Experiments of Fuzzy Real-Time Path Planning for

Proceedings of the 2002 IEEE International Conference on Robotics & Automation Washington, DC • May 2002

Experiments of Fuzzy Real-Time Path Planning for Unicycle-Like Mobile Robots Under Kinematic Constraints Gianluca Antonelli

Stefano Chiaverini

Giuseppe Fusco

Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale Universith degli Studi di Cassino via G. Di Biasio 43, 03043 Cassino (FR), Italy {antonelli, chiaverini, fusco } ~unicas. it

Abstract This paper presents an experimental study concerning the application of a real-time motion planning algorithm to an unicycle-like mobile robot. The desired trajectory to be followed by the mobile robot in presence of bounds on the linear/angular velocities and accelerations is available on-line. Moreover, in the respect of the kinematic constraints, the desired path has to be kept as long as possible. The implemented algorithm is based on a discrete-time kinematic control which implement a warping of the time law based on the definition o / a virtual time. In addition, a fuzzy inference system handles the additional information given by the difference between the virtual and real time in order to exploit knowledge in advance of the desired path. The experimental results confirm the effectiveness o/ the adopted algorithm.

1

Introduction

In several applications the problem of steering a mobile robot with unicycle-like kinematics arise. The vehicle is usually subject to kinematic constraints such as bounded linear and angular velocities and accelerations; moreover path tracking capabilities with a the desired trajectory available in real-time are often required. References [1, 3, 4] deal with the motion control problem for the end effector of a robot whose trajectory is given in real-time; reference [3], however, does not take into account acceleration constraints that are included in references [1, 4].

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In a previous paper [2] a motion planning algorithm for an unicycle-like mobile robot in presence of velocity and acceleration constraints has been presented. This algorithm satisfies the kinematic constraints, given in terms of linear and angular maximum velocities, and dynamic constraints, often given in terms of maximum linear and angular acceleration e.g., as in [8], while keeping the desired path. The algorithm is based on a discrete time kinematic control which implements a warping of the time law that makes use of a virtual time. Moreover, a fuzzy inference system handles the additional information given by the difference between virtual and real time when the virtual time is slowed down. After briefly recalling the main steps of the algorithm, this paper presents an experimental study concerning the application of the considered motion planning algorithm to the Magellan Pro mobile robot manufactured by Real World Interface [7]. The experimental results show the behavior of the considered robot and validate the effectiveness of the adopted procedure.

2

Problem formulation

Let define as Pd(t) lbd (t)

(1) =

dPd (t) dt

(2)

the desired trajectory in 2 dimensions and its time derivative for a unicycle-like mobile robot.

2147

The problem of kinematic control under the following constraints is considered: •

• • • • •

on-line availability of the desired trajectory; linear positive velocity limited in norm by Vmax; linear acceleration limited in norm by i)max; angular velocity limited in norm by Wm~x; angular acceleration limited in norm by (J2max; path tracking requirement.

The kinematics of an unicycle mobile robot is given by: {i

-

vc°s(0) v sin(0)

where x and y are the cartesian coordinates of the point p representing the vehicle position and ~ is its orientation with respect to an inertial frame. The outputs of the motion planning algorithm are the linear and angular velocities v, w to be fed to the low level dynamic motion control. Due to saturation of the actuators, those are limited with their time derivatives.

Overview

of the

Tk -- Tk--1 q- A T k ,

ATk Pd'k--Pd(Tk-1)

~d(t) --Pd(T(t)),

At '

(6)

It is worth noting that the actual desired velocity is decreased with respect to the planned one when ATk is lower than At.

3.1

Scaling

technique

The scaling technique is based on 3 steps denoted with the subscripts A, B and C. The output of this stage are the velocities vc and wc send to the vehicle low level controller. Let first compute the following vector:

algorithm

PA(tk) -- l~d(Tk) -+- Kp (pd('rk_l) - - P ( t k - 1 ) ) .

The adopted algorithm is deeply described in [2]. In the following, an overview of the algorithm will be presented focusing our attention on the implementation issues and the experimental results. The basic idea of the algorithm is to slow down the desired trajectory, by the introduction of a virtual time, when velocity/acceleration constraints are encountered. Moreover, satisfaction of the path constraint is also of concern. When the virtual time is smaller than the real time it is possible to exploit the forward trajectory knowledge given by the difference between real and virtual time giving some predictive actions. Those actions are output by a Fuzzy Inference System (FIS). It is worth noticing that in real applications, such as vehicle motion control, a forward knowledge of the trajectory is always available to correctly drive the vehicle. Let introduce a virtual time T via a time warp in the trajectory generation, i.e.,

(5)

where the increment ATk is computed at each step. Under the assumption that at each step the increment of the desired trajectory Pd iS a linear function of the time increment the following relation holds:

(3)

02

3

The algorithm developed is implemented in discrete time form and generates the time warp on line. To this aim, the value of the virtual time at the k-th instant, Tk, is recursively calculated as

(7)

From this vector it is possible to extract the value of v and w to be fed to the dynamic control (from now on the index k will be dropped out for notation compactness). Notice that, in absence of kinematic bounds, this equation guarantees satisfaction of the path constraint. However, v and w, with their time derivatives, may result larger than their allowed values. If v, ~), w and & are inside their allowed range, the values of v and w can be found as: VA WA

= =

IIPA(tk)lJ

(8)

k~[Zp A(tk) - O(tk-1)]

(9)

where k~ is a control gain. Let first check the linear and angular velocities. The following values for v and w can be computed: VB

(4)

where T(t) is an increasing function such that T(0) = 0 and T(t) ~_ t. The latter condition is imposed by the constraint of on-line availability of the desired trajectory.

2148

-

f Vmax

if VA > Vmax

[

if VA ~_ Vmax

VA

-

(10) ~at

(11)

sign(wA) "Sdma x

(Zs

-

(12) (WB -- w(t~_,)) / A t

(13)

The second step is aimed at satisfying the linear and angular accelerations constraints. This is done as follows: _

VC

if 10~1 > (14)

=

v ( t k - x ) + 'VC " A t

(15)

_

[

if I -I >

[

if I BI _