# Research on Human-Robot Collaboration Method for Parallel Robots Oriented to Segment Docking
## 3 Control Design
### Robust Motion Control
To ensure the robustness of the control law, it is now considered that a control term \(\mathbf{\omega}\) is added to expression (19), and (19) becomes the following:  
\[\mathbf{y}=\mathbf{\tilde{J}}(I)(\ddot{\widetilde{\mathbf{x}}}+\mathbf{K}_{D_{x}}\dot{ \widetilde{\mathbf{x}}}+\mathbf{K}_{p_{x}}\widetilde{\mathbf{x}}-\mathbf{\tilde{J}}(I,\dot{I} )\dot{\mathbf{I}}+\mathbf{\omega}) \tag{23}\]  
To determine the exact value of \(\mathbf{\omega}\), [PERSON]'s second method was used to determine the stability of the system, the following positive definite quadratic form is selected as the Liapunov function:  
\[\mathbf{V}(\mathbf{\xi})=\mathbf{\xi}^{T}\mathbf{A}\mathbf{\xi}>0 \tag{24}\]  
where \(\mathbf{\xi}=\left[\ddot{\widetilde{\mathbf{x}}}\right]^{T}\) is the state variable of the system, \(\mathbf{A}\) is a positive definite symmetric matrix, while the derivative of the function \(\mathbf{V}\) concerning time is as follows:  
\[ \begin{array}{l}\dot{\mathbf{V}}=\dot{\mathbf{\xi}}^{T}\mathbf{A}\mathbf{\xi}+\mathbf{\xi}^{T} \mathbf{A}\dot{\mathbf{\xi}}\\ =\mathbf{\xi}^{T}(\mathbf{H}^{T}\mathbf{A}+\mathbf{A}\mathbf{H})\mathbf{\xi}+2\mathbf{\xi}^{T}\mathbf{A}\mathbf{D}( \mathbf{\eta}^{*}+\ddot{\mathbf{J}}+\ddot{\mathbf{J}}-\mathbf{\omega})\\ =-\mathbf{\xi}^{T}\mathbf{P}\mathbf{\xi}+2\mathbf{\xi}^{T}\mathbf{A}\mathbf{D}(\mathbf{\eta}^{*}+\ddot{ \mathbf{J}}+\ddot{\mathbf{J}}-\mathbf{\omega})\end{array}  \tag{25}\]  
where \(\mathbf{P}=-(\mathbf{H}^{T}\mathbf{Q}+\mathbf{Q}\mathbf{H})\) is a symmetric matrix, \(\mathbf{H}= \begin{bmatrix}\mathbf{0}&\mathbf{I}\\ -\mathbf{K}_{D_{\dot{I}}}&-\mathbf{K}_{D_{\dot{I}}}\end{bmatrix} \), \(\mathbf{D}= \begin{bmatrix}\mathbf{\theta}\\ \mathbf{I}\end{bmatrix} \). To ensure asymptotic stability, a must satisfy \(\dot{\mathbf{V}}<0\). The first term on the right-hand side of the equal sign in (25) is negatively definite, so \(\mathbf{\omega}\) is taken to ensure only that the second term on the right-hand side of the equal sign is less than or equal to zero. Set \(\mathbf{z}=2\mathbf{D}^{T}\mathbf{A}\mathbf{\xi}\) and take \(\mathbf{\omega}\) to be the following value:  
\[\mathbf{\omega}=\frac{\rho}{\|\mathbf{z}\|}\mathbf{Z} \tag{26}\]  
Due to the kinematic constraints of a robot, the following reasonable assumptions can be made:  
\[\begin{array}{l}\left\|\ddot{\mathbf{I}}\right\|\leq\beta<\infty,\forall\mathbf{I}, \dot{\mathbf{I}}\\ \left\|\ddot{\mathbf{I}}\  
The value of \(\rho\) must be satisfied:  
\[\rho>\|\eta^{*}\|+\beta+\mathbf{f} \tag{30}\]  
While the uncertainty term \(\mathbf{\eta}^{*}\) remains unknown, the introduction of the control term \(\mathbf{\omega}\) ensures that \(\dot{\mathbf{V}}\) is a negative definite. Finally, within the sliding subspace of \(\mathbf{z}=0(\mathbf{A}=0)\), the ideal theoretical scenario is for the control signal \(\mathbf{\omega}\) to operate at infinite frequency [26]. Nevertheless, in practice, this signal experiences high-frequency alternations, resulting in torque oscillations. To mitigate this high-frequency exchange (jitter), a specific range \(\varepsilon\), adjacent to the sliding subspace, is defined. Subsequently, the control law that was proposed earlier is modified based on this range as follows:  
\[\mathbf{\omega}=\left\{ \begin{array}{ll}\frac{\rho}{\|\varepsilon\|}\mathbf{Z}& \text{for}\quad\|\mathbf{z}\|\geq\varepsilon\\ \frac{\rho}{\varepsilon}\mathbf{Z}&\text{for}\quad\|\mathbf{z}\|<\varepsilon\end{array} \right. \tag{31}\]  
The fractional-order variable damping admittance control (FVDAC) and the inverse dynamics robust control presented in this section form a tandem relationship in the control system. The FVDAC acts as an upper-level decision algorithm designed to generate a compliant motion plan based on the interaction forces in human-robot collaboration. Then, the inverse dynamics robust control receives this motion plan and controls the parallel robot to execute this compliant motion plan with stability and precision.