Frequency-Based Robust Control for Micro Quadrotors Flying in Narrow Vertical Spaces

Unmanned aerial vehicles (UAVs) are ideal for tasks such as inspections in confined spaces, owing to their maneuverability and independence from terrain constraints, unlike terrestrial robots. However, the aerodynamic effects in confined spaces are greater than that in open areas, because of the propeller-induced flow (PIF). This research addresses this issue as a robust-control challenge and develops a frequency-based robust-control solution. Preliminary experiments reveal that robust control is more suitable than feed-forward control, because of the complex relationships between flight and environmental conditions. To counteract the PIF-based aerodynamic effects in confined spaces, we propose a frequency-based robust controller employing a nonlinear disturbance observer (NDO). The NDO gain design is based on wind-gust frequencies. Comparative experiments, including static and dynamic scenarios, are conducted between the conventional controllers (proportional-integral–derivative and geometric controllers) and proposed robust solution. The results demonstrate improved quadrotor performance compared to that of nonrobust controllers, exhibiting enhanced hovering and navigation within a 0.4 m narrow space under 3 m/s wind when using a Crazyflie. This research provides valuable insights for improved safety and control in confined environments and serves as a guide for developing advanced control strategies and safety protocols for potential real-world applications.

SECTION I. Introduction

In high-altitude areas or small confined spaces, which are potentially dangerous or inaccessible to humans, unmanned aerial vehicles (UAVs) are one of the primary robotic-system choices to conduct inspections and gather essential information. In flights inside tight spaces, proximity effects encompass ground, ceiling, and wall effects that affect the UAVs, potentially leading to poor performance or instability. Researchers have studied and proposed models to compensate for these effects. Cheeseman and Bennett introduced an analytical single-rotor ground-effect model tailored for helicopters [3]. An improved model addressing the increase in thrust was developed by Li et al. [4], while Sanchez considered rotor placements, flow recirculation, and central body lift for a more accurate real-world ground effect [5]. New models for the ground effect in forward flights have been presented and compared favorably to the existing models [6]. Computational fluid dynamics (CFD) analyses were used to study and compensate for the ceiling effect in bridge inspections [7], improving the stability and position accuracy. Proximity effects affecting micro quadrotors were analyzed to balance the model accuracy and safety considerations [8]. Traditional approaches, such as using protective equipment, could be adopted to mitigate these effects [9]. In horizontal confined spaces, optimal forward-flight speeds could mitigate proximity issues and unknown aerodynamic effects [1].

Aerodynamic effects that cause undesired drags, potentially degrading the performance, were studied offline, and lookup tables were developed for hovering simulations [10]. Rigorous aerodynamic modeling, based on the blade-element momentum theory and classical quadrotor dynamics, was supported by wind-tunnel experiments [11]. Notably, these effects arose not only from external wind gusts but also from the movement or altitude changes of the quadrotor itself, posing challenges for complete compensation because of the mismatched conditions [11].

Research on UAV flights in vertical tunnels is currently limited. In [2], a state-estimation-based integral backstepping controller was implemented for robustly maneuvering within a confined space (2.35 m(W) x 2.38 m(H) x 5.9 m(L)) using a 2.3 kg, 0.55 m quadrotor. A subsequent work showcased static thrust experiments and field tests [14]. The static-thrust-experiment results revealed that the change in thrust was the lowest near the center and that tunnel effects could contribute up to a 7% thrust difference between various locations within the tunnel. During the field tests, improved performance was observed with horizontal and vertical errors under 5 and 10 cm, respectively, compared to PID control, which exhibited horizontal and vertical errors exceeding 10 and 15 cm, respectively, when subjected to a 4 m/s outdoor wind disturbance. These results were particularly promising for medium-sized quadrotors, where aerodynamic effects could be less turbulent than that for micro quadrotors.

This novelty of this paper is summarized below.

• First ever specialized solution for safe quadrotor flights in narrow confined spaces - To address wind disturbances, we propose a frequency-based robust controller incorporating a nonlinear disturbance observer (NDO) and prescribed performance control (PPC). Unlike [19], in which a robust controller with NDO and PPC was used, our approach specifically designs the NDO gain within the identified wind-gust frequency range. Additionally, we utilize PPC to limit the maximum error, ensuring controlled motion of the quadrotor inside a walled environment.

• First to address the aerodynamic effects affecting micro quadrotors in narrow confined spaces - This paper focuses on micro quadrotors weighing below 250 g, which are particularly susceptible to aerodynamic effects and external disturbances, owing to their lightweight nature.

