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Understand how a robot works.

Build working knowledge of robotics and machine learning through examples you can change, calculations you can check, and data you can inspect.

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148 lessons to explore

Foundations first

  1. Foundations / 13 min read

    Kinematic singularities: find the tip velocities an arm can produce

    Use a two-link robot arm to distinguish exact rank loss from near-singular conditioning. Calculate the minimum-norm joint rates for a requested tip velocity and identify the component the arm cannot produce.

  2. Foundations / 11 min read

    Manipulability: read a robot’s velocity ellipse

    Map a joint-rate budget into a robot’s possible tool velocities. Read the ellipse’s singular values, compare area with conditioning, and understand singular poses, units, and the limits of force duality.

  3. Foundations / 14 min read

    Numerical inverse kinematics: solve a tool position with local steps

    Use a position Jacobian and damped least squares to refine a two-joint arm toward a target. Inspect accepted steps, compare starting guesses, and distinguish convergence, a stalled solve, and unreachable geometry.

  4. Foundations / 14 min read

    Differential inverse kinematics: turn a tip-velocity command into a joint step

    Calculate damped joint rates for a robot tip-velocity command, measure the resulting speed and direction error, and compare an instantaneous prediction with one finite joint step.

  5. Foundations / 12 min read

    Kinematic redundancy: use the motion a task leaves free

    Split a three-link arm’s joint rates into a primary solution and null-space motion. Check the exact projector, compare damping leakage, and measure why a finite step can move a tool with zero initial velocity.

  6. Foundations / 12 min read

    Joint limits in inverse kinematics: solve a bounded velocity step

    Turn physical joint ranges and speed limits into bounds on a local inverse-kinematics command. Compare a constrained least-squares solution with clipping, and check the resulting finite arm position.

  7. Foundations / 13 min read

    Analytical inverse kinematics: find both arm configurations for a target

    Derive both joint-angle solutions for a two-link robot arm, check them with forward kinematics, and identify unreachable targets and merged boundary branches.

  8. Foundations / 11 min read

    Robot workspaces: derive the reachable position set

    Derive the exact position workspace of a two-link robot with elbow limits. Test targets against its annulus, recover a valid arm configuration, and separate position reach from orientation and path feasibility.

  9. Foundations / 13 min read

    Trajectory time scaling: choose when a robot follows its path

    Separate a robot’s geometric path from its timing. Compare cubic and quintic profiles, derive joint speed and acceleration through the chain rule, and choose a duration that meets explicit limits.

  10. Foundations / 13 min read

    Configuration space: follow joint paths across angle boundaries

    Represent a robot arm as a point in joint space, follow paths across periodic angle boundaries, and distinguish angular distance from workspace motion and collision clearance.

  11. Foundations / 11 min read

    Collision checking: test the motion between endpoints

    Check a translating disk against a circular obstacle, including every point between its endpoints. Derive the closest-point test, expose missed samples, and distinguish broad-phase box overlap from a collision.

  12. Foundations / 12 min read

    Trapezoidal velocity profiles: accelerate, cruise, and stop

    Build a rest-to-rest motion for one linear joint. Derive triangular and trapezoidal velocity profiles, calculate braking distance, and inspect exact position, velocity and acceleration within explicit limits.