Input: What is the problem encountered when building the fuselage sides?

Context:
Probably one of the most frustrating things about building experimental aircraft, especially when starting with a minimum of pre-fabricated parts, is to start building and ending up with an unexpected result. Every builder starts a new project by wanting it to go "perfectly." So when things aren't going well, especially at the beginning, the frustration can lead to an unfinished airplane.
This is the first article in a series dedicated to helping builders of the Rand Robinson KR series planes build a straight and true fuselage -- the first part of the construction process. Borrowing from modern boatbuliding techniques, focus will be on the KR-2S, but the principles apply to the entire lineup of KR-1 & KR-2 series planes.
While building the KR-2(s) a common surprise is encountered by builders when the completed fuselage sides are laid into position to form the fuselage box section. With many hours spent building the sides flat, finding the once straight longerons that now bow up from the building surface, form a most dissatisfying "banana" shape. Especially when using the preformed fiberglass parts, this curve in the top longeron is not acceptable. The builder is left wondering what went wrong and no amount of clamping or brute force forming will solve the problem to any degree of satisfaction. The problem is not the builder's fault. The solution starts by understanding the three dimensional relationship of the assembled parts being built.
First understand that the plans show the finished form of the plane. They show the "projected" form as you would expect to see it if viewing an actual plane from the top, ends and from the side. Since the sides are sloped (flared) outward, looking from the side, the distances given by measuring the profile drawing are "foreshortened" and don't give the proper shape for building the fuselage with a flat top longeron. What needs to be done is to "develop" the "true" distances and shape of the flat panel so that when it is curved into position, the longerons lay flat.
Second, understand that the dimensions called for in the plans put a twist in the sides that tends to work the panel in two directions of curvature. This twist makes the panel "undevelopable" meaning that that shape cannot be unrolled into an equivalent flat shape. This is important when laying out the side and bottom panels onto flat plywood. To illustrate this, try forming a piece of paper around a soda can. The paper can be formed flat around the can either straight or at a diagonal to it's length. It has only one direction of curvature and is by definition "developable". Now try to form the same piece of paper around a baseball. It won't lie flat on the surface without some deformation (folding, wrinkling or tearing) of the paper. The ball has curvature in more that one direction and is a "compounded" shape. Paper (or plywood) can only be readily formed in developable shapes as opposed to aluminum or other metal which can accept in plane deformation. A developable surface is needed to lay out a curved surface when the materials used can't be deformed with any degree of in-plane strain.
Initially, the fuselage sides are laid out flat with reference to the top longeron measured to a straight chalk line. The bowing problem starts when the side panels are bent and sloped to form the fuselage box section. If the sides were not sloped (tumbled home), the section formed would be cylindrical and the longerons would lie flat. Since the sides are tumbled home, the section formed is now conical. When a conical shape is cut with a plane (building surface) not perpendicular to it's axis, the shape formed is elliptical -- exactly what happens with the top longeron. When it's built flat, bent to form a cylindrical section, and sloped to form a conical section, it takes on an elliptical shape firewall to tailstock.
This method borrows heavily from proven techniques used in the marine trades. It should be stressed at this point that although the layout procedure is not complicated, it is important to take your time. If the layout is not going well initially, start over! Better to erase layout errors now than to have them built it and cause surprises later.
Layout to ensure a fair and true fuselage starts by drawing a reference line (baseline) on the building surface. Refer to figures 2 & 3 and use a wire guide to draw a very straight baseline. About 500 lbs. Of tension should be adequate. One could use a chalk line, but we're talking airplanes here, not house framing.
The main layout difference is that the baseline isn't used as a reference for the top longeron. The baseline references the mid point of the firewall for the developed (and true dimensioned) side panel. Although the baseline will still be the reference, the top and bottom longerons will be laid separately.
Layout differences don't end there. Each of the stations (vertical members) will be laid out with a calculated separation so that when the panels are formed into position, they land on the spacing called for in the plans. Another major difference is that the bottom & side panels are applied after forming the fuselage box section. This is mainly to obtain the ability to "fair" the side and bottom surfaces and insure a straight and true shape.
Refer to figure 1 for the layout of the new developed side panel. The firewall (station a) is layed out perpendicular to the baseline. Longitudinal (station) measurements are given along the length of the baseline from the firewall. Vertical dimensions are given to reference the angle and breadths of the station at the baseline.
Notice that the top longeron is bowed outward and that the stations are spaced slightly greater than called out in the plans. When the panels are formed into the box frame section ,they will work into the dimensions specified in the plans.
Strike a centerline, longer than is needed on the building surface using a wire guide. Draw off the firewall line perpendicular to the centerline at one end.
Using the distances listed in the balloons, mark them off on the centerline. Distances are measured to the nearest sixteenth of an inch. Take time to mark them off carefully. Don't mark off the distances in a cumulative fashion. Use the firewall as a common reference.
Using the angles listed at each station, mark off a station line longer than is needed. The angles are measured to the nearest hundredth of a degree. Take time to mark them off carefully.
At each station, start by marking off each short (bottom longeron) line distance from the centerline. Use your set of trammels or beam compass for doing this. Mark the intersection of the short line with the station line.
At each station, mark off each long (top longeron) line distance from the intersection of the short line distance and the station line. Again the trammels or beam compass is best for completing this step. Mark the intersection of the long line distance with the station line.
Using the longeron as a batten, trace out the inside and outside curves of the longeron. After the batten is secure, in between each station, fasten a keeper block inside and outside to preserve the shape of the longeron taking care to avoid potential future interference with the diagonal members to be installed later. The fairing blocks can be removed or left in place if they won't interfere with building. The vertical station members and their diagonals can now be measured and positioned. Remember to refer to the plans for the material thickness direction.
After vertical and diagonal members are cut and fitted, take time to draw their outlines on the building surface to cut down on time and confusion when laying out the opposite side.
Finishing the side panel is accomplished in a manner similar to that called for in the handbook with the exception that the side and bottom skin panels will be attached later.
The next article in the series will discuss jigging and building techniques to ensure alignment and straightness of the flat built side panels. Also covered will be building a "strongback" jig to assure alignment of the side panels when they are formed into their final shape.
Part 3 in the series will cover assembly of the side panels using the jigs. Some joint details will be discussed that will ensure a stronger and more fair fuselage assembly. Also covered will be the layout & attachment of the side and bottom ply skins.
U.S. Mail: Densmore Associates, inc.
