Transformations

The moro.transformations module provides functions for rotations, homogeneous transformations, Euler angles, axis-angle representations, and Denavit-Hartenberg transformations.

Numython R&D, (c) 2026 Moro is a Python library for kinematic and dynamic modeling of serial robots. This library has been designed, mainly, for academic and research purposes, using SymPy as base library.

moro.transformations.axa2quat(k, theta, deg=False)

Convert an axis-angle orientation to a scalar-first unit quaternion.

moro.transformations.axa2rot(k, theta, deg=False)

Build a rotation matrix from an axis-angle representation.

Parameters:

klist, tuple or sympy Matrix

Rotation axis. Accepted formats are a 3-element list, a 3-element tuple, a column matrix of shape (3, 1), or a row matrix of shape (1, 3). The vector is normalized internally to a column matrix. The zero vector is rejected because it does not define a rotation axis.

thetafloat, int or symbolic

Rotation angle in radians by default.

degbool, optional

If True, theta is interpreted in degrees. Default is False.

Returns:

Rsympy.matrices.dense.MutableDenseMatrix

Rotation matrix of shape (3, 3) computed with Rodrigues’ formula.

moro.transformations.dh(a, alpha, d, theta)

Compute the Denavit-Hartenberg homogeneous transformation matrix.

Parameters:

aint, float or symbolic

Link length (DH parameter).

alphaint, float or symbolic

Link twist (DH parameter).

dint, float or symbolic

Link offset (DH parameter).

thetaint, float or symbolic

Joint angle (DH parameter).

Returns:

sympy.matrices.dense.MutableDenseMatrix

Denavit-Hartenberg homogeneous transformation matrix of shape (4, 4).

Examples

With numerical values:

>>> dh(100, pi/2, 50, pi/2)
⎡0  0  1   0 ⎤
⎢            ⎥
⎢1  0  0  100⎥
⎢            ⎥
⎢0  1  0  50 ⎥
⎢            ⎥
⎣0  0  0   1 ⎦

Using symbolic values:

>>> a = symbols("a")
>>> t = symbols("t")
>>> dh(a, 0, 0, t)
⎡cos(t)  -sin(t)  0  a⋅cos(t)⎤
⎢                            ⎥
⎢sin(t)   cos(t)  0  a⋅sin(t)⎥
⎢                            ⎥
⎢  0        0     1     0    ⎥
⎢                            ⎥
⎣  0        0     0     1    ⎦

moro.transformations.eul2rot(phi, theta, psi, seq='zxz', deg=False, intrinsic=True)

Build a rotation matrix from Euler or Tait-Bryan angles.

moro.transformations.htm2rot(T)

Extract the rotation block from a homogeneous transformation matrix.

Parameters:

Tarray-like or sympy Matrix

Homogeneous transformation matrix. It is converted with Matrix(T) and must have shape (4, 4). No full SE(3) membership validation is performed.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Upper-left rotation block of shape (3, 3).

moro.transformations.htm2tra(T)

Extract the translation vector from a homogeneous transformation matrix.

Parameters:

Tarray-like or sympy Matrix

Homogeneous transformation matrix. It is converted with Matrix(T) and must have shape (4, 4). No full SE(3) membership validation is performed.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Translation column vector of shape (3, 1).

moro.transformations.htmrot(theta, axis='z', deg=False)

Return a homogeneous transformation matrix for a pure rotation.

Parameters:

thetafloat, int or symbolic

Rotation angle. By default, the value is interpreted in radians.

axisstr

Rotation axis, "x", "y" or "z". Matching is case-insensitive. Default is "z".

degbool, optional

If True, theta is interpreted as degrees. Default is False.

Returns:

Hsympy.matrices.dense.MutableDenseMatrix

Homogeneous transformation matrix of shape (4, 4).

