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Paint the town red in a Bloody Mary collar !
For the strong but sensitive types.
Absurd webbing collars for are only for larger dogs and come in L and XL. Just as strong as our leather collars, but with the added softness and support of cotton webbing. This collar uses extra strong utility cotton webbing which has been hand stitched into colourful patterned leather. A comfortable collar for that little extra control, designed so that strength can still be luxurious.
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- Hand Stamped 'absurd' Logo
As with any hand made product the colours and patterns will vary from batch to batch. Please bear this in mind when ordering.


text:
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#305 1820 9 ST SW has been sold or is no longer on the market. Here are some properties nearby #305 1820 9 ST SW which might interest you.
#405 929 18 Av Sw
Calgary - Lower Mount Royal - $590,000
Located in desirable Lower Mount Royal just steps away from the shops and restaurants of 17th Ave SW this rare 4 bedroom, 2 storey penthouse condo is the largest unit in the entire concrete building. The main floor boasts an open plan with hardwood flooring and large bright windows that radiates plenty...
Listing courtesy of:
#402 1229 Cameron Av Sw
Calgary - Lower Mount Royal - $550,000
You'll find 1450+ sq ft of living space, 2 bedrooms, 2 bathrooms + 2 indoor parking stalls in this sophisticated Lower Mount Royal condo. Upon entering, you'll be struck by the warmth & coziness within, including an oversized living room with luxurious fireplace feature wall & access to the...
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#404 929 18 Av Sw
Calgary - Lower Mount Royal - $529,900
AMAZING VALUE FOR THIS 2 STOREY CORNER UNIT PENTHOUSE, 2 BEDROOMS, 2.5 BATHS, 2 SIDE BY SIDE UNDERGROUND PARKING STALLS, OVER 1500 SQFT, 3 SEPARATE BALCONIES, AND ALL IN A WELL RUN CONRETE BUILDING! As you enter this large open concept property you will greeted with rich hardwood flooring, an updated...
Listing courtesy of:
#301 1732 9a St Sw
Calgary - Lower Mount Royal - $489,900
Beautifully renovated 2 storey apartment in the highly sought-after community of Lower Mount Royal! This incredible and spacious open concept unit has tons of sunlight on a quiet street steps away from downtown and trendy 17th Ave. It has 2 bedrooms, 2 bathrooms, a loft, two balconies and over 1300 SF...
Listing courtesy of:
#205 1720 10 St Sw
Calgary - Lower Mount Royal - $479,900
Location, location, LOCATION!!! Steps away from 17 Avenue, you can walk to work (or shop) downtown and it's close to ALL the amenities you could ever want! One of the few 'new' condos in the area, it was built in 2015 and still feels brand new! This beautiful 2 bedroom, 2 bathroom condo has a dream kitchen...
Listing courtesy of: RE/MAX FIRST
#303 838 19 Av Sw
Calgary - Lower Mount Royal - $449,000
Lower Mount Royal renovated sub penthouse large open plan with 2 bedrooms and 2 full bathrooms 9 foot ceilings and a gourmet kitchen with 8 foot island quartz counters and stainless steel appliances. Includes underground titled parking, in suite laundry and storage, This corner unit is bright and sunny...
Listing courtesy of:
#203 836 Royal Av Sw
Calgary - Lower Mount Royal - $445,000
Fantastic Inner City Location at UNO on Royal Avenue! Just short of 900 sq ft this spacious and modern 2 bed/2 bath corner unit is perfect for all of your inner city needs. The Kitchen is complete with quartz countertops, top of the line appliances, tons of cabinetry and a wine/beverage fridge. The Master...
Listing courtesy of:
#103 836 Royal Av Sw
Calgary - Lower Mount Royal - $424,900
WELCOME TO UNO..This stylish 2 bed/2 bath unit features wide plank hardwood flooring, trendy kitchen with quartz counter tops, stainless steel appliances and eating bar. Dining with custom built-in shelving, storage with laundry, over sized family room with lots of windows and access to the terrace/garden....
Listing courtesy of: RE/MAX HOUSE OF REAL ESTATE
#303 1732 9a St Sw
Calgary - Lower Mount Royal - $419,900
PRICE REDUCED $10,000! Fully Renovated 2-Storey Loft in the Heart of Lower Mount Royal. LOWEST PRICED UNIT in the building by $80,000! Walk to downtown, 17th Ave. shops and restaurants. This one has it all…2Bed/2Bath (each on separate floors – Master on top floor), open concept layout, Extra-high...
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#101 808 Royal Av Sw
Calgary - Lower Mount Royal - $399,888
Super bright main floor 3 bdrm condo on a lovely tree-lined street in the heart of lower Mount Royal. Renovated top to bottom in cool designer whites & grays; makes for a modern & stylish suite. New flring, knock-down ceiling, paint, appliances; everything is brand new. Nothing to do but move...
Listing courtesy of: RE/MAX HOUSE OF REAL ESTATE
#304 1731 9a St Sw
Calgary - Lower Mount Royal - $389,900
Bright & beautiful super sized 2 bedroom end unit in the heart of Lower Mount Royal. Only steps from 17th Avenue and a quick walk to the downtown core, this condo is truly a GEM! The 1500+ sq.ft. space, flooded with natural light, boasts a wood burning brick faced fire place in the living room, an...
Listing courtesy of:
#108 930 18 Av Sw
Calgary - Lower Mount Royal - $344,900
Spacious 2 bedroom, 2 bath unit with 2 indoor parking stalls. Great location in Lower Mount Royal, just steps from 17 ave. Open concept kitchen/living room, showcasing slate flooring in the entry, kitchen and bathrooms, dark laminate flooring in the living room, and carpet in the bedrooms. Both bedrooms...
Listing courtesy of:
#202 1818 14 St Sw
Calgary - Lower Mount Royal - $339,999
Lower Mount Royal gem: 1 bedroom plus den (or to use as guest bedroom) with 2 full baths. Underground parking, plus designated storage locker. Triple pane windows; L-shape bright office (off Family Room) to balcony. Spacious closets in Master bedroom (walk-in), as well in the Den. Unit upgrades include:...
Listing courtesy of:
#106 824 Royal Av Sw
Calgary - Lower Mount Royal - $334,500
This affordable upscale 2-storey loft is located in the heart of Lower Mount Royal, has high ceilings, a spiral staircase, wood burning fireplace, large windows for natural light, hardwood floors and beautiful finishings. This contemporary condo is the perfect space for entertaining or to cozy up after...
Listing courtesy of:
#100 1904 10 St Sw
Calgary - Lower Mount Royal - $325,000
This condo truly is the best of both worlds! This dog-friendly building is located just steps to 17th Ave in the beautiful and affluent community of Lower Mount Royal. A private 600 sq ft Zen garden is attached to this one bedroom condo with 12ft ceilings and immense natural light! Cook in this upgraded...
Listing courtesy of:
#101 836 Royal Av Sw
Calgary - Lower Mount Royal - $314,500
Welcome to this stylish, spacious, pet friendly 1 bedroom/1 bath in the UNO building of Lower Mount Royal. A large walk-out patio with BBQ gas line opens onto a shared courtyard with a large, sunny deck. Upgraded contemporary finishings include 5 inch plank engineered hardwood and tile flooring throughout,...
Listing courtesy of:
#102 1731 9a St Sw
Calgary - Lower Mount Royal - $309,900
This is the only suite with TWO ASSIGNED PARKING STALLS, one underground and an outside stall as well. SIZE matters and this inner city condo is close to 1100 square feet and includes in suite laundry, a master fit for a King, separate dining area, and attractive brick faced, floor to ceiling, wood burning...
