Using x_pre to add in the current x constraints

User Questions
1

Hello! I’d like to set a constraint, and it’s a path constraint. That is to say, each x(k) is constrained by an equation involving x(k - 1). How can I define this problem? I find that it’s not easy for me to set up the constraint function.

2

I use this strategy to update the parameter p. However, if I do it this way, I won’t be able to solve it all at once:

    for i in range(20):
        # Update triangle parameters
        acados_solver.set(i, "p", np.array([x_prev[0], x_prev[1], theta_prev, L]))

        # Solve the OCP
        status = acados_solver.solve()

        if status != 0:
            print(f"Solver failed at step {i} with status {status}")
            break

        # Retrieve current state and update x_prev and theta_prev
        x_prev = acados_solver.get(i, "x")
        theta_prev = np.arctan2(acados_solver.get(i, "u")[1], acados_solver.get(i, "u")[0])
3

Hi, is the Index k here referring to the stage or the control problem instant in an NMPC scheme?

4

Hi kaethe, Index k refers to the time step or stage within the prediction horizon.
The possible method that occurred to me is to expand the state space and then store xk and xk-1. However, I don’t know how to define this model.

5

Yes that is the main idea!

If you use discrete dynamics, this is very simple. The augmented model is simply \tilde{x}_k = (x_k, x_{k-1}) with \tilde{x}_{k+1} = \begin{bmatrix} f(x_k, u_k) \\ x_k \end{bmatrix}.

If you use an integrator, then you need to reformulate the original dynamics \dot{x} = f(x, u) as follows:
\dot{\tilde{x}} = \begin{bmatrix} v \\ \frac{\partial f}{\partial t}(x, u) \end{bmatrix} where \tilde{x} = (x, v). Often you want to do the same augmentation for the controls (i.e. include a = \dot{u} in your state vector).

Best, Katrin

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6

Hello! Kaethe, thank you for your response.

I am currently facing a dilemma in choosing the type of path constraint: whether to use the h type or the g type. The h type represents nonlinear constraints, while the g type represents linear ones. I noticed that in the OCP interface, the method constraints_set(stage_, field_, value_, api='warn') allows setting parameters for path constraints, where p corresponds to h type constraints and C and D correspond to g type constraints.

So, does this mean that when defining h type constraints, I need to set p to make the constraints different at each step? Similarly, for g type constraints, I need to set C and D.

However, I have another concern. If the constraint is nonlinear and the mathematical model of the constraint differs at each step, it seems impossible to define a new constraint function merely by changing p. This is because con_h_expr is defined globally in the AcadosModel, and it cannot be redefined for a specific stage. Therefore, I am wondering if, in scenarios where the time horizon is divided into segments (e.g., with a step size of 20, where h is h1 for the first 10 steps and h2 for the next 10 steps), this approach would not work.

7

Yes nonlinear constraints might be adapted stagewise by setting the parameters p. If you need structurally different constraints, you can use a multi-phase formulation.

See preprint and example.

8

Thank u ! Kaethe! It is a fantastic framework！

1 Like