Examples

>>> htmrot(pi/2)
⎡0  -1  0  0⎤
⎢           ⎥
⎢1  0   0  0⎥
⎢           ⎥
⎢0  0   1  0⎥
⎢           ⎥
⎣0  0   0  1⎦
>>> htmrot(pi/2, "x")
⎡1  0  0   0⎤
⎢           ⎥
⎢0  0  -1  0⎥
⎢           ⎥
⎢0  1  0   0⎥
⎢           ⎥
⎣0  0  0   1⎦
>>> htmrot(30, "y", True)
⎡0.866025403784439  0         0.5         0⎤
⎢                                          ⎥
⎢        0          1          0          0⎥
⎢                                          ⎥
⎢      -0.5         0  0.866025403784439  0⎥
⎢                                          ⎥
⎣        0          0          0          1⎦
>>> t = symbols("t")
>>> htmrot(t, "x")
⎡1    0        0     0⎤
⎢                     ⎥
⎢0  cos(t)  -sin(t)  0⎥
⎢                     ⎥
⎢0  sin(t)  cos(t)   0⎥
⎢                     ⎥
⎣0    0        0     1⎦

moro.transformations.htmtra(x=0, y=0, z=0)

Calculate the homogeneous transformation matrix of a translation.

Parameters:

xint, float or symbolic, optional

Translation along the x-axis. Default is 0.

yint, float or symbolic, optional

Translation along the y-axis. Default is 0.

zint, float or symbolic, optional

Translation along the z-axis. Default is 0.

Returns:

Hsympy.matrices.dense.MutableDenseMatrix

Homogeneous transformation matrix

Examples

>>> htmtra()
⎡1  0  0  0⎤
⎢          ⎥
⎢0  1  0  0⎥
⎢          ⎥
⎢0  0  1  0⎥
⎢          ⎥
⎣0  0  0  1⎦

>>> htmtra(10,-40,50)
⎡1  0  0  10 ⎤
⎢            ⎥
⎢0  1  0  -40⎥
⎢            ⎥
⎢0  0  1  50 ⎥
⎢            ⎥
⎣0  0  0   1 ⎦

>>> htmtra(z=100)
⎡1  0  0   0 ⎤
⎢            ⎥
⎢0  1  0   0 ⎥
⎢            ⎥
⎢0  0  1  100⎥
⎢            ⎥
⎣0  0  0   1 ⎦

>>> a,b,c = symbols("a,b,c")
>>> htmtra(x=a, y=b, z=c)
⎡1  0  0  a⎤
⎢          ⎥
⎢0  1  0  b⎥
⎢          ⎥
⎢0  0  1  c⎥
⎢          ⎥
⎣0  0  0  1⎦

moro.transformations.invhtm(T, *, tol=1e-09)

Compute the structured inverse of a rigid homogeneous transform.

Parameters:

Tmatrix-like, shape (4, 4)

Homogeneous transformation in SE(3).

tolpositive real, optional

Numerical tolerance used for SE(3) validation.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Structured rigid-body inverse.

moro.transformations.is_homogeneous_transform(T, *, tol=1e-09)

Return True, False, or None according to membership in SE(3).

moro.transformations.is_rotation_matrix(R, *, tol=1e-09)

Return True, False, or None according to membership in SO(3).

moro.transformations.quat2axa(q, deg=False, *, tol=1e-09)

Convert a scalar-first quaternion to principal axis-angle form.

moro.transformations.quat2rot(q, *, tol=1e-09)

Convert a scalar-first quaternion [w, x, y, z] to a rotation matrix.

moro.transformations.rot(theta, axis='z', deg=False)

Return a rotation matrix that represents a rotation of theta about axis.

Parameters:

thetafloat, int or symbolic

Rotation angle. By default, the value is interpreted in radians.

axisstr

Rotation axis, "x", "y" or "z". Matching is case-insensitive. Default is "z".

degbool, optional

If True, theta is interpreted as degrees. Default is False.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Rotation matrix of shape (3, 3).

moro.transformations.rot2axa(R, deg=False, tol=1e-09)

Return the principal axis-angle representation of a rotation matrix.

moro.transformations.rot2eul(R, seq='zxz', deg=False, intrinsic=True, tol=1e-09)

Return Euler/Tait-Bryan angles that reconstruct a rotation matrix.

moro.transformations.rot2htm(R)

Build a homogeneous transformation matrix from a rotation matrix.

