Tutorial

Setup

See the Python installation instructions.

Introduction

A system with position and velocity states and an acceleration input is an example of a double integrator. We want to go from 0 m at rest to 10 m at rest in the minimum time while obeying the velocity limit (-1, 1) and the acceleration limit (-1, 1).

The model for our double integrator is ẍ = u where x is the vector [position; velocity] and u is the acceleration. The velocity constraints are -1 ≤ x(1) ≤ 1 and the acceleration constraints are -1 ≤ u ≤ 1.

Importing required libraries

from jormungandr.optimization import OptimizationProblem
import numpy as np

Initializing a problem instance

First, we need to make a problem instance.

T = 5.0  # s
dt = 0.005  # 5 ms
N = int(T / dt)
 
r = 2.0
 
problem = OptimizationProblem()

Creating decision variables

First, we need to make decision variables for our state and input.

# 2x1 state vector with N + 1 timesteps (includes last state)
X = problem.decision_variable(2, N + 1)
 
# 1x1 input vector with N timesteps (input at last state doesn't matter)
U = problem.decision_variable(1, N)

By convention, we use capital letters for the variables to designate matrices.

Applying constraints

Now, we need to apply dynamics constraints between timesteps.

# Kinematics constraint assuming constant acceleration between timesteps
for k in range(N):
    p_k1 = X[0, k + 1]
    v_k1 = X[1, k + 1]
    p_k = X[0, k]
    v_k = X[1, k]
    a_k = U[0, k]
 
    # pₖ₊₁ = pₖ + vₖt + 1/2aₖt²
    problem.subject_to(p_k1 == p_k + v_k * dt + 0.5 * a_k * dt**2)
 
    # vₖ₊₁ = vₖ + aₖt
    problem.subject_to(v_k1 == v_k + a_k * dt)

Next, we'll apply the state and input constraints.

# Start and end at rest
problem.subject_to(X[:, 0] == np.array([[0.0], [0.0]]))
problem.subject_to(X[:, N] == np.array([[r], [0.0]]))
 
# Limit velocity
problem.subject_to(-1 <= X[1, :])
problem.subject_to(X[1, :] <= 1)
 
# Limit acceleration
problem.subject_to(-1 <= U)
problem.subject_to(U <= 1)

Specifying a cost function

Next, we'll create a cost function for minimizing position error.

# Cost function - minimize position error
J = 0.0
for k in range(N + 1):
    J += (r - X[0, k]) ** 2
problem.minimize(J)

The cost function passed to Minimize() should produce a scalar output.

Solving the problem

Now we can solve the problem.

problem.solve()

The solver will find the decision variable values that minimize the cost function while satisfying the constraints.

Accessing the solution

You can obtain the solution by querying the values of the variables like so.

position = X.value(0, 0)
velocity = X.value(1, 0)
acceleration = U.value(0)

Other applications

In retrospect, the solution here seems obvious: if you want to reach the desired position in the minimum time, you just apply positive max input to accelerate to the max speed, coast for a while, then apply negative max input to decelerate to a stop at the desired position. Optimization problems can get more complex than this though. In fact, we can use this same framework to design optimal trajectories for a drivetrain while satisfying dynamics constraints, avoiding obstacles, and driving through points of interest.