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hot combs—they all obviously benefited from the jolt of electrification. But the eraser? What was so problematic about the humble eraser that it needed electrifying? 1935 patent application for an apparatus for erasing, “Hand held rubbers are clumsy and cover a greater area than may be required.” Aye, there’s the rub, as it were. Lukowski’s cone-tipped electric eraser, he argued, could better handle the fine detail. Consider the careful technique Roscoe C. Sloane and John M. Montz suggest in their 1930 book Elements of Topographic Drawing. To make a correction to a map, these civil engineering professors at Ohio State University recommend the following steps: With a smooth, sharp knife pick the ink from the paper. This can be done without marring the surface. Place a hard, smooth surface, such as a [drafting] triangle, under the erasure before rubbing starts. When practically all the ink has been removed with the knife, rub with a pencil eraser. Erasing was not for the faint of...
a week ago

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Video Friday: RIVR Delivers Your Package

Video Friday is your weekly selection of awesome robotics videos, collected by your friends at IEEE Spectrum robotics. We also post a weekly calendar of upcoming robotics events for the next few months. Please send us your events for inclusion. RoboSoft 2025: 23–26 April 2025, LAUSANNE, SWITZERLAND ICUAS 2025: 14–17 May 2025, CHARLOTTE, NC ICRA 2025: 19–23 May 2025, ATLANTA, GA London Humanoids Summit: 29–30 May 2025, LONDON IEEE RCAR 2025: 1–6 June 2025, TOYAMA, JAPAN 2025 Energy Drone & Robotics Summit: 16–18 June 2025, HOUSTON, TX RSS 2025: 21–25 June 2025, LOS ANGELES ETH Robotics Summer School: 21–27 June 2025, GENEVA IAS 2025: 30 June–4 July 2025, GENOA, ITALY ICRES 2025: 3–4 July 2025, PORTO, PORTUGAL IEEE World Haptics: 8–11 July 2025, SUWON, KOREA IFAC Symposium on Robotics: 15–18 July 2025, PARIS RoboCup 2025: 15–21 July 2025, BAHIA, BRAZIL RO-MAN 2025: 25–29 August 2025, EINDHOVEN, THE NETHERLANDS CLAWAR 2025: 5–7 September 2025, SHENZHEN World Robot Summit: 10–12 October 2025, OSAKA, JAPAN IROS 2025: 19–25 October 2025, HANGZHOU, CHINA IEEE Humanoids: 30 September–2 October 2025, SEOUL CoRL 2025: 27–30 September 2025, SEOUL Enjoy today’s videos! I love the platform and I love the use case, but this particular delivery method is... Odd? [ RIVR ] This is just the beginning of what people and physical AI can accomplish together. To recognize business value from collaborative robotics, you have to understand what people do well, what robots do well—and how they best come together to create productivity. DHL and Robust.AI are partnering to define the future of human-robot collaboration. [ Robust AI ] Teleoperated robotic characters can perform expressive interactions with humans, relying on the operators’ experience and social intuition. In this work, we propose to create autonomous interactive robots, by training a model to imitate operator data. Our model is trained on a dataset of human-robot interactions, where an expert operator is asked to vary the interactions and mood of the robot, while the operator commands as well as the pose of the human and robot are recorded. [ Disney Research Studios ] Introducing THEMIS V2, our all-new full-size humanoid robot. Standing at 1.6m with 40 DoF, THEMIS V2 now features enhanced 6 DoF arms and advanced 7 DoF end-effectors, along with an additional body-mounted stereo camera and up to 200 TOPS of onboard AI computing power. These upgrades deliver exceptional capabilities in manipulation, perception, and navigation, pushing humanoid robotics to new heights. [ Westwood ] BMW x Figure Update: This isn’t a test environment—it’s real production operations. Real-world robots are advancing our Helix AI and strengthening our end-to-end autonomy to deploy millions of robots. [ Figure ] On March 13, at WorldMinds 2025, in the Kaufleuten Theater of Zurich, our team demonstrated for the first time two autonomous vision-based racing drones. It was an epic journey to prepare for this event, given the poor lighting conditions and the safety constraints of a theater filled with more than 500 people! The background screen visualizes in real-time the observations of the AI algorithm of each drone. No map, no IMU, no SLAM! [ University of Zurich (UZH) ] Unitree releases Dex5 dexterous hand. Single hand with 20 degrees of freedom (16 active+4 passive). Enable smooth backdrivability (direct force control). Equipped with 94 highly sensitive touch points (optional). [ Unitree ] You can say “real world manipulation” all you want, but until it’s actually in the real world, I’m not buying it. [ 1X ] Developed by Pudu X-Lab, FlashBot Arm elevates the capabilities of our flagship FlashBot by blending advanced humanoid manipulation and intelligent delivery capabilities, powered by cutting-edge embodied AI. This powerful combination allows the robot to autonomously perform a wide range of tasks across diverse settings, including hotels, office buildings, restaurants, retail spaces, and healthcare facilities. [ Pudu Robotics ] If you ever wanted to manipulate a trilby with 25 robots, a solution now exists. [ Paper ] via [ EPFL Reconfigurable Robotics Lab ] published by [ IEEE Robotics and Automation Letters ] We’ve been sharing videos from the Suzumori Endo Robotics Lab at the Institute of Science Tokyo for many years, and Professor Suzumori is now retiring. Best wishes to Professor Suzumori! [ Suzumori Endo Lab ] No matter the vehicle, traditional control systems struggle when unexpected challenges—like damage, unforeseen environments, or new missions—push them beyond their design limits. Our Learning Introspective Control (LINC) program aims to fundamentally improve the safety of mechanical systems, such as ground vehicles, ships, and robotics, using various machine learning methods that require minimal computing power. [ DARPA ] NASA’s Perseverance rover captured new images of multiple dust devils while exploring the rim of Jezero Crater on Mars. The largest dust devil was approximately 210 feet wide (65 meters). In this Mars Report, atmospheric scientist Priya Patel explains what dust devils can teach us about weather conditions on the Red Planet. [ NASA ]

5 days ago 7 votes
How Dairy Robots Are Changing Work for Cows (and Farmers)

