SCARA pick-and-place cell

What the page models

A SCARA arm (two vertical-axis joints plus a short vertical quill) moves a part from a conveyor stop A to a tray B. The question is the classic production trade-off: a faster arm finishes a move sooner but places the gripper less accurately, so more grasps miss and have to be repeated. Somewhere in between lies a speed that gives the most parts per minute.

Kinematics

For link lengths L1 = 0.5 m, L2 = 0.4 m the tip is at x = L1 cos(t1) + L2 cos(t1 + t2), y = L1 sin(t1) + L2 sin(t1 + t2). The inverse solution uses cos(t2) = (x^2 + y^2 - L1^2 - L2^2) / (2 L1 L2) (elbow bent the same way at both stations).

Motion in time

Both joints move together with a trapezoidal speed profile: accelerate at a = 14 rad/s^2, cruise at v = s * 5 rad/s, decelerate. A joint that has to turn an angle d takes T = d / v + v / a if it reaches cruise speed, or T = 2 sqrt(d / a) if the move is too short (triangular profile). The slower of the two joints sets the move time. The quill needs 0.22 s to go down or up and the jaws 0.18 s to close or open.

Why speed hurts accuracy

The model assumes the gripper lands with a random miss in the plane, each axis normally distributed with spread sigma = sigma0 (1 + kappa s^2): the arm is repeatable at rest (sigma0) and residual vibration grows with the square of the speed fraction s. The part is gripped only if the miss distance is smaller than the jaw tolerance delta. The miss distance of two independent normal axes follows the Rayleigh law, so the grasp success probability is p = 1 - exp(-delta^2 / (2 sigma^2)).

Throughput

A successful cycle takes T_ok = 2 T_AB + 4 T_z + 2 T_g (move to A, lower-grip-lift, move to B, lower-release-lift). Each failed grasp adds T_fail = 2 T_z + T_g + T_s (lower, close on air, lift, settle 0.25 s). The number of attempts is geometric, so the expected time per part is E[T] = T_ok + (1/p - 1) T_fail and the throughput is 60 / E[T] parts per minute. Increasing speed shortens T_AB but lowers p; the green marker on graph 1 shows the best speed for the current settings.

What changes the optimum

Wider jaws (larger delta) or a stiffer, damped arm (smaller kappa or sigma0) push the optimum towards full speed, because misses become rare. A worn arm or narrow jaws push it to slower, careful motion. At very low speed the move times are short compared with the fixed times for the quill and jaws, so the curve flattens.

Simplifications

One part at a time, an always-full feeder, no conveyor timing, no gripper force or slip, no cable or payload dynamics, a speed-only error model (real vibration also depends on acceleration, payload and the mounting) and a fixed move length. The numbers (5 rad/s, 14 rad/s^2, 0.6 mm) are plausible for a small SCARA but are not tied to a specific product. The page is about the shape of the trade-off, not a datasheet.