$$ \begin{aligned} m_{21}(q) &= m_{12}(q), m_{22}(q) = m + m_1 + m_2, m_{23}(q) = 0, m_{24}(q) = (m_1 + m_2)l_1 \cos(\theta_{1y}), \\ m_{25}(q) &= 0, m_{26}(q) = m_2 l_2 \cos(\theta_{2y}) \end{aligned} \qquad (E.3) $$

$$ \begin{aligned} m_{31}(q) &= m_{13}(q), m_{32}(q) = m_{23}(q), m_{33}(q) = (m_1 + m_2)l_1^2 \cos^2(\theta_{1y}), \\ m_{34}(q) &= 0, m_{35}(q) = m_2 l_1 l_2 \cos(\theta_{2y}) \cos(\theta_{1y}) \cos(\theta_{2x} - \theta_{1x}), \\ m_{36}(q) &= m_2 l_1 l_2 \sin(\theta_{2y}) \cos(\theta_{2x}) \sin(\theta_{1x} - \theta_{2x}) \end{aligned} \qquad (E.4) $$

$$ \begin{aligned} m_{41}(q) &= m_{14}(q), m_{42}(q) = m_{24}(q), m_{43}(q) = m_{34}(q), m_{44}(q) = (m_1 + m_2)l_1^2 \\ m_{45}(q) &= m_2 l_1 l_2 \cos(\theta_{2y}) \sin(\theta_{2x}) \sin(\theta_{1y} - \theta_{1x}), \end{aligned} \qquad (E.5) $$

$$ m_{46}(q) = m_2 l_1 l_2 (\sin(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) + \cos(\theta_{2y}) \cos(\theta_{1y})) $$

$$ \begin{aligned} m_{51}(q) &= m_{15}(q), m_{52}(q) = m_{25}(q), m_{53}(q) = m_{35}(q), m_{54}(q) = m_{45}(q) \\ m_{55}(q) &= m_2 l_2^2 \cos^2(\theta_{2y}), \end{aligned} \qquad (E.6) $$

$$ m_{56}(q) = 0 $$

$$ \begin{aligned} m_{61}(q) &= m_{16}(q), m_{62}(q) = m_{26}(q), m_{63}(q) = m_{36}(q), m_{64}(q) = m_{46}(q), m_{65}(q) = m_{56}(q), \\ m_{66}(q) &= m_2 l_2^2 \end{aligned} \qquad (E.7) $$

Using the Christoffel's symbols (see Appendix A.2) allows us to calculate the matrix $C(q, \dot{q})$ possessing the useful property that $\dot{m}_{ij}(q, \dot{q}) = c_{ij}(q, \dot{q}) + c_{ji}(q, \dot{q})$, $1 \le i, j \le 6$ (the argument $(q, \dot{q})$ is dropped):

$$ \begin{aligned} C_{1\bullet} &= (0, 0, -(m_1 + m_2)l_1(\sin(\theta_{1x}) \cos(\theta_{1y})\dot{\theta}_{1x} + \cos(\theta_{1x}) \sin(\theta_{1y})\dot{\theta}_{1y}), \\ &-(m_1 + m_2)l_1(\cos(\theta_{1x}) \sin(\theta_{1y})\dot{\theta}_{1x} + \sin(\theta_{1x}) \cos(\theta_{1y})\dot{\theta}_{1y}), \\ &-m_2 l_2(\sin(\theta_{2x}) \cos(\theta_{2y})\dot{\theta}_{2x} + \cos(\theta_{2x}) \sin(\theta_{2y})\dot{\theta}_{2y}), \\ &-m_2 l_2(\cos(\theta_{2x}) \sin(\theta_{2y})\dot{\theta}_{2x} + \sin(\theta_{2x}) \cos(\theta_{2y})\dot{\theta}_{2y}) \end{aligned} \qquad (E.8) $$

$$ C_{2\bullet} = (0, 0, 0, -(m_1 + m_2)l_1 \sin(\theta_{1y})\dot{\theta}_{1y}, 0, -m_2 l_2 \sin(\theta_{2y})\dot{\theta}_{2y}) \qquad (E.9) $$

$$ \begin{aligned} C_{3\bullet} &= (0, 0, -(m_1 + m_2)l_1^2 \sin(\theta_{1y}) \cos(\theta_{1y}) \dot{\theta}_{1y}, -(m_1 + m_2)l_1^2 \sin(\theta_{1y}) \cos(\theta_{1y}) \dot{\theta}_{1x}, \\ &m_2 l_1 l_2 \sin(\theta_{1x} - \theta_{2x})[\cos(\theta_{1y}) \cos(\theta_{2y}) \dot{\theta}_{2x} + \sin(\theta_{1y}) \sin(\theta_{2y}) \dot{\theta}_{2y}], \\ &-m_2 l_1 l_2 \cos(\theta_{1y})[\cos(\theta_{1x}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2x} + \sin(\theta_{1x}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2y} \\ &+ \cos(\theta_{1x}) \cos(\theta_{2y}) \sin(\theta_{1x}) \dot{\theta}_{2y} - \sin(\theta_{1x}) \cos(\theta_{2x}) \cos(\theta_{2y}) \dot{\theta}_{2y}] \end{aligned} \qquad (E.10) $$

