# Controllable Motion Synthesis and Reconstruction
with Autoregressive Diffusion Models

Wenjie Yin1, Ruibo Tu1, Hang Yin1, Danica Kragic1, Hedvig Kjellström1, Mårten
Björkman1 1Division of Robotics, Perception and Learning, KTH Royal Institute
of Technology, Sweden. {yinw, ruibo, hyin, dani, hedvig<EMAIL_ADDRESS>
###### Abstract

Data-driven and controllable human motion synthesis and prediction are active
research areas with various applications in interactive media and social
robotics. Challenges remain in these fields for generating diverse motions
given past observations and dealing with imperfect poses. This paper
introduces MoDiff, an autoregressive probabilistic diffusion model over motion
sequences conditioned on control contexts of other modalities. Our model
integrates a cross-modal Transformer encoder and a Transformer-based decoder,
which are found effective in capturing temporal correlations in motion and
control modalities. We also introduce a new data dropout method based on the
diffusion forward process to provide richer data representations and robust
generation. We demonstrate the superior performance of MoDiff in controllable
motion synthesis for locomotion with respect to two baselines and show the
benefits of diffusion data dropout for robust synthesis and reconstruction of
high-fidelity motion close to recorded data.

##  I INTRODUCTION

Motion synthesis techniques play an important role in computer animation,
video games, human-robot interaction [1], etc. Recently, significant progress
has been achieved in motion generation and reconstruction by utilizing deep
generative models [2], which can be broadly divided into deterministic and
probabilistic models. Deterministic models [3, 4] frame the motion synthesis
task as a regression problem in which the response and input have exact
relationships, leading to stereotypical results with limited diversity. In
contrast, probabilistic models fit probabilistic distributions to the data
distribution [5, 6], which capture a range of motion patterns and as such
significantly improve the motion diversity and fidelity.

Despite recent advances in deep generative models, motion synthesis still
remains challenging in a number of aspects. For example, capturing complex
relations between body limbs and motion frames requires models that are less
susceptible to failures such as mode collapse. Also, robust and coherent
synthesis is desirable even when long-term generation is conditioned on
imperfect data. The latter is particularly demanding when human skeletal data
are extracted from noisy sensors or previously generated frames. Earlier work
often assume perfect conditioning patterns [5] or manually defined graph
structures [2], and thus cannot satisfactorily mitigate this issue.

Figure 1:  Illustration of the generation process: MoDiff progressively
denoises a noisy motion sequence to get a natural movement, given past motion
and control contexts.

In this paper, we propose MoDiff, a diffusion-based probabilistic model for
high-quality controllable human motion synthesis, as illustrated in Fig. 1.
Diffusion-based approaches have recently gained traction for their superior
performance as probabilistic generative models and found application also in
human motion synthesis [7, 8]. Score-based diffusion models require no special
neural network architectures, such as in the case of flow-based approaches [5,
2], to optimize an exact likelihood. Our work leverages this flexibility by
introducing a cross-modal Transformer-based architecture, which enables richer
representations for encoding past motion frames and control contexts. The
approach also exploits the intermediate representations generated in the
diffusion process for a natural dropout strategy to improve robustness. Our
evaluation on domain standard human locomotion datasets shows that MoDiff
outperforms baseline methods and produces realistic motions conditioning on
control contexts of other modalities. We further demonstrate that the same
framework can be applied for reconstruction of imperfect motion sequences.

In summary, our contributions are:

  * •

We present a flexible neural architecture, MoDiff, that integrates multimodal
transformer and autoregressive diffusion models for motion generation and
reconstruction of missing parts in motion sequences.

  * •

We propose diffusion data dropout, utilizing the forward process to obtain
diffusion-induced motion representations, which can be employed to improve
adherence to the control context.

  * •

The proposed approach achieves superior results for human motion generation
and reconstruction. Applications include, but is not limited to, locomotion
synthesis, text-to-motion, and music-to-dance.

The paper proceeds as follows: Section II gives an introduction to previous
work on human motion synthesis, diffusion models, and data dropout. Section
III formulates the problem and provides a comprehensive description of the
proposed framework for controllable motion synthesis and reconstruction.
Section IV introduces the experimental setup, discusses the results, and
compares these to two baselines. Finally, Section V summarizes the paper and
outlines future directions.

##  II RELATED WORK

In this section, we provide an overview of deep learning-based human motion
synthesis (Section II-A) and denoising diffusion probabilistic models (DDPMs)
(Section II-B), and then describe prior work on data dropout (Section II-C).

###  II-A Human Motion Synthesis

Deep learning approaches have been widely adopted for human motion synthesis
following earlier success in other domains. These approaches can be
categorized into deterministic and probabilistic methods. Most previous works
follow the deterministic approach to yield a fixed output for a given input.
For instance, Fragkiadaki et al. [3] first applied recurrent neural networks
to human motion prediction. Butepage et al. [4] directly fed the most recent
previous frames through an encoder-decoder network to predict future motion
frames. Li et al. [9] introduced a conditioned LSTM for synthesizing long-term
motion patterns, and Martinez et al. [10] employed a sequence-to-sequence
architecture with residual connections for joint prediction.

To synthesize motion patterns with more variety and diversity, probabilistic
methods have also been adopted. Earlier works model distributions over human
motion with Gaussian mixture models [11], and Gaussian Process models [12]. As
for deep neural networks, Variational Autoencoders (VAEs) and their variants,
which optimize a lower bound on the data log-likelihood, have been used in
combination with recurrent structures for controllable motion synthesis [13].
VAE-based approaches have also been utilized in cross-modal synthesis tasks,
such as generating motion sequences from speech [14]. Another significant
branch of methods is based on Generative Adversarial Networks (GAN) [6, 15].
GANs are deemed more powerful and effective for encoding, but are often
difficult to train and evaluate. To this end, Flow-based generative models
gained in popularity, as they enable tractable likelihood evaluation and
efficient model parameterization. MoGlow [5] models motion sequences with
autoregressive normalizing flow and recurrent neural networks. Yin et al. [2]
further integrated MoGlow with graph convolutional networks for motion
reconstruction. Ho et al. [16] presented diffusion models, a new paradigm for
probabilistic generative modeling that allows for greater flexibility in the
choice of architectures. Our method adopts a diffusion model for improved
model capacity and training stability in learning controllable motion
synthesis.

###  II-B Denoising Diffusion Probabilistic Models (DDPMs)

DDPMs [16, 17] are a class of generative models inspired by non-equilibrium
thermodynamics. DDPMs have a forward and a reverse process. The forward
process progressively adds noise to data and the reverse process learns to
construct data samples from the noise. The learning model allows regular
neural networks to be used, which makes the models both analytically tractable
and flexible.

Several works on human motion generation have adopted diffusion models. MID
[18] encodes historical information on behaviors and global signals as an
embedding and devises a Transformer-based diffusion model for human trajectory
prediction. BelFusion [19] is a diffusion model that exploits a behavioral
latent space for human motion prediction. The recent MotionDiffuse [7] is
considered to be the first diffusion model-based text-driven motion generation
framework with instructions on body parts. Similar to MotionDiffuse, MDM [8]
integrates diffusion models and CLIP [20] for text-to-motion generation.
Recently, diffusion models have been utilized to generate dance movements, a
challenging task due to the intricate postures, rhythms, and compositions
involved in dance, often accompanied by music. Alexanderson et al. [21]
pioneer diffusion models with Conformer [22] for audio-driven dance motion
generation. Tseng et al. [23] propose EDGE, a transformer-based diffusion
model that generates dance sequences conditioned on music.

Our proposed framework applies diffusion models for motion generation and
models the temporal information in an autoregressive manner, similar to
TimeGrad [24], with a cross-modal transformer inspired by Li et al. [25] for
controllable generation from various modalities. The difference is that our
framework can impose/alter the control signal on the motion generation process
on the fly while other models need a supplied command, whether it be text-
based or in other forms, before the full sequence can be generated. Moreover,
the design of the autoregressive diffusion model is flexible and can be
extended to tasks that require robust generation, e.g. synthesis and
reconstruction from imperfect motion frames, without additional training,
something that is not featured in the works reviewed above.

###  II-C Data Dropout

Due to an over-reliance on the autoregressive context, autoregressive models
often suffer from poor adherence to the control context and as such have
compromised consistency in controllable generation [5]. Such a phenomenon is
exacerbated with long-term prediction. Natural approaches to counter this
problem includes removing some or all of the conditioning information during
learning. Bowman et al. [26] randomly replace some part of the conditioned
word tokens with a generic unknown word token. They apply this technique to a
decoder, helping the model to capture higher-order statistics. Wang et al.
[27] propose a data dropout strategy that randomly sets the data to zero in
both the training and generation stages, forcing the models to focus on the
control context, thus alleviating this problem. Kovács et al. [28] propose an
input channel dropout scheme, forcing the network to make decisions based on a
subset of channels. To rememdy the poor adherence issue in the original MoGlow
[5], MoGlow applies dropout to entire frames of the autoregressive past motion
context.

In this paper we proposes a new diffusion-induced dropout scheme. We leverage
the Gaussian noise injected in the diffusion forward process and use
intermediate representations as corrupted conditions. Our strategy naturally
removes the requirement of tuning a separate process and is found to benefit
the consistency between the generated motion and the control contexts. In
addition, this also encourages encoders to learn richer and more robust
representations.

##  III METHODOLOGY

This section formulates our target problem and establishes notations used
throughout the paper. Preliminaries about denoising diffusion probabilistic
models are also given, including the training and inference strategies. On the
basis of these, we introduce our contributed framework.

###  III-A Problem Formulation

In our scenario, we treat the human motion sequence as a series of poses, and
the aim is to synthesize the future motion and reconstruct the past imperfect
poses using an autoregressive diffusion model. Formally, a 3D skeleton-based
pose at time step $t$ is denoted as $m^{t}$, with corresponding additional
information $a^{t}$, such as control signals, texts, music pieces, etc. For
the synthesis task, the input of the diffusion framework is the past human
poses $x^{t}=\left\\{m^{t-T_{h}},m^{t+1-T_{h}},\cdots,m^{t}\right\\}$, and the
control input
$c^{t}=\left\\{a^{t-T_{h}},a^{t+1-T_{h}},\cdots,a^{t+T_{p}}\right\\}$, where
$T_{h}$ denotes the length of the observed past poses and $T_{p}$ denotes the
number of predicted frames. The output of the framework is the predicted
future motion frames, written as
$y^{t}=\left\\{m^{t+1},\cdots,m^{t+T_{p}}\right\\}$. For the reconstruction
task, the past human poses are partially observed, e.g. with missing frames or
missing body joints. In such cases, the task is to reconstruct a complete
motion $x^{t}$ from an imperfect input, that we denote $\hat{x}^{t}$, with the
control input $c^{t}$.

