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\section{Introduction}\label{sec:intro}
A major challenge towards self-organizing networks (SON) is the joint
optimization of multiple SON use cases by
coordinately handling multiple configuration parameters. Widely
studied SON use cases include coverage and capacity optimization
(CCO), mobility load balancing (MLB) and mobility robustness
optimization (MRO)\cite{3GPP36902}.\cosl{We need a reference here} However, most of
these works study an isolated single use case and ignore the conflicts
or interactions between the use cases
\cite{giovanidis2012dist,razavi2010self}.
In contrast, this paper considers a joint optimization of two strongly
coupled use cases: CCO and MLB. The objective is to achieve a good
trade-off between coverage and capacity performance, while ensuring a
load-balanced network. The SON functionalities are usually implemented
at the network management layer and are designed to deal with \lq\lq
long-term\rq\rq \ network performance. Short-term optimization of
individual users is left to lower layers of the protocol stack. To
capture long-term global changes in a network, we consider a
cluster-based network scenario, where users served by the same base
station (BS) with similar SINR distribution are adaptively grouped
into clusters. Our objective is to jointly optimizing the following
variables:
\begin{itemize}
\item Cluster-based BS assignment and power allocation.
\item BS-based antenna tilt optimization and power allocation.
\end{itemize}
The joint optimization of assignment, antenna tilts, and powers is an
inherently challenging problem. The interference and the resulting
performance measures depend on these variables in a complex and
intertwined manner. Such a problem, to the best of the authors'
knowledge, has been studied in only a few works. For example, in
\cite{klessig2012improving} a problem of jointly optimizing antenna
tilt and cell selection to improve the spectral and energy efficiency
is stated, however, the solution derived by a structured searching
algorithm may not be optimal.
In this paper, we propose a robust algorithmic framework built on a
utility model, which enables fast and near-optimal uplink solutions and sub-optimal downlink solutions\cosl{Do
we know that this is near-optimal?} by exploiting three properties:
1) the monotonic property and fixed point of the monotone and strictly subhomogenoues (MSS) functions \footnote{Many literatures use the term {\it interference function} for the functions satisfy three condotions, positivity, monotonicity and scalability \cite{yates95}. Positivity is shown to be a consequence of the other two properties \cite{leung2004convergence}, and we use the term {\it strctly subhomogeneous} in place of scalable from a constraction mapping point of view in keeping with some related literature \cite{nuzman2007contraction}.}, 2)
decoupled property of the antenna tilt and BS assignment optimization
in the uplink network, and 3) uplink-downlink duality. The first
property admits global optimal solution with fixed-point iteration for
two specific problems: utility-constrained power minimization and
power-constrained max-min utility balancing
\cite{vucic2011fixed,stanczak2009fundamentals,schubert2012interference,yates95}. The
second and third properties enable decomposition of the
high-dimensional optimization problem, such as the joint beamforming
and power control proposed in
\cite{BocheDuality06,schubert2005iterative,huang2013joint,he2012multi}. Our
distinct contributions in this work can be summarized as follows:\\
1) We propose a max-min utility balancing algorithm for
capacity-coverage trade-off optimization over a joint space of antenna
tilts, BS assignments and powers. The utility defined as a convex
combination of the average SINR and the worst-case SINR implies the balanced performance of capacity and coverage. Load
balancing is improved as well due to a uniform distribution of the
interference among the BSs.\\
2) The proposed utility is formulated based on the MSS functions, which allows us to find the optimal solution by applying
fixed-point iterations.\\
3) Note that antenna tilts are BS-specific variables, while assignments are cluster-specific, we develop two optimization problems with the same objective functions, formulated either as a problem of per-cluster variables or as a problem of per-base variables. We
propose a two-step optimization algorithm in the uplink to iteratively
optimize the per BS variables (antenna tilts and BS power budgets) and the cluster-based variables (assignments and cluster power). Since both problems aim at optimizing the same objective function, the algorithm is shown to be convergent.\\
4) The decoupled property of antenna tilt and assignment in the uplink decomposes the high-dimensional optimization problem and enables more efficient optimization algorithm. We then analyze the uplink-downlink duality by using the Perron-Frobenius theory\cite{meyer2000matrix}, and propose an efficient
sub-optimal solution in the downlink by utilizing optimized variables
in the dual uplink.
\section{System Model}\label{sec:Model}
We consider a multicell wireless network composed of a set of BSs
$\set{N}:=\{1,\ldots, N\}$ and a set of users $\set{K}:=\{1,\ldots,
K\}$. Using fuzzy C-means clustering algorithm \cite{bezdek1984fcm},
we group users with similar SINR distributions\footnote{We assume the
Kullback-Leibler divergence as the distance metric.} and served by the same BS into
clusters. The clustering algorithm is beyond the scope of this
paper. Let the set of user clusters be denoted by
$\set{C}:=\{1,\ldots,C\}$, and let $\bm{A}$ denote a $C\times K$
binary user/cluster assignment matrix whose columns sum to one. The
BS/cluster assignment is defined by a $N\times C$ binary matrix
$\bm{B}$ whose columns also sum to one.
Throughout the paper, we assume a frequency flat channel. The
average/long-term downlink path attenuation between $N$ BSs and $K$
users are collected in a channel gain matrix $\bm{H}\in {\field{R}}^{N\times
K}$. We introduce the cross-link gain matrix $\bm{V}\in{\field{R}}^{K\times
K}$, where the entry $v_{lk}(\theta_j)$ is the cross-link gain
between user $l$ served by BS $j$, and user $k$ served by BS $i$,
i.e., between the transmitter of the link $(j, l)$ and the receiver of
the link $(i, k)$. Note that $v_{lk}(\theta_j)$ depends on the antenna
downtilt $\theta_j$. Let the BS/user assignment matrix be denoted by
$\bm{J}$ so that we have $\bm{J}:=\bm{B}\bm{A}\in\{0,1\}^{N\times K}$,
and $\bm{V}:=\bm{J}^T\bm{H}$. We denote by $\bm{r}:=[r_1, \ldots,
r_N]^T$, $\bm{q}:=[q_1, \ldots, q_C]^T$ and $\bm{p}:=[p_1, \ldots,
p_K]^T$the BS transmission power budget, the cluster power allocation
and the user power allocation, respectively.
%
\subsection{Inter-cluster and intra-cluster power sharing factors}
\label{subsec:powFactor}
We introduce the inter-cluster and intra-cluster power sharing factors
to enable the transformation between two power vectors with different
dimensions. Let $\bm{b}:=[b_1, \ldots, b_C]^T$ denote the serving BSs
of clusters $\{1, \ldots, C\}$. We define the vector of the
inter-cluster power sharing factors to be $\bm{\beta}:=[\beta_1,
\ldots, \beta_C]^T$, where $\beta_c:=q_c/r_{b_c}$. With the
BS/cluster assignment matrix $\bm{B}$, we have $\bm{q}:=\ma{B}_{\ve{\beta}}^T \bm{r}$,
where $\ma{B}_{\ve{\beta}}:=\bm{B}\mathop{\mathrm{diag}}\{\bm{\beta}\}$. Since users belonging to the
same cluster have similar SINR distribution, we allocate the cluster
power uniformly to the users in the cluster. The intra-cluster sharing
factors are represented by $\bm{\alpha}:=[\alpha_1, \ldots,
\alpha_K]^T$ with $\alpha_k=1/|\set{K}_{c_k}|$ for $k\in\set{K}$,
where $\set{K}_{c_k}$ denotes the set of users belonging to cluster
$c_k$, while $c_k$ denotes the cluster with user $k$. We have
$\bm{p}:=\ma{A}_{\ve{\alpha}}^T\bm{q}$, where $\ma{A}_{\ve{\alpha}}:=\bm{A}\mathop{\mathrm{diag}}\{\bm{\alpha}\}$. The
transformation between BS power $\bm{r}$ and user power $\bm{p}$ is
then $\bm{p}:=\bm{T}\bm{r}$ where the transformation matrix
$\bm{T}:=\ma{A}_{\ve{\alpha}}^T\ma{B}_{\ve{\beta}}^T$.
%
\subsection{Signal-to-interference-plus-noise ratio}\label{subsec:SINR}
Given the cross-link gain matrix $\bm{V}$, the downlink SINR of the $k$th user depends on all
powers and is given by
\begin{equation}
\operator{SINR}_k^{(\text{d})}:=\frac{p_k \cdot v_{kk}(\theta_{n_k})}{\sum_{l\in\set{K}\setminus k} p_l \cdot v_{lk}(\theta_{n_l})+\sigma_k^2}, k\in\set{K}
\label{eqn:DL_SINR}
\end{equation}
where $n_k$ denotes the serving BS of user $k$, $\sigma_k^2$ denotes
the noise power received in user $k$. Likewise, the uplink SINR is
\begin{equation}
\operator{SINR}_k^{(\text{u})}:=\frac{p_k \cdot v_{kk}(\theta_{n_k})}{\sum_{l\in\set{K}\setminus k} p_l \cdot v_{kl}(\theta_{n_k})+\sigma_k^2}, k\in\set{K}
\label{eqn:UL_SINR}
\end{equation}
%
Assuming that there is no self-interference, the cross-talk terms can
be collected in a matrix
\begin{equation}
[\tilde{\ma{V}}]_{lk}:=
\begin{cases}
v_{lk}(\theta_{n_l}), & l\neq k\\
0, & l=k
\end{cases}.
\label{eqn:PsiMat}
\end{equation}
Thus the downlink interference received by user $k$ can be written as
$I_k^{(\text{d})}:=[\tilde{\bm{V}}^T\bm{p}]_k$, while the uplink interference
is given by $I_k^{(\text{u})}:=[\tilde{\bm{V}}\bm{p}]_k$.
A crucial property is that the uplink SINR of user $k$ depends on the
BS assignment $n_k$ and the single antenna tilt $\theta_{n_k}$ alone,
while the downlink SINR depends on the BS assignment vector
$\bm{n}:=[n_1,\ldots, n_K]^T$, and the antenna tilt vector
$\bm{\theta}:=[\theta_1, \ldots, \theta_N]^T$. The decoupled property
of uplink transmission has been widely exploited in the context of
uplink and downlink multi-user beamforming \cite{BocheDuality06}\cosl{Reference} and
provides a basis for the optimization algorithm in this paper.
%
The notation used in this paper is summarized in Table \ref{tab:CovCap_notation}.
\begin{table}[t]
\centering
\caption{NOTATION SUMMARY}
\begin{tabular}{|c|c|}
\hline
${\emenge{N}}$ & set of BSs \\
${\emenge{K}}$ & set of users \\
${\emenge{C}}$ & set of user clusters\\
$\bm{A}$ & cluster/user assignment matrix\\
$\bm{B}$ & BS/cluster assignment matrix\\
$\bm{J}$ & BS/user assignment matrix\\
$c_k$ & cluster that user $k$ is subordinated to\\
${\emenge{K}}_{c}$ & set of users subordinated to cluster $c$\\
$\bm{H}$ & channel gain matrix\\
$\bm{V}$ & interference coupling matrix\\
$\tilde{\bm{V}}$ & interference coupling matrix without intra-cell interference\\
$\tilde{\bm{V}}_{\bm{b}}$ & interference coupling matrix depending on BS assignments $\bm{b}$\\
$\tilde{\bm{V}}_{\bm{\theta}}$ & interference coupling matrix depending on antenna tilts $\bm{\theta}$\\
$\bm{r}$ & BS power budget vector\\
$\bm{q}$ & cluster power vector\\
$\bm{p}$ & user power vector\\
$\bm{\alpha}$ & intra-cluster power sharing factors\\
$\bm{\beta}$ & inter-cluster power sharing factors\\
$\bm{A}_{\bm{\alpha}}$ & transformation from $\bm{q}$ to $\bm{p}$, $\bm{p}:=\bm{A}_{\bm{\alpha}}^T\bm{q}$\\
$\bm{B}_{\bm{\beta}}$ & transformation from $\bm{r}$ to $\bm{q}$, $\bm{q}:=\bm{B}_{\bm{\beta}}^T\bm{r}$\\
$\bm{T}$ & transformation from $\bm{r}$ to $\bm{p}$, $\bm{p}:=\bm{T}\bm{r}$\\
$\bm{\theta}$ & BS antenna tilt vector\\
$\bm{b}$ & serving BSs of clusters\\
$b_c$ & serving BS of cluster $c$\\
$\bm{n}$ & serving BSs of the users\\
$n_k$ & serving BS of user $k$\\
$\bm{\sigma}$ & noise power vector\\
$P^{\text{max}}$ & sum power constraint\\
\hline
\end{tabular}
\label{tab:CovCap_notation}
\end{table}
\section{Utility Definition and Problem Formulation}\label{sec:ProbForm}
As mentioned, the objective is a joint optimization of coverage,
capacity and load balancing. We capture coverage by the worst-case
SINR, while the average SINR is used to represent capacity. A cluster-based utility $U_c(\bm{\theta},\bm{r},\bm{q},\bm{b})$ is introduced as the combined function of the worst-case SINR and average SINR, depending on BS
power allocation $\bm{r}$, antenna downtilt $\bm{\theta}$ ,
cluster power allocation $\bm{q}$ and BS/cluster assignment
$\bm{b}$.\footnote{The reader should note that user-specific variables
$(\bm{p},\bm{n})$ can be derived directly from cluster-specific
variables $\bm{q}$ and $\bm{b}$, provided that cluster/user
assignment $\bm{A}$ and intra-cluster power sharing factor
$\bm{\alpha}$ are given.} To achieve the load balancing by distributing the clusters to the BSs such that their utility targets can be achieved \footnote {The assignment of clusters also distributes the interference among the BSs.}, we formulate the following objective
$$\max_{(\bm{r},\bm{\theta},\bm{q},\bm{b})}\min_{c\in\set{C}} \frac{U_c(\bm{r},\bm{\theta},\bm{q},\bm{b})}{\gamma_c}$$
where $\gamma_c$ is the predefined utility target for cluster $c$.
The BS variables $(\bm{r},\bm{\theta})$ and cluster variables $(\bm{q}, \bm{b})$ are optimized by iteratively solving\\
1) Cluster-based BS assignment and power allocation
$\max_{(\bm{q},\bm{b})}\max_{c\in\set{C}} U_c(\bm{q},\bm{b})/\gamma_c$ given the fixed $(\hat{\bm{r}},\hat{\bm{\theta}})$ \\
2) BS-based antenna tilt optimization and power allocation $\max_{(\bm{r},\bm{\theta})}\max_{c\in\set{C}} U_c(\bm{r},\bm{\theta})/\gamma_c$ given the fixed $(\hat{\bm{q}},\hat{\bm{b}})$.
In the following we introduce the utility definition and problem formulation for the cluster-based and the BS-based problems respectively. We start with the problem statement and algorithmic approaches for the
uplink. We then discuss the downlink in Section \ref{sec:Duality}.
%
\subsection{Cluster-Based BS Assignment and Power Allocation}\label{subsec:clusterOpt}
Assume the per-BS variables
$(\hat{\bm{r}}, \hat{\bm{\theta}})$ are fixed, let the interference
coupling matrix depending on BS assignment $\bm{b}$ in
\eqref{eqn:PsiMat} be denoted by $\V_{\ve{b}}$. We first define two utility
functions indicating capacity and coverage per cluster respectively, then we introduce the joint utility as a combination of the capacity and coverage utility. After that we define the cluster-based max-min utility balancing problem based on the joint utility.
%
\subsubsection{Average SINR Utility (Capacity)}\label{subsubsec:LB_A}
With the intra-cluster power sharing factor introduced in Section
\ref{subsec:powFactor}, we have $\bm{p}:=\ma{A}_{\ve{\alpha}}^T \bm{q}$. Define the
noise vector $\bm{\sigma}:=[\sigma_1^2, \ldots, \sigma_K^2]^T$, the
average SINR of all users in cluster $c$ is written as
\begin{align}
\bar{U}_c^{(\text{u},1)}&(\bm{q}, \bm{b}) := \frac{1}{|\set{K}_c|} \sum_{k\in\set{K}_c}\operator{SINR}_k^{(\text{u})}\nonumber\\
&= \frac{1}{|\set{K}_c|} \sum_{k\in\set{K}_c}\frac{q_c \alpha_k v_{kk}}{\left[\V_{\ve{b}} \ma{A}_{\ve{\alpha}}^T \bm{q}+\bm{\sigma}\right]_k}\nonumber\\
&\geq \frac{1}{|\set{K}_c|}\frac{q_c \sum_{k\in\set{K}_c} \alpha_k v_{kk}}{\sum_{k\in\set{K}_c} \left[\V_{\ve{b}} \ma{A}_{\ve{\alpha}}^T \bm{q}+\bm{\sigma}\right]_k}
=U_c^{(\text{u},1)}(\bm{q}, \bm{b})
\label{eqn:CL_cap_1}
\end{align}
The uplink capacity utility of cluster $c$ denoted by $U_c^{(\text{u},1)}$ is
measured by the ratio between the total useful power and the total
interference power received in the uplink in the cluster. Utility
$U_c^{(\text{u},1)}$ is used instead of $\bar{U}_c^{(\text{u},1)}$ because of two
reasons: First, it is a lower bound for the average SINR. Second, it
has certain monotonicity properties (introduced in Section
\ref{sec:OPAlgor}) which are useful for optimization.
