8.8 Control

authored by Dr. Zhu Contributions from Dr. Manchester

Control systems drive the spacecraft toward a desired attitude. There are two different flavors of control: passive and active. Passive control systems act upon the system without any sensory input and take advantage of physics! Some examples of passive control systems are passive magnetic damping and gravity booms discussed earlier in configurations. Active control systems take an attitude estimate, calculate control input, and actuators execute the control input to drive the spacecraft to the desired attitude. This section will discuss the available technology and algorithms to control spacecraft attitude.

Actuators

Magnetic Torquers

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Magnetic torquing is similar to passive magnetic damping in that both types of control use a magnetic field. Magnetic torquers differ from passive magnetic damping in that they can change the magnetic strength of their electromagnetic coils, instead of relying on the constant strength of a permanent magnet. The electromagnetic coil’s field is controlled by switching current flow through the coils. The magnetic dipole generated by the magnetorquer is expressed by the formula:

where n is the number of turns of the wire, I is the current provided, and A is the vector area of the coil. The dipole interacts with the magnetic field generating a torque:

where m is the magnetic dipole vector, B is the magnetic field vector (for a spacecraft it is the Earth’s magnetic field vector), and τ is the generated torque vector.

Magnetorquers are commonly used for coarse attitude control and to desaturate angular momentum build-up of the satellite, particularly for dumping momentum in reaction wheels or control moment gyroscopes. They have often been used in Low Earth Orbit (LEO) satellites. They are useful for initial acquisition maneuvers, like detumbling and nadir-pointing.

Reaction Wheels

Reaction wheels are the most common actuator for active control. They’re highly reactive and offer continuous feedback control. Reaction wheels offer internal torque only, which means that the system still needs an external torque source to dump momentum and/or desaturate the accumulated momentum in the wheels. The control logic is simple for reaction wheels that are mounted in independent (orthogonal) axes but can get trickier with reaction wheels that offer redundancy in a configuration.

Reaction wheels create torque on the spacecraft by creating equal but opposite torques on the reaction wheels, which are flywheels on motors. For three axes of torque, three wheels are necessary, like the image on the left. Usually, four wheels are used for redundancy; the subsequent controller math needs to account for the wheel speed biasing equation. Static & dynamic imbalances can induce vibrations so we suggest mounting reaction wheels on isolators. In an implementation, reaction wheels are usually operated around some nominal spin rate to avoid stiction effects.

Control Moment Gyroscopes

^{Control Moment Gyro Simulator. Video by DSSL Technion}

Control moment gyroscopes are very much like reaction wheels but offer an additional degree of freedom by mounting the reaction wheel onto a gimbal. Instead of producing torque by spinning up the reaction wheel or varying the rotor speed, a control moment gyroscope produces torque by tilting the rotor’s spin axis without necessarily changing its spin speed. CMGs are also far more power-efficient. For a few hundred watts and about 100 kg of mass, large CMGs have produced thousands of newton meters of torque. A reaction wheel of similar capability would require megawatts of power. CMGs require more mass and volume than a reaction wheel to fit in the extra gimbal motor and structural supports.

CMGs come in different design varieties: single-gimbal, dual-gimbal, and variable-speed. The most effective CMGs include only a single gimbal. When the gimbal of such a CMG rotates, the change in direction of the rotor’s angular momentum represents a torque that reacts onto the body to which the CMG is mounted, e.g. a spacecraft. A dual-gimbal  CMG includes two gimbals per rotor. As an actuator, it is more versatile than a single-gimbal CMG because it is capable of pointing the rotor’s angular momentum vector in any direction. However, the torque generated by one gimbal’s motion must often be reacted by the other gimbal on its way to the spacecraft, requiring more power for a given torque than a single-gimbal CMG. Most CMGs hold rotor speed constant using relatively small motors to offset changes due to dynamic coupling and non-conservative effects. Some academic research has focused on the possibility of increasing and decreasing rotor speed while the CMG gimbals. Variable-speed CMGs (VSCMGs) offer few practical advantages when considering actuation capability because the output torque from the rotor is typically much smaller than that caused by the gimbal motion. The ISS employs four double-gimbaled CMGs.

