2.1. Entire design of the AQT-HR

The AQT-HR is designed by considering lightweight, fast mode switching and autonomous movement requirements, which takes the hierarchical structure to avoid conflict among all parts. The robot model comprises the flight pattern, the tilting mechanism, the battery storage room, and the ground movement chassis, as shown in Fig. 2. The flight mode is deployed by making the quadrotor parallel to the ground, the ground movement mode is deployed by tilting the quadrotor and activating the dual-servo mechanism which is mounted on the chassis. The design concept includes three aspects: (1) realizing the dual-mode mobility using one single driving source, (2) reserving a sufficient room for sensor components and making the robot’s scale smaller, and (3) most of the robot’s elements can be installed and disassembled conveniently, and the entire mechanical structure remains solid and impact-resistant.

Figure 2. The entire structure of the AQT-HR. (a) Flight pattern, (b) tilting mechanism, (c) battery storage room, (d) ground movement chassis.

2.2. The design of the flight mode

The flight mode is based on the structure of the quadrotor airframe and the tilting mechanism. Firstly, the airframe structure adopts a multi-hole cleat connection to allow for the mounting of the motors, eliminating the need for a shaft clamping device. In addition, the airframe reduces the weight by hollowing out in several places, meanwhile allowing more airflow underneath, reducing lift losses, and being more compact. Secondly, the quadrotor configuration has been modified to achieve the tilting function, as shown in Fig. 3a. Finally, depending on the tilting mechanism, the quadrotor can rotate in the pitch direction with the shaft as the rotation center. The key point of rotation design is to reasonably realize the axial and circumferential limits of the rotating parts. The axial limit ensures that the rotating parts will not slide around in the axis direction of the rotating shaft, while the circumferential limit provides coaxially between the rotating parts and the rotating shaft. In our design, the limit sleeve, the rotation maintainer, and the shaft retaining block are designed to prevent sliding.

Figure 3. The modifications in the quadrotor. (a) Shaft retaining block, (b) limit sleeve, (c) shaft, (d) rotation maintainer.

In order to ensure that the robot is impact-resistant during take-off and landing, the rotation maintainer needs to have high stiffness and strength. Therefore, a special design of the rotation maintainer is shown in Fig. 3b, which uses a double layer of beams and rounded corners to increase the structural stiffness in the horizontal direction, and two hollowed-out reinforcement bars in the vertical direction. The advantages of the design are that it strengthens the support frame against vertical impact and reduces the weight. In addition, the tilting servo motor is mounted at the bottom of the rotating maintainer. To ensure safety, the rotation maintainer’s finite element analysis simulation is conducted by using SolidWorks.

Figure 4. The analysis of the rotation maintainer.

The total weight of the robot is estimated to be 3 kg. Considering a safety factor, a static force of 40 N was applied to the structural element for simulation testing. The results are shown in Fig. 4a and b. Figure 4a shows the stress distribution in each part of the rotation maintainer after the force is applied, with a maximum value of 2.21 $\times$ 10 $^{6}$ N/m $^{2}$ , which is much smaller than the material’s elastic modulus of 2 $\times$ 10 $^{3}$ MPa. Figure 4b shows the deformation and displacement after the force is applied, with a maximum value of 0.1013 mm. As can be seen, there is not major deformation or structural fracture of the rotation maintainer occurred, which is in line with the design expectations.

The flight pattern consists of the quadrotor configuration and the tilting mechanism, which converts the direction of the driving force, as shown in Fig. 5. The output torque is transmitted to the quadrotor configuration via the servo rod, the linkage, the fixer beneath the quadrotor and the shaft to achieve rotational angle control. The maximum forward tilt angle of the quadrotor is measured to be 37.5 $^{\circ }$ , which indicates that 3/5 of the thrust force can be converted to ground locomotion driving force, improving the ability to move quickly on the ground.

Figure 5. The servo tilting mechanism. (a) Fixer beneath the quadrotor, (b) linkage, (c) servo rod, (d) servo, (e) servo support.

