PairwiseNet: Pairwise Collision Distance Learning for High-dof Robot Systems
Authors: Jihwan Kim
Frank C. Park
Venue: CoRL 2023 Poster (2023)
Keywords: Robot Collision
Collision Distance
Machine Learning

Abstract: Motion planning for robot manipulation systems operating in complex environments remains a challenging problem. It requires the evaluation of both the collision distance and its derivative. Owing to its computational complexity, recent studies have attempted to utilize data-driven approaches to learn the collision distance. However, their performance degrades significantly for complicated high-dof systems, such as multi-arm robots. Additionally, the model must be retrained every time the environment undergoes even slight changes. In this paper, we propose PairwiseNet, a model that estimates the minimum distance between two geometric shapes and overcomes many of the limitations of current models. By dividing the problem of global collision distance learning into smaller pairwise sub-problems, PairwiseNet can be used to efficiently calculate the global collision distance. PairwiseNet can be deployed without further modifications or training for any system comprised of the same shape elements (as those in the training dataset). Experiments with multi-arm manipulation systems of various dof indicate that our model achieves significant performance improvements concerning several performance metrics, especially the false positive rate with the collision-free guaranteed threshold. Results further demonstrate that our single trained PairwiseNet model is applicable to all multi-arm systems used in the evaluation. The code is available at https://github.com/kjh6526/PairwiseNet.

PairwiseNet: Pairwise Collision Distance Learning for High-dof Robot Systems
Jihwan Kim Seoul National University i wiw.akiimrobbtiiee.
Frank Chongwoo Park Seoul National University f fcp0anu. ac.
Abstract: Motion planning for robot manipulation systems operating in complex environments remains challenging problem. It requires the evaluation of both the collision distance and its derivative. Owing to its computational complexity, recent studies have attempted to utilize data-driven approaches to learn the colli- sion distance. However, their performance degrades significantly for complicated high-dof systems, such as multi-arm robots. Additionally, the model must be re- trained every time the environment undergoes even slight changes. In this paper, we propose PairwiseNet, a model that estimates the minimum distance between two geometric shapes and overcomes many of the limitations of current models. By dividing the problem of global collision distance learning into smaller pairwise sub-problems, PairwiseNet can be used to efficiently calculate the global collision distance. PairwiseNet can be deployed without further modifications training for any system comprised of the same shape elements (as those in the training dataset). Experiments with multi-arm manipulation systems of various dof indi- cate that our model achieves significant performance improvements concerning several performance metrics, especially the false positive rate with the collision- free guaranteed threshold. Results further demonstrate that our single trained Pair- wiseNet model is applicable to all multi-ann systems used in the evaluation. The code is available at https:: / //g thub. com/ jh62266 P rwi seNet.
Keywords: Robot Collision, Collision Distance, Machine Learning
Introduction
Motion planning algorithms such as RRT [1, 2, 3] and its many variants [4, 5, 6, 7] all require collision distance the minimum distance between the robot and its nearest obstacle (including other links for self-collision avoidance). Among these, some even require the derivatives of the collision distances. It is well-known that calculating this distance involves finding the minimum distance between each robot link and the obstacles, which can be computationally intensive, especially for high-dof robots with complex geometries.
To alleviate the computational burden, one possible solution is to train a collision distance function using data. By collecting sufficient data consisting robot configurations and their corresponding collision distances, machine learning models such as kernel perceptron models [8], support vector machines (SVM) [9], and neural networks [10, 11, 12, 13, 14], can be used to learn the collision distance function. This learned function can then be used to quickly determine if a given configura- tion is collision-free While these data-driven approaches have demonstrated satisfactory results for low-dof robot systems, often they perform poorly for higher-dof robots. The challenge lies in the fact that the collision distance function for higher-dof robots is complex and highly non-convex.
Another challenge faced by existing data-driven methods is their sensitivity to small environmental changes. For example, the addition of new obstacles or change of the robot's 's base position can lead to completely different collision distance function; for many of these methods, the entire training 7th Conference Robot Learning (CoRL. 2023), Atlanta, USA.

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(a) Robot
(b) Painwise col. distance
(c) Global col. distance
PairwiseNet
min (+)
dy
Collision Distance
network
deol
Bemmnt Pairs
Figure 1: An illustration of the global collision distance estimation through PairwiseNet. (a) Robot environment at given joint configuration. (b) Pairwise collision distances for all element pairs are determined through PairwiseNet. (c) The smallest these distances becomes the global collision distance.
procedure must be repeated, from data collection to model training. (One possible exception is [8], which proposes an efficient model update strategy for dynamic environment updates, but their method is limited to low-dof robot systems and still requires an additional training procedure.)
We present PairwiseNet, a collision distance estimation method that provides a promising alterna- tive to existing data-driven approaches used for predicting the global collision distance. Instead of directly estimating the global collision distance, PairwiseNet focuses estimating the pairwise col- lision distance: the minimum distance between two elements in the robot system. The PairwiseNet model takes as input the point cloud data of two geometric shapes and their relative transformation and outputs the minimum distance between these two shapes. To estimate the global collision dis- tance, PairwiseNet first predicts the minimum distances for every possible pair of elements in the system. then selects the minimum these pairwise distances as the estimate for the global colli- sion distance (see Figure 1). The efficient parallel batch computation of the neural network enables the rapid prediction of minimum distances between pairs of elements.
Compared to the complex and highly non-convex function of the global collision distance, the min- imum distance function between a pair of elements is simpler and easier to train. By breaking down the challenging task of learning the global collision distance into smaller sub-problems of the pair- wise collision distance learning, our PairwiseNet achieves significant performance improvements for high-dof robot systems.
Another advantage of PairwiseNet is its applicability to any system composed of known shape ele- ments (shape elements that are sufficiently trained for estimating pairwise collision distance). The trained PairwiseNet model can be used without need for additional training or modifications in such systems. For example, consider a scenario in which sufficiently large dataset containing pairwise collision distances between the links of Panda robot is available. In this case, the Pair- wiseNet model trained using this dataset robots, regardless of the number of robots their respective positions, as this is possible because the collision distance estimation for such systems can be broken down into pairwise collision distance estimations for each element pair, and these pairwise distances are already known by the trained PairwiseNet model. As long as the system is exclusively comprised of shape elements that have been learned during training, the trained PairwiseNet model applicable to any such system. Even in cases where the system undergoes changes, such as changing the robot's base position adding another robot arm, if the geometric shape elements of the system remain unchanged, the trained PairwiseNet model remains applicable to the changed system without any modifications.
be applied any system consisting of multiple Panda

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Our approach has been evaluated in high-dof multi-am am robot manipulation systems, ranging from the two-arm (14-dof) to four-arm (28-dof) systems, as well as a single-arm robot with obstacles. The results demonstrate that our approach outperforms existing learning- -based methods in terms of collision distance regression error, collision checking accuracy, and notably the False Positive Rate with the collision-free guaranteed threshold (Safe-FPR). Moreover, our approach performs better even when using single trained PairwiseNet model for all multi-arm systems.
2 Related Works
Several machine learning-based methods for collision distance estimation have been proposed due to their computationally efficient inference procedures for collision distances and derivatives. [6] used SVM classifiers identify whether each pair of parts of humanoid robot was in safe or dangerous self-collision status given specific joint configuration. Only the minimum distances of the dangerous pairs of parts were estimated using capsule-based BV algorithm, simplifying the calculation of collision distances and derivatives. [9] also employed an SVM classifier for a 14-dof dual-arm robot manipulation system. The SVM classifier inputs a vector consisting of the positions of all joints in the system and outputs a collision label either <1 for a collision or > I for a collision-free state. In [13], SVM and neural network models were trained to predict the collision label of humanoid manipulation system given joint configuration. Separate collision classifier models were trained for every sub-part pairs, such as the left arm and right leg, resulting in a total of 10 sobemsdels used for collision label predictions.
Similar to 19]. [10] utilized joint positions as inputs for their multi-layer perceptron neural network model. Meanwhile, [11] employed a positional encoding vector of the joint configuration as input for their neural network model. [12] trained neural network model estimate the collision distance using an extended configuration containing both joint and workspace configurations as input, with the model outputting the collision distance of the system. DiffCo [8] is collision classifier model based on kernel perceptron that generates both the collision score and its derivative. DiffCo also utilizes an efficient active learning strategy that adjusts the trained collision score function for dy- namic updates in the environment. Similarly, CollisionGP [15], a Gaussian process-basee collision classifier model, has been proposed. CollisionGP determines the collision query for a given joint configuration and also measures the uncertainty of the model in its prediction. Recently, GraphDis- (Net [14] was proposed The model inputs the information on geometric shapes, which represented as graphs for both the manipulator links and obstacles. GraphDistNet then utilizes the geometric relationship between the two graphs to predict the collision distance.
Graph Neural Network (GNN) model for collision distance estimation.
Similar to our method, some works [16, 17] approached the challenge by decomposing a complex problem into several simpler sub-problems. [16] proposed novel configuration space decomposi- tion method. This method separates the robot into disjoint components and trains classifier for the collision-fii -fre configuration space of each component. Since the components near the base link have a relatively low-dimensional configuration space, training classifiers for these components is easier than training a single classifier for the whole system. [17] trained a collision predictor for generating collision free human poses. They focused on the fact that collisions only affect local regions of the human body. Therefore, they designed set of local body parts, and the collision prediction was accordingly decomposed into these local parts.
The effectiveness of these existing methods has been experimentally demonstrated only for low-dof robot systems; their performance degrades substantially for high-dof systems operating in complex environments. particular, most learning- based collision distance estimation methods establish a collision-free guaranteed threshold to ensure that no collisions occur during actual manipulation. However, existing methods often suffer from high false positive rate, resulting in overly cau- tious collision detection when utilizing the collision-free guaranteed threshold. In comparison, our method demonstrates effective collision distance estimation performance even in high-dof robot systems and maintains low false positive rates when using the collision-free guaranteed threshold.
3

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Robot EnK.
Element 2
x 76
Fully-connected Neural Network
T22
Concat
Encoder
Min. Distances
A Batch size i Element Pairs
of All Pairs
Extract Element Pairs
B of Pointcloud Pairs
LB x:32
min()
Stack Batch
Tiz SE((3
Vectorize
Collision Distance
Tranformations
dgui
[a] Data Prepracessing
(b) Encodor
[c| Regressar
Figure 2: An illustration of estimating the global collision distance via PairwiseNet, our pairwise collision distance learning method.
3 Learning Pairwise Collision Distance
3.1 Problem Formulation
We assume the availability of a simulator environment of the target system, which includes the robot kinematics and geometric shapes of links and obstacles. We aim to determine the optimal model parameter & for the pairwise collision distance estimation model fgx which can predict the collision distance between any pair of geometric shapes. The model takes the point cloud data of two geometric shapes P,, P, (expressed in each corresponding object coordinates) and the relative transformation Tij SE) as input, and outputs the estimated pairwise collision distance d; between the two shapes.
(P,
(1)
After training the model, the global collision distance can be determined by the procedure as shown in Figure 2. First, a set of element pairs and corresponding transformations S(g) {(P,, Py. ky Next, PairwiseNet determines the pairwise collision distance between each element pair in S(q), and the minimum distance found among these taken the global collision distance of the robot system. The global collision distance estimator function F& be expressed in the form of
the given joint configuration is extracted from the target robot system.
desa = Fx S) min (P..P, (P..P,
(2)
J (P,
(3)
where (o)((g) is the estimated global collision distance in the joint configuration q. Using the batch computation of the neural network model, efficiently estimate the minimum distances of element pairs.
3.2 Network Architecture
PairwiseNet consists of two main components: encoder that creates shape feature vector from a point cloud data of geometric shape (Figure 2b), and regressor that predicts the minimum distance between two shape feature vectors and a transformation (Figure 2c). The encoder em- ploys two EdgeConv layers from Dynamic Graph Convolutional Neural Network [18] to extract 32-dimensional shape feature vectors from the point cloud data. The regressor then combines the two shape feature vectors and the transformation into single vector and uses four fully connected layers with hidden state dimensions of (128, 128, 128) output the minimum distance (Figure 2c). The training of PairwiseNet uses the mean-squared error (MSE) between the estimated and actual

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0.5m 120"
d 0.3m
0.5m
Figure 3: Test environments for the collision distance learning performance evaluation. We selected (a) two arms, (b) three arms, and (c) four arms robot systems.
collision distances as the loss function
Σ
(g - d
(4)
Dunin
(P.P: (P.P,, d.
where di E R is the ground-truth pairwise collision distance, and Dirain denotes the training dataset.
3.3 Efficient Inference Strategy of PairwiseNet
Our approach includes an efficient inference strategy for the global collision distance calculation by eliminating the need to run the encoder, deep neural network that transforms the point cloud data into feature vectors. Since the point cloud data of element pairs remains unchanged regardless of the joint configuration, shape feature vectors of element pairs can be calculated and saved once for each robot system before calculating the collision distance. Using these pre-calculated shape feature vectors, PairwiseNet is able to estimate the collision distance only using the regressor, a simple neural network composed of fully-comnected layers. Implemented Py PoTch [19], PairwiseNet is capable of performing collision distance estimation for the joint configuration in less than 0.5ms. Details on the inference time for PairwiseNet can be found in Appendix B.I.
Experiments
4.1 Collision Distance Learning for Multi-arm Robot Systems
Target Systems For the test environments, selected three high-dof multi-arm robot systems as illustrated in Figure 3. We employed 7-dof Franka Emika Panda robot arms for our test environ- ments. For the test dataset of each target environment, we sampled one million joint configurations from a uniform distribution within the joint limits.
Training Dataset for PairwiseNet We collected pairwise collision dis- tance data from dual-arm robot ma- nipulation systems. For the diver- sity of the dataset, utilize dual- arm robot systems with various rel- ative positions between the two arms as illustrated in Figure 4. We sample and o with eight equally spaced val- ues within the range [0, 2 ). and sam- ple with five equally spaced values within the range 1m, 1.0m]. In to- tal, we use 320 different combinations of (R, 0, Ø), resulting in 320 different dual-arm robot systems.
Robot 2
Robot
Robot
0
Robot
Figure 4: An illustration of multi-arm robot systems for gen- erating the training dataset. Training data points are gener- ated from dual-arm robot environments with various relative positions between two arms.

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For each system, we sampled joint configurations uniformly within the joint limits and extracted set of element pairs S(g) at each joint configuration (Figure 2a). To obtain the ground-truth pair- wise collision distance dip E R between the element pair, we use the collision distance estimation algorithm implemented in PyBullet [20]. the two elements collide, the collision distance the negative of their penetration depth (the distance by which one convex object enters into the inte- rior of another during a collision [21]). The resulting training dataset Diriin contains 3 million data points.
Baselines
We trained our method and other existing collision distance estimation methods.
Capsule: A bounding volume method with capsule-shape collision primitives used in [22, 6].
JointNN: fully-connected neural network model that directly uses joint configurations in- puts (the input representation used in [6, 13]).
PosNN: fully-connected neural network model that uses joint positions as inputs (the input representation used in [10, 9J).
jointNERF [11]: A fully-connected neural network model that uses positional embedding vec- tors of joint configurations as inputs.
ClearanceNet [12]: A neural network model that takes joint configurations as inputs and utilizes two fully-connected layers, each followed by dropout layer.
DifíCo [8]: A kernel perceptron model that takes joint configurations as inputs and outputs the collision score.
Existing collision distance learning methods were trained on one million uniformly sampled data points within the joint limits for each target system. For the DiffCo model, we were limited to dataset size of 50,000 data points, as this was the maximum feasible size for kernel perceptron training hardware (AMD Ryzen Threadripper 3960X, 256GB RAM, and NVIDIA GeForce RTX4090 with 24GB VRAM).
Performance Evaluation We evaluate the performance of collision distance learning using four metries: MSE, AUROC, Accuracy, and Safe-FPR. These metrics target both the collision distance regression and collision classification, with a robot configuration being classified as collision if the collision distance is below the threshold (for DiffCo, the collision score above the threshold). MSE represents the mean squared error between the ground truth and estimated global collision distance. AUROC is the area under the receiver operating characteristic curve for the collision classification. Accuracy is the classification accuracy of collisions with the threshold = 0. Lastly, in order for the trained collision distance estimation model to be used in actual path planning tasks, a sufficiently conservative threshold must be used to ensure that collisions cannot occur. However, the more conservative the threshold used, the more false alarms will occur where non-collision robot configurations are incorrectly classified as collisions. Safe-FPR is used to evaluate performance in these situations, representing the false alarm rate when using the least yet sufficient conservative threshold that can classify all collision configurations in the test dataset collisions.
Table presents the evaluation results of PairwiseNet and other existing methods. PairwiseNet outperforms the existing methods in all performance metrics in all three multi-arm robot systems compared the existing methods, with the top-performing metries highlighted in bold. Notably, PairwiseNet even shows better performance metries than Capsule, computationally expensive BV- based collision distance estimation method.
Figure 5 shows the collision distance estimation results for the four-arm robot system during collision-free trajectory. The robot configurations at the start and end points the path are such that they involve all four robot arms intricately intertwined, so the robot arms move through the near-collision region. The ground-truth collision distance and the estimated collision distances by PairwiseNet and other baselines during the trajectory are represented in the bottom-right plot. The performance of PairwiseNet can be confirmed, it is the only method that accurately estimates the complex ground- truth collision distance of the four-an arm robot system.

