# Stochastic approaches: modeling the probability of encounters between
H2-molecules and metallic atomic clusters in a cubic box

Maximiliano L. Riddick, Leandro Andrini
Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas
Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET)
Departamento de Matemática, Fac. de Ciencias Exactas, UNLP
La Plata, Argentina
<EMAIL_ADDRESS>
Enrique E. Álvarez
Instituto de Cálculo, Fac. Ciencias Exactas y Naturales, Ciudad Universitaria,
Pabellón II, UBA (CABA)
Departamento de Fisicomatemática, Fac. de Ingeniería, UNLP
Ciudad Autónoma de Buenos Aires and La Plata, Argentina
&Félix G. Requejo
Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas
Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET)
Departamento de Física, Fac. de Ciencias Exactas, UNLP
La Plata, Argentina

###### Abstract

In recent years the advance of chemical synthesis has made it possible to
obtain “naked”clusters of different transition metals. It is well known that
cluster experiments allow studying the fundamental reactive behavior of
catalytic materials in an environment that avoids the complications present in
extended solid-phase research. In physicochemical terms, the question that
arises is the chemical reduction of metallic clusters could be affected by the
presence of H${}_{{}_{2}}$ molecules, that is, by the probability of encounter
that these small metal atomic agglomerates can have with these reducing
species. Therefore, we consider the stochastic movement of $N$ molecules of
hydrogen in a cubic box containing $M$ metallic atomic clusters in a confined
region of the box. We use a Wiener process to simulate the stochastic process,
with $\sigma$ given by the Maxwell-Boltzmann relationships, which enabled us
to obtain an analytical expression for the probability density function. This
expression is an exact expression, obtained under an original proposal
outlined in this work, i.e. obtained from considerations of mathematical
rebounds. On this basis, we obtained the probability of encounter for three
different volumes, $0.1^{{}^{3}}$, $0.2^{{}^{3}}$ and $0.4^{{}^{3}}$
m${}^{{}^{3}}$, at three different temperatures in each case, $293$, $373$ and
$473$ K, for $10^{{}^{1}}\leq N\leq 10^{{}^{10}}$, comparing the results with
those obtained considering the distribution of the position as a Truncated
Normal Distribution. Finally, we observe that the probability is significantly
affected by the number $N$ of molecules and by the size of the box, not by the
temperature.

_K_ eywords Wiener Process $\cdot$ Probability of encounters $\cdot$ Molecular
Collisions $\cdot$ Atomic-Clusters $\cdot$ Mathematical Rebounds

##  1 Introduction

In the last two decades there has been an important development in clusters
chemistry, and consequently new questions arise on the basis of these
developments [1, 2, 3, 4, 5, 6, 7]. This interest is due to an atomic clusters
containing up to a few dozen atoms exhibit features that are very different
from the corresponding bulk properties and that can depend very sensitively on
cluster size [8]. In particular, many of these transition metal clusters are
used in the field of catalysis [1, 9, 10, 11]. One of the basic principles of
catalysis is that when the smaller the metal particles, the larger the
fraction of the metal atoms that are exposed at surfaces, where they are
accessible to reactant molecules and available for catalysis [1]. It is well
known in chemistry that the encounter between two molecules can give rise to a
chemical reaction, and from the mathematical aspect there are two fundamental
ways to represent these types of situations as continuous, represented by
differential equations whose variables are concentrations, or as discrete,
represented by stochastic processes whose variables are the number of
molecules [12].

Without loss of generality, it can be considered that the molecular
chemisorption is due to the encounter between a molecule and a surface (or a
cluster in this case) with the energy necessary for the phenomenon of
adsorption to occur [13]. Besides, the kinetics of hydrogen chemisorption by
neutral gas-phase metal clusters exhibits a complex dependence on both cluster
size and metal type [14]. For different chemical purposes, for example, in the
case of copper clusters (Cun) is very important to have control of the
chemisorption of hydrogen on these clusters, i.e. the formation of Cun-H2
species [15].

From a reductionist point of view, the molecular chemisorption is a problem of
encounter between bodies: metal clusters and reactant molecules. In our first
approximation (mathematical reduction) we will consider the problem as a
problem of encounter or collisions between bodies. We are interested in
proposing this strategy because we are focused to answer what is the
probability of meeting between $N$ hydrogen molecules ($N$-H2) and a fixed $M$
metallic clusters ($M$-Men), for a given time $t$, where the H2 move freely in
a bounded volume $V$ of $\mathbb{R}^{3}$-space. Under this assumption, we are
going to consider H2-molecules and Men-clusters as rigid spheres of radii
$r_{1}$ and $r_{2}$, respectively. Then, it is considered that there will be a
collision whenever the center-to-center distance between an H2-molecule and a
Men-cluster is equal to $r_{12}=r_{1}+r_{2}$ [16]. Also, in this context we
propose the H2-molecules follow a Brownian motion, namely: (a) it has
continuous trajectories (sample paths) and (b) the increments of the paths in
disjoint time intervals are independent zero mean Gaussian random variables
with variance proportional to the duration of the time interval [17].

The pioneering work of T.D. Gillespie [16, 18, 19] have given rise to a large
number of works that are proposed different algorithms for the calculation for
numerically simulating the time evolution of a well-stirred chemically
reacting system, although despite recent major improvements in the efficiency
of the stochastic simulation algorithm, its drawback remains the great amount
of computer time that is often required to simulate a desired amount of system
time [20]. While our method is a simple reduction to collisions of molecules,
allows to calculate the probability of encounter (scheduled in $R$) for a
large number of molecules ($\approx 10^{6}$) and clusters ($\approx 10^{20}$)
with advantages regarding the cost of calculation, and the effects of first
approximation can provide statistical support to the design of experiments.
This calculation is possible using a stochastic model (Wiener process) in the
context of considerations from the Maxwell-Boltzmann theory.

##  2 A first theoretical approaching

As we announced in the introduction, we will assume that hydrogen molecules
have a random movement, whence let $\mathcal{H}(t)=(X(t),Y(t),Z(t))$ the
random variable which specify the space point where the H2 hydrogen molecule
is at time $t$. Trivially, $\mathcal{H}(t)$ depends on an initially point
$\mathcal{H}(0)=(x_{0},y_{0},z_{0})$. Thus, when the initial starting point is
undefined, $\mathcal{H}(t)=\mathcal{H}(t,x_{0},y_{0},z_{0})$. Our interest is
in how probably is that the distance between $\mathcal{H}(t)$ and a fixed
point $(a,b,c)$ is smaller than $\epsilon$. The fixed point $(a,b,c)$ are the
coordinates for Men.

Let’s consider the random variable $D(t)$ as the variable that measures the
distance between $\mathcal{H}(t)$ and the fixed point $(a,b,c)$. Following the
classical Pythagorean relationship,
$D(t)=\sqrt{(X(t)-a)^{2}+(Y(t)-b)^{2}+(Z(t)-c)^{2}}$, and in general
$D(t)=D(t,x_{0},y_{0},z_{0},a,b,c)$.

Now, given a time window $[0,\tau]$, let

$R_{\tau}:=\left\\{\begin{array}[]{lclcl}1&\mbox{if}\displaystyle\min_{t\in[0,\tau)}\displaystyle
D(t)\leq\epsilon,\\\ \\\ 0&\mbox{otherwise}.\\\ \end{array}\right.$ (1)

So, for a fixed $t_{0}>0$, we define
$G(t_{0})=P(D(t_{0})\leq\epsilon)=\int_{0}^{\epsilon}D(s)ds$. Then,
$P(R_{\tau}=1)=\int_{0}^{\tau}G(t)dt$.

Thus, given $\tau>0$, $R_{\tau}$ depends only on the initial values
$(x_{0},y_{0},z_{0},a,b,c)$. Now, if we have $M$-Men, the probability that the
H2 molecule does not meet with any of the clusters is
$P(R_{\tau_{1}}=0,R_{\tau_{2}}=0,...,R_{\tau_{M}}=0)=p_{A}$, where
$R_{\tau_{i}},i\in\\{1,...,M\\}$, follows the definition given in the eq. 1.

If $N$-H2 molecules are in the environment, let $A_{j}$ the event “the j-th
hydrogen molecule meet with a metallic cluster”. Under random starting points,
we are interested in $P(A_{1}^{C}\cap A_{2}^{C}\cap...\cap
A_{N}^{C})=p_{A}^{N}$ according to the independence among the hydrogen
molecules.

