Direct collocation

Overview

Direct collocation is a way to discretise a continuous-time model inside an optimisation. Between sampling instants the state is represented by polynomials, and the model's dynamics are enforced at a set of collocation points within each period 2606.31889 §4 2606.31889 §5. The paper uses it in a nonlinear model predictive controller, where the finite-horizon problem is solved at every sampling instant 2606.31889 §3 2606.31889 §4.

The only source on this page is one paper, and it uses collocation as a means to an end. It does not study collocation itself. Three points are used per sampling period 2606.31889 §5. The paper names Gauss–Legendre points as one way to obtain such polynomials, but it does not say which node placement its own runs used 2606.31889 §4. The paper also does not state the form of the collocation constraints beyond the model equations it discretises 2606.31889 §2 2606.31889 §4.

Variants

Collocation inside a receding-horizon NMPC

The model is a nonlinear kinematic model with six states and three inputs 2606.31889 §2. Each sampling period of 0.1 s is discretised with three collocation points, and the horizon is 6 s 2606.31889 §5. The controller solves the finite-horizon problem at each step, applies only the first input and repeats 2606.31889 §3 2606.31889 §4. The card for this paper: Tractor-trailer path tracking with NMPC.

The reported results are author-reported and come from simulation only, with no field or hardware test 2606.31889 §5. They show the trailer position error staying below 16 cm along x once moving and below 12 cm along y in headland turns 2606.31889 §5.2. The same paper reports violent oscillation in the inputs in the trailer-controlled case, which the authors propose to damp with filters whose constants are not given 2606.31889 §5.2. These numbers describe the whole controller, not the collocation step alone, so they do not show what collocation contributes.

No other variant is covered by a card on this wiki. Other collocation schemes, such as different node placements or different polynomial orders, are not described by any source here.

When to use which

There is one variant and one source, so this section cannot rank approaches. The constraints that the source does fix are these:

Implementation sketch

The receding-horizon loop below follows the paper's controller. The collocation step is shown as a call, because the paper does not give the form of its constraints. An implementer needs the original paper's equations or a further source before writing that step.

# Receding-horizon NMPC with collocation  [arxiv:2606.31889 §3] [§4]
# Input:  model F(x, u), sampling period T_s = 0.1 s, horizon N  [§5]
# Output: first input of each solved problem, applied in turn

def control_step(state, previous_input, reference):
    # Discretise F over one sampling period with three collocation points  [§4] [§5]
    # The paper does not state the constraint form; this call is a placeholder
    constraints = collocate(F, T_s, points=3)
    # Solve the finite-horizon problem with the cost of eq. 19  [§4]
    # Cost: quadratic in the trailer position error and the inputs  [§4]
    inputs = solve_nmpc(state, reference, previous_input, constraints, N)
    # Apply only the first input and re-solve at the next period  [§3]
    return inputs[0]

Contradictions

None known. The page rests on one source, and that source does not report a conflicting result for collocation.

Papers

• Tractor-trailer path tracking with NMPC: uses three-point collocation inside a nonlinear MPC for a tractor-trailer, with simulation results only.