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_forward.h

@@ -43,21 +43,27 @@ MJAPI void mj_step2(const mjModel* m, mjData* d);
 MJAPI void mj_forward(const mjModel* m, mjData* d);
 
 // forward dynamics with skip; skipstage is mjtStage
-MJAPI void mj_forwardSkip(const mjModel* m, mjData* d,
-                          int skipstage, int skipsensor);
+MJAPI void mj_forwardSkip(const mjModel* m, mjData* d, int skipstage, int skipsensor);
+
 
 
 //-------------------------------- integrators -----------------------------------------------------
 
-// Euler integrator, semi-implicit in velocity
-MJAPI void mj_Euler(const mjModel* m, mjData* d);
-
 // Runge Kutta explicit order-N integrator
 MJAPI void mj_RungeKutta(const mjModel* m, mjData* d, int N);
 
+// Euler integrator, semi-implicit in velocity
+MJAPI void mj_Euler(const mjModel* m, mjData* d);
+
+// Euler integrator, semi-implicit in velocity, possibly skipping factorisation
+MJAPI void mj_EulerSkip(const mjModel* m, mjData* d, int skipfactor);
+
 // fully implicit in velocity
 MJAPI void mj_implicit(const mjModel *m, mjData *d);
 
+// fully implicit in velocity, possibly skipping factorization
+MJAPI void mj_implicitSkip(const mjModel *m, mjData *d, int skipfactor);
+
 
 //-------------------------------- solver components -----------------------------------------------