Dynamic Center-of-Mass Displacement in Aerial Manipulation: An Innovative Platform Design

We assume that the aerial vehicle is symmetric around the main axis of the rigid link along the interaction direction as in Fig. 2(a). By neglecting small friction forces, the problem simplifies to a planar scenario, as illustrated in the free body diagram in Fig. 2(b). We denote $T_{1}$ and $T_{2}$ as the thrust magnitude of the front and back rotors, respectively, and $G_{0}$ as the gravitational force acting on the total system. The component of $T_{2}$ along the interaction direction is $f_{h}$, while $f_{g}$ represents the component of $T_{2}$ opposing the gravitational force. The contact force exerted by the environment, passing through the system’s \accom, is denoted by $f_{c}$. We define $l_{1}$ and $l_{2}$ as the distances between the propellers’ centers and the \accom. Assuming the system is in equilibrium during interaction, with zero net forces and torques, the following relationships hold:

	$\displaystyle f_{g}$	$\displaystyle=\frac{G_{0}l_{1}}{l_{1}+l_{2}}=G_{0}r_{l},$		(1)
	$\displaystyle T_{1}$	$\displaystyle=G_{0}(1-r_{l}),$		(2)
	$\displaystyle f_{c}$	$\displaystyle=f_{h}=\sqrt{T_{2}^{2}-f_{g}^{2}}.$		(3)

where $r_{l}=\displaystyle\tfrac{l_{1}}{l_{1}+l_{2}}\in[0,1]$. The distance between front and back rotors $l_{1}+l_{2}$ is fixed. To investigate the effect of \accom location on the system’s force exertion, we analyze the case with $G_{0}=30$\mathrm{N}$$, $T_{2}=20$\mathrm{N}$$, and $l_{1}+l_{2}=0.27$\mathrm{m}$$. By varying $r_{l}$ within its feasible range, the forces change according to eqs. 1, 2 and 3, as depicted in Fig. 3.

In traditional aerial system designs with a fixed \accom, the configuration typically corresponds to $r_{l}=0.5$. In this setup, the thrust from the back rotor generates a component $f_{g}$ that partially compensates for the gravitational effect, in conjunction with the thrust from the front rotor $T_{1}$. Consequently, the force applied to the environment $f_{h}=f_{c}$ is limited to $\frac{13}{20}=0.65$ of the back rotor thrust $T_{2}$. When the \accom is shifted closer to the back rotor ($r_{l}>0.5$), the force exerted on the environment decreases. As illustrated in Fig. 3, $f_{c}$ approaches zero as $r_{l}$ increases beyond $0.5$, reaching near zero around $r_{l}=0.7$. This demonstrates that moving the \accom closer to the back rotor reduces the system’s overall force exertion capability.

Conversely, moving the \accom closer to the front rotors ($r_{l}<0.5$) enhances force generation on the work surface. When $r_{l}=0$, the system exerts the maximum force on the environment, with $f_{c}=T_{2}$. In this configuration, the back rotor thrust is fully utilized to generate the contact force on the work surface, while the gravitational component $f_{g}$ is zero. The system’s \accom is aligned with the front rotors at $l_{1}=0$ and the front rotor thrust fully compensates for the gravitational force, resulting in $T_{1}=G_{0}$.

The wrench analysis indicates a novel approach to enhance force generation in aerial systems: shifting the system’s \accom closer to the front rotors. Traditional underactuated aerial vehicles with 4-\acdof are known for their energy efficiency, particularly when carrying heavy loads or manipulators. Therefore, we propose a novel aerial system design that functions as a 4-\acdof vehicle during navigation (in navigation mode), with a fixed \accom aligned with the \accog.

For interaction tasks (in interaction mode), the platform features tiltable back rotors and a dynamic \accom displacing configuration. This configuration allows the platform to adjust its \accom position closer to the front rotors during interactions, thereby enabling higher force exertion.

In the following sections, we detail the design, modeling, and control of this aerial manipulation system, specifically addressing the dynamic \accom design.