Listing courtesy of:
#243 1719 9a St Sw
Calgary - Lower Mount Royal - $279,900
Exceptionally designed by Sturgess Architecture, The Block defines the modern style of the urban professional and is located in lower Mount Royal. You will be taken in by the spacious and open concept in this boutique styled condo. Brushed oak engineered hardwood compliments the modern finishings along...
Listing courtesy of:
NOTE: All information on this website are deemed reliable, however, CREB cannot guarantee the accuracy, correctness, or completeness of any information.

subset: commoncrawl


text:
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text: Hi folks, as some of us - Slyfox - now is initiated to the holy circle of Gentoo devs, this seems to be a good chance for a closer cooperation between mc.o + Gentoo. One concrete thing is that Gentoo still has several own patches, which i'd prefer to go away. So my proposal: maintain Gentoo branches/tags (on per-upstream-release basis) which contain all of Gentoo's changes directly (maybe there will also be several litte things which chould be done directly in the sourcetree instead of workarounds in the ebuilds) - this (IMHO) should make long-term maintenance (also for other distros which might be interested in Gentoo's changes). BTW: one thing's a bit strange in the ebuilds: there's an dependency on e2fsprogs @linux. Seems that causes mc to include undelfs support. Better: have a separate useflags for the optional mc-vfs'es. cu -- ---------------------------------------------------------------------- Enrico Weigelt, metux IT service -- http://www.metux.de/ phone: +49 36207 519931 email: weigelt metux de mobile: +49 151 27565287 icq: 210169427 skype: nekrad666 ---------------------------------------------------------------------- Embedded-Linux / Portierung / Opensource-QM / Verteilte Systeme ----------------------------------------------------------------------


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Alexander Bueso WebFG News | 10 Feb, 2017 17:04 - Updated: 21:54 | | |
|20:47 22/11/17
|17:46 22/11/17
|17:46 22/11/17
|16:56 22/11/17
|16:56 22/11/17
|20:48 22/11/17
|17:46 22/11/17
|19:29 22/11/17
|23:58 22/11/17
|21:26 22/11/17
By Jasper Lawler, senior market analyst at London Capital Group
Commerzbank, UniCredit, Mediobanca, Societe Generale and BNP Paribas have all reported Q4 results with Credit Suisse reporting on 15 February followed by UK banks in the weeks to follow.
Commerzbank earnings dipped as it made extra loan provisions but came in ahead of expectations. Italy's UniCredit posted a huge annual loss that was in line with analyst expectations as it wrote off bad loans in an attempt to clean up its balance sheet. Societe Generale posted a smaller-than-expected drop in fourth-quarter net profit. Mediobanca reported net profit almost doubled in the last quarter thanks to improved revenues across most business lines. BNP Paribas reported a rise in profits that missed estimates and promised a new digitisation push to lower costs.
Despite the generally well-received earnings, the below chart shows the Euro Stoxx 600 Banks Index negative on the year.
An improving European and global growth outlook contributed to a rise in bank stocks towards the end of last year. Bank shares have been some of the biggest beneficiaries of higher bond yields, improving global economic growth prospects and probable regulatory reforms in the US.
Donald Trump's presidential election victory has flipped the consensus forecast for bank stocks on its head. Worries about a government bailout of Deutsche Bank have been swept under the carpet under a wave of Trump-led optimism.
ECB stress tests show that European banks are better capitalized than during the European debt crisis but low interest rates and debilitating levels of non-performing loans mean a sustained rise in profitability is still not within reach.
The sensitivity of bank shares to political instability makes us skeptical further gains are on the horizon. The nose dive in European bank stocks in early 2016 after the Deutsche Bank coco bond scare led to very low valuations. Investors went bottom-fishing with European banks in 2016 but are unlikely to hold on for longer term growth.
The chart below shows that since the start of the year UK banks have been outperforming the wider European banking sector.
The Italian rescue package announced in December, Monte dei Paschi avoiding collapse for the umpteenth time and Brexit uncertainty saw European banks outperform those in the UK. This has reversed in 2017 as attention has turned to risks surrounding European elections this year.
The outperformance of UK banks in February relates to US banking regulation. Because of London's pre-eminence as a financial capital and because of the threat of institutions leaving after Brexit, a repeal of the Dodd Frank act in the US is more likely to be matched by a loosening of regulations in the UK than in the rest of Europe.
Recent evidence from Brexit and the US presidential election demonstrate political uncertainty doesn't have to mean lower growth or a fall in the stock market.
In fact, US growth accelerated and stocks hit new highs in the lead up to the November presidential election.
The difference in Europe is that the planned elections could be very disruptive to monetary policy, one of the main driving forces behind the rise in European markets.
France's Marine Le Pen and Holland's Geert Wilders both favour a referendum on membership of the Eurozone.
Any chance of a banking union or capital markets union, both potentially useful to European banks, will be scuppered if populist anti-EU candidates and parties rise to a position of power.
We are positive that the improved sentiment towards the banking sector can continue in pockets, but that investors must be selective.
We favour UK banks over those in Europe and prefer the larger European banks, especially in Spain but would avoid Italy.
00:20 US close: Wall Street mixed ahead of Thanksgiving holiday
Wall Street finished mixed on Wednesday, following the record highs set the day before and amid lighter-than-usual trading volumes ahead of the Thanksgiving holiday on Thursday.
22 Nov Jefferies upgrades Kingfisher to 'buy', raises target price
Analysts at Jefferies upgraded home improvement products retailer Kingfisher (KGF) to 'buy' from its previous 'hold' rating on Wednesday, raising its target price from 310p to 400p along the way.
22 Nov Europe close: German stocks give back gains amid political impasse
Stocks ended the session modestly lower as traders waited for the release of the minutes of the US Federal Reserve's last rate-setting meeting in the evening and monitored the news-flow out of Germany, amid speculation that a 'grand coalition' between Germany's two main political parties, the centre-right CDU/CSU and the Socialist SPD, may yet be possible.
22 Nov Lebanon prime minister Hariri retracts resignation
Saad al-Hariri has revoked his resignation as Lebanon's prime minister, according to a local news report on Wednesday.
22 Nov Budget: Planning probe spooks builders, mixed reaction for stamp duty cut
A range of new housing policies were proposed by Chancellor Philip Hammond in his Budget statement, including more funds for smaller housebuilder, a probe into potential holding of land by major housebuilders and a stamp duty cut for first-time buyers.
22 Nov US open: Shares hold ground ahead of FOMC minutes
Wall Street is holding its ground following the record highs set the day before, albeit amid lighter than usual trading volumes ahead of the Thanksgiving Holiday on Thursday, with investors looking to the release of the minutes of the most recent meeting of the Federal Open Market Committee.
22 Nov London close: Stocks higher on UK budget
Shares closed higher on Wednesday as investors sifted through corporate releases as well as the Autumn Budget.
22 Nov FTSE 250 movers: Qinetiq boosted by Berenberg note as Thomas Cook tanks
The FTSE 250 index was up around 0.50% on the day in which Chancellor Philip Hammond revealed his budget for the upcoming year.
22 Nov Higher margins offset declining sales volumes at Cambria
Motor dealer Cambria Automobiles managed to successfully grow profit and revenue throughout the financial year, as per-unit profits counteracted a slide in sales volumes.
22 Nov First production at Texas asset transforms Pantheon Resources into full-cycle energy and power business
Oil and gas exploration firm Pantheon Resources announced "another year of progress" in its full-year results on Wednesday, as it completed the transition from a pure-play exploration company to a full-cycle energy and power business.


