Mobile robot dynamics and conrol

1. Mobile robot
dynamics and control
Ian Tsybulkin
March 24, Kiev, Projector

2. so large
wheels ?
I have no
more power
hold me tight

3. Create / modify
the model
Find equations
of motion
Develop control
Simulate and
check
1
2
3
4

4. Ex.1: drone stabilization
• Simple example: Vertical stabilization of a
quadrocopter

5. • F ~ U
• z(t=0) = 0;
• dz/dt(t=0) = 0;
• z(t) -> zend;
• |U| < Umax
• U(z, dz/dt, zend) - ?
F1 F2
m∙g

6. • A simple control (proportional to error)
• kU = mg; U0 = mg / k
• U* = U0 + d∙( zend - z(t) ); U = lim(U*)

7. proportional + velocity
• under, over, and critically damping control

8. PID controller
https://en.wikipedia.org/wiki/PID_controller

9. • PID controller is simple and generic approach
• PID controller is not optimal and in some
complicated cases can be unstable
https://github.com/tsybulkin/drone_stab

10. Ex.2: inverted
pendulum
https://github.com/tsybulkin/pend

11. Dynamics
α
mg

12. Phase portret
0

13. Three modes
• Mode 1: Pumping energy until E < Ethr
•
• U1(t) = sign(da/dt)
• Mode 2: No control until |a| > a0
• Mode 3: Balancing control |a| < a0

15. LQR
• Q, R - positive-definite matrices
• u = - Kx; Riccati equation
• scipy.linalg.lqr(A,B,Q,R)
Wikipedia: https://en.wikipedia.org/wiki/Linear-quadratic_regulator

16. LQR
K: [[ 10.09901951 2.78076962]]
eigen vals: [-2.72150939+0.j -5.62079946+0.j]

18. Ex.3:
The CubliETH Zurich

19. Dynamics
R
a
md
mb
2L
https://github.com/tsybulkin/cuba

20. Optimal control by ML
• Agent explores different strategies by trial and error
method
• Optimal policy: maps states —> actions,
maximizing total reward

21. Policy (control) learning
S1
S2
S3
u1
u2
u3

23. Ex.4: hopper
MIT hopper, 1986
α
𝜷
𝑥
𝑥
m
m
𝑥
2-link hopper
https://github.com/tsybulkin/hop2

24. Challenges
• more degrees of freedom (DoF)
• highly non-linear system
• hybrid dynamics:
phase 1 (touches the ground) —> phase 2 (does
not touch the ground) —> phase 1 —> …

25. Lagrange method
α
𝜷
𝑥𝑦
𝑥𝑦1
m1g
m2g
𝑥𝑦2

26. Lagrange equation

27. Hybrid dynamics
touches the
ground
does not
touch the
ground
Transition between states can be tricky