We denote $m_{S},\ m\in\mathbb{R}^{+}$ as the mass of the shifting-mass plate and the total system respectively, where $m_{S}<m$. The shifting-mass plate is connected to a linear actuator and can move along $\bm{x}_{B}$ starting from the system’s \accog up to the alignment frame tip. The location of the shifting-mass plate is represented by its \accog position. We denote $l_{S}$ as the length of the shifting-mass plate along the body axis $\bm{x}_{B}$, as in Fig. 5(b). The shifting-mass plate position along $\bm{x}_{B}$ \acwrt the body frame origin $\bm{O}_{B}$ is denoted as $l(t)\in[0,L+L_{0}-0.5l_{S}]$. Changing the shifting-mass plate position results in a displacement of the system’s \accom along $\bm{x}_{B}$, denoted by $d(t)$. When the system’s \accom is located at $d>L$ outside the rotor-defined area (i.e., over-displaced), the aerial vehicle often flips around the contact area, leading to instability. Therefore, $d(t)$ is restricted by $[0,L]$ to avoid risky scenarios and damage to the platform. Assuming the symmetric mass distribution of the platform around its body axes, the relation between $l(t)$ and $d(t)$ is given by:

$$d(t)=\frac{m_{S}}{m}l(t).$$

(4)

With eq. 4,the maximum \accom displacement $d=L$ corresponding to the case $r_{l}=0$ in Fig. 3 results in a shifting-mass plate position of $l=\displaystyle\tfrac{m}{m_{S}}L>L$. The shifting-mass plate position is thus restricted to $[0,\displaystyle\tfrac{m}{m_{S}}L]$. Neglecting friction forces from the work surface, with $d=L$, only front rotors supply the force for gravity compensation. The back rotors can tilt up to $\alpha=90\degree$ to be fully used for generating the pushing force.

A physical prototype with a total weight of $3.12$\mathrm{kg}$$ has been developed from scratch, utilizing 3D-printed parts and carbon-fiber tubes. This prototype facilitates simple pushing tasks using the alignment frame, as shown in Fig. 5(c). The linear motion of the shifting-mass plate is enabled by a lead screw, a Dynamixel servo, and a screw nut housed in a box rigidly attached to the shifting-mass plate, for details please refer to [30]. This design ensures precise, slow motion of the plate, with a velocity of $0.001$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$, making the dynamics of shifting the \accom negligible. To enhance the interaction quality, foam is attached to the tip of the alignment frame, providing mechanical compliance to reduce impact during initial contact and ensuring smooth interaction.

To simplify the modeling, we assume that the body axes of frame $\mathcal{F}_{B}$ correspond with the system’s principal axes of inertia. Unlike the conventional aerial vehicles with fixed centers of mass and constant moments of inertia, while displacing the system’s \accom along the body axis $\bm{x}_{B}$, the moments of inertia along body axes $\bm{y}_{B}$ and $\bm{z}_{B}$ vary.

To reduce modeling inaccuracies, we present the moment of inertia estimation associated with the shifting-mass plate position $l(t)$. To do so, the geometry, mass, and moments of inertia of the main components used in the physical model in Fig. 1 are captured by realistic CAD models. By measuring, $I_{xx}=0.0444$$\mathrm{kg}\text{\,}{\mathrm{m}}^{2}$ stays constant for different value of $l$, while $I_{yy}$ and $I_{zz}$ change values along with $l$. With the measured data and similarly to [33], linear regression technology is applied to obtain mathematical models of $I_{yy}$ and $I_{zz}$ \acwrt $l$:

$$\begin{split}&I_{yy}(l)=0.49l^{2}+0.0538\ ($\mathrm{kg}\text{\,}{\mathrm{m}}^{2}$),\\ &I_{zz}(l)=0.52l^{2}+0.0795\ ($\mathrm{kg}\text{\,}{\mathrm{m}}^{2}$).\end{split}$$

(5)

The estimated inertia matrix of the designed system is thus given by $\bm{I}(l)=diag(\begin{bmatrix}I_{xx}&I_{yy}(l)&I_{zz}(l)\end{bmatrix})\in\mathbb{R}^{3\times 3}$.

