Nonlinear

The focus of this paper is to solve the tracking control problem for a 2-DOF underactuated mechanism via nonlinear

In the present paper, we address the output tracking control problem in

For nonlinear mechanical systems, tracking control problem is known to be more difficult than stabilization mainly for underactuated systems whose initial conditions are close to an unstable equilibrium point. The central problem in nonminimum-phase underactuated systems, solved here, is the specification and design of output feedback inner-tracking controllers to drive the output (joint position) to a nontrivial reference trajectory in spite of external disturbances.

The prior work on the tracking control of nonminiminum-phase systems includes, among others, the results of Consolini and Tosques [

The method we use for defining a desired trajectory for underactuated system is based on the work of Berkemeier and Fearing [

The above problem is locally resolved within the framework of nonlinear

The aforementioned

In contrast to the standard

The paper is organized as follows. Background materials on time-varying

Consider a nonlinear system of the form

The functions

These assumptions are made for technical reasons. Assumption (A1) guarantees the well-posedness of the above dynamic system, while being enforced by integrable exogenous inputs. Assumption (A2) ensures that the origin is an equilibrium point of the nondriven (

A causal dynamic feedback compensator

Given a real number

The time-varying

Assumptions (A1)–(A3) allow one to linearize the corresponding Hamilton-Jacobi-Isaacs inequalities from [

There exists a bounded positive semidefinite symmetric solution of the equation

There exists a bounded positive semidefinite symmetric solution to the equation

According to the time-varying bounded real lemma [

Differential equations (

Let conditions (C1) and (C2) be satisfied, and let

In what follows, Theorem

Consider the equation of motion of an underactuated mechanical system given by the Lagrange equation

The

We point out that the present formulation is different from typical formulation of output tracking and regulation [

Plot of desired trajectories for Acrobot by selecting several values of

Profile of the frequency and amplitude of oscillations for several values of

Our objective is to design a controller of the form

the output to be controlled is given by

the joint position vector

To begin with, let us introduce the state deviation vector

Thus, the output feedback controller (

The controller performance was studied in simulation by applying the exposed ideas to the Acrobot, depicted in Figure

Schematic diagram of the acrobot where

The control goal was achieved by implementing the nonlinear

Phase portrait of the first joint trajectory and desired trajectory (

Time evolution of the output (

Time evolution of the determinants of the principal minors of the matrix

Time evolution of the determinants of the principal minors of the matrix

The output feedback Nonlinear

The equation motion of Acrobot, described by (

Parameter values for the Acrobot.

In this appendix we provide the computed matrix