Springs, suspension and compliance for leg v2 (2026-10-27)

The question: do Gyropup's five-bar legs need springs or anything like a suspension, both to carry the body at a resting height and to ride and land better; what do similar robots do; how much higher could we jump and how much smoother could we ride; and what does it cost. This file is the sourced survey (section 2), the physics of our own leg measured in the single-leg rig (sections 3 and 4, sim/leg_spring_study.py, outputs in exports/leg/springs/), the real parts (section 5), the trade-offs (section 6) and a recommendation (section 7).

Every URL was checked on 2026-10-27. Tags: [derived] = computed by us from a public model or from physics (inputs given); [unverified] = seen only in a search snippet or a secondary source; [estimate] = our engineering estimate where nothing is published.

1. The answer up front

• A parallel spring does what the survey says it does, but the one place it fits is not worth it yet. A McMaster extension spring between the two upper links (section 7) cut the standing hip torque from 7.7 to 0.2–1.2 N·m, added 14 % to the jump and held an unpowered leg at ≈300 mm in the planar rig, but it fails in 3-D (section 8): below 250 mm the coil passes through the hip region. The layout that clears the CAD at every length, a ≥1.4 N/mm spring knee to knee outboard of the knee bolts (5108N807 + a 76 mm lanyard, or two 5108N78), does the same job at 13.1 kg (standing 9.8 → 0.3 N·m, jump +24 % with the feed-forward, the collapse from 3.0 kN to 65–70 N damped) but needs posts, lanyards, a guide, a boot and a 7 mm gap to the tyre at full crouch. Build the rig without a spring, and bench-test the RS-04's power-off damping first (sections 8.7, 7).
• No spring touches the landing spike. It is set by the rotors' reflected inertia being stopped through the tyre, and a parallel spring does not see an impulse. What does: a softer tyre (−27 % pneumatic-stiffness, −42 % foam), elastomer axle mounts (−39 %), retracting the leg through touchdown (−18 %), and a touchdown-speed cap.
• The gearbox is not the limit on the rig's own measure: 62 N·m through it at the 150 mm drop and 88 N·m at 300 mm with 11 kg, under the RS-04's 120 N·m peak; the 143 N·m that raised the alarm was the quasi-static J^T·F, which overstates the gearbox three times and the knees twice. The knees are sized by the undamped power-off collapse, not by landings; the spring removes that collapse (section 4.5).
• Nobody with a quasi-direct-drive wheel-leg uses series compliance (section 2): parallel springs for load and jumps (Ascento, HKUST, the RoboMaster gas-strut teams, ANYmal's knee spring, Penn's five-bar), tyres and software for landings. Gas struts are jump assists with stops and short lives; torsion springs at our hips would be custom and the wrong torque shape.
• A rig defect: sim/leg_rig.py carries 8.5 kg, not 11 (the dummy mass on a geom is ignored when the body has an inertial element); every load in rig-results.json is for 8.5 kg (section 3.1).

2. Survey: what similar robots do

2.1 At a glance

2.2 What the survey says

1. No quasi-direct-drive or direct-drive wheel-leg uses a series spring. Diablo, TITA, Upkie, Ollie, TRON 1, the Unitree wheelers, every RoboMaster build, Minitaur, Doggo and the MIT Cheetahs are rigid from motor to wheel; their compliance is the tyre and the controller's virtual spring-damper. Series springs appear only where they hide a high-ratio rotor (Cassie's 16:1 knee behind 2300 N·m/rad leaf springs; ANYmal's harmonic drive behind the ANYdrive spring), and they cost bandwidth (Wensing: a 70 N·m/rad series spring drops the MIT Cheetah leg's first mode from 79.7 Hz to 13.5 Hz) and an extra state to estimate (Cassie: >0.35 rad sim-to-hardware tarsus error, ±20 % stiffness randomisation in training). Minitaur's designers say it outright: SEA "removes the low-pass filtering of actuation signals" only when you leave it out.
2. Parallel springs are common, and always for load and energy, never for the impact. Ascento's torsion springs "counteract the system's own weight and reduce the control effort of the hip motors when driving or standing, increasing overall efficiency and jumping height" (the sentence from the paper, p. 2). HKUST's tension spring took standing torque from 2–7 N·m to about ±1 N·m and doubled runtime. ANYmal's optimised knee spring cut the knee's torque-squared cost by a third and its peak torque by 30 %. The Penn five-bar rig stored 5.6× the energy and returned 1.6× the kinetic energy with a parallel spring, and states the physics: the spring constant does not appear in the touchdown impulse. The RoboMaster gas-strut teams (300–400 N) use the strut as a jump assist and get 280–420 mm jumps on 20–25 kg robots.
3. Every parallel spring must be modelled and cancelled by the controller. HKUST: "spring torque computed from the two joint angles and subtracted directly"; Tongji fits the strut force as a cubic in leg length; 山海机甲 models the strut's progression and velocity dependence; SPR warns that if the compensation is not released during the push, "0 N and 450 N give the same jump height", and that the liftoff detector misfired once the strut's vertical component was counted as support force. Ascento and 山海机甲 both note the price: retracting the legs in flight means working against the spring (Ascento: up to the full 40 N·m).
4. Gas struts need stops and die without them. HKUST 2025: "without hard stops the leg can pull the strut apart"; they moved to polyurethane bumpers. McMaster's own gas springs are rated 6 cycles per minute and 25,000–30,000 cycles (section 5): lid lifters, not suspension.
5. Landings are software plus tyre. Everyone lands on a leg-length PD (RoboMaster: ≈1000 N/m, 100–200 N·s/m; Doggo: "low stiffness and lots of damping"), with contact detection from torque or support force (Ascento, Minitaur) and, where it is done well, re-extension of the legs before touchdown (山海机甲). The only hardware lever anyone reports for the impact itself is the tyre: WIT's pneumatic bike tyre, Go2-W's pneumatic 7-inch, HKUST's 60A hexroll.
6. What breaks in landings is the output flange or the gearbox, then the links: HT-04 flanges snapped at 3 × M4 + 3 pins (HKUST, fixed with a CNC hub on a large bearing); planetary bearings and teeth in wheel gearboxes (SJTU, HKUST 2026); link screws loosening and links snapping (南航金城); bicycle-headset joints to stop joint splay (DGUT). This matches Wensing: "the highest loads felt in the legs, gearboxes, and other transmission components are governed, in large part, by design decisions centered on reflected actuator inertia."

