Synthetic-analytic behavior-based control framework: Constraining velocity in tracking for nonholonomic wheeled mobile robots

Abstract

This work presents a genetic programming control design methodology that extends the traditional behavior–based control strategy towards a synthetic-analytic perspective. The proposed approach considers the internal and external dynamics of the system, providing solutions to a general structure, and including analytic functions, which can be studied within the Control Theory framework. The method is illustrated for the tracking control problem under bounded velocity restrictions of a nonholonomic wheeled mobile robot. A classic Control Theory (CT) based controller that solves the tracking problem (but not the velocity constraint requirement) is chosen from the literature; based on its stability properties, a modified structure where the search of suitable analytic basis behaviors, fulfilling both control objectives simultaneously, can be introduced. The proposed framework takes the form of a learning process based on Genetic Programming (GP) which generates a set of nonlinear tracking controllers satisfying pre-specified velocity bounds. A collection of 9113 suitable nonlinear solutions were obtained to augment the ground controller. Simulations and real–time experiments are performed to illustrate the effectiveness of the methodology through the testing of the models with the best performance, as well as those with lower structural complexity.

Introduction

The synthesis of nonlinear controllers where multiple objectives are pursued constitutes a significant challenge to the control research community. Here, the aim is to make a system meet additional constraints without compromising the achievement of a desired motion or behavior. While most state-of-the-art proposals addressing control problems can be classified in two main broad approaches (Control Theory and Soft Computing techniques), there are successful results where these are combined to take advantage of the strengths of each method, [7], [14], [26], [27], [31]. In this work, a methodology for the automated synthesis of nonlinear controllers through an evolutionary process is proposed. The objective is to solve the tracking control problem in nonholonomic wheeled mobile robots while fulfilling a bounded velocity policy.

Control of nonholonomic wheeled mobile robots is challenging since these mechanisms possess more degrees of freedom than available control variables. By definition, they have motion constraints related to their wheels configuration. In other words, this kind of robots is unable to move simultaneously and independently in any arbitrary direction in the horizontal plane and to instantaneously change their orientation (i.e., translational and rotational motion). Therefore control of nonholonomic mobile robots has gathered the attention from various research groups, as evidenced by the literature, [5], [9], [10], [11], [13], [15], [20], [21], [25], [28], [29].

From a practical point of view, kinematic control laws are the most common way to develop motion controllers for wheeled mobile robots. While many proposals from the state-of-the-art provide designs using torque or voltage as the control signal, most commercial and academic prototypes use velocity inputs instead. Such velocity inputs are either specified as velocities for each of the robot’s wheels, or as linear and angular velocities in the kinematic model. However, independently of which speeds are considered in the design, for safety purposes, it is necessary to comply with predefined boundaries in the velocities to avoid skidding and slipping effects in the robot’s motion [24].

To assume that the robot actuators can supply any demanded torque, voltage, or velocity simplifies the design stage of the controllers but, in practice, this supposition is not true. When the signal demanded by the controller exceeds the physical capacities of the actuators, they operate in saturated mode. Keeping the actuators working at the saturation limit decreases their lifespan since they work at their maximum capacity. To preserve the actuators’ integrity, the saturation mode should be avoided whenever it is possible. Then, it follows that, in general, it is necessary to derive controllers such that actuator constraints are considered in the design.

Saturation of the reachable velocities can be addressed by the specific use of saturation functions (such as the hyperbolic tangent or the sign) in combination with CT or soft computing methods to design the controllers. For example, from a CT approach, many works have been proposed to solve one or various of the control problems related to nonholonomic mobile robots (tracking, path following, stabilization, flocking, or formation [5]), take for example, [5], [10], [11], [20], [21], [28]. In [5], the tracking control problem subject to bounded velocity and torque is addressed using two first-order filters. The filters ensure that the torque and velocity constraints are satisfied producing uniformly continuous feedback signals. A backstepping method is used in [10] to solve the path following problem when the actuator velocities are limited. Notably, this strategy is directed at high-speed applications. Jiang et al. [11] present an adaptive controller, based on passivity and normalization, to achieve global stabilization and tracking for a mobile robot subject to input velocity constraints. Nonlinear Model Predictive Control (NMPC) is employed in [20] for trajectory tracking in nonholonomic mobile wheeled robots. NMPC can naturally deal with restrictions; then, the proposal achieves the tracking objective while restricting input velocities. Serrano et al. [28] propose a tracking controller for a wheeled mobile robot subject to saturation in the angular and linear speeds. The approach uses nonlinear programming methods to calculate the controller parameters. A generalized framework for several types of wheeled mobile robots to satisfy the path following requirement while keeping the velocities within acceptable bounds is presented in [21]. The authors use explicit expressions for the input velocities to meet the tracking objective.

