Table of Contents
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Velocity and Acceleration
We modified our MATLAB script to produce velocity and acceleration by differentiating position, enabling us to plot output position, velocity, and acceleration as a function of input position (mechanismAnalysis.m). Our physical robot uses link lengths equal to those shown in the above table multiplied by (50/7), and we use those physical values from here on.
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As expected, we see that during the center half of its range of motion our mechanism's output features a smooth, linear x velocity and a relatively constant y position, exactly what our robot needs. However, this data represents the motion of a single mechanism, whereas our robot contains four parallel mechanisms, two or four of which are in contact with the ground at a given point in time. Therefore, a more meaningful analysis would be a piece-wise combination of these plots.
We modify our MATLAB script again (robotAnalysis.m). Assuming that only the lower leg is in contact with the ground, our robot's position, velocity, and acceleration are functions of the lower leg only. (Keep in mind that these drawings show our mechanism upside down, so the "lower" leg is the one with the higher y position value.)
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| If we place two legs 180º apart and display only the grounded leg ... | we see that our x position is continuously linearly increasing and our y height remains approximately constant, ... | our x velocity is a relatively constant positive value and our y velocity is approximately zero, and ... | our x and y accelerations are always close to zero. (Note that the magnitudes for acceleration are tiny.) |
All of these features fit our robot's needs perfectly. Our combination of mechanisms produces near-constant forward motion at a near-constant vertical height.
Counterweight Analysis
Our balancing system will use a counterweight, moving it periodically to keep it above our robot's support polygon, which is the smallest convex shape containing all of the points at which the robot contacts the ground.
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C. Tunca, N. Pehlivan, N. Ak, B. Arnrich, G. Salur, and C. Ersoy, “Inertial Sensor-Based Robust Gait Analysis in Non-Hospital Settings for Neurological Disorders,” Sensors, vol. 17, no. 4, p. 825, Apr. 2017. |
Our weight needs to rest over the grounded foot throughout the swing phase, then switch quickly to the landing foot during the double support phase. As our position analysis shows, this double support phase is only about 10º wide. If our robot walks at one step every two seconds, that means our counterweight has to shift in a fraction of a second.