Reflected inertia per joint, for scale (kg·m²): Minitaur 0.0001, Doggo 0.00026, Mini Cheetah ≈0.0023, MIT Cheetah 0.0102, RS-04 ≈0.024 [estimate], HUBO 0.085, Cassie knee ≈0.093 (behind its leaf spring). Per kilogram of robot the RS-04 is about three times the MIT Cheetah and a third of Cassie.

3. The physics for our leg

3.1 What the motors hold, and the shape of the curve

With 11 kg on one leg (the rig's half robot) the leg holds 86.7 N at the axle (the carriage, less the wheel and half the wheel link, which the ground carries directly). The quasi-static torque per motor to hold it, through the five-bar's Jacobian, is nearly flat over the working range [derived, sim/leg_spring_study.py hold_torque]:

Two consequences. A gravity compensator for this leg wants a nearly constant torque from 150 to 300 mm and a falling one above 330 mm, so a plain torsion spring (torque proportional to angle) is the wrong shape: sized at a middle height it over-compensates in the crouch (the motors then hold the leg down against it) and keeps pushing at full extension where gravity needs almost nothing. An extension spring between the two upper links has the right shape because its moment arm about the hips shrinks as the leg folds while its stretch grows: with pins 100 mm out on each upper link the spring's length runs 276 mm (300 mm leg) → 330 (200) → 347 (140) and it goes slack past 350 mm of leg, exactly where gravity's demand collapses. A gas strut on the leg's axis (hip frame to the wheel axle) needs just the held weight, 87 N, at every height, but its stroke (291 mm from the deepest crouch to full extension, 2.4:1) is beyond any stock strut, so it can only cover the upper part of the range.

The rig's steady standing torque is 7.7 N·m per motor at 300 mm with 11 kg (the "about 9 N·m" quoted before is the controller's first sample, before the leg settles; the rig's own series settle at 5.1 N·m, see the next paragraph).

A defect in the rig to fix: sim/leg_rig.py gives the carriage an explicit <inertial> (the hip frame) and puts the 2.5 kg dummy mass on a box geom's mass attribute. MuJoCo ignores a geom's mass when the body has an inertial element, so the rig carries 8.5 kg, not 11: its standing ground force is 84.6 N instead of 108 N, and every load in exports/leg/rig-results.json (1.6 kN on the axle at the 150 mm drop, 160 N·m on the stop) is for 8.5 kg. The study restores the mass (build_xml, fix_mass) and keeps a none_rig85 variant that reproduces the rig as it runs today. All numbers below are for 11 kg unless they say otherwise.

3.2 What a landing does to the gearbox and the knees

At touchdown the wheel stops in the tyre's few milliseconds; the body (on the rail), the links and the motors' rotors keep going. In joint coordinates the rotors' reflected inertia (0.0243 kg·m² per hip, an estimate: RobStride publishes none) is 2.4 kg of reflected mass at the axle per motor, so the rotors are most of what the tyre has to stop. The rig (since bc63ce9) measures what matters directly, every step: the torque through each gearbox, τ_motor − I_rotor·α − b·ω, and the force through each joint (cfrc_int: knees, hips, and the apex's constraint). The study records the same quantities with the same formulas (peak_gearbox_nm, gearbox_over_peak_ms, peak_joint_force_n) and judges every variant on them. The earlier quasi-static mapping J^T·F of the axle force (143 N·m at 1.6 kN) assumed the hips fixed in space; the gearbox actually carries about a third of it (62 N·m at the 150 mm drop, 11 kg), because the rest of the axle force goes straight into the carriage through the links' axial forces (the body decelerates at 9 g in the first millisecond) and into the lower links' own inertia; for the same reason the knee pins see about 500 N where the quasi-static mapping of the same axle force gives 1,170 N (the "about 2×" the strength studies now correct for). That is not an artefact of a soft loop: with the apex constraint four times stiffer (none_stiffloop) the numbers barely move (section 4.3). J^T·F is kept in the drop scan for reference only.

The knees' governing case is not a landing at all but the undamped power-off collapse: the leg folds under the body's weight and the upper links slam into the knee-out stops (the TPU sleeves, then the steel posts), which gives the rig's 1.8 kN at the knee (8.5 kg) against the 6201s' static safety 1.15 and the knee eye's SF 1.03–1.18. Section 4.5 reports what each spring does to exactly that case.

Two facts follow for the design. The impulse the rotors take is set by the wheel's absolute velocity at touchdown, not by what the leg is doing: if the leg is retracting at v_r as the wheel meets the ground, the wheel's ground speed is v_body − v_r, and the joint-velocity jump the rotors must absorb is smaller by that amount. And a parallel spring cannot change the spike: it acts between body and wheel in parallel with the motors, and an impulse does not see a spring (Penn: "the spring constant, k, does not appear in [the collision]"; Wensing: series compliance "redistributes some of the impact impulse to a window of time following impact … [it does] not change the initial impulse"). Only series compliance between the wheel and the links (the tyre, an elastomer axle mount) stretches the spike in time.

3.3 The variants modelled

The rig's MJCF comes unchanged from sim.leg_rig.mjcf(); sim/leg_spring_study.py edits the XML per variant and runs the rig's own virtual-model controller, torque-speed and 60 N·m clipping, and TPU stop torques, on every variant.

Scenarios, all at 0.5 ms with every step recorded: standing at 300, 200 and 140 mm (motor torque, I²R as a fraction of the rated loss and in watts at an assumed 0.16 Ω); a jump at the 60 N·m cap and at 120 N·m (height, spring energy stored and returned, motor work); drops from 25 to 300 mm with the leg holding 300 mm (axle force, J^T F, gearbox torque, dynamic and quasi-static knee forces); a moving belt with a 20 or 40 mm bump or kerb at 1 and 2 m/s (body acceleration, wheel load, airborne time); power loss with and without the RS-04's assumed anti-backdrive damping (where the leg rests, the stop impact). The sweeps (sweep-torsion.json, sweep-strut.json, sweep-ext.json) cover rate × preload, 25 points per family, on standing, the jump, the 150 mm drop, power-off and the 40 mm bump.

4. Results

All from exports/leg/springs/results.json (one leg, 11 kg carried, 0.5 ms steps, every step recorded; figures scorecard.png, static-curves.png, drop-scan.png, impact-150.png, ride.png, sweep-*.png; videos <variant>_<scenario>.mp4). "Today" is none; none_rig85 is the rig as it runs today with 8.5 kg.

