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Bipedal spring mass walking sagittal plane theory

Discussion in 'Biomechanics, Sports and Foot orthoses' started by Simon Spooner, Oct 7, 2010.

  1. David Smith

    David Smith Well-Known Member

    Mike

    I don't have your email address to send the paper to. If you send an email to david@foothouse.co.uk I can send it by return

    I'll have to spend a while looking at your equations :eek:

    Dvae
     
  2. m.mouck

    m.mouck Active Member

    I'll do that as soon as possible, Dave, thanks.

    Also, I found an error on Figure 2. In the calculation for the (x(1),y(1).

    It should be x(1)=LLcosA, not LLsinA; and it should be y(1)=LLsinA, not LLcosA.

    Sorry about that. I wish there was a way to change the Figure 2. It`s this type of error in algebra and trigonometry that we really have to be on the lookout for.

    And, keeping track of signs in these equations can be a bit difficult.


    Mike M
     
  3. Dave or Simon, while Mike and Dave are going at it with number and formula that I will never get and they find interesting. How is all this going to help with clinical thinking around Kleg if for walking or running.

    Not trying to say anything negative against the discussion and will continue to read , but more asking how is going to help clinically.
     
  4. Geyer's PhD thesis can be downloaded here:
    http://www.cs.cmu.edu/~hgeyer/Publications/Geyer05PhDThesis.pdf

    Mike W., I don't follow the math either. As far as I can follow, Mike M. and Dave are trying to work out the required input characteristics to the bipedal spring mass model to recreate flat spots in the force time curve. Once we know this we will better understand how changes in Kleg and other factors may result in flat spots.

    I saw a patient yesterday who is a mathematics teacher and was discussing this with him, I might ask him to take a look over the maths that Mike M. is developing when I see him next week. And maybe he'll translate it into English for me.;)

    But let's ask Dave and Mike M.. Dave, Mike (without any more equations for now please, can You keep Mike W. and I onboard here by giving us thickies at the back an update in clinical terms on what you are doing, I've kind of got lost since I went abroad teaching. Since we started filling the pages with equations are we any further forward at working out the cause in lower limb kinematics terms for flat spots in the force/ time curve? Are are we re-inventing the wheel here? If Geyer produced a bipedal spring mass model for human locomotion, why are you not just using his model and plugging in numbers? Why do you seem to be trying to add to Geyer's model by including things like wind and back-pack forces? Surely it would be easier to keep it as simple as possible at the moment, since we are talking about data which is usually collected in a gait lab, why are we interested in wind and back-packs? How is this model accounting for translation of the centre of pressure along the foot during the contact phase? How does this relate clinically to flat-spots in force/ time curves?
     
  5. Sorry to go back several posts, but I've been away. One comment: shouldn't the rear leg be shortening during this period, not lengthening as the diagram figure 1 shows, since this is what happens during normal walking at this point in the gait cycle via increased knee flexion? I know Geyer's diagram shows a lengthening here too, but does this fit with in-vivo observation?
     
  6. m.mouck

    m.mouck Active Member

    Hi guys,

    Thanks very much for the link to Geyer's thesis, Simon, I had no trouble getting it from there.

    I can answer all your questions regarding this work, but first I'd like to go over a few other papers that I've found, to see if there's anything useful.

    As far as the figure, Simon, thank you for pointing out the physical reality, and this will be important when I create some "perfect" data in order to test a few things. When I get a little farther along, hopefully you'll provide some input as to realistic characteristics for COM trajectory.

    But, this is a general picture meant to show how the labels are used, not to define a specific point. I shouldn't have said this picture represented the point of contact of the contralateral leg, since it can represent any sequential time points.

    The real details about how the legs are relatively compressed at each point is something we'll measure, and will be shown in the variations in fractional leg compression (M) and k(leg). I'm creating this to be able to measure any and all movement patterns, even highly irregular, so from this preliminary diagram you should just use it to see the meanings of the labels and the general relative geometry.

    I just got Geyer's thesis, and have only looked over it briefly, but on first glance, it looks like what he's doing is what all others are doing, imposing artificial regularity on the system, which is essential for his derivations. I'll explain this a little more later, and why it's not valid (in my uneducated opinion) after I've had a chance to look more carefully at his equations, in case I'm missing something.