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Table Collision Distance Estimation Performances
Accuracy
Env.
Methods
MSE
AUROC
safe-FPR
(e 0)
Capsule
5.47e-4
0,9995
0.9776
0.0247
JointNN
3.63e-4
0.9955
0.9794
0,3200
PosNN
2.71c-4
0.9970
0,9823
0.1476
Fig. (a)
jointNERF
2.98c-4
0.9962
0,9808
0,2371
(Two arms)
ClearanceNet
1.11e-3
0,9853
0.9621
0,4570
DiffCo*
0.9824
0.9818
0,3141
PairwiseNet (our)
0.24c-4
0.9998
0.9941
0.0200
Capsule
5,96e
0.9993
0.9775
0.0241
JointNN
9.69e-4
0,9902
0.9721
0,2679
PosNN
5.41c-4
0.9951
0,9801
0.1336
Fig. (b)
jointNERE
8.22c-4
0.9920
0,9747
0.2213
(Three arms)
ClearanceNNet
4.63e-3
0.9499
0,9395
0.6067
DiffCo
0.9603
0.9453
0,5858
PairwiseNet (our)
0.24c-4
0,9997
0.9944
0.0189
Capsule
6.66c-4
0.9986
0,9468
0.0694
JointNN
1.59e-3
0.9718
0,9183
0,6988
PosNN
7.18e-4
0.9885
0,9478
0.5371
Fig. 3 (c)
jointNERR
1.30e-3
0,9778
0,9280
0,6260
(Four arms)
ClearanceNet
6.67c-3
0.8738
0,8202
0,9965
DiffCo*
0.8811
0.8306
0.9874
PairwiseNet (our)
0.46e-4
0.9994
0.9858
0.0650
*DiffCo outputs the collision score from (collision-free) to (collision).
Colfation Distnce [m)
0.15
sentNott
2.03
(1)
Time (4)
Figure 5: Collision distance estimation for the four-arm robot system: Images depict the four-arm robots following trajectory, and the plot illustrates ground truth and estimated collision distances of PairwiseNet and other baselines during the trajectory.
Generalizability Existing learning methods have used individual models trained from each sys- tem's individual dataset estimate collision distance the three target robot systems. In contrast, PairwiseNet estimates collision distance with only model for all three target robot systems, and its performance is even better than that of existing methods that use individual models for each sys- tem. Therefore, PairwiseNet, trained with sufficient datasets, can apply the same model without additional training, even the robot base positions or number of robot arms change.
4.2 Collision Detection in Real-world Environment
We perform experiments in real-world environments with a 7-dof Panda robot arm (Figure 6). This workspace is populated with obstacles such as shelves and tables that add complexity the robot arm's operational landscape. To validate the performance, tion with the robot occasionally sweeping close to, and sometimes colliding with, tables and shelves.
provide human-guided demonstra-