Introducing the cluster coupling term $\overline{\ma{G}}_{\ve{b}}^{(\text{u})}:=\bm{\Psi}\bm{A}\V_{\ve{b}}\ma{A}_{\ve{\alpha}}^T$, where $\bm{\Psi}:=\mathop{\mathrm{diag}}\{|\set{K}_1|/g_1, \ldots, |\set{K}_c|/g_C\}$ and $g_c:=\sum_{k\in \set{K}_c}\alpha_k v_{kk}$ for $c\in\set{C}$; and the noise term $\overline{\bm{z}}:=\bm{\Psi}\bm{A}\bm{\sigma}$,
the capacity utility is simplified as
\begin{align}
U_c^{(\text{u},1)}(\bm{q}, \bm{b})&:=\frac{q_c}{\set{J}_c^{(\text{u},1)}(\bm{q}, \bm{b})}\label{eqn:CL_cap_2}\\
\mbox{where } \set{J}_c^{(\text{u},1)}(\bm{q}, \bm{b})&:=\left[\overline{\ma{G}}_{\ve{b}}^{(\text{u})}\bm{q}+\overline{\bm{z}}\right]_c. \label{eqn:CL_cap_inter}
\end{align}
%
\subsubsection{Worst-Case SINR Utility (Coverage)}
Roughly speaking, the coverage problem arises when a certain number of the SINRs are lower than the predefined SINR threshold. Thus, improving the coverage performance is equivalent to maximizing the worst-case SINR such that the worst-case SINR achieves the desired SINR target. We then define the uplink coverage utility for each cluster as
\begin{align}
U_c^{(\text{u},2)}(\bm{q},\bm{b})&:=\min_{k\in\set{K}_c}\operator{SINR}_k^{(\text{u})}=\min_{k\in\set{K}_c}
\frac{q_c\alpha_k v_{kk}}{\left[\V_{\ve{b}} \ma{A}_{\ve{\alpha}}^T \bm{q}+\bm{\sigma}\right]_k}\nonumber\\
&= \frac{q_c}{\max_{k\in\set{K}_c}\left[ \bm{\Phi}\V_{\ve{b}} \ma{A}_{\ve{\alpha}}^T \bm{q}+\bm{\Phi}\bm{\sigma}\right]_k}
\label{eqn:CL_cov_1}
\end{align}
where $\bm{\Phi}:=\mathop{\mathrm{diag}}\{1/\alpha_1 v_{11}, \ldots, 1/\alpha_K v_{KK}\}$. We define a $C \times K$ matrix $\bm{X}:=[\bm{x}_1|\ldots|\bm{x}_C]^T$, where $\bm{x}_c:=\bm{e}^j_K$ and $\bm{e}^j_i$ denotes an $i$-dimensional binary vector which has exact one entry (the j-th entry) equal to 1. Introducing the term $\underline{\ma{G}}_{\ve{b}}^{(\text{u})}:=\bm{\Phi}\V_{\ve{b}} \ma{A}_{\ve{\alpha}}^T$, and the noise term $\underline{\bm{z}}:=\bm{\Phi}\bm{\sigma}$, the coverage utility is given by
\begin{align}
U_c^{(\text{u},2)}(\bm{q},\bm{b})&:=\frac{q_c}{\set{J}_c^{(\text{u},2)}(\bm{q}, \bm{b})}\label{eqn:CL_cov_2}\\
\mbox{where } \set{J}_c^{(\text{u},2)}(\bm{q}, \bm{b}) & := \max_{\bm{x}_c:=\bm{e}_K^j, j\in\set{K}_c} \left[\bm{X}\underline{\ma{G}}_{\ve{b}}^{(\text{u})}\bm{q}+\bm{X}\underline{\bm{z}}\right]_c. \label{eqn:CL_cov_inter}
\end{align}
%
\subsubsection{Joint Utility and Cluster-Based Max-Min Utility Balancing}\label{eqn:LB_maxmin}
The joint utility $U_c^{(\text{u})}(\bm{q}, \bm{b})$ is defined as
\begin{align}
U_c^{(\text{u})}(\bm{q}, \bm{b})&:=\frac{q_c}{\set{J}_c^{\ul}(\bm{q}, \bm{b})}\label{eqn:LB_utility_1}\\
\mbox{where }\set{J}_c^{\ul}(\bm{q}, \bm{b})&:= \mu\set{J}_c^{(\text{u},1)}(\bm{q}, \bm{b})+(1-\mu)\set{J}_c^{(\text{u},2)}(\bm{q}, \bm{b})\label{eqn:LB_utility_2}.
\end{align}
In other words, the joint interference function $\set{I}_c^{(\text{u})}$ is a convex combination of $\set{I}_c^{(\text{u},1)}$ in \eqref{eqn:CL_cap_inter} and $\set{I}_c^{(\text{u},2)}$ in \eqref{eqn:CL_cov_inter}.
The cluster-based power-constrained max-min utility balancing problem in the uplink is then provided by
\begin{problem}[Cluster-Based Utility Balancing]
\begin{equation}
C^{(\text{u})}(P^{\text{max}})=\max_{\bm{q}\geq 0, \bm{b}\in \set{N}^C} \min_{c\in\set{C}} \frac{U_c^{(\text{u})}(\bm{q}, \bm{b})}{\gamma_c}, \mbox{s.t. } \|\bm{q}\|\leq P^{\text{max}}
\label{eqn:LB_OP}
\end{equation}
Here, $\|\cdot\|$ is an arbitrary monotone norm, i.e., $\bm{q}\leq\bm{q}'$ implies $\|\bm{q}\|\leq\|\bm{q}'\|$, $P^{\text{max}}$ denotes the total power constraint.
According to the joint utility in \eqref{eqn:LB_utility_1},\eqref{eqn:LB_utility_2}, the algorithm optimizes the performance of capacity when we set the tuning parameter $\mu=1$ (utility is equivalent to the capacity utility in \eqref{eqn:CL_cap_2}), while with $\mu=0$ it optimizes the performance of coverage (utility equals to the coverage utility in \eqref{eqn:CL_cov_2}). By tuning $\mu$ properly, we can achieve a good trade-off between the performance of coverage and capacity.
\label{prob:LB}
\end{problem}
%
\subsection{BS-Based Antenna Tilt Optimization and Power Allocation}\label{subsec:AO}
Given the fixed $(\hat{\bm{q}},\hat{\bm{b}})$, we compute the intra-cluster power allocation factor $\bm{\beta}$, given by $\beta_c:=\hat{q}_c/\sum_{c\in\set{C}_{b_c}}\hat{q}_c$ for $c\in\set{C}$. We denote the cross-link coupling matrix depending on $\bm{\theta}$ by $\V_{\ve{\theta}}$. In the following we formulate the BS-based max-min utility balancing problem such that it has the same physical meaning as the problem stated in \eqref{eqn:LB_OP}. We then introduce the BS-based joint utility interpreted by $(\bm{r}, \bm{\theta})$.
\subsubsection{BS-Based Max-Min Utility Balancing}\label{subsubsec:AO_maxmin}
To be consistent with our objective function $C^{(\text{u})}(P^{\text{max}})$ in \eqref{eqn:LB_OP}, we transform the cluster-based optimization problem to the BS-based optimization problem:
%
\begin{problem}[BS-Based Utility Balancing]
\begin{align}
C^{(u)}&(P^{\text{max}})=\max\limits_{\bm{r}\geq 0, \bm{\theta}\in\Theta^N} \min\limits_{c\in\set{C}}
\frac{U_c^{(\text{u})}(\bm{r},\bm{\theta})}{\gamma_c}\nonumber\\
&=\max\limits_{\bm{r}\geq 0, \bm{\theta}\in\Theta^N}
\min\limits_{n\in\set{N}}\left(\min\limits_{c\in\set{C}_n}\frac{U_c^{(\text{u})}(\bm{r},\bm{\theta})}{\gamma_c}\right)\nonumber\\
& = \max\limits_{\bm{r}\geq 0, \bm{\theta}\in\Theta^N} \min\limits_{n\in\set{N}}
\widehat{U}_n^{(\text{u})}(\bm{r},\bm{\theta}), \mbox{ s.t. } \|\bm{r}\|\leq P^{\text{max}}
\label{eqn:maxmin_AO}
\end{align}
\label{prob:AO}
\end{problem}
where $\Theta$ denotes the predefined space for antenna tilt configuration.
\subsubsection{BS-Based Joint Utility}\label{subsubsec:AO_joinyUtility}
It is shown in \eqref{eqn:maxmin_AO} that the cluster-based problem is transformed to the BS-based problem by defining
\begin{align}
\widehat{U}_n^{(\text{u})}(\bm{r},\bm{\theta})&:=\min_{c\in\set{C}_n}\frac{U_c^{(\text{u})}(\bm{r},\bm{\theta})}{\gamma_c}= \frac{r_n}{\widehat{\set{J}}_n^{\ul}(\bm{r}, \bm{\theta})}\label{eqn:AO_utility_1}\\
\widehat{\set{J}}_n^{\ul}(\bm{r}, \bm{\theta}) &:= \max_{c\in\set{C}_n} \frac{\gamma_c}{\beta_c} \set{J}_c^{\ul}(\bm{r}, \bm{\theta}),
\label{eqn:AO_utility_2}
\end{align}
where $\set{J}_c^{\ul}(\bm{r}, \bm{\theta})$ is obtained from $\set{J}_c^{\ul}(\bm{q}, \bm{b})$ in \eqref{eqn:LB_utility_2} by substituting $\bm{q}$ with $\bm{q}:=\ma{B}_{\ve{\beta}}^T\bm{r}$, and $\tilde{\ma{V}}_{\bm{b}}$ with $\tilde{\ma{V}}_{\bm{\theta}}$. Note that \eqref{eqn:AO_utility_1} is derived by applying the inter-cluster sharing factor such that $r_n:=q_c/\beta_c$ for $n=b_c$. Due to lack of space we omit the details of the individual per BS capacity and coverage utilities corresponding to the cluster-based utilities \eqref{eqn:CL_cap_1} and \eqref{eqn:CL_cov_1}.
%
%
%
%
\section{Optimization Algorithm}\label{sec:OPAlgor}
We developed our optimization algorithm based on the fixed-point iteration algorithm proposed by Yates \cite{yates95}, by exploiting the properties of the monotone and strictly subhomogeneous functions.
\subsection{MSS function and Fixed-Point Iteration}\label{subsec:contraction}
The vector function $\bm{f}: {\field{R}}_+^K\mapsto {\field{R}}_+^K$ of interest has the following two properties:
\begin{itemize}
\item {\it Monotonicity}: $\bm{x}\leq \bm{y}$ implies $\bm{f}(\bm{x})\leq\bm{f}(\bm{y})$,.
\item {\it Strict subhomogeneity}: for each $\alpha>1, \bm{f}(\alpha \bm{x})<\alpha\bm{f}(\bm{x})$.
\end{itemize}
A function satisfying the above two properties is referred to be {\it monotonic and strict subhomogeneous (MSS)}. When the strict inequality is relaxed to weak inequality, the function is said to be {\it monotonic and subhomogeneous (MS)}.
\begin{theorem}\cite{nuzman2007contraction}
Suppose that $\bm{f}: {\field{R}}_+^K\mapsto {\field{R}}_+^K$ is MSS and that $\bm{h}=\bm{x}/l(\bm{x})$, where $l:{\field{R}}_+^K \mapsto {\field{R}}_+$ is MS. For each $\theta>0$, there is exactly one eigenvector $\bm{v}$ and the associated eigenvalue $\lambda$ of $\bm{f}$ such that $l(\bm{v})=\theta$. Given an arbitrary $\theta$, the repeated iterations of the function
\begin{equation}
\bm{g}(\bm{x})=\theta \bm{f}(x)/l(\bm{f(x)})
\label{eqn:fixedpointiteration}
\end{equation}
converge to a unique fixed point such that $l(\bm{v})=\theta$.
\label{Theoremmapping}
\end{theorem}
The fixed point iteration in \eqref{eqn:fixedpointiteration} is used to obtain the solution of the following max-min utility balancing problem
\begin{equation}
\max_{\bm{p}}\min_{k\in\set{K}} U_k(\bm{p}), \mbox{ s.t. } \|\bm{p}\|\leq P^{\text{max}}
\label{eqn:prob_maxmin_1}
\end{equation}
where the utility function can be defined as $U_k(\bm{p}):= p_k/f_k(\bm{p})$.
\subsection{Joint Optimization Algorithm}\label{subsec:JointOptAlgor}
We aim on jointly optimizing both problems, by optimizing $(\bm{q}, \bm{b})$ in Problem \ref{prob:LB} and $(\bm{r},\bm{\theta})$ in Problem \ref{prob:AO} iteratively with the fixed-point iteration. In the following we present some properties that are required to solve the problem efficiently and to guarantee the convergence of the algorithm.
\subsubsection{Decoupled Variables in Uplink}
In uplink the variables $\bm{b}$ and $\bm{\theta}$ are decoupled in the interference functions \eqref{eqn:LB_utility_2} and \eqref{eqn:AO_utility_2}, i.e., $\set{J}_c^{\ul}(\bm{q}, \bm{b}):=\set{J}_c^{\ul}(\bm{q}, b_c)$ and $\widehat{\set{J}}_n^{\ul}(\bm{r}, \bm{\theta}):=\widehat{\set{J}}_n^{\ul}(\bm{r}, \theta_n)$. Thus, we can decompose the BS assignment (or tilt optimization) problem into sub-problems that can be independently solved in each cluster (or BS), and the interference functions can be modified as functions of the power allocation only:
\begin{align}
\set{J}_c^{\ul}(\bm{q})&:=\min_{b_c\in\set{N}} \set{J}_c^{\ul}(\bm{q}, b_c)\label{eqn:modi_inter_1}\\
\widehat{\set{J}}_n^{\ul}(\bm{r})&:=\min_{\theta_n\in\Theta} \widehat{\set{J}}_n^{\ul}(\bm{r}, \theta_n) \label{eqn:modi_inter_2}
\end{align}
\subsubsection{Standard Interference Function}
The modified interference function \eqref{eqn:modi_inter_1} and \eqref{eqn:modi_inter_2} are \textit{standard}.
Using the following three properties: 1) an affine function $\bm{\set{I}}(\bm{p}):=\bm{V}\bm{p}+\bm{\sigma}$ is standard, 2) if $\bm{\set{I}}(\bm{p})$ and $\bm{\set{I}}'(\bm{p})$ are standard, then $\beta\bm{\set{I}}(\bm{p})+(1-\beta)\bm{\set{I}}'(\bm{p})$ are standard, and 3) If $\bm{\set{I}}(\bm{p})$ and $\bm{\set{I}}'(\bm{p})$ are standard, then $\bm{\set{I}}^{\text{min}}(\bm{p})$ and $\bm{\set{I}}^{\text{max}}(\bm{p})$ are standard, where $\bm{\set{I}}^{\text{min}}(\bm{p})$ and $\bm{\set{I}}^{\text{max}}(\bm{p})$ are defined as $\set{I}_j^{\text{min}}(\bm{p}):=\min\{\set{I}_j(\bm{p}), \set{I}_j'(\bm{p})\}$ and $\set{I}_j^{\text{max}}(\bm{p}):=\max\{\set{I}_j(\bm{p}), \set{I}_j'(\bm{p})\}$ respectively \cite{yates95}, we can easily prove that \eqref{eqn:modi_inter_1} and \eqref{eqn:modi_inter_2} are standard interference functions.
