Optimal-state Dynamics Estimation for Physics-based Human Motion Capture from Videos

Authors: cuong  le
viktor  johansson
manon  kok
bastian  wandt
Venue: NeurIPS_cc_2024_Conference
Published: 2024-04-23 15:24:01

Abstract: Human motion capture from monocular videos has made significant progress in recent years. However, modern approaches often produce temporal artifacts, e.g. in form of jittery motion and struggle to achieve smooth and physically plausible motions. Explicitly integrating physics, in form of internal forces and exterior torques, helps alleviating these artifacts. Current state-of-the-art approaches make use of an automatic PD controller to predict torques and reaction forces in order to re-simulate the input kinematics, i.e. the joint angles of a predefined skeleton. However, due to imperfect physical models, these methods often require simplifying assumptions and extensive preprocessing of the input kinematics to achieve good performance. To this end, we propose a novel method to selectively incorporate the physics models with the kinematics observations in an online setting, inspired by a neural Kalman-filtering approach. We develop a control loop as a meta-PD controller to predict internal joint torques and external reaction forces, followed by a physics-based motion simulation. A recurrent neural network is introduced to realize a Kalman filter that attentively balances the kinematics input and simulated motion, resulting in an optimal-state dynamics prediction. We show that this filtering step is crucial to provide an online supervision that helps balancing the shortcoming of the respective input motions, thus being important for not only capturing accurate global motion trajectories but also producing physically plausible human poses. The proposed approach excels in the physics-based human pose estimation task and demonstrates the physical plausibility of the predictive dynamics, compared to state of the art. The code is available on https://github.com/cuongle1206/OSDCap.

