Korrawe Karunratanakul

Jinlong Yang

Yan Zhang

Michael J Black
email: black@tue.mpg.de

Krikamol Muandet

Siyu Tang
email: siyu.tang@inf.ethz.chjyang

Grasping Field: Learning Implicit Representations for Human Grasps

Figure 1: Ground truth grasps and generated grasps. Each row corresponds to one object. Left three columns show the ground truth grasps, each from three different viewpoints. The middle three columns show one generated example, and the right three columns show another generated example. Note that these objects are never seen during training. See Appendix E (Fig. E.4) for more examples.

Introduction

Capturing and synthesizing hand-object interaction is essential for understanding human behaviours, and is key to a number of applications including augmented and virtual reality, robotics and human-computer interaction. Despite substantial progress, fully automatic synthesis of highly realistic human grasps remains an unsolved problem. The anatomical complexity of the human hand and the variety of manufactured and natural objects make it extremely challenging to pose the hand such that it interacts with the object in a natural and physically plausible way. Recent data-driven approaches explore deep learning technology to learn and leverage powerful object representations, yet they are mainly limited to simple robotic end effectors, such as parallel jaw grippers [69]. In this work, we seek to understand: 1) what is an efficient and expressive representation for modeling hand-object interaction, that can facilitate realistic human grasp synthesis given an unseen 3D object; and 2) how can we learn such a representation from data.

Our key observation is that human grasping is rooted in physical hand-object contact. Through this contact, humans are able to grasp and manipulate objects naturally. To better model hand-object interaction, we must find a way to effectively represent the contact between hands and objects. To this end, we propose a novel interaction representation that is based on regressing a continuous function that we call the Grasping Field. The grasping field maps any 3D point to a 2D space, where each dimension of the 2D space indicates the signed distance to the surface of the hand and the object respectively (see Sec. 3.1 for a formal definition). Inspired by [START_REF] Mescheder | Occupancy networks: Learning 3d reconstruction in function space[END_REF][START_REF] Oechsle | Texture fields: Learning texture representations in function space[END_REF][START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF], we further utilize a deep neural network to parameterize the grasping field and learn it from data. As a result, the learned grasping field serves as a powerful representation to facilitate hand-object interaction modelling.

Based on the grasping field representation, we propose a generative model, in which we generate plausible hand grasps given an object point cloud. We show that our model can produce physically and semantically plausible synthetic grasps, which are similar to the ground truth. Generated grasps on unseen objects are shown in Fig. 1 andFig. E.4.

We further demonstrate the effectiveness of the grasping field representation by considering the task of 3D hand and object reconstruction from a single RGB image. In recent work, Hasson et al. [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF] introduce an end-to-end learnable model to reconstruct 3D meshes of the hand and object simultaneously, producing the state-of-the-art results on several datasets. Physical constraints, such as no interpenetration and proper contact, are enforced during the training. However, there are several drawbacks of their mesh-based representation for hand-object interaction modeling. First, they heuristically pre-define regions of the hand that can be in contact with objects. Second, their object representation is limited to objects of genus zero. Third, the resolution of their contact inference is limited by the resolutions of the hand and object meshes. In contrast, with the grasping field representation, it is not necessary to first compute the hand and object meshes, and then compute the contact region. Instead, one can easily infer the contact region by querying the signed distances of input 3D points. Furthermore, the physical constraints, such as no inter-penetration and proper contact, can be efficiently computed and enforced. As demonstrated in our experiments, our model considerably reduces the interpenetration between the reconstructed hand and the object, and improves the quality of 3D hand reconstruction, compared with [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF].

In summary, our contributions are: [START_REF] Antotsiou | Taskoriented hand motion retargeting for dexterous manipulation imitation[END_REF] We propose the grasping field, a simple and effective representation for hand-object interaction; (2) Based on the grasping field, we present a generative model to yield semantically and physically plausible human grasps given a 3D object point cloud;

(3) We further propose deep neutral networks to reconstruct the 3D hand and object given an RGB input in a single pass; (4) We perform extensive experiments to show that our method outperforms the baseline [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF] on 3D hand reconstruction and on synthesizing grasps that appear natural.

Related work

Human grasp and contact. There is a large body of work on capturing and recognizing human grasps [START_REF] De-An | How do we use our hands? discovering a diverse set of common grasps[END_REF]22,[START_REF] Heumer | Grasp recognition with uncalibrated data gloves -a comparison of classification methods[END_REF][START_REF] Nakamura | The complexities of grasping in the wild[END_REF][START_REF] Pirk | Understanding and exploiting object interaction landscapes[END_REF][START_REF] Yang | Grasp type revisited: A modern perspective on a classical feature for vision[END_REF]. Recently, [22] introduced a stretch-sensing soft glove to capture accurate hand pose without extra optical sensors. Puhlmann et al. [67] utilized a touch screen to facilitate the capturing process of human grasping. As physical contact is fundamental to hand-object interaction, researchers have proposed methods to capture and modeling contact from diverse modalities [5,42], but these often interfere with natural movement. Concurrent to this work, [7] proposed a new dataset of hand-object contact paired with RGB-D images. Our work differs in that our focus is on learning an interaction representation, which is efficient and easy to interface with deep neural networks.

Grasp synthesis. Grasp synthesis is a longstanding problem in robotics and graphics, resulting in an extensive literature [START_REF] Antotsiou | Taskoriented hand motion retargeting for dexterous manipulation imitation[END_REF]4,[START_REF] Brahmbhatt | Contact-Grasp: Functional Multi-finger Grasp Synthesis from Contact[END_REF][START_REF] Elkoura | Handrix: Animating the human hand[END_REF][START_REF] Kim | Physics-based hand interaction with virtual objects[END_REF][START_REF] Kry | Interaction capture and synthesis[END_REF]43,[START_REF] Mahler | Learning ambidextrous robot grasping policies[END_REF][START_REF] Oikonomidis | Full dof tracking of a hand interacting with an object by modeling occlusions and physical constraints[END_REF]65,70,76,[START_REF] Seo | Planar, bimanual, wholearm grasping[END_REF][START_REF] Sridhar | Real-time joint tracking of a hand manipulating an object from RGB-D input[END_REF][START_REF] Zhao | Robust realtime physics-based motion control for human grasping[END_REF]. As early as 1991, Rijpkema and Girard [70] proposed a knowledge-based approach to incorporate the role of the human hand, object, environment and animator for the task of computer-animated grasping. More recent works can be categorized into three types of approaches: analytic, data-driven and hybrid approaches. For the analytic approaches [39,[START_REF] Seo | Planar, bimanual, wholearm grasping[END_REF], the grasps are often synthesized by formulating the problem as a constrained optimization problem that satisfies a set of criteria measuring the stability or other properties of the grasps. The data-driven approaches [START_REF] Pinto | Supersizing self-supervision: Learning to grasp from 50k tries and 700 robot hours[END_REF]69] often employ machine learning methods to learn representations for synthesizing grasps. An excel-lent survey of data-driven grasp generation is presented in [3]. Recent hybrid approaches [START_REF] Mahler | Learning ambidextrous robot grasping policies[END_REF][START_REF] Mahler | Dex-net 3.0: Computing robust robot suction grasp targets in point clouds using a new analytic model and deep learning[END_REF] combine analytic models and deep learning tools to synthesize grasps for various end effectors. Finally, the most related approaches to our work was presented in [START_REF] Corona | Ganhand: Predicting human grasp affordances in multi-object scenes[END_REF] and [START_REF] Taheri | GRAB: A dataset of whole-body human grasping of objects[END_REF], where neural networks are used to predict hand parameters of the MANO hand model [74] given object information. In [START_REF] Corona | Ganhand: Predicting human grasp affordances in multi-object scenes[END_REF], the model learns to predict the best grasp type from the grasp taxonomy [START_REF] Feix | The grasp taxonomy of human grasp types[END_REF] according to the RGB images of the objects. Then, the predicted hands are optimized together with the object meshes to refine the contact points. While in [START_REF] Taheri | GRAB: A dataset of whole-body human grasping of objects[END_REF], the parameters are generated directly from the given Basis Point Set [START_REF] Prokudin | Efficient learning on point clouds with basis point sets[END_REF] of the objects. Our work differs from the previous works in that, by also considering the object distance field, we propose a learnable representation for modelling hand-object interaction that can be used without contact post-processing. Empowered by deep neural networks, the learned representation enables us to synthesize realistic human hand grasping a given object naturally.

