An ideal spherical wrist has three revolute axes intersecting at one point. This calculator evaluates a selected offset/reach angle proxy, compares offset with a selected tolerance, and reports a disclosed proximity index around the θ5=0 collinearity singularity. It does not calculate robot pose error or singularity severity.
Spherical Wrist · Selected Offset-Comparison Ledger
Explore selected wrist-axis offset comparisons and a neutral θ5 singularity-proximity index without treating them as robot accuracy or risk predictions.
Equation Used
- Offset and reach form only a selected right-triangle comparison.
- Tolerance is a user-selected comparison value, not a manufacturer specification.
- The wrist convention places the collinearity singularity at sinθ5=0.
- Ten degrees is a display normalization reference only.
- Full axis geometry, transforms, calibration, Jacobian conditioning, loads and safety are excluded.
The former singularity-risk percentage is relabeled as a normalized proximity index to a disclosed 10° reference.
Selected geometric comparisons
The diagram separates the ideal common wrist centre from a selected displaced axis. Its offset geometry and θ5 orientation are illustrative in scale but preserve the entered values and the known collinearity condition.
What the calculator determines
Select axis offset, comparison reach, tolerance and joint-5 angle. The calculator reports atan(e/L), e/t, t−e, and a normalized |sin θ5| proximity index that reaches 100% at θ5=0 and falls to zero at or beyond the disclosed 10° reference.
Selected comparison ledger
αproxy=atan(e/L); ratio=e/t; margin=t−e; proximity=100[1−min(1,|sinθ5|/sin10°)].
Selected-input example
At e=2 mm, L=500 mm and t=0.1 mm, the angle proxy is about 0.229°, the offset/tolerance ratio is 20 and the signed tolerance difference is −1.9 mm. At θ5=0°, the disclosed proximity index is 100%.
Comparison, not robot accuracy analysis
Actual TCP error depends on axis directions, transforms, tool geometry, calibration and controller models. Singularity conditioning depends on the wrist Jacobian; the 10° curve is only a display reference and is not probability, severity or a safe operating limit.
Questions
Where is the ideal spherical-wrist singularity?
For the common orthogonal 3R wrist convention, sinθ5=0 aligns the first and third wrist axes and the orientation Jacobian loses rank.
Does atan(e/L) predict TCP error?
No. It is only the selected right-triangle angle proxy for offset and reach.
References & Further Reading
- Wikipedia contributors. Robotic arm. Wikipedia
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