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Model predictive control in Python based on quadratic programming

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# qpmpc

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Model predictive control (MPC) in Python for optimal-control problems that are quadratic programs (QP). This includes linear time-invariant (LTI) and time-variant (LTV) systems with linear constraints. The corresponding QP has the form:

```math
\begin{array}{rl}
\underset{x_k, u_k}{\min} \quad & w_{t} \|x_N - x_{\mathit{goal}}\|_2^2 + w_x \sum_{k=0}^{N-1} \| x_k - x^{\mathrm{ref}}_k \|_2^2  + w_u \sum_{k=0}^{N-1} \| u_k - u^{\mathrm{ref}}_k \|^2_2 \\
\mathrm{s.t.} \quad & x_{k+1} = A_k x_k + B_k u_k \\
& C_k x_k + D_k u_k \leq e_k \\
& x_0 = x_{\mathit{init}}
\end{array}
```

This module is designed for prototyping. If you need performance, check out the [alternatives](#alternatives) below.

## Installation

```sh
pip install qpmpc
```

## Usage

This module defines a one-stop shop function:

```python
solve_mpc(problem: MPCProblem, solver: str) -> Plan
```

The [``MPCProblem``](https://scaron.info/doc/qpmpc/usage.html#qpmpc.mpc_problem.MPCProblem) defines the model predictive control problem (LTV system, LTV constraints, initial state and cost function to optimize) while the returned [``Plan``](https://scaron.info/doc/qpmpc/usage.html#qpmpc.plan.Plan) holds the state and input trajectories that result from optimizing the problem (if a solution exists). The ``solver`` string is used to select the backend [quadratic programming solver](https://github.com/stephane-caron/qpsolvers#solvers).

## Example

Let us define a triple integrator:

```python
    import numpy as np

horizon_duration = 1.0  # [s]
    N = 16  # number of discretization steps
    T = horizon_duration / N
    A = np.array([[1.0, T, T ** 2 / 2.0], [0.0, 1.0, T], [0.0, 0.0, 1.0]])
    B = np.array([T ** 3 / 6.0, T ** 2 / 2.0, T]).reshape((3, 1))
```

Suppose for the sake of example that acceleration is the main constraint acting on our system. We thus define an acceleration constraint ``|acceleration| <= max_accel``:

```python
    max_accel = 3.0  # [m] / [s] / [s]
    accel_from_state = np.array([0.0, 0.0, 1.0])
    C = np.vstack([+accel_from_state, -accel_from_state])
    e = np.array([+max_accel, +max_accel])
```

This leads us to the following linear MPC problem:

```python
    from qpmpc import MPCProblem

x_init = np.array([0.0, 0.0, 0.0])
    x_goal = np.array([1.0, 0.0, 0.0])
    problem = MPCProblem(
        transition_state_matrix=A,
        transition_input_matrix=B,
        ineq_state_matrix=C,
        ineq_input_matrix=None,
        ineq_vector=e,
        initial_state=x_init,
        goal_state=x_goal,
        nb_timesteps=N,
        terminal_cost_weight=1.0,
        stage_state_cost_weight=None,
        stage_input_cost_weight=1e-6,
    )
```

We can solve it with:

```python
    from qpmpc import solve_mpc

solution = solve_mpc(problem, solver="proxqp")
```

The solution holds complete state and input trajectories as stacked vectors. For instance, we can plot positions, velocities and accelerations as follows:

```python
    import matplotlib.pyplot as plt

t = np.linspace(0.0, horizon_duration, N + 1)
    X = solution.states
    positions, velocities, accelerations = X[:, 0], X[:, 1], X[:, 2]
    plt.ion()
    plt.plot(t, positions)
    plt.plot(t, velocities)
    plt.plot(t, accelerations)
    plt.grid(True)
    plt.legend(("position", "velocity", "acceleration"))
```

This example produces the following trajectory:

![2022-03-30-172206_1920x1080_scrot](https://user-images.githubusercontent.com/1189580/160871543-3734ec65-fe74-4a6f-8452-a877aa4050b1.png)

The behavior is a weighted compromise between reaching the goal state (weight ``1.0``) and keeping reasonable finite jerk inputs (weight ``1e-6``). The latter mitigate bang-bang accelerations but prevent fully reaching the goal within the horizon. See the [examples](examples/) folder for more examples.

## Areas of improvement

This module is incomplete with regards to the following points:

- Cost functions: can be extended to general linear stage cost functions
- Documentation: there are some undocumented functions
- Test coverage: only one end-to-end test

Check out the [contribution guidelines](CONTRIBUTING.md) if you are interested in lending a hand.

## Alternatives

You can also check out the following open-source libraries:

### Linear model predictive control

| Name                                                       | Systems               | Languages   | License      |
|------------------------------------------------------------|-----------------------|-------------|--------------|
| [Copra (LTV fork)](https://github.com/ANYbotics/copra)     | Linear time-variant   | C++, Python | BSD-2-Clause |
| [Copra (original)](https://github.com/jrl-umi3218/copra)   | Linear time-invariant | C++, Python | BSD-2-Clause |
| [mpc-interface](https://github.com/Gepetto/mpc-interface)  | Linear time-variant   | C++, Python | BSD-2-Clause |
| [pyMPC](https://github.com/forgi86/pyMPC)                  | Linear time-variant   | Python      | MIT          |

### Nonlinear model predictive control

| Name                                                       | Systems               | Languages           | License      |
|------------------------------------------------------------|-----------------------|---------------------|--------------|
| [acados](https://github.com/acados/acados)                 | Nonlinear             | C++, Matlab, Python | BSD-2-Clause |
| [Aligator](https://github.com/Simple-Robotics/aligator/)   | Nonlinear             | C++, Python         | BSD-2-Clause |
| [Crocoddyl](https://github.com/loco-3d/crocoddyl)          | Nonlinear             | C++, Python         | BSD-3-Clause |

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