Actuator Dynamics

2.4 Actuator Dynamics

Spacecraft attitude actuators convert electrical, mechanical, or chemical energy into control torque. Together with the sensors and control algorithms, they form the “muscles” of the attitude control system. This section summarises the main actuator types, how they generate torque, and the limits that matter for modelling and design.

The four most common actuator families are:

• Reaction wheels for fine pointing and momentum storage.
• Control moment gyros (CMGs) for large, fast slews.
• Thrusters for impulsive or continuous torque and orbit manoeuvres.
• Magnetorquers for low-cost torque in low Earth orbit (LEO).

We conclude with generic actuator limits and simple first-order models that can be used in simulations and control design.

2.4.1 Reaction Wheels

Reaction wheels are motor-driven flywheels mounted inside the spacecraft. When the motor accelerates or decelerates the wheel, an equal and opposite torque is applied to the spacecraft body, changing its angular velocity.

Wheel torque generation

Let $I_w$ be the wheel’s polar moment of inertia and $\omega_w$ its spin rate. The wheel angular momentum is

\[ h_w = I_w \omega_w. \]

A motor torque $\tau_m$ produces the wheel acceleration

\[ \dot{\omega}_w = \frac{\tau_m - \tau_{\text{friction}}}{I_w}, \]

where $\tau_{\text{friction}}$ captures bearing friction and cogging losses. By conservation of angular momentum, the spacecraft body experiences an opposite torque

\[ \tau_{\text{body}} = -\tau_m. \]

Arrays of three or four wheels, oriented along different axes, are used to generate arbitrary control torques in the body frame.

Momentum storage, saturation, and desaturation

Because persistent disturbance torques are balanced by wheel torques, the wheel speeds slowly drift away from their nominal values. The total stored wheel momentum

\[ \mathbf{H}_w = \sum_k I_{w,k}\,\omega_{w,k}\,\hat{\mathbf{e}}_k \]

eventually approaches the design limit set by maximum allowable wheel speed. This is called momentum saturation. Once saturated, the wheels can no longer provide additional control torque in some directions.

To avoid this, the spacecraft periodically performs desaturation (or “momentum dumping”) by using an external actuator such as thrusters or magnetorquers to apply torque while commanding the wheels back toward lower speeds.

2.4.2 Control Moment Gyros (CMGs)

Control Moment Gyros use the gyroscopic coupling of a spinning rotor to generate large control torques. Compared with reaction wheels, CMGs can deliver much higher torque for a given electrical power, making them attractive for agile Earth-observation or crewed-vehicle attitude control.

Gyroscopic torque generation

A CMG consists of a rotor spinning with constant angular momentum $\mathbf{h} = h\,\hat{\mathbf{s}}$ mounted in a gimbal. When the gimbal rotates at rate $\dot{\gamma}$ about the axis $\hat{\mathbf{g}}$, the rotor spin axis changes direction and the spacecraft experiences the gyroscopic torque

\[ \boldsymbol{\tau}_{\text{CMG}} = \dot{\mathbf{h}} = \boldsymbol{\Omega}_g \times \mathbf{h} = h\,\dot{\gamma}\,\hat{\mathbf{g}} \times \hat{\mathbf{s}}. \]

For a CMG cluster, this relationship is often written in matrix form

\[ \boldsymbol{\tau} = \mathbf{A}(\boldsymbol{\gamma})\,\dot{\boldsymbol{\gamma}}, \]

where $\boldsymbol{\gamma}$ collects all gimbal angles and $\mathbf{A}$ is the CMG steering matrix.

Singularities and steering laws

When the steering matrix $\mathbf{A}$ loses rank, some torque directions cannot be produced no matter how the gimbals move. These configurations are called singularities. Near a singularity, very large gimbal rates are required to generate modest torques, which may violate hardware limits.

Practical CMG control uses singularity-robust steering laws, which:

• Command gimbal motions that track the desired torque while
• Internally redistributing momentum (“null motion”) to avoid singular directions.

Detailed steering law design is beyond the scope of these notes, but the existence of singularities is crucial when selecting CMGs for a mission.

2.4.3 Thrusters

Chemical or cold-gas thrusters provide impulsive forces used for both orbit control and attitude control. They are particularly useful for:

• Initial acquisition and detumbling after launch.
• Large attitude slews or momentum dumps.
• Missions without continuous-duty momentum-exchange devices.

