Add implicit integrator.

Added analytic derivatives of smooth (unconstrained) dynamics forces, with respect to velocities:
  - Centripetal and Coriolis forces computed by the Recursive Newton-Euler algorithm.
  - Damping and fluid-drag passive forces.
  - Actuation forces.

A new implicit-in-velocity integrator is implemented using the analytic derivatives. This integrator lies between the Euler and Runge Kutta integrators in terms of both stability and computational cost.

PiperOrigin-RevId: 450377010
Change-Id: Ie192b441876c22e732fb749333926f296e0a09cc

test/engine/testdata/derivative/damped_actuators.xml

@@ -0,0 +1,24 @@
+<mujoco>
+  <worldbody>
+    <body name="damping in the joints">
+      <joint type="slide" axis="0 0 1" damping="10"/>
+      <geom size=".03"/>
+      <body>
+        <joint axis="0 1 0" damping=".1"/>
+        <geom type="capsule" size=".01" fromto="0 0 0 .1 0 0"/>
+      </body>
+    </body>
+    <body name="damping in the actuators" pos="0 0.1 0">
+      <joint name="slide" type="slide" axis="0 0 1"/>
+      <geom size=".03"/>
+      <body>
+        <joint name="hinge" axis="0 1 0"/>
+        <geom type="capsule" size=".01" fromto="0 0 0 .1 0 0"/>
+      </body>
+    </body>
+  </worldbody>
+  <actuator>
+    <general joint="slide" biastype="affine" biasprm="0 0 -10"/>
+    <general joint="hinge" biastype="affine" biasprm="0 0 -0.1"/>
+  </actuator>
+</mujoco>

test/engine/testdata/derivative/energy_conserving_pendulum.xml

@@ -0,0 +1,25 @@
+<mujoco>
+  <option integrator="implicit">
+    <flag constraint="disable" energy="enable"/>
+  </option>
+  <worldbody>
+    <light pos="0 0 1"/>
+    <geom type="plane" size="1 1 .01" pos="0 0 -1"/>
+    <body pos="0.15 0 0">
+      <joint type="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 .1 0"/>
+          </body>
+        </body>
+      </body>
+    </body>
+  </worldbody>
+</mujoco>

test/engine/testdata/derivative/tumbling_thin_object.xml

@@ -0,0 +1,17 @@
+<mujoco>
+  <option density="1.225" viscosity="1.8e-5" wind="0 0 1" integrator="implicit"/>
+
+  <worldbody>
+    <light pos="0 0 1"/>
+    <geom type="plane" size="1 1 .01" pos="0 0 -1"/>
+    <body>
+      <freejoint/>
+      <body>
+        <geom type="box" size=".025 .01 0.0001" pos=".025 0 0" euler="20 0 0" mass="1e-4"/>
+      </body>
+      <body>
+        <geom type="box" size=".025 .01 0.0001" pos="-.025 0 0" euler="-19 0 0" mass="1e-4"/>
+      </body>
+    </body>
+  </worldbody>
+</mujoco>