###  2.1 Adaptation to our context

Next, we proceed to realize the analysis according to the Brownian Motion
Theory [17], in which the movement of the particle is independent among
different axis, and we are going to assume that it follows a Wiener process
[21, 22]. Then,

$\displaystyle X(t)=x_{0}+W_{X}(t)$ $\displaystyle Y(t)=y_{0}+W_{Y}(t)$
$\displaystyle Z(t)=z_{0}+W_{Z}(t)$

And we will say that $W_{X}(t)$, $W_{Y}(t)$ and $W_{Z}(t)$ are following a
Wiener processes with $\sigma=\sqrt{\frac{k_{b}T}{m}}$, where $k_{b}$ is the
Boltzmann’s constant, $T$ is the absolute temperature in Kelvin (K) and $m$ is
the H2’s mass in kg. That is, we are imposing a physical behavior that obeys
Maxwell-Boltzmann’s considerations. According this:

$\displaystyle X(t)\sim\mathcal{N}(x_{0},\sigma^{2}t)$ $\displaystyle
Y(t)\sim\mathcal{N}(y_{0},\sigma^{2}t)$ $\displaystyle
Z(t)\sim\mathcal{N}(z_{0},\sigma^{2}t)$

With density function $f_{X}(x,t|x_{0})$, $f_{Y}(y,t|y_{0})$ and
$f_{Z}(z,t|z_{0})$, respectively. Under these assumptions:

$\displaystyle
f_{X}(x,t|x_{0})=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{x-x_{0}}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle
f_{Y}(y,t|y_{0})=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{y-y_{0}}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle
f_{Z}(z,t|z_{0})=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{z-z_{0}}{\sigma\sqrt{t}}\right)^{2}\right\\}$

####  2.1.1 Unbounded conditions

Under unbounded conditions, as it is well known, the density of the particle
position in the space for a fixed $t$ follows the expression:

$\displaystyle f_{XYZ}$
$\displaystyle(x,y,z,t|x_{0},y_{0},z_{0})=f_{X}(x,t|x_{0})\cdot
f_{Y}(y,t|y_{0})\cdot f_{Z}(z,t|z_{0})=$
$\displaystyle=\frac{1}{(\sqrt{2\pi\sigma^{2}t)^{3}}}\exp\left\\{-\frac{1}{2}\left(\frac{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}{\sigma^{2}t}\right)\right\\}$

This function is continuous in the variables $x,y,z,t$, then is also
integrable in a measurable context. Because of this fact, Fubini’s theorem is
aplicable. Now, calling
$\nu=\sqrt{\frac{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}{2\sigma^{2}}}$,
and integrating over the variable $t$ by substitution, results:

$\displaystyle f_{XYZ}(x,y,z,\tau|x_{0},y_{0},z_{0})$
$\displaystyle=\frac{1}{-\nu(\sqrt{2\pi\sigma^{2}})^{3}}\int_{0}^{\tau}\frac{-\nu}{(\sqrt{t})^{3}}\exp\left\\{-\left(\frac{\nu}{\sqrt{t}}\right)^{2}\right\\}dt$
$\displaystyle=\frac{1}{\nu(\sqrt{2\pi\sigma^{2}})^{3}}\int_{\sqrt{\frac{\nu}{\tau}}}^{\infty}e^{-u^{2}}du$

Remembering that the $erfc$ function [23] is defined by:

$erfc(z)=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}dt$

we conclude:

$f_{XYZ}(x,y,z,\tau|x_{0},y_{0},z_{0})=\frac{1}{2\pi\nu(\sqrt{2\sigma^{2}})^{3}}erfc\left(\sqrt{\frac{\nu}{\tau}}\right)$

From the physical-experimental perspective that the problem is lays out, the
unbounded system lacks interest, so we will proceed to study the case of the
bounded system.

####  2.1.2 Bounded conditions

We assume that the experiment takes place into a cubic recipe centered at the
origin. This implies that $X(t)$, $Y(t)$ and $Z(t)$ $\in[-L;L]$, for a fixed
volume $V=L^{3}$ in $\mathbb{R}^{3}$-space.

In a similar issue the traditional way of approaching is by “truncation" [24,
25]. A drawback of this approach is the fact that the truncation does not
represent precisely the reflection on the boundaries. An illustrative and
motivational argument is given by the following example: suppose a random walk
of $N=4$ steps, with starting point at the origin. Then, the walker moves $1$
step at right or left (with equal probability) at each step. Then, after four
steps, the resultant probabilities of the walker position are:

$\displaystyle 0$ $\displaystyle;\mbox{ with probability }3/8,$
$\displaystyle-2\mbox{ or }2$ $\displaystyle;\mbox{ with probability }2/8,$
$\displaystyle-4\mbox{ or }4$ $\displaystyle;\mbox{ with probability }1/16.$

The probability values (under truncation) in the closed interval $[-2,2]$ for
the values $(-2,-1,0,1,2)$ are, respectively:

$(2/7,0,3/7,0,2/7)$

With fixed boundaries, considering reflections at $[-2,2]$, we can construct
the following Markov transition matrix $P$:

$P=$ 0 1 0 0 0 1/2 0 1/2 0 0 0 1/2 0 1/2 0 0 0 1/2 0 1/2 0 0 0 1 0

At the fourth step, after some algebra, we obtain the respectively mass point
probability for the position of the walker. This is provided by the stochastic
vector

$(1/4,0,1/2,0,1/4)$

(given by the third file of $P^{4}$, i.e.: with starting point at the origin).
At this point, is clearly the difference between truncation and “rebounds"
(considering reflection on the boundary).

We must modify the density of the position $\mathcal{H}(t)$ according to the
particle rebounds (see Fig. 1). It is important to note that the rebounds
indicated in the figure in gray colour do not correspond to the physical
rebounds of the particles in the cubic box, but to the contributions of the
displaced distribution considering an infinite behavior.

Figure 1: In red colour an arbitrary normal distribution,
$\mathcal{N}(0,\sigma)$. We observe in gray colour the folding of the normal
distribution at the edge of the box. We could see that $A+B$ = $2L$.

Inside the box, the derived density $f_{B}$ according to the variable
$X(t)\sim\mathcal{N}(x_{0},\sigma^{2}t)$ with density function $f_{X}$ of the
particle position (for each dimension, see Fig. 1) follows the expression:

$f_{B}(x)=\left[f_{X}(x)+f_{B^{+}}(x)+f_{B^{-}}(x)\right]\times I_{[-L;L]}(x)$

where:

$\displaystyle f_{B^{+}}(x)$
$\displaystyle=f(x+2A)+f(x+2A+2B)+f(x+2A+2B+2A)+...=$
$\displaystyle=f(x+2(L-x))+f(x+2(L-x)+2(x-(-L))+...$
$\displaystyle=f(-x+2L)+f(x+4L)+f(-x+6L)+...$
$\displaystyle=\sum_{k=1}^{\infty}f((-1)^{k}x+2kL)$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{((-1)^{k}x+2kL)-x_{0}}{\sigma\sqrt{t}}\right)^{2}\right\\}$

and

$\displaystyle f_{B^{-}}(x)$
$\displaystyle=f(x-2B)+f(x-2B-2A)+f(x-2B-2A-2B)+...=$
$\displaystyle=f(x-2(x-(-L)))+f(x-2(x-(-L))-2(L-x))+...$
$\displaystyle=f(-x-2L)+f(x-4L)+f(-x-6L)+...$
$\displaystyle=\sum_{k=1}^{\infty}f((-1)^{k}x-2kL)$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{((-1)^{k}x-2kL)-x_{0}}{\sigma\sqrt{t}}\right)^{2}\right\\}$

The proof that $f_{B}$ is a density function is straightforward its
definition. Trivially, $f_{B}>0$, and by construction:

$\int_{-\infty}^{\infty}f_{B}(t)dt=\int_{-L}^{L}f_{B}(t)dt=\int_{-\infty}^{\infty}f_{X}(t)dt=1$

For practical purposes, we now try to find an upper bound to this expression.
Looking at in the model proposed, the next constraint is straightforward
$|(-1)^{k}x-x_{0}|\leq 2L$.

Following these constraints:

$\displaystyle f_{B^{+}}(x)$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{((-1)^{k}x+2kL)-x_{0}}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle\leq\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{-2L+2kL}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{2(k-1)L}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{2kL}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\sum_{k=0}^{\infty}\exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}^{k^{2}}$

It is known that $\displaystyle\sum_{k=0}^{\infty}\
r^{k^{2}}=\frac{1}{2}+\frac{1}{2}\Theta_{{}_{E}}[3,0,r]$ where
$\Theta_{{}_{E}}$ is the Jacobi theta elliptic function [23]. So:

$\displaystyle f_{B^{+}}(x)$
$\displaystyle\leq\frac{1}{\sqrt{2\pi\sigma^{2}t}}\sum_{k=0}^{\infty}\exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}^{k^{2}}$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\left(\frac{1}{2}+\frac{1}{2}\Theta_{{}_{E}}\left[3,0,exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}\right]\right)$

and

$\displaystyle f_{B^{-}}(x)$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{((-1)^{k}x-2kL)-x_{0}}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle\leq\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{-2L-2kL}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{-2(k+1)L}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle=\sum_{k=2}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}t}}\exp\left\\{-\frac{1}{2}\left(\frac{-2kL}{\sigma\sqrt{t}}\right)^{2}\right\\}$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\left(\sum_{k=0}^{\infty}\exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}^{k^{2}}-1-\exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}\right)$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma^{2}t}}\left(\frac{1}{2}+\frac{1}{2}\Theta_{{}_{E}}\left[3,0,exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}\right]-1-\exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}\right)$

Then,

$\displaystyle f_{B^{+}}(x)+f_{B^{-}}(x)$
$\displaystyle\leq\frac{1}{\sqrt{2\pi\sigma^{2}t}}\left(\Theta_{{}_{E}}\left[3,0,exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}\right]-\exp\left\\{-\frac{1}{2}\left(\frac{4L^{2}}{\sigma^{2}t}\right)\right\\}\right)$
$\displaystyle=C_{B}$

For each $x\in[-L,L]$, $f_{B}(x)\leq f_{X}(x)+C_{B}$. Besides $C_{B}$ does not
depends on $x$. Consequently, we have a maximum for the density $f_{B}$ which
is equal to $f(x_{0})+C_{B}$.

Calling $P_{B}=(f(x_{0})+C_{B}).2\epsilon$, we can conclude that:

$P(X(t)\in(a_{0}-\epsilon,a_{0}+\epsilon))\leq P_{B},\text{for any $a_{0}$.}$

Analogous, $C_{B}$ is the same for the variables $Y(t)$ and $Z(t)$, and we
know that $f(x_{0})=f(y_{0})=f(z_{0})$. Then, the same result is available for
the variables $Y(t)$ and $Z(t)$. According the bounded $C_{B}$, it is
straightforward the uniform convergence of the series $f_{B^{+}}$ and
$f_{B^{-}}$ (by the M Weierstrass criteria). An important fact to remark is
that $P_{B}$ is not even a probability, but in the case in we are interested,
we know that is a real number bigger than the probability desired, and then,
under certain conditions, we can work with it.

