The beautiful gray cover of the Christmas Village pop up card is tranquil and hints at the scene inside. A light post adorned with a festive red bow and topped with a little pile of snow rests in front of a rustic snow covered fence. Snow delicately falls and lands atop the fence and sidewalk. Once opened a Christmas Village emerges from the center of the pop up card. A beautifully decorated Christmas Tree is at the center and is surrounded by wrapped Christmas presents. Beautifully detailed and reminiscent of collectible ceramic Christmas villages, this pop-up card is breathtaking. One can almost imagine themselves strolling through the village, completing the last of the Christmas shopping, and preparing for the annual lighting of the Christmas tree. This pop up is the quintessential Christmas card. Give one to your favorite Christmas Village collector or to anyone who truly exemplifies the holiday spirit. This is the card for that friend who begins counting down to Christmas in July. This charming pop up card is 5 inches wide by 7 inches tall and comes with one embossed Lovepop envelope.
The cover of this lovely blue card is adorned with an illustration of two daisies with a butterfly fluttering above. Upon opening, a beautiful patch of daisies is revealed with a delicate butterfly landing on one of the flowers. A whimsical butterfly embellishment is laser-cut on the top of the card. We believe there can never be too many designs for those in between occasions because there is never a wrong time to send someone a thoughtful note. Butterflies are universally loved, and who doesn't love to stumble upon a field of wild daisies while on a hike. This design brings that feeling indoors by giving us beautiful daisies to display year round. This card is blank and can be used for any occasion or no occasion at all. Makes a lovely get well or birthday card. A great reminder that spring is near for that friend who struggles during the colder months. Pair this pop-up card with a sincere note and make someone's day. This sweet card is 5 inches wide by 7 inches tall and comes with one embossed Lovepop envelope.
Wonder that what can help you to show your love in his or her birthday? Why not choose the simplest way this time by sending birthday cards. Just express your feeling by words to your love. The Unomor birthday greeting card is design with 3D pop up three layers cake in the centre. You can write down your words in the blank space. Suitable for birthday or anniversary. Pearl paper made. Size: 6.1X5.3 inches / 15.5X13.5 cm Hollow character craftsmanship This UNOMOR product is sold with a one-year full money-back guarantee. If you're dissatisfied with your purchase for any reason, contact us for a replacement or refund.
Use the notebook that keeps up with your professional look and style and meets your everyday note-taking needs.
Printed on high quality paper stock, this handsome Father's Day card features a photographic image of a first-time dad holding his baby's hands, accented with a light blue plaid design and twine tied in a knot near the fold. It's a perfect way for a mother or father to wish their son a Happy Father's Day. Cover message: Wonderful son. Wonderful father. Wonderful adventures ahead. Inside message: On this amazing journey called fatherhood… know we're proud of you, so happy for you, and love you so very much…always. Happy 1st Father's Day. Includes one Father's Day card for son and one envelope. Card size is 5.8 inches wide by 8.3 inches long. All Hallmark Father's Day cards are made with paper from well-managed forests. Hallmark was founded in 1910 when 18-year-old J.C. Hall arrived in Kansas City, Missouri, with a couple of shoeboxes of postcards under his arm and the American dream in his heart. Hallmark products are sold online and in nearly 40,000 retail stores across the globe and people continue to recognize the iconic Hallmark brand as "the very best."
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Turkey reports over 1,600 new infections, 1,000 more coronavirus recoveries
by Anadolu Agency
ANKARA Sep 04, 2020 - 8:01 pm GMT+3
People enjoy the sunset near the Bosporus in front of the Hagia Sophia Grand Mosque and the Blue Mosque in Istanbul, Turkey, Aug. 5, 2020. (EPA Photo)
by Anadolu Agency Sep 04, 2020 8:01 pm
Turkey on Friday reported more than 1,000 new recoveries from the novel coronavirus, with over 1,600 infections over the past 24 hours, the country's health minister said.
Citing Health Ministry data, Fahrettin Koca said 1,612 new cases have raised the country's overall count to 276,555.
He also said 1,021 more patients recovered in the last 24 hours, taking the total to 249,108.
The death toll in the country rose to 6,564 as 53 more people died over the past day, he added.
"The rise in the number of patients in critical condition continues. The number of recovered patients is still less than the number of new patients," Koca said on Twitter.
A total of 117,113 more COVID-19 tests were conducted over the past 24 hours, pushing the total up to over 7.58 million.
According to Health Ministry data, 7.5% of the total patients are suffering from pneumonia, while the number of patients in critical condition stands at 1,076.
Across the world, COVID-19 has claimed over 870,400 lives in 188 countries and regions since last December. The U.S., Brazil, India and Russia are currently the worst-hit countries.
More than 26.39 million infections have been reported worldwide, with recoveries exceeding 17.56 million, according to figures compiled by the U.S.' Johns Hopkins University.
turkey covid-19 pandemic
Aurora borealis: Northern lights grace the skies

redpajama_set_name: RedPajamaCommonCrawl


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Home » Administration » Community invited to Treatment & Recovery Center dedication ceremony and open house on June 23
Community invited to Treatment & Recovery Center dedication ceremony and open house on June 23
Thursday, May 26, 2022 - 12:11pm
A dedication and ribbon cutting ceremony for the Treatment and Recovery Center (TRC) will be at 5:30 p.m. Thursday, June 23, followed by an open house until 7 p.m.
The dedication will include brief remarks by Bert Nash Community Mental Health Center CEO Patrick Schmitz, LMH Health President and CEO Russ Johnson, Douglas County Commission Chair Shannon Reid and Kansas Department for Aging and Disability Services Secretary Laura Howard.
"When we or our loved ones are experiencing a crisis, especially a psychiatric emergency, we require the support of people who know what they are doing and who are compassionate and understanding in doing it," said TRC Executive Director George Thompson. "Our team aims to be trustworthy and competent companions for those who need our services. We want them to leave our facility with their hope and well-being restored and with the support essential to moving forward with their lives. "
The TRC, located at 1000 W. Second St., is part of the Treatment and Recovery Campus of Douglas County and will serve community members with serious mental illness, substance use disorders and addiction issues. The TRC is scheduled to open this summer. The Treatment and Recovery Campus includes the supportive housing facilities, Transitions and The Cottages at Green Lake, which were completed in 2021.
The TRC will provide immediate access to care and connect people to community resources that puts them on the path to recovery. It is the place where community members can come whether they're seeking a life change or find themselves in crisis and need help immediately.
The 20,436-square-foot TRC facility will provide behavioral health crisis services performed by a patient-focused, multi-disciplinary team. Staff will include mental health professionals, peer support specialists, behavioral health specialists, licensed addiction staff, case managers and psychiatric professionals. The crisis recovery center will have an access center, 23-hour observation unit, and crisis stabilization unit.
"We are excited to introduce the groundbreaking Treatment and Recovery Center to our community," said Russ Johnson, LMH Health President and CEO. "The center is a wonderful example of how community partnerships can lead to something amazing. We are grateful to be a part of the vital work of mental health and addiction recovery work and are excited about the positive impact it will have on our community."
"The TRC will be a vital local resource for generations to come, and we have the people of Douglas County community to thank for this critical investment," Douglas County Commission Chair Shannon Reid said. "Service providers, grassroots advocates, and dedicated government staff have been tirelessly working to develop an integrated behavioral health system in our county that is accessible, compassionate, and responsive to the needs of people in crisis. I am incredibly proud of Douglas County for making strides toward better systems, and a culture, of care."
The estimated cost to design, construct and furnish the TRC is $10.6 million and it will be funded through a variety of resources including the quarter-cent sales tax that was approved by voters in November 2018 to improve behavioral health services and facilities in Douglas County. Treanor HL is the architect for the building and Mar Lan Construction provided construction management services.
The Behavioral Health Leadership Coalition partners have been instrumental in planning efforts for the center. These agencies include: LMH Health, Bert Nash Community Mental Health Center, DCCCA, Heartland Community Health Center, Lawrence-Douglas County Public Health, Heartland Regional Alcohol and Drug Assessment Center (RADAC), Lawrence-Douglas County Housing Authority and Kansas Suicide Prevention Headquarters.
"We're very honored and excited to be part of this project, and we are so appreciative of our agency partners who join us in delivering these critical services," Bert Nash Center CEO Patrick Schmitz said. "Because of that support and access to services, the destructive cycle of untreated mental illness will be broken, and countless lives will be changed and improved."
Douglas County Director of Behavioral Health Projects Bob Tryanski said, "It's tempting to focus on all of the work that has been done and all of the people who made this happen, but the real cause for celebration is that lives will be saved and changed for the better. That's what happens when people come together to do what's right, when it's right, because it's right."
Related Document(s):
Event flyer for TRC dedication ceremony and open house
Karrey Britt, Communications Specialist
kbritt@douglascountyks.org
Location & Contact:
1100 Massachusetts Street
Judicial and Law Enforcement Center:
Copyright © 2023 Douglas County Kansas


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Find out if your community recycles cartons:
The Current State of Carton Recycling
We are proud to share that 62.1% of U.S. households can now recycle their cartons through curbside and drop-off programs. That's over 72 million households in 49 states. 82 of the 100 largest communities have access to carton recycling in places like San Francisco, California and Grand Rapids, Michigan.
When the Carton Council was formed in 2009, carton recycling was only accessible to 18% of households. We are thrilled to reach 62.1%, marking a 244% increase since we got to work. Much of this success can be attributed to our growing community of carton recycling partners and friends who work just as tirelessly as we do to make carton recycling a habitual practice in households across the country.
Moving forward, we can't wait to bring carton recycling to millions of more families over the course of the year. We hope you will join us in continuing to raise awareness and spread the message to #RecycleYourCartons!
What We've Learned from 10 Years of Carton Recycling
Lessons Learned from the Greenest Schools in Florida
Why You Should Think Globally and Recycle Locally
How I Encourage My Neighbors to Recycle
Meet the Faces of Carton Recycling
Recycle Right, Recycle Often, Especially Your Cartons
Email: info@recyclecartons.com


The following marinas have boat slips, berths, and moorings in Southern France! The south portion of France is located on the Mediterranean Sea. You have the option of selecting your area of interest directly below or simply scroll down further to view all marinas and yacht clubs in South France. Continue narrowing down your search until you've located the right marina/ yacht club for you with a boat slip/ berth/ mooring that you'll be happy about!


