# Research on Human-Robot Collaboration Method for Parallel Robots Oriented to Segment Docking
## 3 Control Design
### Robust Motion Control  
In admittance control, velocity and acceleration in motion planning are typically time-varying functions. To achieve motion planning generated by fractional-order variable damping admittance control on the Stewart platform, inverse dynamic control must be utilized. The inertia and Coriolis terms in the dynamic model of the Stewart platform remain uncertain due to the presence of unknown interaction forces. Therefore, the robustness of the inverse dynamics control is necessary to ensure that the Stewart platform maintains an accurate tracking performance for the supple trajectories generated by the fractional-order conductor control, even in the presence of unknown interaction forces, and to satisfy the stringent requirements for segmental docking accuracy. The expression of Lagrangian dynamics equations emerges when an external force, \(h_{e}\), acts upon the end of the Stewart parallel platform:  
\[M(I)\ddot{I}+C(I,\dot{I})\dot{I}+n(I,\dot{I})=u_{c}-J^{T}(I)h_{e} \tag{17}\]  
The length \(I\) of each branch motorized cylinder is chosen as a generalized coordinate, \(M(I)\) is the inertia matrix of the Stewart parallel robot, \(C(I,I)\) is the Coriolis force term, and \(n(I,\dot{I})\) is an n-dimensional vector dependent on the state of the system (including nonlinear coupling terms such as the centripetal force, friction, and gravity), while \(u_{c}\) is a control input term, which is modeled as a nonlinear multivariate control system. The following control law can be obtained after compensating for the nonlinearity:  
\[u_{c}=\hat{M}(I)y+\hat{C}(I,\dot{I})\dot{I}+\hat{n}(I,\dot{I})+J^{T}(I)h_{e} \tag{18}\]  
where \(\hat{M}(I)\), \(\hat{C}(I,\dot{I})\) and \(\hat{n}(I,\dot{I})\) are the estimated values of \(M(I)\), \(C(I,\dot{I})\) and \(n(I,\dot{I})\), respectively, and \(y\) is the control input to be designed for the desired motion planning. A common approach is to choose the following input, as described in [25]:  
\[y=\bar{J}(I)(\widetilde{\dot{x}}+\mathbf{K}_{D_{x}}\dot{\widetilde{x}}+\mathbf{K}_{P_{ \widetilde{x}}}\widetilde{x}-\dot{J}(I,\dot{I})\dot{I}) \tag{19}\]  
where \(\mathbf{K}_{D_{x}}\) and \(\mathbf{K}_{P_{\widetilde{x}}}\) are both positive definite diagonal matrices, \(\bar{\bar{J}}(I)\) is the generalized inverse matrix of the Jacobi matrix, and \(\widetilde{\dot{x}}=x_{c}-x\) denotes the position error. Substituting(19) into (18), and then substituting (18) into (17), the closed-loop dynamics model is obtained as follows:  
\[\ddot{I}=\mathbf{y}-[\mathbf{I}-\mathbf{M}^{-1}(I)\hat{\mathbf{M}}(I)]\mathbf{y}+\mathbf{M}^{-1}(I)[ \widetilde{\mathbf{C}}(I,\dot{I})\dot{I}+\widetilde{\mathbf{n}}(I,\dot{I})] \tag{20}\]  
where \(\widetilde{\mathbf{C}}(I,\dot{I})=\hat{\mathbf{C}}(I,\dot{I})-\mathbf{C}(I,\dot{I})\), \(\widetilde{\mathbf{n}}(I,\dot{I})=\hat{\mathbf{n}}(I,\dot{I})-\mathbf{n}(I,\dot{I})\). Take \(\mathbf{y}\) in Equation (19) as the control input and set the following:  
\[\mathbf{\eta}^{*}=\mathbf{J}[\mathbf{I}-\mathbf{M}^{-1}\hat{\mathbf{M}}][\mathbf{y}+\hat{\mathbf{M}}^{-1}( \widetilde{\mathbf{C}}\dot{I}+\widetilde{\mathbf{n}})]-\mathbf{J}\hat{\mathbf{M}}^{-1}( \widetilde{\mathbf{C}}\dot{I}+\widetilde{\mathbf{n}}) \tag{21}\]  
Then  
\[ \begin{array}{l}\mathbf{\eta}^{*}=[\ddot{\widetilde{\mathbf{x}}}+\mathbf{K}_{D_{x}} \dot{\widetilde{\mathbf{x}}}+\mathbf{K}_{p}\widetilde{\mathbf{x}}-\ddot{\mathbf{J}}]+\mathbf{J}[ \mathbf{I}-\mathbf{M}^{-1}\hat{\mathbf{M}}]\hat{\mathbf{M}}+(\widetilde{\mathbf{C}}\dot{\mathbf{I}}+ \widetilde{\mathbf{n}})+\mathbf{J}\ddot{\mathbf{I}}\\ =\ddot{\widetilde{\mathbf{x}}}+\mathbf{K}_{D_{x}}\dot{\widetilde{\mathbf{x}}}+\mathbf{K}_{p_{x }}\widetilde{\mathbf{x}}-\ddot{\mathbf{J}}\dot{\mathbf{I}}-\mathbf{J}\ddot{\mathbf{I}}\end{array}  \tag{22}\]  
It is clear that \(\widetilde{\mathbf{x}}\) is no longer guaranteed to converge to zero due to the presence of \(\mathbf{\eta}^{*}\). Therefore, it is necessary to study the stability of the controller under unknown external forces in depth.