Hand pose estimation. Hand pose estimation is a longstanding problem, and various input modalities have been considered, e.g., RGB images [START_REF] Baek | Weakly-supervised domain adaptation via gan and mesh model for estimating 3d hand poses interacting objects[END_REF][START_REF] Doosti | Hopenet: A graph-based model for hand-object pose estimation[END_REF][START_REF] Iqbal | Hand pose estimation via latent 2.5d heatmap regression[END_REF][START_REF] Mueller | Ganerated hands for realtime 3d hand tracking from monocular rgb[END_REF][START_REF] Panteleris | Using a single rgb frame for real time 3d hand pose estimation in the wild[END_REF][START_REF] Simon | Hand keypoint detection in single images using multiview bootstrapping[END_REF][START_REF] Spurr | Cross-modal deep variational hand pose estimation[END_REF][START_REF] Zimmermann | Learning to estimate 3d hand pose from single rgb images[END_REF] or RGB-D and depth sensors [START_REF] Hamer | Tracking a hand manipulating an object[END_REF]35,[START_REF] Malik | Handvoxnet: Deep voxel-based network for 3d hand shape and pose estimation from a single depth map[END_REF][START_REF] Mueller | Real-time hand tracking under occlusion from an egocentric RGB-D sensor[END_REF][START_REF] Oberweger | Training a feedback loop for hand pose estimation[END_REF]. Due to the lack of large scale 3D ground truth data, synthetic data has often been used for training [START_REF] Dibra | Monocular RGB hand pose inference from unsupervised refinable nets[END_REF][START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF][START_REF] Malik | DeepHPS: End-to-end estimation of 3D hand pose and shape by learning from synthetic depth[END_REF]. Recently, instead of estimating the hand skeleton, recovering the pose and the surface of the hand has become popular using statistical hand models, e.g., the MANO model [74], that can represent a variety of hand shapes and poses [25,[START_REF] Zhang | Endto-end hand mesh recovery from a monocular rgb image[END_REF][START_REF] Zimmermann | Freihand: A dataset for markerless capture of hand pose and shape from single rgb images[END_REF]. Using the template derived from MANO, [41] show that it is also possible to regress hand meshes directly using mesh convolution. In this work, we represent the 3D hand by a signed distance field, instead of a parametric hand model, due to the difficulty of incorporating object interaction into the model parameter space. For fair comparison with the parametric hand model representation, we fit the MANO model [74] into our resultant signed distance fields. The experimental results indicate the advantage of our new interaction representation.

Object model representation. Learning 3D object models using various types of representations has also been explored [9, [START_REF] Groueix | AtlasNet: A papier-mâché approach to learning 3D surface generation[END_REF]33,44,[START_REF] Maturana | VoxNet: A 3D convolutional neural network for real-time object recognition[END_REF][START_REF] Michalkiewicz | Deep Level Sets: Implicit surface representations for 3d shape inference[END_REF][START_REF] Su | A point set generation network for 3d object reconstruction from a single image[END_REF]90,[START_REF] Wu | MarrNet: 3D Shape Reconstruction via 2.5D Sketches[END_REF]. Recently, the community has focused on using the implicit functions such as the Signed Distance Function (SDF) [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF], Occupancy Networks [START_REF] Mescheder | Occupancy networks: Learning 3d reconstruction in function space[END_REF], Implicit Field [START_REF] Chen | Learning implicit fields for generative shape modeling[END_REF], and their derivations [START_REF] Genova | Local deep implicit functions for 3d shape[END_REF][START_REF] Genova | Learning shape templates with structured implicit functions[END_REF], as these can model arbitrary object topology with adjustable resolution. Due to these advantages, we also adopt implicit functions to capture hand-object interaction.

Hand-object interaction. Reconstructing hand and object jointly has been studied with both RGB input and RGB-D input [START_REF] Pham | Handobject contact force estimation from markerless visual tracking[END_REF][START_REF] Rogez | 3d hand pose detection in egocentric rgb-d images[END_REF][START_REF] Rogez | First-person pose recognition using egocentric workspaces[END_REF][START_REF] Romero | Monocular real-time 3D articulated hand pose estimation[END_REF]78,[START_REF] Supančič | Depth-based hand pose estimation: data, methods, and challenges[END_REF][START_REF] Tsoli | Joint 3D tracking of a deformable object in interaction with a hand[END_REF][START_REF] Tzionas | Capturing hands in action using discriminative salient points and physics simulation[END_REF]88,[START_REF] Wang | Learning a generative model for multistep human-object interactions from videos[END_REF]. Recently, Hasson et al. [27,[START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF] achieved promising results on explicitly modeling the contact by combining a parametric hand model MANO [74], with the mesh based representation for the object. As data for hand-object interaction is limited, we opt to use their synthetic dataset, the ObMan dataset [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF], which is sufficiently large for training a neural network. Our work differs from previous hand-object reconstruction work mainly by focusing on the novel representation of contact and learning both hand and object in the signed-distance space, which allows arbitrary shape modelling and easier distance field manipulation. Furthermore, we go beyond the reconstruction task by proposing generative models to synthesize realistic human grasps given a 3D object.

Method

Grasping field

The grasping field (GF) is based on the signed distance fields of the object and the hand, formally defined as a function f GF : R 3 → R 2 , mapping a 3D point to the signed distances to the hand surface and the object surface, respectively. In this way, the contact and inter-penetration relations between the hand and the object can be explicitly and efficiently represented. Specifically, the hand-object contact manifold is given by

C = {x | f GF (x) = 0 for x ∈ R 3 }.
The volume of hand-object inter-penetration is given by

I = {x | f GF (x) < 0 for x ∈ R 3 }.
Inspired by [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF] and [START_REF] Mescheder | Occupancy networks: Learning 3d reconstruction in function space[END_REF], we propose to model f GF using a deep neural network, and learn it from data. Therefore, one can infer hand-object interaction in 3D space without the explicit hand and object surfaces. The learned GF can be considered as an interaction prior, which enables us to infer various grasping poses of the hand, only based on the 3D object. Furthermore, in contrast to previous works, e.g. [START_REF] Hassan | Resolving 3D human pose ambiguities with 3D scene constraints[END_REF][START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF][START_REF] Zhang | Generating 3d people in scenes without people[END_REF], which can only evaluate body-object interactions after obtaining the body and the object meshes, when using GF as the representation in hand-object reconstruction from images, we model the hand, the object, and the contact area by the implicit surfaces in a common space, largely improving the physical plausibility of the reconstruction.

According to the aforementioned merits of the GF, we use it to address two tasks in this paper; i.e. hand grasp generation given 3D objects and hand-object reconstruction from RGB images. Different GF networks are designed specifically for different tasks.

Grasping field for human grasp synthesis

In this section, we show how to use GF to synthesize human grasps. Given an object point cloud, the goal is to generate diverse hand grasps that interact with the object in a natural manner.