text: Create 23/52 June 9, 2013 Inspired by Che and Fidel – one picture of each of my children a week for 52 weeks. Excited about 'Big School', full of questions, usually a dinosaur. Learned the word "why", has no fear, charms everyone she meets. PhotographsProject 52 You Might Also Like Make it: Easy Christmas Envelope Cushion Tutorial November 4, 2013 City & Guilds Textiles – Patchwork & Quilting: Summer School June 28, 2016 Made it: Too Big Blanket* February 19, 2013 No Comments Leave a Reply Cancel Reply Name * Email * Website Notify me of follow-up comments by email. Notify me of new posts by email. Previous Post Two Next Post Drifting


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Posted by Curt Snyder, Tustin CA on March 06, 1998 at 10:42:34:
I carried the torch in Tustin, CA. What a blast it was! I almost torched my hair, but the policeman on the motorcycle next to me cautioned me before I did so. It was amazing to see so many people, all out there cheering me on. For days after I was in a sort of shock from the sheer volume and masses of people. The torch resides in a rack on my wall now and is a wonderful topic of conversation!
I now live in Oceanside CA.
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|Price (PHP)
|Price (PHP)
|Website Redesign using FTP
|We have a website that needs a complete redesign using FTP access.
|PHP, Website Design, Graphic Design, WordPress, HTML
|Nov 23, 2017
|Today6d 20h
|Web development
|I need some changes to an existing website. Asp .net web forms. Work is on university module.
|PHP, Website Design, Graphic Design
|Nov 22, 2017
|Today6d 14h
|install my script from kogaoscript.com
|Hello, ive uploaded all files to ftp but i dont see installscreen like in the installvideo on youtube Here is the youtube video to install [url removed, login to view] here is url [url removed, login to view] then you need install the modules. I need a developer who can fix it in next 1 1,5 hours Thanks
|PHP, Javascript, Website Design, MySQL, HTML
|Nov 22, 2017
|Today6d 11h
|Server Configuration for Windows 2008 R2 IIS7
|I have a Windows 2008 R2 running IIS7 that I need help configuring. I have a website running on the server, DNS pointing to it correctly, the pages serve up without problem.... BUT... my website is programmed in 'Classic ASP' and the 'Session Variable' keep expiring. I have tried to fix it with Google Searches and such, but I fear I have made matters worse. I would like some...
|ASP, System Admin, IIS, Windows Server, Web Hosting
|Nov 22, 2017
|Today6d 6h
|I have just updated woo-commerce on my wordpress installation and it now doesnt seem to load the backend of wordpress. If i remove woocommerce via ftp the back end of wordpress is avaiable. need to get woo-commerce up to date along side admin section working. This needs to be done ASAP
|PHP, WordPress, CSS, HTML, WooCommerce
|Nov 22, 2017
|Today6d 1h
|finish off computer program to capture image, last Dev disappeared
|I need my small program that gets the rstp from my cctv camera and saves it snapshot to a local folder and ftp. The person who 3/4 had it competed then just disappeared? I can send you the program they did upto now.
|C Programming, .NET, Windows Desktop
|Nov 22, 2017
|Help needed with converting php adult site to wordpress...porting over content etc
|Hi guys,ftp etc Have a client who wants to update her site (adult content in nature) but it's an old php site with no cms behind it etc. Want to find her a new wordpress theme and port the old videos and what-not over to the new site. Have the domain, and that won't be changing. Access to the ftp etc handy as well.
|PHP, Website Design, WordPress, MySQL, HTML
|Nov 22, 2017
|Nov 22, 20175d 21h
|Comparison of http and ftp by using opnet tool.
|Comparison of http and ftp by using opnet tool.
|Linux, Cisco, Storage Area Networks
|Nov 22, 2017
|Nov 22, 20175d 19h
|Web development
|I need a new website. I need you to design and build my personal website. I want a website where people can have their antique classic cars collections display for other people.
|PHP, Website Design, Graphic Design
|Nov 21, 2017
|Nov 21, 20175d 17h
|Arabic Speaker .Net Developer (Teaching) 2 hours daily £10 per two hours
|I need Arabic [url removed, login to view] Developer to teach me to develop websites in .Net ASP and SQL 2 hours daily £10 per two hours
|PHP, .NET, C# Programming, ASP.NET, Microsoft SQL Server
|Nov 21, 2017
|Nov 21, 20175d 11h
|exiting php site development import/export php module
|I need a programmer to develop an exiting php base site. I need a new a new module in the admin side. With the new function i need to uplode all existing products on the selected sub menus. The uplode will be from excel spreadsheet and the photos with ftp file transfer. The module need to handle
|PHP, Javascript, HTML
|Nov 21, 2017
|Nov 21, 20175d 5h
|New graphic template for an e-commerce Asp custom made
|We need to apply a Themeforest html template to an existing e-commerce website made in ASP. The job must be done in maximum 3 days.
|ASP, Website Design, Graphic Design, HTML
|Nov 21, 2017
|Nov 21, 20175d 5h
|Here is a brief of what I'm looking for. Please could you get back to me with prices, timescale and an example or two of work you've done. Brief: The product name is Pû Plû (it's a French play on words that roughly translates to "no more stink"). The product is a natural toilet spray, packaged in aluminium spray bottles. The fragrances are floral and natu...
|Graphic Design, Logo Design
|Nov 21, 2017
|Nov 21, 20174d 23h
|Urgent requirement of a classic SRS document for B2B E-commerce web portal
|I need a SRS document based on the requirement that I'll share to the right candidate. looking for a professional technical writer who has experience in writing SRS (Software Requirements Specification) document for B2B E-commerce web portals. Only serious candidate with relevant experience should apply.
|Legal, Project Management, Technical Writing, Proposal/Bid Writing, Business Writing
|Nov 21, 2017
|Nov 21, 20174d 18h
|Urgent requirement of a classic SRS document for B2B E-commerce web portal
|I need a SRS document based on the requirement that I'll share to the right candidate. looking for a professional technical writer who has experience in writing SRS (Software Requirements Specification) document for B2B E-commerce web portals. Only serious candidate with relevant experience should apply.
|Legal, Project Management, Technical Writing, Proposal/Bid Writing, Business Writing
|Nov 21, 2017
|Nov 21, 20174d 18h
|Organizing Adobe Flex / Apache Flex code and ASP.NET code into Project Solution -- Repost
|We currently have a source code for an application that was developed using FLEX and [url removed, login to view] the FLEX source was been compiled with the FLEX framework and was never organized into the Adobe FLEX Development environment (We need to organize it on the latest version of Adobe FLEX that as far as i know is FLEX 4.7). What we need is someone that is very good working with FLEX and...
|XML, Adobe Flash, C# Programming, HTML5, ASP.NET
|Nov 20, 2017
|Nov 20, 20174d 11h
|Desgin ASP Bootstrap website (4 pages)
|I have [url removed, login to view] Bootstrap pages. I need a designer (web designer- CSS- Html) to design 4 pages includes (colors- buttons- gridview- rearrange items)
|Bootstrap, Graphic Design, HTML, PHP, Website Design
|Nov 20, 2017
|Nov 20, 20174d 7h
|Design a cool Retro Golden Age of Hollywood style Movie Studio Logo and Background
|Intro : Guys I am inviting you to produce and submit a graphic or combined illustrated banner, that we can use at the start of our films and for social media. READ ME! Creative Brief : Your graphic should capture the qualities of elegance, glamour, romantic and of classic cinema, a nostalgic quality that brings to mind the 1940s/50s Hollywood Era. Check the added pictures for examples, on...
|Concept Art, Graphic Design, Illustration, Logo Design, Visual Arts
|Nov 20, 2017
|Nov 20, 20175h 19m
|simple website forum
|I am looking for a developer to re-write/update or start over on a small website forum. The basic requirement is for a forum consisting of; post and reply, uploading of image and email notification to both originator and replier (if there is such a word!) There is an existing forum, written using an access database and written in .asp. The existing forums allow anyone to post. There is one generic...