We now introduce the actuation wrenches (i.e., forces and torques) of the system in Fig. 4. Given the tiltable back rotors, the aerial vehicle can have thrust components along the interaction axis $\bm{x}_{B}$. The actuation force vector $\bm{F}_{a}\in\mathbb{R}^{3}$ expressed in the body frame $\mathcal{F}_{B}$ is given by¹¹1For simplicity, $C_{(\cdot)}$ and $S_{(\cdot)}$ denote $\cos(\cdot)$ and $\sin(\cdot)$, while $(\cdot)_{1,2,3,...,n}$ represents the sum of thrust and drag torques, e.g., $T_{3,4,5,8}=T_{3}+T_{4}+T_{5}+T_{8}$:

$$\bm{F}_{a}=\begin{bmatrix}F_{1}\\ 0\\ F_{3}\end{bmatrix}=\begin{bmatrix}T_{3,4,5,8}S_{\alpha}\\ 0\\ T_{3,4,5,8}C_{\alpha}+T_{1,2,6,7}\end{bmatrix}.$$

(6)

The actuation torque vector $\bm{\Gamma}_{a}\in\mathbb{R}^{3}$ \acwrt the system’s \accom expressed in $\mathcal{F}_{B}$ is given by:

$$\leavevmode\resizebox{390.25534pt}{}{ $\bm{\Gamma}_{a}=\begin{bmatrix}\Gamma_{1}\\ \Gamma_{2}\\ \Gamma_{3}\end{bmatrix}=\begin{bmatrix}\big{(}T_{2,7}-T_{1,6}+(T_{3,8}-T_{4,5})C_{\alpha}\big{)}W+\tau_{3,4,5,8}S_{\alpha}\\ T_{3,4,5,8}C_{\alpha}L-T_{1,2,6,7}L\\ (T_{4,5}-T_{3,8})S_{\alpha}W+\tau_{3,4,5,8}C_{\alpha}+\tau_{1,2,6,7}\end{bmatrix}$}.$$

(7)

The system therefore has 5-\acdof actuation provided by 9 system inputs including the rotors’ rotating speed and back rotors’ tilting angle. We denote $\bm{u}=\begin{bmatrix}\alpha&\Omega_{1}&...&\Omega_{8}\end{bmatrix}^{\top}\in\mathbb{R}^{9}$ as the system inputs.

Assuming slow motion of the shifting-mass plate, we neglect the dynamics of the moving plate. The system equations of motion expressed in the body frame $\mathcal{F}_{B}$ can be written as:

$$\begin{bmatrix}\bm{M}&\bm{0}\\ \bm{0}&\bm{I}\end{bmatrix}\begin{bmatrix}\dot{\bm{\upsilon}}\\ \dot{\bm{\omega}}\end{bmatrix}+\begin{bmatrix}\mathbb{S}(\bm{\omega})\bm{M}\bm{\upsilon}\\ \mathbb{S}(\bm{\omega})\bm{I}\bm{\omega}\end{bmatrix}+\bm{G}=\begin{bmatrix}\bm{F}_{a}\\ \bm{\Gamma}_{a}\end{bmatrix}+\begin{bmatrix}\bm{F}_{C}\\ \bm{\Gamma}_{C}\end{bmatrix}.$$

(8)

where $\bm{M}\in\mathbb{R}^{3\times 3}$ is the mass matrix. $\bm{G}\in\mathbb{R}^{6}$ is the gravity term expressed in the body frame. Considering the \accom displacing, we have:

$$\leavevmode\resizebox{346.89731pt}{}{$\bm{G}(l)=\begin{bmatrix}\bm{R}^{\top}g\bm{e}_{3}\\ \mathbb{S}(\bm{R}^{\top}g\bm{e}_{3})\begin{bmatrix}-d(t)&0&0\end{bmatrix}^{\top}\end{bmatrix}=\begin{bmatrix}\bm{G}_{lin}\\ \bm{G}_{ang}\end{bmatrix}$},$$

