2.1. Path planning

Pioneering collision-free path planning for robotic arms engaged in a wide spectrum of tasks, ranging from welding in industrial contexts to delicate medical applications, remains a central focal point in the realm of robotics [Reference Boscariol, Gasparetto, Scalera, Carbone and Laribi4]. path-planning methods can broadly be classified into two categories: classical and learning-based [Reference Tamizi, Yaghoubi and Najjaran5]. Classical path-planning approaches encompass a wide range of techniques, including artificial potential field [Reference Long6], bio-inspired heuristic methods [Reference Tian and Collins7], and sampling-based path planners [Reference Elbanhawi and Simic8]. In contrast, learning-based path planners primarily utilize various machine-learning techniques to plan for the robot’s path. These methods have been gaining popularity in recent years due to their ability to handle complex and dynamic environments.

Sampling-based methods are widely used in path planning. In this regard, they can be divided into two categories: single-query algorithms and multi-query algorithms. While single-query approaches aim to find a path between a single pair of initial and goal configurations, the multi-query ones try to be efficient when there are multiple pairs [Reference Lynch and Park9]. The Rapidly-exploring Random Tree (RRT) [Reference LaValle10] and Probabilistic Road Map (PRM) [Reference Kavraki and Latombe11] are two of the most well-known algorithms for single-query and multi-query approaches, respectively. PRM is a roadmap algorithm that aims to build an exhaustive roadmap of the configuration space to plan the path between any two configurations given to it. While PRM has been used for path planning of robotic arms before [Reference Rodríguez, Montaño and Suárez12, Reference Rosell, Suárez, Rosales and Pérez13], it has several limitations. Firstly, it generates paths that are far from the optimal solution. Secondly, generating the roadmap is computationally expensive [Reference Karaman and Frazzoli14], making it inefficient for path planning in relatively high-dimensional environments, even for static tasks such as bin-picking [Reference Ellekilde and Petersen15].

Furthermore, given an initial and goal configuration pair, RRT grows a tree of waypoints from the initial configuration in order to connect the two configurations. An important characteristic of RRT algorithm is its probabilistic completeness, meaning that it is guaranteed to find a path if there exists one [Reference Lynch and Park9]. Due to RRT’s capability in non-convex high-dimensional spaces, it has been extensively employed in many studies for path planning of robotic arms [Reference Rybus and Seweryn16–Reference Rybus18]. To increase the performance of RRT in different aspects, such as planning time and optimality, there are many variants of this method in the literature. In ref. [Reference Kuffner and LaValle19], the bi-directional RRT (Bi-RRT) is introduced for path planning of the 7-DOF manipulator. Moreover, in ref. [Reference Riedlinger, Tamizi, Tikekar and Redeker20] this algorithm was used for distributed picking application. This method simultaneously builds two RRTs; one of them attempts to find the path from the initial configuration to the final configuration, and the other one attempts to find the path in the opposite direction and tries to connect these two trees in each iteration. Although this method is faster than RRT, they both often produce suboptimal paths in practice and the quality of the path depends on the density of samples and the shape of the configuration space. In order to enhance the performance of the Bi-RRT method for manipulators, a pioneering study by Xu et al. [Reference Xu and Wang21] presents a novel modified version of this technique. The key innovation of their work lies in the integration of a sparse dead point saved strategy, effectively reducing the frequency of collision checks.

RRT* is the optimal version of the RRT and finds the asymptotically optimal solution if one exists [Reference Karaman and Frazzoli14]. While that is a strong feature of RRT*, it is inefficient in high-dimensional spaces and has a relatively slow convergence rate. As a result, it is not a suitable solution for the real-time path planning of a robotic arm. To overcome the problems mentioned above, biased sampling is one of the promising methods which can improve the performance of sampling-based path planners, due to the adaptive sampling of the configuration space of the robot. Batch-informed trees (BIT*) [Reference Gammell, Srinivasa and Barfoot22] and informed RRT* [Reference Gammell, Srinivasa and Barfoot23] are other significant variants of the RRT* that employ an informed search strategy and batch-processing approach to find the optimal path in a more efficient way by focusing the search on promising areas of the tree. Although this method is able to find the optimal solution, it is not suitable for real-time path planning since it requires a large amount of computational resources, which can lead to delays in the system’s response time.

