# Segmented Hybrid Impedance Control for Hyper-Redundant Space Manipulators
## 5 Object Manipulation Control
### Object Position Control
The potential force magnitude is defined as follows:  
\[\mathrm{h_{position_{x}}}=\mathrm{k_{p1_{x}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2 _{x}}}\mathrm{d_{x}}}-1\Big{)}, \tag{40}\]  
\[\mathrm{h_{position_{y}}}=\mathrm{k_{p1_{y}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2_{ y}}}\mathrm{d_{y}}}-1\Big{)}, \tag{41}\]  
\[\mathrm{h_{position_{z}}}=\mathrm{k_{p1_{z}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2_{ z}}}\mathrm{d_{x}}}-1\Big{)}, \tag{42}\]  
where \(\mathrm{k_{p1_{x}}}\), \(\mathrm{k_{p2_{x}}}\), \(\mathrm{k_{p1_{y}}}\), \(\mathrm{k_{p2_{y}}}\), \(\mathrm{k_{p1_{z}}}\) and \(\mathrm{k_{p2_{z}}}\) are control gains that regulate the magnitude and responsiveness of the stabilization force, and ddd represents the distance between the desired and actual positions in each spatial dimension (X, Y, and Z). The control strategy also includes a spring-mass-damping model for the forces applied in each spatial dimension. This approach simulates spring-like behavior to stabilize the system around the desired position while dampening the oscillations through a damping coefficient. For each dimension, the control system computes a force based on the positional deviation and velocity difference, using both proportional control and spring-damping mechanics. Equations (40)-(42) then become as follows:  
\[\mathrm{h_{position_{x}}} =\mathrm{k_{pl_{x}}}\left(\mathrm{e^{-k_{p2_{x}}d_{x}}-1}\right)+ \mathrm{k_{s}.d_{x}}+\mathrm{c_{d}.v_{x}}, \tag{43}\] \[\mathrm{h_{position_{y}}} =\mathrm{k_{pl_{y}}}\left(\mathrm{e^{-k_{p2_{y}}d_{y}}-1}\right)+ \mathrm{k_{s}.d_{y}}+\mathrm{c_{d}.v_{y}},\] (44) \[\mathrm{h_{position_{z}}} =\mathrm{k_{pl_{x}}}\left(\mathrm{e^{-k_{p2_{x}}d_{z}}-1}\right)+ \mathrm{k_{s}.d_{z}}+\mathrm{c_{d}.v_{z}}, \tag{45}\]  
where \(\mathrm{v_{x}}\), \(\mathrm{v_{y}}\) and \(\mathrm{v_{z}}\) are velocity errors in the X, Y and Z directions. \(\mathrm{k_{s}}\) the stiffness of the spring and \(\mathrm{c_{d}}\) is the damping coefficient. The forces vectors can be computed as follows:  
\[\mathrm{f_{position_{x}}} =\frac{\mathrm{h_{position_{x}}}}{\mathrm{d_{x}}}\left(\mathrm{p_ {b_{x}}}-\mathrm{P_{b,j_{x}}}\right), \tag{46}\] \[\mathrm{f_{position_{y}}} =\frac{\mathrm{h_{position_{y}}}}{\mathrm{d_{y}}}\left(\mathrm{p_ {b_{y}}}-\mathrm{P_{b,j_{y}}}\right),\] (47) \[\mathrm{f_{position_{z}}} =\frac{\mathrm{h_{position_{z}}}}{\mathrm{d_{z}}}\left(\mathrm{p_ {b_{z}}}-\mathrm{P_{b,i_{z}}}\right), \tag{48}\]  
The torque exerted by the force is given by:  
\[\mathrm{\tau_{position_{x}}} =\mathrm{M(q)}\mathrm{J_{i}^{T}(q)}\mathrm{f_{position_{x}}}, \tag{49}\] \[\mathrm{h_{position_{y}}}=\mathrm{k_{p1_{y}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2_{ y}}}\mathrm{d_{y}}}-1\Big{)}, \tag{41}\]0 (50) \[\mathrm{h_{position_{y}}}=\mathrm{k_{p1_{y}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2_{ y}}}\mathrm{d_{y}}}-1\Big{)}, \tag{41}\]1  
where \(\mathrm{J_{i}^{T}(q)}\) is the transpose of the Jacobian matrix forming the first joint to \(\mathrm{C_{j}}\). To maintain system stability, the control strategy uses projection matrices to decompose the applied torques into distinct subspaces. The torques along the X- and Y-axes are projected into their respective subspaces to manage the directional stability, while the torque along the Z-axis is handled separately in a dedicated subspace. This approach ensures precise control of each spatial component, reducing the cross-axis interference and enhancing the overall system robustness. The projection matrices can be written as follows:  
\[\mathrm{h_{position_{y}}}=\mathrm{k_{p1_{y}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2_{ y}}}\mathrm{d_{y}}}-1\Big{)}, \tag{41}\]2 \[\mathrm{h_{position_{y}}}=\mathrm{k_{p1_{y}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2_{ y}}}\mathrm{d_{y}}}-1\Big{)}, \tag{41}\]3 (53) \[\mathrm{h_{position_{y}}}=\mathrm{k_{p1_{y}}}\Big{(}\mathrm{e}^{-\mathrm{k_{p2_{ y}}}\mathrm{d_{y}}}-1\Big{)}, \tag{41}\]4  
where \(A_{x}\), \(A_{y}\) and \(A_{z}\) are Pfaffian constraints matrices in the joint space for the X, Y and Z components.