Structure:
    Main wheel
       ↓
    ○───○  (small rollers around center)
   ○ ◯ ○   (blue = roller axis)
    ○───○

Motion:
├─ Forward: Main wheel pushes
├─ Side: Rollers allow sideways glide
├─ Rotate: In place rotation possible
└─ Diagonal: Any direction!

Advantages:
✓ Holonomic motion (any direction)
✓ No turning radius needed
✓ Perfect 90° turns

Disadvantages:
✗ Expensive ($30-60 each)
✗ Complex mechanically
✗ Less traction (roller slip)
✗ Requires 4 wheels minimum

Best for: Soccer robots, navigation in tight spaces

Structure:
  Angled rollers (45°)
     ↙ ↓ ↘
    □─◯─□  (rollers at 45°)
     ↙ ↓ ↘

Motion Pattern (4-wheel):
Forward:   All wheels rotate same direction
Strafe:    Left/right wheels opposite
Rotate:    Front/back opposite pairs
Diagonal:  Combinations of above

Result: Holonomic motion like omni wheels

Advantages:
✓ Holonomic capability
✓ Better traction than omni
✓ Simpler mechanism

Disadvantages:
✗ Expensive
✗ Requires precise alignment
✗ Complex control software

Simple cylinder (rubber or plastic)
     ↓
  [O] = Wheel on axle

Motion:
├─ Forward/backward: Rotates
├─ Steering: Must pivot entire wheel
└─ Sideways: Zero slip (must turn)

Advantages:
✓ Very cheap ($5-15 each)
✓ Simple mechanics
✓ Reliable and robust
✓ Good traction

Disadvantages:
✗ Requires turning radius
✗ Three-point turns needed
✗ Differential steering complex

Best for: Most robots (wheeled platforms)

Advantages: Best grip, durable, all-weather
Disadvantages: Heavy, wear over time, affected by temperature
Best for: Outdoor robots, off-road, high traction required

Advantages: Good grip, durable, consistent
Disadvantages: More expensive, overkill for most robots
Best for: Commercial-grade robots

Advantages: Cheap, light, never wears
Disadvantages: Poor grip, slips on smooth surfaces
Best for: Smooth floor line-followers, indoors only

Small wheels (2-3 inches):
├─ Pro: Faster acceleration, responsive
├─ Con: Slower max speed, bouncy
└─ Best for: Line-followers, tight spaces

Medium wheels (4-6 inches):
├─ Pro: Good balance, stable
├─ Con: Average performance
└─ Best for: General mobile robots

Large wheels (8+ inches):
├─ Pro: Highest speed, smooth over obstacles
├─ Con: Slow acceleration, heavy
└─ Best for: Racing, outdoor terrain

Structure:
    Teeth
      ▲
  ◯─●─◯  (flat teeth)
  ◯─●─◯
    ◯▼◯

Motion:
Input shaft ──→ [Teeth mesh] ──→ Output shaft
          90° rotation (shafts parallel)

Ratio calculation:
Output_speed = Input_speed × (Pinion_teeth / Gear_teeth)
Torque multiplies inversely

Example:
Input: 1000 RPM, 10 teeth
Output gear: 50 teeth
Output: 1000 × (10/50) = 200 RPM, 5× torque

Advantages:
✓ Simple, reliable
✓ Efficient (95%+)
✓ Compact
✓ Wide ratio range

Disadvantages:
✗ Noise at high speed
✗ Shaft misalignment causes wear
✗ Backlash (play) possible

Best for: Motors to wheels, gearbox stages

Structure:
    Shaft B
      ↑
  ◣───┤───◢
  ◢───┤───◣  ← Angled teeth (cone shape)
      ↑
    Shaft A

Motion: Right angle power transfer

Advantages:
✓ Compact 90° connection
✓ Good efficiency

Disadvantages:
✗ Expensive
✗ Complex to align
✗ Harder to replace
✗ Lower efficiency (80-90%)

Best for: Robot arms (elbow joints), differentials

Structure (Epicyclic):
      Sun
       ◯
   ◯ ◢◣ ◯  ← Planets revolve
   ◢ ◯ ◣
      Ring

