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Lets not loose this as it is potentially an important exercise to the understanding of leg stiffness and performance.
Kevin wrote: ... "typically, the male runners have much more stiff legs than do their female counterparts (stiffness being determined by how little time the runners feet spend on the ground for a given running speed)."
I replied:
"If you were just looking at contact times did you normalise the data for body mass? http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1624931/ There is an obvious comeback to this... Anyone? Think about y'all."
What is the "obvious comeback" given that Kevin suggested increased leg stiffness in males compared to females observed via contact times (not necessarily true, but lets run with that for this exercise)? You'll need to understand simple harmonic oscillators, the influence of spring (leg) stiffness and mass on resonant frequency and how this relates to contact time and how contact time relates to performance to answer this.
If you're interested in leg stiffness, tell me why normalising for body mass becomes irrelevant if Prof. Kirby's contention is correct, men being generally heavier than women (it's not, but lets just help his ego here by pretending he's right; like he needs his ego massaging- "I acknowledge that I am guessing, which I, again, am pretty good at doing"- extraordinary )
Lets not loose this as it is potentially an important exercise to the understanding of leg stiffness and performance.
Kevin wrote: ... "typically, the male runners have much more stiff legs than do their female counterparts (stiffness being determined by how little time the runners feet spend on the ground for a given running speed)."
I replied:
"If you were just looking at contact times did you normalise the data for body mass? http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1624931/ There is an obvious comeback to this... Anyone? Think about y'all."
What is the "obvious comeback" given that Kevin suggested increased leg stiffness in males compared to females observed via contact times (not necessarily true, but lets run with that for this exercise)? You'll need to understand simple harmonic oscillators, the influence of spring (leg) stiffness and mass on resonant frequency and how this relates to contact time and how contact time relates to performance to answer this.
If you're interested in leg stiffness, tell me why normalising for body mass becomes irrelevant if Prof. Kirby's contention is correct, men being generally heavier than women (it's not, but lets just help his ego here by pretending he's right; like he needs his ego massaging- "I acknowledge that I am guessing, which I, again, am pretty good at doing"- extraordinary )
Simon:
For two runners, running at the same velocity, one with a support time of 200 msecs and the other one with a support time of 400 msec, why wouldn't the runner with the shorter ground contact time also have a stiffer leg? Wouldn't this be a good qualitative way of measuring relative leg stiffness between one runner and another, having them run at the same speed and measuring the duration of their support phase of running?
In addition, one of the training effects I have noted in runners as they become more fit is that their support time decreases over time, which I assumed was related to them increasing their leg stiffness with increased conditioning. Do you know of any study that has measured leg stiffness in runners during a longitudinal conditioning study?
__________________
Sincerely,
Kevin
**************************************************
Kevin A. Kirby, DPM
Adjunct Associate Professor
Department of Applied Biomechanics
California School of Podiatric Medicine at Samuel Merritt College
(it's not, but lets just help his ego here by pretending he's right; like he needs his ego massaging- "I acknowledge that I am guessing, which I, again, am pretty good at doing"- extraordinary )
Simon:
I believe that any good clinician will have the ability to be "pretty good at guessing", since, in all reality, how many absolute truths do we know about the physiology and biomechanics of the human body? Therefore, the decisions we all make for our patients are based many times on educated guesses.
__________________
Sincerely,
Kevin
**************************************************
Kevin A. Kirby, DPM
Adjunct Associate Professor
Department of Applied Biomechanics
California School of Podiatric Medicine at Samuel Merritt College
For two runners, running at the same velocity, one with a support time of 200 msecs and the other one with a support time of 400 msec, why wouldn't the runner with the shorter ground contact time also have a stiffer leg? Wouldn't this be a good qualitative way of measuring relative leg stiffness between one runner and another, having them run at the same speed and measuring the duration of their support phase of running?
In addition, one of the training effects I have noted in runners as they become more fit is that their support time decreases over time, which I assumed was related to them increasing their leg stiffness with increased conditioning. Do you know of any study that has measured leg stiffness in runners during a longitudinal conditioning study?
The key is in regard to the differing mass- with males generally being heavier than females. As I said there is an obvious come back regarding the requirement to normalise for mass given your statement of the males having shorter contact time than the females. Assuming a heavier mass with a shorter contact time for males, why does normalising for mass for the lighter females become irrelevant? Let the others play, Dr Kirby... but think about spring mass oscillators. Just for you Kevin: what if a lighter individual had a shorter contact time than a heavier individual- would that mean they had a stiffer leg? The answer you are looking for is: not necessarily. So, looking at contact times in isolation might be a rather blunt instrument, I guess.