$$ \begin{aligned} C_{4\bullet} &= (0, 0, (m_1 + m_2)l_1^2 \sin(\theta_{1y}) \cos(\theta_{1y})\dot{\theta}_{1x}, 0, m_2 l_1 l_2 \sin(\theta_{1y})[\cos(\theta_{1x}) \cos(\theta_{2x}) \cos(\theta_{2y})\dot{\theta}_{2x} \\ &+ \sin(\theta_{1x}) \sin(\theta_{2x}) \cos(\theta_{2y})\dot{\theta}_{2x} - \cos(\theta_{1x}) \sin(\theta_{2x}) \sin(\theta_{2y})\dot{\theta}_{2y} + \sin(\theta_{1x}) \sin(\theta_{2y}) \cos(\theta_{2x})\dot{\theta}_{2y}], \\ &m_2 l_1 l_2[\cos(\theta_{1x}) \cos(\theta_{2x}) \cos(\theta_{2y}) \sin(\theta_{1y})\dot{\theta}_{2y} + \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \cos(\theta_{2y})\dot{\theta}_{2y} \\ &- \cos(\theta_{1y}) \sin(\theta_{2y})\dot{\theta}_{2y} - \cos(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y})\dot{\theta}_{2x} + \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{2x})\dot{\theta}_{2x}] \end{aligned} \qquad (E.11) $$

$$ \begin{aligned} C_{5\bullet} &= (0, 0, -m_2 l_2 \cos(\theta_{2y})[\cos(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{1x})\dot{\theta}_{1x} - \cos(\theta_{1x}) \cos(\theta_{1y}) \sin(\theta_{2x})\dot{\theta}_{1x} \\ &+ \cos(\theta_{1x}) \cos(\theta_{1y}) \sin(\theta_{1y})\dot{\theta}_{1y} + \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x})\dot{\theta}_{1y}], \\ &-m_2 l_1 l_2 \cos(\theta_{2y})[\cos(\theta_{1x}) \cos(\theta_{2x}) \sin(\theta_{1y})\dot{\theta}_{1x} + \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x})\dot{\theta}_{1x} \\ &- \cos(\theta_{1x}) \cos(\theta_{1y}) \sin(\theta_{2x})\dot{\theta}_{1y} + \cos(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{1x})\dot{\theta}_{1y}], \\ &-m_2 l_2^2 \sin(\theta_{2y}) \cos(\theta_{2y}) \dot{\theta}_{2y}, \\ &-m_2 l_2^2 \sin(\theta_{2y}) \cos(\theta_{2y}) \dot{\theta}_{2x} \end{aligned} \qquad (E.12) $$
$$
\begin{align}
C_6\bullet &= (0, 0, m_2 l_1 l_2 \sin(\theta_{2y}) [\cos(\theta_{1x}) \cos(\theta_{1y}) \cos(\theta_{2x}) \dot{\theta}_{1x} + \sin(\theta_{1x}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2x} \nonumber \\
&\qquad + \cos(\theta_{1x}) \cos(\theta_{1y}) \sin(\theta_{2x}) \dot{\theta}_{1y} - \sin(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \dot{\theta}_{1y}], \nonumber \\
&\qquad m_2 l_1 l_2 [\cos(\theta_{1x}) \cos(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1y} + \sin(\theta_{1x}) \sin(\theta_{2x}) \sin(\theta_{2y}) \cos(\theta_{1y}) \dot{\theta}_{1y} \nonumber \\
&\qquad - \cos(\theta_{2y}) \sin(\theta_{1y}) \dot{\theta}_{1y} + \cos(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1x} \nonumber \\
&\qquad - \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{2x}) \dot{\theta}_{1x}], \nonumber \\
&\qquad m_2 l_2^2 \sin(\theta_{2y}) \cos(\theta_{2y}) \dot{\theta}_{2x}, 0) \tag{E.13}
\end{align}
$$

Notice that the expressions given in [183, Appendix A] also use the Christoffel's symbols. The gravity potential energy is given by:

$$ U_g(q) = -m_1 g l_1 \cos(\theta_{1x}) \cos(\theta_{1y}) - m_2 g [l_1 \cos(\theta_{1x}) \cos(\theta_{1y}) + l_2 \cos(\theta_{2x}) \cos(\theta_{2y})], \quad (E.14) $$

so that the gravity generalized force is equal to:

$$ -\frac{\partial U_g}{\partial q} = \begin{pmatrix} 0 \\ 0 \\ -(m_1 + m_2)gl_1 \sin(\theta_{1x}) \cos(\theta_{1y}) \\ -(m_1 + m_2)gl_1 \cos(\theta_{1x}) \sin(\theta_{1y}) \\ m_2gl_2 \sin(\theta_{2x}) \cos(\theta_{2y}) \\ m_2gl_2 \cos(\theta_{2x}) \sin(\theta_{2y}) \end{pmatrix} \quad (E.15) $$