For practical purposes, the error through the $C_{B}$ implementation can be
minimized, since the first $S$ terms are available, and the tail can be
compared with

$\displaystyle\sum_{k=0}^{S-1}r^{k^{2}}$
$\displaystyle\leq\sum_{k=0}^{\infty}r^{k^{2}}=\sum_{k=0}^{S-1}r^{k^{2}}+\sum_{k=S}^{\infty}r^{k^{2}}$

And,

$\sum_{k=S}^{\infty}r^{k^{2}}=\sum_{k=0}^{\infty}r^{(k+S)^{2}}=\sum_{k=0}^{\infty}r^{k^{2}+2kS+S^{2}}=r^{S^{2}}\sum_{k=0}^{\infty}r^{k^{2}}r^{2kS}\leq
r^{S^{2}}\sum_{k=0}^{\infty}r^{k^{2}}$

Then,

$\displaystyle\sum_{k=S}^{\infty}r^{k^{2}}$ $\displaystyle\leq
r^{S^{2}}\left(\frac{1}{2}+\frac{1}{2}\Theta_{{}_{E}}[3,0,r]\right)$

Controlling the value of S controls the value of the error made by truncating
the sum. As we said, $C_{B}$ does not depend on $x$, thus, the desired
probability can be estimated with any degree of accuracy, according the
computational cost necessary to this development.

Taking into consideration the Brownian Motion Theory, in the time lapse of 1
second, the particle position under unbounded conditions follows a
$\mathcal{N}(x_{0},\sigma^{2})$ distribution. To discretize the problem, if we
partitioned the time axis of $\tau$ seconds in $\tau$ intervals of 1 second
each one, then:

$\displaystyle P(\mathcal{H}(t)\in B_{\epsilon}(a,b,c))\leq
P(\mathcal{H}(t)\in Q_{\epsilon}(a,b,c))$

where $Q_{\epsilon}(a,b,c)$ denotes the cube centered in $(a,b,c)$ with side
size $2\times\epsilon$. And, considering the independence between $X(t)$,
$Y(t)$ and $Z(t)$, with $X(t)\in(a-\epsilon,a+\epsilon)$,
$Y(t)\in(b-\epsilon,b+\epsilon)$ and $Z(t)\in(c-\epsilon,c+\epsilon)$,
$P(\mathcal{H}(t)\in Q_{\epsilon}(a,b,c))=P(\mathcal{H}\in Q_{\epsilon})$ is

$\displaystyle P(\mathcal{H}\in Q_{\epsilon})=P(X(t))\times P(Y(t))\times
P(Z(t))\leq P_{B}\times P_{B}\times P_{B}=P_{B}^{3}$

For each second $\tau_{j}$ for $\tau_{j}\in\\{1:\tau\\}$, $P(\mathcal{H}(t)\in
Q_{\epsilon}(a,b,c))\leq P_{B}^{3}$. Then, under the Wiener process
formulation, $\mathcal{H}(\tau_{j})\perp\mathcal{H}(\tau_{k}|\tau_{j})$ if
$j\neq k,j\leq k$.

$P(\mathcal{H}(\tau_{j})\in Q_{\epsilon}(a,b,c))\leq
P_{B}^{3},\forall\tau_{j}\in\\{1:\tau\\}$. Calling $F:=$“$\\#$ of
$\tau_{j}\in\\{1:T\\}$ in which $H(\tau_{j})\in Q_{\epsilon}(a,b,c)$", we are
interesting in the event $F=0$.

According its nature, $F$ is a Binomial random variable
$\mathcal{B}(\tau,P_{B}^{3})$. Consequently, the non-collision probability is
$p_{{}_{NC}}=P(F=0)\leq(1-P_{B}^{3})^{\tau}$. At this point, we only can
conclude that the probability of the encounter between a hydrogen molecule and
a Men cluster in a time $\tau$ is less than $p$. We proceed to analyze what
happens when the number of hydrogen molecules and metallic clusters increase.
We emphasize that the H${}_{{}_{2}}$ molecules have a random movement while
the clusters are confined in a fixed region of space. Since $p$ is the
probability that a random hydrogen molecule meets in the cube $Q_{\epsilon}$
in which a Men cluster is, the most unfavorable case with $M$ clusters is when
there is no intersection among the cubes that contain it. In this case:

$\displaystyle p_{A}$ $\displaystyle=P(R_{\tau_{1}}=0,...,R_{\tau_{M}}=0)$
$\displaystyle=1-\bigcup_{i=1}^{M}P(R_{\tau_{i}}=1)$ $\displaystyle\geq
1-\left(\sum_{i=1}^{M}P(R_{\tau_{i}}=1)\right)$ $\displaystyle=1-M\times p$

In view of this analysis, we can conclude that the non-collision probability
is higher than $p_{NC}$.

In regular conditions, when this approach is used, the values of $p_{NC}$ and
$N$ outcomes into a several numerical instability. In this case, the small
value of $p_{NC}$ and the large value of $N$ place us in conditions to use the
Poisson approach to the Binomial distribution (with parameter $\lambda=N\times
p$). Then, $P(X=0)\approx\exp(-\lambda)$. Even in the cases when the
probability is still unavailable, the expected number of collisions is
presented according a time window, and then we can estimate the probability of
collisions in a time window $T$ using the relationship between the Poisson and
Exponential distributions[26].

Next, we present the results of the analysis whit different box dimensions (in
meters) and number of hydrogen molecules ($N$), according to $M=1.9\times
10^{{}^{20}}$ Cu${}_{{}_{20}}$-clusters [27], where the
Cu${}_{{}_{20}}$-clusters have been considered as spheres.

##  3 Results and analysis

###  3.1 Obtaining non-collision probability values

The situation we consider is approximately a “realistic”situation, with
$M=1.9\times 10^{{}^{20}}$ Cu${}_{{}_{20}}$-clusters in a cubic box according
to the standard dimensions of reaction chambers (0.1${}^{{}^{3}}$,
0.2${}^{{}^{3}}$ and 0.4${}^{{}^{3}}$ m${}^{{}^{3}}$), and a variable
$N$-H${}_{{}_{2}}$-molecules “contamination”($10^{{}^{1}}\leq N\leq
10^{{}^{10}}$). It worked with three temperatures, T, 293, 373 and 473 K. The
choice of T is arbitrary, conditioned by the possible reaction temperatures
[28].

In Fig. 2 we observe the results obtained for the simulations, considering the
maximum sum. That is, take $S=10^{{}^{6}}$, perform the sum, and add the
maximum level for the error. Clearly, a greater probability of non-collision,
$p_{{}_{NC}}$, is observed depending on the increase in volume.

For a detailed study, we proceed as follows: we model the data obtained
through a non-linear graphic fitting considering a Boltzmann decrease
function,
$g(x)=A_{{}_{2}}+\frac{A_{{}_{1}}-A_{{}_{2}}}{1+\exp\left(\frac{x-x_{{}_{0}}}{dx}\right)}$
(see Fig. 3). In the Appendix A.2 we show the statistical results for each
parameter in each data fitting.

Under these considerations, we can calculate the critical value
(criticality)[29, 30] of hydrogen molecules, that is “what is the value of N
for which the non-collision probability is greater than $\frac{1}{2}$”, i.e.
the value of the exponent for which $\frac{1}{2}<p_{{}_{NC}}$.

It should be clarified that, in the strict physical sense, there is no abrupt
phase transition to consider “criticality”. As we assumed in the introduction,
we consider that there is a chemical reaction if there is an encounter between
two molecules, and under this assumption we are considering as critical the
level of presence of hydrogen for a chemical reaction to occur. In any case,
it can be demonstrated that there is an “abrupt”transition behavior, for a
well defined interval in the number of molecules. In Fig. 3 we can observe
this behavior.

Figure 2: Results for the non-collision probability, $p_{{}_{NC}}$, vs.
$\ln(N)$ for $L=0.05m$ (V${}_{{}_{1}}$), $L=0.1m$ (V${}_{{}_{2}}$) and
$L=0.2m$ (V${}_{{}_{3}}$), at T = 293 K (blue square), 373 K (black star) and
493 K (red triangle). Figure 3: Data (blue square) modeling using a non-linear
Boltzmann decrease function (green line).

In Table 1 we can see the critical values obtained from the decrease model for
each box and each temperature. For the smallest volumes, V${}_{{}_{1}}$ and
V${}_{{}_{2}}$, it is observed that the critical value of $N$ depends more
strongly on the temperature than in the case of the larger volume
(V${}_{{}_{3}}$). Although it is remarkable the fact of dependence with the
size of the box, it can be seen directly from Fig. 2. In this way, and under
these simplified assumptions, we can obtain control of contaminant molecules
in relation to the volume and temperature parameters. Linear behavior is
evident from the values obtained (Table 1, $N$ vs. temperature). Moreover, as
the volume increases the slope increases from negative values to null value.