Main paper: 
 introduction: 
 Three-dimensional human motion estimation is a long-standing and challenging research goal in computer vision, particularly in monocular scenarios due to inherent depth ambiguities. Previous approaches have incorporated kinematic priors, e.g. by enforcing smoothness, maintaining bone length constancy, or imposing symmetry constraints. However, due to inconsistencies of frame-wise predictions, these solutions do not necessarily lead to physically plausible motions. This has led to the emergence of a new research direction that combines traditional 3D motion estimation with physical models of the human skeleton. Instead of directly predicting a human pose, these approaches estimate the internal joint torques and exterior forces that drive the motion. Consequently, physics simulators are employed to obtain the resulting motion [37, 38, 8, 50, 21] .However, since simulators are never perfect representations of the real world, they introduce inevitable errors, where the complex human body was never fully modelled, only approximation by rigid body dynamics [3] . Moreover, measurements, including "ground truth" recordings, are inherently noisy. To tackle these problems, we propose OSDCap, a state-aware architecture that combines a differentiable physical simulation with our novel neural Kalman filtering approach. Fig. 1 shows our reconstructed poses predicted from noisy kinematics estimates as well as the estimated dynamics from a video.OSDCap is an online filtering and dynamics estimation that can be trained in an end-to-end manner. In detail, our approach consists of two steps, starting from a noisy kinematics reconstruction obtained by an off-the-shelf video-based 3D human pose estimator: 1) a simulation branch that estimates joint torques using a PD controller and computes the resulting motion, 2) an adaptive filtering stage that combines the output of the simulation stage and the video-based kinematics input to produce a refined motion. We follow prior work that utilizes the meta-PD algorithm for torque calculation [38, 21] to simulate plausible motion. However, the effectiveness of the PD algorithm heavily relies on the choice of the P and D gains [38] and on an accurate model of the human kinematic chain, which is generally unknown. Moreover, the measurements from the monocular 3D kinematics pose estimator contain a large amount of noise, which ultimately leads to inaccurate predictions. Shimada et al. [38] mitigate these problem by introducing an additional offset term into the PD controller. While this approach still produces reasonable output motion it is neither physically explainable nor consistent with PD controllers in control theory. We aim to solve this problem at the root by taking inspiration from control theory and propose a solution for processing the imperfect PD calculation by a learnable Kalman filtering method [36] . The proposed Kalman filter takes the simulated motions and the noisy 3D pose estimation as inputs, combines them, and produces an optimal state prediction as the output. The Kalman filter effectively refines the PD controller-based simulated motions into more plausible and realistic motions. While the Kalman filter fixes inaccurate kinematic measurements from the 3D pose estimator it does not take different weight distributions in the human body parts into account. We calculate an initial weight distribution -the inertia matrix -for an average human body shape. However, as for the skeletal structure, these are only approximations that lead to inaccurate simulations. We mitigate this issue by predicting an inertia bias matrix in each time step which is added to the initial inertia matrix. progress, vision-based human 3D pose estimation is still an ill-posed problem, due to the loss of depth information from the monocular setup. Therefore, captured 3D motions often contain different types of implausibility, ranging from unnatural poses, jittering, or unrealistic body artifacts [46, 8] . 
physics-based 3d human motion capture: 
 Recent studies [37, 50, 46, 8, 21] enforce physics as constraints for motion reconstruction, eliminating implausible artifacts created by the monocular estimation, i.e. jittering, ground penetration, and unnatural human poses.Motion imitation using reinforcement learning (RL) is a popular approach for simulating physically plausible results [32, 49, 50, 33, 51] . RL-based methods enforce physics constraints in the reward functions, either from manually-designed formulas, or from physics engines. The bottleneck of RL-based approaches is the low transferability of the learned policies to unseen motions. Motion optimization is another common approach for physics-based human motion capture. However, optimization problems often require a differentiable framework, thus, instead of relying on nondifferentiable physics engines, prior studies [34, 37, 46] adapt simplified motion equations [5] as a dynamics constraint for simulated motions. More recent approaches [13, 9] manage to optimize through non-differentiable simulation using evolutionary optimization methods [11] . Gärtner et al. [8] implement a differentiable version of PyBullet [3] , resulting in an optimizable framework with complex physics engines. However, most optimized motion solutions, similar to RL-based solutions, have limited adaptability to different data distributions, requiring re-optimizing on new sets of action.Utilizing the generalizability of neural network models in an end-to-end manner is still an open line of research, due to the difficulty of finding physical plausibility patterns from data. Rempe et al. [35] utilizes a variational autoencoder architecture for predicting plausible motions, approximating the dynamics simulation by a decoder network. This assumption might result in unrealistic force prediction with respect to biomechanics literature. Li et al. [21] utilize the meta-PD controller with learnable parameters for torque prediction, but with an additional compensation term based on root residual forces. Zhang et al. [52] realize a transformer-based autoencoder to refine kinematics input sequences, while integrating physics constrain inside the latent embedding. However, both Li et al. [21] and Zhang et al. [52] make predictions based on the encoding of the full motion sample, i.e. they require knowledge of past and future motions, therefore, limiting the applicability of the method to offline setups, where future information is available. Shimada et al. [38] also use a meta-PD controller for calculating the optimal joint torques, which in turn generates a simulated motion matching visual kinematics estimation. Despite the plausibility of the estimated pose, the global precision of the motion in world coordinate is limited and not fully addressed.We aim to leverage physics-based approach (with meta-PD controller) on motion data captured by monocular camera systems, in a recursive online setup, and expand the prediction to more complex practical movements such as sports. 
method: 
 This section presents our proposed approach OSDCap in detail. We start by creating an average proxy character B based on the uniform human configuration from the Human3.6M dataset [15] . The character approximates a human body by circles and cylinders. Additionally, we leverage the pretrained neural network TRACE [40] to obtain an initial 3D pose estimation. Without any additional priors, OSDCap aims to predict the joint torques and external forces that drive the proxy character to match the kinematics evidence given by TRACE. Following prior work, we employ a neural network that predicts the parameters of a meta-PD controller which consecutively predicts the joint torques. While related approaches [38, 21] stop here, we note that the quality of the motion given by the PD controllers' prediction highly depends on the realism of the videos-based kinematic estimation and the proxy character. Since a model can always only be an approximation of the real world, this leads to inaccurate predictions which prior work compensates for by adding an additional offset term to the PD controller. Unfortunately, this not only introduces a non-physical assumption but through experimentation we found that this term attributes the major part to the prediction of the PD controller. We aim to maintain the physical plausibility of our approach by introducing a novel filtering approach inspired by a neural Kalman filter [36] to refine and update the motion states. Additionally, at each time step, the foot contact states and ground reaction forces are estimated directly from the motion leading to a full description of the system dynamics. Fig. 2 shows an overview of our method.Figure 2 : The main pipeline of OSDCap. Our approach consists of one neural network model, OSDNet (orange), and three processing components. OSDNet takes the current system state, estimates a Kalman gain matrix, PD gains, external force and an inertia-bias matrix. The optimal pose estimation performs contains a Kalman filter for the current system state and the input kinematics. Yellow refers to the algorithm's state vectors and cyan denotes processing operations. The physics priors block (gray) computes the inertia matrix and non-linear forces using the Composite rigid-body algorithm and Inverse dynamics [5] . Using the PD algorithm and forward dynamics (Eq. 1), the physics simulation block (green) updates the velocity based on the computed optimal pose and physics priors.Similar to previous studies [34, 37, 38, 46] , we enforce physics constraints based on Rigid Body Dynamics [5] , inline with the Newtonian equation of motion. For a total of N keypoints, the full human pose is represented as a vector q ∈ R 6+3N , encoding the global translation and rotation in the first 6 entries, and internal joint angle states in the remaining 3N entries. q ∈ R 6+3N is the corresponding velocity vector. The motion dynamics of the captured human poses should satisfy the Newtonian equation of motion, expressed asEQUATIONwhere M(q) ∈ R (6+3N )×(6+3N ) is the inertia matrix, computed from the proxy character, q ∈ R 6+3N is the acceleration, τ ∈ R 6+3N are the internal joint torques, λ ∈ R 6+3N are the external forces, and h(q, q) ∈ R 6+3N is the non-linear term including gravitational, Coriolis, and centrifugal forces, computed using inverse dynamics with zero acceleration on the