Math is Not a Four Letter Word

2017 FTC Kick-Off

Andy Driesman

FTC4318 Green Machine Reloaded

andrew.driesman@gmail.com

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Goals

• Discuss concept of trade space/studies

• Demonstrate the importance of using math and physics in robot design

• Provide a basic understanding of useful physics and mathematical relationships.

• Provide examples.

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Agenda

• Why calculate rather than build?

• Electric Motors and Servo Specifications

• Gears and Gear Ratios

• Chains Drives

• Using Math and Physics to Design Ball Launcher

• The use of Margin

• Battery Health

• Other types of calculations.

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Why “Do Math”?

• Many FTC design iterations are done by “trial and error”.

▫ Trial and error takes time and money.

• Doing a calculation allows the following without building hardware:

▫ To see if a design will work (or not)

▫ Select components

▫ Calculate performance

▫ Bound a design

▫ Test various design ideas quickly

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Other Options than Building • Modeling and simulation

▫ Example: Algodoo – Physics based simulation tool

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Concept of a Trade Space

• Definition: set of program and system parameters, attributes, and characteristics required to satisfy performance standards.

• Translation: many different solutions can satisfy the same requirements.

• To explore even a simple trade space one needs to employ math and physics

▫ Building (or even prototyping) every solution is impractical.

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Trade Study for Velocity Vortex

Ball Shooter

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Hot Wheels 3 5.0 5 4.0 2 - 3 3.0 3 3.0 4 3.0 4 3.0 3 4.0 89.0

Choo Choo 5 5.0 4 4.0 4 - 4 3.0 5 3.0 3 3.0 3 3.0 4 4.0 102.0

Single Wheel Hot Wheel 3 5.0 5 4.0 2 - 4 3.0 5 3.0 5 3.0 4 3.0 4 4.0 105.0

Fan 1 5.0 5 4.0 1 - 2 3.0 3 3.0 1 3.0 5 3.0 2 4.0 66.0

Cross Box 4 5.0 2 4.0 5 - 2 3.0 4 3.0 1 3.0 2 3.0 1 4.0 59.0

Sling Shot 3 5.0 2 4.0 3 - 3 3.0 4 3.0 3 3.0 3 3.0 3 4.0 74.0

Diving Board 4 5.0 4 4.0 4 - 4 3.0 5 3.0 4 3.0 4 3.0 4 4.0 103.0

Paddle Wheel 3 5.0 3 4.0 3 - 1 3.0 5 3.0 5 3.0 4 3.0 3 4.0 84.0

Airsoft Launcher 5 5.0 4 4.0 5 - 4 3.0 5 3.0 4 3.0 4 3.0 3 4.0 104.0

Example Trade Space from Res-Q

– using a cord to lift the robot

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The Golden Rule

You never ever get something for nothing!!!

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Electrical Motors 101

• Motors are electro-mechanical devices that convert electrical energy to mechanical energy. They have the following characteristics: ▫ Torque – Rotational force.

SI Unit is: Newton*meter (N*m) though sometimes kilogram*centimeter (kgf*cm)

English Unit are: ounce*inches (oz*in) or Pound*feet (lbf*ft)

▫ Motor Speed (velocity) – Rotational rate. SI unit is: radians per sec

English units: Degrees per second or Rotations per minute (RPM) or Rotations per Second (RPS) One rotation = 360°

▫ Speed is inversely proportional to Torque

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Neverrest 40/Tetrix Motor Specifications

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Torque vs Current curves

Torque vs Speed curves

Specifications

Tetrix Motor • No Load Speed = 139 ± RPM

• Output Speed at max power = 76 RPM

• Output Torque at max power (t) = 11.8 kgf*cm=1.2 N*m=10 lbf*in

• Stall Torque = 23.4 kgf*cm (Speed = Zero)=2.3 N*m=19 lbf*in

HS485HB Servo (180°) • No Load Speed = 45 RPM (0.66 sec/180°

rotation)

• Stall Torque = 4.8 kg*cm = 4 lbf*in

Continuous Rotation Servo • No Load Speed = 43 RPM

• Stall Torque = 2.8 kg*cm

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Available Tetrix Gears • 40 tooth (diameter ~34 mm) • 80 tooth (diameter ~66 mm) • 120 tooth (diameter ~97 mm) • Tetrix Gear Specs:

▫ Diametrical Pitch: 32 teeth/in ▫ Pressure Angle: 20 degrees ▫ Face Width: ¼ inch

• Tetrix Ratios: 1.0:2.0:3.0 • Gears from other vendors are

allowed.