2.3. The design of the ground movement mode

The configuration of the mobile chassis is shown in Fig. 6. The ground flexible movement is achieved through a dual-servo steering mechanism. The mobile chassis’s optimized design is X-shaped, with the main processor mounted at the center of gravity, and the four wheels symmetrically distributed. At the same time, the chassis is made of carbon fiber with multiple rounded corners and hollowed-out areas to lighten the weight. The chassis’ shape design and material selection allow the robot to reduce damage caused by unstable falls and offset some landing impacts. The four wheels of the chassis contain two movable wheel components and two fixed wheel components, with the front dual-servo steering mechanism achieving the robot’s ground turning capability, as shown in Fig. 7a. The movable wheel components are mirror symmetrically mounted on the front, with their support parts extending in a specific shape, which connects to the servo rods through linkage to realize the torque transfer from the servo to the movable wheels. The rear wheels are fixed directly to the rear side of the chassis to assist the ground movement. The structure is shown in Fig. 7b.

Figure 6. The ground movement chassis. (a) Dual-servo steering mechanism, (b) chassis plate, (c) main processor, (d) fixed wheel components.

Figure 7. Wheel components of the chassis.

2.4. The hardware architecture of the AQT-HR

The robot’s hardware architecture contains motors, sensors, control boards, and other electronic components. Among these, the thrust force generated by motors and blades is essential in achieving mobility and needs to be considered first. The static thrust calculation [Reference Chen30] of the propeller blades is shown as follows:

(1) \begin{equation} T_{s} = 1.283\times 10^{-12} \times \text{RPM}^{2} \times D^{4} \times \rho \times K_{t} \times K_{g} \end{equation}

$T_s$ : static thrust generated by the propeller blades, N;

Figure 8. The electrical system of the AQT-HR.

Figure 9. The AQT-HR is an autonomous robot designed to fly and drive in various environments.

RPM: motor speed, rpm;

$D$ : the diameter of the propeller blades, inches;

$\rho$ : air density, taking the conventional value of 1.293 kg/m $^3$

$K_t$ : static thrust coefficient, taking the conventional value of 0.73.

$K_g$ : the acceleration of gravity, taking the conventional value of 9.82 m/s $^2$

T-motor 1555 blades and antigravity MN4006 motors are selected as the force generator. Each motor weighs 68 g and can provide a 21.6 N maximum thrust based on the manufacturer’s specification. According to Eq. (1), consider the expected total weight of the robot, which is 3.5 kg. We can calculate the maximum thrust of the hybrid robot will reach 86 N. This indicates that the robot can achieve great acceleration in the vertical direction and is capable of high agility. Furthermore, the ground movement mode needs to be constructed. Therefore, DS3218 20 kg $\cdot$ cm servo was selected as the power output component for the tilting structure, and MG90s 2 kg $\cdot$ cm motor was chosen as the steering servo.

After the actuated components have been determined, the hardware architecture would be built, as shown in Fig. 8. Pixhawk V4 was selected as the flight control board and Arduino Uno as the ground locomotion control board to enhance the robot control performance, respectively. The airborne computer of the AQT-HR is an Nvidia Jetson TX2. The robot outdoor localization system relies on the GPS module to obtain horizontal localization in an open environment and a barometer to get altitude information; the localization accuracy of the GPS device is about 0.2 m. The indoor localization system relies on the T265 binocular camera and deploys the Visual-Inertial Odometry(VIO) algorithm to fuse IMU information with image processing information to obtain the robot’s pose. The realization of VIO is based on an open-source project called “vision $\_$ to $\_$ mavros” and processed in the Nvidia Jetson TX2. This processor is with 256 CUDA cores and 4 GB LPDDR4 memory, which makes the image processing in real-time can be satisfied.

The AQT-HR takes a hierarchical structure so that the main frame comprises panels, using 2 mm carbon fiber to ensure light weight and increased structural stiffness. All the fasteners and transmission parts, including the rods, mounts, and wheels, are manufactured using 3D printing chosen from PLA and with a tire size of 45 mm. For the tilting structure, the ABS material was selected to ensure the structure’s stiffness and strength. The prototype of the AQT-HR is shown in Fig. 9.

