Wheel + Config Analysis

SECTION 1 / 3

2.75″ vs 4″ Wheel Math

Five engineering factors, head-to-head. Both configurations target the same linear speed (~5.3 ft/s) using different gear ratios.

Configuration Comparison (3 Wheel Sizes, 55W Push-Heavy)

All three wheel sizes can hit the Override 5.0-5.5 ft/s sweet spot with appropriate gearing. The differences are in clearance, gear ratio simplicity, and force at the rim.

Spec	2.75″ Wheels	3.25″ Wheels (Spartan V1.5)	4″ Wheels
Wheel radius	1.375″	1.625″	2.0″
Cartridge	Blue 600 RPM	Blue 600 RPM	Blue 600 RPM
Reduction for ~5.2 ft/s	4:3 (36→48)	5:3 (36→60)	2:1 (36→72)
Wheel RPM at that ratio	450	360	300
Linear speed at that ratio	5.40 ft/s	4.97 ft/s	5.24 ft/s
Pushing force per wheel	13.6 lb	14.4 lb	14.0 lb
4-wheel total force at stall	~54 lb	~57 lb	~56 lb
Ground clearance	1.4″ (marginal)	1.6″ (good)	2.0″ (excellent)
Pinion size on motor	8T (sensitive)	12T (forgiving)	12T (forgiving)
Wheel weight (each)	~50 g	~65 g	~85 g
Drivetrain weight (4 wheels)	~200 g	~260 g	~340 g

⭐

3.25″ is a strong middle option. It splits the difference between 2.75″ (light + low) and 4″ (high clearance + simple gearing): 1.6″ clearance is enough for Override debris, the 12T pinion size is forgiving like 4″, and the gear ratio (5:3) is clean. Slightly slower than the other two options at clean ratios (4.97 vs 5.24 ft/s), but the ~5% gap is below driver-perception threshold.

Factor 1: Linear Speed

All three wheel sizes hit the 5.0-5.5 ft/s sweet spot with appropriate gearing. The differences are within driver-perception noise floor:

Speed formula: ft/s = (RPM \times π \times D) / (12 \times 60) 2.75″ at 450 RPM: (450 \times π \times 2.75) / 720 = 5.40 ft/s 3.25″ at 360 RPM: (360 \times π \times 3.25) / 720 = 4.97 ft/s \leftarrow Hero Bot V1.5 4″ at 300 RPM: (300 \times π \times 4) / 720 = 5.24 ft/s

If the team wants the exact 5.24 ft/s with 3.25″ wheels, the gear reduction needs to be 600/(5.24×720/π/3.25) = 1:1.582, achievable with a custom gear pair (e.g., 36T → 57T). The closest stock VEX clean ratio is 36T → 60T = 1:1.667 → 360 RPM → 4.97 ft/s. Trade ~5% speed for clean stock ratios — usually worth it.

Factor 2: Pushing Force at the Rim

All three configs produce nearly identical pushing force when geared for the same speed range. Force = motor torque × gear reduction / wheel radius:

Pushing force at 5+ ft/s gearing: 2.75″ (4:3 reduction): 14 in-lb \times 1.33 = 18.7 in-lb at wheel; force = 18.7 / 1.375 = 13.6 lb/wheel 3.25″ (5:3 reduction): 14 in-lb \times 1.67 = 23.3 in-lb at wheel; force = 23.3 / 1.625 = 14.4 lb/wheel 4″ (2:1 reduction): 14 in-lb \times 2.00 = 28.0 in-lb at wheel; force = 28.0 / 2.000 = 14.0 lb/wheel 3.25″ slightly wins on force per wheel (~3-6% over 2.75″/4″) — the higher reduction overcomes the smaller-than-4″ radius. But the difference is well within material variation between motor batches.

Factor 3: Ground Clearance

1.4″

2.75″ Clearance

Marginal. Will catch on cup fragments and pin debris at midfield.

1.6″

3.25″ Clearance

Sweet spot. Enough to roll over typical Override debris without catching.

2.0″

4″ Clearance

Most generous. Comfortable margin for any field condition.

