Intelligent Control and Automation, 2011, 2, 430449 doi:10.4236/ica.2011.24050 Published Online November 2011 (http://www.SciRP.org/journal/ica) Copyright © 2011 SciRes. ICA Particle Swarm Optimization Algorithm vs Genetic Algorithm to Develop Integrated Scheme for Obtaining Optimal Mechanical Structure and Adaptive C o n troller of a Robot Rega Rajendra, Dilip K. Pratihar Soft Computing Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India Email: regaraj@gmail.com, dkpra@mech.iitkgp.ernet.in Received July 20, 2011; revised August 11, 2011; accepted September 15, 2011 Abstract The performances of Particle Swarm Optimization and Genetic Algorithm have been compared to develop a methodology for concurrent and integrated design of mechanical structure and controller of a 2dof robotic manipulator solving tracking problems. The proposed design scheme optimizes various parameters belong ing to different domains (that is, link geometry, mass distribution, moment of inertia, control gains) concur rently to design manipulator, which can track some given paths accurately with a minimum power consump tion. The main strength of this study lies with the design of an integrated scheme to solve the above problem. Both realcoded Genetic Algorithm and Particle Swarm Optimization are used to solve this complex optimi zation problem. Four approaches have been developed and their performances are compared. Particle Swarm Optimization is found to perform better than the Genetic Algorithm, as the former carries out both global and local searches simultaneously, whereas the latter concentrates mainly on the global search. Controllers with adaptive gain values have shown better performance compared to the conventional ones, as expected. Keywords: Manipulator, Optimal Structure, Adaptive Controller, Genetic Algorithm, Neural Networks, Particle Swarm Optimization 1. Introduction In the age of high precision manufacturing of compo nents and parts, there is a need to develop precisely con trolled manipulator for handling microprecision jobs and machining applications. Also, the manipulator has to traverse a given trajectory as accurately as possible to avoid variations in trajectory causing damage to other parts in assembly or distorted machining of geometric components or inaccurate welding of jobs. Thus, trajec tory tracking is one of the most important tasks to be performed by the manipulator. A user specify motions as the sequences of points through which a tool fixed to the endeffector of a ma nipulator has to pass. The effectiveness of such motion specifying mechanism is greatly increased, if the tool moves in a specified path between the userspecified points. Intermediate points are interpolated along the path at regular intervals of time during the motion, and the manipulator’s kinematics equations are solved to produce the corresponding joint parameter values. The developed path interpolating function offers several ad vantages, including less computational cost and im proved motion characteristics. A second method uses a motion planning phase to precompute enough interme diate points, so that the manipulator may be driven by interpolation of joint parameter values, while keeping the tool on an approximately prespecified path. This tech nique allows a substantial reduction in realtime compu tation. The planning is done by an efficient recursive algorithm, which generates enough intermediate points to guarantee that the tool’s deviation from the path to be tracked stays within a prespecified error bounds. Several attempts were made by various investigators to design and develop suitable controllers for the robots. The following modelbased robot controllers had been
R. RAJENDRA ET AL.431 used: computed torque control, nonadaptive Propor tional Derivative (PD) control, PD control with feed back, and others. Some of those attempts are discussed below. Qu [1] developed PD control scheme to solve trajec tory tracking problem of a manipulator. He concluded that PD control with timevarying gains could guarantee global stability for the trajectory following problem of a manipulator. Homsup and Anderson [2] proposed a modelbased PD control for a 2dof planar manipulator. Its performance was measured using system performance ellipsoids drawn utilizing the information of control pa rameters, manipulator’s Jacobian and inertia matrices. Thus, the performance was dependent not only on the robot’s kinematics and dynamics, but also on the control algorithm. A correlation between trajectory tracking er rors and workspace location was established. Kelly and Salgado [3] developed a design procedure for selecting gain values (that is, and d ) of PD control with computed feedforward of robot dynamics and desired trajectory. The performance of their approach was tested through computer simulations on 2dof manipulator. An evolutionary PD control strategy was used by Ouyang and Zhang [4] to improve trajectory tracking perform ance of a closedloop robot manipulator. It could ensure good trajectory tracking performance without using the knowledge of robot dynamics. The performance of their strategy was tested through computer simulations and found to be better than conventional PD and nonlinear PD control strategies in terms of trajectory tracking per formance and fluctuation in the actuator’s torques. Ravichandran et al. [5] developed a scheme for simul taneous plantcontroller design optimization for a 2dof planar rigid manipulator and nonlinear PD controller. During optimization, a heuristic search technique named evolution strategy and a weightedsum problem formula tion were adopted to take care of multiple objectives and generate a single Paretooptimal solution. The perform ance of the developed scheme was demonstrated on computer simulations, and the potential of the scheme to yield desirable designs of the manipulator and controller was realized. Soft computingbased tools [6] had also been used to design and develop suitable controller for the robots. Some of those studies are discussed here. Ozaki et al. [7] proposed a nonlinear compensator using neural net works for trajectory control of a 2dof manipulator. Its performance was compared with that of adaptive con troller proposed by Craig [8] in compensating unstruc tured uncertainties of the