Substituting \eqref{eqn:modi_inter_1} and \eqref{eqn:modi_inter_2} in Problem \ref{prob:LB} and Problem \ref{prob:AO}, define $U_c^{(\text{u})}(\bm{q}):=q_c/\set{I}_c^{(\text{u})}(\bm{q})$ and $U_n^{(\text{u})}(\bm{r}):=r_n/\widehat{\set{J}}_n^{\ul}(\bm{r})$,
we can write both problems in the general framework of the max-min fairness problem \eqref{eqn:prob_maxmin_1}:
\begin{itemize}
\item[]Problem 1. $\max_{\bm{q}\geq 0}\min_{c\in\set{C}} U_c^{(\text{u})}(\bm{q})/\gamma_c, \|\bm{q}\|\leq P^{\text{max}}$.
\item[]Problem 2. $\max_{\bm{r}\geq 0}\min_{n\in\set{N}} U_n^{(\text{u})}(\bm{r}), \|\bm{r}\|\leq P^{\text{max}}$
\end{itemize}
%
The property of the decoupled variables in uplink and the property of utilities based on the standard interference functions enable us to solve each problem efficiently with two iterative steps: 1) find optimum variable $b_c$ (or $\theta_n$) for each cluster $c$ (or each BS $n$) independently, 2) solve the max-min balancing power allocation problem with fixed-point iteration.
%
\subsubsection{Connections between The Two Problems}
Problem \ref{prob:LB} and Problem \ref{prob:AO} have the same objective $C^{(\text{u})}(P^{\text{max}})$ as stated in \eqref{eqn:LB_OP} and \eqref{eqn:maxmin_AO}, i.e., given the same variables $(\hat{\bm{q}}, \hat{\bm{b}}, \hat{\bm{r}}, \hat{\bm{\theta}})$, using \eqref{eqn:AO_utility_1}, we have $\min_{c\in\set{C}} U_c^{(\text{u})}/\gamma_c=\min_{n\in\set{N}} \widehat{U}_n^{(\text{u})}$. Both problems are under the same sum power constraint. However, the convergence of the two-step iteration requires two more properties: 1) the BS power budget $\bm{r}$ derived by solving Problem \ref{prob:AO} at the previous step should not be violated by the cluster power allocation $\bm{q}$ found by optimizing Problem \ref{prob:LB}, and 2) when optimizing Problem \ref{prob:AO}, the inter-cluster power sharing factor $\bm{\beta}$ should be consistent with the derived cluster power allocation $\bm{q}$ in Problem \ref{prob:LB}.
To fulfill the first requirement, we introduce the per BS power constraint $P_n^{\text{max}}$ for Problem \ref{prob:AO} equivalent to the BS power budget $r_n$ in Problem \ref{prob:LB}. We also propose a scaled version of fixed point iteration similar to the one proposed in \cite{nuzman2007contraction} to iteratively scale the cluster power vector and achieve the max-min utility boundary under per BS power budget constraints, as stated below.
\begin{equation}
q_c^{(t+1)} =\frac{\gamma_c\set{I}_c^{(\text{u})}(\bm{q}^{(t)})}{\|\bm{B}\bm{\set{I}}^{(\text{u})}(\bm{q}^{(t)}) \oslash {\bm{P}^{\text{max}}}^{(t)}\|_{\infty}}
\label{eqn:FP_LB}
\end{equation}
where $\oslash$ denote the element-wise division of vectors, $\|\cdot\|_{\infty}$ denotes the maximum norm, ${\bm{P}^{\text{max}}}^{(t)}:=\bm{r}^{(t)}$.
To fulfill the second requirement, once $\bm{q}^{(n+1)}$ is derived, the power sharing factors $\bm{\beta}$ need to be updated for solving Problem \ref{prob:AO} at the next step, given by
\begin{equation}
\bm{\beta}^{(n+1)}:=\bm{Q}^{-1}\bm{B}^T\bm{r}^{(n)}, \mbox{where } \bm{Q}=\mathop{\mathrm{diag}}\{\bm{q}^{(n+1)}\}
\label{eqn:FP_LB_beta}
\end{equation}
The scaled fixed-point iteration to optimize Problem \ref{prob:AO} is provided by
\begin{equation}
r_n^{(t+1)}= \frac{P^{\text{max}}}{\|\bm{\widehat{\set{I}}}^{(\text{u})}(\bm{r}^{(t)})\|}\cdot \widehat{\set{I}}_n^{(\text{u})}(\bm{r}^{(t)})
\label{eqn:FP_AO_1}
\end{equation}
%
The joint optimization algorithm is given in Algorithm \ref{alg:optim-algor}.
%
\begin{algorithm}[t]\label{alg:optim-algor}
\caption{Joint Optimization of Problem \ref{prob:LB} and \ref{prob:AO}}
\begin{algorithmic}[1]
\STATE broadcast the information required for computing $\bm{V}$, predefined constraint $P^{\text{max}}$ and thresholds $\epsilon_1,\epsilon_2,\epsilon_3$
\STATE arbitrary initial power vector $\bm{q}^{(t)}>0$ and iteration step $t:=0$
\REPEAT[joint optimization of Problem \ref{prob:LB} and \ref{prob:AO}]
\REPEAT[fixed-point iteration for every cluster $c\in\set{C}$]
\STATE broadcast $\bm{q}^{(t)}$ to all base stations
\FOR{all assignment options $b_c \in \set{N}$}
\STATE compute $\set{I}_c^{(\text{u})}(\bm{q}^{(t)}, b_c)$ with \eqref{eqn:LB_utility_2}
\ENDFOR
\STATE compute $\set{I}_c^{(\text{u})}(\bm{q}^{(t)})$ with \eqref{eqn:modi_inter_1} and update $b_c^{(t+1)}$
\STATE update $q_c^{(t+1)}$ with \eqref{eqn:FP_LB}
\STATE $t := t+1$
\UNTIL{convergence: $\bigl| q_c^{(t+1)} - q_c^{(t)}\bigr| / q_c^{(t)} \leq \epsilon_1$}
\STATE update $\bm{\beta}^{(t)}$ with \eqref{eqn:FP_LB_beta}
%
\REPEAT[fixed-point iteration for every BS $n\in\set{N}$]
\STATE broadcast $\bm{r}^{(t)}$ to all base stations
\FOR{all antenna tilt options $\theta_n \in \Theta$}
\STATE compute $\widehat{\set{I}}_n^{(\text{u})}(\bm{r}^{(t)}, \theta_n)$ with \eqref{eqn:AO_utility_2}
\ENDFOR
\STATE compute $\widehat{\set{I}}_n^{(\text{u})}(\bm{r}^{(t)})$ with \eqref{eqn:modi_inter_2} and update $\theta_n^{(t+1)}$
\STATE update $r_c^{(n+1)}$ with \eqref{eqn:FP_AO_1}
\STATE $t := t+1$
\UNTIL{convergence: $\bigl| r_n^{(t+1)} - r_n^{(t)}\bigr| / r_n^{(t)} \leq \epsilon_2$}
\STATE update ${P_n^{\text{max}}}^{(t)}:=r_n^{(t)}$
\STATE compute $l^{(t+1)}:=\min_{n\in\set{N}} \widehat{U}^{(\text{u})}_n(\bm{r}^{(n+1)})$
\UNTIL{convergence: $|l^{(t+1)}-l^{(t)}|/l^{(t)}\leq\epsilon_3$}
\end{algorithmic}
\end{algorithm}
%
\section{Uplink-Downlink Duality}\label{sec:Duality}
We state the joint optimization problem in uplink in Section
\ref{sec:ProbForm} and propose an efficient solution in Section
\ref{sec:OPAlgor} by exploiting the decoupled property of $\bm{V}$
over the variables $\bm{\theta}$ and $\bm{b}$. The downlink problem,
due to the coupled structure of $\bm{V}^T$, is more difficult to
solve. As extended discussion we want to address the relationship
between the uplink and the downlink problem, and to propose a
sub-optimal solution for downlink which can be possibly found through
the uplink solution.
Let us consider cluster-based max-min capacity utility balancing
problem in Section \ref{subsubsec:LB_A} as an example. In the downlink
the optimization problem is written as
\begin{align}
\vspace{-0.2em}
\max_{\bm{q}, \bm{b}}\min_c &\frac{U_c^{(\text{d},1)}(\bm{q}, \bm{b})}{\gamma_c}, \mbox{s.t. } \|\bm{q}\|_1\leq P^{\text{max}}\nonumber\\
\mbox{where } & U_c^{(\text{d},1)} :=\frac{q_c}{[\bm{\Psi}\bm{A}\V_{\ve{b}}^T\ma{A}_{\ve{\alpha}}^T\bm{q}+\bm{\Psi}\bm{z}^{(\text{d})}]}
\label{eqn:LB_dl}
\vspace{-0.2em}
\end{align}
The cluster-based received noise is written as $\bm{z}^{(\text{d})}:=\bm{A}\bm{\sigma}^{(\text{d})}$.
In the following we present a virtual dual uplink network in terms of
the feasible utility region for the downlink network in
\eqref{eqn:LB_dl} via Perron-Frobenius theory, such that the solution
of problem \eqref{eqn:LB_dl} can be derived by solving the uplink
problem \eqref{eqn:LB_ul} with the algorithm introduced in Section
\ref{sec:OPAlgor}.
%
\begin{proposition}
Define a virtual uplink network where the link gain matrix is
modified as
$\bm{W}_{\bm{b}}:=\mathop{\mathrm{diag}}\{\bm{\alpha}\}\V_{\ve{b}}\mathop{\mathrm{diag}}^{-1}\{\bm{\alpha}\}$,
i.e., $w_{lk}:=v_{lk}\frac{\alpha_l}{\alpha_k}$, and the received
uplink noise is denoted by $\bm{\sigma}^{(\text{u})}:=[{\sigma^2_1}^{(\text{u})},
\ldots, {\sigma^2_K}^{(\text{u})}]^T$, where
${\sigma_k^2}^{(\text{u})}:=\frac{\Sigma_{\text{tot}}}{|\set{K}_{c_k}|\cdot
C}$ for $k\in\set{K}$, and assume
$\Sigma_{\text{tot}}:=\|\bm{\sigma}^{(\text{u})}\|_1=\|\bm{\sigma}^{(\text{d})}\|_1$
(which means, the sum noise is equally distributed in clusters,
while in each cluster the noise is equally distributed in the
subordinate users). The dual uplink problem of problem
\eqref{eqn:LB_dl} is given by
\begin{align}
\vspace{-0.2em}
\max_{\bm{q},\bm{b}}\min_c & \frac{U_c^{(\text{u},1)}(\bm{q}, \bm{b})}{\gamma_c}, \mbox{s.t. } \|\bm{q}\|_1\leq P^{\text{max}}\nonumber\\
\mbox{where } & U_c^{(\text{u},1) }:=\frac{q_c}{[\bm{\Psi}\bm{A}\bm{W}_{\bm{b}}\ma{A}_{\ve{\alpha}}^T\bm{q}+\bm{\Psi}\bm{z}^{(\text{u})}]}
\label{eqn:LB_ul}
\vspace{-0.2em}
\end{align}
where $\bm{z}^{(\text{u})}:=\bm{A}\bm{\sigma}^{(\text{u})}$.
\label{prop:Duality}
\end{proposition}
\begin{proof} The proof is given in the Appendix.
\end{proof}
Note that the optimizer $\bm{b}^{\ast}$ for BS assignment in downlink can be equivalently found by minimizing the spectral radius $\bm{\Lambda^{(u)}(\bm{b})}$ in the uplink. Once $\bm{b}^{\ast}$ is found, the associate optimizer for uplink power ${\bm{q}^{(\text{u})}}^{\ast}$ is given as the dominant right-hand eigenvector of matrix $\bm{\Lambda}^{(\text{u})}(\bm{b}^{\ast})$, while the associate optimizer for downlink power ${\bm{q}^{(\text{d})}}^{\ast}$ is given as the dominant right-hand eigenvector of matrix $\bm{\Lambda}^{(\text{d})}(\bm{b}^{\ast})$.
Proposition \ref{prop:Duality} provides an efficient approach to solve
the downlink problem with two iterative steps (as the one proposed in
\cite{BocheDuality06}): 1) for a fixed power allocation
$\hat{\bm{q}}$, solve the uplink problem and derive the assignment
$\bm{b}^{\ast}$ that associated with the spectral radius of extend
coupling matrix $\bm{\Lambda}^{(\text{u})}$, and 2) for a fixed assignment
$\hat{\bm{b}}$, update the power $\bm{q}^{\ast}$ as the solution of
\eqref{eqn:DL_matrixEqua}.
Although we are able to find a dual uplink problem for the
downlink problem in \eqref{eqn:LB_dl} with our proposed utility
functions \emph{under sum power
constraints},
\insl{we are not able to construct a dual network with decoupled properties for the modified problem
\emph{under per BS power constraints} \eqref{eqn:FP_LB}. However,
numerical experiments show that our approach to the downlink
based on the proposed uplink solution does improve the network
performance, although the duality does not exactly hold between the downlink problem and our proposed uplink problem under the per BS power constraints.}
%
%
\section{Numerical Results}\label{sec:Simu}
We consider a real-world urban scenario based on a pixel-based mobility model of realistic collection of BS locations and pathloss model for the city of Berlin. The data was assembled within the EU project MOMENTUM and is available at \cite{MOMENTUM}. We select 15 tri-sectored BS in the downtown area. Users are uniformly distributed and are clustered based on their SINR distributions as shown in Fig. \ref{fig:Berlin} (UEs assigned to each sector are clustered into groups and are depicted in distinct colors). The SINR threshold is defined as -6.5 dB and the power constraint per BS is 46dBm. The 3GPP antenna model defined in \cite{3GPP36942} is applied.
Fig. \ref{fig:convergence} illustrates the convergence of the algorithm. Our algorithm achieves the max-min utility balancing, and improves the feasibility level $C^{(u)}(P^{\text{max}})$ by each iteration step.
In Fig.\ref{fig:cov_cap_mu} we show that the trade-off between coverage and capacity can be adjusted by tuning parameter $\mu$. By increasing $\mu$ we give higher priority to capacity utility (which is proportional to the ratio between total useful power and total interference power), while for better coverage utility (defined as minimum of SINRs) we can use a small value of $\mu$ instead.
Fig. \ref{fig:coverage}, \ref{fig:capacity} and \ref{fig:power} illustrate the improvement of coverage and capacity performance and decreasing of the energy consumption in both uplink and downlink systems by applying the proposed algorithm, when the average number of the users per BS is chosen from the set $\{15,20,25,30,35\}$. In Fig. \ref{fig:capacity} we show that the actual average SINR is also improved, although the capacity utility is defined as a lower bound of the average SINR. Fig. \ref{fig:power} illustrate that our algorithm is more energy efficient when comparing with the fixed BS power budget scenario. Compared to the near-optimal uplink solutions, less improvements are observed for the downlink solutions as shown in Fig. \ref{fig:coverage}, \ref{fig:capacity} and \ref{fig:power}. This is because we derive the downlink solution by exploiting an uplink problem which is not exactly its dual due to the individual power constraints (as described in Section \ref{sec:Duality}). However, the sub-optimal solutions still provide significant performance improvements.
