Planar Two-Link Manipulator Manipulability Calculator

The Yoshikawa manipulability measure is a scalar index proportional to the volume of the velocity ellipsoid.

It quantifies how isotropic and uniform the robot end-effector velocity is in Cartesian coordinates.

The kinematic Jacobian relates joint angular velocities to end-effector Cartesian velocities.

When the elbow angle is 90 degrees, sin(theta2) = 1, achieving maximum manipulability.

When the arm is fully extended (theta2 = 0) or folded (theta2 = 180), sin(theta2) = 0, causing a kinematic singularity.

A higher manipulability index means the arm can accelerate and move flexibly in any direction.

A value near zero indicates kinematic singularity, where infinite joint velocity is needed to produce finite Cartesian speed.

The singularity warning alerts you if the elbow angle is within 10 degrees of complete extension or folding.

• Elbow joint angle: The primary determinant of planar manipulability.
• Link length product: Longer links yield larger velocity ellipsoids for the same joint velocities.
• Condition number: As the ellipse flattens near singularities, directional dexterity deteriorates.

• Trajectory planning to avoid singular postures during path execution.
• Robotic posture optimization for heavy lifting or high-precision welding tasks.

• Applies to 2-DOF planar serial kinematic chains.
• Assumes ideal rigid bodies without joint compliance or gearbox backlash.