manipulator. The neural net workbased approach was found to be effective and effi cient in learning manipulator dynamics, and conse quently could track the trajectory accurately. Ghalia and Alouani [9] designed a fuzzy logicbased controller of a 2dof manipulator to determine appropriate gain values of the compensator for tracking some trajectories accu rately. The performance of their approach was tested on computer simulations. Rueda and Pedrycz [10] proposed a hierarchical fuzzyneuralPD controller for Ndof robot manipulators to solve tracking problems accurately. In their approach, a coordinator was implemented as a fuzzyneural network, whose purpose was to select acti vation levels for local regulators implemented as time varying PD controllers. Park and Asada [11] developed a concurrent design method of determining mechanical structure and suitable controller for a 2dof planar nonrigid manipulator. An attempt was made to achieve high speed positioning by optimizing arm link geometry, actuator locations and feedback gains. In their study, optimal feedback gains minimizing the settling time were obtained as the fluc tuations of structural parameters, which were optimized using a nonlinear programming technique. Based on the obtained optimal design, one prototype robot was built and an outstanding performance was observed. The concept of PSO algorithm was introduced by Kennedy and Eberhart [12] in 1995. It is a popula tionbased search algorithm, which is initialized with the population of random solutions, called particles, and the population is known as swarm. Several modifications in the PSO algorithm had been done by various researchers. Shi and Eberhert [13] introduced a new parameter called inertia weight into the original PSO algorithm, which played an important role in balancing the global and lo cal searches. Clerc and Kennedy [14] analyzed how a particle carries out its search in a complex problem space and modified the original PSO on the basis of this analy sis. Chen et al. [15] improved the PSO algorithm with adaptive inertia weight W and acceleration coefficients in order to maintain population diversity and sustain good convergence capacity to optimize backpropagation neural networks. PSO is simple in concept, as it has a few parameters only to be adjusted. It has found applications in various areas like constrained optimization problems, minmax problems, multiobjective optimization problems and many more. In addition to these application areas, it has been applied to evolve weights and structure of some neural networks (NNs). Han and Jiang [16] proposed an endpoint prediction model of Basic Oxygen Furnace (BOF) steelmaking based on PSOtuned radial basis function neural network. Braik et al. [17] developed a mechanism to improve performance of NN in modeling a chemical process through PSO. Abe and Komuro [18] utilized an NN, tuned by PSO to save energy of the flexible manipulator for pointtopoint motion. Joint an Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL. Copyright © 2011 SciRes. ICA 432 gles generated by the NN suppressed the residual vibra tion and hence, minimized the motor torques, which was kept as the objective function. The authors conducted numerical studies and verified by the same with experi ments and concluded that PSO is an efficient optimizer. Although a considerable amount of work has been carried out in this field of research, there is still a need for an integrated scheme for obtaining optimal mechani cal structure and controllers with adaptive gain values of 2dof manipulator solving tracking problems. The aim of this study is to obtain an optimal mechanical structure of the manipulator along with adaptive controller that en ables high precision positioning. The links of the ma nipulator are treated as rigid bodies. To speed up opera tions, one needs powerful actuators and lightweight arm links. In positioning, however, the major issue is to minimize the settling time of the control system. The settling time depends on a broad range of design pa rameters including mass and stiffness properties of the links, gain values of the compensator, and others. These parameters are coupled to each other and have intricate interactions with respect to the robot’s settling time. For instance, increasing the structural stiffness alone does not always decrease the settling time. All the design pa rameters must be considered in an integrated manner in order to optimize the performance. In the present paper, an integrated scheme for obtaining optimal mechanical structure and controller of a 2dof manipulator solving path tracking problems, has been proposed. Four ap proaches are developed, and their performances have been compared on two path tracking problems. The remaining part of this paper has been organized as follows: Section 2 deals with mathematical formulation of the problem. Tools and techniques used in the present work are discussed and the proposed algorithms have been explained in Section 3. Results are stated and dis cussed in the Section 4. Some concluding remarks are made in Section 5 and the scope for future work is indi cated in Section 6. 2. Mathematical Formulation of the Problem This section deals with mathematical formulation of the problem. It has been posed as a constrained optimization problem 2.1. Trajectory Analysis A two degrees of freedom serial manipulator with hollow circular crosssection links of lengths: 1 and 2 (where 1 > 2) has been considered, as shown in Figure 1. A photograph of the setup has been displayed Figure 2. The links are connected using rotary joints. The endeffector of the manipulator is directed to track one straight and another circular paths separately, start ing from initial position ( L L L L , ii Y) up to final position (, f Y) in time . The forward kinematics equations of the manipulator can be written as follows: t (a) (b) Figure 1. (a) A two degrees of freedom manipulator traversing a straightline trajectory with torques applied at its joints; (b) load distributions on two links of the manipulator .
