The Soft-PVTOL: modeling and control

Abstract

This paper presents, for the first time, the soft planar vertical take-off and landing (Soft-PVTOL) aircraft. This concept captures the soft aerial vehicle's fundamental dynamics with a minimum number of states and inputs but retains the main features to consider when designing control laws. Unlike conventional PVTOL and multi-rotors, where altering position inevitably impacts orientation due to their underactuated design, the Soft-PVTOL offers the unique advantage of separating these dynamics, opening doors to unparalleled maneuverability and precision. We demonstrate that the Soft-PVTOL can be modeled using the Euler-Lagrange equations by assuming a constant curvature model in the aerial robot's arms. Such a mathematical model is presented in detail and can be extended to several constant curvature segments in each Soft-PVTOL arm. Moreover, we design a passivity-based control law that exploits the flexibility of the robot's arms. We solve the tracking control problem, proving that the error equilibrium globally exponentially converges to zero. The controller is tested in numerical simulations, demonstrating robust performance and ensuring the efficacy of the closed-loop system.

Figures (9)

• Figure 1: The concept of the Soft-PVTOL involves two soft arms modeled using the constant curvature approach. We adopt the convention that when the arm tilts below the $x$ axis in the frame, we assign a negative value, as illustrated in the case of $q_l$ in the diagram. This convention is already accounted for in \ref{['eq:coords']}.
• Figure 2: The thrust of the right-hand side motor. From the figure, it is easy to see that the torque generated by the right-hand-thrust $T_r$ is given by $\left(l_r\frac{\sin{qr}}{q_r}\right) \left( T_r\cos{q_r} \right)$. Please note that we have omitted the value of $\epsilon$ for simplicity.
• Figure 3: The block diagram illustrates the closed-loop system, where it is assumed that all system states $(\mathbf{q}, \dot{\mathbf{q}})$ are available for feedback—a standard assumption in aerial robot control.
• Figure 4: Function approximation described in Assumption \ref{['ass:approx']}.
• Figure 5: The $x-z$ plot illustrates the 2D trajectory tracked by the Soft-PVTOL. It is important to note that while in a conventional PVTOL, the $x$ displacements are associated with tilting angles of $\theta$, in this particular configuration, the roll angle remains fixed at zero throughout.

Theorems & Definitions (3)

• Lemma 1
• Lemma 2
• Theorem 1: Passivity-based control Spongbook