This paper combines the advantages of [19] and [20] to improve the flight performance of micro quadrotors inside narrow confined spaces. This paper also provides evidence that quadrotor flight is more challenging in narrow vertical spaces than in open spaces, through CFD visualization and preliminary experiments. Moreover, we demonstrate that the PIF-based aerodynamic effects inside confined spaces are not a deterministic problem, necessitating the development of a robust controller. Other proximity effect compensation methods use a deterministic feed-forward controller, where control parameters characterizing the relationship between the quadrotor and the environment are first identified and then used for the control action. Environmental factors such as tunnel size relative to quadrotor dimensions, flight speed, and hovering positions can further exacerbate the flight conditions in these spaces.

Section III presents the problem formulation, setting three major objectives to prove the need for a robust controller when flying inside vertical confined spaces. Section IV provides a robust control solution by proposing a frequency-based NDO. Section V demonstrates the effectiveness of the proposed robust control through experiments on hovering, trajectory tracking, and response to upstream wind disturbances.

SECTION II. Nomenclature

This paper uses the following nomenclature: unmanned aerial vehicle (UAV), computational fluid dynamics (CFD), propeller-induced flow (PIF), root mean square error (RMSE), and nonlinear disturbance observer (NDO). Controllers include proportional-integral–derivative (PID) and geometric controller (Geo) [17]. Tunnel configurations include fully walled (FW) and lower Wall 1 opened (L1 opened). Proximity effect regions include inside ground effect (IGE) and outside ground effect (OGE). Symbols used are m for the mass of the quadrotor, J for the inertia matrix, g for the acceleration due to gravity, p, v, aacceleration vector, and $\boldsymbol {\omega }$ for the angular velocity. The position vector components are x, y, and z for the respective axial positions, and the attitude angles include $\theta$ , $\phi$ , and $\psi$ for roll, pitch, and yaw, respectively.

SECTION III. Problem Statement

The goal of this section is to justify the need for developing a robust controller, by demonstrating four critical points:

1. The difference in quadrotor flight between confined spaces and open spaces.

2. The complexity of the relationship between the flight conditions [(i) flight altitudes, (ii) flight direction, and (iii) flight speed].

3. The complexity of the relationship between the environmental conditions [(i) different tunnel sizes, (ii) different configurations] in both hovering and dynamic tests.

4. Motors are not yet saturated.

These conditions are essential to justify the need for developing a robust controller for quadrotor flight within narrow confined spaces. The first goal highlights the clear existence of the problem. The second and third goals underscore the complex relationships between the quadrotor flight conditions and environmental conditions. We aim to show that it is difficult to predict exactly how the resulting airflow from the PIF will behave, given that the actual confined spaces can have uneven sizes and random openings and that the quadrotor will fly at different altitudes and speeds. Unlike the compensation for ground effect, the PIF-based aerodynamic effect inside confined spaces is not a deterministic problem where the quadrotor can compensate using a feed-forward controller; rather, a feedback robust controller is more suitable. The fourth goal ensures that there is sufficient control margin available to develop supplementary robust algorithms.

A. Computational Fluid Dynamics

CFD analysis illustrates the fluid dynamics, such as wind flow, within a defined area. In confined spaces, managing wind flow, complicated by recirculation, poses challenges for UAV performance. CFD visually represents these recirculation patterns and their impacts on UAVs. Typically limited to static scenarios, CFD analysis offers deterministic insights.

Using Ansys Workbench, CFD simulations were conducted for a quadrotor hovering at an altitude of 0.7 m. The model was based on the Crazyflie, with its propellers modeled as four thin disks and inlet flow velocities set according to the altitude. The tunnel size matched the experimental setup depicted in Fig. 1, with default wall conditions set for friction.

FIGURE 1.

Experimental setup including (a) lighthouse positioning base stations, (b) Crazyflie micro quadrotor with lighthouse deck and body-fixed $\{B\}$ and inertial frames $\{I\}$ , and (c) narrow vertical experimental setup along with dimensions.

Three cases were analyzed: L1 open, FW, and open space (Fig. 2). In the L1 open case, an imbalance in wind profiles was evident because of the airflow entering and exiting through the top opening of the L1 wall. Conversely, in the FW case, the contained airflow reflected off the UAV’s top, potentially affecting the motors. Even slight roll or pitch variations could disrupt the wind balance, causing undesired motion. In the open-space case, minimal wind vectors affected the rotor disk.

FIGURE 2.

Illustrations of CFD analysis for L1 open, FW, and open-space cases.

These findings demonstrated G1, confirming that navigating confined spaces posed greater challenges owing to recirculation. Furthermore, in accordance with G3.ii, different tunnel configurations impacted the quadrotor flight dynamics differently.

B. System Architecture

To facilitate flight experiments within narrow tunnels, we utilized a commercial Crazyflie quadrotor platform, depicted in Fig. 1.b. The Crazyflie weighed 28 g and had a diagonal distance $w_{q}$ of 0.08 m. Localization was achieved using the lighthouse positioning system [15], employing four V2 base stations as shown in Fig. 1.a. Crazyflies used the STM32F405 ARM Cortex-M4 microcontroller (Cortex-M4, 168MHz, 192kb SRAM, 1Mb flash) and nRF51822 radio and power management microcontroller unit (Cortex-M0, 32Mhz, 16kb SRAM, 128kb flash). These microcontrollers handled all the calculations necessary for flight control, such as sensor fusion, motor control, and communication with other systems. They also utilized a 3-axis accelerometer/gyroscope (BMI088) and high-precision pressure sensor (BMP388). Their main communication module was a 2.4GHz ISM band radio transceiver.

