## Appendix A  
To establish the proof of Theorem 1, it is essential to rely on two widely recognized lemmas:  
**Lemma A1** ([43]). : _Let \(\mathbf{x}(t):[t_{0},\infty)\to\mathbb{R}^{n}\) be a state vector such that \(\mathbf{x}(t)\in\mathcal{L}_{2}\cap\mathcal{L}_{\infty}\) and \(\hat{\mathbf{x}}(t)\in\mathcal{L}_{\infty}\). Then, the following holds:_  
\[\lim_{t\to\infty}\mathbf{x}(t)=\mathbf{0}. \tag{101}\]  
**Lemma A2**. : _[PERSON]'s Lemma ([44]). Consider a differentiable function \(\mathbf{x}(t)\) defined over \([t_{0},\infty)\) such that it remains bounded and converges to a finite limit as \(t\to\infty\). If \(\hat{\mathbf{x}}(t)\) is uniformly continuous, then the time derivative satisfies \(\hat{\mathbf{x}}(t)\to\mathbf{0}\) as \(t\to\infty\)._  
The proof is developed through the following steps:  
1. Given that \(\mathbf{q}_{\text{d}}\) is assumed to remain constant, the PID control laws (6)-(8), in combination with the assumed human dynamics described in (9)-(11), are mathematically equivalent to the formulation in (23)-(25) by applying the relationships in (17)-(18) and the condition given by (12). Furthermore, under the definition (29), the control law presented in (23)-(25) can also be expressed equivalently as (31)-(33), provided that the conditions in (34)-(36) are satisfied. As a result, the closed-loop system dynamics described by (43) are equivalent to those obtained by substituting the PID control laws (6)-(8) and the human dynamics in (9)-(11) directly into the system equations in (1). From this equivalence, the following conclusions can be drawn: \[\mathbf{x} =\mathbf{0} \Rightarrow \hat{\mathbf{e}}_{i}=\mathbf{0},\,\hat{\mathbf{q}}_{i}=\mathbf{0},\] (102) \[\mathbf{e}_{i} =\mathbf{0},\] (103) \[\mathbf{\tilde{\xi}}_{i} =-\mathbf{K}_{ii}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\text{d}}).\] (104) These results hold because, by applying the conditions in (34)-(36), if \(\mathbf{x}=\mathbf{0}\), then it follows from (24), (29), (30), (32)-(33), (41), and (42) that: \[\mathbf{w}_{i} = \mathbf{\Lambda}_{i}\int_{0}^{t}\bigl{(}\mathbf{e}_{i}-\mathbf{\bar{e}}_{i}+ \mathbf{A}_{i}\mathbf{e}_{i}\bigr{)}\text{d}\mathbf{t}=\mathbf{\Lambda}_{i}\mathbf{\tilde{\xi}}_{ i}=-\mathbf{K}_{wi}^{-1}\mathbf{g}_{i}(\mathbf{q}_{\text{d}}).\] (105) It is important to note that this does not represent a change of coordinates. Thus, there is no diffeomorphism between the variables \(\mathbf{\dot{e}}_{i}\), \(\mathbf{\dot{q}}_{i}\), \(\mathbf{e}_{i}\), \(\mathbf{\tilde{\xi}}_{i}\), and the state vector \(\mathbf{x}\). The equivalence established in this step, which connects the PID control laws (6)-(8) and the human dynamics (9)-(11) to the closed-loop system formulation in (23)-(25) and ultimately (43), ensures that the dynamics of the system are well-defined and consistent under the given conditions. This equivalence lays the groundwork for introducing a positive definite candidate function in the subsequent analysis, enabling the examination of the stability properties of the system. 2. To establish that \(\mathbf{x}\to\mathbf{0}\) as \(t\to\infty\), let us analyze the following positive definite function \(V_{i}(\mathbf{x})\), which, although not a Lyapunov function, is constructed with respect to the state variables of system (43): \[V_{i}(\mathbf{x})=\frac{1}{2}\mathbf{x}_{k}^{\text{T}}\mathbf{K}_{wi}\mathbf{x}_{k}+\frac{1}{ 2}\mathbf{e}_{i}^{\text{T}}\mathbf{e}_{i}+\frac{1}{2}\mathbf{s}_{i}^{\text{T}}\mathbf{H}_{i}( \mathbf{q}_{i})\mathbf{s}_{i}+(\mathbf{B}_{i}\mathbf{e}_{i}-\mathbf{\bar{e}}_{i})^{\text{T}}\mathbf{B} _{i}^{-1}\mathbf{K}_{wi2}\mathbf{\Lambda}_{i}(\mathbf{B}_{i}\mathbf{e}_{i}-\mathbf{\bar{e}}_{i}),\] (106) where \(i=1,\text{r}\) and \(k=1\) for \(i=1\) or \(k=4\) for \(i=\text{r}\). The positive definite function \(V_{i}(\mathbf{x})\) is chosen because it encapsulates the energy-like properties of the system, combining contributions from \(\mathbf{e}_{i}\), \(\mathbf{s}_{i}\), and \(\mathbf{\dot{e}}_{i}\).