Sprung Linkage

Also known as a sprung arm.

Sprung LinkageContinuousCascadeCustom PartsAdvanced

An ordinary extension or gas spring drives the lift through a lever arm, linkage, or cam profile. Because the moment arm changes with position, the output force profile can in principle be shaped to match gravity closely — at the cost of geometry design and tuning.

Best for

Offseason projects and advanced teams who want a custom force profile (including near-perfect gravity matching) from cheap springs.

Poor fit for

Mid-season adoption, tight packaging, or teams without the CAD and iteration time to design and tune linkage geometry.

ComplexityHigh

Custom geometry, pivots, and structure; the force profile is an engineered output, not a purchased one.

Force accuracyest.Potentially excellent

In theory the profile can be shaped to match gravity almost exactly; achieved accuracy depends entirely on execution and tuning.

Packagingest.Demanding

Levers and links sweep through volume as the lift moves; the swept envelope must stay clear.

Tuning effortest.High

Geometry changes require remanufactured parts; expect multiple iterations to converge.

In-season riskest.High

Long, hard-to-bound development time makes this a classic offseason experiment.

Click a metric to see why it’s rated that way. Ratings marked est. are editorial estimates awaiting test data.

Quick recommendation

Use this when

You want a force profile no COTS spring provides (e.g. matching a stepped or angled-lift load)
You have offseason time to iterate on geometry and validate with measurements
Structural mounting for linkage pivots and spring loads is available
Your team is comfortable with linkage synthesis or cam profile design in CAD

Avoid this when

It is competition season and the lift must work next week
Packaging around the lift base is already tight — linkages sweep volume
Pivot slop or frame flex would corrupt the designed force profile
Nobody on the team can own the geometry design end-to-end

Mechanism overview

Every other method on this site buys its force profile from a catalog. A sprung linkage designs it: an ordinary extension spring (or gas spring, or torsion spring) acts on the lift through a lever, linkage, or cam, and the geometry — not the spring — decides what force the lift feels at each position.

Annotated diagram — to be addedA lever pivoted at the base with an extension spring attached partway along it, connected to a lift stage. Annotations show the spring's moment arm shrinking as the lever rotates while the spring force grows — the two effects canceling to hold output force constant.

The core idea: as the mechanism moves, the spring stretches (force grows) while its moment arm about the pivot changes. Choose the geometry so those two effects multiply into whatever output profile you want — constant force, a staircase, or a profile matched to an angled lift.

Terminology used on this page

Moment arm — perpendicular distance from a pivot to a force's line of action.
Transmission ratio — output force per unit spring force, set by the ratio of moment arms; varies with position.
Cam profile — a shaped surface a follower rides on: the fully general way to encode a force-vs-position curve.
Zero-free-length spring behavior — the classic trick behind perfect gravity balancers (see derivation below).

Continuous and cascade implementations

The continuous staircase is the hardest target for a linkage: the ideal output force jumps at stage transitions, and smooth linkage geometry can only approximate a jump with a steep ramp.

Typical arrangement: the linkage acts on the first stage or carriage run near the lift base; a cam gives the sharpest achievable transitions.
Force profile: whatever you design — realistically a smoothed staircase with error concentrated at the transitions.
Advantages: the only method that can even attempt exact matching of a stepped load with one spring.
Limitations: every profile change is a part change; friction and pivot slop blur exactly the sharp features you fought for.

Physics and force matching

Moment balance at the pivot

F_{\text{out}}(q)\; a_{\text{out}}(q) \;=\; F_{s}(q)\; a_{s}(q)

In plain terms: At every position q of the mechanism, the output force times its moment arm equals the spring force times the spring's moment arm. Because both moment arms change with position, the ratio between spring force and output force is programmable through geometry.

Assumes: Quasi-static · Frictionless pivots · Rigid links

Design target

F_{\text{out}}(q) \;\overset{!}{=}\; F_{\text{gravity}}(x(q))

In plain terms: The design problem in one line: choose geometry so the linkage's output force equals the gravity load at every corresponding lift extension. The exclamation mark means 'this is the requirement we're solving for,' not an identity.

