power optimization of planar redundant manipulator moving along constrained end trajectory using hybrid techniques

Alexandria Engineering Journal (2017) xxx, xxx–xxx H O S T E D BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com ORIGINAL ARTICLE Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques Mohamed M.Y.B Elshabasy a,b,*, Khaled T Mohamed c, Atef A Ata d a Department of Mechanical Engineering, Alexandria University, Alexandria 21544, Egypt Department of Mechanical Engineering, Male Branch, Jubail University College, Jubail Industrial City 31961, Saudi Arabia c Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt d Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt b Received September 2016; revised 19 January 2017; accepted 23 January 2017 KEYWORDS Planar manipulator; Hybrid optimization; Power genetic algorithm Abstract Optimum trajectory planning for a planar redundant manipulator which consumes minimum power is considered in this paper The equations of motion for the manipulator are derived in matrix form using Lagrange Equation Two hybrid optimization algorithms with constrained variables during the search have been presented and compared to design an optimal trajectory for planar redundant manipulator based on minimum power to be consumed Two trajectories with the same initial angular displacement of the manipulator joints and two different angular positions at the end of the trajectories are studied Both the hybrid technique of the Genetic Algorithm (GA) and the constrained Fmin function (GA-Fmincon) and the hybrid genetic algorithm with the pattern search (GA-PS) subjected to the same constraints give the same results for the trajectory that ends with angular displacements of (p/2, 0, 0) rad with trivial differences in the power values at any arbitrary mission period For the second trajectory, which ends with angular positions of (1, 1, 1) rad, the GA-PS gives a penalty function of smaller value Ó 2017 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction Since robot manipulators play a crucial role in our life nowadays, demands for flexibility and versatility of their * Corresponding author at: Department of Mechanical Engineering, Alexandria University, Alexandria 21544, Egypt E-mail addresses: mohamed_elshabasy@alexu.edu.eg (M.M.Y.B Elshabasy), ktawfik64@alexu.edu.eg (K.T Mohamed), atefa@alexu edu.eg (A.A Ata) Peer review under responsibility of Faculty of Engineering, Alexandria University performance have been increased Redundancy of robot manipulators is essential to increase their flexibility and versatility To a large extent, the optimization techniques and the control algorithms are unrestrictedly proposed and effectively utilized as an advantage of the manipulator redundancy However, these algorithms and techniques should be valid for redundant and non-redundant manipulators Earlier trials of exploiting the manipulator redundancy advantages are those researches conducted by Hanafusa et al [1] and Yoshikawa [2] They tried to obtain the advantages of redundancy through applying of various control algorithms They showed that in some cases the control algorithms and objective functions led http://dx.doi.org/10.1016/j.aej.2017.01.040 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 to undesired arm configurations Thus, they suggested the use of local and global criteria to solve this problem Some researchers have suggested a number of solution techniques for solving the kinematic control problem for redundant manipulators Further, Field and Stepanenko [3], Hirakawa and Kawamura [4], and Hansen et al [5] used different control algorithms, using dynamic models, to find the trajectory of optimal total energy consumption Recently, Chettibi and Lemoine [6] used sequential quadratic programming to balance between the transfer time and the electric power consumed during the transfer of the joints DC motors They used electro-mechanical constraints, which may not be applicable considering technological constraints inherent to the motors and associated amplifiers All previous researches used local optimization methods, which fail in some cases or require some additional constraints on end effector and joint trajectory Until 1990, most relevant literature that has appeared, proposed approaches that were based on the instantaneous or local resolution of redundancy at the velocity level using the manipulator’s Jacobian matrix Also, the proposed global optimization techniques involved increased computational complexity which ruled them out in practical online implementation for continuously modified sensory feedback trajectories Schreiber et al [7] and Kazerounian and Wang [8], developed path planning techniques, for kinematically redundant manipulators, involving the optimization of integral kinematic cost criteria They exhibited numerical solutions for the periodic boundary value problem that results in periodic joint angle time histories In some demonstrated cases the necessary