LQI Control System Design with GA Approach for Flying-Type Firefighting Robot Using Waterpower and Weight-Shifting Mechanism

Abstract

This study proposes a flying robot using waterpower and a novel weight-shifting mechanism, whose purpose is to be applied in firefighting tasks in water areas that are difficult to access and are suppressed by conventional firefighting methods. The sufficient amount of water in the area is used for propelling the robot, as well as fire suppression activity. A weight-shifting mechanism governs the weight distribution of the robot head in order to perform the robot motions. In this paper, the system’s dynamical characteristics are analyzed in detail through mathematical models. A linear-quadratic integrator (LQI) is designed for controlling the system motion. Additionally, the LQI is tuned with the genetic algorithm (GA) so that both system performance and robustness are optimally preserved. Simulation studies are carried out, in which the proposed system is compared with a cascade proportional-integral-derivative (PID) control system. The results validate the feasibility of the design and show the superiority of the proposed control system in motion performances. Moreover, the LQI-GA consumes 2.28% less water and uses 83.85% less kinetic energy of the actuator than the PID control system.

1. Introduction

Fire accidents are unpredictable and can spread quickly. They often cause significant damage to people and property. Therefore, the development of improved and effective fire suppression systems always attracts much interest. Automatic fire-extinguishing and sprinkler systems have been widely used and have proven their efficiency in indoor fire accidents. Nevertheless, in many cases, firetrucks and firefighters need to directly battle flames to preserve property and the lives of humans or animals. The firefighters always have to contend with multiple difficulties—for example, carrying heavy hoses and tools in and around heavy pieces of unsafe structures. The brutal conditions also put the firefighters at risk of loss of life or injury. Moreover, in some tricky conditions, such as fire accidents happening on a narrow bridge, a crowded harbor, or on the sea, it is extremely difficult for the fire trucks and firefighters to approach the location to do their mission.

Over the past decade, unmanned vehicles or remotely controlled robots have been deployed in fire accidents, not only to assist but sometimes to replace the manpower in firefighting missions. Several intelligent firefighting systems with different shapes, features, and extinguishing agents have been developed for different situations. Generally, they can be classified into two main groups according to the method of approaching the flames: terrestrial systems and aerial systems.

In chronological order, the terrestrial systems were the ones that were first studied and used in fire accidents in residential areas, industrial fields, and wildfires [1,2,3]. They can be ground vehicles or wheeled, legged, and even humanoid robots. Commercial firefighting robots [4,5,6,7] are equipped with modern technologies that greatly increase their durability, capability, and feasibility. For instance, the first commercialized firefighting robots, Thermite RS1 and RS3 (Textron, Providence, RI, USA) [4], the AirCore TAF35 (MAGIRUS, Ulm, Germany) [5], and the Shark-robotic Colossus (Shark Robotics, La Rochelle, France) [6], are wheeled robots that carry water hoses and spaying nozzles to attack the fire. They also provide reconnaissance and situational awareness in high-risk areas through camera systems, so they can operate remotely. The power engines and continuous track configurations allow them to carry a huge amount of water, overcome obstacles, and traverse difficult terrain. This enables them to battle the fire from inside or at a closer distance, which is too dangerous for a firefighter. In addition, the Firefighting Robot System (Mitsubishi Heavy Industries, Ltd., Chiyoda City, Tokyo, Japan) [7] includes a firefighting robot equipped with onboard sensors, such as RTK-GPS (SoftBank Group, Minato City, Tokyo, Japan), LiDAR (Velodyne Lidar, San Jose, CA, USA), IMU (Imugene Limited, New South Wales, Australia), and odometry, and a hose extension accessory. The completed system has the ability to self-drive directly to the scene of a fire and perform the firefighting mission effectively. However, the large dimensions and heavy weight of the wheeled robots make them hard to operate in narrow environments or high locations.

Animal-inspired robots—for instance, snake firefighting robots [8] and bug firefighting robots [9]—have been considered in these situations. The snake robot designed by Liljeback et al. [8] works as a self-propeller fire hose that can crawl into a burning narrowed space and extinguish a fire. Dumiak [9] proposed the idea of bug-inspired legged robots going around in nature to detect the location of fire ignition and suppressing it to prevent large-scale wildfires. In addition, Kim et al. [10] proposed deploying the humanoid robot SAFFiR (Robotiq, Lévis, QC, Canada) for firefighting in naval vessels. The animal-inspired and humanoid robots are flexible, so they suit the missions in narrow environments or complex structures. One drawback that would impact their use would be low visibility due to smoke and extremely harsh environments. Additionally, with high-rise buildings, traffic congestion, or water scenarios, these robots are not practical to deploy. Moreover, the operation range of terrestrial firefighting robots is generally limited.

On the other hand, aerial vehicles, especially multi-propeller drones, can easily access these locations, so they can inject fire-extinguishing agents to suppress the flames. The world’s first flying-type firefighting drones, Ehang 216-F (Ehang, Guangzhou, China) [11] and the ZHUN Walkera (Guangzhou Walkera Technology Co., Ltd., Guangzhou, China) [12], used firefighting foam and dry powder, respectively, for the task. They are specially designed for suppressing fires in high-rise buildings. However, a drone can only carry a certain amount of the firefighting agent, so it is not possible to work continuously and be ineffective in serious fire accidents. This can be overcome by continuously conveying the water from a ground hydrant or pump to the drone’s nozzle via a flexible fire hose. Alternate systems such as Aerones (Riga, Latvia) [13] and Goufei Fire-fighting UAV (Goufei Aviation, Chongquing, China) [14] were designed to overcome this issue. One drawback is that the higher the drone flies, the heavier it becomes. The higher power demanded of the propeller drone and its battery’s capacity constraint meant that the fire accident location affects both the flight range and the time of the drone. One other thing to consider is that the airflow generated by the propellers can also spread the flame to neighboring areas.

More recently, flying-type firefighting robots that use water thrust force have been considered, especially for a fire that happens in water areas. The approach was inspired by water-powered flying devices such as flying boards [15,16] and water jetpacks [17,18,19]. In [20,21,22,23], the Dragon Firefighter system generates thrust by speeding up the water flow through a nozzle assembly and uses a servos actuator system to adjust the angle of nozzles, thus adjusting the force directions. This not only lifts the hose and the system, but the water jet actuator also directs the water flow to suppress the flame. Unfortunately, rotational motions are very limited with the system due to its structure. A later flying-type firefighting system in [24] uses four nozzles for maneuvering and a separated sprinkler to suppress the fire. The movement of the proposed system is carried out by changing the flowrate of each nozzle. The development of this system is still in the initial stage. The placement of the actuation system and the location of the water-conveying hose have not been finalized yet.

