A real-world data-free initialization method for neural network controllers: A case study of the multicopter interception control

Abstract

Learning-based controller design methods, such as reinforcement learning and learning-based Lyapunov control, face significant challenges when applied to complex control tasks. Reinforcement learning methods typically demand substantial interaction data during initial training, often exhibiting slow convergence, high sample complexity, and unsafe exploration. Conversely, learning-based Lyapunov control methods, while providing formal stability guarantees, generally require a pre-existing, stable controller or expert demonstrations to bootstrap the learning process, which is often unavailable for novel control problems. To address this fundamental bottleneck, this article proposes a model-based rapid initialization method for neural network controllers. The core idea is to circumvent the need for prior expert knowledge or real-world interaction data. Instead, we leverage the system model to synthesize datasets that inherently satisfy Lyapunov stability conditions. A neural network policy is then trained via supervised learning on this synthetic data, yielding an initially stable controller. This policy serves as a high-quality starting point for subsequent fine-tuning via reinforcement learning or learning-based Lyapunov control methods, significantly improving sample efficiency and safety. We validate the proposed method through a challenging case study: image-based visual servoing control for high-speed multicopter interception. Both simulations and physical experiments demonstrate that the trained policy, without any online reinforcement learning fine-tuning, enables successful interception at a final speed of 15 m/s on a real quadcopter platform.

Keywords

Lyapunov methods visual servoing high-speed interception D-learning supervised learning

Introduction

In recent years, an increasing number of studies have utilized learning-based methods to address control issues (O’Connell et al., 2022; Tutsoy and Barkana, 2021). The advancement of reinforcement learning (RL) has made significant contributions to the application of neural network controllers in the field of robot control (Huang et al., 2023; O’Connell et al., 2022; Wang et al., 2023). Although these research findings demonstrate the outstanding performance brought by neural network controllers, the lack of strict stability guarantees makes these methods difficult to apply in safety-critical scenarios. Particularly, before the neural network controller reaches an ideal state, the iterative training process of the neural network controller may lead to dangerous consequences. To overcome this lack, Lyapunov stability methods (Lyapunov and Walker, 1994) in control theory are integrated with learning-based approaches to provide a definitive analytical and design framework. In the studies by Han et al. (2020) and Du et al. (2023), the Lyapunov function is utilized as a critic function to assess policy performance. In Chow et al. (2018) and Chow et al. (2019), Lyapunov stability conditions are used as additional constraints to guarantee system stability. Ganai et al. (2023) and Chang and Gao (2021) develop the target control policy by incorporating a Lyapunov function into the reward function. Compared to Quan et al. (2025), Shen and Quan (2025), and Liu et al. (2025), these studies incorporate Lyapunov conditions into the RL framework in a relatively straightforward method.

The research in Quan et al. (2025), Shen and Quan (2025), and Liu et al. (2025) proposes learning the Lyapunov function and its derivative (referred to as the D-function) from expert demonstration data while adhering to stability constraints, thus facilitating the development of a control policy that inherently ensures Lyapunov stability. These learning-based Lyapunov control (LLC) methods offer formal stability assurances for the target policies and yield favorable outcomes. However, two key issues limit the application of LLC methods. (1) In order to ensure that the trained policy can be applied in practice, the datasets used in the training process should encompass the region of interest (RoI) (Dai et al., 2021; Dawson et al., 2023), which is exceedingly challenging before obtaining an available control policy. The sparsity of the sampled datasets makes the training process of the neural network challenging. (2) A more critical issue is that a common prerequisite for LLC methods is the availability of a stable expert controller or demonstration trajectories to guide the learning process or to learn a valid Lyapunov function.

These two key issues create a circular dependency: designing a stable expert controller for a complex task is often the very problem we aim to solve with learning-based methods. Furthermore, even in RL, a stable initial policy is highly valuable to mitigate the issues of poor initial exploration and sample inefficiency (Shen and Quan, 2025). This leads to a fundamental yet often overlooked challenge in learning-based control: How can we efficiently obtain an initial, stable control policy for a novel task without relying on pre-existing expert knowledge or costly, potentially unsafe, real-world exploration? While traditional control theory may theoretically build such controllers, it requires extensive domain knowledge and iterative tuning, particularly for systems with high-dimensional perception and complex dynamics.

To bridge this gap created by the two issues, this article introduces a novel, rapid initialization method for neural network controllers that requires neither expert demonstrations nor real-world interaction data. Our key insight is that for robotic systems, an approximate dynamic model is often easier to obtain than expert policies or large-scale real-world data. Leveraging this model, we systematically generate state-action datasets that are guaranteed to satisfy Lyapunov stability conditions by construction. A neural network policy is then trained via supervised learning to mimic these stabilizing actions. The resulting policy provides a provably stable starting point for any learning-based control method (RL or LLC). The main contributions of this work are as follows:

A model-based, data-free framework is proposed to quickly generate stable initial policies without relying on experts or real-world trials.

We validate our method through a high-speed (15 m/s) multicopter interception case study, providing a complete pipeline and demonstrating successful real-world intercepts without any online adaptation.

It is important to note that the primary goal of this method is to efficiently bootstrap a usable, stable policy with satisfactory baseline performance. Further refinement for optimal performance can subsequently be achieved by combining it with RL or LLC techniques.

The article is organized as follows: (1) Section ‘Problem formulation’ outlines the coordinate systems and mathematical models employed in the case study. (2) Section ‘Policy generation method’ shows the rapid methodology for training neural network policies, firstly. Then, the proposed method is applied to the case study of image-based visual servoing control for the multicopter interception. (3) Section ‘Stability verification and simulations’ validates the stability of the trained policies by the almost Lyapunov condition (Liu et al., 2020). Meanwhile, the simple simulation results are given. (4) Section ‘Experiment’ presents the experimental result of the trained policies implemented on quadcopter platforms.

Problem formulation

In this article, the proposed method will be used to obtain the image-based visual servoing policy for multicopter interception. The policy is a neural network with the target position in the image plane coordinates as the input. Here, the desired position of the target point is the center of the image plane (Yang et al., 2025b; Yang and Quan, 2020).

