Modelling slalom handle path as pendulum

[H2OkieNC]

I watched the video on swing arc by @JoelHowley and was inspired to model the handle path as a pendulum being pulled at constant boat speed. I saw the GUT that @Horton posted, but it didn't look like it went this far. Anybody else out there done this, or something similar? I graphed the results and will try to attach screenshot of the model at 34 mph 35 off. For now I assumed the skier just got the handle to 11.5 m width at the peak of the swing arc but this could easily be changed. This may be pretty academic and not super practical, but an ambitious vision would be to use it to calculate the "optimal" path in terms of minimizing speed or acceleration or rope tension.

[Than_Bogan]

As a Certified Math Geek, I have spent some time on this model. I'd be happy to share the "progress" that I made. My main conclusions, however, were: 1) The math to compute an optimal path is INCREDIBLY difficult, resulting in hideous differential equations that likely can only be numerically approximated rather than actually solved. 2) That assumes one can figure out what the true constraints are and (much harder) what parameter is desired to be optimized.* 3) The effort required FAR exceeded the value, even counting the math itself as entertainment value.

 

*And there's waaay more choices there than you may at first realize. Even supposing you pick "rope tension," you have to much more specific, because we're talking about optimizing an entire path. Do you minimize the average rope tension (which theoretically could include an instantaneous burst that is massive) or minimize the maximum rope tension (which theoretically could mean holding near to the max tension for a very high fraction of the time) or do you invent some hybrid that tries to trade off both factors, such as an Ln norm, and if so what n do you pick and why? (The answer "I found a value of n that let me actually solve the differential equation" would be a compelling one. :) )

 

All that said, I did stumble across something very strange that I think any true math geek has to find mildly interesting:

For one exact rope length, which iirc is pretty close to 14.25m, it is theoretically possible to run the slalom course in such a way that the magnitude of the acceleration at the handle is a constant the entire time. (I.e. it only changes in direction in order to achieve a path that meets the constraints.)

 

Nerd Out!

[Obrienslalom]

This is great. Thanks for posting both the results and the modelling.

 

I started something like this in SVG/javascript with the idea of playing around with the same concepts. I didn't get very far with it. I remember working with a 2 force vectors: the rope & the force the ski/skiier was creating (very contrived respresentation). The results looked ok on a graph, but I wasn't completely confident with my math and got lazy. I'll see if I can find where I put the code.

 

I encourage you to keep going. I think it's fun to play with and visualize. Happy to help contribute as well.

[Than_Bogan]

There was one maybe-sorta useful observation that came out of this exercise, although perhaps the real credit for pointing this out goes to Adam Caldwell.

 

When you have a very short rope, width is very deceptive. Instead you need to think about "height" -- i.e. swinging up on the boat.

 

The reason is as the handle swings through from say 70 degrees to 90 degrees, its width actually doesn't change all that much. sin(70) ~= 0.94 meaning that you've already achieved 94% of the maximum possible width when you could still travel 20 degrees "higher" on the boat.

[Zman]

Gotta love a mathematician!

B)

 

[rawly]

You guys must be lost , this is wood shop , the science wing is across campus , way over there.

[skispray]

@Than_Bogan when can we expect a new installment in the GUT series? Would love to see this stuff laid out for a geeky audience to read.

[Bruce_Butterfield]

In addition to estimating the line tension, you would also need to model momentum and drag. You may be able to get a reasonable guess on momentum, but drag varies with the square of speed AND the angle and attitude of the ski.

 

In essence, you need a very complicated free body diagram of all the forces on the handle and how they change with time/position in the course and what the skier does. That should keep an army of geeks busy for the next hundred years or so.

[Than_Bogan]

@skispray Years ago, I started to write it up, but it's a huge pain to make much sense out of it, and the presentation needs to change drastically depending on the reader's background. Adding in the fact that it's incomplete and almost entirely useless, I lost motivation.

 

But I'd be willing to try to share with individuals if you want to contact me and let me know specifically what you're interested in and what your math background is.

 

(Well, assuming I can find all my notes. Once I realized how useless it was, I got a bit less careful about keeping track of it...)

[Than_Bogan]

@Bruce_Butterfield That level of detail is likely impossible. Certainly for me. I was trying to do something FAR simpler ... and failed.

[ToddL]

All of this is modeled on pure glass calm water, right? I've seen pictures of that stuff. LOL.

[AdamCord]

I tried to do a similar model a few years ago. Best I could come up with is that you COULD model this using pendulum math IF you kept gravity in the equation as variable. What's called "gravity" in normal pendulum math is actually caused by wind and the water moving under the skier. Of course this force will vary massively depending on the roll, pitch, and yaw of the ski, as well as the speed of the ski relative to the water and air. You basically have to guess what the gravity is at each point in the skier's swing around the pylon. I really don't see any way to accurately model the gravity variable, although you could potentially deduce what it is at some points in the course using line tension data? That would at least give you some sort of numbers to start with.

