So here is a video of the "Mechanical Turk" arm moving a chess piece across the board using forward kinetics:
Oops, I crashed into the board! So I think I may have a go at calculating the inverse kinetics to avoid the arm from crashing into the board or (more likely) neighboring chess pieces. The arm has to move the end effectors pretty much straight up a few inches to clear all the pieces, then it needs to move above all the pieces until it is directly above the destination square, then it needs to move directly vertically down.
I could probably heuristically do this with forward kinematics (for example, move up and down first by just using the elbow or such), but if I can get the inverse kinematics working OK it would give me a lot more control over the system.