Factor 4: Gear Alignment Difficulty

Wheel size	Common pinion size	Alignment difficulty
2.75″ (4:3 reduction)	8T → 24T (3:1 step-up for half-motor) or 8T → 11T pinions	Hard — 8T pinions skip teeth under load
3.25″ (5:3 reduction)	12T → 20T or 36T → 60T	Easy — 12T pinions are forgiving
4″ (2:1 reduction)	12T → 24T or 36T → 72T	Easy — same forgiving 12T pinions

Factor 5: Weight

3.25″ saves about 80 g of drivetrain weight vs 4″, and adds 60 g vs 2.75″. Negligible for stability calculations but matters for thermal/battery life on long match days. Total ~1-2% of robot mass.

Head-to-Head Verdict

Factor	2.75″	3.25″	4″	Winner
Linear speed at clean ratios	5.40 ft/s	4.97 ft/s	5.24 ft/s	Effectively tied
Pushing force per wheel	13.6 lb	14.4 lb	14.0 lb	Tied (3% spread)
Ground clearance	1.4″	1.6″	2.0″	4″ best, 3.25″ adequate
Gear alignment	8T sensitive	12T forgiving	12T forgiving	3.25″ & 4″ tied
Drivetrain weight	~200 g	~260 g	~340 g	Lighter is marginal
Wheelbase fit (under 18″)	Very flexible	Flexible	Tighter	Smaller wheels free chassis space

✓

For Override 55W push-heavy: 3.25″ and 4″ both work well. 2.75″ is the weakest of the three due to ground clearance and 8T pinion alignment issues.

3.25″ (Spartan V1.5 choice): good clearance, clean 5:3 gearing with 12T pinions, slightly highest pushing force per wheel. Saves ~80 g over 4″.
4″ (alternative): max clearance, clean 2:1 gearing, marginally faster at clean ratios. Heaviest option.
2.75″ (avoid for this build): 8T pinions are alignment-sensitive and known to skip under load.

📐

Cross-reference: See /drivetrain-builds-55w for the complete parts lists for both 2.75″ and 4″ Blue cartridge configs, and /drivetrain-101 §2 for recommended speed targets and weight ranges.

SECTION 2 / 3

Hero Bot V1.5 — Full Mechanical Math

Verifying the V1.5 architecture: drivetrain speed, arm torque + reach, toggle output, claw weight handling, full power budget.

Architecture Recap (Spartan Hero Bot V1.5 — As Built)

Subsystem	Spec	Motors	Power
Drivetrain	4× Blue 11W cartridges, 3.25″ omni wheels	4 (ports 1-4)	44W
Arm (4-bar)	2× Red 100 RPM cartridges, 12T:60T mirrored, rubber band assist	2 (ports 5-6)	22W
Manipulator	1× Standard V5 claw (276-2235), Architecture A	1 (port 7)	11W
Toggle	1× Red 100 RPM cartridge, 1:1 chain to flex wheel	1 (port 8)	11W
Total	—	8	88W (at R10a cap)

📋

The build is committed: 3.25″ omni wheels with 4×Blue 11W motors are already on the chassis. The math below works backward from this fixed configuration to figure out actual speed, force, and reach. Gear ratio choice is still flexible — that's what the team can tune.

Drivetrain Math (3.25″ Wheels)

With 3.25″ wheels and Blue 600 RPM cartridges, the linear speed depends entirely on the gear reduction. Here are the candidate configurations:

Reduction	Gear Pair	Wheel RPM	Linear Speed	Verdict
1:1 (none)	direct	600	8.51 ft/s	Too fast — driver will lose control
4:3	36T → 48T	450	6.38 ft/s	Fast — skills strategy only
5:3	36T → 60T	360	4.97 ft/s	★ RECOMMENDED — sweet spot
2:1	36T → 72T	300	4.25 ft/s	Slow but high-force
3:1	12T → 36T	200	2.84 ft/s	Too slow — push-only strategy

Speed formula recap: ft/s = (RPM \times π \times D) / 720 For 3.25″ wheels at the recommended 5:3 reduction (36T \to 60T) : Wheel RPM = 600 / 1.667 = 360 RPM Linear speed = (360 \times π \times 3.25) / 720 = 4.97 ft/s This sits at the lower end of the 5.0-5.5 ft/s sweet spot from /drivetrain-101 . Slightly slower than the 4″/2:1 alternative (5.24 ft/s), but the gap is below driver-perception threshold.