Network architecture. The network architecture is shown in Fig. 2a; we adopt the encoder-decoder framework. To extract features from point clouds, we use the Point-Net encoder [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF] with residual connection. The encoder is trained jointly with other network layers from scratch. The encoder-decoder network takes a query 3D point, and two point clouds of the hand and the object as input, and produces the signed distances of the query point to the hand and the object surfaces. In addition, the encoded object point cloud feature is fed into the hand point cloud encoder, leading to a hand distribution conditioned on the object. Note that this variational encoder-decoder network only requires both hand and object point clouds during the training. During inference, only the conditioning object point cloud and the query point are required. The hand features are sampled from the learned latent space, as in a standard VAE [START_REF] Kingma | Auto-encoding variational bayes[END_REF].

The training loss consists of the following terms:

The reconstruction loss L rec : For each query point x, the input object point cloud p o and the input hand point cloud p h , the reconstruction loss is designed for the hand and object individually:

L rec = |c(f CGF (x, p o ), δ) -c(SDF p o (x), δ)| + |c(f CGF (x, p h ), δ) -c(SDF p h (x), δ)|, (1)
where f CGF is the grasping field network (Fig. 2a). SDF p h (•) and SDF p o (•) are the ground truth SDF for the hand and object, respectively. In addition, c(s, δ) := min(δ, max(-δ, s)) is a function to constrain the distance s within [-δ, δ]. δ is set to 1cm in all experiments. KL-Divergence L kl : In order to generate new hand grasps, we use a KL-divergence loss to regularize the distribution of hand latent vector h, obtained from the hand point cloud encoder h = E h (p h |p o ), to be a normal distribution. The loss is given by

L kl = KL-div (N (µ(h), σ(h))||N (0, I)) , (2)
where N (0, I) denotes a standard high-dimensional normal distribution, µ and σ denotes mean and standard deviation.

For generation, the hand latent vector h is sampled from a standard normal distribution.

Classification loss L cls : Besides predicting the signed distances of a query point, we also train the network to produce the hand part label of a query point to parse the hand semantically. To achieve this, we introduce a classification loss, which is given by a standard cross-entropy loss. The hand part annotation is based on the MANO model [74] as illustrated in Fig. 2b.

Grasping field for 3D hand-object reconstruction from a single RGB image

Our proposed grasping field is an expressive representation for modelling hand-object interactions in 3D. Here we address the challenging task of 3D hand-object reconstruction from a single RGB image, i.e. f CGF : R 3 × I → R 2 , in which I ∈ I is a 2D image. We model such a conditional GF by two types of deep neural networks and learn their parameters from data. Network architecture. The network architectures are illustrated in Fig. 3, which are designed to recover both hand and object in a single pass. To enable a direct comparison with [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF], the two-branch network is employed (Fig. 3a), which addresses hand and object individually. Similar to [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF], we introduce contact and inter-penetration losses during the training to facilitate a better 3D reconstruction on the contact regions of the hand and the object. To introduce hand-object interactions in early stages, we propose a one-branch network (Fig. 3b), which uses the same image encoder and has the same number of layers with the twobranch model. See Appendix A for architecture details.

The training loss consists of the following terms:

The reconstruction loss L rec : For each query point x and the input image I, the reconstruction loss is designed for the hand and object individually, and is given by The inter-penetration loss L ip : To avoid surface interpenetration between the reconstructed hand and object, we define the inter-penetration loss as

L rec = p∈{p h ,p o } |c(f CGF (x, I), δ)-c(SDF p (x), δ)|, (3)
L ip = x max(-1, f CGF (x, I) , 0), ( 4
)
where 1 is a 2D one-vector, and •, • denotes a dot product. This loss function actually penalizes the negative sum of predicted signed distances to the object and to the hand. If the hand and the object are separate and have no contact, the signed distance sum of every point in 3D space is always positive, and hence is ignored by our inter-penetration loss.

On the other hand, if the hand and the object have interpenetration, then this inter-penetration loss does not only penalize the points in the intersection volume, but also all 3D points in the space, indicating that the predicted hand and object are incorrect. Compared to the inter-penetration methods in [START_REF] Hassan | Resolving 3D human pose ambiguities with 3D scene constraints[END_REF][START_REF] Zhang | Generating 3d people in scenes without people[END_REF], which only penalize the intersection volume, our loss applies stronger constraints.

Contact loss L cont : Our proposed contact loss encourages hand-object contact, and is given by

L C = x min(α|f CGF (x, I)| 2 , 1), (5)
where α is a hyper-parameter. We can see that f CGF (x, I) = 0 corresponds to the hand-object contact surface. Therefore, it ignores points with predicted grasping field |f CGF (x, I)| 2 ≥ 1 α , and only encourages points with |f CGF (x, I)| 2 < 1 α to be the contact points. In our study, we empirically set α = 0.005 based on the hand-object interactions in the training data. Finally, we employ the same Classification loss L cls as the one proposed in Sec. 3.2.

From grasping field to mesh

With the trained grasping field conditioned on images or point clouds, one can compute the signed distances to the hand and object of a query 3D point. To recover the hand, object and their interactions, we first randomly sample a large number of points, and evaluate their signed distances. The point clouds belonging to the hand and the object can be selected, according to point-object signed distances close to zero. Then, the hand mesh and the object mesh are obtain by marching cubes [45].

In addition, the hand mesh can be recovered by fitting the MANO [74] model to the hand point cloud. In this case, we can obtain hand segmentation, hand joint positions, and a compact representation of the hand simultaneously, according to the pre-defined topology in MANO.

Denoting the MANO model by M(β, θ) with the parameter β and θ representing hand shape and pose respectively, we minimize 6 l=1 d(p h , M(β, θ) l ) to recover the hand configuration, in which l denotes the 6 parts of hand, i.e. the palm and the 5 fingers, ph denotes the hand point cloud produced by our model, M(β, θ) l denotes the MANO hand mesh belonging to the hand part l, and d(•, •) denotes the Chamfer distance [START_REF] Fan | A point set generation network for 3d object reconstruction from a single image[END_REF][START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF]. The hand segmentation is shown in Fig. 2b.

The implementation details are thoroughly presented in Appendix A.

Experiments

We demonstrate the effectiveness of the grasping field representation on two challenging tasks: human grasp generation given a 3D object and 3D hand-object reconstruction from a single image.

Dataset.

To train the generative model for human grasp synthesis, we need ground truth 3D meshes of interacting hands and objects. Unfortunately, existing datasets often lack the desired properties.The limitations include small dataset size and lack of 3D ground truth hand pose or shape. For the 3D reconstruction task, we also mainly use the ObMan dataset for training and testing. To test the effectiveness of our network on real-world images, however, we follow the same approach as [28] to train and test on the FHB dataset.

Evaluation metrics. Our goal is to generate physically plausible and semantically meaningful 3D human hand given an object. Therefore, we quantitatively evaluate the generated samples according to physics-based metrics and use large-scale perceptual studies to measure the visual realism of the grasps. For the 3D reconstruction task, we use Chamfer distance and hand joint error. Details of the evaluation metrics are in Appendix C.

(1) Physical metrics: A valid human grasp implies stable hand-object contact without interpenetration. Consequently, we use the following evaluation metrics: a) Intersection volume and depth. The hand and object mesh are voxelized and the interpenetration depth is the maximum distance from all the intersected voxels to the surface of another mesh. b) Ratio of samples with contact. We define a contact between the object and the hand when any point on the surface of the hand is on or inside the surface of the object. We calculate the ratio of samples over the entire dataset that have interpenetration depth more than zero. c) Grasp stability. Using physics simulation [START_REF] Coumans | Bullet physics library. Open source: bulletphysics[END_REF], we hold the hand constant, apply gravity, and measure the average displacement of the object's center of mass during a fixed time period.

(2) Semantic metric: We perform perceptual studies using Amazon Mechanical Turk to evaluate the naturalness of our generated grasps. Details of the study are presented in Appendix C and D.

(3) 3D reconstruction quality: We use the Chamfer distance between reconstructed and ground truth hand surfaces to evaluate the hand reconstruction quality as in [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF]. Hand joint distance is computed following [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF][START_REF] Zimmermann | Learning to estimate 3d hand pose from single rgb images[END_REF].