###  III-B Motion Diffusion Models

Input: data $y_{0}^{t}$, past motion $x^{t}$, and control input $c^{t}$

Repeat __

Initialize $s\sim$ Uniform($(1,\cdots,S)$,
$\epsilon\sim\mathcal{N}(0,\mathbf{I})$

if diffusion dropout, $p\sim\mathcal{U}(0,1)$ do

$x_{d}^{t}=x^{t}_{s}$ if $p<p_{d}$, else, $x_{d}^{t}=x^{t}$

Take gradient step on

$\nabla_{\theta}\left\|\epsilon-\epsilon_{\theta}(\sqrt{\overline{\alpha}_{s}}y_{0}^{t}+\sqrt{1-\overline{\alpha}_{s}}\epsilon,s,x_{d}^{t},c^{t})\right\|$

Until __converged

Algorithm 1 Training for data sample at time $t$

We address the formulated problem with our proposed autoregressive diffusion
model (MoDiff) based on DDPM [16], as illustrated in Fig. 2. We define the
forward diffusion process as $(y_{0},y_{1},\cdots,y_{S})$, where $S$ is the
maximum number of diffusion steps. For the sake of brevity, we omit the
superscript $t$. The forward process is a stochastic process with a fixed
Markov chain that gradually adds Gaussian noise to the ground truth future
motion data $y_{0}=y$ until the distribution of $y_{S}$ is close to a standard
Gaussian distribution:

$\displaystyle q(y_{1:S}|y_{0})$
$\displaystyle:=\prod_{s=1}^{S}q_{\theta}(y_{s}|y_{s-1});$ (1) $\displaystyle
q(y_{s}|y_{s-1})$
$\displaystyle:=\mathcal{N}(y_{s};\sqrt{1-\beta_{s}}y_{s-1},\beta_{s}\mathbf{I}),$

where $\beta_{1},\beta_{2},\cdots,\beta_{S}$ are the fixed variance schedulers
for controlling the noise scale. As shown in [16], the forward diffusion
sample at any diffusion step $s$ can be calculated in one step as:

$q(y_{s}|y_{0}):=\mathcal{N}(y_{s};\sqrt{\overline{\alpha}_{s}}y_{0},(1-\overline{\alpha}_{s})\mathbf{I}),$
(2)

where $\alpha_{s}=1-\beta_{s}$ and
$\overline{\alpha}_{s}=\prod_{i=1}^{s}\alpha_{i}$. In the reverse generation
process, we learn this process as $(y_{S},y_{S-1},\cdots,y_{0})$ and generate
motions by progressively denoising the pose from $y_{S}$ to $y_{0}$. We model
this reverse generation process by parameterizing Gaussian transition
probabilities with the past poses $x$ and the past and current control signal
$c$ as conditioning information. The reverse generation process is formulated
as:

$\displaystyle p_{\theta}(y_{0:S}|x,c)$
$\displaystyle:=p(y_{S})\prod_{s=1}^{S}p_{\theta}(y_{s-1}|y_{s},x,c)$ (3)
$\displaystyle p_{\theta}(y_{s-1}|y_{s},x,c)$
$\displaystyle:=\mathcal{N}(y_{s-1};\mu_{\theta}(y_{s},s,x,c),\Sigma_{\theta}(y_{s},s)),$

where $p(y_{S})$ denotes a prior noise Gaussian distribution and
$\Sigma_{\theta}(y_{s},s)=\beta_{s}\mathbf{I}$. All transitions share the same
parameters.

###  III-C Training and Inference

Input: noise $y_{S}^{t}\sim\mathcal{N}(0,\mathbf{I})$, past motion $x_{t}$,
and control input $c^{t}$

for __ $s=S,\cdots,1$, do

$y_{s-1}^{t}=\frac{1}{\sqrt{\alpha_{s}}}(y_{s}^{t}-\frac{\beta_{s}}{\sqrt{1-\overline{\alpha}_{s}}}\epsilon_{\theta}(y_{s}^{t},s,x^{t},c^{t}))+\sqrt{\beta_{s}}\mathbf{z}$

where $\mathbf{z}\sim\mathcal{N}(0,\mathbf{I})$ if $s>1$, else $\mathbf{z}=0$

end for

Output: $y_{0}^{t}$

Algorithm 2 Inference the future motions $y_{0}^{t}$ Figure 2: Overview of the
MoDiff schematic: a Transformer-based diffusion probabilistic model
synthesizes future motion by the learned reverse diffusion process. The
correlation between the motion and control context is depicted by the cross-
modal transformer encoder. Future motions are predicted autoregressively.