    Also, although I want a closer look, Geyer makes a few assumptions which are highly questionable. I want to look at these more closely. One, in particular, if it's as it appears to be, would show why his approach is not proper for this system.

    But, Geyer is not doing what I'm doing, it's only superficially similar. It's certainly not reinventing the wheel, and we can't take his equations and just plug in numbers. As far as practical application, right now it looks like Geyer's work is of little real value, in my opinion.

    And, it's necessary to keep the external force terms in these equations. Even though it appears to make the equations more complicated, it really doesn't. I think I'd go so far as to say that's it's essential to keep them in. Remember that we're adapting a mathematical model to describe a real process.

    In order to do that, we have to be able to account for deviations from mathematical regularity, as well as relate them to some relevant physical reality. The geometry of the system strictly defines the relative orientations of the force vectors, and we have to be able to account for variations from the ideal model dynamics. This is done using external force terms, but the horizontal is more important than the vertical right now, I think.

    I'll describe exactly what I'm trying to provide for you, clinically, but I'd like to look over the thesis again and a couple of other papers first.

    Believe me, I wouldn't be doing this if I didn't think it would provide you with something useful and unique. In a few days I'll be able to provide a more reasoned comment about this part.

    So far, I've seen nothing in the literature which causes me to change my approach, which is practical, real application of the sagittal biped spring model.

    I'd also like to comment on the COP path, etc., but I'll do that in the next post.


    Mike M
     
  7. m.mouck

    m.mouck Active Member

    Hi guys,

    Sorry for the delay, but I got a bit side-tracked.

    I've been looking at this term, k(leg), but I'm not done yet. I'm trying to determine how this value can be interpreted analytically, with respect to COMtraj, force and body segment movement patterns.

    However, right now I can comment on some of the questions presented previously.

    Generally, what I'm trying to do here is put real, dynamic numbers to the general discussion you've been having on leg stiffness, force, instantaneous changes in COMtraj, etc., but on looking at it for a while, in the next post I'd like to discuss the basic nature of that value, k(leg), as well. (In the next post I'll also discuss what's provided in the literature, which is nothing like what I'm doing, in case you mistakenly believe you already have this.)

    As far as whether the equations progress the discussion, they're essential, otherwise you could never get beyond the theoretical. But, I haven't really done anything yet. I've only listed the known general equations which are relevant, and then put in the labels from the figure. The force equation is just putting in the general equations and rearranging this. This is only the start of the process.

    As far as accounting for the horiz. translation of the COP (COPtrans) during the contact phase, that would be properly incorporated as a secondary element, after the primary equations are developed. It's essential that it's done this way, since if I included COPtrans in the basic analysis, that would bias the entire system.

    I'm trying to provide tools for investigation, so different hypotheses regarding gait can be investigated. If the COPtrans was included as an integral element of the primary equations, not only would it be defining the system based on the assumption that COPtrans is an integral element (which it very likely is, in some way), but also that it's relevance is shown via the geometric relationship described by the model. That is, that the "spring" aspect of the model is properly defined along the line connecting the COP and COM.

    While this is an attractive proposition, and it may be true, no one has proven that, to my knowledge (but, since I can't do effective background research with just the internet, it could have been).

    By leaving it out at first, COPtrans can later be superimposed as a secondary aspect (secondary doesn't mean less important), but other possibilities, like using the ankle joint for the end of the spring and using the foot as a lever, or replacing the spring element with a piston, etc., can also be investigated.

    So, I'd like to provide the means to evaluate the role of COPtrans from a number of perspectives, and allow the comparison of various other options, if desired, within the same technical framework.

    As far as how this would relate clinically to flat-spots on the vFTC, these periods result when the "leg" in single stance, or both legs in double stance, extend or contract in specific ways so that the trajectory of the COM either remains constant (in any orientation), or is changing in a constant way (constant vertical accel.).

    It must be co-ordinated variations in the force supplied by each leg that causes these patterns (obviously), and you propose that the spring model and a correlation of k(leg) (which is being equated to "leg stiffness", I believe) for each leg will illuminate this process.