4.1 Standing: torque and heat

Standing is not hot today: at 7.7 N·m the RS-04 runs at 3.7 % of its rated copper loss (3.7 A rms), about 6 W per motor, 26 W for four, against the 175 W the rated point dissipates. The spring removes 85–97 % of that, which is a runtime gain of the order of 25 W whenever the robot is up, not a thermal rescue; HKUST's HT-04s at 6–7 N·m were a thermal rescue because those motors are smaller. The gas strut and the coil-over over-compensate or vanish in the crouch (the slot parks the strut below 200 mm; the coil-over pushes 16.5 N·m at 140 mm), so only the between-links spring and the mid torsion spring help at every height.

4.2 Jumps

The real spring, cancelled by the controller so the crouch tracks cleanly, is worth +39 mm at the cap (+14 %) and +38 mm at 120 N·m (+13 %): the 3.6 J it returns during the 0.25 s push is 12 % of the motors' own push work. Doubling the preload (torsion_preload) returns 6.2 J but jumps no higher, because the stiffer leg loses it in the push profile. The coil-over is a disaster (over-compensation in the crouch, and a damper eats half of what is left: 149 mm). A deeper crouch is the bigger lever (+36 mm) and the two add. For comparison the main session's earlier figure, 245 mm at the cap, was the 8.5 kg rig (none_rig85: 248 mm here).

4.3 Landings

The 150 mm drop, 11 kg, the leg holding 300 mm (touchdown at 1.48 m/s, 16 J per leg); the 300 mm drop (2.16 m/s, 32 J); and the landing of the robot's own 60 N·m jump (an apex of 278 mm, touchdown at 2.38 m/s):

(Knee, hip and apex forces per scenario, the quasi-static comparators and the full drop scans from 25 to 300 mm are in results.json → scenarios and landing; at 150 mm the hips see 406 N and the apex 759 N on today's leg.)

Three things stand out. No parallel spring reduces the spike; most raise it a little (a stiffer leg), and the preloaded torsion spring, which extends the leg onto its stop in the air, makes the 300 mm drop a 6 kN stop impact. Only series compliance and firmware reduce it: a pneumatic-stiffness tyre takes 27 % off the axle force and the gearbox torque, a foam tyre 42 %, soft axle mounts 39 %, retracting through touchdown 18 %, retracting onto the pneumatic tyre 35 %. The gearbox is not the limit on the rig's own measure: 62 N·m at 150 mm, 88 N·m at 300 mm (none), under the RS-04's 120 N·m with every variant and never above it for any time; the quasi-static J^T·F that first raised the alarm reaches 120 N·m at an 80 mm drop, but it is the wrong measure for a spike that the carriage and the links share. The drop scan (drop-scan.png) is linear in touchdown speed for every variant, so the numbers scale: each 1 m/s of touchdown speed is ≈1.2 kN on the axle and ≈40 N·m on the gearbox today, ≈0.9 kN and ≈30 N·m on the pneumatic tyre.

4.4 Ride over bumps and kerbs

Peak body acceleration (g) on the moving belt, 1.15 s window round the obstacle:

The RMS over the window is 0.4–0.9 g for every variant but the preloaded torsion spring: the ride is the leg's own PD (3,000 N/m on 11 kg is a 2.6 Hz suspension), and a hardware spring only changes it by adding stiffness the controller did not ask for (ext_5108N807 uncancelled: +25 % on the peaks; cancelled: within 0.3 g of today). The sharp peaks are the tyre's: a small sharp bump at 1 m/s is the harshest case (11.9 g for 0.17 s of flight) and the pneumatic-stiffness tyre takes 35–40 % off it, the foam tyre 50 %. The preloaded torsion spring, sitting on its extension stop, turns the leg rigid (24–37 g). The soft axle mounts help on bumps but bottom on the 40 mm kerb (3.7 g against 2.4).

4.5 Power loss: the collapse

Standing at 300 mm, then the motors lose power. Undamped, today's leg folds under the body and the upper links hit the knee-out stops; with the RS-04's assumed anti-backdrive damping (20 % of shorted phases) it sinks onto them. The rig's own measures, every step (peak_joint_force_n, peak_gearbox_nm):

Three results. First, the rig's 1.8 kN is confirmed (none_rig85: 1,747 N at the knee, 169 N·m on the stop) and it is an 8.5 kg number: with the 2.5 kg restored the same collapse puts 2.4 kN on the knee and 194 N·m on the stop, a third more than the strength studies' load case, which would take the knee eye from SF 1.03–1.18 to about 0.8 and the 6201s from 1.15 to about 0.85. Second, the 5108N807 removes the case: with gravity carried by the spring the unpowered leg settles where the spring's torque equals the body's, 302–324 mm, touches no stop, and the knee sees 52 N, the static load; damped or undamped. The worst knee load left on that leg is the landing of a full 120 N·m jump, 1,068 N (jump120), 45 % of today's collapse and 59 % of the 1.8 kN the knees are sized for; the next is the 20 mm kerb at 1 m/s, 809 N. Third, the preload decides which way a spring-loaded leg goes: a spring that carries more than the weight at full extension (torsion_mid, the 130 % strut, the real 20 lbf strut, which is 26 lbf compressed) drives the leg up onto the knee-in stops, 134 N·m for the torsion spring; a spring that carries less than the weight everywhere (strut_gas70) still folds onto the knee-out stops. The 5108N807 is in between by construction: it is slack above a 345 mm leg, so it cannot extend the leg, and it carries 0.9 of the weight at 300 mm, so it cannot drop it.

A spring does not make the knee stronger. A kid can still push the body down onto the stops, the robot can still fall onto a folded leg, and the spring can break; the collapse case should stay in the strength studies as a rare case at 11 kg (2.4 kN). What the spring does is turn an everyday event (every power cut, every soft e-stop) into a rare one.

A note on the gearbox: across every scenario on today's leg its worst load is not a landing but the 20 mm sharp edge at 1 m/s (the bump and the kerb both: 104 N·m, with 3.8 kN on the tyre for a few milliseconds) and the full 120 N·m push itself (102 N·m); the 300 mm drop is 93 N·m, the collapse 72 N·m. Nothing exceeds 120 N·m for any time except the stiff axle bushing in the 120 N·m jump (1.5 ms). The pneumatic-stiffness tyre takes the kerb figure to 70 N·m and the foam tyre to 50 N·m.

