Add efficient finite-difference Jacobians of mj_step.

- Add `qH` and `qHDiagInv` to `mjData` to save factorized modified inertia.
- Add `mj_EulerSkip`, `mj_implicitSkip`, to `engine_forward.c`.
- Using the above functions, implement `mj_stepSkip` in `engine_derivative.c`.
- Add `mjd_stepFD` and `mjd_transitionFD` to `engine_derivative.c` to compute `mj_step` Jacobians.
  - Exploit "Skip" functionality for speed.
  - Correctly handle quaternion derivatives.
  - Handle warmstarts and control limits.

PiperOrigin-RevId: 456584811
Change-Id: Iee8541f11e7b66feb8f431cb102d9bbe65461f79

test/engine/testdata/derivative/damped_pendulum.xml

@@ -0,0 +1,31 @@
+<mujoco>
+
+  <default>
+    <joint damping=".01" frictionloss=".1"/>
+  </default>
+
+  <worldbody>
+    <light pos="0 0 1"/>
+    <geom type="plane" size="1 1 .01" pos="0 0 -1"/>
+    <body pos="0.15 0 0">
+      <joint name="hinge" axis="0 1 0"/>
+      <geom type="capsule" size="0.02" fromto="0 0 0 .1 0 0"/>
+      <body pos="0.1 0 0">
+        <joint type="slide" axis="1 0 0" stiffness="200"/>
+        <geom type="capsule" size="0.015" fromto="-.1 0 0 .1 0 0"/>
+        <body pos=".1 0 0">
+          <joint type="ball"/>
+          <geom type="box" size=".02" fromto="0 0 0 0 .1 0"/>
+          <body pos="0 .1 0">
+            <joint axis="1 0 0"/>
+            <geom type="capsule" size="0.02" fromto="0 0 0 0 0 .1"/>
+          </body>
+        </body>
+      </body>
+    </body>
+  </worldbody>
+
+  <actuator>
+    <motor joint="hinge"/>
+  </actuator>
+</mujoco>

test/engine/testdata/derivative/linear.xml

@@ -0,0 +1,28 @@
+<mujoco>
+  <!--  linear dynamical system with fixed, anaylticaly computable derivatives  -->
+  <default>
+    <joint axis="1 0 0" damping=".01" stiffness=".1"/>
+    <geom type="box" size=".05 .05 .05"/>
+  </default>
+
+  <worldbody>
+    <light pos="0 0 1"/>
+    <body>
+      <joint name="joint0"/>
+      <geom/>
+      <body pos=".15 0 0">
+        <joint name="joint1"/>
+        <geom/>
+        <body pos=".15 0 0">
+          <joint/>
+          <geom/>
+        </body>
+      </body>
+    </body>
+  </worldbody>
+
+  <actuator>
+    <motor joint="joint0" ctrllimited="true" ctrlrange="-1 1"/>
+    <motor joint="joint1" ctrllimited="true" ctrlrange="-1 1"/>
+  </actuator>
+</mujoco>