proxy character [6] . Our goal is to estimate the two vectors τ and λ that produce plausible motion dynamics.The proposed OSDCap consists of three main processing stages: an optimal pose estimation, a physics priors calculation, and a velocity update based on physics simulation. The optimal pose estimation phase is a filtering approach inspired by KalmanNet [36] that estimates an optimal output pose based on the current system state and video-based 3D kinematics inputs. The physics priors calculation computes the current inertia matrix and non-linear forces based on the proxy character from the current system state. The physics simulation phase computes the next velocity state using the PD algorithm and forward dynamics (Eq. 1) from the estimated optimal pose.The required inputs for the optimal pose estimation and physics simulation are estimated by our neural network OSDNet. OSDNet consists of two modules, one predicts Kalman gains for the Kalman filtering, and the other predicts PD gains, external forces and inertia-bias for the physics simulation.Optimal Pose Estimation. As shown in Fig. 2 , the Kalman filtering block takes the current system state and the videos-based kinematics pose as inputs. Following traditional Kalman filters the next predict state is computed from the previous state. We define the state transitioning phase asEQUATIONThe predicted positional state q t+1|t is the physics-constrained body pose. The observation matrix H (Fig. 2 ) maps the predicted states q t+1|t to an observed simulated positional state. C is an adaptation matrix to reduce the gap between observed states from videos, and observed states from physics simulation. H and C are optimized along with OSDNet while training, but stay constant during inference. From the current system states, a Kalman state update process is performed asEQUATIONwhere K t contains the estimated Kalman gains at step t based on the current states and observations. Inspired by [36] , the Kalman gains estimation module is implemented as Gated Recurrent Units, which have the ability to propagate the latent motion dynamics throughout the simulated motion via the hidden states of the GRU. The prediction of Kalman gains requires information about the the system's state dynamics [36] , thus four additional dynamics features need to be feed into OSDNet's GRU input, namely: observation, innovation, forward evolution and forward update. They are calculated as ∆evolution = q t|t -q t-1|t-1 ∆update = q t|t -q t|t-1EQUATIONWe modify the original design from [36] due to the practical reasons of our system. In self-occluded scenarios, the noisy input qt often contains artifacts such as body deformation and they often last for a period of time (approx. 10 frames). The intermediate difference between qt and qt-1 in the original design [36] is not strong enough to model those artifacts, because they could both contain the same incorrect kinematic estimation. We change the calculation of ∆observation as in Eq. 4 better deal with mis-detection cases that would cause large responses in ∆observation. The pose q t|t is the optimal state at step t and inherits the global translation estimation from kinematics observations while retaining the physical plausibility of the human pose from the physics simulation.Physics Simulation. The purpose of the physics simulation stage in Fig. 2 is to update the velocity qt|t that best describes the dynamics of the filtering process. Therefore, the estimated pose q t+1|t+1 can be used as the target signal for the PD algorithm, calculating the joint torque τ t that maps the predict pose q t+1|t to the optimal pose q t+1|t+1 . The joint torque is predicted by the PD algorithmEQUATIONwhere κ P , κ D are proportional and derivative gains respectively. Inspired by [38] , the meta-PD controller was applied at this stage, where κ P , κ D are learnable and estimated from OSDNet. By using the filtered optimal pose as the target, no unrealistic temporal filtering or optimization is needed to refine the noisy kinematics inputs.Additionally, the external forces are also estimated by OSDNet, assuming the source of external forces comes only from contact points and is computed asEQUATIONwhere J c t is Jacobian matrix that maps linear velocity at contact point c to rotational velocity of every other joints, ρ c t and f c t are the contact probability and the linear force vector at contact c. The three vectors are separately estimated by OSDNet.Inertia Estimation. Since we do not have access to the real bone length and mass distribution of the human, there exists a knowledge gap between simulated human character and the real human subject, the inertia tensor computed by the composite rigid-body algorithm is sub-optimal. OSDNet is designed to also estimate an inertia bias term M b t that reduces this knowledge gap. The required acceleration to drive the current simulated pose to the next states is calculated as in Eq. 7.EQUATIONTo update the system state, finite interpolation is applied, using the newly calculated acceleration qt . The update process is given by qt+1|t+1 = qt|t + qt ∆t, (8) where qt+1|t+1 is the updated system state that represents the current system dynamics, under physics constraints from gravity and contact forces. The system now proceeds back to the transitioning phase in Eq. 2, creating a closed loop process that works recursively.To reconstruct the optimal state, we define the overall objective loss L as a weighted sum of multiple loss functions asEQUATIONwhere ω 1 = 0.5, ω 2 = 0.1, ω 3 = 0.7, ω 4 = 0.2, ω 5 = 0.4 are weighting factors. The optimal reconstruction losses L q t+1|t+1 t and L p t+1|t+1 t measure the L1 distance between the estimated optimal pose q t+1|t+1 and its corresponding 3D keypoints (obtained from forward kinematics) with the ground-truth poses q GT t+1 and ground-truth 3D keypoints p GT t+1 . The supervision for predict pose q t+1|t is carried out similarly, ensuring the correct behaviour of the physics simulation. L c t is the contact loss, using Binary Cross Entropy measurement between the predicted contact probabilities ρ c t of two feet with pseudo-ground-truth contact binary labels ρc t . We generate the ground truth contact labels for training based on the foot-ground distances of ground-truth 3D keypoints. The individual losses are computed asEQUATIONBy re-introducing a part of the noisy kinematics measurements into the prediction, an additional regularization loss L reg t is beneficial to ensure smoothness and plausibility of the output motions. The regularization consists of three objectives: 1) L acc t is the acceleration loss, computed as the absolute difference between qt and qt-1 , 2) L vel t is the velocity loss, measuring the distance between the first-order difference of ground-truth motion q GT t+1 and of estimated optimal q t+1 , and 3) The friction loss L f ric t encourages the feet to stay in the same position during ground contact. With the regulator weighting of ω 6 = 0.14, ω 7 = 0.03, ω 8 = 0.28, L reg t is expressed asEQUATION4 ExperimentsQuestion: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? • If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [NA] Justification: There are no obvious negative societal impacts of the work performed. Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
datasets: 
 We evaluate our approach on two human motion benchmark datasets. The first and main dataset is the popular Human3.6M dataset [15] . The dataset contains indoor 3D human motion capture data, including 2D and 3D keypoints, skeleton joint angles, and videos. Seven actors perform 15 different actions. Following previous work [38, 21] , the first five subjects (S1, S5, S6, S7, S8) are used for training, and the last two (S9, S11) for evaluation. According to [37] , only actions that have foot-ground contacts were considered. Details about the selected sequences are found in the supplemental document C.The second database is Fit3D [7] . Fit3D contains indoor motion capture data for a variety of exercises. We split the data by taking samples from the 6 actors (s03, s04, s05, s07, s08, s10) for training, and 2 actors (s09, s11) for evaluation, inspired by the setup from [37] on Human3.6M.Since the scene setting from Human3.6M and Fit3D are very similar, we perform an additional evaluation on the new dataset SportsPose [14] , which consists of video-based sport action sequences with corresponding ground truth 3D keypoints. We use this dataset to show out-of-domain performance, since the 3D kinematics estimator TRACE [40] has not been trained on it.The initial motion observation is generated by TRACE [40] . As suggested by [38, 8] , all extracted motions are down-sampled from 100Hz to 50Hz. The samples are aligned to the world origin in the first frame, then split into 100-frame sub-sequences to utilize batch training and evaluation. The proxy character is created with respect to the provided skeleton metadata in Human3.6M [15] , including the mean bone lengths and joint angles configuration. The inertia matrix and bias force (including gravitational, Coriolis, and centrifugal forces) are calculated online using RBDL [6] , based on the state of the proxy character.We train OSDNet in an end-to-end procedure. OSDNet consists of three fully-connected layers, followed by six different heads for PD gains (κ P , κ D ), inertia bias (M b ), contact probability (ρ c ), linear external force from the ground (λ), and Jacobian matrix (J). These six entries are responsible for the motion simulation phase, following Eq. 1 and 6. The GRU units in the proposed optimal-state prediction module (cf. Fig. 2 ) take current system states, additional dynamics features (Eq. 4), and its hidden state h gru as inputs. The output is the Kalman gain-matrix for the Kalman update process. For a details descriptions of the OSDNet's architecture, please refer to the supplementary document A.OSDNet is trained for 15 epochs with a base learning rate of 5e -4 and a batch size of 64. The learning rates from all training processes are scheduled to reduce by a factor of 10 at epochs 10 and 13. LeakyReLU and Layernorm are used as the activation function and normalization for each linear layer of every module. We also apply a training warm-up strategy on the first 5 epochs by increasing the learning rate by factor of 2 to the base learning rate at epoch 5. This helps