Bipedal spring mass walking sagittal plane theory

I don't think I can touch it yet, but I think we are getting closer. Maybe "flat-spots" was a red herring, and we should attempt to address the other things?

 

Simon,

I think it is time to roll out the tensegrity concept once again. Buckminister Fuller described that when discontinuous compression, continuous tension (tensegrity) systems fail, they "collapse" at 90 degrees to the force applied to them. Complete failure may not be necessary....only subtle variations in the tension to affect the support mechanisms.

Since the biomass tension spring may be the way Geyer describes this event, is this the calculation that the body must make to create symmetry, ie, tense one side and then relax the other? Therefore, can the fascia network, once limbs are extended, offer sufficient tension to support our musculoskeletal system, with only subtle flexions necessary to adjust how this is transferred through the entire body? To my way of thinking, this is the only thing that makes sense. Subtle alterations to tension then create symmetry in ossilations.

Thanks for the stimulating conversation.

Howard

 

I agree with the above, I'm not quite sure how this relates to tensegrity theory though?
Willing to learn as ever.
Do you mean that the leg spring can either be in tension or compression? What about the points of maximal oscillation displacement, i.e highest or lowest point of the oscillation, when it is in neither tension nor compression? Doesn't this create a flaw in the theory?
Also if we model the leg as a spring, when its at it's resting, unloaded length then this should equal the point of zero compression, zero tension.- right?

 

I recall one of Winter's papers looked at kinetic and potential energy of gait and said that the kinetic energy was traded for potnetial and then back again. Specifically, as the body falls forward when it is anterior to the stance leg it is lowering its potential energy and raising its kinetic energy. When the lead leg hits the ground and lowest potential energy is reached, the velocity is highest and the process is reversed as the kinetic energy is used to get the body up over the new stance leg.

I like applying the ideas talked about to the involved anatomy. The leg stiffness idea should correlate with something that can change stiffness like the knee. The change in vastus activation in one leg versus the other can provide a mechanism by which the leg stiffness changes. In looking at tensegrity structures and then looking at the anatomy of the body, I don't see how compression structures are held apart from other compression structures via tension members. In other words, I don't see how tensegrity applies to the anatomy.

Regards,

Eric

 

Eric, thank you for your contribution. I tend to agree. The question is this though: if we model the lower limb as a compression spring and we pull that spring out of the body and onto our desktop, it's is a compression spring, so is that spring a structure of pure tensegrity when a load is applied to it? My gut instinct is, no. So if we put it back into the system, i.e the body, what would make it a tensegrity member? My gut instinct is that it doesn't become a tensegrity member and that this is a diversion from this thread.:drinks:bang:

 

Just re-read this in light of last nights brain-storm. Interesting.
http://jeb.biologists.org/cgi/reprint/201/21/2935.pdf

 

Aha hahahahaha!!:empathy:

 

I don't follow you, Dave
This paper was written without knowledge of the bipedal spring mass walking model. It says to me that 3 factors may be employed / will influence COM displacement: angle of attack of the leg at initial contact; progression of the COP; leg stiffness. What does it tell you?

I think all three of these are inter-related, and describe/ explain the observed gait changes which Howard has reported and others commonly observed in clinic.

What about late midstance pronation? Is this explained thus: COP does not progress forward at sufficient rate; leg spring stiffness is reduced, resulting in an increased period of oscillation for the leg spring, i.e. the leg spring is still compressing when it should be extending, thus the COM is not reaching it's maximal vertical height at midstance? Brain storm time again, Dave. What happens to the leg kinematics, COP progression and COM displacement in someone exhibiting late midstance pronation?

Lets try some common pathologies and see if we can work out theoretical pathways in which alteration in the angle of attack of the leg at initial contact; progression of the COP; or alteration in leg stiffness may result in or be the consequence of increased stress in the pathological tissue. Take your pick...

 

Yeah sorry you had to be there to get the full gist of that expression. It was meant to be an expression of delight in finding another important bit of the jigsaw with 10,000 pieces.

Regards Dave

 

I'll leave it there for now and see how you pick up on my brainstorming thoughts.

Regards Dave

 

Hi folks, been renovating the cabin to get it finished for winter already been minus 6 overnight another reason to look forward to OZ and 3 weeks this christmas, anyway.

Ive been thinking about the COM and rthyms over the last couple of days while putting in new floors. Heres what Ive got, the more I look the more the importance of Rhythms are for body function such as muscle Piper rhythms and the COM following a somewhat constant path a bit like a ticking clock when moving at the same speed. Which why it make sense to me about the longer leg reducing it´s leg stiffness through knee flexed etc, as you folks we discussing last week, but I hit a rather big wall.

If the longer limb reduced its leg stiffness with greater knee flexion through the joint coupling shaft to subtalar joint we should always see greater STJ pronation with the longer leg, but as has been discussed this is a myth.

So this is where I´m at anyone got any ideas ?

ps if it discussed in Simons links Ive not read it yet.

 

Perhaps it's explained via a different cat peeling technique? As I said earlier: "3 factors may be employed / will influence COM displacement: angle of attack of the leg at initial contact; progression of the COP; leg stiffness". Or perhaps through variation in coupling between the forefoot, rearfoot and shank?