%
%
%
%
%
\section{Conclusions and Further Research}\label{sec:con}
We present an efficient and robust algorithmic optimization framework build on the utility model for joint optimization of the SON use cases coverage and capacity optimization and load balancing. The max-min utility balancing formulation is employed to enforce the fairness across clusters. We propose a two-step optimization algorithm in the uplink based on fixed-point iteration to iteratively optimize the per base station antenna tilt and power allocation as well as the cluster-based BS assignment and power allocation. We then analyze the network duality via Perron-Frobenius theory, and propose a sub-optimal solution in the downlink by exploiting the solution in the uplink. Simulation results show significant improvements in performance of coverage, capacity and load balancing in a power-efficient way, in both uplink and downlink. In our follow-up papers we will further propose a more complex interference coupling model and the optimization framework where frequency band assignment is taken into account. We will also examine the suboptimality under more general form of power constraints.
\begin{figure}[t]
\centering
\includegraphics[width=.5\textwidth]{BerlinReceivedSignalStrengthMap_v2}
\caption{Berlin Scenario.}
\label{fig:Berlin}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{convergence}
\caption{Algorithm convergence.}
\label{fig:convergence}
\end{figure}
%
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{cov_cap_mu}
\vspace{-1em}
\caption{Trade-off between utilities depending on $\mu$.}
\label{fig:cov_cap_mu}
\vspace{-1.5em}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{coverage}
\caption{Performance of proposed algorithm: coverage.}
\label{fig:coverage}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=.43\textwidth]{capacity}
\caption{Performance of proposed algorithm: capacity.}
\label{fig:capacity}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=.43\textwidth]{power}
\caption{Performance of proposed algorithm: per-BS power budget.}
\label{fig:power}
\end{figure}
%
\input{appendices}
\subsection*{Acknowledgements}
We would like to thank Dr. Martin Schubert and Dr. Carl J. Nuzman for their expert advice.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
%
\bibliographystyle{IEEEtran}
| 2023-04-23T08:17:26.599Z | 2016-07-19T02:04:46.000Z | redpajama/arxiv | arxiv_0000 | 17 | 6,891 |
61decaa1f993790b6b05ece2d83435830a9a74c1 | \section{Introduction}
Level design is a core feature of what defines a video game. When constructed correctly, it is a main determinant of player experience. From the designer's perspective, level design can be either a tedious, but necessary step in the game's development or a creatively freeing process - sometimes it is both. Most levels are designed with the intent to teach the game's interactable space - the mechanics - to the player in a way that is (ideally) engaging, fun, visually pleasing, intuitive, and informative \cite{rogers2014level,koster2013theory,green2017press}.
Levels designed for tutorial sections of the game create simplistic and low-risk environments. These levels are direct, and sometimes oblique in their intention so the player can grasp the core mechanics of the game as quickly as possible. As the player becomes more familiar with the mechanics and how they work together in the game's system, the levels should also increase in complexity and challenge. The general design of these levels in turn needs to be as complex and engaging both visually and functionally~\cite{khalifa2019intentional,anthropy2014game}.
Most games demonstrate each mechanic at least once throughout the entire level space; combining and ordering them in a way that builds itself based on the player's current skill as they get more familiar with the game \cite{totten_2016,anthropy2014game}. However, designing levels to explore multiple combinations of mechanics is an arduous task to undertake for a level designer. While it is unlikely that a player would play all of these levels, creating this possibility space of levels would allow the level designer to hand-pick and order them in a way that the mechanics feel coherent. Furthermore, an adaptive game with a diverse set of levels would allow the player to explore different combinations of mechanics according to their own pace and preference. For example, if a player is having difficulties with a certain necessary mechanic, such as long jumps in most Super Mario games or the spin attack in most Legend of Zelda games, having a specific subset of levels with a focus on these more challenging mechanics would allow the player to develop their skill with the mechanic better than a single tutorial level~\cite{anthropy2014game}. In contrast, if a player has fully mastered a mechanic, the game could select a level that uses the mechanic in a more challenging situation or a level with an entirely different mechanic space for them to master next. Levels could be selected automatically from the generated level space and dynamically ordered in a way that adapts to the player's current skill level as opposed to a hard-coded level ordering~\cite{yannakakis2011experience}. Generating a diversity of levels that explore multiple mechanic combinations would save time in the design process and allow for more creative flexibility in a way that manually designed levels could not provide.
This paper presents a system that seeks to combine human design and AI-driven design to enable mixed-initiative collaborative game level creation. Users can choose to start from a blank slate with their work while adding their own edits then have an AI back-end evolve their work towards a pre-defined objective. This objective function can be defined by minimalism in design, maximization of game mechanic coverage, overall quality, or any other feature that could contribute to the quality of the level. Alternatively, users may select from a variety of AI suggestions and pre-generated samples to begin their work and then make changes as necessary. This design process is not limited to the initializing step of the level; the user and AI system can switch their roles as designers at any point in the creation process. Concurrently, the AI system will look at what its previous users have created and submitted, and ask new users to design levels that complement what's already there. With this design process, the mechanic space of a game can be fully explored and every combination of mechanics can be represented by a level. With a human-based rating system, the automated system can learn to design levels with better quality and the human users can design levels that are missing from this mechanic combination space.
This project demonstrates the mixed-initiative collaborative process through level design for the independent, Sokoban-like game `Baba is You' - a game whose mechanics are defined and modified by the level design itself and the player's interaction with it. Levels can be made either by users, AI, or a mixed combination of both and uploaded to the level database to be used for future creations and to improve the quality of the AI's objective function.
\subsection{Baba is Y'all v1 (prototype)}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{imgs/level_matrix.png}
\caption{Baba is Y'all Version 1 Main Screen (from April 2020)}
\label{fig:levelMat1}
\end{figure}
The first version of Baba is Y'all (BiY v1) was released officially in March 29th, 2020, and promoted chiefly on Twitter. This version served as a prototype and proof-of-concept system for mixed-initiative AI-assisted game content collaboration specifically for designing levels in the game `Baba is You' (Arvi 'Hempuli' Teikari, 2017).
This system was built on concepts from three different areas of content creation:
\begin{itemize}
\item \textbf{Crowdsourcing:} a model used by different systems that allows a large set of users to contribute toward a common goal provided by the system~\cite{brabham2013crowdsourcing}. For example, Wikipedia users participate to fill in missing information for particular content.
\item \textbf{User content creation:}
allows players to create levels for a game/system and upload them online to the level database for other players to play and enjoy - i.e. Super Mario Maker (Nintendo, 2015), Line Rider (inXile Entertainment, 2006), and LittleBigPlanet (Media Molecule, 2008).
\item \textbf{Quality diversity:} the underlying technique behind our system. It ensures that the levels made from combining the first 2 concepts are of both good quality and diverse in terms of the feature space they are established in~\cite{pugh2016quality}. For this system, the feature space is defined as the potential game mechanics implemented in each level.
\end{itemize}
The Baba is Y'all website (as shown in figure~\ref{fig:levelMat1}) was a prototype example of a mixed-initiative collaborative level designing system. However, the site was limited by the steep learning curve required to interact with the system~\cite{charity2020baba}. Features of the site were overwhelming to use and lack cohesion in navigating the site.
\subsection{Baba is Y'all v2 (updated release)}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{imgs/dark_main.png}
\caption{Baba is Y'all Version 2 Main Screen (as of September 2021)}
\label{fig:levelMat2}
\end{figure}
The second version of Baba is Y'all\footnote{http://equius.gil.engineering.nyu.edu/} (BiY v2) was released on May 27th, 2021 and designed to have a more user-friendly setup. It was similarly promoted via Twitter and on mailing lists. This version includes a cleaner, more compact, and more fluid user interface for the entire website and consolidated many of the separate features from the BiY v1 site onto fewer pages for easier access. Three main webpages were created for this updated system.
Unlike the previous version, which showed all of the mechanic combination levels (both from the database and unmade) in random order, the updated level selection page adds level tabs that separates levels by recently added (New), highest rated (Top), and levels with rules that had not been made yet (Unmade.) A carousel scrolling feature shows 9 levels at a time to not overwhelm the player with choices (as shown in figure~\ref{fig:levelMat2}). The level rating system is also included on the main page as a tab, as well as the search feature. The personal level selection tab allows users to see their previously submitted levels and login to their account to submit levels with their username as the author or co-author.
The updated level editing page consolidates both the user editing with the PCG level evolution onto one page. Users can easily switch between manually editing the level themselves and allowing the PCG back-end system to edit the level while pausing in between. Users can also select rule objectives for the system to evolve towards implementing. To fight the problem of blank canvas paralysis, users can start from a set of different types of levels (both PCG and user-made)~\cite{krall2012artist}. Once a level is successfully solved, users may name the level upon submission - further personalizing the levels and assigning authorship.
A slideshow tutorial is provided for the users and describes every feature and function of the site instead of the walkthrough video that was featured on BiY v1. Users can also play a demo version of the `Baba is You' (Arvi 'Hempuli' Teikari, 2017) game to familiarize themselves with the game mechanics/rule space and how they interact with each other (game dynamics). For quick assistance, a helper tool is provided on the level editing page as a refresher on how to use the editing tool.
In addition to updating the features and collecting more data about the levels created, we conducted a formal user study with 76 participants to gather information about which features they chose to use for their level creation process and their subjective opinion on using the site overall. This user study, as well as the general level statistics collected from the site's database, showed that our new interface better facilitated the user-AI collaborative experience to create more diverse levels.
\section{Background and Related Work}
The Baba is Y'all system uses the following methods in the collaborative level design process: procedural content generation to create new levels from the AI backend, quality diversity to maintain the different kinds of levels produced from the system and show the coverage of game mechanics across each level, crowdsourcing so the AI may learn to create new levels from previously submitted "valid" levels - either those made exclusively by users, the system itself, or a combination of both, and finally mixed-initiative AI so that the user and evolutionary algorithm can develop the level together. Each method is described as the following:
\subsection{Procedural Content Generation}
Procedural content generation (PCG) is defined as the process of using a computer program to create content that with limited or indirect user input \cite{shaker2016procedural}. Such methods can make an automated, quicker, and more efficient content creation process, and also enable aesthetics based on generation. PCG has been used in games from the 1980's Rogue to its descendent genre of the Rogue-likes used in games such as Spelunky (Mossmouth, LLC, 2008) and Hades (Supergiant Games, 2020), as well as games that revolve around level and world generation such as Minecraft (Mojang, 2011) and No Man's Sky (Hello Games, 2016). PCG can be used to build levels such as The Binding of Isaac (Edmund McMillen, 2011), enemy encounters such as Phoenix HD (Firi Games, 2011), or item or weapon generation such as Borderlands (Gearbox Software, 2009). In academia, PCG has been explored in many different game facets for generating assets \cite{ruela2017procedural, gonzalez2020generating}, mechanics \cite{khalifa2019general, togelius2008experiment,browne2010evolutionary}, levels \cite{snodgrass2016learning,charity2020mech}, boss fights~\cite{siu2016programming}, tutorials~\cite{khalifa2019intentional,green2018atdelfi}, or even other generators \cite{kerssemakers2012procedural,earle2021learning,earle2021illuminating,khalifa2020multi}.
A plethora of AI methods underpin successful PCG approaches, including evolutionary search \cite{togelius2010search}, supervised and unsupervised learning \cite{summerville2018procedural,liu2021deep}, and reinforcement learning \cite{khalifa2020pcgrl}. The results of these implementations have led to PCG processes being able to generate higher quality, more generalizable, and more diverse content. PCG is used in the Baba is Y'all system to allow the mutator module to create new `Baba is You' levels.
\subsection{Quality Diversity}
Quality-diversity (QD) search based methods are increasing in usage for both game researchers and AI researchers \cite{pugh2016quality,gravina2019procedural}. Quality-diversity techniques are search based techniques that try to generate a set of diverse solutions while maintaining high level of quality for each solution. A well-known and popular example is MAP-Elites, an evolutionary algorithm that uses a multi-dimensional map instead of a population to store its solutions~\cite{mouret2015illuminating}. This map is constructed by dividing the solution space into a group of cells based on a pre-defined behavior characteristics. Any new solution found will not only be evaluated for fitness but also for its defined characteristics then placed in the correct cell in the MAP-Elites map. If the cell is not empty, both solutions compete and only the fitter solution survives. Because of the map maintenance and the cell competition, MAP-Elites can guarantee a map of diverse and high quality solutions, after a finite number of iterations through the generated population.
The MAP-Elites algorithm has also been extended into Constrained MAP-Elites \cite{khalifa2018talakat, khalifa2019intentional, alvarez2019empowering}, Covariance Matrix Adaptation using MAP-Elites (CMA-ME) \cite{fontaine2020covariance}, Monte Carlo Elites~\cite{sfikas2021monte}, MAP-Elites via Gradient Arborescence~\cite{fontaine2021differentiable}, and etc. For this project, we use the Constrained MAP-Elites algorithm to maintain a diverse population of `Baba is You' levels where the behavior characteristic space of the matrix is defined by the starting and ending rules of a level when it is submitted.
\subsection{Crowdsourcing data and content}
Some, but relatively few, games allow users to submit their own custom creations using the game's engine as most games do not have their source code available or even partially accessible for modifications to add more content in the context of the game. Whether through a built-in level editing system seen in games like Super Mario Maker (Nintendo, 2015), LittleBigPlanet (MediaMolecule, 2008), or LineRider (inXile Entertainment, 2006) or through a modding community that alter the source code for notable games such as Skyrim (Bethesda, 2011) Minecraft (Mojang, 2011,) or Friday Night Funkin' (Ninjamuffin99, 2020), players can create their own content to enhance their experience and/or share with others.
In crowdsourcing, many users contribute data that can be used for a common goal. Some systems like Wikipedia rely entirely on content submitted by their user base in order to provide information to others on a given subject. Other systems like Amazon's MechanicalTurk crowdsource data collection, such as research experiments \cite{buhrmester2016amazon}, by outsourcing small tasks to multiple users for a small wage. An example of a game generator based on crowdsourced data is Barros et al.’s DATA Agent \cite{barros2018killed,green2018data}, which uses crowd-sourced data such as Wikipedia to create a point-click adventure game sourced from a large corpus of open data to generate interesting adventure games.
What differentiates the Baba is Y'all system from other level editing systems or interactive PCG systems is that the Baba is Y'all site has a central goal: populate the MAP-Elites matrix with levels that cover all possible rule combinations. With this system, users may freely create the levels they want, but they may also work towards completing the global goal of making levels with a behavior characteristic that has not been made before. Participation in this task is encouraged by the AI back-end system that keeps track of missing cells in the MAP-Elites matrix.
\subsection{Mixed-Initiative AI}
Mixed-initiative AI systems involve a co-creation of content between a human user and an artificially intelligent system~\cite{yannakakis2014mixed}. Previous mixed-initiative systems include selecting from and evolving a population of generated images \cite{secretan2008picbreeder,bontrager2018deep}, composing music \cite{mann2016ai,tokui2000music}, and creating game levels through suggestive feedback \cite{machado2019pitako}. Mixed-initiative and collaborative AI level editors for game systems have thoroughly been explored in the field as well through direct and indirect interaction with the AI backend system \cite{shaker2013ropossum,liapis2013sentient,butler2013mixed,guzdial2018co,zhou2021toward,bhaumik2021lode,alvarez2019empowering,smith2010tanagra,delarosa2021mixed}.
Since the release of the first Baba is Y'all prototype and paper~\cite{charity2020baba}, the implementation of mixed-initiative systems have grown in the game and AI research field. Bhaumik implemented an AI constrained system with their Lode Encoder level editing tool that only allowed users to edit a level from a set of levels generated by a variational autoencoder - forcing users to only edit from a palette provided by the AI back-end tool~\cite{bhaumik2021lode}. Delarosa used a reinforcement learning agent in a mixed-initiative web app to collaboratively suggest edits to Sokoban levels \cite{delarosa2021mixed}. Zhou used levels generated with the AI-assisted level editor Morai Maker (a Super Mario level editor) to apply transfer learning for level editing to Zelda \cite{zhou2021toward}. These recent developments look more into how the human users are affected through their relationship with collaborating with these AI systems and how it can be improved through examining the dimensionality of the QD algorithm, the evolutionary process, or the human-system interaction itself \cite{alvarez2020exploring}. We look to incorporate these new perspectives into this updated iteration of Baba is Y'all and evaluate the effects through a user study.
\section{System Description}
The updated Baba is Y'all site's features were condensed into 2 main pages to make navigation and level editing much easier and intuitive:
\begin{itemize}
\item \textbf{The Home Screen:} contains the level matrix \textit{Map Module}, the search page, the \textit{Rating Module} page, and the \textit{User Profile} page. From here, users can also change the visuals of the site from light to dark mode, view the tutorial section or the site stats page by clicking on the Baba and Keke sprites respectively at the top of the page, and create a new level from scratch by clicking on various 'Create New Level' buttons placed on various subpages. Figure~\ref{fig:levelMat2} shows the starting page of the home screen.
\item \textbf{The Level Editor Screen:} contains both the \textit{Editor Module} and the \textit{Mutator Module}. Users can also test their levels with themselves or with the Keke solver by clicking on the Baba and Keke icons at the bottom of the canvas. Figure~\ref{fig:editor_screen} shows the starting page of the level editor screen.
\end{itemize}
In the following subsections, we are going to explain the different modules that constitutes these two main screens. Each of the following modules are either being used in the home screen, the level editor screen, or both.
