Mathematical Problems in Engineering

Volume 2015, Article ID 105357, 11 pages

http://dx.doi.org/10.1155/2015/105357

## Finding the Quickest Straight-Line Trajectory for a Three-Wheeled Omnidirectional Robot under Input Voltage Constraints

^{1}1st R&D Institute-3, Agency for Defense Development, 111 Sunam-dong, Yuseong-gu, Daejeon 305-600, Republic of Korea^{2}Department of Computer Science and Engineering, Kwangwoon University, 20 Kwangwoon-ro, Nowon-gu, Seoul 139-701, Republic of Korea^{3}Department of Computer Engineering, College of Information Technology, Gachon University, 1342 Seongnamdaero, Sujeong-gu, Seongnam-si, Gyeonggi-do 461-701, Republic of Korea

Received 27 October 2014; Revised 31 December 2014; Accepted 8 January 2015

Academic Editor: Javier Moreno-Valenzuela

Copyright © 2015 Ki Bum Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We provide an analytical solution to the problem of generating the quickest straight-line trajectory for a three-wheeled omnidirectional mobile robot, under the practical constraint of limited voltage. Applying the maximum principle to the geometric properties of the input constraints, we find that an optimal input vector of motor voltages has at least one extreme value when the orientation of the robot is fixed and two extreme values when rotation is allowed. We can find an explicit representation of the optimal vector for a motion under fixed orientation. We derive several properties of quickest straight-line trajectories and verify them through simulation. We show that the quickest trajectory when rotation is allowed is always faster than the quickest with fixed orientation.

#### 1. Introduction

Currently used in many applications, mobile robots can be characterized by the motions of which they are capable. Unlike nonholonomic differential drive mobile robots [1–3], holonomic mobile robots such as three-wheeled omnidirectional mobile robots (TOMRs) can perform independent translational and rotational motion from any starting configuration [4].

The dynamics of omnidirectional mobile robots have been widely studied, as a prerequisite for effective controller design. Dynamic models of the orthogonal type of omnidirectional mobile robot were developed by [5, 6]. A model in which simplified geometric relationships determine the robot’s position has since been introduced [7]. The effect of slipping is taken into account in a recent model [8], a simple model of the friction between the robot’s wheels and the floor. The general dynamic characteristics of -wheeled omnidirectional mobile robots with control redundancy have also been studied [9]. A fuller model of TOMR dynamics, including motor parameters has recently been introduced in the global coordinate frame [10].

In this paper, we address the time-optimality of TOMR motions. However, it is worth remembering that other characteristics of a motion, such as energy consumption, safety, simplicity of planning, and accuracy, may also be of importance. Nevertheless Balkcom et al. [11] note that time-optimality is a fundamental characteristic of a robot.

Most research on time-optimality of robot has focused on velocity and acceleration constraints [12, 13]. The rapidity of a motion may also be limited by torque or curvature constraints [14, 15]. However, since most mobile robots are battery powered, constraints on voltage may be the most crucial [16].

Many researchers have tried to minimize the motion times of two-wheeled differential-drive mobile robots [12–19]. Recently, a few researchers have made progress with the same problem for TOMRs. The near-optimal strategy described by Kalmár-Nagy et al. [7] is intended to be computationally efficient, making it suitable for real-time applications; however, the input voltage constraint is not fully considered, and the underlying dynamic model omits the Coriolis terms. A different approach to optimizing TOMR times, using a genetic algorithm [20] and nonlinear programming, was suggested by Fu et al. [21]; however, it is computationally intensive, and its numerical nature does not capture the general properties of TOMR motion. Balkcom et al. [11] classified trajectories designed to minimize motion times in terms of spin, circular arc, and tangential motion. They proposed fast trajectories based on “spin,” “roll,” “shuffle,” and “tangent” motions. They did not determine which trajectory is optimal for specific start and end configurations.

The quickest trajectory for a TOMR is not usually a straight line. However, in practice, a path is often expressed as a combination of simple paths, such as straight lines or clothoid arcs. Thus, it is worthwhile to look at straight-line trajectories.

In this paper, using the full dynamics of a TOMR under input voltage constraints in global coordinates, we show how to find the quickest straight-line trajectory. We also consider motions with and without rotation of the robot. When rotation is allowed, we do not specify the final heading of the TOMR. If we were to specify its final heading, we would be faced with a multiobjective optimization: we leave that for future work. We address the problem, with or without rotation, using conventional optimization theory.

The remainder of this paper is organized as follows: in Section 2, we review the dynamics of a TOMR based on the analysis of [10]. In Sections 3 and 4, respectively, we formulate and analyze the problem of minimizing the time required for a straight-line motion without and with rotation. Simulation results are presented in Section 5 and we draw some conclusions in Section 6.

#### 2. Dynamics of Three-Wheeled Omnidirectional Mobile Robots

The TOMR considered in this paper consists of a cylindrical body mounted on three omnidirectional wheels, each driven by its own DC electric motor, located at equal intervals. We assume that the mass of the robot around the circumference of the robot’s bone is uniformly distributed about its center and that the wheels are of the (orthogonal) type [4], meaning that each wheel can roll freely perpendicular to the direction in which it is driven by its axle. This idealized TOMR is depicted schematically in Figure 1, where is the mass of the robot, is the inertia of robot rotation, is the radius of wheel, is the distance between the center of robot and one wheel, and is the force acting on the tangential direction in each wheel [10].