Twenty-one days that build a robot arm library from nothing, and check every number in it against something that did not share its derivation.
This is a book about the mathematics that moves a robot arm, written for Python 3 and NumPy. It starts from a single geometric decision — how to write down where a thing is, and which way it faces — and ends with a six-axis arm drawing a square in space under model-based control, tracking it to 0.0748 mm.
Nothing here imports a robotics library to do any work. You build armkit: fourteen modules and about three thousand lines of plain NumPy covering rotation matrices and SO(3), Euler angles and gimbal lock, axis-angle and the exponential map, unit quaternions and SLERP, homogeneous transforms and SE(3), Denavit-Hartenberg parameters, forward kinematics, the reachable and dexterous workspace, the geometric and analytic Jacobian, singularities and manipulability, closed-form and numerical inverse kinematics, damped least squares and nullspace redundancy, joint-space and Cartesian trajectories, the inertia tensor, the recursive Newton-Euler algorithm, the mass-Coriolis-gravity decomposition, forward dynamics and simulation, and computed-torque control.
Every routine is written out and then tested against an independent oracle: a symbolic Lagrangian against a force recursion, a potential-energy gradient against inverse dynamics, central differences against every analytic derivative, energy conservation against an integrator, a pendulum period against the small-angle formula, and somebody else's solver against this one. The disagreements are reported as numbers, in the text, at the precision they were measured.
So are the mistakes. The recursive Newton-Euler algorithm was wrong twice before it was right, in ways that produced torques of the right size and a plausible dependence on configuration. A thin-rod idealization failed the validity check introduced in the same chapter. A tracking error of 24.72 degrees turned out to be the integrator diverging rather than the controller failing. Each is in the book, with the measurement that found it, because a book that only shows the working version teaches you nothing about how to tell.
WHAT IS IN IT
21 chapters, one a day, in three weeks: Pose, Motion, Dynamics. 238 figures, all generated by scripts in the book. 153 listings, every one a complete runnable program whose printed output was captured from that run. 126 quiz questions and 86 exercises.
WHO IT IS FOR
Developers and engineers who know Python and want to understand robot motion properly, rather than call a library and hope. You need NumPy and secondary-school trigonometry. Everything else is built in front of you.
WHAT YOU WILL BE ABLE TO DO
Describe any rigid body's pose five ways and say what each costs. Compute where an arm's tool is, how fast it moves, and where its geometry collapses. Invert the kinematics exactly where a closed form exists and numerically where it does not. Plan motion in joint space and in the space the tool actually moves in. Give a link a mass and an inertia tensor, compute the torques a motion needs and the motion a torque produces, and write a controller that uses the model — then measure what happens when the model is wrong.
