# Simulation-Guided Path Optimization for Resolving Interlocked Hook-Shaped Components
## 2 Related Works  
The majority of offline motion planners are based on either sampling or numerical optimization. Sampling-based planners [6] sample points from the configuration space of the robot to find a path between them, such as by building roadmaps [7] or incrementally growing a tree toward those sampled points [8]. In contrast, optimization-based algo  
Figure 1: Comparative example between the industrial baseline (**a**) and our framework (**b**) for the separating subtask. The industrial baseline does not account for interlocked components, often resulting in multiple components being picked at once. In contrast, our framework leverages predictions from the physics simulator to update the motion, effectively separating interlocked components. rithms reformulate motion planning as a mathematical program, using cost and constraint functions to capture desired motion properties. For high-dimensional systems, various methods have been developed to address non-linearities, including sequential convex optimization [1], covariant updates with Hamiltonian Monte Carlo [9], and graphs of convex sets [10]. In the context of resolving interlocked components, as addressed in this paper, such planners are particularly appealing because they enable reasoning about the separation process by encoding this task-specific objective into the optimization problem. However, a key limitation of optimization-based planners is their reliance on cost terms and constraint functions that are continuous, differentiable, and ideally convex. In robotic applications involving high-dimensional manipulators, these functions are often non-linear and non-convex, which can lead to local minima and increased computational time. Additionally, modeling contact-rich interactions with differentiable functions is particularly challenging, especially for inexperienced users or in bin-picking scenarios where the shape of interlocked components varies frequently. This variability is also characteristic of the problem addressed in this paper, further complicating the modeling of such contact-rich interactions. An attractive alternative is to use a class of black-box optimization algorithms that do not rely on gradient information but instead employ stochastic strategies. Their main advantage is the ability to handle non-differentiable cost terms, which simplifies the design of cost and constraint functions. [PERSON] et al. presented the STOMP planner [2], which generates noisy trajectory to update the candidate solution that is optimized in each iteration. By leveraging the GPU's ability to perform parallel computing, sampling-based Model Predictive Control (MPC) methods can achieve control rates between 10 Hz and 125 Hz within a finite receding time horizon, while not requiring differentiable costs or linearization. The state-of-the-art control algorithms in this class include Model Predictive Path Integral (MPPI) [11; 12] and Cross-Entropy Method (CEM) [13], which have demonstrated very fast and robust behavior, even in highly dynamic environments such as off-road autonomous driving. The STORM algorithm [14] utilizes Halton B-spline sampling and tensors to accelerate the computation of rollouts within MPPI, while ensuring the smoothness of sampled candidate trajectories. In the context of manipulators and sampling-based optimization, [4] presents the VP-STO algorithm that integrates the CMA-ES solver [15] with the trajectory representation based on optimal basis functions [3]. Our approach utilizes their idea of trajectory representation to simplify the optimization process but incorporates path integral [16] update rules. The path integral formulation has also been used in the reinforcement learning domain as Policy Improvement with path integrals (PI2) to update control policies [17], leading to its other variations, including [18]. The path integral approach has also been used in combination with demonstrations to train visual-motion policies [19].