#  The Cross-lingual Conversation Summarization Challenge

Yulong Chen∗♠♡ , Ming Zhong∗♣ , Xuefeng Bai∗♠♡ , Naihao Deng‡ ,
Jing Li♢ , Xianchao Zhu♢ , Yue Zhang♡
♠ Zhejiang University
♡ School of Engineering, Westlake University
♣ University of Illinois at Urbana-Champaign
‡ University of Michigan, Ann Arbor
♢ Sichuan Lan-bridge Information Technology Co., Ltd.
<EMAIL_ADDRESS><EMAIL_ADDRESS>Equal Contribution.

###### Abstract

We propose the shared task of cross-lingual conversation summarization,
_ConvSumX Challenge_ , opening new avenues for researchers to investigate
solutions that integrate conversation summarization and machine translation.
This task can be particularly useful due to the emergence of online meetings
and conferences. We construct a new benchmark, covering 2 real-world scenarios
and 3 language directions, including a low-resource language. We hope that
_ConvSumX_ can motivate researches to go beyond English and break the barrier
for non-English speakers to benefit from recent advances of conversation
summarization.

##  1 Task Overview

The cross-lingual conversation summarization (ConvSumX) task asks models to
output a salient, concise, fluent and coherent summary in the target language
(_e.g._ , Chinese), given a conversation in the source language (_e.g._ ,
English). In particular, ConvSumX contains 2 tracks: (1) daily dialogue
summarization and; (2) query-based meeting minute. Each track covers 3
language directions: English-to-Chinese (En2Zh), English-to-French (En2Fr) and
English-to-Ukrainian (En2Uk). Figure 1 gives an example for each track in
_ConvSumX_ respectively, where we show summaries (queries) in 4 languages
(including English). Both automatic and manual evaluations are used to measure
the performance of submitted systems, while the evaluation is highly inclined
to human evaluation focusing on cross-lingual consistency (Section 3.4).

##  2 Motivation

Thanks to the availability of large-scale corpora Gliwa et al. (2019); Chen et
al. (2021a); Zhong et al. (2021b), research on conversation summarization has
made great progress Zhong et al. (2021a); Ni et al. (2021); Ghazvininejad et
al. (2021); Lin et al. (2022). However, existing corpora in this area focus on
English while ignoring other languages Feng et al. (2021a). Such English-
dominated corpora lead to a barrier for non-English speakers to benefit from
conversation summarization research, which becomes more severe in the era of
epidemic, where international meetings are held online and participants
communicate in English.

Figure 1: Examples of _ConvSumX_. Given the conversation text (and the
_query_), the task is to generate a _summary_. From top to bottom, the
languages are English (En), Chinese (Zh), French (Fr) and Ukrainian (Uk). The
English summaries (and queries) are from the original datasets.

_ConvSumX_ integrates conversation summarization and machine translation,
involving the language shift from one to another and stylistic shift from long
spoken conversations to concise written monologues. Ideally, using the _first
translate, then summarize_ and vice versa pipelines can solve the task.
However, besides the difficulties that have been discussed in monolingual
conversation summarization Chen et al. (2021b); Feng et al. (2021b), pipelines
methods suffer from problems caused by machine translation systems. For
example, for _translation-first_ pipelines, most existing machine translation
systems show poor performance on conversation text Wang et al. (2021). For
_summarization-first_ systems, translating summaries without considering
conversation context can lead to inconsistent translation, in particular for
polysemous words. Take $\texttt{C}^{\texttt{En}}$2$\texttt{S}^{\texttt{Zh}}$
111The setting means the input conversation text is in English, and the output
summary is in Chinese. for example. The summary “ _Bob is going to the bank._
”, where “ _bank_ ” can be translated into “岸边” (river bank) or “银行”
(financial bank), requires models to determine the proper translation by
considering conversation context. Such issues can be also in end-to-end
systems developed for cross-lingual news summarization and directly using
those methods can lead to error propagation Zhu et al. (2019); Xu et al.
(2020); Liang et al. (2022). Thus, more sophisticated designs that take care
of conversation natures or data selection strategies that can make better use
of silver data are in need.

To this end, _ConvSumX Challenge_ encourages researchers to investigate
different solutions to cross-lingual conversation summarization. First, from
the perspective of downstream applications, _ConvSumX_ is useful for both
business and personal uses. Second, from the perspective of research,
_ConvSumX Challenge_ looks for a general method that can deal with cross-
lingual conversation summarization. Although we present 3 typical target
languages in this work, we hope that _ConvSumX_ can motivate researches to a
broader range of target languages. Third, from the perspective of social good,
_ConvSumX_ aims to break the barrier of accessing information for non-English
speakers and to make them benefit from the advance of conventional English-
dominated conversation summarization technologies.

We hope that _ConvSumX_ can gain interests from both communities of text
summarization and machine translation and also push the progress on related
fields for other languages, including more low-resource languages.

##  3 Task Description

Formally, the task of _ConvSumX Challenge_ asks participants to provide a
system that can output a summary in a target language given the input
conversation text in a source language.

###  3.1 Setting

The _ConvSumX Challenge_ focuses on the low-resource/few-shot setting and
cross-lingual/domain transfer technologies. The low-resource/few-shot here is
stated from the perspective of _lacking large gold training data_. The term
_gold data_ refers to cross-lingual $\langle$conversation-summary$\rangle$
pairs that are annotated by experts.

Track | Data Source | Domain | Query | # Conv. | # Summ. | Train/Dev/Test
---|---|---|---|---|---|---
Track 1 | DialogSum | Daily-life Dialogue | ✗ | 131.0 | 13.8 | 12,460/500/500
Track 2 | QMSum | Product Meeting | ✓ | 6,007.7 | 70.5 | 690/145/151
Academic Meeting | ✓ | 13,317.3 | 52.7 | 259/54/56
Table 1: Statistics of _ConvSumX_. # Conv. and # Summ. are averaged lengths of
conversations and summaries.

The reasons are: (1) gold parallel data are limited as the annotation is
costly, in particular when source conversations involve domain expert
knowledge (_e.g.,_ academic meeting). In contrast, machine translation data
and monolingual summarization data are abundant and useful Perez-Beltrachini
and Lapata (2021); (2) we seek for a general solution that can be applied to
not only the target languages as in this paper, but also other languages.
However, for practical consideration, we provide large silver data generated
by multiple translation systems (_cf. Section 3.3.2_). We encourage
participants to make use of external resources to solve the task.

The above setting is widely adopted by existing cross-lingual summarization
datasets in other domains, such as the first large-scale cross-lingual
summarization corpus, NCLS dataset Zhu et al. (2019) and its succeeding works
Xu et al. (2020); Bai et al. (2021); Liang et al. (2022).

###  3.2 Tracks

The _ConvSumX Challenge_ consists of 2 tracks, focusing on different
scenarios, respectively.

Track 1 focuses on cross-lingual summarization for real-life dialogues. This
track is in line with the INLG 2021 _DialogSum Challenge_ Chen et al. (2021b)
while we extend _DialogSum_ into a cross-lingual setting. _ConvSumX_ can be
particularly useful in scenarios such as travelling or studying abroad where
summarizers can serve as personal assistants for non-English speakers.

Track 2 focuses on cross-lingual meeting minutes. Compared with daily
conversations, meeting conversations are much longer and contain rich topic
switches and more professional knowledge. Generating cross-lingual meeting
minutes can help non-English speakers to quickly access information of their
interest, especially in the case where conferences are mostly held in English.
In particular, Track 2 asks a system to generate a summary in the target
language, given an input meeting text in the source language and a query in
the target language.

###  3.3 Data

####  3.3.1 Data Selection

The data of _ConvSumX_ are derived from two public English datasets, namely
DialogSum (Chen et al., 2021a) and QMSum (Zhong et al., 2021b). We choose
these two datasets for: (1) their high-quality annotation and; (2) that they
are suitable for real-life applications. Table 1 shows the statistics.

