When the links of a planar two-joint arm line up, its tip loses one instantaneous direction of motion. Explore how joint angles and link lengths affect the translational Jacobian and its two singular values.
Exact 2R Jacobian and Singular Values
Set two link lengths and joint angles. The calculator reports the Jacobian determinant magnitude, its two singular values and the tip radius for this one ideal planar model.
Equation Used
- The arm is planar with two ideal revolute joints and rigid links.
- Only planar tip translation is modeled; orientation, dynamics, joint limits and control are excluded.
- Singular values have length units because joint rates are expressed in radians per unit time.
The Jacobian relates joint rates to tip velocity for this ideal two-link arm. Its determinant and singular values describe instantaneous motion capability, rather than the operating limits of a particular robot.
Rank loss in a planar two-link arm
The Jacobian maps joint rates to instantaneous tip velocity. For this two-joint planar arm it is a 2 by 2 matrix. When the links are collinear, its two columns become linearly dependent and one instantaneous tip-velocity direction is lost.
The illustration keeps both link lengths fixed. Its ellipse semiaxes scale with the calculated singular values; the ellipse orientation is schematic and does not give the principal velocity directions.
One teaching model, not a robot safety monitor
Use the calculator to inspect exact 2R geometry and rank loss. It does not model a particular industrial wrist, controller, path, joint limit, payload, speed limit or protective function.
Planar 2R position and Jacobian
The determinant is zero at q2=0 or plus/minus 180 degrees. Singular values come from the eigenvalues of J J transpose and show the maximum and minimum tip-speed scaling per unit joint-rate vector.
Geometry example
For L1=200 mm, L2=160 mm, q1=35 degrees and q2=45 degrees, |det J| is about 22,627 mm squared, the singular values are about 62.14 and 364.13 mm, and the tip radius is about 332.9 mm. These are kinematic quantities only.
Limits of the two-link model
Real singularity handling depends on the complete robot Jacobian, task, controller, joint limits, velocity limits and manufacturer documentation.
Jacobian-singularity questions
Is every small determinant unsafe?
No universal threshold follows from this model, and determinant magnitude also changes with units and link scale.
What does sigma-min equal to zero mean?
For this 2R translation model, it means the Jacobian has lost rank and one instantaneous tip-velocity direction is unavailable.
Does this model a six-axis wrist?
No. It models only a two-link planar arm.
Robotics references
- MIT Robotic Manipulation, Basic Pick and Place: 2R planar kinematics, Jacobian and full-extension rank loss.
- Stanford CS223A, Introduction to Robotics solution: determinant-zero test and collinear-link singular configurations.
- Modern Robotics, Singularities: Jacobian rank and lost end-effector velocity directions.
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