This calculator solves the ideal planar reach of two rigid links joined by a revolute elbow. Its angle convention is explicit: 0 degrees is folded back and 180 degrees is straight.
Jointed Radial Arm Interactive Calculator
Explore current reach, geometric minimum and maximum reach, and the ideal annular workspace under full-rotation assumptions.
Equation Used
- Two positive rigid link lengths and ideal zero-clearance revolute joints.
- Current reach is radial distance from shoulder pivot to tip.
- Annular workspace assumes full 360-degree shoulder rotation and full 0-to-180-degree elbow travel.
- No joint stops, self-collision, obstacles, joint offsets, tool dimensions/orientation, cable routing, gravity, mass, counterbalance, load, reactions, torque, deflection, stability, structure, bearing life, dynamics, control or safety model.
The workspace area assumes the shoulder can rotate through 360 degrees and the elbow can sweep continuously from fully folded to fully straight, with no stops, collisions or obstructions. It is a geometric area, not usable floor coverage or load capacity.
Two-link geometry
A primary link rotates about a fixed shoulder, and a secondary link rotates about the elbow at the primary link tip. The illustration holds both link lengths and the entered elbow angle constant while rocking the shoulder through a small angle.
The included elbow angle theta is 0 degrees when the links fold back in opposite directions and 180 degrees when they form one straight extended line.
Use as an ideal reach check
The calculator can test whether a radial distance lies between the geometric minimum and maximum for two chosen lengths. It cannot determine usable workspace in the presence of stops, collisions, tools, obstacles or required tool orientation, and it cannot size a load-bearing arm.
Law of cosines with a clear convention
With theta defined as the included elbow angle, R²=L1²+L2²−2L1L2 cos theta. This gives R=|L1−L2| at 0 degrees and R=L1+L2 at 180 degrees.
If the shoulder rotates through a full circle and the elbow reaches both limits, the ideal planar workspace is an annulus of area pi(Rmax²−Rmin²)=4piL1L2.
Worked example
For L1=1.8 m and L2=1.2 m, minimum reach is 0.6 m, maximum reach is 3.0 m and ideal annular area is about 27.14 m². At theta=90 degrees, current reach is sqrt(1.8²+1.2²)=2.163 m.
Limits of the reach calculation
Real joint stops, link thickness, tools and nearby obstacles reduce the ideal reachable area. Load capacity requires a separate force and structural calculation, including arm weight and base reactions. The geometric workspace does not establish usable coverage under load.
Questions
Why is 180 degrees fully extended?
That is the calculator's stated included-angle convention; the links point in the same outward direction at 180 degrees.
Is every point in the annulus usable?
Only in the unrestricted ideal geometry. Real stops, collisions and tool orientation remove regions.
Does area imply load capacity?
No. No force, torque, deflection or strength calculation is included.
References & Further Reading
- Wikipedia contributors. Radial arm. Wikipedia
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