Pushing Force at 5:3 Reduction

At 5:3 reduction (360 RPM at wheel): Motor stall torque: 14 in-lb (Blue cartridge output) Torque at wheel = 14 \times 1.667 = 23.3 in-lb at wheel shaft Force at rim = 23.3 / 1.625″ = 14.4 lb push per wheel 4-wheel total at stall: ~57 lb push (theoretical, ignoring traction limits) This is competitive with both 2.75″ (54 lb) and 4″ (56 lb) configs. The 3.25″ + 5:3 actually has a slight edge on force per wheel.

✓

Recommendation: 5:3 reduction (36T pinion → 60T driven gear). Easiest clean ratio that hits the speed sweet spot with 3.25″ wheels and Blue cartridges. Uses standard VEX gears, 12T-equivalent pinion (36T motor pinion mating with 60T driven). Pushing force is competitive with all alternatives. If your team is using a different ratio, run the speed calculation against this table to see if you're on target.

Alternative: 4:3 Reduction (Faster, Less Force)

If your strategy emphasizes cycle speed over pushing power:

At 4:3 reduction (450 RPM at wheel): Linear speed: 6.38 ft/s (above sweet spot — skills/aggressive strategy only) Pushing force: 11.5 lb per wheel (about 20% less than 5:3 ratio) Verdict: Use only if drivers are advanced and strategy is fast-cycle / no-contest

Arm (4-Bar) Math

The V1.5 arm uses 2× Red 100 RPM cartridges with 12T:60T external reduction:

Arm output speed: Red cartridge: 100 RPM at motor output Through 12T:60T (5:1 reduction): 100 / 5 = 20 RPM at arm pivot Angular speed: 20 RPM = 0.33 RPS = 120°/sec Arm output torque (per motor): Red 11W motor stall torque: ~21 in-lb at cartridge output Through 5:1 reduction: 21 \times 5 = 105 in-lb per motor Total arm stall torque (2 motors mirrored): 2 \times 105 = 210 in-lb

Arm Load Check (Standard V5 Claw)

The standard V5 Claw (276-2235) is the manipulator option in Architecture A. Component masses for load analysis:

Component	Mass	Position	Static Torque
4-bar arm structure	~400 g (0.88 lb)	4″ midpoint	3.52 in-lb
End-effector mount + shaft	~150 g (0.33 lb)	8″ end	2.64 in-lb
V5 standard claw + motor	~280 g (0.62 lb)	8″ end	4.96 in-lb
Cup (held in claw)	~30 g (0.07 lb)	9″ end (cup hangs below claw)	0.59 in-lb
Total static load	~860 g	—	~12 in-lb

Arm torque utilization: 12 in-lb (load) / 210 in-lb (stall) = ~6% of stall This is excellent thermal headroom. Each motor at <3% of stall current. Heating \propto current² \approx (3%)² = 0.09% of stall heating per motor Two motors share the load = effectively zero thermal stress Conclusion: arm motors will not overheat under any realistic V5RC use.

Rubber Band Assist

The V1.5 spec mentions "rubber band assist on lift." This shifts the static torque calculation: rubber bands provide ~2-4 in-lb of pre-loaded torque toward the up direction, reducing the arm motor load on lift (and adding load on the down stroke).

📐

Why bother with rubber bands if the arm is at 6% of stall? The bands smooth out the lift motion and let you tune the arm to "hover" without continuous motor current. Without bands, the motors must actively hold the arm against gravity = continuous low-level current draw = battery drain over a 2:00 match. With bands, the arm holds at any angle with motor power off (or near-off). Saves ~5-10% of total match battery consumption.

Arm Reach Math

With 3.25″ omni wheels, the chassis bottom plate sits at 1.625″ above ground (wheel hub level). With 1″ c-channel chassis structure, the top of chassis is at 2.625″ above ground. The 4-bar arm pivot typically mounts on a tower above the chassis top.

