Montgomery Circuit Court Judge Greg Griffin, appointed by the governor in 2014, granted a 10-day TRO against the hearings after arguments Friday.
The Alabama Supreme Court is allowing lawmakers to move forward with impeachment hearings against Gov. Robert Bentley, who has been fighting to stay in office after allegations that he inappropriately used his position to cover up an affair with a top aide. Bentley, who is in his second term, is accused of inappropriate use of state resources. "I have not", Bentley said.
Bentley is seeking a temporary restraining order to block Friday's scheduled release of a report compiled by the special counsel in the impeachment probe. He says he assumed these were communications with Rebekah Mason due to their timing in relation to phone calls between Mason and Governor Bentley, but he never actually saw Mason's name associated with these texts.
The order had halted the impeachment process until hearings could be held on Bentley's claim that lawmakers did not give enough time to present an adequate defence.
Sharman also wrote that Bentley obstructed the legislative investigation by refusing to cooperate and redacting text messages and other material requested by the committee.
The governor's lawyer called the report an "amalgam of hearsay rumor and innuendo".
Republican House Speaker Mac McCutcheon on Friday said that the spectacle has been an embarrassment to the state.
"The people of this state have never asked to be told of or shown the intimate, embarrassing details of my personal life and my personal struggles", Bentley said.
The report included thousands of pages of exhibits, including text messages of Mason and Bentley exchanging declarations of love and affection. The report will be given to the House Judiciary Committee as the committee weighs whether to recommend impeaching Bentley.
Bentley has insisted nothing he did was illegal and said Friday he would not resign.
"That panel says there's probable cause that the governor violated state campaign finance laws".
The controversy erupted a year ago when the former head of state law enforcement, Spencer Collier, a day after being fired by Bentley, publicly accused Bentley of having an affair with his longtime political adviser, Mason.
The governor's lawyers argued the process was unfair to Bentley, while Sharman said impeachment proceedings were a legislative matter outside the court's jurisdiction.


text:
Anopheles letabensis este o specie de țânțari din genul Anopheles, descrisă de Lambert și Maureen Coetzee în anul 1982. Conform Catalogue of Life specia Anopheles letabensis nu are subspecii cunoscute.

Referințe

Legături externe

Anopheles

redpajama_set_name: RedPajamaWikipedia


Your call will be answered by one of our volunteers, the majority of whom have been bereaved by suicide themselves. They will listen to you and answer any questions you may have. If you would like them to, they will post a copy of our support booklet and other information to you.
It can take courage to make that first call – if you find that you hang up, you are welcome to call again. Many people find that they are initially not sure what they would talk about – but after a few minutes they find they are talking freely about what is on their mind.
Is it a counselling session?
No – our services are about self help, connecting people who have been bereaved so that they can support and share with each other. Some people use our helpline as well as counselling sessions. If you want to find out more about counselling services, click here for links to other organisations.
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If our helpline is not open when you want to talk, you may wish to try our email service – though we cannot guarantee an immediate response, many people find that the act of writing down their thoughts is very helpful. If you need to talk with someone and we are not open, you might like to try the Samaritans – their free phone number is 116 123. They do not provide the same service but they will listen to you and provide support.


text:
Canada can learn a lot from China and Europe when it comes to its data strategy, says Deloitte
Buckley Smith
In its quest to form a clear data strategy, Canada should find a balance between the privacy-centric model of the GDPR in Europe and the more business-favored strategy in China, according to Deloitte Touche Tohmatsu Limited.
This was one of the steps suggested by Deloitte in its recently published report about what the Canadian federal government should do to claim a top spot in the global AI ecosystem.
Many have called data the new oil of business, so for Canada to be able to be viewed as a business powerhouse, it must learn to harness this data while still considering the privacy of its citizens, according to Jas Jaaj, partner at Omnia AI, Deloitte.
Jaaj said a clearly defined strategy in regards to data and privacy is especially important because many of the current policies are outdated in relation to the modern world we currently live in.
"What we're seeing is that when it comes to the Canadian policy framework, most of the policies that are in effect were created before the widespread of digitization. Which leaves room for what we call operating risk and legal gray zones for businesses and citizens," Jaaj pointed out when questioned about the current policies in place. "And it's time now, considering where we are in the maturity of AI, to modernize them."
And he points to two specific examples of data strategies that have been successful in their own different ways: Europe and China.
Jas Jaaj, partner at Omnia AI, Deloitte, says that Canada must find a middle ground for its data strategy.
"We always like to strike a balance between the protection aspect of the policy versus policy being used to infuse innovation into an economy. When you look at GDPR, it's rather stringent when it comes to how the data protection framework has been laid out… where they kind of have done an excellent job in terms of protection and the privacy aspect of it. Having said that, it's not as optimized for infusing innovation. When you look at China, on the other hand, when it comes to their digital rights frameworks that they use, they are on the other end, where it's probably not as well protected when it comes to their citizens as it can be but definitely sparks that innovation within the ecosystem."
And he said this is a perfect opportunity for Canada to define the middle ground between the two extremes of data strategies.
"Canada has an opportunity right now to strike that right balance by learning from these different geographies," said Jaaj. "To be able to come up with that optimal mix and then lead the globe when it comes to demonstrating how AI can be used not only to protect and safeguard the rights and values of Canadians but also use it as an innovation engine to grow the economy."
One concept suggested in the report was the idea of data trusts. In such a scenario, an independent organization would be entrusted with the data of multiple organizations (whether they be private corporations or government bodies) and would be responsible for deciding when and how to use the data entrusted to them.
Beyond the privacy concerns that could be allayed through this model of "data custodians", as he calls them, Jaaj also explained that this could lead to opportunities for collaboration between data sets that may not have been possible without a data trust.
"The way in which you get the most value out of data is by combining data sets that typically don't sit together. The trusts become a mechanism to be able to bring them together and then provide that incremental value by finding insights at the intersection of the data."
Sponsor: BlackBerry
The Senior Leader's Guidebook to Emergency Management and Business Continuity
Public Sector data strategy, Deloitte, federal government, GDPR
Hashtag Trending - Facebook announces cryptocurrency; Google Calendar crashes; CIA spied through smart TV's
About Buckley Smith
Staff writer for IT World Canada. Covering the world of technology as it applies to business. Buckley is an avid sports fan who loves travel, food, and music.
Hashtag Trending – AI and the Beatles, British Airways gets fined, Steve Jobs was a 'wizard'
UK regulator proposes fining British Airways a record $300 million over data breach
LogPoint continues to expand into global markets
Canadian Blood Services protecting its data against natural disasters with co-location data centre system

redpajama_set_name: RedPajamaCommonCrawl


text:
Starbucks' Schultz announces halt to stock buybacks, shares fall
By Hilary Russ Posted on April 4, 2022
FILE PHOTO: Starbucks in New York City
NEW YORK (Reuters) -Starbucks Corp will pause billions of dollars of stock buybacks to invest more in employees and stores, longtime former chief executive Howard Schultz said on Monday on his return to lead the global coffee chain for a third time.
Schultz disclosed the plan in a letter to Starbucks workers, customers and shareholders at a time when the company faces growing unionization of its U.S. workforce.
Shares closed more than 3% lower. Analysts predicted the suspension to knock 2 to 3% off earnings in fiscal 2023.
Starbucks spent nearly $12 billion in fiscal 2019 and 2020 combined to buy back its own shares. In October it said it would spend $20 billion over three years on stock repurchases and dividends.
Employees at 10 U.S. Starbucks locations have voted in recent months to join Workers United, an affiliate of the Service Employees International Union.
Baristas at more than 170 U.S. locations have petitioned a federal labor board for union elections since August, when the union drive went public.
Starbucks announced last month that Schultz would take over as interim CEO, following the retirement of Kevin Johnson. Shares of the coffee chain shot up about 5% on March 16, the day of the announcement.
By suspending share buybacks, Schultz is signaling he wants to make bold moves and shift the company's cash flow to invest in growth and employees while trying to fend off unions, said Ivan Feinseth, chief investment officer of Tigress Financial Partners, which holds Starbucks stock in its clients' accounts.
"You have to take the opportunity to invest money where it will get the best return on multiple levels both as a direct investment and to show what the company's current focus is," Feinseth said.
Schultz is practically synonymous with the company he took over in 1987. During his previous four decades as chief executive and chairman, it grew from 11 stores to more than 28,000 in 77 markets worldwide, according to a news release.
Schultz wrote in the letter that he will travel in coming weeks to meet with employees.
"I am returning to the company to work with all of you to design that next Starbucks — an evolution of our company deep with purpose, where we each have agency and where we work together to create a positive impact in the world," he wrote.
(Reporting by Hilary Russ, writing by Stephanie Kelly; Additional reporting by Praveen Paramasivan; Editing by Sam Holmes, Kirsten Donovan and Howard Goller)
Starbucks Chairman and CEO Schultz delivers remarks at the Starbucks 2016 Investor Day in Manhattan, New York
U.S. Senate negotiators reach deal on $10 billion for COVID aid
Biden urges Putin war crimes trial after Bucha killings
Exxon signals record quarterly profit from oil and gas prices
Omicron sub-variant BA.2 makes up 72% of COVID variants in U.S. – CDC


text:
The Salmon Falls (East) Historic District encompasses a node of rural development in western Buxton, Maine.  It includes thirteen historic properties, mainly Greek Revival in character, extending north along Maine State Route 117, north of its junction with Simpson Road, and two on the latter street, that were all built between about 1800 and 1850.  The district was listed on the National Register of Historic Places in 1987.

Description and history
The Salmon Falls area, divided between Buxton and Hollis by the Saco River, developed beginning in the late 18th century as a modest agricultural area, and the falls on the river, for which it is named, were developed in the second quarter of the 19th century with small scale industrial operations.  The road that is now Maine State Route 117 was the main road in the area leading north from the port of Saco, and became the focus for development on the Buxton side of the village.  The area declined economically in the second half of the 19th century, and further growth was consequently quite limited.

The district is centered along a stretch of Route 117 extending south from a bend in the Saco River to the road's junction with Simpson and Elden Roads.  Eleven properties line this piece of road, with another two on Simpson and Elden completing the district inventory.  The oldest house, the Elden family homestead, stands on the south side of Elden Road, and is a fine Federal-style estimated to date to the 1790s.  The Came-Marshall House, located just northwest of the junction, is probably the most impressive house in the district: it is a 2-1/2 story Federal-style brick structure, with an elaborate entry that has sidelight windows and a semi-circular fanlight.  Most of the houses in the district, however, were built between 1825 and 1850, and are Greek Revival in character.  Most are wood-frame structures 1-1/2 or 2-1/2 stories in height, although one is built out of brick.  The latest building in the district is the c. 1868 Darrah Tavern, which anchors the district's northern end.

See also

Salmon Falls (West) Historic District, across the Saco River in Hollis
National Register of Historic Places listings in York County, Maine

References

Federal architecture in Maine
Greek Revival architecture in Maine
Buxton, Maine
Historic districts on the National Register of Historic Places in Maine
National Register of Historic Places in York County, Maine


This event is set up to allow members to order name tags online. We will offer this opportunity several times throughout the year and will provide the opportunity to order with registration for the annual meeting.
The board typically meets evenings on the first Tuesday of the month. All members are welcome and encouraged to attend, only board members can vote.
Election of board members, vote on positions and program for fiscal year 2019-2020.
Learn about Solar and how community purchasing power can make it more affordable.


GA? and yExplore are now Firefox 5 ready.
I received emails from Mozilla yesterday that stated GA? – Is Google Analytics Installed and yExplore are compatible with Firefox 5. So I've updated the version numbers in the extensions and they are ready to go.
Next Twin Cities Twitter weather alerts are working again.
Previous Lily's getting ready to plant the garden.


Explore the Atlantis legend from its birth in the prehistoric world and its transmission through time. Follow the Atlantis tales long journey in history kept alive by ancient lost cultures in their myths and legends and rich wall art, cave paintings and frescoes.
Look out for New book title and future posts on the release date via Amazon in print and Kindle E-book format.