As mentioned above, since collision-checking is computationally expensive and the configuration space gets exponentially bigger with the increase of dimensionality, most of the classical sampling-based methods suffer from high computation and low convergence rates, making them impractical for use in a complex environment with high-dimensional configuration space and real-time applications such as bin-picking [Reference Iversen and Ellekilde24]. Ellekilde et al. [Reference Ellekilde and Petersen15] present a new algorithm for planning an efficient path in a bin-picking scenario based on using lookup tables. Although this method is almost instantaneous and faster than sampling-based planners, it is memory inefficient.

To this end, learning-based path planners have emerged to deal with the limitations of classical methods. Learning-based techniques can solve the long run-time problem of sampling-based methods by providing biased sampling nodes, allowing them to run faster and more efficiently [Reference Cheng, Shankar and Burdick25]. For instance, Qureshi et al. [Reference Qureshi and Yip26] proposed a deep neural network-based sampler to generate nodes for sampling-based path planners. This work shows a significant improvement in the performance of sampling-based path planners in terms of planning time.

Reinforcement learning (RL) is a subcategory of learning-based techniques for path planning in an unknown environment [Reference Aleo, Arena and Patané27–Reference Li, Ma, Ding, Wang and Jin30] where the agent learns the optimal behavior through interactions with the environment and receiving reward signals. RL can be a promising solution when there is limited information regarding the environment. However, especially in the context of path planning, using RL poses several limitations. Due to the trial-and-error nature of RL, the agent needs to strike a good balance between exploration and exploitation during its learning process. Hence, it requires a vast amount of data to learn a viable policy. Therefore, in addition to being sample-inefficient, it requires powerful hardware and significant memory for processing, especially when neural networks with a large number of parameters are employed. Moreover, when using DRL to solve the path-planning problem, the long horizon nature of it poses a great challenge in having an effective learning process. In its basic form, the reward shaping used to solve these problems are sparse functions where the agent receives a positive signal only when it has reached the goal. However, the DRL algorithm using this reward shaping scheme can often exhibit instability during training [Reference Nair, McGrew, Andrychowicz, Zaremba and Abbeel31]. Furthermore, devising dense reward functions requires expert knowledge and engineering and a careful training routine since it can be susceptible to convergence to suboptimal policies [Reference Xie, Shao, Li, Guan and Tan32, Reference Peng, Yang, Li and Khyam33]. To this end, imitation learning is an alternative to these limitations. In imitation learning, the training dataset is provided by an expert during the execution of the task. For instance, in ref. [Reference Rahmatizadeh, Abolghasemi, Behal and Bölöni34], a recurrent neural network is trained by using demonstration from the virtual environment to perform a manipulation task.

Neural planners are relatively new methods in the path-planning context. These planners are based on deep neural networks, and their purpose is to solve the path-planning problem as an efficient and fast alternative to sampling-based methods. Bency et al. [Reference Bency, Qureshi and Yip35] proposed a recurrent neural network to generate an end-to-end near-optimal path iteratively in a static environment. Moreover, in refs. [Reference Qureshi, Simeonov, Bency and Yip36] and [Reference Qureshi, Miao, Simeonov and Yip37], motion planning networks (MPnet) is introduced. This method has a strong performance in unseen environments as the path can be learned directly.

2.2. Collision approximation

Collision detection is a vital component of path planning in robotics and can consume a significant amount of computation time. In fact, it has been reported that it can take up to 90 $\%$ of the total path-planning time [Reference Elbanhawi and Simic8]. There are several methods for performing collision checking in path planning. Analytical methods like GJK (Gilbert-Johnson-Keerthi) algorithm [Reference Gilbert, Johnson and Keerthi38] use mathematical equations and geometric models to determine if two objects will collide. They are often fast and efficient but can struggle with complex shapes and interactions. Grid-based methods like Voxel Grid [Reference Lawlor and Kalée39] divide the environment into a grid of cells and check for collisions by looking at the occupied cells. This is often faster than analytical methods but can result in a loss of precision. Another simple method for collision detection is by approximating objects with overlapping spheres [Reference Hubbard40]. In this approach, the number of spheres used to represent each object is a crucial hyperparameter. Additionally, this method is more instantaneous than other traditional collision detection algorithms.