|PHP, ASP, MySQL, ASP.NET, Microsoft SQL Server
|Nov 20, 2017
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|Web Designing
|Need a web designer with Skills of photoshop, ASP, Wordpress
|ASP, WordPress
|Nov 20, 2017
|Nov 20, 20174d
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Yaromir SteinerView All Tags
Liberty Center Lands 2 Tenants
Liberty Center, the approximately $350 million mixed-use project currently under construction in Liberty Township has signed two more tenants. Flip Sid, a gourmet burger restaurant, and the Funny Bone Comedy Club and Restaurant will now join other tenants such as Dillard's, DICK'S Sporting Goods, Cobb CinéBistro, Brio Tuscan Grille, Cheesecake Factory, Kona Grill and Pie & Pints.Cincinnati 14 February 2015 20:06
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text:
KPL Strategic Plan
We are pleased to announce that we have completed our Strategic Plan for the Kensington Library, our work with a 12-person committee for the past several months. This initiative was begin for two principal reasons—to ensure the library's continuing quality of service to the citizens of Kensington and to help prepare to recognize the library's 125th anniversary in 2020. As you know, the world of books and publishing has significantly changed in the past several years and thus so has the place and purpose of libraries in our communities. The digital revolution has changed the way the public reads and receives information, while at the same time young people and adults are increasingly coming together to explore ideas and discuss issues of common interest, both of which have influenced the look of libraries and the services they provide.
We would love residents of all ages to come together to discuss the plan. We see this as an opportunity to bring the community together and to offer ways in which individuals and groups might help us in our mission to ensure the library remains as vital to Kensington citizens as it has in the past. The goals we've established cannot be achieved through the work of only the library staff and its trustees. Rather, the plan is based on the assumption that the library's future as a significant community resource will best be achieved through a collaborative effort of many Kensingtonians who share in the vision and the mission described in the report.
For this reason, won't you join us on Sunday, December 3 from 3-5 PM in the Library's Sawyer Room to discuss the exciting ideas and directions the committee has envisioned. We hope at that time to amplify the goals outlined in the Strategic Plan, hear reactions from friends of the library and begin more concrete planning for the library's future. Refreshments will be served.
Join the Kensington Public Library as it looks towards its 125th Anniversary and plans for a vital future for our community!
In 2020 the Kensington Public Library will celebrate its 125th year in its present location providing its patrons with free access to information and materials. The library traces its history to 1767 when the first library was formed as "The Social Library of Kensington" and has been serving the community out of the same brick building that Joseph Hilliard donated in 1895.
We have recently initiated a Strategic Planning Process for the future direction of the library and the role it plays in the town. A committee representing a cross-section of the town has been formed to achieve these goals, but we want Kensington residents (both library and non-library users) to help us make the library a place that can be an integral part of the community for all!


text:
|Publication number
|Publication type
|Application number
|Publication date
|Jan 8, 2008
|Filing date
|Feb 1, 2006
|Priority date
|Feb 1, 2005
|Fee status
|Also published as
|Publication number
|Mark A. Kasevich, Boris Dubetsky
|Original Assignee
|The Board Of Trustees Of The Leland Stanford Junior University
|Export Citation
|BiBTeX, EndNote, RefMan
|Patent Citations (6), Referenced by (7), Classifications (11), Legal Events (5)
|External Links: USPTO, USPTO Assignment, Espacenet
This application claims priority from provisional U.S. patent application No. 60/649,302 filed on Feb. 1, 2005 and herein incorporated in its entirety.
This application was supported under grant number W911NF-04-1-0047 awarded by DARPA, thereby affording the government certain rights in this invention.
The present invention relates generally to kinematic navigation employing the quantum effect of atom interference and more particularly the invention allows to navigate by directly using the atom interferometer's phases and without using any accelerometers or gyroscopes.
For the purposes of the present invention atom interferometry refers generally to quantum mechanical interference processes that involve entities having a rest mass. Among other, such entities include atoms, ions and molecules. For a general overview of the quantum mechanical process of matter-wave interferometry the reader is referred to U.S. Pat. No. 3,761,721 to Altshuler and Franz. For a review of how matter-wave interferometry is applied in constructing inertial sensors the reader is further referred to U.S. Pat. Nos. 4,874,942 and 4,992,656 to John F. Clauser and the references cited therein.
Applications of atom interferometry in measuring kinematic properties of the various entities such as their velocities and accelerations have been explored in more detail by Chu et al. in U.S. Pat. Nos. 5,274,231 and 5,274,232. These references teach the use of the Raman process to induce internal energy transitions and thus affect the internal energy state of the entity as well as other kinematic properties such as its momentum or velocity. More precisely, pulses of electromagnetic energy are applied to the entity to stimulate transitions between non-radiative energy levels. The detuning of the electromagnetic pulses, i.e., the difference between the energy levels of the transition and the energy imparted by the electromagnetic pulses allows one to control the distribution of velocities among the entities. Precise control of the velocity distribution permits high accuracy measurements of properties such as velocities and accelerations. In turn, such precise measurements allow one to build more accurate kinematic sensors such as accelerometers and gyroscopes.
For practical inertial or kinematic navigation the kinematic properties, e.g., accelerations and rotation frequencies of the traveling object need not only be measured accurately but also frequently. In fact, for most navigation purposes the kinematic properties of the moving objects should be measured at time intervals on the order of 1 ms. Unfortunately, the best measurement accuracy in atom interferometers is obtained when measurements are performed over time intervals ranging from 10 ms to 500 ms since the accuracy of an atom interferometric measurement increases with measurement time. Hence, prior art atom interferometric sensors designed to serve as accelerometers or gyroscopes are not sufficiently quick for applications in precise kinematic navigation.
In view of the above shortcomings of the prior art, the present invention has as its objective to provide a method and an apparatus that permit direct use of atom interferometric phases, measured in time intervals optimal for the atom interferometry, for inertial or kinematic navigation without measuring kinematic properties either at those times or in between of them.
It is another object of the invention, to provide a method and apparatus that can measure body trajectory with accuracy on the order of 5 m/hr or better, such that it can be used in terrestrial vehicle navigation.
Still other objects and advantages will become apparent from the ensuing description.
The object and advantages of the invention are addressed by a method for determining the coordinates of a body, where the coordinates express the body's inertial or kinematic properties, including, for example, its trajectory. In accordance with the method, atom interferometers are provided in a body frame X of the body whose inertial or kinematic properties are to be studied. During operation, interfering entities used by the atom interferometers are released into a known frame X′ that is decoupled from the body frame X. An optical pulse sequence is then applied to the interfering entities to affect the quantum-mechanical matter-wave phases of the interfering entities as a function of the coordinates. Under these conditions, the coordinates of the body are determined from the phases of the atom interferometers. The optical pulse sequence can comprise a π/2-π-π/2 sequence or some other sequence. The pulse sequence can also be a Raman pulse sequence.
In a preferred embodiment the atom interferometers are triggered at regular time intervals. The interfering entities do not necessarily have to be atoms, and may include any objects exhibiting quantum-mechanical matter-wave interference properties. Thus, in general, the interfering entities may consist of atoms, ions and molecules and clusters. In case of using atoms, it is preferable to release them into the known frame X′ in the form of a gas cloud.
In one specific embodiment, the atom interferometers include basic atom interferometers for operating on interfering entities released at an initial time t0 and a zero initial velocity v0=0 in the body frame X. There are preferably three such basic atom interferometers aligned with three orthogonal axes, e.g., the three axes of a Cartesian coordinate system. Additional atom interferometers are provided for operating on interfering entities released at the initial time t0 and at a non-zero initial velocity v0≠0 in the body frame X. When equipped with basic and additional atom interferometers in this manner, the body rotation matrices R can be determined from phase differences between the basic and the additional atom interferometers and from rotation matrices R restored for preceding measurement times. Body coordinates can also be determined from the phases of the basic atom interferometers, restored rotation matrices R, and body coordinates restored from preceding measurement times and body velocities at the times of releasing the interfering entities.
