A 3-SPS-1S parallel robot-based laser sensing for applications in precision agriculture

Abstract

A second green revolution is needed to double global food production over the next 30 years, and requires the use of the best of the technologies such as precision agriculture, to maximize production with minimum input costs and resources. This document shows advances in the development of prototype to direct acquired and sensing information to management fertilizes and water for applications in precision agriculture. We present the numerical/computational stage of such prototype which is equipped with laser sensors to determine the content of water, fertilizer or any other nutrient in row crops. Determining this information is key in order to know if the crop needs more of the above-mentioned substance. Our application is devoted for micro-crops which need to be swept with high precision as a result of its size. This agricultural problem translates to control engineering as a tracking problem.

Appendix to derive Eq. (7)

From Eq. (5), we obtain that $\delta \mathbf{q}^{T}= \delta \mathbf{x}_{p} ^{T}\,\mathbf{J}_{p}^{T}$ and similarly, from Eq. (6), we get that $\delta \mathbf{x}_{i}^{T}=\updelta \mathbf{x}_{p}^{T}\, \mathbf{J}_{i}^{T}$ and substituting the latter terms in $\delta {{\varvec{q}}}^{T}\tau +\delta {{\varvec{x}}}_p {\hat{{\varvec{F}}}}_p +\sum {\delta {{\varvec{x}}} _i^{T}}{\hat{{\varvec{F}}}}_i =0$ we obtain that

$$\begin{aligned} \left( {\delta {{\varvec{x}}}_p^{T}\,{{\varvec{J}}}_p^{T}} \right) \tau +\delta {{\varvec{x}}}_p\, {\hat{{\varvec{F}}}}_p +\sum {\left( {\delta {{\varvec{x}}} _p^{T}\,{{\varvec{J}}}_i^{T}} \right) } {\hat{{\varvec{F}}}}_i =0 \\ \delta {{\varvec{x}}}_p^{T}\left( {{{\varvec{J}}}_p^{T}\tau +{\hat{{\varvec{F}}}}_p +\sum {{{\varvec{J}}}_i^{T}} {\hat{{\varvec{F}}}}_i } \right) =0 \end{aligned}$$

And from there, Eq. (7) follows from the virtual work principle.