Geometric finite-time inner-outer loop trajectory tracking control strategy for quadrotor slung-load transportation

Abstract

This paper presents a geometric finite-time inner-outer loop control strategy for slung payload transportation using a quadrotor. The underactuated nature of the quadrotor in conjunction with the cable connection makes it impossible to simultaneously and arbitrarily track the load position, cable direction and the quadrotor’s attitude. By exploiting the differential flatness of the system dynamics, it is possible to arbitrarily control the load position, with the cable direction and quadrotor attitude remaining dependent on the trajectory and controller dynamics. A two-loop feedback control strategy is proposed comprising an inner-loop control law addressing quadrotor attitude tracking within finite time, and an outer-loop controller addressing the stabilization of the cable direction and position tracking of the payload trajectory. The control laws are expressed in a coordinate-free setting and are singularity-free. We include the effects of unmodeled dynamics on the load system and design an adaptive estimator to estimate and reject such uncertainties. The performance and robustness of the feedback laws is demonstrated through numerical simulations and experimental results.

Appendices

A Proof of Theorem 1

Let $V_{1}: SO(3) \longrightarrow {\mathbb {R}}$ be the attitude error function constructed as

$$\begin{aligned} V_{1}(\mathbf{R}_{e}) = \text{ trace }(K_{p}(I_{3\times 3} - \mathbf{R}_{e})), \end{aligned}$$

where $K_{p}=\text{ diag }(\lambda _{1},\lambda _{2},\lambda _{3})$ and $\lambda _{1},\lambda _{2},\lambda _{3}$ are positive and distinct. Taking the time derivative of $V_{1}$, we get

$$\begin{aligned} \frac{d V_{1}(\mathbf{R}_{e})}{dt} = - \text{ trace }(\text{ skew }(K_{p}{} \mathbf{R}_{d}^{T}{} \mathbf{R}) [\mathbf{e} _{\varvec{\omega }_{b}}]^{\wedge }) \end{aligned}$$

where $\text{ skew }(A) = \frac{1}{2}(A - A^{T})$. The differential of $V_{1}$ with respect to $\mathbf{R}_{e}$ is denoted by $\mathbf{R}_{e}$ and defined through

$$\begin{aligned} {\dot{V}}_{1}(\mathbf{R}_{e}) = \mathbf{e} _{R}^{T} \mathbf{e} _{\varvec{\omega }_{b}}. \end{aligned}$$

Using the identity $\text{ trace }({\widehat{x}}{\widehat{y}}) = - 2 x^{T} y$ for any $x,y \in {\mathbb {R}}^{3}$, and the breve map notation from Sect. 2, it follows that

$$\begin{aligned} \mathbf{e} _{R} = \sum _{i = 1}^{3} \lambda _{i} \mathbf{R}^{T}{} \mathbf{R}_{d}\hat{\mathbf{e }}_{i} \times \hat{\mathbf{e }}_{i} . \end{aligned}$$

Adding and subtracting the finite-time converging virtual control law

$$\begin{aligned} \phi (\mathbf{R} _{e})= - k_{R} \frac{\varvec{e}_{R}}{(\varvec{e}_{R}^{T} \varvec{e}_{R})^{1-1/p}} \end{aligned}$$

(18)

allows to rewrite the Lyapunov time derivative as

$$\begin{aligned} {\dot{V}}_{1} = - k_{R} ( \mathbf{e} _{R}^{T} \mathbf{e} _{R} )^{\frac{1}{p}} + \mathbf{e} _{R}^{T} ( \mathbf{e} _{\omega } - \phi (\mathbf{R}_{e}) ), \end{aligned}$$

where p is the reaching time parameter and $0< \frac{1}{p} \le 1$. We now define a new backstepping error as

$$\begin{aligned} \mathbf{e} _\omega = \mathbf{e} _{\omega } - \phi (\mathbf{R}_{e}), \end{aligned}$$