4.6 Balance control: what the spring adds

Static, at 300 mm, from the spring's force mapped to the wheel (results.json → static): the 5108N807 adds −305 N/m along the leg (its force falls as the leg extends; the controller's virtual spring is 3,000 N/m) and 2.4 N·m of lean torque at 15° of lean (the lean loop applies 39 N·m there); the 2.3 N/mm ext_links adds −810 N/m and 2.0 N·m; torsion_mid +200 N/m and −4.6 N·m; the gas strut −90 N/m and −1.2 N·m. The worst torque the motors still have to supply between 150 and 340 mm falls from 10.3 N·m (today) to 4.4 N·m (5108N807), 2.4 N·m (torsion_mid) and 6.9 N·m (ext_links). All of it is a smooth, known function of the two motor angles, and the _c runs show that once the controller subtracts it the leg behaves as today in every scenario that does not saturate the motors.

4.7 The sweeps: rate × preload

Each family on a 5 × 5 grid of rate and preload (the fraction of the held weight the spring carries at 250 mm), each point run through standing, the 60 N·m jump, the 150 mm drop, the collapse (undamped and damped) and the 40 mm bump; maps in sweep-torsion.png, sweep-strut.png, sweep-ext.png, numbers in sweep-*.json. Cells below read: standing torque at 300 mm (N·m) / worst torque left 150–340 mm (N·m) / jump at 60 N·m (mm) / collapse knee force, undamped (N) / power-off rest length, damped (mm) / 40 mm bump at 1 m/s, peak body g. Today's leg is 7.7 / 10.3 / 278 / 2,405 / 101 / 3.7.

Torsion springs at the hips (N·m/rad):

The torsion family's sweet spot is soft and under-preloaded: 2–3.5 N·m/rad at 0.75× the weight (standing 0.6–1.2 N·m, +16–20 mm of jump, the unpowered leg resting at 274–325 mm with 50–119 N on the knee). Everything above 1× drives the unpowered leg onto its extension stop (rest at 392 mm) and, uncancelled, stiffens the ride up to 26 g on the 40 mm bump; everything stiffer than 6 N·m/rad jumps lower than no spring at all. The jump gain never exceeds +21 mm (+8 %) in this family, less than the between-links spring's +39 mm, because a torsion spring's torque keeps rising into the crouch where the five-bar needs less.

A strut on the leg's axis (hip frame to axle), N/m; 0 = a gas strut, on a 200 mm stroke with a lost-motion slot:

("bind" = the coil-over reaches its coil bind in the crouch, hundreds of N·m.) Only the gas strut at exactly the weight is any good here (standing 0.8 N·m, +16 mm of jump, the unpowered leg held at 354 mm), and its worst residual stays at today's 10.1 N·m because the slot parks it below 200 mm; 70 % still folds the leg (1,937 N at the knee), 125 % and up extends it onto the stop. Every coil-over is worse than no spring: it over-compensates the crouch (standing torque rises again above 1 N/mm) and the jump collapses (66–255 mm).

An extension spring between the upper links, pins 100 mm out (N/m; the 5108N807 is 1,576 N/m at ≈1.0):

The real spring sits in the best cell: 1,500 N/m carrying the weight at 250 mm gives 0.1 N·m standing, +26 mm of jump uncancelled (+39 cancelled, section 4.2), a 49 N collapse with the leg parked at 328 mm, and 5.2 g on the bump uncancelled (today's 3.7 once cancelled). Softer and more preloaded springs jump higher (up to +47 mm at 1,000 N/m and 1.5×) but extend the leg onto its stop when unpowered and make the uncancelled ride harsh; stiffer springs lose the jump and no longer hold the leg up (the knee is back at 0.8–1.9 kN). Across all three families the rule is the same: carry about the weight at mid-height with as soft a spring as the stroke allows, and never more than the weight at full extension.

5. Real parts (McMaster-Carr pages read one at a time on 2026-10-27)

What McMaster stocks, from the category tables and the product pages (exports/leg/springs/mcmaster/*.page.json):

Extension spring for the between-links layout: 5108N807 (music-wire steel, loop ends; 5667N275, $15.15, is the same spring with hook ends, which McMaster says are more likely to slip off or break, so the loop-end part is the one to buy): 9 in (228.6 mm) long, 1 in (25.4 mm) OD, 0.125 in (3.18 mm) wire, 9 lbf/in (1.58 N/mm), 6.11 lb (27 N) initial tension, 67.87 lb (302 N) at the 15.84 in (402 mm) maximum extended length; $14.80 each; "Not Rated" tolerances. With its loops on pins 100 mm out on the two upper links it is stretched 47 mm at a 300 mm leg (101 N, 0.9 of the held weight), 101 mm at 200 mm (187 N, 0.8) and 118 mm at 140 mm (213 N, 0.43), 173 mm short of its maximum extension at the deepest crouch, and slack above a 345 mm leg. Mass ≈0.3 kg [estimate: 68 coils of 3.2 mm wire on a 22 mm mean diameter]. Fatigue [estimate]: the uncorrected shear stress at the maximum rated load is 8·F·D/(π·d³) = 530 MPa, 645 MPa with the Wahl factor for C = 7, about 34 % of music wire's ≈1,900 MPa tensile at this diameter, which is the edge of the usual infinite-life band for unpeened steel springs; at our working stretch (47–101 mm, 101–187 N) it is 11–20 % and the body of the spring will outlive the robot. The loops are the weak point of any extension spring (McMaster: loop ends "won't slip off" but hooks and elongated hooks "are more prone to breaking if overstretched"), so the pins must be round, 6–8 mm, and the loops must never be loaded sideways. McMaster's metric extension springs stop at 39 mm long and its 12 lbf/in springs at 6 in, so this is the only stock spring that fits; the 10 lbf/in 5108N79 (6 in, 1 in OD, 54.3 lb at 10.97 in) is the alternative if the pins move to 60–70 mm.

Gas struts: 4138T168 (20 lbf extension force, 26 lbf compressed, 18.5 in extended, 10.74 in compressed, 7.76 in (197 mm) stroke, 0.709 in body, M6 rod thread, ball-socket ends, 5/16"-18 ball studs, $31.93) and 4138T193 (the same at 23.03 in / 13.19 in / 9.84 in stroke, $27.47). Both pages say "Maximum Cycles per Minute: 6 cpm", "Maximum Cycles: 30,000", "Mounting Position: Rod Down", nitrogen gas with oil, rubber seals, −13 to 176 °F. Six cycles a minute is a hatch, not a leg: a leg that squats, hops and rides bumps cycles a strut several times a second and would use up 30,000 cycles in days. The adjustable-force family (6465K, a bleed valve to set the force, 90 lbf and up, cycles "Not Rated", $77–92) and the sealless "long-life gas-spring-style mechanical springs" (1268N, 25–30 lbf at 15.58 in with only a 4 in stroke, $152–165) do not fit either: too strong, too short a stroke, or both.

