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

include/mujoco/mjmodel.h

@@ -126,7 +126,8 @@ typedef enum mjtTexture_ {        // type of texture
 
 typedef enum mjtIntegrator_ {     // integrator mode
   mjINT_EULER         = 0,        // semi-implicit Euler
-  mjINT_RK4                       // 4th-order Runge Kutta
+  mjINT_RK4,                      // 4th-order Runge Kutta
+  mjINT_IMPLICIT                  // implicit in velocity
 } mjtIntegrator;
 
 
@@ -568,6 +569,7 @@ struct mjModel_ {
 
   // sizes set after mjModel construction (only affect mjData)
   int nM;                         // number of non-zeros in sparse inertia matrix
+  int nD;                         // number of non-zeros in sparse derivative matrix
   int nemax;                      // number of potential equality-constraint rows
   int njmax;                      // number of available rows in constraint Jacobian
   int nconmax;                    // number of potential contacts in contact list