(9)

with $g=-9.81$$\mathrm{m}\text{\,}{\mathrm{s}}^{-2}$, $\bm{e}_{3}=\begin{bmatrix}0&0&1\end{bmatrix}^{\top}$, and $d=\frac{m_{S}}{m}l$ from eq. 4. $\bm{\upsilon},\bm{\omega}\in\mathbb{R}^{3}$ are the linear and angular velocity vectors of the body frame $\mathcal{F}_{B}$ \acwrt the world frame. Assuming that the interaction wrenches at the end-effector and alignment frame tip are the only sources of external wrenches, we denote $\begin{bmatrix}\bm{F}_{C}\\ \bm{\Gamma_{C}}\end{bmatrix}\in\mathbb{R}^{6}$ as the stacked force and torque vector exerted from the environment to the system. We use $\mathbb{S}(\bm{a})\in\mathbb{R}^{3\times 3}$ to present the skew symmetric matrix such that $\mathbb{S}(\bm{a})\bm{b}=\bm{a}\times\bm{b}$, and we have $\dot{\bm{R}}=\bm{\omega}\times\bm{R}=\mathbb{S}(\bm{\omega})\bm{R}$. We define the operator $(\cdot)^{\vee}$ such that $\bm{a}=\mathbb{S}(\bm{a})^{\vee}$.

We use the geometric controller in [28] to control the system’s attitude dynamics. The low-level controller outputs the desired actuation torques $\bm{\Gamma}_{a}^{*}$ by tracking the system’s orientation and angular velocity in the body frame. The attitude tracking error $\bm{e}_{R}\in\mathbb{R}^{3}$ is defined as $\bm{e}_{R}=\frac{1}{2}(\bm{R}^{*\top}\bm{R}-\bm{R}^{\top}\bm{R}^{*})^{\vee},$ and the angular velocity tracking error $\bm{e}_{\omega}\in\mathbb{R}^{3}$ is defined as: $\bm{e}_{\omega}=\bm{\omega}-\bm{R}^{\top}\bm{R}^{*}\bm{\omega}^{*}$. We define the integral error $\bm{e}_{I}=\int_{o}^{t}\bm{e}_{\omega}(\tau)+c_{2}\bm{e}_{R}(\tau)d\tau$, with $c_{2}$ being a positive constant. A nonlinear attitude dynamics feedback controller is designed as the following:

$$\begin{split}\bm{\Gamma}_{a}^{*}(l)=&-\bm{K}_{R}\bm{e}_{R}-\bm{K}_{\omega}\bm{e}_{\omega}-\bm{K}_{I}\bm{e}_{I}+\bm{\omega}\times\bm{I}(l)\bm{\omega}\\ &+\bm{G}_{ang}-\bm{I}(l)(\mathbb{S}(\bm{\omega})\bm{R}^{\top}\bm{R}^{*}\bm{\omega}^{*}-\bm{R}^{\top}\bm{R}^{*}\dot{\bm{\omega}}^{*}),\end{split}$$

(10)

where $\bm{K}_{R}$, $\bm{K}_{\omega}$, and $\bm{K}_{I}\in\mathbb{R}^{3\times 3}$ are positive-definite matrices.

The high-level selective impedance control outputs the desired actuation forces $\bm{F}_{a}^{*}$ by tracking the body frame origin $O_{B}$ attached to the aerial vehicle’s \accog, instead of the shifting-mass plate in the world frame. The system’s linear dynamics can be re-written as:

$$\bm{M}\dot{\bm{\upsilon}}^{w}+\mathbb{S}(\bm{\omega}^{w})\bm{M}\bm{\upsilon}^{w}+mg\bm{e}_{3}=\bm{F}_{a}^{w}+\bm{F}_{C}^{w},$$

(11)

where $\bm{F}_{a}^{w}=\bm{R}\bm{F}_{a}$ and $\bm{F}_{C}^{w}=\bm{R}\bm{F}_{C}$. Considering the frame transformation and eq. 6, $\bm{F}_{a}^{w}$ is given by:

$$\bm{F}_{a}^{w}=\begin{bmatrix}C_{\theta}C_{\psi}F_{1}+(S_{\theta}C_{\psi}C_{\phi}+S_{\psi}S_{\phi})F_{3}\\ C_{\theta}S_{\psi}F_{1}+(S_{\theta}S_{\psi}C_{\phi}-C_{\psi}S_{\phi})F_{3}\\ -S_{\theta}F_{1}+C_{\theta}C_{\phi}F_{3}\end{bmatrix}.$$

