Press n or j to go to the next uncovered block, b, p or k for the previous block.
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* four-bar-kinematics.ts
*
* Kinematics engine and Grashof classifier for 4-bar planar linkages calibrated in
* LEGO Technic beam sizes (hole units).
*
* 1 LEGO Technic hole pitch = 8.0 mm
*/
export interface FourBarLinks {
/** Ground / base frame link length (holes between base pivots A and D) */
ground: number;
/** Input / crank link length (holes between pivot A and B) */
crank: number;
/** Coupler / bridge link length (holes between pivot B and C) */
coupler: number;
/** Output / follower / rocker link length (holes between pivot D and C) */
follower: number;
}
export interface CouplerPointDef {
/** Distance along the coupler bar BC from B in holes (default: coupler/2) */
alongBar: number;
/** Distance perpendicular to bar BC in holes (positive = counter-clockwise) */
perpendicular: number;
}
export type GrashofClassification =
| 'CRANK_ROCKER'
| 'DOUBLE_CRANK'
| 'DOUBLE_ROCKER'
| 'PARALLELOGRAM'
| 'DELTOID'
| 'NON_GRASHOF_TRIPLE_ROCKER'
| 'IMPOSSIBLE_GEOMETRY';
export interface GrashofAnalysis {
classification: GrashofClassification;
isGrashof: boolean;
canCrankRotate360: boolean;
shortest: number;
longest: number;
sumShortLong: number;
sumOtherTwo: number;
description: string;
simpleExplanation: string;
}
export interface Point2D {
x: number;
y: number;
}
export interface FourBarPose {
/** Fixed base pivot A (motor axis) at (0, 0) */
pA: Point2D;
/** Moving crank pin B */
pB: Point2D;
/** Moving coupler pin C */
pC: Point2D;
/** Fixed base pivot D (follower axis) at (ground, 0) */
pD: Point2D;
/** Optional tracer / tool point on coupler */
pP: Point2D;
/** Crank angle in radians (from AD line) */
theta2Rad: number;
/** Coupler angle in radians */
theta3Rad: number;
/** Follower angle in radians */
theta4Rad: number;
/** Transmission angle mu between coupler and follower (0° to 180°) */
transmissionAngleDeg: number;
/** Quality of transmission angle: 'OPTIMAL' | 'GOOD' | 'WARNING' | 'CRITICAL' */
transmissionQuality: 'OPTIMAL' | 'GOOD' | 'WARNING' | 'CRITICAL';
/** True if links reach closure at this angle; false if jammed/locked */
isValidClosure: boolean;
/** Reason if closure failed */
errorMessage?: string;
}
export interface LinkagePreset {
id: string;
name: string;
subtitle: string;
description: string;
links: FourBarLinks;
couplerPoint: CouplerPointDef;
defaultBranch: 'OPEN' | 'CROSSED';
}
/** Standard LEGO Technic straight beam sizes by hole count */
export const LEGO_STANDARD_BEAM_SIZES = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21] as const;
export const LEGO_STUD_PITCH_MM = 8.0;
/** Convert LEGO Technic holes to millimeters */
export function holesToMm(holes: number): number {
return holes * LEGO_STUD_PITCH_MM;
}
/** Convert millimeters to LEGO Technic holes */
export function mmToHoles(mm: number): number {
return mm / LEGO_STUD_PITCH_MM;
}
/**
* Analyzes the 4-bar linkage using Grashof's Theorem:
* S + L <= P + Q
* where S is shortest link, L is longest link, and P, Q are the other two links.