Say hello to Colin the Commuter, this little guy needs to get to work! Help him along the London Underground in this new retro 8-bit open world adventure.
Commuting in London: there's only so much one can take of staring blankly into space and trying not to make eye contact with the commuter opposite you. There are even stories of the dreaded 'accidental crotch stare'. Rush Hour App will make that a thing of the past.
If there's a signal failure in real life, it's in the game too! Vent your frustrations in-game when the inevitable 'leaves on the line' causes havoc with your travel plans. Heck, it's much more fun than just sighing heavily and staring angrily at your watch. And if for once the tubes are running perfectly (hoorah!), there will be plenty of scenarios to test your skills against.
Think you are the ultimate commuter?
Then pit your skills against other Rush Hour commuters to show who rules those platforms.
Collect B-Coins along the way to get high scores and power ups (a certain Mayor of London might pop up to help you out).
Get from A to B in the quickest time and earn your place on that leader board.
Good Deed Feed: Being the ultimate commuter isn't just about haughtily navigating along the lines, it's also about helping your fellow traveller. Show the true spirit of London and aid those in need.
In Rush Hour, there will be a selection of mini games to play between changing lines and to collect a big stash of B-Coins. Mini games include Tube Roulette, Escalator Rush and Carriage Surfer.
Woahh there tiger, Rush Hour App isn't available to download and play just yet. I know it's looking pretty snazzy already but this is all concept work at the moment. I'm working really hard to get the funding to get it developed, so please enter your email just below to be kept up to date with the latest news. I pinky swear I won't spam you with anything!
This entry was posted on March 9, 2014 by Antonia.


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Submissions featuring "Language"
Barnett, W. et al. (2008) Educational effects of the Tools of the Mind curriculum: A randomized trial (Journal Article)
Title Educational effects of the Tools of the Mind curriculum: A randomized trial
The effectiveness of the Tools of the Mind (Tools) curriculum in improving the education of 3- and 4-year-old children was evaluated by means of a randomized trial. The Tools curriculum, based on the work of Vygotsky, focuses on the development of self-regulation at the same time as teaching literacy and mathematics skills in a way that is socially mediated by peers and teachers and with a focus on play. The control group experienced an established district-created model described as a "balanced literacy curriculum with themes." Teachers and students were randomly assigned to either treatment or control classrooms. Children (88 Tools and 122 control) were compared on social behavior, language, and literacy growth. The Tools curriculum was found to improve classroom quality and children's executive function as indicated by lower scores on a problem behavior scale. There were indications that Tools also improved children's language development, but these effects were smaller and did not reach conventional levels of statistical significance in multi-level models or after adjustments for multiple comparisons. Our findings indicate that a developmentally appropriate curriculum with a strong emphasis on play can enhance learning and development so as to improve both the social and academic success of young children. Moreover, it is suggested that to the extent child care commonly increases behavior problems this outcome may be reversed through the use of more appropriate curricula that actually enhance self-regulation.
Barnett, W. S., Jung, K., Yarosz, D. J., Thomas, J., Hornbeck, A., Stechuk, R., Burns, S.
Academic outcomes, Developmental outcomes, Experimental, Language, Learning, Literacy, Pretend play, Self-regulation
Developmental psychology, Early childhood education, Educational research
Cognitive, Teacher/practitioner, Types of play, Copyright okay
Bauminger‐Zviely, N. et al. (2014) Spontaneous peer conversation in preschoolers with high-functioning autism spectrum disorder versus typical development (Journal Article)
Title Spontaneous peer conversation in preschoolers with high-functioning autism spectrum disorder versus typical development
Bauminger‐Zviely, N., Karin, E., Kimhi, Y., Agam‐Ben‐Artzi, G.
https://onlinelibrary.wiley.com/doi/abs/10.1111/jcpp.12158
Journal of Child Psychology and Psychiatry
Language, Peers play, Social play
Academic, Teacher/practitioner
Bulotsky-Shearer, R. et al. (2012) Peer Play Interactions and Readiness to Learn: A Protective Influence for African American Preschool Children From Low-Income Households (Journal Article)
Title Peer Play Interactions and Readiness to Learn: A Protective Influence for African American Preschool Children From Low-Income Households
Guided by a strengths-based resiliency framework, this article reviews a body of research on the positive influence of interactive peer play for African American preschool children from low-income households. This literature provides evidence for positive associations among interactive peer play experiences at home and in school, and childrens early childhood social and academic skills. It presents the development and validation of three distinct dimensions of interactive peer play with African American children attending Head Start. It reviews research examining associations between these 3 dimensions and childrens academic and social outcomes, as well as evidence-based interventions designed to foster interactive peer play for this population. It highlights challenges and directions for future research, with emphasis on the likely research needed to extend our understanding of interactive peer play experiences for Latino and Asian American children and the complex mechanisms through which positive peer interactions during early childhood may support childrens early learning and development.
Bulotsky-Shearer, R. J., Manz, P. H., Mendez, J. L., McWayne, C. M., Sekino, Y., Fantuzzo, J. W.
Academic outcomes, Cultural context, Developmental outcomes, Language, Literature review, Peers play, Pre-academic skills, Social play, Social-emotional, Socio-economic background
Cultural context
Pre-academic skills
Adjustment, African-american, Cognitive, Head Start, Health, Interactive peer play, Play partners, Resilience, Review, Types of play, 0_abstractgood, Copyright okay
Cabrera, N. et al. (2017) THE MAGIC OF PLAY: LOW-INCOME MOTHERS' AND FATHERS' PLAYFULNESS AND CHILDREN'S EMOTION REGULATION AND VOCABULARY SKILLS (Journal Article)
Title THE MAGIC OF PLAY: LOW-INCOME MOTHERS' AND FATHERS' PLAYFULNESS AND CHILDREN'S EMOTION REGULATION AND VOCABULARY SKILLS
Female, Male, Language, Playfulness
Child language, Child, Preschool, Emotions, Father-Child Relations, Fathers, Humans, Language development, Mother-Child Relations, Mothers, Mütter, Play and Playthings, Sprache, Verspieltheit, Vocabulary, Väter, carácter juguetón, emotionale Regulierung, enjouement ludique, langage, lenguaje, mamás, mères, papás, PLAY, pères, regulación emocional, régulation de l'émotion, الآباء-, الأمهات-, التنظيم الانفعالي-, اللعب-, اللغة, 嬉戲性, 情感調節, 感情調整, 母親, 父親, 言語, 語言, 遊び心
Title The Magic of Play: Low-Income Mothers' and Fathers' Playfulness and Children's Emotion Regulation and Vocabulary Skills
Language, Playfulness
Father, Mütter, Parent/ Guardian play, Sprache, Verspieltheit, Väter, carácter juguetón, emotionale Regulierung, enjouement ludique, Fathers, langage, lenguaje, mamás, Mothers, mères, papás, pères, regulación emocional, régulation de l'émotion, الآباء-, الأمهات-, التنظيم الانفعالي-, اللعب-, اللغة, 嬉戲性, 情感調節, 感情調整, 母親, 父親, 言語, 語言, 遊び心
Conner, J. et al. (2014) A Play and Language Intervention for Two-Year-Old Children: Implications for Improving Play Skills and Language (Journal Article)
Title A Play and Language Intervention for Two-Year-Old Children: Implications for Improving Play Skills and Language
The purpose of this study was to develop an intervention for 2-year-old children to enhance play and language skills. The intervention was implemented over a 4-week period and included components of reading, modeling, and positive reinforcement of language and play. Specifically, children were read a story and played with a matching toy set. Participants included 10 children, all age 2, who attended a child care center. Five participants received the play intervention, and five were used as comparison. All children were assessed using the Play in Early Childhood Evaluation System (PIECES), the Preschool Language Scale (PLS), and a Vocabulary Assessment. The results of this study showed that children who received the intervention increased pretend play more than the comparison group and also increased comprehension and expressive communication skills more than the comparison group. Implications for early childhood educators and parents are discussed.
Conner, J., Kelly-Vance, L., Ryalls, B., Friehe, M.
Journal of Research in Childhood Education
Academic outcomes, Developmental outcomes, Exploratory play, Language, Peers play, Play assessment, Pretend play
Cognitive, Comparative analysis, Early childhood, Language development, PLAY, Play Coding Scheme, Play partners, Preschool Language Scale, Program effectiveness, Storytelling, Teacher/practitioner, Types of play, United States, Copyright okay
Elias, C. et al. (2002) Self-regulation in young children: Is there a role for sociodramatic play? (Journal Article)
Title Self-regulation in young children: Is there a role for sociodramatic play?
This study tested Vygotsky's assumption that sociodramatic play in early childhood contributes importantly to the development of self-regulation. It also explored whether the link between sociodramatic play and self-regulation differs for impulsive and nonimpulsive preschoolers. In a short-term longitudinal design, 51 middle-income 3- and 4-year-olds were observed in their preschool classrooms. Naturalistic observations of total dramatic play, complex sociodramatic (CSD) play, and solitary dramatic play and of self-regulation in two classroom contexts—clean-up periods and group circle time—were gathered at Time 1, in the fall of the school year. To assess development of self-regulation, clean-up and circle time observations were repeated at Time 2, in late winter and early spring. CSD play predicted development of self-regulation during clean-up periods, whereas solitary dramatic play was negatively correlated with improvement in clean-up performance. The CSD play/improved self-regulation relationship was particularly strong for high-impulsive children, nil for low-impulsive children. Findings are consistent with Vygotsky's theory and suggest that sociodramatic experiences may be especially advantageous for impulsive children, who are behind their peers in self-regulatory development.
Elias, C. L., Berk, L. E.
Developmental outcomes, Language, Longitudinal, Peers play, Pretend play, Self-regulation, Solitary play
Cognitive, Play partners, Types of play, Copyright okay
Farver, J. et al. (2000) Within Cultural Differences: Examining Individual Differences in Korean American and European American Preschoolers' Social Pretend Play (Journal Article)
Title Within Cultural Differences: Examining Individual Differences in Korean American and European American Preschoolers' Social Pretend Play
Individual differences in 30Korean American and 30 European American preschoolers' play behavior were examined to understand how intracultural variations in children's skills and behavioral characteristics may be associated with social pretend play in early childhood. Observers recorded the children's social behaviors and play complexity. Teachers rated children's social behavior, parents completed a child rearing question- naire, and children were given the Wechsler Preschool and Primary Scale of Intelli- gence–Revised and the Multidimensional Stimulus Fluency Measure. The findings showed that there were similar patterns predictive of pretend play for both groups.Over- all, children's interactive style, positive social interaction with peers, and creativity scores significantly predicted pretend play. The results suggest that individual factors related to pretend play transcend culture.
Farver, J. A. M., Kim, Y. K., Lee-Shin, Y.
http://jcc.sagepub.com/cgi/doi/10.1177/0022022100031005003
Journal of Cross-Cultural Psychology
Correlational, Creativity, Developmental outcomes, Free play, Language, Pretend play
Cognitive, Cross-cultural, Cultural psychology, Observational data, Types of play, Copyright okay
Fromberg, D. et al. (2006) Play from Birth to Twelve (Book)
Title Play from Birth to Twelve
In light of recent standards-based and testing movements, the issue of play in childhood has taken on increased meaning for educational professionals and social scientists. This second edition of Play From Birth to Twelve offers comprehensive coverage of what we now know about play, its guiding principles, its dynamics and importance in early learning. These up-to-date essays, written by some of the most distinguished experts in the field, help students explore:
all aspects of play, including new approaches not yet covered in the literature
how teachers in various classroom situations set up and guide play to facilitate learning
how play is affected by societal violence, media reportage, technological innovations and other contemporary issues
which areas of play have been studied adequately and which require further research.
Pronin Fromberg, D., Bergen, D.
http://www.tandfebooks.com/isbn/9780203622650
Routledge Ltd - M.U.A.
Cultural context, Developmental outcomes, Games with rules, Humour, Language, Learning, Object play, Outdoor play, Peers play, Physical play, Pretend play, Social play, Social-emotional
0-23 months olds, 11-12 year olds, 24-35 months olds, 3-5 year olds, 6-7 year olds, 8-10 year olds
Cognitive, Cross-cultural, Play development, Play partners, Types of play, Copyright okay