I believe that any good clinician will have the ability to be "pretty good at guessing", since, in all reality, how many absolute truths do we know about the physiology and biomechanics of the human body? Therefore, the decisions we all make for our patients are based many times on educated guesses.
Yeah, I guess it could also be argued that its better to look at the evidence than to guess. But lets not get hung up on this since it was honestly just a joke, Kevin. Goodnight.
Let me ask the question again, but this time, more precisely:
For two runners of the same body mass, running at the same velocity, one with a support time of 200 msecs and the other one with a support time of 400 msec, why wouldn't the runner with the shorter ground contact time also have a stiffer leg?
__________________
Sincerely,
Kevin
**************************************************
Kevin A. Kirby, DPM
Adjunct Associate Professor
Department of Applied Biomechanics
California School of Podiatric Medicine at Samuel Merritt College
Let me ask the question again, but this time, more precisely:
For two runners of the same body mass, running at the same velocity, one with a support time of 200 msecs and the other one with a support time of 400 msec, why wouldn't the runner with the shorter ground contact time also have a stiffer leg?
The key is that you didn't previously specify the body mass, Kevin. Nor, I rather doubt, did you normalise the data for body mass when you looked at contact times in your patients over the last 27 years. So frankly without this information, you could only have been guessing about leg stiffness in looking at contact times. I was trying to be kind by suggesting that males are generally heavier than females therefore your contention could be right because in a spring mass system with a heavier mass and a shorter frequency the spring stiffness might well be higher than in a similar system with a lighter mass and a longer frequency and at the same time I was trying to allow others to explore the physics of simple harmonic oscillators, since this is a good model of the body in running locomotion.
I think you know that I know the answer to your question, Kevin. But differences in foot length might be significant in altering contact times in the presence of identical leg stiffness and identical mass and velocity.
Let me ask the same question again, third time lucky: "Lets take your example with two identical female athletes with the same cortical thickness in their tibias. They follow the same training plan. One runs with high leg stiffness, the other with more compliant leg stiffness. Which one is more likely to develop tibial stress fracture and why?"
Let me ask the same question again, third time lucky: "Lets take your example with two identical female athletes with the same cortical thickness in their tibias. They follow the same training plan. One runs with high leg stiffness, the other with more compliant leg stiffness. Which one is more likely to develop tibial stress fracture and why?
I believe that the injury risk of tibial stress fracture in runners depends more on the direction and point of application of the ground reaction force (GRF) vector relative to the long axis of the tibia than it depends on whether the lower extremity stiffness is different between these two hypothetical runners. Tibial stress fractures are probably most commonly due to abnormal bending moments in the tibia so that if the GRF vector is causing an eccentric load on the tibia, rather than an axial load, than a medial tibial stress fracture will be more likely to occur regardless of the inherent stiffness of the lower extremity.
However, if the point of application of GRF, the 3D location of the GRF vector relative to the long axis of the tibia, the bone density in the tibia, the moment area of inertia along the tibia, the muscle strength in the lower leg muscles, the training habits, and racing frequency are all the same in these two hypothetical runners, then I guess that the runner with the stiffer leg would be more likely to develop a tibial stress fracture due to the greater vertical loading force occurring across the tibia over a shorter period of time.
__________________
Sincerely,
Kevin
**************************************************
Kevin A. Kirby, DPM
Adjunct Associate Professor
Department of Applied Biomechanics
California School of Podiatric Medicine at Samuel Merritt College
I think you know that I know the answer to your question, Kevin.
The correct answer is that if runners of identical mass, are running at the same velocity and one runner has a support period of 200 msecs and the other runner has a support period of 400 msecs, then in all likelihood, the runner with the shorter duration support phase will have the stiffer leg.
I sure do see lots of short duration stance phases in the elite runners....wonder how many of these elite runners get bone vs soft tissue injuries...isn't the Achilles tendon a soft tissue? My guess, is that elite middle distance and long distance runners get just as many soft tissue injuries as bone injuries even though their leg stiffness is probably on the high side compared to the population of runners as a whole.