Fig. 1.c illustrates the narrow vertical test bed, constructed from transparent, sturdy acrylic to facilitate the lighthouse system. The square tunnel had a height of 1.64 m and widths $w_{t}$ of 0.6, 0.5, and 0.4 m, tailored to fit the Crazyflie’s size. The lower walls were 0.75 m high. The experiment was conducted in an enclosed room with minimal air movement, ensuring reliable results without external influences such as open windows or air conditioning.

To ensure safe and consistent performance analysis, all flight trajectories were preplanned, allowing multiple tests under controlled conditions.

C. Hovering Experiments

The following tests were conducted at different heights and with tunnel configurations to demonstrate how significantly the PIF-based aerodynamic effects could affect hovering and near-hovering flights. The default PID cascaded controller was utilized in the experiments; a weaker controller was used to amplify the effects, rather than a stronger controller like Geo. The qualitative tool employed for assessment was the normalized RMSE, calculated as follows:

$$\begin{equation*} \bar {\mathbf {p}} =\frac {\sqrt {\frac {1}{N}\sum _{i=1}^{N} (\mathbf {p}_{di}^{\mathrel {2}} - \mathbf {p}_{i}^{2})^{2}}}{w_{q}} \quad \text {[m/m]} \tag {1}\end{equation*}$$ where $\mathbf {p}_{d,i}$ is the desired position. The intention of normalizing it was to facilitate easy comparison and applicability with other micro quadrotor platforms.

In Fig. 3, the first observation is that quadrotor flights inside confined spaces result in more errors than flights in open spaces, which aligns with the G1. The hovering experiment results for different configurations are presented. The (FW, IGE) case exhibits the worst horizontal and vertical position errors of 0.28 m/m and 0.7 m/m, respectively. In the case of G3.ii, we can also confirm the performance difference between the FW and L1 open cases. The L1 open case has worse altitude performance than the (OGE, FW) cases because of the uneven recirculation of PIF.

FIGURE 3.

(a) Error plots and (b) normalized RMSE magnitudes of x and z positions in two tunnel configurations and open space. Two altitudes were tested—-0.1 m for IGE and 0.3 m for OGE.

In the case of G3.i, hovering experiments for two tunnel sizes and at five altitudes were conducted in the L1 open configuration, as shown in Fig. 4. It was particularly challenging to achieve successful outcomes for experiments in a narrower tunnel when only PID control was used; therefore, experiments were conducted in 0.6 and 0.5 m tunnels. As depicted in Fig. 4, the horizontal error for a narrower vertical space was higher at all altitudes. A and B altitude points are where the UAV hovers in a three-walled space, while C, D and E altitude points are where the UAV flies in a four-walled space. Due to the weakness of the PID control setting, higher altitudes (C, D, and E) have higher vertical error than lower altitudes (A and B). However, altitudes C and D have greater vertical error than altitude E. This demonstrate the thrust changes in different structures of a vertical tunnel. Therefore, in the case of G2.i, hovering at various altitudes can produce different flight performances.

FIGURE 4.

Comparison of hovering performances in two tunnel sizes at five altitudes: A - 0.15 m, B - 0.3 m, C - 1.2 m, D - 1.5 m, and E - 1.65 m.

D. Trajectory Tracking Experiments

Dynamic experiments were also conducted to analyze the effects of narrow vertical spaces on the flight performance of micro quadrotors. Fixed smooth trajectories in polynomial form were generated through waypoints p. The Crazyflie accepts polynomial-based trajectories that can be implemented via an autonomous_sequence_high_level Python file in the Crazyflie-lib-python codes. In relation to G2.ii, two trajectories were tracked, including a clockwise (CW) and a counter-clockwise (CCW) trajectory, to analyze both ascending and descending data inside the narrow vertical space. Constants were set in the ascending and descending parts by defining the time points t based on the desired velocity $v_{d}$ .

$$\begin{equation*} t_{i+1}=t_{i} + (p_{t+1}-p_{t})/v_{d} \tag {2}\end{equation*}$$ Figure 5 presents the results for the CW and CCW trajectories. The Crazyflie follows an ascending trajectory inside the 0.6 m narrow vertical space and descends outside, re-entering to complete another loop. In CCW, the Crazyflie starts inside the narrow vertical space and ascends first to a height of 1.8 m, performing the CCW rotation. Even for dynamic experiments, results in the three-dimensional (3D) plot (Fig. 5.a-b) show a deterioration of tracking performance when flying inside the narrow vertical space, which aligns with G1.

