So here's the story playing all over the world Woman Survives Drinking Her Own Breast Milk – Really? I ask, are you serious? Now before you go crazy on me for being heartless, I am happy Susan O'Brien is OK, I'm glad she made it, it's great to know she is safely back with her family. The thing I object to is the story line.
Susan was on what was stated as a grueling 12.5 mile race through the forest, when at some point she took a wrong turn and became lost. Night set in, it was cold and rainy, she feared for her life. I know that cold and rain are devastating to someone unprepared for it. She just had on her running clothing, so was not ready for any kind of stay in the forest in those conditions. The first question I asked when thinking about her life being saved by her own breast milk, "how long was she stranded?" The answer, a stunning 24 hours! Certainly I would be concerned about her situation, cold, rain and wind, but for goodness sakes she is not going to starve to death in 24 hours. I have lots more fat reserves on me than she does being an athlete, but certainly she could go a week or two.
I don't want to take anything away from her, she tried to do all she could think of to stay alive, but all the press in the world is running the story that she was saved by her breast milk. I think she was saved by being in good physical condition, and by doing what she could to isolate herself from the effects of the elements, and by thinking positively about getting back to her family.
I guess I can't blame the press, the story does make a great headline. I read through all the articles on her that I could find, no one seemed to bother to consider that other factors probably saved her. Readers are left with the understanding that she would have totally run out of energy and died had Susan not as she says, "had a bit of my milk." Susan believes it helped her, and perhaps it did, mentally at least.
I think the stories that ran should at least have raised the possibility that other factors may have contributed to her survival, instead of leaving the false impression that the milk did it. We are not given much detail in what I could find, but she did say she kept covering herself in dirt (and debris I would suppose), most likely to get away from the wind.
Having followed survival stories for the better part of my life, I have heard of some amazing things. It seems to me that when I was younger, the press did this amazing thing called, "investigation" that seems to be lacking these days much of the time. It used to be a reporter would try to find a back story, find some additional detail, nowadays it's just retyping whatever is put out. A reader should be educated in reading a story, otherwise what is the purpose in even writing it?
A reader for example could at least be given the information that in Susan's predicament her most important thing to do for the night at least is to shelter herself the best she could to avoid hypothermia. Wet and cold are bad enough, but add a little wind and things turn deadly in a hurry. It could be mentioned perhaps some simple sheltering she could have done without any tools.


On the record! In preparation of the 2019-2020 school year, the Health Unit will be requesting parents, registering their children for kindergarten, to submit their child's immunization record and information to us directly. Soon you will be receiving posters that can go out to all families who are registering their child for kindergarten. This poster will provide families with information on how the can submit to us their child's immunization information. Please contact us if you have questions, 1-800-660-5853.
New Vision Screening! Spring 2019, a pilot vision screening program for SK students will be offered by trained Health Unit staff at selected schools. If your school is selected you will be contacted in February by letter. There will be 3 non-invasive tests which take roughly 10–15 minutes per child. While, vision screening is part of good visual health it does not replace an annual comprehensive eye exam by an eye doctor. A valid Health Card will cover the cost of an eye exam once a year for children aged 0–19. To learn more visit: https://healthunit.org/health-information/babies-children/visual-health/ or for more information call the Health Unit at 1-800-660-5853.
Parenting matters! Celebrate your family this month by spending some extra time together. Enjoy some winter activities outside – build a snowman, or maybe go sledding or skating. If the weather isn't cooperating, why not play a board game, or make a craft? It doesn't matter what you choose to do, as long as you are doing something you enjoy together. Strong relationships make family life more fun! For more ideas, visit www.triplep-parenting.ca. To register for a local Triple P Parenting session, visit http://healthunit.org/clinics-classes/triple-p-parenting/ or call locally 1-800-660-5853.


text:
William Bwalya Miko (born 1961) is a Zambian abstract painter and art lecturer. He is Head of Department in Fine Arts at the Zambian Open University.

Life
As a student at Evelyn Hone College in the late 1980s Miko helped found the Zambia National Visual Arts Council, serving on the first interim national executive committee. In 1999 he obtained a Masters in Fine Art from Middlesex University.
He is Senior Visual Arts Lecturer in the School of Media, Performing and Fine Arts at the Zambian Open University (ZAOU), which began teaching a Fine Arts degree course in 2010. At the first graduation of ZAOU fine arts graduates in 2014, Miro turned himself into a human canvas, wearing a white suit as students daubed and slashed paint over him in a performance titled 'Correcting the National Anomaly'.

Miko has presented conference papers on Zambian art in other African countries, as well as Europe and the United States.

References

1961 births
Living people
Zambian painters
Academic staff of Zambian Open University
Art educators

redpajama_set_name: RedPajamaWikipedia


text:
Q: How to put  text in a textarea code box I am in need of a way to put <textarea> (as text) inside a <textarea> and then close the textarea inside the text area with a </textarea> but doing so, closes the first textarea. It allows the code <textarea> inside the textarea though
<textarea width=100px>  <!--Text area to appear on screen-->

<title> Test </title>
<textarea placeholder="Hello, Enter your username:">
</textarea>  <!--This bit is supposed to appear inside the text area as text-->

</textarea>  <!--End of the textarea-->

I have tried using:
<script>
<!--
document.write(unescape("%3C/textarea%3E"));
//-->
</script>

instead of </textarea> but it doesnt work

A: Replace < with its equivalent HTML entity - <:
<textarea width=100px>
  <!--Text area to appear on screen-->
  <title>Test</title>
  <textarea placeholder="Hello, Enter your username:">
  </textarea>  <!--This bit is supposed to appear inside the text area as text-->

</textarea><!--End of the textarea-->

Demo. Do not confuse this with URL encoding (%nn), as these are different beasts.

A: You should escape special characters like < > & and " in html:
< = <
> = >
& = &
" =  "
This looks like:
<textarea placeholder="Hello, Enter your username:">
</textarea>

</textarea>  <!--End of the textarea-->

If you're using a serverside script/program to render html, almost all languages have escaping functions for html.

redpajama_set_name: RedPajamaStackExchange


text:
Advertisement - Click to support our sponsors.
R A I N B O W _ S P O R T S
Wallace will
write letter to
concerned group
University of Hawaii officials
have yet to make a formal
statement in the Confederate
flag controversy
By Pat Bigold
Star-Bulletin
There still has been no formal public response from the University of Hawaii to complaints by two civic groups about remarks made about the Confederate flag by men's basketball head coach Riley Wallace.
But a school spokesman indicated yesterday that Wallace will write a letter to the African-American Association.
That group and the Japanese American Citizens' League both expressed concern about Wallace's comments on whether South Carolina should fly the Confederate flag from the statehouse dome. The NCAA has demanded that South Carolina remove the flag.
The African-American Association, which issued its first statement Wednesday, also called for the coach's removal "as soon as contractually possible."
Wallace said in an April 29 Star-Bulletin story that the South Carolina flag issue was none of the NCAA's business, and should be settled by South Carolinians. That state's legislature has since voted to remove the flag.
"We recognize that for many African Americans the Confederate battle flag symbolizes and represents slavery, Jim Crow segregation, and racism," stated JACL chapter President Allison Tanaka.
Outgoing university President Kenneth Mortimer yesterday repeated his stance that Wallace was only exercising his right as a citizen to freedom of speech.
A spokesman said Wallace and Mortimer met Wednesday at Wallace's request. But the content of the meeting was not disclosed.
Faye Kennedy, spokeswoman for the African-American Association, said she'd appreciate a letter from Wallace. "But it still wouldn't really solve the problem," she said, "because the university has to take some position."
She said that a letter alone "would be completely inadequate."
Kennedy said she is disappointed to see the university "circling the wagons" over the issue of Wallace's remarks and believes it could prove costly.
"Now this issue has been on the Internet, and if the university leaves it floating around, it may be weighed by black athletes on the mainland considering coming here," she said.
Tanaka took a similar view in her release yesterday.
"Coach Wallace should be more sensitive to the pain that continuing display of the Confederate flag causes African Americans, including players and his coaching colleagues," she said. "His insensitivity can only hurt the University of Hawaii in its efforts to recruit athletes of all races."
Susan Kreifels also contributed to this report.
UH Athletics
Ka Leo O Hawaii

redpajama_set_name: RedPajamaCommonCrawl


Isn't this the most beautiful table you have ever seen!!! I couldn't have done it without Jill and her perfect vintage finds and help with the styling. The plates and small colored glasses were the most perfect addition to the table top.
All of the foods were baby girl themed. The menu included Bambina Burgers, Petite Grilled Cheese and Tomato Soup, Mini Fruit Skewers, Baby Cake Cupcakes, and more!!!
We used vintage Ball Jars to hold the Peony Flowers. Quite possible my favorite flower!!!


Poland will probably ask its bank creditors to renegotiate principal debt repayments because of a severe shortage of hard currency and pressures from Western creditor governments, banking sources in London said Friday.
Despite the shortage, Poland last month met the first payment of principal due under a 1981 bank debt rescheduling pact.
But it was unable to make a payment of $550 million in interest arrears to governments at the year's end on Tuesday.
Official creditors now want Warsaw to reduce its repayments to Western banks to leave more of Poland's meager hard currency holdings available for them, the sources said.
Poland, with about $29 billion of hard currency debt, will meet official Western creditors - known as the Paris Club - on Jan. 23 and 24.
Bankers said Poland had not yet requested any meetings with representatives of banks that rescheduled the country's debts. Such a meeting could be held shortly before or after the talks with the Paris Club, they added.
Government creditors granted Poland a three-month extension after it failed to make its interest repayments. But bankers said it was uncertain whether Poland could even meet the new March 31 deadline.