Most toddler toys these days constantly beep, blink or buzz. Youngsters typically get caught up in electronics and battery-operated toys by the age of two. It's important to budget your little one's time with electronics by putting activities in their path that encourage development. Have no fear… Svan Ride-Ons are here! These adorable plasma cars don't run on anything other than baby's laughter and energy!
Put out imaginary fires with Svan's Fire Truck Ride-On! This adorable plasma will keep your child moving for hours. Your little fireman or firewoman will love scootering around the house to save the day. Equipped with a removable seat for storage, your little one can pack their supplies for any tough jobs that may arise.
The Svan Ride On Fire Truck is pushed along by your child's feet, helping them develop motor skills, balance and confidence.
Bring the same outdoor physical activities to the comfort of your home with Svan's Ride-Ons. All toys in the collection are constructed from all natural wood using non-toxic paint and BPA-free materials and are fully compliant with American Society for Testing and Materials (ASTM) standards.


There is a new BOGO Snapple Straight Up coupon that you won't want to miss! This is available for the 18.5 oz bottles of tea and is up to a $1.48 value.
Note that this coupon prints with a Walmart logo, but it's a manufacturer so you can use it anywhere! Go ahead and print it to have ready for an upcoming sale.


Kayes Region, Mali. 25mm x 25mm x 20mm.
The original label is included with the specimen, this ex Gunnar Faber Minerals (Germany). Included in the description is the Tsavorite (Green) as well as associated Grossular (Olive Green). An interesting little specimen indeed.


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Radio România Internaţional/ Newsflash/ April 10, 2019
President Klaus Iohannis takes part in Brussels in the European Council meeting, dominated by the issue of the Brexit; Romanian Senate Speaker Calin Popescu-Tariceanu met in Washington with Vice-President Mike Pence and his national security adviser Keith
LEGAL – Romania's Constitutional Court delayed for Thursday the debate on a possible legal conflict between Parliament and the Court of Cassation and Justice on judicial panels specialized in corruption. The motion was filed by the deputy speaker of the Chamber of Deputies, Florin Iordache, during the time when the former speaker, Liviu Dragnea, delegated to him his attributions. On Wednesday, Iordache claimed that the High Court refused to form expert panels for corruption, ignoring the law, which mandated them.
BRUSSELS- Romanian President Klaus Iohannis takes part on Wednesday in Brussels in the meeting of the European Council, dominated by the withdrawal of the UK from the European Union. According to Radio Romania's correspondent in Brussels, EU leaders are debating the request made by PM Theresa May to delay the Brexit until June 30, as well as the proposal made by EC President Donald Tusk to extend by a year the process, with the option of leaving earlier if the Parliament in London ratifies the agreement. Head negotiator Michel Barnier said that this new delay would be extended depending on the plan that the British leader submits to European leaders.
WASHINGTON – The president of the Romanian Senate, Calin Popescu Tariceanu met in Washington on Tuesday the US vice-president Mike Pence and his national security adviser Keith Kellogg. Tariceanu highlighted the major role Romania plays in adjusting and implementing measures of defense and determent in NATO's eastern flank and pleaded for strengthening through diversification the Romania-USA political-dialogue, as well as for boosting the US economic and strategic presence on Romania's territory. The president of the Romanian Senate has invited the US Vice President to pay a formal visit to Romania.
BUCHAREST – Romania's Constitutional Court delayed for Thursday the debate on a possible legal conflict between Parliament and the Court of Cassation and Justice on judicial panels specialized in corruption. The motion was filed by the deputy speaker of the Chamber of Deputies, Florin Iordache, during the time when the former speaker, Liviu Dragnea, delegated to him his attributions. On Wednesday, Iordache claimed that the High Court refused to form expert panels for corruption, ignoring the law, which mandated them.
LABOR - Romania tops the charts in the EU for equality of opportunity for men and women, with a difference in wages of only 3.5%, while the EU average stands at 15%, said Minister of Labor and Social Equality Marius Budai, at the informal meeting of line ministers of the EU. He said that this issue was a priority for Romania as the country holding the presidency of the Council of the European Union. The minister specified that, within three months, Romania reached all its four aims for Union citizens, providing better and safer work conditions, ensuring labor mobility across the EU, equality of opportunity for men and women, and balance between work and private life.
NATO – Romania is host to the biggest sized medical military exercise in the history of the North Atlantic Alliance. 2,000 soldiers, firefighters, physicians, paramedics, nurses, and volunteers from 39 member and partner states are simulating urban warfare interventions.


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ANDREW MWENDA'S BLOG
Andrew M. Mwenda is the founding Managing Editor of The Independent, Uganda's premier current affairs newsmagazine. One of Foreign Policy magazine 's top 100 Global Thinkers, TED Speaker and Foreign aid Critic
Inside Museveni's 34 years
The pros and cons of the President's long rule and what they portend for the country he has rebuilt
THE LAST WORD | ANDREW M. MWENDA | This week, President Yoweri Museveni and his NRM celebrate 34 years in power. Museveni inherited a country whose state had disintegrated and economy collapsed. The country had been plagued by political instability manifest in military coups and civil wars. In the seven years between the overthrow of Idi Amin in April 1979 and Museveni's take over in January 1986, Uganda had seven presidents, an average of one president per year. Uganda was so ungovernable the legendary Lee Kuan Yew of Singapore once said it could not recover even in 100 years.
Posted by Independent UG at 4:54 AM 1 comment:
Museveni opens a Pandora's box
How the president's intervention to halt the procurement of Kampala-Jinja expressway is a disaster
THE LAST WORD | Andrew M. Mwenda | Last week, I had a meeting in Mukono, a town only 20km east of Kampala. The meeting was scheduled for 2pm. Knowing the heavy traffic on Jinja Road, where Mukono is located, I left Kampala City Center at 1pm. This gave me one hour to navigate the traffic jam. Jinja Road is a major artery connecting our landlocked country to the sea. It is congested with long queues of trailers that make traffic jams on that road a nightmare. But Wednesday last week was record breaking. I got to Mukono at 4pm.
Posted by Independent UG at 4:52 AM No comments:
Kampala grinding to a halt
How NRM politics have made Kampala a dysfunctional city and what cannot be done about it
THE LAST WORD | ANDREW M. MWENDA | As we enter 2020, Kampala has become a dysfunctional city. The heavy rains of 2019 have left most of the roads in a horrible state of disrepair. Most streets are filled with potholes, some even with craters. Motorists have to drive at the slowest speed. This makes cars pile behind each other leading to chronic traffic jams that have made movement around the city a nightmare. The only roads that have not been destroyed by the rains are those built under the leadership of Jennifer Musisi. This is not to blame the old Kampala City Council (KCC), which was led and dominated by the opposition Democratic Party (DP). True it was filled with gross corrupt and incompetent. But equally, it was grossly underfunded.
Andrew M. Mwenda
Stories & Interviews
Aljazeera - Africa to Obama: Mind your own business
International New York Times - Don't save congo
TED - Andrew Mwenda on progress in Africa
CNN - Madonna and Africa's 'celebrity saviors'
International New York Times - Celebrities Should Not Victimize Africa
International Herald Tribune - Foreign aid sabotages reform
Gelf Magazine - Fight for press freedom in Uganda
Washington Post - Africa Must Fix Itself
Guardian - Profile: Andrew Mwenda
BBC - Africans on Africa Debate


← That's Not What They Signed Up For!
What if We Encourage Makers of the Components of Solar Power to Open Shop in Puerto Rico?
Tesla could put its cutting-edge storage battery knowledge to work. Puerto Rico already has the demand for electricity.
All you would need would be Tesla, or other solar power batteries, and a factory to build solar panels, so the island could have a whole new electric grid for each community. The Panels could even go on top of the new community buildings and homes.
You wouldn't even need a power company. We lose something like half of our electricity when it travels through lines, more if it travels long distances. (This is what I think China should do as well to get power to remote villages.) If each area produced its own power that would recoup that power, so to speak, because it wouldn't be lost.
We should IMPROVE the US Virgin Islands and Puerto Rico – not just rebuild them. We can easily bring them into the 21st century with elegant city and town planning that includes electricity! England keeps its Virgin Islands in showplace shape because the Crown believes, and rightly so, that they are a key vacation spot for the world and those islands represent Britain. The British Virgin Islands get more holiday makers (vacationers) each year than England does.
This entry was posted in Science and tagged Puerto Rico Solar Power Communities, Rebuild a Better Puerto Rico with Solar Power. Bookmark the permalink.


