# Asymmetric Passive Dynamic Walker

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3 [ ] H11 H12 H = H21 H22 (27) h = (mt sw b2 sw +ms sw (lt sw +b1 sw ))L st sin(q 2 q 1 ) (28) [ ] 0 h q B = 2 (29) h q 1 0 Fig. 2. The actual gait pattern of the walker with the mass location. See the attached video for a further demonstration of the ive dynamic walker asymmetric gaits with several parameters changed. H22 = mt sw b2 2 sw + ms sw lt 2 sw (6) H23 = ms sw lt sw b1 sw cos(q 3 q 2 ) (7) H31 = H13 (8) H32 = H23 (9) H33 = ms sw b1 2 sw () H11 H12 H13 H = H21 H22 H23 (11) H31 H32 H33 h122 = (mt sw b2 sw + ms sw lt sw )L st sin(q 2 q 1 ) (12) h133 = ms sw b1 sw L st sin(q 3 q 1 ) (13) h211 = h122 (14) h233 = H23 = ms sw lt sw b1 sw sin(q 3 q 2 ) (15) h311 = h133 (16) h322 = h233 (17) 0 h122 q 2 h133 q 3 B = h211 q 2 0 h233 q 3 (18) h311 q 1 h322 q 2 0 g 1 = (ms st a1 st + mt st (ls st + a2 st ) + (mh + ms sw + mt sw )L sw ) sin(q 1 )g (19) g 2 = (mt sw b2 sw + ms sw lt sw ) sin(q 2 )g () g 3 = ms sw b1 sw sin(q 3 )g (21) G = [g 1 g 2 g 3 ] T (22) The two-link dynamics are described by the Lagrangian of a double pendulum system. The matrices are as follows: H11 = ms st a1 2 st + mt st (ls st + a2 st ) 2 + (mh + ms sw + mt sw )L 2 t (23) H12 = (mt sw b2 sw + ms sw (lt sw + b1 sw ))L st cos(q 2 q 1 ) (24) H21 = H12 (25) H22 = mt sw b2 2 sw + ms sw (lt sw + b1 sw ) 2 (26) g 1 = (ms st a1 st + mt st (ls st + a2 st ) + (mh + ms sw + mt sw )L sw ) sin(q 1 )g () g 2 = (mt sw b2 sw + ms sw (lt sw + b1 sw )) sin(q 2 )g (31) G = [g 1 g 2 ] T (32) The collision events are derived from conservation of angular momentum about the foot for the whole system and the hip for the swing leg. For those equations and a more in depth description of the dynamics, see [7]. IV. RESULTS For the model to effectively exhibit an asymmetric gait, we had to change the physical parameters that defined it. Starting with an ideal (symmetric) step pattern, we took each variable and incremented it through a range of values while holding the other parameters constant. Parameters were changed on the right leg only to differ its gait pattern from the ideal leg. The changed parameters included: mass of the thigh, mass of the shank, location of the knee, location of the thigh mass, location of the shank mass, and total length of the leg. These parameters do not necessarily have physical analogs to those of a human, but the changed parameters generate gait patterns that may be synonymous with gaits from actual impairments. For each parameter we changed on one leg, we recorded the step length and the limit cycle trajectory for the first steps. The step length is defined as the distance between the legs at heel strike, which easily separates symmetric gait patterns from those with asymmetries. Note that our model is step based instead of time based, which allowed us to keep track of each leg as it es from swing to stance phase. From these measures, we found that the following four step patterns emerged: 1) Symmetric step pattern: This pattern was previously described as the ideal gait. It consists of step lengths that quickly converge to one value. Its uniformity relates to a normal human gait and can be seen in Figure 3. 2) Leg specific single step pattern: This pattern consists of each leg having one unique step length and occurred when the parameters of the model were closer to the ideal values. This pattern was found by changing the location of the right shank mass, which was moved through a range of values from m to 6 m measured from the end of the shank to the shank mass (a1r in Figure 1). Qualitatively, we see the greatest correlation between this gait pattern and a human asymmetrical gait pattern. For this reason, we will to look more in depth at its dynamics in the future. An example of the single step pattern is shown in Figure

5 !" Mass of mtr (kg) (a) Changing the mass of the right thigh. )%!% Mass of msr (kg) (b) Changing the mass of the right shank.!"&\$%!"&(%!"&&%!"&'%!"&%!"#\$% Length Length Fig. 6. Leg specific quadruple step pattern: This interesting gait pattern was found only when the mass of the right thigh (mt R ) was given a value of 74 kg. It is shown that each leg has four distinct step lengths. Looking at the bottom step length diagrams, a comparison between the step lengths for each stance phase is given. Another interesting pattern in the limit cycle trajectories is how the graphs are shifted vertically and horizontally. As seen in Figure 4, the graph is shorter from top to bottom, but has roughly the same starting and ending angles. This indicates a shorter time span for this phase and ultimately leads to a higher angular acceleration. By changing each design parameter and holding the others constant, we deemed the changed design parameter a ing parameter if the walker could stably walk steps. A depiction of this can be seen in Figure 7. The parameter is marked high if it ed and low if it ed. Also, plotted in these figures are the left and right step lengths versus the value of the changed parameter. This plot shows the asymmetry of the gait. The ideal parameter value is located where the plots cross and is the point where the walker exhibits a symmetric gait pattern. Figures 7a and 7b show that the mass of the shank can be stably decreased by 54% and increased by %, yet the mass of the thigh can only be increased or decreased by 5%. Changing the location of the shank mass has a larger range than when changing the position of the thigh mass. From this it is clear that the thigh mass is more sensitive to changes than the shank mass. Changing where the knee is located has a very large range. It can be located as high as the thigh mass or as low as just above the shank mass. In Figure 7f the larger a2r is, the closer the knee is to the shank mass. These results indicate that there are several parameters that can be changed to affect an asymmetric gait on a ive dynamic walker. None of these trials included mixing effects such as a lighter shank and a heavier thigh, but those are likely to evoke a similar range of asymmetric gaits and will be tested in future work !" Distance of msr From the Foot (m) (c) Moving the mass of the right shank.!" Distance of mtr From the Knee (m) (d) Moving the mass of the right thigh !" Length of LR (m) (e) Changing the length of the right leg by changing a1r and b2r Length of a2r (m)!" (f) Moving the knee by changing a2r and b1r (same leg length). Pass/Fail Right Length Left Length Fig. 7. The success and ure of the different parameters and the step lengths of each leg over all the parameters tested. Length Length Length Length 856

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