text:
Pierre Cardin's Bubble Palace For Sale
Supratim Talukder and His Divine Gift of Dance
The Next Crisis Will Come From Cryptocurrencies
Every decade or so, a catalyst is provided for a major economic crisis. The nineties had the dotcom bubble, the noughties had the sub-prime mortgage crisis and the twenties will have cryptocurrencies.
Jul 14, 2021 | By Patrick Tan
Credit: Dmitry Demidko / Unsplash
Living on the banks of the mighty Mekong River that flows from the Tibetan Plateau across thousands of miles through Indochina and into the South China Sea, Tran Nguyen knows how high his home needs to be built above the banks.
Nguyen and his family have farmed and fished the Mekong for generations, through war and strife, they have carved a living for themselves, by reading the signals of the river that gives and takes.
But over the past decade, and thanks to numerous hydroelectric projects upstream on the Mekong, the once bountiful river that provided much-needed nutrients downstream has taken more than it's given.
Crop failures have become more common and fish stocks have grown thinner.
Initially, the changes were too subtle to be of much significance, too slight to command much attention.
But in the past few years, the seasonal floods from the relentless monsoon rains that are part and parcel of life on the Mekong have been getting more extreme.
And last year, despite generations of the water level never rising above where Nguyen built his house, his home and all his belongings were swept away in a massive flood.
According to Nguyen,
"It's either that there isn't enough or water comes too much and floods — nothing about the Mekong is normal now."
"I haven't seen the Mekong change so fast before."
But it's not just the Mekong that's been changing rapidly, the global monetary system has as well and in ways that may provide the perfect setup for the next big financial crisis.
Been there done that, blown that bubble
Credit: Marc Sendra Martorell / Unsplash
For veterans of previous financial bubbles, there is an unmistakable sense of déjà vu.
Stock valuations at their richest since the dot-com bubble of 2000, home prices are back to their pre-financial crisis peak, and some of the diciest companies have no trouble raising debt from the markets, or selling shares.
All this is happening while investors are pouring billions of dollars into everything from green energy to cryptocurrencies.
READ MORE: Champs-Élysées To Undergo Green Transformation
But at the root of all of this seemingly method-less madness, is the U.S. Federal Reserve and whose prints have been on previous bubbles and crashes as well.
It's no big secret that easy monetary policy has regularly fueled financial booms and it's exceptionally easy now.
With the Fed having kept interest rates low since the pandemic first landed in America last March — and even signs of heightened inflation only bringing forward the schedule for raising rates, instead of spurring an actual move to increase borrowing costs — it has signalled to the markets that it has little appetite to tighten liquidity.
It's fixed, let's break it
Credit: Greg Rosenke / Unsplash
Despite signs that the U.S. economy is on the mend, the Fed still continues to soak up both U.S. Treasuries and mortgage-backed securities to the tune of some US$120 billion a month.
And despite a recent spike in bond yields, over the past week, yields have slipped down again and the benchmark U.S. 10-year Treasury yield, from which almost all other debt takes reference, is still well below inflation.
For only the second time in over four decades, real yields remain deeply negative, distorting market incentive mechanisms and asset prices.
But did the Fed or indeed, any other central bank really have any other choice but to flood the markets with liquidity?
A rock and a harder place
Credit: Zoltan Tasi / Unsplash
Students of the Great Depression will know that a reluctance to intervene by both the Hoover administration and the U.S. Federal Reserve made things worse and prolonged the economic malaise unnecessarily.
While the Hoover administration did roll out relief packages, it refused to intervene in the economy directly.
And the Fed at the time restricted liquidity, refusing to prop up failing banks when the financial system needed it the most.
This is why there is no question that the Fed had to act during the pandemic, which at its most intense, threatened far more damage than even the 2008 Financial Crisis.
READ MORE: The Next Financial Crisis Could Be an Opportunity for Cryptocurrencies
But signs that not just the U.S. economy, but the global economy, is on the mend, could undermine the justification for such low interest rates, threatening the underpinnings of market valuations.
Yet the Fed has been here before.
In the late 1990s, the Fed slashed rates in response to the 1997 Asian Financial Crisis and the near-collapse of the hedge fund Long-Term Capital Management.
The move by the Fed was seen by many as an implicit market backstop, and while it shored up markets, it also teed them up for the dot-com bubble that followed almost immediately thereafter.
Once the dot-com bubble had burst, the Fed's low-rate policy in response to that crisis kept liquidity taps flowing to water the ground for the subsequent housing bubble and crisis.
On both occasions, Fed officials vehemently defended their policies, arguing that to raise rates (or not cut them) simply to prevent bubbles, would compromise their main goals of low unemployment and inflation, and do more harm than letting the bubble deflate on its own.
While the Fed of those previous bubbles was at least ostensibly concerned about inflation, the current Fed is now proselytizing its own home-brewed economic theory called "New Monetary Policy," which accepts higher inflation (over 2%) during some periods to "make up" for periods of lower inflation (in the decade after the 2008 Financial Crisis).
And while this may all sound like a bit of voodoo — because who wouldn't want to have their cake and eat it? — Fed officials have been expounding on New Monetary Policy with a straight face, as if inflation is like a dial on a thermostat that you can set.
Credit: Glenn Carstens-Peters / Unsplash
To be sure, the pandemic shutdown last March triggered a hit to U.S. economic output that was initially worse than the 2008 Financial Crisis, but after just two months, economic activity began to recover as restrictions eased and businesses adapted to social distancing.
The Fed initiated new lending programs so numerous that many of them went unused, and Congress passed the US$2.2 trillion Cares Act, making the Emergency Economic Stabilization Act of 2008 look like pocket change in comparison.
Since then, however, coronavirus vaccines have arrived faster than anyone could have hoped for, and the U.S. economy has reopened at breakneck pace, and even though scores remain unvaccinated, the U.S. still leads the rich world in vaccination numbers.
At current rates, the U.S. economy is not only likely to hit its pre-pandemic size before the end of the year, it'll likely be larger and more powerful than it was heading into the pandemic.
READ MORE: The Global Pandemic: Impact on Singapore's Private Residential Market
Yet despite that, both fiscal and monetary taps remain wide open.
In March, the Biden administration won a huge victory by passing a sweeping US$1.9 trillion additional coronavirus relief package and is working to roll out a fresh US$1.2 trillion infrastructure framework by August.
And although bond markets appear to be functioning normally, the Fed is still the 400-pound gorilla that is weighing heavily on bond yields.
This unprecedented injection of both monetary and fiscal stimulus into an economy already rebounding thanks to a broad reopening and pent-up demand is why asset markets are starting to build-up into bubble territory.
S&P 500 stocks are now trading at about 22 times the coming year's profits, according to data from FactSet, a level which was only exceeded at the peak of the dot-com bubble in the year 2000.
Investors have also demonstrated a willingness to accept the lowest yields since 1995 for bonds of junk-rated companies, the narrowest spread above safe U.S. Treasuries since the eve of the 2008 Financial Crisis, according to data from Bloomberg Barclays.
Residential and commercial properties, adjusted for inflation, are also near the peak when the housing bubble was at its fullest in 2006.
And cryptocurrencies, an asset class that didn't exist prior to the 2008 Financial Crisis, are now worth an estimated US$1.37 trillion according to data from CoinMarketCap.com
While stock and real estate valuations may be more justifiable today than they were in say 2000, or 2006, because the returns on U.S. Treasuries are so much lower, it becomes a bit of a stretch to justify other asset prices.
It's Not the Size That Matters, It's What You Use to Measure It
Credit: William Warby / Unsplash
Just as has been the case in previous bubbles, conventional valuation metrics are being dismissed as outdated, especially when it's come to nascent asset classes.
A Bank of America report noted in April that companies with relatively lower carbon emissions and higher water efficiency attracted higher valuations, but not because of superior cashflow or future profits, but because of a flood of new money in so-called ESG funds which invest according to environmental, social and governance criteria.
Conventional valuations are also redundant for cryptocurrencies which for the most part earn no interest, generate no rent, and produce no dividends.
More importantly, cryptocurrencies have yet to firmly establish substantial use cases outside of speculation.
Advocates claim that cryptocurrencies will one day displace fiat currencies, and use decentralized finance or DeFi as an example of how the industry will disintermediate the legacy financial system.
And as recently as May, cryptocurrencies were valued in total at over US$2.4 trillion, more than all the U.S. dollars in circulation.
Whether it's cryptocurrencies or the internet, as with previous episodes of bubble blowing, financial innovation is also at work.
READ MORE: Crypto Crash Investigation — What Caused the Cryptocurrency Crash?
Zero-commission brokers such as Robinhood Markets and SoFi have empowered a previously unserved investor class, with limited experience or education on the machinations of the markets.
Stock splits, by retail investor favourites such as Apple and Tesla, have made their bite-sized chunks more consumable for the mass retail investing market.
Retail investors are now estimated to account for up to 26% of market volume, according to data from Vanda Research, and the Bank of International Settlements, a clearing bank for the world's central bank, estimates that retail investors now influence the overall direction of the markets more than ever before.
Investors also don't need to go too far to find examples of retail investor influence either — just look at GameStop, AMC Entertainment, and the league of extraordinary meme stocks, and the fortunes of "professional investors" like Melvin Capital licking their wounds from their short positions.
Just like traditional metrics, retail investors are also using their own tools for valuation that have nothing to do with what you may have learned in Finance 101, buying stocks and other assets for reasons completely unrelated to their underlying business or prospects.
And while such speculative activity is often blamed on the Fed, U.S. Treasury Secretary Janet Yellen will do well to remind everyone that "correlation does not imply causation."
But the causal link between fiscal stimulus and speculation remained unbroken when last March, flows into both ETFs and mutual funds spiked after the U.S. Treasury started distributing US$1,400 stimulus checks.
So how does it end?
Credit: Emily Morter / Unsplash
It's impossible to say if, or even whether, this all ends.
Expensive stocks could eventually earn the profits necessary to justify their current prices and valuations may eventually normalize, especially if the global economy continues on its current trajectory.
And more extreme pockets of speculation may collapse under their own weight, as profits disappoint, competition emerges or aspirations go unfulfilled.
But for assets across the board to fall would need some sort of larger macroeconomic event, such as a recession, financial crisis, or even runaway inflation.
With the pandemic in retreat, a recession seems unlikely for now.
And with the Fed demonstrating its willingness to raise rates (or at least bring forward the schedule for rate hikes) and supply constraints eventually being ironed out, inflation is less likely as well.
This leaves a crisis linked to some hidden fragility in the system as being the most likely precursor to a wider financial fallout and it looks like Wall Street is laying the pipes to that sewer right now.
Cryptic Business
Credit: Unsplash
Over the past 18 months, the number of big Wall Street names signing up to cryptocurrencies has increased to such a level that it hardly registers anymore in the digital asset markets.
Whereas in the past, an announcement such as the US$55 billion hedge fund of Marshall Wace planning its first forays into the cryptocurrency market would have seen a huge uptick in prices, last week that news received a disinterested yawn instead.
Cryptocurrency markets have become so accustomed to fresh institutional entrants that news of their arrival fails to make waves anymore.
And like the boy who cried wolf one too many times, even Elon Musk's tweets throwing shade on Bitcoin and Ether did nothing to stop them from ascending into the weekend.
From BlackRock to Fidelity Investment Management, Goldman Sachs to Citigroup, Morgan Stanley to JPMorgan Chase, Paul Tudor Jones, MicroStrategy, Stanley Druckenmiller, Ray Dalio, Tesla, Square, Visa, PayPal, Mastercard, the list of cryptocurrency converts keeps growing.
And that all boils down to excess liquidity chasing increasingly sparse opportunities.
It should therefore come as no surprise that financial institutions, for whom a low interest rate environment eats into their profits and trading activity can no longer generate the alpha of last year, are turning to cryptocurrencies.
And each additional institutional name betting on cryptocurrencies, adds to the caché and credibility of the space, which in its current state is unregulated and growing in size and significance.
With the crypto-savvy Gary Gensler heading up the U.S. Securities and Exchange Commission, the likelihood of an inaugural U.S. Bitcoin ETF grows, and then it's off to the races.
Miffed that they have yet to blow up the cryptocurrency bubble to its fullest potential, Wall Street, which was initially dismissive of the digital asset dominion, may now take the lead to apply their creativity to the space the same way they did for internet companies and mortgages.
And if something as pedestrian as a mortgage can be juiced up, what more the exotic nexus of finance and technology that is cryptocurrencies?
Wall Street's signature move has been making money by selling clients the most complex instruments that even they themselves struggle to understand and to that end, cryptocurrencies are the perfect product by melding two already complicated fields — finance and technology.
Even seasoned investor Mark Cuban couldn't avoid taking a hit from his investment in Titan, an algorithmic-linked stablecoin.
The Fed thankfully has already noted the risks, singling out cryptocurrency prices as a potential danger for the first time in its overall assessment of financial system stability.
In a brief comment contained in the Fed's semi-annual Monetary Policy Report, the central bank noted that policymakers are paying more attention to cryptocurrencies, which even on a good day, represents just a sliver of the massive financial system.
U.S. Federal Reserve Chairman Jerome Powell has said that he wants the Fed to play a "leading role" in the development of international standards for digital currencies, which is exactly what Wall Street wants because you can't game the system if there are no rules.
For now, cryptocurrencies are too small and too insignificant in the grand scheme of the global financial system to register concern.
Even at its highest watermark, cryptocurrencies were a fraction of the value of derivatives traded on a daily basis, but Wall Street could change that.
While bubbles aren't blown in a day, there are signs that Wall Street is already in the midst of laying the pipes to blow up a massive sewer bubble in cryptocurrencies.
From proprietary trading desks specializing in cryptocurrencies, to the proliferation of exchange-traded products underpinned by digital assets in Europe, there is a growing sense that the space is maturing, and the groundwork for a massive hype cycle is being laid down.
Like so many other bubbles before, the clues are subtle and the warning signs only obvious with the benefit of hindsight, but cryptocurrencies could potentially cause the next financial crisis, maybe not today, maybe not tomorrow, but someday, and for the rest of its life.
By Patrick Tan, CEO & General Counsel of Novum Alpha
Novum Alpha is the quantitative digital asset trading arm of the Novum Group, a vertically integrated group of blockchain development and digital asset companies. For more information about Novum Alpha and its products, please go to novumalpha.com/ or email: ask@novum.global
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text:
Le Lude () is a commune in the Sarthe department in the region of Pays de la Loire, northwestern France. On 1 January 2018, the former commune of Dissé-sous-le-Lude was merged into Le Lude.

