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

src/engine/engine_derivative.h

@@ -0,0 +1,44 @@
+// Copyright 2022 DeepMind Technologies Limited
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+#ifndef MUJOCO_SRC_ENGINE_ENGINE_DERIVATIVE_H_
+#define MUJOCO_SRC_ENGINE_ENGINE_DERIVATIVE_H_
+
+#include <mujoco/mjdata.h>
+#include <mujoco/mjexport.h>
+#include <mujoco/mjmodel.h>
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+// analytical derivative of smooth forces w.r.t velocities:
+//   d->qDeriv = d (qfrc_actuator + qfrc_passive - qfrc_bias) / d qvel.
+MJAPI void mjd_smooth_vel(const mjModel* m, mjData* d);
+
+// centered finite difference approximation to mjd_smooth_vel
+MJAPI void mjd_smooth_velFD(const mjModel* m, mjData* d, mjtNum eps);
+
+// add (d qfrc_passive / d qvel) to DfDv
+MJAPI void mjd_passive_vel(const mjModel* m, mjData* d, mjtNum* DfDv);
+
+// add forward finite difference approximation of (d qfrc_passive / d qvel) to DfDv
+MJAPI void mjd_passive_velFD(const mjModel* m, mjData* d, mjtNum eps, mjtNum* DfDv);
+
+
+#ifdef __cplusplus
+}
+#endif
+
+#endif // MUJOCO_SRC_ENGINE_ENGINE_DERIVATIVE_H_