Incorporating Jazzit in your firm reduces manual processes, the stress of producing your own templates, and improves client satisfaction, profitably and staff morale.
Now you can focus on more critical tasks and services that generate value for both your clients and your firm.
Jazzit offers three products for CaseWare Working Papers: Jazzit Fundamentals, Jazzit Checklists and Jazzit SCORE, creating a powerful suite of automated solutions for small and medium sized practitioners. Jazzit Fundamentals, our principle product, is an integrated suite of templates that assists public accountants in completing client engagements.
All of our products incorporate customer feedback to create the most comprehensive collection of templates for CaseWare Working Papers.
Why do Accountants rely on Jazzit?
When you open a Jazzit template it automatically compares itself to the newest version in the Resource Centre (a central library) and notifies you when an update is available.
Each template includes a green strip at the top indicating the version that is currently in the client file and the version in the Resource Centre. You no longer have to guess if your template is up to date.
Accountants automate workflow which increases speed of preparation and decreases errors.
Working papers share key information with the financial statements, notes and other working papers dramatically reducing data entry and transfer errors.
For example, the Jazzit long term debt working paper shares the lender, rate, payment, collateral and loan balance with the long term debt note in the Jazzit financial statement template.
Consistency and continuity with clients financial reports and letters from year to year is very easy.
Data automatically moves to appropriate columns when a year-end close is completed in CaseWare.
For example, the property plant, and equipment working paper moves the ending amounts to the opening column and clears out the additions and disposals columns on roll forward.
Increase quality of your service to clients; find and correct errors as they are made.
Increases file accuracy. Errors can be caught before they are sent to the reviewer.
Our diagnostics notify the preparer if an amount on the working paper does not agree to the balance in the CaseWare trial balance. This eliminates the back and forth between preparer and reviewer to correct this type of error.
At the heart of Jazzit's flexibility is the Jazzit Resource Centre. By managing the templates in a centralized Resource Centre (typically located on your server) you are able to make changes to individual templates within the Jazzit suite. You will now have those changes available for all your client engagements.
In any Jazzit working paper, letter or checklist, you will see at the top of the document the version currently in the client file and the most recent version available in the Resource Centre. You can decide whether or not to update this particular template.
In the Jazzit financial statements, every statement page, note and accounting policy is individually date stamped in the Resource Centre. You are able to control which statement pages, notes and policies you wish to update.
Users are automatically notified when a update is available in the Jazzit Resource Centre. This is shown in the green strip at the top of Jazzit templates and within the Home Menu of the financial statements.
Jazzit Administrators have a control document within the Resource Centre which manages updates by reporting the version of all the modules within the Resource Centre and providing a button to download the updates from the Jazzit website. This Resource Centre update tool identifies exactly which modules have updates available. Resource Centre modules can be updated to the latest version with a click of a button. Once the Resource Centre has been updated, preparers are automatically notified of updates as they open Jazzit templates within the client engagements.


text:
Home Architecture of Life
Staff Writer, May 12, 2016 May 12, 2016 , Architecture, Visual arts, 0
The question of what a particular museum aims to be is, today, so often preceded by the question of who's behind its design. The recent construction boom among major American art institutions has resulted in the same kind of name-dropping that typically accompanies an Academy Awards red carpet: the San Francisco Museum of Modern Art, reopening later this month, has been completely made over, its original Mario Botta building transformed by Norwegian firm Snøhetta; the Whitney Museum's much-lauded new home boasts a tiered design by Renzo Piano; and the Broad's eye-catching flagship building comes courtesy Diller Scofidio + Renfro, as does the Museum of Modern Art's sleek future expansion. The Berkeley Art Museum and Pacific Film Archive, which reopened this January in a new building also designed by Diller Scofidio + Renfro, is far more modest by comparison, but similar questions that have been asked of those higher-profile projects, about what a new home means for an institution's identity and how it might recalibrate its relationship to its public, pertain.
JOE IURATO "EVERYTHING BUT THE WHOLE STORY"
Staff Writer, October 15, 2015 October 15, 2015 , Conceptual art, Graffiti, Graphic, Visual arts, 0
This past week Station 16 Gallery in Montreal opened Everything but the Whole Story, a solo show featuring Joe Iurato. Based on...
MAÏMOUNA GUERRESI
Staff Writer, February 15, 2016 February 15, 2016 , Photography, Visual arts, 0
Guerresi's subjects appear mystical, infused with references to Sufi Islam, a religion she converted to from Catholicism when she...
JAMES JEAN "ZUGZWANG"
Staff Writer, November 12, 2015 November 12, 2015 , Painting, Visual arts, 0
James Jean knows chaos. Thrashing waves of lines swirl around his figures, who try to make sense of the...
A FANTASY WORLD MADE BY OLEG OPRISCO
Staff Writer, August 17, 2015 August 17, 2015 , Visual arts, 0
We had never really thought of it before…. but having a BBQ underwater? That's right, when Bruce Mozert starting...
HOME IS WHERE THE ART IS
Staff Writer, May 2, 2016 May 2, 2016 , Photography, Visual arts, 0
Inspiring this renovation is the gift of 260 works from the art compilation of Doris and Donald Fisher, who...
DESTROYING THE WEAK by CLEON PETERSON X CASE STUDYO
Staff Writer, October 25, 2015 October 25, 2015 , Sculpturing, Visual arts, sculpturing, visual art, 0
We just had a good week with Cleon; his shirt sold out in our webstore about as soon as...
CALEB BROWN
Staff Writer, December 22, 2015 December 22, 2015 , Painting, Visual arts, animals, Internet's obsession, paintings, 0
Artist Caleb Brown is one of many brilliant painters featured in our new book, Juxtapoz Wild. Below, the artist...
ORDER AND CHAOSThe Execution of Lady Jane Grey


text:
Commandments: Law and Religion Blog
U.S. Supreme Court Rules for Colorado Baker: Answering Questions About the Masterpiece Cakeshop Case
This month, the U.S. Supreme Court issued its opinion in Masterpiece Cakeshop, Ltd. v. Colorado Civil Rights Commission. The case was closely watched, particularly because the High Court could have answered some tough questions about religious liberty, freedom of speech, and anti-discrimination laws protecting LGBT rights. The Court's opinion, however, dodged some of the harder questions, but gave a victory to Jack Phillips, the Colorado baker. Here, we answer some common questions about the case.
What was the case about again?
Back in 2012, Jack Phillips, the owner of Masterpiece Cakeshop, was visited in his shop by a couple who asked Phillips to create a cake celebrating their wedding. Phillips declined, noting that he did not create cakes for same-sex weddings on the ground that doing so would violate his sincerely held religious beliefs about marriage. He offered to sell the couple other goods, but not a wedding cake. The couple later brought suit against Phillips for discrimination.
How did the couple sue the baker for refusing to create a wedding cake?
Colorado law provided the couple here with the right to make a legal complaint about the refusal of service. The couple brought a discrimination complaint to the Colorado Civil Rights Division (CCRD), claiming a violation of Colorado's public accommodation laws. Most states have public accommodation laws, which generally provide that when people or businesses sell goods and services to the public, they must do so without regard to certain protected characteristics of the customer. Most states protect traits like race or color. In Colorado, sexual orientation is a protected characteristic (this is not the case in all parts of the country).
The Colorado enforcement agency and courts have interpreted the refusal to provide a cake for a same-sex wedding as equivalent to refusing to provide goods/services based on sexual orientation (rather than as a disagreement with the concept of same-sex marriage), on the theory that, for the most part, those seeking a cake for a same-sex wedding will be LGBT, and that disagreeing with same-sex weddings implies disagreement with the whole LGBT lifestyle, which implies discrimination against LGBT persons.
What happened before the case got to the Supreme Court?
The baker lost at every level. The CCRD found a violation of the state's public accommodation law and referred the case to a seven-member panel called the Colorado Civil Rights Commission, which in turn, sent the case to an ALJ for a formal hearing. The ALJ sided with the couple and the CCRD, and the Commission affirmed that decision in full. As would become important to the U.S. Supreme Court's ultimate decision, several Commission members expressed disdain for Mr. Phillips's beliefs. Most notably, one member commented during a public hearing about Phillips's case as follows:
Freedom of religion and religion has been used to justify all kinds of discrimination throughout history, whether it be slavery, whether it be the holocaust, whether it be—I mean, we—we can list hundreds of situations where freedom of religion has been used to justify discrimination. And to me it is one of the most despicable pieces of rhetoric that people can use to—to use their religion to hurt others.1
Meanwhile, the Commission was also hearing a series of cases involving another individual, William Jack, who had unsuccessfully attempted to have several bakeries create cakes with anti-gay marriage messages and Bible verses. Jack brought claims for discrimination on the basis of creed to the CCRD. In considering Jack's cases, the Commission found no discrimination, noting that the bakeries declined because the message was offensive to the bakers.
Phillips next appealed to the Colorado Court of Appeals, which affirmed the decision of the Commission. The Court of Appeals did not see an issue with the treatment Phillips received below, distinguishing the Jack cases. The Colorado Supreme Court declined to review that decision, but the U.S. Supreme Court then agreed to hear the case.
What issues were before the Court?
The Court actually granted certiorari to consider "Whether applying Colorado's public accommodations law to compel Phillips to create expression that violates his sincerely held religious beliefs about marriage violates the Free Speech or Free Exercise Clauses of the First Amendment." (But that is not really the question that the Court answered.)
What did the Supreme Court decide?
While the Masterpiece Cakeshop case presented an opportunity for the U.S. Supreme Court to resolve some of these thorny issues, it did not. Instead, the Court held that the hostility toward Phillips's religious beliefs and the differential treatment he received v. the Jack cases demonstrated that he did not receive a neutral, fair adjudication. For those reasons, the Court concluded that the judgment must reversed.
The Supreme Court ruled 7-2 that the Commission's actions violated the Free Exercise Clause of the First Amendment. The case was decided on religious freedom grounds under a narrow exception to the normal rule that free exercise of religion must typically yield to neutral state laws. Normally, the government needs only a rational basis to burden someone's free exercise of religion (here, applying Colorado law to force Phillips to create a wedding cake in violation of his religious beliefs about marriage). But if the government acts out of hostility to religion in burdening free exercise, the case is reviewed differently. If government does not act neutrally toward religion, it must justify its burden on religion at a much higher level.
Nor did the Court decide the case on free speech grounds. So, it did not decide whether creating a wedding cake is expressive conduct that qualifies for protection under the Free Speech Clause of the First Amendment.
This means that the arguments for or against free speech or free exercise rights can continue to be litigated in courts across the country.
What does Masterpiece mean for religious freedom?
Siding with the baker here sends a message that religious beliefs must be respected and cannot simply be dismissed as bigotry. Justice Kennedy's opinion spends a lot of time explaining how LGBT individuals must be treated with respect in the marketplace, while on the other hand recognizing that disagreement with same-sex marriage on religious grounds must also be respected as a legitimate belief.
The Court seemed to set up a kind of continuum of religious freedom vis-à-vis state protection of LGBT rights, but leaves it up to future cases to decide exactly where each side's rights must yield. The Court remarked that, for instance, it would clearly be unconstitutional for a state to compel clergy to perform a same-sex wedding ceremony if doing so conflicted with the clergy's religious beliefs. But that same theory could not be extended to give every business owner or purveyor of goods/services an exemption to providing goods and services connected with same-sex weddings. The Court, however, leaves the exact line unclear.
Does Masterpiece apply differently for individuals and for ministries?
While the general principle that the government must not be hostile to religion is the same, it is likely added protections will apply. In the case of an individual, the courts can only look at whether the religious beliefs are sincerely held as a beginning point. But in the case of a ministry, the court is generally not allowed to examine decisions made as a matter of religious doctrine, because courts are forbidden to do that under a doctrine called ecclesiastical abstention. So a church or other ministry will likely have more freedom to follow its religious beliefs—as Justice Kennedy indicated when he commented that it would clearly be unconstitutional to force clergy to perform a same-sex wedding ceremony. The key for the ministry would be exactly how it had defined its beliefs.
Does the case mean that similar cases will come out in favor of the business owner?
Not necessarily. Because the decision in this case turned on the unique facts presented, the opinion does not do much by way of creating a rule for future courts to apply in deciding these cases. In fact, there has been one case that has come out since Masterpiece that uses the opinion to support a finding of discrimination. However, if a business owner can show that the state agency had a similar animus toward his or her religious beliefs, Masterpiece will definitely be relevant.
1 Masterpiece Cakeshop, Ltd. v. Colo. Civil Rights Commission, 584 U.S. ___ (2018), slip op. at 13.


