A few years ago, Amazon.com Inc. triggered a robot arms race when it purchased a company called Kiva Systems, maker of automated warehouse robots. Now its would-be rivals are landing bigger and bigger cash injections to try to compete with the e-commerce giant.

Amazon whipped out its wallet and threw down $775 million to purchase these robot legions in 2012. The acquisition effectively gave Jeff Bezos, its 52-year-old chief executive, command of an entire industry.