Available Tetrix Sprockets • 16 tooth (diameter ~36 mm) • 24 tooth (diameter ~52 mm) • 32 tooth (diameter ~68 mm) • Sprockets from other vendors are

allowed. • Ratios 1.0:1.5:2.0 • Compatible w/ #25 Chain

Gears, Gear Ratios and Units

• Gears are used: ▫ To transmit power from one place to another. ▫ Change direction that power is applied. ▫ To transform the motor output:

Increase/decrease rotation (angular) rate Increase/decrease torque ( a measure of “turning” force)

• Rotational Rate or Gear Speed ▫ Measured in Rotations per minute (RPM) ▫ 1 RPM equivalent to 1 minute per rotation ▫ 1 RPM = 360° per minute = 6° per second

• Torque ▫ Torque is a unit of rotational force at a distance. Units are:

N*m, kgf*cm or lbf*inch. ▫ 1 kgf*cm is the equivalent of placing a 1 kg weight at a 1 cm

distance from the axis of rotation.

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Using Gears to Change Speed and Torque Output Speed = Input Speed x Gear Ratio

Output Torque = Input Torque / Gear Ratio

• Gear Ratio

▫ 40 tooth “input”:80 tooth “output”

▫ Ratio is 40:80 or 1:2 or ½.

• Output Speed (RPM) Calculation

▫ Input Speed is multiplied by the gear ratio of ½ to get output speed

▫ 76 RPM* ½ =38 RPM

• Output Torque (T) Calculation

▫ Input torque is divided by the gear ratio to get output torque

▫ 11.8 kg*cm/(½) =23.6 kg*cm

• Gears introduce inefficiency (due to friction), so there is always a loss.

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INPUT (drive)

S=100 RPM

T=40 in*oz

OUTPUT

S=50 RPM

T=80 in*oz

80 tooth gear

OUTPUT

W=100 RPM

T=40 in*oz

40 tooth gears

More Complicated Illustration

• Gears can be “stacked” into a Gear Chain

▫ Overall ratio is calculated by multiplying the individual ratios

▫ 120 tooth:40 tooth= 120:40=3:1=3/1

▫ (3/1)x(3/1)x(3/1)=27:1=27

• Speed (RPM) Calculation

▫ Input Speed is multiplied by the gear ratio of 27/1 to get output speed

▫ 100 RPM* 27 = 2700 RPM

• Torque (T) Calculation

▫ Input torque is divided by the gear ratio to get output torque

▫ 40 oz*in/(27) =1.5 in*oz

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• Small gears are 40 tooth gears

• Large gears are 120 tooth gears

Input

Gear

Output

Gear

Converting Rotational Motion to Linear Motion Linear Force = Torque of final gear / Radius of final gear

Torque = Force x Radius

Radius = Torque/Force • Force = Torque / Radius ▫ Units of Force are Newtons. Kgf

and lbf

▫ Torque = torque of the motor after gearing.

▫ Radius – is the radius of the final gear.

• The bigger the radius, the less force.

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FORCE

TORQUE

Radius (r)

Radius (r)

FORCE

TORQUE

Chains and Sprockets Output Speed = Input Speed x Sprocket Ratio

Output Torque = Input Torque / Sprocket Ratio

• Chains and sprockets calculations are very similar to gears.

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• Sprocket Ratio

▫ 24 tooth “drive”:32 tooth “load”

▫ Ratio is 24:32 or 3:4 or ¾.

• Speed (RPM) Calculation

▫ Input Speed is multiplied by the gear ratio of ¾ to get output speed.

▫ 100 RPM* ¾ =75 RPM

• Torque (T) Calculation

▫ Input torque is divided by the gear ratio to get output torque

▫ 40 in*oz/(¾) =53 in*oz

24 tooth

Drive

Sprocket

32 tooth Load

Sprocket

Velocity Vortex Design

• Requirements

▫ Ball height cannot exceed 6 feet.

▫ Ball repeatedly go through 30” wide vortex

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Trade Study for Ball Shooter

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Hot Wheels 3 5.0 5 4.0 2 - 3 3.0 3 3.0 4 3.0 4 3.0 3 4.0 89.0

Choo Choo 5 5.0 4 4.0 4 - 4 3.0 5 3.0 3 3.0 3 3.0 4 4.0 102.0

Single Wheel Hot Wheel 3 5.0 5 4.0 2 - 4 3.0 5 3.0 5 3.0 4 3.0 4 4.0 105.0

Fan 1 5.0 5 4.0 1 - 2 3.0 3 3.0 1 3.0 5 3.0 2 4.0 66.0

Cross Box 4 5.0 2 4.0 5 - 2 3.0 4 3.0 1 3.0 2 3.0 1 4.0 59.0

Sling Shot 3 5.0 2 4.0 3 - 3 3.0 4 3.0 3 3.0 3 3.0 3 4.0 74.0

Diving Board 4 5.0 4 4.0 4 - 4 3.0 5 3.0 4 3.0 4 3.0 4 4.0 103.0

Paddle Wheel 3 5.0 3 4.0 3 - 1 3.0 5 3.0 5 3.0 4 3.0 3 4.0 84.0

Airsoft Launcher 5 5.0 4 4.0 5 - 4 3.0 5 3.0 4 3.0 4 3.0 3 4.0 104.0

Air-Soft Launcher

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Design Questions

• What ball velocity is required to reach a height of 6 feet?