3.1. Dynamics model of the flight mode

The actuator of the hybrid robot is the quadrotor configuration. Therefore, to improve the stability of locomotion, a dynamics model of the quadrotor will be established for the control method design [Reference Ahmed, Kumar and Patil31–Reference Nemati and Kumar33]. Firstly, we will establish two coordinate frames, the inertial frame $I_{x}I_{y}I_{z}$ , also known as the Earth Centered Earth Fixed coordinate, which is used to express the motion of the quadrotor relative to the ground, where the positive direction of the Z-axis is defined as the opposite of gravity and belongs to ENU coordinate frame. Next, the body-fixed frame $B_{x}B_{y}B_{z}$ will be established, which indicates the angular velocity and attitude information of the body itself, with its origin taken at the geometric center of the quadrotor. The definition of two frames is shown in Fig. 10.

Figure 10. The definition of the two coordinate frames. $ I_{x}I_{y}I_{z}$ is the inertial frame, $B_{x}B_{y}B_{z}$ is the body-fixed frame, $\omega _i$ is the rotation speed of the motor $i$ , and $\phi, \theta, \psi$ is roll, pitch, yaw angle of the quadrotor.

According to the above coordinate frames, we define the position and velocity of the quadrotor as $\boldsymbol{x} = (x, y, z)^{T}$ , $\boldsymbol{v} = (\dot{x}, \dot{y}, \dot{z})^{T}$ on the inertial frame. The attitude of the quadrotor can be defined on the body-fixed frame. Roll, pitch, and yaw angle of the robot can be represented as $\boldsymbol{\theta } = (\phi, \theta, \psi )$ and $\boldsymbol{\dot{\theta }} = (\dot{\phi }, \dot{\theta }, \dot{\psi })$ . Based on the theory of the coordinate transformation, the transformation matrix for angular velocities from the body-fixed frame ( $\boldsymbol{\dot{\theta }}$ ) to the inertial frame ( $\boldsymbol{\omega }$ ) is shown as:

(2) \begin{equation} \boldsymbol{\omega }=\left [\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & -\sin{\theta } \\[5pt] 0 & \cos{\phi } & \cos{\theta } \sin{\phi } \\[5pt] 0 & -\sin{\phi } & \cos{\theta } \cos{\phi } \end{array}\right ] \dot{\boldsymbol{\theta }} \end{equation}

The rotation matrix from the body-fixed frame to the inertial frame is

(3) \begin{equation} \boldsymbol{R}=\left [\begin{array}{c@{\quad}c@{\quad}c} C_{\psi } C_{\theta } & C_{\psi } S_{\theta } S_{\phi }-S_{\psi } C_{\phi } & C_{\psi } S_{\theta } C_{\phi }+S_{\psi } S_{\phi } \\[5pt] S_{\psi } C_{\theta } & S_{\psi } S_{\theta } S_{\phi }+C_{\psi } C_{\phi } & S_{\psi } S_{\theta } C_{\phi }-C_{\psi } S_{\phi } \\[5pt] -S_{\theta } & C_{\theta } S_{\phi } & C_{\theta } C_{\phi } \end{array}\right ] \end{equation}

where $S_{x} = \sin{x}$ , $C_{x} = \cos{x}$ . The rotation matrix $\boldsymbol{R}$ is orthogonal; thus, $\boldsymbol{R}^{-1} = \boldsymbol{R}^{T}$ . According to ref. [Reference Lee, Leok and McClamroch34], the torque vector of the quadrotor $\boldsymbol{\tau }_{B}$ can be derivated as:

(4) \begin{equation} \boldsymbol{\tau }_{B}=\left [\begin{array}{c} L k\left (\frac{\sqrt{2}}{2}\omega _{1}^{2}-\frac{\sqrt{2}}{2}\omega _{2}^{2}-\frac{\sqrt{2}}{2}\omega _{3}^{2}+\frac{\sqrt{2}}{2}\omega _{4}^{2}\right ) \\[5pt] L k\left (\frac{\sqrt{2}}{2}\omega _{1}^{2}+\frac{\sqrt{2}}{2}\omega _{2}^{2}-\frac{\sqrt{2}}{2}\omega _{3}^{2}-\frac{\sqrt{2}}{2}\omega _{4}^{2}\right ) \\[5pt] b\left (\omega _{1}^{2}-\omega _{2}^{2}+\omega _{3}^{2}-\omega _{4}^{2}\right ) \end{array}\right ] \end{equation}

where $L$ represents the distance between the motor and the quadrotor’s center, $k$ is the thrust coefficient, and $b$ is the torque coefficient.

