Dawn M. Tilbury

Stabilization of trajectories for systems with nonholonomic constraints

A new technique for stabilizing nonholonomic systems to trajectories is presented. It is well known that such systems cannot be stabilized to a point using smooth static-state feedback. In this note, we suggest the use of control laws for stabilizing a system about a trajectory, instead of a point. Given a nonlinear system and a desired (nominal) feasible trajectory, the note gives an explicit control law which will locally exponentially stabilize the system to the desired trajectory. The theory is applied to several examples, including a car-like robot. >

Steering three-input nonholonomic systems: the fire truck example

In this article we steer wheeled nonholonomic systems that can be represented in a so-called chained form. Sufficient condi tions for converting a multiple-input system with nonholonomic velocity constraints into a multiple-chain, single-generator chained form via state feedback and a coordinate transfor mation are presented along with sinusoidal and polynomial control algorithms to steer such systems. Our example is the three-input nonholonomic system of a fire truck, or tiller truck. In this three-axle system, the control inputs are the steering velocities of both the first and third (or tiller) axles and the forward driving velocity of the truck. Simulation results are given for parallel parking, left hand turning, right hand turn ing, and changing lanes. Comparison is made to the same vehicle without tiller steering.

Trajectory generation for the N-trailer problem using Goursat normal form

Shows how the machinery of exterior differential systems can be used to help solve nonholonomic motion planning problems. Since the Goursat normal form, for exterior differential systems is dual to chained form for vector fields, the authors solve the problem of steering a mobile robot with N trailers by converting the system into chained form, doing the path-planning in the chained form coordinates, and converting the path back into the original coordinates. Simulations of the N-trailer system parallel parking and backing into a loading dock are included.<<ETX>>

A multisteering trailer system: conversion into chained form using dynamic feedback

This paper examines the kinematic model of an autonomous mobile robot system consisting of a chain of steerable cars and passive trailers, linked together with rigid bars. The state space and kinematic equations of the system are defined, and it is shown how these kinematic equations may be converted into a multiinput chained form. The advantages of the chained form are that many methods are available for the open-loop steering of such systems as well as for point-stabilization; some of these methods are discussed here. Dynamic state feedback is used to convert the system to this multiinput chained form. It is shown how the dynamic state feedback that is used in this paper corresponds to adding, in front of the steerable cars, a chain of virtual axles which diverges from the original chain of trailers. Two different example systems are also presented, along with simulation results for a parallel-parking maneuver.

Steering car-like systems with trailers using sinusoids

Methods for steering car-like robots with trailers are investigated. A connection is demonstrated between Murray and Sastrys (1990, 1991) work of steering with integrally related sinusoids and Sussmann and Lius (1991) recent work on asymptotic behavior of systems with high-frequency sinusoids as inputs. The merits of coordinate transformations, relative to the convergence properties, are discussed. Simulation results for a car-like robot with two trailers are presented.<<ETX>>

A multi-steering trailer system: Conversion into chained form using dynamic feedback

In this paper, we examine in detail the kinematic model of an autonomous mobile robot system consisting of a chain of steerable cars and passive trailers, connected together with rigid bars. We define the state space and kinematic equations of the system, modeling the pair of wheels on each axle as able to roll but not slip. we then investigate how this system of kinematic equations may be converted into multi-input chained form. The advantages of the chained form are that many methods are available for the open-loop steering of such systems as well as for point-stabilization. In order to convert the system to this multi-input chained form, we use dynamic state feedback. We draw some motivation from the very simple example of a kinematic unicycle and the relationships of the angular velocities therein, and we show how the dynamic state feedback that we use corresponds to adding, in front of the steerable cars, a chain of virtual axles which diverges from the original chain of trailers. We briefly discuss how some of the methods which have been proposed for steering and stabilizing two-input chained form systems can be generalized to multi-chained systems. for concreteness, we also present two different example systems: a fire truck (three axles) and a five-axle, two-steering system. Simulation results for a parallel-parking maneuver for the five-axle system are included in the form of margin movies.

On Goursat normal forms, prolongations, and control systems

In this paper, the method of exterior differential systems for analyzing nonlinear systems is presented. Conditions are given for converting Pfaffian systems into Goursat normal form, and for converting control systems into Brunovsky form. All of the existing results on feedback linearization for control systems can be restated in the language of Pfaffian systems, and in addition, new conditions for linearizing control systems using dynamic extension are given.<<ETX>>

Extended Goursat normal forms with applications to nonholonomic motion planning

The theme of this paper is the generalization of Goursat normal forms for Pfaffian systems with co-dimension greater than two. There are necessary and sufficient conditions for the existence of coordinates that transform a Pfaffian system with co-dimension greater than two into an extended Goursat normal form, which is the dual of the multiple-chain, single-generator chained form mentioned in our earlier work (1993). In this paper, we concentrate on how to find such coordinate transformations for multisteering, multitrailer mobile robot systems so that we can use available steering and stabilization algorithms for nonholonomic motion planning. We present a methodology for constructing a coordinate transformation and apply it to the example of a five-axle, two-steering mobile robot.<<ETX>>