Design and Control Strategies of Multirotors with Horizontal Thrust-Vectored Propellers

3.1. Control Allocation Module

3.2. Thrust-Vectoring Position Control in SE(3)

• Tilted Flight Mode: This mode refers to the normal flight mode where the vehicle controls its attitude, i.e., roll, pitch, and yaw, to fulfill the desired position, velocity, and acceleration setpoints. The position control of the UAV platform on the SE(3) controller is based on the geometric controller described in [24]. The tracking errors for the position and velocity are expressed as follows:

  $$\begin{array}{c}\hfill e_p={\mathbf{R}}_{yaw}^T({}^{W}p-{}^{W}p_{sp})\\ \hfill e_v={\mathbf{R}}_{yaw}^T({}^{W}p-{}^{W}p_{sp})\end{array}$$

  (5)

  where ${\mathbf{R}}_{yaw}$ represents a pure rotation about the z-axis by the yaw angle setpoint ${\psi }_{sp}$, as shown below:

  $$R_{yaw}=\left[\begin{array}{ccc}\cos \left({\psi }_{sp}\right)& -\sin \left({\psi }_{sp}\right)& 0\\ \sin \left({\psi }_{sp}\right)& \cos \left({\psi }_{sp}\right)& 0\\ 0& 0& 1\end{array}\right]$$

  (6)

  The integral terms can then be described as follows:

  $$e_i={\int }_0^t(e_v\left(t\right)+c_1{e}_p\left(\tau \right))d\tau$$

  (7)

  where $c_1>0$ is a proportional gain that determines the contribution of the position error to the integral error. The integral error $e_i$ is then saturated by $\sigma >0$, which depends on the UAV’s vertical and horizontal thrust capabilities. A PID controller is applied to obtain the desired acceleration, ${\dot{v}}^{*}$. The gains of the controller depend on the flight mode, since the propellers and the horizontal thrusters have different dynamic effects. The gains $K_p$, $K_i$, and $K_d$ are expressed as diagonal matrices and follow the general form

  $$K=\left[\begin{array}{ccc}{k}_x& 0& 0\\ 0& k_y& 0\\ 0& 0& k_z\end{array}\right]$$

  (8)

  where the respective elements $k_x$, $k_y$, and $k_z$, represent the proportional, integral, and derivative gains for the x-, y-, and z-axes.

  $${\dot{v}}^{*}=-K_p{e}_p-K_d{e}_v-K_i{e}_i+{\dot{v}}_{sp}$$

  (9)

  $$\mathbf{F}=m({\dot{v}}^{*}+{}^{B}g)$$

  (10)

  The force vector is then represented in the inertial frame by the transformation given by

  $${}^W\mathbf{F}={\mathbf{R}}_{yaw}\left(\mathbf{F}\right)$$

  (11)

  The thrust to attitude block uses the derived force vector to calculate the attitude setpoint. The desired orientation is expressed by the unit vectors, ${\widehat{x}}_B$, ${\widehat{y}}_B$, and ${\widehat{z}}_B$, which are calculated as follows:

  $${\widehat{z}}_B=-{\displaystyle \frac{{}^W\mathbf{F}}{\parallel {}^W\mathbf{F}\parallel }}$$

  (12)

  $${\widehat{x}}_{proj}=\left[\begin{array}{ccc}\cos {\psi }_{sp}& \sin {\psi }_{sp}& 0\end{array}\right]$$

  (13)

  where $x_{proj}$ refers to a projection of the x-axis in the $xy$-plane.

  $${\widehat{y}}_B={\widehat{z}}_B\times {\widehat{x}}_{proj}$$

  (14)

  $${\widehat{x}}_B={\widehat{y}}_B\times {\widehat{z}}_B$$

  (15)

  The obtained attitude is then constrained by the maximum allowable tilt of the vehicle. The attitude setpoint can subsequently be expressed in quaternion form as $q_{sp}$. Following this, the force setpoint ${\mathbf{F}}_{sp}$ is expressed using the force vector calculated in Equation (10), as follows:

  $${\mathbf{f}}_{thrust}={}^W\mathbf{F}·\left(\mathbf{R}{e}_3\right)$$

  (16)

  $${\mathbf{F}}_{\mathrm{sp}}={\left[\begin{array}{ccc}0& 0& f_{thrust}\end{array}\right]}^T$$

