d0: position
d1: velocity

sim1: pybullet
sim2: z3 (using equations of motion)

Run the following files in order

1. pendulum_manipulator.py: This file loads 2 single-link URDFs and is run n times. The position and velocity at each time-step is written to a file position_velocity_pybullet_data.txt. The format of the data written is as follows:

{
  "0": {
    "position": 1,
    "velocity": 0
  },
  "179": {
    "position": 1.0125164889847702,
    "velocity": 0.37573403437650427
  },
  "536": {
    "position": 1.0552359296886857,
    "velocity": 0.8064906280500382
  },
  "894": {
    "position": 1.1087021975111322,
    "velocity": 1.143005540634034
  }
}

The position and velocity at time-step 0, 179, 536 and 894 are given above.

2. pendulum_z3_synthesize_mdp.py: This file uses the data in the file generated above and using equations of motion, motion damping proxy is generated. This is done by using the following z3 constraint: The difference between the velocity generated using equations of motions and velocity recorded for the pybullet model should be less than 0.5. Once the mdp is generated, it is stored to a file: mdp_list.txt. The format of this file is as follows:

1.9998504291489958
1.9997380314304503
1.9996091741698674
1.999508508526166
1.9995006679764413
1.9993568921816436
1.999348755695386
1.9992416713598171

3. pendulum_z3_collect_data.py: This file is run to finally collate the data generated in Step 2 and this data is written to a file position_velocity_z3_data.txt. The motor damping proxy generated in step 2 is used to generate the position and velocity of the pendulum by plugging it into the equations of motion. The format of this file is as follows:

{
  "0": {
    "position": 0,
    "velocity": -0.9998504291489958,
    "mdp": 1.9998504291489958
  },
  "179": {
    "position": 179,
    "velocity": 11.656213574036299,
    "mdp": 1.9997380314304503
  },
  "536": {
    "position": 715,
    "velocity": -514.7729785773156,
    "mdp": 1.9996091741698674
  },
  "894": {
    "position": 1609,
    "velocity": 430.9686954800673,
    "mdp": 1.999508508526166
  }
}

Note that the timesteps at which this data is collected is same as the ones at which data was collect from the pybullet models.

4. pendulum_compare.py: Run this file to load the original pybullet single link URDF model using the same data that was stored in position_velocity_z3_pybullet.txt . The model generated using z3 is also loaded from the position_velocity_z3_data.txt.

---

Commands to run all the files:

python3 pendulum_manipulator.py
python3 pendulum_z3_synthesize_mdp.py
python3 pendulum_compare.py