1 Math is Not a Four Letter Word 2017 FTC Kick-Off Andy Driesman FTC4318 Green Machine Reloaded andrew.driesman@gmail.com
2 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.
3 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.
4 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
Other Options than Building Modeling and simulation Example: Algodoo Physics based simulation tool 5
6 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.
7 Trade Study for Velocity Vortex Ball Shooter Consistency of Speed Weight Cycle Time Weight Noise Made Weight Packaging Size 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 Weight # of Motors Weight Simplicity Weight Durability Weight Ease of Maintennce Weight Weighted results
8 Example Trade Space from Res-Q using a cord to lift the robot
9 The Golden Rule You never ever get something for nothing!!!
10 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
11 Neverrest 40/Tetrix Motor Specifications Torque vs Current curves Torque vs Speed curves
12 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 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
13 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.
14 Using Gears to Change Speed and Torque Output Speed = Input Speed x Gear Ratio Output Torque = Input Torque / Gear Ratio OUTPUT W=100 RPM T=40 in*oz INPUT (drive) S=100 RPM T=40 in*oz 40 tooth gears OUTPUT S=50 RPM T=80 in*oz 80 tooth gear 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.
15 More Complicated Illustration Output Gear Small gears are 40 tooth gears Large gears are 120 tooth gears Input Gear 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
Converting Rotational Motion to Linear Motion Linear Force = Torque of final gear / Radius of final gear Torque = Force x Radius Radius = Torque/Force TORQUE Radius (r) 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. TORQUE Radius (r) 16 FORCE FORCE
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. 24 tooth Drive Sprocket 32 tooth Load Sprocket 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 17
18 Velocity Vortex Design Requirements Ball height cannot exceed 6 feet. Ball repeatedly go through 30 wide vortex
19 Trade Study for Ball Shooter Consistency of Speed Weight Cycle Time Weight Noise Made Weight Packaging Size 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 Weight # of Motors Weight Simplicity Weight Durability Weight Ease of Maintennce Weight Weighted results
Air-Soft Launcher 20
21 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?
22 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 time 2 Vertical component of velocity = ball velocity x Sine of launcher angle
Ballistic Trajectory that meets Requirements 23-1.00 2.00 3.00 4.00 5.00 6.00 7.00 0.03 0.14 0.24 0.35 0.45 0.56 0.66 0.76 0.87 0.97 1.08 1.18 1.28 1.39 1.49 1.60 1.70 1.81 1.91 2.01 2.12 2.22 2.33 2.43 2.54 2.64 2.74 2.85 2.95 3.06 3.16 3.26 3.37 3.47 3.58 3.68 3.79 3.89 3.99 4.10 Height (ft) Y (ft)
24 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 stroke 2 )/mass) 1/2
Buying a Spring 25
Launcher Performance 26
27 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.
28 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.
29 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.
30 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
31 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
32 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
33 Tetrix Motor Spec Sheet