DialogSum is a large-scale real-life scenarios dialogue summarization dataset,
consisting of face-to-face spoken dialogues that cover a wide range of daily-
life topics. In particular, DialogSum provides multi-references for each
dialogue in test set. We ask annotators to first choose the best reference
summary from multi-references and then annotate it into the target languages.

QMSum is a query-based meeting minute dataset, covering 3 domains, namely
academic, product and committee. We choose academic meeting and product
discussion meeting for annotation as they are more in line with our
motivation.

####  3.3.2 Annotation

We invite native translators as our annotators and ask them to annotate
summaries in dev and test sets of DialogSum and QMSum into 3 target languages
including Chinese, French and Ukrainian. Note that our data annotation is not
the simple translation of summaries, instead, each annotation needs to take
care of original English conversations to ensure that the annotated cross-
lingual summary is consistent with the input (_cf._ Section 2).

Besides manually annotated dev and test sets, following Zhu et al. (2019), we
construct silver training data using machine translation. In particular, we
translate summaries in target languages using multiple engines, including
Google translate 222https://translate.google.com, NiuTrans
333https://niutrans.com and LanMT 444https://www.dtranx.com. In addition, to
provide resources for pipeline methods, we translate conversation texts using
the same methods.

###  3.4 Evaluation

The evaluation of the _ConvSumX Challenge_ considers both automatic and manual
evaluation metrics.

####  3.4.1 Automatic Evaluation

Following previous cross-lingual summarization work Zhu et al. (2019), we use
Rouge scores Lin (2004) for automatic evaluation. Rouge scores evaluate the
model performance by considering the overlap of $n$-grams in the system-
generated summary against the reference summary. Although recent works claim
that Rouge fails to measure important information regarding factual
consistency Zhang et al. (2020); Fabbri et al. (2021), we choose Rouge
because: (1) it directly reflects model’s ability of obtaining salient
information and; (2) it can be easily applied to multiple languages including
low-resource language.

####  3.4.2 Manual Evaluation

As neural summarizers mostly contain factual errors that cannot be easily
detected by automatic metrics Zhu et al. (2019); Fabbri et al. (2021) and
translated words can be various Freitag et al. (2021), automatic evaluation
such as Rouge can be less accurate. Thus, our evaluation highly relies on
manual evaluation. Given that the _ConvSumX_ integrates conversation
summarization and machine translation, we adopt multiple human evaluation
metrics from both tasks to better measure model performance.

In particular, standard summarization metrics include: Fluency, Consistency,
Relevance and Coherence Kryscinski et al. (2019); standard machine translation
metrics include: Omission, Untranslation, Mistranslation, Addition and
Terminology Mariana (2014). However, except for Fluency, summarization metrics
evaluate generated summaries from the perspective of input documents in the
same language while machine translation metrics evaluate translation from the
perspective of source sentences (the English summary in our case). There can
be an evaluation inconsistency between these two tasks. In addition, there is
overlap between these two groups of metrics. For example, a mistranslated
summary can be regarded as containing consistency errors.

To unify the aforementioned evaluation metrics and obtain fine-grained
evaluations, we propose to evaluate system-generated summaries from following
aspects against source conversation texts.

Fluency evaluates the quality of individual sentence, including the grammar,
word order, etc.

Coherence evaluates the collective quality of all generated sentences.

Relevance evaluates the importance of information in the generated summary.

Consistency evaluates factual alignment of the generated summaries against the
source conversation, including information that is not presented in the
conversation, wrong causal relation, etc.

Terminology evaluates the use of language. For example, the generated word can
be a right translation but is improper in certain domains (_e.g.,_ academic
meeting).

Untranslation judges whether the generated summaries contain untranslated
words.

Overall score measures the overall quality for each summary.

For each metric above, we randomly extract 10% generated summaries and ask
annotators to give scores from $1$ to $5$. The higher, the better.

##  4 Related Work

Zhu et al. (2019) propose the first large scale cross-lingual news
summarization dataset, facilitating the study in this filed using neural
network models. Bai et al. (2021) construct an English-to-German news
summarization dataset using the automatic method of Zhu et al. (2019). Perez-
Beltrachini and Lapata (2021) construct a cross-lingual dataset based on
Wikipedia, focusing on European languages. In particular, Perez-Beltrachini
and Lapata (2021) use the lead paragraph in other languages aligned by
Wikipedia interlanguage links, and the original document to construct
$\langle$document-summary$\rangle$ pairs. Similarly, Ladhak et al. (2020)
construct the WikiLingua dataset based on multi-lingual WikiHow.

Very recently, Wang et al. (2022) and Feng et al. (2022) construct cross-
lingual dialogue summarization datasets. In particular, Wang et al. (2022)
manually translate summaries from SAMSum Gliwa et al. (2019), an online
written dialogue summarization dataset, and 40k data in MediaSum Zhu et al.
(2021) into German and Chinese. Feng et al. (2022) construct MSAMSum by
automatically translating SAMSum into Chinese, French and Russian. Compared
with them, our work focuses on spoken conversation in multiple scenrios, and
cover low-resource language (Ukrainian). In addition, we also focus on query-
based meeting scenarios, which can be more useful in real-world applications.

##  5 Conclusion

We propose the _ConvSumX Challenge_ to address the task of cross-lingual
conversation summarization, with the hope that _ConvSumX_ can encourage
researchers to investigate various methods for conversation summarization
beyond the English, in particular for low and mid-resource languages.

##  6 Acknowledgement

Yue Zhang is the corresponding author. We thank Yang Liu, Da Yin and Jianyu
Wang for insightful discussion and proofreading. Our sincere appreciation goes
to professional translators from Sichuan Lan-bridge Information Technology.

##  7 Ethical Consideration

###  7.1 Copyright and License of Datasets

The _ConvSumX Challenge_ uses cross-lingual $\langle$conversation-
summary$\rangle$ pairs that are annotated on the top of two English
conversation summarization datasets, namely DialogSum and QMSum, to evaluate
models. Both DialogSum and QMSum are free for academic use with the MIT
license, which contains no limitation to use, modification or distribution. We
will also make our annotated data available for the academia.

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# Prospects For Precision Measurements with Reactor Antineutrinos at Daya Bay

Daya Bay Collaboration

(September 27, 2013)

###### Abstract

In 2012 the Daya Bay experiment made an unambiguous observation of reactor
antineutrino disappearance over kilometer-long baselines and determined that
the neutrino mixing angle $\theta_{13}$ is non-zero. The measurements of Daya
Bay have provided the most precise determination of $\theta_{13}$ to date.
This whitepaper outlines the prospects for precision studies of reactor
antineutrinos at Daya Bay in the coming years. This includes precision
measurements of sin22$\theta_{13}$ and $\Delta m^{2}_{ee}$ to $<$3%, high-
statistics measurement of reactor flux and spectrum, and non-standard physics
searches.

##  I Introduction

Since the discovery of a non-zero value for $\theta_{13}$ DYB1 , Daya Bay has
provided the most precise measurement of sin22$\theta_{13}$ of $0.089\pm
0.010\pm 0.005$ with 139 days of 6 antineutrino detector (AD) data DYB2  in
June 2012. Recently, with 217 days of 6-AD data, we have obtained the most
precise measurement of sin22$\theta_{13}=0.090^{+0.008}_{-0.009}$ and the
first measurement of $\Delta m^{2}_{ee}=(2.59^{+0.19}_{-0.20})\times
10^{-3}~{}eV^{2}$.

Two more ADs were installed and have been in operation since October 19, 2012.
We present the expected precision in sin22$\theta_{13}$ and $\Delta
m^{2}_{ee}$ as a function of time in Sections II and III respectively. The
prospects for precise measurement of the reactor antineutrino flux and
spectrum are summarized in Section IV.

