SCARA Retasking: Reach, Cycle Time and Warm-Up Offset
A four-axis SCARA robot with shoulder link 350 mm and elbow link 300 mm is being retasked from one product to another. The new pick point is at x = 420 mm, y = 120 mm and the place point is at x = 150 mm, y = −380 mm, both measured from the shoulder axis in the horizontal plane. The payload is 2 kg. The shoulder joint is limited to 240 °/s and 1200 °/s²; the elbow joint to 360 °/s and 2000 °/s². The current motion profile is trapezoidal with 0.20 s acceleration, 0.50 s cruise and 0.20 s deceleration, giving a 0.90 s move. The cell needs 57 cycles/min, so only 0.15 s remains for gripper actuation and settling. The robot repeats to ±0.02 mm, but the vision system reports a 0.4 mm offset at the new place point after a warm-up. The line engineer wants to increase the joint speeds by 30 %.
1. Check whether both points are within reach and compute the joint angles and joint displacements for an elbow-up solution.
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Reach is r = sqrt(x² + y²). Pick: rA = sqrt(420² + 120²) = sqrt(190,800) = 436.7 mm. Place: rB = sqrt(150² + 380²) = sqrt(166,900) = 408.5 mm. The maximum reach is L1 + L2 = 650 mm and the minimum is |L1 − L2| = 50 mm, so both points are inside the workspace.
For a two-link planar arm, cos θ2 = (r² − L1² − L2²) / (2 L1 L2). At A: cos θ2 = (190,800 − 122,500 − 90,000) / (2 × 350 × 300) = −21,700 / 210,000 = −0.1033, so θ2 = 95.9° (elbow up). θ1 = atan2(y, x) − atan2(L2 sin θ2, L1 + L2 cos θ2) = 15.9° − 43.1° = −27.2°.
At B: cos θ2 = (166,900 − 212,500) / 210,000 = −0.2171, so θ2 = 102.5°. θ1 = −68.4° − 45.8° = −114.2°.
Joint displacements: Δθ1 = 87.0°, Δθ2 = 6.6°. The shoulder does almost all the work; the elbow barely moves. Neither point is near the inner or outer singularity, so the inverse kinematics is well conditioned for this move.
2. Check the 0.90 s trapezoidal move against the joint velocity and acceleration limits. What margin remains, and how should it be used?
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Convert the shoulder displacement to radians: 87.0° × π/180 = 1.52 rad. For a trapezoid with 0.20 s acceleration, 0.50 s cruise and 0.20 s deceleration, the peak velocity is reached after the acceleration ramp. The area relation is distance = v_peak × (total time − acceleration time), because the acceleration and deceleration ramps average to v_peak/2. Thus v_peak = 1.52 / (0.90 − 0.20) = 2.17 rad/s = 124°/s. Acceleration = v_peak / 0.20 = 10.9 rad/s² = 622°/s².
The shoulder limits are 240°/s and 1200°/s², so the move uses about 52 % of the velocity limit and 52 % of the acceleration limit. The elbow is far below its limits. The move is kinematically feasible.
The remaining margin should not automatically be converted into a shorter cycle. It is needed for settling, gripper closure, payload deflection and any trajectory blending. A 57 cycles/min cell gives 1.05 s per cycle; 0.90 s of motion leaves 0.15 s, which is tight but possible if the robot is not vibrating at the end of the move. Verify settling with a dial indicator or the vision system before reducing the move time.
3. Evaluate the proposal to increase joint speeds by 30 % and explain the 0.4 mm warm-up offset. What should be checked before accepting the new cycle?
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A 30 % speed increase raises the shoulder peak to about 161°/s and acceleration to about 808°/s², still below the 240°/s and 1200°/s² limits. Kinematically it fits. Dynamically, torque scales with acceleration, so the shoulder and elbow torques rise by about 30 %, and structural vibration and settling time may rise disproportionately. The 0.15 s settling window could disappear, causing the gripper to close while the arm is still moving. The robot's rated payload and inertia map must be checked at the new acceleration, not just the speed limit.
The 0.4 mm offset after warm-up is not a repeatability problem: repeatability is ±0.02 mm, while absolute accuracy is 0.4 mm. A 0.4 mm error over a roughly 0.4 m reach corresponds to about 0.001 rad, or 0.057°, which is consistent with thermal expansion of the links or a zero/home drift. Check whether the offset appears only after the motors and gearboxes warm up, whether it changes with ambient temperature, and whether re-homing removes it.
Before accepting the faster cycle, verify: payload inertia and joint torque margin, settling at the place point, vision calibration after warm-up, repeatability over 100 cycles, and whether the 0.4 mm offset is stable enough to be compensated. Do not use speed increase to hide a thermal calibration error.