Improve documentation of solver parameters.

PiperOrigin-RevId: 595723619
Change-Id: I00b67eaec02d2e85ce60fb44eb3e582ce036be3a

doc/modeling.rst

@@ -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

test/engine/engine_core_constraint_test.cc

@@ -161,6 +161,62 @@ TEST_F(CoreConstraintTest, WeldRotJacobian) {
   mj_deleteModel(model);
 }
 
+// test formulas for penetration at rest
+TEST_F(CoreConstraintTest, RestPenetration) {
+  constexpr char xml[] = R"(
+  <mujoco>
+    <worldbody>
+      <geom type="plane" size="1 1 1"/>
+      <body pos="0 0 .2">
+        <joint type="slide" axis="0 0 1"/>
+        <geom size=".1"/>
+      </body>
+    </worldbody>
+  </mujoco>
+  )";
+  mjModel* model = LoadModelFromString(xml);
+  ASSERT_THAT(model, testing::NotNull());
+  mjtNum gravity = -model->opt.gravity[2];
+  mjtNum damping_ratio = 0.8;
+  mjData* data = mj_makeData(model);
+
+  for (const mjtNum reference : {-100.0, -10.0, 0.1, 0.01}) {
+    for (const mjtNum impedance : {0.3, 0.9, 0.99}) {
+      // set solimp
+      for (int i=0; i < model->ngeom; i++) {
+        model->geom_solimp[i*mjNIMP + 0] = impedance;
+        model->geom_solimp[i*mjNIMP + 1] = impedance;
+      }
+
+      // set solref
+      for (int i=0; i < model->ngeom; i++) {
+        model->geom_solref[i*mjNREF + 0] = reference;
+        model->geom_solref[i*mjNREF + 1] = reference < 0 ? -10 : damping_ratio;
+      }
+
+      // simulate for 50 seconds
+      mj_resetData(model, data);
+      while (data->time < 50) {
+        mj_step(model, data);
+      }
+
+      mjtNum depth = -data->contact[0].dist;
+      mjtNum expected_depth;
+      if (reference < 0) {
+        expected_depth = gravity * (1 - impedance) / -reference;
+      } else {
+        mjtNum tc_dr = reference * damping_ratio;
+        expected_depth = gravity * (1 - impedance) * tc_dr * tc_dr;
+      }
+
+      EXPECT_THAT(depth, DoubleNear(expected_depth, 1e-10));
+    }
+  }
+
+  mj_deleteData(data);
+  mj_deleteModel(model);
+}
+
 static const char* const kDoflessContactPath =
     "engine/testdata/core_constraint/dofless_contact.xml";
 static const char* const kDoflessTendonFrictionalPath =