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Quaternion Class

The Quaternion class provides a compact, singularity-free representation of 3D rotations for spacecraft attitude determination and control.

Quaternion 

Quaternion(w: float, x: float, y: float, z: float)

Represents a quaternion for 3D rotations.

Quaternions provide a compact, singularity-free representation of rotations. The quaternion is stored as [w, x, y, z] where w is the scalar part and [x, y, z] is the vector part.

Parameters:

Name Type Description Default
w float

Scalar component

 required
x float

X component of vector part

 required
y float

Y component of vector part

 required
z float

Z component of vector part

 required
Example 
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import brahe as bh
import numpy as np

# Create identity quaternion
q = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
print(f"Norm: {q.norm()}")

# Create from array
q_vec = np.array([1.0, 0.0, 0.0, 0.0])
q2 = bh.Quaternion.from_vector(q_vec, scalar_first=True)

# Convert to rotation matrix
dcm = q.to_rotation_matrix()

# Quaternion multiplication
q3 = q * q2

# Normalize
q3.normalize()

Initialize instance.

data property 

data: ndarray

Get the quaternion components as a numpy array [w, x, y, z].

Returns:

Type Description
ndarray

numpy.ndarray: 4-element array containing quaternion components

w property 

w: float

Get the scalar (w) component of the quaternion.

Returns:

Name Type Description
float float

Scalar component

x property 

x: float

Get the x component of the quaternion's vector part.

Returns:

Name Type Description
float float

X component

y property 

y: float

Get the y component of the quaternion's vector part.

Returns:

Name Type Description
float float

Y component

z property 

z: float

Get the z component of the quaternion's vector part.

Returns:

Name Type Description
float float

Z component

conjugate method descriptor 

conjugate() -> Quaternion

Compute the conjugate of the quaternion.

Returns:

Name Type Description
Quaternion Quaternion

Conjugate quaternion with negated vector part
Example 
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import brahe as bh

q = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
q_conj = q.conjugate()

from_euler_angle builtin 

from_euler_angle(e: EulerAngle) -> Quaternion

Create a quaternion from an Euler angle representation.

Parameters:

Name Type Description Default
e EulerAngle

Euler angle representation

 required

Returns:

Name Type Description
Quaternion Quaternion

Equivalent quaternion
Example 
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import brahe as bh

euler = bh.EulerAngle("XYZ", 0.1, 0.2, 0.3, bh.AngleFormat.RADIANS)
q = bh.Quaternion.from_euler_angle(euler)

from_euler_axis builtin 

from_euler_axis(e: EulerAxis) -> Quaternion

Create a quaternion from an Euler axis representation.

Parameters:

Name Type Description Default
e EulerAxis

Euler axis representation

 required

Returns:

Name Type Description
Quaternion Quaternion

Equivalent quaternion
Example 
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import brahe as bh
import numpy as np

axis = np.array([0.0, 0.0, 1.0])
ea = bh.EulerAxis(axis, 1.5708, bh.AngleFormat.RADIANS)
q = bh.Quaternion.from_euler_axis(ea)

from_quaternion builtin 

from_quaternion(q: Quaternion) -> Quaternion

Create a quaternion from another quaternion (copy constructor).

Parameters:

Name Type Description Default
q Quaternion

Source quaternion

 required

Returns:

Name Type Description
Quaternion Quaternion

New quaternion instance
Example 
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import brahe as bh

q1 = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
q2 = bh.Quaternion.from_quaternion(q1)

from_rotation_matrix builtin 

from_rotation_matrix(r: RotationMatrix) -> Quaternion

Create a quaternion from a rotation matrix.

Parameters:

Name Type Description Default
r RotationMatrix

Rotation matrix

 required

Returns:

Name Type Description
Quaternion Quaternion

Equivalent quaternion
Example 
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import brahe as bh
import numpy as np

mat = np.eye(3)
rm = bh.RotationMatrix.from_matrix(mat)
q = bh.Quaternion.from_rotation_matrix(rm)

from_vector builtin 

from_vector(v: ndarray, scalar_first: bool) -> Quaternion

Create a quaternion from a numpy array.

Parameters:

Name Type Description Default
v ndarray

4-element array containing quaternion components

 required
scalar_first bool

If True, array is [w, x, y, z], else [x, y, z, w]

 required

Returns:

Name Type Description
Quaternion Quaternion

New quaternion instance
Example 
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import brahe as bh
import numpy as np

v = np.array([1.0, 0.0, 0.0, 0.0])
q = bh.Quaternion.from_vector(v, scalar_first=True)

inverse method descriptor 

inverse() -> Quaternion

Compute the inverse of the quaternion.

Returns:

Name Type Description
Quaternion Quaternion

Inverse quaternion
Example 
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import brahe as bh

q = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
q_inv = q.inverse()

norm method descriptor 

norm() -> float

Calculate the norm (magnitude) of the quaternion.

Returns:

Name Type Description
float float

Euclidean norm of the quaternion
Example 
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import brahe as bh

q = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
norm = q.norm()

normalize method descriptor 

normalize() -> Any

Normalize the quaternion in-place to unit length.

Example 
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import brahe as bh

q = bh.Quaternion(2.0, 0.0, 0.0, 0.0)
q.normalize()

slerp method descriptor 

slerp(other: Quaternion, t: float) -> Quaternion

Perform spherical linear interpolation (SLERP) between two quaternions.

Parameters:

Name Type Description Default
other Quaternion

Target quaternion

 required
t float

Interpolation parameter in [0, 1]

 required

Returns:

Name Type Description
Quaternion Quaternion

Interpolated quaternion
Example 
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import brahe as bh

q1 = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
q2 = bh.Quaternion(0.707, 0.707, 0.0, 0.0)
q_mid = q1.slerp(q2, 0.5)

to_euler_angle method descriptor 

to_euler_angle(order: str) -> EulerAngle

Convert to Euler angle representation.

Parameters:

Name Type Description Default
order str

Rotation sequence (e.g., "XYZ", "ZYX")

 required

Returns:

Name Type Description
EulerAngle EulerAngle

Equivalent Euler angles
Example 
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import brahe as bh

q = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
euler = q.to_euler_angle("XYZ")

to_euler_axis method descriptor 

to_euler_axis() -> EulerAxis

Convert to Euler axis representation.

Returns:

Name Type Description
EulerAxis EulerAxis

Equivalent Euler axis

to_quaternion method descriptor 

to_quaternion() -> Quaternion

Convert to quaternion representation (returns self).

Returns:

Name Type Description
Quaternion Quaternion

This quaternion

to_rotation_matrix method descriptor 

to_rotation_matrix() -> RotationMatrix

Convert to rotation matrix representation.

Returns:

Name Type Description
RotationMatrix RotationMatrix

Equivalent rotation matrix

to_vector method descriptor 

to_vector(scalar_first: bool) -> ndarray

Convert quaternion to a numpy array.

Parameters:

Name Type Description Default
scalar_first bool

If True, returns [w, x, y, z], else [x, y, z, w]

 required

Returns:

Type Description
ndarray

numpy.ndarray: 4-element array containing quaternion components
Example 
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import brahe as bh

q = bh.Quaternion(1.0, 0.0, 0.0, 0.0)
v = q.to_vector(scalar_first=True)