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Home » News & Events » Lonseth and Milne Lectures
Click for Milne Lectures.
The Lonseth lecture series was established in 1985 to honor Arvid T. Lonseth, Professor Emeritus and former chair of the Mathematics Department at Oregon State University. Professor Lonseth was a superb and devoted scholar and teacher of mathematics. The lecture series is a continuing testimony to Arvid's strong interest and commitment to the mathematical education of students, especially undergraduates. He earned his B.A. in mathematics at Stanford University and his doctorate under Hans Lewy at the University of California, Berkeley, in 1939. His research was principally in integral equations, the calculus of variations, and computational methods. He joined the OSU Mathematics Department in 1948 at the invitation of department chair W. E. Milne and was promoted to full professor three years later. During his tenure as department chair from academic year 1954-55 to March of 1968, Professor Lonseth set the department firmly and successfully on its present course: a department with wide expertise, with a special interest in mathematics of the world around us, and with a dedication to undergraduate education. He retired in 1978, but his interest in teaching and learning never waned. Professor Lonseth attended virtually all of the Lonseth lectures until his death in April 2002. He always viewed video tapes of the lectures he could not attend due to poor health. These lectures remind us of our debt to Arvid.
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04/19/2019 CH2M Hill Alumni Center, Willamette Room Mai Gehrke Laboratoire J.A. Dieudonné, Université Côte d'Azur, France Lonseth Lecture Using abstract mathematical structures to study algorithmic complexity questions Automata are very simple computational models. They are important in applications of computer science but also serve as a laboratory for studying the complexity of algorithms. In this talk we introduce automata and show how finite monoids, certain very abstract algebraic structures, may be assigned...
05/03/2018 Linc 200 Henri Berestycki EHESS Lonseth Lecture Of Predator and Prey The classical Lotka-Volterra system that describes predator – prey interaction and is one of the cornerstones of mathematical ecology. An extension of the original Lotka- Volterra system that aims at showing how territories are formed as a result of strong competition between packs of predators...
05/02/2017 LaSells Stewart Center William Yslas Vélez University of Arizona Lonseth Lecture Lonseth Lecture Series Many different events impact your decisions (note the plural) to continue your studies in mathematics. The solitary work that you perform when you do homework, the interactions with peers and the feedback from your instructors all play a role in your self-assessment. These self-assessments continue...
05/10/2016 Construction and Engineering Hall, LaSells Stewart Center Richard Tapia Rice University Lonseth Lecture The Remarkable Journey of Isoperimetric Problem: From Euler to Steiner to Weierstrass Dr. Richard Tapia, professor in the Department of Computational and Applied Mathematics at Rice University, will present his talk, "The Remarkable Journey of Isoperimetric Problem: From Euler to Steiner to Weierstrass," an overview of the history of the impactful isoperimetric problem. He will...
05/12/2015 LaSells Stewart Center C&E Hall Harold R. Parks Oregon State University Lonseth Lecture Plateau's Problem and the Geometry of Soap Films By dipping a wire frame in a soap solution, one can often produce a soap film spanning the frame. That soap film will have the smallest area among nearby surfaces that also span the frame. The mathematical model for this type of surface is a minimal surface. Plateau's problem is to show the...
05/20/2014 LaSells Stewart Center C&E Hall Dr. David Pengelley New Mexico State University Lonseth Lecture Sophie Germain's grand plan to prove Fermat's Last Theorem Sophie Germain (1776-1831) is the first woman known to have created important new mathematical research. She is best known in number theory for the first general result aimed at proving Fermat's Last Theorem, finally proven only 20 years ago. However, unpublished manuscripts, and a letter to Gauss...
05/21/2013 LaSells Stewart Center C&E Hall Dr. Paul Zorn St. Olaf College Lonseth Lecture Extreme Calculus There is more to elementary calculus than may first meet the eye, even to those of us who have learned (or taught it) at various levels. With appropriate help from graphical, numerical, and algebraic computing, well-worn calculus techniques and topics---polynomials, optimization, root-finding,...
05/15/2012 LaSells Stewart Center C&E Hall David Bressoud Macalester College Lonseth Lecture Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture 2012 Lonseth Lecture Date: Tuesday, May 15, 2012 [awards ceremony 3:30 pm, lecture 4:00 pm, reception following] Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture Abstract: What is the role of proof in mathematics? Most of the time, the search for proof is less about...
05/24/2011 LaSells Stewart Center J. Michael Shaughnessy Portland State University Lonseth Lecture Favorite Chance Encounters Dr. Shaughnessy has taught mathematics content courses and directed professional development experiences for mathematics teachers at all levels, K–12, as well as community college and university. He has authored or coauthored more than 60 articles, books, and book chapters on issues in the teaching...
05/11/2010 Dr. Keith Devlin Stanford University Lonseth Lecture When Mathematics Changed Us At four distinct stages in the development of modern society, a mathematical development changed — in a fundamental, dramatic, and revolutionary way — how people understand the world. (A fifth such change may be taking place during our lifetime, but only history will say if this is really the case...
05/19/2009 LaSells Stewart Center Robert Daverman University of Tennessee, A.M.S. Lonseth Lecture Mysteries of the Cantor Set The Cantor set exhibits captivating and, occasionally, bizarre phenomena in diverse branches of mathematics. And it is a fundamentally important object -- anyone who completely understands the Cantor set is assured of mathematical success. This talk will describe some beguiling Cantor set...
05/27/2008 LaSells Stewart Center John W. Lee Lonseth Lecture Weierstrass Approximation Theorems Weierstrass published his celebrated approximation theorems in July of 1885. I will start with brief speculations on the antecedents of Weierstrass' work and move on to a selective survey of results and/or proofs related to Weierstrass' original theorems. The survey will include results and/or...
05/08/2007 LaSells Stewart Center Professor Jim Douglas, Jr. Purdue University Lonseth Lecture The Role of Capillarity in Multiphase Flow in Porous Media Professor Lonseth was very interested in seeing that mathematics interact with other disciplines to improve the understanding of phenomena in these disciplines, and this lecture will be devoted to showing by three examples the importance of including the effects of capillarity in approximating...
05/02/2006 LaSells Stewart Center Professor Peter Lax Lonseth Lecture Degenerate Symmetric Matrices Abstract: If a finite group G acts on a set X in such a way that each non-trivial element of G fixes a unique point, then they all must fix the same point (i.e. G has a global fixed point which is necessarily unique). We will cover the proof of this result as given in a paper by Max Forester and...
05/10/2005 LaSells Stewart Center Doug Arnold Institute for Mathematics and its Applications, University of Minnesota Lonseth Lecture The New Mathematical Gravitational Astronomy Contemporary understanding of the cosmos is based on on Einstein's amazing insight that gravity is simply a manifestation of curvature. One ineluctable, though subtle, consequence of this theory of general relativity, is that violent cosmic events--imagine two black holes wildly orbiting around...
05/11/2004 Steven G. Krantz Washington University in St. Louis Lonseth Lecture A Matter of Gravity It is a standard topic in any multivariable calculus course to develop the concept of "centroid" or "center of gravity", and to teach the student to calculate this center. Rarely is there any further investigation into properties of the center of gravity. Nonetheless, there are interesting...
04/29/2003 John H. Ewing Executive Director American Mathematical Society Lonseth Lecture The Mathematics Inside Your Computer Computers don't operate using only bits and bytes to perform logic and arithmetic. They use sophisticated mathematics to perform many of the routine tasks you take for granted every time you turn on your machine. This talk will survey a small sample of that sophisticated mathematics, from an...
05/28/2002 Colin Adams Williams College Lonseth Lecture Real Estate in Hyperbolic Space: Investment Opportunities for the New Millennium Have you found the new investment climate a bit on the chilly side? Nervous about stocks, bonds and mutual funds? Afraid of risky investments in Euclidean space? Then real estate in hyperbolic space is for you. We will discuss the enormous potential of this new investment opportunity and describe...
05/16/2000 Constance Reid Noted Mathematical Historian and Biographer Lonseth Lecture The Improbable Life of Richard Courant Almost thirty years after his death, Richard Courant remains a highly controversial figure in mathematics, complex and contradictory; but the message he emphasized throughout his long career was one that he had absorbed in his youth in Gottingen from David Hilbert and Felix Klein--the underlying...
05/18/1999 Kenneth A. Ross University of Oregon Lonseth Lecture The Mathematics of Card Shuffling How many times do you have to shuffle a deck of cards before it is well mixed? What do we mean by well mixed? Questions like this will be discussed and seen to lead to the study of random walks on certain finite groups. This is an expository talk on work by Persi Diaconis and his colleagues, though...
05/05/1998 Philip A. Anselone Oregon State University Lonseth Lecture The Power of Calculus: The legacy of Newton Isaac Newton developed calculus and used it to derive universal laws of motion and gravitation that apply not only on Earth but also to the planets and stars. His laws justify and explain the pervious discoveries of Galileo and Kepler. Newton's laws, particularly force equals mass times...
05/06/1997 Margaret Wright Bell Laboratories, Lucent Technologies Lonseth Lecture Model, Speed up, Optimize, Remodel: Fun and Profit for Mathematics and It's Friends Mathematics plays a major role in formulating and modeling real-world problems--but models are never right the first time. So mathematics also enters in speeding up complicated calculations, optimizing whatever the current model may be, figuring out its defects, and then producing a more realistic...
05/07/1996 Robert Osserman Stanford University Lonseth Lecture The Shape of the Universe Mathematics plays a major role in formulating and modeling real-world problems--but models are never right the first time. So mathematics also enters in speeding up complicated calculations, optimizing whatever the current model may be, figuring out its defects, and then producing a more realistic...
05/25/1995 Ronald L. Graham AT&T Bell Labs Lonseth Lecture Mathematics and Computers: Recent Successes and insurmountable Challenges There is no question that the recent advent of the modern computer has had a dramatic impact on what mathematicians do and how they do it. However, there is increasing belief that many apparently simple problems may in fact be forever beyond any conceivable computer approach. In the talk I will...
05/24/1994 Tsit-Yuen Lam University of California-Berkeley Lonseth Lecture Mistakes We all Made: How Error-Free is Mathematics? Mathematics, as a subject, derives its beauty from its internal consistency and sound logic. It is thus axiomatic that the proofs and argumentations used in the development of mathematics be absolutely accurate and error-free. Yet the history of mathematics is replete with instances of false starts...
04/27/1993 Mary Ellen Rudin University of Wisconsin-Madison Lonseth Lecture Dimension When dealing with topological spaces which are note necessarily metric, we run into a variety of questions. We will discuss several rather nice classes of such spaces as well as some conjectures. We will prove, with the aid of one rather special example, that three of the conjectures are false.
05/19/1992 John Horton Conway Princeton University Lonseth Lecture On the Shape of Things Conway is recognized for his studies in combinatorics and group theory, which is the branch of algebra that studies the properties of symmetries of figures, and how you can go from one symmetry to another. Conway has made some major an fundamental discoveries in this field.
05/14/1991 Ian Stewart University of Warwick Lonseth Lecture Four Encounters With Sierpinski's Gasket Sierpinski's gasket is a fractal, obtained by repeatedly deleting the middle section of a triangle. It shows up in a number of different areas of mathematics, with surprising cross-connections. The talk will describe four occurrences of the gasket: 1. What Sierpinski originally invented it for; 2...
05/01/1990 Serg Lang Yale University Lonseth Lecture A B C Conjecture Recently, there have been some deep new insights into classical old problems, like Fermat's last theorem. Some of these insights can be expressed in terms of fairly elementary mathematics involving polynomials and numbers. I will describe some of these insights.
05/16/1989 George Andrews Pennsylvania State University Lonseth Lecture Ramanujan's Lost Notebook The "Lost" Notebook provides us with a record (probably incomplete) of Ramanujan's discoveries during the last year of his life. A number of his formulas from this document have been proved and analyzed; however, many remain unproved and totally mysterious. We shall survey some of the topics...
05/17/1988 G. D. Chakerian University of California-Davis Lonseth Lecture Cantor Dust Under a Binary Tree This lecture will deal with some of the more paradoxical properties of the real numbers, from a geometrical point of view. In particular, the famous Cantor ternary set will be used to illustrate the idea of a fractal, a set of fractional dimension.
05/19/1987 Gilbert Strang MIT Lonseth Lecture Chaos: Strange Attractors and Fractuals Professor Strang is noted for his illuminating lectures on a wide variety of mathematical topics. His talk should appeal to students and former students of mathematics and also to teachers of mathematics from high school through graduate school.
05/20/1986 Ivan Niven University of Oregon Lonseth Lecture Some Surprising Results in Elementary Mathematics Although the background assumed is modest, the results are ingenious and not widely known. Professor Niven is noted for his lucid presentations of mathematical ideas. His lecture should appeal to students and former students of mathematics and also to teachers of mathematics from high school...