This dairy barn is full of cows, as you might expect. Cows are being milked, cows are being fed, cows are being cleaned up after, and a few very happy cows are even getting vigorously scratched behind the ears. “I wonder where the farmer is,” remarks my guide, Jan Jacobs. Jacobs doesn’t seem especially worried, though—the several hundred cows in this barn are being well cared for by a small fleet of fully autonomous robots, and the farmer might not be back for hours. The robots will let him know if anything goes wrong. more frequently than the twice a day at a traditional dairy farm. Not only is getting milked more often more comfortable for the cows, cows also produce about 10 percent more milk when the milking schedule is completely up to them. Jan Jacobs is the human-robot interaction design lead for Lely, a maker of agricultural machinery. Founded in 1948 in Maassluis, Netherlands, Lely deployed its first Astronaut milking robot in the early 1990s. The company has since developed other robotic systems that assist with cleaning, feeding, and cow comfort, and the Astronaut milking robot is on its fifth generation. Lely is now focused entirely on robots for dairy farms, with around 135,000 of them deployed around the world. Essential Jobs on Dairy Farms The weather outside the barn is miserable. It’s late fall in the Netherlands, and a cold rain is gusting in from the sea, which is probably why the cows have quite sensibly decided to stay indoors and why the farmer is still nowhere to be found. Lely requires that dairy farmers who adopt its robots commit to letting their cows move freely between milking, feeding, and resting, as well as inside and outside the barn, at their own pace. “We believe that free cow traffic is a core part of the future of farming,” Jacobs says as we watch one cow stroll away from the milking robot while another takes its place. This is possible only when the farm operates on the cows’ schedule rather than a human’s. “We were spending 6 hours a day milking,” explains dairy farmer Josie Rozum, whose 120-cow herd at Takes Dairy Farm uses a pair of Astronaut A5 milking robots. “Now that the robots are handling all of that, we can focus more on animal care and comfort.”Lely in just 20 to 30 seconds. The actual milking takes only a few minutes, but with the average small dairy farm in North America providing a home for several hundred cows, milking typically represents a time commitment of 4 to 6 hours per day. Cows are happier with continuous access to food, which means feeding them several times a day. The feed is a mix of roughage (hay), silage (grass), and grain. The cows will eat all of this, but they prefer the grain, and so it’s common to see cows sorting their food by grabbing a mouthful and throwing it up into the air. The lighter roughage and silage flies farther than the grain does, leaving the cow with a pile of the tastier stuff as the rest gets tossed out of reach. This makes “feed pushing” necessary to shove the rest of the feed back within reach of the cow. 68 kilograms of manure a day. All that manure has to be collected and the barn floors regularly cleaned. Dairy Industry 4.0 The amount of labor needed to operate a dairy meant that until the early 1900s, most family farms could support only about eight cows. The introduction of the first milking machines, called bucket milkers, helped farmers milk 10 cows per hour instead of 4 by the mid-1920s. Rural electrification furthered dairy automation starting in the 1950s, and since then, both farm size and milk production have increased steadily. In the 1930s, a good dairy cow produced 3,600 kilograms of milk per year. Today, it’s almost 11,000 kilograms, and Lely believes that robots are what will enable small dairy farms to continue to scale sustainably. Lely But dairy robots are expensive. A milking robot can cost several hundred thousand dollars, plus an additional US $5,000 to $10,000 per year in operating costs. The Astronaut A5, Lely’s latest milking robot, uses a laser-guided robot arm to clean the cow’s udder before attaching teat cups one at a time. While the cow munches on treats, the Astronaut monitors her milk output, collecting data on 32 parameters, including indicators of the quality of the milk and the health of the cow. When milking is complete, the robot cleans the udder again, and the cow is free to leave as the robot steam cleans itself in preparation for the next cow. Lely argues that although the initial cost is higher than that of a traditional milking parlor, the robots pay for themselves over time through higher milk production (due primarily to increased milking frequency) and lower labor costs. Lely’s other robots can also save on labor. The Vector mobile robot handles continuous feeding and feed pushing, and the Discovery Collector is a robotic manure vacuum that keeps the floors clean. At Takes Dairy Farm, Rozum and her family used to spend several hours per day managing food for the cows. “The feeding robot is another amazing piece of the puzzle for our farm that allows us to focus on other things.”Takes Family Farm Marcia Endres, a professor in the department of animal science at the University of Minnesota. Endres specializes in dairy-cattle management, behavior, and welfare, and studies dairy robot adoption. “When we first started doing research on this about 12 years ago, most of the farms that were installing robots were smaller farms that did not want to hire employees,” Endres says. “They wanted to do the work just with family labor, but they also wanted to have more flexibility with their time. They wanted a better lifestyle.” added Lely robots to their dairy farm in Ely, Iowa, four years ago. “When we had our old milking parlor, everything that we did as a family was always scheduled around milking,” says Josie Rozum, who manages the farm and a creamery along with her parents—Dan and Debbie Takes—and three brothers. “With the robots, we can prioritize our personal life a little bit more—we can spend time together on Christmas morning and know that the cows are still getting milked.” Takes Family Dairy Farm’s 120-cow herd is milked by a pair of Astronaut A5 robots, with a Vector and three Discovery Collectors for feeding and cleaning. “They’ve become a crucial part of the team,” explains Rozum. “It would be challenging for us to find outside help, and the robots keep things running smoothly.” The robots also add sustainability to small dairy farms, and not just in the short term. “Growing up on the farm, we experienced the hard work, and we saw what that commitment did to our parents,” Rozum explains. “It’s a very tough lifestyle. Having the robots take over a little bit of that has made dairy farming more appealing to our generation.” Takes Dairy Farm about a third of the adoption rate in Europe, where farms tend to be smaller, so the cost of implementing the robots is lower. Endres says that over the last five years, she’s seen a shift toward robot adoption at larger farms with over 500 cows, due primarily to labor shortages. “These larger dairies are having difficulty finding employees who want to milk cows—it’s a very tedious job. And the robot is always consistent. The farmers tell me, ‘My robot never calls in sick, and never shows up drunk.’ ” The Lely Luna cow brush helps to keep cows’ skin healthy. It’s also relaxing and enjoyable, so cows will brush themselves several times a day.Lely much more relaxed and friendly toward people they meet. Rozum agrees. “We’ve noticed a tremendous change in our cows’ demeanor. They’re more calm and relaxed, just doing their thing in the barn. They’re much more comfortable when they can choose what to do.” Cows Versus Robots Cows are curious and clever animals, and have the same instinct that humans have when confronted with a new robot: They want to play with it. Because of this, Lely has had to cow-proof its robots, modifying their design and programming so that the machines can function autonomously around cows. Like many mobile robots, Lely’s dairy robots include contact-sensing bumpers that will pause the robot’s motion if it runs into something. On the Vector feeding robot, Lely product engineer René Beltman tells me, they had to add a software option to disable the bumper. “The cows learned that, ‘oh, if I just push the bumper, then the robot will stop and put down more feed in my area for me to eat.’ It was a free buffet. So you don’t want the cows to end up controlling the robot.” Emergency stop buttons had to be relocated so that they couldn’t be pressed by questing cow tongues. One of the dirtiest jobs on a dairy farm is handled by the Discovery Collector, an autonomous manure vacuum. The robot relies on wheel odometry and ultrasonic sensors for navigation because it’s usually covered in manure.Evan Ackerman Besides maintaining their dominance at the top of the herd, the current generation of Lely robots doesn’t interact much with the cows, but that’s changing, Jacobs tells me. Right now, when a robot is driving through the barn, it makes a beeping sound to let the cows know it’s coming. Lely is looking into how to make these sounds more enjoyable for the cows. “This was a recent revelation for me,” Jacobs says. ”We’re not just designing interactions for humans. The cows are our users, too.” Human-Robot Interaction Last year, Jacobs and researchers from Delft University of Technology, in the Netherlands, presented a paper at the IEEE Human-Robot Interaction (HRI) Conference exploring this concept of robot behavior development on working dairy farms. The researchers visited robotic dairies, interviewed dairy farmers, and held workshops within Lely to establish a robot code of conduct—a guide that Lely’s designers and engineers use when considering how their robots should look, sound, and act, for the benefit of both humans and cows. On the engineering side, this includes practical things like colors and patterns for lights and different types of sounds so that information is communicated consistently across platforms. Jacobs doesn’t want his robots to try to be anyone’s friend—not the cow’s, and not the farmer’s. “The robot is an employee, and it should have a professional relationship,” he says. “So the robot might say ‘Hi,’ but it wouldn’t say, ‘How are you feeling today?’ ” What’s more important is that the robots are trustworthy. For Jacobs, instilling trust is simple: “You cannot gain trust by doing tricks. If your robot is reliable and predictable, people will trust it.” The electrically driven, pneumatically balanced robotic arm that the Lely Astronaut uses to milk cows is designed to withstand accidental (or intentional) kicks.Lely From Dairy Farmers to Robot Managers With the additional time and flexibility that the robots enable, some dairy farmers have been able to diversify. On our way back to Lely’s headquarters, we stop at Farm Het Lansingerland, owned by a Lely customer who has added a small restaurant and farm shop to his dairy. Large windows look into the barn so that restaurant patrons can watch the robots at work, caring for the cows that produce the cheese that’s on the menu. A self-guided tour takes you right up next to an Astronaut A5 milking robot, while signs on the floor warn of Vector feeding robots on the move. “This farmer couldn’t expand—this was as many cows as he’s allowed to have here,” Jacobs explains to me over cheese sandwiches. “So, he needs to have additional income streams. That’s why he started these other things. And the robots were essential for that.” Besides managing the robots, the farmer must also learn to manage the massive amount of data that the robots generate about the cows. “The amount of data we get from the robots is a game changer,” says Rozum. “We can track milk production, health, and cow habits in real time. But it’s overwhelming. You could spend all day just sitting at the computer, looking at data and not get anything else done. It took us probably a year to really learn how to use it.” A Robotic Dairy A: One Astronaut A5 robot can milk up to 60 cows. After the Astronaut cleans the teats, a laser sensor guides a robotic arm to attach the teat cups. Milking takes just a few minutes. C: The Vector robot dispenses freshly mixed food in small batches throughout the day. A laser measures the height of leftover food to make sure that the cows are getting the right amounts. E: As it milks, the Astronaut is collecting a huge amount of data—32 different parameters per teat. If it detects an issue, the farmer is notified, helping to catch health problems early. F: Automated gates control meadow access and will keep a cow inside if she’s due to be milked soon. Cows are identified using RFID collars, which also track their behavior and health. A Sensible Future for Dairy Robots After lunch, we stop by Lely headquarters, where bright red life-size cow statues guard the entrance and all of the conference rooms are dairy themed. We get comfortable in Butter, and I ask Jacobs and Beltman what the future holds for their dairy robots. feed-pushing robot is equipped with lidar and stereo cameras, which allow it to autonomously navigate around large farms without needing to follow a metal strip bolted to the ground. A new overhead camera system will leverage AI to recognize individual cows and track their behavior, while also providing farmers with an enormous new dataset that could allow Lely’s systems to help farmers make more nuanced decisions about cow welfare. The potential of AI is what Jacobs seems most excited about, although he’s cautious as well. “With AI, we’re suddenly going to take away an entirely different level of work. So, we’re thinking about doing research into the meaningfulness of work, to make sure that the things that we do with AI are the things that farmers want us to do with AI.” Lely is aware of this and knows that its robots have to find the right balance between being helpful, and taking over. “We want to make sure not to take away the kinds of interactions that give dairy farmers joy in their work,” says Beltman. “Like feeding calves—every farmer likes to feed the calves.” Lely does sell an automated calf feeder that many dairy farmers buy, which illustrates the point: What’s the best way of designing robots to give humans the flexibility to do the work that they enjoy? Dairy farms are different. Perhaps that’s because the person buying the robot is the person who most directly benefits from it. But I wonder if the concern over automation of jobs would be mitigated if more companies chose to emphasize the sustainability and joy of work equally with profit. Automation doesn’t have to be zero-sum—if implemented thoughtfully, perhaps robots can make work easier, more efficient, and more fun, too. Jacobs certainly thinks so. “That’s my utopia,” he says. “And we’re working in the right direction.”

