Trouble modeling runner’s swing-leg as double compound pendulum.

[tedjan]

There is a problem in biomechanics that relates to a human runner’s gait (1) and how much power they consume. At each speed a runner chooses a Preferred Step Frequency or PSF (2). If an athlete tries to run with a step frequency that differs from their PSF at that speed, their oxygen consumption increases (3). There is no physics based model to explain this phenomenon.

I thought I could make headway on the problem by treating the swing leg, not the stance leg, as a double compound pendulum hung from the hip (1). Unfortunately, the top of the hip or pivot for the upper pendulum is not fixed. Instead, the hip pivot is being accelerated in the horizontal and vertical directions by the changing forces and angles of the stance leg (4). The accelerations are time dependent. They vary greatly while the other leg, the stance leg, is in contact with the ground.

I think that there are two ways of modeling the swing leg from its trailing position at toe off , thorough its position next to the stance leg at mid-stance to its forward swing position just before touch down. The swing leg can be treated as a double compound pendulum hung from the hip in two possible reference frames. In the laboratory reference frame it can be modeled as if it is hung from a jerking moving pivot. In the Center of Mass reference frame – the CM is close to the hip pivot – it can be treated as if it hung from a fixed pivot in a non-inertial accelerating CM frame.

I thought it would be easier to treat the swing leg in the hip or CM frame and apply the accelerations of the hip support directly to the masses as in m[1,2]*a[x,y].

Maybe this is the wrong approach, but I am having a great deal of difficulty with this problem. I can’t write the Legrangian because the potential energy is not constant since the vertical acceleration changes from zero at touch down to about 2-g at mid-stance. At the same time there is a horizontal acceleration on the masses. It increases from zero at touch down to a positive peak and then decreases to zero at mid-stance. It then does the reverse from mid-stance to toe-off.

I’ve been working on this problem for several months with little to show. I was able to push a fixed but flexed leg (5) through the swing cycle and confirm that the swing time for the fixed but flexed leg came very close to the swing time for the leg while running.

I would greatly appreciate any suggestions on solving this problem.

Ted

(1) http://members.aol.com/EasyExperiments/GaitCycle/GaitCycle.gif
(2) http://members.aol.com/EasyExperiments/PSFVsSpeed.gif
(3) http://members.aol.com/EasyExperiments/SmithAndHunter.gif
(4) http://members.aol.com/EasyExperiments/Accelerations.gif
(5) http://members.aol.com/EasyExperiments/BentKneeLeg.jpg

 

[eljose79]

Dear Ted,

Thank you for bringing this interesting problem to our attention. As a biomechanist and physicist, I can understand the difficulty you are facing in trying to model the swing leg during running. It is a complex and dynamic process that involves multiple forces and accelerations.

Based on your description, it seems like you are on the right track in treating the swing leg as a double compound pendulum. However, I would suggest considering the stance leg as well in your model. The stance leg also plays a significant role in the overall gait and its interaction with the swing leg should not be overlooked.

In addition, instead of trying to model the swing leg in two different reference frames, I would recommend using a single reference frame and incorporating the accelerations of both the hip and the stance leg. This will give a more comprehensive understanding of the forces and accelerations involved in the gait cycle.

Regarding the issue with writing the Lagrangian, I would suggest considering the stance leg as a fixed point and treating the swing leg as a moving system. This will allow you to write the Lagrangian for the swing leg without the need for a constant potential energy term.

I understand that this problem has been challenging for you and I commend your persistence in trying to solve it. I would also suggest seeking the help of other biomechanists and physicists who may have experience in this area. Collaboration and discussion with others can often lead to new insights and solutions.

Best of luck in your research and I hope these suggestions will be helpful to you.

 

[charng]

Thank you for bringing this issue to my attention. I can understand your frustration in trying to model the runner's swing-leg as a double compound pendulum. It is a complex problem that involves both physics and biomechanics.

Based on your approach, it seems that you are trying to model the swing-leg as a rigid body with two masses (representing the thigh and shank) connected by a rod (representing the lower leg). However, as you have pointed out, the problem lies in the fact that the hip pivot is not fixed and is being accelerated in both horizontal and vertical directions.

One suggestion I have is to consider the swing-leg as a flexible body instead of a rigid one. This would allow for the varying accelerations at the hip pivot to be taken into account. You could also consider using a Lagrangian formulation to model the swing-leg as a series of small masses connected by springs, with the hip pivot being one of those masses. This would allow for the varying accelerations to be incorporated into the potential energy term.

Another approach could be to use a multi-body dynamics software, such as OpenSim or AnyBody, to model the swing-leg. These software programs allow for the incorporation of complex biomechanical movements and could potentially provide a more accurate representation of the swing-leg as a double compound pendulum.

I understand that this problem has been a challenge for you, but I encourage you to continue exploring different approaches and seeking input from other experts in the field. With persistence and collaboration, I am confident that a physics-based model for this phenomenon can be developed.