“Move 5 kg, 0.5 m, in 1 second.” That single sentence sets a motion profile, which sizes a motor, which picks a transmission — and the payoff is a Pugh matrix of real, in-stock components with links you can actually click.
A single horizontal axis has to move a 5 kg payload a distance of 0.5 m and settle, all inside 1 second, repeatedly, for years. Add a carriage and tooling and call the moving mass ~8 kg. That is the whole specification, and it is enough to choose a motor, a coupling, a drive, and a rail — and to decide whether to assemble them or buy a finished actuator off a shelf.
The trap is reaching for a parts catalog first. The right first move is the motion profile: how the move is split into accelerate / cruise / decelerate fixes the peak speed and peak acceleration, and everything mechanical hangs off those two numbers.
Each link in the chain — profile → force → torque → motor → transmission — is a short calculation Claude will set up, plot, and sanity-check in one pass. It will also do the part of selection engineers quietly dread: trawling supplier catalogs for parts that hit the spec, pulling the inertia and torque numbers off datasheets, and laying them in a decision matrix. What it cannot do is pick the weights in that matrix — whether this axis cares more about cost, precision, or lead time. That judgment is yours; the rest is legwork.
The panel below is the live sizing tool. The top chart is a seven-segment (jerk-limited) motion profile — the profile real motion controllers run, with smooth acceleration ramps instead of instant jerks. Two knobs shape it: how much of the move is spent ramping (vs. cruising), and how rounded the ramps are. The bottom chart is the transmission optimization. Pick a motor, drag the knobs, and watch what passes.
The move is fixed, so peak velocity is set the moment you choose α — for a symmetric profile the velocity ramp is point-symmetric, so the jerk-smoothing knob β doesn't change the peak speed at all, only how hard the acceleration spikes:
vpk = d / [ T (1 − α) ] apk = vpk / [ αT (1 − β/2) ]Smoothing the jerk (raising β) is not free: it pushes the peak acceleration — and therefore the peak force F = ma and peak torque — up. That is the first real trade: gentler on the mechanics and the payload, harder on the motor.
The bottom chart is the part people get wrong. The classic result says the transmission ratio that minimizes required torque matches the reflected load inertia to the rotor inertia, at radius r* = √(Jmotor / m). Drag the radius slider to that dip and you will find it sits at roughly 1.5–2 mm — a belt pulley you cannot buy, and right inside the shaded zone where the motor over-speeds. For a load this light, torque is never the binding constraint; maximum RPM and what the catalog actually stocks are. So you move off the textbook optimum, to the right, until the radius is something real and the speed drops under the ceiling.
But moving right has its own cost, and the readout shows it: reflected load inertia grows as r², so the inertia ratio climbs fast. Even a small Ø20 mm belt pulley lands you around 40:1 — far above the textbook 10:1. Point-to-point belt axes live with that; a precision or high-bandwidth axis instead drops to a ballscrew, whose tiny effective radius (lead / 2π) pulls the ratio back to single digits, at the price of needing the motor near its RPM limit. Two real constraints — over-speed pushing r up, inertia ratio pushing it down — squeeze the answer between them. Knowing which one binds, and when, is the whole skill.
Sizing narrows each category to a handful of survivors. A
Pugh matrix turns “which one” from an
argument into a table: pick a baseline, score every alternative
+1 / 0 / −1 against it on the criteria this axis
cares about, and read the column totals. The point isn't the arithmetic
— it's that the criteria and their weights are written down where
you can challenge them.
Every part below was checked against its live supplier listing while this page was written. Verified 2026-05-24 — specs and links drift, so treat them as a starting basket and re-confirm before you buy.
Default scores are for a general-purpose, cost-aware, ~1 m/s,
moderate-precision axis, all criteria weighted equally.
+1 / 0 / −1 are relative to the datum column;
the weighted total = Σ (weight × score),
and the highlighted column is whichever currently leads.
† = supplier page blocks automated fetching;
link + specs confirmed via search and a distributor — re-open in a
browser before ordering.
⚙ The weight cells are editable — raise the criteria your axis actually lives or dies on (set the rest to 0) and the totals and the winning column recompute live. That is the whole point of a Pugh matrix: argue about the weights, not the answer.
· Lin Engineering 4118M-06P — NEMA 17, 62.3 oz-in (0.44 N·m), rotor inertia 0.28 oz-in²
· Lin Engineering SilverPak 23CE (CE-5718M-02P) — NEMA 23, motor+driver+controller+encoder, 1.22 N·m holding, closed-loop, 12–40 VDC
· Teknic ClearPath CPM-SDSK-2321S-RLN — 0.7 N·m cont / 3.5 peak, 3170 RPM, integrated drive+encoder, ~$315
The SilverPak 23CE and the ClearPath are both integrated closed-loop drives, so the servo's edge narrows to torque, top speed, and smoothness — it still wins for a demanding axis, but by less than against a bare stepper. The NEMA 17 is left in to show it's only marginal for 8 kg. Push the cost weight up and the integrated stepper takes it.