corresponding to the error between the actual state $\mathbf{e} _{\varvec{\omega }_{b}}$ and virtual control $\phi (\mathbf{R}_{e})$. The initial system (6)–(7) in $(\mathbf{R}_{e},\mathbf{e} _\omega )$ coordinates is,

$$\begin{aligned}&{\dot{\mathbf{R}}}_{e} = \mathbf{R}_{e} [\mathbf{e} _\omega + \phi (\mathbf{R}_{e})]^{\wedge }, \nonumber \\&\dot{\mathbf{z }}_\omega = \dot{\mathbf{e }}_{\omega } - {\dot{\phi }}(\mathbf{R}) = {\dot{\varvec{\omega }}}_{b} + (\varvec{\omega }_{b} \times \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d})\nonumber \\&\qquad - \mathbf{R}_{e}^{T}{\dot{\varvec{\omega }}}_{b}^{d} - {\dot{\phi }}(\mathbf{R}) \end{aligned}$$

(19)

where ${\dot{\phi }}(\mathbf{R}_{e})$ is calculated from (18) as

$$\begin{aligned} \frac{d}{dt}\phi (\mathbf{R}_{e})&= -\frac{k_{R}}{(\mathbf{e} _{R}^{T} \mathbf{e} _{R})^{(1-1/p)}}\\&\quad \times \left[ I_{3 \times 3} - \frac{2(p-1)\mathbf{e} _{R}{} \mathbf{e} _{R}^{T}}{p(\mathbf{e} _{R}^{T} \mathbf{e} _{R})^{(1-1/p)}} \right] \dot{\mathbf{e }}_{R} \end{aligned}$$

and $\dot{\mathbf{e }}_{R}$ is calculated from $\mathbf{e} _{R}$ as

$$\begin{aligned} \frac{d}{dt}{} \mathbf{e} _{R}&= \frac{d}{dt}\left( \sum _{i=1}^{3}\lambda _{i} \mathbf{R}_{e}^{T}\hat{\mathbf{e }}_{i} \times \hat{\mathbf{e }}_{i}\right) \\&= \sum _{i=1}^{3} \lambda _{i}\hat{\mathbf{e }}_{i} \times [\mathbf{e} _{\omega }]^{\wedge } \mathbf{R}_{e}^{T}\hat{\mathbf{e }}_{i}. \end{aligned}$$

Therefore, using the error states, (19) becomes

$$\begin{aligned}&\dot{\mathbf{z }}_\omega = J_{b}^{-1}\left[ \left( \mathbf{e} _\omega + \phi (\mathbf{R}_{e}) + \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d}\right) \right. \\&\quad \left. \times J_{b}(\mathbf{e} _\omega + \phi (\mathbf{R}_{e}) + \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d})\right] \\&\quad + \left( \mathbf{e} _\omega + \phi (\mathbf{R}_{e}) + \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d}\right) \\&\quad \times \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d} - \mathbf{R}_{e}^{T}{\dot{\varvec{\omega }}}_{b}^{d} - \frac{d}{dt}\phi (\mathbf{R}_{e}) + J_{b}^{-1}M. \end{aligned}$$

Continuing with the backstepping methodology, we define a composite Lyapunov function as

$$\begin{aligned} V_{2}(\mathbf{R}_{e},\mathbf{e} _\omega ) = V_{1} + \frac{1}{2} \mathbf{e} _\omega ^{T} J_{b} \mathbf{e} _\omega \end{aligned}$$

The time derivative of $V_{2}$ is

$$\begin{aligned} \frac{d V_{2}}{dt}= & {} - k_{R} ( \mathbf{e} _{R}^{T} \mathbf{e} _{R} )^{\frac{1}{p}} + \mathbf{e} _{R}^{T} \mathbf{e} _\omega + \mathbf{e} _\omega ^{T} J_{b} \dot{\mathbf{z }}_\omega \\= & {} - k_{R} ( \mathbf{e} _{R}^{T} \mathbf{e} _{R} )^{\frac{1}{p}} + \mathbf{e} _\omega ^{T}\\&\quad \times \left[ (\phi (\mathbf{R}_{e}) + \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d}) \times J_{b}(\mathbf{e} _\omega + \phi (\mathbf{R}_{e}) \right. \\&\quad \left. + \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d}) + J_{b}((\mathbf{e} _\omega + \phi (\mathbf{R}_{e})) \times \mathbf{R}_{e}^{T}\varvec{\omega }_{b}^{d})\right. \\&\quad - J_{b}{} \mathbf{R}_{e}^{T}{\dot{\varvec{\omega }}}_{b}^{d} \\&\quad \left. - J_{b}\frac{d}{dt}\phi (\mathbf{R}_{e}) + M + \mathbf{e} _{R} \right] , \end{aligned}$$

and, in closed-loop system with control law (8), becomes

$$\begin{aligned} \frac{dV_{2}}{dt} = - k_{R} ( \mathbf{e} _{R}^{T} \mathbf{e} _{R} )^{\frac{1}{p}} - ( \mathbf{e} _\omega ^{T} L \mathbf{e} _\omega )^{\frac{1}{p}} , \end{aligned}$$

a negative semi-definite function. The set where ${\dot{V}}_{2}=0$ is given by $E \times \{\varvec{e}_{\omega }=0\}$ where

$$\begin{aligned} E= & {} \{ I_{3 \times 3}, \text{ diag }(-1,1,-1), \text{ diag }(1,-1,-1),\\&\text{ diag }(-1,-1,1) \} \end{aligned}$$

is the set of critical points of $\dot{V}_1$. From [20](Lemma 4.1), we know that $V_{1}$ is a Morse function taking values on ${\mathbb {R}}^{+}$, vanishing only at $\mathbf{R}_{e}=I_{3\times 3}$ with the other three elements of E as non-degenerate critical points. Prior research on attitude stabilization and tracking on SO(3) [20, 9, 22, 23], proves that the three non-identity equilibria form stable closed submanifolds of SO(3) and their complement is open and dense in SO(3). Applying similar arguments to [5], we conclude that the closed-loop error system reaches, in finite time, a neighborhood of the origin $(I_{3 \times 3}, {\mathbf {0}})$ where the inequality

$$\begin{aligned} \mathbf{e} _{R}^{T}{} \mathbf{e} _{R} \ge \text{ trace }(K_{p}(I - \mathbf{Q} )) \end{aligned}$$

(20)

is valid, resulting in

$$\begin{aligned} {\dot{V}}_{2} \le - k_{R} \left( V_{1}^{1/p} + (\mathbf{e} _\omega ^{T} J_{b} \mathbf{e} _\omega )^{1/p} \right) . \end{aligned}$$

(21)

Using the inequality $(a+b)^{\alpha } \le a^{\alpha } + b^{\alpha }$, the time derivative can be expressed as a fractional exponent of the original function,

$$\begin{aligned} {\dot{V}}_{2} \le - k_{R} V_{2}^{1/p}, \end{aligned}$$

implying a finite-time convergence within

$$\begin{aligned} t_{e} = \frac{p}{k_{R}(p - 1)}V_{e}^{(1-1/p)} \end{aligned}$$

where $V_{e}$ is the Lyapunov function value when entering the neighborhood of the origin where (20) is valid. Hence, for all initial conditions

$$\begin{aligned} (\mathbf{R}_{e}(0),\mathbf{e} _{\omega }(0)) \in SO(3) \backslash {\mathcal {M}} \times {\mathbb {R}}^{3}, \end{aligned}$$

where ${\mathcal {M}}$ is the union of all the stable closed manifolds, the solutions of the closed-loop system converge to $(I_{3 \times 3},0)$. Noting that $SO(3) \backslash {\mathcal {M}}$ is open and dense in SO(3) and using (21) we can state that the quadrotor attitude closed-loop system with the proposed control laws is almost globally finite-time stable [5, 38] and, equivalently, the closed-loop of the quadrotor attitude system (6)-(7) is finite-time stable around $(\mathbf{R}_{d},\varvec{\omega }_{b}^{d})$.

B Proof of Theorem 2

The proposed attitude controller in Sect. 4.1 through (8) ensures $\mathbf{R}(t) = \mathbf{R}_{d}(t)$ for $t \ge T$, where $T>0$ is finite. More explicitly, we obtain $\mathbf{r} _{3} =\mathbf{R} \mathbf{e} _{3}= \mathbf{r }_{3}^{d}$ for $t \ge T$, hence from (13) and (16) we have

$$\begin{aligned} fR \mathbf{e} _{3} = \varPi _\mathbf{q }^{\parallel }(F) + \varPi _\mathbf{q }^{\bot }(F), \quad \forall t\ge T. \end{aligned}$$

Since there is no finite-time escape of the load-cable subsystem, without loss of generality, we can assume $t > T$ and F can be used a control input for the load-cable subsystem.

We perform the stability analysis for the load-cable subsystem in parts, designing tentative Lyapunov function for the different subsystems, and then combining them in a single Lyapunov function with provable negative semi-definite time derivative.

Part 1: Consider a tentative positive definite Lyapunov function for the load position tracking as

$$\begin{aligned} V_\mathbf{x } = \frac{1}{2} \mathbf{e} _{x}^{T} k_{x} \mathbf{e} _{x} + \frac{1}{2} \mathbf{e} _{v}^{T} \mathbf{e} _{v} + k_{x_{I}} \mathbf{e} _{x}^{T} \mathbf{e} _{v}, \end{aligned}$$

with $k_{x}$ a positive definite matrix and

$$\begin{aligned} 0< k_{x_{I}} < \sqrt{k_x}. \end{aligned}$$

(22)

Taking the derivative along the dynamics (3) and in closed-loop with the control laws (9) and (10), an rearranging terms we get

$$\begin{aligned} \dot{V}_\mathbf{x }= & {} - k_{x}k_{x_{I}} \mathbf{e} _{x}^{T} \mathbf{e} _{x} - (k_{v} - k_{x_{I}}) \mathbf{e} _{v}^{T} \mathbf{e} _{v} - k_{x} k_{v} \mathbf{e} _{x}^{T} \mathbf{e} _{v} \\&+ \frac{1}{m_{{Q}} + m_{\ell }} (k_{x_{I}} \mathbf{e} _{x}+ \mathbf{e} _{v})^{T} \varPi _\mathbf{q }^{\parallel }( \Vert F_c\Vert \mathbf{q} _d + {\mathbf {R}} {\tilde{\varDelta }} ). \end{aligned}$$

Part 2: Define a candidate error function for the cable attitude as

$$\begin{aligned} V_{L} = \varPhi (\mathbf{q} ,\mathbf{q} ^{d}) + \frac{1}{2} m_{{Q}} \ell \mathbf{e} _{\varvec{\omega }_{L}} ^{T} \mathbf{e} _{\varvec{\omega }_{L}} + k_{q_{I}} \mathbf{e} _\mathbf{q }^{T} \mathbf{e} _{\omega _{L}}. \end{aligned}$$

(23)

From the relation

$$\begin{aligned} \Vert \mathbf{e} _\mathbf{q } \Vert = \frac{(2 k_q - \varPhi )\varPhi }{k_p^2}, \end{aligned}$$

the lower bound

$$\begin{aligned} \frac{1}{2} \Vert \mathbf{e} _\mathbf{q } \Vert ^2 \le \varPhi \end{aligned}$$

can be established leading to (23) being positive definite for

$$\begin{aligned} 0< k_{q_I} < \sqrt{k_q m_{{Q}} \ell }. \end{aligned}$$

(24)

