# Attitude and Spacecraft Conventions

Creation date: 2019-02-14 22:18:49 Update date: 2019-02-15 13:33:38

Policy: Expert Review

Authority: CCSDS.MOIMS.NAV

OID: 1.3.112.4.57.4

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13 records in registry

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1.3.112.4.57.4
Attitude and Spacecraft Conventions

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Details Status Name Description And Reference Nomenclature Default Units/Type References OID

Provisional

Euler Angles

Euler angles are used to represent a rotation from an initial frame A to a final frame B as a product of three successive rotations about reference unit vectors, the angles of these rotations are the Euler angles. There are 12 possible sequences. The rotation sequence and the rotation angles are specified when providing the final transformation matrix (direction cosine matrix). For example, $$M_{BA}=M_{312}=M_{3}(\phi)M_{1}(\theta)M_{2}(\psi)$$. $$M_{2}(\psi)$$ is the first rotation of $$\psi$$ about the 2nd axis of the initial frame A. $$M_{1}(\theta)$$ is the second rotation of the angle $$\theta$$ about the 1st axis of the intermediate frame. $$M_{3}(\phi)$$ is the 3rd rotation of the angle $$\phi$$ around the 3rd axis of the second intermediate frame, completing the transformation into the final frame B. Mathematically this is written as
$$M_{312} = \left[ \begin{array}{ccc} \mathrm{cos}(\phi) & \mathrm{sin}(\phi) & 0 \\ -\mathrm{sin}(\phi) & \mathrm{cos}(\phi) & 0\\0 & 0 & 1 \end{array} \right] \cdot$$

$$\left[ \begin{array}{ccc} 1 & 0 & 0\\ 0 & \mathrm{cos}(\theta) & \mathrm{sin}(\theta) \\ 0 & -\mathrm{sin}(\theta) & \mathrm{cos}(\theta) \end{array} \right] \left[ \begin{array}{ccc} \mathrm{cos}(\psi) & 0 & -\mathrm{sin}(\psi) \\ 0 & 1 & 0\\ \mathrm{sin}(\psi) & 0 & \mathrm{cos}(\psi) \\ \end{array} \right]$$

$$M_{312}$$ ($$\phi$$,$$\theta$$,$$\psi$$)

deg

1.3.112.4.57.4.5

Provisional

Inertia

The moment of inertia tensor is a symmetric 3x3 matrix. Expressed in a coordinate frame attached to the center of mass of a spacecraft body. The subscript indicates the frame in which the inertia is resolved.
$$I_{SC}=\left[ \begin{array}{ccc} I_{XX} & -I_{XY} & -I_{XZ}\\ -I_{XY} & I_{YY} & -I_{YZ}\\ -I_{XZ} & -I_{YZ} & I_{ZZ}\end{array}\right]$$

$$I_{SC}$$

kg-m$$^{2}$$

1.3.112.4.57.4.7

Provisional

Direction Cosine Matrix

Represents the orientation of a frame B with respect to a frame A, the coordinate transformation from frame A to frame B.

$$M_{BA}$$

N/A

1.3.112.4.57.4.1

Provisional

Spin Axis

The axis about which a spacecraft is spinning, often closely aligned with the major principal axis of inertia.

N/A

N/A

1.3.112.4.57.4.11

Provisional

Quaternion

The first three elements form the vector part of the quaternion, the fourth is a scalar element. Defined as $$Q=\left[ \begin{array} {c} Q1\\ Q2\\ Q3\\ QC\end{array}\right]=\left[\begin{array}{c}e_{1}\mathrm{sin}(\frac{\phi}{2})\\ e_{2}\mathrm{sin}(\frac{\phi}{2})\\ e_{3}\mathrm{sin}(\frac{\phi}{2})\\ \mathrm{cos}(\frac{\phi}{2})\\ \end{array}\right]$$
where $$e_{1}, e_{2}, e_{3}$$ are the three elements of the Euler rotation axis (unit vector) and $$\phi$$ is the Euler rotation angle. The quaternion represents the coordinate transformation from frame A to frame B.