If 4-bar pivot is at chassis-top level (H_pivot = 2.625″) with 8″ arms: Arm at +30°: claw at H = 2.625 + 8\timessin(30°) = 6.625″ Arm at +45°: claw at H = 2.625 + 8\timessin(45°) = 8.28″ Arm at +60°: claw at H = 2.625 + 8\timessin(60°) = 9.55″ Arm at +90° (vertical): claw at H = 2.625 + 8 = 10.625″ If 4-bar pivot is on a 4″ tower (H_pivot = 6.625″) with 8″ arms: Arm at +30°: claw at H = 6.625 + 4 = 10.625″ Arm at +45°: claw at H = 6.625 + 5.66 = 12.28″ Arm at +60°: claw at H = 6.625 + 6.93 = 13.55″ Arm at +90° (vertical): claw at H = 6.625 + 8 = 14.625″

Goal	Top of Goal	Required Claw Height	8″ arm at chassis	8″ arm on 4″ tower
Alliance goal	3.25″	~6.5″	✓ Easy (+30°)	✓ Easy (-15°)
Short neutral	5.77″	~9″	✓ Reaches (+55°)	✓ Easy (+18°)
Tall center	8.77″	~12″	✗ MARGINAL (10.6″ max)	✓ Reaches (+43°)

⚠️

Critical question for the build team: where exactly is the 4-bar pivot mounted? If it's on a tower 4″+ above the chassis (typical V5 build), the arm easily reaches all three goal heights. If the pivot is at chassis-top level (low tower), the tall goal is marginal. Measure the actual pivot height on your robot before deciding whether the arm needs lengthening.

Pivot height ≥ 5″ above ground: 8″ 4-bar arms reach all goals → build is fine as-is
Pivot height < 4″ above ground: 8″ arms can't reach tall goal → consider 10″ arms OR skip tall goal strategically

Toggle Mechanism Math

V1.5 toggle: 1× Red 100 RPM cartridge, 1:1 chain to a flex wheel mounted on a tower.

Toggle wheel speed: 100 RPM (1:1 ratio, no reduction) Toggle wheel torque: ~21 in-lb at the flex wheel shaft Override toggle: a sliding 2.05″ wide barrel that needs to be pushed left or right Toggle moves linearly when flex wheel contacts it (rolling friction) Required force to slide toggle: ~2-4 lb (estimated, manual doesn't specify) Flex wheel diameter (typical): 4″ (276-7390) Force at flex wheel rim: 21 in-lb / 2.0″ = 10.5 lb Margin: 10.5 lb available vs ~3 lb needed = ~3\times margin. ✓ Comfortable.

Manipulator (Standard V5 Claw) Math

Architecture A uses the standard V5 claw (276-2235). Specs:

Mass: ~280 g
Grip range: opens to 4.5″, closes to 1.5″ — range of 3.0″ travel
Stock motor: 11W motor, typically Green 200 RPM cartridge
Drive: rack-and-pinion via 12T pinion on the claw motor
Grip force: ~5-8 lb at the claw tips (manufacturer spec, varies)

Cup gripping check: Override cup mass: ~30 g (0.07 lb) Cup diameter at narrowest: 3.15″ (well within claw 4.5″ open range) Required grip force: enough to overcome cup's weight while accelerating Worst case: arm rotating at 120°/sec, cup accelerating at ~10 ft/s² (1g) Centripetal force on cup at 9″ radius: F = mω²r = 0.014 kg \times 12.6 rad/s \times 0.229 m = 0.04 N (0.01 lb) Plus gravity component: 0.07 lb Total force needed: <0.1 lb. Claw provides 5-8 lb. ~50\times margin. ✓ Easy. Pin gripping check: Override pin: hex prism, 3.16″ widest, 1.40″ narrowest, ~6.5″ tall Standard claw closes to 1.5″ — can grip the narrowest part of pin (1.40″) only barely Pin gripping is geometrically marginal. The standard claw was designed for cylindrical objects; pin's hex geometry + tapered shape makes it unreliable. Recommendation: add custom rubber/silicone strips inside the claw tips, OR use Architecture B (dual claw) for separate cup + pin handling.

