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How to prevent it to despawn:

If a Cyborg does not have any items inside its inventory, armor slots or utilities inventory, and it is not following any owner, it can despawn. So, be carefull with your tamed Cyborg, always keep at least one item inside its inventory.
 

How to activate a Cyborg:

When you found a Cyborg, it is deactivated because of its battery charge. You need to put a day light sensor on its utility inventory. But this is not enough to charge it, you will need to put Cyborg in the sun. As Cyborgs are found inside caves, you will need to open a hole in the ceiling to the surface, so that the sun's rays can reach it. You can see how is the battery charge level. Make sure it has enough charge before start moving.
 

How to make a Cyborg to follow and protect you:

First you need to activate it and charge its battery. If you want it to follow you, just click at the red button. Your name will appear at Current owner space and the red button will change to a green button, indicating that it will follow you wherever you went. Cyborgs will try to not interfere with you, so it will follow you keeping five blocks away from you. 
Cyborgs will protect you when they are following you. How? If a monster or a player hits you, Cyborg will start to follow your aggressor and he will strike at him until he is dead or another monster or player hits you again.
It is a good idea to wear your Cyborg with some diamond vests. It will protect him from damage. Cyborgs uses the same vests as players uses.
 

Where you can leave your Cyborg:

Never leave the cyborg in the rain. It will starting to short circuit and starting walking randomly. You need to leave the Cyborg in a place with roof and solar light to keep battery charged at its maximum.
 

How to use Cyborg's inventory:

You can save your items inside its inventory. If you are mining, cyborg will get all blocks from the floor for you when your inventory is full.
 

How yo use Cyborg's utilities inventory:

First slot is the sensor for charging. Just put a day light sensor on it and put your Cyborg at the sun to charge it. 
Second slot is the Block Finder slot. If you want to search for diamonds, you will need to harvest a diamond block with a pickaxe with silk enchantment. Put this block in this slot and go walking around in a cave with your cyborg. When he is five blocks near a diamond ore, he will pick this block with his hand and will start jumping and emitting a sound to show you that he found the block! You just need to search where!
Third slot is the gas for jetpack. Cyborg's jetpack uses coal as gas.
Fourth slot is the Jetpack.
Fifth slot is the weapon slot. In fact, the only weapon that is usable here are swords. But you can put everything you want here. If you are using the dynamic lights mod, I think that a Torch here will be useful. If Cyborg is using a sword and the sword breaks, he will looking for another sword inside his inventory and automatically pick it to use!
 

About Jetpack:

Jetpack allows Cyborgs to hover over water, lava and will help them to not take any damage when he fell from above. Jetpacks will help cyborgs to fly big montains to reach you. He will try to follow you in every directions!
 

What happens if Cyborg dies (water, mobs or lava)?

It depends on where and how it died. If a monster killed him, its inventory will drop all over the floor and you need to pick up all!
If a Cyborg went inside water, it's circuits will fail and he will not move anymore. You will need to remove all water from him or he will stay underwater forever.
If a Cyborg dies inside lava, it will be smelted by it. There is no way to save your cyborg BUT!!! Your inventory will stay with him forever because the body's part will be intact inside lava. I don't know how but you will need to remove all lava from him to reach his body and get your items back!