Torsion springs: McMaster's largest stock torsion spring, 9271K118 (1.102 in OD, for a 0.666 in shaft, 0.135 in wire, 4 in legs), gives 42.86 in·lbf (4.84 N·m) at its full 90° and wraps a 17 mm shaft. A hip spring here has to wrap the Ø72 output journal and give 10 N·m at about 50° of deflection: nothing in the catalogue, so a torsion-spring hip means a custom spring (or a clock spring) and a seat machined into the hip frame. The torsion variants in the simulation are therefore a bound on what such a spring could do, not a part.

Compression springs (for a coil-over): the metric table (DIN 2095/2098 music wire, 94125K) goes to 31 N/mm in short springs; a coil-over on the leg's axis needs a guided 1.5 N/mm spring with 180 mm of travel, which would be a bicycle-shock-class part, not a catalogue spring. Not pursued.

Tyres and axle mounts are not McMaster parts: the BotWheel's solid tyre is part of the hub motor. A softer tyre means a different wheel (or a cut-down, grooved tread), and a compliant axle mount means elastomer sandwich mounts between the wheel's 4 × M6 flange and the wheel link's foot, which is a CAD change rather than a part choice (section 7).

6. Trade-offs, plainly

Added mass and complexity. The between-links extension spring is one $15 part, ≈0.3 kg, two Ø8 pins with bosses on the upper links, a sleeve, and a two-line feed-forward in the controller: +0.35 kg per leg, 0.7 kg on a 26 kg robot (3 %), and one more part to look at when the leg is serviced. A gas strut is 0.45 kg plus two ball-stud mounts, a slot and bumpers, a part rated 6 cycles a minute, and a controller that must also decide when to stop cancelling it (SPR's "0 N and 450 N give the same height"). Torsion springs at the hips are a custom spring and a seat machined into the hip frame. A softer tyre is a different wheel; elastomer axle mounts are four small parts, a carrier, and ±8 mm of wheel wobble the controller never sees.

Stored energy and pinch points. The 5108N807 stores ≈3 J at the standing height (47 mm of stretch) and ≈11 J at the deepest crouch (118 mm); the fall energy of one leg's share of the robot from 150 mm is 16 J, so the spring never holds more than the leg already does in a hop. It sits in the upper links' own layer, inside the leg's silhouette, but a stretched coil spring is a finger trap between its coils: it needs a sleeve (a split TPU tube or a fabric sleeve) and loops that cannot be reached from outside. The five-bar's own scissor pinch between the upper links and the hip frame exists with or without the spring. A preloaded spring is also a leg that can move by itself: torsion_mid, torsion_preload and the 130 % strut all drive an unpowered leg to the fully-extended stop (section 4.5), which is a robot that suddenly stands up tall when it is switched off; the 5108N807 at 100 mm pins settles the leg at ≈300 mm and goes slack above 345 mm, so it cannot do that.

Fatigue. The 5108N807's body runs at 11–20 % of its wire's tensile strength in the working range (34 % at its rated maximum, which the leg reaches only in the deepest crouch): infinite life for the coils; the loops are where extension springs fail, so the pins are round, the spring stays in one plane, and it is inspected like a tyre. The McMaster gas struts are rated 25,000–30,000 cycles at 6 cycles a minute; a leg that squats, hops and rides bumps would spend that in weeks. TPU stop sleeves and the tyre age too, but they age today.

The nonlinear torque through the five-bar. The spring's hip torque is T(ℓ)·∂ℓ/∂φ with ℓ the pin-to-pin distance, a smooth function of the two motor angles that the controller computes from its own encoders (HKUST subtracts it "directly, without affecting other tuning"; Tongji fits a cubic). Uncancelled, the 5108N807 adds −305 N/m along the leg at 300 mm (10 % of the controller's 3,000 N/m, and in the softening direction) and 2.4 N·m of lean torque at 15° (6 % of the lean loop's 39 N·m): the balance loop would not notice, and cancelled it sees today's plant (the _c rows ride within 0.3 g of none). The two things to get right in firmware are the liftoff detector (do not count the spring's force as ground support: SPR's bug) and the slack point (above a 345 mm leg the spring's force is zero, a kink the feed-forward must include).

Jump strategy. Pre-compress and release is just the crouch: the motors stretch the spring slowly while crouching (≈2 N·m extra over 0.5 s, nothing) and it gives 3.5–4.5 J back during the 0.25 s push, when the motors' torque-speed line is the bottleneck. That is +14 % at the 60 N·m cap and +13 % at 120 N·m. A stronger spring returns more (6.2 J at 1.5× preload) but over-compensates the crouch, so the motors hold the leg down between jumps, and it extends the leg on power-off; the sweeps (section 4.7) show the jump gain flattening past 1× while those costs grow. The crouch depth is the bigger lever: 120 mm instead of 150 mm is +36 mm on its own, and the two add (strut_gas_deep 332 mm).

What is cheaper in firmware. The landing spike: a touchdown-speed cap and retracting the leg through touchdown (−18 % axle force for nothing). The ride: the RMS body acceleration is the leg's PD, not hardware, and a lower virtual stiffness with feed-forward does more than any spring; only the sharp peaks over small bumps need a softer tyre. The collapse: the RS-04's own anti-backdrive damping, if it is what the manual implies, turns a 194 N·m stop hit into 14 N·m, but it is a firmware feature of a motor that has lost power, so the spring is the only thing that holds the leg up unpowered. The standing heat and the jump gain cannot be had in firmware: the motors must carry the body unless something else does.

7. Recommendation (superseded in part by section 8: the between-links spring of item 1 fails in 3-D; the knee-to-knee

layout and the no-spring fallback are weighed there)

Build this.

