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

src/engine/engine_derivative.h

@@ -24,7 +24,7 @@ extern "C" {
 #endif
 
 // analytical derivative of smooth forces w.r.t velocities:
-//   d->qDeriv = d (qfrc_actuator + qfrc_passive - qfrc_bias) / d qvel.
+//   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
@@ -36,6 +36,12 @@ 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);
 
+// advance simulation using control callback, skipstage is mjtStage
+MJAPI void mj_stepSkip(const mjModel* m, mjData* d, int skipstage, int skipsensor);
+
+// finite differenced state-transition and control-transition matrices dy = A*dx + B*du
+MJAPI void mjd_transitionFD(const mjModel* m, mjData* d, mjtNum eps, mjtByte centered,
+                            mjtNum* A, mjtNum* B);
 
 #ifdef __cplusplus
 }