FIGURE 5.

(a) Error and position plots of micro quadrotor flight following a clockwise (a.1) and counter-clockwise (b.1) vertical rotation pattern at 0.2 m/s starting from inside the narrow vertical space. The flights were divided into 15 regions, with take-off and landing exempted. (a.2) and (b.2) are the 3D plots of (a.1) and (b.1), respectively. (a.3) and (b.3) show the flight performances while flying at various speeds.

To further demonstrate the significance of point G2.iii, experiments with various flight speeds were conducted, the results of which are presented in Fig. 5.a.2 and Fig. 5.b.2. For the CW direction, greater errors occurred when the UAV flew at 0.7 m/s, which could cause collisions. It was also apparent that slow trajectories for both CW and CCW at 0.2 m/s had greater error than trajectories at intermediate speeds. All flight tests crashed for CCW at 0.7 m/s; therefore, that result was not included.

E. Motor Measurements

In relation to G4, we aimed to determine whether the motor system of the Crazyflie could handle a supplementary robust algorithm without causing saturation. Figure 6 shows that in both hovering and trajectory-tracking experiments with different altitudes and flight speeds, the motors operated at an average norm of approximately 80%. Therefore, the Crazyflie motor system could accommodate a supplementary robust algorithm without reaching saturation.

FIGURE 6.

Averaged norm of the four motors’ PWM percentage quadrotor flights at two altitudes and two tunnel sizes for two flight conditions (a) hovering experiments and (b) trajectory tracking experiments.

This section demonstrated the necessity (G1, G2 and G3) and feasibility (G4) of robust control for micro quadrotors flying inside a narrow vertical tunnel. The next section will present the frequency-based robust control solution suitable for micro quadrotors operating in confined spaces.

SECTION IV. Frequency-Based Robust Control Solution

Quadrotor trajectory tracking control (QTTC) involves designing control strategies to enable a quadrotor to accurately follow a predefined trajectory in space. As demonstrated in the previous section, achieving robust performance becomes more challenging for a quadrotor when navigating inside confined spaces, owing to limited space, potential obstacles, and aerodynamic effects. In narrow vertical spaces, external disturbances can be considered auto-generated, wherein wind forces are reflected to the walls from the quadrotor’s own four motors. Although several factors including motion planning and sensor accuracy contribute to successful flights inside narrow vertical spaces, robust control remains a fundamental aspect that we intend to focus on.

A. Dynamic Modelling

This study considers the inertial frame $\{I\}$ and body-fixed frame $\{B\}$ , as shown in Fig. 1.b. The dynamic model is described by Newton’s equations as follows:

$$\begin{align*} \ddot {\mathbf {p}} & = - g \mathbf {e}_{3} + \frac {1}{m}f \mathbf {R} \mathbf {e}_{3} + \mathbf {d}_{p} \\ \dot {\mathbf {R}} & = \mathbf {R}sk(\boldsymbol {\Omega }) \\ \mathbf {J} \dot {\boldsymbol {\Omega }} & = -\boldsymbol {\Omega }\times \mathbf {J}\boldsymbol {\Omega } +\boldsymbol {\tau } +\mathbf {d}_{o} \\ \mathbf {R} & = \begin{bmatrix} c_{\theta } c_{\psi }& \quad s_{\phi } s_{\theta } c_{\psi }- s_{\phi } c_{\psi }& \quad c_{\phi } c_{\theta } c_{\psi }+ s_{\phi } s_{\psi }\\ c_{\theta } s_{\psi }& \quad s_{\phi } s\theta s_{\psi }+ c_{\phi } s_{\psi }& \quad c_{\phi } s_{\theta } s_{\psi }- s_{\phi } c_{\psi }\\ -s_{\theta }& \quad s_{\phi } c_{\theta }& \quad c_{\phi } c_{\theta }\end{bmatrix} \tag {3}\end{align*}$$ where $\mathbf {e}_{3}$ is $\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}^{\top }$ , R is the rotation matrix from $\{B\}$ to $\{I\}$ , and $sk(\boldsymbol {\Omega })\mathbf {v}=\boldsymbol {\Omega } \times \mathbf {v}$ is the skew-symmetric matrix. Lastly, f and $\boldsymbol {\tau }$ are the total thrust and torque vector, acting as the control inputs, and $\mathbf {d}_{p}$ and $\mathbf {d}_{o}$ are the disturbance vectors.