\subsection{Baba is You}
`Baba is You' (Arvi ``Hempuli'' Teikari, 2019) is a puzzle game where players can manipulate the rules of a level and properties of the game objects through Sokoban-like movements of pushing word blocks found on the map. These dynamically changing rules create interesting exploration spaces for both procedurally generating the levels and solving them. The different combinations of rules can also lead to a large diversity of level types that can be made in this space.
The general rules for the `Baba is You' game can be referred to from our previous paper~\cite{charity2020baba}. To reiterate, there are three types of rule formats in the game:
\begin{itemize}
\item \textbf{X-IS-(KEYWORD)} a property rule stating that the game object class `X' has a certain property such as `WIN', `YOU', `MOVE', etc.
\item \textbf{X-IS-X} a reflexive rule stating that the game object class `X' cannot be changed to another game object class.
\item \textbf{X-IS-Y} a transformative rule changing all game objects of class `X' into game objects of class `Y'.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=0.4\linewidth]{imgs/simple_level.png}
\caption{An example of a simple `Baba is You' level.}
\label{fig:simple_map}
\end{figure}
The game sprites are divided into two main different classes: the object class and the keyword class. Sprites in the object class represent the interactable objects in the map as well as the literal word representation for the object. Sprites in the keyword class represent the rules of the level that manipulate the properties of the objects. For example, figure~\ref{fig:simple_map} shows four different object class sprites [BABA (object and corresponding word) and FLAG (object and corresponding word)] and three different keyword class sprites [IS (x2), YOU, and WIN]. The keyword class sprites are arranged in two rules: `BABA-IS-YOU' allowing the player to control all the Baba objects and `FLAG-IS-WIN' indicating that reaching any flag object will make the player win the level. The system has a total of 32 different sprites: 11 object class sprites and 21 keyword class sprites. Because the game allows rule manipulation, object classes are arbitrary in the game as they serve only to provide a variety of objects for rules to affect and for aesthetic pleasure.
\subsection{Game Module}
The game module is responsible for simulating a `Baba is You' level. It also allows users to test the playability of levels either by directly playing through the level themselves or by allowing a solver agent to attempt to solve it. This component is used on the home screen when a user selects a level to play and the editor screen for a user to test their created level.
Because the game rules are dynamic and can be altered by the player at any stage in the solution, the system keeps track of all the active rules at every state. Once the win condition has been met, the game module records the current solution, the active rules at the start of the level, and the active rules when the solution has been reached. These properties are saved to be used and interpreted by the Map module (section~\ref{sec:map_module}).
The activated rules are used as the level's characteristic feature representation and saved as a chromosome to the MAP-Elites matrix.
The game module provides an AI solver called 'KEKE' (based on one of the characters traditionally used as an autonomous 'NPC' in the game). KEKE uses a greedy best-first tree search algorithm that tries to solve the input level. The branching space is based on the five possible inputs a player can do within the game: move left, move right, move up, move down, and do nothing. The algorithm uses a heuristic function based on a weighted average of the Manhattan distance to the centroid distance for 3 different groups: keyword objects, objects associated with the `WIN' rule, and objects associated with the `PUSH' rule. These were chosen based on their critical importance for the user solving the level - as winning objects are required to complete the level, keyword objects allow for manipulation of active rules, and pushable objects can directly and indirectly affect the layout of a level map and therefore the accessibility of player objects to reach winning objects. The heuristic function is represented by the following equation:
\begin{equation}
h = (n + w + p) / 3
\end{equation}
where $h$ is the final heuristic value for placement in the priority queue, $n$ is the minimum Manhatttan distance from any player object to the nearest winnable object, $w$ is the minimum Manhatttan distance from any player object to the nearest word sprite, and $p$ is the minimum Manhatttan distance from any player object to the nearest pushable object.
As an update for this version of the system, the agent can run for a maximum of 10000 iterations and can be stopped at any time. A user may also attempt to solve part of the level themselves and the KEKE solver can pick up where the user left off to attempt to solve the remainder of the level. This creates a mixed-initiative approach to solving the levels in addition to editing the levels. However, even with this collaborative approach, the system still has limitations and difficulty solving levels with complex solutions - specifically solutions that require back-tracking across the level after a rule has been changed. The solver runs on the client side of the site and is limited by the capacity of the user's computational resources. Future work will look into improving the solver system to reduce computational resource. We will also look for better solving algorithms to improve the utility of the solver such as Monte Carlo Tree Search (MCTS) with reversibility compression~\cite{cook2021monte}.
\subsection{Editor Module}
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\linewidth]{imgs/editor_screen.png}
\caption{A screenshot of the level editor screen}
\label{fig:editor_screen}
\end{figure}
The editor module of the system allows human users to create their own `Baba is You' levels in the same vain of Super Mario Maker (Nintendo, 2015). Figure~\ref{fig:editor_screen} shows the editor window that is available for the user. The user can place and erase any game sprite or keyword at any location on the map using the provided tools. As a basis, the user can start modifying either a blank map, a basic map (a map with X-IS-YOU and Y-IS-WIN rules already placed with X and Y objects), a randomly generated map, or an elite level provided by the Map Module. Similar to Super Mario Maker (Nintendo, 2015), the created levels can only be submitted after they are tested by the human player or the AI agent to check for solvability. For testing the level, the editor module sends the level information to the game module to allow the user to test it.
This updated version of the site also includes an undo and redo feature so that users may erase any changes they make. A selection and lasso feature is also available so users can select specific areas of the level and move them to another location. Unlike the previous version, all tiles are available to the user on the same screen and the user may seamlessly transition from the editor module to the mutator module and vice versa for ease of access and better interactivity and collaboration between the AI system and the user.
\subsection{Mutator Module}\label{sec:mutator_module}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{imgs/evolver_screen.png}
\caption{A screenshot of the level evolver page}
\label{fig:evolver_screen}
\end{figure}
The Mutator module is a procedural content level generator. More specifically, the Baba is Y'all system uses an evolutionary level generator that defines a fitness function based on a version of tile-pattern Kullback-Liebler Divergence (ETPKLDiv\footnote{https://github.com/amidos2006/ETPKLDiv}) algorithm~\cite{lucas2019tile}. Figure~\ref{fig:evolver_screen} shows the updated interface used by the evolver. As mentioned before in the previous subsection, this version of the mutator module can interface seamlessly with the other modules to allow the user more ease of access between manual editing and evolutionary editing. The user can easily transfer the level from the editor module to the mutator module and vice versa.
When switching between the editor module and the mutator module, the level loses its pure procedurally generated or pure human-designed quality and becomes a hybrid of the two - thus mixed-initiative interaction between the algorithm and the user.
The evolver interface provides the user with multiple customizations such as the initialization method, stopping criteria, evolution pausing, and an application of a mutation function allowing manual user control. With these features, the user is not directly changing the evolution process itself, but instead guiding and limiting the algorithm towards generating the level they want.
The ETPKLDiv algorithm uses a 1+1 evolution strategy, also known as a hillclimber, to improve the similarity between the current evolved levels and a reference level. The algorithm uses a sliding window of a fixed size to calculate the probability of each tile configuration (called tile patterns) in both the reference level and the evolved level and tries to minimize the Kullback-Liebler Divergence between both probability distributions.
Like Lucas and Volz, we use a window size of 3x3 for the tile selection. This was to maximize the probability of generating initial rules for a level, since rules in `Baba is You' are made up of 3 tiles. However, in our project, we used 2+2 evolution strategy instead of 1+1 used to allow slightly more diversity in the population~\cite{lucas2019tile}. We also modified the fitness function to allow it to compare with more than one level. The fitness value also includes the potential solvability of the level ($p$), the ratio of empty tiles ($s$), and the ratio of useless sprites ($u$).
The final fitness equation for a level is as follows:
\begin{equation}\label{eq:fitness}
fitness_{new} = min(fitness_{old}) + u + p + 0.1 \cdot s
\end{equation}
where $fitness_{old}$ is the Kullback-Lievler Divergence fitness function from the Lucas and Volz work~\cite{lucas2019tile} compared to a reference level. The minimum operator is added as we are using multiple reference levels instead of one and we want to pick the fitness of the most similar reference level.
In the updated version of Baba is Y'all, we recalculate the ratio of useless objects ($u$) used in the original version's equation. The value $u$ is defined as the combined percentage of unnecessary object and word sprites in the level. This is broken up into 2 variables $o$ and $w$ for the objects and words respectively. The $o$ value corresponds to the objects that are not required or predicted to act as a constraint or solution for the level. The value for $o$ can be calculated as follows:
\begin{equation}
o = \frac{i}{j}
\end{equation}
where $i$ is the number of objects sprites initialized in the level without a related object-word sprite and $j$ is the total number of object sprites initialized in the level. While the $w$ value corresponds to the words that have no associated object in the map (this does not apply to keyword class words such as ``KILL'' or ``MOVE''). The value for $w$ can be calculated as follows:
\begin{equation}
w = \frac{k}{l}
\end{equation}
where $k$ is the number of word sprites initialized in the level without a related object-word sprite and $l$ is the total number of word sprites initialized in the level. To combine both variables $o$ and $w$ into the one variable $u$ a constant ratio is applied. In the system, 0.85 is applied to the $o$ variable and 0.15 to $w$. This is to more weight on reducing the number of useless object sprites as opposed to useless word sprites, as word sprites can be used to modify the properties of objects or transform other object sprites.
The $u$ value is implemented in order to prevent noise within the level due to having object tiles that cannot be manipulated in any way or have relevancy to the level. A human-made level may include these ``useless'' tiles for aesthetic purposes or to give the level a theme - similar to the original `Baba is You' levels. However, the PCG algorithm optimizes towards efficiency and minimalist levels, therefore ignoring the subjective aspect of a level's quality (which can be added later by the user).
The playability of the level ($p$) is a binary constraint value that determines whether a level is potentially winnable or not. The value can be calculated as follows:
\begin{equation}
p =
\begin{cases}
1, & \text{has [`X-IS-YOU' rule, `WIN' keyword]} \\
0, & otherwise
\end{cases}
\end{equation}
This is to ensure any levels that are absolutely impossible to play or win are penalized in the population and less likely to be mutated and evolved from in future generations. We used a simple playability constraint check instead of checking for playability using the solver because the solver take time to check for playability. Also, all playable levels by the solver usually end up being easy levels due to the limited search space we are given for the best first algorithm.
The ratio of empty tiles ($s$) is the ratio of empty space tiles to all of the tiles in the level. The equation can be calculated as follows:
\begin{equation}
s = \frac{e}{t}
\end{equation}
where $e$ is the number of empty spaces in the level and $t$ is the total number of tiles found in the level. The value $s$ is multiplied with a value of $0.1$ in equation~\ref{eq:fitness} to avoid heavy penalization for having any empty spaces in a level and to prevent encouragement for levels to mutate towards populating the level with an overabundance of similar tiles in order to eliminate any empty space.
The Mutator module is not run as a back-end process to find more levels, instead it has to be done manually by the user. This is done due to the fact that some generated levels cannot be solved without human input. One might wonder why not generate a huge corpus of levels and ask the users later to test them for the system. This could result in the system generating a multitude of levels that are either impossible to solve or are solvable but not subjectively ``good'' levels - levels the user would not find pleasing or enjoyable. This overabundance of ``garbage'' levels could lead to a waste of memory and a waste human resources. By allowing the user direct control over which levels are submitted from the generation algorithm, it still guarantees that the levels are solvable and with sufficient quality and promote using the tool in a mixed-initiative approach. Future work will explore implementing a fully autonomous generator and associated solver to expand the archive of levels without human input.
\subsection{Objective Module}\label{sec:objective_module}
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\linewidth]{imgs/obj_screen.png}
\caption{A screenshot of the rule objective screen}
\label{fig:objective_screen}
\end{figure}
In conjunction with the Mutator module (section~\ref{sec:mutator_module}), an Objective Module has been implemented to help guide the evolver towards generating levels that match selected objectives - or rules - set by either the Map Module or the user. Like before this will nudge both the user and the evolver back-end towards creating levels with mechanic combinations that have not been made in the site database.
Users can select from the table of mechanics which sets of rules to include in the level - whether initially at the start of the level, at the solution, or either. Initial rules can be found automatically when the user or evolver edits the level, final rules can only be determined at the end of the level - when the solution has been found. Active rules are highlighted with a green backlight in the table and change accordingly when a rule is created or removed.
The evolver also prioritizes levels that match as many of the selected rules as possible. A cascading function is used to rank the generated levels from the chromosome population. The evolver first evaluates how well a generated level corresponds to the selected objectives then looks at the fitness function. With this, the evolver becomes more involved with expanding the level database for the site and actively tries to help the user fill these missing levels.
\subsection{Rating Module}\label{sec:rating_module}
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\linewidth]{imgs/rating_screen.png}
\caption{A screenshot of the rating screen with 2 levels shown}
\label{fig:rating_screen}
\end{figure}
Like the original system, a rating for a single level is determined by comparison to another level within the site database. The user must determine the better level based on two qualities: level of challenge and quality of aesthetic design. A level that is considered `more challenging' could indicate that the solution search space for the level takes longer to arrive at or is not as intuitive or straightforward. A level that is considered to have `better design' represents that the level is more visually pleasing and elegant with its map representation - a quality that is hard to generate automatically with AI. Users can select between the two levels for each feature by shifting a slider towards one level or the other.
\subsection{Map Module}\label{sec:map_module}
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\linewidth]{imgs/map_screen.png}
\caption{A screenshot of the map selection screen}
\label{fig:select_screen}
\end{figure}
The Map module functions as both storing all of the levels in the site database as well as recommending specific levels to the user to use for their own level creation process. The Map module is the core module of the system. To maintain distinguish-ability between quality and diverse levels, we implemented the MAP-Elites algorithm for this module.
\begin{table}[t]
\caption{Chromosome Rule Representation}
\centering
\begin{tabular}{|p{0.2\linewidth}|p{0.7\linewidth}|}
\hline
Rule Type & Definition \\
\hline
\hline
X-IS-X & objects of class X cannot be changed to another class \\
X-IS-Y & objects of class X will transform to class Y \\
X-IS-PUSH & X can be pushed \\
X-IS-MOVE & X will autonomously move \\
X-IS-STOP & X will prevent the player from passing through it\\
X-IS-KILL & X will kill the player on contact\\
X-IS-SINK & X will destroy any object on contact\\
X-IS-[PAIR] & both rules 'X-IS-HOT' and 'X-IS-MELT' are present \\
X,Y-IS-YOU & two distinct objects classes are controlled by the player \\
\hline
\end{tabular}
\label{tab:rrp}
\end{table}
When a level is submitted to be archived, the system uses the list of active rules at the start and the end of the level as behavior characteristic for the input level to determine its location in the map. There are 9 different rules checked for in each level - based on the possible rule mechanics that can be made in the Game module system. Table \ref{tab:rrp} shows the full list of possible rules. Since these rules can be active at the beginning or at the end, it makes the number of behavior characteristics equal to 18 instead of 9 which provide us with a map of $2^{18}$ cells.
The Map Module can recommend levels to start from when designing a new level. Like the Mutator Module (section~\ref{sec:mutator_module}), it also takes the Objective Module (section~\ref{sec:objective_module}) into consideration when selecting its recommendations. The Map Module can provide levels that most similarly match the objectives chosen and provide either other levels the user has previously made or high rated (and intuitively high quality) ``elite'' levels.
In this project we are using a multi population per each cell of the Map-Elites similar to the constrained Map-Elites~\cite{khalifa2018talakat}. The quality of the level is determined by user ratings - performed by the Rating Module.
\subsection{User Profiles}
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\linewidth]{imgs/user_screen.png}
\caption{A screenshot of the user profile screen for the user 'Milk'}
\label{fig:profile_screen}
\end{figure}
The user profiles feature is the newest addition to the Baba is Y'all site. Like the original system, if a user creates a profile through the site's login system and submits a level, they get authorship attributed to the submitted level. Users can also find their previously made levels on the profile page - called ``My Levels'' - and replay them, edit them, or view the level's mechanic combination. A user's personal stats for their level submissions can also be viewed on the page including the number of levels submitted, number of rule combinations contributed, and their top rated level. This feature was implemented to provide more user agency and personalization on the site and give users better access to their own submitted levels.
Through the search page, players can search for specific levels by username or by level name. This creates a sense of authorship over each of the levels, even if the level wasn't designed with any human input (i.e. a level with PCG.js as the author) and encourages the collaborative nature of the site between AI and human. Users may also share links to site levels via the game page.
\section{Results}
The following results were extracted from the entire Baba is Y'all v2 site and includes data from levels made from participants not involved with the study.
