Mathematical Problems in Engineering

Volume 2015, Article ID 590138, 11 pages

http://dx.doi.org/10.1155/2015/590138

## Applying BAT Evolutionary Optimization to Image-Based Visual Servoing

^{1}Departamento de Ingenierías CUTONALA, Universidad de Guadalajara, Avenida N. Periférico No. 555 Ejido San José Tateposco, 48525 Tonalá, JAL, Mexico^{2}Departamento de Sistemas de Información CUCEA, Universidad de Guadalajara, Periférico Norte 799, Los Belenes, 45100 Zapopan, JAL, Mexico

Received 11 October 2014; Accepted 14 May 2015

Academic Editor: Sabri Arik

Copyright © 2015 Marco Perez-Cisneros 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

This paper presents a predictive control strategy for an image-based visual servoing scheme that employs evolutionary optimization. The visual control task is approached as a nonlinear optimization problem that naturally handles relevant visual servoing constraints such as workspace limitations and visibility restrictions. As the predictive scheme requires a reliable model, this paper uses a local model that is based on the visual interaction matrix and a global model that employs 3D trajectory data extracted from a quaternion-based interpolator. The work assumes a free-flying camera with 6-DOF simulation whose results support the discussion on the constraint handling and the image prediction scheme.

#### 1. Introduction

Past decades have witnessed the extensive development of the visual servoing (VS) control. Three fundamental schemes have practically represented most of VS implementations [1, 2]. First, the image-based visual control (IBVS), also known as 2DVS, employs an error computation between the visual features belonging to a target object in a given image and its corresponding features in a target image. Such error is subsequently employed as guidance for the visual control algorithm just as it is carefully detailed in the following. Second, the position-based visual servoing (PBVS), also named as 3DVS, works entirely on the visual computation of geometric poses, whose values are subsequently used to regulate the camera movement. Likewise, a third group is represented by a wide number of hybrid VS approaches that generally profit over a mindful combination between 2DVS and 3DVS advantages.

In particular, the classic IBVS control problem is defined as an exponential minimization of the aforementioned image plane error between the current and target image . In turn, such error can be subjected to a classic minimization procedure assuming a gradient-like approach such that . A well-known relationship between the object’s velocity and its corresponding image plane velocity can thus be defined by stocking each point velocity relationship into a single matrix known as the interaction matrix or the visual Jacobian. Mathematically, the overall velocity relationship can thus be defined (see [1]) as with being the interaction matrix and representing the velocity screw vector over time. A classical feedback control law can thus be defined asIn this case, is the pseudoinverse of the image Jacobian matrix and a negative constant, with being the resulting control signal. Despite the implementation of such VS scheme being fairly simple, some important drawbacks have been highlighted by Chaumette in [3], with unstable behavior arising from the tracking of large displacements and complex rotations, or from the generation of nonfeasible motions. Therefore, the handling of either the 2D constraints or the 3D limitations, as well as the generation of feasible trajectories for a given visual task, must be all appropriately addressed.

Two constraints must be appropriately handled in order to assure an appropriate visual control behavior: first, the well-known visibility constraint that refers to the adequate handling of the control problem in order to assure that visual features always remain within the camera field of view and second the 3D constraint that challenges the generation of convenient visual servoing schemes that yield admissible camera motions within a valid workspace.

The use of optimal control fundamentals for visual servoing has been defined as an appropriate and convenient tool to build visual servoing schemes that carefully considered the aforementioned visual constraints. Actually, several applications have been reported in the literature over the last two decades. First, the seminal works of Hashimoto and Kimura in [4] and Schramm and Morel in [5] that incorporated an LQ-based optimal control scheme and a Kalman filter-based algorithm, respectively, in order to guide the movements of a robotic manipulator.

Other approaches have capitalized the advantages of the LMI approach to build predictive control schemes for visual servoing [6, 7]. Despite the fact that such works have focused over the designing of an appropriate control law for the visual servoing scheme, other proposals have also included optimal schemes for the combination of path planning and trajectory tracking in order to assure the fulfilment of the visibility constraint and the generation of an optimal trajectory for the camera. Excellent examples of such combination can be found in the works of Schramm and Morel in [8] and the use of LMI structures made by Chesi in [9]. In the particular case of path planning, it is important to consider the work of Mezouar and Chaumette [10] and the robust approach proposed later by Kazemi et al. in [11]. In this case, an LMI based algorithm is used to define an optimal path planning solution assuming that not a unique solution for the problem may exist and also that it may not be unique, while the required camera tracking is supplied through a classic image-based visual controller [12].

Other optimal VS control implementations include the use of predictive control to compensate for errors in the tracking task of a visual feedback scheme in case of no prior information about the 3D model being supplied to the visual controller [13] or in the case of using active filtering through predictive control for biomedical applications that support robotized surgery [14].