This uber cool destination is so accessible from the UK. The glorious north is our favourite bit of the island as it is slightly more undiscovered and away from the hectic crowd-filled beaches and bars of the south (altho' it would be a shame not to have a quick peek at all those beautiful people!).
With its hippy markets, international restaurants and beaches Ibiza never fails to impress those who visit for the first time. It has attracted a very eclectic bunch since the 60s when the first hippies arrived.
From simple organic fare for the children to delicious fresh salads & awe-inspiring dishes by International chefs, there is a bit of everything here. The island is humming with fabulous restaurants - chefs have migrated here from Paris, London and New York. Due to the high standards that have been set in recent years the restaurants get better and better. There are excellent Spanish staples including gazpacho, croquetas and paella as well as Ibizan dishes such as Zarzuela and Sofrit Pages (a spicy stew).
There are many organic farms in the interior of the island ensuring that the restaurants are stocked with the very best ingredients. If you've never tried a Spanish vodka and tonic we suggest you charge yourself... the jota b and coke is a favourite among the young. The cocktails are strong but oh so good.
Wonderful hippy markets have been on the island since the early 1970's.
Beaches to die for: the crystal clear waters at Es Cana, Cala Llonga and Playa d'en Bossa will leave you entranced.
Diving the coral reef in Ibiza is an experience not to be missed. There are diving lessons at the excellent diving school in Portinax.
Watch the sunrise come up from the dazzling, open air dance floors...or just dance all day. Frankly, day is night and...night is day.
There is a good choice of different golf clubs on the island, good for all standards. There is so much for children to do in the day, some restaurants have fabulous creches so that the children are just as happy as you.
If it's nightlife you're after, you've come to the right place: top clubs, famous djs, hot bands, a cosmopolitan mix of clubbers, local residents, freaks, mature partygoers, fresh faces, and the full alphabet of celebrities.
But equally stay up north and just enjoy the gentle sound of hippy drums on moonlit beaches in contrast to the crazy south of the island. Dance until dawn wherever you are on the island and finish your night with chocolate and churros as you wander home.
Watching the sun rise and set into the sea is a favourite pastime in Ibiza. However, you mustn't leave the island without your fair share of oggling the people, spend the day at the Blue Marlin Cafe or The Jockey Club and just stare! Es Vedra, the rock in the south, is just off the coast and looks just beautiful day or night. There is loads laid on for children with many restaurants running creches so that you grown ups can enjoy a nice long lunch. There's some culture in Ibiza town which has a very old centre.
Ibiza is famous for many things but clubbing is one of its most infamous. We love the more relaxed side to the island, drumming on the beaches in the north at sunset, shopping at the hippy markets or taking a boat trip over to Formentera and hanging out on the beaches there.
Ibiza is a melting pot of celebrities including Jade Jagger, Kate Moss and Daisy Lowe. It has always attracted beautiful people from Errol Flynn to Katie Price. Sienna Miller is a regular with Jude Law. The Spanish celebrities love it too with Penelope Cruz often there on holiday. Sly Stallone is a big fan and others papped in their bikinis include Kylie, Keira Knightly and Liz Hurley.


text:
\section{\,}
\label{app:proof_lyap_ham}
\begin{proof}
Let $\mathbf{p}_e = \mathbf{p} - \mathbf{p}^*$ be the position error and
$\mathbf{R}_e \coloneqq \mathbf{R}^{*\top} \mathbf{R} =
\begin{bmatrix}
	\mathbf{r}_{e1} & \mathbf{r}_{e2} & \mathbf{r}_{e3}
\end{bmatrix}^\top$
be the rotation error, then the error state can be expressed as $\mathbf{x}_e \coloneqq (\mathbf{\mathfrak{q}}_e, \mathbf{\mathfrak{p}}_e)$, where
\begin{equation*}
    \mathbf{\mathfrak{q}}_e = \begin{bmatrix} (\mathbf{p} - \mathbf{p}^*)^\top\quad 	\mathbf{r}_{e1}^\top \quad \mathbf{r}_{e2}^\top \quad \mathbf{r}_{e3}^\top \end{bmatrix}^\top, \quad \mathbf{\mathfrak{p}}_e = \mathbf{\mathfrak{p}}.
\end{equation*}
Since the system is fully-actuated, i.e. the input matrix $\mathbf{B}_{\boldsymbol{\theta}}(\bf{\mathfrak{q}})$ is invertible. The controller in \eqref{eq:stab_ctrl_law} exists and the resulting closed-loop error dynamics becomes \citep{duong21hamiltonian}:
\begin{equation}
\label{eq:desired_port_Hal_dyn_tracking}
\begin{bmatrix}
\dot{\mathbf{\mathfrak{q}}}_e \\
\dot{\mathbf{\mathfrak{p}}}_e \\
\end{bmatrix}
= \begin{bmatrix}
\bf0 & \mathbf{J} \\
-\mathbf{J}^\top & -\mathbf{K}_d\end{bmatrix}
\begin{bmatrix}
\frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{q}}_e} \\
\frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{p}}_e}
\end{bmatrix},\qquad \mathbf{J} =\begin{bmatrix}
	\mathbf{R}^\top\!\!\!\! & \bf0 & \bf0 & \bf0 \\
	\bf0 & \hat{\mathbf{r}}_{e1}^\top & \hat{\mathbf{r}}_{e2}^\top & \hat{\mathbf{r}}_{e3}^\top
	\end{bmatrix}^\top.
\end{equation}
Using group property, rotation error matrix $\mathbf{R}_e = \mathbf{R}^{*\top}\mathbf{R} \in SO(3)$, therefore, $\mathbf{R}_e$ is a orthogonal matrix. All columns of $\mathbf{R}_e$ are orthonormal and all elements in $\mathbf{R}_e$ are less than $1$, hence, $\tr\prl{\mathbf{I} - \mathbf{R}^{*\top} \mathbf{R}} \geq 0$.
Since $\mathbf{M}$ is a positive definite matrix, it is easy to see that ${\cal H}_d$ is positive definite, and $0$ minimum value is achieved only at $\mathbf{x}^*_e = ({\mathfrak{q}}_e, \bf0)$ with ${\mathfrak{q}}_e = [\bf0^\top, \mathbf{e}_1^\top, \mathbf{e}_2^\top, \mathbf{e}_3^\top]^\top$.
The time derivative can be computed as:
\begin{equation}
\begin{aligned}
\dot{{\cal H}}_d(\mathbf{x}, \mathbf{x}^*) &= \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{q}}_e}^\top\dot{\mathbf{\mathfrak{q}}}_e + \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{p}}_e} ^\top \dot{\mathbf{\mathfrak{p}}}_e\\
              &= \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{q}}_e}^\top \mathbf{J} \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{p}}_e} - \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{p}}_e}^\top \mathbf{J}^\top  \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{q}}_e} - \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{p}}_e}^\top \mathbf{K}_d \frac{\partial \mathcal{H}_d}{\partial \mathbf{\mathfrak{p}}_e}\\
              & = -{\mathfrak{p}}_e^\top \mathbf{M}^{-1}({\mathfrak{q}})\mathbf{K}_d \mathbf{M}^{-1}({\mathfrak{q}}) {\mathfrak{p}}_e.
\end{aligned}
\end{equation}
Hence, $\dot{{\cal H}}_d(\mathbf{x}, \mathbf{x}^*) \leq 0$ for all $\mathbf{x}_e$. It is not hard to show that the only point can stay within set
$\crl{\dot{{\cal H}}_d = 0}$ is at origin. By the LaSalle's invariance principle \citep{khalil2002nonlinear}, the system  \eqref{eq:desired_port_Hal_dyn_tracking} asymptotically stabilizes to desired equilibrium $\mathbf{x}^*_e$, i.e. $\mathbf{x}^* = ({\mathfrak{q}}^*, \bf0)$.
\end{proof}
\section{\,}
\label{app:proof_safe_stab}
\begin{proof}
From Lemma~\ref{lem:lyap_ham}, we know that for any constant $\mathbf{x}^* = ({\mathfrak{q}}^*, \bf0)$, ${\cal H}_d(\mathbf{x}, \mathbf{x}^*)$ is a Lyapunov function to certificate stability of $\mathbf{x}^*$. Hence,
\begin{equation*}
    {\cal S}(\mathbf{x}, \mathbf{x}^*) = \crl{\mathbf{x} \mid \Delta E(\mathbf{x}, \mathbf{x}^*) \geq 0} = \crl{\mathbf{x} \mid H_d(\mathbf{x}, \mathbf{x}^*) \leq \frac{2 \bar{d}{\,}^2(\mathbf{p}^*, {\cal O})}{k_\mathbf{p}}}
\end{equation*}
is forward invariant and control signal ${\boldsymbol{\pi}} \prl{\mathbf{x}, \mathbf{x}^*}$ can steer the state $\mathbf{p}$ of the system towards $\mathbf{p}^*$.  It remains to show that during the convergence, constraints over $\mathbf{p}(t)$ is never violated. In the proof of Lemma~\ref{lem:lyap_ham}, we have shown that the second term of desired Hamiltonian~\eqref{eq:desired_hamiltonian}, $\frac{1}{2} k_{\mathbf{R}}\tr(\mathbf{I} - \mathbf{R}^{*\top}\mathbf{R})$ is non-negative for all $\mathbf{R}^* \in SO(3)$, therefore,
$\frac{2}{k_\mathbf{p}} \mathcal{H}_d \prl{\mathbf{x}, \mathbf{x}^*} \geq (\mathbf{p} - \mathbf{p}^*)^\top (\mathbf{p} - \mathbf{p}^*).$
From \eqref{eq:dyO} and above inequality, we know that $\mathbf{p}(t) \in {\cal F}$ for all $t \geq t_0$, since
\begin{equation*}
d^2(\mathbf{p}^*, {\cal O}) \geq \bar{d}\,^2(\mathbf{p}^*, {\cal O}) \geq \frac{2}{k_\mathbf{p}} \mathcal{H}_d \prl{\mathbf{x}_0, \mathbf{x}^*} \geq \frac{2}{k_\mathbf{p}} \mathcal{H}_d \prl{\mathbf{x}(t), \mathbf{x}^*} \geq d^2 \prl{\mathbf{p}^*, \mathbf{p}(t)}.
\end{equation*}
\end{proof}

\section{\,}
\label{app:proof_safe_output_tracking}
\begin{proof}
    For conciseness, $\Delta E(t) = \Delta E \prl{\mathbf{x}(t), {\boldsymbol{\ell}}(\mathbf{g}(t), \bar{\bfg}(t))}$.
	Initially, $\mathbf{g}_0 = \mathbf{p}_0 = \mathbf{r}(0) \in \mathcal{LS}(\mathbf{x}_0,\mathbf{g}_0)$ and $\Delta E(t_0) > 0$, the local projected goal $\bar{\bfg}$ and associated rotation matrix $\mathbf{R}^*(\mathbf{g}, \bar{\bfg})$ is well defined. As $\bar{\bfg}$ moves along the reference path $\mathbf{r}$, i.e., path parameter $\sigma$ in \eqref{eq:lpg} increases. While $\mathbf{g}$ chasing $\bar{\bfg}$ by \eqref{eq:governor}, system state $\mathbf{x}$ tracks $\mathbf{x}^* = {\boldsymbol{\ell}}(\mathbf{g}, \bar{\bfg})$ using controller ${\boldsymbol{\pi}}(\mathbf{x}, \mathbf{x}^*)$ in \eqref{eq:stab_ctrl_law}.
	During this process, the safety margin $\Delta E(t)$ is fluctuating which regulates behavior of $\mathbf{g}$ through $\bar{\bfg}$. Since the system dynamics are continuous, $\Delta E(t)$ cannot become negative without crossing $0$ from above at some time $T_0$. As $\Delta E(t) \downarrow 0$, the local safe zone will shrink to a point, i.e., $\mathcal{LS}(\mathbf{x}, \mathbf{g}) \downarrow \crl{\mathbf{g}}$ . This immediately stops the movement of governor because $\bar{\bfg} = \mathbf{g}(T_0)$ and $\dot{\mathbf{g}}(T_0) = 0$.
	From Proposition~\ref{prop:safe_stab}, we know that $\mathbf{x}(t)$ will stay within ${\cal S}(\mathbf{x}, \mathbf{x}^*(T_0))$ for $t \geq T_0$ and position constraints will not be violated. As $\mathbf{x} \rightarrow \mathbf{x}^*(T_0)$ because $\dot{H}_d(\mathbf{x}, \mathbf{x}^*(T_0)) < 0$ when $\mathbf{g}$ is static and $\mathbf{x} \neq \mathbf{x}^*(T_0)$, there exists $h > 0$ such that $\Delta E(T_0 + h)$ becomes strictly positive. Hence, the governor is able to  move again towards new $\bar{\bfg}$ getting further along the path as discussed previously. This process continues until the augmented system stabilized at $\prl{{\boldsymbol{\ell}} \prl{\mathbf{r}(1), \mathbf{r}(1)}, \mathbf{r}(1)}$ where $\bar{\bfg}$ stops changing.
\end{proof}
\section{Learning Hamiltonian Dynamics on the $SE(3)$ Manifold}
\label{subsec:ham_dyn_learning}
\subsection{Training a translation-equivariant $SE(3)$ Hamiltonian dynamics model}
\label{subsec:ham_neural_ode}
We obtain the training dataset ${\cal D} = \{t^{(i)}_{0:N}, {\mathfrak{q}}^{(i)}_{0:N},{\boldsymbol{\zeta}}^{(i)}_{0:N}, \mathbf{u}^{(i)}\}_{i = 1}^{D}$ by applying a constant input $\mathbf{u}^{(i)}$ to the system and sampling the generalized coordinates and velocities at times $t_0^{(i)} < t_1^{(i)} < \ldots < t_N^{(i)}$ using an odometry algorithm (\cite{vio_benchmark,OdometrySurvey}) or a motion capture system. The control input $\mathbf{u}^{(i)}$ can be generated by manually driving the robot or using an  existing controller. Since the system dynamics does not change if we shift the position $\mathbf{p}$ to any points in the world frame,  we offset the trajectories in the dataset ${\cal D}$ so that they start from the position $\bf0$ and learn the system dynamics well around the origin. This is sufficient for control purposes, e.g. using the controller design in Sec. \ref{sec:ham_control}, because the control input driving the system from state $\mathbf{x}$ with position $\mathbf{p}$ to the desired state $\mathbf{x}^*$ with position $\mathbf{p}^*$ is the same as the one driving the system from the state $\mathbf{x}$ with position $\bf0$ to the desired state $\mathbf{x}^*$ with the offset position $\mathbf{p}^* - \mathbf{p}$.