 

OK guys let’s try a different perspective. Ballistics of Trajectories:

If we consider a bullet fired from a gun barrel that is parallel to the ground and at an initial height of 10m for example and a similar bullet dropped from 10m height at the same time, then the bullets hit the ground at the same time. The variable here is the muzzle velocity of the bullet (ignore change in velocity caused by air resistance) The varying muzzle velocity changes the distance covered in the time it takes for the bullet to drop 10m when acted on by gravity.

Now apply this to our maximum CoM height and its horizontal velocity at that time, lets take that as mid stance for now.

If we consider the CoM just prior to heel strike when the CoM is at its lowest vertical height it has a certain momentum and its energy equals P and P=1/2mv^2 i.e. kinetic energy = half the mass times the velocity squared. The mass is constant so any change in kinetic energy is relative to the change in velocity.

So, just as in the bullet example, in theory and ignoring the stance leg support, the distance tended to be travelled by the CoM from max vertical height to the next lowest point, i.e. contralateral heel strike, would be determined by the CoM velocity at the time of max CoM height.

In the Lee, Farley paper we see that the conditions of attack angle, leg stiffness and CoP progression distance are the determinants of the relative change in CoM height from initial to maximum height and the difference between attained maximum height and potential maximum height (i.e. compass gait).

So if we assume that symmetry is a desirable objective in gait progression, and so if we can also assume it was desirable to have each CoM trajectory equal in length and time, then to achieve this it may be optimal for the CoM velocity to be equal at both mid stance / max height positions.

To achieve this optimal scenario it would be necessary to moderate or regulate the energy input or the energy consumption.

So consider the example where the CoM trajectory velocity from heel strike to mid stance (max stance height) is impeded. To maintain the CoM velocity it would be necessary to add some more energy, from the push off leg for instance, or lower the Stance leg max height. It can be seen in very pathological gait that adding energy from other trick actions is a usual way of achieving this, e.g. as Ricky(earlier example) does by using the pelvic rotation action. Adding energy may be useful but also may be inefficient physiologically and pathological due to increased accelerations and abrupt ‘jerk’ i.e. changes in accelerations. (NB in engineering jerk is often a condition to be avoided to reduce material or structural damage smooth changes in acceleration are preferred, also ‘jerk’ is a subjective and relative term.) Consider that if the CoM loses velocity it loses energy exponentially by the square of the change in velocity (P=mv^2), so adding energy to the system may be far more costly than conserving velocity and avoiding kinetic energy loss.
Lowering the max CoM height and flattening the trajectory curve would allow conservation of energy because of the decrease in the change of relative CoM height. The advantage here would be that, as explained in the Lee paper, energy lost due to gravity would be recoverable in part in terms of stored elastic energy of the muscles and tendons due to the spring action of the stance leg.

The one snag, or at least one of the snags, here is that lowering the absolute CoM height also reduces the distance travelled for the same angular displacement. One way to avoid loss of distance travelled is to flatten the trajectory arc by increasing the CoP progression length. This can be achieved by making the foot longer i.e. pronating the foot. Could pronation assist CoM progression in the gait with sub optimal biomechanics e.g. that gait resulting in sub optimally impeded CoM progression? Just getting those thought out there to you guys for consideration.

Regards Dave

 

Right still trying to get my head around the angle attack stuff, But if we take the most likely one when thinking about the title of the thread- plantar fasciitis.

We know from earlier reading that soft tissue stress comes from those with reduced leg stiffness. We also understand that Fnhl and maybe Fndl ( Functional digitial limitus , was about just write fnhl, but since earlier discussion should include all digits ) leads to increased tension in the plantarfascia.

So with reduced leg stiffness we will get increased Knee and hip flexion - the Joint coupling from the tibia (shank - not shaft :bang with lead to STJ pronation which then leads to increased tension in the PF, and Plantarflexion moments from the PF on the 1st MTP joint and all MTP joints . If the progression of the COP is slowed and is distial to the 1st MTP joint and all others then moments from GRF will be also plantarflexion further increasing the tension of the plantarfasia. Which may lead to pathology.

???

 

Interesting thoughts, Dave. How much does foot pronation lengthen the posterior - anterior pathway of the CoP by? What is known of the effects of pronation on COP pathway?

 

So, increasing the angle of gait (toe out angle) should reduce the potential for forward displacement in the sagittal plane of the CoP and thus require reduced leg stiffness?

 

Yes I see what your getting at with the toe out but does it require less stiffness or just a lower CoM position?

Dave

 

Dave, Simon and all,

Sorry for the sudden absence. My life seems to get in the way periodically and PA is just not in the time budget.

But I do have time for just a short reply so here goes.

Consider the action of the muscles of the lower extremity. During single support phase, these muscles only fire eccentrically. They are resisting, not creating motion. By their action, it was seem highly plausible that they are affecting the spring tension.

It is particularly important to view the period of time from peak in CoM elevation to the termination of single support. This is when the potential energy used to raise the CoM switches to kinetic energy, as the CoM begins its descent until opposite heel strike. This coordinates with the period of full knee extension and time when hip extension goes from neutral to positive. My observation is that the ease of function during this period is when sagittal plane restriction creates the greatest negative effect. It prevents the efficient use kinetic return, and causes inefficient concentric (as compared to eccentric) muscle contraction as a "backup". When this happens repetitively, we all understand the consequences of chronic overuse. In light of the spring tension thoughts, I thought this would fit rather nicely.


Dave asked "Could pronation assist CoM progression in the gait with sub optimal biomechanics e.g. that gait resulting in sub optimally impeded CoM progression?"