    I've actually already shown the relationship that's necessary for the k(leg) for each leg during double stance, in order to produce a flat-spot, and this is shown in the bottom figure on the left in Fig 3, if you put F(v,T) and F(v,ext) both as constant (which means the numerator is constant and the denominator varies in this term).

    The problem with this equation, of course, is that these terms still have to be put in a form to make it easier to see and understand the dynamic relationships. (We could look at each term in this equation right now, but let's leave that for now.)

    So, in order to understand flat-spots, you have to produce the equations, and then work with them to understand the terms. Without the equations, all you have is supposition and theory, and you could never have any more.

    But it's also critical that the values and methods involved are relevant and valid, and that they can be unambiguously interpreted.

    Ambiguity is, without a doubt and by far, the most significant problem in current kinematic gait methods (ie. interpretation and some data collection). And, the problem is virtually always the use of inappropriate reference systems, which lead to numbers which can't be specifically interpreted.

    A lot of work and money goes in to the development of analytical principles and methods. In order to be sure this isn't wasted, it's essential to look at the terms involved and define how those terms can be related to the real, physical system.

    Taking this term, k(leg), out of its "natural habitat" and applying it verbatim to gait has very specific technical consequences which have to be understood and defined. I'm sure there must have already been a discussion of this, but since I haven't seen one yet, I'll do it.

    So, I'd like to do a more detailed analysis of this term and it's "validity" and potential utility, in my uneducated opinion, with respect to the proposed applications. Right now, I've got a few bones to pick with that term which I'll discuss in the next post (a week or two), but I want to read a bit more first.


    Mike M
     
  8. m.mouck

    m.mouck Active Member

    Hi guys,

    I'd like to discuss the applicability of the spring model to the measurement of dynamic (instantaneous) gait in the sagittal plane. This isn't a discussion about the relevance of "limb stiffness", but rather how this can be interpreted using the spring model.

    One important step is to look at the model, the spring, and determine the consequences of applying the exact equation to the motions of the COM.

    For a spring, k is a constant, and it's a physical property of the spring, described as resistance to compression, or stiffness. So, the force is a function of a single variable (or vice versa), and spring force and COM displacement (COMdispl) (along the spring axis) show a linear relationship.

    In real gait, k(leg) is a variable, and is a function of 2 variables, force and COMtraj, which are not linearly related. Usually k(leg) is converted to a "constant", like the value at max. displacement, or the slope from the force/displacement graph.

    So, k has been changed to a variable and then artificially redefined as a constant, k(leg).

    The next step is to look at the consequence of k(leg) being a function of 2 variables, instead of a constant.

    k(leg) changes due to 2 specific physical processes, resistance to applied force and limb extension (ie. changes in COMtraj). I haven't finished looking at all the papers yet, but it looks like no one tries to distinguish the two, although it's clearly recognised.

    If we freeze a person with a COMdispl of 3cm down, the k(leg) would increase if there was extra force but the COM remained stationary (like if it started at a different angle), and it would increase if the person extended the limb (at any rate) so the COM is at 1cm down, for eg.

    So an increase in k(leg) can come about from 2 different conditions which are not linearly related.

    This is where opinion comes in, and it's yours that's important.

    It seems to me it's necessary to distinguish these. Resisting force and extending the "limb" are distinct processes, aren't they? Do you guys see it that way? It's up to you to decide whether you can tolerate these 2 things being mixed in one term, in undefined proportion (when you don't necessarily have to have it that way).

    Before you decide, though, let's look at how this affects how k(leg) changes. Since this is dynamic measurement, we have to use instantaneous values.

    The primary goal is to find a way to correlate COMdispl (and general trajectory), force data and "limb stiffness", described by the term, k(leg), as well as other gait parameters, like velocity. Unfortunately, in gait, since COMdispl and the vFTC are not linearly related, k(leg) is actually a proportionality variable, whereas in the spring, k is a proportionality constant.

    I think it's sufficient to look at a single condition to show the problem.

    We know there could be many different COMtraj associated with a flat-spot on the vFTC. This flat-spot means constant vertical force (up or down), which means constant vertical accel. (which might be zero). This would also show as a horizontal line on the vertical force(vF)/COMdispl graph, which would mean k(leg)=0 when the slope is used.

