Partial derivatives and gradients: predict a multivariable change
Hold one input fixed to find a partial derivative, combine the partials into a gradient, and compare a directional derivative with the actual change from a finite step.
Writing
Clear accounts of what was tried and learned.
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.
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Product of exponentials: build a robot arm’s forward kinematics
Build a two-joint arm's tool pose from fixed home screw axes and matrix exponentials. Check the result against geometry, inspect multiplication order, and connect space and body formulas.
Projections and least squares: find the closest fit
Project a vector onto a direction, measure its orthogonal residual, and connect that geometry to least squares, regression, and nonunique coefficients.
Pseudoinverse: choose the smallest least-squares solution
Understand the Moore–Penrose pseudoinverse through exact and inconsistent systems. Separate residual error from solution norm, inspect projectors, and see how an SVD cutoff changes the problem.
Rotation matrices: turn vectors, track frames, and check the order
Build rotation matrices that preserve length and handedness. Compare fixed-axis rotations in 3D, distinguish rotating a vector from changing its coordinates, and undo a rotation with its transpose.
Singular value decomposition: directions, gains, and low-rank approximation
Build an SVD from orthogonal directions and nonnegative gains. See a circle become an ellipse, identify lost directions, and measure the error from keeping one singular component.
Taylor expansion and linearization: predict locally and check the error
Build constant, linear, and quadratic approximations around a chosen center. Compare their errors, calculate a Taylor remainder bound, and connect the same idea to gradients, Hessians, and Jacobians.
Twists and screw axes: connect point velocities to rigid motion
Build a six-component twist from a screw axis, calculate point velocities, and compare exact helical motion with a tangent prediction. Separate pitch, accumulated displacement, current rate, and pure translation.
Unit quaternions: compose rotations and understand the sign
Rotate vectors with Hamilton quaternions, check composition order, and see why q and minus q describe the same orientation. Includes an interactive experiment and Python.
Bayesian inference: learning a robot's grasp success rate
Learn Bayesian inference through a robot grasp example. Update a Beta prior with successes and failures, then distinguish uncertainty from prediction.
Conditional probability: reading a robot sensor
Learn conditional probability with an interactive robot sensor example. Work through the formula, Bayes' rule, base rates, and common mistakes using counts.
Cross product: find a normal direction and calculate torque
Calculate a three-dimensional cross product, follow the right-hand rule, and connect its magnitude to area. Explore signed torque with a movable lever arm and force.