Since the mometum ${\mathfrak{p}}$ is not directly available from the dataset ${\cal D}$, we use the time derivative of the generalized velocity, derived from \eqref{eq:momenta_Mtwist}:
\begin{equation}
\label{eq:hamiltonian_zetadot}
\dot{{\boldsymbol{\zeta}}} =  \prl{ \frac{d}{dt} \mathbf{M}^{-1}(\mathbf{\mathfrak{q}}) }\mathbf{\mathfrak{p}} + \mathbf{M}^{-1}(\mathbf{\mathfrak{q}})\dot{\mathbf{\mathfrak{p}}}.
\end{equation}
Eq. \eqref{eq:Ham_sys_pf_nlin} and \eqref{eq:hamiltonian_zetadot} describe the Hamiltonian dynamics of the generalized coordinates and velocities with unknown inverse generalized mass matrix $\mathbf{M}^{-1}({\mathfrak{q}})$, input matrix $\mathbf{B}({\mathfrak{q}})$, and potential energy ${\cal U}({\mathfrak{q}})$, for which we aim to approximate by three neural networks $\mathbf{M}_{\boldsymbol{\theta}}^{-1}({\mathfrak{q}}), \mathbf{B}_{\boldsymbol{\theta}}({\mathfrak{q}})$ and ${\cal U}_{\boldsymbol{\theta}}({\mathfrak{q}})$, respectively, with parameters ${\boldsymbol{\theta}}$.

To optimize for the parameters ${\boldsymbol{\theta}}$, we use the Hamiltonian-based neural ODE framework that encodes the Hamiltonian dynamics \eqref{eq:Ham_sys_pf_nlin} and \eqref{eq:hamiltonian_zetadot} with $\mathbf{M}_{\boldsymbol{\theta}}({\mathfrak{q}}), \mathbf{B}_{\boldsymbol{\theta}}({\mathfrak{q}})$ and ${\cal U}_{\boldsymbol{\theta}}({\mathfrak{q}})$ in the network structure (Fig. \ref{fig:fa_drone_learning}(a)). The forward pass rolls out the Hamiltonian dynamics \eqref{eq:Ham_sys_pf_nlin} and \eqref{eq:hamiltonian_zetadot} with the neural networks $\mathbf{M}_{\boldsymbol{\theta}}({\mathfrak{q}}), \mathbf{B}_{\boldsymbol{\theta}}({\mathfrak{q}})$ and ${\cal U}_{\boldsymbol{\theta}}({\mathfrak{q}})$ using a neural ODE solver (\cite{chen2018neural}) to obtain an predicted sequence $(\bar{{\mathfrak{q}}}^{(i)}_{0:N}, \bar{{\boldsymbol{\zeta}}}^{(i)}_{0:N})$ at times $t_0^{(i)} < t_1^{(i)} < \ldots < t_N^{(i)}$ for each $i = 1,\ldots, D$. The loss function is defined as ${\cal L} = \sum_{i = 1}^D \sum_{n = 1}^N c({\mathfrak{q}}^{(i)}_{0:N},{\boldsymbol{\zeta}}^{(i)}_{0:N}, \bar{{\mathfrak{q}}}^{(i)}_{0:N}, \bar{{\boldsymbol{\zeta}}}^{(i)}_{0:N})$ where the distance metric $c$ is defined as the sum of position, orientation, and velocity errors on the tangent bundle $TSE(3)$ of the pose manifold $SE(3)$:
\begin{equation}
c \prl{{\mathfrak{q}}^{(i)}_{0:N},{\boldsymbol{\zeta}}^{(i)}_{0:N}, \bar{{\mathfrak{q}}}^{(i)}_{0:N}, \bar{{\boldsymbol{\zeta}}}^{(i)}_{0:N}} = c_{\mathbf{p}}(\mathbf{p},\bar{\mathbf{p}}) + c_{\mathbf{R}}(\mathbf{R},\bar{\mathbf{R}}) + c_{{\boldsymbol{\zeta}}}({\boldsymbol{\zeta}},\bar{{\boldsymbol{\zeta}}}),
\end{equation}
with the position error $c_{\mathbf{p}}(\mathbf{p},\bar{\mathbf{p}}) = \| \mathbf{p} - \bar{\mathbf{p}}\|^2_2$, the velocity error $c_{{\boldsymbol{\zeta}}}({\boldsymbol{\zeta}},\bar{{\boldsymbol{\zeta}}}) = \| {\boldsymbol{\zeta}} - \bar{{\boldsymbol{\zeta}}}\|^2_2$, and the rotation error $c_{\mathbf{R}}(\mathbf{R},\bar{\mathbf{R}}) = \|\left(\log (\bar{\mathbf{R}} \mathbf{R}^\top)\right)^{\vee} \|_2^2$. The $\log$-map $\log (\cdot): SE(3) \mapsto \mathfrak{so}(3)$ is the inverse of the exponential map, returning a skew-symmetric matrix in $\mathfrak{so}(3)$ from a rotation matrix in $SE(3)$, and the $\vee$-map $(\cdot)^\vee : \mathfrak{so}(3) \mapsto \mathbb{R}^3$ is the inverse of the hat map $\hat{(\cdot)}$ in Sec. \ref{sec:problem}. The network parameters ${\boldsymbol{\theta}}$ are optimized using gradient descent by back-propagating the loss through the neural ODE solver. This is done efficiently using adjoint method, where the gradient $\partial {\cal L}/\partial {\boldsymbol{\theta}}$ is calculated by a single call to a reverse-time ODE starting from $t = t_N$ at $({\mathfrak{q}}_N^{(i)}, {\boldsymbol{\zeta}}_N^{(i)})$.

\subsection{Evaluation of the $SE(3)$ Hamiltonian dynamics model of a simulated hexarotor}
\label{subsec:ham_dyn_learning_evaluation}

\begin{figure}[t]
\centering
\begin{subfigure}
        \centering
        \includegraphics[height=30mm]{figs/learning/network_arch_l4dc.png}

        \label{fig:net_arch}
\end{subfigure}%
\hfill
\begin{subfigure}
        \centering
        \includegraphics[height=27mm]{figs/drone/hexarotor2.png}

        \label{fig:hexarotor}
\end{subfigure}%

\begin{subfigure}
        \centering
        \includegraphics[height=26mm]{figs/learning/loss_log.pdf}

        \label{fig:loss_log}
\end{subfigure}%
\hfill%
\begin{subfigure}
        \centering
        \includegraphics[height=26mm]{figs/learning/M2_x_all.pdf}

        \label{fig:learned_inertia}
\end{subfigure}%
\hfill%
\begin{subfigure}
        \centering
        \includegraphics[height=26mm]{figs/learning/g_all_x.pdf}

        \label{fig:learned_g}
\end{subfigure}%
\caption{$SE(3)$ Hamiltonian neural ODE network: (a) network architecture, (b) simulated hexarotor for evaluation, (c) training loss, (d) learned inverse inertia $\mathbf{J}_{\boldsymbol{\theta}}({\mathfrak{q}})^{-1}$, and (e) learned input matrix $\mathbf{B}_{\boldsymbol{\theta}}(\frak{q})$ along a test trajectory, evaluated on the simulated hexarotor.}
\label{fig:fa_drone_learning}
\end{figure}

We consider a simulated hexarotor unmanned aerial vehicle (UAV) (Fig. \ref{fig:fa_drone_learning}(b)) with fixed-tilt rotors pointing in different directions (\cite{hexarotor}) modeled as a fully-actuated rigid body with mass $m = 0.027$ and inertia matrix $\mathbf{J} = 10^{-5}\text{diag}([2.4, 2.4, 3.2])$. The robot's ground-truth dynamics satisfy Hamilton's equations in \eqref{eq:Ham_sys_pf_nlin} with generalized mass $\mathbf{M}(\mathbf{\mathfrak{q}}) = \diag(m\mathbf{I}, \mathbf{J})$, potential energy ${\cal U}(\mathbf{\mathfrak{q}}) = mgz$, and the input matrix $\mathbf{B}(\mathbf{\mathfrak{q}})= \mathbf{I}$. The control input $\mathbf{u}$ is a 6-dimensional wrench, including a 3-dimensional force and a 3-dimensional torque.
Since the mass $m$ of the hexarotor can be easily measured, we assume the mass $m$ is known, leading to a known potential energy ${\cal U}({\mathfrak{q}}) = mg\begin{bmatrix}
0 & 0 & 1
\end{bmatrix} \mathbf{p}$, where $\mathbf{p}$ is the UAV position and $g\approx 9.8ms^{-2}$ is the gravitational acceleration. We approximate the inverse generalized mass matrix by $\mathbf{M}^{-1}_{\boldsymbol{\theta}}(\mathbf{\mathfrak{q}}) =\diag(m^{-1}\mathbf{I}, \mathbf{J}_{{\boldsymbol{\theta}}}^{-1}(\mathbf{\mathfrak{q}}))$ and learn the inverse inertia matrix $\mathbf{J}_{{\boldsymbol{\theta}}}(\mathbf{\mathfrak{q}})^{-1}$ and the input matrix $\mathbf{B}_{\boldsymbol{\theta}}({\mathfrak{q}})$ from data.

We mimic manual flights in an area free of obstacles using a PID controller and drive the hexarotor from a random initial pose to a desired poses, generating $18$ $1$-second trajectories. We shift the trajectories to start from the origin and create a dataset $\mathcal{D} = \{t_{0:N}^{(i)},\mathbf{\mathfrak{q}}_{0:N}^{(i)}, {\boldsymbol{\zeta}}_{0:N}^{(i)}, \mathbf{u}^{(i)})\}_{i=1}^D$ with $N = 5$ and $D = 432$. The Hamiltonian-based neural ODE network\footnote{Code: \url{https://thaipduong.github.io/SE3HamDL/}} is trained with the dataset ${\cal D}$, as described in Sec. \ref{subsec:ham_dyn_learning}, for 5000 iterations and learning rate $10^{-3}$. Fig. \ref{fig:fa_drone_learning}(c) shows the loss function during training. Note that if we scale $\mathbf{M}_{\boldsymbol{\theta}}({\mathfrak{q}})$ and the input matrix $\mathbf{B}({\mathfrak{q}})$ by a constant $\gamma$, the dynamics of $({\mathfrak{q}}, {\boldsymbol{\zeta}})$ in \eqref{eq:Ham_sys_pf_nlin} and \eqref{eq:hamiltonian_zetadot} does not change. Fig. \ref{fig:fa_drone_learning}(d) and \ref{fig:fa_drone_learning}(e) plot the scaled version of the learned inverse mass $\mathbf{J}_{\boldsymbol{\theta}}({\mathfrak{q}})^{-1}$ and the input matrix $\mathbf{B}_{\boldsymbol{\theta}}({\mathfrak{q}})$, converging to the constant ground truth values.

