Mechanism Mobility Calculation
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By varying Watt's original six-bar linkage, Mehdigholi and Akbarnejad created a 1 DOF walking mechanism that can be split into two sets: a 4-bar and 5-bar linkage. This separation is shown with two different highlights in the sketch below. Here, the five bar linkage is composed of links 1, 4, 5, 3, and ground but is not standard as it does not have two grounding points. In other words, it actually has two degrees of freedom, and requires a second system to restrict its position, velocity and acceleration. That role is fulfilled by conducting kinematic analysis on the four-bar mechanism of links 1', 3', 2, and 6 (ground). Because of this setup, the conventional method of analyzing a Watt's Six-Bar Linkage will not work–this mechanism is lacking one more grounding point for this methodology. Instead, seeing that theta 2 and 3' are unknowns in the 4-bar linkage and that they can be found using two equations (one real, one imaginary), allows me to then solve for the unknowns in the "five-bar" linkage of theta 4 and 5. (Note: All angles below are measured relative to the horizontal in a counterclockwise direction, as is set out by the convention established for this class.)
From this setup, there are two different vector loop equations:
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Using the relationships found above, the position, velocity, and acceleration of the "end effector"—the foot here or link 5—was plotted using MATLAB below. The input crank (Link 1 above) was set to rotate with a speed of 1.57 rad/s or 15 RPM clockwise. This full range of motion creates a crank-rocker six-bar linkage, as can be seen in the animation. Additionally, the animation features a ternary link, which was simply added to the link assembly in PMKS to illustrate the position of the "foot" as an offset to link 5. Since there is not a rotating slider, there is not a Coriolis acceleration, so instead the resolved acceleration at the third "joint" on link 5 is shown with the orange line in the attached PMKS animation. The position plot successfully resembles the desired trajectory for a "step" described by Mehdigoli and Akbarnejad in their paper. Additionally, some discontinuities are excluded in the velocity and acceleration plots to have a defined and reasonable plot.
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