reducing the impact of unstable physics simulation at the beginning of training, mitigating gradient explosion.There are two standard protocols for the evaluation on Human3.6M [15] . Both of these protocols assess the Mean Per Joint Position Error (MPJPE). This metric represents the average Euclidean distance between the reconstructed joint coordinates and the provided ground truth 3D keypoints. While the first protocol directly calculates the MPJPE for root-aligned poses, the second protocol initially employs a rigid alignment between the poses which is called MPJPE-PA (MPJPE Procrustes Aligned). Since our approach estimates poses in a global coordinate system, we additionally calculate the MPJPE-G in global coordinates which is the MPJPE without frame-wise root alignment. In addition to the different variations of the MPJPE, the Percentage of Correct Keypoints (PCK) measures the percentage of predicted joints that are within a distance of 150mm or less from their corresponding ground truth joint. Unlike the PCK, the CPS measurement [43] determines a pose as correct only if all its joints are estimated correctly according to a threshold value, similar to the PCK. To ensure independence from a specific threshold value, the CPS computes the area under the curve within the 1mm to 300mm threshold range. To evaluate the global translation error, not accounting for the differences between poses, we report the global root position (GRP) error, which calculates the Euclidean distance between only the root joints. We also use the acceleration (Accel) metric from [19] to measure the jitter of the output motions. Accel is computed as the second-order difference between 3D keypoints across all sequence frames.We report the quantitative results of OSDCap and other related work on different metrics in Tab. 1. Due to the novelty of dynamics-based motion capture the evaluation protocols differ significantly across different approaches. Here, we make an effort of consistently structuring approaches with similar evaluation protocols to achieve a fair comparison. To be as comparable as possible we follow the most used protocol introduced by Shimada et al. [37] . We outperform all online approaches in MPJPE, PCK and CPS. For the global error MPJPE-G, we improve upon state of the art by a large margin. Notably, DnD [21] achieves a lower MPJPE-PA. However, DnD's estimation depends on encoding the full action sequence, extracted from temporal convolutions, assuming significantly more knowledge which is not suitable for an online setting. Moreover, AMASS [25] is used as an additional training data source, thereby, not following the standard protocols for Human3.6M. SimPoE [50] achieves best smoothness performance on the Accel metric, due to being constrained by a high-frequency physics engine. However, as the discussion in Sec. Table 2 : Evaluation results on Fit3D [7] and SportsPose [14] . OSDCap improves the kinematics baseline TRACE by a large margin across all metrics. We fine-tune OSDCap (pretrained on Hu-man3.6M) on SportsPose's ground truth keypoints for additional 15 epochs. Even with very noisy inputs from SportsPose, OSDCap still manage to retain the robust estimation thanks to the Kalman filtering process, especially on global translation metrics (MPJPE-G and GRP).pose-wise rigid alignment, is reported for completeness. While being a reasonable metric for singleimage pose estimation, we argue that for physics-based pose estimation, rigid alignments distort the interpretation of the results since they remove all information about global rotation.We additionally evaluate OSDCap on the more challenging motions in the Fit3D dataset [7] . Tab. 2 shows the results. Since Fit3D is recorded in the same setting as Human3.6M, we additionally evaluate on the newer SportsPose [14] dataset to show the generalizability to other motion domains. We improve on the kinematics baseline TRACE by a large margin, especially for global metrics as shown by the MPJPE-G and GRP. Fig. 3 illustrates the benefits of OSDCap. OSDCap significantly reduces the impact of noisy and inaccurate kinematics input when encountering high depth-uncertainty from monocular views, while retaining the correct estimation with respect to the ground truth. The bars on the left represent the predicted Kalman gains, where the y-direction is the direction of the optical axis which indicates a predicted low trust in the kinematics prediction and leads the Kalman filter to prefer the physics simulation. Fig. 3b shows an example (side view) of OSDCap adjusting unnatural leaning into a physically plausible pose by our physics simulation.We conduct an ablation study to verify the impact of the optimal-state estimation process on simulated motions. We sample a subset of data consisting of only the first action class of all subjects in the camera view 60457274 from Human3.6M [15] . S9 and S11 are for evaluation, and the rest for training. This setup creates a suitable challenge to test the proposed method, limiting the types of motion that are seen during training. Tab. 3 shows the original simulated result from straight-foward smoothing methods, PD controller and the improvement by OSDCap. Table 3 : Ablation study on the impact of OSDNet on a subset of Human 3.6M [15] . Naive methods such as median or Gaussian smoothing cannot help with the plausibility of the pose. Without our Kalman filtering process, the PD controller cannot train and estimate the correct dynamics. We also study the effects of the inertia-bias M b and some performance gains has been recorded. Naive approaches for smoothing the noisy input estimation apply temporal filters such as median or Gaussian filter. However, simply filtering the signal does not help the motion to become physically plausible, unnatural poses still prevail. As shown in Tab. 3, naive filtering reduces the jitter of the input motions (reduction in Accel measurements), but does not help with any other metrics.Methods #params. MPJPE ↓ MPJPE-G ↓ MPJPE-PA ↓ PCK ↑ GRP ↓ Accel ↓ [mm] [mm] [mm] [%] [mm] [mm/s 2 ]To ensure the plausible physics constraints of the forward dynamics process (unlike [37, 38] ), we employ external forces into the calculation, which leads to a much more challenging scenario for the PD controller. This can be observed in Tab. 3 where the PD controller struggles to reconstruct the motions, even with temporal filtering on the input signals and increase the number of parameters. By using our optimal-state estimation module, the PD controller has a significantly better performance, leading to the optimal results for online human motion reconstruction.The biggest challenge of using classical Kalman filter for OSDCap is the tuning of unknown noise covariances of both the kinematic input TRACE [40] and the simulated result from PD controller. Our choice of a learnable Kalman filter [36] relieves us from trial-and-error process of finding the correct noise covariance matrices and achieves the best results. We conducted an additional experiment where we replace our learnable filter by a traditional one, the results are shown in Tab. 4.Assuming noise covariances that are constant over time and equal in all directions, the ratio between the noise covariance of the simulated PD controller (process noise) and the noise covariance of the kinematic input TRACE (measurement noise) governs the quality of the Kalman filter estimates. The evaluation results can be seen in floating above the ground is ignored and produces no error, we propose using an additional grounddistance (GD) metric. For foot-skating, we followed DiffPhy to compute the percentage of frames that contain skating artifacts over the whole sequence.We provide additional metrics for physic-based measurements introduced in Sec. 3.3. The results can be seen in Tab. 5. OSDCap helps refining the input kinematics on most of the physics-based metrics. Note that TRACE [40] outperforms our approach in the ground penetration metric. The reason is that in most cases the TRACE predictions float above the ground, which gives a low penetration error but can be seen as equally bad. Thus, we additionally provide a ground distance metric (GD) to reflect the correct foot-ground quality during contact. The value is computed as the mean absolute vertical differences between foot contact points and ground plane during contact duration, expressed asEQUATIONwhere ρ c is the predicted binary label of contact, p OSD c and p GT c are the 3D vertical positions of contact. There are a total of six contact points considered, three contacts in each foot accounting for heel, foot and toe. The joint configuration follow Human 3.6M skeleton [15] , with bone length between joints optimized during training and fixed during inference. 
conclusion: 
 This paper presents OSDCap, a new physics-based approach to reconstruct kinematics-based human motion captured from monocular videos. We found that previous approaches relying only on a physical simulation produce non-optimal motions due to unavoidable imperfections in the physical model and noisy measurements. This led us to introduce a learnable Kalman filtering for refining implausible motions simulated by a PD controller with noisy kinematic evidence as the target. In comparison with related research on physics-based motion capture, the proposed approach achieves state-of-the-art results on the Human3.6M, Fit3D, and SportPose datasets, especially on the global estimation of pose trajectories.Limitations and future work. While taking a step into highly accurate predictions of the full body dynamics, our physical external forces are still not comparable to directly measuring with mechanical force plates. However, our approach only requires a single camera, e.g. from a smartphone, instead of a motion capture studio or other expensive hardware, such as force plates, to estimate meaningful forces. In the future, detailed modeling for the hands, feet, and body shape, will be investigated, targeting more realistic motion reconstruction. 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA]Justification: The work does not have a high risk for misuse such that safeguards are needed.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: The creators or original owners of assets (e.g., code, data, models), used in the paper are properly credited.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. 
 