Assistance of CoM progression is what sagittal plane facilitation is all about. Hip extension, heel rocker, ankle joint dorsiflexion, mtpj dorsiflexion (with windlass) in succession all designed to permit the body to pass rapidly above it while self supporting in the same process. When this fails, pronation does "assist" in CoM progression, but it, in and of itself, is inadequate to prove the same the amount of of motion equivalent to normal sagittal mechanics. Yet the kinetic return occuring simultaneously must be dissipated somewhere. Hence, the posture flexion process (knee, hip, torso) occurs.

Simon had previously asked, why bring up tensegrity. The more I read about tensegrity, the more convinced I become it is the way all biologic structures are constructed. It is true both macro and microscopically. Many believe that individual human cells are tensegrity structures as well. The cell membrane provides the continuous tension, while the microtubules and liquid protoplasm the discontinuous compression components. If we think of the fascia and skin (tension) covering everything continuously, with a discontinuous skeletal (jointed) system, then the tensegrity design fits. Further, considering how tensegrity structures have a "neutral" energy point, then the its energy storage/return advantage becomes a further enhancement.

Hope that this adds to the discussion.

Howard

 

Simon, Howard, Michael and Eric

Perhaps you might consider this.

The CoM or any free mass of 'm' with a velocity of 'x' is decelerated by a constant applied force 'f' when the velocity of ‘m’ reaches zero the direction of force applied, relative to the velocity of mass, is changed 180 degrees (I.E. just like at running midstance where the horizontal force applied changes from braking to propulsion and CoM motion changes from down to up and the force is the same absolute direct but opposite relative direction) and the mass is accelerated by a constant force. Initial velocity is 1 and terminal velocity is 1.

Question: Could the position time displacement graph of mass 'm' be a curve, even tho the Force time graph is a flat line? I.E. the vertical displacement goes thru a positional range of 1(initial) to 1(final) thru zero at mid point (or high , low, high), over time ‘t’.

As proposed earlier a flat line force time graph represents a state of constant acceleration of mass (CoM). Constant acceleration is equal to constant change of velocity but not constant velocity. A constant change in velocity must represent a constant change in positional displacement of the mass.

E.G. Mass ‘m’10kg at initial velocity ‘v’of 200m/s is decelerated by constant force ‘f’ 100N for 5 seconds (t). Positional displacement =(v1 – v2)/t = (a=f/m) = 10m/s so each second the force decelerates the mass by an additional 10m/s and so each second the mass travels or is displaced by the sum of the change in velocity per second. Sorry can’t put it more clearly than that.

So during second (t) 1 the 1s displacement is -10m and during second 2 the 2nd displacement is -20m 3rd is -30m 4th is -40m and 5th is – 50m. The sum of displacement is 50+40+30+20+10 = 150m.

You can see then that the constant acceleration and or deceleration (which is equivalent to a +/- applied force) = a constantly variable change in positional displacement. This is shown on the bottom graph (vertical displacement).

I plotted a simple displacement function of f'(x) = x^2 becasue this returns a perfectly conctant displacement variable over time that satisfies the criteria of a one-zero-one curve. Then I plotted the first and 2nd derivatives with respect to time i.e. velocity and acceleration and then derived force from the acceleration curve and guess what, a curved positional displacement can return a flat force time graph. See graphs below.

NB. The mathematical anomaly of zero has to be removed because any sum multiplied by zero = zero and so you get a zero anomaly on the graph also the nature of the positional bias returns negative values (i.e. 1 zero -1) when the bias of the displacement changes in terms of direction and so this also must be removed in the processing. This begs the question: Could this be what happens to the signal that we see displayed on a FTC graph of plantar pressure when characterised by a plantar pressure measurement device? Pressure mat Manufacturers, answers on a postcard please, this would be useful to know.

PS ignore the axis values they are just for convenience of displacement initial = 100 with a range of 100 to -100 and time was 200.
PPS I'm not 100% sure about my work assumptions so if there is a math bof reading please put me out of my misery as quickly and painlessly as possible.

 

Aren't the two interrelated? Further thoughts: the ratio of resting leg length to foot length should be significant too. i.e. a long leg with short foot should be disadvantageous. Which then brings in to play the concept of the high heel shoe, which will effectively shorten the foot and lengthen the leg.

Dave, I'm sort of following your math. I think you are saying that the pressure mat displaying a flat-line is a spurious function due to the manner in which the calculations are performed- right? What does it mean in terms of feet and legs?

 

Yes I am but what I'm also pointing out (hopefully correctly) is that there can be lots of mass acceleration and a wide range of motion but still get a flat line FTC.

Regards Dave

Ooops I've just realised a flaw in my previous argument, get back to you soon.

 

I didn't plug gravity into the equation for Vcom displacement so the graphs work for horizontal displacements not affected by gravity but the vertical displacement in terms of force and acceleration will be different, when I get time to update it.

 

OK

So if you plug gravity in then the only way the vertical CoM displacement can give a flat line FTC output is if the velocity of the displacement is constant (i.e. like travelling in a lift or stopped, as I mentioned previously) but it is not possible to have a vertical CoM displacement curve that changes direction or bias like that of the horizontal CoM displacement, which requires a constant acceleration. In fact both examples require an absolute constant acceleration but the vCoM displacement is constant acceleration due to gravity (which is a relative constant velocity with respect to the ground) and the hCoM displacement is a constant acceleration due to applied force.