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\section*{INTRODUCTION}
In recent years, intelligent robot arms have been playing increasingly important roles in both industry and people's daily life. Nonetheless, one of the biggest challenges toward the wide adoption of intelligent robot arms still lies in real time motion planning,  i.e., how we can enable a robot to compute its trajectories to achieve its goal in real time~\cite{latombe2012robot}. Regardless of different configurations and sizes of the robots, to work in complex environments, they need to satisfy multiple task constraints such as collision avoidance and contact maintenance, which make the motion planning problem difficult to solve in real time.

Task constrained motion planning for robot arms rises in many situations~\cite{stilman2007task}, for example, when robots are interacting with humans, or when robots are interacting with the environment during door-opening, welding, polishing or grinding. The task constraints can  be divided into two categories: 1) the workspace inequality constraints (e.g., for collision avoidance), and 2) the task equality constraints (e.g., for contact maintenance). The workspace inequality constraints require that the robot does not collide with itself or with the  obstacles in the environment~\cite{mi2019sampling}. The task equality constraints can be interpreted as additional objectives that the robot should satisfy. For example, in a grinding task, the robot needs to maintain contact with the surface being grind.

\subsection*{Related Work}
Conventional task-constrained motion planning uses kinematic control techniques~\cite{klein1983review} by transforming a task-space trajectory (or a Cartesian space trajectory) to the robot configuration space using the inverse of the task Jacobian. Since the configuration space usually has higher dimension than the task space, there is a null space associated with the Jacobian inverse. The motions in the null space can be optimized locally to achieve other objectives, i.e., collision avoidance. There are many methods for local null space optimization. The most widely used approaches are through energy function based methods~\cite{wei2019safe}, which include potential field methods~\cite{khatib1986real}, control barrier functions~\cite{Chiara2019ECC}, safe sets~\cite{HsienChung2017CCTA}, etc. Those methods first define a scalar energy function (also called a potential function, a barrier function, or a safety index) that attains small value in the collision free space and high value in the unsafe space. Then the scalar energy function will serve as the objective function to be minimized for the null space optimization. The resulting robot trajectory will be repulsed from the obstacles.  However, it is challenging and time consuming to generate the scalar energy function in the high-dimensional joint configuration spaces~\cite{bansal2017hamilton}. Moreover, the solution of the local optimization can easily be trapped into local optima, hence creating stability issues or deadlocks.

To address the issues mentioned above, global planning approaches are needed. There are two types of algorithms  for global constrained motion planning: sampling-based methods~\cite{kim2016tangent} and optimization-based methods~\cite{liu2017convex}. Sampling-based methods plan trajectories by generating random joint space displacements until the goal is reached. Representative methods include probability road maps (PRM)~\cite{kavraki1996probabilistic} and rapidly-exploring random tree (RRT)~\cite{lavalle1998rapidly}, both of which can generate trajectories that satisfy the workspace inequality constraints (e.g., collision avoidance). However, the trajectories planned by construction are usually not smooth~\cite{li2019dynamical}. The chance for the random sampled trajectories to satisfy the task equality constraints (e.g., for contact maintenance) is low~\cite{stilman2007task}. Hence, the exploration may be inefficient

Optimization-based methods, on the other hand, generate much smoother trajectories compared to the sampling-based methods. For optimization-based planning, an optimization problem needs to be formulated first, which includes an object function and multiple constraints. A desired trajectory is obtained by solving the optimization problem~\cite{liu2018convex}. For example, the CHOMP algorithm~\cite{ratliff2009chomp} solves motion planning problems that penalize the magnitude of joint velocities and accelerations and the distance towards static obstacles. The ITOMP algorithm~\cite{park2012itomp} divides obstacles into dynamic obstacles and static obstacles, and solves motion planning problems that penalize the distance towards both dynamic and static obstacles.

One major challenge for optimization-based planning algorithm is its computational inefficiency. Since the optimization problem for motion planning is usually highly nonlinear and non-convex, it is computationally expensive to obtain a solution using generic nonlinear optimization solvers such as sequential quadratic programming (SQP)~\cite{boggs1995sequential} and sequential expanded Lagrangian homotopy (SELH) ~\cite{Dharmawan2018JMR}. Those algorithms obtain solutions by solving the Karush-Kuhn-Tucker (KKT) equations. These generic algorithms work poorly on robot motion planning problems since they fail to incorporate domain specific information, e.g., the geometry of the problem.

To improve the performance of optimization solvers and tackle the computation challenge, various methods that directly convexify the optimization problem using domain knowledge have been proposed. For example, Liu \textit{et al}.~\cite{liu2018convex} proposed the convex feasible set (CFS) algorithm to efficiently handle non-convex inequality constraints through iterative convexification of the constraints. E. Todorov \textit{et al}.~\cite{Emanuel2004ICINCO} proposed the iterative LQR algorithm to efficiently handle nonlinear equality constraints through iterative linearization of the constraints. Howell \textit{et al}.~\cite{howell2019altro}~\cite{jackson2020scalable} proposed ALTRO which combines iLQR with an augmented Lagrangian method to handle general state and input constraints. However, the task equality constraints are not considered in ALTRO. To the best knowledge of the authors, there is very limited research on real time optimization algorithms for constrained trajectory generation that can handle both the non-convex workspace inequality constraints and the nonlinear task equality constraints.

\subsection*{Challenges and Contributions}
This paper focuses on developing optimization-based planning algorithms for contact-rich trajectory generation in confined environments, where the robot trajectory in the joint space is subject to workspace inequality constraints for collision avoidance and equality constraints to maintain contact. This algorithm is applied on a weld bead removal task using a six degrees of freedom industrial robot arm. In this application, the workspace inequality constraints require the trajectory to be collision-free and the task equality constraints require the robot end-effector tip to always maintain contact with the weld bead. In addition to weld bead removel, the method for contact-rich trajectory generation in confined environments can also be applied to many other applications, such as robotic welding, rehabilitation, etc.

There are two major challenges regarding the development of optimization-based contact-rich trajectory generation algorithms: 1) real time computation, 2) satisfaction of various constraints. As pointed out earlier, the optimization problem for trajectory generation in a cluttered environment is highly nonlinear and non-convex, which is hard to solve in real time.