conditions were not sufficient to determine a globally optimal solution, but lead to a family of locally optimal solutions that were not homotopically equivalent In addition, Kurtz and Vincent [9] used simultaneous objective functions to obtain the optimum kinematics of a parallel manipulator By the beginning of this century, the global optimization techniques replaced the local ones While enumerative global search methods are overwhelmed by the size of the search space, the GA provides a more robust approach GA-based search and optimization techniques have recently found increasing use in machine learning, robot motion planning, scheduling, pattern recognition, image sensing, etc [10] Yue and Henrich [11] introduced a trajectory planning method for flexible redundant manipulator based on GA to minimize vibration and/or executing time of the point-to-point motion The goal was to achieve fast but low vibration operations Huang et al [12] presented a hybrid method for the optimum kinematic design of two-degree-of-freedom parallel manipulators with mirror symmetrical geometry The global optimal architecture was obtained in terms of a closed-form parametric relationships in which design variables can be reduced Then the specified workplace was generated using the required design parameter to be optimized They optimized the kinematics of a two degree of freedom parallel robot with revolute joints Nearchou [13] used the GA to minimize the end-effector positional error concurrently with the robot joint displacements Almost all the previously discussed researches focused on the optimum kinematic and/or trajectory without energy or torque consideration Garg and Kumar [14], used both the GA and the Simulated Annealing techniques to identify the optimal trajectory based on minimum joint torque requirements The simulated results showed that the Simulated Annealing converges to solution faster than GA Lei and Du M.M.Y.B Elshabasy et al [15], proposed an adaptive algorithm based on the GA for stable, minimum energy trajectory Shinkatu [16] proposed a simple technique based on GA to approximate the joints of the underwater manipulator trajectory as a polynomial The GA searches for the optimal coefficients of the polynomial to minimize the time of fitness of the objective function Yun and Xi [17] introduced an improved GA scheme to achieve the optimum motion planning in the joint space Their approach showed good performance for 2-DOF robots Lee and Lee [18] proposed a genetic an off-line optimum trajectory planning between the end points of the assigned trajectory Tian and Collins [19] developed a GA for 2-DOF robots to search for optimal trajectory in the task space They built the objective function on minimizing the rotation angles of two joints Khadwilard and Pongcharoen [20] applied GA to minimize the rotation angles of a two link planar robot in the existence of obstacles in a two-dimensional workspace They concluded that the cubic interpolation for point-topoint interpolation is the best Elshamli et al [21] applied the GA to solve the path planning problem in stochastic 2D mobile robot space Their algorithm was efficient in solving various tasks in 2D dynamic environments Pires et al [22] used the GA to minimize the space/time ripple in the trajectory without collision At the same time, they tried to enhance the mechanical structure reliability For a three-link manipulator, Kazem et al [23] used a multi-objective GA to minimize traveling time and space They applied constraints on the applied torque to the joints in existence of obstacles in the work space Hansen et al [24] minimized the energy consumption of two axis industrial robots by trajectory optimization They considered friction losses and servo drives and inverters’ losses Their results indicated the intense effect of energy exchange in the robot controller power electronics Ruiz et al [25,26] used the Genetic Algorithm to minimize the driving torques (or force) in redundant planar parallel manipulators They investigated the effect of redundancy, in parallel robots, on the minimization of the consumed energy along predefined trajectory Ata and Myo [27,28], used the GA followed by the Pattern Search to form the Generalized Pattern Search (GPS) to obtain optimal trajectory of a planar redundant manipulator based on minimum energy consumption Ata and Myo [27] combined the Genetic Algorithm and the Pattern Search to form the Generalized Pattern Search (GPS) to obtain the optimal point-to-point trajectory tracking of redundant manipulators They also used the GPS for designing a collision free trajectory for planar redundant manipulator [28] Ata [29] discussed and analyzed the various objectives of the optimal trajectory planning a long with the optimization techniques He [29] concluded that combining the Genetic Algorithms with other classical optimization methods would have a better performance as a generalized optimization technique Chandra and Rolland [30–32] published