\subsection{Reference governor design}
\label{sec:rg_ctrl}

We introduce a virtual system, called a \emph{reference governor} \citep{RG_bemporad1998reference}, which will adaptively track the path $\mathbf{r}$ defined in Sec.~\ref{sec:problem} and provide a time-varying reference $\mathbf{x}^*(t)$ for the actual system in \eqref{eq:cls_nln}. The motion of the governor system needs to be regulated to balance the energy of the Hamiltonian system with the distance to the unsafe set ${\cal O}$, keeping the safety margin in \eqref{eq:pbf} positive. Define the governor as a first-order linear time-invariant system with state $\mathbf{g}(t) \in \mathbb{R}^3$ and dynamics:
\begin{equation} \label{eq:governor}
\dot{\mathbf{g}} = -k_g \prl{\mathbf{g} - \mathbf{u}_\mathbf{g}}
\end{equation}
where $k_g$ is a positive gain and $\mathbf{u}_\mathbf{g}$ is the governor control input. The input $\mathbf{u}_\mathbf{g}$ will be chosen to move the governor system along the reference path without violating the safety condition $\Delta E(\mathbf{x}, \mathbf{x}^*)  \geq 0$ obtained in Proposition~\ref{prop:safe_stab}. Define a \emph{local safe set} $\mathcal{LS}(\mathbf{x},\mathbf{g})$ as a region around the governor state $\mathbf{g}$ that does not violate safety:
\[
\mathcal{LS}(\mathbf{x}, \mathbf{g}) \coloneqq \crl{ \mathbf{q} \in \mathbb{R}^{m} \mid \norm{\mathbf{q} - \mathbf{g}}^2 \leq (1+\epsilon)^{-1} \Delta E(\mathbf{x},\mathbf{x}^*)},
\]
where $\epsilon > 0$ is arbitrarily, ensuring that $\mathcal{LS}(\mathbf{x},\mathbf{g}) \subseteq \text{int}(\calF)$. The size of the local safe set determines how fast the governor can move along the reference path without endangering safety.

\begin{definition} \label{def:lpg}
A \emph{local projected goal} at system-governor state $(\mathbf{x}, \mathbf{g})$ is a point $\bar{\bfg} \in \mathcal{LS}(\mathbf{x},\mathbf{g})$ that is furthest along the reference path $\mathbf{r}$:
\begin{equation} \label{eq:lpg}
	\bar{\bfg} = \mathbf{r}(\sigma^*),  \;\; \sigma^* = \mathop{\arg\max}\limits_{\sigma \in [0,1]} \crl{ \sigma \mid  \mathbf{r}(\sigma) \in \mathcal{LS}(\mathbf{x},\mathbf{g})}.
\end{equation}
\end{definition}

Choosing the governor input as $\mathbf{u}_{\mathbf{g}} = \bar{\mathbf{g}}$ forces the governor to track the reference path adaptively, taking the safety condition $\Delta E(\mathbf{x}, \mathbf{x}^*)  \geq 0$ into account. Given the local projected goal $\bar{\mathbf{g}}(t) \in \mathbb{R}^3$ and the governor state $\mathbf{g}(t) \in \mathbb{R}^3$, we also generate a desired reference state $\mathbf{x}^*(t)$ for the system in \eqref{eq:cls_nln} by lifting $\mathbf{g}(t)$ to $\mathbb{R}^{18}$. We may choose $\mathbf{g}(t)$ as the desired position with zero desired velocity but, to provide guidance on the $SE(3)$ manifold, we need to also generate a desired orientation $\mathbf{R}^*(t)$. We construct a lifting function ${\boldsymbol{\ell}}: {\cal F} \times \text{int}(\calF) \mapsto \mathbb{R}^{18}$ to obtain $\mathbf{x}^* = {\boldsymbol{\ell}}(\mathbf{g},\bar{\mathbf{g}})$ as:
\begin{equation} \label{eq:lift_funcs}
{\boldsymbol{\ell}}(\mathbf{g}, \bar{\bfg}) = \begin{bmatrix}
\mathbf{p}^{*\top} & \mathbf{r}_1^{*\top} & \mathbf{r}_2^{*\top} &\mathbf{r}_3^{*\top}, \bf0^\top, \bf0^\top
\end{bmatrix}^\top,
\end{equation}
where $\mathbf{p}^* = \mathbf{g}$ and $\mathbf{r}_1^*$, $\mathbf{r}_2^*$, $\mathbf{r}_3^*$ are the rows of the matrix:
\begin{equation} \label{eq:rotRg}
	\mathbf{R}^*(\mathbf{g}, \bar{\bfg}) =
	\begin{cases}
	\mathbf{I}  & \text{if } \mathbf{e}_3 \times \mathbf{c}_1 = 0\\
	\brl{\mathbf{c}_1\;\; \mathbf{c}_2 \;\; \mathbf{c}_3} & \text{otherwise},
	\end{cases}
\end{equation}
with $\mathbf{e}_3 = \brl{0, 0, 1}^\top$, $\mathbf{c}_1 = (\bar{\bfg} - \mathbf{g}) / \norm{\bar{\bfg} - \mathbf{g}}$, $\mathbf{c}_2 = (\mathbf{e}_3 \times \mathbf{c}_1) / \norm{\mathbf{e}_3 \times \mathbf{c}_1}$, and $\mathbf{c}_3 = (\mathbf{c}_1 \times \mathbf{c}_2) / \norm{\mathbf{c}_1 \times \mathbf{c}_2}$. If $\mathbf{g} = \bar{\bfg}$, the most recent backup of $\mathbf{R}^*$ or $\mathbf{I}$ may be used.

Our safe tracking control design is visualized in Fig.~\ref{fig:structure}. It consists of two parts: 1) a first-order reference governor system with state $\mathbf{g}$ adaptively following the local projected goal $\bar{\bfg}$ along the path $\mathbf{r}$ and 2) a closed-loop Hamiltonian system tracking the reference signal $\mathbf{x}^* = {\boldsymbol{\ell}}(\mathbf{g}, \bar{\bfg})$. Our main result is summarized in the following theorem.

\begin{figure}[t]
\centering
\begin{subfigure}
    \centering
    \includegraphics[height=37mm]{figs/rg_framework_ham.pdf}

\end{subfigure}%
\hfill%
\begin{subfigure}
    \centering
    \includegraphics[height=35mm]{figs/rotRg_zoom.png}

\end{subfigure}%
\caption{(a): structure of the reference-governor tracking controller: a reference governor with state $\mathbf{g}$ adaptively tracks a point $\bar{\mathbf{g}}$ along the desired path $\mathbf{r}$ and generates a time-varying equilibrium $\mathbf{x}^* = {\boldsymbol{\ell}}(\mathbf{g}, \bar{\bfg})$ for the closed-loop Hamiltonian system. (b): a local projecte goal $\bar{\bfg}$ (purple dot) is generated as the furthest intersection between the local safe set $\mathcal{LS}(\mathbf{x}, \mathbf{g})$ (yellow sphere) and the path $\mathbf{r}$ (blue curve). Given $\mathbf{g}$ and $\bar{\bfg}$, a desired equilibrium $\mathbf{x}^* = {\boldsymbol{\ell}}(\mathbf{g}, \bar{\bfg})$ is generated for the Hamiltonian system with orientation indicated by the red, green, and blue arrows, respectively.}
\label{fig:structure}
\end{figure}

\begin{theorem}\label{thm:main_result}
Given a reference path $\mathbf{r}$, consider the closed-loop Hamiltonian system in \eqref{eq:cls_nln} and the closed-loop reference governor system, $\dot{\mathbf{g}} = -k_g \prl{\mathbf{g} - \bar{\mathbf{g}}}$, with local projected goal $\bar{\mathbf{g}}$ provided in Definition~\ref{def:lpg}. Suppose that the initial state $(\mathbf{x}_0, \mathbf{g}_0)$ satisfies:
	%
	\begin{equation} \label{eq:thm_ic}
	\Delta E \prl{\mathbf{x}_0, {\boldsymbol{\ell}}(\mathbf{g}_0, \bar{\bfg}_0)} > 0,\quad  \mathbf{g}_0 = \mathbf{r}(0) = \mathbf{p}(t_0) \in \text{int}(\calF),
	\end{equation}
	%
where $\Delta E(\mathbf{x}, \mathbf{x}^*)$ is a dynamic safety margin defined in \eqref{eq:pbf}. Then, the joint system state $(\mathbf{x}, \mathbf{g})$ converges to $({\boldsymbol{\ell}}(\mathbf{r}(1), \mathbf{r}(1)), \mathbf{r}(1))$ without violating the constraints, i.e., $\mathbf{p}(t) \in {\cal F}$, $\forall t \geq t_0$.
\end{theorem}

\begin{proof}
See Appendix~C in the extended version \citep{refgovham_extended}.
\end{proof}

\section{Conclusion}
This paper developed a neural ODE network for learning a dynamics model of an $SE(3)$ Hamiltonian system and a tracking controller that enables safe autonomous navigation in unknown environments. Given only a training set of system state-control trajectories, our approach estimates the system dynamics and synthesizes a controller that avoids obstacles based on run-time distance measurements. Our method was demonstrated on a simulated hexarotor aerial robot navigating in complex 3D environments. Future work will focus on capturing model uncertainty and external disturbances in the design and deploying it on a hardware platform.

\section{Safe Tracking using a Reference Governor}

In this section, we first describe a passivity-based regulation controller for arbitrary pose stabilization in Sec.~\ref{sec:ham_control}. We derive sufficient conditions for safety based on an invariant level set of the closed-loop system's Hamiltonian. Finally, in Sec.~\ref{sec:rg_ctrl}, we propose a reference governor control policy to adaptively generate a regulation pose along the desired path and achieve safe navigation.

\subsection{Passivity-based control for learned Hamiltonian dynamics}
\label{sec:ham_control}

Given the learned model of the system dynamics:
\begin{equation} \label{eq:ols_nln}
\dot{\mathbf{x}} = \mathbf{f}_{\boldsymbol{\theta}}(\mathbf{x}) + \mathbf{G}_{\boldsymbol{\theta}}(\mathbf{x}) \mathbf{u}, \quad \mathbf{x}(t_0) = \mathbf{x}_0,
\end{equation}
we want to find a control policy that stabilizes the system to a desired equilibrium $\mathbf{x}^* \coloneqq (\mathbf{\mathfrak{q}}^*, \bf0)$ with the desired generalized coordinates ${\mathfrak{q}}^* = (\mathbf{p}^*, \mathbf{R}^*)$ and zero momentum ${\mathfrak{p}}^* = \bf0$, i.e. zero generalized velocity.
Specifically, we design a control policy $\mathbf{u} = {\boldsymbol{\pi}}(\mathbf{x},\mathbf{x}^*)$ to shape the total energy (Hamiltonian) of the closed-loop system so that it achieves a minimum at the desired state $\mathbf{x}^* = (\mathbf{\mathfrak{q}}^*, \bf0)$. By injecting energy into the system through the controller $\mathbf{u} = {\boldsymbol{\pi}}(\mathbf{x},\mathbf{x}^*)$, we aim to achieve the following desired Hamiltonian:
\begin{equation}
\label{eq:desired_hamiltonian}
    \mathcal{H}_d (\mathbf{x}, \mathbf{x}^*) =  \frac{1}{2}k_\mathbf{p}(\mathbf{p} - \mathbf{p}^*)^\top(\mathbf{p} - \mathbf{p}^*) + \frac{1}{2} k_{\mathbf{R}}\tr(\mathbf{I} - \mathbf{R}^{*\top}\mathbf{R}) + \frac{1}{2}(\mathbf{\mathfrak{p}}-\mathbf{\mathfrak{p}}^*)^\top\mathbf{M}_{\boldsymbol{\theta}}^{-1}(\mathbf{\mathfrak{q}})(\mathbf{\mathfrak{p}}-\mathbf{\mathfrak{p}}^*),
\end{equation}
where $k_\mathbf{p}$ and $k_{\mathbf{R}}$ are positive gains. We use the interconnection and damping assignment passivity-based control (IDA-PBC) approach \citep{van2014port} to obtain matching conditions relating the dynamics in \eqref{eq:ols_nln} to error state dynamics associated with the desired Hamiltonian. Solving the matching conditions, as described in \citet{duong21hamiltonian}, leads to a controller, consisting of an energy-shaping term $\mathbf{u}_{ES}$ and a damping-injection term $\mathbf{u}_{DI}$:
\begin{equation}
    \label{eq:stab_ctrl_law}
    \mathbf{u} =  \underbrace{\mathbf{B}_{\boldsymbol{\theta}}^{\dagger}(\mathbf{\mathfrak{q}})\left(\mathbf{\mathfrak{q}}^{\times\top} \partial V/ \partial \mathbf{\mathfrak{q}} - \mathbf{\mathfrak{p}}^{\times}\mathbf{M}_{\boldsymbol{\theta}}^{-1}(\mathbf{\mathfrak{q}})\mathbf{\mathfrak{p}} - \mathbf{e}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*)\right)}_{\mathbf{u}_{ES}} +\underbrace{\prl{- \mathbf{B}_{\boldsymbol{\theta}}^{\dagger} (\mathbf{\mathfrak{q}})  \mathbf{K}_\mathbf{d}\, \mathbf{M}_{\boldsymbol{\theta}}^{-1}(\mathbf{\mathfrak{q}})\mathbf{\mathfrak{p}}}}_{\mathbf{u}_{DI}},
\end{equation}
where $\mathbf{B}_{\boldsymbol{\theta}}^{\dagger}(\mathbf{\mathfrak{q}}) = \left(\mathbf{B}_{\boldsymbol{\theta}}^{\top}(\mathbf{\mathfrak{q}})\mathbf{B}_{\boldsymbol{\theta}}(\mathbf{\mathfrak{q}})\right)^{-1}\mathbf{B}_{\boldsymbol{\theta}}^{\top}(\mathbf{\mathfrak{q}})$ is the pseudo-inverse of $\mathbf{B}_{\boldsymbol{\theta}}(\mathbf{\mathfrak{q}})$, $\mathbf{K}_\mathbf{d} = \diag(k_\mathbf{v} \mathbf{I}, k_{\boldsymbol{\omega}}\mathbf{I})$ is a damping gain with positive terms $k_\mathbf{v}$, $k_{\boldsymbol{\omega}}$, and  $\mathbf{e}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*)$ is the error between $\mathbf{\mathfrak{q}}$ and $\mathbf{\mathfrak{q}}^*$:
\begin{equation}
\label{eq:coordinate_error}
\mathbf{e}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*) = \begin{bmatrix} \mathbf{e}_{\mathbf{p}}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*) \\ \mathbf{e}_{\mathbf{R}}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*) \end{bmatrix} = \begin{bmatrix} k_\mathbf{p} \mathbf{R}^\top(\mathbf{p} - \mathbf{p}^*) \\ \frac{1}{2}k_{\mathbf{R}}\prl{\mathbf{R}^{*\top}\mathbf{R}-\mathbf{R}^\top\mathbf{R}^{*}}^{\vee}\end{bmatrix}.
\end{equation}

\begin{lemma} \label{lem:lyap_ham}
If the input gain matrix $\mathbf{B}_{\boldsymbol{\theta}}(\bf{\mathfrak{q}})$ of the system in \eqref{eq:ols_nln} is invertible, the control policy $\mathbf{u} = {\boldsymbol{\pi}} (\mathbf{x}, \mathbf{x}^*)$ in \eqref{eq:stab_ctrl_law} always exists and asymptotically stabilizes the system to an arbitrary reference $\mathbf{x}^* = ({\mathfrak{q}}^*, \bf0)$ with Lyapunov function given by the desired Hamiltonian ${\cal H}_d(\mathbf{x},\mathbf{x}^*)$ in \eqref{eq:desired_hamiltonian}.
\end{lemma}

\begin{proof}
See Appendix~A in the extended version \citep{refgovham_extended}.
\end{proof}