⚠️

Standard V5 claw has a geometry problem with pins. The pin's hex shape and 1.40″ neck are at the limit of the claw's closed position. In testing, expect:

~70% reliability on pin pickup (cup pickup will be near 100%)
Higher failure rate on tapered/contoured pins
Need to add grip-enhancing rubber to the claw tips

This is a known limitation of Architecture A. If reliable pin handling matters strategically, consider Architecture B (dual claw) with a separate cup-claw and pin-claw, or Architecture E (pneumatic side-grab) with a custom geometry sized for the pin specifically. See /spartan-hero-bot for the architecture options.

Power Budget Verification

Does V1.5 fit in 88W (R10a) and 55W drivetrain (R11a)?

Subsystem	Motors	Cartridge	Power Budget	R11a Status
Drivetrain	4 × 11W	Blue	44W	11W under R11a cap
Arm (4-bar)	2 × 11W	Red	22W	—
Manipulator	1 × 11W	Green	11W	—
Toggle	1 × 11W	Red	11W	—
TOTAL	8 motors	—	88W	At R10a cap exactly

✓

The V1.5 architecture is legal and at the cap. 88W total = R10a max. 44W drivetrain = 11W under R11a (room to add 1×5.5W half-motor or 1×11W to push to 55W if push-heavy strategy desired). Zero headroom on R10a — any additional motor would require removing one elsewhere.

Path to 55W Drivetrain (Push-Heavy Upgrade)

The V1.5 baseline is 44W drivetrain (under-spec for push-heavy). To upgrade to 55W:

Option 1: Add 1\times11W motor to drivetrain Drivetrain: 5 \times 11W = 55W ✓ Total: 88W + 11W = 99W ✗ EXCEEDS R10a cap Must remove a motor elsewhere — but every other subsystem is already at minimum. Option 2: Add 1\times5.5W half-motor to drivetrain Drivetrain: (4 \times 11W) + (1 \times 5.5W) = 49.5W (still under 55W cap) Total: 88W + 5.5W = 93.5W ✗ EXCEEDS R10a cap Option 3: Add 2\times5.5W half-motors to drivetrain, remove 1 toggle motor Drivetrain: (4 \times 11W) + (2 \times 5.5W) = 55W ✓ Toggle: removed (or merged with manipulator on a chained shaft — though R11b prohibits PTO from drivetrain) Total: 55W + 22W + 11W = 88W ✓ This works, but you lose the dedicated toggle motor. Architecture trade-off.

⚠️

The V1.5 architecture is power-budget-locked at 44W drivetrain. Going to 55W push-heavy requires removing a motor elsewhere (probably the dedicated toggle motor, merging it back with the arm). This is the engineering trade-off teams must accept: pure 55W push-heavy with V1.5's mechanism count is not possible without giving up subsystem independence.

SECTION 3 / 3

Verdict + Recommendations

Bottom-line answers to both engineering questions.

Question 1 Answer: 3.25″ (Already Built) Is a Good Choice

✓

The 3.25″ wheels already on the V1.5 chassis are a solid choice. They sit between 2.75″ (too low ground clearance, alignment-sensitive 8T pinions) and 4″ (heaviest, biggest chassis footprint). For Override 55W push-heavy:

Linear speed at 5:3 reduction: 4.97 ft/s — at the low end of the sweet spot. ✓
Pushing force per wheel: 14.4 lb at 5:3 — slightly higher than 4″ alternative. ✓
Ground clearance: 1.6″ — adequate for cup/pin debris. ✓
Gear alignment: 12T pinions (forgiving). ✓

No reason to switch wheel size. Focus tuning effort on gear ratio + arm reach instead.

Question 2 Answer: V1.5 With 3.25″ Wheels — Math Check

Verdict on the V1.5 architecture as actually built:

V1.5 Aspect	Math Check	Verdict
Drivetrain at 5:3 reduction	360 RPM at wheel = 4.97 ft/s	✓ Sweet spot, recommended
Pushing force at 5:3 reduction	14.4 lb per wheel = ~57 lb total	✓ Competitive with 4″/2.75″
Arm 5:1 reduction, 210 in-lb stall	Correct calculation	✓ Verified
Arm at 6% of stall under load	Verified (12 in-lb / 210)	✓ Comfortable thermal margin
Toggle 1:1, 21 in-lb at flex wheel	Correct calculation	✓ ~3× margin over need
Standard V5 claw handles cup	50× force margin	✓ Easy
Standard V5 claw handles pin	Geometrically marginal	~ Add grip enhancements
Arm reach to 8.77″ tall goal	Depends on tower height	~ Verify pivot height
Power budget = 88W (R10a)	Verified (4×11 + 2×11 + 11 + 11 = 88)	✓ At cap exactly

