Calculator Inputs
Use the responsive three-column grid on large screens, two on smaller screens, and one on mobile.
Formula Used
Target radius: r = √(x² + y²)
Reachability check: |L1 - L2| ≤ r ≤ L1 + L2
Second joint cosine: cos(θ2) = (x² + y² - L1² - L2²) / (2L1L2)
Second joint angle: θ2 = ±acos(cos(θ2))
First joint angle: θ1 = atan2(y, x) - atan2(L2 sin(θ2), L1 + L2 cos(θ2))
Forward verification: x' = L1 cos(θ1) + L2 cos(θ1 + θ2) and y' = L1 sin(θ1) + L2 sin(θ1 + θ2)
Jacobian determinant: det(J) = L1L2 sin(θ2)
This model solves inverse kinematics for a two-link planar arm. It is accurate for position-only calculations in a flat workspace and is widely used for mathematical robotics demonstrations.
How to Use This Calculator
- Enter the first and second link lengths using the same unit system.
- Enter the target X and Y coordinates relative to the robot base.
- Select auto mode to compare elbow-up and elbow-down answers, or choose one branch directly.
- Pick the angle wrap style and decimal precision that match your reporting preference.
- Press Calculate to show results above the form, inspect the graph, and export CSV or PDF files.
Example Data Table
| Link 1 | Link 2 | Target X | Target Y | Reachable | θ1 (deg) | θ2 (deg) |
|---|---|---|---|---|---|---|
| 10 | 8 | 12 | 4 | Yes | 57.65 | -91.43 |
| 14 | 9 | 16 | 6 | Yes | 52.27 | -86.59 |
| 18 | 10 | 20 | 5 | Yes | 43.05 | -89.84 |
FAQs
1. What does this calculator solve?
It solves planar inverse kinematics for a two-link robot arm. Enter link lengths and a target point, and it returns joint angles, reachability, Jacobian behavior, and forward-kinematic verification.
2. Why can one target point produce two answers?
A two-link arm usually reaches the same point using two configurations: elbow-up and elbow-down. Both are mathematically valid unless the arm is fully stretched or folded into a singular case.
3. What makes a point unreachable?
The target radius must stay between the inner workspace radius and the outer workspace radius. If it falls outside that annulus, no real joint-angle solution exists for this robot geometry.
4. Why is the Jacobian determinant shown?
The determinant helps identify singular behavior. When it approaches zero, the robot loses directional control sensitivity, and small target changes can create large joint-angle swings.
5. Can this page solve 3D industrial robots?
This version focuses on a two-link planar model for clean mathematical analysis. Full 3D robots usually require additional joints, wrist orientation handling, constraints, and iterative numerical methods.
6. Do the units matter?
Yes, but only for consistency. Use the same unit for both links and the target coordinates. The angle results stay the same, while position outputs follow your chosen unit label.
7. Why do angles change rapidly near some targets?
Rapid changes often happen near singular positions or workspace boundaries. The robot still reaches the point, but the geometry becomes sensitive, so joint values can vary sharply.
8. How can I verify the result before using it?
Check the forward-kinematic endpoint, confirm the graph touches the requested target, and compare the solution with your real robot joint limits and controller conventions before deployment.