Next, given the closed-loop system:
\begin{equation} \label{eq:cls_nln}
\begin{aligned}
\dot{\mathbf{x}} &= \mathbf{f}_{\boldsymbol{\theta}}(\mathbf{x}) + \mathbf{G}_{\boldsymbol{\theta}}(\mathbf{x}) {\boldsymbol{\pi}} \prl{\mathbf{x}, \mathbf{x}^*}, \quad \mathbf{x}(t_0) = \mathbf{x}_0,
\end{aligned}
\end{equation}
we derive conditions on the initial state $\mathbf{x}_0$ under which the position $\mathbf{p}$ converges to $\mathbf{p}^*$ safely, remaining in the safe set ${\cal F}$. We first define a dynamic safety margin (DSM) $\Delta E(\mathbf{x}, \mathbf{x}^*)$ for the Hamiltonian dynamics \eqref{eq:cls_nln}:
\begin{equation} \label{eq:pbf}
	\Delta E(\mathbf{x}, \mathbf{x}^*) \coloneqq \bar{d}{\,}^2(\mathbf{p}^*, {\cal O}) - \frac{2}{k_\mathbf{p}} \mathcal{H}_d \prl{\mathbf{x}, \mathbf{x}^*},
\end{equation}
where $\bar{d}{\,}^2(\mathbf{p}^*, {\cal O})$ is the truncated distance to the unsafe set ${\cal O}$ in \eqref{eq:dyO}. Given a fixed desired point $\mathbf{x}^*$, the DSM function measures the the trade-off between safety, measured by $\bar{d}{\,}^2(\mathbf{p}^*, {\cal O})$, and system activeness, measured by the Lyapunov function $\mathcal{H}_d \prl{\mathbf{x}, \mathbf{x}^*}$.  and allows us to find a positively forward invariant set ${\cal S}(\mathbf{x}, \mathbf{x}^*)$ in Prop. \ref{prop:safe_stab} such that for any $\mathbf{x}_0 \in {\cal S}(\mathbf{x}, \mathbf{x}^*)$, the position $\mathbf{p}$ converges to $\mathbf{p}^*$ while remaining in the safe set ${\cal F}$.

\begin{proposition} \label{prop:safe_stab}
Controller~\eqref{eq:stab_ctrl_law} renders ${\cal S}(\mathbf{x}, \mathbf{x}^*) \coloneqq \crl{\mathbf{x} \mid \Delta E(\mathbf{x}, \mathbf{x}^*)  \geq 0}$ a positively forward invariant set for system~\eqref{eq:cls_nln}. Furthermore, if $\mathbf{x}_0 \in {\cal S}(\mathbf{x}, \mathbf{x}^*)$, then the state $\mathbf{p}(t)$ converges to $\mathbf{p}^*$ asymptotically without violating the collision, i.e., $d(\mathbf{p}(t), {\cal O}) \geq 0$ for all $t \geq t_0$.
\end{proposition}

\begin{proof}
See Appendix~B in the extended version \citep{refgovham_extended}.
\end{proof}

Based on the results in this section for a fixed equibrium point $\mathbf{x}^*$, we develop a reference governor in Sec. \ref{sec:rg_ctrl} to adaptive change $\mathbf{x}^*$ over time so that the robot can safely track a desired path of robot positions.

\section{Safe Tracking using a Reference Governor}

In this section, we first describe a passivity-based regulation controller for arbitrary pose stabilization in Sec.~\ref{sec:ham_control}. We derive sufficient conditions for safety based on an invariant level set of the closed-loop system's Hamiltonian. Finally, in Sec.~\ref{sec:rg_ctrl}, we propose a reference governor control policy to adaptively generate a regulation pose along the desired path and achieve safe navigation.

\subsection{Passivity-based control for learned Hamiltonian dynamics}
\label{sec:ham_control}

Given the learned model of the system dynamics:
\begin{equation} \label{eq:ols_nln}
\dot{\mathbf{x}} = \mathbf{f}_{\boldsymbol{\theta}}(\mathbf{x}) + \mathbf{G}_{\boldsymbol{\theta}}(\mathbf{x}) \mathbf{u}, \quad \mathbf{x}(t_0) = \mathbf{x}_0,
\end{equation}
we design a control policy to stabilize the system to a desired equilibrium. Let $(\mathbf{\mathfrak{q}}^*, {\boldsymbol{\zeta}}^*)$ be the desired generalized coordinates $(\mathbf{p}^*, \mathbf{R}^*)$ and velocities $(\mathbf{v}^*, {\boldsymbol{\omega}}^*)$. Note that the desired velocity ${\boldsymbol{\zeta}}^*$ is defined in the inertial frame $(\mathbf{p}^*, \mathbf{R}^*)$, while the system's velocity ${\boldsymbol{\zeta}}(t)$ and and momentum ${\mathfrak{p}}(t)$ in \eqref{eq:momenta_Mtwist} are specified in the body-fixed frame $(\mathbf{p}(t),\mathbf{R}(t))$. We define the desired momentum ${\mathfrak{p}}^*$ by transforming ${\boldsymbol{\zeta}}^*$ to the body-fixed frame:
\begin{equation}\label{eq:desired_momentum}
    \mathbf{\mathfrak{p}}^* = \mathbf{M}_{\boldsymbol{\theta}} ({\mathfrak{q}}) \begin{bmatrix} \mathbf{R}^\top \mathbf{R}^* \mathbf{v}^* \\ \mathbf{R}^\top \mathbf{R}^* {\boldsymbol{\omega}}^* \end{bmatrix}.
\end{equation}
We design a control policy $\mathbf{u} = {\boldsymbol{\pi}}(\mathbf{x},\mathbf{x}^*)$ to shape the total energy (Hamiltonian) of the closed-loop system so that it achieves a minimum at the desired state $\mathbf{x}^* \coloneqq (\mathbf{\mathfrak{q}}^*, \mathbf{\mathfrak{p}}^*)$. By injecting energy into the system through the controller $\mathbf{u} = {\boldsymbol{\pi}}(\mathbf{x},\mathbf{x}^*)$, we aim to achieve the following desired Hamiltonian:
\begin{equation}
\label{eq:desired_hamiltonian}
    \mathcal{H}_d (\mathbf{x}, \mathbf{x}^*) =  \frac{1}{2}k_\mathbf{p}(\mathbf{p} - \mathbf{p}^*)^\top(\mathbf{p} - \mathbf{p}^*) + \frac{1}{2} k_{\mathbf{R}}\tr(\mathbf{I} - \mathbf{R}^{*\top}\mathbf{R}) + \frac{1}{2}(\mathbf{\mathfrak{p}}-\mathbf{\mathfrak{p}}^*)^\top\mathbf{M}_{\boldsymbol{\theta}}^{-1}(\mathbf{\mathfrak{q}})(\mathbf{\mathfrak{p}}-\mathbf{\mathfrak{p}}^*),
\end{equation}
where $k_\mathbf{p}$ and $k_{\mathbf{R}}$ are positive gains. We use the interconnection and damping assignment passivity-based control (IDA-PBC) approach \citep{van2014port} to obtain matching conditions relating the dynamics in \eqref{eq:ols_nln} to error state dynamics associated with the desired Hamiltonian. Solving the matching conditions, as described in \citet{duong21hamiltonian}, leads to a controller, consisting of an energy-shaping term $\mathbf{u}_{ES}$ and a damping-injection term $\mathbf{u}_{DI}$:
\begin{equation}
    \label{eq:stab_ctrl_law}
    \mathbf{u} =  \underbrace{\mathbf{B}_{\boldsymbol{\theta}}^{\dagger}(\mathbf{\mathfrak{q}})\left(\mathbf{\mathfrak{q}}^{\times\top} \partial V/ \partial \mathbf{\mathfrak{q}} - \mathbf{\mathfrak{p}}^{\times}\mathbf{M}_{\boldsymbol{\theta}}^{-1}(\mathbf{\mathfrak{q}})\mathbf{\mathfrak{p}} - \mathbf{e}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*) + \dot{{\mathfrak{p}}}^*\right)}_{\mathbf{u}_{ES}} +\underbrace{\prl{- \mathbf{B}_{\boldsymbol{\theta}}^{\dagger} (\mathbf{\mathfrak{q}})  \mathbf{K}_\mathbf{d}\, \mathbf{M}_{\boldsymbol{\theta}}^{-1}(\mathbf{\mathfrak{q}})(\mathbf{\mathfrak{p}}-\mathbf{\mathfrak{p}}^*)}}_{\mathbf{u}_{DI}},
\end{equation}
where $\mathbf{B}_{\boldsymbol{\theta}}^{\dagger}(\mathbf{\mathfrak{q}}) = \left(\mathbf{B}_{\boldsymbol{\theta}}^{\top}(\mathbf{\mathfrak{q}})\mathbf{B}_{\boldsymbol{\theta}}(\mathbf{\mathfrak{q}})\right)^{-1}\mathbf{B}_{\boldsymbol{\theta}}^{\top}(\mathbf{\mathfrak{q}})$ is the pseudo-inverse of $\mathbf{B}_{\boldsymbol{\theta}}(\mathbf{\mathfrak{q}})$, $\mathbf{K}_\mathbf{d} = \diag(k_\mathbf{v} \mathbf{I}, k_{\boldsymbol{\omega}}\mathbf{I})$ is a damping gain with positive terms $k_\mathbf{v}$, $k_{\boldsymbol{\omega}}$, and  $\mathbf{e}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*)$ is the error between $\mathbf{\mathfrak{q}}$ and $\mathbf{\mathfrak{q}}^*$:
\begin{equation}
\label{eq:coordinate_error}
\mathbf{e}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*) = \begin{bmatrix} \mathbf{e}_{\mathbf{p}}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*) \\ \mathbf{e}_{\mathbf{R}}(\mathbf{\mathfrak{q}},\mathbf{\mathfrak{q}}^*) \end{bmatrix} = \begin{bmatrix} k_\mathbf{p} \mathbf{R}^\top(\mathbf{p} - \mathbf{p}^*) \\ \frac{1}{2}k_{\mathbf{R}}\prl{\mathbf{R}^{*\top}\mathbf{R}-\mathbf{R}^\top\mathbf{R}^{*}}^{\vee}\end{bmatrix}.
\end{equation}

\begin{lemma} \label{lem:lyap_ham}
If the input gain matrix $\mathbf{B}_{\boldsymbol{\theta}}(\bf{\mathfrak{q}})$ of the system in \eqref{eq:ols_nln} is invertible, the control policy $\mathbf{u} = {\boldsymbol{\pi}} (\mathbf{x}, \mathbf{x}^*)$ in \eqref{eq:stab_ctrl_law} always exists and asymptotically stabilizes the system to an arbitrary reference $\mathbf{x}^*$ obtained as in \eqref{eq:desired_momentum} with Lyapunov function given by the desired Hamiltonian ${\cal H}_d(\mathbf{x},\mathbf{x}^*)$ in \eqref{eq:desired_hamiltonian}.
\end{lemma}
\begin{proof}
See Appendix~\ref{app:proof_lyap_ham}.\NA{Please make a pass through the proof. Note that $\mathbf{x}^*$ is time-varying due to \eqref{eq:desired_momentum}. Also, the main claim of the theorem is that when $\mathbf{B}_{\boldsymbol{\theta}}(\bf{\mathfrak{q}})$ is invertible, we can stabilize to an \emph{arbitrary} $\mathbf{x}^*$.}
\end{proof}

