Editor’s Note: This article deals with complex mathematics and requires background knowledge to understand fully. As prior reading, the following articles and videos are recommended:

• Khan Academy: Introduction to Reference Frames
• Wikipedia: Rotation of Axes in Two Dimensions
• Khan Academy: Dot Product
• Khan Academy: Dot vs. Cross Products
• Khan Academy: Dot and Cross Products Using Unit Vector Notation

Introduction

The objective of the Attitude Determination and Control System (ADCS) is to maintain the spacecraft’s orientation within the desired limit with respect to a reference frame. To design a control system for a satellite, we need to understand the satellite’s dynamics. This article will introduce Hughes’s Vectrix notation, which is widely used in dynamic systems design.

Vectrix notation was invented by Professor Peter Carlisle Hughes (1940-2023) at the University of Toronto Institute of Aerospace Studies (UTIAS). Initially published in 1986 in the reference book ”Spacecraft Attitude Dynamics” [1], vectrix notation has been continuously taught in university courses and used by many professionals in the field.

In astrodynamics, we have to deal with physical phenomena in multiple reference frames. For example, the sun vector can be located in the Earth-Centered Inertial frame (ECI), but for ADCS, the sun vector must be converted into the satellite body frame, which is a non-inertial frame. Dealing with complex dynamic systems across multiple reference frames can be challenging. Vectrix notations offer a superior approach to representing diverse dynamic phenomena in a specific reference frame, which can enable us to demonstrate how changes in satellite motion can be represented in multiple frames [2].

What is a vectrix?

A vector is a useful way to define a quantity having direction as well as magnitude, this is why vectors are used widely in many branches of physics and engineering. Let us consider a vector \underrightarrow{v} in three-dimensional space. In order to give this vector a magnitude and direction, we must define a reference frame.

We can denote by \mathcal{F}_a a reference frame having basis vector \{\underrightarrow{a_1},\underrightarrow{a_2},\underrightarrow{a_3}\} of unit length. Note that only dextral orthonormal sets of basis vectors – that is, right-handed, orthogonal and normalized as the Cartesian coordinate system – can be used. Using this notation, we can express \underrightarrow{v} as:

\begin{equation}
    \underrightarrow{v} = \underrightarrow{\mathcal{F}_a^T} \bold{v} =  v_{1} \underrightarrow{a_{1}} + v_{2} \underrightarrow{a_{2}} + v_{3} \underrightarrow{a_{3}}
\end{equation}

where

\begin{equation}
    \bold{v} \triangleq \begin{bmatrix}
    v_{1} \\ v_{2} \\ v_{3}
    \end{bmatrix}
\end{equation}

is the column matrix containing the components of vector \underrightarrow{v} as resolved in \mathcal{F}. Therefore, \underrightarrow{\mathcal{F}} is the vectrix associate with \mathcal{F} and defined as

\begin{equation}
    \underrightarrow{\mathcal{F}} \triangleq \begin{bmatrix}
    \underrightarrow{a_1} \\ \underrightarrow{a_2} \\ \underrightarrow{a_3}
    \end{bmatrix}
\end{equation}

Note that the result may seem like a 1 \times 1 matrix because of the multiplication of a 1 \times 3 matrix with a 3 \times 1 matrix, but the result is a vector because \{\underrightarrow{a_1}, \underrightarrow{a_2}, \underrightarrow{a_3}\} are basis vectors so a vectrix is a 3 \times 3 matrix.

Therefore, we can also see the following result using the Vectrix notation.

\begin{equation}
    \underrightarrow{v} = \underrightarrow{\mathcal{F}_a^T} \bold{v} = \bold{v^{T}}\underrightarrow{\mathcal{F}_a}
\end{equation}

We note that the representation of \underrightarrow{v} in a given reference frame makes the vector a 3 \times 1 form that we are all familiar with. Vectrix notation explicitly separates reference frame and vector representation.

Dot Products and Cross Products

This section includes an overview of how the dot and cross products are calculated using vectrix notation.

