Writing
Explanations, tutorials, build notes, and essays grounded in work that can be examined. Explore a concept, try an experiment, and follow the reasoning behind it.
Start with the interactive foundationsShowing 49–60 of 101 posts
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.
Use virtual work and a Jacobian transpose to calculate a robot arm's joint loads. Distinguish external and holding torque, check space and body frames, and interpret zero-torque loads.
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.
Build space and body Jacobians from joint screw axes, recover the physical tool velocity, and compare their ranks with a position-only task. Explore a planar two-link arm and verify its derivatives in Python.
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.
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.
Calculate a force's moment about a chosen origin, include a free couple, and transform a moment-first wrench between frames. Use a worked planar load to check the inverse-transpose rule and power invariance.
Transform angular-first twists between body and space frames. Derive the origin-shift term, distinguish linear twist coordinates from point velocity, and check a planar example with an interactive adjoint matrix.
Rotate a vector around any nonzero axis. Normalize the direction, follow Rodrigues’ parallel and perpendicular terms, and understand the equivalent descriptions at zero and 180 degrees.
Convert a fixed landmark between robot and world coordinates. Learn frame conventions, translation and rotation, inverse transforms, and point versus displacement.
Understand derivatives as local rates, compare secant and tangent slopes, and multiply the correct factors through nested functions. Test a smooth curve and a corner with an interactive experiment.
Explore roll, pitch, and yaw with full rotation matrices. Compare equivalent orientations at ±90° pitch and separate Euler angle rates from angular velocity.