In some embodiments of the method, the velocity of the body at the time of releasing the interfering entities for subsequent atom interferometric measurements is obtained by interpolation from restored body positions. Depending on the specific implementation of the method, the velocity of the body can be measured at predetermined body positions or the velocity of the body can be excluded.
In terrestrial applications, the method is practiced in situations where the known frame X′ is the Earth frame Xe. For reasons of accuracy, it is important that in these applications the initial velocity of the interfering entities released at initial time t0 in the Earth frame Xe be sufficiently small to allow the Earth's gravity potential V(r) to be expressed by the following expression:
V({right arrow over (r)})=V 0 −{right arrow over (x)}�{right arrow over (g)} e −T ik x i x k,
where ge is gravity acceleration and Tik is gravity gradient tensor. Also, especially when tracking the trajectory of the body over longer distances, it is advantageous that the Earth's gravity potential include both a normal gravity potential V(n) (r) and an anomalous gravity potential V(a) (r).
The invention further extends to an apparatus for determining the coordinates of the body. The apparatus has atom interferometers that are mounted in the body frame X, e.g., by being firmly attached to the body at a convenient location. The atom interferometers use interfering entities that are released into the known frame X′ which is decoupled from the body frame X. Furthermore, the apparatus has a source for applying an optical pulse sequence to the interfering entities and thereby affect the matter-wave phases of the interfering entities as a function of the coordinates. The apparatus has a unit that determines the coordinates of the body from the phases of the atom interferometers.
The apparatus can use basic atom interferometers and additional atom interferometers for operating on interfering entities released at zero and non-zero initial velocities v0 in body frame X. The interfering entities can be atoms, ions, molecules or clusters.
The apparatus can be employed in any body, for example a vehicle, that moves on the surface of the Earth, in the Earth's atmosphere, in outer space or indeed any environment that is inertial or non-inertial. In many applications, the vehicle will be earth-bound and therefore be confined to trajectories in the Earth frame.
The specific embodiments of the apparatus and particular applications of the method are described in the below detailed description with reference to the drawing figures.
Vehicle 10 has a unit 12 that houses a number of atom interferometers 14. Atom interferometers 14 are fixed in the non-inertial reference frame or body frame X of vehicle 10. In the present case, body frame X of vehicle 10 is described in Cartesian coordinates x1, x2, x3. It will be understood by those skilled in the art, that depending on the symmetry and the trajectory of body 10 a different coordinate system, e.g., a spherical coordinate system can be employed.
At an initial time t0 interfering entities 16 are released into the known frame X′ that is decoupled from body frame X. The frame into which entities 16 are released can be an inertial or non-inertial frame X′. In outer space virtually no gravitational force acts on entities 16 and thus known frame X′ is an inertial frame in which the motion of entities 16 is rectilinear. It should be noted that interfering entities 16 can be atoms, ions, molecules or any entities of matter that will undergo quantum-mechanical matter-wave interference capable of atom interferometric measurement.
In the present case, entities 16 are released near the Earth 18. Hence, known frame X′ is the Earth frame and it is described in Cartesian coordinates x′1, x′2, x′3. The Earth's gravitational field g and the Earth's rotation Ω determine the forces acting on entities 16 in Earth frame X′. In fact, once released into Earth frame X′ entities 16 follow an inertial trajectory 20 that is governed by gravitational, Coriolis and centrifugal forces as well as gravity-gradient induced forces.
In the present embodiment, interfering entities 16 are atoms that form a gas cloud 17. Atoms 16 are released at an initial velocity v0 as measured in body frame X. Initial velocity v0 can be zero or non-zero. Note that boldface as well as superscripted arrows are used to denote vector quantities throughout the document.
In a preferred embodiment, atoms 16 are released at both a zero initial velocity v0=0 and a non-zero initial velocity v0≠0 such that interferometeric measurements can be performed by atom interferometers 14 on entities 16 with different initial velocities v0 to help in recovering the coordinates in accordance with a preferred method. Although only two atom interferometers 14 having optical pulses propagating along axes x1 and x2 are shown in
Over time, vehicle 10 executes a body motion that is described by a trajectory 22. In addition to following trajectory 22, vehicle 10 also changes its orientation or undergoes rotation as expressed by a rotation matrix R. The trajectory of vehicle 10 can be described by its position in time, or X(t). It is an objective of the method and apparatus of the invention to keep the error ΔX of this navigation method to less than 5 m over the course of one hour or:
ΔX<5 m. Eq. 1
Note that unit 12 is indicated in a dashed line and gas cloud 17 of gas atoms 16 is shown following inertial trajectory 20 that extends outside vehicle 10. This is done for illustrative purposes to clearly show how gas atoms 16 once released into Earth frame X′ follow inertial trajectory 20 instead of the non-inertial trajectory 22 and rotation enforced on plane 10 by its propulsion system and aerodynamic forces. In practice, the method of invention is performed on short time scales such that entities 16 do not travel outside a small unit 12 during the measurement.
Specifically, after release of atoms 16 at initial time t0 an optical pulse sequence 24 is applied to them at times t0+t1, t0+t1+T, t0+t1+2T, where t1 is a possible time delay between the release or launching times of atoms 16 and the application of the first optical pulse and T is the time separation between the pulses of the sequence. In this particular case sequence 24 is a Raman pulse sequence and X1, R1; X2, R2; X3, R3 correspond to the coordinates and the rotation matrices R of body frame X at times t0+t1, t0+t1+T, t0+t1+2T. The Euler angle convention used in rotation matrices R that quantify the rotation of body frame X is illustrated in
Pulse sequence 24 is preferably a π/2-π-π/2 sequence of Raman pulses spaced at equal time intervals T. In other words, the Raman pulses occur with time delay T after an initial time t0=0 at which entities 16 are released into Earth frame Xe and at discrete subsequent times that can be described as t=jT, where j=0, 1, 2. Before considering the complete description of the motion of entities 16 after their release into Earth frame Xe it is useful to first review the basic principles of operation of atom interferometers 14.
In the diagram of
The splitting of inertial trajectory 20 into paths 20A, 20B and different evolution of matter-waves propagating along paths 20A, 20B gives rise to the quantum interference effect used in this invention. The phases of the matter-waves undergo different amounts of change depending on their path as well as applied optical pulse sequence 24. Since body frame X in which optical pulse sequence 24 is generated moves along with body 10 relative to paths 20A, 20B the coordinates of body 10 affect the optical pulses sequence 24 that acts on the matter-waves propagating along paths 20A, 20B. More precisely, optical pulse sequence 24 affects matter-wave phases of interfering entities 16 as a function of the coordinates of body 10.
The application of optical pulse sequence 24 affects the phases of the two interfering matter-wave packets on paths 20A, 20B and yields different measurement results at outputs 26, 28 of interferometer 14A. In other words, different numbers of atoms 16 are measured at output 26 in high energy state |hi> and at output 28 in low energy state |lo> as a function of position and rotation of body frame X. The coordinates and rotation matrices acting on body frame X, or equivalently on body 10, are determined by a unit 30 from the phases of a number of interferometers 14.
Additional information about the use of Raman pulse sequences for atom interferometric measurements on entities in the inertial frame of reference is contained in Kai Bongs, et al., "High-order inertial phase shifts for time-domain atom interferometers", Physics Department, Yale University, 25 Feb. 2003 published on the web at http://xxx.lanl.gov/abs/quant-ph/0204102, which is herein incorporated by reference. It should be noted that any suitable type of optical excitation method may be used, including single photon, two photon (Raman) and multi-photon (3 photon and higher) excitations.