Coordinate systems and models

This article uses five coordinate systems, which are the Earth-fixed coordinate system (EFCS), the body coordinate system (BCS), the gimbal camera coordinate system (GCCS), the strapdown camera coordinate system (SCCS), and the image coordinate system (ICS), as shown in Figure 1. The positive orientation adheres to the right-hand principle. The Earth-fixed coordinate axes ${{}^{e}x, {}^{e}y, {}^{e}z}$ point north, east, and downward to gravity, respectively. The body coordinate axes ${{}^{b}x, {}^{b}y, {}^{b}z}$ point forward, rightward, and downward of the multicopter, respectively. Assuming the camera is connected to the multicopter via a two-axis gimbal (including roll axis and pitch axis), the ${{}^{g}x, {}^{g}y, {}^{g}z}$ axes of the GCCS point to the rightward, downward, and forward directions of the camera’s optical axis, respectively. The GCCS and BCS share the same origin. The orientations of the BCS and GCCS frames differ solely by a tilt angle, which can be decomposed into pitch and roll angles for clarity. Meanwhile, the GCCS and EFCS frames are aligned in roll and pitch but differ only in yaw. In this study, the camera and aircraft are mounted directly without a gimbal, which makes the system more reliable (Yang et al., 2025b). This article describes the installation attitude of the strapdown camera with reference to the BCS. The strapdown camera is installed with only a non-zero pitch angle ( $α \neq 0$ ) as shown in Figure 1, and the positive direction is likewise defined by the right-hand rule. To improve the interception capability of the strapdown camera aircraft, it is necessary to set $α > 0$ (Yang et al., 2025a). Compared with strapdown cameras, gimbal cameras decouple the line of sight from the aircraft’s tilt angle, thereby simplifying the interception problem addressed in this article. Accordingly, based on principles of camera imaging geometry, an equivalent transformation model is established to convert image coordinates from the strapdown camera to the gimbal camera. Similar to the GCCS, the defined SCCS coordinate axes ${{}^{s}x, {}^{s}y, {}^{s}z}$ point rightward, downward, and forward along the camera’s optical axis, respectively. The transformation between these two coordinate systems can be fully described by the roll angle $ϕ_{s}$ and pitch angle $θ_{s}$ of the strapdown camera relative to the EFCS. The image coordinate axes ${{}^{i}x, {}^{i}y}$ point rightward and downward, respectively. The origin of the ICS is the center of the image. The subscript s denotes the ICS axes and coordinates of the strapdown camera, while the subscript g represents the ICS axes and coordinates of the gimbal camera.

Figure 1.

Description of coordinate systems in the multicopter interception problem. The left superscript indicates the abbreviation of the coordinate system, where the letter ‘g’ represents the gimbal camera coordinate system, the letter ‘e’ represents the Earth-fixed coordinate system, the letter ‘s’ represents the strapdown camera coordinate system, and the letter ‘b’ represents the body coordinate system. The installation angle of the strapdown camera is α. The dashed rectangle represents the imaging plane of the camera. ${}^{i}p$ is the position of the interception target in the image coordinate system.

Assuming that the coordinate of the interception target in the ICS of the strapdown camera is denoted as $({}^{i}{\bar{p}}_{x, s}, {}^{i}{\bar{p}}_{y, s})$ , which can be converted to the ICS of the gimbal camera by roll angle $ϕ_{s}$ and pitch angle $θ_{s}$ of the strapdown camera. As shown in Figure 2(a), $({}^{i}{\bar{p}}_{x, 1}, {}^{i}{\bar{p}}_{y, 1})$ is a temporary coordinate value computed based on the roll angle $ϕ_{s}$ . That is

\begin{matrix} \begin{matrix} {}^{i}{\bar{p}}_{x, 1} & = {}^{i}{\bar{p}}_{x, s} \cos ϕ_{s} - {}^{i}{\bar{p}}_{y, s} \sin ϕ_{s} \\ {}^{i}{\bar{p}}_{y, 1} & = {}^{i}{\bar{p}}_{x, s} \sin ϕ_{s} + {}^{i}{\bar{p}}_{y, s} \cos ϕ_{s} \end{matrix} \end{matrix}

(1)

Figure 2.

Geometric relationship between the target point’s coordinates in the strapdown camera image and its coordinates in the gimbal camera image. (a) Temporary coordinates after roll angle $ϕ_{s}$ transformation. (b) Final coordinates in the gimbal camera image after pitch angle $θ_{s}$ adjustment.

According to Figure 2(b), the coordinate in the ICS of the gimbal camera is derived as

\begin{matrix} \begin{matrix} {}^{i}{\bar{p}}_{x, g} & = {}^{i}{\bar{p}}_{x, 1} \\ {}^{i}{\bar{p}}_{y, g} & = \tan (\arctan {}^{i}{\bar{p}}_{y, 1} - θ_{s}) \end{matrix} \end{matrix}

(2)

Then, $({}^{i}{\bar{p}}_{x, g}, {}^{i}{\bar{p}}_{y, g})$ is the coordinate of the interception target in the ICS of the gimbal camera.