Recently, machine-learning techniques have been utilized to overcome the limitations of traditional collision detection methods in robotics. For example, support vector machines [Reference Pan and Manocha41] have been used to calculate precise collision boundaries in configuration space. Gaussian mixture models were applied in another study [Reference Huh and Lee42], resulting in the path being generated five times faster than bi-directional RRT. K-Nearest neighbors is another machine-learning technique used in modeling configuration space for collision detection [Reference Pan and Manocha43]. Fastron, a machine-learning model, was proposed by Das et al. [Reference Das and Yip44] to model configuration space as a proxy collision detector, decomposing it into subspaces and training a collision checker for each region. The majority of previous studies in this field rely on decomposing the configuration space, which requires significant engineering and simplifications.

4.1. Training and dataset generation

The main purpose of the path-planning network is to clone the behavior of an “expert” planner. In cases like this, where it is feasible for an expert to demonstrate the desired behavior, imitation learning is a useful approach. Moreover, while RRT-based path planners are not themselves sequential decision-makers, they can be modeled as:

(3) \begin{equation} q_{t+1} = f(q_t,q_{target}) \end{equation}

The purpose of the neural planner is to learn the function $f$ . Behavior cloning, which focuses on using supervised learning to learn the expert’s behavior, is the most basic type of imitation learning. The process of behavioral cloning is rather straightforward. The expert’s demonstrations are broken down into state-action pairs, which are then used as independent and identically distributed (i.i.d.) training data for supervised learning. An important concern when using supervised learning for behavioral cloning is the risk of encountering out-of-distribution data. During the inference process, the neural planner may make decisions that were not previously observed in the dataset. Consequently, even minor inaccuracies in the replicated behavior can accumulate rapidly, resulting in a substantial failure rate in identifying viable paths. To mitigate this issue, we propose the data aggregation algorithm (DAGGER) in our study [Reference Ross, Gordon and Bagnell45].

The training process of the neural planner and the collision checker is explained in Algorithm 1. At the beginning of the process, some initial demonstrations from the expert planner are required. To do so, $RandomGoalConf(T)$ function samples T goal configurations in the collision-free space of the bin. The generated goal configurations are given to the expert planner to generate the paths using $GeneratePath$ function which requires initial and goal configurations. The expert planner outputs two datasets: the neural planner dataset ( $D$ ) and the collision checker network dataset ( $C$ ). $D$ only consists of the final path that the expert planner generates; however, $C$ includes every segment which was checked by the expert planner using the geometrical collision checker. In the case of an RRT-based algorithm, the final path will be stored in $D$ and the whole RRT tree and all the instances of its corresponding steer function are stored in $C$ .

To train the neural planner, the demonstration dataset generated by the expert planner should be of high quality. To generate a high-quality dataset, post-processing is applied to the paths generated by the expert planner. As shown in Fig. 3, the expert planner generates a collision-free path for a random query. Followed by that, the binary state contraction (BSC) is utilized to remove redundant and unnecessary waypoints, resulting in a shorter overall path. The sparsity of waypoints is desirable as it simplifies the subsequent processing, such as collision checking, communication with the robot controller, and trajectory planning. While the output of BSC is similar to the lazy state contraction presented in ref. [Reference Qureshi, Simeonov, Bency and Yip36], BSC is computationally more efficient and reduces the run time. After BSC, the resampling function is used to ensure the smoothness and efficiency of the planned path. This involves discretizing the waypoints at regular intervals, which are guaranteed to be collision-free as they already exist on the collision-free path. Importantly, these additional waypoints do not increase the overall length of the path but do serve to reduce the mean and variance of the target step magnitudes, $||q_{t+1} - q_t||$ . While larger steps require fewer neural network forward passes and collision checks during the inference phase, smaller steps offer several advantages. Firstly, they minimize the deviation from the planned path by reducing the magnitude of individual errors $||\hat{q}_{t+1} - q_{t+1}||$ . Secondly, they are more likely to be collision-free as they occupy less physical space. Finally, they offer a more normalized training target for the neural network, where the problem is reduced to predicting a direction in the limit. However, BSC is a crucial step before resampling, as it relies on dividing large steps into regular steps, and a dense path of short steps cannot be divided. Fig. 4 illustrates this procedure in a 2D environment, highlighting how this method enhances the smoothness and quality of the path. The $PostProcess$ procedure in Algorithm 1 presents the use of BSC and resampling on the generated paths.

Figure 3.