 Appendix: A Network details Fig. 4 shows the architecture of the proposed OSDNet. The network consists of multiple branches for Kalman gains, PD controller gains, inertia-bias, and external force estimations. The Kalman estimation module takes the combined inputs from the GRU's and current states embedding, outputting the Kalman gains matrix. The diagonal of Kalman gain matrix is initialized to be approximately 0.5 by modifying the bias of the last linear layer. This is due to the instability of the PD branch at the beginning of training, which may cause gradient explosion if the Kalman gains are too low (zero trust in the kinematics stream). The hidden states h GRU of the GRU unit are updated throughout the simulated sequence and used as one of the inputs for the next prediction.Similar to [38] , we scale κ P and κ D differently for global translation, global rotation, and joint angles. The initial scaling are [30.0, 14.0, 1.9] for κ P and [1.5, 0.1, 0.05] for κ D . Notice that our initial gains are much lower than [38] , because we want to explain the global motion by external reaction forces, avoiding the need for "unrealistic" residual force. These scalings are further optimized along with the training of OSDNet. t and external force λ t are estimated. The proposed GRU unit with size 128 takes the dynamics features (mentioned in Sec. 3.2) as input, the Kalman gain matrix K t is estimated from the concatenation of GRU and the state embedding. The hidden state h gru is continuously updated at each time step. For a better estimation of foot-ground contacts and reaction forces, we also feed the feet position and linear velocity as additional inputs.The simulated proxy character is created based on the body configuration of the SMPL model [24] . Bone lengths and weight distribution are the same as in the Human 3.6M metadata of the 'common' human body [15] . Fig. 5 is a visualization of the proxy character, composed of spheres and cylinders. The bone lengths are treated as extra learnable parameters and optimized along with the model during the training process. During testing, the bone lengths are fixed to the ones learned during training, i.e. no ground truth bone lengths are used when testing.Figure 5 : The simulated proxy character used in the paper. The RBDL library [6] is used to extract the inertia matrix M t and bias forces h(q, q).As mention in Sec. 4.1, we evaluate our proposed method on Human3.6M [15] , Fit3D [7] , and SportsPose [14] . For Human3.6M, we follow prior works [37, 38, 21] to consider actions that only involve foot-ground contact (S1, S5, S6, S7, S8) for training and (S9, S11) for testing. For [7] , we apply the same protocol with only foot-ground available actions are used: (s03, s04, s05, s07, s08, s10) for training and (s09, s11) for testing. For SportsPose, we only consider sequences that contain human at time step 0: (S02, S03, S05, S06, S07, S08, S09) for fine-tuning and (S12, S13, S14) for evaluation.TRACE [40] is used to extract the kinematics input from the data. All extracted motions are aligned at the origin in the first time step, eliminating the effect of wrongly calibration process. Each action is then equally split into 100-frame sub-sequences, utilizing batch processing for training the OSDNet.Since there are ground truth contact labels are provided in all three datasets [15, 7, 14] , we generate our own annotations based on the ground truth keypoints. To create contact labels, ground truth feet 3D positions are considered. If a foot position is within 10 cm (already compensated for shoes and inconsistent MoCap sensor placement) above the ground plane and also not moved more than 2cm from the previous frame, it is labeled as a valid contact point. Similar protocol is applied for both left and right foot. The foot-ground contacts are modelled directly by the OSDNet and automatic data annotation, using the ground truth 3D poses from the training data set.To mitigate the Gimbal lock problem in the original Euler representation of Human3.6M [15] , we convert all root rotation (d.o.f 3 rd to 6 th of the state vector q t|t ) into quaternions quat t|t = (x, y, z, w), Figure 6 : Example results on SportsPose [14] test data. Here we show four out of five action classes of SportsPose [14] that have foot-ground contacts. Qualitatively OSDCap matches the provided ground truth much better than the kinematics input TRACE.with the real part at the end. Since quaternions are not a linear representation, the computation of quaternion differences is given asEQUATION)∆quat t|t is the input error term for the PD controller for computing the corresponding root torque by Eq. 5 and Eq. 1. The procedure for finite integration (during the state transitioning stage) from state vector q quat t|t to the predict state q quat t+1|t given the system state vector qquat t in quaternions is expressed asEQUATIONThe proposed pipeline of OSDCap was trained and evaluated on the NVIDIA-A100 GPU with 40Gb of memory. In average, OSDCap requires an additional 0.02 second on top of the processing time of the kinematics estimation [40] on each frame. Each ablation study in 4. Besides, we also train and evaluate OSDCap on multiple random seed values to demonstrate the reproducibility of results. Table 1 presents the means and standard deviations of the evaluation results across multiple random seeds from 0 to 4. We did not report error bars for every other experiment since it would be too computationally expensive. One can refer to our additional supplementary material for a better visualization of the OSDCap reconstructions against the kinematics input and ground truth. Some example footage on the challenging SportsPose dataset can be seen in Fig. 6 .Additional visualization of estimated pose overlayed on 2D input images can be found in Fig. 7 .There is always a trade-off between the reprojection error and the model-based assumptions. In our case the physics simulation uses stronger assumptions than a purely kinematics-based model. On the other hand, it produces more plausible motion as shown in Tab. 1 of the main paper and Tab. 5.Despite not being a training objective during training, the re-projected poses match well with the input humans in the input image as shown in Fig. 7 . The slight offset is due to the mis-match bone length between the proxy character and the actual testing human subjects. An adaptive human shape estimation would be investigated in the future.NeurIPS Paper Checklist Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 