## Appendix F. Dynamics of 3D double-pendulum with varying links' lengths and constant masses

The system is depicted in Fig. 3 (b). The system's kinetic energy is given by:

$$
\begin{align}
T(q, \dot{q}) ={}& \frac{1}{2} m \dot{x}^2 + \frac{1}{2} m \dot{y}^2 + \frac{1}{2} m_1 [\dot{y} + \dot{l}_1 \sin(\theta_{1y}) + l_1 \dot{\theta}_{1y} \cos(\theta_{1y})]^2 \nonumber \\
& + \frac{1}{2} m_1 [\dot{x} + \dot{l}_1 \sin(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1x} \cos(\theta_{1x}) \cos(\theta_{1y}) - l_1 \sin(\theta_{1x}) \sin(\theta_{1y}) \dot{\theta}_{1y}]^2 \nonumber \\
& + \frac{1}{2} m_1 [-\dot{l}_1 \cos(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1x} \sin(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1y} \cos(\theta_{1x}) \sin(\theta_{1y})]^2 \nonumber \\
& + \frac{1}{2} m_2 [\dot{x} + \dot{l}_1 \sin(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1x} \cos(\theta_{1x}) \cos(\theta_{1y}) - l_1 \dot{\theta}_{1y} \sin(\theta_{1x}) \sin(\theta_{1y}) + \dot{l}_2 \sin(\theta_{2x}) \cos(\theta_{2y}) \nonumber \\
& \phantom{+ \frac{1}{2} m_2 [\dot{x} + \dot{l}_1 \sin(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1x} \cos(\theta_{1x}) \cos(\theta_{2y}) - l_1 \dot{\theta}_{1y} \sin(\theta_{2x}) \sin(\theta_{2y})]^2} \nonumber \\
& + \frac{1}{2} m_2 [\dot{y} + \dot{l}_1 \sin(\theta_{1y}) + l_1 \dot{\theta}_{1y} \cos(\theta_{1y}) + \dot{l}_2 \sin(\theta_{2y}) + l_2 \dot{\theta}_{2y} \cos(\theta_{2y})]^2 \nonumber \\
& + \frac{1}{2} m_2 [-\dot{l}_1 \cos(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1x} \sin(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1y} \cos(\theta_{1x}) \sin(\theta_{1y}) \nonumber \\
& \phantom{+ \frac{1}{2} m_2 [-\dot{l}_1 \cos(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1x} \sin(\theta_{1x}) \cos(\theta_{1y}) + l_1 \dot{\theta}_{1y} \cos(\theta_{1x}) \sin(\theta_{2y})]^2} \nonumber \\
& - \dot{l}_2 \cos(\theta_{2x}) \cos(\theta_{2y}) + l_2 \dot{\theta}_{2x} \sin(\theta_{2x}) \cos(\theta_{2y}) + l_2 \dot{\theta}_{2y} \cos(\theta_{2x}) \sin(\theta_{2y})]^2 \tag{F.1}
\end{align}
$$

The mass matrix $M(q) \in \mathbb{R}^{8 \times 8}$ is given as follows, with $q = (x, y, \theta_{1x}, \theta_{1y}, \theta_{2x}, \theta_{2y}, l_1, l_2)^\top$, row by row:

$$
\begin{align}
m_{11}(q) &= m + m_1 + m_2, \quad m_{12}(q) = 0, \quad m_{13}(q) = (m_1 + m_2)l_1 \cos(\theta_{1x}) \cos(\theta_{1y}), \nonumber \\
m_{14}(q) &= -(m_1 + m_2)l_1 \sin(\theta_{1x}) \sin(\theta_{1y}), \quad m_{15}(q) = m_2 l_2 \cos(\theta_{2x}) \cos(\theta_{2y}), \nonumber \\
m_{16}(q) &= -m_2 l_2 \sin(\theta_{2x}) \sin(\theta_{2y}), \quad m_{17}(q) = (m_1 + m_2) \sin(\theta_{1x}) \cos(\theta_{1y}), \nonumber \\
m_{18}(q) &= m_2 \sin(\theta_{2x}) \cos(\theta_{2y}) \tag{F.2}
\end{align}
$$

$$
\begin{align}
m_{21}(q) &= 0, \quad m_{22}(q) = m + m_1 + m_2, \quad m_{23}(q) = 0, \quad m_{24}(q) = (m_1 + m_2)l_1 \cos(\theta_{1y}), \nonumber \\
m_{25}(q) &= 0, \quad m_{26}(q) = m_2 l_2 \cos(\theta_{2y}), \quad m_{27}(q) = (m_1 + m_2) \sin(\theta_{1y}), \quad m_{28}(q) = m_2 \sin(\theta_{2y}) \tag{F.3}
\end{align}
$$
$$
\begin{align}
m_{31}(q) &= m_{13}(q), m_{32}(q) = m_{23}(q) = 0, m_{33}(q) = (m_1 + m_2)l_1 \cos^2(\theta_{1y}), \nonumber \\
m_{34}(q) &= 0, m_{35}(q) = m_2 l_1 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}), \tag{F.4} \\
m_{36}(q) &= m_2 l_1 l_2 \sin(\theta_{2y}) \cos(\theta_{1y}) \sin(\theta_{1x} - \theta_{2x}), m_{37}(q) = 0, \nonumber \\
m_{38}(q) &= m_2 l_1 \cos(\theta_{2y}) \cos(\theta_{1y}) \sin(\theta_{2x} - \theta_{1x}) \nonumber
\end{align}
$$