Motion:
Input: Sun or Ring
Output: Carrier (planets)
High ratio possible in compact space

Ratios: 3:1 to 100:1 in same space as spur

Advantages:
✓ Extremely compact
✓ Very high ratios possible
✓ Smooth operation
✓ Good efficiency

Disadvantages:
✗ Very expensive
✗ Complex mechanics
✗ Difficult to repair
✗ Backlash harder to control

Best for: Compact robot joints, servo gearboxes

Structure:
    Worm shaft
        ↓ (screw-like)
      ▔▔▔▔▔
    ◯ ████ ◯
    ◯ ████ ◯
      ▁▁▁▁▁
        ↑
    Worm wheel

Motion:
Worm rotates 1 complete turn
Worm wheel rotates 1 tooth (high ratio!)

Ratio: 5:1 to 100:1 easily

Advantages:
✓ Extremely high ratio
✓ Compact
✓ Self-locking (no backdrive)
✓ Smooth quiet operation

Disadvantages:
✗ Very expensive
✗ Low efficiency (50-70%)
✗ Heat generation
✗ Limited to low-speed output

Best for: Robot arm joints, precision positioning

Formula:
Ratio = Pinion_teeth / Gear_teeth

Speed change:
Output_RPM = Input_RPM / Ratio

Torque change:
Output_torque = Input_torque × Ratio × Efficiency

Example Mobile Robot:
├─ Motor: 1000 RPM, 0.5 N⋅m
├─ Gearbox: 10:1 ratio, 95% efficiency
├─ Output: 100 RPM, 4.75 N⋅m
└─ Fast enough? Low torque? Adjust ratio!

Structure:
    Sprocket  
    (tooth wheel)
         ▲
    ◯───────◯  ← Chain links
    ◯───────◯
         ▼
    Sprocket

Motion:
Input sprocket → Chain → Output sprocket
Mechanical connection (no slip!)

Advantages:
✓ No slip (positive drive)
✓ Efficient (95%+)
✓ Can span long distances
✓ Tolerates misalignment
✓ Easy ratio changes (swap sprockets)

Disadvantages:
✗ Requires lubrication
✗ Noisy
✗ Heavy
✗ Needs tension adjustment
✗ Wears over time

Best for: Robot wheels, conveyor systems, outdoor robots

Structure:
    Pulley
      ◯
    ┏━━━┓  ← Rubber belt
    ┗━━━┛
      ◯
    Pulley

Motion:
Input pulley → Belt grip → Output pulley
Friction drive (slight slip possible)

Types:
├─ Flat belt (simple, weak)
├─ V-belt (wedge grip, powerful, still smooth)
└─ Timing belt (teeth, positive drive)

Advantages:
✓ Smooth quiet operation
✓ Isolates vibration
✓ No maintenance (no lubrication)
✓ Doesn't need precision alignment

Disadvantages:
✗ Slip possible (especially flat belts)
✗ Speed sensitive (changes with wear)
✗ Tension must be correct
✗ Heat generation possible

Best for: Smooth power transmission, conveyor, fans

Chain/Belt ratio depends on sprocket/pulley size:

Ratio = Driven_size / Driver_size

Example:
Input pulley: 30mm diameter
Output pulley: 90mm diameter
Ratio = 90/30 = 3:1 (3x slower, 3x more torque)

For chains (teeth count):
Input sprocket: 20 teeth
Output sprocket: 60 teeth
Ratio = 60/20 = 3:1

Simple machine examples:

Lever:
    ↓ Force in
    │
────┼────────
    │        ↓ Load out
    
MA = Lever_arm_in / Lever_arm_out

Pulley:
    ☐ ← Fixed
    │
    │ ── Rope
    │  
   Load

MA = Number_of_rope_segments

Incline:
   ╱╲
  ╱  ╲← Length
 ╱────╲
 Height

MA = Length / Height

Example 10:1 gearbox:

Input: 1 newton force (small gear)
Output: 10 newton force (large gear)
Mechanical Advantage = 10

Reality: Losses due to friction
├─ Efficient gearbox: 95% = 9.5× MA
├─ Chain drive: 95% = 9.5× MA
├─ Belt drive: 90-95% = 9-9.5× MA
└─ Worm gear: 60-70% = 6-7× MA

