Another quick search on Inverse Kinematics for Raspberry Pi turned up a project for the meArm that is exactly what I was thinking about. I have forked a copy to my account in case I need to make changes. Once this is implemented I can put X,Y,Z (fake Z) coordinates on the trash and have the arm navigate to the specified location. Once I have this implemented it will complete Phase 1 of the project in being able to detect trash and pick up. The next milestone will be getting the robot to move to trash out of reach, pick up, and navigate to next piece until it looks clean.

https://github.com/RorschachUK/meArmPi