So therefore we can conclude, or at least I will conclude, that a flat FTC is the result of the vCoM displacement either stopping or continuing in the same direction of motion at a constant velocity.

If we consider this in terms of leg stiffness or rather joint stiffness as defined by muscle force and internal moments about the joint, as described in an earlier post, then in the case of the constant upward velocity vCoM displacement, the knee for instance, is less stiff than where the vCoM displacement is accelerating in the same upward direction.

However where the vCoM displacement is downward then the case of constant velocity displacement is more stiff than when the vCoM is decelerating (minus acceleration) in the downward direction.

Earlier I said that it is not possible to have a vertical CoM displacement that changes direction without significant changes in acceleration and so also FTC amplitude. But what about when the change in direction of vertical CoM displacement is thru a shallow curve? (And as far as I remember at this juncture, the relative vCoM displacement curve tends to become shallower in the apropulsive) It may be possible that the Kleg can remain almost constant from braking to propulsive stage thru midstance or rather, because there would be significantly reduced braking and propulsion force peaks due to the compliant nature of the supporting leg, it may be better to say from early stance to late stance thru mid stance

More later

Regards Dave

 

Nice work, Dave.
Which makes the transition periods from double to single and back to double limb support all the more interesting. How does the body maintain a constant vertical stiffness, in the face of changing support (leg) stiffness?

How do we stop the vertical displacement of the CoM?

 

Its taken me till this sentance to understand pretty much anything you have said in the last 5 posts or so. I would not say I understand ,but I think I´m with you. Way above my pay grade.

 

Simon

We don't want to stop it per se because it is also useful, we want to allow it to stop where it is unnecessary outside an optimal range.

Micheal

I bashing this out quickly in spare time so it needs to be read with that caution but I'm pretty sure the physics concepts are right.
Dave

 

You're avoiding the question Mr Smith, why might it want to stop? Presumably for a quick cough and a drag- who wouldn't? It'll stop momentarily at the highest peak and lowest trough of it's oscillation, but I'm guessing we are talking about stopping outside of these points of the oscillation? If it is getting too high or too low, then the body will shut down the COM's vertical displacement for a while to maintain stability/ efficiency?

 

Simon

Damn you caught me out been sitting doing accounts for several hours. I was formulating some thoughts re your question when wife just phoned to see if I woz coming home, so I just put something together quickly to sign off for the evening.

Who mentioned something about thumbs


LoGL Dave

 

Hi guys,

I don't know if you recall, I'm not in the formal gait industry, but have created a new method to measure gait. This comment does not relate to that, though.

Although I don't believe spring models like the one you're discussing will have much utility generally, especially at lower speeds, that doesn't mean I can't help you develop the fundamental equations necessary to apply it and to fully understand the strengths and weaknesses of the proposed model.

However, although I can do it all myself, it would be much, much better if one or a few of you were willing to follow along and provide input when possible. One reason this is so important here, in my opinion, is because the characterisation of the spring model necessarily goes through the correlation of force and COM position.

I don't know how it will progress, but it may be possible to create a way to more effectively correlate the sag.COM and force graphs, as we're figuring out the model dynamics. I don't know if this can be done, but it looks possible, given the nature of the material.

In the meantime, I'd like to make a few a priori comments on flat-spots on the vertical F/t curve.

I would think that the only time flat-spots would be seen is when the COM is travelling with constant vertical acceleration, as Dave mentioned. The COM can have any trajectory and acceleration in the horizontal plane, as well as any vertical velocity and trajectory, as long as it's with constant vertical acceleration.

When vertical acceleration is zero, the vertical COM velocity is constant, but in any orientation relative to the ground. This would show as a straight line (at any angle) on the sagittal plane graph of the COM vs time if the horizontal plane acceleration was 0 (constant velocity), but a curve if the horiz. accel. is not 0.

When vertical accel. is non-zero, velocity is changing in the vertical direction at a constant rate (ie. constant accel.). The shape of the sag. COM trace would be a curve (up or down), which would also be a function of the horizontal velocity and acceleration.

So, I think there could be great (infinite?) variability in the sag. COM trace which could show as a flat spot on the vert. F/t curve. Therefore, I seriously doubt you could recognise whether a specific sag. COM pattern would show as vF/t flat spots just by its shape. (That doesn't mean a specific COM pattern can't be practically linked to a specific force profile, due to the pseudo-regularity and pseudo-symmetry of gait.)

Of course, the value of the force at each time point shows whether the COM is accelerating up or down, but the vF/t curve shape couldn't illuminate the specifics of COM position, in my opinion, since even when the COM is accelerating up (total vGRF greater than standing force), it could still be travelling toward the floor (or in any direction) along any number of trajectories.

I think the F/t curve is best viewed as a reflection of changes in COM trajectory (which is relative), rather than a direct reflection of absolute trajectory. So, a flat-spot on the F/t curve means no change in vert.COM trajectory, but the trajectory (which could even one that is changing, as long as it's a constant rate of change) could be anything.

This doesn't mean there isn't some functional significance to F/t flat-spots, but they have to be evaluated respecting all of the technical specifics.

NOTE that this discussion will include all elements and is strictly technical. When applying calculations to real systems, there may be difficulties and/or simplifications which aren't immediately evident, or which can't (or shouldn't) be immediately taken in to account.

