Exo Ankle Springs Memo V5

ankle springs memo. july 28 2004. rev 5. djp. compare 1. monoarticular spring - Lever vs Pulley? 2. biarticular gastroc-like spring - levers vs pulleys? 3. both mon and biart spring in parallel. - levers vs pulleys? 4. clutched mono and biart springs with common elastic element. 5. solutions with and without positve and negative power weightings. [Abbott et al 1952] negative 1/3 to 1/7 as costly. [DeLooze et al 1994] neg work 0.3 to 0.5 as costly. we will use 0.5 as conservative factor. emg and ankle muscle information from the literature. human walking book. indicates gastrocnemius(gas) and soleus(sol) are both on at the same time. the soleus is on from 11-47% of the cycle and the gas is on from 13-50% of the cycle. this is averaged population data. [hof et al 1983] has raw and filtered EMG data for individual subjects. it doesn’t significantly differ from human walking book but shows more detail of the magnitude changes in time. there is usual a peak at each end for the sol and maybe a peak at the end for the gas but not as consistant. [gottschall and kram 2002] found that when providing a 10% bodyweight propulsive force the EMG of the GAS decreased to 59% it’s original value. the SOL emg did not change. they believed they negated the conclusion of [neptune et al 2001] [neptune et al 2001] simulated the sagittal plane of walking with six muscles in the leg. they determined that the sol function is to propel the body forward and that the gas function is to break the knee to initiate swing. [hof et al 1983] estimate neg work of 0.2-0.35J/kg and positive work of 0.35 to 0.5J/kg at the ankle during walking. ankle [rad],[Nm],[W],[rad/s] curves[bogert 2003]. 70kg. 1.2m/s. 0.9m leg. 1s step. ankle angle

ankle velocity

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pos area = 788 neg area = 224

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if we consider the shape of the angle curve vs the shape of the moment curve we can see that a monoarticular linear torsional spring cannot capture the position/torque dependance. it is important to note that the peaks of the torque and position curves do not line up as well as the fact that they are different shapes. the power curve has approximately 3 times as much positive area as negative.

1. monoarticular ankle spring. theoretically, a monoarticular spring would at best capture the full negative power area(energy) and transfer that into the positive power region. we will define the savings as follows: total = neg power integral + pos power integral = 224+788 = 1013 weighted total = 0.5*negpower integral + pos power integral = 0.5*224 + 788 = 900 savings = 1 - net/total weighted savings = 1 - weighted net/weighted total the best theoretical savings would be savings = 1 - (total-2*224)/total the best theoretical weighted savings is a little more complicated. results monoarticular spring with a pulley attachment: ankle power, weighted net = 0.64584, springNEG = 314.4592, springPOS = 321.6358 200 original net springs

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ankle moment, pulley rad = 0.054875, k-rotary(Nm/rad) = 301.1266 40 original pulley lever net

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results for a lever arm optimized case. ankle power, weighted net = 0.64053, springNEG = 323.6855, springPOS = 331.0272 200 original net springs

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ankle moment, pulley rad = 0.059188, k-rotary(Nm/rad) = 350.316 40 original pulley lever net

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in both the above cases we use the weighted power integral to find the optimal moment arm. in this case it turns out to be about 36% savings for a pulley of about ~55 mm or a lever of ~60mm. [Bogert 2003] found a pulley radius of 60mm in his paper with unweighted net power optimization. This is a K_rot ~ 300 Nm/rad for a 70kg person. ~4.3 Nm/rad per kg. shifting the zero ankle angle by a few data points doesn’t change much. 1-3% for ankle rest angle beingany one of the first 4 ankle data points. starts to trail off after that.

************************** END MONO ANKLE SPRING *********

2+3. biarticular gastroc-like spring and biart+mono. the biart here is a lever attachment at both ends. when we use a pulley the optimizer ends up placing all the load onto the knee joint. it finds moment arms of 1m+ at the knee...

we consider a physical linear extension spring traveling from the thigh to the foot. ankle power 200 original net springs

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in both cases. Below are the permutations and someof the savings and moment arm sizes that were found. K(knee biart moment arm), A(ankle biart moment arm), mA(mono ankle moment arm). pulleys. weighted. %50 savings. A(0.0065m),K(0.35m),mA(0.1564) pulleys.unweighted. %52 savings. A(0.0001m),K(24.6m),mA(0.17) there are multiple solutions with similar results for this case but all have unbelievably large knee moment arms. levers. weighted. 50% savings. A(0.034m),K(3.9m),mA(0.05) this can blow up in K moment arm depending on initial value. levers. unweighted. 45% savings. A(0.042m),K(0.67m),mA(0.0076) Even the smaller moment arms here are a bit unrealistic in size. It seems to me the way to get the good knee power transfer is to have a much larger moment arm at the knee than the ankle.

4. clutched biart and mono. we consider the case where we have two clutches attached to a single elastic element. this is modelled after gastroc and soleus connection to achilles tendon. we consider the effects of the following sequence. gastroc clutches. knee extends and lengthens achilles tendon. soleus engages and gastroc disengages. energy stored in achilles tendon is released through the ankle only.

ankle power, weighted net = 0.23738, springNEG = 126.4582, springPOS = 758.1835 200 original net springs

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ankle torque, maA =0.045 maK =0.045, k-ankle-rotary(Nm/rad) = 202.5

is the net too high?