a series of papers to solve the forward kinematics in various types of parallel manipulators using various population-based metaheuristic methods In paper [30], they solved the forward kinematics of 3RPR planar parallel manipulator using hybrid metaheuristics In the second paper [31], Chandra and Rolland used the Global-local population memetic algorithm for solving the forward kinematics of parallel manipulators The forward kinematics of the 6-6 parallel manipulator was solved by Rolland and Chandra [32] using an evolutionary algorithm based on generalized generation gap with parent-centric crossover Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Power optimization of planar redundant manipulator The kinetic and potential energies of the manipulator are given by [35]: 1 1 K:E: ẳ I1 ỵ Ih1 ịh_ 21 ỵ I2 h_ ỵ h_ ị ỵ Ih2 h_ 22 ỵ m2 V2D 2 2 _ 1 ỵ I3 h1 þ h_ þ h_ Þ þ mE V2E þ Ih3 h_ 23 2   1 P:E: ẳ m1 gl1 sin h1 ỵ m2 g l2 sin h1 ỵ l2 sin h2 2   ỵ m3 g l1 sin h1 ỵ l2 sin h2 ỵ l3 sin h3 Figure Schematic diagram of the planar redundant robot This paper is an extension to the series of paper published by Ata [27–29], as the main objective was to minimize the power to be consumed by a planar manipulator moving along a trajectory of constrained ends The optimization results are obtained using two hybrid optimization techniques In the first technique, the GA is used with the Pattern Search as a hybrid pack-up function to be used in case the GA process discontinued for any reason The constrained minimum function (Fmincon) replaces the PS as another hybrid backup function to the GA function in the second technique The rest of this article is organized as follows The next section discusses the problem formulation The hybrid optimization techniques are discussed in section three The fourth section includes the simulations and discussions In the last section, concluding remarks of the present study are listed Problem formulation In our investigation, a 3R planar redundant manipulator shown in Fig is considered The end-effector is assumed to move along a trajectory of specified end points The objective function was to minimize the power consumed as the end effector move along this trajectory This objective reflects on the reduction in the kinetic energy, which in turn leads to minimize the torque applied to each of the three joints The displacements of the revolute joints are tailored to polynomial functions of the time as will be discussed in the next section Each link i has a mass mi, length li, moment of inertia about its center of rotation Ii and motor hub inertia Ihi The end effector of the robot has mass mE and linear velocity VE The equations of motion for the robot arm can be derived using Lagrange Equation in the form:     d @L @L ẳ Qi ; 1ị dt @ q_ i @qi where L represents the Lagrangian, which equals the difference between the total kinetic energy (K.E.) and the total potential energy (P.E.) of the manipulator Qi refers to the generalized forces acting on the system under investigation The torques applied to the manipulator joints represent these generalized forces ð2Þ ð3Þ Upon substituting of Eqs (2) and (3) into Eq (1) and after some algebraic manipulation, the equations of motion can be given in matrix format as follows: 32 32 _ 3 € h1 M11 M12 M13 C11 C12 C13 T1 h1 76 € 76 _ 7 ¼ þ T M M M C C C 21 21 22 23 54 h2 22 23 54 h T3 € M31 M32 M33 C31 C32 h3 32 3 _ _ R11 R12 R13 G11 h1 h2 76 7 ỵ R21 R22 R23 54 h_ h_ ỵ G22 R31 R32 R33 G33 h_ h_ C33 h_ 23 ð4Þ where M, C, R, and G respectively represent the Inertia matrix, the Centrifugal matrix, the Coriolis matrix, and the gravity matrix [35] The elements of each matrix are listed in details in Appendix Hybrid optimization techniques Genetic Algorithms are search procedures based on the mechanics of natural genetics and natural selection They combine an artificial survival of the fittest with genetic operators abstracted from nature to form a surprisingly robust search mechanism that is suitable for variety of search problems [33] Due to the strength of the GA technique, many researchers have used it to obtain optimal trajectory for redundant manipulators based on some criteria such as minimum time, minimum kinetic energy, and obstacle avoidance The reader is referred to Refs [10–29] which are briefly discussed in the literature review section There are several features of GAs that make them attractive for those kinds of problems [34]: The cost or fitness function used to resolve the redundancy has no requirements for continuity in derivatives So any cost function can be selected There is no need to compute the Jacobian matrix The GA solution needs only the forward kinematics equation, which is easy to obtain The joint rotation limits, which are present in any feasible robot design, are handled directly Thus, any solution obtained by the GAs is physically realizable The GAs work with joint angles represented as digital values, which is more representative for