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_core_smooth.h

@@ -48,10 +48,18 @@ void mj_crbSkip(const mjModel* m, mjData* d, int skipsimple);
 // composite rigid body inertia algorithm
 MJAPI void mj_crb(const mjModel* m, mjData* d);
 
+// sparse L'*D*L factorizaton of inertia-like matrix M, assumed spd
+MJAPI void mj_factorI(const mjModel* m, mjData* d, const mjtNum* M, mjtNum* qLD, mjtNum* qLDiagInv,
+                      mjtNum* qLDiagSqrtInv);
+
 // sparse L'*D*L factorizaton of the inertia matrix M, assumed spd
 MJAPI void mj_factorM(const mjModel* m, mjData* d);
 
 // sparse backsubstitution:  x = inv(L'*D*L)*y
+MJAPI void mj_solveLD(const mjModel* m, mjData* d, mjtNum* x, const mjtNum* y, int n,
+                      const mjtNum* qLD, const mjtNum* qLDiagInv);
+
+// sparse backsubstitution:  x = inv(L'*D*L)*y, use factorization in d
 MJAPI void mj_solveM(const mjModel* m, mjData* d, mjtNum* x, const mjtNum* y, int n);
 
 // half of sparse backsubstitution:  x = sqrt(inv(D))*inv(L')*y