\subsection{User and Author-based Data}
All users on the Baba is Y'all site had the option of registering for a new account to easily find their saved work as well as attribute personal authorship to any levels they submitted. Those who participated in the user study were given pre-made usernames in order to verify the levels they submitted from their responses and to protect their identities. These users only had to provide an email address to register for both the site and the survey.
The site had a total of 727 unique users registered - only 78 (10\%) came from outside of the user study while the rest of the users participated in the survey.
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\linewidth]{imgs/Level-types.png}
\caption{Sample levels generated for the system. The left column is user generated levels, the middle column is evolver module levels, and the right column is mixed-initiative user and evolver levels}
\label{fig:level_types}
\end{figure}
We looked into all the levels created by the users and we divided them based on how the mixed-initiative tool was used to create them. We divided them into three main categories (as shown in figure~\ref{fig:level_types}):
\begin{itemize}
\item \textbf{User-Only levels:} were created from a blank map exclusively by the human user without any AI assistance.
\item \textbf{PCG-only levels:} were created solely by the AI tool without any human input aside from choosing which tool to use and when.
\item \textbf{Mixed-author levels:} involved both the human user as well as the AI tool in the creation process of the level.
\end{itemize}
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c c c|}
\hline
Author Type & Number & \%\\
\hline\hline
User-only & 103 & 66.45 \\
\hline
PCG-only & 16 & 10.32\\
\hline
Mixed-author & 36 & 23.23\\
\hline
\hline
Total & 155 & 100\\
\hline
\end{tabular}
\end{center}
\caption{Authorship for levels submitted}
\label{tab:level_author}
\end{table}
The majority of the levels submitted were user only (66.45\%), however almost a quarter (23.23\%) of the levels submitted had mixed-authorship. Table \ref{tab:level_author} shows the full data for this area. Looking at this table, we notice that the amount of submitted levels are a lot less than total number of users ($155$ levels and $727$ users). This big difference in the numbers is due to releasing the system online with no security measures. This attracted a lot of bots that created multiple accounts so they could fill out the user survey via the link provided, but did not submit any levels.
\subsection{Level-based Data}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{site_graphs/rule_perc.png}
\caption{Site results for the rule distribution across levels submitted}
\label{fig:level_rule_dist}
\end{figure}
Looking into all the $155$ submitted levels, we found only $74$ different cells in the MAP-Elites matrix were covered. This is less than 1\% of the whole number of possible rule combinations ($2^{18}$ possible combinations). Figure~\ref{fig:level_rule_dist} shows the rule distributions over all of the levels submitted. The X-is-KILL rule was used the most in over half of the levels submitted and the X-is-STOP rule was used the second-most at 44.52\%. This may be because these rules create hazards for the player and add more depth to the level and solution. Meanwhile, the X-is-[PAIR] rule was used the least in only 12.9\% of the levels submitted. This is likely due to the lock-and-key nature of the rule combinations that require more intentionally placed word blocks that can also be accomplished with the X-is-SINK or X-is-KILL rule.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c c c c|}
\hline
User Type & \# Rules & Sol. Length & Map Size (\# tiles) \\
\hline\hline
User-only & 2.563 $\pm$ 2.19 & 25.834 $\pm$ 26.11 & 117.883 $\pm$ 50.84\\
\hline
PCG-only & 1.00 $\pm$ 1.17 & 19.062 $\pm$ 14.68 & 95.437 $\pm$ 25.60\\
\hline
Mixed-author & \textbf{2.833 $\pm$ 2.56} & \textbf{26.027 $\pm$ 20.36} & \textbf{127.722 $\pm$ 49.91}\\
\hline
\end{tabular}
\end{center}
\caption{Averaged attributes for different types of created levels}
\label{tab:avg_author}
\end{table}
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\linewidth]{site_graphs/rule_dist.png}
\caption{Rule distributions across the different authored levels}
\label{fig:rule_dist}
\end{figure}
The relation between rules and the different type of authors can be shown in table~\ref{tab:avg_author}. Some levels may use no rules at all (only containing the required X-is-YOU and X-is-WIN rules.) The mixed-author levels has the highest number of average rules per level ($2.833$), while PCG-only levels have the lowest average ($1$). The rule distributions for each author type are shown in Figure \ref{fig:rule_dist}. The PCG-authored levels had the least variability between rules while the Mixed-authored levels had the most variability. Mixed-author levels also had the highest average solution length and highest average level size, with PCG levels having the lowest for both attributes.
\section{User Study}
The following results were extracted from a Google Form survey given to the experiment participants. Users were instructed to play a level already made on the site, create a new level using the level editor, test it, and finally submit it to the site. They were also given the option to go through the tutorial of the site if they were unfamiliar with the `Baba is You' game or needed assistance with interacting with the level editor tool.
Of the $727$ users registered on the site, only a total of $170$ responses were received, however, only $76$ of these responses were valid. These responses were evaluated based on cross-validation and verification between the saved level on the website and the level ID they submitted via the survey that they claimed they authored. Many of these invalid responses contained levels that either did not exist in the database or were claimed to be authored by another user already. The following results are taken from the self-reported subjective survey given to the valid $76$ users.
\subsection{Demographic Data}\label{sec:demographics}
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\linewidth]{hor_survey_graphs/freq_v2.png}
\caption{A. Frequency for playing games; B. Frequency for designing levels for games}
\label{fig:freq_des_play}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\linewidth]{hor_survey_graphs/pref_v2.png}
\caption{Preference for solving or making puzzles}
\label{fig:design_pref}
\end{figure}
Half of the users who completed the survey answered that they frequently played video games (more than 10 hours a week) with around 80\% of the users stating they play for at least 2 hours a week (figure~\ref{fig:freq_des_play}). Conversely, only 28.9\% of users responded that they spend 2 or more hours a week designing levels for games with 40.8\% of users stating they never design levels at all (figure~\ref{fig:design_pref}). When asked if they prefer to solve or make puzzles, 50\% of participants responded that they prefer to solve puzzles, while only 6.6\% preferred the latter. 40.8\% of users were split on the preference for designing and solving puzzles.
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\linewidth]{hor_survey_graphs/experience_v2.png}
\caption{A. Experience playing Sokoban; B. Experience with 'Baba is You'; C. Experience with AI-assisted level editing tools}
\label{fig:exp_graph}
\end{figure}
We asked participants if they had ever played the original game `Baba is You' by Hempuli (either the jam version or the Steam release as both contain the rules used in the Baba is Y'all site), played a Sokoban-like game (puzzle games with pushing block mechanics), and have experience with AI-assisted level editing tools. Figure~\ref{fig:exp_graph} shows the distribution of the users' answers for these questions. Only 30\% of participants had played the game before, meanwhile 22\% had heard of it but had never played it. For the rest, this study would be their first experience with the game. Interestingly enough, 96\% of the participants stated they had played a Sokoban-like game so we can infer that the learning curve would not be too harsh for the new players. Concerning AI-Assisted level editing tools, 75\% of users had never used them before, with 5.3\% stating they were unsure if they had ever used one - thus the learning curve for AI-collaboration would be much higher and new to participants.
\subsection{Self-Reported Site Interactions}
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\linewidth]{survey_graphs/feat.png}
\caption{Survey results for users' reports on the features they used}
\label{fig:feat_report}
\end{figure}
Figure \ref{fig:feat_report} shows the full list of features that participants interacted with on the site. Users were given the optional task to go through the tutorial section of the Baba is Y'all site to familiarize themselves with both the mechanics of the original `Baba is You' game, the AI assisted tools available to them through the level editor, and the site layout and navigation itself. 81.6\% of users went through this tutorial (whether fully or partially was not recorded.) The second task for users was to play a level that was previously submitted to the website database. 100\% of users were able to solve a level by themselves, however 72.4\% of users reported choosing to watch the Keke AI solver complete the submitted level as well. The third and final task for the participants was to submit their own `Baba is You' level using the level editor. Here, users were asked the most about their involvement with the AI system. Some users chose to create more than one level, so they may have multiple experiences and their design choices may not be mutually exclusive (i.e. using a blank level and also using an AI-suggested level.)
For the initial creation of the level, 88.2\% of users chose to start with a blank map. 9.2\% of users started with a level that had already been submitted to the level database - either a level that had been ranked as an elite level or a level created by the user themselves (in the case that they submitted more than one level during this study.) 6.6\% of users started with a level that was suggested from the 'Unmade' page - ideally with the intent to make a level with a rule combination that had not been made yet - thus expanding the MAP-Elites rule combination matrix in the database. Unfortunately, we forgot to ask users in the survey if they started with the random level option that was also provided by the AI assistance tool - so we lack data to report on this statistic.
For editing the level, 81.6\% of users reported editing a level completely by hand without any AI assistance. 27.6\% of users edited the level with help from either the evolver algorithm or the mutator functions provided by the AI assistance back-end. 19.7\% of users reported using the objective table to aid the evolver tool in creating the level. We think this low percentage is attributed the fact that a large population of users were unfamiliar with the system or `Baba is You' game overall. This - as well as the lack of selection for level comparison from the previously submitted levels in the database - made using the evolver tool towards certain goals too steep of a task to accomplish and learn. Finally, when testing the level, 59.2\% of users reported using the Keke solver AI when testing their levels and 72.4\% of users named their levels.
While not required in the tasks given, we also asked participants about any extra site features they chose to explore. 23.7\% of users reported submitting a level rating from the 'Rate' page. 51.3\% of users reported using the 'Search' tool to search for specific levels (what their search criteria was we did not ask.) Finally, 19.7\% of users reported using the 'Share Level' to share a submitted level link with others online.
The least used interactions - 'Started with a database-saved map in the level editor', 'Started with a level suggestion from the Unmade page', and 'Used the objectives table to evolve levels' - were also all related to the AI mixed-initiation of the system. The first could be attributed to a lack of overall levels in the database (at the start of the experiment there were only around 40 available levels) therefore leading to a lack of viable options for the user to choose from. However, the lack of usage for the other two features could be attributed to the opposite problem of having too many options to choose from - again due to lack of levels available to choose from in the database. Trying to make a level with constrained parameters may have also been too steep of a task to accomplish for someone who was totally unfamiliar with the system or even the `Baba is You' game overall. There was also no incentive for a player to create a level suggested by the system as opposed to making a level from scratch. We also didn't explicitly instruct users to make a level from the suggested set, and instead allowed them to make whatever level they wanted with the editor - whether with the prompted ruleset or from their own ideas.
\section{Discussion}
\subsection{Data Analysis}
It is clear from both the submitted level statistics of the site and the self-reported user survey that mixed-authorship is not the preference for users when designing levels. Many users would still prefer to have total control over their level design process from start to finish. For future work, we can look to limit user control and encourage more AI-assistance with the design process similar to the work done by Bhaumik et al.~\cite{bhaumik2021lode}.
The limitations of the AI back-end (both the evolver and solver) may be at fault for the lack of AI interaction. The mutator and evolver system are dependent on previously submitted levels and level ratings in order to ``learn'' how to effectively evolve levels towards high quality design. As a result, the assistant tool is always learning what makes a ``good'' level from human input. If there is a lack of available data for the tool to learn from, the AI will be unable to create quality levels - causing the user to less likely submit mixed-initiative co-created levels, and causing a negative feedback loop.
The fitness function defined for the evolver and mutator tool may be inadequate for level designing. It could produce a level that is deemed ``optimal'' in quality by its internal definition, but may actually be sub-par in quality for a human user. Another flaw in the AI-collaboration system, could be that the users lacked direct control on the evolver and mutator and attempting to use them in middle of creation might have been more problematic as it could destroy some of the level structures that the users were working on. Future work could remedy this problem by giving users various mutation "options" similar to the AI selections in RLBrush \cite{delarosa2021mixed} and Pitako. \cite{machado2019pitako} Finally, the `Keke' AI solver was also lacking in performance as a few participants mentioned that the solver was unable to solve their prototype levels that they themselves could end up solving in just a couple of moves. An improved AI solver would help with the level creation efficiency.
\subsection{User Comments and Feedback}
We gave the participants opportunities to provide open feedback about their experience using the site in order to gather more subjective data about their experience as well as collect suggestions for potential new features.
Almost no users experienced any technical difficulties or bugs that prevented them from using the site. The few that did mentioned formatting issues with site caused by their browser (i.e. icons too close together, loading the helper gifs, font colors.) However, one user mentioned that this issue may have been because they were using the site from their phone (we unfortunately did not provide users with instructions to complete the study on a desktop or laptop.) In the future, we will be sure to exhaustively test the site on as many browsers as possible - both desktop-based and mobile - to be more accessible.
Some users were confused by the tutorial and the amount of information it conveyed for the entire site citing it as ``intimidating'', ``overwhelming'', and ``a bit complex''. However, other users reported the lack of information saying it was ``not detailed'', or had ``sufficient information [...] but could have been delivered in a more comprehensible way.'' To make the game more accessible, we will most likely try to make the tutorial section less intimidating to new users by limiting the amount of information shown (possibly through a ``table of contents'' as suggested by one participant) while still being comprehensible enough to understand the level editor and tools.
For feature suggestions, many users wished for larger maps and vocabulary - like those found in the Steam-release `Baba is You' game. Users also wished for a save feature that would allow them to make ``drafts'' of their level to come back later to edit. Many users also suggested a co-operative multiplayer feature for level editing and level solving - we can assume with another human and not an AI agent.
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\linewidth]{hor_survey_graphs/browser_v2.png}
\caption{User feedback for likelihood to return using the site after the experiment}
\label{fig:reuse_likelihood}
\end{figure}
While the results of the statistics on the levels submitted were disappointing for involvement of the AI assisting tool, we also asked users how likely they would continue using the site after the experiment. 38.2\% of users said they would continue to use the site, while 55.3\% said they would maybe use the site (figure~\ref{fig:reuse_likelihood}). Many users were optimistic and encouraging with the concept of incorporating AI and PCG technologies with level design - citing the project as a ``cool project'', ``a very unique experience'', a ``lovely game and experiment'', and ``very fun.'' At the time of writing, a few users did return, as their 'Keke' assigned usernames were shown as authors on the New page, long after the study was completed. Most notably, the Keke subject user Keke978 who took up the username 'Jme7' and contributed 28 more levels to the site after the study was concluded and currently holds the title for most levels submitted and most rule combinations on the site.
Many users also provided us with constructive feedback for feature implementation, site usability, and suggestions for improvement with how to further incorporate the AI back-end interactivity. As shown in figure~\ref{fig:exp_graph}, 70\% of users who played with the system had never played the game `Baba is You' and 75\% of people had never used an AI-assisted level editor tool before this experiment. Based on this information and retainability of users to complete the survey and provide the constructive feedback, we can extrapolate 2 conclusions: 1. the game stands alone, independent of `Baba is You', as an entertainment system; and 2. for people with even limited AI-gaming experience, as long as they are not completely foreign to gaming, this project has the ability to grasp their attention long enough to understand it, tinker around, and then give constructive feedback.
\section{Conclusion and Future Work}
The results from the user study have demonstrated both the benefits and limitations of a crowd-sourced mixed-initiative collaborative AI system. Currently, users still prefer to edit most of the content themselves, with minimal AI input - due to the lack of submitted content and ratings for the AI to learn from. Pretraining the AI system before incorporating it into the full system would be recommended to create more intelligent systems that can effectively collaborate with their human partners for designing and editing content. This would lead to more helpful suggestions on the evolver's end as well as better designed levels overall. This project is the start of a much longer and bigger investigation into the concept of crowd-sourced mixed initiative systems that can use quality diversity methods to produce content and we have many more ideas to improve upon the Baba is Y’all system.
As suggested by many participants in the user study, we would like to incorporate level design collaborations between multiple users and multiple types of evolutionary algorithms all at once to create levels. Our system would take inspiration from collaboration tools such as LodeEncoder \cite{bhaumik2021lode}, RLBrush \cite{delarosa2021mixed}, and Roblox (Roblox Corporation, 2006). This would broaden the scope and possibilities of level design and development even further to allow more creativity and evolutionary progress within the system. This collaboration setting will open multitude of interesting problems to investigate such as authorship.
Outside of the `Baba is You' game, we would like to propose the development of an open-source framework to allow mixed-initiative crowd-sourcing level design for any game or game clone. Such games could include Zelda, Pacman, Final Fantasy, Kirby, or any other game as long as we have a way to differentiate between levels mechanically and we can measure minimum viable quality of levels. Adding more games to the mixed-initiative framework would allow an easier barrier of entry to players who may have been unfamiliar with the independent game `Baba is You' but is very familiar with triple-A games produced by companies such as Nintendo.