Remark 1. The coordinate symbol with overline is the normalized values. Take the coordinate symbol of the ICS of strapdown camera, namely $({}^{i}{\bar{p}}_{x, s}, {}^{i}{\bar{p}}_{y, s})$ , as example, it is calculated by

\begin{matrix} {}^{i}{\bar{p}}_{s} = {[{}^{i}{\bar{p}}_{x, s}, {}^{i}{\bar{p}}_{y, s}]}^{T} = {[\frac{{}^{i}{p_{x, s}}}{f_{oc}}, \frac{{}^{i}{p_{y, s}}}{f_{oc}}]}^{T} \end{matrix}

(3)

where $f_{oc}$ is focal of length in pixel units, and $({}^{i}{p_{x, s}}, {}^{i}{p_{y, s}})$ is the coordinates of the image point expressed in pixel units. Similarly, the coordinates of the target point in the image plane of the gimbal camera are simply expressed as

\begin{matrix} {}^{i}{\bar{p}}_{g} ≜ ({}^{i}{\bar{p}}_{x}, {}^{i}{\bar{p}}_{y}) \end{matrix}

(4)

The dynamics of the target point ${}^{i}{\bar{p}}_{g}$ in the gimbal camera image plane (Chaumette and Hutchinson, 2006) is

\begin{matrix} {}^{i}{\overset{\cdot}{\bar{p}}}_{g} = L_{s} {}^{g}{\tilde{v}} \end{matrix}

(5)

where the image Jacobian matrix $L_{s}$ is

\begin{matrix} L_{s} = [\begin{matrix} - \frac{1}{c_{z}} & 0 & \frac{{}^{i}{\bar{p}}_{x}}{c_{z}} & {}^{i}{\bar{p}}_{x} {}^{i}{\bar{p}}_{y} & - (1 + {}^{i}{\bar{p}}_{x}^{2}) & {}^{i}{\bar{p}}_{y} \\ 0 & - \frac{1}{c_{z}} & \frac{{}^{i}{\bar{p}}_{y}}{c_{z}} & 1 + {}^{i}{\bar{p}}_{y}^{2} & - {}^{i}{\bar{p}}_{x} {}^{i}{\bar{p}}_{y} & - {}^{i}{\bar{p}}_{x} \end{matrix}], \end{matrix}

(6)

$c_{z}$ is the object distance of the gimbal camera,

\begin{matrix} {}^{g}{\tilde{v}} = {[{}^{g}{v^{T}} {}^{g}{ω^{T}}]}^{T}, \end{matrix}

(7)

${}^{g}v = {[{}^{g}{v_{x}} {}^{g}{v_{y}} {}^{g}{v_{z}}]}^{T}$ is the velocity of the gimbal camera in the GCCS, and ${}^{g}ω = {[{}^{g}{ω_{x}} {}^{g}{ω_{y}} {}^{g}{ω_{z}}]}^{T}$ is the angular rate of the gimbal camera in the GCCS. In this article, the coordinates of the target point in the image plane of the gimbal camera are unaffected by the pitch and roll motion of the body, that is, ${}^{g}{ω_{x}} = 0$ and ${}^{g}{ω_{z}} = 0$ . Then, equation (5) is simplified as

\begin{matrix} {}^{i}{\overset{\cdot}{\bar{p}}} = L_{s}^{'} {}^{g}{\tilde{v}'} \end{matrix}

(8)

where the image Jacobian matrix $L_{s}^{'}$ is

\begin{matrix} L_{s}^{'} = [\begin{matrix} - \frac{1}{c_{z}} & 0 & \frac{{}^{i}{\bar{p}}_{x}}{c_{z}} & - (1 + {}^{i}{\bar{p}}_{x}^{2}) \\ 0 & - \frac{1}{c_{z}} & \frac{{}^{i}{\bar{p}}_{y}}{c_{z}} & - {}^{i}{\bar{p}}_{x} {}^{i}{\bar{p}}_{y} \end{matrix}], \end{matrix}

(9)

and

\begin{matrix} {}^{g}{\tilde{v}'} = {[{}^{g}{v_{x}} {}^{g}{v_{y}} {}^{g}{v_{z}} {}^{g}{ω_{y}}]}^{T} . \end{matrix}

(10)

Control strategy

In order to accomplish the interception mission with the multicopter, the strategy employed in this research is as follows: The multicopter is made to fly at a designated velocity $v_{z, g}$ along the ${}^{g}z$ -axis, which is straightforward for multicopters to achieve. Simultaneously, the designed policy, namely controller, is utilized to regulate the motion velocities of the multicopter along the ${}^{g}x$ -axis and ${}^{g}y$ -axis so that the position of the intercepted object in the gimbal camera image tends to the center of the image. Note that the position of the intercepted object in the gimbal camera image is converted from that in the strapdown camera image through equations (1) and (2).

Policy generation method

In this section, the method to generate a neural network policy for interception will be introduced. First, the policy design process will be briefly described. Second, a neural network policy used for object interception is designed and trained in detail. Finally, the almost Lyapunov condition (Liu et al., 2020) is adopted for control stability verification.

Policy training method

Initialization

Controller designers should create a randomly initialized neural network policy $c_{θ_{0}}$ according to specific control problem, where θ is the weight of neural network $c_{θ_{0}}$ . Meanwhile, the system model should be known. Note that the model originally used for conventional control design can be reused. Here, the system model is expressed as

\begin{matrix} \overset{\cdot}{x} = f (x, u), \end{matrix}

(11)

where $x$ is the system state and the $u$ is system input. To minimize the modeling cost, the artificially constructed models often overlook the physical factors that exert a relatively minor influence on the controlled system. Consequently, the established models do not precisely mirror the actual scenario. Nevertheless, developing models grounded in objective physical laws retains significant practical relevance for control design.

Candidate D-function calculation

Based on the control object, design a Lyapunov function $V (x)$ . Then, the controlled system model (11) is used to obtain the derivative of the $V (x)$ , namely D-function (Quan et al., 2025)

\begin{matrix} D (x, u) = \frac{∂V}{\partial x} f (x, u) . \end{matrix}

(12)

Dataset construction based on Lyapunov conditions

For all the given $x_{j}$ in the RoI, find corresponding $u_{j}$ satisfying

\begin{matrix} \begin{matrix} \min_{η > 0, u_{j}} & | D (x_{j}, u_{j}) + ηW (x_{j}) | \\ s . t . & D (x_{j}, u_{j}) < 0 \end{matrix} \end{matrix}

(13)

where η is a positive number, $W (0) = 0$ , and $W (x_{j}) > 0$ for $\forall x_{j} \in ℝ^{n} ∖ {0}$ . Here, $η > 0$ is a user-specified parameter that determines how negative $D (x_{j}, u_{j})$ should be relative to the positive function $W (x_{j})$ . A larger η typically leads to faster convergence of the state but may require larger control efforts, which must be balanced against actuator limits. In practice, η can be chosen by gradually increasing its value from a small starting point until the desired convergence behavior is achieved.

