Frames

Functions to transform rotational frame quantities

Package Contents

	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
	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)

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/type*	*Description*
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.