To best understand the method of invention in detail, it is instructive to first consider the simplest case in which vehicle 10 is moving in an environment without gravity and undergoes no rotation, as shown in the diagram of
Pulse sequence 24 is a π/2-π-π/2 sequence of Raman pulses spaced at equal time intervals T. In other words, the Raman pulses occur with time delay T after an initial time t0=0 at which entities 16 are released into inertial frame X′ and at discrete subsequent times T and 2T. Under these conditions, the interference between the different atomic passes has a phase φ given by:
φ={right arrow over (k)}[{right arrow over (x)}(t 0)−2{right arrow over (x)}(t 0 +T)+{right arrow over (x)}(t 0+2T)], Eq. 2
where k is an effective wave vector associated with the Raman process,
{right arrow over (x)}(t)={right arrow over (x)}′(t)−{right arrow over (X)}(t) Eq. 3
x(t) and x′(t) are positions of the center of cloud 17 in noninertial body frame X and inertial frame X′. Since in inertial frame X′ entities 16 move with constant velocity v′, trajectory 20 is described by:
{right arrow over (x)}′(t)={right arrow over (x)}′(t 0)+{right arrow over (v)}′(t−t 0), Eq. 4
and thus, in body frame X one finds:
{right arrow over (x)}(t)={right arrow over (x)}(t 0)+{right arrow over (X)}(t 0)−{right arrow over (X)}(t)+{right arrow over (v)}′(t−t 0). Eq. 5
Substituting equation 5 into equation 2, we can see that in the absence of rotation, interferometer phase is velocity insensitive and can be expressed in body frame X by:
φ=−{right arrow over (k)}[{right arrow over (X)}(t 0)−2{right arrow over (X)}(t 0 +T)+{right arrow over (X)}(t 0+2T)]. Eq. 6
Since the interferometers are launched when entities 16 are released at times t0=jT, and the corresponding phases φj are measured, one can express Xj+2 through Xj+1 and Xj and restore step by step trajectory 22 of vehicle 10 at these times when X0 and X1 are known. To accomplish this, we rewrite equation 6 as:
δX j =δX j−1−φj−2 , j≧2, Eq. 7
where δXj=Xj−Xj−1, Xj={right arrow over (k)}�{right arrow over (X)}j, and find its solution,
from which
Changing the order of summation we obtain the expression:
In cases where vehicle 10 acceleration is constant, we know that φj=−{right arrow over (k)}�{right arrow over (a)}T2. In this case, since
where V0 is the platform/body velocity—equation 8 reduces to the simple expression:
We thus see that three interferometers 14A, 14B and 14C having mutually perpendicular wave vectors k1, k2, and k3 as shown in
Although the rotation between phase measurements can be significant, we can assume for simplicity of explanation, that rotation is small during the corresponding motion of vehicle 10. In other words, the rotation matrix can be expressed by:
and Ω(t) is the rotation frequency.
The trajectory 20 of cloud 17 of entities 16, in this case atoms, in inertial frame X′ is still given by equation 4. Meanwhile, in the noninertial body frame X the motion of cloud 17 is described by:
{right arrow over (x)}(t)=[1−{right arrow over (ψ)}(t)�][{right arrow over (x)}′(t)−{right arrow over (X)}′(t)]. Eq. 12
Assuming for simplicity that cloud 17 is launched in the center of the noninertial body frame X, as shown in
{right arrow over (x)}(t)=[1−{right arrow over (ψ)}(t)�][{right arrow over (X)}(t 0)−{right arrow over (X)}(t)+(t−t 0){right arrow over (V)}]+(t−t 0){1+[{right arrow over (ψ)}(t 0)−{right arrow over (ψ)}(t)]�}{right arrow over (v)} 0. Eq. 13
Since unknown rotation angles ψ(t) and position X(t) are coupled in equation 13, three interferometers 14A, 14B, and 14C (see
In the present embodiment, entities 16 for interferometric measurement by the first three interferometers 14A, 14B, and 14C, referred to as basic interferometers (BI) are launched or released into inertial frame X′ at zero initial velocity v=0. In addition, entities 16 for interferometric measurement with interferometers 14D, 14E, and 14F, referred to as additional interferometers (AI) are launched or released into inertial frame X′ at a non-zero initial velocity v≠0. Also, delaying optical pulse sequence 24 of Raman pulses to affect matter-wave phases of interfering entities 16 to time t1 after initial time t0 is beneficial to the method. Pulse sequence 24 is thus a π/2-π-π/2 sequence of Raman pulses spaced at equal time intervals T commencing at time t0+t1 and at discrete subsequent times t0+t1+T and t0+t1+2T. Under these conditions, we can express the phase through X(t) and ψ(t) as:
For a given wave vector k the phase difference between AI and BI is sensitive only to the rotation angles ψ,
and if angles at preceding times t0+t1 and t0+t1+T are already known, then the vector {right arrow over (ψ)}(t0+t1+2T) can be restored using phase difference 15. For example, to get ψ1(t0+t1+2T) one needs to use an AI in which atoms are launched along axis 3 and wave vector directed along axis 2, measure the phase difference between this AI and the BI having the same wave vector and get:
We see that this approach can be repeated with two additional AIs that have velocity vectors directed along axes 3 and 1 and wave vectors directed along axes 1 and 2, respectively to obtain ψ2(t0+t1+2T) and ψ3(t0+t1+2T) in the same manner.
It is important to underline that rotation angles ψ can be restored step by step from phase differences φr only since they are not coupled to velocity V or position X of vehicle 10. In fact, because of the absence of such coupling, it is possible to restore the orientation of vehicle 10 with much greater accuracy than even its position.
After finding the vector ψ(t), one can turn attention to the BI phases as follows:
φ={right arrow over (k)}�{right arrow over (χ)}(t 0 ,t 1,0); where Eq. 17a
{right arrow over (χ)}(t 0 ,t 1,0)=2[1−{right arrow over (ψ)}(t 0 +t 1 +T)�][{right arrow over (X)}(t 0 +t 1 +T)−{right arrow over (X)}(t 0)]−[1−{right arrow over (ψ)}(t 0 +t 1+2T)�][{right arrow over (X)}(t 0 +t 1+2T)−{right arrow over (X)}(t 0)]−[1−{right arrow over (ψ)}(t 0 +t 1)�][{right arrow over (X)}(t 0 +t 1)−{right arrow over (X)}(t 0)]+[2(t 1 T){right arrow over (ψ)}(t 0 +t 1 +T)−(t 1+2T){right arrow over (ψ)}(t 0 +t 1+2T)−t 1{right arrow over (ψ)}(t 0 +t 1)]�{right arrow over (V)}. Eq. 17b
Note that at this point, since we have restored the orientation, the only unknown is body velocity vector V at the initial time t0 when entities 16 were released. There are a number of techniques that can be used to obtain vector V.
The first technique is based on interpolation that employs atom interferometric measurements, which normally cannot be used for this purpose. In accordance with the invention, however, atom interferometric measurements are used to recover V by introducing a time delay between initial time to at which entities 16 are released and the time when first Raman pulse is applied, namely t0+t1, as shown in
Here we assume that both position X of vehicle 10 and rotation matrix R at times 0 and T are known. In this case the rotation angles ψ1, ψ2 and ψ3 can be obtained by applying equation 16 as shown above. Since all clouds 17 are launched at the same time, namely at initial time t0, the vehicle velocity vector V=0 for all of them. Now, by resolving equation 17b with respect to X(t0+t1+2T) one restores body positions at all ni points t=0, . . . (ni−1)T.
Since the last adjacent pair of times are (ni−2)T and (ni−1)T, the latest times when the next set of 6(ni−2) interferometers should be launched is t0=(ni−2)T, but in this case one needs to find body velocity V. For this purpose we fit trajectory 22 of vehicle 10 by an interpolation polynomial. For example, for ni=3 one can use a linear fit:
while for larger number of interpolation points ni, we use a third order interpolation polynomial, for which one finds:
Using this approach one finds positions X for the next ni−2 points reaching time (2ni−3)T and then repeats the process.