 

What I like about @H2OkieNC 's' model above is that it does a good job of showing how much more the skier needs to move DOWN the lake after centerline than across it, which is a key tenant of what we've been working on with GUT.

[rfa]

Both my math and slalom abilities, relative to @Than_Bogan's are pretty similar...way lacking! On a more positive note, as a 32off skier and a post-grad engineering degree (too many years ago) I am able to understand and enjoy the general concepts in these academic discussions.

So, from own experience it is pretty easy to "see" that, at least through 32off, can be accomplished with various ski paths. From observation, I suspect that 35/38 also can allow some variability in ski path (although less than the longer line lengths). When seeing 39/41 skiers, it is less obvious to my naked eyes, whether the ski paths are truly different despite different skier styles. This reminds me of a long ago conversation with Jamie B. where he explained that running 41 required him to stay within a 2 inch wide path (think of a 2" hose, he said).

If these observations are correct, it implies that the modelling efforts described here could potentially help those of us less accomplished skiers, but less so, real short line skiers. On the other hand, we (the less accomplished) do not really need sophisticated modelling (or any modelling) to "optimize" our ski paths because it is pretty obvious when we do it "right" (or at least better).

My curiosity then is whether those of you who have looked in some detail at modelling ski paths, and/or actually run 39/41, are able to confirm (or suggest) that 39/41 can only be run as per Jamie B. "2in. hose". Because if the answer is yes, these modelling efforts, however interesting, have only academic value, since both ends of the line length spectrum cannot "learn" much from it.

But as a "mini-geek" (compared to @Than_Bogan - ha) I enjoy this stuff, so keep it up!

 

 

 

[Than_Bogan]

@rfa If you want to talk path, the first thing I need to point out is that "path" isn't quite an adequate description. Typically one thinks of a path as being a single object that has no time element. What we want here is what could be called a "parametric path." Specifically x and y are both functions of time. We want to know where the skier is at each moment, not just the shape of the curve drawn. This is necessary if we desire to estimate velocity, acceleration, and rope tension for a given "path."

 

There is little doubt in my mind that Jamie was literally incorrect: Many (parameteric) paths are possible for completing -41. However, I think it is very likely that small deviations from the optimal (whatever that is!!) become harder much more quickly than deviations from the optimal on a longer line length.

 

Moreover, each skier is ultimately up against his or her limits. When I ski 34/-28, I purposely do NOT take the easiest path. I (try to) take the path that gives me the largest margin for error. That path requires substantially more effort. I would absolutely love to take a high-margin-for-error path through 34/-38, too, but I cannot. I have insufficient ability. Instead I am forced to (try to) take the easiest path, because I am barely capable of sustaining said easiest path. That comes with a much lesser margin for error. Thus my fun is typically ended by my first mistake.

 

The same is likely true for Jamie, except for him that point is 36/-41.

[Than_Bogan]

@rfa Also, I am quite convinced that there is almost nothing at all to learn from this, at least if we mean "learn to round more buoys." I certainly learned some stuff, such as the generalization of derivative minimization for optimizing the function itself, which is called the Euler-Lagrange Equation. It's a logical extension of the notion that the minima and maxima of a function occur where the derivative is zero, but I was not previously aware that the concept had been generalized to identifying a function that mini/maximizes some criteria.

[rfa]

Thank you @Than_Bogan! "keep it up" was not a literal request to get right back to me with detailed feedback. BUT, much appreciated!

Yes, while I used "ski path" in my post, I was referring to the handle path (which, as modeled by @H2OkieNC, includes x, y, t. Of course, handle vs. ski path are two different things and ski path is what ultimately matters (and makes the "math" so much more challenging).

I think you confirmed my lighthearted general comment that "these modelling efforts, however interesting, have only academic value, since both ends of the line length spectrum cannot 'learn' much from it".

Sorry I didn't see you this summer...but saw a couple of our friends post great scores on C-75s. Nice!

[RGilmore]

Maybe I'm just dumb, but given that the handle path - relative to the pylon - will always be an arc. IMO the only other "variable" required is the rope's angle (relative to centerline) at each turn's apex.

 

The boat speed and elapsed time between apex points are known constants. So the only unknown is the rope's angle when those apex points are reached. But those points will vary greatly between skiers. For example, Chris Rossi, on a 13m line, typically apexes quite wide of the buoy-line, and relatively early (up-course) of the ball. Meanwhile, a skier just learning 13m might be barely getting his ski around the ball at apex. Those two handle paths with be very different, but so will the rope angle-at-apex - which I think primarily defines what each of those handle paths will be.

[Martin1980]

Check out this old thread @H2OkieNC https://www.ballofspray.com/forum#/discussion/4206/explain-to-me-how-a-shorter-rope-is-more-difficult and especially read the posts by @dn .