The datasets

\begin{matrix} S = {s_{1}, s_{2}, \dots, s_{N}} \end{matrix}

(14)

are composed of all the number of data point $s_{j} = (x_{j}, u_{j})$ . The size of the dataset $S$ is determined by the number of possible state $x$ combinations.

Training neural network $c_{θ_{0}}$ with $S$

Train $c_{θ_{0}}$ by minimizing $L (θ_{k})$ as shown in equation (15)

\begin{matrix} L (θ_{k}) = \min_{c_{θ_{k}}} \frac{1}{N} \sum_{j = 1}^{N} ∥ c_{θ_{k}} (x_{j}) - u_{j} ∥_{2}, \end{matrix}

(15)

where k is the training epoch and N represents the number of data items in the datasets $S$ .

The real-world data-free policy training framework has been presented in this subsection. The control policy $c_{θ_{k}}$ , namely controller, is trained from the cost function equation (15) with datasets $S$ . In the following, the training framework will be combined with multicopter interception control problem to validate its effectiveness.

Training of interception policy

Initialization

Model (9) is selected for the neural network policy design. The system state vector is

\begin{matrix} {[{}^{i}{\bar{p}}_{x} {}^{i}{\bar{p}}_{y} {}^{g}{v_{z}} c_{z} {}^{g}{ω_{y}}]}^{T} \in ℝ^{5} . \end{matrix}

(16)

However, only ${}^{i}{\bar{p}}_{x}$ and ${}^{i}{\bar{p}}_{y}$ are concerned state for the interception task. The system input vector $u$ is

\begin{matrix} u = {[{}^{g}{v_{x}} {}^{g}{v_{y}}]}^{T} . \end{matrix}

(17)

This study employs a standard feed-forward neural network architecture, which is well suited to learning the static state-to-action mapping required by our policy. The network’s width and depth were determined through experimental tuning. The network configuration consists of an input layer with three neurons and an output layer comprising a single neuron. The architecture incorporates three hidden layers, each containing 16 neurons. The hyperbolic tangent (tanh) activation function is implemented between all adjacent network layers, which makes the neural network smooth (Chang et al., 2019). The choice of a feedforward neural network is guided by its suitability for learning the static state-to-action mapping required by our policy, while its final architectural parameters (width and depth) were finalized through experimental tuning. According to the section ‘Control strategy’, two separate neural networks with the same architecture are used for generating ${}^{g}{v_{x}}$ and ${}^{g}{v_{y}}$ , namely $c_{θ_{x, k}}$ and $c_{θ_{y, k}}$ .

Candidate D-function calculation

The key to completing the interception task is to make the target position in the gimbal camera image tend towards the image center, that is, ${}^{i}{\bar{p}}_{x} \to 0$ and ${}^{i}{\bar{p}}_{y} \to 0$ . Therefore, the Lyapunov functions for ${}^{i}{\bar{p}}_{x}$ and ${}^{i}{\bar{p}}_{y}$ are, respectively, designed as follows:

\begin{matrix} \begin{matrix} V_{x} & = \frac{1}{2} {}^{i}{\bar{p}}_{x} \cdot {}^{i}{\bar{p}}_{x}, \\ V_{y} & = \frac{1}{2} {}^{i}{\bar{p}}_{y} \cdot {}^{i}{\bar{p}}_{y} . \end{matrix} \end{matrix}

(18)

The final Lyapunov function is

\begin{matrix} V = V_{x} + V_{y} . \end{matrix}

(19)

If $V_{x} \to 0$ and $V_{y} \to 0$ , then it has $V \to 0$ . Here, the policies along ${}^{g}x$ -axis and ${}^{g}y$ -axis are designed separately. With equation (9), the derivatives of $V_{x}$ and $V_{y}$ are

\begin{matrix} \begin{matrix} \frac{\partial V_{x}}{\partial {}^{i}{\bar{p}}_{x}} {}^{i}{\overset{\cdot}{\bar{p}}}_{x} & = {}^{i}{\bar{p}}_{x} (- \frac{{}^{g}{v_{x}}}{c_{z}} + {}^{g}{v_{z}} \frac{{}^{i}{\bar{p}}_{x}}{c_{z}} - (1 + {}^{i}{\bar{p}}_{x}^{2}) {}^{g}{ω_{y}}), \\ \frac{\partial V_{y}}{\partial {}^{i}{\bar{p}}_{y}} {}^{i}{\overset{\cdot}{\bar{p}}}_{y} & = {}^{i}{\bar{p}}_{y} (- \frac{{}^{g}{v_{y}}}{c_{z}} + {}^{g}{v_{z}} \frac{{}^{i}{\bar{p}}_{y}}{c_{z}} - {}^{i}{\bar{p}}_{x} {}^{i}{\bar{p}}_{y} {}^{g}{ω_{y}}) . \end{matrix} \end{matrix}

(20)

Let

\begin{matrix} \begin{matrix} D_{x} ({}^{i}{\bar{p}}_{x}, {}^{g}{v_{x}}, {}^{g}{v_{z}}, c_{z}, {}^{g}{ω_{y}}) & = \frac{\partial V_{x}}{\partial {}^{i}{\bar{p}}_{x}} {}^{i}{\overset{\cdot}{\bar{p}}}_{x}, \\ D_{y} ({}^{i}{\bar{p}}_{y}, {}^{g}{v_{y}}, {}^{g}{v_{z}}, c_{z}, {}^{g}{ω_{y}}) & = \frac{\partial V_{y}}{\partial {}^{i}{\bar{p}}_{y}} {}^{i}{\overset{\cdot}{\bar{p}}}_{y} . \end{matrix} \end{matrix}

(21)