With the formulated forward diffusion process and reverse generation process,
to generate the human pose of future motion frames $y_{0}$, the training
process optimizes the log-likelihood in the reverse generation process by
maximizing the variational lower bound:

$\displaystyle\mathbb{E}[\text{log}p_{\theta}(y_{0})]$
$\displaystyle\geq\mathbb{E}_{q}[\text{log}\frac{p_{\theta}(y_{0:S},x,c)}{q(y_{1:S}|y_{0})}]$
(4)
$\displaystyle=\mathbb{E}_{q}[\text{log}p(y_{S})+\sum_{s=1}^{S}\text{log}\frac{p_{\theta}(y_{s-1}|y_{s},x,c)}{q(y_{s}|y_{s-1})}].$

We utilize the negative bound as the loss function, written as the KL-
divergence between Gaussian distributions:

$\displaystyle\mathbb{E}_{q}[$
$\displaystyle-\text{log}p_{\theta}(y_{0}|y_{1},x,c)+D_{KL}(q(y_{S}|y_{0})||p(y_{S}))$
(5)
$\displaystyle+\sum_{s=2}^{S}D_{KL}(q(y_{s-1}|y_{s},y_{0})||p_{\theta}(y_{s-1}|y_{s},x,c))].$

The first KL-divergence term with $q(y_{S}|y_{0})||p(y_{S})$ has no learnable
parameters and can thus be ignored. $q(y_{s-1}|y_{s},y_{0})$ is conditioning
on $y_{0}$, which is tractable and can be represented as:

$\displaystyle
q(y_{s-1}|y_{s},y_{0})=\mathcal{N}(y_{s-1};\tilde{\mu}_{s}(y_{s},y_{0}),\tilde{\beta}_{s}\mathbf{I}),$
(6)

where $\tilde{\mu}_{s}$ and $\tilde{\beta}_{s}$ is calculated as:

$\displaystyle\tilde{\mu}_{s}(y_{s},y_{0})$
$\displaystyle=\frac{\sqrt{\overline{\alpha}_{s-1}}\beta_{s}}{1-\overline{\alpha}_{s}}y_{0}+\frac{\sqrt{\alpha_{s}}(1-\overline{\alpha}_{s-1})}{1-\overline{\alpha}_{s}}y_{s}$
(7) $\displaystyle\tilde{\beta}_{s}$
$\displaystyle=\frac{1-\overline{\alpha}_{s-1}}{1-\overline{\alpha}_{s}}\beta_{s}\mathbf{I}.$

Ho et al. [16] show that the second KL-divergence term can be calculated as:

$\displaystyle\mathbb{E}_{q}\left[\frac{1}{2\beta_{s}}\left\|\tilde{\mu}_{s}(y_{s},y_{0})-\mu_{\theta}(y_{s},s,x,c)\right\|^{2}\right]+\lambda,$
(8)

where $\lambda$ is a constant value that does not depend on $\theta$. We
choose the reparameterization method shown in [16] that

$\displaystyle\mu_{\theta}(y_{s},s,x,c)=\frac{1}{\sqrt{\alpha_{s}}}(y_{s}-\frac{\beta_{s}}{\sqrt{1-\overline{\alpha}_{s}}}\epsilon_{\theta}(y_{s},s,x,c)),$
(9)

and the objective function is simplified to

$\displaystyle\mathbb{E}_{\epsilon,y_{0},s}\left\|\epsilon-\epsilon_{\theta}(\sqrt{\overline{\alpha}_{s}}y_{0}+\sqrt{1-\overline{\alpha}_{s}}\epsilon,s,x,c)\right\|^{2},$
(10)

where $\epsilon_{\theta}$ is a network that predicts Gaussian noise
$\epsilon\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ and is trained with the MSE
loss. The training is performed for all steps $s\in[1,S]$. The complete
training procedure with the simplified objective function is displayed as
Algorithm 1, with the diffusion dropout later introduced in Section III-D.
After the network $\epsilon_{\theta}$ is trained, given past poses $x$,
control input $c$, and $y_{S}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, we can
synthesize future motions $y_{0}$ with the reverse generation process
$y_{s-1}\sim p_{\theta}(y_{s-1}|y_{s})$,:

$\displaystyle
y_{s-1}=\frac{1}{\sqrt{\alpha_{s}}}(y_{s}-\frac{\beta_{s}}{\sqrt{1-\overline{\alpha}_{s}}}\epsilon_{\theta}(y_{s},s,x,c))+\sqrt{\beta_{s}}\mathbf{z},$
(11)

where $\mathbf{z}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ for $s\in[2,S]$ and
$\mathbf{z}=\mathbf{0}$ when $s=1$. We follow the generation procedure in
Algorithm 2 to predict the next sample
$y_{0}^{t}=\\{m_{0}^{t+1},\cdots,m_{0}^{t+T_{p}}\\}$. Inspired by Li et al.
[25] and Guillermo et al. [1], our model outputs the next $T_{p}$ poses given
the past $T_{h}$ poses to improve model performance. We then only update the
next time step $t+1$, pass it autoregressively to the transformer-based
encoder together with the next control input, and repeat the same procedure
until all desired motion frames are synthesized.

###  III-D Diffusion Data Dropout

Diffusion data dropout is applied during the training stage to improve data
efficiency and model robustness. To be specific, we drop motion context
information through the forward diffusion process without extra effort. A
larger diffusion step $s$ in the diffusion data dropout leads to the past
motion $x^{t}$ being more corrupted and less information is preserved. Similar
to Equation 2, the past motion information with respect to step $s$ can be
calculated in one step as:

$q(x_{s}^{t}|x^{t}):=\mathcal{N}(x_{s}^{t};\sqrt{\overline{\alpha}_{s}}x^{t},(1-\overline{\alpha}_{s})\mathbf{I}).$
(12)

As shown in Algorithm 1, data dropout is done if $p<p_{d}$, where
$p\sim\mathcal{U}(0,1)$ and $p_{d}$ is the diffusion dropout rate. We
stabilize the training by starting with $p_{d}=0$, i.e., with complete motion
context information. A diffusion dropout scheduler
$P_{d}=\\{p_{d1},p_{d2},\cdots,p_{dn}\\}$ is set to increase the dropout rate
during training. As the diffusion dropout rate increases, the denoising
process becomes more focused on the control context and more robust to
corruption of the motion context.