• What spring constant is required to accelerate a ball to such a velocity?

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Calculating Ballistic Trajectory

• Distance (x) = horizontal component of velocity x time

▫ Horizontal component of velocity = ball velocity x Cosine of launcher angle

• Height (y) = initial height + (vertical component of velocity x time) – ½ x gravitational constant x time2

▫ Vertical component of velocity = ball velocity x Sine of launcher angle

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Ballistic Trajectory that meets

Requirements

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Height (ft)

Y (ft)

Calculating Spring Constant

• Velocity required to get 6 foot height was 20 feet per second.

• Spring Constant required to achieve 20 feet/sec.

• Spring Constant = mass of ball and launcher x (initial velocity)2 / (spring stroke)2

• Initial Velocity = ((spring constant x spring stroke2)/mass)1/2

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Buying a Spring

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Launcher Performance

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Thoughts on Velocity Vortex

Solution • Perfect challenge for the application of HS physics and algebra.

▫ Use saved significant robot development time ▫ Got the performance of the launcher spot on trying on the second

spring try. ▫ Launch consistently hit a 6” circle within the vortex.

• However, launcher turned out to not be optimal solution. Team failed to weight system speed (# of balls per sec) high enough.

• So while recycle time was fast (~1 sec), designing for a faster recycle time would have increased scoring.

• Lessons Learned: ▫ Team knew recycle time was not optimal, but thought “good enough”.

Be more aggressive!! ▫ While the design had a huge number of benefits (precision, reliability,

maintainability), team was emotionally attached to it. Maintain objectivity.

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Margin

• Margin is a factor added to a calculation to account for real world conditions

▫ Motors wear out

▫ Batteries are not fully charged

▫ Gear train and chain drives are not 100% efficient.

▫ Imprecise construction techniques.

• Where practical, we use a factor of 2.

▫ For example: if a calculation says, we need a given torque, I multiply it by 2.

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Battery Health • “Nameplate” capacity of battery is 3,000 mAmp*hour (good surrogate for battery

energy). Under “perfect” conditions battery will provide 3 amps for one hour before it is completely discharged. ▫ Batteries do wear out. They slowly loose their capacity. ▫ Measuring the health of a battery requires specialized equipment (e.g.: battery beak).

• A rule of thumb (for batteries used in this environment) is that a battery’s usable capacity is about 30% - 50% of the name-plate or 1,000 mA*Hr to 1,500 mA*Hr.

• How much energy does a 2.5 minute match use? ▫ Dependent on #of motors, servos, activity of play, friction, etc. Hard to calculate

actuals, but easy to calculate maximum. Each battery has a 20A fuse. Meaning max current draw is 20A.

▫ A*Hr used = 20A*2.5min/60 min/hour = 0.83 A*H = 830 mA*Hr. ▫ Good policy to switch out batteries every match or every other match.

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Other things of interest that can be

calculated • Force applied by surgical tubing

▫ Used during “Ring It Up” to calculate the amount of assist provided to a scissor lift

▫ Dependent on the material modulus, inner and outer diameters of the tubing.

• Axle or beam strength ▫ Used in “Block Party” to understand why axles kept bending and

identify a replacement material. ▫ Dependent on Young’s modulus, area and length of axle/beam

and type of support. • Scissor lift forces

▫ Used to calculate the amount of force required to lift a another robot during “Block Party”

▫ Dependent on geometry of scissor lift. • Etc

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Summary of Formulas

• Gears:

▫ Output Speed = Input Speed x Gear Ratio

▫ Output Torque = Input Torque / Gear Ratio

• Chains and Sprockets:

▫ Output Speed = Input Speed x Sprocket Ratio

▫ Output Torque = Input Torque / Sprocket Ratio

• Converting Rotational Motion to Linear Motion

▫ Linear Force = Torque applied to final gear / Radius of final gear

• Converting Linear Motion to Rotational Motion

▫ Torque = Linear Force x Radius

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References

• Understanding DC Motors

▫ http://lancet.mit.edu/motors/motors3.html

• Gears, Chains and Sprockets

▫ https://www.youtube.com/watch?v=D_i3PJIYtuY

▫ http://www.societyofrobots.com/mechanics_gears.shtml

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Tetrix Motor

Spec Sheet

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