See also
Communes of the Sarthe department

References

External links

 Website of Le Lude castle

Communes of Sarthe
Anjou


text:
AgriLink.group Navigation
Benefits Solutions
Pension Solutions
Executive Solutions
Court Decision: When does Employer's duty to accommodate end?
A recent decision by the Ontario Divisional Court has outlined that an employer's duty to accommodate ends when an employee's disability becomes permanent and as a result, the contract of employment cannot be upheld.
The case – Katz v. Clarke
Mr. Clarke (the employee), was employed as a Front Store Manager at a pharmacy owned by the Katz Group ("Katz") where Mr. Clarke had been employed for approximately 13 years at the time of termination. However, he had been off work on a leave of absence for over five years.
Mr. Clarke had been approved for short-term disability, followed by long-term disability.
Great-West Life (the disability insurer) advised the employer (Katz) that based on medical information, Mr. Clarke was not able to perform the essential duties of his position and there was no reasonable expectation that he would be able to perform them in the foreseeable future. Based on this, Katz advised Mr. Clarke that his employment had been frustrated due to his prolonged absence from work, his inability to perform the duties of his position and the lack of reasonable expectation that he would be able to return to perform the duties of his position in the foreseeable future.
Mr. Clarke's legal counsel advised the employer that Mr. Clarke was "working very hard to get well so that he can return to his former employment and perform the essential duties of his position", the employer provided Mr. Clarke with the opportunity to provide updated medical information outlining his projected return to work date. Mr. Clarke failed to provide this information and Mr. Clarke's employment was terminated due to frustration of contract.
Mr. Clarke commenced an action against Katz of wrongful dismissal and damages under the Ontario Human Rights Code. Katz moved for summary judgement to have the claim dismissed. The motion judge dismissed this motion for summary judgement.
The motion judge noted that Katz was aware of Mr. Clarke's desire to return to full employment and it was arguable that Katz's failure to spend time and effort to explore with Mr. Clarke how his desire to return to work could be accommodated resulted in a "rush to judgment" that breached the Code.
Katz appealed the decision to the Ontario Divisional Court.
The Ontario Divisional Court overturned the decision of the motion judge, saying that an employer's duty to accommodate is only triggered when an employee informs an employer not only of their wish to return to work, but also provides evidence of their ability to return to work.
The Court also confirmed that an employer's duty to accommodate ends where the employee is no longer able to fulfill the basic obligations of the the employment relationship.
When the employee's disability is permanent, the employment contract is impossible to uphold and is therefore frustrated. However it can be noted that in this case the employer was successful in establishing frustration of employment after the employee had been absent from work for more than five years.
Katz Group Canada Ltd. v. Clarke, 2019 ONSC 2188,
Benefits Canada: 'Inherently impossible' to accommodate employee unable to work, finds Ontario court
Miller Thomson : Katz et al. v. Clarke: Ontario Divisional Court Upholds Termination of Employee Due to Frustration of Contract
The inception of AgriLink Group started in the spring of 2013 after meeting with a large agricultural client. There was an expressed need for a benefits specialist to serve the agriculture community and the like-minded companies they conduct business with. After multiple meetings with the top benefits providers in Canada, an agreement to offer enhanced coverages to the agricultural community was completed.
The formation of Agri-Link Group was founded with the vision of providing a premier, client -centric benefit solutions. Driven by perfection, our company philosophy embraces innovation, reliability and exceptional service for all our clients and members of their teams. A national agricultural specialist was introduced to the market.et.
Our goal is to make sure our clients and associates have access to the information they need to make smarter decisions - check out our blog for some great resources and industry updates.
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Our goal is to respond immediately to every request - in case you need to reach us quickly, please feel free to reach out directly:
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text:
\section*{Acknowledgment}
This work was funded by grant (IIS-1734361) from the National Science Foundation and Honda Research Institute USA.
\section*{Appendix}
\renewcommand{\thesubsection}{\Alph{subsection}}\label{sec:appendix}

\subsection{Personal Space Definition}\label{sec:ps_defn}

\begin{figure}[th!]
\centering
\includegraphics[width=0.3\linewidth]{figures/personal_space.jpg}
\caption{A sample personal space}
\label{fig:PS}
\end{figure}

An example of the personal space is shown in Fig.~\ref{fig:PS}. Each personal space is first constructed by identifying the variances along the four principle axes to the agent's front, sides and rear, defined respectively as:
\begin{equation}
\begin{split}
    \sigma_{f}^i &= \max(2v^i, 0.5),\\ \sigma_{s}^i &=2\sigma_{f}^i/3\\ \sigma_{r}^i &=\sigma_{f}^i/2\\
\end{split}\mbox{.}
\end{equation}
Based on the four principle axis, the personal space for agent $i$ is represented as a set of boundary points.
\begin{equation}
    \mathcal{P}^i(q^i) = \{b^i(\phi), \phi \in [0,2\pi)\}\mbox{,}
\end{equation}
where
\begin{equation}
\begin{split}
    b^i(\phi) &= s^i + L_{e}(\phi)\left(\begin{matrix}
                        \cos(\theta^i+\phi) \\
                        \sin(\theta^i+\phi)
                   \end{matrix}\right)\mbox{,} \\
    L_e(\phi) &= \sqrt{\frac{C}{\cos^2{\gamma}/(2\sigma_1)+\sin^2{\gamma}/(2\sigma_2)}}\mbox{,} \\
    \gamma &= \mbox{mod}(\phi, \pi/2)\mbox{,} \\
    (\sigma_1, \sigma_2) &= \begin{cases}
        (\sigma_{f}^i, \sigma_{s}^i), &\mbox{if } 0 \leq \phi < \pi/2\\
        (\sigma_{s}^i, \sigma_{r}^i), &\mbox{if } \pi/2 \leq \phi < \pi\\
        (\sigma_{r}^i, \sigma_{s}^i), &\mbox{if } \pi \leq \phi < 3\pi/2\\
        (\sigma_{s}^i, \sigma_{f}^i), &\mbox{if } 3\pi/2 \leq \phi < 2\pi
        \end{cases}\mbox{.}
\end{split}
\label{eq:personal_space}
\end{equation}
$C$ is a constant used to adjust the scale of the personal space.

\subsection{Partial Input Handling}\label{sec:partial_input}

Note that in a dynamic pedestrian scene, we will have frequent occurrences of partial inputs for individual agents or groups due to new agents entering the scene or new groups being formed respectively. Therefore, our prediction model must be able to handle cases in which the input is complete up to a past window $t_{\hat{h}}$ with $t_{\hat{h}} = t -\hat{h}$, $\hat{h} < h$, i.e., $\boldsymbol{\mathcal{G}}_{t_{\hat{h}}:t}$. To handle these cases, for time $t_{h} <\tau < t_{\hat{h}}$, we compute $\mathcal{G}^j_{\tau}$ by making the following membership assumptions:
\begin{itemize}
    \item For any agent $i\in G^j_t$ such that $i\notin G^j_{\tau}$ and for whom we have the complete state history $s^i_{t_h:t}$, we set $g^i_{\tau} = j$. In other words, the prior group membership of any recent members of group $j$ is set to $j$ (despite agent $i$ possibly being a member of another group $j'$ at those instances).

    \item For any agent $i\in G^j_t$ such that $i\notin G^j_{\tau}$ and for whom we only have \emph{partial} state history $s^i_{t_{\hat{h}}:t}$, we take the agent's last known state $s^{i}_t$ and velocity $u^{i}_t$ and back propagate it as $s_{\tau-1}^{i}=s_{\tau}^{i}-u_{\tau}^{i}dt$.

\end{itemize}
Given a small $h$, these assumptions should reflect a close approximation of the group's complete history state, because pedestrian group switching process is gradual and pedestrian movements are smooth and predictive.

\subsection{Prediction Model Details}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/autoencoder.jpg}
    \caption{Our simple encoder-decoder network's architecture. The decoder's deconvolution layers mirror the layout of the encoder.}
    \label{fig:autoencoder}
\end{figure}

Our encoder-decoder network largely leverages [50]'s C3D network. As shown in Fig.~\ref{fig:autoencoder}, the encoder architecture contains the following layers (beginning from the input layer): one $3\times 3\times 3$ convolution layer with $64$ channels, one $1\times 2\times 2$ maxpool layer, one $3\times 3\times 3$ convolution layer with $128$ channels, another $1\times 2\times 2$ maxpool layer, another $3\times 3\times 3$ convolution layer with $128$ channels, one $3\times 3\times 3$ convolution layer with $256$ channels, one $2\times 2\times 2$ maxpool layer, another $3\times 3\times 3$ convolution layer with $256$ channels, one $3\times 3\times 3$ convolution layer with $512$ channels, another $2\times 2\times 2$ maxpool layer, two $3\times 3\times 3$ convolution layers with $512$ channels and another $2\times 2\times 2$ maxpool layer.

We used an initial learning rate of $1e-5$, batch size of $1$ and trained for $200$ epochs. We used Adam optimizer with default PyTorch settings. The data samples are generated by sampling a random segment during the evolution of a group for all groups in all the datasets. The data samples are normalized in scale and positions such that the entire group space sequence fits inside the $224\times 224$ image sequences and the geometric center of the group in the last input sequence frame is at the center of the image. After obtaining the predictions from the model, we filter out pixel predictions with confidence level less than $0.5$. An example is shown in Fig.~\ref{fig:data_sample}. For evaluation on a particular dataset, including both evaluation of the encoder-decoder network's performance and the policies in the navigation setting, we use the model that was trained on the other four datasets.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/data_sample.jpg}
    \caption{Top: An example group space input sequence for our encoder-decoder network. Mid: The ground truth future sequence of the group. Bottom: The predictioned future sequence of the group as outputted by our encoder-decoder network.}
    \label{fig:data_sample}
\end{figure}

In simulated laser scan settings, we do not retrain the group shape prediction models. Instead, we transfer the learned group shape prediction models on perfect perception settings directly into this new setting. We use a nearest neighbor approach based on geometric centers to identify the history sequence of a group in order to predict the group's future states. If the nearest neighbor of a group in the previous frame is more than $0.25m$ away, then we say no prior history of this group is available and use the technique in section~\ref{sec:partial_input} to linearly back-propagate the group's history.

To integrate this encoder-decoder network into our G-MPC framework, we performed the following additional processing steps (a detailed explanation of footnote 1). Because the encoder-decoder network model takes image-based inputs, we first convert the group space convex hulls into images. We use the homography matrix provided by the dataset to convert the coordinates of the convexhull vertices from meters to pixels. Then we draw these convexhulls on empty canvases to obtain the images. We preprocessed these images to normalize the convexhulls' sizes and positions such that these convexhulls fit inside the images throughout the input sequence and are approximately at the center of the image for the last input frame. Once we obtain the output sequence, we take the coordinates of the vertices at the edge of the output shape blobs and convert them back from pixels to meters using the inverse of the homography matrix.