Table 1: Critical values obtained from the decrease model for each box and each temperature, for mathematical robounds. L [m] | 293 K | 373 K | 473 K
---|---|---|---
0.05 | 3.25 | 3.16 | 3.09
0.10 | 4.97 | 4.86 | 4.72
0.20 | 5.96 | 5.96 | 5.96
Figure 4: Results for the non-collision probability, $p_{{}_{NC}}$, vs. $\ln(N)$ for $L=0.05m$ (V${}_{{}_{1}}$), at T = 293 K, 373 K and 493 K. Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical rebound model. Figure 5: Results for the non-collision probability, $p_{{}_{NC}}$, vs. $\ln(N)$ for $L=0.1m$ (V${}_{{}_{1}}$), at T = 293 K, 373 K and 493 K. Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical rebound model. Figure 6: Results for the non-collision probability, $p_{{}_{NC}}$, vs. $\ln(N)$ for $L=0.2m$ (V${}_{{}_{1}}$), at T = 293 K, 373 K and 493 K. Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical rebound model. Table 2: Critical values obtained from the decrease model for each box and each temperature, for truncated normal model. L [m] | 293 K | 373 K | 473 K
---|---|---|---
0.05 | 3.27 | 3.18 | 3.12
0.10 | 5.01 | 4.91 | 4.76
0.20 | 6.00 | 5.98 | 5.96

##  4 Conclusion

By way of conclusion, it can be indicated that considering a Wiener stochastic
process, for thermodynamic-statistical movements of a gas confined in a box,
and considering mathematical rebounds bounded by the physical-geometric
contour of the problem, the analytical expression could be obtained for the
probability density function of encounters between two differentiated species
of molecules (one of the species fixed in the box -solid or liquid- and the
other species is a gas whose molecules move stochastically). In addition, the
function obtained can be calculated numerically or can be bounded. The bounded
process allows to reduce the computational cost, and to limit the error from
cutting the sum in a finite number. In particular, there is an error control
that can be made, and it is possible to refine the process according to the
precision required.

From the physical-chemical point of view, it is observed that both the number
of gas molecules and the dimensions of the box affect the probability of
encounter. For this model, temperature is a parameter that has a lower
incidence on the values of the probability of encounter. At this point some
considerations have to be made. The first is that in a strict sense a chemical
reaction is more than the encounter of two chemical entities. The second is
the exceptional chemical nature of metal clusters, which make them highly
reactive. Despite the simplicity of the model we are proposing, this model can
account in an experiment design about the collision probability between two
chemical entities (and this collision can lead to a chemical reaction).

From the point of view of computation, it is a system that requires less
computational cost (time + memory) than the algorithmic systems developed for
this type of problems, so it contributes as a test method in the design of
experiments.

The comparison with an established method (truncated normal model) was
optimal. In the method of mathematical rebounds the number of molecules needed
for a reaction is less than the number obtained by the truncated normal model.
This is an advantage when strict contamination control is needed.

On the other hand, in terms of obtaining the density function, mathematical
results can be generalized for volumes of rectangular prisms of uneven sides.
In addition, it remains to calculate the first and second order moments of the
density function obtained, work that exceeded the purposes of present
communication.

## Acknowledgments

This was was supported in part by PICT-2019-0784, PICT-2017-3944,
PICT-2017-1220, PICT-2017-3150 (PICT, Agencia Nacional de Promoción de la
Investigación, el Desarrollo Tecnológico y la Innovación) and PPID-I231 (PPID,
Universiad Nacional de La Plata).

## References

  * [1] Bruce C. Gates.  Supported metal clusters: synthesis, structure, and catalysis.  Chemical reviews, 95(3):511–522, 1995.
  * [2] M Arturo López-Quintela.  Synthesis of nanomaterials in microemulsions: formation mechanisms and growth control.  Current Opinion in Colloid & Interface Science, 8(2):137–144, 2003\.
  * [3] Puru Jena and A. Welford Castleman Jr.  Clusters: A bridge across the disciplines of physics and chemistry.  Proceedings of the National Academy of Sciences, 103(28):10560–10569, 2006.
  * [4] Shahana Huseyinova, Joseé Blanco, Feélix G. Requejo, Joseé M Ramallo-López, M Carmen Blanco, David Buceta, and M Arturo Loópez-Quintela.  Synthesis of highly stable surfactant-free cu5 clusters in water.  The Journal of Physical Chemistry C, 120(29):15902–15908, 2016\.
  * [5] Lichen Liu and Avelino Corma.  Confining isolated atoms and clusters in crystalline porous materials for catalysis.  Nature Reviews Materials, 6(3):244–263, 2021.
  * [6] Huixia Luo, Peifeng Yu, Guowei Li, and Kai Yan.  Topological quantum materials for energy conversion and storage.  Nature Reviews Physics, 4(9):611–624, 2022.
  * [7] Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong, Alexander M Dalzell, Ashutosh Kumar, Phillip Helms, Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, et al.  Is there evidence for exponential quantum advantage in quantum chemistry?  arXiv preprint arXiv:2208.02199, 2022.
  * [8] Mingli Yang, Koblar A Jackson, Christof Koehler, Thomas Frauenheim, and Julius Jellinek.  Structure and shape variations in intermediate-size copper clusters.  The Journal of chemical physics, 124(2):024308, 2006.
  * [9] Manfred T Reetz and Wolfgang Helbig.  Size-selective synthesis of nanostructured transition metal clusters.  Journal of the American Chemical Society, 116(16):7401–7402, 1994\.
  * [10] John D Aiken III and Richard G Finke.  A review of modern transition-metal nanoclusters: their synthesis, characterization, and applications in catalysis.  Journal of Molecular Catalysis A: Chemical, 145(1-2):1–44, 1999\.
  * [11] Gareth S Parkinson.  Unravelling single atom catalysis: The surface science approach.  arXiv preprint arXiv:1706.09473, 2017.
  * [12] Michael A Gibson and Jehoshua Bruck.  Efficient exact stochastic simulation of chemical systems with many species and many channels.  The journal of physical chemistry A, 104(9):1876–1889, 2000.
  * [13] David E Brown, Douglas J Moffatt, and Robert A Wolkow.  Isolation of an intrinsic precursor to molecular chemisorption.  Science, 279(5350):542–544, 1998.
  * [14] MR Zakin, RO Brickman, DM Cox, and A Kaldor.  Dependence of metal cluster reaction kinetics on charge state. ii. chemisorption of hydrogen by neutral and positively charged iron clusters.  The Journal of chemical physics, 88(10):6605–6610, 1988.
  * [15] Xiang-Jun Kuang, Xin-Qiang Wang, and Gao-Bin Liu.  A density functional study on the adsorption of hydrogen molecule onto small copper clusters.  Journal of Chemical Sciences, 123(5):743–754, 2011.
  * [16] Daniel T Gillespie.  Exact stochastic simulation of coupled chemical reactions.  The journal of physical chemistry, 81(25):2340–2361, 1977.
  * [17] Zeev Schuss.  Theory and applications of stochastic processes: an analytical approach, volume 170.  Springer Science & Business Media, 2009.
  * [18] Daniel T Gillespie.  A general method for numerically simulating the stochastic time evolution of coupled chemical reactions.  Journal of computational physics, 22(4):403–434, 1976.
  * [19] Daniel T Gillespie.  Concerning the validity of the stochastic approach to chemical kinetics.  Journal of Statistical Physics, 16(3):311–318, 1977.
  * [20] Daniel T Gillespie.  Approximate accelerated stochastic simulation of chemically reacting systems.  The Journal of chemical physics, 115(4):1716–1733, 2001.
  * [21] Ben Leimkuhler and Charles Matthews.  Molecular dynamics.  Interdisciplinary applied mathematics, 36, 2015.
  * [22] Ben Leimkuhler and Charles Matthews.  Numerical methods for stochastic molecular dynamics.  In Molecular Dynamics, pages 261–328. Springer, 2015.
  * [23] Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni.  Formulas and theorems for the special functions of mathematical physics, volume 52.  Springer Science & Business Media, 2013.
  * [24] James J Heckman.  The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models.  In Annals of economic and social measurement, volume 5, number 4, pages 475–492. NBER, 1976.
  * [25] Charles M Stein.  Estimation of the mean of a multivariate normal distribution.  The annals of Statistics, pages 1135–1151, 1981.
  * [26] Jeroen Gerritsen and J Rudi Strickler.  Encounter probabilities and community structure in zooplankton: a mathematical model.  Journal of the Fisheries Board of Canada, 34(1):73–82, 1977.
  * [27] Leandro Andrini, Germán J Soldano, Marcelo M Mariscal, Félix G Requejo, and Yves Joly.  Structure stability of free copper nanoclusters: Fsa-dft cu-building and fdm-xanes study.  Journal of Electron Spectroscopy and Related Phenomena, 235:1–7, 2019.
  * [28] Avelino Corma, Patricia Concepción, Mercedes Boronat, María J Sabater, Javier Navas, Miguel José Yacaman, Eduardo Larios, Álvaro Posadas, M Arturo López-Quintela, David Buceta, Ernest Mendoza, Gemma Guilera, and Álvaro Mayoral.  Exceptional oxidation activity with size-controlled supported gold clusters of low atomicity.  Nature Chemistry, 5(9):775–781, 2013.
  * [29] Per Bak and Maya Paczuski.  Complexity, contingency, and criticality.  Proceedings of the National Academy of Sciences, 92(15):6689–6696, 1995.
  * [30] Terrie M. Williams.  Criticality in stochastic networks.  Journal of the Operational Research Society, 43(4):353–357, 1992\.

Appendix

A.1

Errors in the Boltzmann model for the probability calculated according to
mathematical rebounds.

Program used: Origin 9.1

In all cases, number of points is 10, and degrees of freedon is 6.

