I. In the above article [1], there is a typing error in equation (1), which has to be
$$\begin{aligned} w=\begin{bmatrix} x \\ y\\ \varphi \end{bmatrix} =\begin{bmatrix} L~C_{\theta _{q\!-\!n}}+L~C_{{(\theta _{q\!-\!n}+\theta _{q\!-\!n\!+\!1)}}}+\dots +L C_{{\sum _{i=q\!-\!n}^{q+m} \theta _{i}}} \\ L~S_{\theta _{q-n}}+L~S_{{(\theta _{q-n}+\theta _{q-n+1})}}+\dots +L\! S_{{\sum _{i=q-n}^{q+m} \theta _{i}}}\\ \theta _{q-n}+\theta _{q-n+1}+\dots +\theta _{q+m} \end{bmatrix} \end{aligned}$$
where \(S_a=\sin (a)\) and \(C_a=\cos (a)\).
This also applies to the equation format in the Appendix A, where the following terms have to be defined as
$$\begin{aligned}{} & {} J11=-L\left\{\sin \theta _{q\!-\!n}+~\sin (\theta _{q\!-\!n}+\theta _{q\!-\!n\!+\!1})+\dots + \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\}\\{} & {} J1n=-L\left\{ \sin {\sum _{i=q\!-\!n}^{q-1}} \theta _{i}+ \sin {\sum _{i=q\!-\!n}^{q}} \theta _{i}+\dots + \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} J1(n+1)=-L\left\{ \sin {\sum _{i=q\!-\!n}^{q+1}} \theta _{i}+ \sin {\sum _{i=q\!-\!n}^{q+2}} \theta _{i}+\dots + \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} J1(m+n)=-L \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i} \end{aligned}$$
and
$$\begin{aligned}{} & {} J21=L\left\{\cos \theta _{q\!-\!n}+~\cos (\theta _{q\!-\!n}+\theta _{q\!-\!n\!+\!1})+\dots + \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} J2n=L\left\{ \cos {\sum _{i=q\!-\!n}^{q-1}} \theta _{i}+ \cos {\sum _{i=q\!-\!n}^{q}} \theta _{i}+\dots + \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} J2(n+1)=L\left\{ \cos {\sum _{i=q\!-\!n}^{q+1}} \theta _{i}+ \cos {\sum _{i=q\!-\!n}^{q+2}} \theta _{i}+\dots + \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} J2(m+n)=L \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}. \end{aligned}$$
Similarly, in Appendix B
$$\begin{aligned}{} & {} {\mathfrak {J}}11=-L\left\{\sin \theta _{q\!-\!n}+~\sin (\theta _{q\!-\!n}+\theta _{q\!-\!n\!+\!1})+\dots + \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} {\mathfrak {J}}1n=-L\left\{ \sin {\sum _{i=q\!-\!n}^{q-1}} \theta _{i}+ \sin {\sum _{i=q\!-\!n}^{q}} \theta _{i}+\dots + \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} {\mathfrak {J}}1(n+1)=-L\left\{ \sin {\sum _{i=q\!-\!n}^{q}} \theta _{i}+ \sin {\sum _{i=q\!-\!n}^{q+1}} \theta _{i}+\dots + \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} {\mathfrak {J}}1(m+n)=-L \sin {\sum _{i=q\!-\!n}^{q+m}} \theta _{i} \end{aligned}$$
and
$$\begin{aligned}{} & {} {\mathfrak {J}}21=L\left\{\cos \theta _{q\!-\!n}+~\cos (\theta _{q\!-\!n}+\theta _{q\!-\!n\!+\!1})+\dots + \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} {\mathfrak {J}}2n=L\left\{ \cos {\sum _{i=q\!-\!n}^{q-1}} \theta _{i}+ \cos {\sum _{i=q\!-\!n}^{q}} \theta _{i}+\dots + \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} {\mathfrak {J}}2(n+1)=L\left\{ \cos {\sum _{i=q\!-\!n}^{q}} \theta _{i}+ \cos {\sum _{i=q\!-\!n}^{q+1}} \theta _{i}+\dots + \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i}\right\} \\{} & {} {\mathfrak {J}}2(m+n)=L \cos {\sum _{i=q\!-\!n}^{q+m}} \theta _{i} \end{aligned}$$
II. Equation (11) is to be defined in the form.
$$\begin{aligned} u=J_{\lambda }^{\dagger } [\dot{\mu _d}-K(w-w_d)]+ (I- J_{\lambda }^{\dagger }J)k. \end{aligned}$$
Including \(\mu \) and \(\mu \) is not needed as they contradict with the definition of \(\dot{\mu _d}\). Notice that the final form of equation (11) does not change where
$$\begin{aligned} u=J_{\lambda }^{\dagger } [\dot{w_d}-{\bar{J}} \dot{\theta _q}-K(w-w_d)]+ (I- J_{\lambda }^{\dagger }J)k. \end{aligned}$$
The above form is the finally applied control law that insures convergence of the recovery part’s tip-link.