###  III-E Motion Reconstruction

For motion reconstruction, we use the same framework without any further
training. Our framework allows an imperfect input
$\hat{x}^{t}=\\{\hat{m}^{t-T_{h}},\cdots,\hat{m}^{t}\\}$ since diffusion data
dropout was applied during training. To reconstruct missing body joints or
frames, we first generate a sequence of $T_{h}$ future frames,
$\\{{m}^{t+1},\cdots,{m}^{t+T_{h}}\\}$. Then we reverse the order of this
sequence and the control signals $\\{a^{t-T_{h}},\cdots,a^{t+T_{h}}\\}$ to
$\\{{m}^{t+T_{h}},\cdots,{m}^{t+1}\\}$ and
$\\{a^{t+T_{h}},\cdots,a^{t-T_{h}}\\}$, exploiting the fact that the training
data are augmented by lateral mirroring and time-reversion. We regard the
reversed motion context and control context as conditioning information to
generate poses. The missing information can be reconstructed by filling the
holes with the generated parts. We repeat until all missing parts are
reconstructed.

###  III-F Network Architecture

The proposed MoDiff framework is composed of an encoder and a decoder. The
encoder encodes the past motion context and control context, and the decoder
models the Gaussian transitions in the Markov chain, as depicted in Fig. 2.
For the encoder, we design two-layer transformers that encode the motion
context $x^{t}$ and control context $c^{t}$ separately, with output embeddings
of the same dimension. The motion context is augmented by diffusion dropout.
In the transformers, a position embedding is introduced to emphasize the
positional relation at different time steps. The outputs are then concatenated
together into a six-layers cross-modal transformer. For the decoder, we design
a three-layer Transformer decoder similar to [18] to model spatial and
temporal dependencies. The inputs include the future motion $y^{t}$, the noise
variable $\epsilon$, a diffusion step index $s$, as well as the output
embedding of the cross-model Transformer. In diffusion step $s$, the noised
future motion $y_{s}^{t}$ is concatenated with a diffusion step embedding.
Finally, fully-connected layers downsample the output to the motion dimension.
Like in TimeGrad [24], we can pass autoregressively to the network to repeat
until the desired horizon although TimeGrad primarily relies on recurrent nets
for propagating temporal information.

##  IV EXPERIMENTS AND EVALUATIONS

In this section, We first describe the experimental setting, including the
dataset, ablation settings, and implementation details. We then present the
results on the generation and reconstruction of human locomotion samples. On
the basis of these, we evaluate and discuss the performance of both the
proposed frameworks and the baselines, followed by preliminary results on
various tasks.

Figure 3:  Footstep analysis for generated motion given past information and
control signals. The footstep count $f_{est}$ on tolerance value $v_{tol}$.
The black dots indicate $v_{tol}^{95}$, the first velocity tolerance for
capturing 95% estimated footsteps.  Figure 4:  Example poses with body joints
flying away that are generated by MoGlow with incomplete past poses.

###  IV-A Experimental Setting

Dataset. We evaluate MoDiff on a dataset of human locomotion preprocessed by
[5] that retargets skeletons from the Edinburgh Locomotion MoCap, CMU Motion
Capture, and HDM05 datasets, which includes various human gaits along
different trajectories. The motion context $m\in\mathbb{R}^{63}$ is
represented by the 3D coordinates of 21 body joints. The control context
$a\in\mathbb{R}^{3}$ includes forward, lateral and rotational velocities. We
slice the training data into 4-second clips and downsample these to 20 fps
with 50% overlap. For synthesis and reconstruction with incomplete input, we
generate data with missing parts by setting some joints to zero.

Ablation Settings. To assess the impact of design decisions, we compare our
proposed MoDiff with MoGlow [5] and TimeGrad [24]. MoGlow is an autoregressive
model based on normalizing flows and LSTM for human motion generation.
TimeGrad is an autoregressive diffusion model for general time series
forecasting and reports state-of-the-art performance on real-world datasets,
such as traffic and electricity. Both MoDiff and MoGlow are probabilistic and
designed for controllable motion synthesis. Unlike MoGlow, MoDiff and TimeGrad
are diffusion-based models, but the encoder of TimeGrad is still based on
LSTM, while MoDiff uses transformers. We evaluate the impact of the diffusion
models and transformers architecture by comparing MoDiff with MoGlow and
TimeGrad. To highlight the advantage of the proposed diffusion data dropout,
we applied this dropout strategy to both MoDiff and the baseline models.

Implementation Details. Following the experimental setting in MoGlow [5],
MoDiff is trained with 10-frame time windows, i.e., $T_{h}=10$. In our
experiments, the maximum number of diffusion steps $S$ is 100. The diffusion
dropout rate is $\\{0,0.05,\cdots,0.25\\}$, increasing every 100 epochs after
500 epochs. The Transformer in our experiments is set to 256 dimensions and 4
heads. We use the standard Transformer encoder in PyTorch, with the T5-style
relative positional embedding. We train with Adam optimizer with a learning
rate of 5e-5 and batch size of 64 for 1000 epochs. All experiments were
conducted on a single NVIDIA A100 Tensor Core GPU.

TABLE I: Results of the footsteps and bone-length analysis with complete
input. The number closest to the recorded motion capture is in bold.