L=0.05 m, T = 293 K  Parameter Value Standard Error A1 0.991 0.009 A2 -0.0060
0.0008 x0 4.98 0.02 dx 0.30 0.03

Reduced Chi-Sqr 2.66387 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.0016
Adj. R-Square: 0.99888

L=0.05 m, T = 373 K  Parameter Value Standard Error A1 0.981 0.006 A2 -0.005
0.003 x0 3.16 0.01 dx 0.22 0.02

Reduced Chi-Sqr 7.93743 $\times$ 10${}^{{}^{-5}}$
Residual Sum of Squares: 4.76246 $\times$ 10${}^{{}^{-4}}$
Adj. R-Square: 0.99957

L=0.05 m, T = 473 K  Parameter Value Standard Error A1 0.976 0.008 A2 -0.0011
0.0009 x0 3.10 0.02 dx 0.21 0.03

Reduced Chi-Sqr 1.30175 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 7.8105 $\times$ 10${}^{{}^{-4}}$
Adj. R-Square: 0.99927

L=0.1 m, T = 293 K  Parameter Value Standard Error A1 0.991 0.009 A2 -0.0060
0.0011 x0 4.98 0.02 dx 0.30 0.03

Reduced Chi-Sqr 2.66387 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.0016
Adj. R-Square: 0.99888

L=0.1 m, T = 373 K  Parameter Value Standard Error A1 0.994 0.008 A2 -0.006
0.002 x0 4.88 0.02 dx 0.33 0.02

Reduced Chi-Sqr 1.88888 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.00113
Adj. R-Square: 0.9992

L=0.1 m, T = 473 K  Parameter Value Standard Error A1 0.996 0.005 A2 -0.0043
0.0019 x0 4.73 0.01 dx 0.33 0.01

Reduced Chi-Sqr 7.76599 $\times$ 10${}^{{}^{-5}}$
Residual Sum of Squares: 4.65959 $\times$ 10${}^{{}^{-4}}$
Adj. R-Square: 0.99967

L=0.2 m, T = 293 K  Parameter Value Standard Error A1 0.993 0.003 A2 -0.0082
0.0025 x0 5.97 0.02 dx 0.31 0.03

Reduced Chi-Sqr 2.64593 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.00159
Adj. R-Square: 0.9989

L=0.2 m, T = 373 K  Parameter Value Standard Error A1 0.993 0.008 A2 -0.0082
0.0025 x0 5.97 0.02 dx 0.31 0.03

Reduced Chi-Sqr 2.64587 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.00159
Adj. R-Square: 0.9989

L=0.2 m, T = 473 K  Parameter Value Standard Error A1 0.993 0.008 A2 -0.0082
0.0025 x0 5.97 0.02 dx 0.31 0.03

Reduced Chi-Sqr 2.64587 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.00159
Adj. R-Square: 0.9989

A.2

Errors in the Boltzmann model for the probability calculated according to the
truncated normal model.

L=0.05 m, T = 293 K  Parameter Value Standard Error A1 0.982 0.006 A2 -0.0003
0.0001 x0 3.29 0.01 dx 0.24 0.01

Reduced Chi-Sqr 6.6611 $\times$ 10${}^{{}^{-5}}$
Residual Sum of Squares: 0.000399
Adj. R-Square: 0.99965

L=0.05 m, T = 373 K  Parameter Value Standard Error A1 0.982 0.006 A2 -0.004
0.003 x0 3.19 0.02 dx 0.22 0.02

Reduced Chi-Sqr 6.69947 $\times$ 10${}^{{}^{-5}}$
Residual Sum of Squares: 4.0196 $\times$ 10${}^{{}^{-4}}$
Adj. R-Square: 0.99960

L=0.05 m, T = 473 K  Parameter Value Standard Error A1 0.982 0.005 A2 -0.0004
0.0003 x0 3.19 0.01 dx 0.22 0.02

Reduced Chi-Sqr 6.69508 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 4.01705 $\times$ 10${}^{{}^{-4}}$
Adj. R-Square: 0.99964

L=0.1 m, T = 293 K  Parameter Value Standard Error A1 0.991 0.003 A2 -0.0022
0.0009 x0 5.02 0.07 dx 0.21 0.03

Reduced Chi-Sqr 5.5392 $\times$ 10${}^{{}^{-5}}$
Residual Sum of Squares: 0.00033
Adj. R-Square: 0.99977

L=0.1 m, T = 373 K  Parameter Value Standard Error A1 0.992 0.006 A2 -0.0047
0.0025 x0 4.92 0.01 dx 0.29 0.02

Reduced Chi-Sqr 1.20178 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.000721
Adj. R-Square: 0.9995

L=0.1 m, T = 473 K  Parameter Value Standard Error A1 0.995 0.005 A2 -0.0045
0.0025 x0 4.77 0.05 dx 0.33 0.01

Reduced Chi-Sqr 8.70051 $\times$ 10${}^{{}^{-5}}$
Residual Sum of Squares: 5.2203 $\times$ 10${}^{{}^{-4}}$
Adj. R-Square: 0.99963

L=0.2 m, T = 293 K  Parameter Value Standard Error A1 0.992 0.003 A2 -0.0078
0.0065 x0 6.01 0.02 dx 0.29 0.03

Reduced Chi-Sqr 2.75184 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.00165
Adj. R-Square: 0.99885

L=0.2 m, T = 373 K  Parameter Value Standard Error A1 0.990 0.007 A2 -0.0068
0.0075 x0 5.99 0.02 dx 0.28 0.03

Reduced Chi-Sqr 2.07471 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.00124
Adj. R-Square: 0.99913

L=0.2 m, T = 473 K  Parameter Value Standard Error A1 0.993 0.008 A2 -0.0082
0.0025 x0 5.97 0.02 dx 0.31 0.03

Reduced Chi-Sqr 2.64587 $\times$ 10${}^{{}^{-4}}$
Residual Sum of Squares: 0.00159
Adj. R-Square: 0.9989


# Unsupervised human-to-robot motion retargeting via expressive latent space

Yashuai Yan∗1 and Esteve Valls Mascaró∗1 and Dongheui Lee1,2 1Yashuai Yan and
Esteve Valls Mascaro and Dongheui Lee are with Autonomous Systems, Technische
Universität Wien (TU Wien), Vienna, Austria (e-mail:
{yashuai.yan,esteve.valls.mascaro, dongheui.lee}@tuwien.ac.at).2Dongheui Lee
is also with the Institute of Robotics and Mechatronics (DLR), German
Aerospace Center, Wessling, Germany.

###### Abstract

This paper introduces a novel approach for human-to-robot motion retargeting,
enabling robots to mimic human motion with precision while preserving the
semantics of the motion. For that, we propose a deep learning method for
direct translation from human to robot motion. Our method does not require
annotated paired human-to-robot motion data, which reduces the effort when
adopting new robots. To this end, we first propose a cross-domain similarity
metric to compare the poses from different domains (i.e., human and robot).
Then, our method achieves the construction of a shared latent space via
contrastive learning and decodes latent representations to robot motion
control commands. The learned latent space exhibits expressiveness as it
captures the motions precisely and allows direct motion control in the latent
space. We showcase how to generate in-between motion through simple linear
interpolation in the latent space between two projected human poses.
Additionally, we conducted a comprehensive evaluation of robot control using
diverse modality inputs, such as texts, RGB videos, and key-poses, which
enhances the ease of robot control to users of all backgrounds. Finally, we
compare our model with existing works and quantitatively and qualitatively
demonstrate the effectiveness of our approach, enhancing natural human-robot
communication and fostering trust in integrating robots into daily life.

Figure 1: Robot motion retargeting from various modalities. Our human-to-robot
motion retargeting connects robot control with diverse source modalities, such
as a text description or an RGB video. We project human skeletons to an
expressive latent space that preserves the semantics of the poses while
permitting pose control in the latent space. We are able to retarget human
poses to robot key-poses (green and blue dots) and empower the generation of
smooth transitions between those key-poses through the interpolation in the
latent space (orange dots), which ensures smooth and natural motion
generation.

##  I INTRODUCTION

In recent years, human-robot interaction has gained significant attention,
with robots increasingly becoming integral parts of our daily lives. As robots
become more prevalent, bridging the gap between human intentions and actions
becomes essential, allowing seamless communication and cooperation between
humans and robots. One crucial aspect of this interaction is the ability to
accurately and intuitively retarget human motion onto robots, improving robot
autonomy. This paper addresses the challenge of enabling robots to mimic human
motions while preserving the stylistic nuances of the original movement.

Retargeting human motion to robots is complex due to fundamental differences
between human and robot anatomies, kinematics, and motion dynamics. Unlike
humans, robots possess rigid bodies, constrained degrees of freedom, and
distinct physical limitations. Consequently, directly mapping human motion to
robot actuators often leads to unnatural and suboptimal robot behavior,
undermining the objective of achieving human-like movements. For example, when
retargeting the motion of a human touching their head with their right hand,
it is crucial that the retargeted robot poses also reproduce this touching
behavior in their motion. Solely replicating the specific arm movements could
lead to the robot hand not being close to the head due to the different robot
kinematics. Encoding this semantic in the retargeting task is essential to
ensure that robots have more natural and intuitive behaviors, leading to
better and easier collaboration with humans.

While motion retargeting is a long-standing challenge in the robotic and
animation community, lately, most research has been focused on the exploration
of large human motion capture (MoCap) datasets [1, 2] to learn and synthesize
human motions from different modality inputs: text [3], 3D scene [4], audio[5]
or conditioned by key-poses [6]. Our primary goal in this research is to
develop a novel method that eliminates the reliance on data annotation,
thereby accomplishing the learning of a shared representation space in which
human and robot poses are mutually and integrally represented. A good
representation space ensures that similar poses from both domains are
positioned close to each other while dissimilar poses are far apart. While
previous research [7, 8] requires manually annotating human and robot pairs
performing the same pose to learn this retargeting process, we consider an
unsupervised training technique that learns how to connect human poses to
robots without requiring pairing data. This is essential as it permits
learning human-robot mapping without prior annotations and reduces the
implementation costs for retargeting to new robots. For that, we recognize the
need to preserve the spatial relationships between human joints and the
semantics aspects that reflect the expression of the original human motion. We
incorporate a perceptual loss into the retargeting learning process to ensure
that the robot faithfully follows the movement of the human.