Great discussion, Simon. Thanks for forcing me to learn more about this subject. Certainly seems that gaining a better understanding of leg stiffness has the potential to help clarify some of the mysteries of running biomechanics. Looking forward to speaking further in person with you on this subject over some Belgian waffles, Belgian beer and Belgian chocolate.
__________________
Sincerely,
Kevin
**************************************************
Kevin A. Kirby, DPM
Adjunct Associate Professor
Department of Applied Biomechanics
California School of Podiatric Medicine at Samuel Merritt College
Let me ask the same question again, third time lucky: "Lets take your example with two identical female athletes with the same cortical thickness in their tibias. They follow the same training plan. One runs with high leg stiffness, the other with more compliant leg stiffness. Which one is more likely to develop tibial stress fracture and why?"
I know I do not post enough but I thought I'd give this one a go... please forgive. But it was a thought on this as I read through and thought I should share for clarification by you. (trying to make sure I'm getting things as I read along)
If the tibias were identical and the mass was identical would it not then have to do more with the overall external tie down effect via origin and insertion of the musculotendounous attachments and the relative stiffness to the weightbearing "core" the tibia? (Like guy wires on a very tall antennae)
The load then has to be stabilized in transit otherwise imalance of the passenger unit takes place (positive trendelenburg) optimal stiffness will have a more stabilizing effect when balanced for the direction of travel and the weight applied will be transferred to the ground more directly through the center of the tibia? But then too stiff and tissue has too much energy and leads to pathology of tissue? So one has to stay as close to the zone of optimum stiffness as much as possible to maintain optimum rythm thus minimizing tissue stress??
Then logically the less stiff leg would have the disadvantage of decreased control via this "tethering" via the musculotendonous systems and have greater lateral bending moments applied to the bone leading to more fractuing in the less stiff leg?
I keep hitting edit to want to scrap it but I'll leave it up...only way I'll learn..
I'm not saying that stiffness isn't important, but is optimizing leg and surface stiffness the key to preventing the majority of running injuries?
Missed this last night. It would appear to be a good start since According to Kerdock et al. McMahon claimed a 50% reduction in running injuries when the surface was tuned to runners. Given the ridiculously high incidence of running related injuries reported in the literature, I'd say that if a 50% reduction could be achieved by optimising the surface and leg stiffness characteristics then this should be a worthy goal.
I know I do not post enough but I thought I'd give this one a go... please forgive. But it was a thought on this as I read through and thought I should share for clarification by you. (trying to make sure I'm getting things as I read along)
If the tibias were identical and the mass was identical would it not then have to do more with the overall external tie down effect via origin and insertion of the musculotendounous attachments and the relative stiffness to the weightbearing "core" the tibia? (Like guy wires on a very tall antennae)
The load then has to be stabilized in transit otherwise imalance of the passenger unit takes place (positive trendelenburg) optimal stiffness will have a more stabilizing effect when balanced for the direction of travel and the weight applied will be transferred to the ground more directly through the center of the tibia? But then too stiff and tissue has too much energy and leads to pathology of tissue? So one has to stay as close to the zone of optimum stiffness as much as possible to maintain optimum rythm thus minimizing tissue stress??
Then logically the less stiff leg would have the disadvantage of decreased control via this "tethering" via the musculotendonous systems and have greater lateral bending moments applied to the bone leading to more fractuing in the less stiff leg?
I keep hitting edit to want to scrap it but I'll leave it up...only way I'll learn..
I,m glad I left this up so I could come back and read it again and realize I am lost like a tissue in the wind. I need to review again.
I think I need to explain. I'm likely coming off as what ... a "dolt"... anyway I was looking at this when I wrote that as from my perspective in relation to CP and the child who is quite hypertonic. (that's what was on my mind when I wrote it) So when you talk about stiffness I was also thinking about these kids. Now at that point the stiffness is so high that it is detrimental to just movement. But make no mistake they are able to walk and some quite well. My point is that when they walk they have a great deal of stiffness but in fact we are using it to our advantage to enable stability. Yes it is detrimental but yes you can exploit it. Might as well as there is not much other alternative right? But the hypotonic child has a much different situation. There is no stiffness to speak of. I can stand them but need to do so carefully so as not to give too much bending moments through the legs as they are not ossified well from LACK of forces stimulating ossification. You were talking about healthy young ladies who are runners. Two different worlds. Sorry about that.