R. RAJENDRA ET AL.433 Figure 2. A setup of the 2dof manipulator. 112 12 112 12 cos cos sin sin LtL t YLt Ltt t (1) The joint angles: and can be obtained by carrying out inverse kinematics, as given below: 1t 2t 2 222 12 1 2 12 22 11 1 12 2 cos 2 sin tan tancos XYLL tLL Lt Y t LL t (2) 2.2. Determination of Power Consumption LagrangeEuler formulation [19] is used to determine energy consumption of the manipulator at its two joints. The manipulator’s joints torques: 1t and 2t consist of inertia, centrifugal and Coriolis and gravity terms, as given below. () ,tDtt htt ct 1 2 (3) where , , and represent inertia, centrifugal and Coriolis, and gravity terms, respectively. Now, torques required at two joints of the manipulator (refer to Figure 1) can be determined as follows: Dhc 11 12 11 12 hc DD , 21 22 22 12 hc DD , where 22 111 2 122 2 11 222 212 2 1cos 3 11 34 Lt mm m DLL L mmm Rr L , 2 22 12 2122 2 12 22 11 43 1 cos 2 L mm DD Rr t mLL , 22 2 22 22 2 11 34 Lr mm DR , 12 12 2 12 2 12 22 2 sin 0.5sin t hm LL t mLL , 2 12 22 2 1 0.5sin t hm LL , 111 1 22 12 21 1 0.5cos 0.5cos cos mt gL c mgLt t mgLt , 22212 0.5 coscmgL tt . Now, energy consumed by the manipulator (kW) in tracing the trajectory can be determined as follows: 100 22 11 10.025 100 ij jiij ij Et , (4) where the second part of this equation indicates the en Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL. 434 ergy dissipated by the motors connected at the joints [20]. It is important to mention that the cycle time has been assumed to be equal to 100 seconds and it is divided into 100 instants. 2.3. Structural Analysis: Bending of the Links Structural analysis [21] is confined to Hooke’s Law of analysis, for which the manipulator is used to verify its behavior while tracking the trajectory with minimum error. Due to the weight of the links and motors, there is a chance of bending and deflection of the links and thus, it may deviate from the desired trajectory. To prevent mechanical failure of the links, the developed stress should be less than the allowable stress , that is, . developed allowable developed allowable For minimum consumption of energy by the manipu lator, suitable hollow crosssection of the links possess ing sufficient strength and rigidity to withstand the bending and deflection is to be determined. The links are made up of aluminum. Figure 1(b) displays load distri bution, on two links of the manipulator, where 1 in dicates the weight of the second motor and second link, and load due to 1 W ; 2 denotes the load due to 2 W ; and w represents uniformly distributed load on the links due to selfweight. Link1 can be considered as a beam of hollow circular crosssection, which is subjected to , and a fixed moment of 1 W w 2 12 2 0.5 cos wLt t . Thus, it is to be de signed considering the effect of bending moment 1 , which can be determined as follows: 2 1121 21 11 2 12 2 0.5 cos 0.5cos mwt WwL MLLL wLt t , (5) where 2m represents the weight of second motor. As the two links are assumed to have same crosssection and link1 is more critical compared to the other one, an at tention has been focused on its safe design. The devel oped bending stress on link1 can be determined as fol lows: W max 22 4 π developed M Rr , (6) where and represent the outer and inner radii of the hollow circular link and the maximum bending mo ment Mmax can be determined as follows : R r 22 112 21 max1 2 0.5 m wwL WLLL L . It is important to mention that linear relationships for elastic modulus (E) and yield strength ( ) of aluminum with its density (ρ) have been established according to [2228], as given below. 1.92554 92.56675Ee (7) 7.52288 730638.91191 allowable E (8) It is to be noted that allowable stress allowable has been kept equal to . 2.4. Stability and Response Analysis in Trajectory Tracking The control problem of a robotic manipulator is that of determining the time history of joint inputs required to cause its endeffectors to execute a commanded motion typically specified either as a sequence of endeffector’s positions and orientations or as a continuous path. De pending on the controller design, the joints inputs may be in the form of joint forces, torques or inputs from actua tors. A particular controller design has a significant im pact on the performance of the manipulator and conse quently, on the range of its possible applications (either pointtopoint or continuous). In addition, mechanical design of the manipulator itself will influence the type of control scheme needed. It is to be noted that permanent magnet DC motors with gear reduction are commonly used in robots. The design objective is to choose the compensator (either PD or PID) in such a way that the plant output tracks or follows a desired path. For set points, tracking is a problem of constant or step reference command d that arises in pointtopoint motion [29], [30]. For a PD compensator, the control input U(s) is given in the Laplace domain as follows: d pd UsKKsss (9) The resulting closedloop system response is given by the following expression: pd d KKs Ds ss s , (10) where D(s) is the disturbance on the system (here, it has been assumed to be equal to zero), and is the closedloop characteristic polynomial given by 2 dp JsB KsK (11) The closedloop system will be stable for positive values of , d and bounded disturbances, and the joint error is given by d s . For the PD com pensator given in Equation (9), the step response is de termined by the closedloop natural frequency and damping ratio . The compensator’s gain values: and d can be obtained as follows: 2, 2, p d KJ JB (12) Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.435 where J is the inertia of the links, motor and gear, which can be determined from the relationship: 2 1Js ; represents effective damping of the system [31]; B indicates the natural frequency. Now, the joint errors can be calculated as follows: , where 1,2, d ijterror i itt i (13) The trajectory equations are found to be as follows af ter considering error in motors and deflection of links. 1 11 1 1 11 1 12 2211 2 12 22 11 2 cos , sin , cos , sin . XL t YL t XLt t YYLtt (14) Therefore, the trajectory error can be calculated as fol lows: 22 222 error error 2 XX YYY (15) Total Trajectory error can be determined like the following. total Err 100 100 11 totalerror error ii Err X Y , (16) Now, average error () can be obtained as fol lows: avg Err 2100 total avg Err Err , (17) This problem can be posed as a constrained optimiza tion as given below. Minimize avg EErr , (18) subject to developed allowable and 0.001 0.0029t , 2600.0 2800.00 , 5.0 < ω1 < 20.0, 10.0 < ω2 < 15.0, 1.0 < s1 < 1.0, 1.0 < s2 < 1.0. 