Furthermore, these abovementioned water-powered firefighting systems are generally in a stage of development and completion. Thus, the conceptual designs and mechanical designs are investigated preferentially. The motion control methods, therefore, are fairly simple. In particular, a proportional controller with speed feedback is implemented in a Dragon Firefighter [20,21,22,23]. The simulations and experiments are adopted, and it is indicated that this controller is not sophisticated and cannot control the system precisely [20]. The investigation of the latter mechanical approach helps it works better with the same controller [21,22,23]. However, a more advanced control technique is also initially implemented to evaluate the feasibility of water-powered flying-type firefighting [24]. A sliding mode control technique is proposed to govern the system motions. The simulation tests indicated the robustness in tracking performance. However, the control effort itself is predicted to bring a challenge for practical applications due to the controller’s actions.

Therefore, a flying-type firefighting robot that is water-powered, automatically controlled, and has a novel weight-shifting mechanism for maneuvering is proposed in this study. The proposed robot uses the water thrust as the propulsion force, similar to the abovementioned system. The robot is aimed to be applied in firefighting tasks in water areas that have an unlimited amount of water but are difficult to access and are suppressed by conventional fire-extinguishing methods. The weight-shifting mechanism adjusts the weight distribution on the system, in a similar way to human movement on the fly board, in order to fly around. The target scenarios for the system application are visualized in Figure 1. In particular, a pump and a flexible hose convey water to generate water to the robot head. The head part generates thrust to take off and adjusts the actuator mechanism to approach the flame. The fire-extinguishing sprinkler sprays the water to suppress the fire remotely or automatically. The zoomed figure visualizes the robot head with its thrust nozzles, fire-extinguishing sprinkler, and weight-shifting mechanism.

In this paper, the system conceptual design is introduced. Mathematical models reveal the dynamical characteristics of the system. To demonstrate the maneuverability and feasibility of the proposed system, a linear-quadratic integrator (LQI) is designed such that the water flowrate and the weight-shifting mechanism are controlled to fly the system to the fire area. Moreover, an optimal approach based on the genetic algorithm (GA) tunes the controller’s gains. Thus, both motion performance and system robustness are preserved.

Accordingly, the contributions of this study in comparison to the relevant studies are as follows:

• A novel conceptual design of the flying firefighting robot using waterpower and a weight-shifting mechanism is proposed.
• The mathematical models of the system are formulated, and its characteristics are analyzed.
• An LQI design with a novel GA approach is introduced.
• Comparative simulation studies reveal the feasibility and efficiency of the proposed system.

The remainder of the paper is organized as follows. Section 2 introduces the system configuration and mathematical models. The LQI control system design with the GA approach is explained in Section 3. The simulation results are discussed in Section 4. Finally, the conclusions are drawn in Section 5.

2. System Configuration and Modeling

2.1. System Configuration

The flying firefighting robot flies thanks to the thrust forces generated from the water flow through the robot head part. In particular, a water pump placed on the ground or floating on the water, depending on the fire accident location, provides the water through a flexible hose to the inlet of the head part. The robot’s head distributes the water flow into four nozzle outlets, whose cross-sectional areas are much smaller than that of the inlet. The nozzles are fixed and pointed straight down for maximum thrust. The water flows are sped up by the nozzle and generate thrust forces. On the other side of the head, two weight blocks, WX and WY, are integrated with linear motors that allow them to move along the axes set in the longitudinal and transverse directions. When the weights are out of the central positions, the mass distribution of the head becomes unbalanced. It results in the head tilting and/or rolling, and subsequently, the robot moves transversely in the corresponding direction. In addition, the head also carries a water sprinkler for battling fire flames. The sprinkler is designed with an external nozzle ring, which atomizes water into a fine mist, which also helps to reduce heat from the flame to the devices on the robot’s head. In this conceptual design, a swivel ball joint is proposed as the connection between the water hose and the robot’s head to reduce the effect of the movement of the hose. The conceptual design of the robot head is drawn in Figure 2 below.

2.2. System Modeling

To analyze the motion characteristics of the system, the Earth-fixed reference frame OXYZ and body-fixed frame O_bx_by_bz_b are defined, as shown in Figure 2. The former has an X-axis pointing to the magnetic north and a Z-axis pointing up. The latter is placed at the center of the mass of the head, in which its x_b-axis is along the longitudinal direction of the rigid body and points forward. The y_b-axis follows the lateral direction of the robot’s head, and finally, the z_b-axis points upward. Let ${\left[\begin{array}{ccc}{\omega }_{xb}& {\omega }_{yb}& {\omega }_{zb}\end{array}\right]}^T$ denote the angular velocity of the body-fixed frame orientation. The body axes rotate from the reference frame axes via a series of Euler angle rotations. They are roll, pitch, and yaw, respectively, and they are denoted by the vector ${\left[\begin{array}{ccc}\varphi & \theta & \psi \end{array}\right]}^T$. The kinematics of the designed robot in the reference frame are written in terms of the Euler angles, as follows:

$$\begin{array}{c}\left[\begin{array}{cccccc}\dot{X}& \dot{Y}& \dot{Z}& \dot{\varphi }& \dot{\theta }& \dot{\psi }\end{array}\right]=R{\left[\begin{array}{cccccc}{\dot{x}}_b& {\dot{y}}_b& {\dot{z}}_b& {\omega }_{xb}& {\omega }_{yb}& {\omega }_{zb}\end{array}\right]}^T,\\ R=\left[\begin{array}{cc}{R}_b^E& O_3\\ O_3& T_b^E\end{array}\right]\end{array}$$

(1)

where $O_{m\times p}$ denotes an m-by-p matrix of zeros, $R_b^E$ is the rotational matrix from the body-fixed frame to the reference frame [25], and $T_b^E$ is the transformation matrix between the angular velocity and the Euler angle rate [26], respectively.

For the robot dynamics, the free body diagram of the head part is illustrated in Figure 3. The hose conveying water and the actuating weights are isolated from the head part. The water hose is replaced by its mass being lifted $M_p$ at the center of the head and a reaction force F₀ at the water inlet port. The actuating weights are represented by the reaction forces due to their motions and position. Additionally, the thrust force generated by the ith nozzle outlet is denoted by F_i (i = 1, 2, 3, 4). The masses of the actuating weights WX and WY are m_wx and m_wy, respectively. Meanwhile, the mass of the remaining head part with its containing water is $M_h$. Additionally, when the firefighting robot gets close to the flame and the extinguishing nozzle activates, a reaction force F_s appears and acts on the system.