__________________
Todd
Orthotist
Last edited by Orthican : 28th January 2012 at 10:24 AM.
Reason: further explanation
this is my homble, igornant, ineducated opinon, but if the discussion of "leg stiffness" continues as I've seen in the various threads, you will never get anywhere.
I wonder why you don't just use COM displacement (COMdispl), and get rid of the useless "k(leg)" term altogether, since that's what you're virtually always talking about.
Once you understand and define the nature of COMdispl, it's directly (though not linearly) correlated to force. There's no need to try and incorporate those 2 into a single equation, which isn't even appropriate, since the equation defines a totally different physical system. The human system is not a spring, especially at slower speeds.
The only way you're going to find success with k(leg) is through a strict cookie cutter type analysis. Although, it could be used to provide a consistent measurement reference as well, so with a few modifications it could be useful diagnostically, but that's using it as a grid, not a model. That's 2 very different things.
Please excuse the venting, but I find it hard to watch such vigorous discussion wasted.
"Leg stiffness" is a perfectly valid idea. And, I find it very intuitive as well. But, you chose a terrible way to illustrate that idea in k(leg) and the spring model.
Notwithstanding that, unless I'm missing something, the example of 2 runners with equal ("body") velocity and different contact times is not valid.
In order for that to happen, the runner with shorter contact time would have to make contact closer to the pelvis in the horizontal plane, in order to shorten contact time. That's a different relative gait pattern.
In order to judge relative bone and joint stress, I would think you'd want to normalize the 3d movement of the body and stance leg, and then judge the difference in perceived stress in the horizontal plane when, for eg., the step foot is placed 2cm more outside than normal, or closer to the pelvis, but the subsequent body trajectory maintained.
So, let's say 3 identical subjects are moving exactly the same up to the start point. Then, just as the left foot contacts the ground, one stays the same, but the other 2 deviate in swing foot trajectory to finally plant the next right foot 10cm closer to the pelvis in the horizontal plane in one (but same (h) foot trajectory), and the other plants the foot the same distance from the pelvis, but 2cm outside, compared to normal. The right foot is the step foot, so it would be placed 2cm to the right, relatively.
Would anyone contribute what you would expect with respect to pain, injury, bone stress, short and long term, etc., from a patient consistently showing these 2 deviations, assuming the relative body positions and trajectories were maintained exactly the same, so the body momentum vectors are identical, except for the swing leg only.
Mike M
Last edited by m.mouck : 3rd February 2012 at 01:32 PM.
Reason: add "though not linearly"
It is somewhat funny to me that you should comment on this stiffness thread as follows: "Please excuse the venting, but I find it hard to watch such vigorous discussion wasted."
Why do I find this comment of yours funny? Beacause nearly every post that I have seen you place here on Podiatry Arena is so convuluted and complicated, that, in effect, all your effort is, at least to me, a wasted effort on your part. Maybe there is a lesson to be learned here?
Anyway....while I have long been interested in the concept of stiffness to explain the load-deformation characteristics of the first ray and of the medial and lateral arches of the foot, I believe that the concept of lower extremity stiffness is also a very valuable concept to apply to the running gait pattern, in which lower extremity stiffness is derived generally from the magnitude of ground reaction force (GRF) relative to the magnitude of center of mass (CoM) displacement during running.
I believe the ultimate proof that the idea of lower extremity stiffness during running is, indeed, a valuable concept is that the late Tom McMahon, PhD was able to use his calculations of lower extremity stiffness during running to design a indoor track at Harvard that not only improved racing speeds but also reduced injury rates in the collegiate athletes that run on that track.
One only needs to understand the basic concept of how lower extremity stiffness can be varied during running by the central nervous system and may be manipulated by the central nervous system by running on different surfaces with different stiffnesses to realize that Tom McMahon's idea can, indeed, be quite a powerful one. The question then becomes, though, how much the concept of lower extremity stiffness can help us clinically treat our patients and understand why certain runners get certain injuries. At this time, I think the concept of lower extremity stiffness will be of some use in this regard. However, I also believe it will not replace the many other more important concepts that more directly affect the injury patterns and treatment of runners.
__________________
Sincerely,
Kevin
**************************************************
Kevin A. Kirby, DPM
Adjunct Associate Professor
Department of Applied Biomechanics
California School of Podiatric Medicine at Samuel Merritt College
I can see that you're not understanding the things I'm saying, although I find it odd you see your lack of understanding as funny. I think you should take 2 lessons from this, first, please read more carefully, and, second, if you don't understand something, ask.