Dataset construction based on Lyapunov conditions

The conditions that the datasets need to satisfy are shown in equation (13), where

\begin{matrix} \begin{matrix} W ({}^{i}{\bar{p}}_{x}) & = ∥ {}^{i}{\bar{p}}_{x} ∥^{2}, \\ W ({}^{i}{\bar{p}}_{y}) & = ∥ {}^{i}{\bar{p}}_{y} ∥^{2}, \end{matrix} \end{matrix}

(22)

and $η = 2$ . The two groups of datasets are expressed as

\begin{matrix} \begin{matrix} S_{x} & = {({}^{i}{\bar{p}}_{x, j}, {}^{g}{v_{x, j}}, {}^{g}{v_{z, j}}, c_{z, j}, {}^{g}{ω_{y, j}}), j = 1, \dots, N}, \\ S_{y} & = {({}^{i}{\bar{p}}_{y, j}, {}^{g}{v_{y, j}}, {}^{g}{v_{z, j}}, c_{z, j}, {}^{g}{ω_{y, j}}), j = 1, \dots, N} . \end{matrix} \end{matrix}

(23)

To ensure that the synthesized dataset is evenly dispersed over the RoI, uniform sampling is used to determine sampling points for each state. The RoI of the states and their sample numbers used in this study are shown in Table 1, where ${}^{g}{w_{y}}$ is considered to be small value. Note that the sample numbers of states need to be ensured by practice. While an excessive number of sampling points will result in an enormous dataset that compromises training efficiency, an insufficient number will lead to subpar training results. According to Table 1 and equation (23), the size of the datasets $S_{x}$ and $S_{y}$ all are 753750. The interception task is mainly accomplished through the translational movement of the multicopter.

Table 1.

The RoI of the system states and their sample steps.

State	Range	Sample step (number)	Unit
${}^{i}{\bar{p}}_{x}$	$[- 1 1]$	$0.01 (201)$	-
${}^{i}{\bar{p}}_{y}$	$[- 1 1]$	$0.01 (201)$	-
${}^{g}{ω_{y}}$	$[- 0.2 0.2]$	$0.1 (5)$	rad/s
${}^{g}{v_{z}}$	$[0.1 15]$	$1 (15)$	m/s
$c_{z}$	$[0.5 50]$	$1 (50)$	m

Training neural network $c_{θ_{x,k}}$ and $c_{θ_{y,k}}$

The neural network policies $c_{θ_{x, k}}$ and $c_{θ_{y, k}}$ are trained using the datasets $S_{x}$ and $S_{y}$ , and by solving the optimization problems

\begin{matrix} \begin{matrix} L (θ_{x, k}) & = \min_{c_{θ_{x, k}}} \frac{1}{N} \sum_{j = 1}^{N} ∥ c_{θ_{x, k}} ({}^{i}{\bar{p}}_{x, j}, {}^{g}{v_{z}}, c_{z}) - {}^{g}{v_{x, j}} ∥, \\ L (θ_{y, k}) & = \min_{c_{θ_{y, k}}} \frac{1}{N} \sum_{j = 1}^{N} ∥ c_{θ_{y, k}} ({}^{i}{\bar{p}}_{y, j}, {}^{g}{v_{z}}, c_{z}) - {}^{g}{v_{y, j}} ∥ . \end{matrix} \end{matrix}

(24)

The control policies $c_{θ_{x, k}}$ and $c_{θ_{y, k}}$ are the controllers for the multicopter interception control, that is, ${}^{g}{v_{x}} = c_{θ_{x, k}}$ and ${}^{g}{v_{y}} = c_{θ_{y, k}}$ .

Stability verification and simulations

Before deploying the trained neural network policy, its stability is verified using the almost Lyapunov conditions (Liu et al., 2020), which have also been adopted for stability analysis in Chang and Gao (2021) and Dawson et al. (2022). The neural network controller synthesized by the proposed method cannot be proven to be stable analytically. The approach presented in Liu et al. (2020) for nonlinear systems relaxes the classical Lyapunov stability condition by allowing the derivative of the Lyapunov function to be non‑negative over a sufficiently small set of states, while still guaranteeing system stability. In this work, states within the RoI, as shown in Table 1, are intensively sampled, and the sign of $D_{x}$ and $D_{y}$ is checked at each sample point. While this sampling‑based verification cannot prove that $D_{x} < 0$ and $D_{y} < 0$ hold for every point (including unsampled ones) in the RoI, the almost Lyapunov conditions theory provides a rigorous theoretical foundation and computational feasibility for such a sample‑based approach.

Verifying almost Lyapunov conditions

The verification of the almost Lyapunov conditions is achieved by the sample-based method. First, we need to intensively resample the states within the RoI as shown in Table 1 to construct state datasets. Based on these datasets, we then calculate the derivative of the Lyapunov function:

\begin{matrix} \begin{matrix} sgn (D_{x} ({}^{i}{\bar{p}}_{x}, c_{θ_{x, k}}, {}^{g}{v_{z}}, c_{z}, {}^{g}{ω_{y}})), \\ sgn (D_{y} ({}^{i}{\bar{p}}_{y}, c_{θ_{y, k}}, {}^{g}{v_{z}}, c_{z}, {}^{g}{ω_{y}})), \end{matrix} \end{matrix}

(25)

where $sgn$ is the sign function. A negative value of this function indicates that the closed-loop system under this state tends towards the desired state.