The time derivative of the tentative Lyapunov function is

$$\begin{aligned} \dot{V}_{L}&= {\dot{\varPhi }} + \langle e_{\varvec{\omega }_{L}}, \nabla _{\varvec{\omega }_{L}}{} \mathbf{e} _{\varvec{\omega }_{L}} \rangle + k_{q_{I}}\langle \mathbf{e} _{\varvec{\omega }_{L}}, \mathbf{e} _{\varvec{\omega }_{L}} \rangle \\&\quad + k_{q_{I}}\langle \mathbf{e} _\mathbf{q }, \nabla _{\varvec{\omega }_{L}} \mathbf{e} _{\varvec{\omega }_{L}} \rangle \rangle \\&= d \varPhi \cdot \mathbf{e} _{\varvec{\omega }_{L}} + \langle k_{q_{I}}{} \mathbf{e} _\mathbf{q } + \mathbf{e} _{\varvec{\omega }_{L}}, \nabla _{\varvec{\omega }_{L}} \mathbf{e} _{\varvec{\omega }_{L}} \rangle \\&\quad + k_{q_{I}} \Vert \mathbf{e} _{\varvec{\omega }_{L}} \Vert ^{2}, \\&= k_{q}{} \mathbf{e} _\mathbf{q } \cdot \mathbf{e} _{\varvec{\omega }_{L}} + \langle k_{q_{I}} \mathbf{e} _\mathbf{q } + \mathbf{e} _{\varvec{\omega }_{L}}, {\dot{\varvec{\omega }}}_{L} \\&\quad - (\mathbf{q} \times \nabla _{\dot{\mathbf{q }}}\tau (\mathbf{q} ,\mathbf{q} ^{d})\dot{\mathbf{q }}^{d}) \rangle \\&\quad + k_{q_{I}} \Vert \mathbf{e} _{\varvec{\omega }_{L}} \Vert ^{2}, \end{aligned}$$

Using the dynamics (3) and substituting the control from (12) we have,

$$\begin{aligned} {\dot{V}}_{L} = - k_{q}k_{q_{I}} \mathbf{e} _\mathbf{q } ^{T} \mathbf{e} _\mathbf{q } - (k_{\omega } - k_{q_{I}}) \mathbf{e} _{\varvec{\omega }_{L}}^{T} \mathbf{e} _{\varvec{\omega }_{L}} - k_{q_{I}}k_{d} \mathbf{e} _\mathbf{q }^{T} \mathbf{e} _{\varvec{\omega }_{L}} \\ - \frac{1}{m_{{Q}} \ell }(k_{q_{I}} \mathbf{e} _\mathbf{q } + \mathbf{e} _{\varvec{\omega }_{L}})^{T} \left( \mathbf{q} \times (\varPi _\mathbf{q }^{\bot }({\mathbf {R}} {\tilde{\varDelta }}))\right) . \end{aligned}$$

Part 3: Let us now define a Lyapunov function for the estimator error

$$\begin{aligned} V_{\varDelta } = \frac{1}{2 g_{I}} {\tilde{\varDelta }} ^{T} {\tilde{\varDelta }}. \end{aligned}$$

with $g_{I}>0$. Taking the time derivative in closed-loop with the estimator dynamics (14) leads to

$$\begin{aligned} {\dot{V}}_{\varDelta }&= \frac{1}{g_{I}} {\tilde{\varDelta }}^{T} \mathbf{R}^{T}\left( \varPi _\mathbf{q }^{\parallel }(k_{x_{I}}{} \mathbf{e} _{x} + \mathbf{e} _{v}) + \mathbf{q} \times (k_{q_{I}}{} \mathbf{e} _\mathbf{q } + \mathbf{e} _{\varvec{\omega }_{L}}) \right) \\&= -\varPi _\mathbf{q }^{\parallel }(\mathbf{R} {\tilde{\varDelta }})^{T}(k_{x_{I}} \mathbf{e} _{x} + \mathbf{e} _{v}) - \varPi _\mathbf{q }^{\bot }(\mathbf{R} {\tilde{\varDelta }})^{T} \\&\quad [\mathbf{q} \times (k_{q_{I}} \mathbf{e} _\mathbf{q } + \mathbf{e} _{\omega _{L}})]. \end{aligned}$$

Notice that the estimator dynamics (14) were carefully chosen to cancel out the unknown terms in $V_\mathbf{x }$ and ${\dot{V}}_{L}$ that depend on the estimation error.

