# Segmented Hybrid Impedance Control for Hyper-Redundant Space Manipulators
## 4 System Control
### Hybrid Impedance Control Principle for Robotic Manipulators  
The force/position hybrid control method aims to separate subspaces, allowing simultaneous tracking of the intended trajectories and the desired contact force magnitude. However, its force control aspect typically employs conventional PI control, which can introduce considerable stiffness to the robotic arm. This method relies on error signals in the force or position to generate control inputs, striving to reach the intended values. In environments with high uncertainties where suitable reference trajectories and force signals are unavailable, employing a force/position hybrid controller may still lead to substantial undesired contact forces. Impedance control offers the robotic arm behavior akin to a mass-spring-damper system, adapting interaction forces when uncertainties in contact point positions and environmental characteristics arise. However, this control scheme relies on the dynamic correlation between force and position to regulate the contact forces and lacks the ability to concurrently achieve both trajectory and contact force tracking. In response to this limitation, a proposed solution in the existing literature introduces a hybrid impedance control method that merges impedance control with force/position hybrid control [21; 22]. This method retains the subdivision of subspaces, maintaining trajectory tracking within the position control subspace. The robotic arm's trajectory is updated using the impedance model. Within the force control subspace, the desired inertia and damping are introduced, while setting the desired stiffness to zero ensures precise force control. The expected motion equation along the force control subspace can be written as follows:  
\[\mathrm{M_{d}\dot{X}+B_{d}\dot{X}-F_{d}}=-F_{ext}, \tag{17}\]  
where \(\mathrm{M_{d}}\) and \(\mathrm{B_{d}}\) are the expected inertia and damping parameter matrices, both of which are positive definite matrices; \(\mathrm{F_{d}}\) is the expected contact force between the robotic arm and the external object; \(\mathrm{F_{ext}}\) is the actual contact forces; and \(\dot{X}\) and \(\dot{X}\) are the acceleration and velocity trajectory vectors of the robotic arm. By adjusting the desired inertia and damping parameters, the system's dynamic response can be modified. Additionally, because this model does not introduce stiffness parameters, it will not affect the desired position trajectory, allowing the system to complete trajectory tracking. When the robotic arm is not in contact with external objects, and when the actual contact force \(\mathrm{F_{e}}=0\), the reference acceleration trajectory can be obtained as follows:  
\[\ddot{X}=\mathrm{M_{d}^{-1}}\left(-B_{d}\dot{X}+F_{d}\right), \tag{18}\]  
At the same time, the external contact forces can be expressed in a form related to the contact material. \[\mathrm{F_{ext}}=\mathrm{M_{e}\ddot{X}+B_{e}\dot{X}+K_{e}X}, \tag{19}\]  
where \(\mathrm{M_{e}}\), \(\mathrm{B_{e}}\) and \(\mathrm{K_{e}}\) are the actual inertia, damping and stiffness parameter matrices of the contact system, all of which are positive definite matrices. Combining (15) and (17), we can have the following:  
\[\mathrm{F_{d}}=(\mathrm{M_{d}}+\mathrm{M_{e}})\ddot{X}+(B_{d}+B_{e})\dot{X}+K_ {e}X, \tag{20}\]  
From the above equation, it can be seen that impedance control in the force control direction can be achieved by modifying the desired inertia parameters \(\mathrm{M_{d}}\) and damping parameters \(\mathrm{B_{d}}\) to adjust the response.