For eg., even though some conditions (as discussed above) should theoretically show as a curve, like in the sag.COM (x,y) plot when accel. is not zero, the relative magnitudes of the quantities (one may be so small as to be insignificant) may mean the curve is so slight that it appears as a straight line on the real graph.

These kinds of things can only be recognised and understood after the full primary equations are developed.

If anyone is interested, I think the best place to start would be square 1, but before that, I've got a few questions.

Any takers?


Mike M

 

Mike

Is this true?

Are you speaking relatively (relative to gravity and the ground reference) or absolutely?

Wouldn't it be correct to say - The absolute vertical acceleration is never zero unless the relative downward acceleration is 9.81m/s i.e. it matches acceleration due to gravity.

In relative terms then - when there is zero acceleration and constant velocity then there will be a flat FTC equivalent to body weight, in this case velocity can be zero, which is also a constant. This would be the case if you went over a force plate on a roller skate keeping you body still, i.e. flat FTC at body weight equivalent.
In absolute terms however this is a case of constant acceleration, which is the only way that you could have a constant force or flat FTC recorded, i.e. constant acceleration = constant force.

So if it were possible in gait to accelerate constantly for a given time in a vertical direction at say 12m/s/s
then assuming an 80kg mass this would equal a constant force or weight or flat FTC output at 960N for the time of the event. or at a constant 7m/s/s and record a force output of a constant 560N for a given time.

The curves in a FTC are characteristic not of acceleration, i.e. constant change of velocity over time, but of a change in vertical acceleration, i.e. what has been called 'jerk'. The change can be interpreted as braking and propulsion in terms of gait.

for your kind consideration

Regards Dave

 

Hi Dave,

Thank you for your comments.

I think I realise why you disagree with the statement.

Since I mentioned horizontal accel., you may think that I'm saying the horizontal accel. contributes to the vF/t profile. Of course, this is not what I'm saying, but I should have worded it better.

Horizontal accel. affects the sagittal position of the COM, but it would not be reflected in the vF/t curve. The vF/t graph is only affected by vert. accel., as you said. The changes in line shape I discuss refer to the COM position in the sagittal plane, not the vF/t graph.

And, I found an error in my last post, which I'll edit as soon as possible. I do a lot of cut/paste when I write, and occasionally mix terms from different discussions. The changes to the line traces mentioned are for the sagittal (x,y) (or (x,z)) plot of the COM, not the (x,t) plot. I don't know how many times I read that, but still missed that error.

I can see why you would disagree with this, since only the sag. COM (x,y) plot, not the (x,t) plot, is affected by horizontal accel. of the COM. But, this factor would not be shown in the vertical force profile, anyway, only in the sag. (x,y) plot of COM position. I believe that the (x,y) sagittal plot is the most common representation in the sagittal plane, so I was going to use it as the main reference. This is the one used for sagittal plane stick figures, and it seems this has to be the one used for this model. Is this appropriate?

(When looking at the sagittal COM position, what plot are you looking at? the (x,y) or (x,t).)

I thought it odd, since I agree with pretty much everything you said, so I couldn't understand what you disagreed with. But, on considering your last statement, I see what the problem was. When referring to accel. and the vF/t curve, it's always vert. accel. When considering (x,y) COM position in the sagittal plane, both vert. and horiz. accel. are relevant.

I don't think, though, that the terms "relative" and "absolute" are good to use here. What we are talking about are "net" effects on the mass (since the real movement of the mass is what we're interested in right now), which are determined through a strict vector analysis. Since the force due to gravity acts on the mass and is a vector, it's always included were relevant, as are any other relevant forces, such as a backpack.

Otherwise, I don't see any disagreement, just perhaps that associated with my error in the first post, where it should be sag. (x,y) plots, and not sag. (x,t). When this change is made, all my previous statements are correct, I believe.

A question: Regarding braking and propulsion, do current methods resolve the contribution of the change in horiz. velocity due to the vertical redirection of the velocity vector (which is a function of the vert. force profile), from the contribution due to muscular action resolved in the other plane(s) (which wouldn't show in the vF/t curve)? I would assume they do, but I haven't seen anything so far specifically showing that.

I was hoping to give a hand in developing the equations necessary to apply the double spring model. I believe you tried a bit of modelling, Dave, and hope you find it worthwhile to continue to contribute, as well as all others who are interested.

A couple of comments on the spring model. I'm sure you realise that flat spots on the vF/t curve can never be (accurately and consistently) modelled using simple harmonic motion of the leg springs.

The only way to model flat spots is to introduce a period of non-harmonic motion in one or both springs, which is what you're doing by changing leg stiffness (k(leg)).

So, the actual proposition with respect to this model, if you say it will be able to model flat spots, would be that walking (sagittal COM position) can be modelled as simple harmonic motion of leg springs, interspersed with (variable) periods of non-(simple) harmonic motion, and that the F/t flat spots relate to periods of non-(simple) harmonic motion with respect to the leg springs (not necessarily the COM, though I'm not sure). Comments?

The most complicated period is where both springs are in contact with the ground, and I think the best place to start would be to define the essential aspects of this system, and what it is, exactly, that we're trying to show. The dynamics of the single spring periods should arise, within the appropriate context, from a simplification of the equations for the most complex period.