ANSI "D" size, computer generated plots of all the layout drawings in this series are available from the author for $30 plus postage & handling. Full (true size) scale plots may be made available depending on demand.
"Scarfing" is the practice of splicing plywood so that short pieces of plywood can be used to span long distances. On the KR, it is required on both the fuselage skins and spar webs. The angle of the splice should be 10 to 12 degrees to maintain strength across the joint. Also, joints should coincide with structural members, such as spar webs or fuselage truss members.
This scarfer is made by mating a regular plunge router (this one costs about $50) to a table saw. Obviously, you really only need a table saw to cut the chamfer, but it does make a nice heavy table for scarfing. You could just as easily use a large work table as the base.First, set the table saw for a 5.5 degree cut (for a 1:12 joint, or 6.5 degree cut for a 10:1 joint), and run a 1 x 6 through on edge to chamfer a corner on the board. Then drill the board for three router mounting holes (two are countersunk) and connect the assembly to the table saw with two 1/4 inch bolts. Use a long (2-3 inch) straight cutting bit to do the cutting. Adjust the bit so it doesn't interfere with your table top, and go to town. Keep pressure on the plywood to ensure contact with the table while you're scarfing. Make sure you feed your material from the same end as you would if you were sawing, or the router will take your plywood away from you and put a big dent in your garage door.
In the late 60's Ken Rand and Stuart Robinson were working as flight system engineers for Douglas Avionics. Ken was working as an electrical engineer, having previously worked for Sperry as an autopilots project engineer, while Stu's degree was in aeronautical engineering from Northrop University. They were two of the guys at the end of the DC-8,9, and 10 assembly lines responsible for correcting some of the nits and picks in various systems before delivery to the customer.
They both wanted to build a fast, inexpensive airplane which was also economical to maintain. Several designs were considered, and plans were bought first for the Jeanie's Teenie and then the Taylor Monoplane. The Monoplane was more to their liking, but would require some modification to fit their needs. A cooperative redesign effort ensued, with virtually no dimensions left untouched. Only the basic fuselage structure, airfoil, and powerplant were retained. The tail shape was Stu's, and came directly from the big DC-8s parked on the ramp outside his office window. The landing gear was designed by Ken, after seeing the gear on a Dewey Bird at Santa Paula airport.
Ken was killed in his KR2 a short time later while flying over Cajon Pass in what was apparently a bad weather / low fuel accident. Ken's wife Jeanette became owner of RR overnight, and stepped up to keep the plans and parts coming. Much of the engineering needs are handled by Bill Marcy of Denver, who's been helping out since early '79.
To date, almost 6000 KR1, 9200 KR2, and 760 KR2S plan sets have been sold. 1200 KR2s are estimated to be flying, with 5 KR2Ss now in the air. Much of the development work done on KR's is now done by the builders themselves. KR builders tend to be innovative, which leads to some interesting modifications. Some of the mods that work eventually creep into the plans. The KR2S is a case in point. Many builders who'd heard of the pitch sensitivity and tight cabin of the KR2 began to build an enlarged version, with the length determined by the most commonly available longeron material. The result is a KR2 that is stretched 2" between firewall and main spar, and 14" behind the main spar. Higher gross weights dictated more wing area, with the new standard becoming the Diehl wing skin. Those who plan to carry passengers commonly stretch the cabin width a few inches, although 1.5 inches is the limit if you still want to use RR's premolded parts.
Mike Stearns addresses the KR Forum crowd.
This year's KR Forum featured guest speakers Mike Stearns, Steve Trentman, and Bill Marcey. Mike Stearns spoke on several topics, including the many sources for KR and homebuilding information available on the Internet. He also mentioned KRNet, the list server devoted entirely to KR aircraft, as well as several notable World Wide Web home pages. He also brought a sample of the new Rand Robinson wing skins with him, and discussed their high temperature core prepreg construction. His KR2S will receive the first set, which is currently being installed at Hinson Composites.
Steve Trentman spoke on his turbine installation. It uses a turbine engine which saw duty as an A7 attack jet starter engine. Total weight is about 85 pounds, while putting out around 90 horsepower. There is a small stockpile of these engines available from government surplus. sources. This engine can only be throttled back to 52% power, which leads to some pretty interesting landings. One inflight failure has been logged so far, with very little damage to the aircraft. More on this exciting development in next month's issue of KROnline.
Les Palmer's KR2 N202LP won Best KR2, Best Engine Installation, and People's Choice awards at the 1995 KR Gathering at Columbia, TN. After researching the KR series, and reading Neil Bingham's "A Critical Analysis of the KR2" (Jan 88 Sport Aviation), Les decided to build his as a single seater, stretched 24" in the tail, while maintaining a stock width firewall. His fuselage is made from Douglas fir, which weighs in at 4 lbs heavier than if constructed from spruce. It is skinned with 1/8" birch plywood. Spars are covered with plywoood on both fore and aft sides, ala KR2S. Diehl wing skins provide the lift. Horizontal stabilizer and elevator were stretched 7" longer on each side, while the vertical stabilizer and rudder were stretched 8" taller. . The fuselage to cowling junction was made more graceful by adding 1.5 inches to the height of the firewall end of the fuselage sides.
Les's canopy is a Dragonfly, using a four linkage system to swing forward when opening. The canopy frame fits snugly into a recess in the foward deck, providing an excellent wind and water seal. The fiberglass work is exemplary.
Seating is luxurious for one.
The cowling is also a work of art, and uses NACA ducts for efficiency. Female molds were made for all the fiberglass parts on Les's plane, so he could proabably be persuaded to make more, if demand dictates. Les also machines a multitude of KR aluminum and steel parts which he now offers for sale.
The firewall was reinforced with aluminum brackets and angles bolted between the longerons in anticipation of the 200 lb Subaru EA-81 engine installation. His 100 HP Asian version is outfitted with an American Holley 5200 caburetor and manifold. It uses a PSRU of Les's own design, featuring two spur gears with a 1.69:1 reduction ratio and a toothed belt. Other than tapping the crank for larger bolts to mount the redrive, no other engine modifications were required. Also, this is probably the only air conditioned KR2 on the planet. The prop is a 60/63 Hegy.
Originally built as a taildragger, the fixed gear is made from 4130 steel tubing. Custom cast 6.00x6 aluminum wheels and steel rotors are mated with 6" Cleveland calipers for braking. An early taxi test accident damaged the main gear, and prompted Les to change to tricycle gear. Again, he designed his own fiberglass main gear, and uses a Diehl nose wheel fork with a 4130 strut and 6" wheel up front.
Early tests revealed cooling problems, which prompted a radiator move from the firewall to a lower cowling location.
The first flight was almost a disaster, as test pilot Randy Smith lost power right after takeoff. He managed a 180 with a safe downwind landing with only minor nosewheel pant damage. The culprit proved to be a spark plug with too much reach, which was quickly remedied. Subsequent flights have shown water temp to be about 210 degrees, oil temp is 220-230, and airspeed is about 180 mph.
Shopping for the Partially Built KR.
This story starts about twenty years ago when I first started looking at the KR-2 as the plane I'd like to build. The only problem at that time was a lack of money, lack of knowledge, and a lack of job stability. I liked the design, except for the low ground clearance of the retractable gear and that a KR was going to be a tight fit for me to fly.
