31:T539,intro = md`# IRIS-RFLEX Kinematics The interactive figure below represents a top-down (sky) view of 3 RFLEX v2 assemblies mounted in an IRIS base. The elements shown for each robot are labeled on this side view of the flexure part: - A: thin black line shows upper edge of clamped base - B: thick black line shows lower extent of pusher elbow (with no load) - C: red rectangle shows cross-section of pusher where it makes contact with the roller - D: blue rectangle shows cross-section that defines the keepout envelope for motion planning - E: small blue circle shows location of fiber tip ${await FileAttachment("rflex-outline.jpg").image({width:width*0.7})} Also shown for each robot, but not labeled on the side view are: - magenta circular ring shows the roller with an animated dashed circumference to indicate the rolling contact - gray line connects roller center to R-motor axis - light gray circle shows patrol radius - ticks inset from the patrol radius show theta hard stops with the active hardstop highlighted in red when theta enters the ambiguous zone between the ticks Use the sliders below to adjust the angles of the theta (-${180+geometry.toverlap/2}° to +${180+geometry.toverlap/2}°) and rflex (0° to 180°) output shafts of each robot (A,B,C). Collisions occur whenever two blue rectangles overlap.`32:T157f,function robotModel(robots, {sizeMM=8.5, sizePix=500}={}) { const { abs,sign,sin,cos,PI } = Math; const { ngroup,rtheta,rangle,rbase,wflex,roffset,rolloff,rroller, rlever,rfiber,lambda,xkeep,lkeep,xelbow,pkeep,toverlap } = geometry; if(!robots.length == ngroup) throw new Error("Unexpected number of robots"); // Establish mm to pixel mapping const fx = d3.scaleLinear().domain([-sizeMM,sizeMM]).range([0,sizePix-1]); const fy = d3.scaleLinear().domain([sizeMM,-sizeMM]).range([0,sizePix-1]); const s = fx(1) - fx(0); const svg = d3.create("svg") .attr("width", sizePix) .attr("height", sizePix) .attr("viewBox", [0, 0, sizePix, sizePix]); const g = svg.append("g") .attr("fill", "lightgray") .attr("stroke", "black"); g.selectAll("g.robot") .data(robots, d=>d.id) .join("g") .classed("robot", true) // Translate to each robot's (x,y) and orientation relative to the IRIS origin .attr("transform", d => `translate(${fx(d.x0)},${fy(d.y0)}) rotate(${-d.T0},0,0)`) .call(R => { // Draw patrol radius circle R.append("circle") .attr("cx", 0) .attr("cy", 0) .attr("r", d => s*d.rpatrol) .attr("fill", "none") .attr("stroke", "lightgray") .attr("stroke-opacity", 0.5); // Draw indicator lines for theta overlap region const frac = 0.9; [-toverlap/2, +toverlap/2].forEach(dt => { const alpha = PI/180 * (dt + rangle); const ux = cos(alpha), uy = sin(alpha); R.append("line") .attr("x1", d => s*d.rpatrol*ux) .attr("y1", d => s*d.rpatrol*uy) .attr("x2", d => s*d.rpatrol*ux*frac) .attr("y2", d => s*d.rpatrol*uy*frac) .attr("stroke-width", 3) .attr("stroke", //d => ((dt>0) && (d.T<-180+dt)) || ((dt<0) && (d.T>180+dt)) ? "red" : "lightgray"); d => (d.T - dt) * sign(dt) < -180 ? "red" : "lightgray"); }); R.append("g") .classed("theta", true) // Rotate to this robot's theta motor angle .attr("transform", d => `rotate(${-d.T+rangle},0,0)`) .call(S => { // Draw edge of RFlex fixed base S.append("line") .attr("x1", -s*rbase) .attr("y1", -s*wflex/2) .attr("x2", -s*rbase) .attr("y2", +s*wflex/2); // Draw line from center of this edge to patrol radius as theta indicator S.append("line") .attr("x1", -s*rbase) .attr("y1", 0) .attr("x2", d => -s*d.rpatrol) .attr("y2", 0) .attr("stroke", "lightgray"); // Draw pusher cross section S.append("rect") .attr("x", d => s * d.rpush) .attr("y", -s*wflex/2) .attr("width", s* pkeep) .attr("height", s*wflex) .attr("stroke", "none") .attr("fill", "red") .attr("fill-opacity", 0.5); // Draw pusher elbow (assuming it does not move with flexure) S.append("line") .attr("x1", s*(xelbow - rbase)) .attr("y1", -s*wflex/2) .attr("x2", s*(xelbow - rbase)) .attr("y2", +s*wflex/2) .attr("stroke-width", 2) .attr("stroke", "black"); // Draw fiber displaced by flexure S.append("circle") .attr("cx", d => s * d.rfiber) .attr("cy", 0) .attr("r", 4) .attr("fill", "blue") .attr("fill-opacity", 0.75); // Draw keepout displaced by flexure S.append("rect") .attr("x", d => s * (d.rkeepout - lkeep/2)) .attr("width", d => s * lkeep) .attr("y", -s * wflex/2) .attr("height", s * wflex) .attr("stroke", "none") .attr("fill", "blue") .attr("fill-opacity", 0.25); const RF = S.append("g") // Offset to RFlex motor axis .attr("transform", d => `translate(0,${-s*roffset})`) .append("g") .classed("rflex", true) // Rotate to this robots R motor angle .attr("transform", d => `rotate(${-d.R},0,0)`); // Draw radius from R axis to roller center RF.append("line") .attr("x1", 0) .attr("y1", 0) .attr("x2", -s*rolloff) .attr("y2", 0) .attr("stroke", "gray") .attr("stroke-width", 2); // Draw roller RF.append("circle") .attr("cx", -s*rolloff) .attr("cy", 0) .attr("r", s*rroller*0.75) .attr("fill", "none") .attr("stroke", "magenta") .attr("stroke-opacity", 0.5) .attr("stroke-width", s*rroller*0.5); RF.append("circle") .attr("cx", -s*rolloff) .attr("cy", 0) .attr("r", s*rroller) .attr("fill", "none") .attr("stroke", "magenta") .attr("stroke-width", 2) .attr("stroke-dasharray", [4, 4]) .attr("pathLength", 72) .attr("stroke-dashoffset", d => 36/PI*sin(d.R*PI/180)*rolloff/rroller); }) }); return svg.node(); }33:Tda6,class Robot { static _count = 0; static idFormat = d3.format("04d"); static nextId = () => `M${Robot.idFormat(Robot._count)}`; constructor({id,R=0,T=0,T0=0,x0=0,y0=0,geom=geometry}={}) { // Calculate patrol radius of fiber relative to theta axis const { rbase, rlever, rollof, rroller, rolloff, rfiber, lambda } = geom; const dmax = rbase - rlever + rolloff + rroller; this.rpatrol = -(rfiber - rbase - lambda * dmax); this.rpatrol2 = this.rpatrol ** 2; // Calculate constants for R <-> rfiber conversion this.A = rfiber - (1 + lambda) * rbase + lambda * (rlever - rroller); this.B = lambda * rolloff; // Calculate constants for keepout geometry this.hx = geometry.lkeep/2; this.hy = geometry.wflex/2; // radial offset of keepout center from fiber tip this.drk = geometry.xkeep + this.hx - geometry.rfiber; Object.assign(this, {T0,x0,y0,geom}); this.R = R; this.T = T; // Save id, using auto-generated one if necessary this.id = id || Robot.nextId(); Robot._count++; } get R() { return this._R } set R(newR) { this._R = newR; const { rolloff,rroller,rbase,rlever,lambda,rfiber,xkeep,lkeep } = this.geom; // Calculate distance of pusher face from theta axis this.rpush = -rolloff * Math.cos(newR*Math.PI/180) + rroller; // Calculate displacement of pusher from its unloaded position const delta = rbase - rlever + this.rpush; // Calculate amplified displacement of fiber tip const dfiber = lambda * delta; // Calculate distance of fiber tip from theta axis this.rfiber = rfiber - rbase - dfiber; //const r = this.A + this.B * Math.cos(newR*Math.PI/180); //const RR = Math.acos((r - this.A) / this.B) * 180/Math.PI; //console.log("rfiber", this.rfiber, r, newR, RR, this.A, this.B); // Calculate distance of keepout center from theta axis this.rkeepout = xkeep - rbase - dfiber + lkeep/2; } isReachable(x, y) { return (x - this.x0) ** 2 + (y - this.y0) ** 2 <= this.rpatrol2; } getTR(x, y) { const {PI,atan2,acos,hypot,min} = Math; let T = atan2(y - this.y0, x - this.x0) * 180/PI - this.T0 - this.geom.rangle; while(T < -180) T += 360; while(T > 180) T -= 360; // Small |r| values are not uniquely mapped to (T,R) so we assume r < 0. const r = -min(this.rpatrol, hypot(x - this.x0, y - this.y0)); const R = acos((r - this.A) / this.B) * 180/PI; return { T, R }; } getXY(T, R) { const { PI,cos,sin } = Math; const theta = (T + this.geom.rangle + this.T0) * PI/180; const r = -(this.A + this.B * cos(R*PI/180)); const x = this.x0 + r * cos(theta); const y = this.y0 + r * sin(theta); return { x, y, r }; } getKeepout({T, R, pad=0.2}={}) { // Use current angles if none provided T = T === undefined ? this.T : T; R = R === undefined ? this.R : R; // Calculate radius of keepout center at this new location const rk = -(this.A + this.B * Math.cos(R*Math.PI/180)) - this.drk; // Calculate orientation of keepout at this new location const tk = (T + this.geom.rangle + this.T0) * Math.PI/180; // Calculate unit vector components along +x const ux = -Math.cos(tk), uy = -Math.sin(tk); // Calculate keepout center const x = this.x0 - rk * ux, y = this.y0 - rk * uy; // Return rectangle as (x,y) of center, (ux,uy) unit vector along +x and half-sizes (hx,hy) return { x, y, ux, uy, hx:this.hx+pad, hy:this.hy+pad }; } }34:T4b2,class Layout { constructor(robots, {maxNeighborDist=15, extentPad=7.1, boundary}={}) { this.robots = robots; this.boundary = boundary; // Index robots by id (and check they are unique). this.index = d3.index(robots, d=>d.id); // Calculate the (x,y) coordinate extent of this layout. const pad = 7.1; this.extent = { xlo: d3.min(robots, d=>d.x0-extentPad), xhi: d3.max(robots, d=>d.x0+extentPad), ylo: d3.min(robots, d=>d.y0-extentPad), yhi: d3.max(robots, d=>d.y0+extentPad) }; // Identify neighbors of each robot. this.neighborPairs = []; const rsqMax = maxNeighborDist**2; const nrobots = robots.length; for(let i1=1; i1 rsqMax) continue; this.neighborPairs.push([robot2.id, robot1.id]); robot1.neighbors = robot1?.neighbors ?? []; robot2.neighbors = robot2?.neighbors ?? []; robot1.neighbors.push(robot2.id); robot2.neighbors.push(robot1.id); } } } }35:T41e,geometry = ({ // Distances in mm, angles in degrees. ngroup: 3, // number of RFlex units on an IRIS mount rtheta: 3.5796, // radius of each theta relative to IRIS center rfiber: 3.9017, // distance of fiber from RFlex base edge with no load rlever: 1.8501, // distance of RFlex lever surface from base edge with no load rangle: -90, // azimuthal angle of theta=0 CW relative to +x toverlap: 10, // size of theta overlap region around +/-180 deg rbase: 2.7221, // distance of RFlex base edge from theta axis roffset: 0.4, // parallel offset of R axis from T axis rolloff: 1.0, // offset of roller center from R axis rroller: 0.628, // radius of roller xkeep: 3.1796, // distance from base edge to unloaded keepout near edge lkeep: 2.8443, // length of keepout xelbow: 5.0991, // distance from base edge to unloaded pusher elbow pkeep: 1.2721, // dy thickness of pusher at R=180 (2.5mm displacement) wflex: 2.0, // RFlex thickness lambda: 1.96068,// ratio of push to fiber displacement via flexure })36:[{"id":"73dd3dc45e2d1bd3d443643ae88755256133df78c5cfdf5e50cdd96ec67d0cce1c9eb94c2fe059d0140defc23db7c10a12a7d11eb77c0486903c39fb3ff70de9","href":"https://static.observableusercontent.com/files/73dd3dc45e2d1bd3d443643ae88755256133df78c5cfdf5e50cdd96ec67d0cce1c9eb94c2fe059d0140defc23db7c10a12a7d11eb77c0486903c39fb3ff70de9","name":"rflex-outline.jpg","type":"image/jpeg","lastModified":1784759429447.317,"size":39120}] 2f:{"title":"IRIS-RFLEX Kinematics","cells":[{"id":0,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":32,"value":"dwg = robotModel(robots)","pinned":false,"hidden":false,"mode":"ojs"},{"id":203,"value":"viewof angles = Inputs.form(\n d3.range(geometry.ngroup).map(k => {\n const c = String.fromCharCode(65 + k);\n const dtheta = geometry.toverlap / 2;\n return [\n Inputs.range([-180-dtheta,180+dtheta], { label:`T${c}`, step:0.1, value: 0 }),\n Inputs.range([0,180], { label:`R${c}`, step:0.1, value: 0 }),\n ];\n }).flat(),\n {\n template: (inputs) => html`

${Object.values(inputs)}`\n }\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":275,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":313,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":319,"value":"layout = new Layout(robots)","pinned":false,"hidden":false,"mode":"ojs"},{"id":27,"value":"robots = {\n const r = geometry.rtheta;\n const { cos,sin,PI } = Math;\n const x = r*cos(PI/6), y = r*sin(PI/6);\n return [\n new Robot({ T:angles[0], R:angles[1], x0: 0, y0: -r, T0:-90 }),\n new Robot({ T:angles[2], R:angles[3], x0: x, y0: y, T0:30 }),\n new Robot({ T:angles[4], R:angles[5], x0: -x, y0: y, T0:150 }),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W36","stdlib":"1","titling":"auto"}