Evaluation: Human grasps generation

Baseline. To our knowledge, there is no previous model that learns to synthesize natural human grasps given a 3D object. Rather than randomly placing the hand around the object, we trained a strong baseline model for grasp generation. Specifically, we replace the decoder (i.e. the grasping field) of our conditional VAE model (Fig. 2a) with fully connected layers to regress MANO hand parameters. Then given a 3D object point cloud and a random sample, our baseline model generates MANO parameters directly. Generated grasps from the baseline are shown in Appendix E. Results. We show the systematic quantitative evaluation of the generative method in Tab. 1 and qualitative results in and tested extensively on the objects from the ObMan test set, FHB and HO3D. Our proposed GF performs substantially better than the baseline on ObMan and achieves comparable quality as the ground truth grasps. When the model is tested on the FHB objects, which are never seen during training, it achieves a comparable perceptual score compared to the ground truth grasps. Surprisingly, on HO3D, our synthesized grasps are judged more realistic than the ground truth grasps of real humans (3.29 vs 3.18). These perceptual studies suggest that our method makes an important step towards the fully automatic synthesis of realistic human grasps.

Regarding the physical plausibility, we observe that our model achieves a better contact ratio and grasp stability (physics simulation) than the ground truth grasps on FHB and HO3D. This is likely due to the GF results having a larger intersection volume. One reason is that there are a very limited number of objects in these two datasets. Some of the test objects are very different from the training objects, resulting in more inter-penetration for the generated grasps. Overall, the combination of visual realism and grasp stability suggests that our results are approaching the level of natural human grasps.

Evaluation: 3D hand-object reconstruction

Apart from serving as a powerful representation for the synthesis task, the proposed GF also facilitates the 3D reconstruction task. In the following, we analyze the different network architectures and training losses proposed in Sec. 3.3. We compare with the baseline method [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF] on the ObMan dataset and [28] on the FHB dataset. The results are summarized in Tab. 2. Due to many limiting factors of the real-world datasets such as FHB and HO3D (see Appendix B for detailed data analysis), learning a reasonable model for joint object and hand reconstruction is extremely challenging. Instead, to evaluate the effectiveness of the GF representation for 3D reconstruction on real-world images, we follow the setting of the latest work [28], where the object 3D model is given as input. Note that this is a com-monly used setting in previous works (e.g. [28, 34,[START_REF] Tekin | H+o: Unified egocentric recognition of 3d hand-object poses and interactions[END_REF]).

Network design. We first analyze the two different network architectures presented in Fig. 3 denoted with and without '2De' respectively. Both architectures achieve comparable performance for hand reconstruction, however differ significantly for intersection error, where the one decoder model achieves considerably better performance, due to the efficient joint modeling of hand-object interaction. Compared with the baseline [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF], the intersection volume and depth are reduced from 6.25 and 1.20 to 0.65 and 0.32, respectively. The contact ratio are comparable among two architectures and baseline model. All our models considerably improve the quality of hand reconstruction, compared with [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF]. The object reconstruction quality is behind hand quality for all model variations including the baseline model. Note that the ObMan dataset contains more than 1600 objects from 8 different classes. The object reconstruction performance is decreased as it remains unclear how to learn the implicit representations to reconstruct a large variety of object classes with a single model [START_REF] Mescheder | Occupancy networks: Learning 3d reconstruction in function space[END_REF][START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF] Training losses. The effect of the contact and interpenetration loss (+L) is shown in Tab. 2 (a), when the loss is imposed during the training of the two-decoder network, the intersection volume and depth are reduced and the overall quality of the interaction is considerably improved. In contrast, for the one-decoder model, our observation is that, for a large portion of 3D points, the signed distances to the object and to the hand are highly correlated, the model that jointly predicts both signed distance values does not need to enforce this auxiliary training loss.

MANO fitting. As shown in Tab. 2 (a), MANO fitting (indicated by GF-MANO) does not have a substantial influence on the reconstruction quality. This implies that on the one hand, the reconstructed hand of our GF model is realistic enough without a statistical model to regularize it, and that on the other hand, the output hand part labels are accurate Hand reconstruction on real-world images. To analyse the effectiveness of the proposed grasping field representation for the 3D reconstruction task on real-world images, we compare our method with the latest work [28] on the FHB dataset. Compared to [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF], the key difference in [28] is that the object is given as part of the input. We explore the same network architecture as [28] and only replace the decoder part with the grasping field. The implementation details are presented in Appendix A (Fig.

A.3).

As stated in [28], definition of hand joint locations vary between datasets. Without hand surface annotation in the FHB dataset, it is difficult to train an accurate regressor that maps between the FHB markers and the MANO joints. Assuming that the joints are identically defined, we fit the MANO model to the FHB markers by minimizing the distance between the MANO joints and the FHB markers. Then the MANO joints obtained in such way are considered as our pseudo ground truth joints, and the obtained MANO surface is used to supervise the training.

We compare the predicted MANO joints with the pseudo ground truth joints as well as the original FHB markers assuming identical joints. As our model is not trained to optimise for the FHB marker locations, the reconstruction error is larger than [28] as shown in Tab. 2 (b). When we evaluate our prediction on the pseudo ground truth MANO joints, the reconstruction error decreases from 2.94cm to 2.6cm. This suggests that the proposed grasping field representation is effective for the task of 3D hand reconstruction from a single image, achieving comparable performance with respect to the start-of-the-art.

Conclusion and Discussion

In this work, we propose a novel representation for handobject interaction, namely the grasping field. Learning from data, the GF captures the critical interactions between hand and object by modeling the joint distribution of hand and object shape in a common framework. To verify the effectiveness, we address two challenging tasks: human grasp generation given a 3D object and shape reconstruction given a single RGB image. The experiments show that the generated hand grasps appear natural and are physically plausible while the hand reconstruction achieves comparable performance as the state-of-the-art.

A limitation of our work is that there is no explicit modeling of the object functionality and human action in the current grasping field representation. In reality, a person holds an object differently based on different intentions. For instance, using a knife or passing it to someone else result in completely different human grasps. One promising future research direction is to incorporate human intention and object affordances into the grasping field for action specific grasps generation. Furthermore, we believe the proposed grasping field representation opens up avenues for several other future research directions. For instance, 3D human hand generation given only an object image and synthesizing the motion of hand-object interactions.

Grasping Field: Learning Implicit Representations for Human Grasps **Appendix**

A. Implementation Details

In Sec. 3.2 and Sec. 3.3, we present the neural networks that are used for human grasps generation and reconstruction, respectively. Here we discuss the implementation details.

A.1. Architecture

In this section, we explain the network architectures used in our experiments. The same decoder architecture is used in our image reconstruction and the hand generation tasks. We change the encoder architectures according to the input type. In our experiments, both the encoder and decoder are jointly trained end-to-end. For image reconstruction, we use the ResNet18 [START_REF] He | Deep residual learning for image recognition[END_REF] model pretrained on the ImageNet dataset [START_REF] Russakovsky | Imagenet large scale visual recognition challenge[END_REF] as an encoder. We change the last layer of the encoder to produce a latent vector of size 256 for the decoder.

For the point cloud input, we use two separated PointNet encoders [START_REF] Fan | A point set generation network for 3d object reconstruction from a single image[END_REF] with additional pooling and expansion layers presented in [START_REF] Mescheder | Occupancy networks: Learning 3d reconstruction in function space[END_REF]. In each encoder, 3D points are first mapped to 512-dimension feature vectors followed by 5 ResNet-blocks, producing a latent vector of size 256. The latent codes for hand and object are then concatenated to make a 512-dimension latent code.