Considering the head part without the two actuating weights as a rigid body, thanks to the isolation of the free-body diagram, the dynamic equations for translational and rotational motions in the body-fixed frame are well-known as:

$$\begin{array}{c}\left(M_h+M_p\right)\left({\dot{v}}_b+{\omega }_b\times v_b\right)=F_b\\ J{\dot{\omega }}_b+{\omega }_b\times J{\omega }_b={\tau }_b\end{array}$$

(2)

where $v_b$ and ${\omega }_b$ are the velocities of the translational and rotational motions in the body-fixed frame. The mass in the translational motions consists of the mass of the robot’s head $M_h$ and that of the water pipe being lifted $M_p$. For the sake of simplicity, $M_p$ is assumed to linearly vary with the robot’s flight altitude via a correction factor k such that $M_p=k\rho AZ$. $\rho$ is the density of the water, and $A$ is the internal cross-sectional area of the hose. $\rho AZ$ is the mass of a water cylinder of area A and height Z. This assumption was also experimentally verified in the literature [19]. Moreover, the correction factor can be validated based on later experiments. For the rotational motions, J is the inertia matrix of the head, which is assumed to be diagonal to reflect the objective of the technical design of the fabricated model. That is, $J=diag\left(J_{xx},\hspace{0.17em}\hspace{0.17em}{J}_{yy},\hspace{0.17em}\hspace{0.17em}{J}_{zz}\right)$. In Equation (2), $\left(M_h+M_p\right){\dot{v}}_b$ and $J{\dot{\omega }}_b$ are the differentiation of the system linear momentum and the moment-of-momentum, respectively. The cross-product terms ${\omega }_b\times v_b$ and ${\omega }_b\times J{\omega }_b$ represent the Coriolis phenomenon.

On the right-hand side, the total forces and torques acting on the robot’s head are given by $F_b$ and ${\tau }_b$, respectively. They include the gravitation force, the water thrust (F_t, τ_t), actuating forces ($-F_{wx},-F_{wy}$) and torques ($-{\tau }_{wx},-{\tau }_{wy}$) from the weight-shifting mechanism, reaction forces from the hose $-F_h$ and the water sprinkler $-F_s$, and other external disturbances ($F_d,{\tau }_d$), such as the wind gust, ceiling, and ground effect. That is:

$$\begin{array}{c}{F}_b=F_t-F_{wx}-F_{wy}+\left(M_h+M_p\right){\left(R_b^E\right)}^T{\left[\begin{array}{ccc}0& 0& g\end{array}\right]}^T-F_h-F_s+F_d\\ {\tau }_b={\tau }_t-{\tau }_{wx}-{\tau }_{wy}+{\tau }_d\end{array}$$

(3)

The water thrust can be obtained by applying Newton’s second law on the water flowing through the head part. That is, the time rate of change of the linear momentum and moment-of-momentum of the water equals the summation of external forces and torques, respectively, acting on the water [24]. To obtain simple but sufficiently reliable dynamics of the fluid, assume that the water flow through the head is frictionless and steady [27]. The water thrust force and torque are written as follows:

$$\begin{array}{c}{F}_t=\left[\begin{array}{c}0\\ 0\\ \frac{1}{\rho A}{\dot{m}}_0^2+\frac{1}{\rho a}{\displaystyle \sum _{i=1}^4{\dot{m}}_i^2}+p_{p}A\end{array}\right]+{\displaystyle \sum _{i=1}^4{\dot{m}}_i{\omega }_b\times r_i}\\ {\tau }_t=-{\displaystyle \sum _{i=1}^4{\dot{m}}_i{r}_i\times \left({\omega }_b\times r_i\right)}\end{array}$$

(4)

In which ${\dot{m}}_0$ and ${\dot{m}}_i$ are the mass flowrates at the water inlet port and the ith nozzle outlet, respectively. a is the cross-sectional of each nozzle outlet. The term $p_{p}A$ represents the water force acting at the water inlet due to the gauge pressure $p_p$ at this area. Furthermore, the vector r_i determines the position of the ith nozzle outlet in the robot’s body frame.

On the other hand, the weight-shifting mechanism affects the robot’s head by the reaction forces and torques resulting from the motion of its two weight blocks. Equations of their motion are written as follows:

$$\begin{array}{c}{m}_{wj}\left({\dot{v}}_{wj}+{\omega }_b\times v_{wj}\right)=F_{wj}+m_{wj}{\left(R_b^E\right)}^T{\left[\begin{array}{ccc}0& 0& g\end{array}\right]}^T,\\ J_{wj}{\dot{\omega }}_b+{\omega }_b\times J_{wj}{\omega }_b={\tau }_{wj}+m_{wj}{r}_{wj}\times {\left(R_b^E\right)}^T{\left[\begin{array}{ccc}0& 0& g\end{array}\right]}^T,\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=x,y$$

(5)

The translational motion of each weight consists of its displacement ($r_{wx}$ and $r_{wy}$) observed from the robot’s head and the motion of the head itself. Thereby, the translational velocities of the weight WX and WY are given by:

$$\begin{array}{c}{v}_{wx}={\left[\begin{array}{ccc}{\dot{r}}_{wx}& 0& 0\end{array}\right]}^T+v_b+r_{wx}\times {\omega }_b,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_{wx}={\left[\begin{array}{ccc}{r}_{wx}& 0& h_{wx}\end{array}\right]}^T,\\ v_{wy}={\left[\begin{array}{ccc}0& {\dot{r}}_{wy}& 0\end{array}\right]}^T+v_b+r_{wy}\times {\omega }_b,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_{wy}={\left[\begin{array}{ccc}0& r_{wy}& h_{wy}\end{array}\right]}^T\end{array}$$

(6)

where ${\dot{r}}_{wx}$ and ${\dot{r}}_{wy}$ are their respective relative velocities, and $h_{wx}$ and $h_{wy}$ are their corresponding vertical distances from the coordinate origin.

On the order hand, the weights rotate with the rotation of the head part carrying them. Their moments of inertia taken at the body coordinate vary with their relative displacements. In other words:

$$J_{wx}=m_{wx}\left[\begin{array}{ccc}{h}_{wx}^2& 0& -h_{wx}{r}_{wx}\\ 0& h_{wx}^2+r_{wx}^2& 0\\ -h_{wx}{r}_{wx}& 0& r_{wx}^2\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{J}_{wy}=m_{wy}\left[\begin{array}{ccc}{h}_{wy}^2+r_{wy}^2& 0& 0\\ 0& h_{wy}^2& -h_{wy}{r}_{wy}\\ 0& -h_{wy}{r}_{wy}& r_{wy}^2\end{array}\right]$$

(7)

Moreover, consider that four nozzles are perfectly symmetric, and because the head is kept stabilized in operation, neglect the nozzles’ elevation difference. The continuity equation indicates that the water flows through the nozzles are equal:

$${\dot{m}}_1={\dot{m}}_2={\dot{m}}_3={\dot{m}}_4=\frac{1}{4}{\dot{m}}_0$$

(8)

The thrust force of the system is generated by these flows, so the different behaviors of the nozzle—for example, when the head tilts or rolls—are always able to be considered as input disturbances.