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bb
(d)
0.32
0.
Dutaaoo Ind
2.00
1.04
2.85
0.00
Time (5]
T
[5
Timme 151
Time 15)
Ground Truth
Painwisn
Clearanceheet
JointhN
Figure 6: Collision distance estimation for the 7-dof Panda robot arm environment with obstacles (tables and shelves). The top images display human-guided robot arm, while the corresponding plots at the bottom illustrate the ground truth and estimated collision distances from PairwiseNet and other baselines at time = (a) 22.9s, (b) 39.6s, (c) 53.7s, and (d) 61.6s, respectively.
The training procedure of PairwiseNet and other baselines, as well as the dataset and model archi- tectures used, are identical to those employed in the experiments for multi-arm robot systems. In the case of PairwiseNet, the dataset consists of element pairs representing tables and shelves, which are further divided into sub-parts. Specifically, a table divided into a tabletop and four legs, while shelf is divided into six plates. The pairwise collision distances between these sub-parts and the robot links are calculated by PyBulee for the training dataset.
The collision distance estimation results are presented in Figure 6. Note that Figure 6 displays only four snapshots from the demonstration; the complete video can be accessed in the supplementary 'video. Compared to the evaluated baselines, PairwiseNet consistently exhibits the highest accuracy in estimating the actual collision distance. reliably detects collisions between the robot arm and obstacles in the majority of cases. In contrast, the other baselines either fail to trigger collision alanms when collision occurs (Figure (b)) produce false collision alarms when collision is present (Figure 6 (c), (d)). PairwiseNet stands out by consistently and accurately identifying collisions between the robot and obstacles.
5 Conclusions
In this paper, present PairwiseNet, a novel collision distance estimation method that estimates the minimum distance between a pair of elements instead of directly predicting the global collision distance of the robot system. By simplifying the problem into smaller sub-problems, our approach achieves significant performance improvements for high-dof robot systems compared to methods that directly predict the global collision distance. Additionally, PairwiseNet is capable of handling environmental changes such robot base repositioning without requiring additional training fine- tuning. We evaluate and compare the collision distance estimation performance of PairwiseNet for both high-dof multi-arm robot systems and single-arm systems in the presence of obstacles, and validate its accurate collision distance estimation and generalization to environmental changes.
Limitations and Future works The generalizability PairwiseNet currently limited to sys- tems that exclusively consist of known shape elements. While PairwiseNet demonstrates robust performance in estimating collision distances within this scope, its applicability environments with unknown (untrained) shape elements has not been fully investigated. Future work is aimed at enhancing the generalizability of PairwiseNet to accommodate systems that include previously un- seen shape elements. This could involve, e.g., expanding the training dataset to incorporate wider range of environmental variations, and incorporating techniques to handle unknown or novel shape elements [23].
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Acknowledgments
This work supported in part by ITTP-MSIT grant 2021-0-02066 (SNU AT Innovation Hub), ITTP- MSIT grant 2022-0-00488 (Training and Inference Methods for Goal-Oriented AI Agents), KIAT grant P0020536 (HRD Program for Industrial Innovation), ATC+ MOTIE Technology Innovation Program grant 20008547, SRRC NRF grant RS-2023-00208052, SNU-AIIS, SNU-IAMD, SNU BK21+ Program in Mechanical Engineering, and SNU Institute for Engineering Research.
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Appendix
A Further Experimental Details
A.1 Strategy for Decomposing System Elements
PairwiseNet achieves robust collision distance estimation performance by dividing the robot sys- tem into multiple elements and calculating the pairwise collision distance between these elements. Therefore, to apply PairwiseNet, the robot system must be divided into these elements nary step. Each element must be a rigid body with an unchanging shape, and in the case of the Panda robot arm used in our experiments, each link was treated as a separate element. Since PairwiseNet does not require each element to convex, non-convex links of the Panda robot arm can be used without additional decomposition.
prelimi-
However, PairwiseNet's superior performance is attributed to the fact that the pairwise collision distance functions between elements are much easier to learn than the global collision distance function. Therefore, if an individual element's shape is complex and highly non-convex, learning the corresponding pairwise collision distance may become difficult, potentially diminishing Pair- wiseNet's effectiveness. Hence, well-considered balance must found between the complexity of decomposing the system and the performance of PairwiseNet.
In our real robot experiments, we decomposed tables into a tabletop and four legs, and shelves into six plates, This strategy was employed not only to simplify complex obstacles and make the pairwise collision distance more tractable but also to leverage the symmetry of the obstacle structure. For example, the bottom, middle, and top plates of the shelf have the same shape, so dividing them into separate elements allows for more efficient use of training data. Furthermore, the fact that the decomposed elements of the tables and shelves take the form of flat rectangular shapes can be beneficial to the learning process.
A.2 Capsule-based Bounding Volume Method for a Panda Robot Arm
We construct capsule- shaped collision primitives for a Panda robot arm baseline collision dis- tance estimation method (see Figure 7). Our approach follows similar methodology of [22], which formulates optimization problem as follows:
4 [|a bel bellmr? +
min artbert
3 mr?
(5)
s.t. dist(p, a,b;) Si, for all p E
(6)
Here, denotes the link index of the Panda robot arm, M; represents the vertices of the 4th link mesh, and a, b; and r; refer the two endpoints and the radius of capsule, respectively, and a,b; represents the line segment connecting the two endpoints. This formulation results in the creation of minimal volume capsules that encapsulate all vertices of the link meshes. The collision distance of the multi-arm robot systems can be estimated through the minimum distance calculation between capsules.
A.3 The Collision-free Guaranteed Threshold
The collision-free guaranteed threshold <sae refers to predefined distance value that is established in collision distance estimation methods. This threshold is set to ensure that during testing or actual operation, the estimated collision distance remains above this threshold for all valid configurations or movements of the robot system. In other words, if the estimated collision distance between the robot and any obstacles remains above the collision-free guaranteed threshold (dam)(g) > (atee). considered safe and collision-free. In our experiments, set the collision-free guaranteed threshold to the least conservative value that allows classify all the collision configurations in the test dataset collisions. These thresholds then utilized for measuring the Safe-FPR.

---

si?
Figure 7: Illustrations of the Panda robot arm with capsule-shape collision primitives.
Table 2: The Collision-free Guaranteed Thresholds
Methods
Two arms
Three arms
Four arms
Capsule
0.0
0.0
0.0
JointNN
0.2111
0.2015
0,2231
PosNN
0.1141
0.1189
0.1756
jointNERF
0.1661
0.1734
0,2001
ClearanceNet
0.2840
0.3713
0,4944
DifCCo
-1.2789
-1.4672
-0,9535
PairwiseNet
0.0150
0.0152
0.0184
A.4 Hyperparameters
Table 3 shows hyperparameters employed in our experiments.
Table 3: Hyperparameters
hyperparameter
value
batch size, learning rate, epoch for PairwiseNet
1000, Ic-3, 2000 191. 1.75e-4, 400 10000. 1e-3, 10000
batch size, learning epoch for ClearanceNet batch size, learning rate, epach for other NN baselines
for k-nearest neighbor of EdgeConv layers of points in the point cloud data of an shape element hidden nodes of EdgeConv layers
100
64
B Additional Experimental Results
B.1 Inference Time Comparison between PairwiseNet and Standard Collision Distance Estimation Method
PairwiseNet incorporates efficient inference strategy using only the regressor network during the inference process, and computing multiple element pairs as a single batch. Thus, despite cal- culating collision distances pairwise like traditional non-data-driven methods, it demonstrates an inference speed as fast as other existing data-driven approaches. We have compared the inference speed of PairwiseNet with that of standard non-data-driver methods and displayed the results in Table 4. The standard collision distance calculation algorithms used for comparison are from the Flexible Collision Library (FCL) and its extended implementation (HPP-FCL).
The test environments were the same three multi-arm robot systems used to evaluate the collision distance estimation performance. The Panda robot arm provides a simplified convex mesh, rep- resenting the original link mesh with only 1/60 of the number of vertices, allowing for efficient collision distance estimation. In our experiments, we measured the inference time using both the original complex mesh and the simplified mesh. The inference time measured as the time taken to estimate the collision distance for total of 1,000 joint configurations.
12