$$
\begin{align}
m_{41}(q) &= m_{14}(q), m_{42}(q) = m_{24}(q), m_{43}(q) = m_{34}(q) = 0, m_{44}(q) = (m_1 + m_2)l_1^2, \nonumber \\
m_{45}(q) &= m_2 l_1 l_2 \sin(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{2x} - \theta_{1x}), m_{46}(q) = m_1 l_1 l_2 \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \nonumber \\
&\quad + m_2 l_1 l_2 [\cos(\theta_{1y}) \cos(\theta_{2y}) + \cos(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y})], m_{47}(q) = 0, \nonumber \\
m_{48}(q) &= m_2 l_1 [\cos(\theta_{1y}) \sin(\theta_{2y}) - \cos(\theta_{2y}) \sin(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x})] \tag{F.5}
\end{align}
$$

$$
\begin{align}
m_{51}(q) &= m_{15}(q), m_{52}(q) = m_{25}(q), m_{53}(q) = m_{35}(q), m_{54}(q) = m_{45}(q), \nonumber \\
m_{55}(q) &= m_2 l_2^2 \cos^2(\theta_{2y}), m_{56}(q) = 0, m_{57}(q) = m_2 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}), \tag{F.6} \\
m_{58}(q) &= 0 \nonumber
\end{align}
$$

$$
\begin{align}
m_{61}(q) &= m_{16}(q), m_{62}(q) = m_{26}(q), m_{63}(q) = m_{36}(q), m_{64}(q) = m_{46}(q), m_{65}(q) = m_{56}(q), \nonumber \\
m_{66}(q) &= m_2 l_2^2, m_{67}(q) = m_2 l_2 [\cos(\theta_{2y}) \sin(\theta_{1y}) - \sin(\theta_{2y}) \cos(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x})], \tag{F.7} \\
m_{68}(q) &= 0 \nonumber
\end{align}
$$

$$
\begin{align}
m_{71}(q) &= m_{17}(q), m_{72}(q) = m_{27}(q), m_{73}(q) = m_{37}(q), m_{74}(q) = m_{47}(q), \nonumber \\
m_{75}(q) &= m_{57}(q), m_{76}(q) = m_{67}(q), m_{77}(q) = m_1 + m_2, \tag{F.8} \\
m_{78}(q) &= m_2[\sin(\theta_{1y}) \sin(\theta_{2y}) + \cos(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x})] \nonumber
\end{align}
$$

$$
\begin{align}
m_{81}(q) &= m_{18}(q), m_{82}(q) = m_{28}(q), m_{83}(q) = m_{38}(q), m_{84}(q) = m_{48}(q), m_{85}(q) = m_{58}(q), \tag{F.9} \\
m_{86}(q) &= m_{68}(q), m_{87}(q) = m_{78}(q), m_{88}(q) = m_2 \nonumber
\end{align}
$$

Notice that the conclusion in item 5 of Lemma 1 still holds for the submatrix $M_{ll}(q) = \begin{pmatrix} m_{77} & m_{78} \\ m_{87} & m_{88} \end{pmatrix}$, which is singular at the vertical posture and if $m_1 = 0$. The nonlinear inertial generalized forces are defined from the matrix $C(q, \dot{q})$ of Christoffel's symbols as (given row by row):

$$
\begin{align}
C_{1\bullet}(q, \dot{q}) = {}& (0, 0, c_{13} = (m_1 + m_2)[-l_1 \sin(\theta_{1x}) \cos(\theta_{1y}) \dot{\theta}_{1x} - l_1 \cos(\theta_{1x}) \sin(\theta_{1y}) \dot{\theta}_{1y} + \cos(\theta_{1x}) \cos(\theta_{1y}) \dot{l}_1], \nonumber \\
& c_{14} = -(m_1 + m_2)[l_1 \cos(\theta_{1x}) \sin(\theta_{1y}) \dot{\theta}_{1x} + \sin(\theta_{1x}) \sin(\theta_{1y}) \dot{l}_1 + l_1 \sin(\theta_{1x}) \cos(\theta_{1y}) \dot{\theta}_{1y}], \nonumber \\
& c_{15} = m_2[-l_2 \sin(\theta_{2x}) \cos(\theta_{2y}) \dot{\theta}_{2x} - l_2 \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2y} + \cos(\theta_{2x}) \cos(\theta_{2y}) \dot{l}_2], \nonumber \\
& c_{16} = -m_2[l_2 \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2x} + l_2 \sin(\theta_{2x}) \cos(\theta_{2y}) \dot{\theta}_{2y} + \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{l}_2], \nonumber \\
& c_{17} = (m_1 + m_2)[\cos(\theta_{1x}) \cos(\theta_{1y}) \dot{\theta}_{1x} - \sin(\theta_{1x}) \sin(\theta_{1y}) \dot{\theta}_{1y}], \ c_{18} = m_2[\cos(\theta_{2x}) \cos(\theta_{2y}) \dot{\theta}_{2x} \nonumber \\
& \phantom{c_{18} = } - \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2y}] \tag{F.10}
\end{align}
$$