For the hand generation task, we change the first layer in the hand encoder to produce a 256-dimension vector for each point then concatenate it with the 256-dimension object latent vector. For image reconstruction with known objects, we assume that the object mesh in the normalized pose is given. We sample surface points from the given object and use a PointNet encoder to compute object latent vector of size 128. The object latent vector is then concatenated with a hand latent vector of size 128 from ResNet18 encoder, producing a latent code of size 256 for the decoder. The overview of the network is shown in

A.2. Data preparation

To prepare the sampled 3D points and their distances to the hand and object surfaces for training, we follow the point sampling method provided by [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF]: For each pair of the hand and object meshes, we translate both meshes such that the hand root joint is at the origin then scale them to fit in a unit cube. The scaling factor is the same for the entire dataset to ensure the hands are normalized across dataset. After that, 40,000 points are sampled in a unit cube. Following [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF], 95% of the total points were sampled near the surface to capture the details of both meshes. For the Chamfer distance calculation, we sample 30,000 points from the surface of the ground truth mesh and reconstructed mesh following [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF]. In case the reconstructed mesh contains more that one connected component, only the largest watertight connected component is retained.

A.3. Training

The contact loss is disabled in the beginning. When computing the reconstruction loss L rec , hand points to the object surface, and object points to the hand surface, are not considered until the contact loss is enabled. In our trials, we observe dramatic degradation when such a mask is not used or when the contact loss is enabled in the beginning.

For the generative GF network conditioned on an object point cloud, the KL loss, L kl , is employed in an annealing scheme; the loss weight is kept at 0 in the first 200 epochs and then linearly increased to 0.1 over the next 200 epochs. We find that such a annealing scheme is essential in our trials. Applying the KL loss in the beginning causes our generative network posterior to collapse.

In all experiments, we use Adam optimizer [37] with learning rate of 10 -4 and decay it to 5 × 10 -5 at after 600 epochs. We train the models for 1,200 epochs without hand-part classification loss and another 100 epochs with the classification loss. Weight decay is used in all layers in the decoder.

A.4. Inference

During inference, we use Marching Cube with resolution 128 to obtain hand and object meshes. As the object can vary in size, we use a two-stage approach to dynamically scale the cube size in the Marching Cube algorithm. First, to find the boundary of the reconstructed meshes, we query equally space points in a unit cube centered at the origin point to locate the The latent code from the encoder is concatenated with 3D point query then given to the decoder. The same latent vector is concatenated again at the middle of the decoder following [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF], with the concatenated vector R having size 253 and 509, respectively. Every "FC" layer except the last layer is followed by a ReLu activation and a dropout layer with drop rate 0.2. The last layer produces the distance to object surface and the distance to hand surface along with hand part classification scores negative signed-distance values which indicate the inside of the mesh. Then, we query again with a cube that covers every negative-value point. Using this approach, no mesh is produced if no negative point is found in the first stage.

B. Dataset Analysis

In this section, we provide detailed analyses on the FHB [19] and the HO-3D [25] datasets. Although these datasets considerably contribute to the studies of hand-object interactions with detailed 3D annotation, our analyse shows that they might not be suitable for learning human grasps and modelling the accurate contact relation between hand and object. First, the number of objects and the types of grasps are limited. As shown in Tab. 5, the number of object is 3 in the FHB dataset and 10 in the HO3D dataset. Second, the (pseudo) ground truth meshes of the interacting hand and object exhibit frequent interpenetration. Sampled ground truth meshes from the HO-3D dataset are illustrated in We evaluate the interpenetration between hand and object meshes quantitatively. We use the same evaluation metric as the one presented in the main paper, namely, the intersection volume (cm 3 ) and depth (cm) (Sec. 4). The results are shown in Table 5.

For the HO-3D dataset, 91.94% of the training examples exhibit hand-object contact. However, among these training examples, the average intersection volume and depth are 10.91 cm 3 and 1,56 cm respectively.

For the FHB dataset, we use the similar subset as the previous work [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF], namely, we exclude the milk bottle related examples and the examples where the distance from hand joints to the object mesh is more than 1cm. We refer to this dataset as FHB c . We further fit the MANO hand model with the provided joint location. For FHB c , 97.1% of the training examples have hand-object contact. However, similar level of intersection between hand and object meshes can be observed in Table . 5.

Overall, the evaluation shows considerable intersection volume and depth of the training data. Therefore we use the ObMan dataset as our main training dataset, where the ground truth quality of the contact regions is more suitable for Furthermore, as shown in the experiment section (Tab. 1), our generated grasps that are learned from the ObMan dataset obtain a higher perceptual score than the ground truth grasps from the HO3D data in the perceptual study, suggesting that the physical plausibility, i.e. no interpenetration and proper contact, plays an important role on the naturalness of human grasps.

C. Details of the evaluation metrics

Evaluation Metrics. For human grasps synthesis, our goal is to generate physically plausible and semantically meaningful 3D human hand given an object. Therefore, we propose to quantitatively evaluate the generated samples using physics metrics and a large-scale perceptual study to measure the perceptual fidelity. In addition, for the quantitative evaluation of our reconstruction networks, we use Chamfer distance and hand joint error.

(1) Physical metric: A valid human grasp implies hand-object contact without interpenetration. Naturally we propose Intersection volume and depth. We follow [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF] to report intersection volume and depth. The hand and object mesh are voxelized using a voxel with edge length of 0.5cm. The interpenetration depth is the maximum distance from all points on the interpenetrated surface to another surface. If the meshes do not overlap, the interpenetration depth is defined as 0.

Ratio of samples with contact. We define a contact between object and hand when any point on the surface of hand is on or inside the surface of the object. To measure the performance of models on hand-object contact quality, we calculate the ratio of samples over the entire dataset that have interpenetration depth more than zero. As all of the samples in the dataset should have contact between hand and object, the best ratio of frames with contact is 100%. A good hand and object reconstruction model should have high ratio of contact and small interpenetration volume and depth.

Simulation displacement. Following [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF], we use physics simulation to evaluate the stability of the grasps. In the simulated environment [START_REF] Coumans | Bullet physics library. Open source: bulletphysics[END_REF], we fix the hand and measure the average displacement of the mass center of the object in a give time period. Small displacement suggests a stable grasp.

(2) Semantic metric: We perform perceptual studies on Amazon Mechanical Turk to evaluate the authenticity of our generated grasps. For each randomly generated sample, we render images from 6 different views, and request participants to score from 1 (low fidelity) to 5 (high fidelity).

(3) 3D reconstruction quality: We use the Chamfer distance between reconstructed and ground truth hand surfaces to evaluate the hand reconstruction quality. Surface distance is approximated by mean square point cloud Chamfer distance (cm 2 ) as implemented in [START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF]. The MANO wrist is sealed to form a watertight mesh for fair comparison. Joins distance is computed following [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF][START_REF] Zimmermann | Learning to estimate 3d hand pose from single rgb images[END_REF]. After MANO parameters are recovered from the predicted hand mesh as described in Sec. 3.4, we compute mean Euclidean distance over 21 joints following [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF][START_REF] Zimmermann | Learning to estimate 3d hand pose from single rgb images[END_REF]. Note, since scale and global translation can not be determined by a single image, for each predicted hand, we optimize the scale and global translation to match the ground truth by minimizing the Chamfer distance between them. Similarly to hand, we also use Chamfer distance as measurement of object surface quality. The predicted object mesh is transformed according to the corresponding predicted hand transformation estimated from the above to align with the ground truth object mesh. Figure E.1 shows the reconstruction results on the test images of the ObMan dataset [START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF]. We observe that our model can recover hand meshes with proper interaction with the object.

D. Details of the perceptual study

E. Qualitative results

Figure E.2 shows the comparison between reconstructed mesh before and after MANO fitting. The hand meshes also come from the single image reconstruction task. We observe that the MANO fitted meshes match the inferred meshes, even in the case where the rasterized hand mesh has merged fingers.