\subsection{Parameter Details}

For the parameters of eq. (1), we picked $\epsilon_s, \epsilon_\theta, \epsilon_v$ such that the grouping outcomes match our qualitative inspection of human grouping in the datasets similarly to our prior work [13]. For ETH, HOTEL, ZARA1 and ZARA2 we set $\epsilon_s = 2.0m, \epsilon_\theta = 30^\circ, \epsilon_v=1.0 m/s$. Because UNIV is more crowded than the other four datasets, group formations are tighter and we set $\epsilon_s = 1.5m, \epsilon_\theta=15^\circ, \epsilon_v=0.5m/s$.

For the parameter of eq.~\eqref{eq:personal_space}, we selected $C$ under the assumption that closely-interacting pedestrians walk around the boundaries of each other's personal space. For ETH, HOTEL, ZARA1 and ZARA2, we set $C=0.35$. Again because UNIV has denser crowds, we set $C=0.25$. If at any given time the robot enters a social group space, we incrementally reduce $C$ by $0.1$ with a minimum value of $0.05$ until the robot is outside of the group space.

For the time horizon parameter $f$ and the history window parameter $h$ from section 4.2, we set $f=8$ and $h=8$ to ensure our MPC formulation's compatibility with the SGAN models.

For the weight parameter $\lambda$ in the cost function in equation (10), we perform a full parameter sweep to tune $\lambda$. We test $\lambda$ with values from $0.1$ to $0.9$ with increments of $0.05$ on randomly sampled $100$ test cases. We then select $\lambda$ that results in high success rates (at least $90\%$) for both agent based and group based policies without predictions and that the success rates of the two policies are the closest to each other. For trials with non-reactive agents, we set $\lambda = 0.65$. For trials with reactive agents, we set $\lambda = 0.3$. Note that we want the weight parameter to be the same for both pedestrian-based and group-based policies because the distance from the pedestrians to the boundaries of the social space are the same in both settings. Keeping the same weight allows fair evaluations of these two types of policies.

For the number of control rollouts $R$ in equation (14), we set $R = 12$.

\subsection{Numeric Results of Fig. 4 and Fig. 5}\label{sec:data}

\begin{table*}
\centering
\caption{Performance per scene under the \textit{Offline} condition.}
\resizebox{\columnwidth}{!}{%
\addvbuffer[0pt 2pt]{
\begin{tabular}{lccccccccccc}
\toprule
\multicolumn{1}{l}{Scene} &
\multicolumn{1}{l}{} &
 \multicolumn{2}{c}{ETH} & \multicolumn{2}{c}{HOTEL} & \multicolumn{2}{c}{ZARA1} &
\multicolumn{2}{c}{ZARA2} &
\multicolumn{2}{c}{UNIV}\\
\midrule

\multicolumn{1}{l|}{Task} &
\multicolumn{1}{l|}{Metric} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{ped-nopred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$91.38$} &
\multicolumn{1}{c|}{$75.86$} &
\multicolumn{1}{c|}{$95.35$} &
\multicolumn{1}{c|}{$95.45$} &
\multicolumn{1}{c|}{$96.00$} &
\multicolumn{1}{c|}{$85.71$} &
\multicolumn{1}{c|}{$97.64$} &
\multicolumn{1}{c|}{$96.90$} &
\multicolumn{1}{c|}{$83.96$} &
\multicolumn{1}{c|}{$83.33$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$56.9$} &
\multicolumn{1}{c|}{$24.14$} &
\multicolumn{1}{c|}{$81.4$} &
\multicolumn{1}{c|}{$63.64$} &
\multicolumn{1}{c|}{$72.0$} &
\multicolumn{1}{c|}{$46.43$} &
\multicolumn{1}{c|}{$82.68$} &
\multicolumn{1}{c|}{$70.54$} &
\multicolumn{1}{c|}{$68.87$} &
\multicolumn{1}{c|}{$64.91$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.29$} &
\multicolumn{1}{c|}{$0.97$} &
\multicolumn{1}{c|}{$1.35$} &
\multicolumn{1}{c|}{$1.14$} &
\multicolumn{1}{c|}{$1.31$} &
\multicolumn{1}{c|}{$1.03$} &
\multicolumn{1}{c|}{$1.39$} &
\multicolumn{1}{c|}{$1.27$} &
\multicolumn{1}{c|}{$0.95$} &
\multicolumn{1}{c|}{$0.98$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$27.16$} &
\multicolumn{1}{c|}{$16.93$} &
\multicolumn{1}{c|}{$19.31$} &
\multicolumn{1}{c|}{$10.55$} &
\multicolumn{1}{c|}{$20.75$} &
\multicolumn{1}{c|}{$8.80$} &
\multicolumn{1}{c|}{$20.87$} &
\multicolumn{1}{c|}{$8.92$} &
\multicolumn{1}{c|}{$22.96$} &
\multicolumn{1}{c|}{$17.38$}\\ \midrule

\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{ped-linear}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$94.83$} &
\multicolumn{1}{c|}{$96.55$} &
\multicolumn{1}{c|}{$97.67$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$96$} &
\multicolumn{1}{c|}{$96.43$} &
\multicolumn{1}{c|}{$94.49$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$86.79$} &
\multicolumn{1}{c|}{$99.12$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$75.86$} &
\multicolumn{1}{c|}{$74.14$} &
\multicolumn{1}{c|}{$86.05$} &
\multicolumn{1}{c|}{$88.64$} &
\multicolumn{1}{c|}{$84.0$} &
\multicolumn{1}{c|}{$89.29$} &
\multicolumn{1}{c|}{$88.19$} &
\multicolumn{1}{c|}{$95.35$} &
\multicolumn{1}{c|}{$79.25$} &
\multicolumn{1}{c|}{$90.35$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.42$} &
\multicolumn{1}{c|}{$1.28$} &
\multicolumn{1}{c|}{$1.28$} &
\multicolumn{1}{c|}{$1.25$} &
\multicolumn{1}{c|}{$1.53$} &
\multicolumn{1}{c|}{$1.36$} &
\multicolumn{1}{c|}{$1.53$} &
\multicolumn{1}{c|}{$1.46$} &
\multicolumn{1}{c|}{$1.01$} &
\multicolumn{1}{c|}{$1.09$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$27.31$} &
\multicolumn{1}{c|}{$18.08$} &
\multicolumn{1}{c|}{$19.28$} &
\multicolumn{1}{c|}{$10.72$} &
\multicolumn{1}{c|}{$20.29$} &
\multicolumn{1}{c|}{$9.48$} &
\multicolumn{1}{c|}{$20.84$} &
\multicolumn{1}{c|}{$9.16$} &
\multicolumn{1}{c|}{$22.93$} &
\multicolumn{1}{c|}{$17.06$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{ped-sgan}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$93.1$} &
\multicolumn{1}{c|}{$94.83$} &
\multicolumn{1}{c|}{$95.35$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$96$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$96.06$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$86.79$} &
\multicolumn{1}{c|}{$100$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$79.31$} &
\multicolumn{1}{c|}{$70.69$} &
\multicolumn{1}{c|}{$81.4$} &
\multicolumn{1}{c|}{$79.55$} &
\multicolumn{1}{c|}{$80.0$} &
\multicolumn{1}{c|}{$92.86$} &
\multicolumn{1}{c|}{$89.76$} &
\multicolumn{1}{c|}{$95.35$} &
\multicolumn{1}{c|}{$73.58$} &
\multicolumn{1}{c|}{$86.84$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.45$} &
\multicolumn{1}{c|}{$1.27$} &
\multicolumn{1}{c|}{$1.34$} &
\multicolumn{1}{c|}{$1.23$} &
\multicolumn{1}{c|}{$1.46$} &
\multicolumn{1}{c|}{$1.35$} &
\multicolumn{1}{c|}{$1.50$} &
\multicolumn{1}{c|}{$1.47$} &
\multicolumn{1}{c|}{$0.98$} &
\multicolumn{1}{c|}{$1.08$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$27.05$} &
\multicolumn{1}{c|}{$17.99$} &
\multicolumn{1}{c|}{$19.20$} &
\multicolumn{1}{c|}{$10.10$} &
\multicolumn{1}{c|}{$20.51$} &
\multicolumn{1}{c|}{$9.66$} &
\multicolumn{1}{c|}{$20.83$} &
\multicolumn{1}{c|}{$9.21$} &
\multicolumn{1}{c|}{$23.05$} &
\multicolumn{1}{c|}{$17.22$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{group-nopred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$94.83$} &
\multicolumn{1}{c|}{$87.93$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$88.64$} &
\multicolumn{1}{c|}{$92$} &
\multicolumn{1}{c|}{$75$} &
\multicolumn{1}{c|}{$96.06$} &
\multicolumn{1}{c|}{$92.25$} &
\multicolumn{1}{c|}{$93.4$} &
\multicolumn{1}{c|}{$93.86$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$82.76$} &
\multicolumn{1}{c|}{$79.31$} &
\multicolumn{1}{c|}{$97.67$} &
\multicolumn{1}{c|}{$86.36$} &
\multicolumn{1}{c|}{$88.0$} &
\multicolumn{1}{c|}{$71.43$} &
\multicolumn{1}{c|}{$92.91$} &
\multicolumn{1}{c|}{$89.92$} &
\multicolumn{1}{c|}{$90.57$} &
\multicolumn{1}{c|}{$88.6$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.70$} &
\multicolumn{1}{c|}{$1.61$} &
\multicolumn{1}{c|}{$1.8$} &
\multicolumn{1}{c|}{$1.61$} &
\multicolumn{1}{c|}{$1.83$} &
\multicolumn{1}{c|}{$1.59$} &
\multicolumn{1}{c|}{$1.68$} &
\multicolumn{1}{c|}{$1.59$} &
\multicolumn{1}{c|}{$1.18$} &
\multicolumn{1}{c|}{$1.22$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$29.52$} &
\multicolumn{1}{c|}{$23.32$} &
\multicolumn{1}{c|}{$21.98$} &
\multicolumn{1}{c|}{$14.38$} &
\multicolumn{1}{c|}{$22.86$} &
\multicolumn{1}{c|}{$13.02$} &
\multicolumn{1}{c|}{$23.47$} &
\multicolumn{1}{c|}{$11.17$} &
\multicolumn{1}{c|}{$28.57$} &
\multicolumn{1}{c|}{$20.61$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{group-pred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$96.55$} &
\multicolumn{1}{c|}{$86.21$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$90.91$} &
\multicolumn{1}{c|}{$96$} &
\multicolumn{1}{c|}{$82.14$} &
\multicolumn{1}{c|}{$96.85$} &
\multicolumn{1}{c|}{$94.57$} &
\multicolumn{1}{c|}{$89.62$} &
\multicolumn{1}{c|}{$93.86$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$82.76$} &
\multicolumn{1}{c|}{$81.03$} &
\multicolumn{1}{c|}{$100.0$} &
\multicolumn{1}{c|}{$88.64$} &
\multicolumn{1}{c|}{$92.0$} &
\multicolumn{1}{c|}{$82.14$} &
\multicolumn{1}{c|}{$92.91$} &
\multicolumn{1}{c|}{$93.02$} &
\multicolumn{1}{c|}{$86.79$} &
\multicolumn{1}{c|}{$92.98$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.67$} &
\multicolumn{1}{c|}{$1.90$} &
\multicolumn{1}{c|}{$1.83$} &
\multicolumn{1}{c|}{$1.65$} &
\multicolumn{1}{c|}{$1.87$} &
\multicolumn{1}{c|}{$1.75$} &
\multicolumn{1}{c|}{$1.77$} &
\multicolumn{1}{c|}{$1.67$} &
\multicolumn{1}{c|}{$1.19$} &
\multicolumn{1}{c|}{$1.32$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$29.51$} &
\multicolumn{1}{c|}{$23.17$} &
\multicolumn{1}{c|}{$21.88$} &
\multicolumn{1}{c|}{$13.63$} &
\multicolumn{1}{c|}{$23.01$} &
\multicolumn{1}{c|}{$11.95$} &
\multicolumn{1}{c|}{$23.45$} &
\multicolumn{1}{c|}{$11.13$} &
\multicolumn{1}{c|}{$27.82$} &
\multicolumn{1}{c|}{$20.06$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{laser-group-pred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$94.83$} &
\multicolumn{1}{c|}{$84.48$} &
\multicolumn{1}{c|}{$97.67$} &
\multicolumn{1}{c|}{$95.54$} &
\multicolumn{1}{c|}{$96$} &
\multicolumn{1}{c|}{$75$} &
\multicolumn{1}{c|}{$93.7$} &
\multicolumn{1}{c|}{$86.05$} &
\multicolumn{1}{c|}{$83.02$} &
\multicolumn{1}{c|}{$89.47$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$81.03$} &
\multicolumn{1}{c|}{$82.76$} &
\multicolumn{1}{c|}{$97.67$} &
\multicolumn{1}{c|}{$90.91$} &
\multicolumn{1}{c|}{$92.0$} &
\multicolumn{1}{c|}{$75.0$} &
\multicolumn{1}{c|}{$92.13$} &
\multicolumn{1}{c|}{$86.05$} &
\multicolumn{1}{c|}{$82.08$} &
\multicolumn{1}{c|}{$85.09$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.75$} &
\multicolumn{1}{c|}{$2.21$} &
\multicolumn{1}{c|}{$1.86$} &
\multicolumn{1}{c|}{$1.70$} &
\multicolumn{1}{c|}{$1.92$} &
\multicolumn{1}{c|}{$1.81$} &
\multicolumn{1}{c|}{$1.8$} &
\multicolumn{1}{c|}{$1.76$} &
\multicolumn{1}{c|}{$1.25$} &
\multicolumn{1}{c|}{$1.40$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$29.57$} &
\multicolumn{1}{c|}{$23.99$} &
\multicolumn{1}{c|}{$22.42$} &
\multicolumn{1}{c|}{$15.45$} &
\multicolumn{1}{c|}{$23.50$} &
\multicolumn{1}{c|}{$12.26$} &
\multicolumn{1}{c|}{$23.63$} &
\multicolumn{1}{c|}{$11.58$} &
\multicolumn{1}{c|}{$29.49$} &
\multicolumn{1}{c|}{$22.36$}\\