Regarding, " One only needs to understand the basic concept of how lower extremity stiffness can be varied during running by the central nervous system and may be manipulated by the central nervous system by running on different surfaces with different stiffnesses to realize that Tom McMahon's idea can, indeed, be quite a powerful one.", I don't know if you realize it, but this statement means absolutely nothing. It doesn't support any idea.
I urge you to go to the link you provided, thank you very much by the way, and actually look at what was done. And just look at the nature of the data.
I don't know what you think that proves. It doesn't affect anything I've said. Obviously, just by looking at the data I can see that the 3d gait patterns must have changed. In order to judge why there was less injury, for eg., you'd have to look at how the patterns have changed. How are they placing the feet with respect to the body, and then how are they managing the body trajectory while vaulting on the planted leg. You're reading way too much in to that work, and it doesn't prove or justify anything you're doing.
I ask you to re-read my above comments, and you should see that I don't say the idea of "leg stiffness" isn't useful. I said it's a perfectly valid idea, and I find it intuitive as well. So, I'm not challenging your general use of that concept.
Please (re-)read my 6 determinants comments, and you may realize that k(leg) is only one way to illustrate/define stiffness, or rather the physical elements you're generalizing to stiffness.
Also note that I said you could find success with a "cookie cutter" type analysis. I can't remember if I ever defined what that was, and this is my own definition, but it's when you try to control the conditions of the test so that, basically, the measurements you're taking can be directly related to how you want to interpret them.
Running is one way, hopping is another, and there are others, which attempt to generalize the unknown conditions so that the differences are a result of fewer extraneous things. You're trying to obliterate the (known and unknown) minor variations, so only the most "significant" elements, according to your interpretation, are reflected in changes in the numbers.
So, applying k(leg) to running and hopping are ways to apply a cookie cutter, and how useful it is can only be determined through application, it can't be predicted. You can only hope you're controlling conditions suitably.
If success is observed, that's excellent, and I hope the work is continuing. It's certainly not ultimate proof of anything. It's very, very far from being anywhere near that. A lot of different approaches can produce similar pictures. And, the McMahon thing is quite a bit different from the simple application of the spring, and, again, look at the data itself, especially the results. There's been a fundamental change in the 3d body movements associated with the gait pattern, and that's what you have to look at.
Just to clarify, I've been talking about k(leg) in the linear equation relating force and COMdispl, which you're trying to use to help interpret real vertical force patterns (as a model). To do that you have to be able to independently compare sequential time points. k(leg) won't give you a relevant comparison in those conditions. It simply won't work as stands. Other applications of k(leg) have to be evaluated independently, but all applications should be viewed with great caution.
Don't jump to the conclusion that I'm saying surface stiffness isn't a factor or isn't important, I am not saying that at all. I'm saying you're not looking at what needs to be looked at in order to properly define stresses, etc.
If anyone is willing to interact, on another thread I'd like to start a discussion where I believe I can show you that the horizontal plane is by far the superior perspective to see and judge physical stresses, so it should be a great asset in podiatry. No figures, no numbers, unless you want them. General discussion over coffee.
If any of you are serious about wanting to create methods that are useable and will stand the test of time, this would probably be a valuable exercise, since it would necessarily entail technical logical reasoning and a discussion of the most basic requirements in the development of analytical methods. You may come to realize that you don't even need models when you have a proper way to measure, and why I say the things I do about k(leg).
Gait isn't actually that complicated, as far as measuring the physical movements (ie. kinematics), once you have good organization.
You don't need any prior knowledge at all, but you would have to come out of your conceptual cage and put the sagittal plane and the idea of models on the shelf for a while. I know the sagittal is your main perspective in your work, so the road will ultimately go there, but it can't start there. It's just talk, and it would help both of us. This should also help in the real interpretation of force patterns.
Is anyone interested in that?
To give you a taste, the first discussion might be labelled something like, "understanding gait", from a perspective that someone like Aristotle might have taken. Although I haven't written it yet. From there it might progress to "in what way should that be observed", etc.
I can see that you're not understanding the things I'm saying, although I find it odd you see your lack of understanding as funny. I think you should take 2 lessons from this, first, please read more carefully, and, second, if you don't understand something, ask.
