Three-dimensional modeling of a tethered UAV–buoy system with relative-positioning and directional surge velocity control

Abstract

This work presents the nonlinear dynamical model and motion controller of a system consisting of an unmanned aerial vehicle (UAV) that is tethered to a floating buoy in the three-dimensional (3D) space. Detailed models of the UAV, buoy, and the coupled tethered system dynamics are presented in a marine environment that includes surface water currents and oscillating gravity waves, in addition to wind gusts. This work extends the previously modeled planar (vertical) motion of this novel robotic system to allow its free motion in all three dimensions. Furthermore, a directional surge velocity control system (DSVCS) is proposed to allow both the free movement of the UAV around the buoy when the cable is slack, and the manipulation of the buoy’s surge velocity when the cable is taut. Using a spherical coordinates system centered at the buoy, the control system commands the UAV to apply forces on the buoy at specific azimuth and elevation angles via the tether, which yields a more appropriate realization of the control problem as compared to the Cartesian coordinates, where the traditional x-, y-, and z-coordinates do not intuitively describe the tether’s tension and orientation. The proposed robotic system and controller offer a new method of interaction and collaboration between UAVs and marine systems from a locomotion perspective, and offers a low-cost and low-logistics alternative to crewed vessels for marine missions. It has potential applications in search-and-rescue missions, monitoring and sensing operations, and floating objects manipulation and retrieval. The system is validated in a virtual high-fidelity simulation environment, which was specifically developed for this work, while considering various settings, operating conditions, and wave scenarios.

Appendices

Appendix A: Buoy inertia and damping matrices

The buoy’s inertia matrix in the world frame $\mathcal {W}$, $\mathbf {M}_{\mathrm {b}}$, is symmetric (i.e., $M_{ij} = M_{ji}$, for $i\ne j$), and has its nonzero elements explicitly given as:

$$\begin{aligned} M_{\mathrm {b},11}&= m_{\mathrm {b}} + a_{11} (c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})^2 \nonumber \\&\quad + a_{22} (c_{\phi _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})^2 + a_{33} (s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})^2, \nonumber \\ M_{\mathrm {b},22}&= m_{\mathrm {b}} +a_{11} (c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})^2 \nonumber \\&\quad + a_{22} (c_{\phi _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})^2 + a_{33} (s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})^2, \nonumber \\ M_{\mathrm {b},33}&= m_{\mathrm {b}} + a_{11} (c_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}})^2 \nonumber \\&\quad + a_{22} (s_{\phi _{\mathrm {b}}})^2 + a_{33} (c_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}})^2, \nonumber \\ M_{\mathrm {b},12}&= - a_{22} (c_{\phi _{\mathrm {b}}}^2 s_{\psi _{\mathrm {b}}}c_{\psi _{\mathrm {b}}})\nonumber \\&\quad +a_{11} (c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})(c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}}) \nonumber \\&\quad + a_{33} (s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})(s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}}), \nonumber \\ M_{\mathrm {b},13}&= - \, a_{11} (c_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}}) (c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}}) \nonumber \\&\quad - a_{22} (c_{\phi _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} s_{\phi _{\mathrm {b}}}) \nonumber \\&\quad + a_{33} (\;\;\, c_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}}) (s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}}), \nonumber \\ M_{\mathrm {b},23}&=a_{11} (c_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}}) (c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}}) \nonumber \\&\quad + a_{22} (c_{\phi _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} s_{\phi _{\mathrm {b}}}) \nonumber \\&\quad + a_{33} (c_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}}) (s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}}), \nonumber \\ M_{\mathrm {b},44}&= (J_{\mathrm {b},xx} + a_{44}) \nonumber \\&\quad + \big [ (J_{\mathrm {b},zz} + a_{66}) - (J_{\mathrm {b},xx} + a_{44}) \big ] s_{\theta _{\mathrm {b}}}^2, \nonumber \\ M_{\mathrm {b},55}&= (J_{\mathrm {b},yy} + a_{55}) \nonumber \\&\quad + \big [ (J_{\mathrm {b},xx} + a_{44}) s_{\theta _{\mathrm {b}}}^2 + (J_{\mathrm {b},zz} + a_{66}) c_{\theta _{\mathrm {b}}}^2 \big ] t_{\phi _{\mathrm {b}}}^2, \nonumber \\ M_{\mathrm {b},66}&= (J_{\mathrm {b},zz} + a_{66}) \nonumber \\&\quad + \big [ (J_{\mathrm {b},xx} + a_{44}) - (J_{\mathrm {b},zz} + a_{66}) \big ] s_{\theta _{\mathrm {b}}}^2 / c_{\phi _{\mathrm {b}}}^2, \nonumber \\ M_{\mathrm {b},45}&= \big [ (J_{\mathrm {b},xx} + a_{44}) - (J_{\mathrm {b},zz} + a_{66}) \big ] s_{\theta _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} t_{\phi _{\mathrm {b}}}, \nonumber \\ M_{\mathrm {b},46}&= \big [ (J_{\mathrm {b},zz} + a_{66}) - (J_{\mathrm {b},xx} + a_{44}) \big ] \nonumber \\&s_{\theta _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} / c_{\phi _{\mathrm {b}}}, \nonumber \\ M_{\mathrm {b},56}&= - \big [ (J_{\mathrm {b},xx} + a_{44}) s_{\theta _{\mathrm {b}}}^2 + (J_{\mathrm {b},zz} + a_{66}) c_{\theta _{\mathrm {b}}}^2 \big ] \nonumber \\&t_{\phi _{\mathrm {b}}} / c_{\phi _{\mathrm {b}}}. \end{aligned}$$