##  II Precision Measurement of sin22$\theta_{13}$

Daya Bay’s currently published measurements of sin22$\theta_{13}$ are still
statistics limited. With the data collected with 8 ADs as of October 2013 the
statistical error will reach the current systematic uncertainty of 0.005 DYB2
. We have assessed improvements in the systematic uncertainty based on
calibration data and increased precision that is statistical in nature (such
as the uncertainties in estimating the 9Li/8He and fast neutron backgrounds).
Based on the projected systematic uncertainty we estimate that the total
uncertainty in sin22$\theta_{13}$ can be reduced from 0.008 to 0.006 in one
year and 0.003 after 3 to 4 years as shown in Fig. 1.

Figure 1: Projected total uncertainty in sin22$\theta_{13}$ for the Daya Bay
experiment. The current systematic uncertainty of 0.005 is used at the
beginning of the 8AD run. Figure 2: Significance with which CP violation can
be observed, by NOvA+T2K (left) and NOvA+T2K+LBNE (right), as a function of
the value of $\delta_{CP}$. Observation of CP violation is equivalent to
measuring $\delta_{CP}\neq 0,\pi$. The significance is calculated by
minimizing over normal and inverted hierarchies, as the hierarchy is assumed
to be unknown. The impact of $\theta_{13}$ precision is shown.

Daya Bay’s short-baseline antineutrino disappearance measurement is the most
precise reactor measurement of sin22$\theta_{13}$ and will likely remain the
most precise measurement of this fundamental parameter for the foreseeable
future. Precise measurement of this quantity may shed light on symmetries
between quarks and leptons at a fundamental level, and test unitarity of the
neutrino mixing matrix. Comparison of future long baseline accelerator
measurements of $\theta_{13}$ to Daya Bay will allow precision tests of the
neutrino-standard model interpretation of neutrino oscillations with
sensitivity to non-standard neutrino interactions and sterile neutrino
scenarios. The expected improvement in sin22$\theta_{13}$ by Daya Bay will
make $\theta_{13}$ the most precisely measured neutrino mixing angle and
extend the CP reach (shown in Fig. 2) and $\theta_{23}$ octant determination
of long baseline accelerator experiments.

##  III Precision Measurement of $\Delta m^{2}_{ee}$

Our discovery of a large value for $\theta_{13}$ enables a measurement of the
effective mass splitting $\Delta m^{2}_{ee}$ with a precision comparable to
the MINOS measurement of $\Delta m^{2}_{\mu\mu}$. The disappearance of
electron antineutrinos over km-long baselines observed at Daya Bay is a
combination of antineutrino oscillations with mass splittings $\Delta
m^{2}_{31}$ and $\Delta m^{2}_{32}$.

Complementary to $\Delta m^{2}_{\mu\mu}$ determined by accelerator-based
experiments, Daya Bay will provide a precise measurement of $\Delta
m^{2}_{ee}$, as shown in Fig. 3, better than 3% after five years of data
collection with the 8-AD configuration. In addition to the neutrino mixing
angles and the Dirac CP-violating phase, the mass-squared differences are
crucial for understanding the nature of neutrinos. First, independent
determination of the three mass-squared differences will validate the sum rule
of neutrino mixing, $\Delta m^{2}_{21}+\Delta m^{2}_{32}+\Delta m^{2}_{13}=0$.
Any deviation from this sum rule could signal the existence of neutrinos
beyond three generations. In addition, comparison of $\Delta m^{2}_{ee}$ and
$\Delta m^{2}_{\mu\mu}$ will provide direct constraint to the neutrino mass
hierarchy (sign of $\Delta m^{2}_{31}$).

Figure 3: Expected Daya Bay uncertainty in $\Delta m^{2}_{ee}$ as a function
of running time with 8 ADs. The current systematic uncertainty is used at the
beginning of the run. The present error of the 6-AD data set is $0.2\times
10^{-3}~{}eV^{2}$.

Since the fundamental parameters $\Delta m^{2}_{32}$ and $\Delta m^{2}_{31}$
are directly related to the effective mass splitting $\Delta m^{2}_{ee}$ in
the L/E region for Daya Bay, the impact of the Daya Bay measurements in both
sin22$\theta_{13}$ and $\Delta m^{2}_{ee}$ with three years of 8-AD data is
illustrated in Fig. 4. Only statistical uncertainties are shown. The relative
systematic uncertainty, which is relevant for the oscillation analysis, is
still smaller than the expected statistical uncertainty of the far site. The
overall normalization is not constrained in the fit, and the best-fit
normalization is used in Fig. 4.

Figure 4: Expected L/E distribution of Daya Bay with 8 ADs in three
experimental halls, EH1, EH2, and EH3 after three years of running. Only
statistical uncertainties are shown.

##  IV Precision Measurement of the Reactor Antineutrino Flux and Spectrum

Daya Bay collects reactor antineutrino data at a tremendous rate which enables
a precision measurement of the reactor antineutrino spectra in the four near
site detectors. For the data sets published to date based on six detectors,
Daya Bay has collected $>$300k antineutrino events. Two additional detectors,
one at the Ling Ao near site and one at the far site, have been operational
since October 2012. The Daya Bay experimental configuration allows spectral
and flux measurements as close as 360m from the reactor cores at the Daya Bay
site and 480m at the Ling Ao site. At these detector locations the measured
reactor antineutrino flux from the nearest reactor cores remains largely non-
oscillated in the standard 3-neutrino oscillation framework. The contribution
from the more distant reactors at $\sim$900m is approximately 20% (7.5%) of
the total event rate at Daya Bay (Ling Ao). The oscillation effects of
antineutrinos from the far reactor and resulting distortions in the measured
antineutrino spectrum can be corrected for in any spectral analysis assuming a
3-neutrino framework. With this unique configuration of multiple, functionally
identical antineutrino detectors at various baselines Daya Bay will be able to
report the following measurements and physics analyses:

###  IV.1 High-precision measurement of the reactor antineutrino spectrum

Using data from the near and far sites Daya Bay will make a measurement of the
reactor antineutrino spectrum with high precision. Amongst all running reactor
experiments Daya Bay will collect the largest sample of antineutrino events
and achieve a $<$1% statistical uncertainty in a 2-year run over a large range
of energies at the near sites.

###  IV.2 Test of the reactor antineutrino spectrum vs predictions and search
for new interactions

Using known reactor operation data such as thermal power output and fission
fraction evolution from reactor core simulations Daya Bay can predict the
expected non-oscillated energy spectrum of reactor antineutrinos emitted from
each reactor. The Daya Bay collaboration works closely with the reactor
company and relevant operation data are provided to the collaboration on a
regular basis. A precise comparison of the predicted reactor antineutrino
spectrum with the spectrum measured at Daya Bay will test our understanding
and calculations of antineutrino emitted from reactors. This is particularly
relevant in the context of recent discussions of our understanding of reactor
flux calculations. Discrepancies in the spectral shape may point to (a)
missing nuclear physics in the reactor spectrum predictions or (b) new physics
beyond the 3-neutrino framework including non-standard interaction (NSI)
effects, and sterile neutrinos. This spectral shape test is independent of the
absolute flux normalization and the uncertainties in the predicted total
antineutrino rate at Daya Bay. Due to the high statistics of the Daya Bay
measurement, the statistical uncertainty in the 2011-2012 Daya Bay data set is
already below the flux conversion uncertainty on the spectrum. Daya Bay’s
measurement of the reactor antineutrino spectrum will be ultimately limited by
our understanding of energy scale uncertainties and detector effects. Detector
studies and simulations are ongoing to improve the energy response model of
the Daya Bay detectors.

