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        "%matplotlib inline"
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      "source": [
        "\nExample demonstrating approaches for adding and handling model noise\n"
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        "import matplotlib.pyplot as plt\n\nfrom progpy.models.thrown_object import ThrownObject\n\ndef run_example():\n    # Define future loading\n    def future_load(t=None, x=None):  \n        # The thrown object model has no inputs- you cannot load the system (i.e., affect it once it's in the air)\n        # So we return an empty input container\n        return m.InputContainer({})\n\n    # Define configuration for simulation\n    config = {\n        'threshold_keys': 'impact', # Simulate until the thrown object has impacted the ground\n        'dt': 0.005, # Time step (s)\n        'save_freq': 0.5, # Frequency at which results are saved (s)\n    }\n\n    # Define a function to print the results - will be used later\n    def print_results(simulated_results):\n        # Print results\n        print('states:')\n        for (t,x) in zip(simulated_results.times, simulated_results.states):\n            print('\\t{:.2f}s: {}'.format(t, x))\n\n        print('outputs:')\n        for (t,x) in zip(simulated_results.times, simulated_results.outputs):\n            print('\\t{:.2f}s: {}'.format(t, x))\n\n        print('\\nimpact time: {:.2f}s'.format(simulated_results.times[-1]))\n        # The simulation stopped at impact, so the last element of times is the impact time\n\n        # Plot results\n        simulated_results.states.plot()\n\n    # Ex1: No noise\n    m = ThrownObject(process_noise = False)\n    simulated_results = m.simulate_to_threshold(future_load, **config)\n    print_results(simulated_results)\n    plt.title('Ex1: No noise')\n\n    # Ex2: with noise - same noise applied to every state\n    process_noise = 15\n    m = ThrownObject(process_noise = process_noise)  # Noise with a std of 0.5 to every state\n    print('\\nExample without same noise for every state')\n    simulated_results = m.simulate_to_threshold(future_load, **config)\n    print_results(simulated_results)\n    plt.title('Ex2: Basic Noise')\n\n    # Ex3: noise- more noise on position than velocity\n    process_noise = {'x': 30, 'v': 1}\n    m = ThrownObject(process_noise = process_noise) \n    print('\\nExample with more noise on position than velocity')\n    simulated_results = m.simulate_to_threshold(future_load, **config)\n    print_results(simulated_results)\n    plt.title('Ex3: More noise on position')\n\n    # Ex4: noise- Ex3 but uniform\n    process_noise_dist = 'uniform'\n    model_config = {'process_noise_dist': process_noise_dist, 'process_noise': process_noise}\n    m = ThrownObject(**model_config) \n    print('\\nExample with more uniform noise')\n    simulated_results = m.simulate_to_threshold(future_load, **config)\n    print_results(simulated_results)\n    plt.title('Ex4: Ex3 with uniform dist')\n\n    # Ex5: noise- Ex3 but triangle\n    process_noise_dist = 'triangular'\n    model_config = {'process_noise_dist': process_noise_dist, 'process_noise': process_noise}\n    m = ThrownObject(**model_config) \n    print('\\nExample with triangular process noise')\n    simulated_results = m.simulate_to_threshold(future_load, **config)\n    print_results(simulated_results)\n    plt.title('Ex5: Ex3 with triangular dist')\n\n    # Ex6: Measurement noise\n    # Everything we've done with process noise, we can also do with measurement noise.\n    # Just use 'measurement_noise' and 'measurement_noise_dist' \n    measurement_noise = {'x': 20}  # For each output\n    measurement_noise_dist = 'uniform'\n    model_config = {'measurement_noise_dist': measurement_noise_dist, 'measurement_noise': measurement_noise}\n    m = ThrownObject(**model_config) \n    print('\\nExample with measurement noise')\n    print('- Note: outputs are different than state- this is the application of measurement noise')\n    simulated_results = m.simulate_to_threshold(future_load, **config)\n    print_results(simulated_results)\n    plt.title('Ex6: Measurement noise')\n\n    # Ex7: OK, now for something a little more complicated. Let's try proportional noise on v only (more variation when it's going faster)\n    # This can be used to do custom or more complex noise distributions\n    def apply_proportional_process_noise(self, x, dt = 1):\n        x['v'] -= dt*0.5*x['v']\n        return x\n    model_config = {'process_noise': apply_proportional_process_noise}\n    m = ThrownObject(**model_config)\n    print('\\nExample with proportional noise on velocity')\n    simulated_results = m.simulate_to_threshold(future_load, **config)\n    print_results(simulated_results)\n    plt.title('Ex7: Proportional noise on velocity')\n\n    print('\\nNote: If you would like noise to be applied in a repeatable manner, set the numpy random seed to a fixed value')\n    print('e.g., numpy.random.seed(42)')\n    plt.show()\n\n# This allows the module to be executed directly \nif __name__=='__main__':\n    run_example()"
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