Parameters:

Rarray-like or sympy Matrix

Rotation block. It is converted with Matrix(R) and must have shape (3, 3). No full SO(3) membership validation is performed.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Homogeneous transformation matrix with zero translation and shape (4, 4).

moro.transformations.rot2quat(R, tol=1e-09)

Convert a rotation matrix to a scalar-first unit quaternion.

moro.transformations.rot2rotvec(R, tol=1e-09)

Return the principal rotation vector of a rotation matrix.

moro.transformations.rotvec2rot(phi)

Convert a three-component rotation vector to a rotation matrix.

moro.transformations.rotx(theta, deg=False)

Calculates the rotation matrix about the x-axis

Parameters:

thetafloat, int or symbolic

Rotation angle (given in radians by default)

degbool

If True, theta is interpreted as degrees. Default is False.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Rotation matrix in SO(3).

Examples

>>> rotx(pi)
⎡1  0   0 ⎤
⎢         ⎥
⎢0  -1  0 ⎥
⎢         ⎥
⎣0  0   -1⎦
>>> rotx(60, deg=True)
⎡1          0                  0         ⎤
⎢                                        ⎥
⎢0         0.5         -0.866025403784439⎥
⎢                                        ⎥
⎣0  0.866025403784439         0.5        ⎦

moro.transformations.roty(theta, deg=False)

Calculates the rotation matrix about the y-axis

Parameters:

thetafloat, int or symbolic

Rotation angle (given in radians by default)

degbool

If True, theta is interpreted as degrees. Default is False.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Rotation matrix in SO(3).

Examples

>>> roty(pi/3)
⎡         √3 ⎤
⎢1/2   0  ── ⎥
⎢         2  ⎥
⎢            ⎥
⎢ 0    1   0 ⎥
⎢            ⎥
⎢-√3         ⎥
⎢────  0  1/2⎥
⎣ 2          ⎦

>>> roty(30, deg=True)
⎡0.866025403784439  0         0.5       ⎤
⎢                                       ⎥
⎢        0          1          0        ⎥
⎢                                       ⎥
⎣      -0.5         0  0.866025403784439⎦

moro.transformations.rotz(theta, deg=False)

Calculate the rotation matrix about the z-axis.

Parameters:

thetafloat, int or symbolic

Rotation angle. By default, the value is assumed to be given in radians.

degbool, optional

If True, theta is interpreted as degrees. Default is False.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Rotation matrix in SO(3).

Examples

Using angle in radians:

>>> rotz(pi/2)
⎡0  -1  0⎤
⎢        ⎥
⎢1   0  0⎥
⎢        ⎥
⎣0   0  1⎦

Using symbolic variables:

>>> x = symbols("x")
>>> rotz(x)
⎡cos(x)  -sin(x)  0⎤
⎢                  ⎥
⎢sin(x)   cos(x)  0⎥
⎢                  ⎥
⎣  0        0     1⎦

Using angles in degrees:

>>> rotz(45, deg=True)
⎡0.707106781186548  -0.707106781186547  0⎤
⎢                                        ⎥
⎢0.707106781186547   0.707106781186548  0⎥
⎢                                        ⎥
⎣        0                  0           1⎦

moro.transformations.rt2htm(R, p)

Build a homogeneous transformation matrix from rotation and translation.

Parameters:

Rarray-like or sympy Matrix

Rotation block. It is converted with Matrix(R) and must have shape (3, 3). No full SO(3) membership validation is performed.

plist, tuple or sympy Matrix

Translation vector. Accepted formats are a 3-element list, a 3-element tuple, a column matrix of shape (3, 1), or a row matrix of shape (1, 3). The vector is normalized internally to a column matrix.

Returns:

sympy.matrices.dense.MutableDenseMatrix

Homogeneous transformation matrix of shape (4, 4).

moro.transformations.skew(u)

Return the skew-symmetric matrix associated with a 3D vector.

Parameters:

ulist, tuple or sympy Matrix

Vector. Accepted formats are a 3-element list, a 3-element tuple, a column matrix of shape (3, 1), or a row matrix of shape (1, 3). The vector is normalized internally to a column matrix.

Returns:

Ssympy.matrices.dense.MutableDenseMatrix

Skew-symmetric matrix of shape (3, 3).

moro.transformations.vex(S, *, tol=1e-09)

Return the vector associated with a 3x3 skew-symmetric matrix.