The results of equation (25) are decided by multiple state inputs. To facilitate the presentation and verification of the results, some relatively unimportant states are fixed here. Let ${}^{g}{v_{z}} = 15 m ∕ s$ and ${}^{g}{ω_{y}} = 0 rad ∕ s$ . Figure 3 presents the domain of attraction of the interception system when using the trained policies $c_{θ_{x, k}}$ and $c_{θ_{y, k}}$ , where $k = 5$ . The values of $V_{x}$ and $V_{y}$ are distinguished by using different shades of background color. The positive values of $D_{x}$ and $D_{y}$ are denoted by green points, while the negative values are denoted by gray points. From the figure on the left, it can be seen that only when $c_{z} < 10 m$ and ${}^{i}{\bar{p}}_{x} = 0$ , there is a situation where $D_{x} > 0$ . This indicates that in most cases $V_{x}$ tends to 0, that is, ${}^{i}{\bar{p}}_{x}$ tends to 0. The same is true for the figure on the right. This means that when the aircraft approaches the target, there will be a static error on ${}^{i}{\bar{p}}_{x}$ and ${}^{i}{\bar{p}}_{y}$ . This is because, near ${}^{i}{\bar{p}}_{x} = 0$ , there are cases where $D_{x} > 0$ and $D_{y} > 0$ . Then, ${}^{i}{\bar{p}}_{x}$ and ${}^{i}{\bar{p}}_{y}$ will converge to the position where $D_{x} = 0 ({}^{i}{\bar{p}}_{x} \neq 0)$ and $D_{y} = 0 ({}^{i}{\bar{p}}_{y} \neq 0)$ , respectively.

Figure 3.

The region of attraction for the interception system with the trained neural network policy when ${}^{g}{v_{z}} = 15 m ∕ s$ and ${}^{g}{ω_{y}} = 0 rad ∕ s$ . Grey points denote negative values, while green points denote positive values.

Remark 2. Given that the target is not a point mass, interception can still be accomplished even if ${}^{i}{\bar{p}}_{x} \neq 0$ , provided that ${}^{i}{\bar{p}}_{x}$ converges to a location sufficiently near 0. The greatest permissible value of $| {}^{i}{\bar{p}}_{x} |$ for successfully intercepting the target is contingent upon the dimensions of the target itself.

Considering a more general case, the pixel distance of the target from the center of the image coordinate system is defined as

\begin{matrix} {}^{i}p = \sqrt{{}^{i}{p_{x}^{2}} + {}^{i}{p_{y}^{2}}} . \end{matrix}

(26)

As shown in Figure 4, assuming that the object distance is much greater than the focal length, the image distance is approximately equal to the focal length. Then, the following relationship exists:

\begin{matrix} \frac{r}{c_{z}} = \frac{{}^{i}{p_{\max}}}{f_{oc}} = {}^{i}{\bar{p}}_{\max} \end{matrix}

(27)

Figure 4.

Geometric condition for stationary target interception. Assuming the target is stationary, to achieve target interception, the relationship between the target size and the distance of the target from the center of the camera image.

In this context, r denotes the radius of the inscribed circle of the intercepted target’s contour, ${}^{i}{p_{\max}}$ signifies the maximum permissible ${}^{i}p$ for target interception (assume the target remains stationary), and ${}^{i}{\bar{p}}_{\max}$ represents the normalized value of ${}^{i}{p_{\max}}$ .

Let $c_{z} = 1 m$ and satisfy the relationship $c_{z} ≫ f_{oc}$ , then it has

\begin{matrix} {}^{i}{\bar{p}}_{\max} = r \end{matrix}

(28)

Equation (28) denotes that when the distance from the camera to the target object is 1 m, if the target’s distance from the center of the camera’s image coordinate system, ${}^{i}{\bar{p}}$ , is less than or equal to r, it can be inferred that if the aircraft persists in its current trajectory along the optical axis, it will successfully touch the target.

Simplified simulation

To verify the policies, we employed a simplified simulation model assuming instantaneous execution of velocity commands by the multicopter. In the study, a neural network controller for ${}^{g}{ω_{y}}$ is not constructed. According to equation (8), both ${}^{g}{ω_{y}}$ and ${}^{g}{v_{x}}$ are able to ensure that the target position in the ICS of the gimbal camera tends to the image center, that is, ${}^{i}{\bar{p}}_{x} \to 0$ and ${}^{i}{\bar{p}}_{y} \to 0$ . In this subsection, the yaw rate ${}^{g}{ω_{y}}$ is set to 0 to highlight the efficiency of the policy $c_{θ_{x, k}}$ . Based on equation (8), the simplified simulation model is equation (29):

\begin{matrix} [\begin{matrix} ċ_{z} \\ {}^{i}{\overset{\cdot}{\bar{p}}}_{x} \\ {}^{i}{\overset{\cdot}{\bar{p}}}_{y} \end{matrix}] = [\begin{matrix} 0 & 0 & - 1 \\ - \frac{1}{c_{z}} & 0 & \frac{{}^{i}{\bar{p}}_{x}}{c_{z}} \\ 0 & - \frac{1}{c_{z}} & \frac{{}^{i}{\bar{p}}_{y}}{c_{z}} \end{matrix}] [\begin{matrix} {}^{g}{v_{x}} \\ {}^{g}{v_{y}} \\ {}^{g}{v_{z}} \end{matrix}], \end{matrix}

(29)

where ${}^{g}{v_{x}}$ is determined by the neural network $c_{θ_{x, k}}$ , while ${}^{g}{v_{y}}$ is generated by $c_{θ_{y, k}}$ . The value of ${}^{g}{v_{z}}$ is fixed at 15 m/s. Simulations were conducted using six distinct sets of initial states.

The two top plots in Figure 5 present the simulation results under different initial states. Purple regions indicate coordinates where $D_{x} < 0$ and $D_{y} < 0$ , with darker shades corresponding to smaller (more negative) D values. States ${}^{i}{\bar{p}}_{x}$ and ${}^{i}{\bar{p}}_{y}$ exhibit faster decay rates (indicating accelerated convergence) in darker purple regions. Furthermore, as $c_{z} \to 0$ , the convergence ${}^{i}{\bar{p}}_{x} \to 0$ and ${}^{i}{\bar{p}}_{y} \to 0$ holds.

Figure 5.

Simulated flight trajectories comparing a non-fixed (top) and a fixed (bottom) $c_{z}$ condition for the policies $c_{θ_{x, k}}$ and $c_{θ_{y, k}}$ . The two top plots show the flight trajectories using model (29) with ${}^{g}{v_{z}} = 15 m ∕ s$ and ${}^{g}{ω_{y}} = 0$ , the two bottom plots show the flight trajectories in the same simulation but with $c_{z} \equiv 10 m$ for the policies $c_{θ_{x, k}}$ and $c_{θ_{y, k}}$ .