a week ago 7 votes
Protecting Robots in Harsh Environments with Advanced Sealing Systems

This is a sponsored article brought to you by Freudenberg Sealing Technologies. The increasing deployment of collaborative robots (cobots) in outdoor environments presents significant engineering challenges, requiring highly advanced sealing solutions to ensure reliability and durability. Unlike industrial robots that operate in controlled indoor environments, outdoor cobots are exposed to extreme weather conditions that can compromise their mechanical integrity. Maintenance robots used in servicing wind turbines, for example, must endure intense temperature fluctuations, high humidity, prolonged UV radiation exposure, and powerful wind loads. Similarly, agricultural robots operate in harsh conditions where they are continuously exposed to abrasive dust, chemically aggressive fertilizers and pesticides, and mechanical stresses from rough terrains. To ensure these robotic systems maintain long-term functionality, sealing solutions must offer effective protection against environmental ingress, mechanical wear, corrosion, and chemical degradation. Outdoor robots must perform flawlessly in temperature ranges spanning from scorching heat to freezing cold while withstanding constant exposure to moisture, lubricants, solvents, and other contaminants. In addition, sealing systems must be resilient to continuous vibrations and mechanical shocks, which are inherent to robotic motion and can accelerate material fatigue over time. Comprehensive Technical Requirements for Robotic Sealing Solutions The development of sealing solutions for outdoor robotics demands an intricate balance of durability, flexibility, and resistance to wear. Robotic joints, particularly those in high-mobility systems, experience multidirectional movements within confined installation spaces, making the selection of appropriate sealing materials and geometries crucial. Traditional elastomeric O-rings, widely used in industrial applications, often fail under such extreme conditions. Exposure to high temperatures can cause thermal degradation, while continuous mechanical stress accelerates fatigue, leading to early seal failure. Chemical incompatibility with lubricants, fuels, and cleaning agents further contributes to material degradation, shortening operational lifespans. Friction-related wear is another critical concern, especially in robotic joints that operate at high speeds. Excessive friction not only generates heat but can also affect movement precision. In collaborative robotics, where robots work alongside humans, such inefficiencies pose safety risks by delaying response times and reducing motion accuracy. Additionally, prolonged exposure to UV radiation can cause conventional sealing materials to become brittle and crack, further compromising their performance. Advanced IPSR Technology: Tailored for Cobots To address these demanding conditions, Freudenberg Sealing Technologies has developed a specialized sealing solution: Ingress Protection Seals for Robots (IPSR). Unlike conventional seals that rely on metallic springs for mechanical support, the IPSR design features an innovative Z-shaped geometry that dynamically adapts to the axial and radial movements typical in robotic joints. Numerous seals are required in cobots and these are exposed to high speeds and forces.Freudenberg Sealing Technologies This unique structural design distributes mechanical loads more efficiently, significantly reducing friction and wear over time. While traditional spring-supported seals tend to degrade due to mechanical fatigue, the IPSR configuration eliminates this limitation, ensuring long-lasting performance. Additionally, the optimized contact pressure reduces frictional forces in robotic joints, thereby minimizing heat generation and extending component lifespans. This results in lower maintenance requirements, a crucial factor in applications where downtime can lead to significant operational disruptions. Optimized Through Advanced Simulation Techniques The development of IPSR technology relied extensively on Finite Element Analysis (FEA) simulations to optimize seal geometries, material selection, and surface textures before physical prototyping. These advanced computational techniques allowed engineers to predict and enhance seal behavior under real-world operational conditions. FEA simulations focused on key performance factors such as frictional forces, contact pressure distribution, deformation under load, and long-term fatigue resistance. By iteratively refining the design based on simulation data, Freudenberg engineers were able to develop a sealing solution that balances minimal friction with maximum durability. Furthermore, these simulations provided insights into how IPSR seals would perform under extreme conditions, including exposure to humidity, rapid temperature changes, and prolonged mechanical stress. This predictive approach enabled early detection of potential failure points, allowing for targeted improvements before mass production. By reducing the need for extensive physical testing, Freudenberg was able to accelerate the development cycle while ensuring high-performance reliability. Material Innovations: Superior Resistance and Longevity The effectiveness of a sealing solution is largely determined by its material composition. Freudenberg utilizes advanced elastomeric compounds, including Fluoroprene XP and EPDM, both selected for their exceptional chemical resistance, mechanical strength, and thermal stability. Fluoroprene XP, in particular, offers superior resistance to aggressive chemicals, including solvents, lubricants, fuels, and industrial cleaning agents. Additionally, its resilience against ozone and UV radiation makes it an ideal choice for outdoor applications where continuous exposure to sunlight could otherwise lead to material degradation. EPDM, on the other hand, provides outstanding flexibility at low temperatures and excellent aging resistance, making it suitable for applications that require long-term durability under fluctuating environmental conditions. To further enhance performance, Freudenberg applies specialized solid-film lubricant coatings to IPSR seals. These coatings significantly reduce friction and eliminate stick-slip effects, ensuring smooth robotic motion and precise movement control. This friction management not only improves energy efficiency but also enhances the overall responsiveness of robotic systems, an essential factor in high-precision automation. Extensive Validation Through Real-World Testing While advanced simulations provide critical insights into seal behavior, empirical testing remains essential for validating real-world performance. Freudenberg subjected IPSR seals to rigorous durability tests, including prolonged exposure to moisture, dust, temperature cycling, chemical immersion, and mechanical