Thanks to the large human motion dataset [2], we are able to construct an
expressive latent space that remains tractable via simple operations. For
instance, thanks to the unsupervised contrastive training, we are able to
generate smooth robot motions between key-poses by simply using linear
interpolation in the latent space. This intuitive behavior facilitates motion
control and also showcases the robustness of our learned latent space.
Moreover, we use a decoder network to translate latent representations to
robot motion control commands. Our pipeline allows for the seamless
translation of human skeleton data into robot motion control.

Additionally, our model can be easily integrated into existing pre-trained
deep learning architectures to accommodate robot motion control from various
modalities, such as texts [9, 3] or 3D scenes [4], enabling flexible and
intuitive control over robot behavior. By addressing this challenge, we
anticipate significant advancements in human-robot collaboration and
interaction. Our research has broad applications, including robot-assisted
therapy, entertainment, teleoperation, and industrial robotics. Enabling
robots to replicate human motion and their intention opens up new
possibilities for intuitive and natural human-robot communication, enhancing
user experience and fostering trust and acceptance in integrating robots into
our daily lives.

In the subsequent sections of this paper, we will discuss the related work,
present our proposed deep learning model, describe the pipeline for human-to-
robot motion retargeting, and demonstrate the effectiveness and applicability
of our approach through extensive experiments and evaluations. We summarize
the contributions of our work next.

  1. 1.

Unsupervised deep learning approach to learn human-to-robot retargeting
without any paired human and robot motion data.

  2. 2.

Direct translation from human skeleton to robot control with simple control
from different modalities: text, video, or conditioned by key-poses.

  3. 3.

Expressive and tractable latent space to generate complex robot motion control
through simple linear interpolation.

##  II RELATED WORK

Existing literature on human-to-robot motion retargeting techniques is
reviewed next, highlighting limitations and the need for advancements in
translating human motion’s overall expressivity and naturality.

###  II-A Motion retargeting in animation

Human motion retargeting onto animated characters has been a long-standing
challenge in the computer graphics community. By bridging the gap between
human motion and animation, motion retargeting enhances the quality and
naturality of character animation, opening up possibilities for various
applications in fields such as film, gaming, and virtual reality.

Classical motion retargeting approaches [10, 11, 12, 13] involved manually
defining kinematic constraints and simplifying assumptions to map human motion
onto animated characters. These methods can be limited in their ability to
handle complex motions and may not accurately capture human movement’s
nuances. However, with the increased availability of motion capture data [1,
2], data-driven approaches have emerged as a more attractive alternative.
These approaches offer the potential to overcome the limitations of classical
methods and achieve more natural and nuanced motion transfer. [7, 8] learned a
share latent representation to translate motions between different kinematic
agents. However, they required paired training data, which is costly and
specific for each robot. To cope with the cost of pairing data, [14] used a
recurrent neural network to learn motion retargeting without pair data using
adversarial training and cycle consistency. [15] showed that disentangling
pose from movement in the retargeting process leads to more natural outcomes.
However, these data-driven approaches required the same source and target
kinematics. Inspired by the intuition that different kinematics can be reduced
to a common primal skeleton, [16] proposed explicitly encoding the different
skeleton topologies and projecting those into a shared latent space without
pairing data. [16] adopts a latent consistency loss to ensure that the
retargeted poses remain faithful to the source. However, their method is
limited to similar kinematics. Our work is inspired by their consistency idea,
but we construct a more expressive shared latent space through a contrastive
loss which improves the retargeting outcome. Recently, [17, 18] have focused
on the motion retargeting but considering the mesh constraints of the animated
characters, and thus adjusting motions to reduce interpenetration and
feasibility of the motions. Despite this topic being essential in robotics,
our work focuses more on preserving the source motion semantics and considers
the feasibility (i.e., self-collision) issue as future work. [18] proposes a
Euclidean distance matrix to account for the motion semantics. While this
approach works for skeletons with similar proportions, it underperforms when
the targets have different trunk-to-arms ratios. On the contrary, we propose
to encode this semantic relation through global rotations, which precisely
capture the semantics in the retargeting task,

###  II-B Motion retargeting in robotics

Despite the great success of motion retargeting for character animation, their
community has only been considering the feasibility of the movements in terms
of physical constraints recently [17, 18]. Apart from the motion’s
feasibility, robotics research also requires adequate control of the
appropriate robot based on the source motion. [19] considers constrained
optimization algorithms to retarget a human motion in a simulated robot but
requires learning a given trajectory and can not fastly overcome new
variations.[20] proposes Bayesian optimization and inverse kinematics to
tackle natural retargeting, but their approach requires manually selecting
joints of interest and is constrained to a few specific motions. Deep learning
has become a solution to ensure generalization in the retargeting process
while being efficient. [21] constructs a shared latent space to retarget human
motion to humanoid robots, and [22] extends the approach by guaranteeing
collision-free robot poses. However, both works require pair data to learn the
shared representation. [22] proposes a process to generate pairing data
automatically, but this process fails to well-preserve the semantics when
projecting relaxed human poses to robot poses.

##  III METHODOLOGY

In this section, we present an overview of our proposed framework for
unsupervised human-to-robot motion retargeting via expressive latent space.
First, we formulate the human-to-robot retargeting task. Then, we describe our
encoder-decoder deep learning architecture, illustrated in Figure 2.

Figure 2: Model overview. Both human data $\mathbf{x}_{h}$ and robot data
$\mathbf{x}_{r}$ are encoded into a common latent space by $Q_{h}$ and
$Q_{r}$, respectively. The triplet loss $\mathcal{L}_{triplet}$ encourages
better clustering of data points in the latent space via bringing similar
samples (e.g., $z^{j}$ and $z^{k}$) together and pushing dissimilar samples
(e.g., $z^{j}$ and $z^{i}$) apart. The decoder $D$ maps latent variables $z$
to robot data. If $z$ is encoded from robot data (e.g. $z^{k}$), the
reconstruction is enforced via $\mathcal{L}_{rec}$. If $z$ is from human data
(e.g., $z^{j}$), the latent consistency between the original latent
representation and that translated by $Q_{r}$ is achieved by
$\mathcal{L}_{ltc}$. During the inference phase, human data is first encoded
into latent variables using $Q_{h}$. From them, we predict robot joint angles
to retarget human motions to the robot effectively.

###  III-A Problem Formulation

Let $\mathbf{x}_{h}=[x_{h,1},\cdots,x_{h,J_{h}}]\in\mathbb{R}^{J_{h}\times n}$
be a human pose composed by $J_{h}$ joints. Similarly,
$\mathbf{x}_{r}=[x_{r,1},\cdots,x_{r,J_{r}}]\in\mathbb{R}^{J_{r}\times n}$
represents a robot pose. Then, the task of human motion retargeting can be
formulated as finding a function $f$ that maps a $\mathbf{x}_{h}$ to
$\mathbf{x}_{r}$ ($f:\mathbf{x}_{h}\longmapsto\mathbf{x}_{r}$) so that
$\mathbf{x}_{r}$ preserves the human-like naturality of the pose
$\mathbf{x}_{h}$. However, human and robot joints usually have different
configurations; for example, a human joint (e.g., wrist joint) can have more
than 1DoF, while one robot joint usually has only 1DoF. We use different
representations of human and robot poses to cope with such differences. This
paper describes each human joint $x_{h,j}$ as its quaternion representation
referring to its parent ($n=4$). While each robot joint $x_{r,j}$ (i.e.,
revolute joint) is described as its joint angle ($n=1$).

In our particular case, and contrary to all works focusing on character
animation, we are interested in the direct control of a robot. Robots can be
controlled via their joint angles. As joint angles for robots and humans have
different configurations, it makes little sense to compare joint angles to
measure their similarity. Instead, we propose to use the global rotation of
body links to compare the similarity between human and robot poses. The
similarity metric is defined in Section III-B.

Previous works [21, 22] rely on the acquisition of a dataset of mapped motions
between the human and the robot to retarget, which we describe as a
$\\{\mathbf{x}_{h},\mathbf{x}_{r}\\}$ pair. These works learn the retargeting
function $f$ in a supervised manner. On the contrary, we consider the
retargeting task without collecting the correct
$\\{\mathbf{x}_{h},\mathbf{x}_{r}\\}$ pair and learn without supervision how
to approximate $f$ better. To this end, our model first learns to project
human $\mathbf{x}_{h}$ and robot $\mathbf{x}_{r}$ poses to the same
representation space. Then, we decode the learned representation to robot
joint angles, which allows us to control the robot directly.

###  III-B Cross-domain similarity metric

To define a similarity metric between humans and robots, we propose to use the
global rotation of body links over using local quaternions [16] or the
relative XYZ position of the end effector [18], as the global rotation can
better preserve visual similarity, as it captures the complete 3D orientation
and remains invariant to coordinate systems and articulation variations. Let
$q_{h,j}$ and $q_{r,j}$ represent the global quaternions of the same semantic
links (e.g., shoulder-to-elbow, elbow-to-wrist, etc.) of a human pose
$\mathbf{x}_{h}$ and a robot pose $\mathbf{x}_{r}$. The distance between the
two poses can be computed as shown in Equation 1, where $<,>$ denotes the dot
product between two vectors.

$D_{GR}(\mathbf{x}_{h},\mathbf{x}_{r})=\sum_{j}(1-<q_{h,j},q_{r,j}>^{2})$ (1)

$D_{GR}$ is employed to measure the similarity between two poses used for
contrastive learning in Section III-C.

