I suspect you don't want either Pic or Arduino, but instead want an Arm based microprocessor or more like, a single board based computer. I would think about something like a Rasberry Pi which typically runs Linux and has 256MB or 512MB of memory, perhaps coupled with an Arduino to control the robot. Beagle bone, mbed, etc. are other boards that are meant for the person doing one off designs.
To highlight what Riva says, the PROGMEM in Arduino is for data only. It is memory located in the flash memory that holds the instructions, and is meant to allow you to have larger read only data tables (such as strings for output) without consuming the rather limited read/write data memory.