Figure 2 :

 2 Figure 2: (a) Illustration of the generative grasping field network conditioned on an object point cloud. The red dashed arrow denotes sampling from a distribution. Network details are in Appendix A. (b) Illustration of hand part labels. Left is our hand part annotation on the MANO model. Right is an example of our predicted surface points with hand part labels.

Figure 4 :

 4 Figure 4: Generated grasps conditioned on objects from the HO3D dataset. Each pair shows the sampled grasp from two viewpoints. The model is trained only on the ObMan [29] dataset.

Fig. 1 ,

 1 4 and Appendix E (Fig. E.3, E.4). The baseline and the GF model are only trained on the ObMan training set,

  and such a task is beyond the focus of this work. Please see Appendix E (Fig. E.1) for visualization.

  Figure A.1 illustrates the one-branch decoder with 8 fully-connected layers used in all tasks.

  Figure A.2 shows the details of the point cloud model.

  Figure A.3.

Figure A. 1 :

 1 Figure A.1: Decoder architecture.The fully-connected layers are denoted as "FC" in the diagram. The latent vector C has 256 and 512 dimensions in the image reconstruction and conditional hand mesh generation task, respectively. The latent code from the encoder is concatenated with 3D point query then given to the decoder. The same latent vector is concatenated again at the middle of the decoder following[START_REF] Park | DeepSDF: Learning continuous signed distance functions for shape representation[END_REF], with the concatenated vector R having size 253 and 509, respectively. Every "FC" layer except the last layer is followed by a ReLu activation and a dropout layer with drop rate 0.2. The last layer produces the distance to object surface and the distance to hand surface along with hand part classification scores

  Fig. B.1.

Figure A. 2 :

 2 Figure A.2:The encoder architecture for the conditional VAE. Each box represents a vector obtained from applying the layer written above. For the object encoder, we use the same architecture as the model for point cloud completion used in[START_REF] Mescheder | Occupancy networks: Learning 3d reconstruction in function space[END_REF]. The hand encoder is conditioned on the latent code from the object decoder. The combined latent codes of hand and object are then concatenated with a 3D point and passed to the decoder

Figure A. 3 :

 3 Figure A.3: The architecture of the GF conditioned on the image, given a known object.

Figure B. 1 :

 1 Figure B.1: Ground truth meshes from the HO-3D dataset

Figure D. 1

 1 Figure D.1 shows the user interface for evaluating the generated grasps on the Amazon Mechanical Turk (AMT). Users are asked to rate the plausibility of the hand-object interactions individually. Each entry consists of images from six different views and is rated by three different users.

Figure E. 5

 5 Figure E.5 shows the generated grasps from our baseline VAE model conditioned on the object surface point cloud. The MANO parameters are directly predicted by the decoder.FigureE.1 shows the reconstruction results on the test images of the ObMan dataset[START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF]. We observe that our model can recover hand meshes with proper interaction with the object.FigureE.2 shows the comparison between reconstructed mesh before and after MANO fitting. The hand meshes also come from the single image reconstruction task. We observe that the MANO fitted meshes match the inferred meshes, even in the case where the rasterized hand mesh has merged fingers.FigureE.3 shows randomly sampled grasps from our VAE model conditioned on the object surface point cloud. We observe that our model can generate a variety of grasps given an object. Figure E.4 shows the generated results of the same model conditioned on the objects from HO3D datset. It should be note that this model is only trained on the ObMan dataset and have never seen these objects before.

  Figure E.5 shows the generated grasps from our baseline VAE model conditioned on the object surface point cloud. The MANO parameters are directly predicted by the decoder.FigureE.1 shows the reconstruction results on the test images of the ObMan dataset[START_REF] Hasson | Learning joint reconstruction of hands and manipulated objects[END_REF]. We observe that our model can recover hand meshes with proper interaction with the object.FigureE.2 shows the comparison between reconstructed mesh before and after MANO fitting. The hand meshes also come from the single image reconstruction task. We observe that the MANO fitted meshes match the inferred meshes, even in the case where the rasterized hand mesh has merged fingers.FigureE.3 shows randomly sampled grasps from our VAE model conditioned on the object surface point cloud. We observe that our model can generate a variety of grasps given an object. Figure E.4 shows the generated results of the same model conditioned on the objects from HO3D datset. It should be note that this model is only trained on the ObMan dataset and have never seen these objects before.

Figure D. 1 :

 1 Figure D.1: Amazon Mechanical Turk user interface for grasping evaluation

Figure E. 2 :

 2 Figure E.2: MANO fitting results on the reconstructions from RGB images. Each row shows a set of ground truth hand mesh, reconstructed hand, and MANO fitted prediction, from two different views. The last two rows demonstrate the robustness of our fitting method where the reconstructed meshes from the estimated SDF values are less satisfactory. However, even with merged fingers, we can still recover reasonable hand mesh.

Figure E. 3 :

 3 Figure E.3: Randomly selected hands generated from the conditional VAE. Each row shows hand and object ground truth followed by three sets of sampled hand meshes, before and after MANO fitting, all from the same view. The samples presented on the two bottom rows are less satisfactory as the generated SDFs have artifacts and we can observe interpenetration between the fitted MANO hand meshes and the object meshes.

Figure E. 4 :

 4 Figure E.4: Hands generated from the VAE conditioned on objects from HO3D dataset. The model is trained only on the ObMan dataset.

Figure E. 5 :

 5 Figure E.5: Each row shows five randomly sampled hands given an object from the baseline conditional VAE. We can observe interpenetration between the hand meshes and the object meshes. In some cases, the hands are not in contact with the objects.

Table 1 :

 1 Evaluation of the grasp synthesis on the objects from the ObMan test set, FHB and HO3D. GT* indicates that the ground truth grasps are obtained by fitting the MANO model to the data. Best results except the ground truth are shown in boldface.

			ObMan			FHB			HO3D
		GT	Baseline	GF	GT*	Baseline	GF	GT	Baseline	GF
	Contact ratio (%) ↑	-	66.89	89.4	92.2	48.8	97.0	93	44.3	90.1
	Intersection vol. (cm 3 ) ↓	-	14.46	6.05	16.6	9.65	21.9	10.5	5.86	14.9
	Intersection depth (cm) ↓	-	0.94	0.56	1.99	1.77	2.37	1.47	1.01	1.46
	Physics simulation (cm) ↓	1.66	4.56	2.07	6.69	8.59	4.62	4.31	8.25	3.45
		± 2.42	± 4.57	± 2.81 ± 5.48	± 3.67	± 4.48 ± 4.42	± 4.18	± 3.92
	Perceptual score {1...5} ↑	3.24	2.40	3.02	3.49	2.43	3.33	3.18	2.03	3.29

Table 2 :

 2 3D reconstruction results on ObMan (a) and FHB (b). 2De refers to the 2 decoder model; L indicates the corresponding model is trained with the contact and interpenetration loss; GF-MANO refers to the MANO hand obtained by fitting the MANO model to the SDF.

	Models	Hand error	Joints Intersection Contact Object Error
		Mean Med (cm) Vol Depth (%)	Mean Med	Models	Hand Joints
	GF	0.419 0.283 -	0.65 0.32 90.8	12.8 6.4	(cm)
	GF+L	0.400 0.261 -	0.00 0.00 5.63	14.2 6.8	GF-MANO
	GF-2De	0.408 0.262 -	8.56 1.01 99.6	11.1 5.7	(MANO joints)	2.60
	GF-2De+L	0.384 0.237 -	0.23 0.20 69.6	11.7 5.9	GF-MANO
	GF-MANO	0.405 0.272 1.13 0.59 0.27 83.3	-	-	(FHB marker)	2.94
	GF-2De-MANO 0.419 0.276 1.14 5.75 0.87 98.9	-	-	Hasson et al.* [28]	2.74
	Hasson et al. [29] 0.533 0.415 1.13 6.25 1.20 94.8	6.7	3.6	(b)
			(a)
	enough for us to fit the MANO model and retrieve hand
	joints or shape parameters for applications that need these
	without undermining the shape and contact estimation. A
	qualitative illustration is presented in Fig. E.2.