What I'm ultimately looking for out of this is to show the required dynamic relationship between the spring constants of the 2 legs (relative leg stiffness), which will give a certain profile of COM accel. (and, hence, profile of COM position) in the sagittal plane (which includes both vert. and horiz. movement (which is antero-posterior, with respect to the body)). So, this should allow both modelling and the characterisation of real patterns (with respect to the model specifics).

But, the analysis of this system requires the definition of specific starting conditions, and I think it would be much better if we use numbers during the development which are near those which would be seen in a "normal" adult. The equations could be developed purely algebraically, but I've found that it's much better to plug in "real" numbers periodically.

I've attached a preliminary diagram of what I see as the essential aspects of this system, Fig 1.

The required initial input is as follows:

1) resting length of the rear leg
2) fractional compression of rear leg at start (ie. at rear-leg1).
3) horiz. accel. of COM
4) start horiz. vel. of COM
5) vert. accel. of COM
6) start vert. vel. of COM
7) rear-leg/floor angle at start (angle A)
8) front leg resting length (front-leg1)(which defines front leg/floor angle (angle B) based on initial conditions, since for now it's assumed contact is made at the front-leg resting length)
9) standard time period

I think this is all. Everything else is defined by these conditions.

Can anyone say what are "normal" values for these, somewhere near the start of double support?


Mike M

 

Hi guys,

I can't seem to edit my first post, and would like to reiterate that there was an error. The discussion of the line shapes refer to the sag. COM (x,y) plot, not the sag. COM (x,t) plot. The (x,t) plot is not a function of horiz. accel., while the (x,y) plot is.

So, just to clarify. The comments were regarding flat-spots on the vF/t curve, and whether there would be characteristic COM trajectories in the sagittal plane which could be consistently associated with them. I'm just using (x,y) for the sag. grid, with x axis as left/right and y axis as up/down. Also, there are 2 main types of plots, (x,y) and (x,t), where t=time.

Flat-spots on the vF/t curve are due to periods of constant vertical force, which means constant vertical acceleration. There are 2 main conditions with constant accel., when accel.=0 and when it's not 0.

When vertical accel. is 0, vertical velocity is constant, but the velocity vector could have any orientation, like at 45deg to the floor. This would show as a straight line on the (x,t) plot (at any angle), but the line shape on the sag. COM (x,y) plot is also a function of the horiz. accel. On the (x,y) plot, a flat spot on the vF/t curve, with vert. accel. equal zero, would show as a straight line if horiz. accel. was 0 (constant horiz. vel.), but a curve if horiz. accel. is not zero.

When vert, accel. is not zero, but still constant (in order to produce the flat-spot on the vF/t)), vertical velocity is changing at a constant rate, but the velocity vector could have any orientation. In the (x,t) plot, this would show as a curve, and on the (x,y) plot, it would show as a curve, which is also a function of the horiz. accel.

And, any curves may be so slight that they appear as straight lines on the real graph.

Note that the forces associated with horiz. accel. would not show in the vF/t curve. Horiz. accel. is a only a factor in COM position on the (x,y) plot.

So, there could be great (theoretically infinite, but practically limited, of course) variability in the sag. COM (x,y) and (x,t) plots related to the flat-spots on the vF/t.

Just a couple of general comments on the development of technical methods.

You guys are obviously having some trouble defining the relevant technical aspects of this system, but you don't need a math person, which I'm not (really), you need a problem-solver, which I am.

You very likely all have the required math skills, you just need to look at the system and organise it in an appropriate and effective way. But, you also have to keep in mind that during the development, your eye can't be just on the identified prize, which is a way to model flat-spots using harmonics.

For eg., I stated that flat-spots on the vF/t curve could never be (accurately and consistently) modelled with simple harmonics, using the biped spring model. However, that doesn't mean these periods can't be modelled based on definable deviations from simple "harmonicity" (is this a word?), as you propose with varying k(leg).

Or, it might be that one leg maintains relative harmonicity, while the other could be defined using a specific deviation (which could be harmonic) superimposed on the normal harmonics (which would appear as non-harmonic).

In order to study this, though, the core system has to be properly defined already. And, it should be defined in such a way as to allow the investigation of different (currently undefined) hypotheses, not just the one of current interest.

So, if you're serious about wanting to develop this model to the point of application, fully defining Figure 1 in my last post is the first step, I believe, no matter how the rest of the work progresses. I think this figure is relevant for all discussions regarding the technical characterisation of the biped spring model. Once I get some input for the initial conditions, I'll start to label Figure 1.

Oh yes, another start condition needed is the mass. 80kg seems to be a popular number, but this would have to mesh with the other values.


Mike M

 

Howdy,

I found another foolish error in the discussion on vF/t curve and COM traj, and it will haunt me until it's corrected.

The reference frame defines the y-axis as up/down, and that's what we're talking about, so the (x,t) should be (y,t). I hope this didn't cause confusion. In the discussion, I knew that I meant vertical, I don't know why it didn't click that x was horizontal.

Using a number different reference frames makes me a bit blind. I very humbly apologize for this error, and hope you still thoughtfully consider the discussion.

Also, I may have hi-jacked the discussion, if you didn't want it to go down this road. If no one responds with respect to initial conditions, I won't post anything else, and please excuse the interference.


Mike M

 

Mike I think this problem requires someone with a good maths education, I tried looking at the harmonics in terms of a Fourier transform as did Murphy and Dananberg. I got the same outcomes as those found by these two did on TekScan but could find no real application to this problem but then the maths ran out anyway.