In summary, by substituting F_t and τ_t from Equation (4), F_wx, F_wy, τ_wx, and τ_wy from Equations (5)–(7), and ${\dot{m}}_i$ from Equation (8), the equations of motion in all directions are obtained as follows:

$$\begin{array}{c}{\ddot{x}}_b=g\sin \theta +\frac{1}{a_1}h{\dot{m}}_0{\omega }_{yb}+d_1,\\ {\ddot{y}}_b=-g\cos \theta \sin \varphi -\frac{1}{a_1}h{\dot{m}}_0{\omega }_{xb}+d_2,\\ {\ddot{z}}_b=\frac{1}{\rho a_1}\left(\frac{1}{A}+\frac{1}{4a}\right){\dot{m}}_0^2-g\cos \theta \cos \varphi +d_3,\\ {\dot{\omega }}_{xb}=\frac{1}{{\sigma }_1}\left[\left(m_{wx}{h}_x+m_{wy}{h}_y\right)g\cos \theta \sin \varphi -m_{wy}{r}_{y}g\cos \theta \cos \varphi -{\omega }_{xb}\left(h^2+l^2\right){\dot{m}}_0\right]+w_1,\\ {\dot{\omega }}_{yb}=\frac{1}{{\sigma }_2}\left[m_{wx}{r}_{x}g\cos \theta \cos \varphi +\left(m_{wx}{h}_x+m_{wy}{h}_y\right)g\sin \theta -{\omega }_{yb}\left(h^2+l^2\right){\dot{m}}_0\right]+w_2,\\ {\dot{\omega }}_{zb}=\frac{1}{{\sigma }_3}\left[-2l^2{\omega }_{zb}{\dot{m}}_0\right]+w_3\end{array}$$

(9)

where:

$$\begin{array}{c}{a}_1=M+m_{wx}+m_{wy},\hspace{0.17em}\hspace{0.17em}{\sigma }_1=J_{xx}+m_{wx}{h}_x^2+m_{wy}\left(r_y^2+h_y^2\right),\\ {\sigma }_2=J_{yy}+m_{wy}{h}_y^2+m_{wx}\left(h_x^2+r_x^2\right),\hspace{0.17em}\hspace{0.17em}{\sigma }_3=J_{zz}+m_{wx}{r}_x^2+m_{wy}{r}_y^2\end{array}$$

(10)

and:

$$\begin{gathered} \begin{array}{c}{d}_1=-\frac{1}{a_1}{m}_{wx}{\ddot{r}}_{wx}+{\dot{y}}_b{\omega }_{zb}-{\dot{z}}_b{\omega }_{yb}+\frac{1}{a_1}\left[-{\dot{\omega }}_{yb}\left(m_{wy}{h}_y+m_{wx}{h}_x\right)+m_{wy}{\omega }_{zb}{\dot{r}}_{wx}+{\dot{\omega }}_{zb}{m}_{wy}{r}_{wy}-F_{hx}-F_{sx}+F_{dx}\right],\\ d_2=-\frac{1}{a_1}{m}_{wy}{\ddot{r}}_{wy}+{\omega }_{xb}{\dot{z}}_b-{\omega }_{zb}{\dot{x}}_b+\frac{1}{a_1}\left[+{\dot{\omega }}_{xb}\left(m_{wx}{h}_x+m_{wy}{h}_y\right)-m_{wx}{\omega }_{zb}{\dot{r}}_{wy}-m_{wx}{\dot{\omega }}_{zb}{r}_x-F_{hy}-F_{sy}+F_{dy}\right],\\ d_3={\omega }_{yb}{\dot{x}}_b-{\omega }_{xb}{\dot{y}}_b+\frac{1}{a_1}\left[m_{wx}\left({\omega }_{yb}{\dot{r}}_{wx}+{\dot{\omega }}_{yb}{r}_{wx}\right)-m_{wy}\left({\omega }_{xb}{\dot{r}}_{wy}+{\dot{\omega }}_{xb}{r}_{wy}\right)+p_{p}A-F_{hz}-F_{sz}+F_{dz}\right],\\ w_1=\frac{1}{{\sigma }_1}\left[\begin{array}{c}\left(m_{wx}{h}_x+m_{wy}{h}_y\right){\ddot{y}}_b-m_{wy}{r}_y{\ddot{z}}_b+{\dot{\omega }}_{zb}{m}_{wx}{r}_x{h}_x+{\omega }_{xb}{\omega }_{yb}{m}_{wx}{r}_x{h}_x-h{\displaystyle \sum _{i=1}^4{\dot{m}}_i}{\dot{y}}_b+m_{wy}{h}_y{r}_y\left({\omega }_{yb}^2-{\omega }_{zb}^2\right)\\ +{\omega }_{yb}{\omega }_{zb}\left(m_{wx}\left(h_x^2+r_x^2\right)-m_{wx}{r}_x^2+m_{wy}{h}_y^2-m_{wy}{r}_y^2-J_{zz}+J_{yy}\right)+{\tau }_{dx}\end{array}\right],\end{array} \\ \begin{array}{c}{w}_2=\frac{1}{{\sigma }_2}\left[\begin{array}{c}-\left(m_{wx}h+m_{wy}{h}_y\right){\ddot{x}}_b+{\omega }_{zb}{\omega }_{xb}\left(-m_{wx}{h}_x^2+m_{wx}{r}_x^2+m_{wy}{r}_y^2-m_{wy}\left(h_y^2+r_y^2\right)-J_{xx}+J_{zz}\right)\\ +m_{wx}{r}_x{h}_x\left({\omega }_{zb}^2-{\omega }_{xb}^2\right)+m_{wx}{r}_x{\ddot{z}}_b+a_6+{\dot{\omega }}_{zb}{m}_{wy}{h}_y{r}_y+{\dot{x}}_{b}h{\displaystyle \sum _{i=1}^4{\dot{m}}_i}-{\omega }_{xb}{\omega }_{yb}{m}_{wy}{h}_y{r}_y+{\tau }_{dy}\end{array}\right],\\ w_3=\frac{1}{{\sigma }_3}\left[\begin{array}{l}{\omega }_{xb}{\omega }_{yb}\left(-m_{wx}\left(h_x^2+r_x^2\right)+m_{wx}{h}_x^2+m_{wy}\left(h_y^2+r_y^2\right)-m_{wy}{h}_y^2-J_{yy}+J_{xx}\right)+{\omega }_{xb}{\omega }_{zb}\left(m_{wy}{h}_y{r}_y-m_{wx}{r}_x{h}_x\right)\\ -m_{wx}{r}_x{\ddot{y}}_b+m_{wy}{r}_y{\ddot{x}}_b+m_{wx}{\dot{\omega }}_{xb}{r}_x{h}_x+m_{wy}{\dot{\omega }}_{yb}{r}_y{h}_y-m_{wy}{r}_{y}g\sin \theta -m_{wx}{r}_{x}g\cos \theta \sin \varphi +{\tau }_{dz}\end{array}\right]\end{array} \end{gathered}$$