[Than_Bogan]

@RGilmore That is not dumb, but you've overlooked something important. The only constraint you've identified is that the value of theta (the rope displacement angle) has to get the rope handle to the buoy at a certain moment in time. That is a super-important constraint, but there are still an infinite number of theta-vs.-time functions that can meet that constraint, and each will draw a different path (both in shape and in time).

 

One such function the small-angle-approximation for a pendulum, of the form theta = ksin(at), where k and a are (in this case) dictated by the rope length, boat speed, and course geometry.

 

But there is no reason to believe that that function is what anyone desires to do when running a slalom course. (Or at least I have no reason to believe that.)

[Than_Bogan]

@dn's work is super-interesting, but, like any exploration, should not be taken as gospel.

There is some disagreement within the Slalom&Math Nerd community regarding some of @dn's conclusions.

[Martin1980]

@Than_Bogan what conclusions does @dn make that you do not agree with?

[Than_Bogan]

I don't have the energy to try to conduct an academic debate in a public forum. Especially since none of this helps anyone to run more buoys anyhow!

[Martin1980]

@Than_Bogan understandable but I do think however that these debates can help some of us run more buoys. Being new to BOS and reading a lot of old posts my mind has been enlightened in regards to how things change as the line shortens. Before reading BOS I thought that the overall speed increased as the line shortens but can now see that the overall speed is very likely LESS as the line shortens, the max speed behind the boat is what increases. I also now see that as the line shortens the effective work zone to get to the same buoy width is less which tells me I have to work harder for less time as the line shortens. Both of these things can potentially help me run more buoys. :)

[Hallpass]

I'm just impressed that there are people out there who remember all this stuff more than 5 minutes after leaving their college final.

[Than_Bogan]

If it helps you run more buoys, do you care if it's true? :) I can't count the number of times I've coached someone (in multiple sports) along the lines of "you won't literally do this, but you need to TRY to do it in order to achieve what you want."

[Than_Bogan]

@Hallpass Don't forget that almost all math was created to solve problems, not to torture students. That was just a bonus side effect!! :)

[RGilmore]

@Than_Bogan I don't want to be disagreeable, but I disagree. You interpreted my "formula" as suggesting the skier "... has to get the rope handle to the buoy at a certain moment in time.", but that is exactly opposite of what I proposed - or meant to propose.

 

I think we are saying the same thing in different terms; my point is that the apex of the skier's turn does not necessarily occur at a "certain moment in time", but will actually vary [in time] from skier to skier. So Rossi's apex will occur at a very different moment in time than my apex (for example) and therefore the rope angle when I reach apex will be quite different from when Rossi reaches apex [assumes same speed and rope length]. The end result is that our handles describe notably different paths because our turns reached apex a notably different points, both in space and in time.

[Than_Bogan]

@RGilmore I still think I read it right, but naturally I may have just misinterpreted the same way twice... The rope angle that must be achieved to round the ball is set by the length of the rope and the skier's reach. That constrains the path. But it doesn't fully constrain the path. There are many (parametric) paths that are theoretically possible that still reach the required rope angle at the buoy.

[bsmith]

Wouldn't any model for handle path have to also account for the possibility of small amounts of slack line as the skier rounds a buoy? Or at 41 off, is there never any slack to the handle on a pass that is run complete?

[Than_Bogan]

@bsmith My experience in engineering has strongly taught me: Always begin with the simplest possible model and see if you can make anything out of that. Only add in the next level of complexity when you have a working version of the simplest model.

 

For the specific case of slack, however, I feel fairly confident that the optimal path does not include any slack, so for the specific goal of trying to figure out something regarding the optimal path, I think slack can actually be ignored forever.

 

But if you want to know what is actually happening to actual skiers ... well then measure it! It's usually silly to try to measure from a model what you could instead measure directly in the real world!

[Deke]

:) Just when I thought the whole idea of GUT was finally simplifying everything I previously thought about slalom this thread has to come along... Stop already!!!

[Than_Bogan]

@Horton That reminds it that it would super-cool to have a "mark this thread as read forever" button, to effectively ignore a topic that is refusing to die but that you have no interest in. Happens to me all the time. And also I cause it to happen to others...

[RGilmore]

@Than_Bogan Many years ago Steve Schnitzer wrote an article about what it takes to ski through 39-off. One of the things he advocated was to ski just enough to get your ankles around the ball, no matter how long or short the line. If you ever watched him ski you'd know he absolutely practiced what he preached.

 

However, many top skiers will do the opposite, skiing wide and early around each ball at their longer lines - Rossi and the late GOAT Andy Mapple being classic examples. Without any question, the rope angles - and therefore the serpentine handle path through the course - must be quite different between the two approaches (wide-and-early vs ball-to-ball). So, IMO, it's going to be very hard to math out the handle path in any meaningful way without absolutely taking into account the specific skiing style (and therefore, rope angles) of any given skier. That was my point.