protect structure, actuators, and sensors. Therefore many spacecraft controllers use saturation and limiter ideas:

• small errors → smooth, approximately linear behavior
• large errors → torque/rate limiting to prevent actuator overdrive and undesirable dynamics

11. Variable-speed CMGs (VSCMGs) — why they matter

Traditional CMGs assume near-constant rotor speed (large stored momentum magnitude). Variable-speed CMGs add an extra control degree of freedom by allowing rotor speed to change.

• can improve momentum shaping and singularity avoidance in some cases
• introduces additional coupling between rotor-speed control and gimbal steering

12. Engineering design checklist

12.1 When reaction wheels are usually enough

• small satellites
• high-precision pointing, modest slew requirements
• tight mass/power budgets
• standard architecture: wheels + magnetorquer unloading

12.2 When CMGs are justified

• fast retargeting / agile imaging
• large spacecraft or high inertia axes
• repeated large-angle slews with limited propellant budget
• willingness to implement steering + singularity management properly

12.3 Always design for momentum management (not just control)

• ensure unloading authority exists (magnetic or thruster)
• define operational modes (fine pointing vs dumping vs maneuver)
• include fault tolerance (wheel failures, gimbal failures, degraded steering)

Summary: What D.4 established

• Reaction wheels generate torque by changing wheel speed (momentum exchange).
• CMGs generate higher torque by steering a large stored momentum vector (gyroscopic precession).
• External disturbances inject momentum continuously, so internal actuators drift toward limits.
• Saturation is inevitable without unloading; unloading uses magnetorquers/thrusters to restore margin.
• CMG singularities are geometry-driven loss of authority: $\operatorname{rank}(A(\delta))<3$.
• Steering logic (pseudoinverse, damping/regularization, null-space bias, plus constraint handling) is central to safe CMG operation.
• For high agility, designs often use torque/rate limiters and may consider variable-speed CMGs for extra flexibility.