Pick & Place Arm 2D: Throughput vs Joint-Speed Limit

This 2D companion swaps the 3D delta robot's three overhead arms for a flat, top-down two-link (SCARA-style) arm, but keeps the same underlying throughput model: every pick→place→home cycle must fit inside a fixed cycle time, and the two joints are solved live with exact two-link inverse kinematics rather than an approximation. Set a target pick rate, a mechanical joint-speed limit and the distance between the pick and place points, and this sim works out the peak angular speed each joint would need under a trapezoidal velocity profile to make that cycle time — the same 1.5×Δθ/segment-time relationship used by the 3D delta-robot version. Push the rate or the spacing high enough relative to the speed limit and the required peak speed exceeds what the joints can deliver, so the arm starts missing picks — the exact throughput ceiling that sizes a real pick-and-place cell's motors.