*/
export function analyzeGrashof(links: FourBarLinks): GrashofAnalysis {
const { ground, crank, coupler, follower } = links;
// Validation: all lengths must be positive
if (ground <= 0 || crank <= 0 || coupler <= 0 || follower <= 0) {
return {
classification: 'IMPOSSIBLE_GEOMETRY',
isGrashof: false,
canCrankRotate360: false,
shortest: 0,
longest: 0,
sumShortLong: 0,
sumOtherTwo: 0,
description: 'Invalid dimensions: All link lengths must be greater than zero.',
simpleExplanation: 'Bars must have a positive length to build a real mechanism!',
};
}
// Geometric feasibility check: The longest link cannot be strictly greater than the sum of the other three
const arr = [
{ name: 'ground', len: ground },
{ name: 'crank', len: crank },
{ name: 'coupler', len: coupler },
{ name: 'follower', len: follower },
];
arr.sort((a, b) => a.len - b.len);
const S = arr[0];
const L = arr[3];
const P = arr[1];
const Q = arr[2];
const sumShortLong = S.len + L.len;
const sumOtherTwo = P.len + Q.len;
if (L.len > S.len + P.len + Q.len) {
return {
classification: 'IMPOSSIBLE_GEOMETRY',
isGrashof: false,
canCrankRotate360: false,
shortest: S.len,
longest: L.len,
sumShortLong,
sumOtherTwo,
description: `Impossible linkage: The longest bar (${L.len} holes) is too long to close a quadrilateral with the other three (${S.len + P.len + Q.len} holes).`,
simpleExplanation: 'Uh oh! The bars are too short to reach across the longest bar.',
};
}
// Check Parallelogram: opposite links are equal
const isParallelogram =
Math.abs(ground - coupler) < 1e-4 && Math.abs(crank - follower) < 1e-4;
const isDeltoid =
(Math.abs(ground - crank) < 1e-4 && Math.abs(coupler - follower) < 1e-4) ||
(Math.abs(ground - follower) < 1e-4 && Math.abs(crank - coupler) < 1e-4);
// Grashof condition: S + L <= P + Q
const isGrashof = sumShortLong <= sumOtherTwo + 1e-6;
if (isParallelogram) {
return {
classification: 'PARALLELOGRAM',
isGrashof: true,
canCrankRotate360: true,
shortest: S.len,
longest: L.len,
sumShortLong,
sumOtherTwo,
description: 'Parallelogram Linkage (Opposite bars are identical in length).',
simpleExplanation: 'The Parallel Lifter! The floating bar stays at the exact same angle the entire time (like a forklift or desk lamp).',
};
}
if (isDeltoid && isGrashof) {
return {
classification: 'DELTOID',
isGrashof: true,
canCrankRotate360: true,
shortest: S.len,
longest: L.len,
sumShortLong,
sumOtherTwo,
description: 'Kite / Deltoid Linkage (Adjacent pairs of equal links).',
simpleExplanation: 'A folding kite mechanism! Spins smoothly with rapid speed changes.',
};
}
if (isGrashof) {
// S is shortest
if (S.name === 'crank') {
return {
classification: 'CRANK_ROCKER',
isGrashof: true,
canCrankRotate360: true,
shortest: S.len,
longest: L.len,
sumShortLong,
sumOtherTwo,
description: 'Crank-Rocker Mechanism (The crank does a full 360° spin while the follower rocks back and forth).',
simpleExplanation: 'The Windshield Wiper! Your motor can spin non-stop in circles, while the output arm wags back and forth.',
};
} else if (S.name === 'ground') {
return {
classification: 'DOUBLE_CRANK',
isGrashof: true,
canCrankRotate360: true,
shortest: S.len,
longest: L.len,
sumShortLong,
sumOtherTwo,
description: 'Double-Crank / Drag Link Mechanism (Both grounded bars can make complete 360° revolutions).',
simpleExplanation: 'The 360° Spinner! Both bars can spin in full circles at varying speeds.',
};
} else {
// S is coupler or follower
return {
classification: 'DOUBLE_ROCKER',
isGrashof: true,
canCrankRotate360: false,
shortest: S.len,
longest: L.len,
sumShortLong,
sumOtherTwo,
description: 'Grashof Double-Rocker (The coupler can rotate 360° relative to adjacent bars, but crank and follower rock).',
simpleExplanation: 'The Playground Swing! The motor cannot spin full circles—the arms rock back and forth.',
};
}
}
// Non-Grashof (Triple-Rocker): S + L > P + Q
return {
classification: 'NON_GRASHOF_TRIPLE_ROCKER',
isGrashof: false,
canCrankRotate360: false,
shortest: S.len,
longest: L.len,
sumShortLong,
sumOtherTwo,
description: 'Non-Grashof Triple-Rocker (S + L > P + Q: No link can make a complete 360° revolution).',
simpleExplanation: 'The Double Swinger! No bar can do a full 360° spin. Every bar swings back and forth between two walls.',
};
}
/**
* Solves the position of all 4 joints for a given crank angle theta2 (in radians).
