Hybrid ODE Simulator

Hybrid ODE Sim is a Python package designed to solve systems of ordinary differential equations (ODEs) that involve both continuous and discrete dynamics. That is, dynamics of the forms:

$$ \dot{x} = f(t, x) \tag{(1) Continuous} $$

$$ x_{k+1} = f(t_k, x_k) \tag{(2) Discrete} \quad $$

This solver is particularly useful for simulations that require handling events or state changes at specific points in time while integrating continuous dynamics in between these events. This commonly occurs in simulating robots and their control/planning systems. The package is meant to be very simple and leverages common numerical integrators to provide accurate and efficient solutions for hybrid systems.

Contents

The package contains four main building blocks: ContinuousTimeModel, DiscreteTimeModel, ModelGraph, and Simulator.

Examples

For examples of how to use different kinds of models and the simulator class, please see the test notebooks.

Discrete Time Models

Each DiscreteTimeModel is evaluated at an integral sample rate, say 5 times per second / 5 Hz. Implementing a DiscreteTimeModel requires overriding the function:

def discrete_dynamics(self, t: float, y: Any) -> Any:
    y_next = ...
    return y_next

DiscreteTimeModel constructors are provided with an initial state y0, a string name, their integral evaluation frequency sample_rate, an optional params object, and an optional logging level.

Continuous Time Models

Between consecutive evaluations of a DiscreteTimeModel, all ContinuousTimeModel instances are integrated simultaneously using a numerical integrator like RK4 (other integrators are available as well). Implementing a ContinuousTimeModel requires overriding the function:

def continuous_dynamics(t: float, y: np.ndarray) -> np.ndarray:
    ydot = ...
    return ydot

ContinuousTimeModel constructors are provided with an initial state y0, a string name, an optional params object, and an optional logging level.

The Model Graph

When multiple DiscreteTimeModel instances are to be evaluated at the same timestep (e.g. two discrete models both run at 100 Hz), the ModelGraph imposes the evaluation order. To indicate that a particular model model_A should receive feedback from / have access to the state model_B, we have two conventions:

1. We use the convention model_B.inputs_to(model_A), where model_B, model_B are DiscreteTimeModel, to indicate that the directed edge model_B -> model_A exists in the topological sorting of the ModelGraph in the case that model_B and model_A are evaluated during the same timestep. The .inputs_to function is only to be used when model_A and model_B are both DiscreteTimeModel.

2. We use the convention model_C.feedback_from(model_D) to get access to model_D's state from the dynamics function of model_C, but this does not impose any evaluation order.

Think of inputs_to as a forward edge in a Simulink model, while feedback_from is a reverse edge.

You may use the self.input_models instance member to access inputs' states. This is a dict with keys as model names (strings) and values being that model's current/most recent state.

Note: There is a zero-order hold on discrete system states between consecutive evaluations if the discrete model's state is queried by other systems in that in-between time.

The Simulator

The simulator ties everything together into one class which takes as input a model graph and an option for the integrator to be used. The Simulator class exposes the simulate function which takes as input the time range (with start and end times as rational numbers via the Fraction type) to simulate.

Installation

You can install hybrid_ode_sim as a local pip package and use the provided library as you would any other.

Setup

Clone the repository locally:

git clone https://github.com/micahreich/hybrid_ode_sim.git  # for https
git clone git@github.com:micahreich/hybrid_ode_sim.git      # for ssh

Install pip requirements from the requirements.txt

Now, when you wish to use hybrid_ode_sim, ideally from within a virtual environment, just run:

pip install -e /path/to/hybrid_ode_sim # install `hybrid_ode_sim` as a pip package along with requirements

And that's it!

Available ODE Solvers

Adaptive Runge-Kutta Methods

An instance of an adaptive Runge-Kutta implementation can be provided as the integrator to a simulator. Currently implemented are RK23 and RK54, but adding other methods is as easy as providing the Butcher Tableau. Other integrators can be implemented by inheriting from the Integrator class.

Fixed-Step Adaptive Runge-Kutta Methods

An instance of a fixed-step Runge-Kutta implementation can be provided as the integrator to a simulator. Currently implemented is RK4, but adding other methods is as easy as providing the Butcher Tableau. Other integrators can be implemented by inheriting from the Integrator class.

Rendering

The package also includes some rendering tooling which animates the results of simulations. Matplotlib animations can be created by creating and rendering PlotElements which grab data from system instances. Multiple plot elements can be combined into one PlotEnvironment, rendered, and also saved as a video or GIF.

TODO

• Implement zero-crossing detection for early stopping
• Benchmark performance against scipy.solve_ivp