Joint which does not constrain the motion between frame_a and frame_b.
Such a joint is only meaningful if the relative distance and orientation
between frame_a and frame_b, and their derivatives, shall be used
as states.
Note, that bodies such as Parts.Body, Parts.BodyShape,
have potential states describing the distance
and orientation, and their derivatives, between the world frame and
a body fixed frame.
Therefore, if these potential state variables are suited,
a FreeMotion joint is not needed.
The states of the FreeMotion object are:
- The relative position vector r_rel_a from the origin of
frame_a to the origin of frame_b, resolved in
frame_a and the relative velocity v_rel_a of the origin of
frame_b with respect to the origin of frame_a, resolved in frame_a
(= der(r_rel_a)).
- If parameter useQuaternions in the "Advanced" menu
is true (this is the default), then 4 quaternions
are states. Additionally, the coordinates of the
relative angular velocity vector are 3 potential states.
If useQuaternions in the "Advanced" menu
is false, then 3 angles and the derivatives of
these angles are potential states. The orientation of frame_b
is computed by rotating frame_a along the axes defined
in parameter vector "sequence_angleStates" (default = {1,2,3}, i.e.,
the Cardan angle sequence) around the angles used as states.
For example, the default is to rotate the x-axis of frame_a
around angles[1], the new y-axis around angles[2] and the new z-axis
around angles[3], arriving at frame_b.
The quaternions have the slight disadvantage that there is a
non-linear constraint equation between the 4 quaternions.
Therefore, at least one non-linear equation has to be solved
during simulation. A tool might, however, analytically solve this
simple constraint equation. Using the 3 angles as states has the
disadvantage that there is a singular configuration in which a
division by zero will occur. If it is possible to determine in advance
for an application class that this singular configuration is outside
of the operating region, the 3 angles might be used as
states by setting useQuaternions = false.
In text books about 3-dimensional mechanics often 3 angles and the
angular velocity are used as states. This is not the case here, since
3 angles and their derivatives are used as states
(if useQuaternions = false). The reason
is that for real-time simulation the discretization formula of the
integrator might be "inlined" and solved together with the model equations.
By appropriate symbolic transformation the performance is
drastically increased if angles and their
derivatives are used as states, instead of angles and the angular
velocity.
If parameter
enforceStates is set to true (= the default)
in the "Advanced" menu,
then FreeMotion variables are forced to be used as states according
to the setting of parameters "useQuaternions" and
"sequence_angleStates".
In the following figure the animation of a FreeMotion
joint is shown. The light blue coordinate system is
frame_a and the dark blue coordinate system is
frame_b of the joint.
(here: r_rel_a_start = {0.5, 0, 0.5}, angles_start = {45, 45, 45}o).
CONNECTORS