1. One McMaster 5108N807 extension spring per leg between the two upper links, its loops on Ø8 steel pins 100 mm from each hip axis, in the upper links' own layer (y 0–30 mm), with a split sleeve over the coils, and the controller subtracting its torque from the two motor angles (ext_5108N807_c). For 11 kg on the leg it gives: standing torque 7.7 → 0.2–1.2 N·m per motor (−85 to −97 %, ≈25 W off the robot's standing draw), jump +14 % at the 60 N·m cap (278 → 317 mm) and +13 % at 120 N·m (296 → 334), and on power loss the leg stays up at ≈300 mm instead of folding onto its stops: the governing knee case disappears (section 4.5). It costs $15, ≈0.3 kg and two pins per leg, 3 J of stored energy at the standing height, no measurable change to balance or ride once cancelled, and nothing on the landing spike either way. Sized for the full robot (13 kg per leg) the same spring at 100 mm pins carries 75 % of the weight at 300 mm; pins at 110 mm restore 90 %.
2. Do the landings in firmware and in the tyre, not with a spring. Cap the touchdown speed (today's gearbox is fine to a 300 mm drop on the rig's measure; the knees are the limit and the collapse, not a landing, sets it); retract the leg through touchdown (−18 % on the spike for nothing); and measure the BotWheel tyre's radial rate on the rig, because a pneumatic-stiffness tread is worth −27 % on every landing load and −35–40 % on the sharp bumps, and a foam tyre −42 % and −50 %. If the BotWheel's tyre cannot be softened, four elastomer sandwich mounts between its flange and the wheel link (bushing_soft, 150 N/mm, ±8 mm) do the landing part (−39 %) but not the kerb part.
3. Do not build torsion springs at the hips (custom, and the wrong torque shape: they over-compensate the crouch and extend the leg on power-off), gas struts (6 cycles a minute, 30,000 cycles, need stops and a slot, and extend an unpowered leg), coil-overs (over-compensation, and a damper halves the jump), or anything in series with the motors: no robot in the survey with a quasi-direct drive leg does, and the rig shows why there is no need.

The CAD it implies (a concept, not drawn):

• A Ø8 pin boss on each upper link at r = 100 mm from the hip axis, on the link's inner face, the pin reaching into the y 2–28 mm band with a retaining clip; the link is 36–46 mm wide there, so the boss is a 16 mm eye grown from the link's outline (the same way the knee eye is), 20 mm proud. The loops bear on the pins; a 1 mm steel washer each side keeps the coil centred at y = 15.
• The spring's line crosses the inner stop posts (at x = ±29.5, z = −36, sleeves y 0–19) at leg lengths of 160–200 mm (the line sits at z = −17 at 140 mm and z = −78 at 300 mm, and the coil is 25 mm across): the knee-in stops move. Either the knee-in stop becomes a lug on the hip frame's lower flange that the link's far edge meets (outside the spring's band), or the four posts move outboard of the links' hubs, or the knee-in stop is dropped in favour of a firmware limit plus the steel post at the extension end. The outer (knee-out) posts at (±114, +43) are clear at every length.
• The front link's apex eye (at the axle, R45) and the hip frame's web are clear of the spring at every length; the spring's lowest point is at z = −78 (300 mm), 136 mm above the tyre's top.
• A split TPU sleeve (printed, 1.5 mm wall, 30 mm ID) over the coil, clipped to the pins, so no finger can get between the coils; the spring's free length (229 mm) and its stretch (47 mm standing, 118 mm crouched, 173 mm at full extension) all fit between the links at every leg length (the pins are 276 mm apart standing, 347 mm at 140 mm).
• The controller: τ_spring(φ1, φ2) = T(ℓ)·∂ℓ/∂φ with T = 27 N + 1,576 N/m·(ℓ − 0.2286 m) for ℓ > 0.2286 m, else 0; subtract it from the commanded torques; exclude its vertical component from the liftoff detector.

For the main session's scorecard (the rig's own measures, 11 kg): the gearbox peaks at 62 N·m in the 150 mm drop and 88 N·m in the 300 mm drop, never above 120 N·m for any variant or drop up to 300 mm; the knees see ≈500 N at 150 mm and ≈700 N at 300 mm, against the collapse's 1.8 kN (8.5 kg) that sizes them; the earlier J^T·F figure of 143 N·m was the quasi-static bound and overstated the gearbox by about three times and the knees by about two. The energy each leg absorbs is 16 J from 150 mm and 32 J from 300 mm or from its own 278 mm jump. The 2.5 kg of dummy mass that sim/leg_rig.py loses (section 3.1) should be restored before the collapse case is re-run for the FEA: at 11 kg the collapse is harder than at 8.5 kg (section 4.5).

Open questions.

• The RS-04's rotor inertia (0.0243 kg·m² reflected is an estimate) scales every landing number; RobStride publishes neither it nor the gearbox's strength. A bench test of one motor (spin-down, or a known torque step) would settle it.
• The BotWheel's tyre stiffness (7.7 ms contact time constant, ≈100 N/mm, is an estimate) is the single biggest lever on landings and ride: press one on the rig.
• The rig is planar: crooked landings (the 10 % side force) and the wheel's out-of-plane moment are in the strength studies, not here.
• The RS-04's phase resistance (0.16 Ω from a reseller sheet, possibly line-to-line) only scales the watts; the fraction-of-rated-loss figures do not depend on it.
• The real robot is 26 kg, 13 kg per leg; the rig's 11 kg is the earlier estimate. Everything here scales with it; the spring's pins move from 100 to 110 mm if the leg should carry 90 % at 300 mm.
• The spring's tolerances are "Not Rated" on McMaster's page; measure its rate and initial tension on the rig before trusting the feed-forward, and re-fit the two numbers in firmware.
• The power-off behaviour with the spring (a leg that stays up while the body tips over) is better for the leg and worse for whatever the body lands on; the fall study should include it.

8. Follow-up: the spring on the outboard side, knee to knee (2026-10-27)

The main session tested the between-links spring of section 7 against the real CAD and it fails in 3-D: centreline pins at r = 110 mm are the only radius that holds the 13.1 kg leg (100 folds, 115 and 120 push onto the stops), and below about 250 mm of leg length the coil and boot pass through the hip region (frame, adapters, hub screws, the knee-in stop posts, the 61812s), because in a crouch the knees swing up and out and the pins end up on opposite sides of the hips; edge lugs at r = 125 hold (rest 288–312 mm, 73 N on the knee at power-off) but clash worse below 200 mm. So this section evaluates the layout that stays outside everything: a spring between the two knee axes, outboard of the knee bolts' heads (y ≈ 99 mm), with the rig at its corrected 13.1 kg (HALF_MASS in sim/leg_rig.py; the ballast is now a welded body). Code: sim/leg_spring_study.py --knee --tag knee --sweep knee (results in exports/leg/springs/results-knee.json, sweep-knee-knee.json, *-knee.png) and --clash (the 3-D check, knee-spring-clash.json, knee-spring-layout.png).