Based on Newton’s Second Law, the mechanical analysis of the quadrotor can be represented as:

(5) \begin{equation} \begin{aligned} m \ddot{\boldsymbol{x}} &=\left [\begin{array}{c} 0 \\[5pt] 0 \\[5pt] m \boldsymbol{g} \end{array}\right ]+\boldsymbol{R} \boldsymbol{T}_{\boldsymbol{B}}+ \boldsymbol{F}_{D} \\[5pt] \boldsymbol{T}_{\boldsymbol{B}} &=\sum _{i=1}^{4} T_{i}=k\left [\begin{array}{c} 0 \\[5pt] 0 \\[5pt] \sum \omega _{i}^{2} \end{array}\right ] \end{aligned} \end{equation}

where $\ddot{\boldsymbol{x}}$ represents the accelerated velocity, $\boldsymbol{T}_{B}$ represents the thrust of the quadrotor on the body-fixed frame, and $\boldsymbol{F}_{D}$ represents the drag force which can be ignored at a low speed.

In addition, the dynamics model of the quadrotor contains the torque analysis according to Euler’s Theorem:

(6) \begin{equation} \boldsymbol{I} \dot{\boldsymbol{\omega }}+\boldsymbol{\omega } \times (\boldsymbol{I} \boldsymbol{\omega })=\boldsymbol{\tau }_B \end{equation}

For an intuitive representation, the torque analysis can be derivated as:

(7) \begin{equation} \dot{\boldsymbol{\omega }}=\left [\begin{array}{l} \tau _{\phi } I_{x x}^{-1} \\[5pt] \tau _{\theta } I_{y y}^{-1} \\[5pt] \tau _{\psi } I_{z z}^{-1} \end{array}\right ]+\left [\begin{array}{c} \frac{I_{y y}-I_{z z}}{I_{x x}} \omega _{y} \omega _{z} \\[5pt] \frac{I_{z z}-I_{x x}}{I_{y y}} \omega _{x} \omega _{z} \\[5pt] \frac{I_{x x}-I_{y y}}{I_{z z}} \omega _{x} \omega _{y} \end{array}\right ] \end{equation}

where $\boldsymbol{\omega }$ is the vector of angular velocity, and $I_{xx}, I_{yy}, I_{zz}$ are the rotational inertia of the body on the $x, y$ , and $z$ axes, respectively.

Combining Eqs. (2), (5), and (7), the analysis model of the quadrotor can be transformed to a form of state space:

(8) \begin{equation} \begin{aligned} &\dot{\boldsymbol{x}}_{1}=\boldsymbol{x}_{2} \\[5pt] &\dot{\boldsymbol{x}}_{2}=\left [\begin{array}{c} 0 \\[5pt] 0 \\[5pt] -g \end{array}\right ]+\frac{1}{m} \boldsymbol{R} \boldsymbol{T}_{\boldsymbol{B}} \\[5pt] &\dot{\boldsymbol{x}}_{3}=\left [\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & -\sin{\theta } \\[5pt] 0 & \cos{\phi } & \cos{\theta } \sin{\phi } \\[5pt] 0 & -\sin{\phi } & \cos{\theta } \cos{\phi } \end{array}\right ]^{-1} \boldsymbol{x}_{4} \\[5pt] &\dot{\boldsymbol{x}}_{4}=\left [\begin{array}{c} \tau _{\phi } I_{x x}^{-1} \\[5pt] \tau _{\theta } I_{y y}^{-1} \\[5pt] \tau _{\psi } I_{z z}^{-1} \end{array}\right ]+\left [\begin{array}{c} \frac{I_{y y}-I_{z z}}{I_{x x}} \omega _{y} \omega _{z} \\[5pt] \frac{I_{z z}-I_{x x}}{I_{y y}} \omega _{x} \omega _{z} \\[5pt] \frac{I_{x x}-I_{y y}}{I_{z z}} \omega _{x} \omega _{y} \end{array}\right ] \end{aligned} \end{equation}

where $\boldsymbol{x}_{1}$ represents the position of the quadrotor, $\boldsymbol{x}_{2}$ represents the velocity, $\boldsymbol{x}_{3}$ represents the attitude angle of the quadrotor, $\boldsymbol{x}_{4}$ represents the angular velocity, and the drag force $\boldsymbol{F}_{D}$ is ignored.