Over the past twenty years I've owned a number of planes, but still always wanted to build my own. I needed one that would fit me, my budget requirements, and have the speed and performance that I wanted. When "KITPLANES" published the article featuring Roy Marsh's new KR-2S, it was the first I had heard of any major modifications or improvements to the same old KR design. I believe that article and Roy Marsh's workmanship have probably been the greatest boon to Rand Robinson (RR) in the last twenty years. It certainly caught my eye! Here was the same design I had decided I wanted to build twenty years ago, with all of the improvements I wanted. It was sitting on fixed gear with some reasonable ground clearance. It had the capability to be built large enough to accommodate me. It has enough prefab parts available that it didn't have to be 100% scratch built if I decided to hurry the project along. And it had the speed I wanted. I knew that Roy's published speeds were probably not realistic expectations for the average KR, but after knocking around for the last three years in my Champ, anything over 90 mph seems pretty fast to me.
After purchasing the info kit and the sales video from Rand Robinson, the next step after deciding for sure to build this plane was to order the KR-2 plans and the KR-2S addendum. I finally got my plans and was putting together my first order to start the plane, when my partner in the Champ pointed out that there was a partially completed KR-2S for sale in Trade-a-plane. My initial answer was "No, I don't even want to look at it. I want to build my own from scratch." My partner insisted that for the advertised price and the fact that it wasn't too far away, I ought to at least give the guy a call and investigate it. "No, I don't think I want to buy someone else's problems," I persisted. That night I went home and crunched up some numbers on the calculator and finally came to the conclusion that for the sake of my budget for the next several years, I really should give this guy a call.
Three days later, I flew to his place about 400 miles away to take a look at his project. At this point I should probably mention that I consider myself to be fairly knowledgeable about airplane construction, although the vast majority of my experience is with tube and fabric. The rest of this article deals with what I looked for and more importantly what I missed and have had to repair in the last year since I purchased the project.
When we went to the seller's house, I found that the left wing was built using the Dan Diehl wing skins and the right wing skins were leaning against the wall inside the house. Also the canopy was in the house with the canopy covered with paper and tape. I wanted to inspect the fuselage first, so off we went to the shop.
There I found a fuselage sitting on it's gear painted in primer gray. The first step was to inspect the quality of workmanship of what could be seen as it sat. The interior of the fuselage looked as if it had been built with a great deal of care. The fit and finish of all of the interior wood was very nice. Even the gussets looked like they had been painstakingly perfectly fitted. The glass work on the turtle back also looked very precise and clean. It was evenly faired into the vertical and horizontal stabs. The tail also appeared to be well built with the exception of a depression directly over the front and rear spars in the horizontal stabs. He explained that when he moved recently, that he had shot the plane with gray primer to protect it from the weather since he wouldn't have ready access to a shop to put it in right away. It ended up sitting out in the hot south Texas summer sun for a few weeks before he got a shop rented to work in. That caused the glass (or possibly the foam inside the horizontal stab) to swell, except that it held onto the spar, so it was slightly ballooned in front of and behind the spars. His recommendation was to fill it back smooth with micro.
I also found a small linear crack in the lower left wing spar cap on the left wing stub. It appeared to be from over tightening the rear spar wing attach fitting bolts. His explanation was that the crack wasn't important because the rear spars only job is to keep the wings from folding back. I also noticed that the holes for attaching the outer wing to the wing stub were badly rounded out on the rear spar. He explained that the Diehl wing skins require the rear spar to be swept slightly more forward than the stock wings. This won't allow you to use the rear spar attach fittings from RR and that I would need to fabricate a new set of rear spar attach fittings.
I also found that the aileron bellcranks were not built or installed as per plans, but found that they looked professional. I couldn't check for function since the right bellcrank and sheeve wasn't installed, the left wing also wasn't installed, and the right wing didn't exist yet.
Next we pulled the inspection panels off of the fuselage and tail and looked at everything I could see with a good flashlight. I didn't find anything else that might be questionable about the fuselage except for a cracked elevator trim tab that was damaged when it fell off it's hanging place on the wall.
Next we spent some time going over his builders log and builders photo album. I still hadn't seen anything that would dissuade me from buying this project.
At this point it was starting to get late and my ride down needed to get airborne for the flight home. I needed to make a decision about whether I wanted this project or not, but I hadn't inspected the wings and canopy yet. I took a cursory look at the left wing and saw lots on micro built up on it and some bubbles in the leading edge, but nothing that looked seriously wrong to my amateur eye. The right wing was only a set of spars in the shop and the Diehl wing skins in the house, so there wasn't much to look at there. The canopy was wrapped in paper and tape, so there wasn't much to look at there either. I decided that even if there were serious problems in the wing that was built, I would be money ahead to go ahead and buy the project. For the advertised price, I could build a new set of wings and still be way ahead financially. We negotiated a final price, shook hands, took my ride to the airport, and started off in search of a U-haul to haul the project home.
Now, at this point, some of you are thinking about what I surely must have forgotten to inspect and why didn't I take a local A & P or EAA member along for the ride. First of all, I don't know any mechanics locally that have any experience with glass and our EAA chapter of which I am VP is woefully lacking in fiberglass knowledge. Secondly, as you will see, I missed plenty. Some by ignorance, some by just not looking close enough.
Now for a list of the problems that I found over the last year and a few of the fixes that I came up with.
I found that the lower set of rear spar attach fittings on the left rear spar were installed backwards with the longer spaced hole towards the fuselage. Since this is the same place that also had the cracked spar cap, it required a major change. Also in the same area he had drilled through the rear spar with a hole saw to create a place for the aileron cable to pass through and managed to cut out the second from the outside vertical brace in the spar. Then he chose to install the aileron bellcranks in front of the rear spar, and cut another hole through the rear spar for the aileron push rod. He also managed to cut out the outside vertical brace in the spar. Since the holes were already drilled through the spar, the choices were to either cut out that section of spar cap and scarf a new piece in, cut the whole rear spar carrythrough out of the fuselage including ruining the left lower wing skin, or do something else creative to reinforce the spar cap and install a custom built set of attach fittings.
I also found that after I built and installed the right side wing stub ribs and skin that the aileron bellcrank setup would not work as installed. The cable that crosses between the two bellcranks had a sharp uphill from the sheeve to the bellcrank in the last 12 inches on either side. This combined with the radius that the bellcranks turn caused the cross cable to pull up tight when the ailerons were pushed to either end of their travel, but allowed the cables to go very slack when the ailerons were centered. Also the Aileron pushrods needed to pass directly through the lower set of rear wing attach fittings to attach to the aileron. This whole rear spar and aileron bellcrank setup was going to either have to be redesigned or cut out and built to plans. The bottom line is that the problems I observed when I inspected this part were much more serious than expected when I had to fix it.