Dot Product

Let us consider two vectors \underrightarrow{u} and \underrightarrow{v} expressed in the same reference frame \mathcal{F}.

\begin{equation}
    \underrightarrow{u} = \underrightarrow{\mathcal{F}^T} \bold{u} = \bold{u^T} \underrightarrow{\mathcal{F}} 
\end{equation}

\begin{equation}
    \underrightarrow{v} = \underrightarrow{\mathcal{F}^T} \bold{v}
\end{equation}

The dot product (inner product) of \underrightarrow{u} and \underrightarrow{v} is denoted by

\begin{equation}
    \underrightarrow{u} \cdot \underrightarrow{v} =  \bold{u^T} \underrightarrow{\mathcal{F}} \cdot \underrightarrow{\mathcal{F}^T} \bold{v} 
\end{equation}

We can see that by matrix multiplication and distributing the dot product operator to the elements of \underrightarrow{\mathcal{F}} \cdot \underrightarrow{\mathcal{F}^T} and invoke the dot products for a pair of basis vectors, we get the identity matrix \bold{I}.

\begin{equation}
    \underrightarrow{\mathcal{F}} \cdot \underrightarrow{\mathcal{F}^T} = \bold{I}
\end{equation}

This also further serves as a formal statement of orthonormality. Finally, we can find the dot product for two vectors.

\begin{equation}
    \underrightarrow{u} \cdot \underrightarrow{v} = \bold{u^T}\bold{v} = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}
\end{equation}

Therefore, we can see the result of the dot product is a scalar.

Cross Product

Let us also consider the cross product of two vectors \underrightarrow{u} and \underrightarrow{v}.

\begin{equation}  
\underrightarrow{u} \times \underrightarrow{v} =  \bold{u^T} \underrightarrow{\mathcal{F}} \times \underrightarrow{\mathcal{F}^T} \bold{v}
\end{equation}

To avoid going into lengthy proof, we will define the cross operator (\cdot)^{\times}. We can apply this operator on a 3 \times 1 \bold{u} as:

\begin{equation}
    \bold{u}^{\times} = \begin{bmatrix}
        0 & - u_{3} & u_{2} \\
        u_{3} & 0 & -u_{1} \\
        -u_{2} & u_{1} & 0
    \end{bmatrix}
\end{equation}

This gives us the cross product of \underrightarrow{u} and \underrightarrow{v} as follows:

\begin{equation}
\underrightarrow{u} \times \underrightarrow{v} = \underrightarrow{\mathcal{F^{T}}} \bold{u}^{\times} \bold{v} =\underrightarrow{\mathcal{F^{T}}} \begin{bmatrix}
        
     u_{2}v_{3}  - v_{2}u_3\\ 
    u_{3}v_{1} - v_{3} u_{1}\\ u_{1}v_{2} - v_{1}u_{2} \end{bmatrix} 
\end{equation}

The cross-product gives us a vector in the \underrightarrow{\mathcal{F}} reference frame.

Rotation Matrices

The vector \underrightarrow{v} can have many equivalent expressions in different reference frames.

\begin{equation}
\underrightarrow{v} = \underrightarrow{\mathcal{F}_a^T} \bold{v_a} = \underrightarrow{\mathcal{F}_b^T} \bold{v_b}
\end{equation}

We can find \bold{v_a}from \bold{v_b} by changing \bold{v_b} from frame \underrightarrow{\mathcal{F}_b} to frame \underrightarrow{\mathcal{F}_a}.

\begin{equation}
\bold{v_a} = \underrightarrow{\mathcal{F}_a} \cdot \underrightarrow{\mathcal{F}_b^{T}} \bold{v_b}
\end{equation}

We can define \bold{C_{ab}}, the rotation matrix from frame \underrightarrow{\mathcal{F}_b} to frame \underrightarrow{\mathcal{F}_a}, as:

\begin{equation}
\bold{C_{ab}} \equiv \underrightarrow{\mathcal{F}_a} \cdot \underrightarrow{\mathcal{F}_b^{T}}
\end{equation}