Model | $f_{est}$ | $v_{tol}^{95}$ | $\mu$ | $\sigma$ | RMSE
---|---|---|---|---|---
Recorded Data | 289 | 5 | 0.315 | 0.263 | 0
MoGlow (no drop) | 314 | 6 | 0.222 | 0.134 | 0.315
MoGlow (drop) | 253 | 5 | 0.341 | 0.250 | 0.250
TimeGrad (no drop) | 284 | 6 | 0.331 | 0.338 | 0.124
TimeGrad (drop) | 281 | 5 | 0.289 | 0.241 | 0.115
MoDiff (no drop) | 280 | 5 | 0.289 | 0.239 | 0.089
MoDiff (drop) | 284 | 5 | 0.334 | 0.286 | 0.072

TABLE II: Footsteps and bone-length results of generation and reconstruction
with incomplete input. The number closest to the recorded motion capture is in
bold.

Model | $f_{est}$ | $v_{tol}^{95}$ | $\mu$ | $\sigma$ | RMSEge | RMSEre
---|---|---|---|---|---|---
Recorded Data | 289 | 5 | 0.315 | 0.263 | - | -
MoDiff (no drop) | 299 | 5 | 0.375 | 0.347 | 0.105 | 0.116
MoDiff (drop) | 293 | 5 | 0.358 | 0.287 | 0.101 | 0.108

Figure 5: Example sequences generated by MoDiff given different past
information and control signals.  Figure 6: An example sequence reconstructed
by MoDiff. The right arm is missing in the past frames.

###  IV-B Results and Discussions

Results. For quantitative analysis, we evaluate the quality of motion
generation with footstep analysis and bone-length analysis, which are widely-
used methods to evaluate artifacts related to heel sliding and limb stretching
[5]. Footstep analysis compares the estimated footsteps of the generated
motion and the recorded motion. The average number of footsteps $f_{est}$ can
be detected at intervals where the horizontal velocity of the heel joints is
below a tolerance value $v_{tol}$. For evaluation, we compare the estimated
number of footsteps at the first velocity tolerance for capturing 95% steps,
denoted as $v_{tol}^{95}$. The estimated number of footsteps on different
$v_{tol}$ in the generated motions are displayed in Fig. 3, the detected
$v_{tol}^{95}$ is shown as black dot in this figure. We can observe that the
curves of MoDiff and TimeGrad, with and without diffusion dropout, are
consistently closer to the curve of the recorded motion compared to MoGlow.
Table I shows the results given complete context, include the estimated
footsteps$f_{est}$, the tolerance value $v_{tol}^{95}$, the mean $\mu$ and
standard deviation $\sigma$ of the step duration, as well as the results of
bone-length analysis. The bone-length analysis is for evaluating bone-
stretching artifacts. We compare the RMSE ($cm$) of the bone length. Table II
shows the results of generation and reconstruction given incomplete context.

Discussion. MoDiff can qualitatively and quantitatively generate recorded
realistic motion with good diversity on the locomotion task, as demonstrated
by the snapshots of generated motion sequences in Fig. 5. In contrast, the
native MoGlow suffers from poor adherence and generates motion with noticeable
foot-sliding artifacts. As can be seen from the results in Table I, the
diffusion-based models, TimeGrad and MoDiff, perform significantly better in
terms of both footstep and bone-length analysis compared to MoGlow which is
based on normalizing flow. However, unlike the models the use LSTM, i.e.
MoGlow and TimeGrad, MoDiff benefits from transformer-based encoders to
further improve on bone-length analysis. The attention mechanisms in MoDiff
discern specific features over a longer period, which is essential for
learning consistent gaits under conditioning signals. Additionally, the
attention-based transformer is effective at extracting meaningful information
from intermediate diffusion steps with roughly corrected skeleton poses, which
is not as feasible with standard LSTM models.

MoDiff can be extended to reconstruct incomplete body joints or frames without
extra training, as illustrated by the reconstruction results in Table II,
results that do not significant diverge from those in Table I. When the past
input is incomplete, MoGlow exhibits unstable outputs, i.e. skeletons with
joints flying away, as shown in Fig. 4. The results from MoDiff are closer to
the recorded motion capture, illustrating the robustness of the diffusion-
based architecture. From the example shown in Fig. 6, we observe that the
reconstructed body joints (the right arm in blue) fit the original skeleton
well. As can be seen from both Table I and Table II, the proposed data dropout
strategy improves the performance of all three methods, which confirms its
effectiveness.

Extra Applications. MoDiff is a task-agnostic framework that can be applied to
other cross-modal generation tasks, such as text-to-motion and music-to-dance.
We show generated samples in the attached multimedia materials link on
YouTube:youtu.be/qeAs9eF3pbs. Note that the framework was not intended to be
text-based even if it is agnostic to the input modality. The text here is just
to generate a command token that can be dynamic and from other control
modalities, e.g., keyboard strokes. It is thus more appropriate to compare
MoDiff with other works for controllable locomotion generation.

##  V CONCLUSIONS

We propose MoDiff, a Transformer-based diffusion model, to tackle the
challenge of controllable and robust human motion synthesis and reconstruction
under imperfect conditions. We introduce a novel diffusion data dropout
strategy utilizing the diffusion forward process, which improves data
efficiency and model robustness. The comparison results on the locomotion
dataset with state-of-the-art baselines demonstrate the superiority of our
MoDiff. MoDiff can be applied to various multimodal synthesis tasks. In the
future, we plan to extend the MoDiff framework for classifier-guided
conditional generation and apply it for more challenging dance motions.

