WOLFRAM SYSTEMMODELER

Frames

Functions to transform rotational frame quantities

Package Contents

Quaternions

Functions to transform rotational frame quantities based on quaternions (also called Euler parameters)

TransformationMatrices

Functions for transformation matrices

Internal

Internal definitions that may be removed or changed (do not use)

Orientation

Orientation object defining rotation from a frame 1 into a frame 2

orientationConstraint

Return residues of orientation constraints (shall be zero)

angularVelocity1

Return angular velocity resolved in frame 1 from orientation object

angularVelocity2

Return angular velocity resolved in frame 2 from orientation object

resolve1

Transform vector from frame 2 to frame 1

resolve2

Transform vector from frame 1 to frame 2

resolveRelative

Transform vector from frame 1 to frame 2 using absolute orientation objects of frame 1 and of frame 2

resolveDyade1

Transform second order tensor from frame 2 to frame 1

resolveDyade2

Transform second order tensor from frame 1 to frame 2

nullRotation

Return orientation object that does not rotate a frame

inverseRotation

Return inverse orientation object

relativeRotation

Return relative orientation object

absoluteRotation

Return absolute orientation object from another absolute and a relative orientation object

planarRotation

Return orientation object of a planar rotation

planarRotationAngle

Return angle of a planar rotation, given the rotation axis and the representations of a vector in frame 1 and frame 2

axisRotation

Return rotation object to rotate around an angle along one frame axis

axesRotations

Return fixed rotation object to rotate in sequence around fixed angles along 3 axes

axesRotationsAngles

Return the 3 angles to rotate in sequence around 3 axes to construct the given orientation object

smallRotation

Return rotation angles valid for a small rotation and optionally residues that should be zero

from_nxy

Return fixed orientation object from n_x and n_y vectors

from_nxz

Return fixed orientation object from n_x and n_z vectors

from_T

Return orientation object R from transformation matrix T

from_T2

Return orientation object R from transformation matrix T and its derivative der(T)

from_T_inv

Return orientation object R from inverse transformation matrix T_inv

from_Q

Return orientation object R from quaternion orientation object Q

to_T

Return transformation matrix T from orientation object R

to_T_inv

Return inverse transformation matrix T_inv from orientation object R

to_Q

Return quaternion orientation object Q from orientation object R

to_vector

Map rotation object into vector

to_exy

Map rotation object into e_x and e_y vectors of frame 2, resolved in frame 1

axis

Return unit vector for x-, y-, or z-axis

Wolfram Language

In[1]:=
Click for copyable input
SystemModel["Modelica.Mechanics.MultiBody.Frames"]
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Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

Package Frames contains type definitions and functions to transform rotational frame quantities. The basic idea is to hide the actual definition of an orientation in this package by providing essentially type Orientation together with functions operating on instances of this type.

Content

In the table below an example is given for every function definition. The used variables have the following declaration:

   Frames.Orientation R, R1, R2, R_rel, R_inv;
   Real[3,3]   T, T_inv;
   Real[3]     v1, v2, w1, w2, n_x, n_y, n_z, e, e_x, res_ori, phi;
   Real[6]     res_equal;
   Real        L, angle;
Function/typeDescription
Orientation R; New type defining an orientation object that describes
the rotation of frame 1 into frame 2.
res_ori = orientationConstraint(R); Return the constraints between the variables of an orientation object
(shall be zero).
w1 = angularVelocity1(R); Return angular velocity resolved in frame 1 from orientation object R.
w2 = angularVelocity2(R); Return angular velocity resolved in frame 2 from orientation object R.
v1 = resolve1(R,v2); Transform vector v2 from frame 2 to frame 1.
v2 = resolve2(R,v1); Transform vector v1 from frame 1 to frame 2.
v2 = resolveRelative(v1,R1,R2); Transform vector v1 from frame 1 to frame 2 using absolute orientation objects R1 of frame 1 and R2 of frame 2.
D1 = resolveDyade1(R,D2); Transform second order tensor D2 from frame 2 to frame 1.
D2 = resolveDyade2(R,D1); Transform second order tensor D1 from frame 1 to frame 2.
R = nullRotation() Return orientation object R that does not rotate a frame.
R_inv = inverseRotation(R); Return inverse orientation object.
R_rel = relativeRotation(R1,R2); Return relative orientation object from two absolute orientation objects.
R2 = absoluteRotation(R1,R_rel); Return absolute orientation object from another absolute
and a relative orientation object.
R = planarRotation(e, angle, der_angle); Return orientation object of a planar rotation.
angle = planarRotationAngle(e, v1, v2); Return angle of a planar rotation, given the rotation axis
and the representations of a vector in frame 1 and frame 2.
R = axisRotation(axis, angle, der_angle); Return orientation object R to rotate around angle along axis of frame 1.
R = axesRotations(sequence, angles, der_angles); Return rotation object to rotate in sequence around 3 axes. Example:
R = axesRotations({1,2,3},{pi/2,pi/4,-pi}, zeros(3));
angles = axesRotationsAngles(R, sequence); Return the 3 angles to rotate in sequence around 3 axes to
construct the given orientation object.
phi = smallRotation(R); Return rotation angles phi valid for a small rotation R.
R = from_nxy(n_x, n_y); Return orientation object from n_x and n_y vectors.
R = from_nxz(n_x, n_z); Return orientation object from n_x and n_z vectors.
R = from_T(T,w); Return orientation object R from transformation matrix T and its angular velocity w.
R = from_T2(T,der(T)); Return orientation object R from transformation matrix T and its derivative der(T).
R = from_T_inv(T_inv,w); Return orientation object R from inverse transformation matrix T_inv and its angular velocity w.
R = from_Q(Q,w); Return orientation object R from quaternion orientation object Q and its angular velocity w.
T = to_T(R); Return transformation matrix T from orientation object R.
T_inv = to_T_inv(R); Return inverse transformation matrix T_inv from orientation object R.
Q = to_Q(R); Return quaternion orientation object Q from orientation object R.
exy = to_exy(R); Return [e_x, e_y] matrix of an orientation object R,
with e_x and e_y vectors of frame 2, resolved in frame 1.
L = length(n_x); Return length L of a vector n_x.
e_x = normalize(n_x); Return normalized vector e_x of n_x such that length of e_x is one.
e = axis(i); Return unit vector e directed along axis i
Quaternions Package with functions to transform rotational frame quantities based on quaternions (also called Euler parameters).
TransformationMatrices Package with functions to transform rotational frame quantities based on transformation matrices.