\bottomrule
\end{tabular}
}
}
\label{tab:results}
\end{table*}

\begin{table*}
\centering
\caption{Performance per scene under the \textit{Online} condition (simulated pedestrians powered by ORCA\cite{ORCA}).}
\resizebox{\columnwidth}{!}{%
\addvbuffer[0pt 2pt]{
\begin{tabular}{lccccccccccc}
\toprule
\multicolumn{1}{l}{Scene} &
\multicolumn{1}{l}{} &
 \multicolumn{2}{c}{ETH} & \multicolumn{2}{c}{HOTEL} & \multicolumn{2}{c}{ZARA1} &
\multicolumn{2}{c}{ZARA2} &
\multicolumn{2}{c}{UNIV}\\
\midrule

\multicolumn{1}{l|}{Task} &
\multicolumn{1}{l|}{Metric} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}} &
\multicolumn{1}{c|}{\textbf{Flow}} & \multicolumn{1}{c|}{\textbf{Cross}}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{ped-nopred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$96.55$} &
\multicolumn{1}{c|}{$98.28$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$97.73$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$97.64$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$88.68$} &
\multicolumn{1}{c|}{$99.12$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$50.0$} &
\multicolumn{1}{c|}{$63.79$} &
\multicolumn{1}{c|}{$79.07$} &
\multicolumn{1}{c|}{$77.27$} &
\multicolumn{1}{c|}{$60.0$} &
\multicolumn{1}{c|}{$64.29$} &
\multicolumn{1}{c|}{$65.35$} &
\multicolumn{1}{c|}{$79.84$} &
\multicolumn{1}{c|}{$60.38$} &
\multicolumn{1}{c|}{$78.95$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$0.93$} &
\multicolumn{1}{c|}{$1.05$} &
\multicolumn{1}{c|}{$0.94$} &
\multicolumn{1}{c|}{$1.04$} &
\multicolumn{1}{c|}{$0.92$} &
\multicolumn{1}{c|}{$0.97$} &
\multicolumn{1}{c|}{$0.98$} &
\multicolumn{1}{c|}{$0.99$} &
\multicolumn{1}{c|}{$0.89$} &
\multicolumn{1}{c|}{$0.98$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$24.02$} &
\multicolumn{1}{c|}{$13.73$} &
\multicolumn{1}{c|}{$15.75$} &
\multicolumn{1}{c|}{$8.50$} &
\multicolumn{1}{c|}{$16.71$} &
\multicolumn{1}{c|}{$6.13$} &
\multicolumn{1}{c|}{$17.16$} &
\multicolumn{1}{c|}{$7.32$} &
\multicolumn{1}{c|}{$17.20$} &
\multicolumn{1}{c|}{$13.82$}\\ \midrule

\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{ped-linear}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$94.83$} &
\multicolumn{1}{c|}{$98.28$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$97.73$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$98.43$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$89.62$} &
\multicolumn{1}{c|}{$99.12$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$43.1$} &
\multicolumn{1}{c|}{$70.69$} &
\multicolumn{1}{c|}{$83.72$} &
\multicolumn{1}{c|}{$90.91$} &
\multicolumn{1}{c|}{$64.0$} &
\multicolumn{1}{c|}{$85.71$} &
\multicolumn{1}{c|}{$74.8$} &
\multicolumn{1}{c|}{$80.62$} &
\multicolumn{1}{c|}{$61.32$} &
\multicolumn{1}{c|}{$88.6$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$0.94$} &
\multicolumn{1}{c|}{$1.04$} &
\multicolumn{1}{c|}{$0.97$} &
\multicolumn{1}{c|}{$1.03$} &
\multicolumn{1}{c|}{$0.94$} &
\multicolumn{1}{c|}{$0.99$} &
\multicolumn{1}{c|}{$0.98$} &
\multicolumn{1}{c|}{$0.99$} &
\multicolumn{1}{c|}{$0.90$} &
\multicolumn{1}{c|}{$0.98$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$23.83$} &
\multicolumn{1}{c|}{$13.25$} &
\multicolumn{1}{c|}{$15.43$} &
\multicolumn{1}{c|}{$8.32$} &
\multicolumn{1}{c|}{$16.54$} &
\multicolumn{1}{c|}{$6.22$} &
\multicolumn{1}{c|}{$17.03$} &
\multicolumn{1}{c|}{$6.74$} &
\multicolumn{1}{c|}{$16.87$} &
\multicolumn{1}{c|}{$13.53$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{ped-sgan}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$96.55$} &
\multicolumn{1}{c|}{$98.28$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$97.73$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$98.43$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$89.62$} &
\multicolumn{1}{c|}{$99.12$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$39.66$} &
\multicolumn{1}{c|}{$63.79$} &
\multicolumn{1}{c|}{$88.37$} &
\multicolumn{1}{c|}{$88.64$} &
\multicolumn{1}{c|}{$64.0$} &
\multicolumn{1}{c|}{$82.14$} &
\multicolumn{1}{c|}{$77.95$} &
\multicolumn{1}{c|}{$82.95$} &
\multicolumn{1}{c|}{$62.26$} &
\multicolumn{1}{c|}{$86.84$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$0.93$} &
\multicolumn{1}{c|}{$1.04$} &
\multicolumn{1}{c|}{$0.95$} &
\multicolumn{1}{c|}{$1.04$} &
\multicolumn{1}{c|}{$0.94$} &
\multicolumn{1}{c|}{$0.99$} &
\multicolumn{1}{c|}{$0.98$} &
\multicolumn{1}{c|}{$0.99$} &
\multicolumn{1}{c|}{$0.90$} &
\multicolumn{1}{c|}{$0.98$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$23.85$} &
\multicolumn{1}{c|}{$13.20$} &
\multicolumn{1}{c|}{$15.63$} &
\multicolumn{1}{c|}{$8.14$} &
\multicolumn{1}{c|}{$16.54$} &
\multicolumn{1}{c|}{$6.18$} &
\multicolumn{1}{c|}{$17.06$} &
\multicolumn{1}{c|}{$6.72$} &
\multicolumn{1}{c|}{$16.90$} &
\multicolumn{1}{c|}{$13.53$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{group-nopred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$93.1$} &
\multicolumn{1}{c|}{$98.28$} &
\multicolumn{1}{c|}{$95.35$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$96$} &
\multicolumn{1}{c|}{$100$} &
\multicolumn{1}{c|}{$97.64$} &
\multicolumn{1}{c|}{$98.45$} &
\multicolumn{1}{c|}{$89.62$} &
\multicolumn{1}{c|}{$100$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$75.86$} &
\multicolumn{1}{c|}{$79.31$} &
\multicolumn{1}{c|}{$81.4$} &
\multicolumn{1}{c|}{$88.64$} &
\multicolumn{1}{c|}{$88.0$} &
\multicolumn{1}{c|}{$71.43$} &
\multicolumn{1}{c|}{$90.55$} &
\multicolumn{1}{c|}{$85.27$} &
\multicolumn{1}{c|}{$68.87$} &
\multicolumn{1}{c|}{$88.6$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.15$} &
\multicolumn{1}{c|}{$1.15$} &
\multicolumn{1}{c|}{$1.17$} &
\multicolumn{1}{c|}{$1.16$} &
\multicolumn{1}{c|}{$1.11$} &
\multicolumn{1}{c|}{$1.03$} &
\multicolumn{1}{c|}{$1.21$} &
\multicolumn{1}{c|}{$1.11$} &
\multicolumn{1}{c|}{$0.94$} &
\multicolumn{1}{c|}{$1.04$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$25.36$} &
\multicolumn{1}{c|}{$15.37$} &
\multicolumn{1}{c|}{$16.62$} &
\multicolumn{1}{c|}{$9.72$} &
\multicolumn{1}{c|}{$18.16$} &
\multicolumn{1}{c|}{$7.50$} &
\multicolumn{1}{c|}{$18.36$} &
\multicolumn{1}{c|}{$8.56$} &
\multicolumn{1}{c|}{$18.98$} &
\multicolumn{1}{c|}{$15.15$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{group-pred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$94.83$} &
\multicolumn{1}{c|}{$98.28$} &
\multicolumn{1}{c|}{$97.67$} &
\multicolumn{1}{c|}{$97.73$} &
\multicolumn{1}{c|}{$92$} &
\multicolumn{1}{c|}{$96.43$} &
\multicolumn{1}{c|}{$99.21$} &
\multicolumn{1}{c|}{$99.22$} &
\multicolumn{1}{c|}{$93.4$} &
\multicolumn{1}{c|}{$98.25$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$74.14$} &
\multicolumn{1}{c|}{$87.93$} &
\multicolumn{1}{c|}{$81.4$} &
\multicolumn{1}{c|}{$95.45$} &
\multicolumn{1}{c|}{$88.0$} &
\multicolumn{1}{c|}{$89.29$} &
\multicolumn{1}{c|}{$91.34$} &
\multicolumn{1}{c|}{$93.8$} &
\multicolumn{1}{c|}{$80.19$} &
\multicolumn{1}{c|}{$91.23$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.13$} &
\multicolumn{1}{c|}{$1.24$} &
\multicolumn{1}{c|}{$1.16$} &
\multicolumn{1}{c|}{$1.21$} &
\multicolumn{1}{c|}{$1.17$} &
\multicolumn{1}{c|}{$1.13$} &
\multicolumn{1}{c|}{$1.18$} &
\multicolumn{1}{c|}{$1.11$} &
\multicolumn{1}{c|}{$0.93$} &
\multicolumn{1}{c|}{$1.07$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$25.19$} &
\multicolumn{1}{c|}{$14.99$} &
\multicolumn{1}{c|}{$16.45$} &
\multicolumn{1}{c|}{$9.14$} &
\multicolumn{1}{c|}{$17.93$} &
\multicolumn{1}{c|}{$7.62$} &
\multicolumn{1}{c|}{$18.22$} &
\multicolumn{1}{c|}{$7.88$} &
\multicolumn{1}{c|}{$18.71$} &
\multicolumn{1}{c|}{$14.74$}\\

