This will be a tad hard to explain, so bear with me. Taking into account only the legs what would be an accurate definition of the position of the upper legs, lower legs and feet with respect to time?

Let's say we have a person running on the spot (Not moving anywhere), from the reference point (Say the spot they're running on, or perhaps their waist) how can we define the position and angle of each of the upper legs, lower legs and feet with respect to time (let's say 0 to 1 for a single cycle of running)

I'm sure there's plenty of documentation on the subject but I'm unable to get anything useful from google.

The purpose is computer animation in an environment that doesn't support key-framing

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    $\begingroup$ I did in fact read the FAQ and my question didn't seem to be in violation of any part of it, I would imagine the dynamics of running are indeed of interest to mathematicians. In addition it would appear that you yourself have not read the FAQ or at least glanced over the 'Be Nice' category as evidenced by your snide and unhelpful comments. $\endgroup$ – user2418 Dec 8 '09 at 22:33
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    $\begingroup$ This question may be a little off the beaten track for this site, but it's an interesting question with a mathematical component. fpqc, I have known algebraic geometers who would think about related issues (related to motion of robotic arms). Perhaps nobody here knows the answer. That's no cause to dismiss the question. $\endgroup$ – moonface Dec 8 '09 at 22:56
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    $\begingroup$ fpqc I'd encourage you to step back a bit and think about whether you're contributing positively with the way you're trying to enforce your ideas of the rules on the site. Thanks! $\endgroup$ – Noah Snyder Dec 8 '09 at 23:23
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    $\begingroup$ @fpqc: I'd encourage you to carefully consider Noah's suggestion. I'm sure you have noticed that every comment reprehending you for being impolite is voted up. Also somebody systematically downvoted all of your answers and questions just a few days back. I'd take a hint from all of this about what people think of your behavior. $\endgroup$ – Alberto García-Raboso Dec 8 '09 at 23:45
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    $\begingroup$ Here's why I modded this down: I work with people whose job is to do things like this. Technical animators basically make up mathematical expressions that look right by combining standard functions, eg. cycles from trig functions and various families of "s-curves" from [0,1]->[0,1] to warp the shape of the functions. It's an ad hoc and primarily artistic skill. (With rare exceptions) there's very little intrinsic mathematical interest. Correctness of a solution is judged by how nice the animation looks. And answering such questions opens the floodgates to a many other questions just like this. $\endgroup$ – Dan Piponi Dec 9 '09 at 0:30

This is probably not directly relevant, but there's a really delightful webpage made by Dror Bar Natan's masters student Dori Eldar about linkages. There's a lot of stuff about the topology of linkages (if I remember correctly, any manifold can be realized as the configuration space of a linkage). The positions of the lower legs (two thighs, two shins, two feet, joined at hips, knees, and ankles) is some nice closed manifoled. You want to then look at some smaller space inside there (for example, there's a limit to how much each joint can move). Physics is going to give you some flow on this space (e.g. the affects of gravity) and there's a choice of movement away from that flow given by moving muscles. Looking now as a subset of space-time you see there's a manifold that's the configuration space of all possible gaits. And you then want to look at the "how fast do you actually move" function on that space and hope that there's a local maximum that looks like running...

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    $\begingroup$ @Whoever downvoted Noah's answer: when downvoting you should leave a comment explaining your reasons. It really helps all of us in creating a better MO. $\endgroup$ – Alberto García-Raboso Dec 8 '09 at 23:49
  • $\begingroup$ @Noah Yes, you do remember that correctly. canyon23.net/math/1985thesis.pdf (found Kevin Walker's thesis via the E.A.T. book of Robert Ghrist. Dr Ghrist has done some work on robots not bumping into each other [braid theory] which may be what you had in mind?) $\endgroup$ – isomorphismes Sep 30 '14 at 0:51

For what you want you need to be looking at medical journals.

I would recommend anything from the Journal of Biomechanics, for example this paper on Procrustes Motion Analysis (PMA):

Abstract: The ability to quantify and compare the movements of organisms is a central focus of many studies in biology, anthropology, biomechanics, and ergonomics. However, while the importance of functional motion analysis has long been acknowledged, quantitative methods for identifying differences in motion have not been widely developed. In this article, we present an approach to the functional analysis of motion and quantification of motion types. Our approach, Procrustes Motion Analysis (PMA) can be used to distinguish differences in cyclical, repeated, or goal-directed motions. PMA exploits the fact that any motion can be represented by an ordered sequence of postures exhibited throughout the course of a motion. Changes in posture from time step to time step form a trajectory through a multivariate data space, representing a specific motion. By evaluating the size, shape, and orientation of these motion trajectories, it is possible to examine variation in motion type within and among groups or even with respect to continuous variables. This represents a significant analytical advance over current approaches. Using simulated and digitized data representing cyclical, repeated and goal-directed motions, we show that PMA correctly identifies distinct motion tasks in these data sets.

  • $\begingroup$ This answer is not math-related at all, again demonstrating how bad this topic actually is. I'm not downvoting, just commenting. $\endgroup$ – Harry Gindi Dec 9 '09 at 17:59

There are many mathematical perspectives one could take on running, many of them I think are more interesting than the narrow question you posed. (Since it's a graphics question another SX site might have been better.)

(The second one is more about swimming than running.)

You might want to (or not) think about putting a sensor on each knee, each foot, each toe, etc., and consider the paths traced out by each sensor. You could use the language of diffeomorphisms and elastic deformations to talk about "small" (or large) deviations. You could also invoke some functional analysis to be a bit more specific about how the paths can deform.

There are a lot of other perspectives you could take--like what about the forces that come up through the heels/metatarsals/toes and travel through both bone and soft matter? Or, finally getting back to what you brought up: the Lie algebra of parameter space which all the angles of the joints. There you're interested in questions that might be answered—or perhaps they'll lead you toward new questions instead—in an introductory differential-geometry or algebraic-topology text. (Spivak DG v1 or Hatcher AT will do.)

But really what you want, I think, are some practical measurements—science derived from kinesiology—rather than pure-mathematics stuff. Ball-and-socket joints move in like a deformed disk; elbows and knees allow motion in a unit interval; and all of this is tied together with a product that's more complicated than Cartesian (you can't put your hand through your chest, for example). Sort of boring, mathematically; that's the stuff I mentioned above that's covered in an introductory DG or AT text. The more relevant information for you, maybe, will be in empirical/scientific specifics of real bodies.


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