Despite their extreme power efficiency, gimbal motion can lead to relative orientations that produce no usable output torque along certain directions. These orientations are known as singularities and are related to the kinematics of robotic systems that encounter limits on the end-effector velocities due to certain joint alignments. Avoiding these singularities is naturally of great interest, and several techniques have been proposed. David Bailey and others have argued (in patents and in academic publications) that merely avoiding the “divide by zero” error that is associated with these singularities is sufficient.

Thrusters

Thrusters or jets can be used to control attitude but at the cost of consuming fuel. Thrusters use consumables, such as Cold Gas (Freon, N2) or Hydrazine (N2H4), that must be toggled on or off. Continuous and proportional control is usually not feasible but can be closely replicated with pulse width modulation. Thrusters are fast and powerful attitude control systems. At least 6 thrusters are necessary to control all three degrees of freedom, as thrusters can only push on the spacecraft, not pull. Thrusters contribute dynamics that are coupled in attitude and translation. Redundancy is usually required, which makes the system more complex and expensive. Like magnetorquers, thrusters may be used to “unload” accumulated angular momentum on reaction-wheel-controlled spacecraft.

_{Mercury RCS Testing by Mark Gray}

Actuator Design Process and Drivers

• Much like the sensor selection discussion, the control actuators must be chosen to fulfill attitude control requirements in up to 3 degrees of freedom. That means at least one or a combined set of the following minimal sets
  • 3 orthogonally mounted magnetorquers, or
  • 3 orthogonally mounted reaction wheels, or
  • 3 orthogonally mounted single-gimbal control moment gyroscopes, or
  • 3 antiparallel pairs of orthogonally mounted thrusters
• Sources of internal torque, like reaction wheels and control moment gyroscopes, only use electricity and do not add additional angular momentum to the spacecraft system. This means that momentum control systems cannot dump the spacecraft’s accumulated angular momentum. But! These sources don’t rely on the space environment or rely on consumables. These technologies guarantee angular momentum as long as there is electricity.
• Sources of external torque, like magnetorquers and thrusters, use consumables or rely on a specific space environment, like the presence of a magnetic field. But! Being able to interact with the space environment is immensely important to get rid of unwanted motion in a transient way.
• To get the best of both worlds, spacecraft that stay in space typically have one internal torque source and external torque source.
  • For smaller spacecraft, this combination is typically magnetorquers and reaction wheels. If you scroll back up to common configurations, you’ll see that both cubesat ADCS packages use a combination of reaction wheels and magnetorquers.
  • For larger spacecraft, the ISS for example, control moment gyroscopes are the main method of attitude control and thrusters augment or back up the system.
• Magnetorquers and reaction wheels scale down well, but only to a point. Thrusters and control moment gyroscopes scale up well, but also to a point.

Control Algorithms

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The controller’s job is to generate commands for the actuators to make the spacecraft point in the right direction based on mission requirements for accuracy and slew rate. To use the information from sensors and continuously adjust actuator commands, the controller must be smart. It has to know what’s happening and decide what to do next. To do this right, the controller has to keep track of

• What’s happening now
• What may happen in the future
• What happened in the past

Knowing what’s happening now is pretty easy—the controller simply asks the sensors to find the current attitude. It then compares this to the desired attitude. The difference between the measured and desired attitude is the error signal. Based on this error signal, the controller steers in the direction of the proper orientation. This is called feedback control. That is, if the attitude is 10° off, the controller commands a 10° change. This is known as proportional control and is used in some form in virtually all closed-loop control systems.

