udaan.manif package

Submodules

• udaan.manif.S2 module
  • S2
  • TS2
• udaan.manif.SO3 module
  • Rot2Eul()
  • SO3
  • TSO3
• udaan.manif.utils module
  • expm_taylor_expansion()
  • hat()
  • rodrigues_expm()
  • vee()

Module contents

udaan.manif.Rot2Eul(R)

class udaan.manif.S2

Bases: object

Unit vector on the 2-sphere S2.

Wraps a 3-vector (unit norm). Supports:

q1 - q2 -> TS2 (configuration error as tangent vector) q + w -> S2 (geodesic step via exponential map, w is a TS2)

__init__(q=None)

property arr: ndarray

Plain numpy array (copy).

config_error(other)

Scalar configuration error: 1 - q^T q_other.

Return type:

float

error_vec(other, version=2)

Configuration error vector on the tangent space.

Parameters:

• other – The other S2 point.

• version – Error formula variant. 2 (default): hat(q)^2 @ q_other 1: q_other x q (cross product)

Return type:

ndarray

static from_spherical(phi=0.0, th=0.0)

Point on S2 from spherical coordinates (azimuth phi, polar th).

step(omega_dt=None)

Geodesic step on S2 via the exponential map.

Parameters:

omega_dt – angular velocity scaled by dt (3-vector).

Returns a new S2 element: q_next = expm(hat(omega_dt)) @ q.

class udaan.manif.SO3

Bases: object

Rotation matrix on the Special Orthogonal group SO(3).

Wraps a 3x3 rotation matrix. Supports:

R1 @ R2 -> SO3 (group composition, if both SO3) R @ v -> ndarray (rotate a vector) R1 - R2 -> TSO3 (configuration error in the Lie algebra) R + w -> SO3 (exponential map step, w is a TSO3) R.T -> SO3 (transpose, preserves type)

property T: SO3

Transpose (same as inverse for SO3).

__init__(R=None)

property arr: ndarray

Plain numpy array (copy).

config_error(other)

Scalar configuration error: 0.5 * tr(I - other^T @ self).

Returns 0 when self == other, approaches 2 for 180-degree error.

Return type:

float

static from_angle_axis(eta)

Construct SO3 from an angle-axis vector via the exponential map.

Parameters:

eta (ndarray) – 3-vector whose direction is the rotation axis and magnitude is the rotation angle (radians).

Return type:

SO3

static from_tilt_yaw(tilt, yaw)

Construct SO3 from a tilt vector and a yaw angle.

Parameters:

• tilt (ndarray) – 3-vector (angle-axis) applied to the upright orientation. The resulting b3 axis is expm(hat(tilt)) @ e3.

• yaw (float) – heading angle in radians. b1 hint is [cos(yaw), sin(yaw), 0].

Return type:

SO3

static from_two_vectors(b3, b1)

Construct SO3 from a primary axis (b3) and a heading hint (b1).

Builds an orthonormal frame [b1’, b2, b3] where b3 is preserved exactly, b1’ is the closest vector to b1 orthogonal to b3, and b2 = b3 x b1’. Handles the singularity when b3 is parallel to b1.

This is the standard construction for desired attitude from thrust direction (b3) and heading direction (b1).

Return type:

SO3

inv()

Inverse rotation (transpose, since R^{-1} = R^T for SO(3)).

Return type:

SO3

step(Omega_dt=None)

Integrate angular velocity via the exponential map.

Parameters:

Omega_dt – angular velocity scaled by dt (3-vector or TSO3).

Returns a new SO3 element: R_next = R @ expm(hat(Omega_dt)).

class udaan.manif.TS2

Bases: object

Tangent vector to the 2-sphere S2.

Wraps a 3-vector representing angular velocity or configuration error on the sphere. Supports:

v1 - v2 -> TS2 (tangent vector difference) v1 + v2 -> TS2 (tangent vector sum) v * s -> TS2 (scalar multiplication) v.transport(q) -> TS2 (project onto tangent space at q)

__init__(vector=None)

property arr: ndarray

Plain numpy array (copy).

property norm: float

Magnitude of the tangent vector.

transport(q)

Transport this tangent vector to the tangent space at q.

Computes -hat(q)^2 @ self, which projects self onto T_q S2 with the correct sign so that ew = w - wd.transport(q).

Return type:

TS2

property vector: ndarray

Raw 3-vector as a plain np.ndarray.

class udaan.manif.TSO3

Bases: object

Element of the Lie algebra so(3) — tangent vector to SO(3).

Wraps a 3-vector representing angular velocity or rotation error. Supports:

w1 - w2 -> TSO3 (tangent vector difference) w1 + w2 -> TSO3 (tangent vector sum) w * s -> TSO3 (scalar multiplication) w.hat() -> 3x3 (skew-symmetric matrix in so(3)) w.transport(R_from, R_to) -> TSO3 (frame transport)

__init__(vector=None)

property arr: ndarray

Plain numpy array (copy).

hat()

Return the 3x3 skew-symmetric matrix in so(3).

Return type:

ndarray

property norm: float

Magnitude of the tangent vector.

transport(R_from, R_to)

Transport this tangent vector from R_from’s body frame to R_to’s.

Computes R_to^T @ R_from @ self, which maps an angular velocity (self) expressed in R_from’s frame into R_to’s frame.

Return type:

TSO3

Example

eOm = Om - Omd.transport(Rd, R) # angular velocity error

property vector: ndarray

Raw 3-vector as a plain np.ndarray.

udaan.manif.expm_taylor_expansion(M, order=2)

udaan.manif.hat(vector)

udaan.manif.rodrigues_expm(vector)

Closed-form matrix exponential for so(3) via Rodrigues’ formula.

Exact for 3-vectors (skew-symmetric generators) and ~1.5x faster than scipy.linalg.expm over the hat map. Uses Taylor expansion for small angles to avoid division-by-zero and maintain accuracy.

udaan.manif.vee(matrix)