How does the Spacecraft Dynamics block work?

In Simulink (Aerospace Blockset) there is a Spacecraft Dynamics block. How does he work? I can’t look inside it (Look Under Mask option is inactive). The description page shows the equations that are used in the block, but the numbers given are clearly not given in full (it seems to me). https://www.mathworks.com/help/aeroblks/getting-started-with-the-spacecraft-dynamics-block.html

In addition, the equations themselves raise some questions, for example, the PDEs are given. Simulink can now solve PDEs in real time?

I want to understand how this block works in order to learn how to write my own to suit my own needs.

Answer 1

[Sam Chak]

Hi Andrew,

The Spacecraft Dynamics block simultaneously models both the Orbital (translational) and Attitude (rotational) dynamics of an orbiting satellite.

This block offers a relatively realistic set of equations of motion for a two-body system, yet it is technically sophisticated enough to include the following features:

• Mass flow rate,
• Rate of change of inertia tensor matrix,
• Six Keplerian orbital elements
• Effect of Oblate ellipsoid ()
• Julian date
• G.A.M.S. Disturbance torques (Gravity gradient , Atmospheric drag , Magnetic field , & Solar radiation pressure )
• Optional Third Body Gravity Effects

I have seen some students struggling to design satisfactory model-based controllers to stabilize the satellite because of the Spacecraft's equations of motion are not directly given. The control designer has to assume that the user-derived model is the exactly same as the internal model of the Spacecraft Dynamics block.

[Andrew Sol]

And what is the best thing to do? I could use theory to write my block and controller, but I don't know a good source for that.

[Sam Chak]

Hi @Andrew Sol

You can start designing simple controllers and running simulations by writing code in MATLAB. Once you believe that you are ready to handle the full dynamics of the satellite, then consider modeling the system using the Spacecraft Dynamics block in Simulink.

%% Satellite Attitude Dynamics
function dx = ode(t, x)
    % Inertias
    Ix  = 4;
    Iy  = 5;
    Iz  = 3.2;
    
    % Control torques
    Tcx = - 1*x(1) - 2*x(4);
    Tcy = - 1*x(2) - 2*x(5);
    Tcz = - 1*x(3) - 2*x(6);
    
    % Equations of motion
        % Kinematics
    dx  = [x(4) + sin(x(1))*tan(x(2))*x(5) + cos(x(1))*tan(x(2))*x(6)
                  cos(x(1))*          x(5) - sin(x(1))*          x(6)
                  sin(x(1))*sec(x(2))*x(5) + cos(x(1))*sec(x(2))*x(6)
        % Dynamics
          (Tcx + (Iy - Iz)*x(2)*x(3))/Ix
          (Tcy + (Iz - Ix)*x(3)*x(1))/Iy
          (Tcz + (Ix - Iy)*x(1)*x(2))/Iz];
end

%% Run the simulation
tspan   = [0, 30];
x0      = deg2rad([3, 6, 9, 0, 0, 0]);
[t, x]  = ode45(@ode, tspan, x0);
plot(t, rad2deg(x(:,1:3)), 'linewidth', 1.5), grid on
xlabel('t'), ylabel('Angle (degree)')
title('Time responses of Euler angles')
legend('\phi, roll', '\theta, pitch', '\psi, yaw', 'fontsize', 12)

[Andrew Sol]

Hi, @Sam Chak

I worked quite a lot in Matlab and Simulink, and all these things are familiar to me. I am not familiar with the theory of astrodynamics and spacecraft dynamics. I think that the options that are available in the Spacecraft Dynamic block are there for a reason - they are of great practical importance.

[Sam Chak]

Hi @Andrew Sol

I understand that you are an experienced user and believe you are ready to utilize the Spacecraft Dynamics block. This block is designed to alleviate the user's burden of researching and modeling the underlying dynamics from scratch. The primary purpose of the Spacecraft Dynamics block is to allow users to simply enter the desired numeric values in the corresponding parameter fields, without the need to manually derive the complex dynamics.

There are numerous books available on Astrodynamics and Spacecraft Dynamics. However, there is no single comprehensive text that is universally considered the "best." Some books may provide the governing equations, but may lack standardized reference tables, such as those detailing Atmospheric density versus Altitude or Magnetic field versus Altitude. This type of supplementary data is often found in dedicated Spacecraft Engineering or Aerospace Engineering Handbooks.

[Andrew Sol]

Hi @Sam Chak

Thank you! I think I've collected some information and will look into it.