Therefore,

\begin{equation}
\bold{v_a} = \bold{C_{ab}} \bold{v_{b}}
\end{equation}

We can say that \bold{C_{ab}} describes the orientation or attitude of \mathcal{F}_a with respect to \mathcal{F}_b. We can expand equation 15 to obtain equation 17, in which \theta_{ij} is the angle between \underrightarrow{a_{i}} and \underrightarrow{b_{j}}.

\begin{equation}
    C_{ab} \equiv \begin{bmatrix} a_{1} \cdot b_{1} & a_{1} \cdot b_{2} & a_{1} \cdot b_{3} \\
    a_{2} \cdot b_{1} & a_{2} \cdot b_{2} & a_{2} \cdot b_{3} \\
    a_{3} \cdot b_{1} & a_{3} \cdot b_{2} & a_{3} \cdot b_{3}
    \end{bmatrix} \equiv \begin{bmatrix}
        \cos{\theta_{11}} & \cos{\theta_{12}} & \cos{\theta_{13}} \\
        \cos{\theta_{21}} & \cos{\theta_{22}} & \cos{\theta_{23}} \\
        \cos{\theta_{31}} & \cos{\theta_{32}} & \cos{\theta_{33}}
    \end{bmatrix}
\end{equation}

We can see that the rotation matrix is composed of the directional cosines between the two sets of basis vectors.

Principal Rotations

We can consider a few special cases of \underrightarrow{\mathcal{F}_a} being displaced from \underrightarrow{\mathcal{F}_b} by a rotation through an angle $\theta$ (using the right-hand rule) about one of the principal axes. This is referred to as a principal rotation.

First let’s consider the rotation about \underrightarrow{b_1}. This will keep \underrightarrow{b_1} parallel to \underrightarrow{a_1} and the rotation matrix reduces to:

\begin{equation}
    C_{ab} = \begin{bmatrix} 1 & 0 & 0 \\
    0 & \cos{\theta} & \cos{(\frac{\pi}{2}-\theta)} \\
    0 & \cos{(\frac{\pi}{2}+ \theta)} & \cos{(\theta)} 
    \end{bmatrix} \equiv \begin{bmatrix}
        1 & 0 & 0 \\
        0  & \cos{(\theta)} & \sin{(\theta)} \\ 0 & -\sin{(\theta)} & \cos{(\theta)}
    \end{bmatrix} \equiv C_1
\end{equation}

This defines a principal rotation and, since it is a rotation about the 1-axis, we can denote it by C_1. The other principal rotations about the 2-axis (\underrightarrow{b_2}) and 3-axis (\underrightarrow{b_3}) can likewise be written as:

\begin{equation}
C_{2} \equiv
\begin{bmatrix}
\cos{(\theta)} & 0 & -\sin{(\theta)} \\
        0  & 1 & 0 \\ \sin{(\theta)} & 0 & \cos{(\theta)} \end{bmatrix}, \: \: C_{3} \equiv
\begin{bmatrix}
\cos{(\theta)} & \sin{(\theta)} & 0 \\
        -\sin{(\theta)}  & \cos{(\theta)} & 0 \\ 0 & 0 & 1 \end{bmatrix}
\end{equation}

These are the three principal rotation matrices.

Euler Rotation

Euler’s Theorem provides a way to represent multiple rotational transformations as a single rotation about a unique axis passing through a fixed point, offering a concise and widely used framework for describing rotations in three-dimensional space. Specifically, it states that any general displacement of a rigid body with one point fixed can be represented as a rotation about an axis through that point.

Directional cosine matrices can be expressed as Euler-axis rotations about a unit-normal axis \underrightarrow{a} and a rotation angle \phi.

\begin{equation}
    C(a,\phi) = \cos\phi \bold{I} + (1-\cos \phi)aa^{T} - sin\phi \; a^{\times}
\end{equation}

The rotation axis \underrightarrow{a} can be expressed in either the initial or final frame of the transformation.