Funny, have you ever thought that maybe your overly verbose and obtuse writing style, where you insinuate that you have all the answers about the biomechanics of gait while no one else knows anything are somehow partially to blame?
Quote:
Originally Posted by m.mouck
Is anyone interested in that?
I'm certainly not interested.
Mike, you should have listened more closely to Dr. Spooner over a year ago when he said:
Quote:
Originally Posted by Simon Spooner
Everyone else is wrong and you're right, Mike. Like I said, good luck with your future. Why do you think people aren't jumping up to try and help you?
**************************************************
Kevin A. Kirby, DPM
Adjunct Associate Professor
Department of Applied Biomechanics
California School of Podiatric Medicine at Samuel Merritt College
Really Kevin, "insinuate that you have all the answers about the biomechanics of gait while no one else knows anything," -these kinds of comments are suited for the playground, not the research arena.
I don't blame you for your response, since I was fairly harsh to you previously. I was harsh because you threw up a couple of straw men, and I saw that as an attempt to deceive. I despise liars, but on thinking I realise you may not have meant it that way.
Since you've chosen to unjustly attack my gait work, naturally I have to respond to that.
So you're another that likes to imply error where none exists, simply because you don't understand.
That's a very destructive habit. There's no place for that in intellectual discussions. I doubt you would find anyone that does research that wouldn't agree with that, no matter what kind.
And that's what this is, to me anyway, research. The points that are brought forward are discussed in order to advance knowledge. No one is expected to just believe anything. There are no experiments, it's just talk.
If there's an error, point it out. Your lack of understanding, even if it's due to the difficult nature of my writing, does not equate to error. Let me put it another way, it's like spitting in someone's face because you think they're "uppity". Would you take that?
Note that the comments about k(leg) and the spring, etc. have nothing to do with my gait work, which you certainly aren't in a position to intelligently comment on.
Please consider the following scenario. A person comes in to an office with a right knee contracture leading to right foot swing abduction and hip-hiking, which would also show as a deviation in the COM trajectory.
The condition is treated, COM devations disappear and the person reports improvement in gait. The doctor then uses that as proof that minimising COM deviations improves gait.
It's easy to see the fallacy for such a stark example as this, but take it back to more subtle changes and conditions and there comes a point where the link to the physical reality is lost.
For McMahon's work, 2 critical elements in gait aren't considered, the pattern of relative foot placements and the body trajectory (not just COM) management used to travel along the path. Changes in relative foot placement alone could account for all the observations, which could easily have resulted from a different feel of the track. There are actually a couple of other little issues as well.
Not saying it did, but it could. They likely didn't consider them because they weren't looking at things in those terms. And, the slight increase in speed they observed can't be properly explained with their model, despite what they say. They never say how the speed is imparted to the system, they just assume it's there, as is usual when using the spring idea.
Interesting that he says "good" runners can increase speed slightly. And, McMahon subsequently co-authored a paper, "Energetics and mechanics of human running on surfaces of different stiffnesses," J Appl Physiol 92: 469–478, 2002, where they stated the mechanisms for the performance enhancement are not understood.
And here's where I thought my work would help, since it could tell exactly what did change and how, in great, very relevant physical detail. You just have to get used to the different perspective, which is looking from overhead and tracking the movements of the feet and pelvis as the main elements, then working from there, and including anything that's useful, from any perspective. Unfortunately, 3d marker data is required, but even looking at theoretical movement patterns from that perspective is revealing.
If no one's interested in that, I'm dissappointed, but that's fine. You're the perfect group to discuss this with.
Just don't imply there's something wrong with my work because you don't like me or the way I write, that has to be shown with specifics.
Needless suffering, potentially billions of dollars and thousands upon thousands of research hours could be saved with the application of my gait work, whether you understand how or not. It's far too important to be frivolously and unjustly maligned. If there's a problem, point it out. What you're doing is saying 2+3 doesn't equal 5, just because you haven't taken the time to learn what the "+" sign means.
In fact, I defy anyone in the world to prove that any of the statements I've made regarding my gait work are not true. I can back up everything I say.
Please note again that my previous comments were regarding k(leg) and the runner example, which have nothing to do with the gait work.
Mike M
PS McMahon used the definition of "step length" as the distance the body moves during foot contact. This is the first time I've seen this particular definition, is that how step length is defined in podiatry?
Re: Stiffness
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