## ACKNOWLEDGMENT

This study has received funding from the European Commission Horizon 2020
research and innovation program under grant agreement number 824160
(EnTimeMent). This work benefited from access to the HPC resources provided by
the Swedish National Infrastructure for Computing (SNIC), partially funded by
the Swedish Research Council through grant agreement no. 2018-05973.

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a considerable volume fraction is warmed past the 70-100 K threshold that
allows the release of impurities. This means that for these comets the
interior will be significantly devoid of super-volatile impurities, which
would have been released already during the early processing of amorphous
water or CO2 ices, and subsequently transported close to the surface. After
activity erodes the super-volatile-rich zones which were closer to the
surface, a comet might not be able to resume strong activity in their absence.
This might lead to the dichotomy between small and large comets, if the former
never lost their bulk impurities in substantial amounts, and thus they can
remain active irrespective of how much of the nucleus erodes.

For completion, however, we also note that comet splitting offers an
independent explanation for increased activity by smaller comets. Comets are
known to split into several active fragments during their journey through the
Solar System (Boehnhardt, 2002). Splitting can expose fresh areas, which would
enhance the overall activity. After some time, however, the volatile-rich
surface of the ensuing fragments might again be replaced by a dust-rich
surface, and the activity pattern might return to similar pre-splitting
levels. It is therefore not clear if splitting alone can explain the observed
trend, and in particular why the turn-off radius is 2 km and not some other
value. Although Tancredi _et al._ (2006) provide one possibility, we
additionally suggest that radial compositional differentiation may be
complementary in explaining the observed trend.

###  5.9 Permittivity

The Consert instrument onboard the Rosetta orbiter and the Philae lander
measured a permittivity contrast between shallow layers (0…25 m depth;
$\varepsilon=1.70-1.95$) and deeper layers of comet 67P/C-G (25…150 m depth;
$\varepsilon=1.20-1.32$; see Kofman _et al._ , 2020). It was already argued by
Kofman _et al._ (2020) that there are at least two mechanisms that in
principle could explain this permittivity contrast: (i) a decreased porosity
in the sub-surface due to vapour transport induced by insolation; or (ii)
composition changes with depth. The latter might be caused by radiogenic-heat-
induced mobility of the volatiles as investigated in this paper. However, as
described in Section 5.2, the re-condensation of volatiles will occur around
the pristine layer, depending on their respective condensation temperatures.
As we have chosen an exemplary 70 K as a proxy for the depth of the pristine
layer (see Figure 25), water ice will condense at greater depths, super-
volatiles will condense at the base of this layer and hyper-volatiles might
condense closer to the surface, where they might or might not be retained
depending on the comet’s incipient orbit.

Bearing this in mind, a comparison to Figure 25 shows that condensation-
enhanced densities as shallow as observed by Consert (0…25 m) are unlikely to
be caused by water ice. However, super-volatiles (e.g., $\mathrm{CO_{2}}$)
might have been (a) accumulated in the observed depths after being relegated
from the comet’s inner portions, where (b) they are now in turn depleted, and
(c) the pristine surface layer also retains its original super-volatiles.
Taking into account that comet 67P/C-G has already been eroded by previous
perihelion passages at its current orbital elements, if comet 67P/C-G formed
at or close to its current size and as early as 4-5 Myrs after CAI (or $\sim$1
Myr for 5% mineral fraction), the above scenario could be plausible. A
quantitative comparison between the permittivities predicted by our
planetesimal-evolution model and the observations of Consert are, however,
beyond the scope of this work and also require detailed vapour-transport
treatment.

How could the vapour transport induced by radiogenic heating be observed in
future missions to primitive bodies of the Solar System? As shown by Consert,
radar techniques are capable of penetrating hundreds of meters deep through
highly porous granular matter (Kofman _et al._ , 2020). The permittivity of
_solid, non-porous_ water ice (at 273 K) has a plateau at $\varepsilon\approx
3$ for frequencies between $\sim 10^{5}$ Hz and $\sim 10^{12}$ Hz (see figure
1 in Artemov, 2019), but rises strongly towards smaller frequencies. The
static permittivity of low-temperature water ice is $\varepsilon>200$ for
$T=100$ K (see figure 4 in Artemov, 2019). As the bulk permittivity of the
refractory porous material of comets is $\varepsilon\approx 1.3$ (Kofman _et
al._ , 2020), high-frequencies radar faces the challenge to detect a rather
small permittivity contrast when entering the ice-enriched layer of the body.
Ultra-low frequency radar might be a better choice, because of its higher
penetration depth and the much higher permittivity contrast between water ice
and the refractory components. With such a method, the greater depths of the
pristine surface layers of comets could be probed and compared to
thermophysical planetesimal-evolution models like the one described in this
paper.

##  6 Summary

In this paper we bring forth a novel framework for the evolution of comets,
under the assumption that they are initially accreted from pebbles. The
ultimate effect of pebble structure is to greatly reduce the effective thermal
conductivity. We use established empirical relations in order to determine the
conductivity, while the comet undergoes thermal and compositional changes as
it evolves from its natal state to the present day. Our model also newly
considers two effects which were not included in past studies: (a) the
compression of pebble inter-porosity by self gravity, for which we performed a
dedicated lab experiment, and (b) the full collapse of pebble inter-porosity
by aqueous alterations (when applicable), while accounting for the energetic,
physical and thermal consequences of the ensuing chemical hydration reactions
between liquid water and initially anhydrous rock.