I decided that I had to remove the rear fittings from the left wing to be replaced with the new set that my neighborhood machinist was cutting out for me. When I put the wing on the work bench to start removing the rear fittings, I thought I had better take a closer look at the bubbles in the leading edge. I found that as I pushed on the leading edge, it delaminated between the glass lay-up on top and the upper and lower wing skin edges that were floxed together underneath. I concluded that that area had to come apart and took a belt sander to the leading edge. What I found was that the leading edge had been floxed together and glassed over, but the mold release had never been scrubbed off the leading edge of the wing. It peeled apart for rebuild quite easily.
When I got back to removing the rear spar attach fittings, I noticed that the woodwork inside the wing looked awfully dull. The reason was that the wing had been closed up without varnishing any of the woodwork. This was rectified with a small hole saw, a number of extensions and a modified undercoating sprayer.
I also found that the aluminum drain fitting in the bottom of the left wing tank had been glassed into place upside down. The tapered pipe threads were tapered the wrong way to install the draincock into the tank. Retapping the fitting the right direction seemed to be a good fix for that problem.
When I finally got around to attaching the wing to the fuselage, I found that the front spar attach fittings were badly misaligned. Although they could be forced into alignment, I didn't think I needed that kind of preload on the main spar fittings. This problem was fixed by calling on my local neighborhood machinist to build me an aligning fixture and reaming the attach holes to the next larger size and ordering the new sized bolts.
On the fuselage I found that although it had new Cleveland wheels and brakes on it, one of the brakes had a severe wobble to it. I must complement the manufacturers for taking care of that problem. One call to the Cleveland factory and they shipped me a new set of wheels and brakes even though the receipt for this set was over four years old and in the original builders name. Their only concern was that this set had never been placed in service yet.
I chose to sand the load of micro off the left wing to see what it was covering. When I got down to the glass, I found that there was no glass for the aft inch and a half of the underside of the wing in front of the aileron hinge. With the Diehl wing skins, you build the wings, then cut the ailerons out of trailing edge of the wing. He had mismeasured and cut too much material off the bottom side of the trailing edge in front of the aileron. It was filled by floxing a piece of spruce into the gap to fill the space between the back edge of the fiberglass and the aileron mount. I chose to wrap the trailing edge of that wing, and the other wing to match with a couple of lay-ups of glass.
When I sanded the primer off the aforementioned damaged trim tab, I found that the hinge was floxed to the leading edge of the foam insides of the tab, but not the glass. I also chose to wrap the front of the trim tab with a lay-up of glass.
I decided to pull the paper off the canopy and take a look at it before I'm ready to bolt it on and fly. The original builder had blown his own canopy and after some of the previous problems, I was beginning to have some concerns about not having looked it over closely enough. The canopy turned out to have been blow a little too large. It ended up with a little larger bubble for headroom, which I didn't object to. However, it had more headroom on the right side than the left. Yes, it was just a little bit lopsided. The main problem was that the canopy is stretched thin enough that it can be easily pushed in with one hand when the weather is warm.. My fear was that this is just thin enough that it may decide to lay on my head or in my lap when flying on a warm day. It will have to be replaced.
I'm sure that many that are reading this could see several of the potential problems before I mentioned them, but some others may not have and I'm sure that there could have been many other problems that didn't but could have existed on this project. This is also not intended to be critical of the gentleman that started this project as many parts of it, especially the wood work are better than I could have done and much of his work is outstanding. I prefer to think that I'll end up with a better plane with his woodwork combined with my glasswork. This article is intended to feature some of the problems that you may run into in buying someone else's project.
The final question is, knowing what I have found over the past year, would I have still purchased this project. The answer is yes, but primarily because the price was right in that I am still money and work ahead of where I would be if I had started the project from scratch. There are a few things that I would have done differently, but nothing that I can't live with. Although I won't be able to say that I built it all from scratch, I have built and rebuild enough of the plane that I should have no problem qualifying under the 51% rule.
You can send comments directly to the author via e-mail at "jscott@LANL.GOV".
Here is an brief explanation of how I built my turtledecks. The jig was constructed from scrap plywood and a few 1x4s that I ripped into stringers. I made two temporary bulkheads from the plywood, one for each end. Remember the forward bulkhead needs to be shaped in a way that will closely match the aft end of your canopy frame. Make an aft bulkhead by placing a straight edge at the top of your forward bulkhead and the trailing edge of your horizontal stabilizer. This will give you an idea of how tall your aft bulkhead needs to be. As far as location, I placed my aft bulkhead just forward of the lower/front of my vertical fin. I constructed the jig on the fuselage, it is glued together with automotive bondo.
After the bulkheads were bondoed to the fuselage I used the stringers that I ripped from the 1x4s and bondoed them to the bulkheads. This gave me a male form to cover with thin plastic or posterboard. I stapled two layers of posterboard to the jig(thin plastic would work better). The posterboard wraps down two inches onto the fuselage. After I was satisfied with the way it looked, I then covered the entire thing with duct tape (fiberglass will not stick to duct tape) On top of this I wetout one layer of tri-ply cloth (22oz) that I had left over from an earlier project, and one layer of 8oz. bid. Remember to mask off your fuselage so you don't get epoxy on it. If you are not familiar with composite lay-ups, you should plan on razor cutting your lay-ups 4 to 6 hours after wetout while the lay-up is still soft enough to cut with a razorblade.
After the lay-up cured (2 or 3 days) it was removed from the jig, and the jig was removed from the fuselage and discarded. (be careful, the bondo sticks very well to the spruce, you could splinter your wood during removal) I now have a fiberglass skin that tends to hold the shape of the jig but is still flexible enough to work with. I made two bulkheads out of 1/4 last-a-foam (AS&S) using the plywood formers from the jig as a guide. I covered these foam bulkheads with one 8oz layer of glass on each side, with a glass to glass edge on the bottom. After cure these bulkheads were bondoed into place (to the fuselage)and the fiberglass skin was pulled down tight and floxed to the bulkheads. When the flox cured the bondo joints were broken, again being careful not to harm the wood. The turtledeck was removed from the fuselage and 2 inch tapes added to the bulkheads inside and out.
At this point the turtledeck looked great and only weighed about 5lbs. but I noticed you could deform the skin by pushing hard on the outside. So I flipped the turtledeck over and from 1/4 inch last-a-foam, I cut two inch wide strips that would run the entire length, forward and aft inside the turtledeck. In effect these would act as composite stringers, I made enough of these two inch wide strips to make up three stringers. One down the center (sort of a backbone) and one on each side of the "backbone" half the distance to the edge of the turtledeck. I sanded the edge of the foam so that when covered with a layer of bid @ 45degrees there would be a nice transition from the turtledeck skin up onto the foam and then back onto the turtledeck I scuff sanded and glued the foam stringers in with micro. I covered the foam stringers with one layer of 8oz bid @ 45degrees.