###  IV.3 Absolute measurement of the reactor flux

In addition to a measurement of the spectral shape, Daya Bay will measure the
absolute reactor flux from the six Daya Bay and Ling Ao reactor cores. The
absolute flux measurement tests our understanding of reactor flux models
within the theoretical uncertainties of the predictions and the experimental
uncertainties associated with the absolute detection efficiencies of the Daya
Bay detectors. An absolute flux measurement in the antineutrino detectors at
Daya Bay will provide unique data points at the baselines of the Daya Bay
experiment (360-2,000 m) and will further our understanding if there is a
deficit in the measured reactor neutrino flux at short baselines, also known
as the “reactor anomaly”. An analysis of past reactor experiments compared
with predictions has revealed a discrepancy of about 5.7% in the absolute
antineutrino flux. While Daya Bay has demonstrated superb relative detector
uncertainties, an absolute measurement will be systematics limited. A
statistical precision of 0.1% will be achievable in the Daya Bay data set.
Improvements in the analysis may eventually reduce absolute detection
uncertainties to $\sim$1%. An absolute reactor flux measurement can test the
theoretical flux prediction with the uncertainty currently estimated at about
2.7%. We can compare the Daya Bay’s measurements to previous reactor flux
measurements, for example, the absolute Bugey-4 measurement with an
uncertainty of 1.4%. Daya Bay’s measurement of the absolute flux and reactor
antineutrino spectrum will provide important input to our understanding of the
“reactor anomaly”.

###  IV.4 Study of the time-evolution of the reactor antineutrino flux

The large reactor antineutrino event rate measured and the unique combination
of baselines and reactor cores at Daya Bay allows a detailed study of the time
variation of the reactor antineutrino flux. This contains information on the
operation of the reactors as well as the evolution and isotopic composition of
the core’s fuel. Correlating the measured antineutrino flux with reactor
operations is of interest to reactor monitoring, the safeguard community, and
applied neutrino science. With six reactors and 4 near-site detectors Daya Bay
will provide the largest data set on reactor flux variations as a function of
time.

## References

  * (1) F. P. An et al., Phys. Rev. Lett. 108, 171803 (2012).
  * (2) F. P. An et al., Chin. Phys. C37, 011001 (2013).


# Covariance Steering for Uncertain Contact-rich Systems

Yuki Shirai†, Devesh K. Jha‡, and Arvind U. Raghunathan‡ † Yuki Shirai is with
the Department of Mechanical and Aerospace Engineering, University of
California, Los Angeles, CA, USA 90095<EMAIL_ADDRESS>K. Jha
and Arvind U. Raghunathan are with Mitsubishi Electric Research Laboratories
(MERL), Cambridge, MA, USA 02139<EMAIL_ADDRESS>
###### Abstract

Planning and control for uncertain contact systems is challenging as it is not
clear how to propagate uncertainty for planning. Contact-rich tasks can be
modeled efficiently using complementarity constraints among other techniques.
In this paper, we present a stochastic optimization technique with chance
constraints for systems with stochastic complementarity constraints. We use a
particle filter-based approach to propagate moments for stochastic
complementarity system. To circumvent the issues of open-loop chance
constrained planning, we propose a contact-aware controller for covariance
steering of the complementarity system. Our optimization problem is formulated
as Non-Linear Programming (NLP) using bilevel optimization. We present an
important-particle algorithm for numerical efficiency for the underlying
control problem. We verify that our contact-aware closed-loop controller is
able to steer the covariance of the states under stochastic contact-rich
tasks.

##  I Introduction

Contacts lead to discontinuous dynamics and thus, planning through contacts
requires careful treatment of constraints arising due to these
discontinuities. Complementarity constraints offer an efficient way of
modeling contact systems. However, uncertainty in contact systems could lead
to stochastic complementarity systems [1]. Even though complementarity systems
are well studied, stochastic complementarity systems are not well understood.
The state and complementarity variables are implicitly related via the
complementarity constraints – uncertainty in one leads to stochastic evolution
of other. This makes uncertainty propagation challenging. Furthermore,
multiplicity of solutions to the complementarity variables also makes it
difficult to characterize the stochastic evolution. In this paper, we present
an approximate treatment of stochastic complementarity systems using
particles. We present the design and evaluation of a contact-aware stochastic
controller for covariance control of the underlying uncertain system. An
important-particle algorithm is presented for an efficient solution to the
resulting stochastic optimization problem.

Chance-constrained optimization (CCO) has been extensively studied in the
control of uncertain systems [2, 3, 4, 5, 6, 7]. It allows us to plan using
the uncertainty in the model by propagating the uncertainty which can be then
used to design a controller for desired performance constraints of the system.
However, in practice, the CCO techniques, based on the analytical form of
chance constraints, impose restrictive assumptions of Gaussian uncertainty and
linear constraints. Further, state uncertainty increases with time and thus
finding a controller for satisfying tighter state constraints could be
infeasible over a long planning horizon. This is often the case in control of
nonlinear systems with large uncertainty. This problem is aggravated for
contact-rich systems due to the presence of discontinuities in system
dynamics.

To circumvent these challenges, we consider particle-based method for
uncertainty propagation and explicit covariance control of our contact-rich
system during optimization.

Contributions.

  1. 1.

We present a novel formulation of covariance steering for complementarity
systems using feedforward and feedback controller design.

  2. 2.

An important-particle algorithm is proposed for numerical efficiency and we
evaluate the proposed method on several examples.

While our motivation is to design robust feedback controllers for manipulation
[8, 9], the full problem is out of the scope of the current formulation. Thus,
in this current paper, we limit the scope to linear complementarity systems
with uncertainty.

##  II Related Work

Our proposed stochastic optimization problem is mainly related to three major
areas of work. The first major area is optimization with complementarity
constraints. This topic has been well studied in optimization and robotics
literature [10, 11, 12, 13]. This approach has been shown to work well for
generating trajectories for manipulation and locomotion problems. However, it
cannot be trivially extended to stochastic complementarity systems to
introduce robustness. More recently, contact-aware feedback controllers for
contact-rich systems have been proposed [14] for linear complementarity
systems. However, it cannot also be extended to consider stochastic
complementarity constraints to provide stochastic guarantees.

Using stochastic complementarity constraints for planning robust manipulation
is not so well understood in literature. Some of the recent work could be
found in [15, 1]. However, the problem with these approaches is that the
uncertainty needs to be very small otherwise the optimization might be
infeasible. Consequently, these approaches could fail to provide robust plans
for uncertain contact systems. Furthermore, uncertainty propagation for
stochastic complementarity systems is not properly modeled in these
approaches. One of the reasons is the implicit relationship between contact
and state variables in complementarity constraints. As a consequence, most of
the known approaches (e.g., extended Kalman filter [16], unscented Kalman
filter [17], moment-based [18, 19]) for uncertainty propagation can not be
used for stochastic complementarity systems.

Since open-loop CCO would lead to quite conservative solutions to satisfy
chance constraints, covariance steering methods have gained attention to deal
with long-horizon planning for uncertain systems [20, 21, 22]. Covariance
steering methods are able to design feedforward and feedback gains
simultaneously to satisfy chance constraints. However, these cannot be
directly applied to contact-rich systems since they assume (in general) linear
dynamics with Gaussian additive noises.

In this paper, we present an approach for planning in stochastic contact-rich
systems which formulates a robust controller design by considering covariance
steering during planning. To understand uncertainty evolution in stochastic
complementarity systems, we use particle-based control formulation [4, 23, 24,
25] to get approximate uncertainty propagation. To the best of our knowledge,
this is the first time that we have shown covariance steering with chance
constraints for complementarity systems.

##  III Problem Formulation

In this section, we describe preliminaries of the method proposed in the
current work.