3. Proposed Algorithm The said constrained optimization problem has been solved using a realcoded Genetic Algorithm (GA) [32], as the variables are real in nature. Moreover, a realcoded GA has some advantages over the binarycoded GA, such as its ability to provide more precise solutions and there is no hamming cliff problem [6]. A GA is a popula tionbased search and optimization technique, which works based on the mechanics of natural genetics and principle of natural selection [33,34]. It starts with a population of solutions created at random, which are further modified using the operators like reproduction, crossover and mutation. In the present study, tournament selection, polynomial mutation and simulated binary crossover have been used. Interested readers may refer to [6] for a detailed description of the algorithm. Particle Swarm Optimization (PSO), introduced by Kennedy and Eberhart in 1995 [12] is a stochastic popu lationbased evolutionary computation technique, which has also been used to solve the said optimization problem. It can be linked to bird flocking, fish schooling or socio logical behavior of a group of people. It has been used to solve a variety of optimization problems. In PSO, the population of solution is known as swarm. It uses a number of agents known as particles that constitute a swarm moving around in the search space looking for the best solution. Each particle is treated as a point in an Ndimensional space, which adjusts its flying according to its own flying experience and that of other particles. Each particle keeps track of its coordinates in the solu tion space, which are associated with the best solution in terms of fitness that has been achieved so far by it. This value is called personal best, Pbest. Another best value that is tracked by the PSO is the best value obtained so far by any particle lying in its neighborhood. This value is called global best, Gbest. The basic concept of PSO lies in accelerating each particle toward its Pbest and the Gbest locations, with a random weighted acceleration at each time step. Four approaches have been developed as explained below. 3.1. Approach 1 Optimizati o n u si ng a re al coded GA only Figure 3 shows the working cycle of a realcoded GA. A GAsolution carries real values of six variables, such as t , ρ, 1 , 2 , 1 , 2 . The constrained optimization problem has been solved using a penalty function ap proach [6]. As it is a minimization problem, a fixed posi tive penalty of P 100 has been added for each vio lated constraint (if any). Thus, the fitness (f) of the GAsolution can be represented as follows: = avg fEErr P (19) The GA tries to find optimal solution iteratively. 3.2. Approach 2 Optimization using a combined neural network and Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL. Copyright © 2011 SciRes. ICA 436 Figure 3. A schematic view representing working cycle of realcoded GA (that is, approach 1). realcoded GA In approach 1, one GAsolution supplies one set of values of 1 , 2 , 1 , 2 for the entire cycle time of 100 seconds. However, it may be required to adopt dif ferent sets of the said parameters for different instants of the cycle time to track the trajectory accurately. It has been tried in this approach. For each duration, error and error are determined by comparing the calculated values of X Y and with their corresponding target values, and these are fed as inputs to a feedforward neural network (refer to Appendix A), which consists of three layers, namely input, hidden and output layers. There are two and four neurons in the input and output layers of the network, as decided by the number of inputs and outputs, respectively. A thorough parametric study has been carried out to determine the numbers of hidden neurons, which has come to be equal to four. The first and third output neurons have logsigmoid transfer func tion, and the second and fourth neurons are assumed to have tansigmoid transfer function. The neurons of input and hidden layers are assumed to have linear and tansigmoid transfer functions, respectively. In this ap proach, the GA carries information of thickness of the links (t), density of the link material ( Y ) and thirty real variables related to the neural network, such as eight and sixteen connecting weights between the input and hidden layers (that is, [V]) and hidden and output layers (that is, [W]), respectively; bias and coefficients of transfer func tion of hidden neurons; four different coefficients of transfer functions of the output neurons. The values of 1 , 2 , 1 and 2 are determined as the outputs of the neural network. Figure 4 displays the flowchart of approach 2. The fitness of a solution has been calculated using Equation (19). 3.3. Approach 3 Optimization using Particle Swarm Optimization (PSO) only Similar to other populationbased optimization meth ods, such as GAs, the PSO starts with the random ini tialization of population particles in the search space. The PSO algorithm works based on the social behaviour of the particles in the swarm. Therefore, it finds the global best solution by simply adjusting the trajectory of each individual toward its own best location and toward the best particle of the entire swarm at each time step (generation) [12,14]. In PSO algorithm, the trajectory of each individual in the search space is adjusted by dy namically changing the velocih particle, accord ty of eac
