The problem of boundary following by a unicycle-like robot with rigidly mounted sensors

Abstract

We consider the problem of reactively navigating an unmanned Dubins-like robot along an equidistant curve of an environmental object based on the distance to its boundary measured perpendicularly to the robot centerline and the angle of incidence of this perpendicular to the boundary. Such a situation holds if e.g., the measurements are supplied by range sensors rigidly mounted to the vehicle body at nearly right angles, or by a single sensor scanning a nearly perpendicular narrow sector. A sliding mode control law is proposed that drives the robot at a pre-specified distance from the boundary and maintains this distance afterwards. This is achieved without estimation of the boundary curvature and holds for boundaries with both convexities and concavities. Mathematically rigorous analysis of the proposed control law is provided, including an explicit account for the global geometry of the boundary. Computer simulations and experiments with real wheeled robots confirm the applicability and performance of the proposed guidance approach.

Introduction

Safe navigation of a mobile robot along the boundary of an environmental object is a fundamental problem in unmanned vehicle research. This navigation task is of immediate interest for border patrolling and structure inspection [1], bottom following by autonomous underwater vehicles [2], lane following by autonomous road vehicles [3] and other missions, where traveling along the boundary at a pre-specified distance from it is an essential requirement. Furthermore, temporarily following the boundary of an obstacle is a standard method employed by many obstacle avoidance algorithms.

Many approaches to solving this problem have been proposed in the literature. They can be broadly classified based on the type and amount of the required sensor data. If full information about the surrounding area is available, the general approach is to navigate the vehicle towards the edge of the observable part of the obstacle, as is exemplified in TangentBug-like algorithms [4], [5]. There are several modifications of this method supporting different degrees of model and sensor sophistication. For example in [6], the robot picks a probational target point based on a series of obstacle detections. Many path planning and model predictive control algorithms also fall in this category. They generate a probational path from the obstacle representation based on the sensor data at each time step and under appropriate circumstances, do ensure boundary following; see e.g. [7], [8]. A combination of model predictive control with sampled obstacle information has also been proposed [9]. However many of these methods do not care about maintaining a given distance to the boundary. Furthermore, these methods typically require relatively bulky multi-directional range sensors and extensive data pre-processing, which may not be suitable for e.g., miniature robots. The focus on real-time implementation with minimization of consumed sensing and computational resources have motivated intensive interest to reactive controllers for which the current control is a simple reflex-like response to the current observation.

Reactive path following has an extensive literature. However many of the proposed feedback control laws are fed by an input variable whose computation may require wide-aperture sensing and intensive pre-processing. For example, reactive (i.e., mapless) vision navigation, for which excellent surveys are available in [10], [11], employs the capability of the visual sensor to capture and memorize a whole chunk of the environment and is typically based on extraction of certain image features and estimation of their motion within a sequence of images, which requires intensive image processing. Some other examples of perceptually and computationally demanding input variables not confined to the area of visual navigation include the closest point on the obstacle boundary, the distance to this point, or the value of another function determined by the entire boundary; see e.g. [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24] for representative samples. Another such variable employed by many proposed controllers, see e.g., [22], [23], [25], is the boundary curvature, which is particularly sensitive to corruption by measurement noises since this is a second derivative property.

In this paper, we consider the problem of boundary following with a pre-specified margin by a non-holonomic planar vehicle based on only the distance along and the reflection angle of the ray perpendicular to the vehicle centerline. Such a situation holds if e.g., the measurements are supplied by several range sensors rigidly mounted to the vehicle body at nearly right angles from its centerline or by a single sensor scanning a nearly perpendicular narrow sector. This perception scheme is used in some applications to reduce the complexity, cost, weight, and energy consumption of the sensor system and to minify detrimental effects of mechanical external disturbances on the measurements. However the related deficit of sensor data makes most of known navigation solutions, along with their analysis, inapplicable in this case. Moreover, this gives rise to special challenges, like the inability to detect a threat of head-on collision in certain situations (see Fig. 3) or strong sensitivity of the overall output of the sensor system to the vehicle posture.

To the best knowledge of the authors, some approaches to the above problem have been proposed in [25], [26]. However, the control law from [26] is aimed at pure obstacle avoidance, with no objective to follow the boundary with a pre-specified margin. Boundary following with a given margin was addressed in [25] via a hybrid strategy of switching between Lyapunov-based highly nonlinear control laws in order to overcome singularities caused by concavities of the tracked curve. In doing so, restrictions on the steering control were neglected by assuming that the vehicle is capable of making arbitrarily sharp turns, and estimates of the boundary curvature were essentially employed. Theoretical analysis ignored the issue of possible front-end collisions with the boundary during transients due to deficiencies of side-view sensors (see Fig. 3), as well as jumps of sensor readings (up to loss of view at the obstacle) possible when the rotating ‘ray of view’ of the sensor first becomes tangential to a convex part of the boundary and then loses touch with it (or vice versa).

This paper offers a mathematically rigorous non-local convergence result for a control saturated and under-actuated non-holonomic Dubins-car type vehicle. It travels forward at a constant speed over planar curves of bounded curvatures and is controlled by upper limited angular velocity. Restrictions on the vehicle turning radius essentially complicate the controller design by, e.g., giving birth to states from which future collision with the boundary is unavoidable. We propose a novel sliding-mode navigation strategy that does not employ curvature estimates and homogeneously handles both concavities and convexities of the followed boundary, as well as transitions between them. This strategy asymptotically steers the vehicle to the pre-specified distance to the boundary and afterwards ensures stable maintenance of this distance. Mathematically rigorous justification of non-local convergence of the proposed strategy is offered. In doing so, possible abrupt jumps of the sensor readings are taken into account. Furthermore, much attention is given to revealing requirements to the global geometry of the boundary that make it possible to avoid front-end collisions with it based on only side-view sensors, thus making extra front-view sensors superfluous. The convergence and performance of the proposed navigation and guidance law are confirmed by computer simulations and real world tests with a Pioneer P3-DX robot, equipped with a SICK LMS-200 Lidar sensor.

The remainder of the paper is organized as follows. Section 2 offers the problem setup, whereas Section 3 discusses assumptions and tuning of the controller parameters. Section 4 presents the main theoretical results, whose proofs are given in Section 8. Sections 5 Simulation tests, 6 Experiments with a real robot are devoted to the results of computer simulations and experiments with a real wheeled robot, respectively. Section 7 offers brief conclusions.