How are microcontrollers designed?
An experienced engineer could hack this together in an FPGA fairly quickly using a hard or soft core for the micro. A good design would take longer because of documentation, CDRs  and other parts of a proper design process. 

The design of the control algorithms is more mathematically challenging than the chip that runs them, especially if you have to calculate inverse kinematics and such like. 

Once proven in an FPGA it could be transferred to an ASIC process at a foundry, given enough time and money. That would be more of a business decision where the volume has to justify the up-front costs. 

Software would be the FPGA tools such as Vivado, and MATLAB/Simulink for the control algorithm design (there exist open source alternatives to the latter, which is very expensive).