(12)

The position tracking error is given by $\bm{e}_{p}^{w}=\bm{p}^{w}-\bm{p}^{w*}$, where $\bm{p}^{w}\in\mathbb{R}^{3}$ represents the position of the origin $O_{B}$ \acwrt $O$ expressed in the world frame. The velocity tracking error can then be displayed as $\bm{e}_{v}^{w}=\bm{\upsilon}^{w}-\bm{\upsilon}^{w*}$ with $\bm{\upsilon}^{w*}=\dot{\bm{p}}^{w*}$. The desired close loop dynamics of the system with the selective impedance control is:

$$\bm{M}\dot{\bm{e}}_{v}^{w}+\bm{D}\bm{e}_{v}^{w}+\bm{K}\bm{e}_{p}^{w}=\bm{F}_{C}^{w},$$

(13)

where $\bm{D}$ and $\bm{K}\in\mathbb{R}^{3\times 3}$ are positive-definite matrices representing the desired damping and stiffness of the system. Combining eq. 11 and eq. 13, the desired actuation force vector expressed in the world frame is defined as:

$$\begin{split}\bm{F}_{a}^{w*}&=\begin{bmatrix}F_{1}^{w*}&F_{2}^{w*}&F_{3}^{w*}\end{bmatrix}^{\top}\\ &=\bm{M}\dot{\bm{\upsilon}}^{w*}-\bm{D}\bm{e}_{v}^{w}-\bm{K}\bm{e}_{p}^{w}+\bm{\omega}^{w}\times\bm{M}\bm{\upsilon}^{w}+mg\bm{e}_{3}.\end{split}$$

(14)

The system introduces coupling between its roll angle $\phi$ and its linear motion in the world frame not having a thrust component along the body axis $\bm{y}_{B}$. Again, combining eq. 12 and eq. 14 the desired roll angle $\phi^{*}$ is calculated via:

$$\begin{split}\phi^{*}=&atan2\big{(}S_{\psi}F_{1}^{w*}-C_{\psi}F_{2}^{w*},S_{\theta}C_{\psi}F_{1}^{w*}+...\\ &S_{\theta}S_{\psi}F_{2}^{w*}+C_{\theta}F_{3}^{w*}\big{)}.\end{split}$$

(15)

Furthermore, by using trigonometric calculations with eq. 12, the desired actuation force vector $\bm{F}_{a}^{*}=\begin{bmatrix}F_{1}^{*}&0&F_{3}^{*}\end{bmatrix}^{\top}$ expressed in the body frame is given by:

	$$F_{1}^{*}=C_{\theta}C_{\psi}F_{1}^{w*}+C_{\theta}S_{\psi}F_{2}^{w*}-S_{\theta}F_{3}^{w*},$$		(16a)
	$$\begin{split}F_{3}^{*}=&\big{(}(S_{\psi}F_{1}^{w*}-C_{\psi}F_{2}^{w*})^{2}+(S_{\theta}C_{\psi}F_{1}^{w*}+...\\ &S_{\theta}S_{\psi}F_{2}^{w*}+C_{\theta}F_{3}^{w*})^{2}\big{)}^{\frac{1}{2}}.\end{split}$$		(16b)

The desired roll angle $\phi^{*}$ is then fed to the low-level attitude controller in Sec. V-A with the attitude references $\psi^{*}$ and $\theta^{*}$. Finally, $\bm{F}_{a}^{*}$ and $\bm{\Gamma}_{a}^{*}$ are fed to the control allocation to derive the desired system inputs $\bm{u}^{*}$, as in Fig. 6.