$$
\begin{align}
C_{2\bullet}(q, \dot{q}) = {}& (0, 0, 0, c_{24} = (m_1 + m_2)[-l_1 \sin(\theta_{1y}) \dot{\theta}_{1y} + \cos(\theta_{1y}) \dot{l}_1], 0, c_{26} = -m_2 l_2 \sin(\theta_{2y}) \dot{\theta}_{2y} \nonumber \\
& + m_2 \cos(\theta_{2y}) \dot{l}_2, c_{27} = (m_1 + m_2) \cos(\theta_{1y}) \dot{\theta}_{1y}, c_{28} = m_2 \cos(\theta_{2y}) \dot{\theta}_{2y}) \tag{F.11}
\end{align}
$$
$$
\begin{aligned}
C_{3\bullet}(q, \dot{q}) = {}& (0, 0, c_{33} = -(m_1 + m_2)l_1 \cos(\theta_{1y}) \sin(\theta_{1y}) \dot{\theta}_{1y} + \frac{1}{2}(m_1 + m_2) \cos^2(\theta_{1y}) \dot{l}_1, \\
& c_{34} = -(m_1 + m_2)l_1 \cos(\theta_{1y}) \sin(\theta_{1y}) \dot{\theta}_{1y}, c_{35} = m_2 l_1 l_2 \cos(\theta_{1y}) \cos(\theta_{1y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2x} \\
& -m_2 l_1 l_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2y} + m_2 l_1 \cos(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{l}_2, \\
& c_{36} = \frac{1}{2}(m_2 - m_1)l_1 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1y} \\
& -m_2 l_1 l_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2x} + m_2 l_1 l_2 \cos(\theta_{2y}) \cos(\theta_{1y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2y} \\
& +m_2 l_1 \sin(\theta_{2y}) \cos(\theta_{1y}) \sin(\theta_{1x} - \theta_{2x}) \dot{l}_2, c_{37} = \frac{1}{2}(m_1 + m_2) \cos^2(\theta_{1y}) \dot{\theta}_{1x}, \\
& c_{38} = m_2 l_1 \cos(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2x} + m_2 l_1 \sin(\theta_{2y}) \cos(\theta_{1y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2y}
\end{aligned}
\tag{F.12}
$$

$$
\begin{aligned}
C_{4\bullet}(q, \dot{q}) = {}& (0, 0, c_{43} = (m_1 + m_2)l_1 \cos(\theta_{1y}) \sin(\theta_{1y}) \dot{\theta}_{1y}, c_{44} = m(1+m_2)l_1 \dot{l}_1, \\
& c_{45} = m_2 l_1 l_2 \sin(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2x} - m_2 l_1 l_2 \sin(\theta_{1y}) \sin(\theta_{2y}) \sin(\theta_{2x}) \cos(\theta_{1x}) \dot{\theta}_{2y} \\
& + \frac{m_1+m_2}{2} l_1 l_2 \sin(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2y} + m_2 l_1 \sin(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{2x} - \theta_{1x}) \dot{l}_2, \\
& c_{46} = \frac{m_1+m_2}{2} l_1 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1x} + \frac{1}{2} m_1 l_1 l_2 \sin(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2x} \\
& - \frac{1}{2} m_2 l_1 l_2 \sin(\theta_{1y}) \sin(\theta_{2y}) \sin(\theta_{2x} - \theta_{1x}) \dot{\theta}_{2x} + m_1 l_1 l_2 \sin(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2y} \\
& - m_2 l_1 l_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \dot{\theta}_{2y} + m_2 l_1 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \cos(\theta_{2y}) \dot{\theta}_{2y} \\
& + \frac{m_1-m_2}{2} l_2 \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \dot{l}_1 + \frac{m_1+m_2}{2} l_1 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{l}_2 \\
& + m_2 l_1 \cos(\theta_{1y}) \cos(\theta_{2y}) \dot{l}_2 + m_2 l_1 \cos(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{l}_2, c_{47} = (m_1 + m_2)l_1 \dot{\theta}_{1y} \\
& + \frac{m_1-m_2}{2} l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{\theta}_{2y}, c_{48} = m_2 l_1 \cos(\theta_{1y}) \cos(\theta_{2y}) \dot{\theta}_{2y} \\
& + m_2 l_1 \sin(\theta_{2y}) \sin(\theta_{1y}) \cos(\theta_{1x}) \cos(\theta_{2x}) \dot{\theta}_{2y} + \frac{m_1+m_2}{2} l_1 \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{2y}
\end{aligned}
\tag{F.13}
$$

