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.axa2rot(k, theta)[source]

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.

Returns:

Rsympy.matrices.dense.MutableDenseMatrix: Rotation matrix of shape (3, 3) computed with Rodrigues’ formula.

moro.transformations.dh(a, alpha, d, theta)[source]

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)[source]

Build a rotation matrix from proper Euler angles.

Parameters:

phiint, float or symbolic: First Euler angle.
thetaint, float or symbolic: Intermediate Euler angle.
psiint, float or symbolic: Third Euler angle.
seqstr, optional: Proper Euler sequence. Supported sequences are "xyx", "xzx", "yxy", "yzy", "zxz" and "zyz". Matching is case-insensitive. Tait-Bryan sequences such as "xyz" are not supported here.
degbool, optional: If True, the input angles are interpreted as degrees and converted to radians before constructing the matrix.

Returns:

sympy.matrices.dense.MutableDenseMatrix: Rotation matrix.

Notes

This function uses column vectors and active rotations. For a sequence seq="abc", the convention is defined by the matrix product R = R_a(phi) @ R_b(theta) @ R_c(psi), where each elementary rotation is produced by rot(). For proper Euler sequences, a == c.

Examples

>>> eul2rot(pi/2, pi/3, pi/4, seq="zxz")
⎡-√2   -√6         ⎤
⎢────  ────   √3/2 ⎥
⎢ 4     4          ⎥
⎢                  ⎥
⎢-√2    √6         ⎥
⎢────   ──   -1/2  ⎥
⎢ 4     4          ⎥
⎢                  ⎥
⎢ √6    √2         ⎥
⎢ ──    ──    1/2  ⎥
⎣ 4     4          ⎦

>>> eul2rot(pi/6, pi/4, pi/3, seq="xyx")
⎡√2              √2        ⎤
⎢──      √6/4    ──        ⎥
⎢2               4         ⎥
⎢                          ⎥
⎢√2    3/8 + √3  1   3⋅√3 ⎥
⎢──    ────────  ─ - ──── ⎥
⎢4        4      8    8   ⎥
⎢                          ⎥
⎢-√6   1   3⋅√3  √3   3/8⎥
⎢────  ─ + ────  ── - ───⎥
⎣ 4    8    8    4     4 ⎦

moro.transformations.htm2rot(T)[source]

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)[source]

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)[source]

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)[source]

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)[source]

Compute the structured inverse of 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: Inverse homogeneous transformation matrix computed from the rigid-body structure, using R.T and -R.T*p instead of a general matrix inverse.

moro.transformations.rot(theta, axis='z', deg=False)[source]

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)[source]

Return the axis-angle representation of a rotation matrix.

Parameters:

Rsympy Matrix: Rotation matrix in SO(3).
degbool, optional: If True, the angle is returned in degrees. Default is False.
tolfloat, optional: Positive tolerance used to validate numeric rotation matrices, classify angles close to 0, classify angles close to pi, and tolerate small floating-point errors in trigonometric quantities. Default is 1e-9.

Returns:

ksympy.matrices.dense.MutableDenseMatrix: Axis of rotation, a 3D vector.
thetafloat, int or symbolic: Rotation angle in radians by default, or in degrees when deg=True.

moro.transformations.rot2eul(R, seq='zxz', deg=False, tol=1e-09)[source]

Calculate proper Euler angles from a rotation matrix.

Parameters:

Rmatrix-like, shape (3, 3): Rotation matrix. The function validates only that the input has shape (3, 3); it does not yet perform a full SO(3) membership check.
seqstr, optional: Proper Euler sequence. Supported sequences are "xyx", "xzx", "yxy", "yzy", "zxz" and "zyz". Matching is case-insensitive.
degbool, optional: If True, returned angles are converted from radians to degrees.
tolfloat, optional: Positive numerical tolerance used only for floating-point classification near the singularities theta = 0 and theta = pi and for clipping small numerical excursions of cos(theta) outside [-1, 1].

Returns:

list of tuple: In the general case, returns two equivalent solutions [(phi1, theta1, psi1), (phi2, theta2, psi2)]. In singular cases, returns a single representative solution with psi = 0.

Notes

The convention matches eul2rot(): column vectors, active rotations and R = R_a(phi) @ R_b(theta) @ R_a(psi) for seq="aba". Euler angle representations are not unique; both general-case solutions reconstruct the same matrix, the second solution may contain a negative intermediate angle, and no additional range normalization is applied. At singularities, phi and psi are not independently determined; setting psi = 0 is only a representative convention.

moro.transformations.rot2htm(R)[source]

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.rotx(theta, deg=False)[source]

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)[source]

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)[source]

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)[source]

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)[source]

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).