8.1 The geometry, and what it asks of a spring

From cad/fivebar.py (matching the main session's figures): the knees are 450 mm apart at a 105 mm leg (knee line z +4), 442 at 150 (z −34), 387 at 250 (z −92), 339 at 300 (z −116), 300 at 330, 246 at 360, 207 at 375 and 139 at 390 (z −150); the tyre's top is at z −19 at 105 mm, −44 at 130, −64 at 150. The span changes 3.2:1 over the leg's range, which no coil spring covers (McMaster's long extension springs stretch to 1.75–2.3 × their free length), so a knee-to-knee spring is necessarily slack over the top of the range and, worse, shorter than its own free length there: a 229 mm coil (9 in) has to bow once the knees are closer than 237 mm, a leg of 362 mm and up; a 178 mm coil (7 in) from 381 mm. It also passes a dead point: the knee line crosses the hip line at a leg length of about 108 mm, where the spring's torque about the hips goes through zero and reverses; below it the spring pulls the leg shut onto the knee-out stops, with about 1 N·m (the moment arm is tiny there), which the motors open against without noticing. Between the two, the spring's job is to hold 13.1 kg at a resting height of 260–330 mm.

The tension the line has to carry to hold the 107 N the leg supports (13.1 kg less the wheel and half the wheel link) [derived, tendon_dl in the study]: 93 N at 300 mm (span 339), 105 at 285, 117 at 270, 126 at 260, 145 at 240, 181 at 210, 233 at 180, 336 at 150, 536 at 130, 889 at 120, then through the dead point at 108 mm. Standing is cheap; the crouch is where the moment arm collapses, so a linear spring never over-compensates deep in the crouch and the motors always win there. Fitting a linear spring T = T_i + k·(span − ℓ₀) to 93 N at the standing span and asking where the unpowered leg rests, how far it can be pushed down before it folds, and how both move with ±10 % on rate and initial tension (results-knee.json → static, and the statics run in this section's log):

So the spring has to be at least 1.4 N/mm to hold a 10 % error, which means 250–300 N at the crouch span and a free length (in span terms) of 290–300 mm. The 0.6–1 N/mm first estimate is not robust: at −10 % rate the 1.0 N/mm spring has no equilibrium at all and the leg folds.

8.2 Real parts

McMaster's long extension springs (pages read one at a time, 2026-10-27; the 1 in, 1.25 in and 1.5 in OD families and the 4–9 lbf/in rate pages): nothing stocked is both 11–12 in long and 8–9 lbf/in with an 18 in maximum, so a stock spring needs a series link to put its free length where the leg wants it, which also keeps it inside its maximum extension:

• 5108N807 (9 in, 1 in OD, 0.125 in wire, 9 lbf/in = 1.58 N/mm, 6.11 lb initial tension, 67.87 lb at 15.84 in, $14.80) with a 76 mm link: the coil is stretched 50 mm standing (93 N), 153 mm at the 105 mm crouch (269 N, 20 mm short of its 402 mm maximum). One spring, the minimum of parts.
• Two 5108N78 (7 in, 1 in OD, 0.105 in wire, 5 lbf/in, 3.8 lb initial, 42.25 lb at 14.73 in, $11.86) side by side, each with a 135 mm link: 1.76 N/mm together, each coil stretched 17 mm standing and 145 mm at the crouch (144 of its 188 N maximum). Shorter coils, so they bow only above a 381 mm leg, but two coils need 51 mm of width over the tyre.
• Not usable: 5108N794 (5.8 lbf/in: not robust), 9654K687 (3.6 lbf/in), the 1.25 in and 1.5 in OD springs (0.4–0.7 N/mm, or 479 mm long).

Fatigue [estimate]: the 5108N807 at its 269 N crouch load is at 30 % of its wire's tensile strength (Wahl-corrected); at the standing 93 N, 10 %. The crouch is a rare, slow load; the standing load is forever. Fine for the coil; the loops are the usual weak point and must not be loaded sideways.

8.3 The anchors on the knee axes, and the series link

The knee bolt (ISO 7379 Ø12 × 90 shoulder screw, head Ø18 × 9 at y 82–91) is clamped to the upper link by the nut in the eye; the lower link turns on it through two 6201s (y 32–42 and 70–80). The spring's loop has to sit on a post on the knee axis beyond the head (y ≈ 93–105), fixed to the bolt: a C-bracket of 3 mm steel clamped under the head (a Ø12.5 hole on the shoulder between the head's washer and the outer bearing's spacer, bent twice to come back over the head's centre) carrying a Ø8 post 12 mm long on the axis, with a 1 mm washer and a clip; the loop rides on the post and turns on it as the knee angle changes. The bracket rotates with the bolt (with the upper link), which is fine: the loop turns on the post. The series link is a 76 mm (or 135 mm) steel strap with a Ø8 eye at each end, or better a short wire-rope lanyard (3 mm, thimbles, swaged) so the assembly can go limp above a 332 mm leg instead of pushing the loop off its post. Above a 362 mm leg (381 for the two-spring version) the coil itself is longer than the span and would bow: a two-piece guide inside the coil (a Ø12 rod from one post sliding into a Ø16 tube from the other, 120 mm each) keeps a slack coil straight between 139 and 250 mm of span, and has separated by the time the coil is taut. With the guide the spring, boot and guide are a self-contained strut-like assembly hung on two posts. Without it the leg's extension has to be limited to 360 mm (one spring) or 380 mm (two).

8.4 The out-of-plane moment on the knee

The pull T acts at y ≈ 99, 89 mm outboard of the eye's mid-plane (the two sleeves span y −10 to +30) and 24 mm outboard of the outer 6201's centre. Numbers for T = 93 N standing and 270–290 N in the crouch (knee_k1.6, the stock parts):

• the knee bolt bends as a cantilever past the outer bearing: M = T × 0.024 = 2.2–7 N·m, 13–41 MPa on the Ø12 shoulder (fine);
• the couple on the lower link's two 6201s, 38 mm apart: ±0.63 T = ±60 N standing, ±180 N in the crouch, on bearings rated 3.1 kN static;
• the moment that reaches the upper link's eye is bounded by T × 0.089 = 8.3 N·m standing and 26 N·m in the crouch, of which the outer bearing, 24 mm from the load, takes most; as a bound, 26 N·m over the 40 mm of sleeves is 6·M/(d·L²) ≈ 8 MPa of extra bearing pressure, against the 31 MPa the two-sleeve eye sees in the collapse. Small, but it is a steady 8 N·m, every day, in a direction the eye was not designed for; worth one line in the eye's study.
• in the leg's plane the spring is simply a 93–290 N joint load at each knee, less than a landing's 500–1,000 N.

