1. Today's Topics Art of Lego Design
Notes from Jason Geist, Carnegie Mellon University
Differential Drive
Notes from Zachary Dodd, Harvey-Mudd College

2. Goals: Build better robots Minimize mechanical breakdowns
Build robots that are easy to control
Encourage good design strategy
Strive for elegant, clever solutions

3. Know your Materials Quick facts:
Plastic bricks since 1949 (wooden blocks prior)
On average, 2100 different parts each year
Manufacturing tolerance: 1/1000 of an inch
Number of ways of combining six 8-stud bricks: 102,981,500
Widely used by scientists and engineers as a rapid prototyping tool

4. Geometry Three plates = 1 brick in height
1-stud brick dimensions: exactly 5/16” x 5/16” x 3/8” (excluding stud height 1/16”),
This is the base geometry for all LEGO components

5. Structure
Common pitfall when trying to increase mechanical robustness:

6. Structure
The right way:

7. Structure
The right way:

8. Connector pegs
Black pegs are tight-fitting for locking bricks together.
Grey pegs turn smoothly in bricks for making a pivot

9. Structure LEGO bricks are finicky: They HATE duct tape.
They HATE hot glue.
They HATE super glue.
They HATE epoxy.
You should never need adhesives to build reliable LEGO structures

10. Drivetrain
LEGO Gears
40T 8T
16T
Bevel
1T Worm
24T
24T
Crown

11. Worm Gears 3 1 2 • Result is a 24:1 gearbox • Not back driveable! 4
Pull one tooth per revolution 3 1 2
• Result is a 24:1 gearbox
• Not back driveable! 4

12. Motors
9V Gear Motor
~ 150 mA
300 RPM (no load)

13. Mounting Motors
Note Bulge under motor

14. Mounting Motors
Add a gear:

15. Mounting Motors
Use special 2x1 stud plate with “wing”

16. Build for good control
Slow vs. fast?
Gear backlash
Stability
Skidding

17. Design Strategy Incremental Test components parts as you build them
Drivetrain
Sensors, sensor mounting
Structure
Don’t be afraid to redesign
KISS

18. Design Strategy
Drivetrain driven
Chassis/structure driven
Modular?

19. Testing Don’t wait until you have a final robot to test
Interaction of systems
Work division (work concurrently)
Develop test methods
Repeatability

20. Philosophy Have fun Be creative, unique
Strive for cool solutions, that work!
Aesthetics: it’s fun to make beautiful robots!

21. Differential drive Most common kinematic choice
- difference in wheels’ speeds determines its turning angle
All of the miniature robots…
Khepera, Braitenberg VL VR

22. Questions (forward kinematics)
Differential drive
Most common kinematic choice
- difference in wheels’ speeds determines its turning angle
All of the miniature robots…
Khepera, Braitenberg
Questions (forward kinematics)
Given the wheel’s velocities or positions, what is the robot’s velocity/position ? VL
Are there any inherent system constraints? VR

23. Questions (forward kinematics)
Differential drive
Most common kinematic choice
- difference in wheels’ speeds determines its turning angle
All of the miniature robots…
Khepera, Braitenberg
Questions (forward kinematics)
Given the wheel’s velocities or positions, what is the robot’s velocity/position ? VL
Are there any inherent system constraints? VR
1) Specify system measurements
2) Determine the point (the radius) around which the robot is turning.
3) Determine the speed at which the robot is turning to obtain the robot velocity.
4) Integrate to find position.

24. Differential drive y x 1) Specify system measurements
- consider possible coordinate systems VL x q l VR
(assume a wheel radius of 1)

25. Differential drive y x 1) Specify system measurements
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. VL x q l VR
(assume a wheel radius of 1)

26. Differential drive y x 1) Specify system measurements
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning.
- to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles VL x
- each wheel must be traveling at the same angular velocity q l VR
ICC “instantaneous center of curvature”
(assume a wheel radius of 1)

27. Differential drive y x 1) Specify system measurements
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles VL x
- each wheel must be traveling at the same angular velocity around the ICC q l VR
ICC “instantaneous center of curvature”
(assume a wheel radius of 1)

28. Differential drive y x 1) Specify system measurements
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and its linear velocity l VR
ICC
w(R+l/2) = VL R
w(R- l/2) = VR
robot’s turning radius
(assume a wheel radius of 1)

29. Differential drive y x 1) Specify system measurements
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and then linear velocity l VR
ICC
w(R+d) = VL R
w(R-d) = VR
robot’s turning radius
Thus,
w = ( VR - VL ) / l
R = l ( VR + VL ) / ( VR - VL )
(assume a wheel radius of 1)

30. Differential drive y x 1) Specify system measurements
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and then linear velocity l VR
ICC
w(R+d) = VL R
w(R-d) = VR
robot’s turning radius
Thus,
w = ( VR - VL ) / l
R = l ( VR + VL ) / ( VR - VL )
So, the robot’s velocity is
V = wR = ( VR + VL ) / 2

31. Differential drive y x 4) Integrate to obtain position
Vx = V(t) cos(q(t))
w(t)
Vy = V(t) sin(q(t)) VL l VR
ICC
R(t)
robot’s turning radius
with
w = ( VR - VL ) / l
R = l( VR + VL ) / ( VR - VL )
What has to happen to change the ICC ?
V = wR = ( VR + VL ) / 2

32. Differential drive y x 4) Integrate to obtain position
Vx = V(t) cos(q(t))
w(t)
Vy = V(t) sin(q(t))
Thus,
x(t) = ∫ V(t) cos(q(t)) dt VL
y(t) = ∫ V(t) sin(q(t)) dt l
q(t) = ∫ w(t) dt VR
ICC
R(t)
robot’s turning radius
with
w = ( VR - VL ) / l
R = l ( VR + VL ) / ( VR - VL )
V = wR = ( VR + VL ) / 2

33. Differential drive y x Kinematics 4) Integrate to obtain position
Vx = V(t) cos(q(t))
w(t)
Vy = V(t) sin(q(t))
Thus,
x(t) = ∫ V(t) cos(q(t)) dt VL
y(t) = ∫ V(t) sin(q(t)) dt l
q(t) = ∫ w(t) dt VR
ICC
Kinematics
R(t)
robot’s turning radius
with
w = ( VR - VL ) /l
R = l ( VR + VL ) / ( VR - VL )
What has to happen to change the ICC ?
V = wR = ( VR + VL ) / 2