R. RAJENDRA ET AL.437 Figure 4. Flowchart of approach 2. ing to its own flying experience and that of other parti cles in the search space. The position and velocity vec tors of particle in the ddimensional search space can be represented as th i 1,, ii id xx and ii id , respectively. The value of i V vector can be varied in the range of 1,, Vv v max max ,vv to reduce the tendency of particles to leave the search space. The value of ma is usually chosen to be equal to maxx v,kx where [35]. According to a user defined fitness function, let us say that the best position of each particle (which corresponds to the best fitness value ob tained by that particle at time t) is 1ii and the fittest particle found so far at time t is 0.1 1k.0 ,, id pPp 1,, gg Pp pd . The new velocities and positions of the particles for the next fitness evaluation are calculated using the following two equations: 1 , id id gd id t bRandpxt andpx t 2 1 id id vt Wv bR (20) 11 id id xt vt id xt , (21) where id is the velocity of dth dimension of the ith particle, W is a constant known as inertia weight [36], 1 and denote the acceleration coefficients, and v 2 b b rand and Rand are two separately generated uniformly distributed random numbers lying in the range Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL. 438 of 0,1 . The first part of Equation (20) represents the previous velocity, which provides the necessary mo mentum for the particles to roam across the search space. The second part of Equation (20) is known as the cogni tive component that represents the personal thinking of each particle. The cognitive component encourages the particles to move toward their own best positions found so far. The third part of Equation (20) is known as the social component, which indicates the collaborative ef fect of the particles in finding the global optimal solution. The social component always pulls the particles toward the global best particle found so far. The PSO is becom ing very popular due to its simple architecture, ease of implementation and ability to quickly converge to a rea sonably good solution. The flowchart of PSO algorithm is shown in Figure 5. The values of six variables, namely t , , 1 , 2 , 1 and 2 are supplied by the PSO during optimization and the fitness of a solution Figure 5. The flowchart of PSO algorithm. Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL. Copyright © 2011 SciRes. ICA 439 has been determined using Equation (19). 3.4. Approach 4 Optimization using combined neural network and PSO algorithm In approach 3, one PSOsolution gives one set of val ues of ω1, ω2, s1, s2 for the entire cycle time of 100 sec onds. However, it may be required to choose separate sets of the said parameters for different instants of the cycle time to track the trajectory accurately. It has been attempted in this approach. For each duration, error and error values are obtained by comparing the calcu lated values of X Y and with their corresponding target values, and these are fed as inputs to a feedfor ward neural network (refer to Appendix A), which con sists of three layers, namely input, hidden and output layers. The input and output layers of the network con tain two and four neurons, respectively. A detailed pa rametric study has been carried out to decide the num bers of hidden neurons, which has turned out to be equal to four. The first and third output neurons have logsig moid transfer function, and the second and fourth neu rons are assumed to have tansigmoid transfer function. The neurons of input and hidden layers are assumed to have linear and tansigmoid transfer functions, respec tively. In this approach, the PSO solution carries infor mation of the thickness of the links (t), density of the link material (ρ) and thirty real variables related to the neural network, such as eight and sixteen connecting weights between the input and hidden layers (that is, [V]) and hidden and output layers (that is, [W]), respectively; bias and coefficients of transfer function of hidden neurons; four different coefficients of transfer functions of the output neurons. The values of ω1, ω2, s1, and s2, are ob tained as the outputs of the neural network. The fitness of a solution has been determined using Equation (19). Y 4. Results and Discussion The performances of the developed four approaches have been tested through computer simulations. The lengths of two links: 1 and 2 are assumed to be equal to 0.3m and 0.2 m, respectively. The terms: B (effective damping) and L L (damping ratio) [31] of Equation (12) have been set equal to 0.02 and 1.0, respectively. The outer radius of two circular links is considered as R = 0.03 m. The motor connected to the second link weighs 0.568 kg. Simulations are conducted on a PIV PC. Results of four developed approaches are stated and discussed below for tracking of one straight path and another circular path separately. 4.1. Straight Path Tracking The 2dof manipulator will have to track a straight path as accurately as possible after consuming minimum power and ensuring enough mechanical strength. Inverse kinematics calculations have been carried out using Equation (2) to determine joint angles corresponding to the movement of the endeffector along a straight path in Cartesian coordinate system. A fourthorder polynomial is found to be suitable to represent the trajectory of 1t as follows: 23 1101112131 taatatatat 4 4 , where the values of the coefficients have turned out to be as a10 = 0.183987352, a11 = −0.008541301 a12 = 0.000193741, a13 = −0.000001798 and a14 = 0.000000007. Similarly, the trajectory function 2t is seen to be a cubic polynomial as follows: 23 212223 atat 22 0 ta at , where a20 = 2.279365512, a21 = −0.011757362, a22 = 0.000026833, a23 = −0.000000209. Angular velo cities and accelerations have been determined for both the joints by differentiating angular displacement with respect to time (t) for once and twice, respectively. Re sults of the developed four approaches are stated, dis cussed and compared below. 4.1.1. Results of Approach 1 As the performance of a GA depends on its parameters, a thorough parametric study is carried out to determine the values of optimal GAparameters. Figure 6 shows the results of the said parametric study, in which one parameter has been varied at a time keeping the others fixed. Thus, the following GAparameters are found to yield the best results: c = 0.76, m = 0.0055, Population size = 150, maximum number of generations max = 150. The optimized values of the parameters: P P P G t , ρ, 1 , 1 , 2 and 2 are found to be equal to 0.001m, 2600.00 kg/m3, 5.002, −0.02612, 10.00 and −0.0141, respectively. The obtained results will be discussed in Figures 7 to 11, at the end of this section. 4.1.2. Results of Approach 2 The set of optimal GAparameters has been obtained using an approach explained earlier. The following GAparameters are found to give the best results: c = 0.86, m = 0.0055, Population size = 220, Maxi mum number of generations max = 240. During opti mization, the ranges of P P P G t and ρ have been kept the same with those of approach 1. The connecting weights: V, W are varied in the range of −1.0 to 1.0. The coefficients of transfer functions of the hidden neurons (1_ hid) and those of first through fourth neurons of out ut layers (that is, , , , ) have a p 2_out a3_out a 4_out a5_out a