Decision: The paper proposes an approach to estimate the 3D pose of a single-person from a monocular camera image. The approach uses a differentiable Kalman-filter that combines a kinematic motion estimator a differentiable physics formulation, which can be trained using a supervised learning approach. Doing so, the approach addresses limitations of previous work (such as jittering). Moreover, the approach outperforms the existing methods in the same online estimation setting while each technical component was validated through ablation studies. 

The reviewers mainly raised issues considering related work and baselines, as well as technical obscurities. However, the authors were able to clarify the open points and provide the missing quantitative results during the rebuttal period. 

Decision:
After a fruitful discussion all reviewers voted for acceptance. I therefore recommend to accept the paper and encourage the authors to use the feedback provided to improve the paper for its final version.

Reviews: review: paper_summary: paper_summary: The kinematic motion estimation suffers from inconsistencies of frame-wise predictions while the physics-based method suffers from the gap between the simulator environment and the real-world ground truth. This paper proposes a method to taking advantages of both by connecting them by a learnable kalman filter network. The proposed method OSDCap is a new physics-based human motion and dynamics estimation method. The proposed Kalman filter takes the simulated motions and the 55 noisy 3D pose estimation as inputs, combines them, and produces an optimal state prediction as the 56 output.  It also has a learnable inertia prediction for weights distribution, that produces plausible motion as well as valuable estimates of exterior forces and internal torques.
strength_weakness: strength_weakness: - Some related works are missing in the experiments, especially considering that kalman filter is widely used for human pose tracking. Moreover, as the authors proposed a learnable module to replace the role of Kalman filter (by predicting the gains), it would be helpful to compare the proposed method and a baseline with the classic kalman filter parametric model.
- There are some missing baselines, especially the kinematic ones in the Table 1, for example PointHMR (CVPR’2023) and KTPFormer (CVPR2024), considering that the papers are available before the submission of this work, is there any reason that some recent works are not listed in the comparison?
questions: questions: I am overall satisfied with the novelty of the proposed method with no previous work having implemented the straightforward but well-connected idea as presented in this work as far as I know. However, I will need more evidence about the experimental significance to adjust my rating. Or, could the authors clarify the reason of the missing of more recent baseline works? I would need more evidence to recognize that the kalman filter (at least the learnable one in this work does work generalizability  and is more superior than the classic parametric KF).
limitations: limitations: The authors have discussed the limitations in the paper. They use the widely distributed datasets with human faces but they are controlled in the studio environment and under consent.
paper_summary: paper_summary: This paper introduces a novel method to estimate 3D human motion from a single camera, aiming for physical plausibility and accuracy. The authors use a differentiable Kalman-filtering approach in an online setting that balances kinematics estimated from an off-the-shelf algorithm (TRACE) and the physics simulation. They built OSDNet, which consists of fully connected and GRU layers, for the given optimal state human dynamics capture task. They show their method outperforms the existing methods in the same online estimation setting while each technical component was validated through ablation studies.

In general, the method sounds reasonable and novel. However, there are several things that need to be cleared.
strength_weakness: strength_weakness: It is not clear how they constructed the character. They mention that they used metadata of Human3.6M dataset. Does it mean they use the ground-truth bone lengths of the subject?
– If so, they should state their method with automatic character generation (similar to what SimPoE did) for a fair comparison. It is not clear if their outperformance is coming from the known limb length assumption or the method itself.
– If not, please state more clearly how they generate the character in the physics simulator.

It is not clear why they selected TRACE as a default method. From Table 1, TRACE seems to significantly underperform VIBE, which has been widely used as a baseline kinematics estimator (e.g., SimPoE). The major advantages of using TRACE are its capability to track a person's identity and the robustness of the camera motion. However, all training and evaluation were done in a fixed camera coordinate system.

In the paper, they mention that since TRACE's output is expressed in the first frame global coordinate, their method is agnostic to the initial calibration (line 473). I disagree with this line. Isn't the initial pose of camera essential to know the gravitational direction? Will this method be robust to the camera which is highly tilted? 

In Table 3, conventional PD methods are showing catastrophic performance. I am doubtful about this, maybe the training configuration was not set properly? This is telling us that neural PD control is destroying the performance which does not align with existing literature. 

Many of the state-of-the-art video-based kinematics estimation models are not stated in the paper including HybrIK (CVPR 2021), CLIFF (ECCV 2022), PMCE (ICCV 2023), ReFit (ICCV 2023), MotionBERT (ICCV 2023), and HMR2.0 (ICCV 2023). To the best of my knowledge, all these methods (not limited to) have better performance on Human3.6M dataset.
questions: questions: [Some are duplicated with the Weakness section]
– How did you construct a physics-based 3D person model?
– Why did you use TRACE? IF you value the potential of future use case with moving camera, this needs to be stated.
– Why do neural PD controllers perform so badly in your test case?
– I guess the GRU unit is uni-directional, am I correct?
limitations: limitations: –
paper_summary: paper_summary: This work focuses on tackling the problem of single-person motion estimation from a monocular video. Current approaches produce temporal artifacts such as jittering. Most approaches are entirely kinematic while others that combine physics, do it by re-simulating the kinematic inputs by using automatic PD controllers. These methods, however, require simplifying assumptions that compromise the motion realism.This method combines information from two sources to produce an optimal estimate of single-person 3D human motion. One source comes from an off-the-shelf kinematic motion estimation pipeline and the other from a differentiable physics formulation. These two "measurements" about the motion are combined in a Kalman-filter to generate an optimal output.  Here, the authors propose to selectively incorporate the physics models with the kinematics observations in an online setting, taking as inspiration the neural Kalman-filter. The method uses a meta-PD controller and a physics-based simulation step. The Kalman filter is realized via a recurrent neural network which aims to balance the kinematic inputs with the simulation. 