Alternatively, Soft Computing techniques can be used to achieve an adequate performance of the robot while enforcing limits in the velocity signals. In [29], formation control is dealt with using adaptive Radial Based Function Neural Networks (RBFNN) to approximate the actuator’s saturation, where the saturated input is the torque. Additionally, the estimation and tracking errors are bounded by saturation functions. It should be noted that the saturation bounds are explicitly incorporated in the controller design. A Takagi–Sugeno fuzzy controller is proposed in [25] to solve the trajectory tracking problem of a unicycle mobile robot which is subject to velocity and actuator saturation. In their work, the saturation nonlinearity is approximated through a set of fuzzy rules, rather than by specific saturation functions. The authors demonstrate the stability of the closed-loop system employing Lyapunov functions. They present three experiments to demonstrate the approach: one positioning challenge, and two tracking experiments following a line and an eight shape curve.

In all of the works mentioned above, it is assumed that the control signal is either the angular or the linear velocity of the robot, with its corresponding boundary. This restriction means that this boundary must be calculated for the wheel configuration and the physical parameters of the actuators. This work, in particular, focuses on solving the control problem where constraints in the velocity of each of the robot’s wheels are considered.

Under the requirements stated above (kinematic restrictions and input velocity bounds), it becomes clear that a hybrid framework will be best to exploit the advantages of different approaches to solve such a complex task. Specifically, the tracking control problem can greatly benefit from the neural network framework of Evolutionary Robotics [8], where the aim is to endow the robot with natural language processing or deductive reasoning to generate behaviors that can solve complex tasks. A behavior can be defined as an independent action resulting from the direct interaction of the system with its environment. This concept was introduced to represent intelligence in artificial systems [3]. Furthermore, a system can exhibit complex behaviors when such activity producers are intertwined and executed in parallel. The works of Matarić et al. [17], [18] and the developments proposed by Arkin [1], led to the evolution of the concept of behaviors into an approach called Behavior–based control. This approach aims to solve control problems within the robotics field; it proposes the development of a process where a set of actions, or modules, named basis behaviors, are combined to achieve desired features of the system. This method was originally developed for situated robots that need to adapt to the dynamics of real-world environments without considering (a) the internal dynamics of the system, or (b) abstract representations of knowledge and reality [19].

Recently, an analytic behavior–based framework was proposed by Clemente et. al [6] for obstacle avoidance with bounded velocity for the position control problem in omnidirectional mobile robots. The proposed methodology takes advantage of the Potential Fields approach to derive a forced behavior and an evolutionary search for adaptive behaviors as the attractive and the repulsion functions. In [22], this methodology was extended to solve the tracking problem in a double integrator system; and in [23] for second-order dynamical systems. A traditional PD with inverse dynamics is proposed to generate the forced behavior, while the learned responses are searched by an evolutionary approach for a bounded flow variable. The present work extends and develops the ideas introduced by the authors in [6], [22], [23]. It aims to control a nonholonomic wheeled mobile robot under velocity constraints using a PI-type controller.

The analytic behavior–based framework has three significant features that distinguish it from traditional Behavior–based control. First, it defines the basic behaviors as analytic functions, which can also be nonlinear; second, it uses of a model of the system to include the internal dynamics; and third, it integrates a CT-based controller with a learning process applying GP to generate the learned behaviors. Moreover, the advantage of employing a CT-based controller is that it can be analyzed in terms of the control theory framework, thus guaranteeing the performance of the system; this is possible through the concept of stability of the equilibrium points. An equilibrium point is a coordinate of the state space such that, whenever the mobile robot starts at it, it will remain at that point for all future time. The stability of a system is proven through a qualitative analysis of the trajectories, or solution curves of the system. The most critical stability criterion within the CT approach is the Lyapunov stability theorem.