vibration. Throughout these tests, IPSR seals consistently achieved IP65 certification, demonstrating their ability to effectively prevent environmental contaminants from compromising robotic components. Real-world deployment in maintenance robotics for wind turbines and agricultural automation further confirmed their reliability, with extensive wear analysis showing significantly extended operational lifetimes compared to traditional sealing technologies. Safety Through Advanced Friction Management In collaborative robotics, sealing performance plays a direct role in operational safety. Excessive friction in robotic joints can delay emergency-stop responses and reduce motion precision, posing potential hazards in human-robot interaction. By incorporating low-friction coatings and optimized sealing geometries, Freudenberg ensures that robotic systems respond rapidly and accurately, enhancing workplace safety and efficiency. Tailored Sealing Solutions for Various Robotic Systems Freudenberg Sealing Technologies provides customized sealing solutions across a wide range of robotic applications, ensuring optimal performance in diverse environments. Automated Guided Vehicles (AGVs) operate in industrial settings where they are exposed to abrasive contaminants, mechanical vibrations, and chemical exposure. Freudenberg employs reinforced PTFE composites to enhance durability and protect internal components. Delta robots can perform complex movements at high speed. This requires seals that meet the high dynamic and acceleration requirements.Freudenberg Sealing Technologies Delta robots, commonly used in food processing, pharmaceuticals, and precision electronics, require FDA-compliant materials that withstand rigorous cleaning procedures such as Cleaning-In-Place (CIP) and Sterilization-In-Place (SIP). Freudenberg utilizes advanced fluoropolymers that maintain structural integrity under aggressive sanitation processes. Seals for Scara robots must have high chemical resistance, compressive strength and thermal resistance to function reliably in a variety of industrial environments.Freudenberg Sealing Technologies SCARA robots benefit from Freudenberg’s Modular Plastic Sealing Concept (MPSC), which integrates sealing, bearing support, and vibration damping within a compact, lightweight design. This innovation optimizes robot weight distribution and extends component service life. Six-axis robots used in automotive, aerospace, and electronics manufacturing require sealing solutions capable of withstanding high-speed operations, mechanical stress, and chemical exposure. Freudenberg’s Premium Sine Seal (PSS), featuring reinforced PTFE liners and specialized elastomer compounds, ensures maximum durability and minimal friction losses. Continuous Innovation for Future Robotic Applications Freudenberg Sealing Technologies remains at the forefront of innovation, continuously developing new materials, sealing designs, and validation methods to address evolving challenges in robotics. Through strategic customer collaborations, cutting-edge material science, and state-of-the-art simulation technologies, Freudenberg ensures that its sealing solutions provide unparalleled reliability, efficiency, and safety across all robotic platforms.

a week ago 7 votes
The Tiniest Flying Robot Soars Thanks to Magnets

A new prototype is laying claim to the title of smallest, lightest untethered flying robot. At less than a centimeter in wingspan, the wirelessly powered robot is currently very limited in how far it can travel away from the magnetic fields that drive its flight. However, the scientists who developed it suggest there are ways to boost its range, which could lead to potential applications such as search and rescue operations, inspecting damaged machinery in industrial settings, and even plant pollination. One strategy to shrink flying robots involves removing their batteries and supplying them electricity using tethers. However, tethered flying robots face problems operating freely in complex environments. This has led some researchers to explore wireless methods of powering robot flight. “The dream was to make flying robots to fly anywhere and anytime without using an electrical wire for the power source,” says Liwei Lin, a professor of mechanical engineering at University of California at Berkeley. Lin and his fellow researchers detailed their findings in Science Advances. 3D-Printed Flying Robot Design Each flying robot has a 3D-printed body that consists of a propeller with four blades. This rotor is encircled by a ring that helps the robot stay balanced during flight. On top of each body are two tiny permanent magnets. All in all, the insect-scale prototypes have wingspans as small as 9.4 millimeters and weigh as little as 21 milligrams. Previously, the smallest reported flying robot, either tethered or untethered, was 28 millimeters wide. When exposed to an external alternating magnetic field, the robots spin and fly without tethers. The lowest magnetic field strength needed to maintain flight is 3.1 millitesla. (In comparison, a refrigerator magnet has a strength of about 10 mT.) When the applied magnetic field alternates with a frequency of 310 hertz, the robots can hover. At 340 Hz, they accelerate upward. The researchers could steer the robots laterally by adjusting the applied magnetic fields. The robots could also right themselves after collisions to stay airborne without complex sensing or controlling electronics, as long as the impacts were not too large. Experiments show the lift force the robots generate can exceed their weight by 14 percent, to help them carry payloads. For instance, a prototype that’s 20.5 millimeters wide and weighing 162.4 milligrams could carry an infrared sensor weighing 110 mg to scan its environment. The robots proved efficient at converting the energy given them into lift force—better than nearly all other reported flying robots, tethered or untethered, and also better than fruit flies and hummingbirds. Currently the maximum operating range of these prototypes is about 10 centimeters away from the magnetic coils. One way to extend the operating range of these robots is to increase the magnetic field strength they experience tenfold by adding more coils, optimizing the configuration of these coils, and using beamforming coils, Lin notes. Such developments could allow the robots to fly up to a meter away from the magnetic coils. The scientists could also miniaturize the robots even further. This would make them lighter, and so reduce the magnetic field strength they need for propulsion. “It could be possible to drive micro flying robots using electromagnetic waves such as those in radio or cell phone transmission signals,” Lin says. Future research could also place devices that can convert magnetic energy to electricity onboard the robots to power electronic components, the researchers add.