computer-controlled systems The trajectory of each joint of the three degree-of-freedom planar redundant manipulator can be designed as a 4th order polynomial as follows: Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 M.M.Y.B Elshabasy et al hi tị ẳ t4 ỵ bi t3 ỵ ci t2 ỵ di t ỵ ei 5ị where ; bi ; ci ; di ; and ei are coefficients to be determined based on the end-conditions as well as the cost function to be minimized The constant ei always represents the initial angle while the constant di can be determined from the endvelocity values For the rest-to-rest maneuvering which is applicable to most of the industrial robots nowadays, the value of di is always zero The other initial conditions are as follows: h1 tf ị ẳ p2 rad; h2 tf ị ẳ 0; and h3 tf ị ẳ 0; hi 0ị ẳ rad; h_ i 0ị ẳ 0; and h_ i tf ị ẳ where i ẳ 1; 2; and and tf represents the time at the end of mission The main objective of the current paper was to move the end-effector of the redundant manipulator along a trajectory, of specified end locations based on the joint displacements’ constraints listed above, with minimum consumed power of the whole mission The Penalty function is given by [10]: X X jh_ i Ti ị j 6ị Penalty Function ẳ jxi Ti ị2 j ẳ Figure where i refers to the joint number which varies between and The summation sign in the previous equation denotes that the quantities at each instant of the total duration are being summed The cost function contains the torques applied at the manipulator joints multiplied by the joint velocity at an arbitrary instant during the mission Each joints’ torque depends on the first and second derivatives of the joints’ displacements with respect to time as indicated in Eq (4) The coefficients of the displacements’ polynomials, as in Eq (5), are the design variables of the implemented optimization techniques Simulation and discussion The simulations of the manipulator under investigation, based on two GA-based hybrid optimization techniques are presented and discussed in this section Using these techniques in the current investigation is the main objective as Ata [29] concluded that these methods give better results than using GA only Comparing the performance of the adopted technique in this paper with other state-of-the-art GA algorithms The program flowchart and the pseudo code of the combined GA-PS and GA-Fmincon optimization techniques Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Power optimization of planar redundant manipulator and other advanced versions of Evolutionary Algorithms is out of the scope of the current investigation In both implemented techniques, the GA is the primary optimization routine that is supported with secondary routines The secondary routine is either PS or Fmincon The secondary routine is used as a pack-up in case the GA is terminated during the optimization process and to refine the polynomials’ coefficients of the utmost minimum penalty function As we are looking for the utmost minimum consumed power mission, the two techniques are compared such that a recommendation for one of these two techniques will be stated The flowchart of the main program and the pseudo code of the two techniques (GA-PS and GA-Fmincon) are shown in Fig to clearly show how the GA and the selected classical optimization technique (PS or Fmincon) work in a hybrid mode The parameters of the manipulator are as follows: m1 ¼ kg; m2 ¼ kg; m3 ¼ kg; L1 ¼ 0:2 m L2 ¼ 0:4 m L3 ¼ 0:2 m Ih1 ¼  I1 Ih2 ¼  I2 Ih3 ¼  I3 Table The parameters techniques used with the optimization Population size Initial range of population values Generation The fraction of genes to be swapped between individuals Hybrid function Compute fitness functions of a population in parallel Reproduction Mutation Crossover Migration 400 [0;0.3] 100 Cross over heuristic = 0.4 Fmincon or PS Always Elite count = 2, crossover rate = 1.2 Gaussian (scale = 1, shrink = 1) Scattered function Forward (fraction 0.2, interval = 20) The simulation results have been conducted using the MATLAB 7.1 on a PC GB ram and 3.1 GHz processor of MB cache The parameters of optimization techniques defined in the MATLAB programs are listed in Table As the mission time is playing a crucial rule in controlling the consumed power during the mission, the total time of the mission will be varied between s and 20 s For each optimization technique, the joints’ displacements of the manipulator are not constrained as they can be any value between and 2p Two trajectories are presented with the same initial point of the end effector, where the joints angles are (0, 0, 0) and different end points No intermediate points on the trajectory were assigned The joint angles, h1, h2, and h3 of the first traÀ Á jectory are p2 ; 0; rad respectively and those of the second trajectory are (1, 1, 1) rad accordingly, and four MATLAB programs were run For each of the four cases, the relation between the optimum consumed power and the mission period is presented in Fig As could be noticed from the two