###  III-A Stochastic Discrete-time Linear Complementarity Systems

In this work, we consider the Stochastic Discrete-time Linear Complementarity
Systems (SDLCS):

$\displaystyle x_{k+1}=$
$\displaystyle{A}_{k}(\xi)x_{k}+B_{k}u_{k}+{C}_{k}(\xi)\lambda_{k+1}+{g}_{k}(\xi)$
$\displaystyle+w_{k}(\xi)$ (1a) $\displaystyle 0\leq\lambda_{k+1}\perp$
$\displaystyle{D}_{k}(\xi)x_{k}+E_{k}u_{k}+{F}_{k}(\xi)\lambda_{k+1}+{h}_{k}(\xi)$
$\displaystyle+l_{k}(\xi)\geq 0$ (1b)

where $k$ is the time-step index, $x_{k}\in\mathbb{R}^{n_{x}}$ is the state,
$u_{k}\in\mathbb{R}^{n_{u}}$ is the control input, and
$\lambda_{k}\in\mathbb{R}^{n_{c}}$ is the algebraic variable (e.g., contact
forces). We define
$x=[x_{1},\ldots,x_{T}],u=[u_{0},\ldots,u_{T-1}],\lambda=[\lambda_{1},\ldots,\lambda_{T}]$.
The parameter $\xi\thicksim\Xi$ is the uncertain parameter with distribution
$\Xi$. In addition, ${A}_{k}(\xi)\in\mathbb{R}^{n_{x}\times n_{x}}$,
$B_{k}\in\mathbb{R}^{n_{x}\times n_{u}}$,
${C}_{k}(\xi)\in\mathbb{R}^{n_{x}\times n_{c}}$,
${g}_{k}(\xi)\in\mathbb{R}^{n_{x}}$, ${D}_{k}(\xi)\in\mathbb{R}^{n_{c}\times
n_{x}}$, $E_{k}\in\mathbb{R}^{n_{c}\times n_{u}}$,
${F}_{k}(\xi)\in\mathbb{R}^{n_{c}\times n_{c}}$, and
${h}_{k}(\xi)\in\mathbb{R}^{n_{c}}$ are all dependent on the uncertain
parameter $\xi$. For simplicity, we abbreviate $\xi$ from these matrices for
the discussion in the following sections. The notation $0\leq a\perp b\geq 0$
denotes the complementarity constraints $a\geq 0,b\geq 0,ab=0$. The initial
state of the system $x_{0}(\xi)$ is also assumed to be uncertain.
$\left\|x\right\|_{Q}^{2}$ means a quadratic term with a weighting matrix $Q$.

In the following, we make the assumption that $F_{k}(\xi)$ is a P-matrix [26]
for all $k$ and $\xi$. Under this assumption, there is an unique solution
$\lambda_{k+1}$ to (1b) for each $\xi$ and any $u_{k},x_{k}$. From this it is
easy to infer that there exists an unique trajectory $x$ and $\lambda$ for any
realization of uncertainty $\xi\thicksim\Xi$ and controls $u$ from every
initial condition $x_{0}(\xi)$. In other words, we can define functions
$\mathbf{x}:\Xi\times\mathbb{R}^{n_{u}(T-1)}\rightarrow\mathbb{R}^{n_{x}T}$
and $\boldsymbol{\lambda}:\Xi\times\mathbb{R}^{n_{u}T}$ that provides the
unique trajectory given a realization of uncertainty, and the controls
trajectory. Note that we do not show explicit dependence on initial condition
due to the dependence of $x_{0}$ on the uncertain parameter $\xi$.

###  III-B Stochastic Control for Contact-Rich Systems

In this work, we aim at finding a robust controller that satisfies chance
constraints over SDLCS. To realize this, the following optimization problem
can be formulated:

$\displaystyle\min_{u}$
$\displaystyle\;\sum_{k=1}^{T}\left\|\mathbb{E}_{\xi\thicksim\Xi}\left[\mathbf{x}_{k}(\xi,u)\right]-x_{d}\right\|_{Q}^{2}+\sum\limits_{k=0}^{T-1}\left\|u_{k}\right\|_{R}^{2}$
(2a) s.t. $\displaystyle\;u_{k}\in\mathcal{U}$ (2b)
$\displaystyle\;\text{Pr}_{\xi\thicksim\Xi}\left(\mathbf{x}(\xi,u)\in\mathcal{X}\right)\geq\Delta$
(2c)

where $Q=Q^{\top}$ is positive semidefinite, $R=R^{\top}$ is positive
definite, $\mathcal{U}$ is a convex polytope consisting of a finite number of
linear inequality constraints. $x_{d}$ is the target state at $t=T$. The set
$\mathcal{X}$ represents a convex safe region where the entire state
trajectory has to lie in. We assume that
$\mathcal{X}=\\{x\in\mathbb{R}^{n_{x}T}\,|\,g_{i}(x)\leq
0\,\forall\,i=1,\ldots,n_{g}\\}$. Pr denotes the probability of an event and
$\Delta$ is the user-defined minimum safety probability, where the probability
of satisfying constraints is at least greater than $\Delta$.

We propose to obtain an approximate solution to (2) using the Sample Average
Approximation (SAA) introduced in [23, 24]. We explain more details in Sec IV.

##  IV Covariance Steering for Contact-Rich Systems

This section presents our proposed framework of stochastic optimal control for
contact-rich systems. Our framework approximates the distribution of the state
and algebraic variables using particles. Under the assumption that $\bar{F}$
is P-matrix, our method can capture stochastic evolution of SDLCS such that we
can formally guarantee the violation of states and design a closed-loop
controller for SDLCS (i.e., covariance steering for SDLCS).

We first present our open- and closed-loop controller formulation for SDLCS
using particles and then present a computationally beneficial approach based
on the active-point method [27] to accelerate the resulting optimization.

###  IV-A Particle-based Control for Contact-Rich Systems

We propose to solve (2) approximately using SAA by sampling the uncertainty.
In particular, we obtain $N$ realizations of the uncertainty
$\Xi^{N}=\\{\xi^{1},\ldots,\xi^{N}\\}$ by sampling the distribution $\Xi$. In
other words, we approximate the distribution $\Xi$ using a finite-dimensional
distribution $\Xi^{N}$ which follows an uniform distribution on the samples.
Accordingly, the SAA for (2) is given as

$\displaystyle\min_{u}$
$\displaystyle\;\sum_{k=1}^{T}\left\|\mathbb{E}_{\xi\thicksim\Xi^{N}}\left[\mathbf{x}_{k}(\xi,u)\right]-x_{d}\right\|_{Q}^{2}+\sum\limits_{k=0}^{T-1}\left\|u_{k}\right\|_{R}^{2}$
(3a) s.t. $\displaystyle\;u_{k}\in\mathcal{U}$ (3b)
$\displaystyle\;\text{Pr}_{\xi\thicksim\Xi^{N}}\left(\mathbf{x}(\xi,u)\in\mathcal{X}\right)\geq\Delta.$
(3c)

Note that the distribution $\Xi$ has been replaced with the finite-dimensional
$\Xi^{N}$ in the above to simplify the computation of the expectation in the
objective and chance constraint. However, there still remains the implicit
function $\mathbf{x}(\xi,u)$ which requires us to simulate the SDLCS for every
realization of $\xi\in\Xi^{N}$. We opt to remove this difficulty by replacing
the implicit functions with the corresponding trajectories $x^{i},\lambda^{i}$
for each $\xi^{i}\in\Xi^{N}$.

Our proposed computational formulation using $N$ particles is given by:

$\displaystyle\min_{x^{i},u,\lambda^{i}}$
$\displaystyle\;\sum_{k=1}^{T}\left\|\frac{1}{N}\sum\limits_{i=1}^{N}x^{i}_{k}-x_{d}\right\|_{Q}^{2}+\sum_{k=0}^{T-1}\left\|u_{k}\right\|_{R}^{2}$
(4a) s.t.
$\displaystyle\;x_{k+1}^{i}={A}^{i}_{k}x_{k}^{i}+B_{k}u_{k}+{C}^{i}_{k}\lambda_{k+1}^{i}+{g}^{i}_{k}+w_{k}^{i}$
(4b)
$\displaystyle\;0\leq\lambda_{k+1}^{i}\perp{D}^{i}_{k}x_{k}^{i}+E_{k}u_{k}+{F}^{i}_{k}\lambda_{k+1}^{i}$
$\displaystyle+{h}^{i}_{k}+l_{k}^{i}\geq 0$ (4c)
$\displaystyle\;x_{0}^{i}=x_{0}(\xi^{i})$ (4d)
$\displaystyle\;u_{k}\in\mathcal{U}$ (4e)
$\displaystyle\;\frac{1}{N}\sum_{i=1}^{N}\mathbb{I}\left(x^{i}\in\mathcal{X}\right)\geq\Delta$
(4f)

where $\mathbb{I}(\cdot)$ is an indicator function returning $1$ when the
conditions in the operand are satisfied and $0$ otherwise. Note that
$x^{i},\lambda^{i}$ represent the state and algebraic variable trajectory,
respectively, propagated from a particular set of particles
$x_{0}^{i},\theta_{k}^{i}$ where
$\theta_{k}^{i}=[{A}^{i}_{k},{C}^{i}_{k},{g}^{i}_{k},{D}^{i}_{k},{F}^{i}_{k},{h}^{i}_{k},w_{k}^{i},v_{k}^{i}]$.
Using $N$ trajectories obtained from $N$ particles, we approximate mean of
random variables as
$\mathbb{E}_{\xi\thicksim\Xi}[\mathbf{x}_{k}(\xi,u)]\approx\frac{1}{N}\sum_{i=1}^{N}x_{k}^{i},\mathbb{E}_{\xi\in\Xi}[\boldsymbol{{\lambda}}_{k}(\xi,u)]\approx\frac{1}{N}\sum_{i=1}^{N}\lambda_{k}^{i}$.
In (4), we approximate (2a) using the mean variable as shown in (4a). Chance
constraints (2c) can be also approximated as (4f) using $N$ realization
trajectories, which can be formulated as integer constraints (see [4]).

In this work, we consider the following controllers:

$\displaystyle\textbf{feedforward}:u_{k}=v_{k}$ (5a)
$\displaystyle\textbf{feedback}:u_{k}=v_{k}+K_{k}(x_{k}-\bar{x}_{k})+L_{k}(\lambda_{k}-\bar{\lambda}_{k})$
(5b)

where $K_{k},L_{k}$ are feedback gains to control covariance. For brevity, we
use
$\bar{x}_{k}=\frac{1}{N}\sum_{i=1}^{N}x_{k}^{i},\bar{\lambda}_{k}=\frac{1}{N}\sum_{i=1}^{N}\lambda_{k}^{i}$.
We emphasize that controlling both states and contact variables is critical
for contact-rich systems and thus we also introduce
$L_{k}(\lambda_{k}-\bar{\lambda}_{k})$ to (5b) to stabilize the system. Here,
we focus on discussing feedback controller (5b) for (4). The optimization
formulation for covariance steering of SDLCS using particles would be:

$\displaystyle\min_{x^{i},v,K,L,\lambda^{i}}\sum_{k=1}^{T}||\bar{x}_{k}-x_{d}||_{Q}^{2}+\sum_{k=0}^{T-1}\left\|u_{k}\right\|_{R}^{2}$
(6a) s. t. $\displaystyle
x_{k+1}^{i}=({A}^{i}_{k}+B_{k}K_{k})x_{k}^{i}+B_{k}v_{k}$
$\displaystyle+({C}^{i}_{k}+B_{k}L_{k})\lambda_{k+1}^{i}+\bar{g}^{i}_{k}$
$\displaystyle-B_{k}K_{k}\bar{x}_{k}-B_{k}L_{k}\bar{\lambda}_{k+1}+w_{k}^{i}\
$ (6b) $\displaystyle
0\leq\lambda_{k+1}^{i}\perp({D}^{i}_{k}+E_{k}K_{k})x_{k}^{i}$
$\displaystyle+E_{k}v_{k}+({F}^{i}_{k}+E_{k}L_{k})\lambda_{k+1}^{i}$
$\displaystyle+{h}^{i}_{k}-E_{k}K_{k}\bar{x}_{k}-E_{k}L_{k}\bar{\lambda}_{k+1}+l_{k}^{i}\geq
0$ (6c)
$\displaystyle(\ref{x0_condn}),(\ref{bounds_variables1}),(\ref{chance_const1})$
(6d)

To solve (6), we need to take care of, (6b), (6c) and (4f). One method is
mixed-integer programming. It is possible that binary variables can be used to
deal with integer constraints (4f) using Big-M formulation. Also, bilinear
terms in (6b) and (6c) can be approximated using McCormick envelopes, leading
to additional binary variables. As a result, a number of binary variables are
introduced and we observed that it is almost impossible to obtain a single
feasible solution. Instead, in this work, we use NLP which can solve (6b) as
nonlinear constraints and (6c) as complementarity constraints. We describe how
we solve (4f) using NLP through complementarity constraints in Sec IV-B.

###  IV-B Bilevel Optimization for Particle-based Control

To solve (6) using NLP, we need to solve integer constraints (4f) in NLP
fashion. To achieve this, we propose the following bilevel optimization
problem.

$\displaystyle\min_{x^{i},v,K,L,\lambda^{i},t^{i},z^{*}}\sum_{k=1}^{T}\left\|\bar{x}_{k}-x_{d}\right\|_{Q}^{2}+\sum_{k=0}^{T-1}\left\|u_{k}\right\|_{R}^{2}$
(7a) $\displaystyle\text{s. t.
}(\ref{slcp1_p111_cov}),(\ref{compl_milp}),(\ref{bounds_variables1})$ (7b)
$\displaystyle\forall j=1,\ldots,n_{g},\;g_{j}(x)\leq t^{i},$ (7c)
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}z^{i,*}\geq\Delta$ (7d)
$\displaystyle\forall
i=1,\ldots,N,\;z^{i,*}=\operatorname*{arg\,min}_{z^{i}}t^{i}z^{i}|0\leq
z^{i}\leq 1$ (7e)

We introduce time-invariant parameter $t^{i}\in\mathbb{R}^{1}$ for each set of
trajectory realization $i$. If $x^{i}\in\mathcal{X}$, $t^{i}\geq-\epsilon$
with $\epsilon\geq 0$. In contrast, if $x\not\in\mathcal{X}$, $t^{i}\geq 0$.
This condition is encoded in (7c). We have in total $N$ lower-level
optimization problems (7e), where each optimization is formulated as linear
programming. $z^{i}\in\mathbb{R}^{1}$ is the decision variable used in $i$ -th
lower-level optimization problem.

The purpose of (7e) is to count the number of trajectory realizations that are
inside $\mathcal{X}$. The optimal solution of (7e) can be as follows:

$z^{i}=\begin{cases}1,&t^{i}<0\\\ \left[0,1\right],&t^{i}=0\\\
0,&t^{i}>0\end{cases}$ (8)

If $t^{i}<0$, (7c) argues that $x^{i}\in\mathcal{X}$ and thus we count this
$i$-th trajectory propagated from $i$-th particles as one. If $t^{i}=0$, (7c)
argues $x^{i}\in\mathcal{X}$ ($x^{i}$ lies on the boundary of $\mathcal{X}$)
and thus we count this $i$-th trajectory propagated from $i$-th particles as
one. If $t^{i}>0$, then $x^{i}$ is not within $\mathcal{X}$, and thus we count
it as zero. Then (7d) considers the approximated chance constraints.

Since the upper-level optimization decision variable $t^{i}$ can be influenced
by other upper-level decision variables, we need to solve these two problems
simultaneously, leading to a bilevel optimization problem. Since the lower-
level optimization problems are formulated as $N$ linear programming problems,
we can efficiently solve the entire bilevel optimization problem using the
Karush-Kuhn-Tucker (KKT) condition as follows:

$\displaystyle\min_{x^{i},v,K,L,\lambda^{i},t^{i},z^{i,*},w^{i}_{+},w^{i}_{-}}(\ref{exp_cost11_bilevel})$
(9a) $\displaystyle\text{s. t.
}(\ref{slcp1_p111_bilevel}),(\ref{param_safe}),(\ref{approx_bilevel_chance})$
(9b) $\displaystyle\forall i=1,\ldots,N,\;0\leq z^{i,*}\leq
1,w^{i}_{+},w^{i}_{-}\geq 0$ (9c) $\displaystyle
w^{i}_{+}(z^{i,*}-1)=0,w^{i}_{-}(z^{i,*})=0,$ (9d) $\displaystyle
t^{i}+w^{i}_{+}-w^{i}_{-}=0$ (9e)

where $w^{i}_{+},w^{i}_{-}$ are Lagrange multipliers associated with
$z^{i}-1\leq 0$, $-z^{i}\leq 0$, respectively. In conclusion, we obtain a
single-level nonlinear programming problem with complementarity constraints,
which can be efficiently solved using an off-the-shelf solver such as IPOPT
[28].