3.2. Dynamics model of the ground movement mode

The ground movement mode contains the turning and forward motion of the hybrid robot. The output variables of the designed controller are horizontal positions $x, y$ , while there are three input variables which are the tilting angle $\gamma$ , the quadrotor’s thrust $F_{T}$ , and the steering angle $\alpha$ . Two outputs correspond to three inputs, belonging to a redundantly driven control model. To reduce the difficulty of control, an assumption is considered as follows:

According to Theoretical Mechanics, a total thrust will be generated at the center of the quadrotor, as shown in Fig. 11. Therefore, based on Fig. 11 and considering the robot’s forward motion on the ground, the force analysis of the hybrid robot can be derivated as:

(9) \begin{equation} \begin{aligned} &F_{sx}=F_{T} \sin{\gamma } \\[5pt] &F_{sz} = F_{T} \cos{\gamma } \\[5pt] &F_{sx}-\mu (N_{1}+N_{2})-F_{D} = 0\\[5pt] &mg - (N_{1}+N_{2}+F_{sz})=0 \\[5pt] \end{aligned} \end{equation}

where $\mu$ is the coefficient of rolling friction, and $F_{D}$ is the air resistance.

Figure 11. Dynamics model of the forward motion. $F_{f}$ is the thrust generated by the two former motors, $F_{b}$ is the thrust generated by the two rear motors, $F_{T}$ is the total thrust, $M_{s}$ is the capsizing moment, $h$ represents the distance between the center of the quadrotor and the geometric center of the chassis, and $d$ represents the distance from the wheels to the geometric center of the chassis.

The titling angle $\gamma$ is selected as the control variable in the ground movement mode. It causes a gyroscopic moment $M_{\text{gyro}}$ which is generated by the relative motion of the quadrotor’s tilting and the motor’s rotation. While the tilting motion occurs only on the Y-axis of the quadrotor, which means the gyroscopic moment can be determined as:

(10) \begin{equation} M_{\text{gyro}} = J_{xy} \vec{\omega }_{y} \times (\vec{\omega }_{1} - \vec{\omega }_{2} +\vec{\omega }_{3} - \vec{\omega }_{4}) \end{equation}

where $J_{xy}$ is the rotational inertia, $\vec{\omega }_{y}$ is the vector of tilting angular velocity, $\vec{\omega }_{i}$ is the rotational vector of each motor, and the direction of $M_{\text{gyro}}$ is around the X-axis, satisfying the right-hand rule. According to the Assumption 1, the rotational speed of each motor remains the same so that the gyroscopic moment will be zero.

The capsizing moment $M_{s}$ caused by the thrust’s horizontal component is hazardous, while it can be offset by the torque from the ground support force, as shown below:

(11) \begin{equation} M_{s} +N_{2}d - N_{1}d = 0, M_{s} = F_{T} h \sin\gamma \end{equation}

where, when the rear support force $N_{2}$ turns to zero, the robot is in the critical condition of flipping over, which must be avoided. Therefore, combined with the balance condition Eq. (9), the thrust and the support force can be calculated as:

(12) \begin{equation} \begin{aligned} 2 N_{1}d +F_{T}(d \cos\gamma - h \sin\gamma ) - mgd &= 0 \\[5pt] F_{T} (h \sin\gamma +d \cos\gamma ) -mgd &= 0 \end{aligned} \end{equation}

to achieve the normal ground movement, and the following requirements should be satisfied:

(13) \begin{equation} \begin{aligned} F_{T} \sin\gamma -\mu (N_{1} + N_{2}) &\geq 0 \\[5pt] mg - F_{T} \cos\gamma -N_{1}-N_{2} &= 0 \end{aligned} \end{equation}

Considering the friction of the ground, the critical condition of the robot’s ground movement can be represented as:

(14) \begin{equation} \begin{aligned} \frac{umg}{\mu \cos\gamma +\sin\gamma } \leq F_{T} & \leq \frac{mgd}{h \sin\gamma +d \cos \gamma } \\[5pt] 0 \leq N_{1} & \leq \frac{mgd - F_{T}(d\cos\gamma - h\sin\gamma )}{2d} \end{aligned} \end{equation}

After that, the dynamics model of the robot’s forward motion can be derivated:

(15) \begin{equation} \begin{aligned} m \ddot{x} &= F_{sx} - \mu (N_{1}+N_{2}) -F_{D} \\[5pt] &= F_{T} \sin{\gamma } - \mu (mg - F_{T} \cos{\gamma }) -F_{D} \end{aligned} \end{equation}

and the state space representation can be represented as:

(16) \begin{equation} \begin{aligned} &\dot{x}_{1} = x_{2} \\[5pt] &\dot{x}_{2} = \frac{F_{T}}{m}(\sin{\gamma } + \mu \cos{\gamma }) - \mu g - \frac{F_{D}}{m} \end{aligned} \end{equation}

where $x_{1}$ represents the position of the robot, and $x_{2}$ represents the velocity of the robot.

Figure 12. The steering model of the hybrid robot.

The driving force of the ground steering structure comes from the thrust’s horizontal component, the point of force application is in the center of the chassis, and the steering model is a front steering mid-drive configuration, as shown in Fig. 12. Based on a symmetrical chassis configuration and the information in Fig. 12, the steering radius can be calculated as follows:

(17) \begin{equation} R_{c} = \frac{L_c}{2 \sin{\frac{\alpha }{2}}} \end{equation}

Following the development of the flight dynamics model and the ground mode dynamics model, the control method for the dual-modal motion of the robot will be proposed in the next part.

3.3. The design of the control method

The control method aims to improve the control accuracy and stability of the robot’s aerial locomotion and reduce the risk of falling. This paper proposes a model-feedforward PID control method for implementing the robot’s aerial locomotion, and PD control is adopted to achieve the robot’s ground locomotion.

The model-feedforward PID approach is intended to accommodate the motion characteristics of the hybrid robot. From structural consideration, the AQT-HR system can be considered as a quadrotor with a hanging under-chassis. Compared to the control problem of a normal quadrotor, the downward of the robot’s center of gravity and the increase in gravitational torque make control more difficult and its parameters harder to adjust. Therefore, the traditional PID algorithm is improved by introducing the dynamics model into it, reducing parameter adjustment complexity while increasing flight control’s efficiency. To facilitate the design of the controller, the robot’s flight dynamics model is linearized, assuming that pitch and yaw alter tiny and the total thrust is approximate to the robot’s weight during the flight, with the following mathematical form:

(18) \begin{equation} \sin{\phi } \approx \phi, \cos{\phi } \approx 1, \sin{\theta } \approx \theta, \cos{\theta } \approx 1, T_{B} \approx mg \end{equation}

According to the mathematical form and the dynamics model (8), a horizontal position control method for the robot can be obtained as follows:

(19) \begin{equation} \left [\begin{array}{c} \ddot{p}_{x}\\[5pt] \ddot{p}_{y} \end{array}\right ] = g \left [\begin{array}{c@{\quad}c} \cos{\psi } & -\sin{\psi } \\[5pt] \sin{\psi } & \cos{\psi } \end{array}\right ] \left [\begin{array}{c@{\quad}c} 0 & 1\\[5pt] -1 & 0 \end{array}\right ] \left [\begin{array}{c} \phi \\[5pt] \theta \end{array}\right ] \end{equation}

where $\ddot{p}_{x}$ is the second derivative of the position on the robot’s x-direction, $\ddot{p}_{y}$ is the second derivate of the position on the robot’s y-direction, $\phi$ determines the roll angle, and $\theta$ determines the pitch angle. The inputs of the controller are the angle of attitude, and there will be a transformation:

(20) \begin{equation} \left [\begin{array}{c} \phi \\[5pt] \theta \end{array}\right ] = \frac{1}{g} \left [\begin{array}{c@{\quad}c} \sin{\psi } & -\cos{\psi } \\[5pt] \cos{\psi } & \sin{\psi } \end{array}\right ] \left [\begin{array}{c} \ddot{p}_{x} \\[5pt] \ddot{p}_{y} \end{array}\right ] \end{equation}