Other Properties of Rotation Matrices

The inverse transformation is physically a rotation through the same angle in the opposite direction. The inverse of a rotation matrix is also its transpose, therefore, we can transfer from \underrightarrow{\mathcal{F}_a} back to \underrightarrow{\mathcal{F}_b} by taking the transpose of the rotation matrix.

\begin{equation}
    C_{ab}^{-1} \equiv C_{ba}  \equiv C_{ab}^{T} 
\end{equation}

If we introduce another reference frame \underrightarrow{\mathcal{F}_c} obtained from \underrightarrow{\mathcal{F}_b} via another rotation, then the components of \underrightarrow{v} in this new frame in terms of its components in \underrightarrow{\mathcal{F}_a} are given by:

\begin{equation}
    \bold{v_{c}} = C_{cb}v_{b} = C_{cb}C_{ba}v_{a} = C_{ca}v_{a}
\end{equation}

Where C_{ca} = C_{cb}C_{ba} is the rotation from \underrightarrow{\mathcal{F}_a} to \underrightarrow{\mathcal{F}_c}. Note that the order of rotation is important.

In conclusion, vectrix notation provides a means of representing a vector within a specific reference frame, and through the application of linear algebra, we can define dot and cross products within that same reference frame. Moving forward, the application of vectrix notation is especially valuable when working with dynamic systems in multiple reference frames, such as satellites or spacecraft orbiting Earth or another celestial body.

References

1. P. C. Hughes, “Spacecraft Attitude Dynamics,” Wiley, New York, 1986.
2. G. M. T. D’Eleuterio and G. R. Heppler, Newton’s Second Law And All That. Toronto, Ontario: University of Toronto, 2019.

What is ADCS?

The Attitude Determination and Control System (ADCS) is a fundamental part of every satellite mission. The objective of the ADCS is to maintain the spacecraft’s orientation within the desired limit with respect to a reference frame. The capability to actively control a spacecraft’s attitude empowers engineers to utilize a wide range of payloads.

The technology for ADCS has undergone significant expansion since the launch of Sputnik 1 on October 4, 1957. ADCS has evolved from simple spin gyroscope stabilization methods to complex optimization control strategies utilizing powerful onboard computers. The tremendous growth in ADCS technology has enabled the development of CubeSats capable of performing complex tasks comparable to those of larger satellites.

Most satellites are equipped with some form of ADCS, which can be either passive or active. Active attitude control involves the use of sensors and internal actuators, while passive control relies on natural forces, such as the Earth’s gravity or magnetic field, to orient the satellite. Satellites that rely solely on passive control are typically used for technology demonstrations, while those with active control have a wider range of applications, including earth observation, communication, scientific research, and technology demonstration.

A simple onboard ADCS for active control is shown below in Figure 1.

The design of ADCS for active control satellites typically involves a range of components, including sensors, actuators, control laws, attitude estimators, and space environment models. Additionally, a means of determining the satellite’s position in orbit is necessary; an error in position estimation will translate to errors in attitude estimation. A mission evaluation must be conducted to determine the appropriate hardware and software. This evaluation should also include pointing budget analysis, actuator sizing and simulation in order to ensure optimal performance. Mission evaluation will be discussed further in its own section.

ADCS Sensor Hardware

A satellite’s ADCS relies on onboard sensors to determine its attitude. CubeSats commonly use magnetometers and sun sensors, but gyroscopes and star trackers can be added when higher accuracy is needed.

A three-axis magnetometer is usually used to measure the local magnetic field vector and compare it with an onboard International Geomagnetic Reference Field (IGRF) ephemeris for attitude estimation. Some satellites have their magnetometers on protruding booms to provide magnetic isolation from the spacecraft body, but most satellites can have magnetometers placed internally without any detrimental effects. Satellites rely on fine sun sensors to measure the sun vector and combine it with an onboard Astronomical Almanac Sun Vector (AASV) ephemeris to facilitate attitude estimation. A very small number of satellites utilize solar panels to measure the sun vector, but this approach is only feasible when a sufficient number of solar panels are present, and its accuracy is not as high as that of sun sensors. By combining sun sensor and magnetometer measurements, full three-axis attitude information can be determined.

