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

include/mujoco/mujoco.h

@@ -1000,7 +1000,7 @@ MJAPI int mju_cholUpdate(mjtNum* mat, mjtNum* x, int n, int flg_plus);
 MJAPI int mju_eig3(mjtNum eigval[3], mjtNum eigvec[9], mjtNum quat[4], const mjtNum mat[9]);
 
 
-//---------------------- Miscellaneous --------------------------------------------------
+//---------------------- Miscellaneous -------------------------------------------------------------
 
 // Muscle active force, prm = (range[2], force, scale, lmin, lmax, vmax, fpmax, fvmax).
 MJAPI mjtNum mju_muscleGain(mjtNum len, mjtNum vel, const mjtNum lengthrange[2],
@@ -1080,6 +1080,13 @@ MJAPI char* mju_strncpy(char *dst, const char *src, int n);
 MJAPI mjtNum mju_sigmoid(mjtNum x);
 
 
+//---------------------- Derivatives ---------------------------------------------------------------
+
+// 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);
+
+
 #if defined(__cplusplus)
 }
 #endif