\midrule
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{laser-group-pred}}} &

\multicolumn{1}{c|}{$\mathcal{S} (\%)$} &
\multicolumn{1}{c|}{$91.38$} &
\multicolumn{1}{c|}{$98.28$} &
\multicolumn{1}{c|}{$95.35$} &
\multicolumn{1}{c|}{$97.73$} &
\multicolumn{1}{c|}{$96$} &
\multicolumn{1}{c|}{$89.29$} &
\multicolumn{1}{c|}{$97.64$} &
\multicolumn{1}{c|}{$97.67$} &
\multicolumn{1}{c|}{$84.91$} &
\multicolumn{1}{c|}{$92.11$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{C} (\%)$} &
\multicolumn{1}{c|}{$70.69$} &
\multicolumn{1}{c|}{$84.48$} &
\multicolumn{1}{c|}{$88.37$} &
\multicolumn{1}{c|}{$93.18$} &
\multicolumn{1}{c|}{$88.0$} &
\multicolumn{1}{c|}{$85.71$} &
\multicolumn{1}{c|}{$88.98$} &
\multicolumn{1}{c|}{$94.57$} &
\multicolumn{1}{c|}{$60.38$} &
\multicolumn{1}{c|}{$78.07$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{D} (m)$} &
\multicolumn{1}{c|}{$1.18$} &
\multicolumn{1}{c|}{$1.33$} &
\multicolumn{1}{c|}{$1.26$} &
\multicolumn{1}{c|}{$1.28$} &
\multicolumn{1}{c|}{$1.18$} &
\multicolumn{1}{c|}{$1.12$} &
\multicolumn{1}{c|}{$1.23$} &
\multicolumn{1}{c|}{$1.14$} &
\multicolumn{1}{c|}{$0.93$} &
\multicolumn{1}{c|}{$1.07$}\\

\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\mathcal{L} (m)$} &
\multicolumn{1}{c|}{$25.40$} &
\multicolumn{1}{c|}{$16.27$} &
\multicolumn{1}{c|}{$16.81$} &
\multicolumn{1}{c|}{$9.82$} &
\multicolumn{1}{c|}{$18.72$} &
\multicolumn{1}{c|}{$8.54$} &
\multicolumn{1}{c|}{$19.07$} &
\multicolumn{1}{c|}{$8.57$} &
\multicolumn{1}{c|}{$21.39$} &
\multicolumn{1}{c|}{$16.46$}\\

\bottomrule
\end{tabular}
}
}
\label{tab:results-RVO}
\end{table*}

Tab.~\ref{tab:results} and Tab.~\ref{tab:results-RVO} are the numerical results of Fig. 4 and Fig. 5. $\mathcal{S}$ is the success rate. $\mathcal{C}$ is percentage of trials in which the robot does not enter any group space (collisions also count as group intrusions). $\mathcal{D}$ is the average minimum distance to pedestrians. $\mathcal{L}$ is the average path length.

\section{Conclusion}

We introduced a methodology for generating group-based representations and predicting their future states. Through an extensive evaluation over the flow and crossing scenarios drawn from 10 different real-world scenes from 2 different human datasets with both reactive and non-reactive agents, we demonstrate the value of group-based prediction in enabling safe and socially compliant navigation. Through experimentation with simulated laser scans, our model displays promising potential to process noisy sensor inputs without much performance downgrade.

Several improvements to our framework are possible. For example, we could incorporate state-of-the-art oracles in the form of advanced video prediction models~\citep{Guen-2020} or incorporate inter-group interaction modeling. Additional considerations such as the set of rollouts or the cost functions could possibly increase performance. We could also integrate our prediction model into alternative control frameworks such as reinforcement learning policies.

Finally, we plan on validating our findings on a real-world robot to fully test the capability of G-MPC to handle noisy sensor inputs. We also plan to investigate ways to improve computation time to enhance our approach's real-world applicability. These include simplifying group representation geometry and predicting future group states in metric space instead of in image space.

\section{Evaluation}\label{sec:evaluation}

We evaluate our framework through a simulation study in which the robot performs a navigation task (a transition between two points) within a crowds of dynamic agents in a set of scenes.

\subsection{Experimental Setup}

We consider a set of realistic pedestrian scenes, drawn from the ETH~\citep{ETH} (ETH and HOTEL scenes) and UCY~\citep{UCY} (ZARA1, ZARA2 and UNIVERSITY scenes) datasets, which often serve as benchmarking testbeds in the motion prediction and social navigation literature~\citep{socialGan, socialLstm,  SRLSTM,dynamic-channel,biswas2021socnavbench}. In each scene, we define two navigation tasks (see~\figref{fig:test_cases}): \textit{Flow}: in which the robot navigates along the crowd flow and \textit{Cross} in which the robot intersects vertically with the traffic flow. For each task, we generate a set of trials by segmenting the scene recording into blocks involving challenging interactions. We define a challenging interaction to be a segment involving at least $5$ pedestrians inside the test region drawn in black in~\figref{fig:test_cases}. This process provided us with a distribution of trials as shown in table~\tabref{tab:trialsperscene}. Across all trials, we keep the robot's maximum speed at $1.75 m/s$.

\begin{table}
    \centering
        \caption{Number of trials per task and scene.}
            \resizebox{.6\columnwidth}{!}{%
    \begin{tabular}{c|c|c|c|c|c}
        \toprule
        Task & ETH & HOTEL & ZARA1 & ZARA2 & UNIV \\
        \midrule
        Flow & 58 & 43 & 25 & 127 & 106\\
        Cross & 58 & 44 & 28 & 129 & 114\\
        \bottomrule
    \end{tabular}
    }
    \label{tab:trialsperscene}
    \vspace{-2em}
\end{table}

We consider two experimental conditions: an \textit{Offline} and an \textit{Online} one. In the \textit{Offline} one, the robot navigates among a crowd moving according to a recording of a human crowd. Under this condition, pedestrians act as dynamic obstacles that do not react to the robot, a situation which could arise in cases where robots are of shorter size and could thus be easily missed by navigating pedestrians. In the \textit{Online} one, the robot navigates among a crowd\footnote{For consistency, the agents in the crowd start and end at the same spots as the agents in the recorded crowd from the Offline condition.} moving by running ORCA~\citep{ORCA}, a policy that is frequently used as a simulation engine for benchmarking in the social navigation  literature~\citep{Everett18_IROS,dynamic-channel,mavrogiannis_etal2021-core-challenges}.

\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/test_cases.jpg}
\caption{Trajectories of all pedestrians in the datasets. The red dots represent the task start and end locations. The red lines represent the task paths. The black box represents the test region to check for non-trivial tasks.}
\label{fig:test_cases}
\vspace{-2em}
\end{figure}

To investigate the value of G-MPC, we develop three variants of it. \textbf{group-pred} is a G-MPC in which the encoder-decoder network has a history $h=8$ and a horizon $f = 8$. \textbf{group-nopred} is a variant that features no prediction at all --it just reacts to observed groups at every timesteps and it is equivalent to the framework of~\citet{Yang-2019-groupnav}. Finally, \textbf{laser-group-pred} is identical to \textbf{group-pred} but instead of using ground-truth pose information, it takes as input noisy lidar scan readings. We simulate this by modeling pedestrians as $1m$-diameter circles and lidar scans as rays projecting from the robot. We refer to the spec sheet of a SICK LMS511 2D lidar for simulation parameters. We further inject noise into the readings according to the spec sheet. Under this simulation, pedestrians may only be partially observable or even completely occluded from the robot.

We compare the performance of these policies against a set of MPC variants using mechanisms for individual motion prediction. \textbf{ped-nopred} is a vanilla MPC that reacts to the current states of other agents without making predictions about their future states. \textbf{ped-linear} is a vanilla MPC that estimates future states of agents by propagating agents' current velocities forward. This baseline is motivated by recent work showing that constant-velocity models yield competitive performance in pedestrian motion prediction tasks~\citep{constant-velocity-model}. Finally, \textbf{ped-sgan} is an MPC that uses S-GAN~\citep{socialGan} to extract a sequence of future state predictions for agents based on inputs of their past states. We selected S-GAN because it is a recent highly performing model. To ensure a fair comparison, all the MPC policy variants are integrated with the same MPC controller evaluated at $dt=0.1$.

We measure the performance of the policies with respect to four different metrics: a) \textit{Success rate}, defined as the ratio of successful trials over total number of trials; b) \textit{Comfort}, defined as the ratio of trials in which the robot does not enter any social group space over the total number of trials; c) \textit{Minimum distance to pedestrians}, defined as the smallest distance between the robot and any agent per trial; d) \textit{Path length}, defined as the total distance traversed by the robot in a trial.

To track the performance of G-MPC, we design hypotheses targeting aspects of safety and group space violation which we investigate under both experimental conditions, i.e., offline and online:

\textbf{H1}: To explore the benefits of group based representations alone, we hypothesize that \textbf{group-nopred} is safer than \textbf{ped-nopred} while achieving similar success rates but worse efficiency.

\textbf{H2}: To explore the full benefit of group based formulation, we hypothesize that \textbf{group-pred} is safer than \textbf{ped-linear} and \textbf{ped-sgan} while achieving similar success rates but worse efficiency.

\textbf{H3}: To explore how our formulation handles imperfect inputs, we hypothesize that \textbf{laser-group-pred} achieves similar safety to \textbf{group-pred} while achieving similar success rate and efficiency.

\textbf{H4}: To check that our formulation is socially compliant, we hypothesize that \textbf{group-nopred}, \textbf{group-pred} and \textbf{laser-group-pred} violate agents' group space less often than the baselines.

\subsection{Results}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/main_rst_noreact.jpg}
    \caption{Performance per scene under the \textit{Offline} condition. Horizontal lines indicate statistically significant results corresponding to different to hypotheses.}
    \label{fig:rst_noreact}
    \vspace{-1em}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/main_rst_react.jpg}
    \caption{Performance per scene under the \textit{Online} condition (simulated pedestrians powered by ORCA~\cite{ORCA}). Horizontal lines indicate statistically significant results corresponding to different to hypotheses.}
    \label{fig:rst_react}
    \vspace{-1em}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/qualitative_example1_2.jpg}
    \caption{Qualitative performance difference between approaches leveraging pedestrian-based (top) and group-based (bottom) representations. Left: non-reactive agents. Right: reactive agents. In all cases, the robot starts from the top and attempts to navigate downward. The green line shows the robot's traversed trajectory.}
    \label{fig:rst_examples}
    \vspace{-2em}
\end{figure}

\textbf{Quantitative Analysis.} Fig.~\ref{fig:rst_noreact} and Fig.~\ref{fig:rst_react} contain bar charts representing the performance of G-MPC compared with its baselines under Offline and Online settings respectively. Bars indicate means, errorbars indicate standard deviations, ``F" and ``C" are flow and cross scenarios respectively, and the number of asterisks indicates increasing significance levels: $\alpha=0.05, 0.01, 0.001$ according to two-sided Mann-Whitney U-tests.