The integration of the CT method with the GP approach allows the implementation of a learning stage seeking fulfillment of additional features in the behavior of the robot. The GP technique allows for the construction of a syntactical tree to represent a solution given in the form of nonlinear controllers; such solutions are composed of mathematical operators and analytic functions derived from the CT approach.

In this work, we extend the application of the analytic behavior-based framework by addressing the tracking problem in nonholonomic mobile robots. Constrained velocities in the robot’s wheels are also considered. In contrast to previous works using this framework, this proposal takes advantage of the Lyapunov stability conditions to derive the learning stage in the mobile robot. A modified structure of a classical state-of-the-art tracking controller is introduced to seek behavior modifiers. Such modifiers simultaneously fulfill the convergence to the desired trajectory while keeping the velocities within some constant boundary value. Moreover, the GP approach guarantees that the speeds in each wheel will never reach the saturation bounds for several given scenarios. Its strength lies in the automation of the synthesis of nonlinear controllers, giving rise to a big set of solutions that can be studied by the CT approach.

The methodology presented in this work permits the automatic design of nonlinear controllers, which are in general hard to derive. This approach could be easily extended to the design of controllers for plants with multiple constraints. The intrinsic characteristics of the method allow the user to explore alternative solutions to the task at hand. The solutions found by the proposed method can be more effective than those found by traditional means by optimizing variables like total energy, settling time, convergence time, etc. These characteristics can be advantageous in a variety of systems, such as in robotics, mechanics, electronics, and in general, in multi-parametric plants, to mention a few.

The rest of the paper is organized as follows. Section 2 contains the problem statement. Section 3 describes the synthesis of the nonlinear controllers for the basic behaviors, the parameters for the GP, and the statistical analysis of the attained solutions. Section 4 describes the discovered nonlinear controllers together with numerical results. Section 5 presents an application example. Finally, the conclusions are given in Section 6.

This paper introduces the development of nonlinear control laws based on an analytic behavior-based framework which integrates Control Theory with the Genetic Programming approach. The aim is to solve the tracking control problem in nonholonomic mobile robots operating with bounded velocity. In contrast with other works found in the literature, this paper seeks the satisfaction of the velocity bounds, individually, for each of the mobile robot’s wheels. The proposed method relies on the Genetic Programming approach to keep the velocity inputs within the saturation bounds by creating control laws that implicitly enforce this behavior.

The contributions of this work can be summarized as follows.

• The present work proposes a novel approach that combines Control Theory and Genetic Programming concepts to deliver a controller design methodology. The proposed strategy generates analytic solutions, in contrast with those obtained from other soft computing techniques, like neural network or fuzzy control systems, such as Mamdami models. The advantage of dealing with analytic solutions is that, on the one hand, the cost associated to the implementation of an analytic solution is lower than that of nonanalytic ones, this means that the solutions obtained using the proposed methodology can be implemented in real-time. On the other hand, the proposal in this paper makes it possible to utilize existent analysis techniques developed within the Control Theory framework, by its analytic properties.

• A set of 9113 solutions were found by the proposed algorithm, thus automating the design process. The automation of such a process implies a reduction in manual design time.

• The methodology in this work produces several solutions of different complexity. The user can choose among them, according to specific requirements of the controlled system, or the computational and implementation cost associated with each application. Also, the automation of the learning process aiming to discover suitable nonlinear controllers reveals insights for the synthesis of new controllers.

• In this paper, the proposed scheme is demonstrated for a nonholonomic wheeled mobile robot. However, it can be generalized to other applications and control problems.

• The learning process was performed in simulation using only one desired trajectory, and a given set of parameters and initial conditions for the robot. In practice, the controllers were tested on a robot with different settings and paths, which demonstrates the applicability of the proposed strategy.