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Table 4: Inference Time Comparison
Inference time for 1000 joint poses (s)
Methods
Two arms
Three arms
Four arms
(64 pairs)
(192 pairs)
(384 pairs)
FCL w/ original mesh
13.87
43.97
92.63
FCL w/ simplified mesh HPP-FCL w/ simplified mesh
2.479
9.079
17.25
0,3920
0,8560
1.9490
CPU
GPU
CPU
GPU
CPU
GPU
ClearanceNet
0.1450
0.0919
0.1484
0.0818
0.1493
0.0825
ClearanceNet (Batch)
0.0299
0.0001
0.0250
0.0001
0,0222
0.0001
PairwiseNet
0.1054
0.0988
0.1590
0.0998
0,2025
0.1045
PairwiseNet (Batch)
0.0362
0.0207
0.1070
0.0224
0.2121
0.0253
Additionally, PairwiseNet is capable of receiving multiple joint configurations as a single batch in- put: it can simultaneously compute the collision distances for all the joint configurations. This is an efficient feature for tasks that require simultaneous collision checks for multiple joint configurations, such campling-based path planning methods like RRT control techniques such Model Pre- dictive Control (MPC). In our experiments, also measured the time taken to calculate the collision distances for 1,000 joint configurations the PairwiseNet (Batch) entry in Table 4. PairwiseNet was implemented using PyTorrch [19]. FCL and HPP-FCL were implemented using the python fcl and hpp- fcl libraries, respectively. Thus, all the time measurements are conducted within Python code. The experiments were carried out an environment equipped with an AMD Ryzen 9 7950X (16 cores, 32 threads), NVIDIA RTX 4090, and 125GB of RAM.
through PairwiseNet, and this is represented under
PairwiseNet demonstrates an inference speed that is at least 20 times and to more than 150 times faster than FCL. As the number of element pairs requiring collision distance calculation increases, the gap between the two widens. This is because in the case of FCL, the time taken for inference increases as the number of element pairs grows, whereas, with PairwiseNet, there is little change in inference time even as the number of pairs increases. HPP-FCL shown to 6~99 times faster than the original FCL. However, PairwiseNet still demonstrates an inference speed that is 4~~18 times faster than HPP-FCL. Moreover, the inference time of HPP-FCL increases the number of pairs increases, which is not the case with PairwiseNet. Even when performing inference through a CPU, the inference time does not increase significantly. For batch calculation, where 1,000 joint configurations are computed at once, PairwiseNet can estimate 1,000 collision distances in as little as 25 milliseconds.
B.2 Training Complexity of PairwiseNet
We utilized 3 million data points (3 million shape element pairs with their corresponding distances) to train PairwiseNet for collision distance estimation in multi-arm robot systems and for single-arm systems with obstacles. For more detailed description of the complexity of PairwiseNet's training process, additional information is presented in Table 5.
Unique element pairs refers the count of element pairs in the system, excluding those with duplicate shapes (for example, the pair of the robot arm's seventh link with the top shelf plate and the pair with the middle shelf plate considered the same since the shapes of the top and middle plates are identical). The more unique element pairs, the greater the number of pairwise collision distances PairwiseNet must learn, resulting in higher training complexity.
Gradient step refers to the number of times the learnable parameters were updated to minimize the loss during the training process.
Training time elapsed refers to the total time taken to complete the training.
The table also includes Validation loss to represent each training result.
13

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The training time was measured on environment with AMD Ryzen 9 7950X (16 cores, 32 threads), NVIDIA RTX 4090, and 125GB of RAM environment.
Table 5: Training Complexity of PairwiseNet
Unique
Training
Gradient
Training elapsed (h)
Validation loss (MSE)
Training Env.
element pairs
data points
steps
Multi-arm
36
1,000,0000
2,860,000
24.8
1.46e-5
Multi-arm
36
3,000,000
4,286,000
35.6
1.43e-5
Single w/ obstacles
70
3,000,000
4,286.000
36.1
8.03e-6
Two arms w/ obstacles
1,000,000
2.860.000
23.3
9.82e-6
We conducted additional experiments analyze the training complexity of PairwiseNet. Initially, for the existing multi-arm robot systems, while we began with million data points for training PairwiseNet, also conducted experiments using fewer data points (1 million) and fewer gradient steps. While the training time decreased, the learning results were comparable to those with the original 3 million data points.
Next, we examined the previously mentioned single-arm system with obstacles. Although there were many unique element pairs (70), since all obstacle elements were rectangular, they were relatively easier learn. Additionally, we trained PairwiseNet with two-arm robot system, adding non- rectangular household objects as obstacles (as shown in Figure 8). Despite using merely total of million data points and fewer gradient steps in this scenario compared to the original PairwiseNet, the validation loss was successfully reduced to 9.82e-6, confirming successful learning.
Gan
Mig
Rnd
Figure 8: A two-arm robot system with four household objects
B.3 Evaluating PairwiseNet in Multi-arm Robot Systems with Various Base Positions
We extended validation of PairwiseNet to include robot systems arranged asymmetrically and in irregular manner, in addition to the three multi-arm robot systems used in our experiments. These additional systems are illustrated in Figure 9. In each systems, the robot arms are arranged in more irregular and complex manner than in the previously used systems, resulting in a greater variety relative positional relationships between the arms.
We used the trained PairwiseNet model that originally used for multi-arm robot systems. Although these new robot systems have complex arrangements of robot arms, they still consist of Panda robot arms, which have been sufficiently trained. Therefore, PairwiseNet can be applied directly to these systems without the need for additional training.
We have presented the results of PairwiseNet's collision distance estimation performance in Table 6. Upon examining these results, we observed that PairwiseNet performed well across all robot sys- tems and metrics, regardless of the complexity of the arrangement of robot arms. The consistency in performance across these varying configurations demonstrates the robustness of PairwiseNet, af-
14