Another technique to obtain vector V can measure it directly by using 3�6 interferometers. In still another approach one can simply exclude V. This may be used when the time delays between the optical pulses is a multiple of T.
We once again require that:
ΔX<5 m, Eq. 1
and that the time during which this inequality is to be obeyed is:
t≈1 hr. Eq. 2
For simplicity, vehicle 10 and atom interferometers 14 are not shown in this drawing and only unit 12 is indicated. We consider navigation of vehicle 10 moving in the normal gravity field V(n)(r) of rotating Earth 18, where the gravity potential is given by:
where θ is the angle between vector r and the North direction (z-axis), P21(cos θ) is the Legendre polynomial, GM=3.986004415�1014 m3s−2 is the geocentric gravitational constant, aE=6378136.3 m is the equatorial radius, e2=f(2−f) is the first eccentricity, f is the polar flattening (1/f=298.25765), and J2=1.0826267�10−3 is the dynamic form factor of the Earth. Entities 16 (only one shown in
{right arrow over (r)}={right arrow over (X)}+{right arrow over (x)}, where x<<X. Eq. 4
This constraint is important for accuracy of the method and can be ensured with a small initial velocity v0 of interfering entities 16 in the body frame X and/or sufficiently small velocity V of vehicle 10.
With the constraint of entities 16 moving in small vicinity 32 during the atomic interferometric measurement, the normal gravity potential can be reformulated with the aid of the following expansions:
where n=X/X, z is a unit vector in the North direction, and Θ is the angle between vectors n and z, as shown in
V (n)({right arrow over (r)})=V 0 −{right arrow over (x)}�{right arrow over (g)} e −T ik x i x k, Eq. 7
where ge is the gravity acceleration and Tik is a gravity gradient tensor. These are given by:
Near the North and South poles the coefficients a, b, d, e and f have an uncertainty in this expansion and this needs to be accounted for by considering small angle θ near 0 and π radians. A standard expansion can be used to accomplish this, as will be appreciated by those skilled in the art.
In addition to the normal gravity potential of equation 3, there exists a gravity anomaly potential V(a)(r) that reflects the Earth's deviation from a perfect geoid. For the purposes of navigating vehicle 10 more accurately and over long trajectories, the gravity anomaly can be either measured locally or obtained from a map of the anomaly and added to the normal gravity potential. For computational purposes, the anomaly can be modeled on the assumption that it has a certain magnitude, a spatial extent and a zero average value. The simplest way to satisfy these requirements is to assume that the anomaly potential V(a)(r) consists of a set of harmonic terms as follows:
Formally, entities 16 move in a hypoeutectic medium having a density ρ=−4πΔV(a)(r)/G≠0 and could be negative. The choice in equation 10 is used for navigation purposes of the present invention. In fact, in an alternative embodiment, a map of the anomaly field can be used instead, if available and as practicable.
We characterize the gravity anomaly by two parameters, an average over space squared of the gravity anomaly accelerations ga and the scale of the harmonics in equation 10 of period sga. The wave vector components are then:
where r is a random number, amplitudes haj are normal random variables normalized to the given value of ga squared, namely:
Let us now return to entity 16 released at initial time t0 and initial velocity v0 in body frame X into Earth frame Xe. Entity 16 has a velocity ve in Earth frame Xe and its position r changes with time in the Earth's rotating frame as a function of its initial velocity and the gravitational, Coriolis, centrifugal and gravity-gradient induced forces. Now, in accordance with the invention, initial velocity v0 is sufficiently small that entity 16 moves in small vicinity 32 during the interferometric measurement allowing the application of equation 7 to describe the Earth's gravity potential V(n)(r). Therefore, the trajectory of entity 16 described by xe(t) evolves in accordance with the following expression:
{umlaut over (x)} ei =g i +q ik x ek+2εiklΩel {dot over (x)} ek, Eq. 13
where each superscripted dot denotes a time derivative, Ωe=Ωez, Ωe=7.292115�10−5s−1 is the rate of the Earth's rotation, and the total acceleration g and gradient tensor qik consist of the normal, anomaly and centrifugal terms:
where ge and Tik are given by equations 8a and 8e.
Now, when navigating vehicle 10 over longer periods of time, it is important not to use the approximate expression for xe(t), because after numerous steps of navigation, error builds up and violates inequality 1 in several minutes. For this reason, it is important to obtain precise expression for the trajectory xe(t) of entity 16. TO do this, we start out by noting that the eigen frequency ω of equation 13 obeys the equation:
|A(ω)|=0, Eq. 15a
A ik(ω)=q ik+ω2δik−2iωε iklΩel, Eq. 15b
which, for the symmetric tensor qik, contains only even powers of ω, i.e.,
Since the problem is reduced to the third order equation 16a in ω2, one can obtain an analytic expression for the trajectory. The vector:
evolves as:
The solution of this equation is given by:
where yi(t) is an eigen vector of the homogeneous part of equation 18 associated with eigen frequency ωi. Evidently,
where xi(t) is an eigen vector of the homogeneous part of equation 13, i.e., the solution of equation A(ωi)xi=0. One can group the six roots of equation 15a into two sub-groups {ω1,ω2,ω3} and {−ω1,−ω2,−ω3}, where
At this point it is convenient to introduce 3�3 matrices x�={{right arrow over (x)}� (1),{right arrow over (x)}� (2),{right arrow over (x)}� (3)}, where {right arrow over (x)}� (i) is an eigen vector associated with eigen frequency �ωi. Using these matrices, one can express the 6�6 matrix of equation 19c as:
to find that:
Representing matrix 19b by 3�3 blocks,
one obtains the following relevant blocks:
Φ11(t)=[x + e − {tilde over (ω)}t +x − e i{tilde over (ω)}t{tilde over (ω)}−1 x − −1 x + {tilde over (ω)}][x + +x −{tilde over (ω)}−1 x − −1 x +{tilde over (ω)}]−1, Eq. 25a
Φ12(t)=i[x + e −i{tilde over (ω)}t x + −1 x + −x − e i{tilde over (ω)}t ][x − {tilde over (ω)}+x + {tilde over (ω)}x + −1 x −]−1, Eq. 25b
Thus using representation 17 we derive expressions 25 which can be integrated over t′ in equation 19a to arrive at the following final result:
{right arrow over (x)} e(t)=x x(t){right arrow over (x)} e(0)+x v(t) {right arrow over (v)} e(0)+x g(t){right arrow over (g)}, Eq. 26a
x x(t)=[x + e −i{tilde over (ω)}t +x − e i{tilde over (ω)}t{tilde over (ω)}−1 x − −1 x + {tilde over (ω)}][x + +x −{tilde over (ω)} −1 x − −1 x +{tilde over (ω)}]−1, Eq. 26b
x v(t)=i[x + e −i{tilde over (ω)}t x + −1 x − −x − e i{tilde over (ω)}t ][x − {tilde over (ω)}+x + {tilde over (ω)}x + −1 x −]−1, Eq. 26c
x g(t)=[x +(1−e −i{tilde over (ω)}t){tilde over (ω)}−1 x + −1 x − +x −(1−e i{tilde over (ω)}t){tilde over (ω)}−1 ][x − {tilde over (ω)}+x + {tilde over (ω)}x + −1 x −]−1. Eq. 26d
In the absence of the gravity gradient terms equation 15a becomes degenerate. In principle, one can get the solution in this case from equation 26 in the limit:
Tik→0. Eq. 27
A simpler way involves using the Hamiltonian equations of motion instead of Lagrangian equation 13. In the present case, for the trajectory xe(t) of entity 16 in Earth frame Xe absent the gravity gradient we obtain:
where we assume that xe(0)=0.