(11)

Equation (11) lists the remaining terms of the system model in addition to Equation (9), including the accelerations of the actuation system, the higher-order terms, the input-output coupling terms, the effects of the water sprinkler and the water hose, and the external disturbance ${\left[F_d{}^T\hspace{0.17em}\hspace{0.17em}{\tau }_d{}^T\right]}^T$. It is worth noting that if the control inputs are the water flowrate and the position of the two weights, the yaw rotation is uncontrollable. Fortunately, it is stable, and the system can still operate with uncontrolled yaw rotation—for example, as in [28]. Furthermore, the effect of the acceleration of two weight blocks leads the system to a non-minimum phase phenomenon. That is, when a weight block moves in one direction, the robot initially moves backward before tilting and moving in that same direction.

3. Control System Design

3.1. LQI Control Law Design

The objective of the control system is to drive the robot from an initial position to follow the desired trajectory with a small tracking error, even in the presence of external disturbances and non-minimum phase phenomena. A servo control scheme with state feedback and an integral action is able to fulfill both requirements. The LQI is the servomechanism with optimal gain matrices that minimize the quadratic cost of the closed-loop control system. The cost function consists of the system states, integrator outputs, and control inputs with the corresponding coefficient matrices that can be easily tuned to reflect the weight of each element. Therefore, the LQI control system is designed based on the linearization of the system model in (9)–(11). Select a local equilibrium state at the robot that is unrotated and stabilized about the desired altitude ${\overline{z}}_b$. It is easily seen that the robot has to inject a constant flowrate ${\dot{\overline{m}}}_0$ such that the thrust force is equal to the total gravitational force at this position. Meanwhile, the two weights in the weight-shifting mechanism remain in their original position. Additionally, the yaw motion is uncontrollable but stable. Therefore, by denoting the state vector as $X_b={\left[\begin{array}{cccccccccc}{x}_b& y_b& z_b-{\overline{z}}_b& {\phi }_x& {\phi }_y& {\dot{x}}_b& {\dot{y}}_b& {\dot{z}}_b& {\omega }_{xb}& {\omega }_{yb}\end{array}\right]}^T$, with ${\phi }_x$ and ${\phi }_y$ angular positions about x_b- and y_b-axes, and the control input vector as $U={\left[\begin{array}{ccc}{\dot{m}}_0-{\dot{\overline{m}}}_0& r_{wx}& r_{wy}\end{array}\right]}^T$, the system can be linearized as follows:

$$\begin{array}{l}{\dot{X}}_b=A{X}_b+BU+D,\\ Y_b=C{X}_b={\left[\begin{array}{ccc}{x}_b& y_b& z_b-{\overline{z}}_b\end{array}\right]}^T\end{array}$$

(12)

The matrices A, B, and C are given by:

$$\begin{gathered} A=\left[\begin{array}{c}\begin{array}{cc}{O}_5& I_5\end{array}\\ \begin{array}{cccccccccc}0& 0& 0& 0& g& 0& 0& 0& 0& \frac{h{\dot{\overline{m}}}_0}{a_1}\\ 0& 0& 0& -g& 0& 0& 0& 0& \frac{-h{\dot{\overline{m}}}_0}{a_1}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \frac{m_{wx}{h}_{x}g+m_{wy}{h}_{y}g}{{\sigma }_1}& 0& 0& \frac{-h{\dot{\overline{m}}}_0}{{\sigma }_1}& 0& \frac{-{\dot{\overline{m}}}_0\left(h^2+l^2\right)}{{\sigma }_1}& 0\\ 0& 0& 0& 0& \frac{m_{wx}{h}_{x}g+m_{wy}{h}_{y}g}{{\sigma }_2}& \frac{h{\dot{\overline{m}}}_0}{{\sigma }_2}& 0& 0& 0& \frac{-{\dot{\overline{m}}}_0\left(h^2+l^2\right)}{{\sigma }_2}\end{array}\end{array}\right], \\ B=\left[\begin{array}{c}{O}_{7\times 3}\\ \begin{array}{ccc}\frac{2\left(\frac{1}{\rho A}+\frac{1}{4\rho a}\right){\dot{\overline{m}}}_0}{\overline{M}+m_{wx}+m_{wy}}& 0& 0\\ 0& 0& \frac{-m_{wy}g}{J_{xx}+m_{wx}{h}_x^2+m_{wy}\left({\overline{r}}_y^2+h_y^2\right)}\\ 0& \frac{m_{wx}g}{J_{yy}+m_{wy}{h}_y^2+m_{wx}\left(h_x^2+{\overline{r}}_x^2\right)}& 0\end{array}\end{array}\right], \\ C=\left[\begin{array}{cc}{I}_3& O_{3\times 7}\end{array}\right] \end{gathered}$$

(13)

$I_n$ is the identity matrix size n and D is the vector that contains the remaining nonlinear elements and unmodelled disturbances.

Moreover, given a desired trajectory $r={\left[\begin{array}{ccc}{X}_d& Y_d& Z_d\end{array}\right]}^T$ in the Earth-fixed frame, from the kinematic relationship in (1), the trajectory can be represented in the robot body frame as follows:

$$r_b={\left(R_b^E\right)}^{T}r-{\displaystyle \underset{0}{\overset{t}{\int }}{\left({\dot{R}}_b^E(\tau )\right)}^{T}r(\tau )d\tau }$$

(14)

An augmented variable of tracking error is defined as follows:

$${\dot{X}}_i=r_b-Y_b$$

(15)

Then, the system model, in combination with the augmented variable, is written as:

$$\underset{\dot{\tilde{X}}}{\underbrace{\left[\begin{array}{c}{\dot{X}}_b\\ {\dot{X}}_i\end{array}\right]}}=\underset{\overline{A}}{\underbrace{\left[\begin{array}{cc}A& O_{10\times 3}\\ -C& O_3\end{array}\right]}}\underset{\tilde{X}}{\underbrace{\left[\begin{array}{c}{X}_b\\ X_i\end{array}\right]}}+\underset{\overline{B}}{\underbrace{\left[\begin{array}{c}B\\ O_3\end{array}\right]}}U+\underset{\tilde{D}}{\underbrace{\left[\begin{array}{c}D\\ r_b\end{array}\right]}}$$

(16)

A state-feedback control law is designed in the form of:

$$U=-\left[\begin{array}{cc}{K}_f& K_i\end{array}\right]\tilde{X}$$

(17)

where the optimal matrix gains are computed by the LQI control method:

$$\underset{K}{\underbrace{\left[\begin{array}{cc}{K}_f& K_i\end{array}\right]}}=R^{-1}{\overline{B}}^{T}P$$

(18)

such that the control law (17) minimizes the following quadratic cost function:

$$\underset{U}{\min }{J}_U={\displaystyle \underset{0}{\overset{\infty }{\int }}\left({\tilde{X}}^{T}Q\tilde{X}+U^{T}RU\right)dt}$$

(19)

Q and R are positive-definite weighting matrices of the state and input, respectively. P is the positive definite and symmetric matrix, which satisfies the algebraic Riccati Equation (20):

$${\overline{A}}^{T}P+P\overline{A}-P\overline{B}{R}^{-1}{\overline{B}}^{\mathrm{T}}P+Q=0$$

(20)

The schematic drawing of the designed control system is depicted in Figure 4.