Next, given the closed-loop system:
\begin{equation} \label{eq:cls_nln}
\begin{aligned}
\dot{\mathbf{x}} &= \mathbf{f}_{\boldsymbol{\theta}}(\mathbf{x}) + \mathbf{G}_{\boldsymbol{\theta}}(\mathbf{x}) {\boldsymbol{\pi}} \prl{\mathbf{x}, \mathbf{x}^*}, \quad \mathbf{x}(t_0) = \mathbf{x}_0,
\end{aligned}
\end{equation}
we derive conditions on the initial state $\mathbf{x}_0$ under which the position $\mathbf{p}$ converges to $\mathbf{p}^*$ safely, remaining in the safe set ${\cal F}$. We first define a dynamic safety margin (DSM) $\Delta E(\mathbf{x}, \mathbf{x}^*)$ for the Hamiltonian dynamics \eqref{eq:cls_nln}:
\begin{equation} \label{eq:pbf}
	\Delta E(\mathbf{x}, \mathbf{x}^*) \coloneqq \bar{d}{\,}^2(\mathbf{p}^*, {\cal O}) - \frac{2}{k_\mathbf{p}} \mathcal{H}_d \prl{\mathbf{x}, \mathbf{x}^*},
\end{equation}
where $\bar{d}{\,}^2(\mathbf{p}^*, {\cal O})$ is the truncated distance to the unsafe set ${\cal O}$ in \eqref{eq:dyO}. Given a fixed desired point $\mathbf{x}^*$, the DSM function allows us to find a positively forward invariant set ${\cal S}(\mathbf{x}, \mathbf{x}^*)$ in Prop. \ref{prop:safe_stab} such that for any $\mathbf{x}_0 \in {\cal S}(\mathbf{x}, \mathbf{x}^*)$, the position $\mathbf{p}$ converges to $\mathbf{p}^*$ while remaining in the safe set ${\cal F}$.

\begin{proposition} \label{prop:safe_stab}
Controller~\eqref{eq:stab_ctrl_law} renders ${\cal S}(\mathbf{x}, \mathbf{x}^*) \coloneqq \crl{\mathbf{x} \mid \Delta E(\mathbf{x}, \mathbf{x}^*)  \geq 0}$ a positively forward invariant set for system~\eqref{eq:cls_nln}. Furthermore, if $\mathbf{x}_0 \in {\cal S}(\mathbf{x}, \mathbf{x}^*)$, then the state $\mathbf{p}(t)$ converges to $\mathbf{p}^*$ asymptotically without violating the collision, i.e., $d(\mathbf{p}(t), {\cal O}) \geq 0$ for all $t \geq t_0$.
\end{proposition}
\begin{proof}
    See Appendix~\ref{app:proof_safe_stab}.\NA{Note that, due to \eqref{eq:desired_momentum}, $\mathbf{x}^*$ is not constant but $(\mathbf{\mathfrak{q}}^*, {\boldsymbol{\zeta}}^*)$ is. Revise according.}
\end{proof}

Based on the results in this section for a fixed equibrium point $\mathbf{x}^*$, we develop a reference governor in Sec. \ref{sec:rg_ctrl} to change $\mathbf{x}^*$ over time so that the robot can safely track the path.

\section{Introduction}
Designing controllers that handle safety constraints and guarantee system stability is an important problem in safety-critical applications, such as autonomous driving \citep{CBF_ames2014control, shalev2016safe}, walking robots \citep{CBF_ames2014rapidly} or medical robots \citep{yip2014model}. Safety depends on the system states, governed by the system dynamics, and the environment constraints. This leads to two requirements for designing provably safe controllers: 1) the availability of an accurate dynamics model of the system and 2) the satisfaction of time-varying safety constraints that are only known at runtime.

The first requirement has motivated many data-driven dynamics learning approaches recently, where machine learning techniques are used to learn dynamical systems, e.g. based on Gaussian processes \citep{deisenroth2015gp,kabzan2019learning} or neural networks \citep{raissi2018multistep,chua2018deep}. For physical systems, recent works \citep{lutter2019deepunderactuated, zhong2020symplectic, duong21hamiltonian} design the model architecture to encode Lagrangian or Hamiltonian formulation of robot dynamics \citep{lurie2013analytical,HolmBook}, which a black-box model might struggle to infer. For Hamiltonian formulation, \cite{zhong2020symplectic} use  a differentiable neural ODE solver \citep{chen2018neural} to generate predicted state trajectory. A loss function is back-propagated through the ODE solver to update its parameters. \cite{duong21hamiltonian} extend this approach by imposing both Hamiltonian formulation and the $SE(3)$ constraints on the ODE structure. A Hamiltonian-based model architecture also simplifies the design of a stable regulation or tracking controller by energy shaping \citep{zhong2020symplectic, duong2021learning, duong21hamiltonian}. The key idea of energy-based controller, e.g. interconnection and damping assignment passivity-based control design (IDA-PBC) \citep{van2014port}, is to inject additional energy via the control input into the system to achieve a desired total energy, minimized at a desired set point.

The second requirement has gained significant attention in planning and control. Model predictive control (MPC) methods \citep{borrelli_MPC_book,NMPC_book,MPC_Automatica_Camacho2006_nlin,MPC_mayne2000constrained} include safety constraints in an optimization problem, which is typically solved by discretizing time on linearized dynamics.
Meanwhile, reachability-based techniques \citep{herbert2017fastrack, Funnel_lib, RTD_kousik2018bridging} directly work for nonlinear systems and
offer strong safety guarantees  but have high computation cost and scalability issues for high-dimensional systems.
Control barrier function (CBF) with quadratic programming (QP) \citep{CBF_ames2014control,CBF_ames2017TAC,CBF_ames2019ECC} offers an
elegant and efficient framework for realtime safe control synthesis. However, it is challenging to construct a valid CBF \citep{CBF_ames2019ECC} that guarantees the feasibility of the QP problem \citep{CBF_xu2018constrained}. Given a stabilizing regulation controller, reference governor techniques \citep{RG_bemporad1998reference,RG_kolmanovsky2014ACC_tutorial,RG_garone2016_ERG} maintain a virtual system to adaptively generate the regulation point so that the stabilized system can follow reference command safely. Recent work \citep{RG_Omur_ICRA17, Li_SafeControl_ICRA20} can achieve safe navigation in unknown environments, but it is hard to apply them for nonlinear system unless it is feedback linearizable.

In this paper, we consider both requirements for rigid-body systems, such as unmanned ground, aerial and underwater vehicles, whose states are described by their $SE(3)$ pose and generalized velocity. We assume that the robot dynamics are unknown but, as a physical system, satisfy Hamilton's equations of motion over the $SE(3)$ manifold. Instead, we are given a set of state-control trajectories, from past experiments or collected by a human operator, and seek to safely track a desired path of robot positions with safety constraints obtained online from sensor measurements, e.g. distances to obstacles from a depth sensor. We first learn a Hamiltonian model of the system dynamics using a physics-guided neural ODE networks \citep{duong21hamiltonian}. As the robot dynamics are equivariant to translation, we offset the trajectories to start from the origin and train a translation-equivariant Hamiltonian neural ODE model. The Hamiltonian structure of the learned model offers an energy-based regulation controller with the total energy of the system viewed as a Lyapunov function. This, in turn, enables us to enforce safety constraints using reference governor techniques without the needs to linearize the system dynamics. Inspired by constraint embedding techniques using a Lyapunov function \citep{RG_garone2016_ERG,RG_Omur_ICRA17}, we impose the safety constraints, based on sensor measurements, on the Lyapunov function. We use the trade-off between safety (distance from constraint violation) and system activeness (measured by the Lyapunov function) to regulate the reference governor and achieve safe and stable position tracking in an unknown environment.

\textbf{Contributions.}
In summary, the contributions of this paper are 1) a neural ODE network approach for learning translation-equivariant $SE(3)$ Hamiltonian system dynamics from state-control trajectory data and 2) a tracking control design for $SE(3)$ Hamiltonian systems with stability and safety guarantees. Our dynamics learning and tracking control techniques are demonstrated on a simulated hexarotor robot using a depth sensor to navigate in unknown complex environments.

\section{Evaluation}
\begin{figure}[t]
	\centering
	\includegraphics[width=0.49\linewidth]{figs/warehouse_l4dc/LMOG_10320.png}%
	\hfill%
	\includegraphics[width=0.49\linewidth]{figs/warehouse_l4dc/fas3d_hexcopter_energy_based_rbt_path_.png}%
	\\
	%
	\includegraphics[width=0.49\linewidth]{figs/warehouse_l4dc/LMOG_10920.png}%
	\hfill%
	\includegraphics[width=0.49\linewidth]{figs/warehouse_l4dc/fas3d_hexcopter_energy_based_safety_stats.pdf}%
	\\
	%
	\includegraphics[width=0.49\linewidth]{figs/warehouse_l4dc/LMOG_11240.png}%
	\hfill%
	\includegraphics[width=0.49\linewidth]{figs/warehouse_l4dc/fas3d_hexcopter_energy_based_dist_vec.pdf}%
	\caption{Safe tracking control for a hexarotor aerial robot in an unknown cluttered environment. The hexarotor (green dot) needs to navigate from a start location (red star) to a goal location (green star) while avoiding obstacles. The obstacles are sensed by a simulated LiDAR sensor (red points). On the left are snapshots of the environment showing the robot (green dot), lidar scans (red dots), and the unsafe set ${\cal O}$ (black surfaces) at different times. The reference path (blue curve) is re-planned online from the governor position (blue dot) to the goal (green star) using an $A^*$ algorithm. The local projected goal $\bar{\bfg}$ (purple dot) is computed based on the obstacle distance (gray ball) and the local safe set (yellow ball).
	%
	On the upper right, projections of the robot's heading (purple arrow) are plotted along the path. The middle right figure shows the dynamic safety margin $\Delta E$ in \eqref{eq:pbf} and the scaled desired Hamiltonian $2 {\cal H}_d(\mathbf{x}, \mathbf{x}^*)/k_\mathbf{p}$ while the lower right figure shows the distance to the obstacles $d(\mathbf{p}(t), {\cal O})$ (green curve), indicating that the safety constraints are never violated.}

	\label{fig:hexarotor_sim_mesh}
\end{figure}

This section evaluates our safe tracking controller on a simulated hexarotor UAV using the learned Hamiltonian dynamics in Sec. \ref{subsec:ham_dyn_learning_evaluation}. The task is to navigate from a start postion to a goal in an environment without colliding with the obstacles. The following control gains were used with the regulation controller in Sec. \ref{sec:ham_control}: $k_p = 0.25$, $k_\mathbf{R} = 125\mathbf{J}$, $k_v = 0.125$, $k_{\boldsymbol{\omega}} = 10\mathbf{J}$ in \eqref{eq:desired_hamiltonian} and $k_\mathbf{g} = 1.0$ with the governor control in \eqref{eq:governor}. We test the controller in a challenging unknown 3D environment with complex obstacles ${\cal O}$. A simulated LiDAR scanner provides point cloud measurements ${\cal P}(t) := \crl{\mathbf{y}_i(t)}$ of the surface of the unsafe set ${\cal O}$, depending on the system pose, with a maximum sensing range of $\beta = 30$. The distance from the governor $\mathbf{g}(t)$ to the unsafe set ${\cal O}$ is approximated via $\bar{d}(\mathbf{g}(t); {\cal O}) \approx \min_{\mathbf{y} \in {\cal P}(t)} \norm{\mathbf{g}(t) - \mathbf{y}}$. The point clouds ${\cal P}(t)$ are used to construct an occupancy grid map online and a reference path $\mathbf{r}$ is replanned periodically using the A* algorithm to ensure that $\mathbf{r}(\sigma) \in \text{int}(\calF)$.