Torque from off-center forces

A thruster producing force $\mathbf{F} = F\,\hat{\mathbf{d}}$ at position $\mathbf{r}$ relative to the spacecraft center of mass generates the torque

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

Attitude thrusters are arranged in pairs or quads so that net translation is cancelled while rotation about a desired axis is produced.

Pulse-width modulation and minimum impulse bit

Most thrusters are either fully on or fully off. Fine control is achieved by firing short pulses and varying their duration or repetition rate. The smallest achievable change in angular momentum is the minimum impulse bit (MIB),

\[ \Delta H_{\min} \approx F\,\Delta t\,r, \]

where $\Delta t$ is the minimum valve open time. If the required control torque is smaller than implied by the MIB, the controller must use deadbands or dithering, which limits pointing precision.

Dynamic and modelling considerations

Key non-idealities include:

• Valve opening delays and rise times.
• Uncertainties in thrust magnitude and direction.
• Mass depletion, which slowly shifts the centre of mass and inertia matrix.

In simulations, thrusters are often modelled as ideal impulses plus small stochastic errors to capture these effects.

2.4.4 Magnetorquers

Magnetorquers are current-carrying coils or rods that generate a magnetic dipole moment. In low Earth orbit they interact with Earth’s magnetic field to produce control torque without consuming propellant, making them popular for small satellites.

Magnetic dipole and torque

A coil with $N$ turns, area $A$, and current $I$ has magnetic dipole moment

\[ \mathbf{m} = N\,I\,A\,\hat{\mathbf{n}}, \]

where $\hat{\mathbf{n}}$ is the coil normal. In the geomagnetic field $\mathbf{B}$ the torque is

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

Because this torque is always perpendicular to $\mathbf{B}$, there is no control authority along the field direction. The system is therefore under-actuated at any instant and attitude control relies on the time-varying orientation of $\mathbf{B}$ along the orbit.

Orbit dependence and limitations

The magnitude of $\mathbf{B}$ is strongest at low altitudes and high latitudes. As the orbit altitude increases, magnetic torques weaken rapidly and eventually become negligible. Additional limitations include:

• Limited torque authority compared with wheels or thrusters.
• Coupling between attitude and orbit: local field direction changes along the orbit.
• Saturation in dipole ($m_{\max}$) due to current and voltage limits.

Despite these constraints, magnetorquers are excellent for detumbling, wheel desaturation, and coarse Earth-pointing for small LEO spacecraft.

2.4.5 Actuator Limits & Simple Dynamic Models

Regardless of the technology, all actuators must obey physical limits. Good control design includes these limits explicitly in simulation models.

Torque and rate saturation

Each actuator has a maximum deliverable torque $\tau_{\max}$ and often a maximum rate of change. We can represent these as

\[ \|\boldsymbol{\tau}\| \le \tau_{\max}, \qquad \|\dot{\boldsymbol{\tau}}\| \le \dot{\tau}_{\max}. \]

Saturation can cause integrator wind-up and loss of stability if not handled carefully in the controller.

Dead zones and thresholds

Many actuators exhibit regions where small commands produce no response:

• Static friction in reaction wheels.
• Minimum impulse bit for thrusters.
• Finite resolution of current drivers for magnetorquers.

These effects are often modelled with dead-zone or quantisation blocks in the control loop.

First-order actuator dynamics

A simple way to capture actuator lag is with a first-order model:

\[ \dot{u}(t) = -\frac{1}{\tau_a} u(t) + \frac{1}{\tau_a} u_{\text{cmd}}(t), \]

where $u_{\text{cmd}}$ is the commanded torque or gimbal rate, $u$ is the actual delivered value, and $\tau_a$ is the actuator time constant. More detailed models can be added for specific hardware (e.g. flexible wheel mounts or valve dynamics).

Coupling with spacecraft dynamics

Actuator dynamics are not independent of the spacecraft:

• Reaction wheels and CMGs add internal angular momentum, altering the effective inertia seen in some manoeuvres.
• Thruster firings change both attitude and orbit and slowly modify the mass properties.
• Magnetorquer effectiveness depends on the current orbit and local magnetic field.

High-fidelity simulations for mission design therefore model the combined, coupled dynamics of the spacecraft and its actuators rather than treating actuators as ideal torque sources.