Are you saying it is possible that your proposed model using initial conditions could make an extrapolation from the vertical force time data to CoM positional in the saggital plane x,y displacement?

Regards Dave

 

I've been away teaching in Portugal and I'm pretty tired today, I'll read and digest and make comment later. In the meantime you and Dave seem to be moving along nicely.

 

Yes this sounds like a reasonable proposal (last few paragraphs) we need some references to work out from to see where we are going. It also need the right skills applied.

 

Late edit

Ooops! Just noticed that should say 3rd btw.

Dave

 

Thanks for the comments, Dave.


Are you saying it is possible that your proposed model using initial conditions could make an extrapolation from the vertical force time data to CoM positional in the saggital plane x,y displacement?

Absolutely. And, vice versa. It should also be able to model using purely theoretical data, and output COMtraj, force (vertical and horizontal), etc. I don't see any reason this can't be done, since this is a strict vector system.


{When vertical accel. is 0, vertical velocity is constant, but the velocity vector could have any orientation, like at 45deg to the floor. This would show as a straight line on the (x,t) plot (at any angle), but the line shape on the sag. COM (x,y) plot is also a function of the horiz. accel. On the (x,y) plot, a flat spot on the vF/t curve, with vert. accel. equal zero, would show as a straight line if horiz. accel. was 0 (constant horiz. vel.), but a curve if horiz. accel. is not zero.}
I think this statement is wrong - in terms of a FTC the vertical acceleration can never be zero if the velocity is constant since constant velocity must equal/concomitant with 9.81m/s/s acceleration and be represented as normal body weight * gravity on the force time curve.
Therefore at a constant vertical velocity the force time curve is flat at body weight and the CoM could be moving in any direction at any speed that returns a constant v velocity. A vCoM displacement with constant acceleration above or below the baseline of 9.81m/s/s (i.e.measured acceleration +gravity or absolute acceleration) would return flat lines at the relative force. Only a change of position with respect to time in the 3rd derivative would produce a curve or slope i.e. changing acceleration. (da/dt)

I think my statement is correct, once you replace (x,t) with (y,t). But, in the last sentence I should have put, "On the COM (x,y) plot..."

Or it would have been better as, " A flat spot on the vF/t curve, when vert. accel. equals zero, would show on the COM (x,y) plot as a straight line (at any angle) if horiz. accel. was 0, but a curve if horiz. accel. is not zero."

The differences in line shapes refer to the COM plots, not the vFTC. The vFTC is showing a flat period for all of these descriptions.

But, I'm not sure if I understand exactly what you're saying. Constant velocity means 0 accel. by definition, and for the primary analysis the reference frame has to be net effects on the COM.

I think this is just a mixing of reference frames, and there's no conceptual disagreement. Your use of "relative" and "absolute" are based on an interpretation for a specific analysis.

I'm not judging it in any way, but this can't be built in to the core equations. However, it can be readily applied once the essential characteristics are defined (ie. there should be no problem incorporating that perspective, as a secondary analysis, at a later time).


I believe we're on the same page, just a problem with terminology.

Before we dig in, I think a few general comments would be useful.

There's no way to know how this kind of thing will go, and there are a number of different ways to look at it. I've only looked at it very superficially up till now. My modus operandi is to incorporate maximum versatility, so the end result is a tool(s) for investigation which isn't limited by a specific interpretation.

But, of course, anyone can take any road they wish. I'm going to be creating a spreadsheet (SS) for this, and each person can make their own, or I'll give you mine periodically, if you like.

For this project, there are 2 main applications, the measurement of real gait patterns and the modelling of gait patterns. Each of these applications requires different input, but I'm going to incorporate both in a single SS program.

This will make it more difficult at first, but in the end, what I expect to have (barring unforeseen complications) is a program that can take the real COM and floor contact point co-ordinate data, and output force, accel., etc. based on the model details. Then, this output can be checked with respect to measured force, accel., etc., in order to determine what has to be done to increase the accuracy (often by incorporating caveats), and to evaluate and improve the efficacy of the model.

Also, we'll have a SS which will take specific input data sets, like accel., vel., etc., and output a graph of the resulting COM trajectory (and other relevant values). For this application, since each term in each equation is defined in a SS column, the values for one or more terms can be manipulated in a host of ways (manually or by use of equations), without changing any of the SS calculations.

And, the values for each term can be graphed in a couple of different ways, in order to help evaluate the physical significance of each term.

So, in the end, we'll have a SS program that not only allows the measurement of real patterns, with resulting data presented within the context of the model, but also a method to evaluate how changes in each term change the observed results. In short, we'll have the core elements for a global gait analysis system based on the sagittal biped spring model.

Whether this will be of any functional value or not is something only real application can determine. I have a number of ideas to facilitate and advance the application, but these would only cloud the issue right now.

The most important thing to remember overall is that this is a strict vector system.

Trigonometry and algebra are fully relevant and the first step is to show, on Figure 1, all of the relationships which we know, as well as the equations which are relevant (keeping in mind that we're looking at sequential time points, so the COM trajectory is linear). Also, each variable should be defined in as many ways as possible.

And, we don't need to use differentials, Fourier transforms, etc., at this stage, since first we have to determine the nature of the equations.

It will take a few days for me to work up the next figure, since there are a lot of labels and equations.

One last comment for this post.