$$
\begin{aligned}
C_{5\bullet}(q, \dot{q}) = {}& (0, 0, c_{53} = -m_2 l_1 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} - m_2 l_1 l_2 \sin(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1y} \\
& + m_2 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{l}_1, \\
& c_{54} = -m_2 l_1 l_2 \sin(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} + m_2 l_1 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{2x} - \theta_{1x}) \dot{\theta}_{1y} \\
& + \frac{m_2 - m_1}{2} l_1 l_2 \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{2x}) \dot{\theta}_{2y} + m_2 l_2 \sin(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{2x} - \theta_{1x}) \dot{l}_1, \\
& c_{55} = -m_2 l_2^2 \cos(\theta_{2y}) \sin(\theta_{2y}) \dot{\theta}_{2y} + m_2 l_2 \cos^2(\theta_{2y}) \dot{l}_2, \\
& c_{56} = \frac{m_2 - m_1}{2} l_1 l_2 \sin(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{2x}) \sin(\theta_{1x}) \dot{\theta}_{1y} - m_2 l_2^2 \cos(\theta_{2y}) \sin(\theta_{2y}) \dot{\theta}_{2x}, \\
& c_{57} = m_2 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} - m_2 l_2 \sin(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1y}, \\
& c_{58} = -m_2 l_2 \cos^2(\theta_{2y}) \dot{\theta}_{2x}
\end{aligned}
\tag{F.14}
$$
$$
\begin{align}
C_{6\bullet}(q, \dot{q}) = {}& (0, 0, c_{63} = m_2 l_1 l_2 \sin(\theta_{2y}) \cos(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} - m_2 l_1 l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \cos(\theta_{2x}) \dot{\theta}_{1y} \nonumber \\
& + \frac{1}{2}(m_1 + m_2) l_1 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1y} + m_2 l_2 \sin(\theta_{2y}) \cos(\theta_{1y}) \sin(\theta_{1x} - \theta_{2x}) \dot{l}_1, \nonumber \\
& c_{64} = -m_2 l_1 l_2 \sin(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{1y}) \dot{\theta}_{1x} + \frac{m_1+m_2}{2} l_1 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1x} \nonumber \\
& + m_1 l_1 l_2 \sin(\theta_{1x}) \cos(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1y} - m_2 l_1 l_2 \sin(\theta_{1y}) \cos(\theta_{2y}) \dot{\theta}_{1y} \nonumber \\
& + m_2 l_1 l_2 \cos(\theta_{1x}) \cos(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1y} + \frac{m_1-m_2}{2} l_1 l_2 \sin(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{2x}) \sin(\theta_{1x}) \dot{\theta}_{2x} \nonumber \\
& + m_2 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{l}_1 + m_2 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \dot{l}_1 \nonumber \\
& + \frac{m_1+m_2}{2} l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{l}_1 + \frac{m_1-m_2}{2} l_1 \sin(\theta_{1x}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{2y}) \dot{l}_2, \nonumber \\
& c_{65} = \frac{m_1-m_2}{2} l_1 l_2 \sin(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{2x}) \sin(\theta_{1x}) \dot{\theta}_{1y} + m_2 l_2^2 \cos(\theta_{2y}) \sin(\theta_{2y}) \dot{\theta}_{2x}, \nonumber \\
& c_{66} = m_2 l_2 \dot{l}_2, \quad c_{67} = m_2 l_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} + m_2 l_2 \cos(\theta_{2y}) \cos(\theta_{1y}) \dot{\theta}_{1y} \nonumber \\
& + m_2 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1y} + \frac{m_1+m_2}{2} l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{\theta}_{1y} \nonumber \\
& c_{67} = m_2 l_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} + m_2 l_2 \cos(\theta_{2y}) \cos(\theta_{1y}) \dot{\theta}_{1y} \nonumber \\
& + m_2 l_2 \cos(\theta_{1x}) \sin(\theta_{1y}) \cos(\theta_{2x}) \sin(\theta_{2y}) \dot{\theta}_{1y} + \frac{m_1+m_2}{2} l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{\theta}_{1y}, \nonumber \\
& c_{68} = m_2 l_2 \dot{\theta}_{2y} + \frac{m_1-m_2}{2} l_1 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{\theta}_{1y} \tag{F.15}
\end{align}
$$

$$
\begin{align}
C_{7\bullet}(q, \dot{q}) = {}& (0, 0, c_{73} = -\frac{1}{2}(m_1 + m_2) \cos^2(\theta_{1y}) \dot{\theta}_{1x}, c_{74} = -(m_1 + m_2) l_1 \dot{\theta}_{1y} \nonumber \\
& - \frac{m_1 - m_2}{2} l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{\theta}_{2y}, \nonumber \\
& c_{75} = -m_2 l_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2x} - m_2 l_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2y} \nonumber \\
& + m_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{1x} - \theta_{2y}) \dot{l}_2, \quad c_{76} = \frac{m_2 - m_1}{2} l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x}) \sin(\theta_{2x}) \dot{\theta}_{1y} \nonumber \\
& - m_2 l_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2x} - m_2 l_2 \sin(\theta_{2y}) \sin(\theta_{1y}) \dot{\theta}_{2y} \nonumber \\
& - m_2 l_2 \cos(\theta_{2y}) \cos(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2y} + m_2 \cos(\theta_{2y}) \sin(\theta_{1y}) \dot{l}_2 \nonumber \\
& - m_2 \sin(\theta_{2y}) \cos(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x}) \dot{l}_2, \quad c_{77} = 0, c_{78} = m_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2x} \nonumber \\
& - m_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{2y} \tag{F.16}
\end{align}
$$