(A1)

Similarly, its damping matrix in $\mathcal {W}$, $\mathbf {D}_{\mathrm {b}}$, is symmetric (i.e., $D_{ij} = D_{ji}$, for $i\ne j$), and has its nonzero elements explicitly given as:

$$\begin{aligned} D_{\mathrm {b},11}&= b_{11} (c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})^2 + b_{22} (c_{\phi _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})^2 \nonumber \\&\quad + b_{33} (s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})^2, \nonumber \\ D_{\mathrm {b},22}&= b_{11} (c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})^2 + b_{22} (c_{\phi _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})^2 \nonumber \\&\quad + b_{33} (s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})^2, \nonumber \\ D_{\mathrm {b},33}&= b_{11} (c_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}})^2 + b_{22} (s_{\phi _{\mathrm {b}}})^2 + b_{33} (c_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}})^2, \nonumber \\ D_{\mathrm {b},12}&= b_{22} (-c_{\phi _{\mathrm {b}}}^2 s_{\psi _{\mathrm {b}}}c_{\psi _{\mathrm {b}}}) \nonumber \\&\quad + b_{11} (c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}})(c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}}) \nonumber \\&\quad + b_{33} (s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}})(s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}}), \nonumber \\ D_{\mathrm {b},13}&= b_{22} (-c_{\phi _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} s_{\phi _{\mathrm {b}}}) \nonumber \\&\quad + b_{11} (-c_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}}) (c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}}) \nonumber \\&\quad + b_{33} (\;\;\, c_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}}) (s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}}), \nonumber \\ D_{\mathrm {b},23}&= b_{22} (c_{\phi _{\mathrm {b}}} c_{\psi _{\mathrm {b}}} s_{\phi _{\mathrm {b}}}) \nonumber \\&\quad + b_{11} (c_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}}) (c_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} + s_{\phi _{\mathrm {b}}} s_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}}) \nonumber \\&\quad + b_{33} (c_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}}) (s_{\theta _{\mathrm {b}}} s_{\psi _{\mathrm {b}}} - s_{\phi _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} c_{\psi _{\mathrm {b}}}), \nonumber \\ D_{\mathrm {b},44}&= b_{44} + \big [ b_{66} - b_{44} \big ] s_{\theta _{\mathrm {b}}}^2, \nonumber \\ D_{\mathrm {b},55}&= b_{55} + \big [ b_{44} s_{\theta _{\mathrm {b}}}^2 + b_{66} c_{\theta _{\mathrm {b}}}^2 \big ] t_{\phi _{\mathrm {b}}}^2, \nonumber \\ D_{\mathrm {b},66}&= b_{66} + \big [ b_{44} - b_{66} \big ] s_{\theta _{\mathrm {b}}}^2 / c_{\phi _{\mathrm {b}}}^2, \nonumber \\ D_{\mathrm {b},45}&= \big [ b_{44} - b_{66} \big ] s_{\theta _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} t_{\phi _{\mathrm {b}}}, \nonumber \\ D_{\mathrm {b},46}&= \big [ b_{66} - b_{44} \big ] s_{\theta _{\mathrm {b}}} c_{\theta _{\mathrm {b}}} / c_{\phi _{\mathrm {b}}}, \nonumber \\ D_{\mathrm {b},56}&= - \big [ b_{44} s_{\theta _{\mathrm {b}}}^2 + b_{66} c_{\theta _{\mathrm {b}}}^2 \big ] t_{\phi _{\mathrm {b}}} / c_{\phi _{\mathrm {b}}}. \end{aligned}$$