We would like to also propose a competition for the online `Keke' solver algorithm for the challenging levels. In this competition, users would submit their own agent that can solve the user-made and artificially created `Baba is You' levels. Ideally, this improve the solver of the `Baba is Y'all' system but also introduce a novel agent capable of solving levels with dynamically changing content and rules - an area that has not been previously explored in the field. Development for this framework for this competition has already begun at the time of writing this paper.
Finally, we would like to propose the creation of a fully autonomous level generator and solver that can act as a user to our system.
This generator-solver pair would work parallel to the current system's mixed-initiative approach, but with a focus on coverage to exhaustively find and create levels for every combination of mechanics. With a redefined fitness function and updated solver (possibly from the Keke Solver Competition,) this could be more efficient than having users manually submit the levels, while still using content created by human users to maintain the mixed-initiative approach.
There are many new directions we can take the Baba is Y'all system and the concept of crowd-sourced collaborative mixed-initiative level design as a whole and this project will hopefully serve as a stepping stone into the area and provide insight on how AI and users can work together in a crowd-sourced website to generate new and creative content.
\section*{Acknowledgment}
The authors would like to thank the Game Innovation Lab, Rodrigo Canaan, Mike Cook, and Jack Buckley for their feedback on the site in its beta version as well as the numerous users who participated in the study and left feedback.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| 2023-04-23T08:17:28.014Z | 2022-03-07T02:05:04.000Z | redpajama/arxiv | arxiv_0000 | 61 | 10,632 |
28aefb059d9f6613e5ff09536977455486e365d4 | \section{Introduction}
\label{sec:intro}
Robots supporting people in their daily activities at home or at the workplace need to accurately and robustly perceive objects, such as containers, and their physical properties, for example when they are manipulated by a person prior to a human-to-robot handover~\cite{Sanchez-Matilla2020,Medina2016,Rosenberger2021RAL,Ortenzi2021TRO,Yang2021ICRA}. Audio-visual perception should adapt -- on-the-fly and with limited or no prior knowledge -- to changing conditions in order to guarantee the correct execution of the task and the safety of the person. For assistive scenarios at home, audio-visual perception should accurately and robustly estimate the physical properties (e.g., weight and shape) of household containers, such as cups, drinking glasses, mugs, bottles, and food boxes~\cite{Sanchez-Matilla2020,Ortenzi2021TRO,Liang2020MultimodalPouring,Modas2021ArXiv,Xompero2021_ArXiv}. However, the material, texture, transparency and shape can vary considerably across containers and also change with their content, which may not be visible due to the opaqueness of the container or occlusions, and hence should be inferred through the behaviour of the human~\cite{Sanchez-Matilla2020,Modas2021ArXiv,Xompero2021_ArXiv,Mottaghi2017ICCV,Duarte2020ICDL_EpiRob}.
In this paper, we present the tasks and the results of the CORSMAL challenge at IEEE ICASSP 2022, supporting the design and evaluation of audio-visual solutions for the estimation of the physical properties of a range of containers manipulated by a person prior to a handover (see Fig.~\ref{fig:avsamples}).
The specific containers and fillings are not known in advance, and the only priors are the sets of object categories ({drinking glasses}, {cups}, {food boxes}) and filling types ({water}, {pasta}, {rice}). The estimation of the mass and dimensions of the containers are novel tasks of this challenge, and complement the tasks of its previous version~\cite{Xompero2021_ArXiv}, such as the estimation of the container capacity and the type, mass and amount of the content. We carefully defined a set of performance scores to directly evaluate and systematically compare the algorithms on each task. Moreover, to assess the accuracy of the estimations and visualise the safeness of human-to-robot handovers, we implemented a real-to-simulation framework~\cite{Pang2021ROMAN} that provides indirect high-level evaluations on the impact of these tasks (see Fig.~\ref{fig:challengetasksdiagram}). The source code of the entries to the challenge and the up-to-date leaderboards are available at
\mbox{\url{http://corsmal.eecs.qmul.ac.uk/challenge.html}}.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{challenge_image.png}
\caption{Sample video frames and audio spectrograms of people manipulating objects prior to handing them over to a robot.}
\label{fig:avsamples}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{diagram_tasks.eps}
\caption{The challenge tasks feeding into the CORSMAL simulator~\cite{Pang2021ROMAN} to evaluate the impact of estimation errors. Given video frames and audio signals from the CORSMAL Containers Manipulation (CCM) dataset~\cite{Xompero2021_ArXiv,Xompero_CCM}), the results of T1 (filling level), T2 (filling type), and T3 (container capacity) are used to compute the filling mass, which is added to T4 (container mass) for estimating the mass of the object (container + filling). The estimated dimensions (T5) are used to visualise the container. The simulator also uses object annotations, such as 6D poses over time, the true weight (container + filling), a 3D mesh model reconstructed offline with a vision baseline~\cite{Pang2021ROMAN}, and the frame where the object is ready to be grasped by the simulated robot arm, for performing and visualising the handover.
}
\label{fig:challengetasksdiagram}
\end{figure*}
\section{The tasks}
\label{sec:tasks}
In the scope of the challenge and based on the reference dataset~\cite{Xompero2021_ArXiv,Xompero_CCM}, containers vary in shape and size, and may be empty or filled with an unknown content at 50\% or 90\% of its capacity.
We define a configuration as the manipulation of a container with a filling type and amount under a specific setting (i.e., background, illumination, scenario).
The challenge features five tasks (Ts), each associated with a physical property to estimate for each configuration $j$.
\begin{description}
\item[Filling level classification (T1).] The goal is to classify the filling level ($\tilde{\lambda}^j$) as empty, 50\%, or 90\%.
\item[Filling type classification (T2).] The goal is to classify the type of filling ($\tilde{\tau}^j$), if any, as one of these classes: 0 (no content), 1 (pasta), 2 (rice), 3 (water).
\item[Container capacity estimation (T3).] The goal is to estimate the capacity of the container ($\tilde{\gamma}^j$, in mL).
\item[Container mass estimation (T4).] The goal is to estimate the mass of the (empty) container ($\tilde{m}_{c}^j$, in g).
\item[Container dimensions estimation (T5).] The goal is to estimate the width at the top ($\tilde{w}_t^j$, in mm) and at the bottom ($\tilde{w}_b^j$, in mm), and height ($\tilde{h}^j$, in mm) of the container.
\end{description}
Algorithms designed for the challenge are expected to estimate these physical properties to compute the mass of the filling as
\begin{equation}
\tilde{m}_f^j = \tilde{\lambda}^j \tilde{\gamma}^j D(\tilde{\tau}^j),
\label{eq:fillingmass}
\end{equation}
where $D(\cdot)$ selects a pre-computed density based on the classified filling type. The mass of the object $\tilde{m}$ is calculated as the sum of the mass of the empty container and the mass of the content, if any.
\section{The evaluation}
\label{sec:evaluation}
\subsection{Data}
CORSMAL Containers Manipulation (CCM)~\cite{Xompero2021_ArXiv,Xompero_CCM} is the reference dataset for the challenge and consists of 1,140 visual-audio-inertial recordings of people interacting with 15 container types: 5 drinking cups, 5 drinking glasses, and 5 food boxes.
These containers are made of different materials, such as plastic, glass, and cardboard. Each container can be empty or filled with water, rice or pasta at two different levels of fullness: 50\% or 90\% with respect to the capacity of the container. In total, 12 subjects of different gender and ethnicity\footnote{An individual who performs the manipulation is referred to as \textit{subject}. Ethical approval (QMREC2344a) was obtained at Queen Mary University of London, and consent from each person was collected prior to data collection.} were invited to execute a set of 95 configurations as a result of the combination of containers and fillings, and for one of three manipulation scenarios. The scenarios are designed with an increasing level of difficulty caused by occlusions or subject motions, and recorded with two different backgrounds and two different lighting conditions to increase the visual challenges for the algorithms. The annotation of the data includes the capacity, mass, maximum width and height (and depth for boxes) of each container, and the type, level, and mass of the filling. The density of pasta and rice is computed from the annotation of the filling mass, capacity of the container, and filling level for each container. Density of water is 1 g/mL.
For validation, CCM is split into a training set (recordings of 9 containers), a public test set (recordings of 3 containers), and a private test set (recordings of 3 containers). The containers for each set are evenly distributed among the three categories.
The annotations are provided publicly only for the training set.
\subsection{Real-to-sim visualisation}
The challenge adopts a real-to-simulation framework~\cite{Pang2021ROMAN} that complements the CCM dataset with a human-to-robot handover in the PyBullet simulation environment~\cite{coumans2019pybullet}. The framework uses the physical properties of a manipulated container estimated by a perception algorithm.
The handover setup recreated in simulation consists of a 6 DoF robotic arm (UR5) equipped with a 2-finger parallel gripper (Robotiq 2F-85), and two tables.
The simulator renders a 3D object model reconstructed offline by a vision baseline in manually selected frames with no occlusions~\cite{Pang2021ROMAN}. The weight of the object used by the simulator is the true, annotated value. We manually annotated the poses of the containers for each configuration of CCM every 10 frames and interpolated the intermediate frames. We also annotated the frame where the person started delivering the object to the robot arm. We use the annotated and interpolated poses to render the motion of the object in simulation and control the robot arm to approach the object at the annotated frame for the handover. If the robot is not able to reach the container before the recording ends, the last location of the container is kept for 2~s.
When reaching the container, the simulated robot arm closes the gripper to 2~cm less than the object width to ensure good contact with the object, and applies an amount of force determined by the estimated weight of the object to grasp the container. Note that in the scope of the challenge, we avoid simulating the human hands so that the object is fully visible and can be grasped by the robot arm. The simulator visualises whether the estimations enable the robot to successfully grasp the container without dropping it or squeezing it. After grasping the container, the robot delivers it to a target area on a table via a predefined trajectory.
\subsection{Scores}
To provide sufficient granularity into the behaviour of the various components of the audio-visual algorithms and pipelines, we compute {13 performance scores} individually for the public test set (no annotations available to the participants), the private test set (neither data nor annotations are available to the participants), and their combination. All scores are in the range $[0,1]$. With reference to Table~\ref{tab:scores}, the first 7 scores quantify the accuracy of the estimations for the 5 main tasks and include filling level, filling type, container capacity, container width at the top, width at the bottom, and height, and container mass.
Other 3 scores evaluate groups of tasks and assess filling mass, joint filling type and level classification, joint container capacity and dimensions estimation. The last 2 scores are an indirect evaluation of the impact of the estimations (i.e., the object mass) on the quality of human-to-robot handover and delivery of the container by the robot in simulation.
\textbf{T1 and T2.} For filling level and type classification, we compute precision, recall, and F1-score for
each class $k$ across all the configurations of that class, $J_k$. \textit{Precision} is the number of true positives divided by the total number of true positives and false positives for each class $k$ ($P_k$). \textit{Recall} is the number of true positives divided by the total number of true positives and false negatives for each class $k$ ($R_k$). \textit{F1-score} is the harmonic mean of precision and recall, defined as
\begin{equation}
F_k = 2\frac{P_k R_k}{P_k + R_k}.
\end{equation}
We compute the weighted average F1-score across $K$ classes as,
\begin{equation}
\bar{F}_1 = \sum_{k=1}^K \frac{J_k F_k}{J},
\label{eq:wafs}
\end{equation}
where $J$ is the total number of configurations (for either the public test set, the private test set, or their combination). Note that $K=3$ for the task of filling level classification and $K=4$ for the task of filling type classification.
\textbf{T3, T4 and T5.} For container capacity and mass estimation, we compute the relative absolute error between the estimated measure, $a \in \{\tilde{\gamma}^j, \tilde{m}_c^j \}$, and the true measure, $b \in \{\gamma^j, m_c^j \}$:
\begin{equation}
\varepsilon(a, b) = \frac{|a - b |}{b}.
\label{eq:ware}
\end{equation}
For container dimensions estimation, where $a \in \left\{\tilde{w}_t^j,\tilde{w}_b^j,\tilde{h}^j\right\}$ and $b$ is the corresponding annotation, we use the normalisation function $\sigma_1(\cdot,\cdot)$~\cite{Sanchez-Matilla2020}:
\begin{equation}
\sigma_1(a,b)=
\begin{cases}
1 - \frac{|a - b |}{b} & \text{if} \quad | a - b | < b, \\
0 & \text{otherwise}.
\end{cases}
\end{equation}
For filling mass estimation\footnote{Note that an algorithm with lower scores for T1, T2 and T3, may obtain a higher filling mass score than other algorithms due to the multiplicative formula to compute the filling mass for each configuration.}, we compute the relative absolute error between the estimated, $\tilde{m}_{f}^j$, and the true filling mass, $m_{f}^j$, unless the annotated mass is zero (empty filling level),
\begin{equation}
\epsilon(\tilde{m}_f^j, m_f^j) =
\begin{cases}
0, & \text{if } m_f^j = 0 \land \tilde{m}_f^j=0, \\
\tilde{m}_f^j & \text{if } m_f^j = 0 \land \tilde{m}_f^j \neq 0, \\
\frac{|\tilde{m}_f^j - m_f^j |}{m_f^j} & \text{otherwise}.
\end{cases}
\label{eq:ware2}
\end{equation}
With reference to Table~\ref{tab:scores}, we compute the score, $s_i$, with \mbox{$i=\left\{3,\dots,8\right\}$}, across all the configurations $J$ for each measure as:
\begin{equation}
\noindent
s_i =
\begin{cases}
\frac{1}{J}\sum_{j=1}^{J}{ \mathds{1}_j e^{-\varepsilon(a, b)}} & \text{if} \, a \in \left\{\tilde{\gamma}^j, \tilde{m}_c^j\right\},\\
\frac{1}{J}\sum_{j=1}^{J}{\mathds{1}_j \sigma_1(a,b)} & \text{if} \, a \in \left\{\tilde{w}^j,\tilde{w}_b^j,\tilde{h}^j\right\},\\
\frac{1}{J}\sum_{j=1}^{J}{ \mathds{1}_j e^{-\epsilon(a, b)}} & \text{if} \, a=\tilde{m}_f^j.\\
\end{cases}
\end{equation}
The value of the indicator function, \mbox{$\mathds{1}_j \in \{0,1\}$}, is 0 only when \mbox{$a \in \left\{ \tilde{\gamma}^j, \tilde{m}_c^j, \tilde{w}_t^j,\tilde{w}_b^j,\tilde{h}^j, \tilde{m}_f^j \right\}$} is not estimated in configuration $j$. Note that estimated and annotated measures are strictly positive, $a>0$ and $b>0$, except for filling mass in the empty case (i.e., $\tilde{\lambda}^j = 0$ or $\tilde{\tau}^j = 0$).
\begin{table*}[t!]
\centering
\scriptsize
\renewcommand{\arraystretch}{1.2}
\setlength\tabcolsep{1.3pt}
\caption{Results of the CORSMAL challenge entries on the combination of the public and private CCM test sets~\cite{Xompero2021_ArXiv,Xompero_CCM}. For a measure $a$, its corresponding ground-truth value is $\hat{a}$. All scores are normalised and presented in percentages. $\bar{F}_1(\cdot)$ is the weighted average F1-score. Filling amount and type are sets of classes (no unit).