###  III-C Human-to-Robot shared representation

We formulate the task of motion retargeting as the translation of two domains.
We adopt two encoders ($Q_{h}$, $Q_{r}$) to project the human and robot poses,
respectively, to a shared representation space. This way, our encoders can
project $\mathbf{x}_{h}\in\mathbb{R}^{J_{h}\times n}$ to $z\in\mathbb{R}^{d}$
and $\mathbf{x}_{r}\in\mathbb{R}^{J_{r}\times n}$ to $z\in\mathbb{R}^{d}$.
Given a human pose $\mathbf{x}_{h}$, our shared latent space is used as a
bridge to generate $\mathbf{x}_{r}$ with conserving its similarity.

We propose to learn the retargeting function
$f:\mathbf{x}_{h}\longmapsto\mathbf{x}_{r}$ without any paired human and robot
motion data. We adopt a similar strategy inspired by the recent success of
contrastive learning methods (e.g., CLIP [23]). Contrastive learning is a
training technique to learn valuable representations from unlabeled data by
comparing and contrasting different instances. It involves training a neural
network to maximize the agreement between positive pairs (similar instances)
and minimize the agreement between negative pairs (dissimilar instances).

Let us assume a large set of data that contains feasible human poses
$\mathbf{x}_{h}$ and robot poses $\mathbf{x}_{r}$. Our method randomly selects
triplets of projections from these data instances. Then we select an anchor
$z_{o}$, a positive sample $z_{+}$, and a negative $z_{\\_}$ sample based on a
global rotation distance $D_{GR}$ in Equation 1. The positive sample $z_{+}$
has a smaller rotation distance to the anchor $z_{o}$ than the negative
instance $z_{\\_}$.

We adopt the Triplet Loss [24] that pulls similar samples close while
simultaneously pushing dissimilar samples away in the latent space. This helps
create a more discriminative and expressive representation space, where
similar instances are clustered together and dissimilar instances are pushed
apart. Equation 2 shows the Triplet Loss $\mathcal{L}_{triplet}$ used in our
scenario, where $\alpha=0.05$.

$\mathcal{L}_{triplet}=max(||z_{o}-z_{+}||_{2}-||z_{o}-z_{-}||_{2}+\alpha,0)$
(2)

###  III-D Shared representation to robot control

Our proposed encoders allow us to project a human pose $\mathbf{x}_{h}$ and a
robot pose $\mathbf{x}_{r}$ into a shared representation space $z$. Therefore,
the next step is to learn how to decode the representation $z$ into robot
joint angles $\mathbf{x}_{r}$ that can directly be used to control the robot.
We use a decoder $D$ to decode the projected robot or human poses $z$ to the
joint angle representation of the robot pose $\mathbf{x}_{r}$. We employ a
standard reconstruction loss over the joint angles in the robot domain, as
shown in Equation 3. To ensure that the decoded joint angles from projected
human poses are feasible, we adopt the latent consistent loss shown in
Equation 4 to encourage robustness in the retargeting process, similar to
[16].

$\mathcal{L}_{rec}=||x_{r}-D(Q_{r}(x_{r}))||_{1}$ (3)

$\mathcal{L}_{ltc}=||Q_{h}(x_{h})-Q_{r}(D(Q_{h}(x_{h})))||_{1}$ (4)

Our approach employs an end-to-end training strategy, enabling the encoder to
learn an expressive shared representation space for both human and robot poses
in an unsupervised manner while ensuring that this representation space is
reconstructible to robot control through our decoder. The total loss employed
during training is a weighing sum as described in Equation 5, where
$\lambda_{triplet}=10,\lambda_{rec}=5$.

$\mathcal{L}=\lambda_{triplet}\mathcal{L}_{triplet}+\lambda_{rec}\mathcal{L}_{rec}+\mathcal{L}_{ltc}$
(5)

##  IV EXPERIMENTS

The experimental setup and datasets used to evaluate the performance of our
model are presented, along with the metrics and benchmarks employed to assess
the accuracy and fidelity of the retargeted robot motions.

###  IV-A Experiment Settings

The hyperparameter configurations used in our framework are listed in this
subsection. The network consisting of two encoders and one decoder is trained
end-to-end with a learning rate of $0.001$ and batch size of $256$. The
encoder and decoder have the same structure; they have $6$ hidden layers, each
with $128$ units. The shared latent space is of 8 dimensions. Adam [25], a
momentum-based method, is utilized to optimize the loss function during
training. We use a Ubuntu 22.04 and RTX A4000 Graphic card for our experiment.

Additionally, we employ a bi-manual Tiago++ robot that integrates two 7-DoF
arms. In this paper, we focus on the motion of the upper and lower arm parts.
We ignore the motion of the two hands, as the HumanML3D human motion dataset
[2] used does not contain hand motions. Therefore, the rotation distance
$D_{RD}$ in Equation 1 is defined on four links: left should-to-elbow, left
elbow-to-wrist, right shoulder-to-elbow, and right elbow-to-wrist. We are
using the Gazebo simulator to simulate the robot, and the environment’s
implementation involves using the MOVEit package, which takes care of the
execution required for internal joint movement in the robot.

###  IV-B Data collection

We present a robot pose generation procedure that requires only the robot’s
kinematic information. First, we sample the robot joint angles from its
configuration space. The robot pose can be computed by following its forward
kinematics. In such a way, we collect around 15M poses from the Tiago++ robot
by randomly sampling angles per joint. For human motions, we use the HumanML3D
dataset [2] that consists of 14616 motions with a total length of 28.59 hours.
In HumanML3D, a human pose is represented by its skeletons. Considering only
the upper-body motion retargeting in this paper, the upper-body pose
consisting of 14 skeletons is selected as the network’s input.

###  IV-C Baseline

We implement S3LE [22] as our baseline. To train S3LE, we use a similar method
as mentioned in [22] to generate paired data. We generate 200K paired data by
selecting the pairs with minimal rotation distance measured by Equation 1. The
paired data is only used to train our baseline method.

###  IV-D Quantitative evaluation

To evaluate the performance of each retargeting method, we annotated 11
distinct expressive motions that were not observed while training. The
annotated motions serve as the ground truth for our evaluation. We employ the
Mean Square Error (MSE) of joint angles between ground truth and predicted
results to quantify our proposed method. Furthermore, our method endeavors to
address motion retargeting in real-time scenarios. We thoroughly evaluated the
computational efficiency and speed at which our model operates.

Table I compares our method with the baseline in Section IV-C. Our method
outperforms the baseline in terms of MSE of joint angles. Furthermore, our
novel approach demonstrates a notable increase in operational efficiency,
surpassing the baseline by more than a factor of three. With an impressive
speed of 1.5kHz, our method readily fulfills the requirements of most advanced
robot control systems.

TABLE I: Performance of our proposed method and the baseline. The Mean Square Error (MSE) of joint angles between ground truth and predicted results are compared here. Bold fonts indicate better results. | Joint Angles | Control Frequency (kHz)
---|---|---
Baseline | 0.44 | 0.4
Ours | 0.21 | 1.5

###  IV-E Qualitative evaluation

Visually compelling examples and comparisons between the original human and
retargeted robot motions are showcased in Figure 3. For the selected human
motions, we annotated their ground truth shown in the last row. Our method
accurately retargets the motion when the input skeleton lifts hands above his
head, lifts hands to his chest, or performs T-pose, whereas the baseline
fails.

Figure 3: Human Retargeting comparison for different key-poses. Various human
skeleton key-poses are retargeted to the Thiago robot. Our model captures the
initial pose’s visual similarity and is closely related to the manually
annotated ground-truth poses.

###  IV-F Ablation Study

An ablation study is conducted to systematically analyze the impact of
individual loss components in our proposed model. We utilize three loss
components in our approach to optimize retargeted motions. When analyzing the
results in Table II, it becomes apparent that the removal of the latent
consistency loss $\mathcal{L}_{ltc}$ results in a slight reduction in the
performance of our method. On the contrary, the Triplet loss
$\mathcal{L}_{triplet}$ is indispensable for the optimization process. As
supported by the experimental results, eliminating $\mathcal{L}_{triplet}$
significantly increases the loss value, rising from 0.21 to 0.57. This
underscores the crucial role played by $\mathcal{L}_{triplet}$ in achieving
improved optimization outcomes, contrary to all previous works that do not
explore our contrastive training.

TABLE II: Ablation study of proposed loss components. Mean Square Error (MSE) of joint angles between ground truth and predicted results. Bold fonts indicate better results. Loss Component | Joint Angles
---|---
$\mathcal{L}_{triplet}+\mathcal{L}_{rec}+\mathcal{L}_{ltc}$ | 0.21
$\mathcal{L}_{triplet}+\mathcal{L}_{rec}$ | 0.24
$\mathcal{L}_{rec}+\mathcal{L}_{ltc}$ | 0.57

###  IV-G From RGB videos to robot motions

The proposed method can generate natural and visually similar motions from RGB
videos. We use, MultiPerson [26], the state-of-the-art 3D pose estimation
algorithm, to obtain human 3D skeletons from RGB images. Since there is no
ground truth, we only show snapshots of reference images and corresponding
Tiago poses in Figure 4 for qualitative evaluation. More videos can be found
in the supplementary materials.

Figure 4: Video-to-Motion. We leverage state-of-the-art off-the-shelf 3D human
pose estimators, MultiPerson [26], to translate RGB images into human
skeletons. Then we employ our proposed method to achieve direct motion control
from human skeletons.

###  IV-H From texts to robot motions

Text is an essential modality for human motions. Using a pre-trained motion
synthesis model, Text-to-Motion Retrieval [27], our method can generate robot
motions with texts. To this end, We first retrieve human motions from texts
with Text-to-Motion Retrieval and then retarget human motions to the Tiago++
robot. Figure 5 shows two examples of retargeting motion from texts. More
examples can be found in the supplementary materials.