Fig.~\ref{fig:hexarotor_sim_mesh} and Fig.~\ref{fig:sim_forest} show the behavior of the closed-loop hexarotor system in an two different environments. The reference governor follows the projected goal $\bar{\mathbf{g}}$ and generates a time-varying equilibrium $\mathbf{x}^* = {\boldsymbol{\ell}}(\mathbf{g},\bar{\mathbf{g}})$ for the hexarotor. The dynamic safety margin $\Delta E(\mathbf{x},\mathbf{x}^*)$ fluctuates during this process but, as can be seen in the figures, it never becomes negative. The augmented system $(\mathbf{x}, \mathbf{g})$ is controlled adaptively, slowing down when the dynamic safety margin decreases (e.g., when the robot is close to an obstacle and has large total energy ${\cal H}_d$) and speeding up otherwise (e.g., when the robot is far away from the obstacles or has small total energy ${\cal H}_d$). The simulations show that our control policy successfully drives the system from the start to the end of the reference path while avoiding sensed obstacle online, i.e., $d(\mathbf{p}, {\cal O})$ remains positive throughout the motion.

\begin{figure}[t]
\centering
\begin{subfigure}
        \centering
        \includegraphics[width=0.45\linewidth,trim=10mm 10mm 10mm 10mm, clip]{figs/forest_580_new.png}

\end{subfigure}%
\hfill
\begin{subfigure}
        \centering
        \includegraphics[width=0.45\linewidth,trim=10mm 10mm 10mm 10mm, clip]{figs/forest_920_new.png}

\end{subfigure}%
\\
\begin{subfigure}
        \centering
        \includegraphics[height=20mm]{figs/forest_l4dc/fas3d_hexcopter_energy_based_safety_stats.pdf}

\end{subfigure}%
\hfill
\begin{subfigure}
        \centering
        \includegraphics[height=20mm]{figs/forest_l4dc/fas3d_hexcopter_energy_based_dist_vec.pdf}

\end{subfigure}%
\caption{Safe navigation of a hexarotor robot in a forest is demonstrated in (a) and (b). The bottom plots (c) and (d) show the dynamic safety margin $\Delta E$, the scaled desired Hamiltonian $2 {\cal H}_d(\mathbf{x}, \mathbf{x}^*)/k_\mathbf{p}$, and the distance to the obstacles $d(\mathbf{p}(t), {\cal O})$, indicating that the safety constraints are never violated.}
\label{fig:sim_forest}
\end{figure}
\section{Problem Statement}
\label{sec:problem}

Consider a robot modeled as a rigid body with position $\mathbf{p} \in \mathbb{R}^3$, orientation $\mathbf{R}\in SO(3)$, body-frame linear velocity $\mathbf{v} \in \mathbb{R}^3$, and body-frame angular velocity ${\boldsymbol{\omega}} \in \mathbb{R}^3$. Let ${\mathfrak{q}} = [\mathbf{p}^\top\;\; \mathbf{r}_1^\top\;\; \mathbf{r}_2^\top\;\; \mathbf{r}_3^\top]^\top$ $\in \mathbb{R}^{12}$ denote the robot's generalized coordinates, where $\mathbf{r}_1$, $\mathbf{r}_2$, $\mathbf{r}_3 \in \mathbb{R}^3$ are the rows of the rotation matrix $\mathbf{R}$. Let ${\boldsymbol{\zeta}} = [\mathbf{v}^\top\;\;{\boldsymbol{\omega}}^\top]^\top \in \mathbb{R}^6$ denote the robot's generalized velocity. The generalized momentum ${\mathfrak{p}}$ of the system is defined as:
\begin{equation}
\label{eq:momenta_Mtwist}
{\mathfrak{p}} = \mathbf{M}({\mathfrak{q}}){\boldsymbol{\zeta}} \in \mathbb{R}^6,
\end{equation}
where $\mathbf{M}({\mathfrak{q}}) \succ 0$ denotes a positive-definite $6 \times 6$ generalized mass matrix. Let $\mathbf{x} = ({\mathfrak{q}}, {\mathfrak{p}}) \in \mathbb{R}^{18}$ be the robot state. The Hamiltonian, $\mathcal{H}(\mathbf{\mathfrak{q}}, \mathbf{\mathfrak{p}})$, captures the total energy of the system as the sum of the kinetic energy ${\cal T}(\mathbf{\mathfrak{q}}, \mathbf{\mathfrak{p}}) =  \frac{1}{2}\mathbf{\mathfrak{p}}^\top \mathbf{M}(\mathbf{\mathfrak{q}})^{-1} \mathbf{\mathfrak{p}}$ and the potential energy ${\cal U}({\mathfrak{q}})$:
\begin{equation}
\label{eq:hamiltonian_def}
\mathcal{H}(\mathbf{\mathfrak{q}}, \mathbf{\mathfrak{p}}) = {\cal T}(\mathbf{\mathfrak{q}}, \mathbf{\mathfrak{p}}) +{\cal U}(\mathbf{\mathfrak{q}}) =  \frac{1}{2}\mathbf{\mathfrak{p}}^\top \mathbf{M}(\mathbf{\mathfrak{q}})^{-1} \mathbf{\mathfrak{p}} + {\cal U}(\mathbf{\mathfrak{q}}).
\end{equation}
As a mechanical system, the time evolution of the state $\mathbf{x}$ is governed by Hamilton's equations of motion \citep{lee2017global}:
\begin{equation} \label{eq:Ham_sys_pf_nlin}
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + \mathbf{G}(\mathbf{x}) \mathbf{u}, \quad \mathbf{x}(t_0) = \mathbf{x}_0,
\end{equation}
where $\mathbf{u} \in \mathbb{R}^p$ is the control input, $\mathbf{f}(\mathbf{x}) = \begin{bmatrix}
\bf0 & \mathbf{\mathfrak{q}}^{\times} \\
-\mathbf{\mathfrak{q}}^{\times\top} & \mathbf{\mathfrak{p}}^{\times}
\end{bmatrix}
\begin{bmatrix}
\frac{\partial \mathcal{H}(\mathbf{\mathfrak{q}}, \mathbf{\mathfrak{p}})}{\partial\mathbf{\mathfrak{q}}} \\
\frac{\partial \mathcal{H}(\mathbf{\mathfrak{q}}, \mathbf{\mathfrak{p}})}{\partial\mathbf{\mathfrak{p}}}
\end{bmatrix}$, $\mathbf{G}(\mathbf{x}) = \begin{bmatrix} \bf0 \\ \mathbf{B}(\mathbf{\mathfrak{q}}) \end{bmatrix}$, and $\mathbf{B}(\mathbf{\mathfrak{q}})\in \mathbb{R}^{6\times p}$ is an input gain matrix. The operators $\mathbf{\mathfrak{q}}^{\times}$, $\mathbf{\mathfrak{p}}^{\times}$ and the hat map $\hat{\mathbf{w}}$ for $\mathbf{w} \in \mathbb{R}^3$ are defined as:
\begin{equation*}
\mathbf{\mathfrak{q}}^{\times} = \begin{bmatrix}
\mathbf{R}^\top\!\!\!\! & \bf0 & \bf0 & \bf0 \\
\bf0 & \hat{\mathbf{r}}_1^\top & \hat{\mathbf{r}}_2^\top & \hat{\mathbf{r}}_3^\top
\end{bmatrix}^\top\!\!\!\!, \quad \mathbf{\mathfrak{p}}^{\times} = \begin{bmatrix} \mathbf{\mathfrak{p}}_{\mathbf{v}}\\\mathbf{\mathfrak{p}}_{{\boldsymbol{\omega}}}\end{bmatrix}^{\times} \!\!\!\!= \begin{bmatrix}
\bf0 & \hat{\mathbf{\mathfrak{p}}}_{\mathbf{v}}\\
\hat{\mathbf{\mathfrak{p}}}_{\mathbf{v}} & \hat{\mathbf{\mathfrak{p}}}_{{\boldsymbol{\omega}}}
\end{bmatrix}, \quad \hat{\mathbf{w}} = \begin{bmatrix}
0 & -w_3 & w_2 \\
w_3 & 0 & -w_1 \\
-w_2 & w_1 & 0
\end{bmatrix}.
\end{equation*}

We consider the case that the parameters of the Hamiltonian dynamics model in \eqref{eq:Ham_sys_pf_nlin}, including the mass $\mathbf{M}({\mathfrak{q}})$, potential energy ${\cal U}({\mathfrak{q}})$, and input matrix $\mathbf{g}({\mathfrak{q}})$, are unknown. Instead, we are given a trajectory dataset ${\cal D} = \{t^{(i)}_{0:N}, {\mathfrak{q}}^{(i)}_{0:N},{\boldsymbol{\zeta}}^{(i)}_{0:N}, \mathbf{u}^{(i)}\}_{i = 1}^{D}$ consisting of $D$ sequences of generalized coordinates and velocities $({\mathfrak{q}}^{(i)}_{0:N},{\boldsymbol{\zeta}}^{(i)}_{0:N})$ at times $t_0^{(i)} < t_1^{(i)} < \ldots < t_N^{(i)}$, collected by applying a constant control input $\mathbf{u}^{(i)}$ to the system with initial condition $({\mathfrak{q}}^{(i)}_{0},{\boldsymbol{\zeta}}^{(i)}_{0})$. We aim to learn the dynamics from the data set ${\cal D}$ and design a control policy $\mathbf{u} = {\boldsymbol{\pi}}(\mathbf{x})$ such that the robot follows a desired reference path without violating safety constraints in an unknown environment. Let ${\cal O}$ and ${\cal F} \coloneqq \mathbb{R}^{3} \setminus {\cal O}$ denote the unsafe (e.g., obstacle) set and the safe (obstacle free) set, respectively. Denote the interior of ${\cal F}$ as $\text{int}(\calF)$. We assume that ${\cal O}$ is not known a priori but the robot can sense the distance $\bar{d}(\mathbf{p}, {\cal O})$ from its position $\mathbf{p}$ to ${\cal O}$ locally with a limited sensing range $\beta > 0$:
\begin{equation} \label{eq:dyO}
\bar{d}(\mathbf{p} , {\cal O}) \coloneqq \min \crl{d(\mathbf{p}, {\cal O}), \beta},
\end{equation}
where $d(\mathbf{p}, {\cal O}) \coloneqq \inf_{\mathbf{a} \in {\cal O}} \norm{\mathbf{p} - \mathbf{a}}$ denotes the Euclidean distance from $\mathbf{p}$ to the set ${\cal O}$. The safe autonomous navigation problem considered in this paper is summarized below.

\begin{problem*}
\label{prob:main_prob}
Let ${\cal D} = \{t^{(i)}_{0:N}, {\mathfrak{q}}^{(i)}_{0:N},{\boldsymbol{\zeta}}^{(i)}_{0:N}, \mathbf{u}^{(i)}\}_{i = 1}^{D}$ be a training dataset of state-control trajectories obtained from a robot with unknown Hamiltonian dynamics in \eqref{eq:Ham_sys_pf_nlin}. Let $\mathbf{r}: \brl{0,1} \mapsto \text{Int}\prl{{\cal F}}$ be a piecewise-continuous function specifying a desired position reference path for the robot. Assume that the reference path starts at the initial robot position at time $t_0$, i.e., $\mathbf{r}(0) = \mathbf{p}(t_0) \in \text{Int}\prl{{\cal F}}$. Using local distance observations $\bar{d}(\mathbf{p}(t),{\cal O})$ of the unsafe set ${\cal O}$ in an unknown environment, design a control policy ${\boldsymbol{\pi}}(\mathbf{x})$ so that the position $\mathbf{p}(t)$ of the closed-loop system converges asymptotically to $\mathbf{r}(1)$, while remaining safe, i.e., $\mathbf{p}(t) \in {\cal F}$ for all $t \geq t_0$.
\end{problem*}

redpajama_set_name: RedPajamaArXiv


text:
Greek Debt Crisis: Latin America's Leftist Leaders Side With Tsipras
By Tim Marcin @TimMarcin
Greek Prime Minister Alexis Tsipras casts his ballot at a polling station in Athens, Greece July 5, 2015. Photo: Reuters
On the heels of Greece's historic referendum vote, Prime Minister Alexis Tsipras is receiving a flood of praise from Latin America. Many are hailing his win Sunday as a victory for leftist ideals after Tsipras' Syriza party took a stance against austerity measures offered by European creditors.
Perhaps most notably, Cuban President Raul Castro said the vote suggested the people of Greece firmly back Tsipras' leadership and his party's ideals. "That result demonstrates the majority support by the Greek people of the courageous policy of the government you lead," Castro said, according to the Associated Press. The message to Tsipras was printed on the front page of official newspapers in Cuba on Monday.
More than 60 percent of voters in Greece rejected bailout terms offered by the country's creditors, which failed to include debt relief, a major sticking point. Tsipras had called for his country to vote "no" on the terms despite the suggestion that it might result in Greece's exit from the Eurozone and the loss of the euro as a currency.
"I call upon you to ignore the sirens, the scaremongering. Decide for Greece, proud Greece in a democratic Europe," Tsripras said to large crowds before the vote, according to the Telegraph.
The debt crisis was sparked by Greece's missed payment of $1.73 billion to the International Monetary Fund last Tuesday. By voting "no" the country called for less harsh bailout terms.
Venezuelan President Nicolas Maduro hailed the results. He tweeted on Sunday, "The 'no' vote in Greece is a victory against the financial terrorism carried out by the International Monetary Fund (IMF)," according to teleSUR.
Tsipras and his Syriza party have been tied with Venezuela, with both governments favoring socialism and disdaining "economic neoliberalism," according to the Telegraph. Tsipras attended the 2013 funeral of former Venezuelan leader Hugo Chavez, and was called a "fresh political air for Europe," by the Venezuelan government upon taking office this year, reported the Telegraph.
Argentine President Cristina Kirchner also sent words of support to Tsipras and Greek people, whom she called "brave," for their vote. She also referenced her own country's financial struggles in the early 2000s. "The Greek people have said 'NO' ... to the impossible and humiliating conditions imposed upon them for the restructuring of their foreign debt," she wrote in a series of tweets. "We Argentines know what this is about. We hope that Europe and its leaders understand the message of the polls ... Nobody can be asked to sign their own death certificate."
Tsakalotos Named Greek Finance Minister
Greeks 'Have Nothing To Lose'


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