In the design of section ‘Training of interception policy’0.0, the neural network policies $c_{θ_{x, k}}$ and $c_{θ_{y, k}}$ require the value of the object distance $c_{z}$ to obtain the speed commands ${}^{g}{v_{x}}$ and ${}^{g}{v_{y}}$ . In practice, obtaining $c_{z}$ requires the design of additional algorithms. Here, we attempt to set a fabricated object distance, $c_{z} = 10 m$ , for the policies for further verification. As shown in the two bottom plots in Figure 5, as $c_{z} \to 0$ , ${}^{i}{\bar{p}}_{x} \to 0$ and ${}^{i}{\bar{p}}_{y} \to 0$ still hold. The two bottom plots in Figure 5 provide improved convergence results. By comparing the distribution of D values and the trend of state trajectories in these two figures and combining with the constraint conditions (13), it can be seen that setting a larger η or designing more reasonable constraint conditions is useful for improving the convergence speed and steady-state error of the state. It should be noted that in the simulation, the dynamic process required for the quadcopter to execute the speed command is not considered. Therefore, in the training process, η cannot be unlimited and set to larger values. A more reasonable approach is to improve the constraint conditions (13) of the datasets.

Experiment

Quadcopter setup

We validated the interception policies on a quadcopter. The neural network policies are run on the onboard computer¹ at 30 Hz, and a yolo-v5 target recognition instance equipped with the 1280×720@10 Hz camera is also run on the onboard computer at 10 Hz. The target coordinate in the ICS of the strapdown camera, namely $({}^{i}{p_{x, s}}, {}^{i}{p_{y, s}})$ , will be resolved by the yolo instance. In our experiments, a red balloon served as the visual interception target. Then, the normalized target coordinate $({}^{i}{\bar{p}}_{x, s}, {}^{i}{\bar{p}}_{y, s})$ is converted into the ICS of the gimbal camera, that is, $({}^{i}{\bar{p}}_{x}, {}^{i}{\bar{p}}_{y})$ . The relevant formulas are equations (1) to (3). Meanwhile, the velocity information ${}^{g}{v_{z}}$ related to the quadcopter is sent to the onboard computer from the PX4 flight control system. Combining with the target coordinate and the velocity information, the interception controller running on the onboard computer will generate the velocity commands and the angular rate command ${{}^{g}{v_{x, cmd}}, {}^{g}{v_{y, cmd}}, {}^{g}{v_{z, cmd}}, {}^{g}{ω_{y, cmd}}}$ sending to PX4 flight control system, as shown in Figure 6. In detail, the velocity commands ${{}^{g}{v_{x, cmd}}, {}^{g}{v_{y, cmd}}, {}^{g}{v_{z, cmd}}}$ need to be transformed into EFCS based on the current yaw angle before sending to the PX4 flight controller. Note that unlike the simulations based on equation (29), the gimbal camera’s yaw rate ${}^{g}{ω_{y}}$ in physical experiments was directly controlled by proportional feedback of ${}^{i}{\bar{p}}_{x}$ . Consequently, the four control inputs driving the quadcopter are

\begin{matrix} \begin{matrix} {}^{g}{v_{x, cmd}} & = c_{θ_{x, k}} ({}^{i}{\bar{p}}_{x}, {}^{g}{v_{z}}, c_{z}), \\ {}^{g}{v_{y, cmd}} & = c_{θ_{y, k}} ({}^{i}{\bar{p}}_{y}, {}^{g}{v_{z}}, c_{z}), \\ {}^{g}{v_{z, cmd}} & = 15 m ∕ s, \\ {}^{g}{ω_{y, cmd}} & = 0.002 \cdot {}^{i}{\bar{p}}_{x} . \end{matrix} \end{matrix}

(30)

where $c_{z}$ is set as $10 m$ in the experiment same as the simulations in Figure 5. The attitude control was handled by the PX4 flight control system, which responded to the commands from the onboard computer.

Figure 6.

Architecture of the neural network–based interception control system for a quadcopter.

The quadcopter has a total mass of $1.1 kg$ . Each rotor has a time constant of $0.03 s$ and can generate a maximum thrust of $11.5 N$ . The distance from the center of the vehicle to each rotor, known as the arm length, is $0.20 m$ . Its moments of inertia are $0.0076 kg \cdot m^{2}$ about the ${}^{b}x$ -axis, $0.0089 kg \cdot m^{2}$ about the ${}^{b}y$ -axis, and $0.0137 kg \cdot m^{2}$ about the ${}^{b}z$ -axis. The camera installed pitch angle is $α = 15 \deg$ .

Experimental result

Figure 7 presents the video snapshots captured by the strapdown camera during the flight, covering the images from the moment the quadcopter just entered the interception flight state until the moment it was about to hit the target. When the interception state was just entered, due to the change in pitch attitude, the target appeared on the upper edge of the image. In the subsequent images, the target gradually moved towards the center of the image. The first four subplots in Figure 8 show velocity commands issued by the onboard computer and the actual response. The last two subplots in Figure 8 track two metrics during interception: (1) the bounding box area of the target in image space (represented by red boxes in Figure 7) and (2) the values of ${}^{i}{\bar{p}}_{x}$ and ${}^{i}{\bar{p}}_{y}$ . Figure 9 shows the scatter plot of the quadcopter’s position during the interception. The ${}^{g}x$ -axis velocity command exhibits fluctuations, causing lateral sway during flight, correlating with ${}^{i}{\bar{p}}_{x}$ variations in Figure 8. Along the ${}^{g}y$ -axis, an initial 3 m/s fluctuation occurs due to abrupt target displacement when pitching forward. Although strapdown-to-gimbal coordinate conversion decouples the target position from the attitude, residual attitude sensitivity persists from unsynchronized image capture and attitude computation. This aligns with ${}^{i}{\bar{p}}_{y}$ dynamics in Figure 8.