\textbf{H1}: We can see from both Fig.~\ref{fig:rst_noreact} and Fig.~\ref{fig:rst_react} that G-MPC achieves statistically significantly larger minimum distances to pedestrians across all scenarios, often with $p<0.001$. This illustrates that the group representation is in itself capable of upgrading a simple MPC with no prediction. As expected, we observe that the price G-MPC pays for that is a larger average path length. We also see that success rates are comparable. Overall, we conclude that H1 holds.

\textbf{H2}: When future state predictions are considered, G-MPC obtains statistically significant results in most scenes supporting its attributes of being safer at the cost of worse efficiency. Thus H2 is partially confirmed. In offline scenarios, G-MPC has lower success rates in crossing scenarios. Upon closer inspection, most failure cases are due to timeouts from G-MPC's conservative behavior. However, in online scenarios where pedestrians react to the robot, G-MPC achieves high success rates. In real-world situations, to cross dense traffic, the robot needs to plan its actions with expectations of reactive pedestrians. Otherwise, the robot will most probably run into \textit{the freezing robot problem}~\citep{trautmanijrr}.

\textbf{H3}: Group-based representations have the potential to robustly account for imperfect state-estimates. Overall, we observe that with simulated imperfect states, G-MPC does not perform statistically significantly worse in terms of safety, but in dense crowds of the UNIV scenes it has worse efficiency and worse success rates in online cases. This shows that H3 holds in terms of safety and, in moderately dense human crowds, holds in terms of efficiency. Future work on better group representation is needed to achieve better efficiency in high-density human crowds given imperfect states.

\textbf{H4}: From Fig.~\ref{fig:rst_noreact} and Fig.~\ref{fig:rst_react}, we can see that G-MPC often has fewer group-space intrusions than its baselines. While this relationship is not always statistically significant, we do see a general trend of the group-based approaches to respect group spaces more often than individual ones. Thus, we conclude that H4 is partially confirmed.

We additionally observe a general trend that \textbf{group-pred} is better than \textbf{group-nopred} in terms of higher success rates, lower chances of group intrusions, longer minimum distances to pedestrians and shorter path lengths. This shows that our group prediction model offers benefits to the robot's navigation. However, in a few scenarios \textbf{group-nopred} performs better. We largely attribute this to the finite inaccuracies of future group predictions and the freezing robot problem that accompanies the robot's more conservative behavior in \textbf{group-pred} than in \textbf{group-nopred}.

\textbf{Qualitative Analysis.} Qualitatively, it is a more common occurrence for regular MPC to perform aggressive and socially inappropriate maneuvers than G-MPC. As shown in the two examples in Fig.~\ref{fig:rst_examples} executed by \textbf{ped-sgan} and \textbf{group-pred} agents, we can see that in offline conditions, the MPC agent aggressively cuts in front of the two pedestrians to the left before proceeding headlong into the cluster of pedestrians, only managing to avoid the deadlock by escaping through the narrow gap that opens up. While for G-MPC, it tracks the movements of the two pedestrian groups coming from the left. When the two pedestrian groups merge, the agent turns around and reevaluates its approach to cross. In the online condition, we observe that the MPC agent cuts through a pedestrian group to reach the other side, forcing a member of the group to stop and yield as indicated by the pedestrian's shrinking personal space, which is proportional to its speed. In the same situation, the G-MPC agent chooses to circumvent behind the social group.
\section{Group-based Prediction}\label{sec:prediction}

We introduce a group representation building on prior work~\citep{wang-split-merge} and a model for group-based prediction that is amenable for use in decentralized multiagent navigation.

\subsection{Group Representation}

Define as $\theta^i\in [0, 2\pi)$ the orientation of agent $i\in \mathcal{N}$ which is assumed to be aligned with the direction of its velocity $u^i$, extracted via finite differencing of its position over a timestep $dt$ and denote by $v^i = ||u^i||\in \mathbb{R}^{+}$ its speed. We define an augmented state for agent $i$ as $q^i = (s^i, \theta^i, v^i)$.

We treat a social group as a set of agents who are in close proximity and share similar motion characteristics. Assume that a set of $J$ groups, $\mathcal{J} = \{1,\dots, J\}$ navigate in a scene. Define by $g^i\in\mathcal{J}$ a label indicating the group membership of agent $i$. We then define a group $j\in\mathcal{J}$ as a set $G^j = \{i \in \mathcal{N} \mid g^i = j \}$ and collect the set of all groups in a scene into a set $\boldsymbol{G} = \{G^j\mid j\in\mathcal{J}\}$.

\label{sec:group-membership}\textbf{Extracting Group Membership.} We define the combined augmented state of all agents as $\boldsymbol{q} = \cup_{i=1:n}q^i$. To obtain group memberships for a set of agents $\mathcal{N}$, we apply the Density-Based Spatial Clustering of Applications with Noise algorithm (DBSCAN)~\citep{dbscan} on agent states:
\begin{equation}
\label{eq:dbscan}
    \boldsymbol{G}\longleftarrow \texttt{DBSCAN}(\boldsymbol{q} \mid \epsilon_s, \epsilon_\theta, \epsilon_v)
\end{equation}
Where $\epsilon_s, \epsilon_\theta, \epsilon_v$ are respectively threshold values on agent distances, orientation and speeds.

\label{sec:groupspacerep}\textbf{Extracting the Social Group Space.} For each group $G^j$, $j\in\mathcal{J}$, we define a \emph{social group space} as a geometric enclosure $\mathcal{G}^j$ around agents of the group. For each agent $i\in G^j$, we define a personal space $\mathcal{P}^i$ as a two-dimensional asymmetric Gaussian based on the model introduced by \citet{kirby-2010}. Refer to Appendix A for detailed descriptions.

Given the personal spaces $\mathcal{P}^i$, $i\in G^j$, of all agents in a group $j$, we extract the social group space of the whole group as a convex hull:
\begin{equation}
    \mathcal{G}^j=\mbox{Convexhull}(\{\mathcal{P}^i\mid i\in G^j\})\mbox{.}
\end{equation}
The shape described by $\mathcal{G}^j$ represents an obstacle space representation of a group containing agents in close proximity with similar motion characteristics. For convenience, let us collect the spaces of all groups in a scene into a set $\boldsymbol{\mathcal{G}} = \{\mathcal{G}^j\mid j\in \mathcal{J}\}$.

\subsection{Group Space Prediction Oracle}\label{sec:prediction-oracle}

Based on the group-space representation of Sec.~\ref{sec:groupspacerep}, we describe a prediction oracle that outputs an estimate of the future spaces occupied by a set of groups $\boldsymbol{\mathcal{G}}_{t:t_f}$ up to a time $t_f = t+ f$, where $f$ is a future horizon given a past sequence of group spaces $\boldsymbol{\mathcal{G}}_{t_h:t}$ from time $t_h = t -h$ where $h$ is a window of past observations:
\begin{equation}
    \boldsymbol{\mathcal{G}}_{t:t_f} \leftarrow \mathcal{O}(\boldsymbol{\mathcal{G}}_{t_h:t}) = \cup_{j=1:J}\mathcal{O}_j(\mathcal{G}^j_{t_h:t})\mbox{,}\label{eq:oracledef}
\end{equation}
where $\mathcal{O}_j$ is a model generating a group space prediction for group $G^j$. Refer to Appendix B for detailed description of partial input handling.

We implement the oracle $\mathcal{O}_j$ of eq.~\eqref{eq:oracledef} using a simple encoder-decoder network. The encoder follows the 3D convolutional architecture in \cite{Tran-2015} whereas the decoder mirrors the model layout of the encoder. The encoder-decoder network takes as input a sequence\footnote{The oracle input sequence is first converted into image-space coordinates using the homography matrix of the scene. We also preprocess inputs to have normalized scale and group positions. The autoencoder output is converted back into Cartesian coordinates using the inverse homography transform.} $\mathcal{G}_{t_h:t}$ and outputs a sequence $\mathcal{G}_{t+1:t_f}$ which we pass through a sigmoid layer. We supervise the encoder-decoder network's output using the binary cross entropy loss.

We verified the effectiveness of our encoder-decoder network on the 5 scenes of our experiments by conducting a cross-validation comparison against a baseline. The baseline predicts the future shapes by linearly translating the last social group shape using its geometric center velocity. We use Intersection over Union (IoU) as our metric. Between the ground truths and the predictions, this metric divides the number of overlapped pixels by the number of pixels occupied by either one of them. As shown in Table \ref{tab:autoencoder_perf}, our encoder-decoder network outperforms the baseline.

\begin{table}
    \centering
        \caption{Prediction Model Performance}
    \resizebox{.8\columnwidth}{!}{%
    \begin{tabular}{c|c|c|c|c|c|c}
        \toprule
        & Metric & ETH & HOTEL & ZARA1 & ZARA2 & UNIV \\
        \midrule
        \multirow{2}{*}{Baseline} & $\mbox{mIoU}$ (\%) & 83.52 & 90.37 & 88.04 & 89.30 & 85.32 \\
        & $\mbox{fIoU}$ (\%) & 76.32 & 85.38 & 82.14 & 83.88 & 77.24\\
        \midrule
        \multirow{2}{*}{Encoder-decoder Network} & $\mbox{mIoU}$ (\%) & 86.66 & 92.10 & 89.97 & 90.94 & 87.52 \\
        & $\mbox{fIoU}$ (\%) & 78.64 & 86.83 & 83.77 & 85.09 & 78.55\\
        \bottomrule
    \end{tabular}
    }
    \label{tab:autoencoder_perf}
    \vspace{-1em}
\end{table}

\section{Introduction}

Over the past three decades, there has been a vivid interest in the area of robot navigation in pedestrian environments~\citep{minerva-robot,kruse13-navSurvey,kretzschmar_ijrr16,trautmanijrr,Mavrogiannis19}. Planning robot motion in such environments can be challenging due to the lack of rules regulating traffic, the close proximity of agents and the complex emerging multiagent interactions. Further, accounting for human safety and comfort as well as robot efficiency add to the complexity of the problem.

To address such specifications, a common~\citep{kretzschmar_ijrr16,trautmanijrr,learning-social-robot,Kim2016,Everett18_IROS} paradigm involves the integration of a behavior prediction model into a planning mechanism. Recent models tend to predict the individual interactions among agents to enable the robot to determine collision-free candidate paths~\citep{kretzschmar_ijrr16,trautmanijrr,MavBluKne_IROS2017}. While this paradigm is well-motivated, it tends to ignore the structure of interaction in such environments. Often, the motion of pedestrians is coupled as a result of social grouping. Further, the motion of multiple agents can often be \emph{effectively} grouped as a result of similarity in motion characteristics. Lacking a mechanism for understanding the emergence of this structure, the robot motion generation mechanism may yield unsafe or uncomfortable paths for human bystanders, often violating the space of social groups.

Motivated by such observations, we draw inspiration from human navigation to propose the use of group-based prediction for planning in crowd navigation domains. We argue that humans do not employ detailed individual trajectory prediction mechanisms. In fact, our motion prediction capabilities are short-term and do not scale with the number of agents. We do however employ effective grouping techniques that enable us to discover safe and efficient paths among motions of crowd networks. This anecdotal observation is aligned with gestalt theory from psychology~\citep{koffka35} which suggests that organisms tend to perceive and process \emph{formations of entities}, rather than individual components. Such techniques have recently led to advances in computer vision~\citep{Desolneux07} and computational photography~\citep{vazquez11}. Similarly, we envision that a robot could reason the formation of effective groups in a crowded environment and react to their motion as an effective way to navigate safely.

