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WOLFRAM SYSTEM MODELER

TransformationMatrices

Functions for transformation matrices

Package Contents

	Orientation 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
	orientationConstraint Return residues of orientation constraints (shall be zero)
	angularVelocity1 Return angular velocity resolved in frame 1 from orientation object and its derivative
	angularVelocity2 Return angular velocity resolved in frame 2 from orientation object and its derivative
	resolve1 Transform vector from frame 2 to frame 1
	resolve2 Transform vector from frame 1 to frame 2
	multipleResolve1 Transform several vectors from frame 2 to frame 1
	multipleResolve2 Transform several vectors from frame 1 to 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 one frame axis
	axesRotations Return rotation object to rotate in sequence around 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 orientation object from n_x and n_y vectors
	from_nxz Return orientation object from n_x and n_z vectors
	from_T Return orientation object R from transformation matrix T
	from_T_inv Return orientation object R from inverse transformation matrix T_inv
	from_Q Return orientation object T 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 T
	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

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[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/type*	*Description*
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[1]:=

SystemModel["Modelica.Mechanics.MultiBody.Frames.TransformationMatrices"]

Out[1]:=