1. Control Design and Analysis of Chained Systems
Application to path following with mobile robots.
Claude Samson

2. Overview Motivation for using smooth time varying feedback
Conversion to Chain form
Conversion to Skew Symmetric form
Implementing two propositions to show:
Application and Simulation
X2 = 0
X = 0 (convergence to a designated path)

3. Remarks about Car-like vehicles
Differentiable or continuous, pure-state feedback cannot be used to stabilize such systems.
Fast (exponential) convergence is not compatible with smooth time varying feedback . The integral
must diverge so that can asymptotically converge to zero.
Other methods of control include sinusoidal, non-smooth, and using other canonical forms. i.e.. “Power Forms”

4. Chain Form Systems
Any nonholonomic system with two inputs and n  4 states can be converted to chain form.
Example:
Remark: This system can be made into a single input, linear system. u1 becomes a function of time.

5. Skew Symmetric Chain Form
Skew symmetric form allows for easy analysis and later helps design optimal control laws.
It is a linear change of coordinates in

6. The new chained system thus becomes:
where

7. Proposition 1: Global Stability of X2 (Z2)
Choose u1 such that and are bounded and does not asymptotically tend to zero
Let w2 be of the form:
To show asymptotic convergence of Z2, choose a positive function V(Z2) such that:
tends to zero.

8. Proposition 2: Stabilizing z1
u1 will become time dependent and will dictate the choice for w2.
u1 must take the form of
The uniformly bounded h function is known as the “heat function”.
Partial derivatives are bounded and h(0,t)=0

9. Application to a Car with n-trailers
Let n = 4 P2 P3 L0 L1 L2 L3 P1 P0 P4

10. Let r be the radius of the front car’s steering wheel
Let be the angular velocity of the wheel about the horizontal axis.
The system’s control variable will then be and
From the position differential Eq.s, the system in terms of individual car velocity and relative angles becomes

11. Path Following
The final set of Eq.s comes from the first car’s motion with respect to the path.
We need the curvature and the distance from P0 to a point P0’ on the curve such that:
Let Then when the first car is following the path, becomes zero.

12. y P0 P1 P0 P1
rmin

13. The equations for the first car become:
The state variables can be
expressed as:

14. Chain form:
The first term on the right side was added to accommodate the choice of u 1

15. Skew Symmetric form:

16. In Search of u2 To find an appropriate feedback law use time scaling.
Since x1 changes monotonically with time, differentiate with respect to x1.
The other state variables can be rewritten in the form:
Then u2 becomes:

17. Applying Proposition 1 Let
Since and are bounded, by proposition 1, v0V asymptotically converges to zero, which implies Z2 is asymptotically stable.

18. Applying Proposition 2 Consider the time varying control
hs is function mapping R into a bounded interval on R. hs(0) = 0 and
When complemented with the control law w2, the entire system (Z) is stabilized at the origin.

19. Simulation with a Circular Path
red = , green = y, blue = s
red = , green = , blue =
Remarks: distance y oscillates over time, but remains less than rmin. As desired, tends to zero as does

20. The End
MR T.
LYAPUNOV
VS.