In this paper, we propose a group-based representation coupled with a prediction model based on the group-space approximation model of \citet{wang-split-merge}. This model groups a crowd into sets of agents with similar motion characteristics and draws geometric enclosures around them, given observation of their states. The prediction module then predicts future states of these enclosures. We conduct an extensive empirical evaluation over 5 different human datasets~\citep{ETH,UCY}, each with a flow following and a crossing scenario. We further conduct a same set of evaluations with agents powered by ORCA~\citep{ORCA} that share the start and end locations in the datasets. Last but not least, we conducted evaluation given inputs in the form of simulated laser scans, from which pedestrians are only partially observable or even completely occluded. We compare the performance of our group-based formulation against three individual reasoning baselines: a) a reactive baseline with no prediction; b) a constant velocity prediction baseline; c) one based on individual S-GAN trajectory predictions~\citep{socialGan}. We present statistically significant evidence suggesting that agents powered by our formulation produce safer and more socially compliant behavior and are potentially able to handle imperfect state estimates.

\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{figures/teaser_new.jpg}
\caption{Based on a representation of social grouping~\citep{wang-split-merge}, we build a group behavior prediction model to empower a robot to perform safe and socially compliant navigation in crowded spaces. (a) Example of our representation overlayed on top of a scene from a real-world dataset~\citep{ETH}. (b) A model predictive controller equipped with our prediction model is able to navigate around the group socially (right) as opposed to the baseline that cuts through the group (left). }
\label{fig:personal_space}
\vspace{-1em}
\end{figure}

\section{Model Predictive Control with Group-based Prediction}\label{sec:mpc}

We describe G-MPC, a model predictive control (MPC) framework for navigation in multiagent environments that leverages the group-based prediction oracle of Sec.~\ref{sec:prediction}.

We describe our group-prediction informed MPC, or G-MPC. At planning time $t$, given a (possibly partial) augmented world state history $\boldsymbol{Q}_{t_{\hat{h}} : t}$, we first extract a sequence of group spaces $\boldsymbol{\mathcal{G}}_{t_{h}:t}$ based on the method of Sec.~\ref{sec:groupspacerep}. Given these, the robot computes an optimal control trajectory $\boldsymbol{u}^* = u^*_{1:K}$ of length $K$ by solving the following optimization problem:
\begin{align}
\left(\boldsymbol{s}^*, \boldsymbol{u}^{*}\right) =&  \arg\min_{u_{1:K}} \sum_{k=1:}^{K} \gamma^{k}J(s_{k+1}, \boldsymbol{\mathcal{G}}_{k+1}, s_T)\label{eq:cost}\\
 s.t.\:
         & \boldsymbol{\mathcal{G}}_{2-h:1} \leftarrow \boldsymbol{\mathcal{G}}_{t_{h}:t}\label{eq:initgroups}\\
         & s_1 \leftarrow s_t\label{eq:initrobot}\\
         & \boldsymbol{\mathcal{G}}_{k+1:k_{f}} = \mathcal{O}(\boldsymbol{\mathcal{G}}_{k_{h}:k})\label{eq:updategroup}\\
         & u_k\in\mathcal{U}\\
         & s_{k+1} = s_k + u_k \cdot dt\label{eq:statetransition}
   \mbox{,}
\end{align}
where $\gamma$ is the discount factor and $J$ represents a cost function, eq.~\eqref{eq:initgroups} initializes the group space history ($k = 2-h$ is the timestep displaced a horizon $h$ in the past from the first MPC-internal timestep $k=1$), eq.~\eqref{eq:initrobot} initializes the robot state to the current robot state $s_t$, eq.~\eqref{eq:updategroup} is an update rule recursively generating a predicted future group sequence up to timestep $k_f = k+f$ given history from time $k_h = k - h$ up to time $k$, $\mathcal{O}$ represents a group-space prediction oracle based on Sec.~\ref{sec:prediction}, and eq.~\eqref{eq:statetransition} is the robot state transition assuming a fixed time parametrization of step size $dt$.

We employ a weighted sum of costs $J_g$ and $J_d$, penalizing respectively distance to the robot's goal and proximity to groups:
\begin{equation}
    J(s_{k}, \boldsymbol{\mathcal{G}}_{k}, s_T) = \lambda J_{g}(s_{k}, s_T) +(1-\lambda)J_{d}(s_{k}, \boldsymbol{\mathcal{G}}_k)
    \label{eq:begin_detail_mpc}\mbox{,}
\end{equation}
where $\lambda$ is a weight representing the balance between the two costs and
\begin{equation}
    J_{g}(s_{k}) =\begin{cases}
    0, & \mbox{if}\: s_{k}\in \boldsymbol{\mathcal{G}}_{k}\\
    ||s_{k-1} - s_T||, & \mbox{else,}
    \end{cases}
\end{equation}
penalizes a rollout according to the distance of the last collision-free waypoint to the robot's goal. Further, we define $J_d$ as:
\begin{equation}
    J_{d}(s_{k}, \boldsymbol{\mathcal{G}}_k) = \exp(-\mathcal{D}\left(s_{k+1}, \boldsymbol{\mathcal{G}}_k)\right)\mbox{,}
\end{equation}
where
\begin{equation}
    \mathcal{D}(s_{k}, \boldsymbol{\mathcal{G}}_k) = \begin{cases}
                \min_{j\in \mathcal{J}} D\left(s_k - \mathcal{G}_k^j\right), \:
                & \mbox{if}\: s_{k}\notin \mathcal{G}^j_{k}\\
                -\min_{j\in \mathcal{J}} D\left(s_k - \mathcal{G}_k^j\right), \:
                &\mbox{else}\mbox{,}
               \end{cases}
    \label{eq:end_detail_mpc}
\end{equation}
where $D(s_k - \mathcal{G}_k^j)$ returns the minimum distance between the robot state and the space occupied by group $j$ at time $k$. Using $D$, function $\mathcal{D}$ computes the minimum distance to any group for a given time. In most cases, the robot lies outside of groups, i.e., $s_{k}\notin \mathcal{G}^j_{k}$ --therefore, the cost $J_d$ tries to maximize the distance $\mathcal{D}$. Sometimes, the robot might end up entering the group space $\boldsymbol{\mathcal{G}}$ --in those cases, $J_d$ tries to minimize $\mathcal{D}$, to steer the robot towards the direction of quickest escape from the group. In case that the robot is inside a group to begin with, we shrink the group sizes in Sec.~\ref{sec:groupspacerep} until the robot is outside the groups again.

To solve eq.~\eqref{eq:cost}, we search over a finite set $\boldsymbol{\mathcal{U}}$ of control trajectories of horizon $K$. With the assumption that the robot is holonomic and is not under any kinematic constraints, we use a set of $R$ control rollouts $\boldsymbol{\mathcal{U}} = \{\boldsymbol{u}^1,...,\boldsymbol{u}^R\}$ with three levels of tangential speeds and a set of turning speed, i.e.,
\begin{equation}
    \label{eq:rollout}
    u_{1:K}^r = (v\cos\psi, v\sin\psi, \omega),\: \psi=\frac{2\pi r}{R},
    v\in\left\{\frac{1}{3}v_{max}, \frac{2}{3}v_{max}, v_{max}\right\},
    \omega\in\left\{0, \pm\frac{\pi}{2}\right\}
\end{equation}
To ensure compatibility between our group-based prediction model and our MPC formulation, we set the control rollout time horizon to be the prediction model's prediction horizon, or $K=f$.

\section{Problem Statement}

Consider a robot navigating in a workspace $\mathcal{W}\subseteq\mathbb{R}^2$ amongst $n$ other dynamic agents. Denote by $s\in\mathcal{W}$ the state of the robot and by $s^i\in\mathcal{W}$ the state of agent $i\in\mathcal{N} = \{1,\dots, n\}$. The robot is navigating from a state $s_{0}$ towards a destination $s_{T}$ by executing a policy $\pi:\mathcal{W}^{n+1}\times\mathcal{U}\to\mathcal{U}$ that maps the assumed fully observable world state $\boldsymbol{S} = s \cup_{i = 1:n} s^i$ to a control action $u\in\mathcal{U}$, drawn from a space of controls $\mathcal{U}\subseteq\mathbb{R}^2$. We assume that the robot is not aware of agents' destinations $s^{i}_T$ or policies $\pi_i:\mathcal{W}^{n+1}\times\mathcal{U}^i\to\mathcal{U}^i$, $i\in\mathcal{N}$. In this paper, our goal is to design a policy $\pi$ that enables the robot to navigate from $s_0$ to $s_T$ safely in a socially compliant fashion.
\section{Related Work}

Over the recent years, a considerable amount of research has been placed to the problem of robot navigation in crowded pedestrian environments~\citep{kretzschmar_ijrr16,trautmanijrr,Everett18_IROS,Mavrogiannis18,chen2019crowd,zhi2020anticipatory,Somani2013,Cai2021,Kiss2021}. Such environments often comprise groups of pedestrians, navigating as coherent entities.
\citet{Sochman-2011} suggests that $50$-$70\%$ of pedestrians walk in groups. Many works exist in group detection. One popular area in such domain is static group detection, often leveraging F-formation theories~\citep{Kendon-1990-Fformation}. However, dynamic groups often dominate pedestrian-rich environments and exhibit different spatial behavior~\cite{Yang-2019-groupnav}. Among dynamic group detection, one common approach is to treat grouping as a probabilistic process where groups are a reflection of close probabilistic association of pedestrian trajectories~\citep{bazzani-2012, Chang-2011-group, Gennari-2004, Pellegrini-2010-group, Zanotto-2012}. Others use graph models to build inter-pedestrian relationships with strong graphical connections indicating groups~\citep{chamveha-2013, Khan-2015}. The social force model~\citep{social-force} also inspires \citet{Mazzon-2013, Sochman-2011} to develop features that indicate groups. Clustering is another common group of technique to group pedestrians with similar features into groups~\citep{solera-2016, Ge-2012-group, taylor-2020-group, Chatterjee1}. For our formulation, it is sufficient to employ a simple clustering-based grouping method proposed by \citet{Chatterjee1}. Other grouping methods will simply result in different group membership assignments.

Applications on groups often focus on a specific behavior aspect. For example, one focus in this area is how a robot should behave as part of the group formation~\citep{Cuntoor-2012}. On dyad groups involving a single human and a robot, some researchers examined socially appropriate following behavior~\citep{Gockley-2007, Granata-2012, Jung-2012, Zender-2007} and guiding behavior~\citep{nanavati-dyad, Feil-Seifer-2011, Pandey-2009}. In works that do not include robots as part of pedestrian groups, some research teams studied how a robot should guide a group of pedestrians~\citep{Garrell-2010, Shiomi-2007, Martinez-Garcia-2005}. From navigation perspective, Yang and Peters~\citep{Yang-2019-groupnav} leverage groups as obstacles, but their group space only involves occasional O-space modeling from F-formation theories. Without the engineered occurrence of O-space, their representation reduces to one of our baselines. Katyal et al.~\citep{Katyal-2020-groupnavRL} introduce an additional cost term that leverages robot's distance to the closest group in a reinforcement learning framework. They model groups using convex hulls directly generated from pedestrian coordinates instead of taking personal spaces into consideration.
In our work, we additionally explore the capabilities of groups in handling imperfect sensor inputs. While our focus is on analysing the benefits of groups, our group based formulation can be easily incorporated into the work of~\citet{Katyal-2020-groupnavRL}'s framework.

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