Part 4: Finally, the definite Lyapunov function for the load-cable subsystem can be written as

$$\begin{aligned} V = V_\mathbf{x } + V_{L} + V_{\varDelta }, \end{aligned}$$

(25)

a positive-definite function under conditions (22) and (24). Its time derivative, gathering the previously computed $\dot{V}_\mathbf{x }$, $\dot{V}_{L}$, and $\dot{V}_{\varDelta }$, is established as

$$\begin{aligned} \dot{V}&= - k_{x}k_{x_{I}} \mathbf{e} _{x}^{T} \mathbf{e} _{x} - (k_{v} - k_{x_{I}}) \mathbf{e} _{v}^{T} \mathbf{e} _{v} - k_{x} k_{v} \mathbf{e} _{x}^{T} \mathbf{e} _{v} \\&\quad + \frac{\Vert F_c\Vert }{m_{{T}}} (k_{x_{I}} \mathbf{e} _{x}+ \mathbf{e} _{v})^{T} \varPi _\mathbf{q }^{\parallel }( \mathbf{q} _d ) - k_{q}k_{q_{I}} \mathbf{e} _\mathbf{q } ^{T} \mathbf{e} _\mathbf{q }\\&\quad - (k_{\omega } - k_{q_{I}}) \mathbf{e} _{\varvec{\omega }_{L}}^{T} \mathbf{e} _{\varvec{\omega }_{L}} - k_{q_{I}}k_{\omega } \mathbf{e} _\mathbf{q }^{T} \mathbf{e} _{\varvec{\omega }_{L}} \\&\le - {\mathbf {z}} ^{T} Q {\mathbf {z}}, \end{aligned}$$

with $Q =$

$$\begin{aligned} \begin{bmatrix} k_{x}k_{x_{I}}{\mathbf {I}} &{} k_{x_I} k_{v}/2{\mathbf {I}} &{} - k_{x_I} \frac{\Vert F_c\Vert }{2m_{{T}}} \widehat{ {\mathbf {q}}} &{} {\mathbf {0}}\\ k_{x_I} k_{v}/2 {\mathbf {I}} &{} k_{v} - k_{x_{I}}{\mathbf {I}} &{} - \frac{\Vert F_c\Vert }{2m_{{T}}} \widehat{ {\mathbf {q}}} &{} {\mathbf {0}} \\ - k_{x_I} \frac{\Vert F_c\Vert }{2m_{{T}}} \widehat{ {\mathbf {q}}} &{} - \frac{\Vert F_c\Vert }{2m_{{T}}} \widehat{ {\mathbf {q}}} &{} k_{q}k_{q_{I}}{\mathbf {I}} &{} k_{q_{I}}k_{\omega }/2 {\mathbf {I}} \\ {\mathbf {0}} &{} {\mathbf {0}} &{} k_{q_{I}}k_{\omega }/2 {\mathbf {I}} &{} k_{\omega } - k_{q_{I}}{\mathbf {I}} \end{bmatrix} \end{aligned}$$

(26)

a matrix that can be rendered positive definite with appropriate choice of gains and

$$\begin{aligned} {\mathbf {z}} = \left[ \mathbf{e} _{x},\, \mathbf{e} _{v},\, \mathbf{e} _\mathbf{q },\, \mathbf{e} _{\varvec{\omega }_{L}} \right] ^{T}. \end{aligned}$$