To address the aforementioned challenges, we propose an iterative convex optimization approach to efficiently handle the constraints and achieve real time computation.
To handle the constraints, we first leverage the CFS algorithm to iteratively
transform the non-convex inequality constraints into a sequence of convex inequality constraints called the convex feasible sets~\cite{liu2017convex}. Then we leverage the approach in iLQR to iteratively linearize the nonlinear equality constraint. Hence the original nonlinear and non-convex optimization problem can be transformed into convex problems and solved iteratively. In the following discussion, we will call the proposed approach as iterative convex optimization for planning (ICOP).
Experimental results show that ICOP can generate desired trajectories that satisfies all the constraints in real time.

The contributions of this paper are summarized as follows:
\begin{enumerate}

    \item We propose the ICOP framework, which can generate high quality trajectories with significantly reduced computation time compared to conventional methods.
    \item We conduct real world robot polishing experiments to demonstrate the applicability of our proposed method in advanced robotic manufacturing.
\end{enumerate}

The remainder of the paper is organized as follows: In Section II, the problem for contact-rich trajectory generation in confined environments is formulated; in Section III, the proposed approach will be introduced; in Section IV, the performance of the proposed algorithm will be illustrated under a weld bead removal task and will be compared against conventional methods; Section V concludes the paper.

\section*{PROBLEM FORMULATION}
This paper  focuses on contact-rich trajectory generation in confined environment. The robot state is denoted as $x \in X \subset \mathbb{R}^n$, where $X$ is the configuration (state) space and  $n$ is its dimension, which specifies the degree of freedom of the robot. The robot state at a discrete time step $t$ is denoted as $x_t$. A trajectory is defined to be the sequence of states from time 1 to time T: $\bold{x} = [x_1;x_2;...;x_T] \in \mathbb R^{n \times T}$.

\subsection*{Safety Specification in Confined Environments}
The workspace inequality constraints are defined so that the robot should be collision-free with the surrounding obstacles. Suppose the area occupied by the obstacle in the Cartesian space is $O\subset\mathbb{R}^3$, the safety constraint for robot with state $x_t$ at time step $t$ is defined as:
\begin{equation}\label{eq: safety}
d(x_t, O) > 0,
\end{equation}
where the function $d(\bullet): \mathbb{R}^n \times \mathbb{R}^3 \to \mathbb{R} $ is a signed distance function that computes the distance from the robot to the obstacle in the Cartesian space. To ensure safety, we require the distance to be greater than zero.

\begin{algorithm*}[h]
  \caption{Iterative Convex Optimization for Planning}\label{ICOP}
  \begin{algorithmic}[1]
    \Procedure{Iterative Convex Optimization for Planning}{$\mathbb{C}_{target},O,T,x_{ini},\xi$}
      \State \textbf{Inputs:}
      \State $\mathbb{C}_{target} \in \mathbb{R}^{m\times T}$: Pre-defined $j$-th robot body point trajectory in Cartesian space
      \State $O$: Environment obstacles
      \State $T$: planning horizon
      \State $x_{ini}$: initial robot joint state
      \State $\xi$: equality threshold

      \State \textbf{Initialization:}
      \State $x_{pre} = x_{ini}$

	\State \textbf{Iteration:}
      \For{$t=0,1,2,..., T$} \Comment{Outer Loop}
      	\State Set $C_{next}\leftarrow\mathbb{C}_{target}(t)$, $x_{ref} \leftarrow x_{pre}$, and $C_{ref} \leftarrow \Gamma(x_{pre})$
	\While{$\| C_{next} - C_{ref} \| > \xi$ \OR $d(x_{ref}, O) < 0$} \Comment{SafeTrack Inner Loop}

		\State Find convex feasible set using \eqref{eq8}
		\State Linearize the nonlinear equality constraint using \eqref{eq9}
		\State Solve the convex optimization problem from \eqref{eq10}, whose solution is ${\hat{x}}_{ref}$;
		\State Update $x_{ref} \leftarrow {\hat{x}}_{ref}$ and $C_{ref}\leftarrow \Gamma(\hat{x}_{ref})$
	\EndWhile
	\State Record $\bm{x}(t) \leftarrow x_{ref}$ and $x_{pre} \leftarrow x_{ref}$
      \EndFor
      \State \textbf{return} reference trajectory $\bm{x}$
    \EndProcedure
  \end{algorithmic}
  \label{alg1}
\end{algorithm*}

\subsection*{Contact-Rich Task Specification}
The contact-rich task equality constraints are defined so that the robot body is in contact with the specified targets. Mathematically, the constraint can be written as:
\begin{equation}
\Gamma_j(x_t) = p_j^t,
\label{2}
\end{equation}
where the $\Gamma_j(\bullet): \mathbb{R}^n \to \mathbb{R}^m $ is a generalized function to project the $j$-th point on the robot body (e.g., tool tip) to the constrained task space given the robot configuration $x_t$. $m\in\{1,2,\ldots, 6\}$ is the dimension of the task space. $m=2$ means the task is constrained in a 2-dimensional plane, e.g., wiping a surface. $m=3$ means the task constraints are defined in a 3-dimensional Cartesian space, e.g., welding on a predefined trajectory. And $m=6$ means the task has 3 translation constraints and 3 angular constraints. $p_j^t \in \mathbb{R}^m$  is predefined target for the $j$-th point on the robot body at time step $t$. Note that the mapping $\Gamma_j$ involves highly nonlinear forward kinematics. This paper mainly considers the case $m=3$.

\subsection*{Optimization Objective}
Instead of optimizing the trajectory as a whole, we propose to generate the contact-rich reference trajectory step by step. To avoid the potential instability and deadlocks issues for local optimization, we may perform the step-by-step optimization from time 1 to time T multiple rounds.
The optimization problem at time step $t$ is formulated as:
\begin{equation}
\begin{aligned}
& \underset{x}{\text{min}}
& & J(x_t,x_{t-1}) = \| x_t - x_{t-1}\|_Q^2 \\
& \text{s.t.} & &  \Gamma_j(x_t) = p_j^t, \forall j = 1,2,3,...,\\
& & &  d(x_t, O) > 0,\\
& & &  x_{min} \leq x_t \leq x_{max},\\
\end{aligned}
\label{eq:opt}
\end{equation}
where $J$ is the cost function; $\| x_t - x_{t-1}\|_Q^2 = (x_t - x_{t-1})^T Q (x_t - x_{t-1})$ penalizes the differences of states between two time steps; $Q$ is the weight matrix; $x_{t-1}$ is the reference configuration state from the last time step; $x_{min},x_{max}\in\mathbb{R}^n$ are joint limits.  The intuition behind \eqref{eq:opt} is that we want to minimize the norm difference between the joint position in the last step and the joint position in the current step to ensure that the robot joint states evolve smoothly. The three constraints are: 1) the equality constraints to maintain contact, 2) the inequality constraints to ensure safety, and 3) the robot joint limits.

\section*{METHODOLOGY}

This section discusses the methodology to solve \eqref{eq:opt}. We will first introduce iterative linearization of the nonlinear equality constraints, then discusse the convex feasible set algorithm to handle the nonconvex inequality constraints, and finally introduce the proposed  ICOP framework.

\subsection*{Iterative Equality Linearization Approximation}
The equality constraint in \eqref{2} encodes the task requirements, but is highly nonlinear and expensive to resolve. In this paper, we assume $\Gamma$ is a twice continuously differentiable function whose second derivative exists and is continuous.
To speed up the computation, we propose to iteratively consider the first order approximation of the nonlinear equality constraint until the solution converges.

Suppose the initial joint state is $x_0$, the corresponding $j$-th robot body point is located at $C_0$. Our target is to find $x_1$ that satisfies $\Gamma_j(x_1) = C_1$, where $C_1$ is the next desired $j$-th robot body point. Suppose the distance between $C_1$ and $C_0$ is less than a small positive constant. Then we can use first-order linear approximation to represent $C_1$ as:
\begin{align}
C_1 &= \Gamma_j(x_0) + \nabla\Gamma_j(x_0)\cdot(x_1 - x_0) + \sigma, \\
	&= C_0 + \nabla\Gamma_j(x_0)\cdot(x_1 - x_0) + \sigma, \label{eq5}
\end{align}
where $\sigma$ is an error term, and $\sigma \to 0$ as $\| x_1 - x_0 \|_2^2 \to 0 $.  Denote $\nabla \Gamma_j(x)$ as $Jac(x, \theta)$, which is the generalized Jacobian matrix at $x$ with respect to a robot feature vector $\theta$ such as the DH parameters. Now we rewrite \eqref{eq5} as:
\begin{equation}
Jac(x_0,\theta)\cdot x_1 = Jac(x_0,\theta)\cdot x_0 + C_1 - C_0 - \sigma,
\label{iela}
\end{equation}
which is a linearized equality constraint for \eqref{2}. Inspired by iLQR, it is reasonable to assume that solving the optimization \eqref{eq:opt} with respect to the iterative approximation of nonlinear equality constraints in \eqref{iela} will lead to converging results~\cite{Emanuel2004ICINCO}. The formal convergence proof is beyond the scope of this paper, which is left for future work.

\subsection*{Convex Feasible Set Algorithm}
To deal with the nonlinear inequality constraint \eqref{eq: safety} for safety in confined environments, we leverage the Convex Feasible Set Algorithm (CFS)~\cite{liu2018convex} to efficiently search the non-convex feasible space for solutions by solving a sequence of convex optimizations constrained in the convex feasible sets.

The CFS algorithm handles problems that satisfy the following two assumptions: 1) The cost function $J$ is strictly convex and smooth, which is satisfied by \eqref{eq:opt}. 2)  The nonlinear safety inequality constraints can be written as $x\in \Lambda$ where $\Lambda = \cap_i \Lambda_i$, and $\Lambda_i = \{x : \phi_i(x) \geq 0\}$ where $\phi_i$ is a continuous, piecewise and semi-convex smooth function, which is satisfied by \eqref{eq: safety}.

Given a reference point $x^r$, we compute a convex feasible set $\mathcal{F} := \mathcal{F}(x^r) \subset \Lambda$ around $x^r$.   Note that the convex feasible set with respect to a reference point is not unique. For each constraint $\Lambda_i$, We will find a convex feasible set $\mathcal{F}_i$ and construct the convex feasible set as $\mathcal{F}(x^r) = \cap_i\mathcal{F}_i(x^r)$. The rules of finding $\mathcal{F}_i$ are summarized below:

\textit{Case 1: $\Lambda_i$ is convex:}
Define $\mathcal{F}_i = \Lambda_i$.