*
* Coordinate system:
* Joint A (motor pivot): (0, 0)
* Joint D (follower pivot): (ground, 0)
* Joint B (crank pin): (crank * cos(theta2), crank * sin(theta2))
* Joint C (coupler pin): intersection of circle(B, coupler) and circle(D, follower)
*/
export function solveFourBarPose(
links: FourBarLinks,
theta2Rad: number,
branch: 'OPEN' | 'CROSSED' = 'OPEN',
couplerPoint?: CouplerPointDef,
prevPointC?: Point2D
): FourBarPose {
const { ground, crank, coupler, follower } = links;
const pA: Point2D = { x: 0, y: 0 };
const pD: Point2D = { x: ground, y: 0 };
const bx = crank * Math.cos(theta2Rad);
const by = crank * Math.sin(theta2Rad);
const pB: Point2D = { x: bx, y: by };
// Special exact case for Parallelogram in OPEN branch
const isParallelogram =
Math.abs(ground - coupler) < 1e-4 && Math.abs(crank - follower) < 1e-4;
if (isParallelogram && branch === 'OPEN') {
const cx = bx + ground;
const cy = by;
const pC: Point2D = { x: cx, y: cy };
const theta3Rad = 0; // Coupler stays perfectly horizontal
const theta4Rad = theta2Rad; // Follower angle matches crank angle
const muDeg = 90; // Parallelogram transmission angle is well-behaved
const transmissionQuality = 'OPTIMAL';
let pP: Point2D = { x: (bx + cx) / 2, y: (by + cy) / 2 };
if (couplerPoint) {
const alongFrac = coupler > 0 ? couplerPoint.alongBar / coupler : 0.5;
const perpLen = couplerPoint.perpendicular;
pP = {
x: bx + alongFrac * ground,
y: by + perpLen,
};
}
return {
pA,
pB,
pC,
pD,
pP,
theta2Rad,
theta3Rad,
theta4Rad,
transmissionAngleDeg: muDeg,
transmissionQuality,
isValidClosure: true,
};
}
// Distance from B to D
const dx = ground - bx;
const dy = -by;
const dBD = Math.hypot(dx, dy);
// Check triangle closure between B, C, D
if (dBD > coupler + follower + 1e-5) {
return createFailedPose(
pA,
pB,
pD,
theta2Rad,
`Linkage cannot close: Distance BD (${dBD.toFixed(2)}h) exceeds coupler + follower (${(coupler + follower).toFixed(2)}h).`
);
}
if (dBD < Math.abs(coupler - follower) - 1e-5) {
return createFailedPose(
pA,
pB,
pD,
theta2Rad,
`Linkage cannot close: Distance BD (${dBD.toFixed(2)}h) is less than |coupler - follower| (${Math.abs(coupler - follower).toFixed(2)}h).`
);
}
if (dBD < 1e-5) {
return createFailedPose(pA, pB, pD, theta2Rad, 'Singularity: Pivot B coincides with Pivot D.');
}
// Circle intersection:
// a1 is distance along line BD from B to the chord connecting intersection points
const a1 = (coupler * coupler - follower * follower + dBD * dBD) / (2 * dBD);
const hSq = coupler * coupler - a1 * a1;
const h = Math.sqrt(Math.max(0, hSq));
// Unit vector along BD
const ux = dx / dBD;
const uy = dy / dBD;
// Normal vector perpendicular to BD (+90 deg counter-clockwise)
const vx = -uy;
const vy = ux;
// Candidate solutions