R. RAJENDRA ET AL. Copyright © 2011 SciRes. ICA 440 (a) (b) (c) (d) Figure 6. Results of GAparametric study: (a) fitness vs Pc; (b) fitness vs Pm; (c) fitness vs population size P; (d) fitness vs maximum number of generations Gmax. been optimized in the ranges of (0.1, 1.0), (2.0, 3.0), (1.0, 2.0), (1.0, 2.0) and (1.0, 2.0), respectively. Moreover, the bias value: b has been varied from 0.000001 to 0.0001. The optimized values of the variables, which are ob tained using this approach, are shown in Table 1. Results have been discussed and compared with those of other approaches with the help of Figures 7 to 11. 4.1.3. Results of Approach 3 In this approach, the following PSOparameters are found to yield the best results: number of particles inter acting with each particle = 3; dimension of the search space d = 6; number of runs =100; number of executions =1000. The optimized values of the parame ters: , ρ, 1 k t , 1 , 2 , and 2 are found to be equal to 0.001m, 2600.00 kg/m3, 13.715, 0.078, 14.271, and 0.00556, respectively. Results of this approach have been explained in detail with the help of Figures 7 to 11, at the end of this section. 4.1.4. Results of Approach 4 The following PSOparameters are seen to give the best results: = 3; d = 36; number of runs =100; number of executions = 1000. The optimal values of the parameters: k t , ρ, 1 , 1 , 2 and 2 are found to be equal to 0.001 m, 2600.00 kg/m3, 11.715, 0.078, 10.271 and 0.00551, respectively. Table 2 shows the optimized val ues of other variables obtained by the PSO algorithm in approach 4. Results of this approach have been compared with that of other approaches in Figures 7 to 11. 4.1.5. Comp arisons The paths tracked and energy consumed by the manipu lator using approaches 1, 2, 3 and 4 are compared here. The values of energy (kW) consumption, and average absolute percent deviation in trajectory tracking by four approaches are found to be equal to 0.241371, 0.241371, 0.238906, 0.23859, and 0.000000763, 0.000000009, .000000585, 0.00000000128, respectively, while trac 0
R. RAJENDRA ET AL. Copyright © 2011 SciRes. ICA 441 (a) (b) Figure 7. Variations of PD controller’s gain values: (a) Kd; (b) Kp for Joint 1. Table 1. Optimized values of the variables obtained by the GA in approach 2 for straight path tracking. Variable Range Optimized value Variable Range Optimized value t' 0.001, 0.03 0.001 W23 −1.0, 1.0 −1.00 ρ 2600.0, 2800.0 2600.0 W24 −1.0, 1.0 −0.99 V11 −1.0, 1.0 −1.00 W31 −1.0, 1.0 −0.99 V12 −1.0, 1.0 −1.00 W32 −1.0, 1.0 −1.00 V13 −1.0, 1.0 −1.00 W33 −1.0, 1.0 −0.99 V14 −1.0, 1.0 −0.99 W34 −1.0, 1.0 0.89 V21 −1.0, 1.0 −1.00 W41 −1.0, 1.0 −0.99 V22 −1.0, 1.0 −1.00 W42 −1.0, 1.0 −0.99 V23 −1.0, 1.0 −1.00 W43 −1.0, 1.0 −0.99 V24 −1.0, 1.0 −1.00 W44 −1.0, 1.0 −0.99 W11 −1.0, 1.0 −0.99 a1_hid 0.1, 1.0 0.099 W12 −1.0, 1.0 −0.99 a2_out 2.0, 3.0 1.999999 W13 −1.0, 1.0 −0.99 a3_out 1.0, 2.0 0.99999 W14 −1.0, 1.0 −1.00 a4_out 1.0, 2.0 1.000 W21 −1.0, 1.0 −0.99 a5_out 1.0, 2.0 0.999999 W22 −1.0, 1.0 −0.99 b 0.000001 to 0.0001 0.00000023 ing the straight path. Figures 7 and 8 show the variations of d , values for joint 1 and joint 2, respectively, while tracing the given straight trajectory. Comparisons of these approaches are shown on the entire trajectory in Figure 9(a), whereas Figure 9(b) displays the same comparison on a finer scale within the smaller ranges of and . Figures 10 and 11 display the variations of percent deviation in predictions of Y and Yvalues in a cycle, respectively, for the four approaches. The devia tions in tracking the trajectory are found to be less in approaches 2 and 4 compared to that in approaches 1 and 3. Approach 4 is found to perform better than other ap proaches in terms of accuracy in path tracking and power consumption. The better performance of approach 4
R. RAJENDRA ET AL. 442 Table 2. Optimized values of the variables obtained by the PSO in approach 4 for straight path tracking. Variable Range Optimized value Variable Range Optimized value t' 0.001, 0.03 0.001 W23 −1.0, 1.0 1.00 ρ 2600.0, 2800.0 2600.0 W24 −1.0, 1.0 −0.7125 V11 −1.0, 1.0 −0.38296 W31 −1.0, 1.0 −0.9599 V12 −1.0, 1.0 0.1 W32 −1.0, 1.0 0.911465 V13 −1.0, 1.0 0.1 W33 −1.0, 1.0 0.62298 V14 −1.0, 1.0 1.00 W34 −1.0, 1.0 −0.67631 V21 −1.0, 1.0 1.00 W41 −1.0, 1.0 −0.049 V22 −1.0, 1.0 0.622 W42 −1.0, 1.0 0.955 V23 −1.0, 1.0 0.761 W43 −1.0, 1.0 −0.84 V24 −1.0, 1.0 0.761 W44 −1.0, 1.0 0.8364 W11 −1.0, 1.0 −0.84 a1_hid 0.1, 1.0 0.2437 W12 −1.0, 1.0 −0.44 a2_out 2.0, 3.0 2.1013 W13 −1.0, 1.0 −0.85 a3_out 1.0, 2.0 2.00 W14 −1.0, 1.0 −0.35 a4_out 1.0, 2.0 2.00 W21 −1.0,1.0 −0.71 a5_out 1.0, 2.0 1.744 W22 −1.0, 1.0 −0.6924 b 0.000001 to 0.0001 0.000023 (a) (b) Figure 8. Variations of PD controller’s gain values: (a) Kd; (b) Kp for Joint 2. could be due to the use of a PSO algorithm in place of a GA. The latter is a potential tool for global search only but its local search capability is poor, whereas the former carries out both the global and local searches simultane ously. Moreover, PSO is a greedier algorithm compared to the GA. 4.2. Circular Path Tracking The endeffector of the manipulator will have to trace a circular path in Cartesian coordinate system. The corre sponding joint angle values are determined using Equa tion (2). A secondorder polynomial is found to be suit Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.443 (a) (b) Figure 9. Comparisons of the paths tracked by the robot using approaches 1, 2, 3 and 4: (a) entire trajectory; (b) a part of the trajectory shown on the finer scales. Figure 10. Variations of percent deviation in prediction of X values in a cycle for four approaches. able to represent the trajectory of as , where 10 = −0.39713758259, 11 = 0.01657138221, 12 = −0.0000095. Similarly, the trajectory function of 2 has been expressed using another secondorder polynomial as 20 21 2, where a20 = 0.92207514343, a21 = 0.00000008916, a22 = −0.00000000085. 