# Compliant Detachment of Wall-Climbing Robot Unaffected by Adhesion State
## 2 Strategy of Compliant Detachment
### Online Impedance Adaptation Controller  
A simple proportional-derivative control can realize the adjustment of the peeling angle, but it lacks online impedance modulation, leading to a lack of adaptability to the adhesion and non-adhesion states. In addition, due to the lack of a bias term, this control method has poor trajectory tracking performance on non-adhesion state. To address this problem, an online impedance adaptation controller [24] is applied to the variable stiffness control to improve motion stability. The overall control strategy is shown in Figure 2b. The robot's target foot trajectory \(p_{d}(t)\) is shown in Figure 2b. The whole trajectory is divided into three stages, namely detachment (red trajectory), attachment (green trajectory), and stretching (blue trajectory). The foot trajectory of an adhesive wall-climbing robot has been studied in References [13; 25]. However, these foot trajectory plannings are both aimed at the anisotropic adhesive materials, and the peeling angle is small in the detachment stage, so they cannot be applied to the isotropic adhesive materials. According to Equation (1), \(\theta\) directly affects adhesion strength. Thus, peeling angle \(\theta\) can be initially set to a large  
Figure 2: (**a**) Force analysis on the foot, the peeling is driven by \(F_{ad}\) and \(F_{limb}\). (**b**) Motion control strategy diagram. Taking a 3-DOF limb, for example: The expected foot end coordinate of the single limb \(p_{d}\) is given by the trajectory. The expected angle \(q_{d}\) and angular velocity \(q_{d}\) of the joints are obtained through the inverse kinematic of the single limb, and the feedback of the current angle \(q\) and angular velocity \(\dot{q}\) is obtained. The error \(e\) is input to online impedance adaptive controller, and the joints torque are output to control the single limb. The origin position (the angles of joints 1–3 are set to 0) of the foot end is set as the coordinate original point (0, 0, 0). value so that the robot can enter the detachment state as soon as possible. In the middle stage, to accelerate detachment, the change rate of the foot on Z direction can be increased to decrease peeling angle \(\theta\). There is still a constraint relationship between the climbing surface and the adhesion foot, and, thus, the adhesion force does not change rapidly. In the last stage of detachment, to reduce the influence of the detachment force on robot stability, peeling angle \(\theta\) is increased. A semi-elliptic trajectory is a convenient scheme for the attachment stage. It allows the middle section to preload and the last section to stably stretch, making adhesion reliable. This trajectory can be expressed by a trigonometric function in the X and Z directions. The corresponding velocity and acceleration are continuous. To maintain fuselage stability during climbing, a straight-line function is used in the stretching stage. The mathematical expressions of the detachment, attachment, and stretching trajectories are, respectively, given by:  
\[ \begin{split} z=a_{mid}H+k_{mid}x+\frac{4(1-2a_{mid})H}{L^{2}}x^ {2}+\frac{2(-4k_{mid}L+5H)}{L^{3}}x^{3}\\ -\frac{8(1-2a_{mid})H}{L^{4}}x^{4}+\frac{8(2k_{mid}L-3H)}{L^{5}} x^{5}(x\in[-\frac{L}{2},\frac{L}{2}],y=-l_{1})\end{split} , \tag{3}\]  
\[\frac{(x-\frac{L}{2})^{2}}{L^{\prime 2}}+\frac{(z-\frac{H}{2})^{2}}{\frac{H^{2}}{2} }=1(x\in[\frac{L}{2},\frac{L}{2}+L^{\prime}],y=-l_{1}), \tag{4}\]  
\[z=0(x\in[-\frac{L}{2},\frac{L}{2}],y=-l_{1}), \tag{5}\]  
where \(x,y,z\) are the three dimensional coordinates of the foot with respect to the base coordinates of the shoulder, \(L\) and \(H\) are the length and height of a single step, \(a_{mid}H\) and \(k_{mid}\) are the height and slope of the constraint point of detachment trajectory, \(l_{1}\) is the length of the robot's thigh, and \(L^{\prime}\) is the extra forward distance in semi-elliptic trajectory. From the designed trajectory, the desired joint angle and angular velocity are obtained via inverse kinematics.