VTOLSmoothTransitionFlight

Official repository of "Smooth Reference Command Generation and Control for Transition Flight of VTOL Aircraft Using Time-Varying Optimization" ¹.

• Paper: https://arxiv.org/abs/2501.00739

Abstract

Vertical take-off and landing (VTOL) aircraft pose a challenge in generating reference commands during transition flight. While sparsity between hover and cruise flight modes can be promoted for effective transitions by formulating ℓ1-norm minimization problems, solving these problems offline pointwise in time can lead to non-smooth reference commands, resulting in abrupt transitions. This study addresses this limitation by proposing a time-varying optimization method that explicitly considers time dependence. By leveraging a prediction-correction interior- point time-varying optimization framework, the proposed method solves an ordinary differential equation to update reference commands continuously over time, enabling smooth reference command generation in real time. Numerical simulations with a two-dimensional Lift+Cruise vehicle validate the effectiveness of the proposed method, demonstrating its ability to generate smooth reference commands online.

Summary

Formulation

For transition flight of VTOL aircraft, ℓ1-norm minimization problems can be formulated to encourage the sparsity between (vertical) rotor thrust, $T_{r}$, and (horizontal) pusher thrust, $T_{p}$ ^{1 2}:

$$\begin{aligned} \min_{T_{r}(t), T_{p}(t), \theta_{\text{ref}}(t), \delta_{e}(t)} &\vert T_{r}(t) \vert + \vert T_{p}(t) \vert\\ \text{s.t.} &T_{r}(t) \geq 0 \\\ &T_{p}(t) \geq 0 \\\ &\theta_{\text{ref}}(t) \in [\underline{\theta}, \overline{\theta}] \\\ &\delta_{e}(t) \in [\underline{\delta}_{e}, \overline{\delta}_{e}] \\\ &T_{p}(t) = F_{\text{des,x}}(t) + D_{1}(t) \\\ &-T_{r}(t) = F_{\text{des,y}}(t) + L(t) - D_{2}(t) \end{aligned}$$

Methods

1. OL-Opt: open-loop optimization-based method ².

   • OL-Opt solves optimization problems (ℓ1-norm minimization) for each time and then interpolate solutions over time.
   • OL-Opt is an offline method, i.e., it generates reference commands in advance.
   • OL-Opt does not consider the time dependence in the optimization problem. The interpolation can lead to the generation of non-smooth reference commands.

2. CL-TVOpt (Proposed): closed-loop time-varying optimization-based method ¹.

   • CL-TVOpt solves a time-varying optimization problem by continuously updating the reference command forward in time.
   • CL-TVOpt is an online method, i.e., it generates reference commands in-flight.
   • CL-TVOpt explicitly consider the time dependence in the optimization problem. It generates smooth reference commands as a solution to an ordinary differential equation (ODE).

OL-Opt	CL-TVOpt (Proposed)

Scenarios

Scenario 1: Hover-to-cruise

Scenario 1 highlights the effect of non-smooth transition.

OL-Opt

• OL-Opt generates non-smooth pitch angle reference command $\theta$, which causes a slight position tracking error $U$ (up). This tracking error is accumulated due to control input saturation.

CL-TVOpt (proposed)

• CL-TVOpt generates smooth pitch angle reference command $\theta$, avoiding undesirable position tracking error due to non-smoothness during transition.

Scenario 2: Cruise-to-hover

Scenario 2 highlights the effect of initial tracking error.

OL-Opt

• OL-Opt is an offline method, which typically implies an inevitable initial tracking error (e.g. in pitch angle $\theta$). A small initial tracking error can be accumulated due to control input saturation, which can cause an abrupt transition flight.

CL-TVOpt (proposed)

• CL-TVOpt is an online method that updates the reference commands continuously forward in time, avoiding an abrupt transition flight due to initial tracking error.

Computation time

Roughly, the mean elapsed time of the CL-TVOpt is 10 times less than that of OL-Opt. It is because OL-Opt requires iterations to converge for optimization at each time, while CL-TVOpt requires solving an ODE for one time step at each time, exploitating the time dependence via time-varying optimization formulation. For more details, readers are referred to the paper ¹.

Code

Data

This repo provides data used in the paper ¹ as follows:

• ./data_plot contains data for plotting figures.
• ./data_time contains data to calculate the mean elapsed time. Note that the data are generated with Julia 1.9.3 on M1 Macbook Air.

Methods

:opt and :tvopt stand for i) OL-Opt and ii) CL-TVOpt (proposed).

Scenarios

Two transition scenarios were considered: i) :h2c (hover-to-cruise) and ii) :c2h (cruise-to-hover).

How to reproduce?

Manifest.toml contains all information to re-create the environment for reproducing results. First, run julia. It is recommended to specify the version, e.g., julia +1.9.3, corresponding to Manifest.toml.

Then,

include("main.jl")

To reproduce simulation results

Run the following commands:

sim(:opt, :h2c)
sim(:opt, :c2h)
sim(:tvopt, :h2c)
sim(:tvopt, :c2h)

Data will be stored in ./data. To plot figures, see function plot_result in ./main.jl for more details.

To calculate elapsed times

Run the following commands (with specific scenario (:h2c or :c2h) and method (:opt or :tvopt) that you want to test. For example,

[sim(:opt, :h2c; savestep=Inf) for i in 1:10]
# [sim(:opt, :c2h; savestep=Inf) for i in 1:10]
# [sim(:tvopt, :h2c; savestep=Inf) for i in 1:10]
# [sim(:tvopt, :c2h; savestep=Inf) for i in 1:10]

Data will be stored in ./data. To avoid startup time of Julia, we recommend running the above command twice consecutively and then remove the data generated by the first command execution. After that, move data from ./data to ./data_time and then run the following commands to calculate the total elapsed time. See average_elapsed_time for more details.

average_elapsed_time(:opt, :h2c)

Note that savestep=Inf is used to exclude the calculation time for logging data. To obtain the average mean elapsed time, the total elapsed time should be divided by the number of points $N = (t_{f}-t_{0}) / \Delta t$. In this code, $t_{0}=0$ s, $t_{f}=125$ s. OL-Opt (opt) is solved for each 0.1 s and CL-TVOpt (tvopt, proposed) is solved for each 0.01 s. Therefore, $N=1,250$ for OL-Opt and $N=12,500$ for CL-TVOpt, respectively. Note that this code only provides the total elapsed time.

Animation

Run the following command with e.g. your_data_path = "./data_plot/tvopt_h2c_tsit5_2024-12-20T18:24:26.726.jld2":

animate(your_data_path)

Cited Works

Footnotes

1. J. Kim, J. L. Bullock, S. Cheng, N. Hovakimyan, "Smooth Reference Command Generation and Control for Transition Flight of VTOL Aircraft Using Time-Varying Optimization", AIAA SciTech 2025 Forum, Orlando, FL, Jan. 2025. ↩ ↩² ↩³ ↩⁴ ↩⁵

2. J. L. Bullock, S. Cheng, A. Patterson, M. J. Acheson, N. Hovakimyan, and I. M. Gregory, “Reference Command Optimization for the Transition Flight Mode of a Lift Plus Cruise Vehicle,” in AIAA SciTech 2024 Forum, Orlando, FL, Jan. 2024. doi: 10.2514/6.2024-0721. ↩ ↩²