01 · Frames, transformations and robot kinematics

Assumptions, implementation & acceptance

Rᵀ is the inverse only for orthonormal R; proper rigid rotations also have Use right-handed frames and matching length units; homogeneous transforms act on column vectors.

Formula conditions, units and worked arithmetic reviewed. The implementation isolates one numerical mechanism; it does not train a policy or control real hardware.

Before running: predict a boundary case

Round trip recovers p. A nonorthogonal R cannot use transpose as inverse.

Minimal executable example

Save the example and linked modules in one folder. Use Node.js 22 or newer, then run node frames-and-kinematics.mjs. No third-party runtime dependencies.

const solve=()=>{
const R=[[0,-1],[1,0]], t=[1,2], p=[1,0];
const world=R.map((row,i)=>row.reduce((s,x,j)=>s+x*p[j],0)+t[i]);
const recovered=p.map((_,i)=>R.reduce((s,row,j)=>s+row[i]*(world[j]-t[j]),0));return [...world,...recovered];
};
const actual=solve();
const expected=[1,3,1,0];
if(JSON.stringify(actual)===JSON.stringify(expected)){console.log('PASS',actual)}else{
const a=[actual].flat(), b=[expected].flat();
if(a.length!==b.length||a.some((v,i)=>!Number.isFinite(v)||Math.abs(v-b[i])>1e-10))throw Error('Numerical mismatch');
console.log('PASS within 1e-10',actual);
}

[
  1,
  3,
  1,
  0
]

Test and transfer

Round trip recovers p. A nonorthogonal R cannot use transpose as inverse.

Record input, code version, expected result and observed result in the notebook below. Then construct a counterexample that violates the assumption above and explain why the original conclusion no longer follows.