---

d2
d2
d 0.5mm, 45"
0.5m
dy, d2. 0 0.5m, 0.7m,
d,, d2, 0 0.5m, 0.3m, 45"
(a)
(b)
(c)
(d)
Figure 9: Illustrations of the top views of various base positions within multi-arm robot systems.
firming that there is no significant difference in the quality of collision distance estimation between these different scenarios.
Table 6: Collision distance estimation performances of PairwiseNet for various multi-arm systems
Accuracy
Env.
MSE
AUROC
safe-FPR
(c = 0)
Fig. 9 (a) (Three arms)
0.24e-4
0.9997
0.9943
0.0341
Fig. 9 (b) (Three arms)
0.17e-4
0.9999
0.9987
0.0048
Fig. 9 (c) (Four arms)
0.43e-4
0.9991
0.9859
0.0611
Fig. 9 (d) (Four arms)
0.27e-4
0.9995
0.9931
0.0292
B.4 Comparing the Scalability to High-dof Systems of PairwiseNet with GraphDistNet
Without directly comparing the collision distance estimation performance, we can highlight the novelty of PairwiseNet relative to GraphDistNet [14]. A key difference lies in scalability to high-dof systems. GraphDistNet's performance was only verified in systems with a maximum of 7-dof, and that within the synthetic planar robot systems where simple graph representations Additionally, experiments with real robots were only conducted with 3-dof. Since GraphDistNet employs a complex GNN structure that takes the graph itself as the input, the model's inference speed is inevitably sensitive to the complexity of the graph. In examining the experimental results, we observed that the inference speed slows down as the graph becomes more complex, being up 120 times slower than ClearanceNet [12]. In contrast, PairwiseNet maintains inference speeds comparable ClearanceNet, even in complex systems with 28 DOF across 4 robot arms.
possible.
15