    So, you see why you can't use the slope of the vF/COMdispl curve to define dynamic k(leg). The mass could be accelerating up, the instantaneous k(leg) would be increasing due to decreased COMdispl, but k(leg) from the slope would be 0 whenever there is constant force. This is a significant problem.

    This conflict arises because, again, force is not a linear function of displacement and k(leg) is a variable, not a constant. This is the main problem. The spring equation is being taken out of context, and then treated as if it were still in that context. It simply isn't the same physical system, and you can't verbatim use the same equations and analytical processes.

    COMtraj may resemble periodic spring motion, but the equation isn't really relevant for dynamic gait, in my opinion. But, keep in mind that the over-riding desire here is to produce useful and valid tools for investigation, and not necessarily to precisely define gait.

    In the literature, they try to get around problems by taking the values at a single point, ie. force at max. COMdispl, using the slope of the vF/COMdispl curve, and/or constraining the COMtraj. This is an attempt to impose linearity (more generally, regularity).

    And, it may be useful to do that. Some papers suggest that applying the spring model has helped, like for designing shoes and running, I think, and if this is the case, that's good. It may well have applications in robotics, etc. Definitely, use it for everything you can. Flog it till you can't get anything else out of it. Just don't over-generalise or over-interpret that process. This leads to ambiguity, even more than already exists due to the basic nature of the system.

    Frankly, I don't know how you could come to any valid or consistent conclusions with that system. But, the pseudo-regularity and pseudo-symmetry of gait complicate the evaluation of validity, since under similar conditions, there will likely be similar results for the same or different subjects. This can very easily be mistaken for "validity", but it certainly isn't.

    For dynamic gait, in my opinion, the ambiguity in k(leg) can't be tolerated. It's just not a proper model, by which I mean k(leg) (in its current form) isn't a suitable term on which to base a secondary analysis related to dynamic limb stiffness, etc.

    Does this make sense? Do you see what I mean regarding the ambiguity in k(leg)? When you compare 2 values of k(leg), or look at the graphs, how useful is the system if you have to guess why they differ (was it limb extension, a change in force pattern, or some combination).

    Of course, you could look at the vF and COMdispl, but then why do you need k(leg), since it's nothing but a proportionality variable for 2 values which are not linearly related. Having k(leg) provides no advantage for dynamic measurement, since it has to be correlated to both input variables in order to be interpreted.

    Again, I'm not saying the "spring" idea is useless. The above comments relate specifically to technical relevance for dynamic (instantaneous) gait measurement and is not an evaluation or judgement of the potential utility of the model for other specific applications (or this one in a different form or way).

    And, of course, it's only my opinion, based on what I've seen so far.

    I'll wait for comments on the above discussion, then I have a couple of other comments.


    Mike M
     
  9. m.mouck

    m.mouck Active Member

    Hi guys,

    I hope I haven't lost everyone.

    In the last post, do you see what I'm saying about k(leg)? that it's a useless term since you have to look at COMdispl and vF anyway, and the term itself doesn't correlate to anything known (because force and COMdispl are not linearly related).....Agree.....Disagree.....?

    Just to round out the discussion, it's fair to question what (if anything) can be done to apply the spring model in a valid way. As a secondary analysis it can be applied at any time, and is valid now, assuming there is already a primary analysis. To apply it independently, as a primary analysis, this requires an understanding of the nature of k(leg), and how it relates the 2 terms, vF and COMdispl.

    Right now I don't know how k(leg) relates the 2 terms, but this would have to be defined, in order to be able to evaluate k(leg) independently of the input variables. Basically, what you need is a term or equation which shows how the instantaneous value of k(leg) relates to the slope of the COMdispl(x) vs vF(y) graph.

    Instantaneous k(leg) and "slope" k(leg) are 2 different values, but they must be related somehow, and it's only by defining this relationship (in a suitable way) that the spring model could be applied as a primary analysis method, upon which secondary analyses could be based.

    This isn't as simple as it may sound. This term or equation would have to incorporate aspects of the COMtraj as well as at least the first derivative of the force (I think). Since COMtraj is involved, there would have to be at least one angle, so you have to introduce at least one other variable.

    k(leg) appears to be unsuitable as the basis for secondary analyses, but that doesn't mean it can't be incorporated as a secondary analysis. The spring idea is being applied in other areas, and it would be valuable to be able to accommodate it in any sagittal measurement system.