$$
\begin{align}
C_{8\bullet}(q, \dot{q}) = {}& (0, 0, c_{83} = -m_2 l_1 \cos(\theta_{2y}) \cos(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} - m_2 l_1 \cos(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{2x} - \theta_{1x}) \dot{\theta}_{1y} \nonumber \\
& + m_2 \cos(\theta_{2y}) \cos(\theta_{1y}) \sin(\theta_{2x} - \theta_{1x}) \dot{l}_1, \quad c_{84} = m_2 l_1 \cos(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} \nonumber \\
& - m_2 l_1 \sin(\theta_{1y}) \sin(\theta_{2y}) \dot{\theta}_{1y} - m_2 l_1 \cos(\theta_{2y}) \cos(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1y} + \frac{m_2 - m_1}{2} l_1 \sin(\theta_{1x}) \sin(\theta_{1y}) \nonumber \\
& \sin(\theta_{2x}) \sin(\theta_{1y}) \dot{\theta}_{2y} + m_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \dot{l}_1 - m_2 \cos(\theta_{2y}) \sin(\theta_{1y}) \cos(\theta_{1x} - \theta_{2x}) \dot{l}_1, \nonumber \\
& c_{85} = m_2 l_2 \cos^2(\theta_{2y}) \dot{\theta}_{2x}, \quad c_{86} = \frac{m_2 - m_1}{2} l_1 \sin(\theta_{2y}) \sin(\theta_{1y}) \sin(\theta_{2x}) \sin(\theta_{1x}) \dot{\theta}_{1y} - m_2 l_2 \dot{\theta}_{2y}, \nonumber \\
& c_{87} = -m_2 \cos(\theta_{1y}) \cos(\theta_{2y}) \sin(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1x} + m_2 \cos(\theta_{1y}) \sin(\theta_{2y}) \dot{\theta}_{1y} \nonumber \\
& - m_2 \cos(\theta_{2y}) \sin(\theta_{2y}) \cos(\theta_{1x} - \theta_{2x}) \dot{\theta}_{1y}, \quad c_{88} = 0 \tag{F.17}
\end{align}
$$

The gravity potential energy is given by:

$$
U_g(q) = -m_1 g l_1 \cos(\theta_{1x}) \cos(\theta_{1y}) - m_2 g [l_1 \cos(\theta_{1x}) \cos(\theta_{1y}) + l_2 \cos(\theta_{2x}) \cos(\theta_{2y})], \quad (F.18)
$$
so that the gravity generalized force is equal to:

$$ -\frac{\partial U_g}{\partial q} = \begin{pmatrix} 0 \\ 0 \\ -(m_1 + m_2)gl_1 \sin(\theta_{1x}) \cos(\theta_{1y}) \\ -(m_1 + m_2)gl_1 \cos(\theta_{1x}) \sin(\theta_{1y}) \\ m_2gl_2 \sin(\theta_{2x}) \cos(\theta_{2y}) \\ m_2gl_2 \cos(\theta_{2x}) \sin(\theta_{2y}) \\ (m_1 + m_2)g \cos(\theta_{1x}) \cos(\theta_{1y}) \\ m_2g \cos(\theta_{2x}) \cos(\theta_{2y}) \end{pmatrix} \qquad (F.19) $$

The vector of torque inputs is $Q = (F_x, F_y, 0, 0, 0, 0, F_{l_1}, F_{l_2})^\top$, where $F_{l_1}$ corresponds to a winch mechanism at the attachment joint between link 1 and the cart, while $F_{l_2}$ corresponds to a winch mechanism mounted in joint with mass $m_1$ (see Fig. 3 (b)). As alluded to in Sec. 2.2, an even more complete model incorporates the 3D dynamics of the payload (considered as a rigid body). This adds three orientation angles (Euler or else), yielding an 11-degree-of-freedom system. The analytical calculations of the matrix $C(q, \dot{q})$ using Christoffel's symbols thus involve $11^3 \times 3 = 3993$ partial derivatives $\frac{\partial m_{ij}}{\partial q_k}$. Certainly the use of a formal calculus tool becomes mandatory at this stage.

## Appendix G. Some technical results for control

The above calculations are not useful for numerical simulation sake, since available multibody toolboxes allow the construction of such dynamical systems automatically. However, they may be necessary to extend the stability analyses relying on zero-state detectability, LaSalle's invariance principle, in order to extend the results obtained for $N = 1$ (in 2D and 3D) and $N = 2$ (in 2D), see Sec. 3.3.4. Consider Appendix A. Let us further split the matrix $M_{x\theta l}(q) = \begin{pmatrix} M_{xl}(q) \\ M_{\theta l}(q) \end{pmatrix}$, so that:

$$ M(q) = \begin{pmatrix} M_{xx} & \bar{M}_{x\theta} & M_{xl} \\ \bar{M}_{x\theta}^\top & M_{\theta\theta} & M_{\theta l} \\ M_{xl}^\top & M_{\theta l}^\top & M_{ll} \end{pmatrix} \qquad (G.1) $$

Item 4 in Lemma 1 allows us to perform Spong's transformation for actuated and unactuated coordinates [206]. Indeed $\ddot{\theta} = M_{\theta\theta}^{-1}(-\bar{M}_{x\theta}^\top\ddot{x} - M_{\theta l}\ddot{l} - NL(q, \dot{q}))$, where $NL(q, \dot{q})$ stands for generic nonlinearities. Hence:

$$ \underbrace{(M_{xx} - \bar{M}_{x\theta} M_{\theta\theta}^{-1} \bar{M}_{x\theta}^\top)}_{\triangleq \mathcal{M}_x} \ddot{x} + \underbrace{(M_{ll} - \bar{M}_{x\theta} M_{\theta\theta}^{-1} M_{\theta l}) \ddot{l}}_{\triangleq \mathcal{M}_{xl}} - \bar{M}_{x\theta} M_{\theta\theta}^{-1} NL(q, \dot{q}) + NL(q, \dot{q}) = F_x \qquad (G.2) $$