$$Q=\left[\begin{array}{c}Q1\\ Q2\\ Q3\\ QC\end{array}\right]$$

N/A

1.3.112.4.57.4.2

Provisional

Torque Vector

Torque ($$T$$) is the rate of change of angular momentum. For example, in the spacecraft body frame (SC)
$$T_{SC}=\dot{H}^{C}_{SC} + \omega_{SC}\times H^{C}_{SC}$$

$$T_{SC}$$

N-m

1.3.112.4.57.4.9

Provisional

Angular Momentum

Defined for a rigid body as the product of the inertia ($$I$$) and the angular velocity ($$\omega$$). If a spacecraft contains devices which contribute angular momentum they are added to the momentum generated by the spacecraft body. The subscript indicates the coordinate frame in which the momentum is resolved. The superscript indicates what elements are included in the momentum. For example, $$W$$ indicates a reaction wheel, $$B$$ indicates just the spacecraft body, $$C$$ indicates the total system momentum, momentum about the system center of mass.
$$H^{B}_{SC}=I_{SC}\omega_{SC}$$
$$H^{C}_{SC}=I_{SC}\omega_{SC}+H^{W}_{SC}$$
$$H^{W}_{SC}=M_{SC,W}I_{W}\omega_{W}$$
Note that in the 2nd equation above the inertia ($$I_{SC}$$ ) includes the inertia of the wheels transverse to their spin axes, but not the inertia along the spin axes. The second term is the momentum contribution of the wheels along their spin axes only, the momentum along the transverse direction is included in the first term. $$M_{SC,W}$$ in the third equation is the direction cosine matrix (see above) defining the transformation from the wheel frame to the spacecraft body frame, $$I_{W}$$ is the wheel spin axis inertia, and $$\omega_{W}$$ is the angular velocity in the spin direction.

$$H^{B}_{SC}, H^{C}_{SC}, H^{W}_{SC}$$

N-m-s

1.3.112.4.57.4.8

Provisional

Phase

Rotation angle about the Spin Axis.

$$\Phi$$

1.3.112.4.57.4.13

Provisional

Nutation

The angle between a spacecraft principal moment of inertia axis and the angular momentum vector.

$$\theta$$

1.3.112.4.57.4.10

Provisional

Euler Rates

The time derivatives of the Euler angle representation. They represent the rotation rates of the individual transformations represented in the three angle Euler angle rotation sequence. The transformation between Euler rates and angular velocity is not orthogonal. The angles are written in the same order as the Euler angle sequence, with a dot to indicate differentiation.

$$\left[\begin{array}{ccc}\dot{\phi} & \dot{\theta} & \dot{\psi}\end{array}\right ]$$

1.3.112.4.57.4.6

Provisional

Spin Rate

The rotation rate about the Spin Axis.

$$\omega$$

1.3.112.4.57.4.12

Provisional

Angular Velocity

The rotational rate of frame B with respect to frame A. The vector direction is the instantaneous axis of rotation of frame B with respect to frame A and the vector magnitude is the instantaneous rate of this rotation. The subscript indicates the frame in which the angular velocity is resolved. Here SC refers to the spacecraft body frame.

$$\omega_{sc}=\left[\begin{array}{c} \omega_{x}\\ \omega_{y}\\ \omega_{z}\end{array}\right]$$

1.3.112.4.57.4.4

Provisional

Quaternion Derivative

Rate of change of the quaternion. The quaternion evolves in time according to $$\dot{Q}=\frac{1}{2}\Omega(\omega)\mathrm{Q}$$ where
$$\Omega(\omega)=\left[\begin{array}{cccc} 0 & \omega_{z} & -\omega_{y} & \omega_{x}\\-\omega_{z} & 0 & \omega_{x} & \omega_{y}\\ \omega_{y} & -\omega_{x} & 0 & \omega_{z}\\ -\omega_{x} & -\omega_{y} & -\omega_{z} & 0 \end{array}\right]$$
and $$\omega$$ is the angular velocity.

$$\dot{Q}=\left[\begin{array}{c}\dot{Q1}\\ \dot{Q2}\\ \dot{Q3}\\ \dot{QC} \end{array}\right]$$

s$$^{-1}$$

1.3.112.4.57.4.3