###  IV-C Important-particle Method for Particle-based Control

One limitation of our method in Sec IV-B is that the computation can be
demanding with many particles to capture the evolution of uncertainty. In this
section, we present an approximate algorithm which samples important particles
which might be most informative for constraint violation. To decrease the
computational burden, we employ an important-particle method (see Alg. 1)
which starts from a relatively small number of particles and keeps adding
particles if the chance constraints are not satisfied due to the lack of the
accurate approximation of variables. Since we start from a small number of
particles, it is possible that our optimization could quickly find a feasible
solution which works over testing data set. However, in the case when the
problem is infeasible for some particles, we add the particles which
experience maximum constraint violation to our set. Thus, we call our proposed
method ”important-particle” method– the worst particles specify the boundary
of feasible sets.

The pseudocode of our important-particle method for covariance steering is
shown in Alg. 1. Param is the collection of parameters such as $Q,R$.
$\alpha,\beta$ represent the number of particles for training and testing the
controller, respectively. $\gamma$ is the number of initial particles our
method uses during its first iteration. $\eta$ is the number of particles our
methods adds to (9) for each iteration.

As shown in Alg. 1, our method keeps adding more particles unless either it
runs more than MAX-ITER or converges to user-defined $\Delta$ given threshold
$\Delta_{\text{th}}$. For each iteration, we run (9). If the obtained solution
is feasible, we do Monte Carlo simulation (MC simulation) over the training
data set with $\alpha$ particles and calculate the empirical safe probability
$\Delta_{\alpha}$. If this $\Delta_{\alpha}$ is close to or greater than
$\Delta$, we terminate the while loop and run the obtained controller over the
testing data set with $\beta$ particles. Otherwise, we choose the $\eta$ worst
particles based on how much they violate the chance constraints and add them
to $\theta$. If we obtain the infeasible solution or the ”restoration phase
failed” solution in IPOPT, we randomly choose the $\eta$ particles.

Algorithm 1
$\operatorname{ImportantParticle}(\text{Param},\alpha,\beta,\gamma,\eta)$

1: $j=0$, $\theta=\gamma$, $\Delta_{\alpha}=0$

2: while $j\leq\text{MAX-ITER}$ and
$(\Delta-\Delta_{\alpha})^{2}\geq\Delta_{\text{th}}$ and
$\Delta>\Delta_{\alpha}$; do

3: Run (9) with $N=\theta$

4: if The obtained solution from (9) is feasible then

5: Run MC simulation with $\alpha$ particles and calculate $\Delta_{\alpha}$.

6: Choose the $\eta$ worst particles that violate chance constraints.

7: else

8: Choose the random $\eta$ particles.

9: $\theta=\theta+\eta$

10: Run MC simulation with $\beta$ particles and calculate $\Delta_{\beta}$.

11: return
$x^{i,*},v^{*},K^{*},L^{*},\lambda^{i,*},t^{i,*},z^{i,*},w^{i}_{+},w^{i}_{-},\Delta_{\beta}$

##  V Results

In this section, we present numerical results for our proposed approach and
compare them against some baselines. In particular, we would like to highlight
and understand the following questions:

  1. 1.

Does uncertainty in complementarity constraints lead to uncertainty in state
trajectory?

  2. 2.

How does the proposed controller perform of variance of states for SDLCS?

We implement our method using IPOPT [28] with PYROBOCOP [11]. The optimization
problem is implemented on a computer with Intel i7-12700K processor. We set
$\alpha=250,\beta=1000$ for Alg. 1. For $\gamma$ and $\eta$ in Alg. 1, we use
the different values for different applications as shown in Table II and Table
III. When we run (9) alone without using Alg. 1, we use 1000 samples to
calculate the empirical probability of failure to evaluate the satisfaction of
chance constraints.

Here we explain how we simulate trajectories (i.e., perform MC simulation for
SDLCS, see [1] for more details). We propagate the dynamics by finding the
roots of the complementarity system with sampled parameters given the control
sequence obtained from optimization. We run each case for 1000 trials with
different sampled parameters to estimate the probability of failure. Note
that, unlike the continuous-domain dynamics, we cannot rollout the dynamics
for SDLCS with the given control sequences since we do not have the access to
$\lambda_{k+1}$.

###  V-A Uncertainty Propagation for SDLCS

Figure 1: (a): cartpole with softwalls. (b): acrobot with soft joints. Figure
2: Uncertainty propagation for cartpole system. Here only uncertainty arises
from stiffness parameters $k_{1},k_{2}$.  Figure 3: Simulated trajectories for
cartpole system using ERM-based controller. $\Delta=0.2$ and
$\Delta_{\text{test}}=0.083$. Red lines show boundaries specified in chance
constraints.

We show uncertainty evolution for SDLCS. We demonstrate this for a cartpole
system with softwalls (see [14] for more details). Here we consider both
$k_{1}$ and $k_{2}$ follows uniform distributions where upper bound of uniform
distribution for $k_{1}$ and $k_{2}$ is 14, 12, respectively, and the lower
bound is 5 for both $k_{1}$ and $k_{2}$. In this experiment, we do not run any
controller: we simply propagate SDLCS given uncertain parameters in order to
show how the SDLCS behaves.

Fig. 2 shows the evolution of uncertainty for the aforementioned system. At
$t=0$ s, there is no uncertainty for state $\theta_{t=0}$. However, because we
provide uncertainty with $k_{1}$ and $k_{2}$, $\lambda_{t=0.1}$ has
uncertainty. This is again because given realization of uncertain parameters,
complementarity constraints give a realization of $\lambda$ and $y$, resulting
in uncertainty in $\lambda$ and $y$. This stochastic $\lambda_{t=0.1}$ brings
uncertainty in $\theta_{t=0.1}$ based on (1). As shown in Fig. 2, both state
and complementarity variables are stochastic. This can not be captured in
approximations like Expected Residual Minimization (ERM) [15].

###  V-B Cartpole with Softwalls

We demonstrate our open- and closed-loop controllers for cartpole with
softwalls system. $x$ is the cart position and $\theta$ is the pole angle.
$u_{1}$ is the control and $\lambda_{1},\lambda_{2}$ are the reaction forces
at from the wall 1, 2, respectively. We have the following deterministic
physical parameters. $g=9.81$ is the gravitational acceleration,
$m_{p}=0.1,m_{c}=1.0$ are the mass of the pole, cart, respectively. $l=0.5$ is
the length of the pole and $d=0.15$ is the distance from the origin of the
coordinate to the walls. We assume that the uncertainty arises from the
${k_{1}},{k_{2}}$ and use the same distribution in Sec V-A. We set $dt=0.1$
for the explicit Euler integration and $T=6$.

Figure 4: Simulated trajectories for cartpole system using our open-loop
controller. $\Delta=0.2$ and $\Delta_{\text{test}}=0.190$ where $\Delta$ is
input of optimization and $\Delta_{\text{test}}$ is the empirically obtained
success rate from MC simulation. Red lines show boundaries specified in chance
constraints. Figure 5: Simulated trajectories for cartpole system using our
closed-loop controller. Top: $\Delta=0.6$ and $\Delta_{\text{test}}=0.510$,
bottom: $\Delta=0.2$ and $\Delta_{\text{test}}=0.188$, where $\Delta$ is input
of optimization and $\Delta_{\text{test}}$ is the empirically obtained success
rate from MC simulation. Red lines show boundaries specified in chance
constraints.