$$ (M_{xl}^\top - M_{\theta l}^\top M_{\theta\theta}^{-1} \bar{M}_{x\theta}^\top) \ddot{x} + \underbrace{(M_{ll} - M_{\theta l}^\top M_{\theta\theta}^{-1} M_{\theta l}) \ddot{l}}_{\triangleq \mathcal{M}_l} - M_{\theta l}^\top M_{\theta\theta}^{-1} NL(q, \dot{q}) + NL(q, \dot{q}) = F_l \qquad (G.3) $$

where $\mathcal{M}_x \succ 0$ and $\mathcal{M}_l \succ 0$ from the Schur complement Theorem (but, in view of Lemma 1, $\mathcal{M}_l$ becomes singular at the vertical posture and if the payload is much heavier than the cable and the hook). Rearranging the matrix in (G.1) and applying again the Schur complement Theorem, it follows that:

**Lemma 2.** Let us consider (G.2) (G.3), then:

$$ \mathcal{M}(q) \triangleq \begin{pmatrix} \mathcal{M}_x(q) & \mathcal{M}_{xl}(q) \\ \mathcal{M}_{xl}^\top(q) & \mathcal{M}_l(q) \end{pmatrix} = \mathcal{M}^\top(q) \succ 0 \qquad (G.4) $$
<table><caption>Table G.16: Main parameters of the HOISTING TOOLBOX</caption><thead><tr><th>Variable</th><th>Value</th></tr></thead><tbody><tr><td colspan="2">file name main_run.m: running the main simulation and updating the results</td></tr><tr><td>Method</td><td>The default value is "all" meaning that the simulations will be done for all controllers. Alternatively, by selecting an integer value for this variable it is possible to conduct the simulation for a single controller. The integer value corresponds to the row of Table 13</td></tr><tr><td>model_select</td><td>The default value is "rigid" meaning that the lumped mass model introduced in Sec. 5 will be used. Alternatively, it is possible to replace this value with "flexible" to use the built-in MATLAB flexible beam model for the cable (this should be used with caution).</td></tr><tr><td colspan="2">file name parameters.m: used to modify the parameters of the simulation</td></tr><tr><td></td><td>traj_type: used to select a trajectory, load_type: used to select the payload type, matched_disurbance_flag and load_disturbance_flag: select disturbances, initial_sway_type: select initial sway SNR_value: SNR of feedback noise</td></tr></tbody></table>

Consequently the system in (G.2) (G.3) with state $(x, \dot{x}, l^\top, \dot{l}^\top)^\top$ and input $(F_x, F_l)^\top$ is controllable (if some basic conditions hold to guarantee $M_{ll} > 0$), linearizable by state feedback. Then the crucial point concerns the integrability properties of the system of unactuated coordinates [206]:

$$ \ddot{\theta} = M_{\theta\theta}^{-1} (-\bar{M}_{x\theta}^{\top} \ddot{x} - M_{\theta l} \ddot{l} - NL(q, \dot{q})) \quad (G.5) $$

Usually such dynamics are nonintegrable and thus can be interpreted as second-order nonholonomic constraints, which do not reduce the state-space dimension [206]. From Lemma 1 the term $M_{\theta l}(q)$ vanishes at the vertical cable's posture and is proportional to $\theta_i$ and sums of angles in a neighborhood of it. Thus around the vertical posture only $\bar{M}_{x\theta}^{\top} \ddot{x}$ remains available as a control input to this subdynamics, and some control action may also exist through $NL(q, \dot{q})$. Said otherwise:

**Proposition 2.** Let $q = (x, \theta_1, \dots, \theta_N, l_1, \dots, l_N)^\top$. The only way to control the $\theta$ dynamics (G.5) in the neighborhood of the cable's vertical posture, is through $\ddot{x}$ and/or some nonlinear velocity couplings.

It is not difficult to extend these results to the case where only $l_1$ varies, and also to the 3D double-pendulum in Appendix F. It is noteworthy that the nonlinear forces in (D.4) could be interesting for control purposes, by choosing suitable $\frac{\partial m_1}{\partial l_1}$ when a winch is mounted on a fixed reference frame outside the OC's moving structure. To the best of the authors' knowledge, this is an open issue.

## Appendix H. Hoisting Toolbox

A set of scripts and files have been developed in this work to conduct all the necessary numerical simulations and optimizations<fnref n="2" />. The controllers listed in Table 13 are implemented in the toolbox and a

<footnote n="2">²A MATLAB installation is required in order to use the toolbox. The toolbox is available on https://github.com/Mojallizadeh/HoistingToolbox under GPL license, and therefore it is possible to modify it to get customized results. All the results presented in this paper are easily reproducible thanks to this toolbox.</footnote>
set of optimization algorithms have been used in the toolbox in order to optimize the parameters of the controllers. Moreover, it can generate and compare the results using plots and tables compatible with LATEX. The toolbox contains many functions and scripts and files. But, for a normal usages, user only needs to consider two script files named main_run.m and parameters.m and modify it according to Table G.16. In addition, other parameters can be modified in the toolbox to get customized results as explained in detail in the corresponding files.

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