# Compliant Detachment of Wall-Climbing Robot Unaffected by Adhesion State
## 2 Strategy of Compliant Detachment
### Mechanism of Detachment Force Regulation  
An isotropic adhesive can repeatedly adhere to a surface without leaving a trace and can maintain a certain normal adhesion strength without tangential adhesion strength, which is beneficial for adhesion reliability. However, this can also lead to an instant change in the adhesion force when the adhesion breaks. To reduce the adhesion strength during the detachment of the adhesive, detachment is generally carried out via peeling, where a peeling force in a certain direction is applied to one end of the adhesive. The Kendall model [21], shown in Figure 1 and Equation (1), gives the steady-state relationship among peeling force \(F\), elastic modulus of the adhesive pad \(E\), thickness of the adhesive pad \(d\), width of the adhesive failure area \(b\), peeling angle \(\theta\), and peeling energy constant \(R\), which is related to the peeling speed. The influence of peeling speed on adhesion strength has been studied in Reference [23]. The adhesion strength can be regulated mainly via peeling  
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angle \(\theta\) and \(R\). Because adjusting the peeling speed would prevent rapid peeling, this study regulates adhesion strength by adjusting \(\theta\). \[\left(\frac{F}{b}\right)^{2}\frac{1}{2 dE}+\frac{F}{b}(1-\cos\theta)-R=0 \tag{1}\]  
because the elastic term is generally small, the normal and tangential adhesion forces \(F_{N}\) and \(F_{T}\), respectively, are given by:  
\[F_{N}=bR\frac{\sin\theta}{1-\cos\theta},F_{T}=bR\frac{\cos\theta}{1-\cos \theta}(\theta\in(0,\pi]). \tag{2}\]  
At the detachment end state, the detachment force instantly decreases to 0, which affects the motion state of the robot. Therefore, a smaller adhesion strength of the detachment end state is beneficial for robot motion stability. According to Equation (2), \(F_{N}\) and \(F_{T}\) are negatively correlated with peeling angle \(\theta\). When \(\theta\) is close to \(\pi\), \(F_{N}\) is close to 0, which is the ideal detachment situation. Therefore, increasing the peeling angle to near \(\pi\) can effectively reduce the probability of instability of the robot during wall climbing. The peel-zone model [20] with friction considered gives a similar conclusion with Kendall model. In a practical situation, the actual adhesion state (related to \(R\) and \(b\)) is affected by such factors as the stiffness of the adhesive feet and the size of the adhesion pre-pressure. We conducted experimental verification for this phenomenon. The experiment method and the results are shown in Appendix A. It is not difficult to find that the above factors both influence the adhesion area. In addition, under the same experimental conditions, the adhesion area will still be slightly different in multiple experiments. So, the attachment process is an unpredictable process for robots, making it difficult to find a desired peeling force or design an optimal foot trajectory online. Therefore, we regulate the peeling angle rather than the unpredictable peeling force. Most existing wall-climbing robots, such as Stickybot III [13] and AnyClimb [15; 16], realize joint motion using position control that tracks a trajectory. With this kind of trajectory tracking control (comparison method in Section 4 and the Supplementary Materials' Video S1), it is difficult to design an optimal foot trajectory for specific adhesion situations because of the lack of feedback. Because position control can output a large moment, the adhesive sheet at the detachment end state under trajectory tracking control is prone to overall adhesion failure rather than peeling, which leads to uncontrollable adhesion strength. Thus, the traditional  
Figure 1: Adhesion film peeling force analysis. trajectory tracking control algorithm is not applicable for wall climbing. This study proposes an online adaptive impedance control where the stiffness and damping are tuned online for adhesive attachment and detachment. The force exerted by the robot foot is shown in Figure 2a.