Assume that the vector of disturbances and unmodelled dynamics $\tilde{D}$ is bounded by a non-negative value $\gamma$ such that ${‖\tilde{D}‖}_2\le \gamma$. To analyze the robustness of the closed-loop system, a Lyapunov function candidate is chosen as follows:

$$V={\tilde{X}}^{\mathrm{T}}P\tilde{X}$$

(21)

Taking the time-derivative of V and applying the control law in (17) and the algebraic Riccati equation in (20) results in the following:

$$\begin{array}{l}\dot{V}\hspace{0.17em}={\tilde{X}}^T\left({\left(\overline{A}-\overline{B}K\right)}^{T}P+P\left(\overline{A}-\overline{B}K\right)\right)\tilde{X}+{\tilde{D}}^{T}P\tilde{X}+{\tilde{X}}^{T}P\tilde{D}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-{\tilde{X}}^T\left(P\overline{B}{R}^{-1}{\overline{B}}^{\mathrm{T}}P+Q\right)\tilde{X}+{\tilde{D}}^{T}P\tilde{X}+{\tilde{X}}^{T}P\tilde{D}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le \left[-{\lambda }_{\min }\left(P\overline{B}{R}^{-1}{\overline{B}}^{\mathrm{T}}P+Q\right){‖\tilde{X}‖}_2+2\gamma {\lambda }_{\max }\left(P\right)\right]{‖\tilde{X}‖}_2\end{array}$$

(22)

where ${\lambda }_{\min }(.)$ and ${\lambda }_{\max }(.)$ indicate the minimum and maximum eigenvalues of the corresponding matrix. In order to preserve the system stability, $\dot{V}$ must be negative-definite. From Equation (22), one can see that $\dot{V}<0$ whenever the state satisfies the following condition:

$${‖\tilde{X}‖}_2>\frac{2{\lambda }_{\max }\left(P\right)}{{\lambda }_{\min }\left(P\overline{B}{R}^{-1}{\overline{B}}^{\mathrm{T}}P+Q\right)}\gamma$$

(23)

which implies the boundedness of $\tilde{X}$.

3.2. GA Approach for Optimally Tuning the Controller Parameters

One can see that minimizing the right-hand side of Equation (23) would result in the smallest boundedness of the tracking errors, thus enhancing the robot’s performance and stability. However, better performance usually comes along with larger control energy. In the case of the proposed firefighting robot, the control energy includes the water pump energy and, especially, the kinetic energies of the two weight blocks in the weight shifting mechanism. From Equation (11), it is worth noting that the fast movement of the actuating weights makes the non-minimum phase phenomenon become worse and even causes the system to be unstable. In fact, if the acceleration of the actuating weights is separated from the vector $\tilde{D}$ of the unmodelled dynamics, the boundedness $\gamma$ of $\tilde{D}$ can be considered as follows:

$$\begin{array}{c}\gamma \le {\gamma }_w+{\gamma }_d,\\ {\gamma }_w=\frac{1}{a_1}{‖\left[\begin{array}{c}{m}_{wx}{\ddot{r}}_{wx}\\ m_{wy}{\ddot{r}}_{wy}\end{array}\right]‖}_2, {\gamma }_d={‖\tilde{D}-\frac{1}{a_1}{\left[\begin{array}{ccc}{m}_{wx}{\ddot{r}}_{wx}& m_{wy}{\ddot{r}}_{wy}& O_{1\times 1}\end{array}\right]}^T‖}_2\end{array}$$

(24)

where ${\gamma }_w$ contains the weights’ acceleration and ${\gamma }_d$ bounds the remainder. Minimizing the effect of ${\gamma }_w$ while minimizing the tracking error strikes a balance between enhancing the system performance and reducing the non-minimum phase phenomenon. In other words, let a cost function be proposed as follows:

$$\underset{Q,R}{\min }{J}_{Q,R}=\left\{\begin{array}{cc}\frac{2{\lambda }_{\max }\left(P\right)}{{\lambda }_{\min }\left(P\overline{B}{R}^{-1}{\overline{B}}^{\mathrm{T}}P+Q\right)}\left[1+\beta \left(m_{wx}{\displaystyle \underset{0}{\overset{t}{\int }}{\ddot{r}}_{wx}^2(\tau )d\tau }+m_{wy}{\displaystyle \underset{0}{\overset{t}{\int }}{\ddot{r}}_{wy}^2(\tau )d\tau }\right) \right] & \left|r_{wx}\right|\le l,\left|r_{wy}\right|\le l; 0\le {\dot{m}}_0\le {\overline{\dot{m}}}_{\max }\hfill \\ \infty & \mathrm{otherwise}\hfill \end{array}\right.$$

(25)

in which $\beta ={\gamma }_w/{\gamma }_d$ is a trade-off between the performance and the non-minimum phase phenomenon. The conditions $\left|r_{wx}\right|\le l,\hspace{0.17em}\hspace{0.17em}\left|r_{wy}\right|\le l,$ and $0\le {\dot{m}}_0\le {\overline{\dot{m}}}_{\max }$ consider the saturation of the control input of the actual system. In particular, the left term of the cost is globally valid. Meanwhile, the other depends on the operating scenario. One can see that they are largest when the robot has to follow step-type references. Therefore, this scenario is going to be taken in the optimization process.

The objective of the optimization process is to find the controller matrices Q and R such that the cost function in (25) is minimized in the mentioned scenarios. The GA [29] is implemented for solving the optimization problem. The process is depicted in the following steps, as well as in Figure 5.

Step 1: Initialization

The first step of optimization is to create an initial population of the Q and R matrices. A pair ($Q_i$, $R_i$) of matrices, where $i\in \left[1,N\right]$, N is the population size, is an individual.

Step 2: Fitness evaluation

Each individual from the previous step is substituted in Equations (18) and (20) to calculate the P and K matrices, respectively. A simulation with the step-type trajectory is conducted with the above matrices, and the cost function (25) is obtained after finishing. A fitness function, which is an evaluation tool to determine the existence of the individual after each optimization generation, can also be defined as the inverse of the cost.