The control allocation problem for the examined case can be divided into finding the desired tilting angle $\alpha^{*}$ value,

$$\alpha^{*}=\frac{\pi}{2}-atan2\big{(}F_{3}^{*}L+\Gamma_{2}^{*}(l),2LF_{1}^{*}\big{)},$$

(17)

and retrieve the corresponding motor velocities $\Omega_{i}$. To handle the redundancy of the system, we introduce a virtual control input vector $\bm{\lambda}=\begin{bmatrix}\Omega_{1}^{2}&\Omega_{2}^{2}&...&\Omega_{8}^{2}\end{bmatrix}^{\top}\in\mathbb{R}^{8}$. By applying eq. 4, eq. 6, and eq. 7 the desired virtual control vector is calculated via pseudo-inverse:

$$\bm{\lambda}^{*}=\bm{H}(\alpha^{*})^{{\dagger}}\cdot\begin{bmatrix}\bm{F}_{a}^{*}\\ \bm{\Gamma}_{a}^{*}\end{bmatrix},$$

(18)

where $\bm{H}(\alpha^{*})\in\mathbb{R}^{6\times 12}$ is the allocation matrix depending on the desired tilting angle $\alpha^{*}$ and $(\cdot)^{{\dagger}}$ is the associate pseudo-inverse operator. Finally, we can compute the desired real system inputs as $\bm{u}^{*}=\begin{bmatrix}\alpha^{*}&\sqrt{\bm{\lambda}^{*}}\end{bmatrix}$. This approach minimizes the motors’ rotating speed with guaranteed positive values [29].

The entire control architecture is implemented via a customized PX4 firmware [32] enabling the support for the servo motor to tilt back rotors and a ROS2 controller (in C++) implementing the designed control law. A separate ROS2 controller is used to move the shifting-mass plate, in which the shifting-mass plate position $l$ is mapped into the servo position. The C++ control node operates at $250$ Hz matching the standard PX4 controller frequency. The following gain values are used during the experiments: $\bm{K}=diag([22.0,\ 22.0,\ 80.0])$, $\bm{D}=diag([10.0,\ 10.0,\ 45.0])$, $c_{2}=0.8$, $\bm{K}_{R}=diag([5.0,\ 5.0,\ 3.0])$, $\bm{K}_{\omega}=diag([1.0,\ 1.4,\ 0.25])$, and $\bm{K}_{I}=diag([0.0,\ 3.25,\ 0.5])$. The codes run on an onboard Lattepanda Delta V3 which communicates with the flight controller via DDS and ROS2. An OptiTrack system is used to track the platform’s position and orientation. A vertical wooden board is set as the work surface, and safety ropes are used to prevent platform damage, see Fig. 1. In practice, the mass distribution across the platform is not symmetric around the body axes. Directly moving the shifting-mass plate to $l=\displaystyle\frac{m}{m_{S}}L$ as introduced in Sec. III-A may lead to over-displaced \accom resulting in risky situations for the platform. Therefore, to avoid platform damage due to over-displaced \accom, we tested a maximum shifting-mass plate position of $0.18$\mathrm{m}$\ll\frac{m}{m_{S}}L=0.48$\mathrm{m}$$.

The first experiments evaluated trajectory tracking capabilities in Cartesian space. Firstly, desired position set-points $x_{des}$, $y_{des}$, and $z_{des}$ were sent to the aerial vehicle, and the tracking errors are displayed in Fig. 7(a). Due to the under-actuation of the system, the desired roll set points $\phi_{des}$ were generated from the corresponding linear motion of the platform as in eq. 15. Later, desired pitch and yaw set points $\theta_{des}$ and $\psi_{des}$ were sent to the aerial vehicle while hovering. The attitude tracking errors are displayed in Fig. 7(b). The testing results show the designed system’s promising orientation and position control during free flight with the proposed control law.

In this scenario, we conducted a series of pushing tasks on a vertical work surface to evaluate the feasibility and effectiveness of the system’s dynamic \accom configuration. The high-level impedance controller in Sec. V treats the system as a mechanical admittance. To perform the pushing task, the desired contact position (i.e., the setpoint) was intentionally set “inside” the work surface, as per the approach outlined in [5]. We denote $\delta p$ as the distance from the work surface to the setpoint along the interaction axis $\bm{x}_{B}$, with $\delta p>0$ indicating positions “inside” the surface.

