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A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with therespective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumedthat the initial heights of bamboos are zero. The robotic gardener maintainingthe garden regularly attends bamboos and trims them to height zero according tosome schedule. The Bamboo Garden Trimming Problem (BGT) is to design aperpetual schedule of cuts to maintain the elevation of the bamboo garden aslow as possible. The bamboo garden is a metaphor for a collection of machineswhich have to be serviced, with different frequencies, by a robot which canservice only one machine at a time. The objective is to design a perpetualschedule of servicing which minimizes the maximum (weighted) waiting time forservicing.  We consider two variants of BGT. In discrete BGT the robot trims only onebamboo at the end of each day. In continuous BGT the bamboos can be cut at anytime, however, the robot needs time to move from one bamboo to the next.  For discrete BGT, we show tighter approximation algorithms for the case whenthe growth rates are balanced and for the general case. The former algorithmsettles one of the conjectures about the Pinwheel problem. The generalapproximation algorithm improves on the previous best approximation ratio. Forcontinuous BGT, we propose approximation algorithms which achieve approximationratios $O(\log \lceil h_1/h_n\rceil)$ and $O(\log n)$.

Perpetual maintenance of machines with different urgency requirements