(A2)

Appendix B: UAV inertia matrix

The UAV’s inertial matrix in $\mathcal {W}$ is symmetric, with elements explicitly expressed as:

$$\begin{aligned} M_{\mathrm {u},11}&= m_{\mathrm {u}}, M_{\mathrm {u},22} = m_{\mathrm {u}}, M_{\mathrm {u},33} = m_{\mathrm {u}}, \nonumber \\ M_{\mathrm {u},12}&= 0, M_{\mathrm {u},13} = 0, M_{\mathrm {u},23} = 0, \nonumber \\ M_{\mathrm {u},44}&= J_{\mathrm {u},xx} + \big [ J_{\mathrm {u},zz} - J_{\mathrm {u},xx} \big ] s_{\theta _{\mathrm {u}}}^2, \nonumber \\ M_{\mathrm {u},55}&= J_{\mathrm {u},yy} + \big [ J_{\mathrm {u},xx} s_{\theta _{\mathrm {u}}}^2 + J_{\mathrm {u},zz} c_{\theta _{\mathrm {u}}}^2 \big ] t_{\phi _{\mathrm {u}}}^2, \nonumber \\ M_{\mathrm {u},66}&= J_{\mathrm {u},zz} + \big [ J_{\mathrm {u},xx} - J_{\mathrm {u},zz} \big ] s_{\theta _{\mathrm {u}}}^2 / c_{\phi _{\mathrm {u}}}^2, \nonumber \\ M_{\mathrm {u},45}&= \big [ J_{\mathrm {u},xx} - J_{\mathrm {u},zz} \big ] s_{\theta _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} t_{\phi _{\mathrm {u}}}, \nonumber \\ M_{\mathrm {u},46}&= \big [ J_{\mathrm {u},zz} - J_{\mathrm {u},xx} \big ] s_{\theta _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} / c_{\phi _{\mathrm {u}}}, \nonumber \\ M_{\mathrm {u},56}&= - \big [ J_{\mathrm {u},xx} s_{\theta _{\mathrm {u}}}^2 + J_{\mathrm {u},zz} c_{\theta _{\mathrm {u}}}^2 \big ] t_{\phi _{\mathrm {u}}} / c_{\phi _{\mathrm {u}}}. \end{aligned}$$

(B3)

Appendix C: Spherical coordinates details

The spherical coordinates transformation to Cartesian coordinates, $\mathbf {R}_{\mathrm {S2C}}$, is made by two ordered rotations, $\mathbf {R}_{\alpha }$ then $\mathbf {R}_{\varphi }$, where:

$$\begin{aligned} \mathbf {R}_{\alpha } = \begin{bmatrix} c_{\alpha } &{} -s_{\alpha } &{} 0 \\ 0 &{} 0 &{} 1 \\ s_{\alpha } &{} \;\;\,c_{\alpha } &{} 0 \end{bmatrix}, \end{aligned}$$

(C4)

and:

$$\begin{aligned} \mathbf {R}_{\varphi } = \begin{bmatrix} c_{\varphi } &{} -s_{\varphi } &{} 0 \\ s_{\varphi } &{} \;\;\,c_{\varphi } &{} 0 \\ 0 &{} 0 &{} 1 \end{bmatrix}. \end{aligned}$$

(C5)

By differentiating $\mathbf {r}$ with respect to time, we find the derivatives of the spherical unit vectors to be:

$$\begin{aligned} \begin{aligned} \dot{\hat{\mathbf {e}}}_{\mathrm {r}}&= \dot{\alpha } \hat{\mathbf {e}}_{\alpha } + c_{\alpha } \dot{\varphi } \hat{\mathbf {e}}_{\varphi }, \\ \dot{\hat{\mathbf {e}}}_{\alpha }&= - \dot{\alpha } \hat{\mathbf {e}}_{\mathrm {r}} - s_{\alpha } \dot{\varphi } \hat{\mathbf {e}}_{\varphi },\\ \dot{\hat{\mathbf {e}}}_{\varphi }&= - c_{\alpha } \dot{\varphi } \hat{\mathbf {e}}_{\mathrm {r}} + s_{\alpha } \dot{\varphi } \hat{\mathbf {e}}_{\alpha }, \end{aligned} \end{aligned}$$

