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Mobile robot
dynamics and control
Ian Tsybulkin
March 24, Kiev, Projector
so large
wheels ?
I have no
more power
hold me tight
Create / modify
the model
Find equations

of motion
Develop control
Simulate and
check
1
2
3
4
Ex.1: drone stabilization
• Simple example: Vertical stabilization of a
quadrocopter
• 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
• A simple control (proportional to error)
• kU = mg; U0 = mg / k
• U* = U0 + d∙( zend - z(t) ); U = lim(U*)
proportional + velocity
• under, over, and critically damping control
PID controller
https://en.wikipedia.org/wiki/PID_controller
• 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
Ex.2: inverted
pendulum
https://github.com/tsybulkin/pend
Dynamics
α
mg
Phase portret
0
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
Robots dynamics and control
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
LQR
K: [[ 10.09901951 2.78076962]]
eigen vals: [-2.72150939+0.j -5.62079946+0.j]
Robots dynamics and control
Ex.3: 

The CubliETH Zurich
Dynamics
R
a
md
mb
2L
https://github.com/tsybulkin/cuba
Optimal control by ML
• Agent explores different strategies by trial and error
method
• Optimal policy: maps states —> actions,
maximizing total reward
Policy (control) learning
S1
S2
S3
u1
u2
u3
Robots dynamics and control
Ex.4: hopper
MIT hopper, 1986
α
𝜷
𝑥
𝑥
m
m
𝑥
2-link hopper
https://github.com/tsybulkin/hop2
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 —> …
Lagrange method
α
𝜷
𝑥𝑦
𝑥𝑦1
m1g
m2g
𝑥𝑦2
Lagrange equation
Hybrid dynamics
touches the
ground
does not
touch the
ground
Transition between states can be tricky

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Robots dynamics and control