In our previous work [16], we presented a research review that highlighted the current trends and gaps in quadrotor tracking control research. We objectively identified leading controllers that had achieved outstanding performances through historical insights, data-driven analyses, and performance-based comparisons. Proportional-derivative (PD) control and model predictive control emerged as leading controls and were successfully deployed. Reinforcement learning could also be a good solution to the confined space problem if there existed a simulator that could mimic actual aerodynamic effects in confined spaces (which unfortunately does not) or if it were possible to train in an actual confined space (which again, is not). Moreover, the review mentioned the most-cited paper in quadrotor control, with 2371 citations (last checked 02/29/2024 in Google Scholar), written by D. Mellinger and V. Kumar [17]. This paper proposed a controller design and trajectory generation for quadrotor maneuvering in three dimensions in a tightly constrained setting typical of indoor environments. Its controller design was based on the earlier publication of T. Lee [18], where geometric control was introduced to achieve almost global stability (i.e., it demonstrated exponential stability when the initial attitude error was below 90 degrees and achieved nearly global exponential attractiveness when the initial attitude error was less than 180 degrees). Its simple but effective control is as follows:

$$\begin{align*} f\mathbf {R}\mathbf {e}_{3} & = mg\mathbf {e}_{3} + m \ddot {\mathbf {p}}_{d} - \mathbf {K}_{p} \mathbf {e}_{p} - \mathbf {K}_{d} \dot {\mathbf {e}}_{p}, \\ \boldsymbol {\tau } & = -\mathbf {k}_{R}\mathbf {e}_{R} - \mathbf {k}_{\Omega }\mathbf {e}_{\Omega }+ \boldsymbol {\Omega }\times \mathbf {J}\boldsymbol {\Omega } \\ & \quad -\mathbf {J}(sk(\boldsymbol {\Omega })\mathbf {R}^{\top }\mathbf {R}_{d}\boldsymbol {\Omega }_{d}-\mathbf {R}^{\top }\mathbf {R}_{d}\dot {\boldsymbol {\Omega }}_{d}), \\ \mathbf {R}_{d}& = \begin{bmatrix} \mathbf {R}_{d1} & \quad \mathbf {R}_{d2}& \quad \mathbf {R}_{d3} \end{bmatrix} \tag {4}\end{align*}$$ where $$\begin{align*} \mathbf {R}_{d3} & = \frac {f\mathbf {R}\mathbf {e}_{3}}{\lVert f\mathbf {R}\mathbf {e}_{3} \rVert }, \\ \mathbf {R}_{d2} & = \mathbf {R}_{d3} \times \begin{bmatrix} c_{\psi _{d}} & \quad s_{\psi _{d}} & \quad 0 \end{bmatrix}^{\top }, \\ \mathbf {R}_{d1} & = \mathbf {R}_{d2} \times \mathbf {R}_{d3}, \tag {5}\end{align*}$$ where $\mathbf {K}_{p}$ and $\mathbf {K}_{d}$ are the PD position-control gains, $\mathbf {k}_{R}$ and $\mathbf {k}_{\Omega }$ are the PD attitude gains, and $\diamond _{d}$ is the reference value of the state $\diamond$ . The position error vector $\mathbf {e}_{\xi }$ is simply described as: $$\begin{equation*} \mathbf {e}_{p} = \mathbf {p} - \mathbf {p}_{d}. \tag {6}\end{equation*}$$ whereas the attitude error vectors are described as $$\begin{align*} \mathbf {e}_{R} & = \frac {1}{2}(\mathbf {R}_{d}^{\top }\mathbf {R} - \mathbf {R}^{\top }\mathbf {R}_{d})^{\vee }, \\ \mathbf {e}_{\Omega }& = \boldsymbol {\Omega } - \mathbf {R}^{\top }\mathbf {R}_{d} \boldsymbol {\Omega }_{d}, \tag {7}\end{align*}$$ where $\mathbf {\diamond }^{\vee }$ is the inverse of the $sk()$ operation.

B. Nonlinear Disturbance Observer

As presented in our review paper [16], many papers have adopted the works of [17] and [18], and one such paper is [19], where the proposal is complemented by a prescribed performance control (PPC) and an NDO.

$$\begin{align*} f\mathbf {R}\mathbf {e}_{3} & = mg\mathbf {e}_{3} + m \ddot {\mathbf {p}}_{d} - \mathbf {K}_{p} \mathbf {e}_{p} - \mathbf {K}_{d} \dot {\mathbf {e}}_{p} \\ & \quad - \mathbf {u}^{PPC}\mathbf {e}_{p} - \hat {\mathbf {d}}_{p}, \\ \boldsymbol {\tau } & = -\mathbf {k}_{R}\mathbf {e}_{R} - \mathbf {k}_{\Omega }\mathbf {e}_{\Omega }- \hat {\mathbf {d}}_{o} + \boldsymbol {\Omega }\times \mathbf {J}\boldsymbol {\Omega } \\ & \quad -\mathbf {J}(sk(\boldsymbol {\Omega })\mathbf {R}^{\top }\mathbf {R}_{d}\boldsymbol {\Omega }_{d}-\mathbf {R}^{\top }\mathbf {R}_{d}\dot {\boldsymbol {\Omega }}_{d}). \tag {8}\end{align*}$$ This robust control is deemed suitable to address the challenges inside narrow vertical spaces because the NDO can effectively compensate for auto-generated and other external wind disturbances. Additionally, the PPC is employed to mitigate large errors and enhance the safety of the flight.