(C6)

From the definition of the spherical coordinates system and from (C6), the position, velocity, and acceleration vectors of $\mathcal {O}_{\mathrm {u}}$ are given by:

$$\begin{aligned} \mathbf {r}&=\mathrm {r}\hat{\mathbf {e}}_{\mathrm {r}}, \end{aligned}$$

(C7a)

$$\begin{aligned} \dot{\mathbf {r}}&= \dot{\mathrm {r}} \hat{\mathbf {e}}_{\mathrm {r}} +\mathrm {r} \dot{\alpha } \hat{\mathbf {e}}_{\alpha } +\mathrm {r}c_{\alpha } \dot{\varphi } \hat{\mathbf {e}}_{\varphi }, \end{aligned}$$

(C7b)

$$\begin{aligned} \ddot{\mathbf {r}}&= (\ddot{\mathrm {r}} -\mathrm {r}\dot{\alpha }^2 - \mathrm {r} c_{\alpha }^2 \dot{\varphi }^2 ) \hat{\mathbf {e}}_{\mathrm {r}}\nonumber \\&\quad + ( \mathrm {r} \ddot{\alpha } + 2\dot{\mathrm {r}} \dot{\alpha } + \mathrm {r} s_{\alpha } c_{\alpha } \dot{\varphi }^2) \hat{\mathbf {e}}_{\alpha } \nonumber \\&\quad + ( \mathrm {r} \ddot{\varphi } c_{\alpha } + 2 \dot{\mathrm {r}} \dot{\varphi } c_{\alpha } - 2 \mathrm {r} \dot{\alpha }\dot{\varphi } s_{\alpha })\hat{\mathbf {e}}_{\varphi }. \end{aligned}$$

(C7c)

The components of the UAV’s thrust vector in the spherical frame are explicitly realized from (26) as:

$$\begin{aligned} \begin{aligned} \texttt {u}_{\mathrm {r}}&= \texttt {u}_1 \big [ c_{\alpha } c_{\varphi } ( s_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} + s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} ) \\&\quad + c_{\alpha } s_{\varphi } ( s_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} - s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} ) + s_{\alpha } (c_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}}) \big ], \\ \texttt {u}_{\alpha }&= \texttt {u}_1 \big [ - s_{\alpha } c_{\varphi } ( s_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} + s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} ) \\&\quad - s_{\alpha } s_{\varphi } ( s_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} - s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} ) + c_{\alpha } (c_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}}) \big ], \\ \texttt {u}_{\varphi }&= \texttt {u}_1 \big [ - s_{\varphi } ( s_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} + s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} ) \\&\quad + c_{\varphi } ( s_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} - s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} ) \big ]. \end{aligned} \end{aligned}$$

(C8)

With the taut cable condition in (30), we have $\mathrm {r}=l$, and the spherical coordinates of the UAV with respect to the buoy’s center of mass, $\mathcal {O}_{\mathrm {b}}$, reduced to:

$$\begin{aligned} \begin{aligned} \mathbf {r}_{\mathrm {u}}&= \mathbf {r}_{\mathrm {b}} + \mathbf {R}_{\mathrm {1,b}} \mathbf {r}_{\mathrm {GH,b}} + \mathbf {R}_{\mathrm {S2C}} [l,0,0]^{\intercal }\\&\quad + \mathbf {R}_{\mathrm {1,u}} (-\mathbf {r}_{\mathrm {GH,u}}). \end{aligned} \end{aligned}$$

(C9)

Based on Assumption 5, we can simplify (C9) to:

$$\begin{aligned} \mathbf {r}_{\mathrm {u}} = \mathbf {r}_{\mathrm {b}} + \mathbf {R}_{\mathrm {S2C}} [l,0,0]^{\intercal }, \end{aligned}$$

(C10)

which then expands to:

$$\begin{aligned} \begin{aligned} x_{\mathrm {u}}&= x_{\mathrm {b}} + l c_{\alpha } c_{\varphi },\\ y_{\mathrm {u}}&= y_{\mathrm {b}} + l c_{\alpha } s_{\varphi },\\ z_{\mathrm {u}}&= z_{\mathrm {b}} + l s_{\alpha }, \end{aligned} \end{aligned}$$