Table 5 :

 5 Characteristics of the FHB c and HO-3D dataset. The intersection volume and depth are calculated from the training sets and are considered when there is contact between hand mesh and object mesh. For the FHB c dataset, the evaluation is done on the pseudo-ground truth meshes.

	Dataset # of frames # of objects	Intersection
		(train/test)		Vol(cm 3 ) Depth(cm)
	FHB c	5082 / 5658	3	10.59	2.34
	HO-3D 66034 / 11524	10	10.91	1.56

Acknowledgement.

We sincerely acknowledge Lars Mescheder and Michael Niemeyer for the detailed discussions on implicit function, Dimitrios Tzionas, Omid Taheri, and Yana Hasson for insightful discussions on hand interaction, Partha Ghosh and Qianli Ma for the help with VAE. Disclosure. MJB has received research gift funds from Intel, Nvidia, Adobe, Facebook, and Amazon. While MJB is a part-time employee of Amazon, his research was performed solely at MPI. He is an investor in Meshcapde GmbH.


Olivier Pinaud

Olivier Chadebec

Laure Line Rouve

Jean Michel Guichon

Andrea Vassilev

EHICLE electrification is widely promoted to answer environmental issues and reduce the reliance on fossil fuels. However, electric vehicles require a significant number of electrical devices within a quite small space (electric motor, power electronics, power battery, power cables, and so forth). Therefore, it may appear inside the cabin, electromagnetic fields due to these components. International Commission on Non-Ionizing Radiation Protection (ICNIRP) gives guidelines to limit exposure to static [START_REF] Icnirp | ICNIRP guidelines on limits of exposure to static magnetic fields[END_REF] and time-varying [START_REF] Icnirp | ICNIRP guidelines for limiting exposure to time-varying electric and magnetic fields[END_REF] magnetic fields. Some studies [START_REF] Karabetsos | EMF measurements in hybrid technology cars[END_REF], [START_REF] Dietrich | Survey and assessment of electric and magnetic field public exposure in the transportation environment[END_REF] have provided measurement results inside electric vehicles. Nevertheless, few works are dedicated to the identification and evaluation of this electromagnetic field by simulation. This goal is a great challenge because electric vehicle is a complex system to model. Providing tools and methods enabling electromagnetic fields computation will be useful for health protection assessment. Moreover, this tool could be applied to electromagnetic compatibility analysis for industrials manufacturers who need to predict possible issues during design process.

II. PRELIMINARY STUDY

Several electric cars have been studied to evaluate the contribution (magnetic levels and frequencies) of each on-board electrical device. This paper leads to an exposure calculation according to ICNIRP recommendations [START_REF] Icnirp | ICNIRP guidelines for limiting exposure to time-varying electric and magnetic fields[END_REF]. Results show that even if each electric device produces a magnetic field far below the ICNIRP level (see Fig. 1), when considering all the contribution together during a drive test, the exposure criterion (that must remain under 100%), can reach ∼20% at startup.

For every car, the strongest field values are reached near the battery pack or power cables. Depending on their location inside the vehicle, the magnetic field inside the passenger cabin may increase. For example, in recent cars, the battery is often located under the passenger cabin. Even if battery pack is shifted at the rear (in the boot) or in the front (under the bonnet), power cables could locally increase the exposure.

Therefore, our measurements illustrates that for exposure purpose the whole contribution of the on-board equipment must be computed. Our study also shows that the devices in which strong currents (or Ampere turns) flow are likely to generate the highest amplitude to electromagnetic fields.

For the studied cars, high currents are distributed inside batteries and along cables and correspond to extremely low frequencies (lower than a few hertz). For this bandwidth, recommendation levels are far higher than the measured ones. Nevertheless, this level goes down to 27 µT in the 3-30 kHz bandwidth. This frequency range corresponds to new power wireless systems designed to reduce the size of the battery pack and supply energy to the vehicle when driving (see on line electric vehicle in Korea [START_REF] Shin | Contactless power transfer systems for on-line electric vehicle (OLEV)[END_REF]). Inside these structures, lots of Ampere turns are developed, which may generate strong magnetic fields inside the car cabin.

For all these reasons, fields created by high currents must be carefully investigated. In this paper, the first sources we focus on are batteries and cables in a static approach because of the associated frequencies. We will show that their contribution inside the vehicle is strongly correlated to the current arrangement inside the battery, outside the battery, and to the shielding effect of the magnetic parts (chassis and frame). We provide a numerical tool and measurement methodology to assess the source level inside the car. Finally, the numerical model is applied to predict and study the reduction of this kind of contribution inside the cabin.

III. VEHICLE ANALYSIS AND ASSUMPTIONS

A three-door-utility electric vehicle from the late '90s like in Fig. 2 has been chosen for this paper. This car is mainly made of ferromagnetic materials whereas recent vehicles may integrate more aluminum, plastic or carbon fiber parts for lightening purpose.

The vehicle ferromagnetic parts acquire a magnetization when subjected to local magnetic fields. This magnetization may produce a shielding effect inside the car with respect to these magnetic fields sources. Shielding effect must be characterized because most electric devices on-board vehicle are located outside the cabin and thus, their magnetic fields are attenuated by the car ferromagnetic body. As noticed, recent cars should integrate less ferromagnetic parts like plastic and thus the shielding effect could be reduced.

As explained in Section II, we will mainly focus on modeling sources like batteries and cables, which are static or extremely low-frequency weak field sources. In a first step, the shielding effect is characterized inside the earth magnetic field (weak field of ∼40 A/m in France) because this characterization is easier in a uniform field. Then, fields created by high currents (few hundreds of amperes) flowing inside the power batteries and associated power cables will be investigated. In both cases, the proposed modeling is magneto static.

IV. NUMERICAL MODELING

In our approach, the geometry of the ferromagnetic parts of the vehicle and the associated surface meshing are created in Flux3-D [START_REF]Flux 3D-Cedrat[END_REF] software. The problem is solved with an integral formulation developed in our own software Locapi [START_REF] Chadebec | A review of magnetostatic moment method[END_REF], better adapted than finite-element method to compute accurately the magnetic field in air region. As the method is associated to a fully dense matrix system, an adaptive multilevel fast multipole method is used to limit memory requirement and to increase the computation speed. The magnetic moment method is precisely presented in [START_REF] Nguyen | Ships magnetic anomaly computation with integral equation and fast multipole method[END_REF].

In the following parts, a field applied along the longest dimension of the car is called longitudinal, transverse if along the width and vertical if along the height.

A. Induced Magnetization due to Earth Magnetic Field

Earth magnetic field varies on the globe surface but can be assumed homogeneous and constant on the zone where a car is moving. Depending on the car heading evolution, the local magnetic field varies in the interior. Then, the shielding effect is characterized for longitudinal and transverse directions (vertical earth field is considered constant and vertical magnetization variations negligible). The characterization in the earth field presents two main advantages: earth field is simple to model because it is homogeneous and it corresponds to reversible magnetization that is associated to an isotropic constant relative permeability. Due to shape effect, the reversible permeability has a minor influence on the internal field measured at different points and thus is set to µ r = 150. The same value will be used when the inductor field will be due to the power currents of the battery and the cables, showing that the low field assumption is still valid. As the depth of the different magnetic parts is not precisely known, they are determined iteratively by studying the fitting between the measured and the computed field inside the vehicle.