The results using ERM and our controller for the open-loop trajectory are
shown in Fig. 3, Fig. 4. We observed that the proposed open-loop controller
shows the better satisfaction of chance constraints compared to the ERM-based
method. This is because our method explicitly considers propagation of
uncertainty for SDLCS while the ERM-based method is unable to consider. Also,
we observe that the gap between the commanded $\Delta$ used in our
optimization and $\Delta_{\text{test}}$ obtained from MC simulation over
testing dataset is smaller the gap between the commanded $\Delta$ used in ERM
method and $\Delta_{\text{test}}$ obtained from MC simulation over testing
dataset. Again this is because our method could capture the evolution of
uncertainty for SDLCS. However, even our open-loop controller does not show
the much better performance than the ERM. To show the higher
$\Delta_{\text{test}}$, we need to input the higher $\Delta$ as an input of
optimization. It is quite difficult especially for long-horizon planning
problems since uncertainty keeps evolving, which can be observed from both
Fig. 3 and Fig. 4.

Next, we discuss the difference among our proposed contact-aware closed-loop,
the non-contact-aware (i.e., $L_{k}=0,\forall k$ in (5b)) closed loop, and the
open-loop controllers. We observed that in Table I, (9) for open-loop
controller with high $\Delta$ was unable to find feasible solutions but (9)
for closed-loop controller could find feasible solutions. Since the closed-
loop controller can change feedback gains to satisfy chance constraints, it
could find feasible solutions with high $\Delta$. Also, Table I shows that the
contact-aware closed-loop controller could find the feasible solution with
high $\Delta=0.8,0.7$ but the non-contact-aware controller (i.e.,
$L_{k}=0,\forall k$ in (5b)) could not. For SDLCS, introducing feedback to
both states and forces is important to realize the robust motion. The MC
simulation results using our contact-aware closed-loop controller are shown in
Fig. 5. In contrast to Fig. 3 and Fig. 4, the closed-loop controller could
bound the distribution of the states because it controls covariance.

We discuss computational results. Firstly, we observe that our important-
particle method converges and the gap between $\Delta_{\text{train}}$ and
$\Delta_{\text{test}}$ is small once it finishes its third time iteration. It
means that our important-particle method could successfully find feasible
trajectories with relative small number of particles. Secondly, in Table II
the important-particle method shows the higher $\Delta_{\text{train}}$ as the
number of particles used in optimization increases. The proposed important-
particle method shows better convergence (in total 208 s to have
$\Delta_{\text{train}}\geq 0.49$) than the naive method (620 s with 50
particles to have $\Delta_{\text{test}}\geq 0.49$) since our important-
particle method keeps choosing the worst-case particles which break chance
constraints.

TABLE I: Comparison of feasibility for cartpole system among open-, non-contact-aware closed, and contact-aware-closed controllers with different $\Delta$. $\circ$ and $\times$ show if optimization finds a feasible solution or not, respectively. $\Delta$ | $0.8$ | $0.7$ | $0.6$ | $0.4$ | $0.2$
---|---|---|---|---|---
Open-loop | $\times$ | $\times$ | $\times$ | $\circ$ | $\circ$
Non-contact-aware closed-loop | $\times$ | $\times$ | $\circ$ | $\circ$ | $\circ$
Contact-aware closed-loop | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
TABLE II: Comparison of safe probability and runtime for cartpole system
between important-particle method (top) with $\gamma=10,\eta=10$ and naive
method (bottom) with $\Delta=0.6$ for designing the closed-loop controller.
$T$ represents runtime for each iteration and $n_{p}$ is the number of
particles.

iter | $1$ | $2$ | $3$
---|---|---|---
$\Delta_{\text{train}}$ | 0.2708 | 0.09 | 0.592
$\Delta_{\text{test}}$ | N/A | N/A | 0.588
T [s] | 25 | 35 | 148
$n_{p}$ | 10 | 20 | 30

Case | 1 | 2 | 3 | 4
---|---|---|---|---
$\Delta_{\text{test}}$ | 0.277 | 0.376 | 0.451 | 0.499
T [s] | 25 | 26 | 55 | 620
$n_{p}$ | 10 | 20 | 30 | 50

###  V-C Acrobot with Soft Joints

Figure 6: Simulated trajectories for acrobot using our open- and closed-loop
controllers. Top: closed-loop controller with $\Delta=0.8$ and
$\Delta_{\text{test}}=0.771$, bottom: open-loop controller with $\Delta=0.4$
and $\Delta_{\text{test}}=0.366$. Red lines show boundaries specified in
chance constraints. The reader should note that open-loop controller solution
was infeasible for $\Delta=0.8$, and thus we show results for $\Delta=0.4$.
TABLE III: Comparison of safe probability and runtime for acrobot system
between important-particle method (top) with $\gamma=4,\eta=4$ and naive
method (bottom) with $\Delta=0.8$. $T$ represents runtime for each iteration
and $n_{p}$ is the number of particles.

iter | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---
$\Delta_{\text{train}}$ | 0.426 | 0.485 | 0.562 | 0.625 | 0.363 | 0.593 | 0.763
$\Delta_{\text{test}}$ | N/A | N/A | N/A | N/A | N/A | N/A | 0.771
$T$ [s] | 31 | 97 | 557 | 887 | 698 | 2450 | 779
$n_{p}$ | 4 | 8 | 12 | 16 | 20 | 24 | 28

Case | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
$\Delta_{\text{test}}$ | 0.009 | 0.103 | 0.159 | 0.541 | 0.670 | 0.553 | 0.539
$T$ [s] | 31 | 15 | 229 | 260 | 944 | 3993 | 901
$n_{p}$ | 4 | 8 | 12 | 16 | 20 | 24 | 28

We also demonstrate our controller for acrobot with soft joints system (see
[14] for more details). $\theta_{1}$ is the first joint angle and $\theta_{2}$
is the second joint angle. $u_{1}$ is the control at the second joint and
$\lambda_{1},\lambda_{2}$ are the reaction forces at from the wall 1, 2,
respectively. We have the following deterministic physical parameters.$g=9.81$
is the gravitational acceleration, $m_{1}=0.5,m_{2}=1.0$ are the mass of the
pole, cart, respectively. $l_{1}=0.5$ is the length of the rod from the first
to the second joint. $d=0.2$ is the angle limit of $\theta_{1}$. We consider
the stochastic physical parameters $k$ and $l_{2}$ where $k$ is the stiffness
of the walls and $l_{2}$ is the length of the second rod. We assume that $k$
follows uniform distribution where the upper bound and the lower bound of the
distribution is 1.6 and 0.6, respectively. We assume that $l_{2}$ follows a
truncated Gaussian distribution where we set the mean to 1.0, variance to
0.01, the upper bound of the interval is 1.3, and the lower bound of the
interval is 0.7, respectively. We set $dt=0.04$ for the explicit Euler
integration and $T=15$.

The open- and closed-loop trajectories are shown in Fig. 6. We observed that
both controller could satisfy chance constraints over the testing dataset and
the closed-loop controller shows the better performance. Table III shows that
the important-particle method shows the higher $\Delta_{\text{test}}=0.771$
than the naive method with the same number of particles used in optimization.

##  VI Discussion

Stochastic complementarity systems are not well understood in literature. This
paper presents a study of SDLCS to perform covariance steering using
particles. Under the assumption of uniqueness of trajectory ($\bar{F}$ is
P-matrix) for complementarity systems, the proposed method is able to compute
covariance controller for SDLCS. We presented an important-particle method to
alleviate computational complexity of the resulting optimization problem. It
is shown that our work could design open- and closed-loop controllers with
chance constraints by appropriately considering the evolution of uncertainty
for SDLCS.

In the future, we would like to study more general manipulation systems by
relaxing the assumption on uniqueness of trajectory for SDLCS. Another
limitation of this work is that the computation is still demanding. Thus, we
would like to employ distributed optimization techniques such as ADMM [29].

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