The authors propose an end-to-end model for this purpose which is not trivial to accomplish. The method is capable of capturing accurate global trajectories and, at the same time, producing physically plausible human poses.
strength_weakness: strength_weakness: ### **Presentation**
The qualitative results could be better presented as it is sometimes hard to have a good sense of the pose estimated by the kinematic approach (TRACE) both in Fig. 6 and in the supplementary .gif images. The way the poses and the original video are visualized can be improved. First, I would suggest making the kinematic skeleton more visible as it is “obscured” by OSDCap results. Even better, it would be nice to have SMPL visualizations of GT, kinematics and OSDCap as it is presented in Fig. 1. In my opinion, changing visualization styles within the paper can reduce the presentation quality. I also advise the authors to focus on video results with several examples. If possible I would like to see this as part of the rebuttal, if not, then this should be present for the camera ready version of the paper. 

### **Physics-based metrics**
Sec 4.3: It would be interesting to show the results for more physics-based metrics other than the Acceleration metric, for example, foot skating and ground penetration, which should be corrected by the physics formulation. As authors are modeling contacts and have physics-based losses (e.g., friction, velocity), it would be interesting to know these metrics in comparison with the SoTA or at least the baseline used (TRACE).

### Minor
- (Typo) Fig.2: 4th line: performs contains-->contains.
- (Typo) L295: HMDCap -->OSDCap?

### References
I would advise the authors to include most recent papers that combine kinematic and physics estimates for pose/motion estimation, either in the introduction or related works. For example:
- Ugrinovic et. al., “MultiPhys: Multi-Person Physics-aware 3D Motion Estimation” (CVPR 2024).
- Zhang et. al. “PhysPT: Physics-aware Pretrained Transformer for Estimating Human
Dynamics from Monocular Videos” (CVPR 2024).
questions: questions: - L38: To be clear, what the authors refer to as a "differentiable physics simulation" is the use of rigid body dynamic equations or is it an actually differentiable simulator, e.g.,  Tiny Differentiable Simulator from PyBullet, similar to Gartner et. al?
- L116: Authors create a proxy character (shown in appendix B). I wonder if it is possible to use the humanoid generated by SimPOE or KinPoly for the same end? This latter humanoid seems to be the most realistic representation of a 3D human body for simulation. I would like what the authors think about this and why they chose this specific form.
- Sec. 3.2: How are the contacts modeled? Are they modeled directly with the GRU and data annotations or is there also a specific physics formulation to account for these contacts?
limitations: limitations: Yes.
paper_summary: paper_summary: The paper presents a novel physics-based human motion capture method that is physically explainable, conforming to the PD control theory and rigid body dynamics. The key designs of the method involve an integration of a kinematic Kalman filter and Newtonian equation-based physics simulation, and learnable Kalman gains, PD gains, external forces and robot inertia biases. The method outperforms existing kinematics-based and physics-based motion capture methods on keypoint accuracies.
strength_weakness: strength_weakness: (1) The contact modeling only considers foot-ground contact, ignoring full-body contact that commonly appears in human-object and human-scene interaction scenarios. Besides, the contact on each foot is represented as a force vector on a pre-defined contact point, ignoring changes in the contact point and the resultant torque of the contact.

(2) The method updates the inertia matrix $M$ online. However, the inertia matrix is the attribute of the robot and should be fixed values during the whole motion capture process for better physical interpretability.

(3) To fully examine the generalizability of the proposed method, existing physics-based methods should also be compared on datasets Fit3D and SportsPose.
questions: questions: Typos:

* In Figure 2, "$q_{t|t-1}$" -> "$q_{t+1|t}$"
* In the caption of Figure 2, "performs" is redundant
* In Equation 6, "$\sum_c^2$" -> "$\sum_{c=1}^2$"
* In line 295, "HMDCap" -> "OSDCap"
* In line 478, "the" is redundant
limitations: limitations: One limitation is that the human dynamic model is formulated as a connection of circles and cylinders, which neglects the modeling of geometric details and wearings of humans.
paper_summary: paper_summary: This paper proposes a method to capture physically plausible human motions from monocular videos. The paper builds on top of NeuralPhys and introduces Kalman filtering to alleviate noise from kinematic motions. The framework combines Kalman filtering and physics simulation to be fully differentiable, thus enabling training in a supervised manner. The experimental results show that the proposed approach can achieve competitive performance.
strength_weakness: strength_weakness: - The simulation and filtering refine the initial kinematic poses with dynamic constraints, which do not consider the image observations. The simulation may remove some high-frequency motions; thus, the refined poses may not be consistent with 2D poses. The authors should overlay the 3D pose onto 2D images and show more visualization results so that we can evaluate the discrepancy.
- Although the method predicts contact information, the framework cannot prevent penetrations between humans and the floor since scene dynamics are not considered.
- The optimal pose is predicted from Kalman filtering, while the velocity is obtained from simulation. In previous works, the final body pose is integrated from simulated velocity as in Eq. 2, which can guarantee physical plausibility. However, the optimal pose in this work is further updated with Kalman filtering, which may not be consistent with simulated velocity. Therefore, the final poses may not obey physical laws.
- The work relies on external force, which may result in some unnatural states (e.g., body leaning).
questions: questions: - Do $q_t$, $\dot{q}_t$ in the output still follow rigid body dynamics as in Eq. 1?
- The work builds on top of NeuralPhys with additional Kalman filtering. What is the performance of applying Kalman filtering on simulated or kinematic motion only?
- Humans may penetrate the ground plane due to depth ambiguity. How does the method address penetration between humans and the ground plane?
- The body shape in this work is fixed. Can the trained models be generalized to different skeleton shapes?
limitations: limitations: The limitations are discussed.

Comments: title: Thank you for your rebuttal
Feedback
comment: We thank the reviewer for the additional references and the productive discussion.

Kalman filtering prevails due to its simple yet effective design in tracking problems. The field of tracking is evolving, new challenges and environmental setups have been posed, i.e. 3D human motion tracking from videos. Therefore, it is worthwhile for deeper investigations of Kalman filter and its newer variances to address those problems optimally in real-world applications.
We thank the reviewer for the productive discussion and the upgrade of the rating.
Thanks for the clarifications.
Thanks for the promise from the authors.

I have seen many papers in the past years claiming to use learnable motion modules to beat classic linear filtering-based motion modeling for tracking or forecasting. However, after investigating the new methods on different data patterns and removing the noise from post-processing tricks, etc, I seldom find methods capable of consistently beating the Kalman filter in the area of visual tracking. This is why the most widely adopted multi-target tracking arts in practice is still a powerful detector + a Kalman filter-based tracker, such as SORT[1]/OC-SORT[2]/ByteTrack[3] even in 2024. 

