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math_utils module

Utilities module for attitude determination and control system.

This module contains a number of utility methods to simplify mathematical calculations performed elswhere. It also implements the TRIAD algorithm.

math_utils.cross(v1, v2)

Computes the cross product of two vectors

NOTE: this function only exists because it outperforms numpy’s
cross function for small vectors. Using it enables a ~2x speedup of the overall simulation
Parameters:
  • v1 (numpy ndarray) – an Nx1 vector
  • v2 (numpy ndarray) – an Nx1 vector
Returns:

the cross product of the input vectors
Return type:

numpy ndarray
math_utils.dcm_to_quaternion(dcm)

Converts a Direction Cosine Matrix (DCM) to a quaternion

Parameters:dcm (numpy ndarray) – a 3x3 transformation matrix that parameterizes the attitude of a satellite
Returns:
the equivalent right-handed quaternion (4x1) with the
scalar part as the last entry
Return type:numpy ndarray
math_utils.get_DCM_i2NED(r)

Computes the inertial to NED (North-East-Down) DCM

Parameters:r (numpy ndarray) – inertial position
Returns:
the 3x3 DCM representing the transformation from the
inertial to NED frame
Return type:numpy ndarray
math_utils.normalize(vector)

Normalizes a vector so that its magnitude is 1

Parameters:vector (numpy ndarray) – an Nx1 vector of arbitrary magnitude
Returns:the normalized vector
Return type:numpy ndarray
math_utils.quaternion_multiply(q1, q2)

Multiplies two quaternions and returns the result

Parameters:
  • q1 (numpy ndarray) – a right-handed quaternion (4x1) with the scalar part as the last entry
  • q2 (numpy ndarray) – a right-handed quaternion (4x1) with the scalar part as the last entry
Returns:

the quaternion product of the quaternion multiplication
Return type:

numpy ndarray
math_utils.quaternion_to_dcm(q)

Converts a quaternion to a Direction Cosine Matrix (DCM)

Parameters:q (numpy ndarray) – a right-handed quaternion (4x1) with the scalar part as the last entry
Returns:
the equivalent (3x3) Direction Cosine Matrix for the
attitude parameterized by the input quaternion
Return type:numpy ndarray
math_utils.skew_symmetric(v)

Returns a skew-symmetric matrix for the input vector

Parameters:v (numpy ndarray) – an Nx1 vector
Returns:the skew-symmetric form of the vector (for purposes of cross-product computation)
Return type:numpy ndarray
math_utils.t1_matrix(angle)

Returns the transformation matrix of a rotation about the 1-axis

Parameters:angle (float) – the angle of rotation about the axis (in radians)
Returns:the transformation matrix for the rotation
Return type:numpy ndarray
math_utils.t2_matrix(angle)

Returns the transformation matrix of a rotation about the 2-axis

Parameters:angle (float) – the angle of rotation about the axis (in radians)
Returns:the transformation matrix for the rotation
Return type:numpy ndarray
math_utils.t3_matrix(angle)

Returns the transformation matrix of a rotation about the 3-axis

Parameters:angle (float) – the angle of rotation about the axis (in radians)
Returns:the transformation matrix for the rotation
Return type:numpy ndarray
math_utils.triad(d_1_b, d_1_i, d_2_b, d_2_i)

Uses TRIAD to get the DCM from inertial to body

Parameters:
  • d_1_b (numpy ndarray) – the first (better) direction measurement in body coordinates (what your simulated sensors measure in the body frame)
  • d_1_i (numpy ndarray) – the first (better) direction measurement in inertial coordinates (what your model says the direction should be)
  • d_1_b – the second (worse) direction measurement in body coordinates (what your simulated sensors measure in the body frame)
  • d_1_i – the second (worse) direction measurement in inertial coordinates (what your model says the direction should be)
Returns:

the 3x3 DCM representing the transformation from the

inertial to body frame
Return type:

numpy ndarray
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