const cand1 = { x: bx + a1 * ux + h * vx, y: by + a1 * uy + h * vy };
const cand2 = { x: bx + a1 * ux - h * vx, y: by + a1 * uy - h * vy };
let pC: Point2D;
if (prevPointC) {
// Continuity tracking: Pick candidate closest to previous pose
const dist1 = Math.hypot(cand1.x - prevPointC.x, cand1.y - prevPointC.y);
const dist2 = Math.hypot(cand2.x - prevPointC.x, cand2.y - prevPointC.y);
pC = dist1 <= dist2 ? cand1 : cand2;
} else {
// Branch 1 (OPEN): +h * v
// Branch 2 (CROSSED): -h * v
pC = branch === 'OPEN' ? cand1 : cand2;
}
const cx = pC.x;
const cy = pC.y;
// Calculate angles
const theta3Rad = Math.atan2(cy - by, cx - bx);
const theta4Rad = Math.atan2(cy - 0, cx - ground);
// Calculate Transmission Angle mu (angle between coupler BC and follower DC)
// Vector BC = (cx - bx, cy - by)
// Vector DC = (cx - ground, cy - 0)
const vBCx = cx - bx;
const vBCy = cy - by;
const vDCx = cx - ground;
const vDCy = cy;
const dot = vBCx * vDCx + vBCy * vDCy;
const magBC = Math.hypot(vBCx, vBCy);
const magDC = Math.hypot(vDCx, vDCy);
const cosMu = Math.max(-1.0, Math.min(1.0, dot / (magBC * magDC)));
let muDeg = (Math.acos(cosMu) * 180) / Math.PI;
// Standard transmission angle is acute angle between lines (0 to 90) or 0 to 180
// In mechanism design, mu is defined as angle between BC and DC (ideal = 90 deg)
let transmissionQuality: 'OPTIMAL' | 'GOOD' | 'WARNING' | 'CRITICAL' = 'GOOD';
const devFrom90 = Math.abs(muDeg - 90);
if (devFrom90 <= 20) {
transmissionQuality = 'OPTIMAL';
} else if (devFrom90 <= 45) {
transmissionQuality = 'GOOD';
} else if (devFrom90 <= 60) {
transmissionQuality = 'WARNING';
} else {
transmissionQuality = 'CRITICAL';
}
// Calculate coupler point P
let pP: Point2D = { x: (bx + cx) / 2, y: (by + cy) / 2 };
if (couplerPoint) {
const alongFrac = coupler > 0 ? couplerPoint.alongBar / coupler : 0.5;
const perpLen = couplerPoint.perpendicular;
// Base point along BC
const basePx = bx + alongFrac * (cx - bx);
const basePy = by + alongFrac * (cy - by);
// Perpendicular unit vector to BC
const uBCx = (cx - bx) / magBC;
const uBCy = (cy - by) / magBC;
const normBCx = -uBCy;
const normBCy = uBCx;
pP = {
x: basePx + perpLen * normBCx,
y: basePy + perpLen * normBCy,
};
}
return {
pA,
pB,
pC,
pD,
pP,
theta2Rad,
theta3Rad,
theta4Rad,
transmissionAngleDeg: muDeg,
transmissionQuality,
isValidClosure: true,
};
}
function createFailedPose(
pA: Point2D,
pB: Point2D,
pD: Point2D,
theta2Rad: number,
errorMessage: string
): FourBarPose {
return {
pA,
pB,
pC: { x: pB.x, y: pB.y },
pD,
pP: { x: pB.x, y: pB.y },
theta2Rad,
theta3Rad: 0,
theta4Rad: 0,
transmissionAngleDeg: 0,
transmissionQuality: 'CRITICAL',
isValidClosure: false,
errorMessage,
};
}
/**
* Precomputes the trajectory path (coupler curve) traced by point P over a full 360 degree sweep.