End-to-end training process of PPCNet: top row: the imitation learning and data aggregation processes for training the planner network. In this process, based on random queries and the data aggregation process, the planning network dataset is constructed. The dataset outputs the tuple $(q_t,q_T,q_{t+1})$ indicating the current, goal, and next configurations. Finally, the loss function $L_{planner}$ is used to backpropagate through the network’s weights. bottom row: population-based probability estimation and collision checker network training processes. In this process, for calculating the population-based probability, KD-tree is employed. Furthermore, using the collision checking network dataset and the respective loss function ( $L_{binary}$ or $L_{Population})$ , the network is updated.

Figure 4.

Post-processing procedure: (a) The generated path by the planner, (b) Path after binary state contraction, (c) Path after resampling.

4.1.1. Path planner training formulation

Once the dataset has been generated, it is time to train the network. The goal of the network is to predict the next configuration, $\hat{q}_{t+1}$ , which should be as close as possible to the actual next configuration, $q_{t+1}$ , in the training dataset. The network uses the dataset to learn the underlying patterns and relationships between the input and output data.

To achieve the goal of accurate prediction, the network must minimize the difference, or loss, between the predicted and actual configurations. This is done by comparing the predicted output $\hat{q}_{t+1}$ to the actual output $q_{t+1}$ and adjusting the network’s parameters to minimize the difference. This process is known as backpropagation and it is typically done using optimization algorithms such as stochastic gradient descent.

(4) \begin{equation} L_{planner} = \frac{1}{N_t} \sum _{j=1}^{N_d}\sum _{i=1}^{T} || \hat{q}_{j,i} - q_{j,i}||^2 \end{equation}

In this equation, $N_t$ is the number of trajectories and $N_d$ is the number of all individual waypoints in the training dataset.

4.1.2. Collision checker training formulation

To ensure the safety of the robot, we experimented with two different approaches for training the collision checker and evaluated their performance. As mentioned earlier, the collision checker network takes the current and next configurations as input and estimates the probability of the segment being in collision. Therefore, the training set for this network should include segments and corresponding labels indicating whether they are in collision or not. The optimization function utilized to train the collision checker for both of the approaches is called binary cross-entropy loss:

(5) \begin{equation} L(\hat{p},p_{true})=\hat{p}\log{p_{true}}+(1-\hat{p})\log{(1-p_{true})} \end{equation}

where $\hat{p}$ is the probability estimation made by the neural network and $p_{true}$ is the true probability. The reason for using binary cross-entropy is due to the fact that, in essence, the collision checker is a binary classification network. Therefore, the standard approach in the literature is to use cross-entropy due to it giving a better probability distribution over the training data.

Binary labels

The most basic approach to tackle the optimization of the network is to use the true labels given by the analytical collision checker. In that case, $p_{true}$ in Eq. (5) becomes either 0 (collision-free segment) or 1 (in-collision segment). In Fig. 3, this approach has been named as $L_{binary}$ . The binary cross-entropy loss function measures the dissimilarity between the network’s predicted probability distribution and the true probability distribution in the binary classification problem.

Population-based labels

The sparsity of the binary labels hinders the performance of the collision checker since it skews the output of the network towards the more common label. The motivation for this method is to make the binary labels continuous. In addition to the mentioned reasoning, making the labels continuous can help the network be better optimized in corner cases. For example, a collision-free segment inside the bin is mostly surrounded by in-collision segments. Hence, the collision-free segment will not be of much help in optimizing the network compared to the massive existence of the in-collision segments. Making the labels continuous will amplify the effect of the collision-free segment. To make the binary labels continuous, we give a regional estimation of the probability of a specific segment being collision-free by retrieving the segments similar to it whose centers lie within the hypersphere (with a specific radius) surrounding the center of the segment. Finally, the population-based probability (denoted as $\hat{p}_{ref}$ to indicate it being a reference probability) estimates the probability of a segment being collision-free by dividing the number of collision-free segments by the total number of segments:

(6) \begin{equation} \hat{p}_{ref}=\frac{\sum _{k=1}^K\unicode{x1D7D9}[p_{true}^{(k)}==0]}{K} \end{equation}

where $\unicode{x1D7D9}$ is the identity-indication function. The advantage of the proposed approach is that the probability labels generated by this method can provide a well-behaving continuous function where near the obstacles, the probability gets near zero and the further it gets from them, it approaches one. In practice, we utilize KD-Tree method (built with the centers of the segments) to find similar segments. Moreover, the more data this approach gets, the better the estimation will be. Therefore, our proposed algorithm stores every segment that was checked by the analytical collision checker during the path-generation process with the expert planner. In the case of an RRT-based expert planner, this corresponds to storing the whole tree and all of the in-collision segments generated by it. While this approach may have higher computational cost compared to the binary labels approach, our results indicate that it can achieve better prediction performance. For this approach, $\hat{p}_{ref}$ is used instead of $p_{true}$ in Eq. (5), and the corresponding loss function is called $L_{population}$ .