Step 3: Decimal Encoding

Each individual ($Q_i$, $R_i$) is encoded into a series of chromosomes s_i by using a decimal encoder.

Step 4: Natural selection

These fitness values are arranged in ascending order based on the linear ranking selection, in which the best individual is placed last and the worst one is in the first position. The jth individuals are selected for the next step via its accumulative probability p_j, as described in Algorithm 1a. The higher a fitness value, the greater the probability of the individual being selected

Step 5: Crossover

Several individuals are randomly chosen to carry out the crossover. The number of conducted individuals is determined by a predefined crossover probability κ. Two crossover points in the chromosome series are selected randomly. A two-point crossover method, as in Algorithm 1b, swaps the chromosomes between two points from one individual to another.

Step 6: Mutation

As in Algorithm 1c, the mutation process arbitrarily changes one or more chromosomes, with a small probability η, in an individual to avoid premature convergence.

Step 7: Decode

The series of chromosomes s_i is decoded and returns the corresponding matrices $Q_i$ and $R_i$.

Step 8: Fitness evaluation

The dependent matrices are computed in the same manner as in Step 2. The simulation is carried out to evaluate the best fitness value.

Step 9: Stop condition

The optimization process will be stopped if there is no change to the best fitness for a specified number of generations. Otherwise, the new loop will be continued.

Algorithm 1. The evolutionary process within the GA optimization

a. Linear Ranking Selection

for k = 1:N    r = rand(0,N);      if (pj–1 ≤ r < pj) then Ik = Ij      end    end

b. Two-point Crossover

for m = 1:N      k1 = rand(0,l); k2 = rand(0,l)      if (κ > rand (0,1)) then      Im(1:k1) := p1(1:k1); Im(k1 + 1:k2) := p2(k1 + 1:k2); Im(k2 + 1:k) := p1(k2 + 1:k)      end    end

c. Uniform Mutation

for m = 1:N      for k = 1:l        if (η > rand(0,1)) then Im(k) = rand(0,9)        end      end    end

4. Simulation Results and Discussion

4.1. Simulation

To evaluate the dynamical characteristics of the designed firefighting robot and validate the feasibility of the proposed LQI control system, two scenarios reflecting two common motion trajectories are considered. One is a step-type trajectory, which is similar to the case where the firefighting system moves from one point to another to battle individual flames. The other is a continuous flying path replicating the way in which the system overcomes obstacles and approaches the fire site. In simulation studies, each scenario is carried out with and without the influence of external forces and torques. These external terms represent the unformulated effects in Section 2, including the effects of the water sprinkler $-F_s$ and the water hose $-F_h$ and external disturbances ${\left[F_d{}^T\hspace{0.17em}\hspace{0.17em}{\tau }_d{}^T\right]}^T$. The latter include the effects of the wind gust, ceiling, ground effect, etc. in the practical operation of the system, but they are neglected in the modeling process. This is to validate the robustness of the proposed control system when operating the fire-extinguishing sprinkler or facing disturbances. The formulations of the external forces and torques are adopted from the previous publication of Nguyen and Hong [30], where they were also used to test the robustness of an aerial system. As mentioned in the previous section, the step-type trajectory is also used in the optimization process of finding the controller’s parameters.

Moreover, a cascade proportional—integral—derivative (PID) control structure from [24] is also compared to the proposed LQI control with the GA approach, in which the outer PID controllers use the horizontal tracking errors in X- and Y-directions to compute the robot’s required roll and tilt rotations, respectively. They are the references for the inner controllers, where the control inputs $r_{wx}$ and $r_{wy}$ are correspondingly obtained. In addition, another PID controller computes the required flowrate ${\dot{m}}_0$ from the altitude control error of the system. Those control laws are mathematically represented in the following formulation:

$$\begin{array}{c}{U}_{PID}=K_{p}e+K_I{\displaystyle \underset{0}{\overset{t}{\int }}e}d\tau +K_D\frac{d}{dt}e,\\ e={\left[\begin{array}{ccccc}{x}_d& {\theta }_d& y_d& {\varphi }_d& z_d\end{array}\right]}^T-{\left[\begin{array}{ccccc}x& \theta & y& \varphi & z\end{array}\right]}^T,\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}{U}_{PID}={\left[{\theta }_d\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_{wx}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\varphi }_d\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_{wy}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{m}}_0\right]}^T\end{array}$$

(26)

The system model given by (9)–(11) is carried into the simulation. The parameters of the designed concept are shown in

Table 1. Parameters of the proposed firefighting robot.

Parameter	Description	Value	Unit
$M_h$	Mass of the head part	3	kg
J	Moment of inertia of the head part	$diag\left\{0.0071,0.0085,0.0133\right\}$	Kg·m2
A	Inner diameter of the water hose	0.05	m
a	Inner diameter of the nozzle outlets	0.01	m
k	Correction factor of the water hose	1.5	-
l, h	Dimensions of the head part	0.25, 0.1	m
mwx, hx	Weight and vertical location of WX	0.5, 0.13	kg, m
mwy, hy	Weight and vertical location of WY	0.5, 0.08	kg, m

. The equilibrium point for linearization is at ${\overline{z}}_b=10\hspace{0.17em}\hspace{0.17em}[\mathrm{m}]$ height with respect to an injected flowrate ${\dot{\overline{m}}}_0=2.7\hspace{0.17em}\hspace{0.17em}[\mathrm{kg}/\mathrm{s}]$ to maintain this altitude for the system. The weighting matrices Q and R are found by the optimization process with the GA after 331 generations. The convergence of the cost function during the process is depicted in Figure 6. The feedback control gains K_i and K_f are computed as in Equations (18) and (20) with the previously obtained Q and R. They are shown in

Table 2. Controllers’ gains.

Controller	Controller’s Gains
LQI-GA	$\begin{array}{c}{K}_f=\left[\begin{array}{cccccccccc}0& 0& 0& 0& 2.1669& 2.7294& 0& 0& 0& 0\\ 1.7038& 3.5594& 0& 0& 0& 0& 0& 0& 40.6451& 28.6725\\ 0& 0& 1.6960& 3.8174& 0& 0& -46.4869& -39.9021& 0& 0\end{array}\right],\\ K_i=\left[\begin{array}{ccc}0& 0& -0.6355\\ -0.3487& 0& 0\\ 0& -0.3276& 0\end{array}\right]\end{array}$
PID	$\begin{array}{c}{K}_P=diag\left\{\begin{array}{ccccc}0.01,& 60,& -0.01,& -60,& 5\end{array}\right\},\\ K_I=diag\left\{\begin{array}{ccccc}0.0001,& 0.05,& -0.0001,& -0.01,& 0.27\end{array}\right\},\\ K_D=diag\left\{\begin{array}{ccccc}0.075,& 150,& -0.075,& -150,& 14\end{array}\right\}\end{array}$

, where the PID controllers’ gains are also presented.