a week ago 12 votes

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Analytic Combinatorics -- A Worked Example

Another day, another blog post that starts with “I was on mse the other day…”. This time, someone asked an interesting question amounting to “how many unordered rooted ternary trees with $n$ nodes are there, up to isomorphism?”. I’m a sucker for these kinds of combinatorial problems, and after finding a generating function solution I wanted to push myself to get an asymptotic approximation using Flajolet–Sedgewick style analytic combinatorics! I’ve never actually done this before, so I learned a lot, and I want to share some of the things I learned – especially how to do this stuff in sage! Now, you might be thinking – didn’t you write a very similar blog post three years ago? Yes. Yes I did. Did I also completely forget what was in that post? Yes. Yes I did, haha. For some reason I was getting it mixed up with this other post from even more years ago, which isn’t nearly as relevant. Thankfully it didn’t matter much, since I’m fairly sure what I wanted to do wouldn’t work in the framework of that post anyways, and now I have a better idea of what this machinery is actually doing which is really exciting (since I’ve been wanting to understand this stuff for years). Let’s start with a warmup problem! How might we count the number of rooted ordered ternary trees with $n$ nodes? These are the kinds of “ternary trees” that you might remember from a first class on algorithms and data structures. Such a tree is either a leaf or an internal node with 1,2, or 3 children. Crucially these children come with an order, because the datatype keeps track of which child is on the left/right (in the case of 2 children) or on the left/middle/right (in the case of 3 children). After all, from a CS perspective, we need to remember this in order to traverse the tree! This means that the following two trees are distinct for our purposes, even though they’re isomorphic as graphs: In a functional programming language, you might describe this datatype by T z = Leaf z | Unary z (T z) | Binary z (T z) (T z) | Ternary z (T z) (T z) (T z) and this translates immediately to give a functional equation for the generating function of $t_n$, which counts the number of rooted ordered ternary trees with $n$ nodes: We can rearrange this as so that we can compute the power series expansion of $T$ by lagrange inversion: = QQbar[[]] # power series ring T = (z/(1 + z + z^2 + z^3)).reverse() show(T.O(10)) Incredibly, this agrees with A036765, which is the “number of ordered rooted trees with n non-root nodes and all outdegrees <= three”, as we hoped! You might reasonably ask if there’s a closed form for these numbers, and this is too much to ask for (indeed, it’s already too much to ask for a closed form for fibonacci numbers, and this is more complicated). But like the fibonacci numbers, we can come up with an excellent approximation: where $C = \frac{(0.8936\ldots)}{2 \sqrt{\pi}}$ is a constant. Indeed, this gives fantastic results! Let’s plot the ratio of this approximation to the true value so we can see just how good this approximation gets as $n$ gets large. Note that it respects the promised big-oh error bound too! = PowerSeriesRing(QQbar, default_prec=N) T = (z/(1 + z + z^2 + z^3)).reverse() return T.coefficients()[n-1:] def approx(n,N): # very very magic C = 0.8936373911078061 / (2 * sqrt(pi)) w = 3.610718613276040 return [(C * w^k * (1/k^1.5)).n() for k in range(n,N+1)] show(html("Plot the error ratios in the range [n,N]")) @interact def _(n=input_box(100, width=20, label="$n$"), N=input_box(120, width=20, label="$N$"), auto_update=False): actuals = actual(n,N) approxs = approx(n,N) ratios = [(actuals[i]/approxs[i]) for i in range(N+1-n)] show(html("ratio at n={}: {}".format(n, ratios[0]))) show(html("ratio at N={}: {}".format(N, ratios[-1]))) text_options = {'horizontal_alignment': 'right', 'vertical_alignment': 'bottom', 'fontsize': 'large'} G = Graphics() G += scatter_plot([(n+i,r) for (i,r) in enumerate(ratios)]) G += plot(1, (x,n,N)) G += plot(1-1/x, (x,n,N)) G += text('$y = 1$', (N, 1), **text_options) G += text('$y = 1-1/x$', (N, 1-1/N), **text_options) G.show() This is extremely cool, but where the hell did this approximation come from? The answer is called Singularity Analysis, and can be found in Chapter 2 Section 3 of Melczer’s excellent An Invitation to Analytic Combinatorics, or Chapters VI and VII of Flajolet and Sedgewick’s tome. See especially Theorem VII.3 on pg 468. Like seemingly every theorem in complex analysis, this is basically an application of the Residue Theorem. I won’t say too much about why it works, but I’ll at least gesture at a proof. You can read the above references if you want something more precise. First, we recall where $C$ is any contour containing the origin inside a region where $T$ is holomorphic. Then we draw the most important picture in complex analysis: Here the obvious marked point is our singularity $\omega$, and we’ve chosen a branch cut for $T$ (shown in blue1) so that $T$ is holomorphic in the region where the pink curve lives. We’ll estimate the value of this integral by estimating the contribution from the big circle, the little circle, and the two horizontal pieces2. It turns out that the two horizontal pieces basically contribute $O(1)$ amount to our integral, so we ignore them. Since the big circle is compact, $T$ will attain a maximal value on it, say $M$. Then the integral along the big circle (of radius $\omega + \epsilon$, say) is bounded by $2 \pi (\omega + \epsilon) \frac{M}{(\omega + \epsilon)^{n+1}} = O((\omega+\epsilon)^{-n})$ To estimate the integral around the little circle, it would be really helpful to have a series expansion at $\omega$ since we’re staying really close to it… Unfortunately, $\omega$ is a singular point so we don’t have a taylor series, but fortunately there’s another tool for exactly this job: a Puiseux Series! I won’t say much about what these are, especially since Richard Borcherds already put out such a great video on the topic. What matters is is that Sage can compute them for us3, so we can actually get our hands on the approximation! We compute the integral around the little circle to be roughly: In step $(1)$ we approximate $T$ by the first term of its puiseux series, in step $(2)$ we apply the generalized binomial theorem, so that in step $(3)$ we can apply the residue theorem to realize this integral as the coefficient of $z^{-1}$ in our laurent expansion. This gives us something which grows like $\widetilde{O}(\omega^{-n})$, dominating the contribution $O((\omega + \epsilon)^{-n})$ from the big circle, so that asymptotically this is the only term that matters. If we were more careful to keep track of the big-oh error for the puiseux series for $T$ we could easily sharpen this bound4 to see This looks a bit weird with the $(-1)^n$, but remember that $\binom{\alpha}{n}$ also alternates sign. Indeed, asymptotics for $\binom{\alpha}{n}$ are well known so that we can rewrite this as Which finally shows us where our magic numbers came from! Sage happily tells us that the dominant singularity for $T$ is at $\omega = (0.2769\ldots)$ and that a puiseux expansion for $T$ at $\omega$ is so that for us $\alpha = 1/2$, and $C_{1/2} = -(0.8936\ldots)$. Finally, since $\Gamma(-1/2) = -2 \sqrt{\pi}$ and $(0.2769\ldots)^{-1} = (3.6107\ldots)$ we see as promised! Indeed, here’s how you could actually get sage to do all this for you! = QQbar[] S. = R[] # our defining polynomial. # we want to solve for T as a function of z P = z*T^3 + z*T^2 + (z-1)*T + z # following Example 2.12 in Melczer, # we compute the dominant singularity w. w = min([r for (r,_) in P.discriminant().roots() if r != 0], key=lambda r: r.abs()) # to work with abelfunctions we need P # to be a polynomial in two variables R. = QQbar[] P = R(P) # compute the puiseux expansion for T at w. # I know from trial and error that the correct # branch to pick is entry [1] in the list. You # just need to check which one gives you # positive real coefficients for T at 0. s = abelfunctions.puiseux(P,w)[1] c = -sqrt(-s.x0/s.xcoefficient) As a fun exercise, you might modify this code (using s.add_term()) to compute a longer puiseux series and get asymptotics valid up to a multiplicative error of $O \left ( \frac{1}{n^2} \right )$ for the number of rooted ternary trees. Try modifying the previous code block to see that our current approximation is not accurate to $O \left ( \frac{1}{n^2} \right )$, and then check that your approximation is! Ok, now that we understand the warmup, let’s get to the actual problem! How many unordered rooted ternary trees are there, up to isomorphism? Now we’re counting up to graph isomorphism, so that our two trees are now considered isomorphic. It’s actually much less obvious how one might pin down a generating function for something like this, but the answer, serendipitously, comes from Pólya-Redfield counting! If this is new to you, you might be interested in my recent blog post where I talk a bit about one of its corollaries. Today, though, we’ll be using it in a much more sophisticated way. Say you have a structure $X$ acted on by a group $G$ and a collection $C$ of “colors”. How can we count the number of ways to give each $x \in X$ a color from $C$, up to the action of $G$? We start by building the cycle index polynomial of the action $G \curvearrowright X$, which we call $P_G(x_1, \ldots, x_n)$. Then we can plug all sorts of things into the variables $x_i$ in order to solve various counting problems. For example, if $C$ is literally just a finite set of colors, we can plug in $x_1 = x_2 = \ldots = x_n = |C|$ to recover the expression from my recent blog post. But we can also do much more! Say that $C$ is a possibly infinite family of allowable “colors”, each of a fixed “weight”. (For us, our “colors” will be trees, and the “weight” will be the number of nodes). Then we can arrange them into a generating function $F(t) = \sum c_i t^i$, where $c_i$ counts the number of colors of weight $i$5. Then an easy argument (given in full on the wikipedia page) shows that $P_G(F(t), F(t^2), \ldots, F(t^n))$ is a generating function counting the number of ways to color $X$ by colors from $C$, counted up to the $G$ action, and sorted by their total weight6. Check out the wikipedia page for a bunch of great examples! For us, we realize an unordered rooted ternary tree is either empty, or a root with 3 recursive children7. In the recursive case we also want to count up to the obvious $\mathfrak{S}_3$ action permuting the children, so by the previous discussion we learn that where \(P_{\mathfrak{S}_3}(x_1, x_2, x_3) = \frac{1}{6}(x_1^3 + 3x_1 x_2 + 2 x_3)\) is the cycle index polynomial for the symmetric group on three letters. Plugging this in we see Now, how might we get asymptotics for $t_n$ using this functional equation? First let’s think about how our solution to the warmup worked. We wrote $F(z,T)=0$ for a polynomial $F$, used the implicit function theorem to get a taylor series for $T$ at the origin, then got a puiseux series near the dominant singularity $\omega$ which let us accurately estimate the taylor coefficients $t_n$8. We’re going to play the same game here, except we’ll assume that $F$ is merely holomorphic rather than a polynomial. Because we’re no longer working with a polynomial, this really seems to require an infinite amount of data, so I’m not sure how one might get an exact solution for the relevant constants… But following Section VII.5 in Flajolet and Sedgewick we can get as precise a numerical solution as we like! We’ll assume that the functions $T(z^2)$ and $T(z^3)$ are already known analytic functions, so that we can write $F(z,w) = -w + 1 + \frac{z}{6} \left ( w^3 + 3 T(z^2) w + 2 T(z^3) \right )$. This is an analytic function satisfying $F(z,T) = 0$. Now for a touch of magic. Say we can find a pair $(r,s)$ with both $F(r,s)=0$ and $\left . \frac{\partial F}{\partial w} \right |_{(r,s)} = 0$… Then $F$ is singular in the $w$ direction at $(r,s)$ so this is a branch point for $T$. Since both $F$ and $F_w = \frac{\partial F}{\partial w}$ vanish at $(r,s)$ the taylor series for $F$ at $(r,s)$ starts Since we know $F(z,T)$ vanishes, we estimate up to smaller order terms so that a puiseux series for $T$ at $(r,s)$ begins If we write $\gamma = \sqrt{\frac{2r F_z(r,s)}{F_{ww}(r,s)}}$ and are slightly more careful with our error terms, the same technique from the warmup shows See Flajolet and Sedgewick Theorem