Variable definitions
Symbol	Quantity	Unit	Notes
$q$	Mechanism position	rad or m	Lever angle or slider position
$a_{s}(q)$	Spring moment arm	m	Changes with position — the design knob
$a_{\text{out}}(q)$	Output moment arm	m	—
$F_{s}(q)$	Spring force	N	k·(stretch + pretension), itself position-dependent

The zero-free-length trick: why perfect balance is possible at all

The classic result behind gravity balancers: hang a lever at angle φ with a weight torque W·L·sin φ. Attach a spring from a frame point directly above the pivot (height h) to a point on the lever (distance d from the pivot). If the spring's force is proportional to its total length (a "zero-free-length" spring), its torque about the pivot works out to k·h·d·sin φ — the same sin φ shape as gravity. Set k·h·d = W·L and the balance is exact at every angle, not just one.

Real extension springs aren't zero-free-length, but pretensioned springs, cable-over-pulley arrangements, or an offset anchor can approximate one. This is the deep reason lamp arms float — and the existence proof that a sprung mechanism can match gravity exactly, given faithful execution.

Worked example: Constant-force lever for a cascade first stage

Target ≈40 N constant at the first stage over a 25 cm stroke, lever length 15 cm, spring anchored per the zero-free-length layout.

Using the layout from the derivation with h = 10 cm anchor height and spring attachment d = 8 cm along the lever: required k·h·d = F·L gives k = 40 × 0.15 / (0.10 × 0.08) = 750 N/m — a stiff but ordinary extension spring. The output stays constant only as well as the spring approximates zero-free-length behavior; expect to tune anchor height on the real mechanism.

Numbers are illustrative — the geometry must be designed and checked in CAD against your stroke mapping from lever angle to lift extension.

Interactive force graph

The model below abstracts execution quality into a single "geometry match" parameter — at 100% the linkage tracks the gravity profile exactly; lower values add the smoothing and ripple that real pivot slop, friction, and imperfect synthesis produce.

Force vs. extension — Sprung linkage

Illustrative model, not measured data

RMS error 0.2 NNormalized RMS 1%Max |error| 0.3 N

Stringing architecture

Carriage / outtake mass1.20 kg

Mass per stage0.45 kg

Stages3

Travel per stage30 cm

Lift angle from horizontal90 °

Geometry match quality85%

View chart data as a table

Extension (cm)	Gravity (N)	Counterspring (N)	Motor (N)
0.0	11.8	11.8	0.0
7.5	11.8	12.0	-0.2
15.0	11.8	12.1	-0.3
22.5	11.8	12.1	-0.3
30.0	11.8	12.1	-0.3
37.5	16.2	15.9	0.3
45.0	16.2	15.9	0.3
52.5	16.2	16.1	0.1
60.0	16.2	16.5	-0.3
67.5	20.6	20.7	-0.1
75.0	20.6	20.9	-0.3
82.5	20.6	20.8	-0.2
90.0	20.6	20.6	0.0

Design and sizing

Lever-arm force calculator (single position)

Spring force

Spring moment arm

Load moment arm

Output force at this position

20.0 N

Leverage ratio

0.33 : 1

This computes output force at ONE position. In a real linkage both moment arms change as the mechanism moves — that changing ratio is the whole design space. Plot the profile across travel in CAD before committing.

A realistic design loop:

Fix the target profile — gravity load vs. extension from your CAD masses (same measurement discipline as constant force).
Choose a topology — plain lever, four-bar, or cam + follower. Prefer the simplest topology that can hit the profile; every extra joint adds slop.
Synthesize geometry — sweep the mechanism in CAD, plot output force vs. extension, iterate anchor points until the curve lies on the target.
Check structure — pivots and anchors see the full spring force continuously; frame flex shifts the profile.
Plan adjustability — slotted anchors or multiple mounting holes; you will tune on the robot.

CAD, manufacturing, and assembly

CAD embed — to be addedSprung-linkage counterspring reference build — No verified reference CAD is available yet. A documented lever or cam balancer belongs here.

