# Segmented Hybrid Impedance Control for Hyper-Redundant Space Manipulators
## 4 System Control
### Impedance Control Principle for Robotic Manipulators  
The impedance control method aims to impart specific mass-stiffness-damping characteristics to the manipulator when it interacts with the external environment. Within the control methods for the manipulator, using position control reflects the system's high impedance characteristics, meaning that external force disturbances have minimal impact on the manipulator's motion variations. Conversely, employing force control reflects the system's low impedance characteristics, where changes in the force due to motion errors are minimal. However, when the manipulator links and the external objects possess significant stiffness, using only force/position hybrid control may result in substantial force tracking errors caused by minor motion variations in the force control direction. By employing impedance control, it is possible to establish a dynamic relationship between the contact force and the motion of the manipulator arm in the joint space as follows:  
\[M_{\text{etm}}\dot{u}_{\text{eA}}+C_{\text{etm}}u_{\text{eA}}+K_{\text{etm}}X _{\text{eA}}=F_{\text{etm}}^{\text{C}}, \tag{11}\]  
where \(F_{\text{etm}}^{\text{C}}\) is the contact force on the end-effector of the manipulator; and \(M_{\text{etm}}\), \(C_{\text{etm}}\) and \(K_{\text{etm}}\) are positive definite parameter matrices, corresponding to the mass, damping and stiffness characteristics of the system, respectively. In this paper, we focus on the case where they are constant matrices. At the same time, \(\dot{u}_{\text{eA}}\), \(u_{\text{eA}}\) and \(X_{\text{eA}}\) can be written as the correction quantities of the corresponding parameters, that is, \(\Delta\dot{u}_{\text{eA}}\), \(\Delta u_{\text{eA}}\) and \(\Delta X_{\text{eA}}\). The impedance control equation is equally applicable in the joint space of the robotic arm, representing the external contact force in the joint space of the robotic arm as follows:  
\[\mathrm{F_{GC,q}}=\mathrm{J_{q}^{T}F_{ext}}, \tag{12}\]  
where \(\mathrm{F_{ext}}\) is the measured contact force. \(\mathrm{J_{q}^{T}}\) is the transpose of a six-dimensional vector function, known as the Jacobian matrix. Then, the impedance control equation in the joint space can be written as follows:  
\[\mathrm{M_{A}\Delta\ddot{q}_{A}+C_{A}\Delta\dot{q}_{A}+K_{A}\Delta q_{A}=F_{GC, q}}, \tag{13}\]  
where \(\Delta\ddot{q}_{A}\), \(\Delta\dot{q}_{A}\) and \(\Delta q_{A}\) are the expected trajectory correction amounts of the manipulator joints. It is worth noting that the impedance control represented by (11) is a generalized impedance control, and its application mode can be adjusted in practical use depending on the control variables. In this research, the robotic arm engages in multi-point and multiple-contact collisions with external objects, requiring force measurement and tracking. Therefore, in this context, a position-based impedance control (admittance control) strategy is employed. This strategy provides the correction of the joint angle acceleration trajectories as follows:  
\[\Delta\ddot{q}_{A}=\mathrm{M_{A}^{-1}}\big{(}\mathrm{F_{GC,q}}-C_{A}\Delta \dot{q}_{A}-K_{A}\Delta q_{A}\big{)}, \tag{14}\]  
Based on the motion of the end-effector actuator, the desired trajectory for the joint angle acceleration of the manipulator arm can be expressed as follows:  
\[\dot{q}_{d,\theta}=\mathrm{J_{\theta}^{T}(q)}\big{(}\ddot{x}_{d}-\dot{J}_{ \theta}(q)\dot{q}\big{)}, \tag{15}\]  
The corrected desired trajectory for the manipulator arm joints is:  
\[ \begin{cases}\ddot{q}_{A}=\Delta\ddot{q}_{A}+\ddot{q}_{d,\theta 1}\\ \dot{q}_{A}=\Delta\dot{q}_{A}+\dot{q}_{d,\theta 1},\\ q_{A}=\Delta q_{A}+q_{d,\theta 1}\end{cases}  \tag{16}\]  
The structure of the position-based impedance controller (admittance controller) is shown in Figure 4. In this paper, the hyper-redundant space manipulator adopts a segmented approach in the joint space using a force/position hybrid control method.