Two Issues to Verify with the Build Team

Issue 1: Confirm Drivetrain Gear Reduction

If the team isn't already at 5:3 (36T → 60T), make this change. Reasoning:

5:3 hits 4.97 ft/s — at the bottom of the sweet spot, push-heavy without being slow
Uses standard 36T motor pinions and 60T driven gears — clean part availability
14.4 lb per wheel pushing force — competitive with 4″/2.75″ alternatives
If currently at 4:3 (450 RPM, 6.4 ft/s): too fast, 20% less force, swap to 5:3
If currently at 2:1 (300 RPM, 4.25 ft/s): slightly slow, swap to 5:3 for better speed match
If currently at 3:1 or higher: way too slow, definitely change

Issue 2: Verify Arm Pivot Tower Height

The 8″ 4-bar arms reach the tall goal (~12″ claw height) ONLY if the pivot is mounted 5″+ above the ground. Quick check:

Measure from ground to the 4-bar pivot shaft on the actual robot.
If ≥ 5″: arm reach is fine, no change needed.
If 3-5″: arm reaches tall goal at full vertical extension — workable but tight.
If < 3″: arm cannot reliably reach tall goal — three options:
- Add a 4″ tower extension below the pivot. Most efficient solution.
- Lengthen 4-bar arms to 10″. Adds CoG impact.
- Skip the tall goal strategically. Score on alliance + short neutral only.

Recommended V1.5 Configuration (with 3.25″ wheels confirmed)

⚙️

Recommended setup after this analysis:

Drivetrain: 4× Blue 11W + 3.25″ omni + 5:3 reduction (36T → 60T) → 360 RPM at wheel → 4.97 ft/s
Arm: 2× Red 100 RPM + 12T:60T mirrored + rubber bands + 8″ 4-bar arms (verify pivot tower height)
Manipulator: Standard V5 claw with rubber grip enhancements for pin handling
Toggle: 1× Red 100 RPM + 1:1 chain to flex wheel (no change)
Power budget: 88W exactly at R10a cap (no change)
Drivetrain power: 44W (11W under R11a cap — room for half-motor upgrade if push-heavy strategy demands)

Build Order This Week

Verify drivetrain gear ratio. If not at 5:3 (36T → 60T), swap gears. Test with stopwatch on 8 ft straight line — target ~1.6 seconds at full throttle (= 5.0 ft/s).
Measure 4-bar pivot tower height. Document the actual height above ground. If < 5″, plan a tower extension or strategy adjustment.
Test claw reach to all 3 goal heights. Use the actual goals from the field if available, or 3D printed mock-ups. Document which heights are achievable.
Add rubber grip pads to claw tips. Cut from common silicone or rubber sheets, attach with M3 screws or strong adhesive. Test pin grip reliability — target ≥80% pickup success.
Verify CoG with calculator at /center-of-gravity. Stability angle ≥ 50° with arm extended carrying a cup.
Document on engineering notebook slides 25 (mechanism comparison), 28-29 (decision matrix with this analysis), 41 (build log with measurements: actual gear ratio, actual tower height, actual reach), 47 (electronics + CoG).

Cross-References

The full Hero Bot V1.5 spec: /spartan-hero-bot
2.75″ and 4″ parts lists with gear diagrams: /drivetrain-builds-55w
Drivetrain learning path: /drivetrain-101
CoG calculator (verify stability with new arm length): /center-of-gravity
Lift selection (chain bar vs DR4B vs cascade): /lift-selection
V2 design study (4-bar + chain bar + pneumatic claw — future build): /chassis-stacked-lift-geometry
Override game element specs: /override-manual-summary

What This Page Answers

Question 1: 2.75″ or 4″ Wheels for 55W Push-Heavy?

Question 2: Does the Hero Bot V1.5 Math Check Out?

Why This Matters Now