Using the experimental data, the pushing force perpendicular to the surface was estimated offline via the wrench estimation method detailed in [34, 17], utilizing the estimated thrust of each propeller. The required aerodynamic coefficients for thrust estimation were measured with an RC Benchmark Thrust Stand, with more details available in [30]. To initiate contact with the surface, a setpoint of $\delta p=0.6$\mathrm{m}$$ was provided to the high-level impedance controller, resulting in the platform exerting approximately $13$\mathrm{N}$$ of pushing force, see Fig. 8. While maintaining contact, the shifting-mass plate moved actively toward the work surface, reaching its maximum allowable position at $l^{*}=0.18$\mathrm{m}$$, thus dynamically displacing the \accom. This process is demonstrated in the first part of the accompanying video¹¹1The video is available at https://youtu.be/NXEnCS1ZLEg..

With the shifting-mass plate at $l^{*}=0.18$\mathrm{m}$$, setpoints of $\delta p=0.8,\ 1.0,\ 1.2$\mathrm{m}$$ were then provided to command progressively higher pushing forces. The platform stably pushed against the vertical surface with forces of $19$\mathrm{N}$$, $23$\mathrm{N}$$, and $28$\mathrm{N}$$, respectively (see Fig. 8). By moving the \accom closer to the front rotors, as described in Sec. II-A, the system was able to achieve stable pushing forces nearly equal to its gravitational force by tilting only the back rotors.

Similar to Scenario 1, a series of setpoints with varying $\delta p$ values were used to conduct pushing tasks. However, in this case, the system’s \accom remained static, with the shifting-mass plate fixed at $l=0$, as depicted in Fig. 8. Throughout the tests, the platform exhibited more pronounced oscillations compared to Scenario 1, as shown in Fig. 8 and the second part of the attached video¹¹footnotemark: 1. Initially, a setpoint of $\delta p=0.2$\mathrm{m}$$ was provided, resulting in an exerted pushing force of approximately $10$\mathrm{N}$$. Subsequently, the platform applied a pushing force of about $15$\mathrm{N}$$ with a setpoint of $\delta p=0.6$\mathrm{m}$$. However, when the setpoint was increased to $\delta p=0.8$\mathrm{m}$$, the system impulsively generated a force of $20$\mathrm{N}$$, leading to risky conditions.

In this section, we compare the force generation capability of the proposed system with various state-of-the-art (SOTA) aerial manipulation platforms. To evaluate force exertion on vertical surfaces during pushing tasks, we consider key parameters for each system: total system mass ($m_{0}$), the number of actuation \acdof, the number of tilting rotor axes engaged for pushing ($n_{t}$), and the maximum achievable pushing force ($f_{p}$). Additionally, we introduce a comparison factor that accounts for both the platform’s mass and the number of tilting axes used for pushing:

$$h_{f}=\frac{f_{p}}{9.8\times m_{0}n_{t}},$$

(19)

where a higher value of $h_{f}$ indicates superior force exertion ability. Table I shows the comparison of our proposed system with SOTA platforms. System s2 [13], a 5-\acdof platform, tilts both front and back rotors to generate pushing forces. It exerts a pushing force of $40$\mathrm{N}$$ with a mass of $3.3$\mathrm{kg}$$ and two tilting axes. System s3 [20], a 6-\acdof platform, generates up to $30$\mathrm{N}$$ of stable pushing force, with a mass of $4.0$\mathrm{kg}$$ and one tilting axis for pushing. System s4 [6], featuring 4-\acdof actuation, can apply up to $15$\mathrm{N}$$ on a vertical surface with a weight of $1.5$\mathrm{kg}$$. It uses both front and back rotors for pushing, similar to the configuration in system s2 [13], with $n_{t}=2$.

In comparison, our proposed system, with only one tilting axis (the back rotors) and a lightweight structure ($3.12$\mathrm{kg}$$), can exert a stable pushing force of $28$\mathrm{N}$$ when the \accom is dynamically displaced closer to the front rotors as in Scenario 1. With a fixed \accom as in Scenario 2, it can push up to $15$\mathrm{N}$$ with more oscillations.

This quantitative comparison highlights the effectiveness of our approach, demonstrating that the proposed system achieves competitive force generation with fewer tilting axes thanks to the dynamic \accom displacing configuration.