}
\begin{tabular}{ccccclllllccrrrrrrrrr}
\specialrule{1.2pt}{3pt}{0.6pt}
T1 & T2 & T3 & T4 & T5 & Description & Unit & Measure & Score & Weight & Type & R2S & RAN & AVG & \cite{Donaher2021EUSIPCO_ACC} & \cite{Liu2020ICPR} & \cite{Ishikawa2020ICPR} & \cite{Iashin2020ICPR} & \cite{Apicella_GC_ICASSP22} & \cite{Matsubara_GC_ICASSP22} & \cite{Wang_GC_ICASSP22} \\
\specialrule{1.2pt}{3pt}{1pt}
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & Filling level & & $\lambda^j$ & $s_1 = \bar{F}_1(\lambda^1, \ldots, \lambda^J, \hat{\lambda}^1, \ldots, \hat{\lambda}^J)$ & $\pi_1 = 1/8$ & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 37.62 & 33.15 & \textbf{80.84} & 43.53 & 78.56 & 79.65 & \multicolumn{1}{c}{--} & 65.73 & 77.40 \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & Filling type & & $\tau^j$ & $s_2 = \bar{F}_1(\tau^1, \ldots, \tau^J, \hat{\tau}^1, \ldots, \hat{\tau}^J)$ & $\pi_2 = 1/8$ & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 24.38 & 23.01 & 94.50 & 41.83 & 96.95 & 94.26 & \multicolumn{1}{c}{--} & 80.72 & \textbf{99.13} \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & Capacity & mL & $\gamma^j$ & $s_3 = \frac{1}{J} \sum_{j=1}^{J} \mathds{1}_j e^{-\varepsilon^j(\gamma^j, \hat{\gamma}^j)}$ & $\pi_3 = 1/8$ & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 24.58 & 40.73 & \multicolumn{1}{c}{--} & 62.57 & 54.79 & 60.57 & \multicolumn{1}{c}{--} & \textbf{72.26} & 59.51 \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & Container mass & g & $m_c^j$ & $s_4 = \frac{1}{J} \sum_{j=1}^{J} \mathds{1}_j e^{-\varepsilon^j(m_c^j, \hat{m}_c^j)}$ & $\pi_4 = 1/8$ & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 29.42 & 22.06 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & 49.64 & 40.19 & \textbf{58.78} \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & Width at top & mm & $w_t^j$ & $s_5 = \frac{1}{J}\sum_{j=1}^{J}{\mathds{1}_j \sigma_1(w_t^j, \hat{w_t}^j)}$ & $\pi_5 = 1/24$ & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 32.33 & 76.89 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & 69.09 & \textbf{80.01} \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & Width at bottom & mm & $w_b^j$ & $s_6 = \frac{1}{J}\sum_{j=1}^{J}{\mathds{1}_j \sigma_1(w_b^j, \hat{w_b}^j)}$ & $\pi_6 = 1/24$ & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 25.36 & 58.19 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & 59.74 & \textbf{76.09} \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & Height & mm & $h^j$ & $s_7 = \frac{1}{J}\sum_{j=1}^{J}{ \mathds{1}_j \sigma_1(h^j, \hat{h}^j)}$ & $\pi_7 = 1/24$ & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 42.48 & 64.32 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & 70.07 & \textbf{74.33} \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & Filling mass & g & $m_f^j$ & $s_8 = \frac{1}{J} \sum_{j=1}^{J} \mathds{1}_j e^{-\epsilon^j(m_f^j, \hat{m}_f^j)}$ & $\pi_8 = 1/8$* & I & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 35.06 & 42.31 & 25.07 & 53.47 & 62.16 & 65.06 & \multicolumn{1}{c}{--} & \textbf{70.50} & 65.25 \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & Object mass & g & $m^j$ & $s_9 = \frac{1}{J}\sum_{j=1}^{J}{\mathds{1}_j \psi^j(m^j, \hat{F}^j)}$ & $\pi_9 = 1/8$* & I & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & 56.31 & 58.30 & 55.22 & 64.13 & 66.84 & 65.04 & 53.54 & 60.41 & \textbf{71.19} \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & Pose at delivery & (mm, $^\circ$) & ($\alpha^j$,$\beta^j$) & $s_{10} = \frac{1}{J}\sum_{j=1}^{J}{\Delta_j(\alpha^j,\beta^j,\eta,\phi)}$ & $\pi_{10} = 1/8$* & I & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & 72.11 & 70.01 & 73.94 & 78.76 & 72.91 & \textbf{80.40} & 60.54 & 73.17 & 79.32 \\
\midrule
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \multicolumn{3}{l}{Joint filling type and level} & $s_{11} = \bar{F}_1(\lambda^1, \tau^1, \ldots, \hat{\lambda}^1, \hat{\tau}^1, \ldots)$ & \multicolumn{1}{c}{--} & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 10.49 & 8.88 & 77.15 & 24.32 & 77.81 & 76.45 & \multicolumn{1}{c}{--} & 59.32 & \textbf{78.16} \\
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \multicolumn{3}{l}{Container capacity and dimensions} & $s_{12} = {s_3}/{2} + (s4 + s5 + s6)/{6}$ & \multicolumn{1}{c}{--} & D & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=white] (1,1) circle (0.5ex);} & 28.99 & 53.60 & \multicolumn{1}{c}{--} & 31.28 & 27.39 & 30.28 & \multicolumn{1}{c}{--} & \textbf{69.28} & 68.16 \\
\specialrule{1.2pt}{3pt}{1pt}
\protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & \protect\raisebox{1pt}{\protect\tikz \protect\draw[black,fill=black] (1,1) circle (0.5ex);} & Overall score & & & $S = \sum_{l=1}^{10} \pi_l s_l$ & \multicolumn{1}{c}{--} & I & \multicolumn{1}{c}{--} & 39.11 & 44.51 & 31.52 & 35.89 & 47.04 & 48.35 & 9.05 & 66.16 & \textbf{73.43} \\
\specialrule{1.2pt}{3pt}{1pt}
\multicolumn{21}{l}{\scriptsize{Best performing results for each row highlighted in bold. Results of tasks not addressed shown with a hyphen (--).}}\\
\multicolumn{21}{l}{\scriptsize{For $s_9$ and $s_{10}$, configurations with failures in grasping and/or delivering the containers in simulation using true physical properties as input are annotated and discarded.}}\\
\multicolumn{21}{l}{\scriptsize{For fairness, the residual between 100 and the scores obtained with true measures of the physical properties are added to $s_9$ and $s_{10}$ to remove the impact of the simulator.}}\\
\multicolumn{21}{l}{\scriptsize{KEY -- T:~task, D:~direct score, I:~indirect score, R2S:~measured in the real-to-simulation framework, RAN:~random estimation, AVG:~average from the training set.}}\\
\multicolumn{21}{l}{\scriptsize{* weighted by the number of performed tasks.}}\\
\end{tabular}
\label{tab:scores}
\vspace{-10pt}
\end{table*}
\textbf{Object safety and accuracy of delivery.}
Object safety is the probability that the force applied by the robot, $\tilde{F}$, enables a gripper to hold the container without dropping it or breaking it~\cite{Pang2021ROMAN}. We approximate the force required to hold the container as
\begin{equation}
\hat{F} \approx \frac{\hat{m} (g + a_{max})}{\mu},
\label{equ:grasp_force_theoretical}
\end{equation}
where $\hat{m}$ is the annotated object mass; $g=9.81$~m/s$^2$ is the gravitational earth acceleration; $a_{max}$ is the maximum acceleration of the robot arm when carrying the object; and $\mu$ is the coefficient of friction between the container and the gripper ($\mu=1.0$~\cite{Pang2021ROMAN}).
The value of the force applied by the robot to grasp the object is calculated with Eq.~\ref{equ:grasp_force_theoretical} using the predicted object mass $\tilde{m}$. We compute object safety as an exponential function that accounts for the difference between the applied normal force $\tilde{F}^j$ (measured in simulation) and the required normal force, $\hat{F}^j$:
\begin{equation}
\psi^j = e^{\frac{| \tilde{F}^j - \hat{F}^j |}{\hat{F}^j} \ln{(1-c)}} = {\ln{(1-c)}}^{\frac{| \tilde{F}^j - \hat{F}^j |}{\hat{F}^j}},
\label{eq:forcesafety}
\end{equation}
where the normal force is the component of the contact force perpendicular to the contact surface and $c$ controls the sensitivity of $\psi^j$~\cite{Pang2021ROMAN}. A negative difference represents a higher probability of dropping the container, and a positive difference represents a higher probability of breaking the container.
We quantify the accuracy of delivering the container upright and within the target area as
\begin{equation}
\Delta_j =
\begin{cases}
1 - \frac{\alpha}{\eta} & \text{if } (\alpha < \eta) \text{ and } (\beta < \phi), \\
0 & \text{otherwise}, \\
\end{cases}
\end{equation}
where $\alpha$ is the distance from the centre of the base of the container to the target location $\boldsymbol{d}$; $\eta$ is the maximum distance allowed from the delivery target location; $\beta$ is the angle between the vertical axis of the container and the vertical axis of the world coordinate system; and $\phi$ is the value of $\beta$ at which the container would tip over.
We compute the score for object safety, $s_9$, as
\begin{equation}
s_{9} = \frac{1}{J}\sum_{j=1}^{J}{ \mathds{1}_j \psi^j(m^j, \hat{F}^j)},
\label{eq:totobjsafetyscore}
\end{equation}
where the value of the indicator function, $\mathds{1}_j$, is 0 only when either the filling mass or the containers mass is not estimated for each configuration $j$;
and the score for the delivery accuracy, $s_{10}$, as
\begin{equation}
s_{10} = \frac{1}{J}\sum_{j=1}^{J}{ \Delta_j(\alpha^j,\beta^j,\eta,\phi)}.
\label{eq:deliveryscore}
\end{equation}
The scores $s_9$ and $s_{10}$ are partially influenced by the simulator conditions (e.g, friction, contact, robot control), but we aimed at making the simulated handover reproducible across different algorithms through the annotated object trajectory, starting handover frame, and reconstructed 3D model.
{\bf Group tasks and overall score.} For joint filling type and level classification ($s_{11}$), estimations and annotations of both filling type and level are combined in $K=7$ feasible classes, and $\bar{F}_1$ is recomputed based on these classes. For joint container capacity and dimensions estimation, we compute the following weighted average:
\begin{equation}
s_{12} = \frac{s_3}{2} + \frac{s4 + s5 + s6}{6}.
\end{equation}
Finally, the overall score is computed as the weighted average of the scores from $s_1$ to $s_{10}$. Note that $s_8$, $s_9$, and $s_{10}$ may use random estimations for either of the tasks not addressed by an algorithm.
\subsection{IEEE ICASSP 2022 Challenge entries}
Nine teams registered for the IEEE ICASSP 2022 challenge; three algorithms were submitted for container mass estimation (T4), two algorithms were submitted for classifying the filling level (T1) and type (T2), and two other algorithms were submitted for estimating the container properties (T3, T4, T5) by three teams. We refer to the submissions of the three teams as A1~\cite{Apicella_GC_ICASSP22}, A2~\cite{Matsubara_GC_ICASSP22}, and A3~\cite{Wang_GC_ICASSP22}.
A1 solved only the task of container mass estimation (T4) using RGB-D data from the fixed frontal view and by regressing the mass with a shallow Convolutional Neural Network (CNN)~\cite{Christmann2020NTNU}. To increase the accuracy, A1 extracted a set of patches of the detected container from automatically selected frames in a video, and averaged their predicted masses.
To classify the filling level (T1) and type (T2), A2 used Vision Transformers~\cite{Dosovitskiy2021ICLR}, whereas A3 used pre-trained CNNs (e.g., Mobilenets~\cite{Howard2017Arxiv}) combined with Long Short-Term Memory units or majority voting~\cite{Hochreiter1997LSTM}. Only audio or audio with visual information (RGB) from the fixed, frontal view is preferred as input. To estimate the container properties (T3, T4, T5), A2 used RGB data from the three fixed views, and A3 used RGB-D data from the fixed frontal view.
A2 used a modified multi-view geometric approach that iteratively fits a hypothetical 3D model~\cite{Xompero2020ICASSP_LoDE}.
A3 fine-tunes multiple Mobilenets via transfer learning from the task of dimensions estimation (T5) to the tasks of container capacity (T3) and mass (T4) estimation~\cite{Wang_GC_ICASSP22}.
These Mobilenets regress the properties using patches extracted from automatically selected frames where the container is mostly visible~\cite{Wang_GC_ICASSP22}. To overcome over-fitting of the limited training data and improve generalisation on novel containers, these Mobilenets are fine-tuned with geometric-based augmentations and variance evaluation~\cite{Wang_GC_ICASSP22}. Overall, A3 is designed to process a continuous stream (online), thus being more suitable for human-to-robot handovers.
Table~\ref{tab:scores} shows the scores of the submissions on the combined CCM test sets. As reference, we provide the results for the properties estimated by a pseudo-random generator (RAN), by using the average (AVG) of the training set for container capacity, mass, and dimensions; or by the algorithms of four earlier entries to the challenge~\cite{Donaher2021EUSIPCO_ACC,Liu2020ICPR,Ishikawa2020ICPR,Iashin2020ICPR}. A3 achieves the highest $\bar{F}_1$ for filling type classification ($s_2 = 99.13$), and joint filling type and level classification ($s_{11} = 78.16$). A3 is also the most accurate in estimating the container mass ($s_4 = 58.78$), followed by A1 ($s_4=49.64$), and the container dimensions.
A2 is the most accurate in estimating the capacity ($s_3 = 72.26$). A2 is also the most accurate for filling mass ($s_8 = 70.50$). A3 has a high accuracy for filling level and type classification, but is affected by its lower accuracy
for capacity estimation.
Among the entries of the challenge at IEEE ICASSP 2022, A3 achieves the best score for object safety ($s_9 = 71.19$) and delivery accuracy ($s_{10} = 79.32$). In conclusion, A3 reaches the highest overall score ($S = 73.43$), followed by A2 ($S = 66.16$).
\section{Conclusion}
\label{sec:conclusion}
Recent, fast advances in machine learning and artificial intelligence have created an expectation on the ability of robots to seamlessly operate in the real world by accurately and robustly perceiving and understanding dynamic environments, including the actions and intentions of humans.
However, several challenges in audio-visual perception and modelling humans with their hard-to-predict behaviours hamper the deployment of robots in real-world scenarios.
We presented the tasks, the real-to-simulation framework, the scores and the entries to the CORSMAL challenge at IEEE ICASSP 2022 .
These new entries complement the algorithms previously submitted to the challenge~\cite{Donaher2021EUSIPCO_ACC,Liu2020ICPR,Ishikawa2020ICPR,Iashin2020ICPR}.
\bibliographystyle{IEEEbib}
| 2023-04-23T08:17:28.124Z | 2022-03-07T02:01:33.000Z | redpajama/arxiv | arxiv_0000 | 66 | 5,570 |
30a0701b2d7f2d3f82acdac04bc19081a3cb2f11 | "\\section{Introduction}\\label{sec:intro}\n\nTime-dependent manipulation of few and many-particle q(...TRUNCATED) | 2023-04-23T08:17:28.423Z | 2022-03-07T02:00:45.000Z | redpajama/arxiv | arxiv_0000 | 75 | 19,470 |
817c40acb6ae629307ae69a4b2a307daa30935a1 | "\\section{Introduction}\n\\label{sec_introduction}\n\nThe Epoch of Reionisation (EoR) marks the sec(...TRUNCATED) | 2023-04-23T08:17:28.729Z | 2022-09-30T02:08:43.000Z | redpajama/arxiv | arxiv_0000 | 91 | 22,647 |
9e8386c8b2d8ef7fb38a3da43ade44dedad91b03 | "\\section*{Introduction}\n\n\n\n\n\nMaterials crystallizing in the perovskite structure attract att(...TRUNCATED) | 2023-04-23T08:17:29.351Z | 2020-12-17T02:13:51.000Z | redpajama/arxiv | arxiv_0000 | 110 | 2,798 |
e8ec9d76bc3ac42a3d8b950b7352da6828031d53 | "\\section{Introduction}\n\nReplicating human movements and behaviour in humanoid robots is a formid(...TRUNCATED) | 2023-04-23T08:17:29.751Z | 2020-12-17T02:13:30.000Z | redpajama/arxiv | arxiv_0000 | 123 | 10,974 |
5f0da58d356fd1293ba86da9658e24c27197f3a4 | "\\section{Introduction}\nThe solar corona is filled with dynamic magnetized plasma, and hosts a var(...TRUNCATED) | 2023-04-23T08:17:31.675Z | 2022-03-31T02:15:33.000Z | redpajama/arxiv | arxiv_0000 | 180 | 3,527 |
34bd255fa58d72b26374511e3125e38769a36db4 | "\\section{Introduction}\nIn recent years, many computer vision (CV) researchers make efforts to des(...TRUNCATED) | 2023-04-23T08:17:31.726Z | 2022-05-10T02:09:10.000Z | redpajama/arxiv | arxiv_0000 | 186 | 6,771 |
30bdbf208d83041979feab2b093cfc0761128235 | "\\section{Introduction}\r\n\r\nSolid-state superconducting circuit studies the interaction between\(...TRUNCATED) | 2023-04-23T08:17:31.942Z | 2022-03-30T02:27:04.000Z | redpajama/arxiv | arxiv_0000 | 191 | 4,258 |
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