Example: Gearbox (10:1) + Belt (3:1) + Chain (2:1)

Total ratio: 10 × 3 × 2 = 60:1

Efficiency:
├─ Gearbox: 95%
├─ Belt: 92%
├─ Chain: 95%
└─ Total: 0.95 × 0.92 × 0.95 = 83% efficiency

So: 60:1 mechanical advantage × 83% efficiency
     = 50× useful output

Equation:
Torque_required = (Friction_coefficient × Weight × gravity × Radius)
                + (Weight × gravity × sin(angle) × Radius)
                + (Mass × Radius × Acceleration)

Example (flat ground):
├─ Weight: 10 kg
├─ Wheel radius: 0.05 m (5 cm)
├─ Friction coefficient: 0.1
├─ Torque = 0.1 × 10 × 9.81 × 0.05 = 0.49 N⋅m

Two motors (front wheel drive):
├─ Each motor needs: 0.49 N⋅m / 2 = 0.245 N⋅m
└─ Standard motor: 0.3 N⋅m works fine

Wheel speed = (Motor_RPM / Gear_ratio) × Wheel_circumference

Example:
├─ Motor: 300 RPM (after gearing)
├─ Wheel diameter: 0.1 m (10 cm)
├─ Circumference: π × 0.1 = 0.314 m
└─ Speed: 300 × 0.314 / 60 = 1.57 m/s = 5.7 km/h

Is this fast enough for competition? Depends on requirements!

Motor power → Losses → Final mechanical power

Typical breakdown (10W motor):
├─ Motor: 90% (inherent losses)
├─ Gearbox: 95% (friction in gears)
├─ Chain: 95% (friction, drive loss)
└─ Wheels: 97% (rolling resistance)

Total: 0.90 × 0.95 × 0.95 × 0.97 = 78% efficient

So: 10W motor × 78% = 7.8W useful mechanical power

Where power goes (10W motor example):

Motor: 1W lost as heat
Gearbox: 0.5W lost (metal-to-metal friction)
Chain: 0.25W lost (link friction)
Wheels: 0.3W lost (rolling resistance)
Total losses: 2.05W
Useful: 7.95W

Solutions to improve:
├─ Better quality gears (reduce friction)
├─ Lubricate chain regularly
├─ Lower rolling resistance tires
├─ Reduce weight
└─ Use high-efficiency motor

System:
Motor → Gearbox (10:1) → Wheel

Specs:
├─ Motor: 300 RPM, 0.5 N⋅m
├─ Gearbox: 10:1, 95% efficiency
├─ Output: 30 RPM, 4.75 N⋅m
├─ Wheel (5cm): Tangential force = Torque/Radius
│  = 4.75 N⋅m / 0.05 m = 95 N (enough to move!)
└─ Speed: 30 RPM × 2π × 0.05 / 60 = 0.157 m/s

For 1 m/s speed needed?
Required RPM: 1 m/s × 60 / (2π × 0.05) = 191 RPM
Gearbox ratio: 300 / 191 = 1.57:1 (lower reduction)

System:
Motor → Planetary gear (20:1) → Link arm

Specs:
├─ Motor: 200 RPM, 0.2 N⋅m
├─ Gearbox: 20:1, 92% efficiency
├─ Joint output: 10 RPM, 3.68 N⋅m
├─ Arm length: 0.3 m
├─ Load: Can hold 1.2 kg at end
│  (3.68 / 0.3 / 9.81 = 1.25 kg) ✓
└─ Speed: 10 RPM allows precise positioning

Tradeoff: Slow but strong (good for arms!)

System:
Motor → Minimal reduction → Pulley

Specs:
├─ Motor: 1500 RPM, 0.1 N⋅m
├─ Reduction: 2:1 (just for extra torque)
├─ Output: 750 RPM, 0.19 N⋅m
├─ Pulley radius: 0.05 m
├─ Belt speed: 750 RPM × 2π × 0.05 / 60 = 3.93 m/s
├─ Carrying capacity: Good for light items
└─ Efficiency: 98% (direct drive, very efficient)

Tradeoff: Fast but weak (good for conveyor!)