Attitude Dynamics Visualization

Loading visualization...

Quaternion (scalar-first)

Direction Cosine Matrix

How to Use This Visualization

Mouse Interaction

Click and drag to rotate Sputnik freely
Any mouse interaction will stop spinning if active

Quaternion Controls

Enter values for the scalar-first quaternion [q₀, q₁, q₂, q₃]
Click "Apply Quaternion" to update attitude
Use "Zero All" or "Reset to Identity" for quick settings

Direction Cosine Matrix

Enter values in the 3x3 DCM grid
Click "Apply DCM" to update attitude
Use the rotation generators for common DCMs
Try different angles for X, Y, and Z axis rotations

Angular Velocity (Spin)

Scroll down in the controls panel to find spin options
Enter values in deg/s for rotation rates around each axis
Use preset buttons for common rotation patterns
Click "Start Spin" to begin continuous rotation
DCM and quaternion values update in real-time during spin

Spacecraft Attitude Dynamics Concepts

Quaternions

Quaternions are a 4-parameter representation of attitude that avoids the singularities found in Euler angles.

$q = q_{0} + q_{1} i + q_{2} j + q_{3} k = [q_{0}, \vec{q}]$

Where $q_{0}$ is the scalar part and $\vec{q} = [q_{1}, q_{2}, q_{3}]$ is the vector part. For rotation, these relate to axis-angle as:

$q_{0} = \cos \frac{θ}{2}, \vec{q} = \hat{n} \sin \frac{θ}{2}$

Where $\hat{n}$ is the rotation axis and $θ$ is the rotation angle.

Direction Cosine Matrix (DCM)

DCM is a 3×3 rotation matrix that transforms vectors from one coordinate system to another.

$C = [\begin{matrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{matrix}]$

The columns represent the body coordinate axes expressed in the inertial frame.

From quaternions, the DCM can be computed as:

$C = (q_{0}^{2} - {\vec{q}}^{T} \vec{q}) I_{3 \times 3} + 2 \vec{q} {\vec{q}}^{T} - 2 q_{0} [\vec{q} \times]$

Torque-Free Motion

In the absence of external torques, angular momentum is conserved. For a rigid body, this means:

$\dot{H} = 0$

Where $H$ is the angular momentum. In body coordinates, this leads to:

$I \dot{ω} + ω \times (I ω) = 0$

Where $I$ is the moment of inertia tensor and $ω$ is the angular velocity.

Constant Angular Velocity

For bodies with special inertia properties or specific initial conditions, $ω$ can remain constant in body coordinates. Our simulation implements:

$\dot{ω} = 0$

This means the spacecraft rotates around a fixed axis with constant speed, even though the orientation changes continuously.

Quaternion Kinematics

For a rotating body with angular velocity $ω$ , the quaternion evolution is given by:

$\dot{q} = \frac{1}{2} q \otimes [\begin{matrix} 0 \\ ω \end{matrix}]$

Where $\otimes$ is quaternion multiplication. This differential equation describes how the orientation changes with time.

DCM Kinematics

The equivalent differential equation for the DCM is:

$\dot{C} = - [ω \times] C$

Where $[ω \times]$ is the skew-symmetric matrix formed from angular velocity:

$[ω \times] = [\begin{matrix} 0 & - ω_{3} & ω_{2} \\ ω_{3} & 0 & - ω_{1} \\ - ω_{2} & ω_{1} & 0 \end{matrix}]$

Euler's Rotation Theorem

Any rotation can be expressed as a single rotation about some axis. This is the principle behind the axis-angle representation.

$R (\hat{n}, θ) = I + \sin θ [\hat{n} \times] + (1 - \cos θ) [\hat{n} \times]^{2}$

This is the foundation of both quaternion and DCM representations.

Body Frame Axes

The red, green, and blue axes in the visualization represent the body-fixed coordinate frame:

X-axis (red) - Body forward direction
Y-axis (green) - Body upward direction
Z-axis (blue) - Body right direction

This right-handed coordinate system rotates with the spacecraft.

Attitude Propagation

For the constant angular velocity case, the quaternion at time $t$ is:

$q (t) = q (t_{0}) \otimes \exp (\frac{1}{2} Ω Δ t)$

Where $Ω$ is a 4×4 matrix formed from angular velocity components and $Δ t = t - t_{0}$ .

Polhode Motion

In torque-free rotation, the angular velocity vector traces a path called a polhode when viewed from the body frame.

For principal axis rotation (as simulated here), the polhode simplifies to a single point, meaning $ω$ is constant in body coordinates.