\textit{Case 2: The complementary of $\Lambda_i$ is convex:}
In this case, we can design a convexified $\phi_i(x)$, so that $\phi_i(x) \geq \phi_i(x^r) + \nabla\phi_i(x^r)(x - x^r)$. If $\phi_i$ is not differentiable, we choose $\nabla \phi_i$ as a sub-gradient so that the steepest descent of $J$ in the set $\Lambda$ is always included in the convex set $\mathcal{F}$.  With respect to a reference point $x^r$, the convex feasible set $\mathcal{F}_i$ is defined as
\begin{equation}\label{eq: cfs case 2}
    \mathcal{F}_i(x^r) = \{ x : \phi_i(x^r) + \nabla\phi_i(x^r)(x - x^r) \geq 0 \}.
\end{equation}

There is a third case considering all other situations, which is not listed here since it is not used in the proposed algorithm. As will be introduced in the experiment section, the safety specification in this paper considers the distance between robot arms and 3-dimensional planes or capsules. Although the complement of the corresponding feasible set in robot configuration space is not always convex, we approximate the convex feasible set using \eqref{eq: cfs case 2} (which might include infeasible points), with the understanding that the approximation error will be minimized when we are approaching the optimal solution. This approach worked successfully in practice and can efficiently find optimal solutions that are strictly feasible, as will be demonstrated in the results section. Nonetheless, we will investigate the feasibility and convergence guarantees of this approximation, as well as consider other sound convexification methods (which do not include infeasible points) in the future.

The CFS algorithm is guaranteed to converge to local optima if we iteratively minimize the cost function in the convex feasible set and use the intermediate solutions as the reference points to generate the next convex feasible sets. For more details about the convergence and feasibility of the CFS algorithm, the readers are referred to~\cite{liu2018convex}.

\subsection*{Iteractive Convex Optimization for Planning}
\begin{figure}
  \centering
  \includegraphics[scale=0.05]{setup.jpg}
  \caption{Real weld polishing experiment setup: the blue object is the 6DOF Motoman GP50 robot, while the orange object is the positioner. The silver object is the workpiece to be polished and the weld is inside the workpiece.}
  \label{setup}
\end{figure}
Our goal is to generate contact-rich trajectories in confined environments in real time to safely follow a predefined $m$-dimensional trajectory. We use the CFS algorithm to tackle the workspace inequality constraints and apply \eqref{iela} to linearize the task equality constraints. Then we reduce the original nonlinear and non-convex optimization problem \eqref{eq:opt} to a quadratic program (QP) and solve it iteratively. The proposed motion planning framework, iterative convex optimization for planning (ICOP), is summarized in \cref{alg1}.

The ICOP motion planning algorithm is an incremental method, which solves for a desired trajectory step by step.

For each outer loop iteration, we get the next desired Cartesian position $C_{next}$, and the configuration $x_{pre}$ from the last step. Next, we use $x_{pre}$ to initialize the temporary reference configuration $x_{ref}$ and the corresponding Cartesian position $C_{ref}$, which will be updated in multiple rounds. Then we enter the SafeTrack inner loop procedure. At each inner loop iteration, we first  compute the corresponding convex feasible set $\mathcal{F}$ in the configuration space. Since the constraint is not convex, according to Case 2 in CFS, we obtain a linear inequality constraint:
\begin{equation}
     \nabla d(x_{ref}, O) x \geq \nabla d(x_{ref}, O) x_{ref} - d(x_{ref},O).
     \label{eq8}
\end{equation}

Secondly, we use \eqref{iela} to linearize the nonlinear contact equality constraint such that the Cartesian position should be $C_{next}$, the resulting equality constraint is:
\begin{equation}
     Jac(x_{ref},\theta)\cdot x = Jac(x_{ref},\theta)\cdot x_{ref} + C_{next} - C_{ref} - \sigma.
     \label{eq9}
\end{equation}

Finally, we define the optimization objective to minimize the difference of the configuration states between the current reference and last time step. Then we solve the following QP problem with respect to \eqref{eq8} and \eqref{eq9} and update the reference configuration accordingly:
\begin{equation}
\begin{aligned}
& \underset{x}{\text{min}}
& & J(x,x_{ref}) = \| x - x_{ref}\|_Q^2, \\
& \text{s.t.} & &  \nabla d(x_{ref}, O) x \geq \nabla d(x_{ref}, O) x_{ref} - d(x_{ref},O),\\
& & &  Jac(x_{ref},\theta)\cdot x = Jac(x_{ref},\theta)\cdot x_{ref} + C_{next} - C_{ref} - \sigma,\\
& & &  x_{min} \leq x \leq x_{max}.\\
\end{aligned}
\label{eq10}
\end{equation}

 Once the stopping criteria for the SafeTrack procedure are satisfied (line 13 in \cref{alg1}), i.e. the robot at configuration $x_{ref}$ is collision-free and the difference between the the $j$-th robot body point and the desired location is less than a threshold, we add $x_{ref}$ to the planned trajectory $\bm{x}$ and update the configuration from last step $x_{pre}$ to be $x_{ref}$. Note that in practice we set the objective for \eqref{eq10} as $\| x - x_{ref}\|_Q^2$ instead of $\| x - x_{pre}\|_Q^2$, since we observe deadlock issue in terms of optimization result when choosing the latter objective. The underlying reason for causing that deadlock issue is left for future work.

\section*{RESULTS}

\subsection*{Experimental Setup}
This section demonstrates the effectiveness of the proposed algorithm on a weld grinding task using a 6DOF robot manipulator. The application requires contact-rich trajectory generation in confined environments given 1) workspace inequality constraints that the trajectory should be collision free and 2) task equality constraints that the robot end-effector tip should follow the pre-defined welding trajectory.
\Cref{setup} shows the experiment setup. A YASKAWA Motoman GP50 robot is mounted on the ground. The GP50 robot needs to grind the weld bead inside the workpiece mounted on the positioner.

We first build a simulation environment as shown in \Cref{sim}. This simulator aims to mimic the real world polishing experimental setup. Note that the real world robot links and workpiece have complex shapes. To reduce the computational complexity in the simulation, we use the simplified geometry presentations for both the robot links and the workpiece. In particular, we use six capsules $\Pi_{j=1,2,..,6}$ to wrap the robot links as shown in the left part of \Cref{sim}. To approximate the workpiece, we use $8$ limited-area planes that intersect with each other to construct a polish tunnel that GP50 robot together with the polishing tool should go through. The plane parameters are denoted as $\mathbb{P} \in \mathbb{R}^{4\times 8}$. We need  four parameters to define a single plane. The approximated workpiece is shown in \Cref{planes}. Given the capsules and the limited-area plane representation, we can further define the motion planning constraints as shown below.

 \begin{figure}
  \includegraphics[scale=0.5]{fig/simulator.pdf}
  \caption{The polishing experiment simulator. The gray transparent capsules are computer aided design models to wrap the robot links. The yellow object on the right is the mounted workpiece. The blue dots inside the workpice are weld points to be polished.}
  \label{sim}
\end{figure}

\subsection*{Task Equality Constraints}

We solve the motion planning problems following the ICOP framework, thus the reference trajectory is generated in an incremental manner. Therefore, at time step $t$, the task equality constraint can be expressed in the following way: the end-effector tip position computed by forward kinematics with respect to the joint configuration $x_t$ should co-locate with the next desired end-effector tip position (weld point) $C_{next}$:
\begin{equation}
\mathbb{FK}(x_t) = C_{next},
\end{equation}
where $\mathbb{FK}(x)$ is the forward kinematics function to get the end-effector tip position in Cartesian space. $C_{next} \in \mathbb{R}^3$  is the desired positions of the weld points in the Cartesian space.

\begin{figure}
  \centering
  \includegraphics[scale=0.4]{fig/planes1.pdf}
  \caption{The workpiece obstacle is approximated using $8$ limited-area planes. The red segments denote the intersection lines of different planes on the workpiece entrance surface.}
  \label{planes}
\end{figure}

 \begin{table*}[htbp]
 \caption{ Performance comparison among our proposed methods, SQP and interior point in terms of generating collision-free polishing trajectory on four workpiece mounting configurations. $C1,C2,C3,C4$ denote configurations of ($l=1$,$\alpha=0.2\pi$), ($l=135$,$\alpha=0.2\pi$), ($l=15$,$\alpha=0$), and ($l=18$,$\alpha=0.125\pi$), respectively. The average TCP distance between end-effector trajectory and weld path, motion planning computation time, and the average closest distance (safe distance) between robot and obstacle are reported. Bold results highlight the best performance.}
\centering
\begin{tabular}{@{}p{0.04\textwidth}*{9}{L{\dimexpr0.1\textwidth-2\tabcolsep\relax}}@{}}
\toprule
& \multicolumn{3}{c}{TCP distance ($cm$)} &
\multicolumn{3}{c}{Computation time ($s$)} & \multicolumn{3}{c}{Safe distance ($m$)}\\
\cmidrule(r{4pt}){2-4} \cmidrule(l){5-7} \cmidrule(l){8-10}
& proposed & interior point & SQP &  proposed & interior point & SQP & proposed & interior point & SQP\\
\midrule
$C1$ & \bf{0.0031}& 0.0925 & 5.5980 & \bf{12.09$\pm$0.90} & 78.61$\pm$4.51 & 23.88$\pm$2.28 & \bf{0.0350} & 0.0198 & 0.0083\\
$C2$ & \bf{0.0041} & 0.0125 & 3.0945 & \bf{11.92$\pm$1.07} & 6181$\pm$1.16 & 23.79$\pm$0.43 & 0.0202 & \bf{0.0390} & 0.0226\\
$C3$  & \bf{0.0048} & 5.8355 & 9.6341 & \bf{11.30$\pm$1.08} & 32.29$\pm$0.87 & 25.92$\pm$2.40 & \bf{0.0250}  & 0.0026 & 0.0104 \\
$C4$  & \bf{0.0045} & 0.0137 & 3.0955 & \bf{{9.60$\pm$0.37}}  & 47.63$\pm$0.04 & 21.08$\pm$0.58 & \bf{0.0401}  & 0.0354 & 0.0280\\
\bottomrule
\end{tabular}
 \label{summarize}
\end{table*}