The time derivative of the Lyapunov function is negative semi-definite if

$$\begin{aligned} k_{x}&> 0 \end{aligned}$$

(27)

$$\begin{aligned} k_{x_I}&< \frac{4 k_{x} k_v}{4k_{x}+k_v^2} \end{aligned}$$

(28)

$$\begin{aligned} k_q k_{q_I}&> \frac{(k_{x}-k_{x_I}^2)\Vert {\mathbf {F}}_c \Vert _{\max }^2}{m_{{T}}^2\left( 4k_{x} k_v-k_{x_I} k_v^2-4k_{x_I} k_{x}\right) } \end{aligned}$$

(29)

$$\begin{aligned} \det (Q)&> 0 \end{aligned}$$

(30)

which can be achieved with sufficiently high gains $k_{x}$, $k_v$, $k_{q}$, $k_{\omega }$ and sufficiently low cross-term gains $k_{x_I}$ and $k_{q_I}$. From Theorem 1, the quadrotor attitude tracks the desired attitude (15) in finite time and, since there is not finite-time escape of the states, the force F generated by the quadrotor can be taken as an input for the load and cable subsystems. Let us consider the Lyapunov function (25), which is positive definite on conditions (22) and (24). At this point, we first assume $\dot{V}$ has non-positive derivative, limiting the tracking errors, in particular,

$$\begin{aligned} \Vert \mathbf{e} _{x}\Vert _{\max } = \sqrt{\frac{2 V(0)}{k_x}}, \quad \Vert \mathbf{e} _{v}\Vert _{\max } = \sqrt{2 V(0)}. \end{aligned}$$

With these established, we can derive a bound on the control force as

$$\begin{aligned} \Vert F_c\Vert < m_{{T}}&\left( \lambda _{\max }(K_p) \Vert \mathbf{e} _{x}\Vert _{\max } + k_v \Vert \mathbf{e} _{v}\Vert _{\max } \right. \\&\qquad \qquad \left. + g + \Vert \dot{\mathbf{v }}_{L}^{d}\Vert _{\max }\right) \end{aligned}$$

and select control and estimation gains such that (27)–(30) are verified, rendering matrix Q positive definite and ensure the function ${\dot{V}}$ is actually negative semi-definite for all time. The tracking errors $\left( \mathbf{e }_{x}, \mathbf{e }_{v}, \mathbf{e }_\mathbf{q }, \mathbf{e }_{\varvec{\omega }_{L}} \right)$ are then bounded and, from the system dynamics and control laws, so are their closed-loop derivatives, implying that ${\ddot{V}}$ is bounded. In other words, all the state errors are smooth and proper, i.e., $(\mathbf{e} _{x},\mathbf{e} _{v},\mathbf{e} _\mathbf{q },\mathbf{e} _{\varvec{\omega }_{L}}, {\tilde{\varDelta }}) \in L_{2}$ and the estimators $\varPi _\mathbf{q }^{\parallel }({\bar{\varDelta }}),\varPi _\mathbf{q }^{\bot }({\bar{\varDelta }})$ are uniformly continuous and bounded since ${\tilde{\varDelta }}$ is bounded, from V, which implies $(\dot{\mathbf{e }}_{x},\dot{\mathbf{e }}_{v},\mathbf{e} _{\dot{\mathbf{q }}}, \dot{\mathbf{e }}_{\omega _{L}}) \in L_{\infty }$. From Barbalat’s Lemma, all the tracking errors and their time derivatives asymptotically converge to zero. The convergence of all errors to zero results in all states tracking their desired values with the possible exception of the cable direction, which has zero error for $\mathbf{q} = \pm \mathbf{q} ^{d}$. A local analysis of the closed-loop system, for initial errors such that $(\mathbf{x} _{L}^{d},\mathbf{v} _{L}^{d},\mathbf{q} ^{d},\varvec{\omega }_{L}^{d})$ is the only reachable equilibrium point, leads to the conclusion that the closed-loop slung load system is asymptotically stable around the desired equilibrium, that is, the system is asymptotically stable around $(\mathbf{x} _{L}^{d},\mathbf{v} _{L}^{d},\mathbf{q} ^{d},\varvec{\omega }_{L}^{d})$.