The desired transition process for the robot’s horizontal position control is:

(21) \begin{equation} \begin{aligned} &e_{p_{x}} = p_{xd}-p_{x} \\[5pt] &e_{p_{y}} = p_{yd} - p_{y} \\[5pt] &\ddot{e}_{p_{x}} =\ddot{e}_{p_{xd}}-\ddot{e}_{p_{x}} = k_{p} e_{p_{x}}+k_{i} \sum _{i=0}e_{p_{x}} dt +k_d \frac{d{e}_{p_{x}}}{dt} \\[5pt] &\ddot{e}_{p_{y}} =\ddot{e}_{p_{yd}}-\ddot{e}_{p_{y}} = k_{p} e_{p_{y}}+k_{i}\sum _{i=0} e_{p_{y}} dt +k_d \frac{d{e}_{p_{y}}}{dt} \end{aligned} \end{equation}

where $e_{p_{x}}$ determines the error between the desired position $p_{xd}$ and the actual position $p_{x}$ , and the same applies to $e_{p_{y}}$ . For position control, the desired acceleration of the robot is 0, so combining Eqs. (20) and (21), the mathematical form of the horizontal position controller for the hybrid robot can be derivated as:

(22) \begin{equation} \begin{aligned} &\phi = \frac{1}{g}(\sin{\psi }(k_{p} e_{p_{x}}+k_{i} \sum _{i=0}e_{p_{x}} dt +k_d \frac{d{e}_{p_{x}}}{dt}) -\cos{\psi }(k_{p} e_{p_{y}}+k_{i} \sum _{i=0}e_{p_{y}} dt +k_d \frac{d{e}_{p_{y}}}{dt}))\\[5pt] & \theta = \frac{1}{g}(\cos{\psi }(k_{p} e_{p_{x}}+k_{i} \sum _{i=0}e_{p_{x}} dt +k_d \frac{d{e}_{p_{x}}}{dt}) +\sin{\psi }(k_{p} e_{p_{y}}+k_{i} \sum _{i=0}e_{p_{y}} dt +k_d \frac{d{e}_{p_{y}}}{dt})) \end{aligned} \end{equation}

where $k_{p}, k_{i}, k_{d}$ are the parameters of the PID controller and should be adjusted to achieve a proper output. The actual control process requires the conversion of the Euler angles into quadratic values, which are solved by the processor to achieve the horizontal position control.

Next, the robot’s height control will be designed. As the same as the design principle of the horizontal position controller, the simplified linear model of the height controller is:

(23) \begin{equation} \ddot{p}_{z} = g - \frac{f}{m} \end{equation}

The thrust force $f$ is the control input corresponding to the robot’s acceleration. In addition, a P-controller for the outer loop velocity is added to avoid the motion oscillation, which results in a height controller of the robot:

(24) \begin{equation} \begin{aligned} &v_{zd} = k_{pv} (p_{zd} - p_{z}) \\[5pt] &e_{zv} = v_{zd} - v_{z} \\[5pt] &f = mg + m(k_{pf}e_{vz} + k_{if}\sum _{i=0} e_{vz} dt +k_{df} \frac{de_{vz}}{dt}) \end{aligned} \end{equation}

where $k_{pv}$ is the coefficient of the P-controller, and $v_{zd}$ is the desired velocity on the z-axis, which is generated by the P-controller. After that, the design of the model-feedforward PID controller is completed, whose parameters are shown in Table I. The motion controller for the hybrid robot’s flight mode is shown in Fig. 13.

Table I. The parameters of the model-feedforward PID control method.

Figure 13. The motion controller for the hybrid robot’s flight mode, $\boldsymbol{p}_{d}=(p_{xd}, p_{yd}, p_{zd})$ , $\boldsymbol{p}=(p_{x}, p_{y}, p_{z})$ , $\boldsymbol{v}=(v_{x}, v_{y}, v_{z})$ and $\boldsymbol{q} = (\phi, \theta, \psi )$ .