VII.3 on page 468 for a more careful proof of this theorem. Now how do we use this? We can approximate $T$ by its taylor series at the origin, then numerically solve for the unique9 positive real solution to the system $F(r,s) = F_w(r,s) = 0$ using this approximation: = B[[]] # cycle index polynnomials S. = QQ[] P3 = (x1^3 + 3*x1*x2 + 2*x3)/6 # to start, the taylor coefficients of T are variables T = sum([t[i] * z^i for i in range(n)]) + O(z^n) lhs = T rhs = 1 + z*P3.subs(x1=T(z), x2=T(z^2), x3=T(z^3)) # formally expand out the rhs and set the coefficients # equal to each other. This gives a recurrence relation # for t[n] in terms of the t[i] for i If we run this code locally with $N=20$, we get the approximation so that we expect and indeed this seems to work really well! Our taylor expansion for $T$ agrees with A000598, as we expected, and comparing our approximation to our taylor expansion gives: = B[[]] S. = QQ[] P3 = (x1^3 + 3*x1*x2 + 2*x3)/6 T = sum([t[i] * z^i for i in range(N)]) + O(z^N) lhs = T rhs = 1 + z*P3.subs(x1=T(z), x2=T(z^2), x3=T(z^3)) I = B.ideal([lhs.coefficients()[i] - rhs.coefficients()[i] for i in range(N)]) return [I.reduce(t[i]) for i in range(n,N)] show(html("Plot the error ratios in the range [n,N]")) show(html("This is likely to time out if you make N too large online,")) show(html("so you might want to play around with it locally instead!")) @interact def _(n=input_box(10, width=20, label="$n$"), N=input_box(30, width=20, label="$N$"), auto_update=False): actuals = actualList(n,N) approxs = approxList(n,N) ratios = [actuals[i]/approxs[i] for i in range(N-n)] show(html("ratio at n={}: {}".format(n, ratios[0].n()))) show(html("ratio at N={}: {}".format(N, ratios[-1].n()))) text_options = {'horizontal_alignment': 'right', 'vertical_alignment': 'bottom', 'fontsize': 'large'} G = Graphics() G += scatter_plot([(n+i,r) for (i,r) in enumerate(ratios)]) G += plot(1, (x,n,N)) G += plot(1-1/x, (x,n,N)) G += text('$y = 1$', (N, 1), **text_options) G += text('$y = 1-1/x$', (N, 1-1/N), **text_options) G.show() I’m pretty proud of this approximation, so I think this is a good place to stop ^_^. As a fun exercise, can you write a program that outputs the number of cyclic rooted ternary trees on $n$ vertices? For these we consider two trees the same if they’re related by cyclicly permuting their children. Compare your solution to A000625 For bonus points, can you check that the number of such trees is, asymptotically, Wow! It’s been super nice to be writing up so many posts lately! Like I said in one of my other recent posts, I’ve had a lot of time to think about more bite sized problems and mse stuff while waiting for my DOI generating code to run, so I’ve had more things that felt quick to write up on my mind. My research is actually going quite well too! I have a few interesting directions to explore, and at least one project that might be wrapping up soon. Of course when that happens I’ll be sure to talk about it here, and I’m still planning out a series on fukaya categories, hall algebras, skein algebras, and more! That’s a pretty long one, though, so it’s easy for me to deprioritize, haha. It’s already heating up in Riverside, consistently in the 80s (Fahrenheit), so I can really feel the summer coming. I’m enjoying the sunny days, though, and it’s been nice to spend time working outside and under trees. Thanks for hanging out, all! Take care of each other, and I can’t wait to chat soon ^_^. Just like $w = \sqrt{z}$, a solution to $w^2 = z$ has branches, so too does $T$, a solution to $T = z + zT + zT^2 + zT^3$. ↩ Really these are circles with a little arc cut out, say by integrating from $\epsilon$ to $2 \pi - \epsilon$… But we’ll end up taking $\epsilon \to 0$ and we’re all friends here, so let’s not worry about it. ↩ Though at time of writing you need the abelfunctions package to do it. There is a builtin implementation of puiseux series, but it won’t actually compute a series expansion for you. ↩ Of course, it’s easy to see how to extend this technique to get better asypmtotics. The first approach is to just keep more terms of the puiseux series of $T$ at $\omega$. Then apply the generalized binomial formula multiple times for each term you kept. You can also do better by keeping track of more of the singularities. Build a contour with multiple keyholes in order to get sharper lower order asymptotics: Of course, you can also combine these approaches to keep track of both more singularities and more puiseux coefficients at each singularity. ↩ In fact we can push things even farther and work with multivariate generating functions, but we won’t need that here. ↩ This is why we plug $F(t^k)$ into $x_k$. Because $x_k$ is responsible for $k$-cycles, so we choose a single color for each $k$ cycle, but we have to count it $k$-many times towards our total weight. See the proof on the wikipedia page for more information! ↩ This actually isn’t the version of the recurrence I used in my mse answer. There I used the convention that a rooted tree had to be nonempty, since… you know… it has a root. But allowing possibly empty trees makes the recurrence much simpler, which in turn allows for much easier to analyze asymptotics. Hilariously this exact example is on the wikipedia page for the Pólya-Redfield theorem, which could have saved me a lot of time writing up that answer. I was a bit worried at first about doing these asymptotics by myself, since this was my first serious attempt at using analytic combinatorics, but serendipitously this exact example was also worked out in Flajolet and Sedgewick VII.5, though slightly more tersely than I would have liked, haha. ↩ Officially we have to check that the choice of puiseux series matches up with our choice of taylor series (since there’s multiple branches to our function). But this is easy to arrange for us by choosing the branch of the puisuex series that leads to all our coefficients being positive reals. If you want to do this purely analytically you need to solve a “connection problem”. See figure VII.9 in Flajolet and Sedgewick, as well as the surrounding text. ↩ Under mild technical conditions this pair $(r,s)$ is unique. See Flajolet and Sedgewick Theorem VII.3. ↩

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