I will be looking forward to the investigation from the authors. 

> [1]: "Simple Online and Realtime Tracking"

> [2]: "Rethinking SORT for Robust Multi-Object Tracking"

> [3]: "ByteTrack: Multi-Object Tracking by Associating Every Detection Box"
We thank the reviewer for the feedback and positive assessment.

We are expecting the evaluation of the related works to be ready soon.
Thanks for the authors' detailed rebuttal. I understand that exploring full-body contact and increasing the interpretability of the inertia matrix could be future work due to their challenges, and I am looking forward to the evaluations on existing physics-based methods. Nevertheless, I believe this work is theoretically valuable and useful to the research field of human motion capture.
We thank the reviewer for the feedback.

**Model-image alignment**

There is always a trade-off between the reprojection error and the model-based assumptions. In our case the physics simulation uses stronger assumptions than a purely kinematics-based model. On the other hand, it produces more plausible motion as shown in Table 1 of the main paper and Table*2 in the rebuttal pdf above.

Despite not being a training objective during training, the re-projected poses match well with the input humans in the input image as shown in Figure*1. The slight offset is due to the mis-match bone length between the proxy character and the actual testing human subjects. An adaptive human shape estimation would be investigated in the future.

**Contact modeling**

We modelled contact through automatic data annotations and predictions from a neural network. Sophisticated contact modelling (i.e. from physics engines) would limit the calculation of a gradient for an end-to-end approach such as ours. However, our simple contact modeling still demonstrates the ability to increase the physical plausibility of the input kinematics as can be observed in Table 1 and Table*1. We will add a discussion to the main paper.

**Additional reference**

Thank you for the additional reference. The Butterworth filtering method used in the paper [A] from Xie and Kry is a good idea for filtering out noisy kinematics inputs before computing inverse dynamics. However, the method requires an empirical selection of cutoff thresholds for each type of motion separately, thus limiting the scalability of the method to different data domains and environmental setups. In contrast, OSDCap provides an end-to-end approach to effectively integrate dynamics information for refining noisy kinematics in an online manner, which is very valuable for real-world applications.

We would like to highlight again that our proposed approach introduces a novel integration of learnable Kalman filters, thereby mitigating problems of previous approaches in physics-based motion capture and producing state-of-the-art results. While we agree with the reviewer’s contact modelling comments, we like to encourage a weighting of the novelty against the non-perfect contact model.
We thank the reviewer for the feedback and final evaluation.

We performed a smaller study in on different parametrizations for a standard Kalman filter in the response to all reviewers above. A more comprehensive study on different variances of Kalman filter family on human motion capture will be investigated as a natural progression of the paper. We are happy to include another parametrization if desired.
Thanks for the rebuttal, which has addressed most of my concerns. However, the rendered images show that the model-image alignment is inferior to purely kinematics-based methods. I hope the authors can discuss the impact of dynamics and filtering on model-image alignment and joint accuracy.

The proposed method cannot guarantee 100% physical plausibility due to the lack of contact modeling and the use of additional filtering. These limitations should be included in the paper.

Additionally, there are existing works [A] that adopt filters to improve dynamics-based methods. The differences between these approaches should be discussed.

[A] Xie, Kaixiang, and Paul G. Kry. "Inverse Dynamics Filtering for Sampling‐based Motion Control." Computer Graphics Forum. Vol. 40. No. 6. 2021.
I appreciate the efforts by the authors to address my concerns. 

I agree that the hyperparameter setting in the classic Kalman filter formulation can be tricky to adjust especially when there are combined factors to consider. Considering the covariance for kinematic input and PD controller outcomes, Multivariate Kalman filtering can be a parametric solution that may be worth future study. Kalman filtering family has been studied for decades but it is still one of the go-to solution in modern tracking and other time series analysis tasks. A comprehensive comparison with different settings and members of this family can significantly improve the convincingness of this paper's soundness.

I am in general satisfied with the responses from the authors and maintain my positive rating towards the acceptance of this paper.
We thank the reviewer for the response and suggestions. We would like to provide the answers to the additional comments below.

**Presentation.**

We also think that separating the visualization into two different figures is a better approach to mitigate the cluttering effects. We will change this in the paper by trimming the uninformative background to make space for the two poses.

**Public code.**

Yes, the implementation for training and testing will be made publicly available upon acceptance as promised in the paper.

**Differentiable engines.**

Integrating an unconstrained physics simulation into our approach is possible. To our experience, it would bring little to no benefits due to the similar rigid body modelling and numerical integration. 
However, the most valuable aspect of a physics simulator is the ability to model contacts and object-interaction effectively. These sophisticated contact modeling results in non-differentiable constrained simulation, leading to an intractable gradient for end-to-end learning.

Promising progress has been made recently to enable differentiability in physics simulations [G, H], and we are currently working towards integrating these newer simulations into physics-based human motion estimation.

[G] Werling et al., NimblePhysics: Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact, RSS'2021.

[H] Hu et al., DiffTaichi: Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact, ICLR'2020.

**Dynamic predictions.**

In this context, we refer dynamic predictions to the estimates of the joint torques and ground reaction forces.
We thank the reviewer for the assessment and the upgrade of the rating.
Dear Authors,

Thank you so much for your solid rebuttal on my review. After carefully reading your response, I would like to increase my initial score to "5".
Thanks for the authors’ response and the rebuttal document. I have some additional comments/questions. 

#### **Presentation Improvement.**
I see what you did in Figure*2, in my opinion this looks much better. This is more of a nuance, I would further suggest breaking each sequence in two images: one image corresponding to the baseline vs. GT and the other for the proposed method vs. GT, this way the comparison is even clearer. Otherwise you could be masking failures of either the baseline or your method behind the clutter. 

#### **Physics-based metrics.**
The results presented in Table*2 of the rebuttal look good. The use of the GD metric makes sense to me as it is true that predicted motions that contain floating can falsely show very good GP metrics. Also, the results presented are quite good for the proposed method which makes the work potentially more impactful. Will authors release the code so that these results can be reproduced?

#### **Differentiable simulation in OSDCap.** 
I see, thanks for the clarification. This is more of a curious inquiry: I would like to know the authors’ opinion/intuition on how would these fully featured differential simulators work with their approach, if they use it to replace the rigid body dynamics equations and Euler's integration? Could this framework allow for such a change?

#### **Selection of proxy characters.**
For clarity, I would like to know what do the authors exactly mean by “dynamic predictions” in this context.