Reviews:
[{"review_id": "q0KniRR0YV", "reviewer": "Reviewer_xJsm", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper proposes an architecture for estimating the distance between links in a robot model. The key novelty of this paper is that they split up the global robot model into links and link poses and estimate the pairwise distance between links.", "summary_of_recommendation": "This paper presents a novel approach for collision distance estimation and demonstrates improvement over a comprehensive set of baselines. The paper would benefit from some experiments regarding space and memory tradeoffs between learned and classic collision detection in motion planners.", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Highly relevant to robotics but no hardware experiments", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "This approach consumes more GPU memory than the global approaches because the link distances have to be computed in parallel. If GPU memory isn't a bottleneck, couldn't you just compute pairwise distances between every pair of points in the pointclouds of the two robots in parallel? This would result in perfect performance and generalization. ", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents a novel approach and demonstrates improvement over a comprehensive set of baselines. \n\nGiven that the intended use case for this is motion planning, it would be good to see some speed comparisons between a standard collision checking approach and this network.\n\nAdditionally, this approach doesn't extend to the types of problems that collision detection is most often needed for (detecting collisions between the robot and an arbitrary environment geometries).", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}, {"review_id": "laYIVpmGdW", "reviewer": "Reviewer_uPAS", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The paper presents a learning based method for collision distance prediction and its' application in collision detection for motion planning of manipulators. The algorithm uses a pairwise collision distance computation for multi-part objects to address the computational complexity of high-dimensional systems. The learning architecture uses an encoder to compute the shape feature vector from the point cloud of each shapes and then a regressor that uses the shape feature vector along with the transformation between the pairs of geometric shape to predict the minimum distance between them. Experiments evaluate the performance of 7-dof panda robot with varying the number of robots and distance between them in each environment. The performance show the proposed method excels in accuracy, safe-FPR, MSE and AUROC when compared to bounding volume method, Joint and position nn, joint nerf, clearance net and Diffco. \n", "summary_of_recommendation": "Overall, the paper presents an interesting idea of computing collision distance between objects by decomposing it into pairwise distance computation. This is evaluated for high-dimensional systems and compared with other known collision distance method computations.  ", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "- The authors should clearly state or experimentally demonstrate how similar the shape elements need to be in the training set to avoid retraining. Can a model trained with UR3 be used to problems using UR5? The shapes are similar but differ in dimensions. Does the model needs complete retraining or pre-trained models can be used with additional layers? This would also clearly indicate what ``slight changes'' the model can handle without retraining.\n- As retraining is required if the system contains dissimilar objects from training set, the authors should evaluate the training set volume required for planning problems including variety of shape elements such as multi-robot system with different robots and workspace obstacles not rectangular in shape. \n- Although the authors have compared their work with other notable work in clearance estimation, the authors should provide empirical comparison with GraphDistNet, ref. 14. GraphDistNet also decomposes multi-body object in form of a graph to apply it to learning based model. \n- The authors show the experimental analysis with multi-robot systems but the multi-robot systems do not include any workspace obstacles and the robot bases are equidistant to each other. The authors should include varying workspace obstacles in the evaluation and vary the distances of the bases. The authors need to also state whether any prior decomposition of the shape obstacles in the training set is required as a pre-processing step.\n- The authors should cite and consider the following works:\n Y. Han, W. Zhao, J. Pan and Y. -J. Liu, \"Configuration Space Decomposition for Learning-based Collision Checking in High-DOF Robots,\" 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Las Vegas, NV, USA, 2020, pp. 5678-5684, doi: 10.1109/IROS45743.2020.9341526.\nTan, Qingyang, Zherong Pan, and Dinesh Manocha. \"Lcollision: Fast generation of collision-free human poses using learned non-penetration constraints.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 35. No. 5. 2021.\nJ. Muñoz, P. Lehner, L. E. Moreno, A. Albu-Schäffer and M. A. Roa, \"CollisionGP: Gaussian Process-Based Collision Checking for Robot Motion Planning,\" in IEEE Robotics and Automation Letters, vol. 8, no. 7, pp. 4036-4043, July 2023, doi: 10.1109/LRA.2023.3280820.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The major strengths of these work are:\n- Computation of collision distance as a pairwise problem for multi-body system is intuitive. This is how even deterministic method of computation of collision distance is performed. \n- Separation of encoder and the regressor in the system architecture is useful in reducing training effort especially in multi-robot scenarios. \n- The authors provide comparative evaluation with other learning based methods such as ClearanceNet. \n- The authors also provide a variety of evaluation metrics and demonstrate the performance for high-dimensional robotic system (28-dof).\n- The authors provide a real-world demonstration of their framework.  \n\nThe areas of improvements are:\n- Clearly state the changes in the system the model can handle without the need for retraining.\n- Empirical comparison with GraphDistNet\n- Variety in the obstacles and multi-robot systems by changing distance of the bases and workspace obstacles of different shapes other than rectangular and their impact on the training effort.\n- Inclusion of other learning based collision detection work in related work. See the list in questions for rebuttal.", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning."}, "extra_text": {}}, {"review_id": "kpE3nJdg9_4", "reviewer": "Reviewer_fxzY", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper aims to develop a data-driven collision-distance estimator that finds the distance between multiple manipulators. However, this is challenging due to the complexity of high-degree-of-freedom robots and the environmental variabilities. To figure it out, this paper proposes PairwiseNet, a collision distance estimation method that finds the pair-wise distance between two sub and known geometries of robots. The main contributions of this paper is as follow:\n-\tEfficient distance estimation formulating easily solvable smaller sub-problems  \n-\tNo need for additional training or modification\n", "summary_of_recommendation": "The proposed methodology shows good accuracy in the setup. However, the method requires strong assumptions and missing comparisons with major methods in the field. I would like to recommend improving this paper. ", "technical_quality": "Fair: Significant additional evidence required to support the claims.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Additional details required", "questions_for_rebuttal": "-\tThis paper needs to show the proposed method is faster and/or more accurate than the non-data-driven algorithm such as the GJK method.\n-\t L182, the penetration depth is hard to define depending on the shape of the geometries. What is the definition of penetration depth?\n-\tMost methods resulted in good performance. How significant result does PairwiseNet make?\n", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Strengths – The main strength of this paper is the decomposition of the distance estimation with complex geometry into parallel sub-problems. \n\nWeaknesses – The method requires known geometries that cannot be scalable and generalizable to new setups. Further, there is no comparison with conventional/representative distance estimation methods such as the GJK method.\n\nThis paper needs to clarify and improve a few things:\n- A major concern is the scope of this work and its limitation. This work assumes the known geometry of robots. The pairwise distance estimation seems to be a good idea, but current distance computation already performs in the same manner. Further, the existing algorithms such as GJK algorithm, which has been used in Flexible Collision Library, are quite efficient if simple geometries are given. Unlike the proposed method, the other methods are scalable/generalizable to variants of geometries that can be given by sensors. Thus, although this proposed method shows efficiency in the defined setup, it is hard to know if we really need this without further evaluation with other distance computation methods.\n", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}, {"review_id": "0mfqusszHui", "reviewer": "Reviewer_vQwN", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The paper proposes a method that views collision detection of articulated bodies as the computation of the distance between the robot's links pairs with a transformation. Once trained, this method is able to extend to high DOF and multi-robot scenes. Experiments also show the efficiency and effectiveness of the method.", "summary_of_recommendation": "I think this paper's idea is interesting. However, after the problem decomposition, I think the subproblem may be easily solved by traditional methods.\n", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Additional details required", "questions_for_rebuttal": "I am interested in one additional experiment. What is the performance of directly computing the cage (maybe a combination of several convex shapes) of the shape and using GJK to compute the shapes' distance after transformation?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Strengths:\n\n- Using a neural network to encode two known shapes with any transformation for distance is interesting. It is efficient and can be extended to high DOF robots. \n- The experiments show the efficiency and effectiveness of the method.\n\nWeaknesses:\n\n- The training complexity of this method may increase quadratically as DOF increases, e.g. every link pair of a humanoid robot will be considered\n- Each robot link has relatively simple geometry and may be represented by several convex shapes. Real-world experiments also decomposite obstacles to several convex shapes.\n\nI think this paper's idea is interesting, decompose collision checking into subproblems and consider using a neural network to encode the relative transformation of shapes and get their distance. However, by this kind of decomposition, the shapes are nearly convex. I am curious what is the performance of directly computing the cage (maybe a combination of several convex shapes) of the shape and using GJK to compute the shapes' distance after transformation.", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers find this work to be both interesting and novel. The comparison to GJK nicely shows the improvement of using the proposed approach over common analytic approaches to collision checking.

I generally agree with the reviewers that comparing to GraphDistNet would improve the paper and I thank the authors for their attempts to do so and understand the difficulties in making this happen. If this could be included for the final version of the paper it would definitely increase its value and quality.
Decision: Accept (Poster)