For a planar two-revolute SCARA arm, target radius is r = √(x²+y²). A target is reachable when |L₁−L₂| ≤ r ≤ L₁+L₂. The two elbow branches use θ₂ = s acos[(r²−L₁²−L₂²)/(2L₁L₂)] and θ₁ = atan2(y,x) − atan2[L₂sinθ₂,L₁+L₂cosθ₂].
Ideal Planar 2R SCARA Inverse Kinematics
Solve ideal planar two-link SCARA joint angles and reachability for either elbow branch.
Equation Used
- Both links are rigid and move in one horizontal plane.
- Shoulder and elbow revolute axes are parallel and vertical.
- Angles are unrestricted and the selected elbow branch is maintained.
- Joint limits, collision, calibration, dynamics, payload and structural effects are excluded.
A negative reach margin means the displayed arm is a boundary solution, not that the target is reached.
Rigid planar links and parallel axes
The animation shows two rigid horizontal links joined by shoulder and elbow revolute joints whose axes are parallel and vertical. The tool moves from a presentation home pose to the selected inverse-kinematic solution.
When the target lies outside the annular workspace, it remains marked while the arm displays the corresponding nearest radial boundary solution.
What the calculator determines
Select two link lengths, target X/Y coordinates and elbow branch. The calculator returns shoulder angle, elbow angle, target radius and signed reach margin.
It does not calculate the vertical prismatic axis, joint limits, collision, singularity conditioning, speed, torque, payload, stiffness or safe operation.
Planar two-link inverse kinematics
c₂ = (x²+y²−L₁²−L₂²)/(2L₁L₂), θ₂ = s acos(clamp(c₂,−1,1)), and θ₁ = atan2(y,x) − atan2(L₂sinθ₂,L₁+L₂cosθ₂). Signed reach margin is min(L₁+L₂−r, r−|L₁−L₂|).
Selected-input example
For L₁ = L₂ = 300 mm and target (300,300) mm, r ≈ 424.3 mm. The elbow-positive branch gives shoulder 0° and elbow 90°, with a positive reach margin of about 175.7 mm.
Scope and design boundary
This is exact ideal planar position geometry. Real SCARA design additionally requires vertical-axis travel, tool orientation, joint limits, collision zones, singularity handling, calibration, dynamics, controls and structural analysis.
Unsupported payload, accuracy, speed, brand, maintenance and application claims are removed.
Questions
Why are there two solutions?
The elbow-positive and elbow-negative branches reach the same interior target with opposite elbow bend.
What does negative reach margin mean?
The target lies outside the outer radius or inside the inaccessible inner radius.
Does clipping make an unreachable target reachable?
No. It only provides a boundary pose while retaining the negative margin.
Is this a complete SCARA model?
No. The vertical and tool-orientation axes and all dynamics are excluded.
Technical references
- Modern Robotics, Degrees of Freedom — robot joint and configuration concepts.
- NIST SP 811 — units.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
