Knee Spring Memo 14062004

jb. left side.7 trials. time spacing = twenty four Hz

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Figure 1: Sagittal plane view of leg joint centers during a single step, heelstrike to heelstrike. Markers are the knee indicate different phases of the gait and are referred to in further plots.

Exoskeleton Memo. 14.06.2004.v1. Knee Springlike Behavior During Walking. dan paluska. june 14, 2004.

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Analysis Part 1. Fitting mono-articular linear springs to 5 subjects.

Gait data from 5 subjects was analyzed. The subjects were walking at self selected walking speeds ranging from 1.2 - 1.4 m/s. The subjects ranged in size and weight from 165 to 192 cm and from 57 to 83 kg. The linear spring constants found for this region were mean 351 Nm/rad , with a standard deviation of 40 Nm/rad. Normalized across the subjects using body weight(BW) and leg length(LL) gives a resulting ’stiffness’ of 0.56 with a standard deviation of 0.12. Figure 1 shows sagittal plane snapshots of the leg during a single walking cycle. Four sections of the gait are labeled with different markers at the knee joint. The first part of the gait is labeled with blue circles at the knee joint. This corresponds to initial stance when the knee is flexing. The next stage of the gait is the mid stance when the knee is extending and the leg is pivoting over the foot. The next two stages take the leg through the rest of the stance and swing phases. In this analysis we are primarily concerned with the first two phases of the gait cycle. These are highlighted in the position and moment graphs of Figure 2. The linear spring at the knee can also be viewed as a physically linear spring between the hip and ankle with a non-linear mathematical force-length relationship.

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subject JC. averages. position vs. knee torque.

subject JB. averages. position vs. knee torque.

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Figure 2: Knee angle vs moment averages for 2 of the five gait subjects. The black solid like represents a linear spring fit to the knee behavior during early stance.

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knee position(radians). positive motion is anterior rotation.

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region of interest

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knee torque(Nm). positive moment generates anterior acceleration. 40 30 20 10 0 -10 -20

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Figure 3: Joint position and moment data for the human knee during a single stride. From Bogert ’exotendons’ paper. Highlighted area is early stance and indicates region of interest for the following graphs.

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Analysis Part 2. Resultant moments from various spring models

The data from the Bogert ’Exotendons’ paper was used to analyze several different spring types at the knee. We are considering the knee only in the initial part of the stance phase as highlighted in Figure 2. We use the position and torque conventions used in the Bogert paper. Positive torques about the joints result in anterior accelerations of the distal segment. In the case of the knee, this means a positive torque will serve to straighten the knee joint. We input the position data into various spring models and calculated the torque that would result if that spring was actually at the knee. Moment arm or spring constant values were then optimized with ’fminsearch’ in MATLAB. The cost function used was the integral of the absolute value of the net remaining torque. Code is available to those interested. The four spring models were as follows: 1. Rotary spring with constant linear stiffness. This is equivalent to a linear constant stiffness spring at a fixed radius(acting about a pulley). m1 = Krotary ∗ theta 2. Linear spring with a constant stiffness acting on a lever(moment arm changing with angle). m2 = Klinear ∗ r2 ∗ sin(theta) ∗ (Lo − L) (where L is calculated using the law of cosines) 3. A linear hardening spring acting on a pulley(constant radius). m3 = Klinear ∗ r3 ∗ (Lo − L) ∗ abs(Lo − L) 4. A linear hardening spring acting on a lever(variable moment arm length). m4 = −delta. ∗ abs(delta). ∗ gain4. ∗ sin(pi/2 − angleK) Results from these models are shown in Figure 2. The original knee moment from the human data is shown in solid blue. The other four spring models are shown as indicated in the figure legend. It is important to note that all of the spring models have quite similar behavior. All resultant torques lead the actual torque by a bit as well. This is due to the fact that the actual position leads the actual torque. Any spring of 3

knee joint moment(Nm). real and simulated spring models.

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real m1 m2 m3 m4

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Figure 4: A plot of knee moments. Original knee data is shown in solid blue. All four spring models(m1-m4) are shown as well. These values are calculated in MATLAB as described in the section.

actual joint power and net joint power for various spring models.

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power net1 net2 net3 net4

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Figure 5: Knee power and net remaining knee power when spring models(m1-m4) are implemented. Net remaining powers are calculated by multiplying the net moment by the original velocity data.

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Leg Length(m) Body Weight(N) Kknee (Nm/rad) Max deflection (rad) Kknee /(BW ∗ LL) Eq Klinear (N/m)

Sub 1 0.86 560 361 0.25 0.749

Sub 2 0.98 809 357 0.22 0.45

Sub 3 0.92 640 284 0.27 0.482

Sub 4 0.89 732 392 0.33 0.602

Sub 5 0.93 752 361 0.29 0.516

Mean

Standard Dev

351 0.38 0.5598

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Table 1: Human subjects, their measured parameters and calculated quantities. the style F = kx or F = kx2 will result in the position leading the torque as shown here. Future work should include looking into a nonlinear spring with non-zero initial stiffness F = kx(x + of f set). For this analysis, we are assuming springs only act in one direction. For a bidirectional quadratic spring the formula is F = kx|x|. It is also important to note all spring moments start and end at zero. This is a requirement is the springs are clutched on and off. When clutched on, the springs have zero energy, which means deflection must be zero. When clutched off, the springs could have non-zero energy but that energy would have to be dumped somewhere. Figure 2 shows the knee power from the human data as well as the remaining knee power as calculated for each of the spring models. The net power for each model is calculated by multiplying the net moment times the original joint velocity. Notice that net1(the simple constant K rotary spring) is the one spring model which require net negative power from the knee joint. At least qualitatively, all spring models seem to leave about the same work for the knee, just in different patterns.

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About the Human Data

The data from the five subjects used in the first part of the analysis was received in .csv form from Spaulding gait laboratory. Java code(originally written by AH and modified by DP) was used to extract columns of interest and heelstrike-to-heelstrike data sets. Table 3 shows the data for the various subjects and the calculated means.

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