(C11)

and similarly, its velocity can be obtained as:

$$\begin{aligned} \dot{\mathbf {r}}_{\mathrm {u}} = \dot{\mathbf {r}}_{\mathrm {b}} + \mathbf {R}_{\mathrm {S2C}} \dot{\mathbf {r}}, \end{aligned}$$

(C12)

which expands to:

$$\begin{aligned} \begin{aligned} \dot{x}_{\mathrm {u}}&= \dot{x}_{\mathrm {b}} + (-l s_{\alpha } c_{\varphi }) \dot{\alpha } + (-l c_{\alpha } s_{\varphi }) \dot{\varphi },\\ \dot{y}_{\mathrm {u}}&= \dot{y}_{\mathrm {b}} + (-l s_{\alpha } s_{\varphi }) \dot{\alpha } + (l c_{\alpha } c_{\varphi }) \dot{\varphi },\\ \dot{z}_{\mathrm {u}}&= \dot{z}_{\mathrm {b}} + (l c_{\alpha }) \dot{\alpha }. \end{aligned} \end{aligned}$$

(C13)

Appendix D: Euler–Lagrange formulation

The Kinetic energy of the tethered UAV–buoy system, $\mathcal {K}$, includes the rotational and translational energies of the UAV and that of the buoy while neglecting the cable’s energy, and is explicitly described as:

$$\begin{aligned} \mathcal {K}&= \frac{1}{2} \Big \{ \dot{x}_{\mathrm {b}} ( M_{\mathrm {b},11} \dot{x}_{\mathrm {b}} + M_{\mathrm {b},12} \dot{y}_{\mathrm {b}} + M_{\mathrm {b},13} \dot{z}_{\mathrm {b}} ) \nonumber \\&\quad + \dot{y}_{\mathrm {b}} ( M_{\mathrm {b},21} \dot{x}_{\mathrm {b}} + M_{\mathrm {b},22} \dot{y}_{\mathrm {b}} + M_{\mathrm {b},23} \dot{z}_{\mathrm {b}} ) \nonumber \\&\quad + \dot{z}_{\mathrm {b}} ( M_{\mathrm {b},31} \dot{x}_{\mathrm {b}} + M_{\mathrm {b},32} \dot{y}_{\mathrm {b}} + M_{\mathrm {b},33} \dot{z}_{\mathrm {b}} ) \nonumber \\&\quad + \dot{\phi }_{\mathrm {b}} ( M_{\mathrm {b},44} \dot{\phi }_{\mathrm {b}} + M_{\mathrm {b},45} \dot{\theta }_{\mathrm {b}} + M_{\mathrm {b},46} \dot{\psi }_{\mathrm {b}} ) \nonumber \\&\quad + \dot{\theta }_{\mathrm {b}} ( M_{\mathrm {b},54} \dot{\phi }_{\mathrm {b}} + M_{\mathrm {b},55} \dot{\theta }_{\mathrm {b}} + M_{\mathrm {b},56} \dot{\psi }_{\mathrm {b}} ) \nonumber \\&\quad + \dot{\psi }_{\mathrm {b}} ( M_{\mathrm {b},64} \dot{\phi }_{\mathrm {b}} + M_{\mathrm {b},65} \dot{\theta }_{\mathrm {b}} + M_{\mathrm {b},66} \dot{\psi }_{\mathrm {b}} ) \nonumber \\&\quad + m_{\mathrm {u}} (\dot{x}_{\mathrm {u}}^2 + \dot{y}_{\mathrm {u}}^2 + \dot{z}_{\mathrm {u}}^2 ) \nonumber \\&\quad + \dot{\phi }_{\mathrm {u}} ( M_{\mathrm {u},44} \dot{\phi }_{\mathrm {u}} + M_{\mathrm {u},45} \dot{\theta }_{\mathrm {u}} + M_{\mathrm {u},46} \dot{\psi }_{\mathrm {u}} ) \nonumber \\&\quad + \dot{\theta }_{\mathrm {u}} ( M_{\mathrm {u},54} \dot{\phi }_{\mathrm {u}} + M_{\mathrm {u},55} \dot{\theta }_{\mathrm {u}} + M_{\mathrm {u},56} \dot{\psi }_{\mathrm {u}} ) \nonumber \\&\quad + \dot{\psi }_{\mathrm {u}} ( M_{\mathrm {u},64} \dot{\phi }_{\mathrm {u}} + M_{\mathrm {u},65} \dot{\theta }_{\mathrm {u}} + M_{\mathrm {u},66} \dot{\psi }_{\mathrm {u}} ) \Big \} \nonumber \\&= \frac{1}{2} \Big \{ \dot{x}_{\mathrm {b}} \big [ (M_{\mathrm {b},11} + m_{\mathrm {u}}) \dot{x}_{\mathrm {b}} + M_{\mathrm {b},12} \dot{y}_{\mathrm {b}} + M_{\mathrm {b},13} \dot{z}_{\mathrm {b}} \big ] \nonumber \\&\quad + \dot{y}_{\mathrm {b}} \big [ M_{\mathrm {b},21} \dot{x}_{\mathrm {b}} + (M_{\mathrm {b},22} + m_{\mathrm {u}}) \dot{y}_{\mathrm {b}} + M_{\mathrm {b},23} \dot{z}_{\mathrm {b}} \big ] \nonumber \\&\quad + \dot{z}_{\mathrm {b}} \big [ M_{\mathrm {b},31} \dot{x}_{\mathrm {b}} + M_{\mathrm {b},32} \dot{y}_{\mathrm {b}} + (M_{\mathrm {b},33} + m_{\mathrm {u}}) \dot{z}_{\mathrm {b}} \big ] \nonumber \\&\quad + m_{\mathrm {u}} \big [ (l \dot{\alpha })^2 + (l c_{\alpha } \dot{\varphi })^2 \nonumber \\&\quad - 2(l s_{\alpha } c_{\varphi } \dot{\alpha } \dot{x}_{\mathrm {b}}) - 2(l c_{\alpha } s_{\varphi } \dot{\varphi } \dot{x}_{\mathrm {b}}) \nonumber \\&\quad - 2(l s_{\alpha } s_{\varphi } \dot{\alpha } \dot{y}_{\mathrm {b}}) + 2(l c_{\alpha } c_{\varphi } \dot{\varphi } \dot{y}_{\mathrm {b}}) + 2(l c_{\alpha } \dot{\alpha } \dot{z}_{\mathrm {b}}) \big ] \nonumber \\&\quad + \dot{\phi }_{\mathrm {b}} ( M_{\mathrm {b},44} \dot{\phi }_{\mathrm {b}} + M_{\mathrm {b},45} \dot{\theta }_{\mathrm {b}} + M_{\mathrm {b},46} \dot{\psi }_{\mathrm {b}} ) \nonumber \\&\quad + \dot{\theta }_{\mathrm {b}} ( M_{\mathrm {b},54} \dot{\phi }_{\mathrm {b}} + M_{\mathrm {b},55} \dot{\theta }_{\mathrm {b}} + M_{\mathrm {b},56} \dot{\psi }_{\mathrm {b}} ) \nonumber \\&\quad + \dot{\psi }_{\mathrm {b}} ( M_{\mathrm {b},64} \dot{\phi }_{\mathrm {b}} + M_{\mathrm {b},65} \dot{\theta }_{\mathrm {b}} + M_{\mathrm {b},66} \dot{\psi }_{\mathrm {b}} ) \nonumber \\&\quad + \dot{\phi }_{\mathrm {u}} ( M_{\mathrm {u},44} \dot{\phi }_{\mathrm {u}} + M_{\mathrm {u},45} \dot{\theta }_{\mathrm {u}} + M_{\mathrm {u},46} \dot{\psi }_{\mathrm {u}} ) \nonumber \\&\quad + \dot{\theta }_{\mathrm {u}} ( M_{\mathrm {u},54} \dot{\phi }_{\mathrm {u}} + M_{\mathrm {u},55} \dot{\theta }_{\mathrm {u}} + M_{\mathrm {u},56} \dot{\psi }_{\mathrm {u}} ) \nonumber \\&\quad + \dot{\psi }_{\mathrm {u}} ( M_{\mathrm {u},64} \dot{\phi }_{\mathrm {u}} + M_{\mathrm {u},65} \dot{\theta }_{\mathrm {u}} + M_{\mathrm {u},66} \dot{\psi }_{\mathrm {u}} ) \Big \}. \end{aligned}$$

(D14)

The kinetic energy is differentiated with respect to each state rate then with respect to time. Only the buoy’s translational states and tether’s orientation states are of interest, since the UAV and the buoy’s rotational states feature no dependencies on other states, which can be seen by inspection of the expanded $\mathcal {K}$. Their corresponding results are expressed as:

(D15)

The differentiation results of the UAV and buoy’s rotational states are omitted as they result in similar components to the rotational dynamics already introduced in (27) and (21).

