Who's gonna define rubbish?

Tailko Jiao
Anything in orbit around the planet that is no longer functioning or under power. Its not rocket science.

Why doesn't the CleanSpace One push the dead satellite into the atmosphere thereby pushing himself to a higher orbit, ready for the next dead satellite? or does it?

Because of the relative speeds and different orbits of the various items of space debris, it makes it very difficult to capture more than one item at a time. This solution looks very expensive if you can only capture one items at a time!

Good question, Taiko Jiao. Obviously it won't really look like those arcade game claws. They might work on a cubesat, but that's about it. They would be as ineffective against a bolt or wrench as they are against stuffed animals in a mall. A version large enough to bring down a real disabled satellite could also be viewed as an anti-satellite weapon.
The problem with space trash has always been ownership. You can only bring down your own trash, most of it isn't Swiss, and other countries would have to contract for this. How much would someone pay to 'return-and-burn' an old satellite? So far, not much. De-orbit propellant costs only, and sometimes only enough propellant for a parking orbit.
This might change if the orbits get so full that they want to replace an old junker bus-sized satellite with a new $200 million model in geosynchronous orbit, but I don't see that capability any time soon.

Absolute insanity. Until we deploy a space-net capable of capturing many pieces of debris and/or disabled whole devices (and either yanking them into decay or jettisoning them into space), we are only kidding ourselves. The second and probably more important item on the agenda is the fixing of responsibility for costs. The UN and WTO should be able to fix that up in short order, no matter what arguments are put forth by nations (and now private parties) re: the ownership of, necessity to and responsibility for removing space junk.
By the way, one launched recovery vehicle could certainly deploy more than one net, each with its own final thruster...and before anyone suggests that only too-heavy Kevlar would be strong enough to withstand variable speed impacts by various debris, think for a moment about both matching speeds and capturing slower velocity junk first.

in the next major war between superpowers a few gigantic nuclear weapons will immidiately be set off in the thermosphere and exosphere.
even 'hardened' satellites that can supposedly withstand high energy electrons cannot withstand too many micro-fragments striking them.
how many 'de-orbiting' devices can 'clear' low earth orbit. you would need too many. the 'solution' is simply flying more sattelites at higher orbits. it costs more energy but it WORKS and it keeps sattelites safe from debri. the real issue then becomes, where can a sattelite be kept safe from high energy electrons , ions and radiation, without getting whipped by the radiation belts.
the inner van allen belt can enter the thermosphere under the right conditions, but is , according to wiki confined to a lower limit of 600 miles. so between 250 and 600. current geostationary sattelite orbit well within the far more diffuse, outer van allen belt, at around 22k miles.
putting sattelites in super high orbits, over 10,00 miles would be far more costly, but you wouldn't really have the micro-fragment problem of lower orbits.
its that simple from a practical perspective. especially if you are keeping human beings in outerspace.

Our orbit must be Kept clean, if not it will be uninhabitable by man or machine, possibly making space travel,satellites or anything else we try to put up their look like swiss cheese. On the other hand it might guard us from meteors as they will not be able to reach earth as they will be destroyed by space trash before they get the chance.

This is a great idea. I'm also very happy to hear that a new, slimmer, autonomous version of the Space Shuttle is being reborn in the ESA. Somebody needs to do it.
It seems to me that a cleaner device could be a maneuverable vehicle that could plant sticky cherry-bomb-like devices on dead satellites and other debris that would be sufficient to propel the trash into the atmosphere where it could burn up on reentry. The cleaner device could stay in space for months (or even longer) and put the tiny thrusters on hundreds of larger chunks of debris (maneuvering to put the thruster on the side opposite Earth) while scooping up smaller debris into its onboard trash can. After its fuel was almost exhausted, it's belly full of little pieces of space junk, and all of its thrusters deployed, it could then be commanded to dive toward the Earth and burn itself up.
The whole legal BS would have to be solved first, of course, with some agency (perhaps a UN Space Agency) having authority to approve removal of particular pieces of space junk by anyone who happens by. The current situation is impossible because of childish squabbling about, "That's mine. You can't touch it!" Imagine how our highways would look if you had to get permission from the owner before picking up a discarded hamburger wrapper. I realize that putting stuff in space is expensive and people don't want their stuff bothered, but really. This junk is becoming hazardous to us.

I bet there's a bunch of classified stuff still up there that will upset various government spook agencies if they fell into the "wrong" hands. Or maybe that's what this is really intended to do?