Algorithm 1.

Training process of PPCNet

4.1.3. End-to-end training process

In this section, an overview of the end-to-end process of training the integrated model which consists of both the planner network and the collision checker network will be discussed. The training procedure is outlined in Algorithm 1.

Initially, a set of trajectories are sampled using the expert planner as the initial demonstration, and the planner and collision dataset are filled. In each iteration, the path planner’s policy is trained using the demonstrations from an expert planner. Consequently, DAGGER process (as explained in Section 4.1) is executed which results in the expansion of both of the datasets. Finally, the collision-checking dataset is trained and the model’s performance is evaluated in the $TestPolicy$ function. The function uses some random goal configurations and attempts to path plan between them (following the process in Algorithm 2). If the success rate of the model is satisfying, the path planner and the collision checker network weights are returned.

Algorithm 2.

Path-generation process using PPCNet

4.1.4. Path-planning process

In this section, the end-to-end path-planning process of the PPCNet is explained. As illustrated in Fig. 2, the two networks, the planning network and collision checking network, do the decision-making sequentially. In the context of the bin-picking application, given the start, pick, and place configurations, the proposed model picks the object from the bin and places it in the place configuration. As outlined in Algorithm 2, in each iteration of the decision-making process the path-planning network outputs the next configuration that the robotic arm must transition into. The current configuration and the next configuration are then checked with the collision-checking network. To do so, the $Steer$ function is used. As explained in Algorithm 3, the path between the two waypoints is discretized into equal segments with lengths close to the resolution step size of the expert planner. This is done in order to have the segments be close to the collision checker training dataset’s distribution. Furthermore, in each segment, the collision-free probability of the segment is checked. In order to decide whether a segment is collision-free or not, a safety threshold is needed. If the probability is more than the threshold, the algorithm continues to the next segment for collision checking. If the probability is less than the threshold, then the segment is deemed to be in collision. In that case, if the in-collision segment is the first segment of the path, then the planner has failed to output a good waypoint. Otherwise, if the in-collision segment is not the first segment, the planning network was successful and the tail of the last collision-free segment will be the output of the $steer$ function. Moving on, if the $steer$ function outputs success, then the next waypoint toward the goal configuration is added to the path. In the case that the steer function outputs failure, the planner network attempts to generate a new waypoint. For that purpose, stochasticity is needed inside the network. Therefore, dropout layers are used during inference to have the network be able to recover from failure cases [Reference Qureshi, Simeonov, Bency and Yip36]. Furthermore, in each iteration, if it is possible to directly connect the current configuration to the target configuration, then the target configuration is added to the path and it will be the final path generated by the PPCNet. At this point, the feasibility of the path is checked with the geometrical collision checker using $IsFeasible$ function. If the path is collision-free, it will be returned; otherwise, the algorithm will attempt to patch the in-collision segments using the expert planner. To this end, the $InCollision$ function outputs the segments in the path that are in collision. It is important to note that if some consecutive segments were in collision, all of the segments will be counted as one bigger in-collision segment and the head and tail of the bigger segment will be considered. This is done in order to have lesser expert planner calls and be more computationally efficient. Furthermore, for each segment, the expert planner attempts to generate an alternative path between the two configurations. If for all of the segments, the patching was done with success, then the proposed path planner would be successful and the path would be returned.

Algorithm 3.

Steer Function $(\boldsymbol{q}_{\boldsymbol{current}},\boldsymbol{q}_{\boldsymbol{next}})$

5.1. Comparative study

Table II presents a comparative analysis of the performance of PPCNet against other planning methods. Our experiments demonstrate that PPCNet exhibits remarkable performance in terms of planning time in all three environments (More than $70 \%$ improvement). Fig. 6 provides a visual illustration of the planning time comparison. The hyperparameters for the competing RRT-based algorithms were tuned such that they produce paths of comparable length to PPCNet, while ensuring that they find a path for each planning task in less than 10 s and 1000 iterations. However, the results show that BIT* and Informed RRT* methods take longer to plan and have lower success rates than PPCNet.