Figure 7.

Video screenshots from the strapdown camera of the quadcopter during the experiment. Each screenshot is labeled with the video timestamp. See supplemental video.

Figure 8.

The response of the quadcopter to the command from the onboard computer.

Figure 9.

Scatter trajectory of the quadcopter during interception.

In the previous stability verification, we did not take into account the changes in yaw angular velocity, thus focusing more on the relationship between the changes of ${}^{i}{\bar{p}}_{x}$ ( ${}^{i}{\bar{p}}_{y}$ ) and the system stability. The experiment indicates that the quadcopter’s yaw angular velocity attained a significant control magnitude during the initial 2.5 s of the interception process, subsequently stabilizing at $\pm 5$ deg/s. This indicates that the neglect of the influence of yaw angular rate in the stability verification did not have a fatal impact on the actual deployment results.

The trained interception control policies successfully executed missions at target speeds during deployment. This confirms our method’s ability to rapidly develop the mediocre control policy via offline training using only a posteriori knowledge and synthetic datasets without real flight data.

Summary of results

This study proposed and validated a novel fast initialization method for neural network controllers. The primary experimental findings are as follows. First, the method successfully generated a stable initial control policy offline using only synthetic datasets constructed from the system model, without requiring real-world flight data. Then, when deployed on a quadcopter platform for an image-based visual servoing interception task, the trained policy achieved a final successful interception velocity of 15 m/s. Finally, the controller effectively guided the target from the edge of the image plane toward its center, as evidenced by the video snapshots and the convergence of the normalized image coordinates ${}^{i}{\bar{p}}_{x}$ and ${}^{i}{\bar{p}}_{y}$ .

Discussion and conclusion

The proposed rapid initialization method provides a pathway to obtain a stable initial neural network policy without real-world interaction data. However, the performance and robustness of this policy in real-time deployment are inevitably influenced by various uncertainties inherent in practical engineering systems. Acknowledging and analyzing these uncertainties is crucial for understanding the limitations of the current approach and guiding future improvements.

In control systems, uncertainties generally originate from internal system characteristics and external disturbances. For our image-based visual servoing application on a multicopter, these can be categorized as follows: (1) Parametric uncertainties: These arise from inaccuracies in the system model parameters used for dataset generation. In our case, this includes the object distance $c_{z}$ , which was set to a fixed value $10 m$ during policy execution in both simulation and experiment. In reality, $c_{z}$ is time-varying and estimated with error. Other parameters include the multicopter’s mass, inertia, and aerodynamic coefficients, which were simplified or assumed constant in the model described by equation (11). (2) Unmodeled dynamics: The mathematical model such as equations (8) and (29) necessarily simplifies the complex, high-fidelity dynamics of the multicopter. It neglects factors such as complex aerodynamic effects, motor dynamics, and image delays. (3) External disturbances: During outdoor flight, the system is subject to wind gusts and turbulence, which act as external, non-parametric disturbances not captured in the offline training phase. (4) Measurement and perception noise: The target pixel coordinates $({}^{i}{\bar{p}}_{x}, {}^{i}{\bar{p}}_{y})$ are derived from a yolo detector, introducing pixel-level detection noise. Furthermore, state information such as velocity from the flight controller is subject to estimation noise and latency. These sensor imperfections add another layer of uncertainty to the control loop.

These uncertainties affect the control system at multiple levels. (1) Training phase discrepancy: The synthetic datasets $S$ are generated from an idealized model. Consequently, the trained policy $c_{θ}$ learns a mapping that is optimal for the nominal model but may not generalize perfectly to the real plant with discrepancies. This is a common challenge in moving from simulation to reality. (2) Stability guarantees: The almost Lyapunov stability verification in the section 0.0‘Verifying almost Lyapunov conditions’ is conducted using the idealized model. Significant uncertainties can violate the condition $D (x, c_{θ} (x)) < 0$ in the RoI, potentially compromising the theoretical stability guarantees in practice. (3) Performance degradation: As observed in the experimental results (Figure 8), the oscillatory behavior in the velocity commands can be attributed to the combined effect of the aforementioned uncertainties.

In summary, this method addresses the cold-start problem in learning-based control by providing a stable initial policy without requiring prior interaction data. Simulations and experiments of image-based visual servoing for the multicopter interception validate the method’s effectiveness. While the method’s offline, model-based nature inherently limits robustness against real-world uncertainties, the initialized policy serves as a high-quality starting point for subsequent fine-tuning via RL or LLC to enhance its performance and robustness, which is one of the future works. Another future work will incorporate comprehensive models and advanced neural networks (e.g. RNNs) to enhance robustness.

Supplemental Material

Footnotes

Acknowledgements

A preprint (Ke et al., 2025) of this work is available online .

ORCID iD

Chenxu Ke

Ethical considerations

This article does not contain any studies with human or animal participants.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Zhejiang Province Human Resources and Social Security Department (grant number ZJ2025198).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Supplemental material

Supplemental material for this article is available online.

Notes

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A real-world data-free initialization method for neural network controllers: A case study of the multicopter interception control

Abstract

Keywords

Introduction

Problem formulation

Coordinate systems and models

Control strategy

Policy generation method

Policy training method

Initialization

Candidate D-function calculation

Dataset construction based on Lyapunov conditions

Training neural network c θ 0 with S

Training of interception policy

Initialization

Candidate D-function calculation

Dataset construction based on Lyapunov conditions

Training neural network c θ x,k and c θ y,k

Stability verification and simulations

Verifying almost Lyapunov conditions

Simplified simulation

Experiment

Quadcopter setup

Experimental result

Summary of results

Discussion and conclusion

Supplemental Material

Footnotes

Acknowledgements

ORCID iD

Ethical considerations

Funding

Declaration of conflicting interests

Data availability statement

Supplemental material

Notes

References

Training neural network $c_{θ_{0}}$ with $S$

Training neural network $c_{θ_{x,k}}$ and $c_{θ_{y,k}}$