1t a 2 10 1112 1taatat a 22 taata a 2 t t 4.2.1. Results of Approach 1 The following GAparameters (obtained through a care ful parametric study as explained earlier) are found to Figure 11. Variations of percent deviation in prediction of Y values in a cycle for four approaches. yield the best results: c = 0.96, m = 0.0055, Popu lation size = 180, Maximum number of generations ma = 240. The optimized values of the parameters: P P P x Gt , ρ, 1 , 1 , 2 and 2 are seen to be equal to 0.001m, 2600.00 kg/m3, 7.017, −0.02, 10.00, and −0.0116, re spectively. 4.2.2. Results of Approach 2 The best results have been obtained with the following GAparameters: c = 0.96, m = 0.0055, Population size = 150, Maximum number of generations max = 240. The optimized values of the parameters: P P PG t , ρ, Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL. 444 1 , 1 , 2 and 2 are seen to be equal to 0.001 m, 2600.00 kg/m3, 7.017, −0.02, 10.00, and −0.0116, re spectively. The optimized values of the variables ob tained by the GA are shown in Table 3. 4.2.3. Results of Approach 3 The following PSOparameters (obtained through a careful parametric study as explained earlier) are found to yield the best results: = 3; d = 6; number of runs = 100; number of executions = 1000. The optimized values of the parameters: , ρ, 1 k t , 1 , 2 and 2 are seen to be equal to 0.001 m, 2600.00 kg/m3, 20.00, −0032556, 10.00 and 0.004085, respectively. 4.2.4. Results of Approach 4 The following PSOparameters (obtained through a sys tematic study as explained earlier) are found to yield the best results: k = 3; d = 36; number of runs = 100; number of executions = 7000. The optimized values of the vari ables obtained by the PSO algorithm are shown in Table 4. 4.2.5. Comp arisons Results of the above four approaches have been com pared here. The variations of d and values for the joints: 1 and 2 are displayed in Figures 12 and 13, respectively. The manipulator is found to consume 0.864591, 0.771185, 0.755222, 0.752206 kW of energy using approaches 1, 2, 3 and 4, respectively. Figure 14(a) shows the circular path tracked by the manipulator using the above four approaches. In order to clearly distinguish the path traced by the robot, a segment of this path has been displayed in Figure 14(b) The values of percent deviation in prediction of X and Yvalues have been de termined as obtained by the said four approaches and these are shown in Figures 15 and 16, respectively. Ap proaches 1, 2, 3 and 4 have yielded the values of average absolute percent deviation in prediction of the path as 0.000001103, 0.00000032, 0.00000038, and 0.00000023, respectively. Once again, approach 4 has proved its su premacy over other approaches in terms of accuracy in path tracking and energy consumption. It has happened so, due to the reasons explained above. 4.3. Comparisons with Others’ Studies Path tracking problems of a 2dof manipulator had been solved by various researchers. In this connection, the studies of Qu [1], Homsup and Anderson [2], Kelly and Salgado [3] and Ouyang and Zhang [4] are worth men tioning. Soft computingbased approaches had also been developed for the said purpose [7,9,10]. This study also deals with trajectory tracking problems of a manipulator. In the present paper, an integrated scheme has been de Table 3. Optimized values of the variables obtained by the GA in approach 2 for circular path tracking. Variable Range Optimized value Variable Range Optimized value t' 0.001, 0.03 0.001 W23 −1.0, 1.0 −1.00 ρ 2600.0, 2800.0 2600.00 W24 −1.0, 1.0 −1.00 V11 −1.0, 1.0 −0.906785 W31 −1.0, 1.0 −1.00 V12 −1.0, 1.0 −0.861631 W32 −1.0, 1.0 −1.00 V13 −1.0, 1.0 −1.00 W33 −1.0, 1.0 −1.00 V14 −1.0, 1.0 −1.00 W34 −1.0, 1.0 −1.00 V21 −1.0, 1.0 −1.00 W41 −1.0, 1.0 −1.00 V22 −1.0, 1.0 −1.00 W42 −1.0, 1.0 −1.00 V23 −1.0, 1.0 −1.00 W43 −1.0,1.0 −1.00 V24 −1.0, 1.0 −1.00 W44 −1.0, 1.0 −1.0 W11 −1.0,1.0 −1.00 a1_hid 0.1, 1.0 0.1 W12 −1.0, 1.0 −1.00 a2_out 2.0, 3.0 2.00 W13 −1.0,1.0 −1.00 a3_out 1.0, 2.0 0.999999 W14 −1.0, 1.0 −1.00 a4_out 1.0, 2.0 1.00 W21 −1.0, 1.0 −1.00 a5_out 1.0, 2.0 0.999999 W22 −1.0, 1.0 −1.00 b 0.000001 to 0.0001 0.000003 Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.445 Table 4. Optimized values of the variables obtained by the PSO in approach 4 for circular path tracking. Variable Range Optimized value Variable Range Optimized value t' 0.001, 0.03 0.001 W23 −1.0, 1.0 −0.99 ρ 2600.0, 2800.0 2600.00 W24 −1.0, 1.0 −0.99 V11 −1.0, 1.0 −0.999 W31 −1.0, 1.0 −1.00 V12 −1.0, 1.0 −0.98 W32 −1.0, 1.0 −1.00 V13 −1.0, 1.0 −0.99 W33 −1.0, 1.0 −1.00 V14 −1.0, 1.0 −0.99 W34 −1.0, 1.0 −0.99 V21 −1.0, 1.0 −0.99 W41 −1.0, 1.0 −0.99 V22 −1.0, 1.0 −0.99 W42 −1.0, 1.0 −0.99 V23 −1.0, 1.0 −1.00 W43 −1.0,1.0 −1.00 V24 −1.0, 1.0 −0.99 W44 −1.0, 1.0 −1.00 W11 −1.0,1.0 −0.99 a1_hid 0.1, 1.0 0.10 W12 −1.0, 1.0 −0.99 a2_out 2.0, 3.0 2.00 W13 −1.0, 1.0 −1.0 a3_out 1.0, 2.0 1.00 W14 −1.0, 1.0 −.0.99 a4_out 1.0, 2.0 1.00 W21 −1.0, 1.0 −0.99 a5_out 1.0, 2.0 1.00 W22 −1.0, 1.0 −0.99 b 0.000001 to 0.0001 0.000 0001691 (a) (b) Figure 12. Variations of PD contr oll er’s gain values: (a) Kd; (b) Kp for Joint 1. veloped to obtain optimal mechanical structure and PD controllers simultaneously for a 2dof manipulator, so that it can track the trajectories accurately after consum ing the minimum power. To the best of the authors’ knowledge, this study is unique for a rigid link manipu lator, although an attempt (not exactly the same) was made by Park and Asada [11] for a nonrigid link ma nipulator. It had been reported in Abe et al. [18], Braik et al. [17], Chen et al. [15] that computational cost of the PSO is less compared to that of a GA. The performance of PSO algorithm has been compared with that of a GA, in the present study. Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL. Copyright © 2011 SciRes. ICA 446 (a) (b) Figure 13. Variations of PD contr oll er’s gain values: (a) Kd; (b) Kp for Joint 2. (a) (b) Figure 14. Comparisons of the circular paths tracked by the robot in approaches 1, 2, 3 and 4: (a) entire trajectory; (b) a segment of the trajectory shown in finer scales. 5. Conclusions An integrated scheme for obtaining optimal mechanical structures and adaptive PD controller for a 2dof ma nipulator has been developed and its performance has been tested through computer simulations on two trajec tory tracking problems. The robot studied in this paper is a simple one. However, the main strength of this study lies with the design and development of the above inte grated scheme. The robot should be able to track the tra jectory accurately, after consuming the minimum power and ensuring no mechanical failure of the same. Four approaches have been developed. In approaches 1 and 3, natural frequency and stability locus point have been kept constant throughout the cycle, whereas these values are selected adaptively in the cycle in approaches 2 and 4. Approach 4 has outperformed other three ap proaches in terms of both power consumption as well as accuracy in trajectory tracking due to the reasons ex plained earlier. Moreover, approach 4 has yielded a more stable system compared to other approaches. The better performance of the PSO algorithm than that of the GA