(Crickey I hope I'm right here I'm pretty sure I am otherwise I'm going to look a right plonker (doubts creeping in))

Whether you're right or wrong, Dave, by putting your thoughts out for discussion even though you're not sure, what I see is a right good researcher.

This isn't an easy project. I don't know how long it will take or what will come out of it. I do know, though, that there will be lots of mistakes. Conflicts in terminology, errors in labelling, conceptual misunderstandings, etc.

Sometimes what makes something wrong isn't immediately evident, it just "feels" wrong. We can't wait until we're sure about things before bringing up issues.

If there's even a hint of a suggestion of a possibility of a chance that there's an error or problem, it has to be immediately investigated, exposed and analysed to determine, first, if there is an error, and, second, if we find an error, how it affects the rest of the material.

In the end, it's conceivable that we could produce a method which will be used by others for a long time. If this does happen, the method has to be rock solid, with respect to the technical aspects, and that has to start at second 1 with constant critical (but fair) evaluation.

The most significant advantage in this type of system is that we have a 2d figure which defines a vector system, and which totally defines the physical system of the model (regardless how the model relates to real gait). That means that all (or the vast majority, at least) of the equations we're going to need should be derivable from a detailed geometric (vector) analysis of the main figure.

I'll let you go there, and start working on the figure. I hope you do this yourself as well, and then we can compare and correlate what we come up with. But you might want to wait for mine for this first one (a week or so), since there are a lot of arbitrary labels, and it would be much better if we're all using the same ones. If there's any problem with the labels I use, they can be changed if necessary.

This first step is only to identify all the geometric, etc. relationships we can, including force, accel., etc., as well as list any relevant general equations (fully expanded), like for constant accel. And, we might as well include kinetic and potential energies as well. Who knows where we'll find insights.


Mike M

 

Mike

This paper attached will provide a lot of the data you require I think.

Dave

 

Thanks for the paper, Dave. Unfortunately, when I try to open it, it says that it's damaged and can't be opened. This also happened when I tried to download Geyer's thesis, from Simon's earlier post, although I had no trouble with Geyer's paper in the same post.

I'm sure this is a problem on my end, since I've been having nothing but trouble with the computer lately. It usually freezes several times a day now. I have no trouble with e-mails, if you could e-mail it to me.

I've attached a preliminary figure, Figure 2, outlining a number of equations and calculations which I just brainstormed. I haven't judged anything, and included even the most simple ones. All algebra and trigonometry relationships and reductions are also relevant, though not shown.

Let me discuss a bit how I'm approaching this.

When looking at any new problem, the first thing I do is try to become more comfortable with the information. For this, it's by looking at all the equations, and making sure that I fully understand each one. For many equations, this is trivial, like for the constant accel. equations.

But, we're going to be creating new equations (not that many, but a few), and these have to be fully understood, every term as well as how the different terms are associated. Remember that the language for this process is equations, and it will be through a recognition of the physical significance of each term, as well as the entire equation, that will allow us to make advancements outside the confines of the original proposition.

Absolutely every term and every equation, ipso facto, has a physical meaning (at least within the technical framework of the model, but possibly also more generally), regardless whether we understand it.

Some equations may be complex at the beginning, so the physical correlation is very difficult. That's why we have to play around with the equations, using established mathematical methods (since this is a strict vector system), in order to put them in to forms (there aren't that many options) where we can more easily recognise the physical relevance.

So, it's essential to keep in mind that at this stage it's all observation (and we have to be very good, creative observers), since we're not doing anything new yet.

In Figure 2, the vertical force equation shown in the box at the bottom is one that we need to understand fully. This equation is (partially) expanded in Figure 3.

Note that this is only an example of the process, and is far from a complete analysis of this equation. Also note that in the general force equations, I've added an extra term to account for external forces, like being pushed, the wind or a backpack, etc.

If we include that now it provides the means to account for any external forces without changes to the SS calculations. Even if there are multiple external forces, they can be resolved to a single vector in another SS, and incorporated via this term. There's also one or two other ways to use this, but that's for a later time.

Also, we're modelling a dynamic system which has alternating periods of single and double support. A very good way to keep checking the equations is to periodically insert values for specific time points, to make sure they reduce to the appropriate equations.

An example is the last equation on the left in Figure 3. This is (one form of) the total vertical force equation for the period of double support. By putting the fractional compression term (M) for the right leg equal to zero, since this means there is no force coming from the right leg, this equation should also appropriately describe the period of single support.

I won't go through the algebra, but you'll find this equation reduces to the total vertical force equation, without the term for the force from the right leg, since it's zero. And, this is as it should be. But, this doesn't prove anything, except that there hasn't been an error in the algebra.

Also, since most of the equations should be multi-variable, we should be on the lookout for ways to simplify and/or impose standard (relative) values on some of the variables, in order to isolate a single variable so we can better evaluate how things change when only that one changes. This would be time dependent, and isn't as simple as it sounds for this system.

This is all pretty much independent of the SS program. The calculations necessary for the SS program aren't an easy thing to discuss. I'm not even going to start that for a week or so, in order to become more familiar with the physical system and general equations.

In the meantime I'm going to kick around these equations, especially the force equations, since these are directly related to acceleration, and, hence, changes in COMtraj.

The force equations should be expanded and rearranged in as many ways as possible, and each equation along the way analysed individually, with respect to the nature of each term, how it varies, as well as the position of the term in the equation.


Mike M