How to implement a minimum-time OCP with auxiliary algebraic variables appearing in constraints (DAE form)?

ebu · October 13, 2025, 8:16pm

Hi everyone

I’m using the Python interface of acados and would like to implement a minimum-time optimal control problem that includes auxiliary algebraic variables appearing in the constraints.

In this problem, the final time T should be an optimisation variable solved by ACADOS — not a fixed parameter.

The setup is inspired by “Optimal Path Planning of CAVs (Connected and Autonomous Vehicles)”, where additional algebraic variables (λ, s) are introduced to handle collision-avoidance constraints.

I want acados to optimise the final time T together with the state, control, and algebraic variables, i.e.

The algebraic variables z only appear in the constraint equations (e.g., contact or collision-avoidance constraints), not in the system dynamics.

What I need help with

1. How can I make `T` an optimisation variable?

Should it be treated as an algebraic variable, a control variable with zero dynamics, or handled through a time-scaling trick?

2. How can I include algebraic variables (`z`) that appear only in constraints?

Do they have to be part of the implicit DAE formulation (f_impl_expr), or can they be declared separately as additional optimisation variables?

3. Are there any examples or templates showing this kind of setup?

Ideally, something that combines a minimum-time objective with auxiliary algebraic constraints (DAE or implicit form).

Thanks a lot for your help.

kaethe · October 14, 2025, 8:53am

Hi

You will need to use time scaling. Here is an example

github.com/acados/acados

examples/acados_python/unconstrained_ocps/hour_glass_p2p_motion/time_optimal_simple_bicycle_model.py

main

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Yes you need to use f_impl_expr to define the DAE. Here is an example using algebraic variables

github.com/acados/acados

examples/acados_python/rsm_example/main.py

acc8199c1


      
          # set up states
          psi_d = SX.sym('psi_d')
          psi_q = SX.sym('psi_q')
          x = vertcat(psi_d, psi_q)
          
          # set up controls
          u_d = SX.sym('u_d')
          u_q = SX.sym('u_q')
          u = vertcat(u_d, u_q)
          
          # set up algebraic variables
          i_d = SX.sym('i_d')
          i_q = SX.sym('i_q')
          z = vertcat(i_d, i_q)
          
          # set up xdot
          psi_d_dot = SX.sym('psi_d_dot')
          psi_q_dot = SX.sym('psi_q_dot')
          xdot = vertcat(psi_d_dot, psi_q_dot)
          
          # set up parameters

Hope this is helpful!

Katrin