\subsection*{Workspace Inequality Constraints}
\begin{figure*}
     \centering
    \begin{subfigure}[t]{0.48\textwidth}
        \raisebox{-\height}{\includegraphics[width=\textwidth]{fig/1_02pi.pdf}}
        \caption{$l=1$, $\alpha=0.2\pi$}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.48\textwidth}
        \raisebox{-\height}{\includegraphics[width=\textwidth]{fig/135_02pi.pdf}}
        \caption{$l=135$, $\alpha=0.2\pi$}
    \end{subfigure}

    \begin{subfigure}[t]{0.48\textwidth}
        \raisebox{-\height}{\includegraphics[width=\textwidth]{fig/15_0.pdf}}
    \caption{$l=15$, $\alpha=0$}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.48\textwidth}
        \raisebox{-\height}{\includegraphics[width=\textwidth]{fig/18_0125pi.pdf}}
    \caption{$l=18$, $\alpha=0.125\pi$}
    \end{subfigure}

     \caption{
     Collision free polishing trajectory generation solution using iterative convex optimization for planning framework on four mounting setting.}
     \label{swipe}
\end{figure*}
To ensure the trajectory is collision-free, at time step $t$, the task inequality constraint can be posed as: the closest distance between workpiece and robot capsules at configuration $x_t$ should be greater than zero.
\begin{equation}
d(x_t,O) = \min_j(D(\Pi_j^t, O)) > 0,
\end{equation}
where $O$ is the workpiece, $\Pi_j^t$ denotes the capsules at time step $t$, with $j=1,2,...,6$. In the polishing scenario, the safety specification requires that the robot link capsules should not intersect with the limited-area planes. Note that the end-effector tip does not need to follow the safety constraint since it needs to maintain contact with the weld bead.

For the capsule that does not cross the workpiece entrance surface (\textbf{case 1}), the safety specification requires that the closest distance between the capsule and the tunnel fringe to be greater than zero. As shown in \Cref{planes}, the tunnel fringe can be represented by the collection of segments $\Sigma$, which are intersection lines of different planes on the workpiece entrance surface. Similarly, for the capsule that crosses the workpiece entrance surface (\textbf{case 2}), the safety specification requires the closest distance between the capsule and $\mathbb{P}$ to be greater than zero. Therefore, by replacing obstacle $O$ with $(\mathbb{P},\Sigma)$, we rewrite the distance function as:
\begin{equation}
D(\Pi_j^t,\mathbb{P},\Sigma)=
\begin{cases}
    \mathcal{D}_1(\Pi_j^t, \Sigma),& \text{if \textbf{case 1}}\\
    \mathcal{D}_2(\Pi_j^t, \mathbb{P}),     & \text{if \textbf{case 2}}
\end{cases},
\end{equation}
where $\mathcal{D}_1(\bullet)$ denotes the function that calculates the closest distance between two line segments. $\mathcal{D}_2(\bullet)$ denotes the function that calculates the smallest distance between line segments and planes. Suppose we define the capsule segments that cross the workpiece entrance surface as working segments. To enforce that robot sticks the polishing tool through the workpiece tunnel, $\mathcal{D}_2(\bullet)$ defines a negative distance value for the working segment that is outside of the workpiece tunnel, and the inequality constraints require the distance value to be strictly positive.

\subsection*{Comparison}

To demonstrate the effectiveness of the proposed method in motion planning given different target and obstacle configurations, we evaluate the proposed algorithm on a collection of four different scenarios. In those scenarios, the workpiece is mounted in different locations relative to the robot. We consider two types of workpiece mounting parameters, translation $l$ along the $x$ axis of the world frame and rotation $\alpha$ about the $y$ axis of the world frame. The four different mountings are illustrated in \Cref{swipe}.
To verify our algorithm, we compare it with the state-of-the-art nonlinear optimization algorithm interior point and SQP. Both the interior point and SQP algorithm solves the nonlinear equality and inequality constraints using MATLAB \verb+fmincon+ function. The proposed algorithm, interior point and SQP are all implemented using an incremental planning manner. They have the same target and obstacle settings, and the termination conditions and optimization objectives are set to be the same. All the experiments are performed on the MATLAB 2019 platform with a $2.3 GHz$ Intel Core i7 Processor.

We evaluate the performance of task constrained motion planning algorithms using the criteria of the average distance between end-effector tip and desired weld point (TCP distance), the computation time to find the feasible solution, and the average closest distance between the robot and the obstacle (safe distance). The comparison results among interior point, SQP and our method are summarized in \Cref{summarize}. The planning horizon for all the four experiments ($C1,C2,C3$ and $C4$) is $43$, and the equality threshold $\xi$ is set to be 1e-4.

 Compared with interior point and SQP, our method takes the shortest computation time to find the feasible solution, which is generally $4$ to $7$ times faster than interior point and $2$ times faster than SQP. Specifically, for each outer loop iteration, it only takes $3\pm2$ inner loop iteration (SafeTrack procedure in \Cref{alg1}) for our method to find the configuration that satisfies both the equality and inequality constraints, with each inner loop iteration only costs $0.1128\pm0.0893$ seconds. On the other hand, it takes $8\pm2$ inner loop iterations for SQP, and takes $30\pm15$ inner loop iterations for interior point to find a feasible configuration solution. The inner loop iteration time costs are $0.0741\pm0.001$ seconds and $0.0657\pm0.0015$ seconds for SQP and interior point, respectively. Therefore, given the highly nonlinear and non-convex constraints, our proposed iterative convex optimization takes significantly less iterations to find a feasible solution, although the computation cost for each iteration is slightly higher than those of SQP and interior point. It is worth mentioning that our method can find a feasible solution within $3$ seconds when implemented using C++.

We also observe that our method has the smallest TCP distance across all the four experiments, which demonstrates that our method can better satisfy the task equality constraints. On the contrary, interior point cannot always exactly satisfy the equality constraints or it fails to find the feasible solution. Noteworthy example is in experiment $C3$ where the average TCP distance of interior point is $5.8355 cm$, which means interior point sacrifices the equality constraints satisfaction in order to find the feasible solution. Similarly, we can observe that SQP fails to satisfy the equality constraints across all four experiments. Furthermore, we can observe that our method maintains a better average safe distance cross the majorities of the experiments ($C1, C3$ and $C4$).

 \begin{table}
\caption{Computation time of our proposed methods in terms of different planning horizon with mounting configuration in $C4$.}
 \centering\footnotesize
   \begin{tabular}{p{2.9cm}p{0.5cm}p{0.4cm}p{0.4cm}p{0.6cm}p{0.7cm}p{0.5cm}}
   \hline
   Planning horizon & 14 & 21 & 43 & 82 & 123 & 164\\
   \hline
   Computation time ($s$) & 3.23 & 4.91 & 9.60 & 17.12 & 2189 & 26.84\\
\hline
   \end{tabular}

 \label{horizon}
\end{table}

To demonstrate the scalability of our method in terms of planning horizon and equality threshold, we conduct additional testing using the workpiece mounting configuration in $C4$. We augment the pre-defined end-effector trajectory (planning horizon) using linear interpolation, then down-sample the augmented trajectory to generate cases of different planning horizons. The computation time for different horizons is shown in \cref{horizon}. We observe that the computation time of our proposed method scales linearly with respect to the planning horizon. Next, we fix the planning horizon as $43$ and summarize the computation time for different equality thresholds in \cref{xi}. We observe that the computation time increases as the equality threshold decreases. A smaller equality threshold means higher precision for the end-effector tracking. Thus a good equality threshold should be carefully tuned to meet both the tracking precision and the computation efficiency requirements in practice.

 \begin{table}
\caption{Computation time of our proposed method in terms of different equality threshold $\xi$ with mounting configuration in $C4$ and planning horizon of $43$.}
 \centering
 \footnotesize
   \begin{tabular}{p{3cm}p{0.7cm}p{0.7cm}p{0.7cm}p{0.7cm}p{0.7cm}}
   \hline
   Equality threshold ($m$) & 1e-2 & 1e-3 & 1e-4 & 1e-5 & 1e-6\\
   \hline
   Computation time ($s$) & 3.74 & 4.96 & 9.60 & 15.13 & 18.91\\
\hline
   \end{tabular}

 \label{xi}
\end{table}

The planned results using our method in the four mounting configurations are visualized in \cref{swipe}. We can observe that the planned trajectory is collision-free and the task equality constraints are satisfied. In summary, the proposed method is able to generate trajectories that satisfy the task equality and workspace inequality constraints for the contact-rich trajectory generation problem in confined environments. Moreover, the numerical comparison results demonstrate that, compared with traditional nonlinear optimization methods, our proposed method has the best computational efficiency which satisfies the real time requirements for industry motion planning problems.

\section*{CONCLUSION}
This paper presented an iterative convex optimization motion planning method that can efficiently handle the constraints in motion planning for redundant robotic systems subject to workspace inequality constraints and task equality constraints. The proposed method can generate reference trajectory with high tracking precision and large safety distance in real time. Our planner guarantees the satisfaction of the task constraints and robust planning performance is maintained across different planning scenarios. Planning experiments using YASKAWA Motoman GP50 robot on weld grinding have been presented to demonstrate the effectiveness of the proposed method and the possible application in the industry.
One direction for future work will be to incorporate the adaptive hyper-parameter tuning on the optimization problem for motion planning. Note that the hyper-parameters in the optimization problem is crucial for the success of the optimization-based motion planning algorithm, e.g., the weight matrix in the objective function.
Other extensions of the present approach will be aimed at relaxing the assumption of static obstacle positions. In particular, our ultimate goal is to devise an online version of the present planner under different complex and stochastic environments with time-varying constraints.

\bibliographystyle{asmems4}

redpajama_set_name: RedPajamaArXiv


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