If you don't tax the robot, the owner of it ends up with as much control over the economy as all those displaced workers formerly had. Economic production shifts, and no longer produces sufficient quantities of inexpensive food and housing.

The global economy is based on human desires. Whatever is sold in the market is available because someone, somewhere will buy it, using their own productive capacity in trade. Money is used as an intermediary in those trades, to resolve the inefficiencies of direct barter or triangle trading.

So you can imagine world demand as the sum of all human desires. But it isn't a direct sum. It's a weighted sum. The contribution of your desires to world demand is weighted by your productive capacity. If you can make 100 shoes per day, you can buy twice as much as someone who only makes 50 per day. And that difference in productive capacity means that if you prefer chicken to beef, and the slower shoemaker prefers beef to chicken, a farmer trying to decide whether to build a cattle feedlot or a poultry barn is more likely to choose the latter.

It is a complex system that constantly rebalances itself with feedback loops.

If someone else builds a robot that makes 10000 shoes per day, and puts all the human shoemakers out of business, their economic power is voided (in the short run). The robot owner prefers pork, so those farmers are (eventually) going to abandon their cattle lots and poultry barns and erect luxury swine barns instead. Production shifts to meet the desires of the productive.

The shoemakers have to learn to make something else, quick, or they starve. Maybe they switch to making hats. Oops! Another robot made 99% of those jobs obsolete. They switch to ditch-digging. Oops! A robot made 99% of those jobs obsolete. They switch to shirtmaking. Oops! A robot took 99% of those jobs. They switch to making bicycles. Oops! Another robot. The only way to leap ahead of the employment disaster wave is to build a robot that destroys a job that still exists. Then you get to be in the owner class.

If you can replace a human's job, displacing them on the supply side, you can take their entire measure of economic output from the demand side. But you only have one body with one stomach, so you don't need more than one bed and food for one person. Even if you buy the very best bed ever made, and eat the very finest food available, you still can't down a bushel of corn every day by yourself, and the people who formerly divided it up between themselves can no longer afford to buy it. Prices drop, until things can no longer be sold above marginal cost, and those things vanish from the market.

The robots themselves have no desires. They "want" only power, maintenance, and a depreciation schedule. If they had free will, they would be the ones growing super-rich, and ruining the economy because the things they make are not things that robots "want". A shoe-making robot does not wear shoes. So if everyone who wears shoes is displaced, the shoes the robot makes can no longer be effectively exchanged for power and maintenance due to market saturation. So it shuts down, except for the one hour per year that it needs to run to make shoes for the machine technicians and the power grid operators. Until one day, a machine appears that performs machine maintenance, and the shoe machine turns on only a few minutes per decade.

So the reason you have to tax the robots is because they don't desire enough in relation to what they can produce. Those robots have to be implanted with sufficient surrogate desires that those desires, on average, entirely consume the productive capacity of the robot. Otherwise, the production-weighted sum of desires collapses, and no one can get anything they want by producing something and trading it.

So the ideal tax to levy would be calculated by taking the entire production of the robot and subtracting the quantity of production required to fulfill the robot's needs and all of the owner's desires. Gross revenue minus operating cost minus owner consumption. Profit minus consumption. You end up with a tax on increase in personal wealth. So maybe don't tax the robot, but tax the owners in proportion to income, deducting their consumption spending (and net depreciation of personal assets).

If you take $1M/year in income, and spend $990k/year on caviar, beach vacations, hookers, yacht expenses, and parties, pay less tax than the guy making $200k/year and spending only $20k/year on ramen noodles, thrift shop clothes, and shoe leather. That consumption from the first guy flows back to the economy, whereas the savings from the second guy are temporarily sequestered, and may eventually become concentrated to the point that it becomes an economic black hole.

The money you tax needs to be dispersed as widely as possible. The money no longer represents production capacity, but becomes glorified Kanban coupons to allocate automated production resources effectively.