Other sensors can also be used in place of or in addition to magnetometers and sun sensors, though they are less common. Star Trackers are optical sensors that take a star picture to estimate the satellite’s attitude in all three axes, but require a satellite to be largely stable (i.e. not spinning without control). Earth horizon sensors are available, but they are not utilized as frequently as star trackers due to their limited accuracy. Gyroscopes can be used to more accurately determine the angular rotation of a satellite than measuring the change in the magnetic field with a magnetometer. Adding these advanced sensors can enable more missions with more challenging requirements, and can also increase redundancy if some of them fail during the satellite’s lifetime.

For satellites that require a high degree of pointing accuracy, Global Navigation Satellite System (GNSS) technology is often utilized for determining position, while other satellites relied on ground tracking and orbital propagator software. Although the orbital state is not an attitude parameter per se, the addition of GNSS technology improves the ability to perform attitude determination since sensors such as sun sensors and magnetometers rely on space environment models that are dependent on the satellite’s current position. An error in position estimation can lead to an error in attitude estimation. While incorporating GNSS technology can enhance a satellite’s performance, there are regulations that exist concerning the integration of GNSS systems on board.

ADCS Actuator Hardware

To control the orientation of a spacecraft, the ADCS requires actuators. Magnetic control is a common approach used by many satellites, which involves generating a local magnetic field that interacts with Earth’s magnetic field to control the spacecraft’s attitude. Passive magnetic control using permanent magnets or active magnetic control using magnetorquers are the most common methods.

Reaction wheels are also frequently used in satellites that require precise control. They work by storing momentum and using the principle of conservation of angular momentum to stabilize the spacecraft. Control Moment Gyroscopes (CMGs) use the same principle as reaction wheels for attitude control. CMGs are more commonly used for larger satellites, but there has been a trend toward miniaturizing CMG technology for use in CubeSats. Momentum dumping is necessary for both reaction wheels and CMGs to prevent saturation, which happens when the wheels or gyroscope reach their maximum rotational speed. Momentum dumping is often handled using magnetorquers on CubeSats.

Other actuator technologies, such as propulsion systems, gravity gradient booms, and solar sails, are rarely used due to their complexity and cost.

ADCS Software

Onboard software governs the ADCS, making all decisions related to spacecraft attitude including mode transitions, executing external commands, conducting attitude estimation, and evaluating control laws. Typically, sensor measurements and ground station commands serve as input to the software, while output commands are directed toward actuators.

The flight code for ADCS must incorporate sensor processing software, which receives and filters measurements from sensors. The software then converts the filtered measurements into a measured ephemeris vector. The ephemeris vector is utilized by an attitude estimator to compare with an onboard astrodynamical model to get the spacecraft’s estimated attitude.

Attitude estimation algorithms such as the Attitude Extended Kalman Filter (AEKF), QUEST and TRIAD are commonly used for CubeSats. AEKF is one of the most commonly use attitude estimation algorithms because it can process non-linear errors with low computation cost. Attitude estimation methods such as the AEKF linearize the satellite equations of motion about a discrete trajectory and propagate it through time.

The main purpose of an attitude controller is to evaluate the appropriate control torque. The control’s torque then gets transmitted to the actuators to control the attitude and the body’s angular rates. The choice of control law depends on the mission objective but there is a trade-off between computation time (accuracy) and power consumption. Tuning the control gains (parameters) and implementing them in a discrete fashion is required to achieve optimal operation. The system bandwidth, inherent errors, and actuator saturation limits place constraints on overall performance, emphasizing the importance of selecting appropriate hardware and software for a given mission.

What to Expect From ADCS Articles

In the upcoming articles, I will be discussing the mathematics involved in dynamic system satellites, as well as exploring different reference frames and timekeeping methods. We will also delve into the space environment and operational conditions, including topics such as disturbance torque. Afterwards, I will cover critical aspects of mission planning including how to set ADCS requirements, mission analysis and modes of operation. I will also discuss the design of GNC software and the selection of appropriate hardware. Lastly, we’ll delve into the crucial process of ADCS system integration and testing, ensuring a comprehensive understanding of this complex topic.