The PPC is based on a barrier Lyapunov function that limits the position error to a prescribed value.

$$\begin{equation*} \mathbf {u}^{PPC} = \mathbf {k}_{\beta }\frac {\boldsymbol {\epsilon }^{2}}{\boldsymbol {\epsilon }^{2} - \mathbf {e}_{p}^{2}}, \tag {9}\end{equation*}$$ where $\mathbf {k}_{\beta }$ is a positive parameter and $\boldsymbol {\epsilon }$ is the desired limit of the position error $\mathbf {e}_{p}$ . The control action $\mathbf {u}^{PPC}$ increases the compensation power of the control input when the absolute value of the position error approaches the prescribed limit.

The NDO is formulated as

$$\begin{align*} \hat {\mathbf {d}}_{p} & = \mathbf {e}_{3} + k_{3}m\mathbf {e}_{2}, \\ \hat {\mathbf {d}}_{o} & = \mathbf {e}_{f} + {k_{f}1}\mathbf {J}e_{\boldsymbol {\Omega }}, \tag {10}\end{align*}$$ where $$\begin{align*} \mathbf {e}_{2} & = \dot {\mathbf {e}}_{p} + \mathbf {K}_{\xi }\mathbf {e}_{p}, \\ \dot {\mathbf {e}}_{3} & = -k_{3}(f\mathbf {R}\mathbf {e}_{z} -mg\mathbf {e}_{3} {-} m \ddot {\mathbf {p}}_{d} +m \mathbf {K}_{\xi }\mathbf {e}_{p} \\ & \quad + \hat {\mathbf {d}}_{p}) + \mathbf {e}_{2}, \\ \dot {\mathbf {e}}_{f} & = -k_{f1}[\boldsymbol {\tau } + \hat {\mathbf {d}}_{o} - \boldsymbol {\Omega }\times \mathbf {J}\boldsymbol {\Omega } \\ & \quad -\mathbf {J}(sk(\boldsymbol {\Omega })\mathbf {R}^{\top }\mathbf {R}_{d}\mathbf {\Omega _{d}}-\mathbf {R}^{\top }\mathbf {R}_{d}\dot {\boldsymbol {\Omega }}_{d})] \\ & \quad +k_{f2}\mathbf {J}^{-1}\mathbf {e}_{R} +\mathbf {e}_{\Omega }, \tag {11}\end{align*}$$ where $k_{3}$ is the NDO position gain, $k_{f1}$ and $k_{f2}$ are the NDO attitude gains, and $\mathbf {K}_{\xi }$ is the P gain in the position control from [19] that can be recalculated from the PD gains in [17] and [18] using the quadratic formula as follows: $$\begin{equation*} \mathbf {K}_{\xi } = \frac {\mathbf {K}_{d} \pm \sqrt {\mathbf {K}_{d}^{2} -4m \mathbf {K}_{p}}}{2m} \tag {12}\end{equation*}$$ In our previous work [20], we proposed a frequency-based wind-gust estimation using fuzzy NDO where the cut-off frequency was set considering the wind-gust frequency for a more accurate estimation. In this study, we propose to design the NDO position gain $k_{3}$ to consider the wind-gust frequency and accurately compensate for the wind disturbance.

Substituting f from (3) into $\dot {e}_{3}$ from (10) yields:

$$\begin{equation*} \dot {\mathbf {e}}_{3} = -k_{3}(m \ddot {\mathbf {e}}_{p}+m \mathbf {K}_{\xi } \dot {\mathbf {e}}_{p}+ \hat {\mathbf {d}}_{p}-\mathbf {d}_{p}) + \dot {\mathbf {e}}_{p} + \mathbf {K}_{\xi } \mathbf {e}_{p}. \tag {13}\end{equation*}$$ Using the Laplace transform of the j-th position, we obtain: $$\begin{align*} {e}_{j3} & = -k_{3} m e_{jp} s+(1-k_{3}m {K}_{j\xi }) {e}_{jp} \\ & \quad + ({K}_{j\xi } {e}_{jp} -k_{3}(\hat {d}_{jp}-{d}_{jp}))s^{-1} \\ \hat {d}_{jp} & = {e}_{jp} + ({K}_{j\xi } {e}_{jp} -k_{3}(\hat {d}_{jp}-{d}_{jp}))s^{-1}. \tag {14}\end{align*}$$

If $e_{\xi }$ is driven to 0 by the controller, the relationship of the estimated disturbance to the actual disturbance can be expressed as a low-pass filter:

$$\begin{align*} \frac {\hat {d}_{jp}}{d_{jp}} = \frac {k_{3}}{s+k_{3}}, \\ k_{3} = 2\pi f_{c}, \tag {15}\end{align*}$$ where $f_{c}$ is the cut-off frequency that can be designed to accurately compensate for the wind disturbance.

