The `transformations` module

Numython R&D, (c) 2024 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]: Given a R^3 vector (k) and an angle (theta), return the SO(3) matrix associated.

moro.transformations.compose_rotations(*rotations)[source]

Composes rotation matrices w.r.t. fixed or movable frames

Parameters

:param rotations: A tuple that contains (angle, axis, frame, deg)
:type rotations: tuple

Returns

return: Rotation matrix ..

:rtype:class:sympy.matrices.dense.MutableDenseMatrix

Examples

>>> compose_rotations((45, "z", "fixed", True), (30, "x", "local", True))
⎡0.707106781186548  -0.612372435695794  0.353553390593274 ⎤
⎢                                                         ⎥
⎢0.707106781186547  0.612372435695795   -0.353553390593274⎥
⎢                                                         ⎥
⎣        0                 0.5          0.866025403784439 ⎦

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

Calculates Denavit-Hartenberg matrix given the four parameters.

Parameters

:param a: DH parameter
:type a: int, float or symbol
:param alpha: DH parameter
:type alpha: int, float or symbol
:param d: DH parameter
:type d: int, float or symbol
:param theta: DH parameter
:type theta: int, float or symbol

Returns

return: Denavit-Hartenberg matrix (4x4) ..

:rtype:class:sympy.matrices.dense.MutableDenseMatrix

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.eul2htm(phi, theta, psi, seq='zxz', deg=False)[source]

Given a set of Euler Angles (phi,theta,psi) for specific sequence this function returns the homogeneous transformation matrix associated. Default sequence is ZXZ.

Parameters

phi: int, float, symbol: First angle of the set
theta: int, float, symbol: Second angle of the set
psi: int, float, symbol: Third angle of the set
deg: bool: This parameter is True if phi, theta, and psi are given in degrees, by default it’s assumed to be False (angles in radians).

Returns

H: sympy.matrices.dense.MutableDenseMatrix: A homogeneous transformation matrix (SE(3)).

Examples

>>> eul2htm(90,90,90,"zxz",True)
⎡0  0   1  0⎤
⎢           ⎥
⎢0  -1  0  0⎥
⎢           ⎥
⎢1  0   0  0⎥
⎢           ⎥
⎣0  0   0  1⎦

>>> eul2htm(pi/2,pi/2,pi/2)
⎡0  0   1  0⎤
⎢           ⎥
⎢0  -1  0  0⎥
⎢           ⎥
⎢1  0   0  0⎥
⎢           ⎥
⎣0  0   0  1⎦

>>> eul2htm(0,pi/2,0,"zyz")
⎡0   0  1  0⎤
⎢           ⎥
⎢0   1  0  0⎥
⎢           ⎥
⎢-1  0  0  0⎥
⎢           ⎥
⎣0   0  0  1⎦

moro.transformations.eul2rot(phi, theta, psi, seq='zxz', deg=False)[source]

moro.transformations.htm2eul(H, seq='zxz', deg=False)[source]

Given a homogeneous transformation matrix this function return the equivalent set of Euler Angles.

If “deg” is True then Euler Angles are converted to degrees.

>>> H = htmrot(pi/3,"y")*htmrot(pi/4,"x")
>>> H
⎡      √6   √6    ⎤
⎢1/2   ──   ──   0⎥
⎢      4    4     ⎥
⎢                 ⎥
⎢      √2  -√2    ⎥
⎢ 0    ──  ────  0⎥
⎢      2    2     ⎥
⎢                 ⎥
⎢-√3   √2   √2    ⎥
⎢────  ──   ──   0⎥
⎢ 2    4    4     ⎥
⎢                 ⎥
⎣ 0    0    0    1⎦
>>> htm2eul(H)
⎛    ⎛√3⎞                     ⎞
⎜atan⎜──⎟, atan(√7), -atan(√6)⎟
⎝    ⎝2 ⎠                     ⎠
>>> htm2eul(H, deg=True)
(40.8933946491309, 69.2951889453646, -67.7923457014035)

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

Return a homogeneous transformation matrix that represents a rotation “theta” about “axis”.

Parameters

thetafloat, int or symbolic: Rotation angle (given in radians by default)
axisstr: Rotation axis
degbool: ¿Is theta given in degrees?

Returns

Hsympy.matrices.dense.MutableDenseMatrix: Homogeneous transformation matrix

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(*args, **kwargs)[source]

Calculate the homogeneous transformation matrix of a translation

Parameters

*argslist, tuple, int, float: Translation vector or components
**kwargsfloat, int: dx, dy and dz keyword arguments

Returns

Hsympy.matrices.dense.MutableDenseMatrix: Homogeneous transformation matrix

Examples

>>> htmtra([50,-100,30])
⎡1  0  0   50 ⎤
⎢             ⎥
⎢0  1  0  -100⎥
⎢             ⎥
⎢0  0  1   30 ⎥
⎢             ⎥
⎣0  0  0   1  ⎦

>>> a,b,c = symbols("a,b,c")
>>> htmtra([a,b,c])
⎡1  0  0  a⎤
⎢          ⎥
⎢0  1  0  b⎥
⎢          ⎥
⎢0  0  1  c⎥
⎢          ⎥
⎣0  0  0  1⎦

Using float/integer arguments:

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

Using keyword arguments:

>>> htmtra(dz=100,dx=300,dy=-200)
⎡1  0  0  300 ⎤
⎢             ⎥
⎢0  1  0  -200⎥
⎢             ⎥
⎢0  0  1  100 ⎥
⎢             ⎥
⎣0  0  0   1  ⎦

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

moro.transformations.rot2axa(R, deg=False)[source]: Given a SO(3) matrix return the axis-angle representation.

moro.transformations.rot2eul(R, seq='zxz', deg=False)[source]

moro.transformations.rotx(theta, deg=False)[source]

Calculates the rotation matrix about the x-axis

Parameters

:param theta: Rotation angle (given in radians by default)
:type theta: float, int or `symbolic`
:param deg: ¿Is theta given in degrees?, False is default value.
:type deg: bool

Returns

return: sympy.matrices.dense.MutableDenseMatrix ..
rtype: Rotation matrix (SO3) ..

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

:param theta: Rotation angle (given in radians by default)
:type theta: float, int or `symbolic`
:param deg: ¿Is theta given in degrees?, False is default value.
:type deg: bool

Returns

return: sympy.matrices.dense.MutableDenseMatrix ..
rtype: Rotation matrix (SO3) ..

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]

Calculates the rotation matrix about the z-axis

Parameters

:param theta: Rotation angle (given in radians by default)
:type theta: float, int or `symbolic`
:param deg: ¿Is theta given in degrees?, False is default value.
:type deg: bool

Returns

return: sympy.matrices.dense.MutableDenseMatrix ..
rtype: Rotation matrix (SO3) ..

Examples

Using angle in radians,

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

Or 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.skew(u)[source]: Return skew-symmetric matrix associated to u vector

The transformations module

The `transformations` module