Now, to obtain the phases we consider that entity 16 is launched at moment t0 from the point d with velocity v and exposed to optical pulses 24 at times:
t=t 0 +t 1 ,t 0 +t 1 +T,t 0 +t 1+2T Eq. 29
using the π/2-π-π/2 sequence of Raman pulses having the effective wave vector k. The phase of this atom interferometer is thus given by:
correspondingly, and x(t) is the atomic trajectory in body frame X. Since after launching entity 16 is decoupled from vehicle 10 and affected only by the Earth gravitational and inertial forces, it is easier to work with atom trajectory in the Earth frame Xe with consequent transition to body frame X only at times when Raman pulses 24 are applied, i.e., at the times of equation 29. Besides avoiding the consideration of inertial forces in body frame X, this also allows us to express the interferometer phase directly through vehicle 10 position and orientation.
Assuming that X(t) is the position of the origin of body frame X of vehicle 10, R(t) is the rotation matrix describing the orientation of the axes of body frame X, and xe(t−t0) is the position of entity 16 in Earth frame Xe with respect to origin of body frame X at the time of launching, X(t0), then the trajectory of entity 16 in body frame X is given by:
{right arrow over (x)}(t)=R(t)[{right arrow over (x)} e(t−t 0)+{right arrow over (X)}(t 0)−{right arrow over (X)}(t)]. Eq. 32
Taking into account the initial condition that entity 16 is launched at moment to from the point d, i.e., x(t0)=d, we find:
{right arrow over (x)} e(0)=R −1(t 0){right arrow over (d)}. Eq. 33
The velocity vector ve of entity 16 in Earth frame Xe can be obtained now by differentiating equation 32 at t=t0 and using the fact that {right arrow over ({dot over (x)}(t0)={right arrow over (v)}0, to find that at the time of launching:
{right arrow over (v)} e(0)={right arrow over (V)}(t 0)+R −1(t 0)[{right arrow over (v)} 0+{right arrow over (Ω)}(t 0)�{right arrow over (d)}]. Eq. 34
Using the above expressions and employing an alternative expression for entity 16 trajectory in Earth frame Xe as follows:
{right arrow over (x)} e(t−t 0)=R −1(t 0){right arrow over (d)}+[x v(t){right arrow over (v)} e(0)+x g(t){right arrow over (g)}] {right arrow over (X)}={right arrow over (X)}(t
which yields the final expression for the interferometer phase:
The method of invention can be used to restore the rotation matrix for any given body trajectory and rotation under the influence of noise. Since we can restore body orientation independently, as described above, we first examine the performance of rotation matrix R(t) restoration under a relative inaccuracy defined as:
where R(t) is the exact rotation matrix. The recurrent relation causes the roundoff errors for rotation matrix R(t) to increase slowly in time when one performs the calculations with quadrupole precision. Examples of the dependencies ΔR(t) are shown in
From the results, we see that the roundoff error (for δφ=0) does not exceed:
ΔR(t)≦1.9�10−21, Eq. 39
while the rather moderate level in phase inaccuracy δφ=10−3 leads to an orientation inaccuracy of:
ΔR(t)≦3.9�10−8=2.2 μdeg. Eq. 40
To restore body position X(t) we varied time delay T between pulses from 1 ms to 100 ms with a step of 1 ms. For each value of T we varied the number of points ni from 3 to 20. For any given T and ni we restored body position and rotation matrix in time until the requirement of ΔX(t)≦5 m was violated or up to a maximum time of 1.5 hours. We determined an optimum number niopt(T) and maximum value of time tmax(T).
We considered two types of body motion; oscillatory motion and pulsed motion. Examples of oscillatory trajectories as restored are shown in
T≈6-10 ms, ni≈4-17. Eq. 41
Example of exact body trajectory and that restored with the method of invention is shown in
In another example, it was assumed that the body was accelerated and rotated only by a pulse applied at the start. For the pulse of acceleration we chose the Lorentz shape:
which corresponds to the smooth part of the body trajectory:
For a pulse in the Euler angle α we chose:
and the same pulses for angles β and γ, except that the maximum value of β did not exceed π (βm∈{0,π}).
As in the case of oscillatory motion, numerical calculation shows reliable restoration of body trajectory during 1 hour for scaling factor sc=0.1. The relationship between the optimum number of interpolation points niopt(T) and maximum time tmax(T) after which the error in body position surpassed 5 m is shown in
One concludes that for the pulsed trajectory the time delay T between Raman pulses and the number of interpolation points can be on the order of:
T≈150-200 ms, n i≈3-20. Eq. 45
For the case of a body moving in the Earth frame Xe, simulations of the navigation process have been performed using equations 26 and 36 with basic and additional interferometers. Typical examples of simulations for five different trajectories are shown in
Based on the above embodiments and specific examples, it will be clear to a person skilled in the art that the method and apparatus of invention admit of many variations. Moreover, the method can be practiced in various types of known frames in addition to those described above. Therefore, the scope of invention should be judged based on the appended claims and their legal equivalents.
|Cited Patent
|Filing date
|Publication date
|Jul 6, 1972
|Sep 25, 1973
|Trw Inc
|Matter wave interferometric apparatus
|Oct 26, 1987
|Oct 17, 1989
|Clauser John F
|Rotation, acceleration, and gravity sensors using quantum-mechanical matter-wave interferometry with neutral atoms and molecules
|Aug 10, 1989
|Feb 12, 1991
|Clauser John F
|Rotation, acceleration, and gravity sensors using quantum-mechanical matter-wave interferometry with neutral atoms and molecules
|Apr 14, 1992
|Dec 28, 1993
|Board Of Trustees, Leland Stanford Jr. University
|Method and apparatus for manipulating atoms, ions or molecules and for measuring physical quantities using stimulated Raman transitions
|Apr 14, 1992
|Dec 28, 1993
|Board Of Trustees, Leland Stanford Jr. University
|Method and apparatus for manipulating atoms, ions or molecules and for measuring physical quantities using stimulated raman transitions
|Jul 2, 2004
|Nov 28, 2006
|Northrop Grumman Corporation
|Method for the processing of non-continuous atom interferometer intertial instrument measurements and continuous wide bandwidth instrument measurements with a gravity database
|Citing Patent
|Filing date
|Publication date
|May 1, 2009
|Mar 29, 2011
|The United States Of America As Represented By The Secretary Of The Navy
|Single-shot spatially-resolved imaging magnetometry using ultracold atoms
|Mar 11, 2009
|Feb 12, 2013
|Cold atom interferometry sensor
|Jan 7, 2013
|Sep 15, 2015
|Cold atom gravity gradiometer
|Mar 15, 2014
|Feb 14, 2017
|The Charles Stark Draper Laboratory, Inc.
|Ring architecture for sequential operation of an atomic gyroscope
|May 5, 2015
|Feb 14, 2017
|King Fahd University Of Petroleum And Minerals
|Method and apparatus for estimation of center of gravity using accelerometers
|May 1, 2009
|Nov 5, 2009
|The Government Of The United States Of America, As Represented By The Secretary Of The Navy
|Single-Shot Spatially-Resolved Imaging Magnetometry using Ultracold Atoms
|Mar 11, 2009
|Mar 31, 2011
|Centre National De La Recherche Scientifique (Cnrs)
|Cold atom interferometry sensor
|U.S. Classification
|250/251, 702/152, 701/494
|International Classification
|Cooperative Classification
|European Classification
|Apr 26, 2006
Owner name: BOARD OF TRUSTEES OF THE LELAND STANFORD JUNIOR UN
Free format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNORS:KASEVICH, MARK A.;DUBETSKY, BORIS;REEL/FRAME:017822/0478
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