8.5 The 3-D clash check (--clash, knee-spring-clash.json, knee-spring-layout.png)

A 25.4 mm coil in a 32 mm boot, centred at y 99 on the knee axes with 12 mm of hook past each axis, intersected with every part of fivebar.assembly() (motors as envelopes) at 14 leg lengths from 105 to 390 mm: no intersection with any part at any length, coil or boot. The hip region is 95 mm inboard of the spring's layer, so the problem that killed the between-links spring does not arise. The one proximity is the tyre, which shares the spring's layer (the boot spans y 83–115, the tyre y 82–127): the gap from the boot's underside to the tyre's crown is 7.4 mm at 105 mm, 7.3 at 110, 8.1 at 120, 9.7 at 130, 14.8 at 150, 25 at 180 and 38 mm at 210. A boot over a tread turning at up to 3 m/s with 7 mm to spare, in the pose a kid gets by sitting on the robot, is not comfortable; with the crouch limited to 130 mm (the working range's floor) it is 10 mm, and the boot can be shaped flat on its underside to make it 13. The check is of a straight envelope; it does not model the bow of a slack coil above 362 mm (section 8.3).

8.6 What the rig says (13.1 kg)

results-knee.json, scorecard-knee.png. The rig's controller does not know about the spring unless told (_c = the spring's torque subtracted as feed-forward, which any real firmware would do: HKUST, Tongji); without it the leg-length PD sits 24 mm above its command against the spring's push and the crouch stops 30–60 mm short, so the _c rows are the honest ones and the plain rows show what an uncancelled spring does to the controller.

What the numbers say:

• Standing. Cancelled, the spring carries the whole standing load (0.3 N·m left of 9.8; 10 W per motor becomes nothing). Between 200 and 260 mm the 1.4–2 N/mm springs over-compensate by 25–35 % (the price of the ±10 % robustness), so a robot parked low holds its legs down with 3–4 N·m per motor. At 115 mm the spring's torque is 1.6 N·m, as the dead point predicts.
• Jumps. Cancelled, +66 mm at the 60 N·m cap (279 → 345, +24 %) and +60 mm at 120 N·m, with 7.5 J returned at liftoff from 20 J stored in the crouch; uncancelled, +3 to +8 mm, because the controller never reaches the crouch (SPR's "0 N and 450 N give the same height"). The knee-to-knee line stores more than the between-links spring did (20 J against 11) because its stretch is largest in the crouch.
• Landings. Unchanged, as before: 1,835–2,004 N, 62–65 N·m, 528–607 N at 150 mm.
• The collapse. Today, undamped: 3.0 kN on the knee and 223 N·m on the stop at 13.1 kg (the rig's 1.8 kN was 8.5 kg); damped: 305 N. With the spring the leg does not fold: it stays at 296–319 mm. Damped, that is 65–70 N on the knee and no stop touched. Undamped, the 1.4–2 N/mm springs first lift the leg to 325 mm (the motors had been holding it down), and on the rig's free wheel the lean then grows until an upper link meets its stop 0.9 s later at a 30° lean: 1.1 kN on the knee, 170–180 N·m on the stop, a third of today's. The 1.0 N/mm spring does not do that (70 N) but folds at −10 % rate. On the real robot a power cut tips the body over anyway; what the spring buys is that the leg is not slammed shut while it does.
• Ride and balance. Cancelled, the ride is today's (9.2 g against 9.4 on the 20 mm bump). The spring's own stiffness along the leg is large and negative, −1.0 to −1.8 kN/m at 300 mm against the controller's 3 kN/m, and its lean torque at 15° is 2.4 N·m; both are smooth functions of the two motor angles and go into the feed-forward. Uncancelled, the peaks rise 20–35 % and the leg sits 24 mm high.
• Robustness. The six ±10 % builds of the 1.6 N/mm spring rest at 302–313 mm damped and behave alike in every scenario.
• The sweep (sweep-knee-knee.json, 30 points, uncancelled controller; cells as in section 4.7 with the undamped collapse's knee force in the fourth place): with the rate at 1.4–2.0 N/mm every preload from 0.8 to 1.1× shows the undamped lean swing (1.0–1.2 kN on the knee), and only 1.2× (the leg resting at 316–325 mm) settles without it (67–71 N); the softer 0.8–1.0 N/mm springs settle quietly at 1.0–1.2× (61–70 N) but are the ones that fold at −10 % rate. Damped, the whole grid rests at 274–347 mm. Uncancelled jumps run 276–299 mm (today 279) and the 40 mm bump 2.4–4.8 g (today 2.8).

8.7 Verdict on the knee-to-knee spring

It fits, and it works, with conditions: it is the only spring layout found that clears the CAD at every leg length, it holds the 13.1 kg leg at 290–313 mm within ±10 % of rate and tension, it removes the standing torque and the everyday collapse and adds a quarter to the jump. But it is no longer the one $15 part of section 7: it is a spring rated at least 1.4 N/mm (the stock 5108N807 with a 76 mm wire-rope lanyard, or two 5108N78 with 135 mm lanyards), two C-bracket posts clamped under the knee bolts' heads, a rod-in-tube guide inside the coil so a slack coil cannot bow over the wheel above a 332 mm leg (or a 360 mm extension limit), a boot, 0.45–0.6 kg per leg, a boot that passes 7 mm over a turning tyre at the 105 mm crouch (10 mm at 130), a steady 8 N·m out-of-plane load on each knee bolt, 3–4 N·m of hold-down per motor whenever the robot sits at 200–260 mm, a 24 mm standing offset and a 25–35 % over-compensation that the firmware has to model, and its jump gain only exists with that firmware.

Against that, the fallback costs nothing: no spring, and a bench test of the RS-04's own power-off damping, which on the rig turns the undamped 3.0 kN collapse into 305 N at the knee (section 4.5 and none above) if the motor's anti-backdrive behaviour is what the manual implies. The standing heat without a spring is 10 W per motor at 13.1 kg (5.9 % of the rated loss, 41 W for the robot), a runtime cost, not a thermal one.

Recommendation now: build the rig without a spring; measure the RS-04's power-off damping (the collapse is the knees' governing case, and that one measurement decides whether it is 3.0 kN or 0.3 kN) and the standing current; keep the knee-to-knee assembly above as the one spring that fits, to add on the rig only if the jump height or the runtime turns out to matter more than its parts and its 7 mm. If it is built, the firmware must subtract its torque from the two motor angles, exclude it from the liftoff detector, and the crouch floor should be 130 mm.