  (17)

  where $\mathbf{R}$ is the rotation matrix defining the orientation of the body frame relative to the inertial frame, and ${\mathbf{e}}_3={[0,0,1]}^T$ is the unit vector along the z-axis of the body frame.
• XY Thruster Flight Mode: This flight mode is specifically designed for configurations with actuators capable of directly generating forces along the x- and y-axes of the vehicle. The roll and pitch angles are kept close to zero degrees, while the horizontal thrusters are used for planar motion. The control of the UAV platform with horizontal thrusters follows a similar formulation to that of the tilted flight mode. However, horizontal and vertical propellers have different effects on the UAV’s movement. Therefore, the matrix K includes gains associated with the respective thrusters, as shown below.

  $$K=\left[\begin{array}{ccc}{k}_{thx}& 0& 0\\ 0& k_{thy}& 0\\ 0& 0& k_z\end{array}\right]$$

  (18)

  where the respective elements $k_{thx}$ and $k_{thy}$ represent the proportional, integral, and derivative gains for thruster control, while $k_z$ corresponds to the gain used in the tilted flight mode. The individual force components $(F_x,F_y,F_z)$ are obtained from Equation (10), and the desired thrust setpoint ${\mathbf{F}}_{sp}$, is constructed as follows:

  $${\mathbf{F}}_{\mathrm{sp}}={\left[\begin{array}{ccc}{F}_x& F_y& F_z\end{array}\right]}^T$$

  (19)
  The attitude setpoint $q_{sp}$ is derived from the calculated unit vectors representing the desired orientation. For planar flight, the unit vector $z_B$ is aligned downward, parallel to the z-axis of the reference frame ${\mathcal{F}}_I$, shown as follows:

  $${\widehat{z}}_B={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T$$

  (20)
  The remaining unit vectors ${\widehat{x}}_B$ and ${\widehat{y}}_B$, will be rotated in the xy-plane by the yaw angle setpoint ${\psi }_{sp}$, as shown below:

  $$\begin{array}{c}\hfill {\widehat{x}}_B={\left[\begin{array}{ccc}\cos {\psi }_{sp}& \sin {\psi }_{sp}& 0\end{array}\right]}^T\\ \hfill {\widehat{y}}_B={\left[\begin{array}{ccc}-\sin {\psi }_{sp}& \cos {\psi }_{sp}& 0\end{array}\right]}^T\end{array}$$

  (21)
• X Thruster Flight Mode: This flight mode integrates control strategies from both tilted flight mode and xy thruster flight mode, enabling planar flight along the positive or negative x-axis, while controlling its roll and pitch for the remaining motions. Only the gains in the x-axis correspond to the thruster, while y and z use the gains of the tilt flight mode. Therefore, matrix K is then expressed as follows:

  $$K=\left[\begin{array}{ccc}{k}_{thx}& 0& 0\\ 0& k_y& 0\\ 0& 0& k_z\end{array}\right]$$

  (22)
  As before, the individual force components $(F_x,F_y,F_z)$ are obtained from Equation (10). Since the vehicle uses both attitude control and horizontal thrusters to achieve motion, the desired thrust vector ${\mathbf{F}}_{sp}$ must be separated into two components. The first component, ${\mathbf{F}}_x$, corresponds to the force distributed to the horizontal thrusters and is aligned with the x-direction. The second component, comprising the remaining force components $(F_y,F_z)$, are grouped into the vector ${\mathbf{F}}_{yz}$. This vector is used for altitude and attitude control and is defined as follows:

  $$\begin{array}{c}\hfill {\mathbf{F}}_{\mathbf{yz}}={\left[\begin{array}{ccc}0& F_y& F_z\end{array}\right]}^T\end{array}$$

  (23)
  With these force components, the desired thrust setpoint ${\mathbf{F}}_{sp}$ is constructed as follows:

  $${\mathbf{f}}_{thrust}={\mathbf{F}}_{\mathbf{yz}}·\left(\mathbf{R}{e}_3\right)$$

  (24)

  $${\mathbf{F}}_{\mathrm{sp}}={\left[\begin{array}{ccc}{F}_x& 0& {\mathbf{f}}_{thrust}\end{array}\right]}^T$$

  (25)
  The attitude setpoint $q_{sp}$ is then derived from the force vector ${\mathbf{F}}_{yz}$, by calculating the unit vectors of the desired orientation. The body-frame z-axis unit vector, ${\widehat{z}}_B$, is computed as follows:

  $$\begin{array}{c}\hfill {}^W\mathbf{F}_{\mathbf{y}\mathbf{z}}={\mathbf{R}}_{yaw}\left({\mathbf{F}}_{\mathbf{yz}}\right)\end{array}$$

  (26)

  $${\widehat{z}}_B=-{\displaystyle \frac{{}^W\mathbf{F}_{\mathbf{y}\mathbf{z}}}{\parallel {}^W\mathbf{F}_{\mathbf{y}\mathbf{z}}\parallel }}$$

  (27)
  For planar motions in the x-axis of the body frame, the pitch angle is set to 0 deg. Consequently, ${\widehat{x}}_B$ and ${\widehat{y}}_B$ are defined as follows:

  $${\widehat{x}}_B=\left[\begin{array}{ccc}\cos {\psi }_{sp}& \sin {\psi }_{sp}& 0\end{array}\right]$$

  (28)

  $${\widehat{y}}_B={\widehat{z}}_B\times {\widehat{x}}_B$$

  (29)
• Y Thruster Flight Mode: The thrust and attitude calculations follow the same logic as the x thruster flight mode, with the key difference being that the horizontal forces are constrained to the positive or negative y-axis. The gain matrix for this mode is given by

  $$K=\left[\begin{array}{ccc}{k}_x& 0& 0\\ 0& k_{thy}& 0\\ 0& 0& k_z\end{array}\right]$$

  (30)
  The desired thrust vector, ${\mathbf{F}}_{sp}$, is decomposed into two components: the horizontal thrust, ${\mathbf{F}}_y$, and ${\mathbf{F}}_{xz}$, which is used for altitude and attitude control.

  $$\begin{array}{c}\hfill {\mathbf{F}}_{\mathbf{xz}}={\left[\begin{array}{ccc}{F}_x& 0& F_z\end{array}\right]}^T\end{array}$$

  (31)

  $${\mathbf{f}}_{thrust}={\mathbf{F}}_{\mathbf{xz}}·\left(\mathbf{R}{e}_3\right)$$

  (32)

  $${\mathbf{F}}_{\mathrm{sp}}={\left[\begin{array}{ccc}0& F_y& {\mathbf{f}}_{thrust}\end{array}\right]}^T$$

  (33)
  The attitude setpoint $q_{sp}$ is then derived from the force vector ${\mathbf{F}}_{xz}$. The unit vectors can then be calculated as follows:

  $$\begin{array}{c}\hfill {}^W\mathbf{F}_{\mathbf{x}\mathbf{z}}={\mathbf{R}}_{\mathbf{yaw}}\left({\mathbf{F}}_{\mathbf{xz}}\right)\end{array}$$

  (34)
  For planar motions along the y-axis of the body frame, the roll angle is set to 0 deg. The unit vectors ${\widehat{x}}_B$ and ${\widehat{y}}_B$ are defined as follows:

  $${\widehat{z}}_B=-{\displaystyle \frac{{}^W\mathbf{F}_{\mathbf{x}\mathbf{z}}}{\parallel {}^W\mathbf{F}_{\mathbf{x}\mathbf{z}}\parallel }}$$

  (35)

  $${\widehat{y}}_B={\left[\begin{array}{ccc}-\sin {\psi }_{sp}& \cos {\psi }_{sp}& 0\end{array}\right]}^T$$

  (36)

  $${\widehat{x}}_B={\widehat{y}}_B\times {\widehat{z}}_B$$

  (37)

3.3. Attitude and Rate Control

4.1. Moment Evaluation of Previous Configurations

4.2. Thrust-Vectored Tilted Propeller Flight

4.3. Vertical Offset Experiments