## 4 Stability Analysis  
In this section, we will analyze the stability of the proposed control, the observer, and the system parameter adaptation scheme described in Section 3. The estimation errors of dynamic and kinematic parameters are defined as follows  
\[\overline{\theta}_{d}=\hat{\theta}_{d}-\theta_{d} \tag{15}\]  
\[\overline{\theta}_{k}=\hat{\theta}_{k}-\theta_{k} \tag{16}\]  
and the vector  
\[y=\left[ \begin{array}{cc}e_{x}^{T}&e_{v}^{T}&\overline{x}^{T}\end{array}  \right]^{T} \tag{17}\]  
Now, we can state the following theorem for the adaptive tracking control, the velocity observer, and parameter adaptation in the task space. **Theorem 1**. : _For dynamic system (1), using adaptive tracking control (11), parameter update law (12) and (13), and observer (14), the design parameters \(\alpha\), \(\lambda\); feedback gains \(k_{v}\), \(k_{p}\); and adaptation matrices \(\Gamma_{d}\), \(\Gamma_{k}\) can be chosen such that the vector \(y\) globally asymptotically converges to zero._  
**Proof. ** In order to perform stability analysis, we consider the Lyapunov function candidate  
\[V=\frac{1}{2}s^{T}M(q)s+\frac{1}{2}e_{x}^{T}\Big{(}k_{p}+\alpha k_{p}\Big{)}e_ {x}+\frac{1}{2}\overline{\theta}_{d}^{T}\Gamma_{d}^{-1}\overline{\theta}_{d}+ \frac{1}{2}\overline{\theta}_{k}^{T}\Gamma_{k}^{-1}\overline{\theta}_{k}+ \frac{1}{2}\ \overline{x}^{T}\overline{x} \tag{18}\]  
Taking the time derivative of \(V\) gives  
\[\dot{V}\ =s^{T}M(q)\dot{s}+\frac{1}{2}s^{T}\dot{M}(q)s+e_{x}^{T}\Big{(}k_{p}+ \alpha k_{v}\Big{)}\dot{e}_{x}+\overline{\theta}_{d}^{T}\Gamma_{d}^{-1}\hat{ \theta}_{d}+\overline{\theta}_{k}^{T}\Gamma_{k}^{-1}\hat{\theta}_{k}+ \overline{x}^{T}\dot{\overline{x}} \tag{19}\]  
Figure 1: Adaptive tracking control scheme. Using the definition of \(s\), we have  
\[ \begin{array}{ll}M(q)\dot{s}&=M(q)\Big{(}\ddot{q}-\ddot{q}_{r}\Big{)}\\ &=\tau-\Big{[}M(q)\ddot{q}_{r}+C\Big{(}q,\dot{q}\Big{)}\dot{q}_{r}+G(q)\Big{]}-C \Big{(}q,\dot{q}\Big{)}s\\ &=\tau-Y_{r}\theta_{d}-C\Big{(}q,\dot{q}\Big{)}s\end{array}  \tag{20}\]  
Using the control in (11) and the definition in (15), we have  
\[M(q)\dot{s}\ =-\dot{J}^{T}\Big{(}k_{v}e_{v}+k_{p}\;e_{x}\Big{)}+Y_{r}\widetilde{ \theta}_{d}-C\Big{(}q,\dot{q}\Big{)}s \tag{21}\]  
Substituting \(M(q)\dot{s}\) from (20) with Property 2, the adaptive update law (12) and (13), the observer (14) into (19), and using the definition of \(\widetilde{\theta}_{k}\) in (16) with Property 4, we have  
\[\dot{V}=-s^{T}\dot{J}^{T}\Big{(}k_{v}e_{v}+k_{p}\;e_{x}\Big{)}+e_{x}^{T}\Big{(} k_{p}+\alpha k_{p}\;e_{v}-\;\lambda\widetilde{x}^{T}\overline{x} \tag{22}\]  
From the definition (9) and (5)-(7), we can compute \(s\) as follows:  
\[s=\dot{J}^{-1}(e_{v}+\alpha\;e_{x}-\lambda\;\overline{x}\;) \tag{23}\]  
Substituting s from (23) in (22), we have  
\[ \begin{array}{ll}\dot{V}&=-e_{x}^{T}\alpha k_{p}e_{x}-e_{v}^{T}k_{v}e_{v}+e_ {x}^{T}\lambda k_{p}\overline{x}+e_{v}^{T}\lambda k_{v}\overline{x}-\;\lambda \widetilde{x}^{T}\overline{x}\\ &=-y^{T}Ay\end{array}  \tag{24}\]  
where  
\[A=\left[ \begin{array}{ccc}\alpha k_{p}&0&-\frac{\lambda k_{p}}{2}\\ 0&k_{v}&-\frac{\lambda k_{p}}{2}\\ -\frac{\lambda k_{p}}{2}&-\frac{\lambda k_{v}}{2}&\lambda\end{array} \right] \tag{25}\]  
We can obtain the following sufficient condition for stability:  
\[\alpha>\frac{\lambda k_{p}}{4-\lambda k_{p}} \tag{26}\]  
Then, it follows from (24) that  
\[\dot{V}\leq-\lambda_{m}(A)\|y\|^{2} \tag{27}\]  
Using Lemma 1, we can conclude that \(y=\left[ \begin{array}{cc}e_{x}^{T}&e_{v}^{T}&\overline{x}^{T}\end{array}  \right]^{T}\) globally asymptotically converges to zero. This completes the proof of Theorem 1. **Corollary 1**. : _The estimation error in velocity \(\dot{\overline{x}}\) asymptotically converges to zero._  
**Proof.