WOLFRAM SYSTEM MODELER

# TransformationMatrices

Functions for transformation matrices # Package Contents

 Orientation type defining rotation from a frame 1 into a frame 2 with a transformation matrix der_Orientation (1/s) New type defining the first time derivative of Orientation Return residues of orientation constraints (shall be zero) Return angular velocity resolved in frame 1 from orientation object and its derivative Return angular velocity resolved in frame 2 from orientation object and its derivative Transform vector from frame 2 to frame 1 Transform vector from frame 1 to frame 2 Transform several vectors from frame 2 to frame 1 Transform several vectors from frame 1 to frame 2 Transform second order tensor from frame 2 to frame 1 Transform second order tensor from frame 1 to frame 2 Return orientation object that does not rotate a frame Return inverse orientation object Return relative orientation object Return absolute orientation object from another absolute and a relative orientation object Return orientation object of a planar rotation Return angle of a planar rotation, given the rotation axis and the representations of a vector in frame 1 and frame 2 Return rotation object to rotate around one frame axis Return rotation object to rotate in sequence around 3 axes Return the 3 angles to rotate in sequence around 3 axes to construct the given orientation object Return rotation angles valid for a small rotation and optionally residues that should be zero Return orientation object from n_x and n_y vectors Return orientation object from n_x and n_z vectors Return orientation object R from transformation matrix T Return orientation object R from inverse transformation matrix T_inv Return orientation object T from quaternion orientation object Q Return transformation matrix T from orientation object R Return inverse transformation matrix T_inv from orientation object R Return quaternion orientation object Q from orientation object T Map rotation object into vector Map rotation object into e_x and e_y vectors of frame 2, resolved in frame 1

# Information

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

Package Frames.TransformationMatrices contains type definitions and functions to transform rotational frame quantities using transformation matrices.

#### Content

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

```   Orientation T, T1, T2, T_rel, T_inv;
Real     v1, v2, w1, w2, n_x, n_y, n_z, e, e_x, res_ori, phi;
Real     res_equal;
Real        L, angle;
```
Function/typeDescription
Orientation T; New type defining an orientation object that describes
the rotation of frame 1 into frame 2.
der_Orientation der_T; New type defining the first time derivative of Frames.Orientation.
res_ori = orientationConstraint(T); Return the constraints between the variables of an orientation object
(shall be zero).
w1 = angularVelocity1(T, der_T); Return angular velocity resolved in frame 1 from orientation object T
and its derivative der_T.
w2 = angularVelocity2(T, der_T); Return angular velocity resolved in frame 2 from orientation object T
and its derivative der_T.
v1 = resolve1(T,v2); Transform vector v2 from frame 2 to frame 1.
v2 = resolve2(T,v1); Transform vector v1 from frame 1 to frame 2.
[v1,w1] = multipleResolve1(T, [v2,w2]); Transform several vectors from frame 2 to frame 1.
[v2,w2] = multipleResolve2(T, [v1,w1]); Transform several vectors from frame 1 to frame 2.
D1 = resolveDyade1(T,D2); Transform second order tensor D2 from frame 2 to frame 1.
D2 = resolveDyade2(T,D1); Transform second order tensor D1 from frame 1 to frame 2.
T= nullRotation() Return orientation object T that does not rotate a frame.
T_inv = inverseRotation(T); Return inverse orientation object.
T_rel = relativeRotation(T1,T2); Return relative orientation object from two absolute orientation objects.
T2 = absoluteRotation(T1,T_rel); Return absolute orientation object from another absolute
and a relative orientation object.
T = planarRotation(e, 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.
T = axisRotation(i, angle); Return orientation object T for rotation around axis i of frame 1.
T = axesRotations(sequence, angles); Return rotation object to rotate in sequence around 3 axes. Example:
T = axesRotations({1,2,3},{90,45,-90});
angles = axesRotationsAngles(T, sequence); Return the 3 angles to rotate in sequence around 3 axes to
construct the given orientation object.
phi = smallRotation(T); Return rotation angles phi valid for a small rotation.
T = from_nxy(n_x, n_y); Return orientation object from n_x and n_y vectors.
T = from_nxz(n_x, n_z); Return orientation object from n_x and n_z vectors.
R = from_T(T); Return orientation object R from transformation matrix T.
R = from_T_inv(T_inv); Return orientation object R from inverse transformation matrix T_inv.
T = from_Q(Q); Return orientation object T from quaternion orientation object Q.
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(T); Return quaternion orientation object Q from orientation object T.
exy = to_exy(T); Return [e_x, e_y] matrix of an orientation object T,
with e_x and e_y vectors of frame 2, resolved in frame 1.

# Wolfram Language

In:= `SystemModel["Modelica.Mechanics.MultiBody.Frames.TransformationMatrices"]`
Out:= 