Figure 5: Text-to-Motion. Our model is able to connect as a pipeline to pre-
trained motion synthesis models. In this case, we first use Text-to-Motion
Retrieval [27] to get human motion in skeleton representation. Then, we
utilize our proposed method to translate the motion into robot control
commands (i.e., joint angles) to mimic it. Our results demonstrate that our
model is able to capture the expressiveness of the text, despite being
retargeted from human to robot kinematics.

###  IV-I From key-poses to robot motions

Thanks to the large human and robot motion datasets that our method is trained
on, the learned latent space is able to cover expressive motions. Furthermore,
the contrastive loss $\mathcal{L}_{triplet}$ makes similar poses close and
dissimilar poses far away in latent space. In such a way, our proposed method
learns an expressive and smooth latent space, which enables us to interpolate
motions between key-poses. In Figure 6, we show three key-poses: A, B, and C,
and the interpolated in-between motions. For interpolation, two key (e.g., A
and B) poses are mapped into two points in latent space, and intermediate
steps can be linearly interpolated in between them. The in-between motions are
decoded from these interpolated steps.

Figure 6: Keyposes-to-Motion. The proposed method enables motion generation
with key-poses. Given distinct key-poses, natural in-between motions can be
generated by linearly interpolating key-poses in our learned latent space. Our
results provide the potential for direct motion control in latent space

###  IV-J Future work

Our next step is to implement our method in a real robot. For that, we will
consider avoiding self-collision by combining our work with previous works
[28, 29, 30, 31].

##  V CONCLUSIONS

In this paper, we presented an unsupervised motion retargeting method that
ensures an expressive latent space for motion generation. To this end, we use
contrastive learning combined with deep latent space modeling to incorporate
human and robot motion data. To construct a shared representation of human and
robot motion, we define a cross-domain similarity metric based on the global
rotation of different body links. Similar motions are clustered together, and
dissimilar motions are pushed apart while constructing the latent space.
Furthermore, our decoder maps the shared representation to robot joint angles
to control a robot directly without any additional optimization process. Our
human-to-robot retargeting framework permits the generation of robot motions
properly while preserving the human source’s semantics and naturality.
Additionally, we connect our model with existing pre-trained models to achieve
motion retargeting from different modalities, such as controlling the robot
with given texts or retargeting from RGB videos. Moreover, our learned latent
space remains tractable and allows for the generation of smooth motion
inbetweening between two distinct key-poses through linear interpolation in
the projected latent space. We showcase all results and the robustness of our
model through various experiments, both quantitatively and qualitatively.

## ACKNOWLEDGMENT

This work is funded by Marie Sklodowska-Curie Action Horizon 2020 (Grant
agreement No. 955778) for project ’Personalized Robotics as Service Oriented
Applications’ (PERSEO).

## References

  * [1] Catalin Ionescu, Dragos Papava, Vlad Olaru, and Cristian Sminchisescu.  Human3.6m: Large scale datasets and predictive methods for 3d human sensing in natural environments.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 2014\.
  * [2] Chuan Guo, Shihao Zou, Xinxin Zuo, Sen Wang, Wei Ji, Xingyu Li, and Li Cheng.  Generating diverse and natural 3d human motions from text.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 5152–5161, June 2022.
  * [3] Guy Tevet, Sigal Raab, Brian Gordon, Yoni Shafir, Daniel Cohen-or, and Amit Haim Bermano.  Human motion diffusion model.  In The Eleventh International Conference on Learning Representations, 2023.
  * [4] Jingbo Wang, Yu Rong, Jingyuan Liu, Sijie Yan, Dahua Lin, and Bo Dai.  Towards diverse and natural scene-aware 3d human motion synthesis.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 20460–20469, 2022.
  * [5] Simon Alexanderson, Rajmund Nagy, Jonas Beskow, and Gustav Eje Henter.  Listen, denoise, action! audio-driven motion synthesis with diffusion models.  ACM Trans. Graph., 42(4):1–20, 2023.
  * [6] Esteve Valls Mascaro, Shuo Ma, Hyemin Ahn, and Dongheui Lee.  Robust human motion forcasting using transformer-based model.  In 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 10674–10680, 2022.
  * [7] Brian Delhaisse, Domingo Esteban, Leonel Rozo, and Darwin Caldwell.  Transfer learning of shared latent spaces between robots with similar kinematic structure.  02 2017.
  * [8] Hanyoung Jang, Byungjun Kwon, Moonwon Yu, Seong Kim, and Jongmin Kim.  A variational u-net for motion retargeting.  pages 1–2, 12 2018.
  * [9] Mathis Petrovich, Michael J Black, and Gül Varol.  Tmr: Text-to-motion retrieval using contrastive 3d human motion synthesis.  arXiv preprint arXiv:2305.00976, 2023.
  * [10] Michael Gleicher.  Retargetting motion to new characters.  Proceedings of the 25th annual conference on Computer graphics and interactive techniques, 1998.
  * [11] Jehee Lee and Sung Yong Shin.  A hierarchical approach to interactive motion editing for human-like figures.  In Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’99, page 39–48, USA, 1999. ACM Press/Addison-Wesley Publishing Co.
  * [12] Christian Ott, Dongheui Lee, and Yoshihiko Nakamura.  Motion capture based human motion recognition and imitation by direct marker control.  In Humanoids 2008 - 8th IEEE-RAS International Conference on Humanoid Robots, pages 399–405, 2008.
  * [13] Dongheui Lee, Christian Ott, and Yoshihiko Nakamura.  Mimetic communication model with compliant physical contact in human—humanoid interaction.  The International Journal of Robotics Research, 29(13):1684–1704, 2010.
  * [14] Ruben Villegas, Jimei Yang, Duygu Ceylan, and Honglak Lee.  Neural kinematic networks for unsupervised motion retargetting.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 8639–8648, 2018.
  * [15] Jongin Lim, Hyung Jin Chang, and Jin Young Choi.  Pmnet: learning of disentangled pose and movement for unsupervised motion retargeting.  In Proceedings of the 30th British Machine Vision Conference (BMVC 2019). British Machine Vision Association, BMVA, September 2019.  30th British Machine Vision Conference (BMVC 2019) ; Conference date: 09-09-2019 Through 12-09-2019.
  * [16] Kfir Aberman, Peizhuo Li, Dani Lischinski, Olga Sorkine-Hornung, Daniel Cohen-Or, and Baoquan Chen.  Skeleton-aware networks for deep motion retargeting.  ACM Transactions on Graphics (TOG), 39(4):62–1, 2020.
  * [17] Ruben Villegas, Duygu Ceylan, Aaron Hertzmann, Jimei Yang, and Jun Saito.  Contact-aware retargeting of skinned motion, 2021.
  * [18] Jiaxu Zhang, Junwu Weng, Di Kang, Fang Zhao, Shaoli Huang, Xuefei Zhe, Linchao Bao, Ying Shan, Jue Wang, and Zhigang Tu.  Skinned motion retargeting with residual perception of motion semantics & geometry.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 13864–13872, 2023.
  * [19] Waldez Gomes, Vishnu Radhakrishnan, Luigi Penco, Valerio Modugno, Jean-Baptiste Mouret, and Serena Ivaldi.  Humanoid whole-body movement optimization from retargeted human motions.  In 2019 IEEE-RAS 19th International Conference on Humanoid Robots (Humanoids), pages 178–185, 2019.
  * [20] Sungjoon Choi and Joohyung Kim.  Towards a natural motion generator: a pipeline to control a humanoid based on motion data.  In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 4373–4380, 2019.
  * [21] Sungjoon Choi, Matthew Pan, and Joohyung Kim.  Nonparametric motion retargeting for humanoid robots on shared latent space.  07 2020.
  * [22] Sungjoon Choi, Min Jae Song, Hyemin Ahn, and Joohyung Kim.  Self-supervised motion retargeting with safety guarantee.  In 2021 IEEE International Conference on Robotics and Automation (ICRA), pages 8097–8103. IEEE, 2021.
  * [23] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever.  Learning transferable visual models from natural language supervision, 2021.
  * [24] Elad Hoffer and Nir Ailon.  Deep metric learning using triplet network, 2018.
  * [25] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  arXiv preprint arXiv:1412.6980, 2014.
  * [26] Junuk Cha, Muhammad Saqlain, GeonU Kim, Mingyu Shin, and Seungryul Baek.  Multi-person 3d pose and shape estimation via inverse kinematics and refinement.  In ECCV, 2022.
  * [27] Mathis Petrovich, Michael J. Black, and Gül Varol.  TMR: Text-to-motion retrieval using contrastive 3D human motion synthesis.  arXiv preprint, 2023.
  * [28] Mikhail Koptev, Nadia Figueroa, and Aude Billard.  Real-time self-collision avoidance in joint space for humanoid robots.  IEEE Robotics and Automation Letters, 6(2):1240–1247, 2021.
  * [29] Hisashi Sugiura, Michael Gienger, Herbert Janssen, and Christian Goerick.  Real-time self collision avoidance for humanoids by means of nullspace criteria and task intervals.  In 2006 6th IEEE-RAS International Conference on Humanoid Robots, pages 575–580, 2006.
  * [30] Agostino De Santis, Alin Albu-Schaffer, Christian Ott, Bruno Siciliano, and Gerd Hirzinger.  The skeleton algorithm for self-collision avoidance of a humanoid manipulator.  In 2007 IEEE/ASME international conference on advanced intelligent mechatronics, pages 1–6, 2007.
  * [31] Hisashi Sugiura, Michael Gienger, Herbert Janssen, and Christian Goerick.  Real-time collision avoidance with whole body motion control for humanoid robots.  In 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 2053–2058, 2007.