# Simulation-Guided Path Optimization for Resolving Interlocked Hook-Shaped Components
## 5 Results
### Real-World Experiments  
In the real-world experiment, we compare our framework against three different demonstrations, each providing a different strategy for separating objects: _Demonstration 1_ and _Demonstration 2_ shown in Figure 6, and _Demonstration 3_ that we also used as an initial guess in our framework, shown in Figure 4. To evaluate sampled candidate paths, we process a batch of 500 candidates in parallel using 500 robots, as shown in Figure 7. Each candidate path consists of 4 \(q_{\text{via}}\) waypoints, which are interpolated to 20 evaluation waypoints. The simulation timestep was set to 0.1 s. After \(J=5\) optimization iterations, we interpolated the optimized waypoints to 50 reference waypoints for the low-level controller. The weights were selected based on empirical tuning and set to the following values: \(\alpha_{\text{smooth}}=10,\alpha_{\text{lim}}=1000,\alpha_{\text{sep}}=50\). During development, we observed that a chain of interlocked components does not behave significantly differently from a pair, as additional components typically fall along with the one closest to the grasped object. In fact, resolving a longer chain can sometimes be easier, since the added weight of the extra components can assist in the separation process. We also encountered challenges with the perception pipeline, which struggled when multiple components were clustered closely together. This degradation in perception performance introduced an unintended bias in the motion planner's behavior. To ensure a fairer comparison with minimal bias, we chose to use only two components in the comparative experiments reported in Table 1. In realistic scenarios, such as when multiple components are present on a table or in a bin, the perception pipeline often fails to detect all component poses due to occlusions caused by the top layer of the detected components. To prevent detected objects from falling in the simulation and deviating from the current state of the real-world environment, we manually added a supporting plane beneath them. \begin{table}
\begin{tabular}{c|c|c|c} \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Success Rate [\%]**} & \multicolumn{2}{c}{**Computational Time**} \\ \cline{3-4}  & & **5 Iterations [s]** & **1 Iteration [ms]** \\ \hline Our & 80 & 2.090 \(\pm\) 0.003 & 417.2 \(\pm\) 0.2 \\ \hline Demo 1 & 30 & - & - \\ \hline Demo 2 & 30 & - & - \\ \hline Demo 3 & 50 & - & - \\ \hline \end{tabular}
\end{table} 
Table 1: Statistics from 10 trials involving only two interlocked objects comparing the proposed framework against three different demonstration strategies. Figure 6: Demonstrations used for comparative experiments. Each strategy represents a different approach to separating objects with respect to the robot’s frame: (**a**) tilting the object to the right, (**b**) tilting the object to the left. The success rate was evaluated over 10 trials, where each run was considered successful only if the robot successfully separated the object while adhering to joint limits and avoiding collisions with the environment. In each trial, the objects were manually interlocked and randomly placed within the shared workspace of the scanner and the robot to specifically investigate the separating subtask. The comparison results are shown in Table 1. We do not present the computational time for demonstrations, as the computational complexity involves only a few matrix multiplications and is therefore considered negligible. The proposed framework outperforms all other demonstrations, as each of them relies on a single strategy for resolving interlocking and therefore fails to generalize across diverse interlocked object configurations. In contrast, our framework evaluates 500 candidate motions, all originating from Demo 3. The observed failure cases were specific to our method and were primarily due to object configurations that were difficult to separate even manually, or to sim-to-real discrepancies affecting the object's physical behavior.