Theory
Here we discuss the high-level strategy, define that strategy using dynamics and an assumed friction model, and how that model was validated.
High-Level Strategy
Given a controllable moving surface, and a part that is to be transported, the operating principle is relatively simple and similar to that of horizontal transport1. First, the part is transported upwards against gravity by sticking to the moving surface. The moving surface then quickly accelerates downward in order to slip relative to the part, catching the part at a lower position. This sticking-slipping cycle repeats to have net motion of the part upwards, as shown in the below graphic.
Fig. 1: Alternating sticking and slipping phases achieve net part transport against gravity.
Dynamics
While many friction models exist, we used the well-known Coulomb friction model. The equations of motion are shown below:
\begin{equation}
\quad \text{sticking:} \quad \dot{z}_P = \dot{z}_S, \quad -\frac{\mu_s F_n}{m_P} -g \leq \ddot{z}_S \leq \frac{\mu_s F_n}{m_P} - g \label{eq:sticking}
\end{equation}
\begin{equation}
\quad \text{slipping:} \quad \ddot{z}_P = \frac{\mu_k F_n}{m_P} \text{sgn}(\dot{z}_S - \dot{z}_P) - g, \quad \dot{z}_P \neq \dot{z}_S \label{eq:slipping}
\end{equation}
Vertical Vibratory Transport Is Difficult
From $\eqref{eq:sticking}$, the mimimun required normal force to prevent the part from accelerating downwards when the surface is stationary ($\small \dot{z}_S = 0$) is the following:
\begin{equation} F_n > \frac{m_p g}{\mu_s} \label{eq:minimum_normal} \end{equation}
From $\eqref{eq:sticking}$ and $\eqref{eq:minimum_normal}$ the minimum required peak acceleration from the surface actuator, $a_{max}$, is the following:
\begin{equation}\label{eq:a_max_lower_bound} a_{max} > \mu_s F_n/m_P + g > 2g \end{equation}
The above equations detail the challenges which upward vertical vibratory transport presents compared to horizontal transport. In most practical cases, we have $\mu_s < 1$, so \eqref{eq:minimum_normal} means that the part $P$ must be squeezed with a force $F_n$ exceeding its own weight. Equation $\eqref{eq:sticking}$ shows that gravity reduces the maximum upward part acceleration during sticking, and that overcoming this limitation requires squeezing the part harder still. However, from \eqref{eq:a_max_lower_bound}, squeezing with higher normal forces requires more powerful actuators to reach higher surface accelerations, which already need to exceed $2g$ (compared to $a_{max} > \mu_s g$ for the horizontal case). Finally, \eqref{eq:slipping} shows that during slipping, the part accelerates faster down than up.
Experimental Validation
To validate our dynamics model described by $\eqref{eq:sticking}$ and $\eqref{eq:slipping}$ we recorded the interaction of a moving surface and a transported part. The recording was then processed by the free software Tracker in order to extract surface and part motion. The surface motion was then used as input to a Simulink model to predict the resulting part motion, which was compared with the experimentally observed part motion. A sample trial (Trial #5) is shown below (there were 10 total trials).
Fig. 2: Simulated (red) and experimental (black) part positions, along with the periodic surface motion (blue).
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Quaid, A. E. (1999, May). A miniature mobile parts feeder: Operating principles and simulation results. In Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No. 99CH36288C) (Vol. 3, pp. 2221-2226). IEEE. ↩