Journal of Robotics

Volume 2015 (2015), Article ID 919073, 15 pages

http://dx.doi.org/10.1155/2015/919073

## Bioinspired Tracking Control of High Speed Nonholonomic Ground Vehicles

Center for Dynamic Systems Modeling and Control, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, USA

Received 5 January 2015; Revised 19 May 2015; Accepted 24 June 2015

Academic Editor: Maki K. Habib

Copyright © 2015 Adam Shoemaker and Alexander Leonessa. 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

The behavior of nature’s predators is considered for designing a high speed tracking controller for nonholonomic vehicles, whose dynamics are represented using a unicycle model. To ensure that the vehicle behaves intuitively and mimics the biologically inspired predator-prey interaction, saturation constraints based on Ackermann steering kinematics are added. A new strategy for mapping commands back into a viable envelope is introduced, and the restrictions are accounted for using Lyapunov stability criteria. Following verification of the saturation constraints, the proposed algorithm was implemented on a testing platform. Stable trajectories of up to 9 m/s were achieved. The results presented show that the algorithm demonstrates significant promise in high speed trajectory tracking with obstacle avoidance.

#### 1. Introduction

With the development of driverless technologies, there is an increasing demand for high speed systems capable of dealing with unstructured environments. Many commonly used obstacle avoidance methods focus on generating a path and forcing the vehicle directly to the desired trajectory [1–7]. Unfortunately, in many cases, such as potential field methods, the desired route is rarely smooth and does not sufficiently account for the desired velocity or vehicle capability [3, 8–13].

To find a more intuitive behavior, we look to biology for inspiration, specifically a generic predator-prey interaction. A cheetah chasing its prey, for example, does not directly mimic the path of its target. Instead, it creates a relatively smooth path, although its target may be moving somewhat chaotically [14]. The smooth path, in turn, allows the cheetah to maintain much of its high speed and still meet the desired goal [14]. As will be shown, the proposed algorithm mimics this behavior by introducing a reference system for the vehicle to follow, analogous to how a predator chases its prey.

Due to the high speed nature of our goal, this study focuses on Ackermann steering based platforms. Because of its simplicity and ability to capture nonholonomic constraints, we begin by considering a unicycle model, which will be shown to fit within the confines of an Ackermann platform by adding saturation constraints. While there exists extensive research in the control of unicycle type robotic systems [15–21], the study begins with the framework laid out in [21, 22] as it introduces a relationship similar to the predator-prey interaction. In particular, when considering the path of the cheetah, its trajectory can be thought of as a filter to the prey’s path; this is possible due to the cheetah maintaining a certain following distance from its prey. The larger the distance is, the more filtered the path becomes. In [22], the authors provide a control algorithm that mimics this behavior. By controlling velocity and angular rate, an algorithm is designed in [22] which theoretically tracks a virtual target at a time-varying distance. However, while capturing certain constraints, the unicycle model, and by consequence the work of [22], are not comprehensive. Much as the cheetah is unable to immediately turn independently of speed, neither are many ground vehicles. This paper focuses on extending preceding work to better satisfy the desired bioinspired model.

This work is organized as follows. In Sections 2 and 3 we introduce the algorithm laid out in [22] along with additional restrictions placed on the system structure and following distance parameters, which are aimed at bolstering controller reliability. Moreover, we evaluate several limitations of the base algorithm in the context of the biologically inspired predator-prey relationship and high speed ground vehicles. In Section 4, we introduce constraints on angular rate and velocity commands, which are based on Ackermann steering platform kinematics. In this section, we develop a new strategy for mapping commands into a viable control space that reflects these kinematic restraints. We additionally provide constraints on the following distance parameters to ensure stability. The algorithm is shown to satisfy the Lyapunov stability criteria, while still meeting all of these requirements.

The proposed algorithm was ultimately implemented on a testing platform, the results of which are presented in Section 5. To satisfy high-level commands, low-level PI controllers were used to control commanded velocity and angular rate. The reference system was manipulated using a basic potential field methodology of strictly attractors and repulsors similar to that presented in [23–25]. Using an Extended Kalman Filter to provide reliable state information, the full algorithm was used to achieve stable trajectories up to 9 m/s, the top speed of the platform. Further work will focus on accounting for actuator dynamics and extending the method to multiple platforms.

#### 2. System Definition

We begin with a unicycle model as it adequately captures the nonholonomic constraints of our Ackermann steering platform. Moreover, the simplicity of the model directly lends itself to controller design. While typically used for differential drive vehicles, we note that this relationship is not exclusive, and by developing saturation constraints, we can force the unicycle model to operate in the confines of an Ackermann steering platform. The unicycle model used is as follows,where is the vehicle’s position, is the longitudinal velocity, is the heading, and is the vehicle’s angular velocity. A reference system is then introduced to represent a virtual target,where is the reference system’s position. In the interest of continuous controller commands, we must guarantee that the reference system is composed of class functions in time. For convenience, the linear velocity of the reference system is defined aswhere

In the following section, we will see that the controller design will guarantee the system described by (1) and (2) will converge to a sufficiently small neighborhood around the reference system, described by (3), such that , as , where is a user defined nominal following distance. By guaranteeing that the vehicle follows the reference system in this manner, we enforce our bioinspired predator-prey interaction mentioned previously.