40 original net tA - CLUTCHED t2 - SOL tauA - GAS

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this results in 76% savings with realistic moment arms. this is using pulleys not levers. the lever version hasn’t been fully tested yet but it’s not expected to be significantly different. the knee power ends up doubling and the total knee and ankle cost stays basically the same(1.12 times greater). this makes since because we’re using passive elements to tranfer the power from the knee to the ankle. but we lose a bit from overshoots. this results in a strain of about 2% in the achilles tendon. it is clear from the torque curve much of the residual torque is positive which indicates it is torque which would have to be generated by the tibialis anterior. this is possibly unrealistic given that it is much weaker(~10x?) than the plantarflexors. in the above graph, tA is the end result ankle torque(red with + marks). t2 is the torque from the mono ankle spring, and tau_A from the biarticular gastroc like element. At ~40 on the X axis is where the clutch switches and you can see tA switch from tau_A to t2. we can see thatt2 is shallower than tau_A which gets us closer to the actual torque curve. the monoarticular clutch switches off at t=62(toe off). sensitivity to switch time: switch time 34 -> switch time 36 -> switch time 38 -> switch time 40 -> switch time 42 ->

67% savings 72% savings 75% savings 76% savings 75% savings

switch time 44 -> switch time 46 ->

72% savings 66% savings

it seems roughly symmetric and some sort of exponential type drop off in savings. +/- 4% of cycle still gives us really good results.

additional optimizing. if we run the optimizer on the knee and ankle moment arms it finds more savings when the moment arm at the knee is 0.0522 and the ankle moment arm is 0.0415. ankle power, weighted net = 0.17823, springNEG = 105.3197, springPOS = 797.1364 200 original net springs

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ankle torque, maA =0.0415 maK =0.0522, k-ankle-rotary(Nm/rad) = 172.225 40 20

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this results in an over 80% savings(weighted) at the ankle(1.16 total energy unweighted) but the declutching of the SOL spring at t=62 results in a torque jump, or a dumping of spring energy. this may be okay if the energy is coming from powered hip motors but not an ideal case if we’re trying for a purely passive system. the take home lesson here is the knee and ankle moment arms need to be the same length. or at least the knee can’t be longer than the angle. both the knee and the ankle have roughly the same excursion so in orcer for the ankle to dump all the energy the knee builds up, it needs to be able cause the same amount of displacement in the opposite direction. this is dependant on the angular travel and moment arm.

another option is to have the longer knee moment arm but clutch over before it maxes out. this works but gives us less savings than the equal moment arm case. to really evaluate this procedure we will need to run it on many sets of data and determine how these sensitivities will play out on the variances in person to person movement.

we can optimize for total ankle and knee cost ankle power, weighted net = 0.63149, springNEG = 301.5171, springPOS = 327.4916 200 original net springs

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ankle torque, maA =0.0542 maK =0.0013, k-ankle-rotary(Nm/rad) = 293.764 40 original net tA - CLUTCHED t2 - SOL tauA - GAS

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this results in basically the same knee energy, 42% ankle unweighted savings(37% weighted) and 22% knee+ankle unweighted savings. maximum strain are on the order of 2-5% for reasonable moment arms.

non-linear springs stress-strain curves vs force length curves. [pioletti et al 1998] tendon viscoeleastic behavoir. k changes by factor of 6 roughly from low to high strain for fast movements. this happens over roughly 5% strain. we could define k(x) = k_initial*e^(36*(x-x0)). e^1.8 ~ 6. this would make k travel from k_initial to 6*k_initial from 0 to 5% strain. the figure below compares the force resulting in two different GAS springs. one spring has a constant K, the other has a spring constant that varies from K/1.8 to 3.3K over 5% strain. this was chosen so the maximums would

roughly be the same magnitude. we can see that the non-linear spring has a sharper profile than the constant K spring. biarticular force with linear and non-linear springs 2000 linear spring exp spring

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at first pass, a non-linear spring decreases the energy storage in the tendon. this can be seen in a force length plot of linear vs hardening springs. the area under the curve is larger for the constant K spring given the same maximum force.

Table 1: Energy absorption of various materials.[Gordon 1978] Material

Max

Ancient Iron Modern spring steel Yew wood Tendon Rubber

Strain[%]

Max

Mod of Density Stress[MPa] Toughness[MJ/m3]

Max Energy [kg/m3]

[J/kg]

0.03 0.3 0.3 8.0 300

70 700 120 70 7

7,800 7,800 600 1,100 1,200

1.3 130 900 2,500 8,000

0.01 1.0 0.5 2.8 10.0

getting out what you put in how much efficiency can we expect from springs? [Brown and Zeglin 1998] got 70% energy recovery for a fiberglass bow spring hopping robot. [Czerniecki et al 1991] found flex foot prosthetic feet to return 84% of the input energy and seattle prosthetic to return 52%.

possibilities the RF and VAS muscles also end together in the patellar tendon.