Answer: The longerons bow up from the building surface, forming a "banana" shape.


Input: What is the effect of accounting for path preference on the robot's belief update?

Context:
Paper Info

Title: Incorporating Human Path Preferences in Robot Navigation with Minimal Interventions
Publish Date: 16 Mar 2023
Author List: Oriana Peltzer, Dylan Asmar, Mac Schwager, Mykel Kochenderfer

Figure

Hyperplane arrangement of a twodimensional space containing two obstacles (colored in gray).The robot is located inside the pink polytope, surrounded by three adjacent obstacle-free polytopes.Each hyperplane on the boundary of the robot's polytope corresponds to one of the nonredundant constraints in eq.(4).(b)Graph derived from the hyperplane arrangement.The nodes on the graph designate polytopes, and edges designate transitions to adjacent polytopes.To estimate the human's preference, the robot updates a posterior over the goal and over which of the graph transitions φ 1 , φ 2 and φ 3 is preferred by the human.(c)Example preference defined over the graph.The location of the goal is indicated in yellow in the lower right polytope.For each node, the outgoing pink arrow designates the edge on the graph corresponding to the preferred transition between polytopes.
Simple, 10 × 10, 8 polytopes.(b) Map 2: Office, 10 × 10, 56 polytopes.(c) Map 3: Classroom, 20 × 20, 73 polytopes.(d) Sampled observations and robot's executed trajectories.
Fig.5: Maps used for simulating the robot navigation problem with path preferences.In (d), the heading angles observed are indicated with arrows.The goal is indicated with a pink circle, and the orange robot corresponds to the starting location.The blue robot follows a policy that accounts for path preference, while the green robot does not.The opacity of the robots increases with time.
Map 1 problem setup and example realizations for goal-only (green) and path preference (blue) solution methods.The robot starts at the lower left corner of the environment, and the goal of the task (pink circle) is in the upper left area.The robot does not know which goal, among 10 options (shown in light blue squares), is the correct goal.The human provides noisy observations, indicated by arrows, at each iteration.The green robot selects actions according to the goal-only baseline, and the blue robot uses our proposed method to infer path preferences.The polytopes composing G are drawn in blue.Probability of correct goal.WLPHVWHS +J (c) Entropy of goal distribution g.
Fig. 6: Probability of the correct goal, fig.6b, and entropy of the goal belief distribution P (g), fig.6c, for the same problem setup, fig.6a.In this problem instance, the human's preference is to go to the goal by passing on the right side of the obstacle.Results are averaged over 50 runs and the area filled represents one standard deviation above and below the mean value.The goal-only baseline shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference.
Success rates in the simple environment (Map 1).The results are averaged over 6 randomly sampled problem instances (start location, goal location, and goal possibilities), and over 50 runs per problem instance.∆T is the number of time steps separating two consecutive human inputs.The robot's mission time is Tmax = 30 time steps.We selected γ h = 1.5, corresponding to relatively noisy human inputs and making the problem more difficult to solve for the robot.
Computation times for Goal Only and Path Preference methods on Map 1 (fig.5a),Map 2 (fig.5b), and Map 3 (fig.5c),averaged over 100 runs with randomly sampled problem instances.The 95 % confidence interval is provided with the mean.We evaluate computation time at the first iteration of each run (where the search depth takes on its highest value Tmax).

abstract

Robots that can effectively understand human intentions from actions are crucial for successful human-robot collaboration. In this work, we address the challenge of a robot navigating towards an unknown goal while also accounting for a human's preference for a particular path in the presence of obstacles.
This problem is particularly challenging when both the goal and path preference are unknown a priori. To overcome this challenge, we propose a method for encoding and inferring path preference online using a partitioning of the space into polytopes. Our approach enables joint inference over the goal and path preference using a stochastic observation model for the human.
We evaluate our method on an unknown-goal navigation problem with sparse human interventions, and find that it outperforms baseline approaches as the human's inputs become increasingly sparse. We find that the time required to update the robot's belief does not increase with the complexity of the environment, which makes our method suitable for online applications.