Although our understanding of comet nuclei has grown tremendously over the
last few years, primarily through the numerous discoveries made by the Rosetta
spacecraft and its array of instruments – there is still a great deal of
uncertainty in the actual relations governing the inner workings of comets.
Hence, we explore a large space of model realisations, including various
pebble sizes and permeability coefficients. We also raise the possibility
that, in similarity to comet 67P/C-G, of all the refractory materials inside a
comet, only the minerals may contribute to radiogenic heating, potentially
lowering the fraction of radionuclides. Lastly, we consider comets that
greatly vary in size and in formation time. These last three parameters are
pivotal, and generally determine the course of the evolution.

This parameter space is explored self-consistently via SEMIO, a 1-dimensional
thermo-physical code for evolving icy bodies, now newly modified to include
the aforementioned pebble-related aspects and empirical relations. We broadly
find that when composed of pebble piles, comets may attain much higher
temperatures than in any previously published study (when comparing to the
same formation time and mineral fraction).

The primary conclusion of our study is that only extremely small, sub-
kilometre comets are truly pristine (in the sense that they may not mobilize
anything but hyper-volatiles). This conclusion is robust for any combination
of the free parameters in our study, as long as the formation time is not too
small. Larger than kilomtre-sized comets may still be pristine, however it
requires (in addition to a large formation time) that their mineral fraction
is negligible, underscoring its importance. The only available data regarding
the isotopic composition of cometary dust exclude that 26Al was ever present
in comet 81P/Wild 2 (Levasseur-Regourd _et al._ , 2018), consistent with this
requirement. On the other hand, if the rocky materials contribute
radionuclides in similar proportions to meteoritic abundances (as
_ubiquitously_ assumed in all previous thermophysical studies of comets and
other icy objects), pristine comets must both be sub-kilometre and form very
late ($>9-10$ Myr). This work emphasises the need to better constrain the
mineral fraction in comets and other icy bodies across the Solar system. As we
investigate a range of comet sizes and formation times, our simulations often
result in the processing and mobilization of at least super-volatiles
materials, if not water vapour, and eventually even liquid water. The common
assumption that comets probe the primordial state of the Solar nebula is
therefore often violated.

By this parameter study, we identify four evolutionary archetypes, attempting
to categorise pebble-accreted comets according to general classes: (a)
pristine comets, which are extremely small and relatively late to form, and
which do not experience internal processing of anything but hyper-volatiles;
(b) somewhat larger, but still relatively small comets that are capable of at
most mobilizing super-volatile ices, or release impurities inside CO2 and
amorphous ice; (c) intermediate-sized or relatively large comets which may
attain sufficient temperatures in order to mobilize water vapour; and (d)
large or extremely large comets that may attain the water melting temperature.
These objects undergo major physical, compositional and energetic changes.

There are several potential implications of pebble structure for interpreting
the formation of comets in the Solar system. Comets are typically small
objects, rarely exceeding radii above 10 km. If pebble comets form through
gravitational collapse, they may be either primordially small or else
collisional products of larger comets (with radii exceeding $\sim$50 km
initially). Our study hints that the first scenario is the most probable,
since we show that extremely large comets are typically aqueously altered, and
have a dense nucleus, such that their collisional products will not have the
observational properties of present-day comets.

We also discuss the early spin-up or spin-down of comets as a result of
physical and compositional modifications triggered by radiogenic heating,
which change their moment of inertia. Spin-down is deemed to be of typically
negligible importance while spin-up seems to be a robust signature of large
aqueously altered comets when we assume that their pebble structure collapses,
roughly halving their rotation period, which may even result in comet
splitting if their initial rotation is already fast.

More implications of pebble structure shed light on the present day
composition of comets. We discuss the significance of the internal
redistribution of volatiles. The first implication is that the measured dust-
to-volatile ratio, even if it is integrated over an entire orbit, does not
provide us with the bulk, primordial dust-to-volatile ratio of the comet. This
is because volatiles can migrate outwards and concentrate closer to the
surface, essentially modifying the initial, radially uniform distribution.
However without knowledge of how quickly the comet formed, the exact abundance
of radiogenic material and especially how much of the nucleus’ external layers
have already been eroded by past activity, we cannot attempt to guess the
primordial ratio based on the comet’s differentiation history as derived from
detailed thermo-physical modeling. The second implication is that volatile-
depleted cometary cores may eventually lead to cessation of comet activity
(i.e. the ongoing ejection of pebbles stops and a lag deposit forms),
rendering the comet perpetually extinct.

In conclusion, this study emphasises the importance of pebble structure to the
formation and development of comet nuclei. We conclude by suggesting that the
delineation of volatiles might be observable by measuring the permittivity of
comet nuclei, in similarity to comet 67P/C-G. We also advocate focusing future
research efforts to the smallest possible comets, as they might be the only
true primordial cometary relics in the Solar system.

##  7 Acknowledgements

We wish to thank Marco Fulle for a thoughtful review of this manuscript,
greatly improving its quality. We also thank Noria Brecher for her help in
performing the ice-pebble compression experiment. This work was funded by the
Niedersächsisches Vorab in the framework of the research cooperation between
Israel and Lower Saxony under grant ZN 3630. HBP acknowledges support from the
Minerva center for life under extreme planetary conditions. AL is funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
grant 278211407 in the framework of the research unit "Debris Disks in
Planetary Systems" (FOR2285).

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