Abstract
A kinetic shape (KS) is a smooth two- or three-dimensional shape that is defined by its predicted ground reaction forces as it is pressed onto a flat surface. A KS can be applied in any mechanical situation where position-dependent force redirection is required. Although previous work on KSs can predict static force reaction behavior, it does not describe the kinematic behavior of these shapes. In this article, we derive the equations of motion for a rolling two-dimensional KS (or any other smooth curve) and validate the model with physical experiments. The results of the physical experiments showed good agreement with the predicted dynamic KS model. In addition, we have modified these equations of motion to develop and verify the theory of a novel transportation device, the kinetic board, that is powered by an individual shifting their weight on top of a set of KSs.
Highlights
Throughout ancient and modern history, the mechanism of rolling has been both useful and fascinating
The rolling kinematics (d2θ/dt2) of a circular wheel is created by a decreasing rolling surface height with distance traveled, while the same rolling motion is created for a rounded object by a decreasing shape radius around the object
These curves have several variations, the spirals, such as the Archimedes spiral, are unique because they represent a curve of radius that increases with angle in polar coordinates. This allows a spiral to roll on a flat surface much like a circular wheel rolls down an incline (Figure 1)
Summary
Throughout ancient and modern history, the mechanism of rolling has been both useful and fascinating. The rolling kinematics (d2θ/dt2) of a circular wheel is created by a decreasing rolling surface height with distance traveled (dy/dx), while the same rolling motion is created for a rounded object by a decreasing shape radius around the object (relative to the axle point) (dR/dθ). This generalized comparison is seen in d2θ dt. Conic sections are curves created by the intersection of cones, often producing interesting, smooth, and asymmetric curves (e.g., Archimedean spiral, logarithmic spiral, Cortes’ spiral, lituus, etc.) These curves have several variations, the spirals, such as the Archimedes spiral, are unique because they represent a curve of radius that increases with angle in polar coordinates. This study demonstrates passive rolling of a rounded shape, it lacks an analytical solution for this motion and how it is affected by the robot weight or ground reaction forces while rolling
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