    In order to do this, once the primary method is defined, there would have to be a correlation to the spring system. Although I haven't even looked at anything yet, I guarantee this can be done, since it's just a matter of geometry.

    I'm not saying abandon the spring idea, just put it in its proper place, as a secondary analysis. The procedure can't be to define gait as a spring (which is an interpretation that may not be true, and by my observations isn't), but rather to define gait so you can see when the "limb" is acting like a spring.

    Another thing that comes to mind is in the evaluation of validity. I always refer to technical validity, but I'm not sure how others view that term. In my analysis, k(leg) is not a technically valid term.

    In gait analysis, one of the most severe examples of a technically invalid kinematic measure in common use is when foot angle (eg. right) is measured using the "line of progression", when it's defined as the line connecting sequential contact points of the other foot (left) (ie. the stride vector). A description of why this term is not valid may illuminate why I say k(leg) is not valid, if you don't see it yet.

    But, if no one is interested, I won't bore you with it.

    I'd like to ask for some comments on the k(leg) discussion. Do you see why I say it's not suitable as a primary analysis?

    Moving forward, I believe the over-riding goal(s) was to develop a method to correlate force patterns, COMdispl (and traj), "limb stiffness", velocity, etc. And, it was suggested that the spring model (specifically k(leg)) could be the basis on which to build that method. I hope no one's going to curl up and start crying like a little girl just because the first idea may not have worked out like you hoped it would.

    First, you have to decide whether you agree with my analysis of k(leg), and how it's unsuitable for the proposed application, being a useless term in its present form. It would be better to just look at vF and COMdispl as individuals, since that's what you have to ultimately do anyway, in order to interpret k(leg).

    Second, you have to decide whether you want to keep using k(leg) anyway, just give up on developing a model, or continue with the development by considering other ways to define the sagittal system, and other ways to incorporate the spring idea (if desired).

    You can see how analytical method development involves subtleties which are difficult to see and define, if you're not used to it. Judging by your reaction to the previous figures and equations I posted, I'm fairly sure no one would be interested in looking at and evaluating the technical aspects of other models (which means more equations). And that's understandable.

    Fortunately, I have a bit of time right now, and would be willing to work on something for you. I believe I understand the things you'd like to be able to do within the paradigm, and have a few ideas. This is what separates the real researchers from the "fly by night". Do you have the fortitude to continue?, or are you going to give up.

    My fear is that I would do it, but then no one would be willing to discuss it. Then, I'd be left with another gait analysis method which no one even tries to understand.

    I can take care of all the technical development, but when it comes to application, that's were there has to be input from you. And, application is the priority.

    So, I'd like to ask outright, if I undertake this project, are there a few of you who would be willing to provide input when I post something? This would mainly be in the form of specific questions, usually regarding details of your work, like what you have available and how you use different methods, as well as how you view things from your perspective.

    If there are any takers, I've got a few more general comments.

    As far as I know, you may have the feeling that it's the spring or nothing, or you may not even believe what I'm saying about k(leg). So, if no one is interested, I won't do it.


    Mike M
     
  10. Mike M

    You seem to want to reinvent the Wheel - Kleg - leg stiffness is already a legitimate module for looking at running gait.

    What is now being looked at is how this relates to injury - Read this thread and the research papers to get an idea - Leg Stiffness thread

    As for this comment, Not going to cry you assume a lot ;)

    Kleg has already been discussed and accepted in peer review journals.
     
  11. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ........:morning:
     
  12. m.mouck

    m.mouck Active Member

    m weber

    Thank you for your comments.

    It's not a matter of reinventing the wheel, it's inventing the wheel. All you have now is a different wheel, which doesn't fit. And, I have to disagree that Kleg is a "legitimate" module. From what I've seen so far, no one even tries to evaluate the model itself, so how could legitimacy be evaluated. Simply applying a model and producing numbers does not make it legitimate. But, we may disagree on the definition of legitimate; I'm referring to technical legitimacy, which is a reflection of accuracy and validity; while you seem to be referring to familiarity related to common use.