*/
export function generateCouplerPath(
links: FourBarLinks,
branch: 'OPEN' | 'CROSSED' = 'OPEN',
couplerPoint?: CouplerPointDef,
steps: number = 180
): Point2D[] {
const points: Point2D[] = [];
const dTheta = (2 * Math.PI) / steps;
let prevC: Point2D | undefined = undefined;
for (let i = 0; i < steps; i++) {
const theta = i * dTheta;
const pose = solveFourBarPose(links, theta, branch, couplerPoint, prevC);
if (pose.isValidClosure) {
points.push(pose.pP);
prevC = pose.pC;
}
}
return points;
}
/**
* Curated FLL and LEGO Technic mechanism presets for students.
*/
export const POPULAR_FLL_LINKAGE_PRESETS: LinkagePreset[] = [
{
id: 'parallel_lifter',
name: 'Parallel Forklift / Cargo Lifter',
subtitle: 'Keeps attachment perfectly level while lifting',
description:
'Opposite bars have identical lengths (9 holes and 5 holes). As the motor arm swings up, the cargo tray stays perfectly level without tipping your game pieces!',
links: { ground: 9, crank: 5, coupler: 9, follower: 5 },
couplerPoint: { alongBar: 4.5, perpendicular: 3 },
defaultBranch: 'OPEN',
},
{
id: 'windshield_wiper',
name: 'Windshield Wiper / Flipper (Crank-Rocker)',
subtitle: 'Spins 360° continuously while sweeping back & forth',
description:
'A compact 3-hole motor crank drives a 7-hole follower arm through an 8-hole coupler. The motor can spin non-stop in circles, sweeping an attachment back and forth to clear obstacles!',
links: { ground: 9, crank: 3, coupler: 8, follower: 7 },
couplerPoint: { alongBar: 4, perpendicular: 0 },
defaultBranch: 'OPEN',
},
{
id: 'chebyshev_walker',
name: 'Chebyshev Walking Leg (Straight-Line Motion)',
subtitle: 'Converts rotary motor spin into flat walking steps',
description:
'Famous 4-bar linkage invented by mathematician Pafnuty Chebyshev. The midpoint of the coupler moves in a near-perfect flat straight line along the floor, then lifts up and steps forward!',
links: { ground: 8, crank: 4, coupler: 10, follower: 10 },
couplerPoint: { alongBar: 5, perpendicular: -4 },
defaultBranch: 'OPEN',
},
{
id: 'high_reach_dumper',
name: 'High-Reach Dumping Bucket',
subtitle: 'Reaches up and tilts to dump mission elements',
description:
'The coupler extends past the follower. When the motor lifts the arm, the bucket stays upright during the climb, then suddenly tilts sharply at the top to dump cargo into a mission tray!',
links: { ground: 10, crank: 5, coupler: 12, follower: 8 },
couplerPoint: { alongBar: 12, perpendicular: 4 },
defaultBranch: 'OPEN',
},
{
id: 'quick_return_shifter',
name: 'Quick-Return Snap Lever',
subtitle: 'Moves forward with high power, returns quickly',
description:
'The forward push stroke takes more time than the return snap. Great for high-torque pushing of mission levers followed by a fast reset to save competition match time!',
links: { ground: 11, crank: 4, coupler: 10, follower: 7 },
couplerPoint: { alongBar: 5, perpendicular: 0 },
defaultBranch: 'OPEN',
},
{
id: 'double_rocker_swing',
name: 'Double-Rocker Swing Gate',
subtitle: 'Large swing arc without full continuous spin',
description:
'Non-Grashof linkage where neither arm can spin all the way around. Both arms swing through wide protective arcs to open and close gate barriers.',
links: { ground: 13, crank: 8, coupler: 7, follower: 9 },
couplerPoint: { alongBar: 3.5, perpendicular: 2 },
defaultBranch: 'OPEN',
},
];
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