Pivot quality is the product: bushings or bearings at every joint; slop at the pivot is noise multiplied through the transmission ratio.
Link manufacturing: printed links for the first geometry iterations; laser-cut or machined for the final — geometry that creeps isn't geometry.
Spring anchors: springs fail at their hooks; use proper spring anchors or captive loops, not screws through the end coil.
Swept volume: model the mechanism's full sweep as a solid in CAD and keep it reserved — the linkage will claim it in every match.
Pinch guards: a loaded lever that snaps through its sweep is a finger hazard; guard it.

Robot photo — to be addedAssembled lever-and-spring counterspring on a lift base, showing bearing pivots, slotted spring anchor for tuning, and the guard covering the sweep volume.

Testing and validation

Profile measurement: motor disconnected, pull the lift through travel with a luggage scale in both directions. Plot the result against the designed profile — this is the report card for your geometry.
Release test: float points, climb regions, settle regions — same interpretation as the constant-force procedure.
Slop audit: wiggle every joint before and after cycle testing; log growth in free play.
Powered testing: current, speed, thermal, and cycle testing per the standard protocol.
Re-measure after tuning: every anchor adjustment changes the whole curve — re-run the profile measurement after each change.

Real-world examples

Sprung arms are common on rotating robot mechanisms (a future rotational section will cover them properly). Documented sprung linkages counterspringing linear lifts are rare, and none are verified on file yet.

Your team here

Sprung Linkage

— · architecture unrecorded

A documented linkage or cam balancer on a linear lift — with the measured force profile and the iteration story — belongs here.

Placeholder card — a real, verified team example belongs here. Contributions welcome.

Common failure modes

Profile drifts as pivots wearMedium severity

Cause: Slop accumulating at each joint, multiplied by the transmission ratio.
Symptoms: A lift that used to float now settles or climbs; audible knock at direction changes.
Prevention: Bearings/bushings at every pivot; slop audit in the maintenance routine.
Fix: Replace worn pivots; re-tune anchors afterward.

Spring hook failureHigh severity

Cause: Extension spring loaded through a screw or sharp hook, fatiguing the end coil.
Symptoms: Sudden total loss of counterspring; spring found across the shop.
Prevention: Proper anchors or captive loops; inspect end coils at events.
Fix: Replace the spring and the anchor detail that killed it.

Linkage sweeps into another mechanismMedium severity

Cause: Swept volume not reserved in CAD, or a late mechanism added into it.
Symptoms: Intermittent jams at specific lift heights.
Prevention: Model the sweep as a solid keep-out in the assembly.
Fix: Re-route the offending mechanism — not the linkage, whose geometry is the product.

Never converges during tuningHigh severity

Cause: Topology can't produce the target profile, or friction swamps the geometry.
Symptoms: Each anchor tweak fixes one region and breaks another.
Prevention: Verify the profile is reachable in CAD sweep analysis before building; minimize joint count.
Fix: Change topology (often: add a cam) or accept a looser target.

How this compares

Bungee

Bungee is what you build when force accuracy doesn't matter; a linkage is what you build when it matters enough to engineer for. Between those extremes, the COTS spring methods usually win.

Guide

Constant Force

Constant-force springs buy 90% of the accuracy for 10% of the design effort on continuous lifts. Choose the linkage only when no catalog rating fits — unusual profiles, angled lifts, or force levels between ratings.

Guide

Constant Torque

Both are advanced methods; constant torque delivers one fixed level from a catalog part, the linkage delivers any level and any shape from custom geometry. Catalog part for constant targets, linkage for shaped ones.

Guide

For the full matrix, force-curve overlay, and decision summary, see Compare Methods.

References

Spring-balancer literature: zero-free-length gravity compensation (to be added)Engineering source
The classical statics result behind perfect spring balancing; standard mechanism-design references cover it.
counterspringing.com planning notesEngineering source
Framing source: methods list and offseason-vs-in-season guidance.
Verified implementations (to be added)Robot example
None on file for linear lifts.