INTRODUCTION

Collaboration between humans and robots has become increasingly important and one key aspect of this collaboration is the ability for robots to adapt to human decisions. In many scenarios, such as a robot navigating through a busy room to deliver an item, it is important for the robot to take into account human preferences.
For instance, humans may prefer a specific path that would allow their colleagues to notice the item being delivered, but this preference may change dynamically based on various factors such as changes in the environment or unforeseen circumstances. While some preferences can be incorporated into the path-planning process, accommodating dynamic user preferences in real-time remains challenging.
In this paper, we propose a way to enable robots to adapt to human preferences dynamically by leveraging real-time feedback to inform decision-making. In this work, we tackle the problem of robot navigation in which the robot cannot observe the goal or the preferred path to the goal, but must make navigation decisions that are influenced by humans through recommended actions.
Prior work has explored how to adapt to a human's preference through feedback, but such approaches often require a high level of intervention, which can be time-consuming and impractical in real-world scenarios. To optimize the use of human input and quickly infer the human's preference, Fig. : An autonomous robot navigates in a simulated classroom towards a goal location (pink circle).
At the start of its mission, it receives direction indications (arrows) from a human that indicate which path it should take to get to the goal. In this scenario, the human wants the robot to go around the desks on the right side of the classroom. A robot that does not reason over path preferences (green) will take the shortest path to the goal regardless of the human's input.
Our method (blue) infers the human's path preference from these indications and adapts to their recommendations. we propose an approach that leverages probabilistic representations of human preference and incorporates real-time feedback. Previous research by Bajcsy et al. considered an online adaptation problem in a manipulation task, where the person can apply forces to the robot to indicate their preferences.
By allowing the robot to continue its task while taking into account a probabilistic representation of human preference, their approach does not require frequent inputs. Building on this idea, we adopt a similar approach to adapt to a human's preference in the context of a robot autonomously navigating through a known environment, such as a cluttered office space.
Specifically, we focus on allowing the human to influence the robot's trajectory with respect to obstacles, by providing guidance on preferred routes or paths, while the robot continues to execute its task. Paths can be represented using homotopy classes . However, homotopies can pose computational challenges when used to encode and infer human preferences.
When the robot maintains a belief over homotopy classes, the inference problem can become exponentially complex with the number of obstacles in the space. Additionally, when the goal is unknown, the number of variables increases with the number of candidate destinations. This complexity can render the decision-making problem intractable.
Our solution is to encode path preference based on a partitioning of the environment into polytopes . This representation allows path preferences to be expressed as sets of preferred transitions between adjacent polytopes. Paths belonging to different homotopy classes correspond to different sequences of transitions.
By leveraging conditional independence assumptions, we can make the Bayesian inference problem tractable. These assumptions exploit the fact that human actions provide information about the path in a piece-wise manner. For example, indicating a preference for navigating around a particular obstacle only provides information about the local area and not the entire path.
Finally, after updating its belief representation over the human's preference, the robot can adapt to indications by replanning online. Our contributions are as follows. • We formulate the human-robot collaboration problem as a Partially Observable Markov Decision Process (POMDP) where both the goal of the task and the human's path preference are unknown random variables.
• We propose an encoding of a human's path preference using a partitioning of the environment into polytopes, along with conditional independence assumptions that make the Bayesian inference problem tractable to infer the task goal and path preference online. • Through simulations in two environments of different sizes and complexity, we show that our method is effective for solving problems where the robot must reach a goal that is unknown a-priori while simultaneously adapting to a human's indications.
Our method shows higher success rates compared to baseline approaches when the human inputs are sparse. Our approach enables a robot to make effective navigation decisions in collaboration with a human, even when the goal and path preference are not known in advance, and with minimal human input. In recent years, there has been a growing interest in shared autonomy and interactive systems, where humans and robots work together to accomplish tasks.
Several approaches have been proposed to address the challenge of enabling effective collaboration between human and robot agents while still achieving high task performance. Losey et al. and Jeon, Losey, and Sadigh propose a framework where a human operator is given control of a task-relevant latent action space while an autonomous system handles the rest.
Dragan and Srinivasa present a formalism for arbitrating between a user's input and a robot's policy when both human and robot share control of the same action space. Cognetti et al. [7] provide a method for real-time modifications of a path, . . . Fig. : We model the intent inference problem with the above diagram.
At each step in time, the robot receives an observation ot from the human conditioned on its current location st, the intended goal g, and the human's path preference θ. The robot updates its belief over g and θ and transitions to a next location st+1. while Hagenow et al. present a method that allows an outside agent to modify key robot state variables and blends the changes with the original control.
However, a common challenge of these approaches is the high level of intervention required from humans. Best and Fitch propose a method for predicting an agent's intended trajectory from observations. Rather than maintaining a belief over the agent's future path, they infer the agent's intended goal among a set of candidate locations at the boundary of the space.
This approach provides information on where the agent is heading and generates a distribution of candidate future trajectories for the agent. Inferring the goal of the task among a discrete set of candidates is also relevant to the area of shared autonomy. Javdani, Srinivasa, and Bagnell propose a formalism for shared control of a robotic arm, where the robot must assist the human in picking up an object but needs to infer which object the human has chosen from joystick inputs.
Planning with homotopy class constraints is useful in problems where the robot's requirements are given with respect to obstacles, and Yi, Goodrich, and Seppi consider topological constraints provided by human operators. Bhattacharya propose an efficient algorithm for solving pathplanning problems under homotopic constraints.
However, the number of homotopy classes for a given problem can be infinite, and as the robot changes location and updates its representation of the world, carrying out inference over homotopy classes in a dynamic environment requires recomputing the set of homotopies at every iteration, making the belief update challenging.
Prior work has addressed the challenge of shared autonomy by considering how robots can infer a human's intended goal, or how they can infer the preferred path to a goal. However, we argue that inferring the goal and the path as separate problems can lead to over-confidence in incorrect beliefs about the user's preferences.
To illustrate this point, consider the following scenario: a robot and a human are collaborating to move an object from one end of a room to Fig. : Using the hyperplanes composing the H-representation of each obstacle, we construct a hyperplane arrangement of the obstacle-free space (a). We define the human's preference for the robot's one step action choices as the posterior distribution (given all human input up to that point) over transitions from the current to the neighboring polytopes, i.e. edges on the graph.
Each time the robot transitions to a new polytope, the set of neighbor polytopes and the distribution over human preferences are updated. another, but there is an obstacle in the way. The human would like the robot to take a path around the obstacle on the left, even though the goal is on the right. If the robot only infers the goal from the human's inputs, it may incorrectly assume that the goal is on the right, and become over-confident in this belief.
On the other hand, if the robot only infers the preferred path, it may mistakenly assume that the goal is on the left, leading to a failure in completing the task. To overcome these challenges, our work proposes a joint inference approach that considers both the human's intended goal and their preferred path to that goal.
Specifically, we model the human's preference over different homotopy classes and leverage a conditional independence assumption to provide a tractable solution. In our approach, we assume that the human's inputs are noisily rational conditioned on both the goal and the preference. By jointly inferring the goal and path preference, we can avoid over-confidence in incorrect beliefs about the user's preferences, leading to improved system performance.
We consider the problem of robot navigation in a known environment to an unknown destination, where a human can intervene and provide a heading direction to the robot using a joystick or force cues. The human also has a preference on which path the robot should take with respect to obstacles, and our objective is for the robot to understand the human's intentions and execute the task with minimal interventions.
Let g be a discrete random variable denoting the goal of the task, belonging to a set of candidates Ω g , and let θ be a discrete-valued random variable representing the human's path preference, belonging to a set of possible preferences Θ. The physical location of the robot at time index t is denoted by s t ∈ R 2 , and the robot's action at time index t, belonging to some action space A, is denoted by a t .
The transition model T (s t+1 | s t , a t ) is deterministic, meaning the robot has full control over its future location. At any time step, the human may provide an observation to the robot. When the human intervenes, the robot receives a direction (heading angle) that can be mapped to a future location in space.
More specifically, we map the direction to an intended location, which is the resulting robot location after advancing in the indicated direction for one time step. For simplicity, we consider that the robot directly makes an observation o t of the location indicated by the human. We assume that the robot has a stochastic observation model for the human P (o t | s t , g, θ) that is conditioned on both the goal of the task g and the human's preferred path θ.
We further assume that having chosen a goal and path preference, the human takes actions to noisily minimize a cost function C g,θ that measures the cost of moving from the robot's current location to the goal along the preferred path. For example, C g,θ (s t , o t ) can be the length of the shortest path from location s t to the goal g after taking a first step to o t , and constrained by path preference θ.
We use C g,θ to induce a probability distribution over observations, given by: where γ h is a hyperparameter that designates the rationality coefficient. This model assumes the human will pick the lowest cost action with the highest probability and the likelihood of an action decreases exponentially with the increase in cost .
Our inclusion of the path preference θ sets our approach apart from . The model is shown in fig. represented as a Bayesian Network.

Inference

At each time step where the human provides an observation, the posterior P (g, θ) is given through the Bayesian update We note that the number of Bayesian updates required at each iteration to update the belief is equal to the cardinality of Ω g × Θ. In addition, each Bayesian update involves computing C g,θ ( .
, . ) in eq. ( ), which involves solving an optimization problem (such as a shortest path problem). In section IV, we propose a specific encoding of preference θ for resolving eq. ( ), while ensuring the number of computations of the cost C g,θ (., .) per update does not grow exponentially with the number of obstacles.

Decision Making

We consider a navigation problem where the robot receives reward according to the model R(s t , g, θ, a t ). We wish to find the optimal policy π that maximizes the expected discounted sum of future rewards, with discount factor γ. The above problem is a Partially Observable Markov Decision Process (POMDP) .
In this section, we propose an encoding of human's path preference θ for computing the posterior in eq. ( ). Devifrom the concept of homotopy classes, we define the preference according to a partitioning of the environment into polytopes, as shown in fig. , creating a hyperplane arrangement of the space.
Hyperplane arrangements have been used by Vincent and Schwager in the context of Neural Network verification. In our setting, we leverage this representation to define path preferences as preferred transitions between adjacent regions of the space.

Hyperplane Arrangement

We assume a two-dimensional environment composed of m polytopic obstacles, each defined by their half-space representation (H-representation) where A i ∈ R di×2 and b i ∈ R di , and where d i is the number of edges (hyperplanes) composing polytope i. Let n = i d i be the total number of hyperplanes. We leverage each obstacle's H-representation to construct a hyperplane arrangement of the environment as shown in fig.
.e. a partitioning of the space into polytopes. More specifically, each location in space belongs to a polytope j for which we can write an H-representation of the form where α j i ∈ {−1, 1} di is a vector specific to polytope j and obstacle i corresponding to the relative position of any point in the set with respect to each hyperplane in O i .
Fig. : Intent inference model in a hyperplane arrangement of the obstacle free space. We spatially decompose the preference θ into a set of preferred neighboring polytopes per region of the space. Within each polytope j, the human preference pj is a discrete distribution over the preferred neighbor in N (j).
We assume that for a location st belonging to polytope j, and given goal g and preference pj, the observation ot and any other preference p i,i =j are conditionally independent. Concatenating elements from each obstacle's Hrepresentation, we can write polytope j's H-representation as where Some of the constraints in eq. ( ) (corresponding to rows of A, b and α j ) are redundant, i.e. the set P j does not change upon their removal.
We can further reduce the Hrepresentation of a polytope to include only non-redundant constraints. By removing the rows corresponding to redundant constraints, we obtain new matrices A j e , b j e and α j e such that we can write the polytope's reduced H-representation as The non-redundant constraints correspond to edges of the polytope.
In other words, as the robot continually moves in space, the first hyperplane that it will cross upon exiting the polytope will correspond to one of the polytope's nonredundant constraints. Vincent and Schwager outline an iterative method for removing redundant constraints by solving n linear programs.
We use this method in practice for computing α j e for each polytope. We can now characterize each polytope by a vector α j e ∈ {−1, 1} n j e , where n j e ≤ n is the number of essential constraints of the polytope. The polytopes P j partition the environment into a hyperplane arrangement.

Path Preference

In this section, we provide a definition of preference θ according to a graphical representation of the environment based on the hyperplane arrangement. Under this representation, a path preference corresponds to a set of preferred transitions. In other words, for each polytope in the space, the human will have a preference to which neighboring polytope they wish to transition.
Let G := (V, E) be an undirected graph, where vertices are obstacle-free polytopes, and edges connect two adjacent polytopes. Each polytope is described by a unique vector α j as defined in eq. ( ). Two polytopes are adjacent if they share non-redundant constraints (rows in eq. ( )) corresponding to the same hyperplane (i.e. they are on opposite sides of the hyperplane).
Let N (v) be the set of neighbors of a vertex v. For each vertex, we denote p v the discrete-valued random variable describing which edge in N (v) the human intends to transition to. Using this formalism, we define a path preference as the set of preferred transitions over all nodes in the graph, Let m θ = v∈V |N (v)| be the cardinality of Θ, and m g = |Ω g | the number of possible goals.
A priori, the number of Bayesian updates required to update the belief at every iteration should be m θ × m g . Now, let us assume the conditional independence relationships described by the new problem diagram in fig. . More specifically, we introduce the assumption that conditioned on a robot location s t , the goal g, and the preference for the corresponding vertex p v in the graph, the observation o t and the preference for any other vertex are conditionally independent.
In other words, the observations the human provides can be defined conditioned only on the robot location, the goal, and the human's preference for its current vertex p v . By introducing this assumption, each update step only requires updating the joint (p v , g), reducing the number of cost computations to |N (v)| × m g .
We can notice that by introducing this assumption, we removed the direct relationship between the number of polytopes in the environment and the complexity of the Bayesian update in eq. ( ). In practice, components of θ are not mutually independent. For example, if the human preference at a vertex v 1 is
, it is unlikely that the human will also prefer p v2 = (v 2 , v 1 ) (turning back). We can improve our model by assuming a dependent relationship between preferences for adjacent edges, which does not significantly increase the complexity of the inference problem. An interesting property of our encoding is that any two paths that belong to different homotopy classes will cross different sequences of polytopes, i.e. they correspond to a different sequence of edges on G.
This can be proved by contradiction. Let us suppose that two continuous trajectories ξ 1 and ξ 2 , with the same start and end points and that do not intersect any obstacle, traverse the same regions in G in the same order. From the construction of the hyperplane arrangement, each polytope that the paths traverse through is obstacle-free.
Therefore, within each polytope, there is no obstacle in the area located in between the portions of ξ 1 and ξ 2 that belong to the region. A smooth transformation of ξ 1 into ξ 2 can be obtained by transforming each portion of ξ 1 belonging to the polytopes it intersects into the corresponding portion of ξ 2 for the same polytopes, where the extremities of the trajectory portions are connected to one another along the polytope's edges (where the same edge is crossed by both paths).
Along this transformation, the paths do not intersect any obstacle, and therefore ξ 1 and ξ 2 belong to the same homotopy class.

EXPERIMENTS

We evaluate our model on a simulated navigation task where the robot must reach a goal that is unknown a priori while respecting the path preferences indicated by a human. The robot navigates in a grid world containing obstacles. The transition model is deterministic: the robot selects an adjacent location on the grid to reach at the next time step.
The robot is also allowed to take diagonal actions. Each location s t in the map can be mapped to a vertex v t ∈ G. Therefore, the actions leading to locations mapped to different vertices correspond to edges on the graph. We note f (s t , a t ) the edge crossed by taking action a t from location s t .
The robot is given a mission time limit T max for reaching the goal. In this problem, we assume that the human selects actions to noisily minimize a cost function C g,θ , where θ is defined as per eq. ( ), corresponding to the length of the shortest path to the goal constrained by the preference (where the robot is only allowed to make transitions on G along preferred edges).
More specifically, where δ(s t , g | o t , p vt ) designates the length of the shortest path from s t to g passing by o t and constrained by preference p vt . This is a slight variant of the cost function proposed by Best and Fitch , where we add in a conditioning on the path preference. We compute costs by running the A path planning algorithm on the environment maps (grid worlds with diagonal actions) and impose preference constraints by pruning invalid transitions from the search tree.
Reward model. At each step in time, the robot receives a reward which is a sum of three components: a goal-specific reward a preference-specific reward or penalty We compute solutions to the POMDP defined in section III-B with the online solver POMCP , and with the particularity that within the rollouts, the robot does not expect to collect human inputs.
Each time a solution is computed, the robot takes an action and may receive an observation. If it does, it updates its belief distribution over the unknown problem variables and resolves the POMDP over a receding horizon.

Baselines

• Goal only. The robot solves the POMDP while ignoring the effects of path preference. Similarly to , we assume the human is taking action to minimize a goaldependent cost C g (s t , o t ) = δ(s t , g | o t ), where the conditioning on the preference is removed. We also omit the path preference's contribution to the reward R pref .
• Compliant. The robot complies with the human input, but does not take an initiative. If the user stops providing information, the robot continues in the last direction indicated for 5 time steps (conserving its momentum), then stops. • Blended. We designed an arbitration function to decide between our proposed policy (accounting for path preferences) and the user's recommendation when the robot receives inputs.
Our metric to evaluate confidence in the robot's prediction for the purpose of arbitration is the entropy of the intention distribution H(g, p i ), where p i denotes the preferred neighbor for the current region. Because our representation of the world is discrete, the arbitration is given by a step function.
Denoted by U , the action corresponding to the human's input, and P , the robot's prediction for the optimal action, we write the policy where we chose h = 1.6 as the confidence threshold.

Results

When evaluating the algorithm, we consider that a run is successful if the robot reached the goal within its allocated mission time T max and only made transitions between graph vertices corresponding to the human's preferences. We vary the time delay between human inputs, from constant guidance (∆ T = 1) to only a single observation (∆ T ≥ T max ).
Success rates. Table I reports the success rates for experiments conducted over six randomly sampled problem instances and 50 runs per instance in Map 1 (fig. ). When the human provides inputs at every iteration, the compliant policy shows the highest success rates. However, as ∆ T increases, the compliant robot is not able to accomplish the task within the allotted time as it does not receive sufficient inputs to do so, and performance decreases compared to the autonomous baselines.
We find that in these runs, accounting for path preference consistently improves performance compared with the goal-only baseline. Results also show that blending the user's input with the robot's policy (Path Preference + Blend) when the human provides information leads to improved performance. Belief entropy.
Figure shows a challenging problem instance where the directions the human provides do not align directly with the shortest path to the goal. By ignoring the effects of preferences in the problem model (goal only), the robot quickly infers from observations that the upper left goal is less likely than others (P (g) drops).
The strong decrease in entropy shows that the robot becomes overconfident in this prediction. Overconfidence in an incorrect goal will prevent the agent from finding the correct goal once the human's indications directly align with it, as it needs to correct for the wrong predictions, as shown in the path realization (fig.
). In this realization, the goal-only method (green robot) fails to search the upper left area within the allotted time. By accounting for path preferences in its model, the blue robot's entropy over the goal distribution decreases more steadily, allowing for it to leverage the human's latest observations and reach the goal successfully.
shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference. Computation time. In table II we provide the time required to solve the POMDP, and the time required to update the robot's belief as it receives new observations.
We compute solutions on three maps: a simple 10 × 10 grid world with 8 polytopes (fig. ), a 10 × 10 grid world with 56 polytopes (fig. ), and a 20×20 grid world with 73 polytopes (fig. ). The latter environment being larger, we increase the mission time and the depth of the search tree in POMCP from T max = 30 (Map 1 and Map 2) to T max = 60 (Map 3).
We do not notice an increase in the time required to update the robot's belief with an increase in problem complexity, which is consistent with our observation that the complexity of the Bayesian update should not increase with the number of obstacles or polytopes. On the contrary, the belief update time on Map 2 and Map 3, containing more obstacles, is reduced compared to the first map.
More obstacles result in fewer iterations when solving the constrained shortest path problem with A . Adding constraints due to the obstacles and polytopes reduces the size of the A search tree. C. Limitations Simulation environments. In our simulations, we hardcoded the preference policy over the maps (e.g. in Map 1, go around the table counter-clockwise).
We randomly sampled problem instances (start and goal locations, and goal options) to reduce the bias introduced by these preference choices. To best evaluate and compare the different approaches, it would be best to sample preferences among a distribution of preferences chosen by a human (for example, from benchmarks resulting from a collection of data).
Creating such a benchmark is an interesting direction for future work. Hyperplane arrangement construction. The main limitation of our approach is that the size and geometry of each polytope depends strongly on the geometry of the obstacles, as seen in fig. . Because of this, the robot can make predictions over preferences that are too refined compared with the topology of the environment.
A direct consequence is that when the size of the polytopes is small, the information provided by the human can be incorrectly interpreted as a preference on the robot's immediate action. Our method can be improved by changing the structure of the hyperplane arrangement so that it relies on the topology of the environment, but does not vary strongly with the geometry of the features in the environment.
For this purpose, topometric maps and region construction algorithms are promising directions. We presented an approach for encoding and inferring a human's path preference in an environment with obstacles. By leveraging a partitioning of the space into polytopes and a stochastic observation model, our method allows for joint inference over the goal and path preference even when both are unknown a-priori.
Our experiments on an unknown-goal navigation problem with sparse human interventions demonstrate the effectiveness of our approach and its suitability for online applications. The time required to update the robot's belief does not increase with the complexity of the environment, which further highlights the practicality of our method.

Answer: The belief entropy decreases more steadily.