However, predicting what’s going to happen and remembering what’s happened in the past can be just as important. For example, if you need to stop at a stop sign, you need to know not only where you are, but also how fast you’re going, so you can hit the brakes in time. Similarly, to hit the desired attitude, the spacecraft controller must monitor the attitude rate, as well as the current attitude. For you calculus buffs, you may recognize this rate of change calculation as a derivative. In this case, by knowing the rate of change or “speed” of attitude, the controller can more accurately determine how to command the actuators to achieve better accuracy. This process is called derivative control.

Sometimes we can be more precise by keeping track of how close we’re getting to the desired result. One way to do this is for the controller to monitor the angular difference between the measured and desired attitude, ∆θ. When the spacecraft reaches the desired attitude, this difference, ∆θ, will be zero. If the system stops commanding the actuators at this point, the attitude will immediately begin to drift due to disturbance torques. A really smart controller, however, won’t just look at the instantaneous ∆θ. Instead, it would keep a running tally, summing the ∆θ over time. The result would always be some value other than zero and would tell the controller how much torque to add in a “steady-state” mode to compensate for the disturbance torques. In calculus, this process is called integration, so we call this type of control integral control. Designers use it for highly accurate pointing.

Regardless of the exact scheme used, the controller combines its memory with its current measurements and the ability to predict future behavior to decide how to command the actuators. This section will review a few control schema and derive the input commands.

Detumbling

The act of tumbling and the control of detumbling is very common, especially on small satellites like CubeSats. Even if the spacecraft is using some other method for attitude stabilization, we need to get rid of initial angular momentum. A very simple and effective control law is the “B-dot” or algorithm. The only required hardware to implement this control scheme is a 3-axis magnetometer and 3-axis torque coils. The control policy centers around derivative control: the derivative of , the magnetic field, is proportional to , the angular velocity in the body frame:

Assuming the inertial magnetic field  changes slowly as the spacecraft moves around the Earth, typically there will be a residual spin roughly at the orbit frequency due to

Back to the control policy in the body frame, is conveniently perpendicular to the:

Torque from the coils is given by:

A reasonable control law is:

Where k is a scalar gain. A small k applies gentle control and a large k applies aggressive control. We can also implement “bang-bang” control, which is turning maximum control effort on and off. The control law is:

But this can lead to chattering due to noise in measurements (especially if using finite differencing).

Since varies over the orbit, we’ll eventually be able to zero out all 3 components’ angular momentum . Typically, this takes a few orbits.

Momentum Dumping

Momentum dumping is widely used, for example on geostationary communication satellites. Reaction wheels build up angular momentum over time and eventually saturate. An external torque is needed to get rid of this momentum. Thrusters can be used, but that requires valuable fuel. The typical approach is to keep the reaction wheel attitude controller running to maintain pointing while pulsing torque coils.

Assuming spacecraft is inertially pointing, . We know the angular momentum of the wheels exactly from wheel speeds. The coil moment command is then:

Where k is a scalar gain.

Torque from the torque coils is antiparallel to the spacecraft body’s angular momentum. Averaged over an orbit, we can zero out all components of the body’s angular momentum.

Actuators Jacobians

Recall we wrote out dynamics in terms of internal angular momentum and external torque :

We need mappings between actuators’ commands (reaction wheel torques, thruster forces) and and . It turns out this mapping is always linear.

The Jacobian for thrusters:

While 3 reaction wheels are needed for full control authority, most spacecraft have 4 or more for redundancy. Four wheels can be arranged in various ways to provide higher performance and single-fault tolerance. Some common configurations are:

The Jacobian of reaction wheels is thus:

Where the wheel torque is dictated by the following equation:

And the control input is the derivative of wheel momenta:

To produce the desired torque from either the thruster or reaction wheel configuration, the control policy is:

This is our ultimate solution! We transform the desired torque into a control input using the relevant actuator Jacobian. Note that the pseudoinverse gives a minimum 2-norm solution (usually corresponds to minimum overall power or fuel consumption). Other choices (e.g. 1-norm) are possible.