    As I stated in the last post, that discussion was not about "leg stiffness", but about how the value k(leg) can be used in order to evaluate leg stiffness. What are k(leg)'s limitations as an analytical term. When you see 2 values, 20 kN/m and 22 kN/m, what does that mean to you. Can you say anything more concrete than, "the 22 leg is stiffer".

    Leg stiffness as an idea is valid and should be able to be formalised to produce useful numbers (and involves 2 primary components, but likely involves one or a few other factors currently undefined), evaluating it using k(leg) is what isn't valid (since the 2 components are mixed in an undefined, complex way, and there are other variables which aren't accounted for).

    I believe I clearly described why it's not valid, if there's an error in the reasoning, please point it out. Technical validity is not a matter of opinion.

    I've already looked at the leg stiffness thread and most of the papers (I couldn't download some). The input I'm talking about would be answers to very specific questions which have to wait for the method development before seeing what they are. But, I see how you've been discussing leg stiffness, in an attempt to evaluate pathology (which is also my focus), and I'd simply say that you don't need the spring model to carry out that discussion, or apply that idea.

    And, being published in a peer-reviewed journal doesn't make anything valid. Peer reviewing is not meant to judge the validity of new ideas, unless there's some obvious conflict which might require clarification. (Although maybe podiatry is different from the molecular sciences, but I doubt it.)

    These papers define the system as a spring, and hence can not evaluate the model. It's the model itself, and the meaning of the term, k(leg), that's the issue. You could write a million papers using it, and produce a trillion numbers, but that doesn't make the model itself any more valid, only more familiar.

    And, it may be useful under specific, well-defined conditions. This is the cookie cutter approach to analysis. It's usually a very poor way to do it, and is used when you have nothing better, like when working with a horrendously deficient measurement paradigm such as this. But, some good things can come out of it, although a lot of bad can as well.

    k(leg) will not be valid as a primary reference until the relationship between the instantaneous k(leg) and slope k(leg) is defined, because that's what's going to show you the real relationship between the vF and COMdispl. Application of the spring model requires the assumption that vF and COMdispl follow a linear relationship, but nature doesn't care about our equations.

    We can't simply define gait, we have to observe gait. But, we have to be able to observe it in a way that doesn't define it. This would be biased, the same as if COPtrans was included in the primary equations, which removes the possibility of evaluating its role.

    vF and COMdispl are not related by a linear relationship, which is a fundamental, essential element of the spring model, and the term k(leg) is a complex unknown variable which can't be interpreted dynamically without correlating to both input variables. This is not an opinion, it's an observation.

    The idea of "limb stiffness" doesn't require the spring model, and recognising the unsuitability of the spring model is totally independent of the idea of limb stiffness. You shouldn't equate the two, which it seems like you're doing (but I could be wrong there).

    Disregarding problems doesn't help the situation, and can result in a lot of wasted money and effort.

    So, if you say the spring is a proper model for dynamic gait, I think you have to account for the arguments presented previously, otherwise I don't see the basis for a dispute. Whether or not others have "accepted" it or not is irrelevant. If anyone who has accepted this model would like to discuss the issues, I'd be glad to more thoroughly outline the reasoning. I doubt there'll be any takers, since that position doesn't have a leg to stand on.

    But, I see you guys don't care about any technical issues. I've shown you, clearly I think, why k(leg) and the spring model are not adequate for the analysis of dynamic (instantaneous) gait, but apparently that doesn't matter.

    Since method analysis is "my thing", and since it appears you're simply disregarding my comments without any attempt to discuss the issues (on this and the other thread), despite the clear description of significant (ie. fatal) conceptual problems, I realise that there's no way I can contribute to this forum.

    I hope there's no hard feelings, since I've only been honestly and fairly doing my best to try and help you. As a dedicated, very hard core thinker, I can't knowingly promote error, which is what I'd be doing if I didn't bring to the forefront the serious problems with the spring model, related to the proposed dynamic applications. This would be welcomed as a breakthrough in a normal research situation, and would lead to advancement, but it seems to be the opposite here.

    I'll only post again if someone comments on my existing posts or tries to disparage me in any way, so I hope you'll continue your discussion as if I hadn't butted in.


    Mike M
     
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