R. RAJENDRA ET AL.447 Figure 15. Variations of percent deviation in prediction of X values in a cycle for four approaches. Figure 16. Variations of percent deviation in prediction of Y values in a cycle for four approaches. could be due to its inherent ability to carry out the global and local searches simultaneously. On the other hand, the GA is a potential tool for global search, although it may not be so much powerful in local search. 6. Scope for Future Work In the present study, the performances of developed ap proaches have been tested through computer simulations. However, it will be more interesting to test their per formances in realexperiments. An improved version of PSO algorithm [37,38] may also be used in future to solve the said problem. The authors are working on these issues. 7. Acknowledgements Rega Rajendra thanks financial support of the All India Council of Technical Education (AICTE), New Delhi, India, under the Quality Improvement Programme (QIP) Scheme, to carry out this study. 8. References [1] Z. Qu, “Global Stability of Trajectory Tracking of Robot under PD Control,” Dynamics and Control, Vol. 4, No. 1, 1994, pp. 5971. doi:10.1007/BF02115739 [2] W. Homsup and J. N. Anderson, “PD Control Perform ance of Robotic Mechanisms,” Proceedings of American Control Conference, Minneapolis, 1012 June 1987, pp. 472475. [3] R. Kelly and R. Salgado, “PD Control with Computed FeedForward of Robot Manipulators: A Design Proce dure,” IEEE Transactions on Robotics and Automation, Vol. 10, No. 4, 1994, pp. 566571. doi:10.1109/70.313108 [4] P. R. Ouyang and W. J. Zhang, “A Novel Evolutionary PD Control and Application for Trajectory Tracking,” Proceedings of 7th International Conference on Control, Automation, Robotics and Vision, Singapore, 25 De cember 2002, pp. 13371342. [5] T. Ravichandran, D. Wang and G. Heppler, “Simultaneous PlantControl Design Optimization of a TwoLink Planar Manipulator,” Mechatronics, Vol. 16, No. 34, 2006, pp. 233242. doi:10.1016/j.mechatronics.2005.09.008 [6] D. K. Pratihar, “Soft Computing,” Narosa Publications, New Delhi, 2008 [7] T. Ozaki, T. Suzuki, T. Furuhashi, S. Okuma and Y. Uchikawa, “Trajectory Control of Robotic Manipulators Using Neural Networks,” IEEE Transactions on Indus trial Electronics, Vol. 38, No. 3, 1991, pp. 195202. doi:10.1109/41.87587 [8] J. J. Craig, “Adaptive Control of Mechanical Manipula tors,” AddisonWesley, Reading, 1988. [9] M. B. Ghalia and A. T. Alouani, “A Robust Trajectory Tracking Control of Industrial Robot Manipulators Using Fuzzy Logic,” Proceedings of 27th South Eastern Sym posium on System Theory (SSST’95), Mississippi, 1214 March 1995, pp. 268271. [10] A. Rueda and W. Pedrycz, “A Hierarchical FuzzyNeu ralPD Controller for Robot Manipulators,” Proceedings of IEEE Conference on Computational Intelligence, Or lando, 2629 June 1994, pp. 673677. [11] J. H. Park and H. Asada, “Concurrent Design Optimiza tion of Mechanical Structure and Control for High Speed Robots,” Journal of Dynamic Systems, Measurements, and Control, Vol. 116, No. 3, 1994, pp. 344356. doi:10.1115/1.2899229 Copyright © 2011 SciRes. ICA
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R. RAJENDRA ET AL.449 List of Symbols and Abbreviated Terms 10 14 ,,aa Coefficients of fourthorder polynomial 20 32 ,,aa Coefficients of cubic polynomial 1_ ahid Coefficient of transfer function of hidden neurons 2_ aout Coefficient of transfer function for first output neuron 3_ aout Coefficient of transfer function for sec ond output neuron 4_ aout Coefficient of transfer function for third output neuron 5_ aout Coefficient of transfer function for fourth output neuron b Bias value 12 ,bb B Acceleration coefficients Effective damping value 12 ,cc d Gravity terms of torque Dimension of search space D Inertia term of torque Ds E Disturbance on the system Energy, kW E Elastic Modulus, N/m2 total Err Err Total error avg Fitness Average error f Acceleration due to gravity, m/s2 Gbest Swarm global best solution max Dynamic Coriolis term G h Maximum number of generations Moment of inertia, kg.m2 k Number of particles interacting with each particle jterr Joint error Inertia of links, motor and gear, kg·m2 d Derivative gain Proportional gain 12 L m ,L Length of the links, m Mass of the link, kg 1 Bending moment of first link Moment, Nm c P P Probability of crossover i ith Particle best fitness P Swarm’s global best fitness m P P Probability of mutation Population size P Penalty term Pbest Particle best solution R Outer radius of hollow circular link r Inner radius of hollow circular link Stability locus point t Time, s t Thickness, m vid Particle’s updated velocity in dth dimen sion xid Particle’s updated position in dth dimen sion w Uniformly distributed load, N/m U(s) Control input Vi Velocity Vector V Connecting weights between input and hidden layers W Connecting weights between hidden and output layers W Inertia weight 1 W W Concentrated load acting on first link, N 2 Concentrated load acting on second link, N 2m W Weight of second motor i Position Vector , ii Y , Coordinates of initial position f Y, Coordinates of final position Damping ratio System response in Laplace Transform Joint angle, rad Angle in Laplace domain, rad Density, kg/m3 Yeild Stress, N/m2 Torque, Nm Natural frequency Closedloop characteristic polynomial GA Genetic Algorithm PD Proportional Derivative PID Proportional Integral Derivative PSO Particle Swarm Optimization SBX Simulated binary crossover Rand( ) Random number generator in the range of (0,1) Copyright © 2011 SciRes. ICA