The kinetic energy of the system is differentiated with respect to the states, ($\frac{\partial \mathcal {K}}{\partial \varvec{\eta }}$), where the resulting nonzero elements are obtained as:

$$\begin{aligned} \begin{aligned} \frac{\partial \mathcal {K}}{\partial \alpha }&= m_{\mathrm {u}} l ( - l s_{\alpha } c_{\alpha } \dot{\varphi }^2 - c_{\alpha } c_{\varphi } \dot{\alpha } \dot{x}_{\mathrm {b}} + s_{\alpha } s_{\varphi } \dot{\varphi } \dot{x}_{\mathrm {b}} \\&- c_{\alpha } s_{\varphi } \dot{\alpha } \dot{y}_{\mathrm {b}} - s_{\alpha } c_{\varphi } \dot{\varphi } \dot{y}_{\mathrm {b}} - s_{\alpha } \dot{\alpha } \dot{z}_{\mathrm {b}} ) , \\ \frac{\partial \mathcal {K}}{\partial \varphi }&= m_{\mathrm {u}} l ( s_{\alpha } s_{\varphi } \dot{\alpha } \dot{x}_{\mathrm {b}} - c_{\alpha } c_{\varphi } \dot{\varphi } \dot{x}_{\mathrm {b}} - s_{\alpha } c_{\varphi } \dot{\alpha } \dot{y}_{\mathrm {b}} \\&- c_{\alpha } s_{\varphi } \dot{\varphi } \dot{y}_{\mathrm {b}} ). \end{aligned} \end{aligned}$$

(D16)

Notice that most elements of $\frac{\partial \mathcal {K}}{\partial \varvec{\eta }}$ cancels out with elements of $\frac{d}{dt}\Big (\frac{\partial \mathcal {K}}{\partial \dot{\varvec{\eta }}} \Big )$.

The system’s potential energy was given in (38), and its external forces and moments vector can be formulated based on (22) and (28) as:

$$\begin{aligned} \begin{aligned}&\tau _1 = \texttt {u}_1 (s_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} + s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} ), \\&\tau _2 = \texttt {u}_1 (s_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} - s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} ), \\&\tau _3 = \texttt {u}_1 c_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} + \rho _{\mathrm {w}} \textit{g} \curlyvee _{\mathrm {im}},\\&\tau _4 = \texttt {u}_1 l \big [ - s_{\alpha } c_{\varphi } ( s_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} + s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} ) \\&\qquad - s_{\alpha } s_{\varphi } ( s_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} - s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} ) + c_{\alpha } (c_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}}) \big ],\\&\tau _5 = \texttt {u}_1 l \big [ - s_{\varphi } ( s_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} + s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} ) \\&\qquad + c_{\varphi } ( s_{\theta _{\mathrm {u}}} s_{\psi _{\mathrm {u}}} - s_{\phi _{\mathrm {u}}} c_{\theta _{\mathrm {u}}} c_{\psi _{\mathrm {u}}} ) \big ], \end{aligned} \end{aligned}$$

(D17)

Additionally, we have: $\varvec{\tau }_{6-8} = \varvec{\tau }_{2,\mathrm {u}}$ and $\varvec{\tau }_{9-11} = \varvec{\tau }_{2,\mathrm {b}}$. The potential energy’s derivative with respect the states rates then to time is obtained as:

$$\begin{aligned} \frac{d}{dt}\Big (\frac{\partial \mathcal {U}}{\partial \dot{\varvec{\eta }}}\Big ) = [0,0,0,0,0,0,0,0,0,0,0]^{\intercal }, \end{aligned}$$

(D18)

and its first derivative with respect to states is obtained as:

$$\begin{aligned} \frac{\partial \mathcal {U}}{\partial \varvec{\eta }} = [0,0,(m_{\mathrm {b}} + m_{\mathrm {u}}) \textit{g} , m_{\mathrm {u}} \textit{g} l c_{\alpha }, 0,0,0,0,0,0,0]^{\intercal }. \nonumber \\ \end{aligned}$$

(D19)