4.2. Results and Discussion

In the first scenario, the firefighting robot takes off, up to a height of 10 [m], and then moves horizontally to a length of 2 [m] in each of the X- and Y-directions, respectively. The geometry of the desired path is shown in Figure 7, along with the tracking performances provided by the two considered controllers: the proposed LQI-GA and the cascade PID. Subsequently, Figure 8 and Figure 9 depict the system response in every direction. The corresponding control inputs are given in Figure 10. For all of them, the left panels show the results when there is no disturbance and vice versa for the ones on the right.

In particular, Figure 7 shows that the designed robot can perform flying tasks as required. Moreover, the proposed LQI-GA manages the robot to closely follow the trajectory. The proposed controller is also robust against the disturbances, as the tracking performance is insignificantly affected in Figure 7b. The time responses in the positioning of the LQI-GA control system are also smooth, without any overshoot. The zoomed figures in Figure 8a prove that the non-minimum phase phenomena are smaller than those in the PID control system; even the PID controllers provide much slower performances. The external disturbances also heavily influence the PID-controlled system, as seen in the figures on the right. Figure 9 indicates that the PID controllers give the robot’s head less tilting and rolling. Nevertheless, the orientations are smoother with the LQI-GA controller, either with or without the presence of the disturbances. The yaw rotation of the robot is generally stable, without any control effort needed.

The smooth control inputs of the proposed controller explain the smooth response and the reduction of the non-minimum phase phenomena in the system. The smooth displacements r_wx and r_wy from the LQI-GA controller indicate the small acceleration of the weight blocks, thus resulting in smaller non-minimum phase phenomena. In contrast, the rough movements of the blocks in the PID control system lead to the fluctuation in orientations and the significant non-minimum phase phenomena.

In the second scenario, the firefighting robot follows a continuous trajectory while also maintaining system stabilization. The complex trajectory, in this case, is given by:

$$r=\left[\begin{array}{c}10e^{\frac{t}{100}}\sin \left(0.1u\right)\cos \left(0.01u\right)\\ 0.1t\sin \left(0.05t\right)\\ 0.1t\end{array}\right]$$

(27)

The simulation results are presented in Figure 11, Figure 12, Figure 13 and Figure 14 in the same manner as the previous scenario. The tracking paths in Figure 11 show that the proposed control system remains the better one. Its resulting motion is closed to the reference, even with the presence of disturbances. Meanwhile, the PID controllers give a lot of overshoots and are easily influenced by external forces and torques. Interestingly, the PID control system follows the trajectory in real-time, while the LQI-GA is lagging behind, as seen in their time responses in Figure 12. However, Figure 13 and Figure 14 clarify that the PID control system requires a lot of rotational corrections and control efforts to achieve this result. They are important factors in evaluating the feasibility of the system in a real application. The integral of square values (ISVs) presented in

Table 3. Control effort in the second scenario.

ISV	Without Disturbances		With Disturbances
	PID	LQI-GA	PID	LQI-GA
${\displaystyle \underset{0}{\overset{t}{\int }}{\dot{m}}_0{}^{2}dt}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}[{\mathrm{kg}}^2/{\mathrm{s}}^2]$	8.0152 × 105	7.8164 × 105	8.0024 × 105	7.7944 × 105
${\displaystyle \underset{0}{\overset{t}{\int }}{\dot{r}}_{wx}{}^{2}dt}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}[{\mathrm{m}}^2/{\mathrm{s}}^2]$	0.4738	0.0851	0.4760	0.00850
${\displaystyle \underset{0}{\overset{t}{\int }}{\dot{r}}_{wy}{}^{2}dt}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}[{\mathrm{m}}^2/{\mathrm{s}}^2]$	0.4738	0.0851	0.4859	0.0938

shows the consumed kinematic energies of the water and the weight-shifting actuators, in which the proposed LQI-GA is significantly more efficient than the other.

Mathematical analyses of the stability and performance of the proposed system have been given in the previous Section. In particular, the robustness of the closed-loop system has been proved via the Lyapunov function candidate (21), and the boundedness of the control error has been given in (23). The GA-based tuning controller gains minimize the cost function (25), which is equivalent to minimizing the control error, even in the presence of disturbances, and minimizing the variation of the weight-shifting actuation system. The proper performances in all simulation scenarios verify the theoretical analyses. The water flows smoothly, especially in comparison with the turbulent flow, due to the control of the nozzles’ flow in the study [24]. This is a great feature since it helps prevent the flutter instability of the water-conveying hose and enhances the total stability of the system.

Meanwhile, PID is a popularly standard control technique, and the cascade control structure has been adopted for aerial systems in many studies—for example, in the references [19,24,30] of the manuscript. The controlled quality depends on the PID controller parameters, which are not easily tuned in cascade control. The result is that the PID-controlled system seems to be slow in tracking the step-type reference but fast and overshot in tracking the sinusoid trajectory. Factors such as the saturation of the inputs and the objective of the reduction of the non-minimum phase phenomenon are difficult to incorporate into the design of the PID controllers. Thus, the PID control system requires a lot of rotational corrections to maintain the tracking performance, and noticeable chattering appears in the PID control inputs. That leads to significant control efforts and consumed water in the PID control system. On average, the LQI-GA consumes 2.28% less water and uses 83.85% less kinetic energy of the actuator than the PID control system.

5. Conclusions

This paper proposed a flying-type firefighting robot that is operated by waterpower and the weight shifting mechanism. The aim of this robot is to be efficiently used for accidents in water areas and other places where conventional firefighting methods are ineffective and the water sources are inexhaustibly available. The mathematical model of the flying firefighting robot was retrieved in detail.

The LQI law with the GA approach was proposed for the autonomous operation of the firefighting system. The LQI preserved the system’s stability. The GA optimization helps to find the gain matrices that minimize both the tracking error and the non-minimum-phase phenomenon. The simulation tests carried two scenarios, where the LQI-GA control system was compared to the cascade PID one from the previous study. The simulation results show the superiority of the proposed control system in terms of performance, robustness, and control effort.

At present, this conceptual design itself found the limits of a lack of generated yawing torque and an increase in the total mass due to two weights of the weight-shifting mechanism. For future works, the limits will be overcome by adding an actuator that is responsible for yaw motion control and using a unique weight for both lateral and horizontal motions. A prototype model is then going to be fabricated and experimented with for further validation. Other approaches for maneuvering the water-powered system are also being considered, such as regulating the injected flowrates or the direction of the nozzle outlets. Moreover, the effect of the water hose will be considered in further research to observe a system’s behavior fully. The system configuration can also be applied for aerial vehicles and other applications.