Fig. 3 shows the obtained model of the car. The chassis frame (light blue) is 3 mm thick and the body shell (gray) is 2 mm thick. Other elements have a local influence and a specific effect on longitudinal or transverse field direction, depending on their shape. In the studied car, there is a vertical grid separating passengers from the rear: modeled (green) with a 1 mm equivalent thickness, it is only sensitive to a transverse magnetic field. Seats also have a ferromagnetic skeleton that affects locally the magnetic field. They are described by a simple shape (blue) of 1 mm thickness and a thin surface of 0.5 mm equivalent thickness (red) for the grid supporting the filling part of the seat. The seats add a local sensitivity to both longitudinal and transverse magnetic fields, for the body and only to transverse magnetic field for the head. Finally, the battery pack is fixed under the rear plate on transverse spars. These spars modeled (magenta) with a U-shape, are 2 mm thick and present local sensitivity to transverse magnetic field.

B. Magnetic Field Created by Currents Inside the Power Battery and Associated Cables

Knowledge and analysis of the cell setup inside the battery casing and of the power cables allow building an equivalent current model of the power circuitry (Version 01 in Fig. 4).

In this section, the half battery pack, located in the rear part of the vehicle, under the boot, is presented (Version 01 in Fig. 4). Li-ion cells are arranged in series to create lines along This rear battery pack model and a part of the associated power cabling are added to the previous vehicle model (Fig. 3) to evaluate the magnetic field generated inside the car. Module of the magnetic field is computed (Fig. 5) along a centered line running from the rear to the front of the vehicle. This line is located 150 mm above the boot plate (i.e., 160 mm above the battery pack casing) and results are for a total current of 100 A.

As expected, the ferromagnetic parts act as a magnetic shield. Peak-to-peak amplitude module is ∼43 µT when the battery pack is alone in the air and reduced to only 25 µT when the vehicle magnetic frame is present. Transverse field component is negligible on the line (amplitude is <2 µT without ferromagnetic material). The longitudinal field component is reduced by a factor three: from ∼45 µT amplitude to 15 µT. Vertical component is less shielded: from 43 to 25 µT.

The other half battery pack is located in the motor compartment and follows the same geometrics rules as for the rear one. The whole power circuitry is added to the vehicle model (Fig. 3) and Fig. 6 shows the material magnetization (in A/m) of the vehicle frame, due to a 100 A current flowing in the whole dc power circuit composed of the rear battery pack connected in series with the front battery pack through power cables (Version 01 in Fig. 4).

V. COMPARISON WITH MEASUREMENTS

As mentioned, some measurements have been made onboard the vehicle at different points: passenger location (see the measurement system inside the car on Fig. 7 and characteristics in Table I) and above the rear plate, near the rear battery.

For longitudinal and transverse directions, Table II lists the ratio between the field component inside the vehicle and the earth magnetic field. Table III lists the magnetic field variation due to the electric power current variation. For both cases, a good correlation is obtained between model and measurements.

A first forward model has been built and validated for magnetostatic sources inside an electrical vehicle-like structure. Besides, this numerical model allows the study of any current distribution (e.g., the cell distribution inside the battery pack or power cables for connection) to minimize its magnetic stray fields inside the vehicle cabin.

VI. APPLICATION TO THE POWER CIRCUIT FIELD REDUCTION

In the studied vehicle, the original battery pack has been updated with Li-ion cells but the battery casing has been kept. On this casing, the battery connectors are separated and located on opposite lateral sides (refer to Version 01 in Fig. 4). As current goes into the battery on one side and goes out on the other side, cables create a quite large current loop that generates a magnetic field along the vertical direction. Therefore, a simple cabling modification should drastically reduce this field; both connectors are then gathered together into a single bipolar connector on one side (see Version 02 in Fig. 4). Electrical current cell distribution has also been modified to minimize their contribution because of electrical loops management. Li-Ion cells remain in series to form lines along the width of the vehicle but for each line, we tried to have its neighbors with an opposite electric current. Then, a line is connected in series with its direct neighbor below to form small electrical loop and obtain the good voltage. The final distribution is composed of 10 columns, each counts four vertical loops and the next column has its loops inverted. Every loop is connected in parallel to obtain the good energy.

Simulation is processed and the field is computed (Fig. 8) on the same line as in Section IV. Results are presented for the rear battery pack and a part of the associated power cabling, for a total current of 100 A and without any ferromagnetic part.

Peak-to-peak amplitude is reduced to ∼7.5 µT when power cables are gathered together and again divided by a factor three (∼2.5 µT) when electrical current cell distribution is also optimized (compared with 43 µT with the original configuration).

The same modification principle is applied to the front battery pack and cables. The whole modified power circuitry (Version 02 in Fig. 4) is added to the vehicle model (Fig. 3) and Fig. 9 shows how the material magnetization (in A/m) of the vehicle frame has been also reduced when 100 A current flows in the whole optimized dc power circuit. VII. CONCLUSION Electric vehicle investigation allows to assess the contribution of each device found on-board (magnetic levels and frequencies dependencies).

A simple vehicle-like structure has been described with Flux3-D software to describe geometry and mesh, and simulation has been processed with an integral formulation. For both uniform (magnetic earth field) and nonuniform (created by electrical current distribution) magnetostatic field, the model has been validated because of the measurements. Thus, a useful modeling tool has been obtained that quantify the field inside an electrical vehicle by considering the shielding effect due to ferromagnetic parts with respect to on-board sources. This shielding effect may correspond to an attenuation of two or three.

We mainly focused on sources carrying high currents across the vehicle like the batteries and connecting conductors because our study has shown that they were mainly responsible for the higher fields inside the car. Finally, a complementary study of the electric current distribution demonstrates that simple rules lead to the strong minimization of the created magnetic field (∼18×).

This paper is a first important step in willing to characterize and model magnetic fields inside electric vehicle for exposure or CEM purposes. In further work, it may be applied to any new structure that develops strong current and special attention should be paid to the modeling of higher frequency ranges.

Fig. 1 .

 1 Fig. 1. Magnetic field level and bandwidth encountered compared with ICNIRP recommendation.

ForwardFig. 2 .

 2 Fig. 2. Studied three-door-utility electric vehicle.

Fig. 3 .

 3 Fig. 3. Numerical model of the vehicle main influence magnetic part.

Fig. 4 .

 4 Fig. 4. Equivalent electrical current model of the power circuit.

Fig. 5 .

 5 Fig. 5. Magnetic induction created by the original rear battery pack in which 100 A flows at 160 mm above the battery casing.

Fig. 6 .

 6 Fig. 6. Magnetization in A/m due to 100 A inside the whole power circuitry.

Fig. 7 .

 7 Fig. 7. Non-magnetic mannequin equipped with triaxes sensors.

Fig. 8 .

 8 Fig. 8. Magnetic induction created by different versions of the rear battery pack in which 100 A flows at 160 mm above the battery casing.

Fig. 9 .

 9 Fig. 9. Magnetization in A/m due to 100 A inside the modified power circuitry.

TABLE I SENSOR CHARACTERISTICS TABLE II ATTENUATION

 ICHARACTERISTICSII FACTOR OF THE EARTH MAGNETIC FIELD COMPONENT

TABLE III MAGNETIC

 III FIELD VARIATION DUE TO ELECTRIC CURRENT VARIATION

Electrical Engineering Laboratory of Grenoble, University of Grenoble, Grenoble 38400, France

CEA-Leti, Grenoble Cedex 9

38054, France Magnetostatic field inside a simplified electric vehicle-like geometry is analyzed. The developed model allows evaluation of the shielding effect of the magnetic chassis and frame. It also proposes an equivalent model for the power battery and cables. The study of different configurations leads to the field reduction inside the car.Index Terms-Electric vehicle, magnetic field measurement, magnetic field simulation, magnetic shielding, magnetostatics.

ACKNOWLEDGMENT

The authors would like to thank Institute Carnot for its support and for the contributions of CEA Liten and members of the EM safety project consortium. Further information can be found at: www.sintef.no/Projectweb/EM-Safety.