Recently, the strategy to incorporate the handling of both visual constraints, that is, the visibility and the feasible motion constraint, within the visual control structure has been focused on expressing the overall visual task from a nonlinear optimization perspective. Therefore, this paper presents a novel optimization scheme that employs an evolutionary optimization method to handle both constraints through a visual predictive control scheme. Under such circumstances, 3D constraints can be considered as state variables while the visibility constraint can be assumed just as a constraint within the output space, just as it has been done in [12]. In order to provide an appropriate model prediction agent, two options are to be considered following the proposal of Allibert and Courtial in [15]. First, a local model uses the classic image Jacobian matrix while a second test uses a quaternion-based 3D trajectory generator. As it will be carefully discussed, the optimization algorithm uses prediction to improve the overall visual servoing performance by means of a predictive control structure that has been specifically designed to fit within the visual control scheme.

Just as it has been widely demonstrated, the use of optimization within the visual servoing control scheme has delivered some relevant contributions in particular for the image-based schemes that naturally handle the most important visual constraints at the same time control signals are generated with remarkable examples being found in [12, 15–17]. However all these solutions use classic optimization methods in order to minimize an objective function since the goal of an optimization scheme is to find an acceptable solution of a given objective function that is defined over a given search space; novel methods that are known as Evolutionary Methods have been proposed as a handy alternative.

In particular, evolutionary algorithms (EA), which are considered as stochastic optimization methods, have been developed by a combination of rules and randomness that mimics several natural phenomena that include some evolutionary processes such as the evolutionary algorithm (EA) proposed by Fogel et al. [18], De Jong [19], and Koza [20], the Genetic Algorithm (GA) proposed by Holland [21] and Goldberg [22], the Artificial Immune System proposed by de Castro and bon Zuben [23], and the Differential Evolution Algorithm (DE) proposed by Storn and Price [24]. Some other methods which are based on physical processes include the Simulated Annealing proposed by Kirkpatrick et al. [25], the Electromagnetism-Like Algorithm proposed by Birbil and Fang [26], and the Gravitational Search Algorithm proposed by Rashedi et al. [27]. Also, there are other methods based on the animal-behavior phenomena such as the Particle Swarm Optimization (PSO) algorithm proposed by Kennedy & Eberhart [28], the Ant Colony Optimization (ACO) algorithm proposed by Dorigo et al. [29], and the BAT algorithm proposed by Yang [30], which is of special importance for this paper.

In particular, this paper approaches the IBVS from an optimization-like perspective that naturally supports the inclusion of visual constraints in the implementation of the vision-based control scheme. As a result, the overall performance of the visual servoing scheme is improved at the same time that the aforementioned constraints are carefully taken into consideration.

The paper has been developed as follows. Section 2 presents an overview of the overall optimization strategy, the control scheme, and its mathematical formulation, as well as the management of image-based constraints that support the optimal IBVS approach. Section 3 focuses on the principles of the BAT optimization algorithm and its basic operational principles. Section 4 discusses the local and global models that are required in the image-prediction scheme, which in turn are represented by the classic IBVS control algorithm and the quaternion-based guidance. Section 5 presents some simulation of the free flying 6-DOF camera in order to demonstrate the active contribution of the the algorithm’s tracking performance and discuss the differences between using the local or the global model for prediction. The last section draws some final conclusions.

#### 2. An Optimization Approach to IBVS

##### 2.1. Structure of the Control Scheme

One of the most successful strategies to incorporate optimization into a feedback control scheme is beyond any doubt of the predictive control. In turn, one of the most well-known structures for predictive control is the internal model control approach [31], whose basic structure has been customized for the image-based visual servoing in the work of Allibert [12]. The basic structure is reproduced in Figure 1 where the robot and its attached camera are modelled inside the plant block. The control input to the system is represented by while the output has been marked as which represents the image plane coordinates of four selected features to track in the image of the object of interest. However, as it is typical in IBVS, the scheme requires the definition of desired (target) locations for the object features in the image, typically represented by . By making use of the error model for IBVS from (1), the predictive control is based upon a generalized error that is defined by the difference between the current plant output at time and the corresponding model output. Define such generalized error as the difference of the system’s output and the predicted model output , yielding , at time . The algorithm should assure that a desired trajectory of visual features on the image plane follows an adequate sequence of points in order to guarantee the fulfillment of both visual constraint that have been mentioned earlier. Therefore, an easy definition for the required trajectory can be defined as the difference between the target feature locations and the preregistered plant-model error , which in turn generates the following expression:The overall error that includes the plant-model difference at time can be included yielding: A very interesting fact emerges as the overall equation is rewritten as follows:This last expression holds a key issue for the optimization approach of IBVS schemes. The minimization of the difference between the desired visual features location and the system’s output corresponds to minimizing the difference between the required visual trajectory and the model output . Actually, the last fact supports the operation of the optimization algorithm that is to be completed if an objective function and some operative rules are defined as it is discussed below.