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Improve documentation of solver parameters.

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PiperOrigin-RevId: 595723619
Change-Id: I00b67eaec02d2e85ce60fb44eb3e582ce036be3a
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Yuval Tassa
2024-01-04 09:16:52 -08:00
committed by Copybara-Service
parent 63ec34f9f8
commit 7ce05f4957
2 changed files with 72 additions and 7 deletions
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doc/modeling.rst
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@@ -272,6 +272,7 @@ approximately
.. math::
\ac + d \cdot (b v + k r) = (1 - d)\cdot \au
:label: eq:constraint
Again, the parameters that are under the user's control are :math:`d, b, k`. The remaining quantities are functions of
the system state and are computed automatically at each time step.
@@ -343,12 +344,9 @@ Next we explain the setting of the stiffness :math:`k` and damping :math:`b` whi
.. admonition:: Intuitive description of the **reference acceleration**
The *reference acceleration* :math:`\ar` determines the **motion that constraint is trying to achieve** in
order to rectify violation. For example, consider a contact between a motionless free body pulled down by gravity
onto a static plane geom. Since there is no motion, the penetration will be entirely determined by the impedance
while the reference has no effect. Now imagine that the body is dropped onto the plane. Upon impact the constraint
will generate a normal force which attempts to rectify the penetration using a particular motion; this motion is
the reference acceleration.
The *reference acceleration* :math:`\ar` determines the **motion that constraint is trying to achieve** in order to
rectify violation. Imagine a body dropped onto the plane. Upon impact the constraint will generate a normal force
which attempts to rectify the penetration using a particular motion; this motion is the reference acceleration.
Another way of understanding the reference acceleration is to think of the unmodeled deformation variables
described in the :ref:`Computation chapter<soPrimal>`. Imagine two bodies pressed together, leading to deformation at
@@ -393,7 +391,12 @@ and the damping ratio is ignored. Equivalently, in the direct format, the :math:
can go unstable. This is enforced internally, unless the :ref:`refsafe<option-flag-refsafe>` attribute of :ref:`flag
<option-flag>` is set to false. The :math:`\text{dampratio}` parameter would normally be set to 1, corresponding to
critical damping. Smaller values result in under-damped or bouncy constraints, while larger values result in
over-damped constraints.
over-damped constraints. Combining the above formula with :eq:`eq:constraint`, we can derive the following result.
If the reference acceleration is given using the positive number format and the impedance is constant
:math:`d = d_0 = d_\text{width}`, then the penetration depth at rest is
.. math::
r = \au \cdot (1 - d) \cdot \text{timeconst}^2 \cdot \text{dampratio}^2
Next we describe the direct format where the two numbers are :math:`(-\text{stiffness}, -\text{damping})`. This
allows direct control over restitution in particular. We still apply some scaling so that the same numbers can be
@@ -406,6 +409,12 @@ and the damping ratio is ignored. Equivalently, in the direct format, the :math:
k &= \text{stiffness} \cdot d(r) / d_\text{width}^2 \\
\end{aligned}
Similarly to the above derivation, if the reference acceleration is given using the negative number format and the
impedance is constant, then the penetration depth at rest is
.. math::
r = \au \cdot (1 - d) \cdot \text{stiffness}
.. tip::
In the positive-value default format, the :math:`\text{timeconst}` parameter controls constraint **softness**.
It is specified in units of time and means "how quickly is the constraint trying to resolve the violation". Larger