Algorithm 1 presents the complete process of the proposed controller, making it easier for readers to understand. Similar to any NDO and as stated in [19, eq. (29)], the proposed algorithm is limited to a certain amount of disturbances, depending on the quadrotor size and power. Following the differential flatness property of quadrotors, the reference trajectory is sufficiently smooth or four times differentiable, as it should be possible to compute the input using the fourth-order time derivatives of the given trajectory.

Algorithm 1 Proposed Controller Algorithm

1:

Initialize: Model parameters, PD gains $\mathbf {K}_{p}$ and $\mathbf {K}_{d}$ , NDO cut-off frequency $f_{c}$ , PPC desired limit $\boldsymbol {\epsilon }$ , NDO altitude gains $k_{f1}$ and $k_{f2}$

2:

Input: System states [p, $\phi$ , $\theta$ , $\psi$ ] and desired trajectory [$\mathbf {p}_{d}$ , $\psi _{d}$ ]

3:

Output: Robust control inputs f and $\boldsymbol {\tau }$

4:

STEP 1: Calculate $k_{3}$ using the appropriate cut-off frequency $f_{c}$ using eq. (15).

5:

STEP 2: Calculate the PPC input using eq. (9). Calculate the estimated disturbance using eq. (10).

6:

STEP 3: Calculate the robust control inputs eq. (8).

SECTION V. Experimental Verification

The experimental setup is identical to that used in the preliminary experiments. Four controllers are available in the Crazyflie firmware: 1) cascaded PID, 2) Mellinger’s geometric control (Geo) used in [17], 3) Smeur’s incremental nonlinear dynamic inversion (INDI), and 4) Brescianini’s quaternion-based controller (Bres). These controllers can be switched by changing the parameter stabilizer.controller.

Experiments using the INDI and Bres controllers yielded unsuccessful results; thus, only PID, Geo, and the proposed controller were compared. PID control is a cascaded PID control where the position references are given to the position loop and the control loop output becomes the reference input of the succeeding loops. Geo follows the work [17]. The proposed controller is easy to implement because it is based on Geo, which is already integrated into the Crazyflie firmware. PID control is a weaker control than Geo, as designed by the developers. However, Geo strictly requires the trajectory to be smooth. Therefore, the proposed controller also requires a smooth trajectory. About the computational cost and delay, as also mentioned in [19], this robust controller is computationally simple. No significant delays were noted while conducting the experiments.

For implementing the proposed algorithm, the cut-off frequency must be designed. In [20], the wind frequency range was specified as 0.005—0.1389 Hz. According to the experimental results shown in Fig. 7, the highest peaks of various representative flight are consistent to be below 0.1 Hz. Hence, the proposed controller should be designed with $f_{c}$ at less than 0.1 Hz. The prescribed limits of the position error were set at $\boldsymbol {\epsilon } = \begin{bmatrix} 0.04 & 0.04 & 0.08 \end{bmatrix}$ to emphasize better horizontal control performance and avoid collisions with the walls. Other controller parameters were $k_{\beta } = 0.00033$ , $k_{f1} = 0.05$ , and $k_{f2} = 0.01$ . The gains for all controllers were left unchanged as they were optimized by Bitcraze developers.

FIGURE 7.

Frequency response of $G_{x_{d} \to x} = \frac {\mathscr {L}\{x(t)\}}{\mathscr {L}\{x_{d}(t)\}}$ where the disturbance affecting x position is below 0.1 Hz ($10^{-}{1}$ ).

Other robust controllers, like sliding-mode and adaptive controllers, were not compared in this study. The novelty of the proposed frequency-based robust control does not lie in proposing a new controller. Geo with NDO and PPC has already been documented in [19]. Instead, the main contribution of the frequency-based robust control is the appropriate tuning of the NDO gain and innovative use of PPC in the walled experimental area. Here, we compare the effects of this gain tuning.

Good hovering performance is crucial for inspection tasks that involve camera-based image capture. Similar to the preliminary experiments, all controllers were tested to hover at five altitudes inside a 0.5 m tunnel. The results are summarized in Tab. 1, where the RMSE values indicate tracking precision and max values denote the safety margin before collision. Because we utilized a square tunnel, we considered the norm of x and y position RMSE, similar to (1).

TABLE 1 Evaluation of Hovering Experiments, [mm]. Performance Improvement Percentage (PIP) is Compared to That of Geo

PID exhibited larger errors (ranging from 0.1 to 0.6 m) in the z direction because of the low settings of the default gains. Geo performed similar to the controller without the proposed robust controller (RC). Compared to Geo, RC exhibited RMSE and max values reduced by up to 66% and 75%, respectively, at hovering point C, indicating improved tracking performance and safety. As depicted in Fig. 8, the proposed frequency-based robust controller outperformed PID and Geo when hovering at an altitude of 1.2 m. The fainter color range in Fig. 8 represents the variations across five tests. The absolute position error did not exceed the prescribed limit $\boldsymbol {\epsilon }$ .