Table II.

Planning time and path length comparison of the proposed method and BI-RRT for 500 random pick-and-place queries.

PPCNet has shown its ability to generate paths with comparable length and success rates compared to other algorithms, which is particularly important in evaluating the effectiveness of path-planning methods. Additionally, the post-processing method employed in dataset generation has allowed PPCNet to outperform its expert planner (Bi-RRT) in terms of path length. The BSC function plays a critical role in eliminating redundant and unnecessary waypoints, resulting in a reduction in path length and an improvement in path quality. The combination of the BSC function and resampling has significantly enhanced the quality of paths generated by PPCNet. Furthermore, our findings show that while the addition of the collision checking to the PPCNet has caused significant improvement compared to the MPNet in the expense of the marginal reduction in the PPCNet’s success rate.

The success rate falling short of 100 $\%$ in a known environment can be attributed to two primary factors. First, the limitations inherent to function approximation become apparent when the planner’s output lacks precision, especially when navigating near obstacles. The paths generated by the expert planner occasionally skirt very close to an obstacle’s edge. Replicating such precision in the planner network becomes challenging, leading to deviations from the expert’s path and resulting in failures. Secondly, our specific objectives, which revolves around planning time and path length, have prompted us to establish defined thresholds. For instance, in pick-and-place tasks, we have set a limit of $300 {\rm ms}$ for overall planning time and a maximum of $100$ waypoints. Any result exceeding these thresholds is classified as a failure. Adjusting these thresholds can potentially improve our success rates, although it is essential to balance them with the efficiency of our approach.

Finally, the study’s results indicate that the population-based training method for the collision-checking network performed better than the binary classification approach, resulting in a more accurate collision-checking model. This is evident from the decrease in the planning time due to a reduction in the need for patching. Hence, the tradeoff between the computational cost of training the population-based collision checker against the accuracy it offers can be observed.

5.2. Real-world implementation

In this section, the motion profile generated by the Kinova Gen3 robot based on the path created by the PPCNet in the real-world bin-picking example is discussed. Due to the path-planning architecture of our approach, the main trajectory-planning process for deploying the robot in the real world relies on the specific software platform, parametric settings, and the architecture of the particular robot in use. Hence, to demonstrate the practical applicability of our method, and to underscore its potential for real-world robotic operations, we made use of the Kinova Kortex API [49] for our Kinova Gen3 robot to perform bin-picking operations. This facilitates the conversion of our established paths into actionable trajectories. For safety considerations during the implementation phase on the Kinova Gen3 robotic arm, we capped the maximum velocity for all six actuators at $36 ^{\circ }/s$ . This constraint served as a determinant to compute the trajectory time for each distinct path segment. Figs. 7 and 8 present motion profiles from a randomly selected bin-picking scenario. As evident, the robot’s motion, guided by the generated path, exhibits a combination of smoothness and safety. The Cartesian space motion profile illustrates that the generated collision-free path maintains a safe distance from the bin, ensuring its safety. Furthermore, there are no abrupt changes in the robot’s joint velocity and acceleration in all of our randomly selected real experiments on Kinova Gen3 robotic arm.

Figure 6.

Planning time comparison between different methods.

Figure 7.

Cartesian space path for a randomly selected bin-picking scenario with the Kinova Gen3 robotic arm. In this scenario, the robot should pick three different objects and place them on the table.

Figure 8.

Motion profile for a randomly selected bin-picking scenario with the Kinova Gen3 robotic arm. top row: joints angles plot middle row: joints velocities plot bottom row: joints accelerations plot.

While our approach did not involve explicit trajectory optimization algorithms, our results suggest that the path generated by PPCNet exhibits the potential for conversion into a smooth trajectory.Footnote 1 The field of trajectory planning offers various methods and techniques to optimize different aspects of the trajectory, as documented in the literature [Reference Boscariol, Caracciolo and Richiedei50–Reference Ichnowski, Danielczuk, Xu, Satish and Goldberg52]. Leveraging these established methods in conjunction with PPCNet could lead to enhanced trajectory refinement and further improvements in the overall performance.