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Lesson 56 of 78 · Geometry & Kinematics

Inverse Kinematics, Jacobians & Singularities

Inverse kinematics (IK) asks which joint configurations produce a desired tool pose. There may be no solution, one boundary solution, several branches, or infinitely many for a redundant robot.

A two-link robot shows elbow-up and elbow-down inverse-kinematic solutions, a reachable annulus, and a stretched singular configuration
A reachable pose may have several joint solutions, while a singular configuration loses motion or force capability in a task direction. Credit: StudyCorner original diagram · CC BY 4.0 · Source

For the planar two-link arm, distance \(r=\sqrt{x^2+y^2}\) must satisfy

\[ |l_1-l_2|\le r\le l_1+l_2. \]

The law of cosines gives

\[ \cos q_2=\frac{x^2+y^2-l_1^2-l_2^2}{2l_1l_2}, \]

typically yielding elbow-up and elbow-down branches. Select using joint limits, collision, cable routing, stiffness, continuity from current pose, and downstream task—not angle size alone.

The Jacobian

For joint vector \(q\) and task velocity \(\dot x\),

\[ \dot x=J(q)\dot q. \]

Static wrench relationship is \(\tau=J^T F\). The same geometry that maps velocity maps force in a dual way. Near a singularity, modest Cartesian velocity can require extreme joint speed, and some force directions become weak or uncontrollable.

Singularity-aware planning

Watch Jacobian rank, determinant for square cases, singular values, or condition number. Set operational thresholds before numerical inversion becomes explosive. Damped least squares trades perfect task tracking for bounded joint motion near singularity.

Workspace is more than reach

Separate reachable position workspace, dexterous workspace with required orientations, payload-rated workspace, collision-free workspace, and process-qualified workspace. A catalog reach sphere does not prove the robot can weld, insert, or inspect at every point within it.

Exercise

For a two-link arm, plot or tabulate five target points. Solve both IK branches, check joint limits, calculate a Jacobian condition measure, and choose one configuration with a written rationale. Add a keep-out region and see whether the best kinematic branch changes.

Source trail

References

Further reading
  • CS223A / ME320 - Introduction to Robotics. Stanford University. verifiedCurrent physics-based syllabus covering spatial transformations, kinematics, Jacobians, dynamics, motion and force control, and vision-based control.
  • Stanford Engineering Everywhere - CS223A Introduction to Robotics. Stanford University. verifiedFree lecture videos, transcripts, handouts, and assignments on robot kinematics, Jacobians, planning, dynamics, and control.
  • Introduction to Robotics. MIT OpenCourseWare. verifiedMechanisms, kinematics, planning, dynamics, controls, actuators, sensors, networks, interfaces, embedded software, laboratories, and a team robot project.

Check your understanding

  1. What does the manipulator Jacobian relate locally?
  2. What happens at a kinematic singularity?