· Ruland zero-backlash jaw couplings† — press-fit spider, damps, 3–45 mm bore
· Ruland MBC bellows (metric, clamp)† — zero-backlash, high stiffness, 3–25 mm bore
Both are zero-backlash and wildly over-rated for ~0.3 N·m; pick the smallest bore-matched size. The jaw's damping and price win on a belt drive. Raise the stiffness weight (i.e. go to a ballscrew) and the bellows takes it.
· ServoCity GT2 20T pulley — 2 mm pitch, Ø12.7 mm PD, ~$8
· SDP/SI GT2 20-groove pulley & GT3 belt — industrial-grade
· HIWIN R16 ballscrew (10/20 mm lead) — ~€177, cut + end-machined to length
A small belt pulley wins for ~1 m/s: cheap, no critical-speed worry, low RPM demand — you accept a ~16–40:1 inertia ratio. Raise the stiffness / repeatability weights (a precision axis) and the ballscrew takes it.
· HIWIN MGN12H block — dynamic load C = 3.72 kN (vast margin on ~80 N) + 600 mm rail
· igus drylin W WS-10-80 — dry-running polymer, maintenance-free, cut to length
For a general, cost- and maintenance-sensitive axis the polymer rail edges it. Raise the stiffness or repeatability weight and it flips hard to the profile rail — the MGN12's 3.72 kN rating is enormous headroom.
· igus drylin ZLW belt actuator — stroke to 3000 mm, to 5 m/s, turnkey
· OpenBuilds V-Slot NEMA 23 belt actuator — GT3 belt, 250–1500 mm, budget DIY
Buying the turnkey actuator wins when time-to-deploy and support matter more than the last bit of cost or stiffness — you still bolt on the servo. Raise the cost or performance-ceiling weight and assemble-from-parts comes back.
A solver proves itself by conserving energy; an experiment proves itself by validity. A component selection proves itself against the datasheet. A choice is only real if it clears every one of these with margin — and they are exactly the checks Claude can compute for each candidate in the matrix.
Resist opening a catalog. Nail vpk and apk from the move first; they are the only inputs the motor sizing needs. A trapezoidal profile minimizes peak speed; a triangular one minimizes time-at-speed; the seven-segment S-curve trades a little extra peak acceleration for smoothness the mechanics and payload will thank you for.
Force becomes torque and linear inertia becomes rotary inertia through the transmission. For a belt, the lever arm is the pulley pitch radius; for a screw it is lead / 2π. Reflected load inertia is m r². Size on peak torque and RMS torque and the inertia ratio — not just whichever one is easiest to look up.
The inertia-matching radius minimizes torque, but torque is cheap here. At r* the motor screams past its RPM limit and the radius maps to hardware that doesn't exist. Let the binding constraint — speed and buildability — choose the radius, then confirm torque still has margin (it will). Optimize the thing that's actually scarce.
Two free levers worth setting. Turn the reasoning effort up
for the hard part — Claude Code's /effort (see
Feed it documents for the
model and effort controls); a transcription wants it low, an analysis like
this one wants it high. And end your prompt with an explicit
self-check — “before you finish, re-check the peak-torque, RMS-torque, RPM, and inertia-ratio margins”
— which is exactly why the prompt above asks Claude to verify itself.
Naming the oracle is the highest-value line in the prompt. And keep the
expensive model's context light — route transcription and formatting
to cheaper tools (see Spend
tokens well).
For the general-purpose axis, the matrices converge on a clean, buildable combo. Set the panel to the NEMA 23 servo and drag the radius to about 6.4 mm (a Ø12.7 mm GT2 20-tooth pulley — the ServoCity / SDP-SI part):
At that point the readout passes every rung: motor speed ~1100 RPM against a 3170 ceiling, peak torque ~0.17 N·m against a 3.5 N·m peak (over 20× margin), RMS torque a fraction of the 0.7 N·m continuous rating. The servo is barely working — which is exactly right: this is a speed-and-inertia problem, not a torque problem.
The honest footnote: the inertia ratio at Ø12.7 mm is ~16:1, above the textbook 10:1. That is normal for a belt drive and fine for point-to-point moves; go to a Ø20 mm pulley and it climbs to ~40:1. If this were a precision axis, you'd take the HIWIN ballscrew (20 mm lead → ~4:1 ratio) and a bellows coupling instead, accepting the motor running near 2200 RPM. And if time-to-deploy beats everything, you skip the build entirely: bolt the ClearPath onto an igus drylin ZLW actuator and you're moving by Friday.
You've chosen the parts. Now explore the trade or deliver the decision — see explore in HTML, deliver in PDF for the one-time setup. For this axis, the two prompts: