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

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)

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”

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.

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

The new chained system thus becomes: where

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.

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

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

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

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.

y P0 P1 P0 P1 rmin

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

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

Skew Symmetric form:

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:

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

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.

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

The End MR T. LYAPUNOV VS.