You can take a look at PLCOpen Motion. This can be thought of a motion control SDK for programming languages used in the automation world. This is general purpose and well established.
Furthermore, you can take a close look at all Robot Controller programming languages (like Kuka KRL, ABB Rapid, Fanuc Karel, Staubli Val3, etc.) These are programming languages build explicitly for motion control applications involving robots, so less general purpose then PLCOpen. The motion control related things are built in the language and do not have an explicit SDK or API character.
For CNCs there is G-Code. It is a language for motion control, but its syntax is a direct successor of the punch-card system used before textual programming languages, so it is really bad in the syntax sense, but very good in the features which it offers for motion control of machine tools.
If you want to have an example of motion control and a modern programming language, you can look at Kuka's Sunrise platform, which uses Java to program robots. I have never used it, so I do not know if it is any good.
In general purpose motion control PLCOpen is the closest you get to a standard, basically anything else is manufacturer specific (event the standardized G-Code has manufacturer specific syntax and features).