#### 3. Control Design

The overall controller is presented in two distinct steps. The first step involves the basic design, while the second step includes saturation algorithms based on Ackermann steering kinematics. In this first step, we assume that the reference system can be controlled to follow a desired trajectory. By ensuring that the actual system converges to a small neighborhood of the reference system, we ensure that the system follows the desired trajectory as well. At the same time, a sufficiently large separation between reference system and vehicle allows a buffer for the vehicle to track the reference, without violating its nonholonomic constraints.

In order to properly design the following distance, it is necessary to define the error between the vehicle and reference system. We start by assuming that the origins of the body fixed coordinate frame, , and of the global inertial coordinate frame, , coincide with the center of mass of the vehicle in the horizontal plane. An orthonormal transformation from to is then defined,Using this transformation along with (1) and (3), the position error can be expressed in the body coordinate frame aswhere . This position error can be thought of as a vector expression of longitudinal and lateral error in the body frame. Next we introduce the commanded distance vector,where is the commanded following distance. By guaranteeing that as , the longitudinal error will converge to while the lateral error goes to zero.

To provide this behavior, we introduce the following control law ([22]),whereand and are scalar tuning constants used to provide bounded signals to the commanded longitudinal and angular velocities, respectively. This control law, which is more thoroughly derived in [22], is based on Lyapunov stability theory. In [22], the authors show it to be the natural result that guarantees a negative definite Lyapunov time derivative, given the Lyapunov function (12) introduced in the following theorem.

In order to ensure that is defined for all , we must guarantee that for all . In practice, if , the vehicle will follow the reference system but in reverse, such that . As such, we restrict the commanded following distance to . For design purposes, a more rigorous constraint of , for all , is enforced, where are tuning parameters used to place a minimum bound on . The commanded following distance, , is designed such that as , provided that as well.

At this point, we note the intuitive form of (8). For instance, from the term, we see that the reference velocity, projected along the vehicle’s longitudinal axis, positively contributes to the commanded velocity, . Likewise the reference velocity, projected along the lateral axis, contributes to the commanded angular rate, . In examining the term, we see that an increasing distance negatively impacts the commanded velocity, as would be expected. Lastly, with regard to the term, we see that the difference between longitudinal error and the commanded distance, , contributes positively to the commanded velocity. Along the same lines, lateral error contributes positively to the angular rate command.

With regard to the term of (8), which is a componentwise operation, we note that while it is unnecessary for theoretical stability, it adds a layer of tuning to the controller. This layer proves particularly useful in experimentation. The function is chosen for its natural saturation capability. Based on the constants and , the function directly contributes bounded signals to commanded longitudinal and angular velocities, which are based on the difference between the tracking error and distance vector .

Theorem 1. *Consider the system described by (1) and (2), the reference system described by (3), and the feedback controller described by (8). If the nominal distance is restricted such that , for all , and the distance is updated according to**where**with tuning constant , then the distance , between the vehicle and reference system, converges to the desired distance, , while guaranteeing that . Meanwhile, the tracking error converges to the distance vector .*

*Proof. *Consider the Lyapunov function candidatewhereIn order to examine the derivative of the Lyapunov function along the system trajectories, the time derivative of the tracking error, , given by (6), is computed as follows,The time derivative of the orthonormal transformation, , is given bywhereBy substituting (1) and (15) into (14), the error dynamics can be simplified toBy substituting the control law given in (8), the error dynamics are then further simplified toThe error dynamics given by (18) are then substituted into the time derivative of the Lyapunov function,If we consider the case in which , we then substitute (10), (11), and (18) into (19) to obtainFrom (20), it is trivial to show that again for all , .

The Lyapunov time derivative is again examined when . Again, we substitute (10), (11), and (18) into (19) to obtainThe commanded following distance, , is restricted such that by design. This behavior is the result of as . As decreases below but still remains larger than , the additional corrective term pushes back toward , eventually guaranteeing as . As such, the corrective term for . Additionally, since , we find that . Since the remainder of the Lyapunov time derivative terms are the same as given in (20), we can conclude that for all .

##### 3.1. Simulation Results

The behavior of the control algorithm is examined prior to the introduction of saturation constraints on commanded longitudinal and angular velocities. Before simulation, it is necessary to choose a nominal following distance. By choosingfor and , we enforce a behavior that tends toward the bioinspired predator-prey model discussed earlier. For example, with higher reference velocities, we intuitively allow a greater distance to respond to instantaneous changes in reference direction. Likewise, (22) draws the vehicle closer to tightly follow slow trajectories. Analogously, large separation allows for a cheetah to easily maneuver while maintaining high speed, whereas a small separation lends itself to tighter tracking.

To evaluate the performance of the controller, we consider the situation in which the reference system is defined as two decoupled first-order systems,where and are the desired reference trajectories in and , respectively. Figures 1–3 show the results of simulation using the parameters given in Table 1.