FIGURE 8.

Comparison of hovering experiments at 1.2 m altitude using the proposed frequency-based robust control (RC) and baseline controllers (PID and Geo (without RC)) with three designed cut-off frequencies. (a) Motor PWM (b) tracking performance.

Regarding the choice of cut-off frequency, tuning at 0.02 Hz was optimal as lower (0.008 Hz) and higher (0.08 Hz) values slightly degraded the flight performance, by approximately 25%. Higher values were also attempted, but resulted in crashes during experiments. Flight performances with RC mostly stayed within the 0.08 m or $\bar {xy}=1$ , normalized by the size of the quadrotor, demonstrating effective evaluation based on size. We verified that none of the controllers caused motor saturation, as shown in Fig. 8.a.

As illustrated in Fig. 9, dynamic experiments were conducted and made more challenging by adding a constant wind disturbance introduced by an electric fan, generating approximately 3 m/s wind at a distance, with the tunnel size reduced to 0.4 m. Fig. 9 illustrates the CFD analysis results presenting the effects of wind disturbance from L1.

FIGURE 9.

Actual experiment area with a CFD analysis illustration of a micro quadrotor affected by a 3 m/s wind from L1. The red region exhibits higher wind speeds than the set value of 3 m/s, attributed to the tube effect caused by the L1 opening measuring 0.75 m x 0.4 m.

Despite the challenging conditions, RC significantly improved the tracking performance even with large disturbances affecting the micro quadrotor, as presented in Fig. 10 and Tab. 2. PID exhibited larger variations of up to 118 and 310 mm in the xy and z directions, respectively. Properly tuned RC outperformed both Geo (No RC) and RC with a cut-off frequency of 0.08 Hz in all evaluation metrics. Owing to extreme aerodynamic effects within the confined space, some flight performances with RC exceeded 0.08 m or $\bar {xy}=1$ , but remained within 1.4 m. The range of the quadrotor flight is indicated by the fainter color in the Fig. 10. It is also worth noting the influence of the structure change, structure from a three-walled space to a four-walled space, by marking the points when the UAV reaches the ABCDE altitudes. It can be seen that $e_{x}$ is positive at A and B altitude points where the UAV transverses a three-walled space, while it is negative at C, D, and E altitude points when the UAV is flying in a four-walled space. This shows how significantly aerodynamic effects change in different tunnel structures. The flight with RC (0.02 Hz) has less error compared to the other controllers.

TABLE 2 Evaluation of Trajectory Tracking Experiments [mm]

FIGURE 10.

Comparison of trajectory tracking experiments using PID, Geo, and the proposed frequency-based robust control (RC) with two designed cut-off frequencies. (a) Motor PWM, (b) tracking performance.

Figure 10.a shows that none of the controllers caused motor saturation under these conditions. Figure 11 shows high estimated disturbances in the x and z directions, due to the wind entering the L1 opening. A CFD illustration is presented in Fig. 9. A comparison of two designed cut-off frequencies shows that the higher cut-off frequency (0.08 Hz) is more sensitive to faster variations (noise and collisions) than 0.02 Hz, which is not beneficial inside narrow confined spaces.

FIGURE 11.

Comparison of estimated disturbance from NDO for two designed cut-off frequencies.

SECTION VI. Conclusion and Future Work

This paper analyzed micro quadrotor flights inside narrow vertical spaces and proposed a frequency-based robust control solution based on NDO. Preliminary flight experiments and CFD analyses illustrated the necessity for robust control, because of the increased aerodynamic effects inside such confined spaces, which could degrade the tracking performance. Moreover, the nondeterministic relationship between flight conditions and environmental factors underscored the need for a robust controller instead of a simple feed-forward approach. The proposed scheme with frequency-based gain tuning demonstrated improved flight performance in a 0.4 m tunnel, even under external upstream wind conditions.

As a future work, we plan to conduct a detailed analysis of the aerodynamic effects inside narrow confined spaces. Furthermore, we intend to explore motion planning and trajectory generation within narrow vertical spaces. Our preliminary experiments indicate that an optimal speed range exists, and we are particularly interested in analyzing the effects of corners in both horizontal and vertical tunnels. Additionally, conducting trajectory testing in uncontrolled environments or field tests is a critical avenue for our upcoming investigations. These efforts aim to further enhance the robustness and applicability of quadrotor flight in confined and dynamic environments.

ACKNOWLEDGMENT

This paper is based on results obtained from a project, JPNP22002, commissioned by the NewEnergy and Industrial Technology Development Organization (NEDO).