figures that the larger the mission time, the smaller the power to be consumed during the mission It is also noticed that the curves of the two optimization techniques are coincident for the both trajectories As the variation in the consumed power with the trajectory period is trivial between and 20 s, the optimum trajectory time is considered equal s for compromising between the fastness of mission and the power curtailing Table contains the values of the consumed powers for both trajectories and both optimization techniques at two trajectory periods, and 20 s For the trajectories, the polynomial coefficients that satisfy the penalty function according to each of the two hybrid optimization techniques are listed down in Table As mentioned before, according to the values of the joints’ displacement and velocities at the end points of the trajectories, the values of ei’s and di’s coefficients are zeros For each technique, the program was run with constraints on the joints’ angles ðÀp=2 hi p=2Þ For the first trajectory, it is noticed that the optimization techniques give almost the same polynomial coefficients In the second trajectory, the corresponding coefficients resulting from the two optimization techniques are different as shown in Table The listed values of the Figure Total consumed powers during using Hybrid GA-Fmincon and Hybrid GA-PS techniques (a) the first mission and (b) the second mission Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 M.M.Y.B Elshabasy et al Table The minimum power at tf = s and tf = 20 s using Ga-Fmincon and Ga-PS techniques First trajectory (GA-Fmincon) tf = s tf = 20 s Power = 6.2798 N m/s Power = 1.0048 N m/s Second trajectory (GA-Fmincon) Power = 5.8575 N m/s tf = s tf = 20 s Power = 0.7176 N m/s Table Coefficient First trajectory (GA-PS) tf = s tf = 20 s Power = 6.2798 N m/s Power = 1.0202 N m/s Second trajectory (GA-PS) tf = s tf = 20 s Power = 4.4847 N m/s Power = 0.7176 N m/s The coefficients of the angular displacements polynomials using two hybrid optimization techniques Mean value Standard deviation Coefficient Mean value Standard deviation Coefficient Mean value Standard deviation First trajectory (GA-Fmincon) a1 2.0614 * 10À4 2.0156eÀ04 a2 À5.1 * 10À4 4.9842eÀ04 a3 67.709 * 10À4 0.0066 b1 b2 b3 À9.434 * 10À3 8.162 * 10À3 À108.3 * 10À3 0.0032 0.008 0.1057 c1 c2 c3 8.683 * 10À2 À3.266 * 10À2 43.334 * 10À2 0.0129 0.0319 0.4230 First trajectory (GA-PS) 2.0625 * 10À4 a1 À5.1 * 10À4 a2 a3 67.710 * 10À4 2.2082eÀ04 5.4604eÀ04 0.0072 b1 b2 b3 À9.436 * 10À3 8.165 * 10À3 À108.3 * 10À3 0.0035 0.0087 0.1158 c1 c2 c3 8.683 * 10À2 À3.266 * 10À2 43.334 * 10À2 0.0141 0.0349 0.4633 Second trajectory (GA-Fmincon) 2.394 * 10À4 2.5479eÀ04 a1 a2 4.193 * 10À4 2.4816eÀ04 a3 À69.41 * 10À4 0.006 b1 b2 b3 À7.736 * 10À3 À10.62 * 10À3 107.1 * 10À3 0.0041 0.004 0.0957 c1 c2 c3 6.2195 * 10À2 7.371 * 10À2 À39.74 * 10À2 0.0163 0.0159 0.383 Second trajectory (GA-PS) 2.827 * 10À4 2.7911eÀ04 a1 a2 À0.893 * 10À4 2.7185eÀ04 53.222 * 10À4 0.0066 a3 b1 b2 b3 0.618 * 10À3 À2.477 * 10À3 À89.06 * 10À3 0.0045 0.0043 0.1049 c1 c2 c3 2.8777 * 10À2 4.1159 * 10À2 38.750 * 10À2 0.0179 0.0174 0.4195 Figure The torques applied to the joints during the first trajectory based on GA-Fmincon optimization technique coefficients are the mean values of runs of the program with values of standard deviations listed beside each coefficient As the difference between the corresponding coefficients of the polynomials of the joint displacements in the first trajectory is trivial, the simulation results based on any of the two optimization techniques are the same These results are pre- Figure The angular displacements of the joints during the first trajectory based on GA-Fmincon optimization technique sented in Figs 4, 5, 6, and Fig shows the torques applied to the three joints during the mission The variations in the three displacement joints during the whole trajectory are shown in Fig 5, while the corresponding angular velocities of the joints are represented in Fig The first trajectory and the manipulator configurations at each selected end- Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Power optimization of planar redundant manipulator Figure The angular velocities of the joints during the first trajectory based on GA-Fmincon optimization technique Figure The first trajectory and the manipulator configuration based on GA-Fmincon optimization technique effector location are shown in Fig In the second trajectory, it is also noticed that the total consumed power in the GA-PS method is smaller than that of the GA-Fmincon as shown in Table Thus, the simulation results of the GA-PS method are adopted and presented in Figs 8–11 Fig shows the torques applied to the three joints during the mission The three displacements joints are shown in Fig 9, while the corresponding angular velocities are presented in Fig 10 The loci of the end-effector along with the corresponding manipulator configurations are shown in Fig 11 From Fig 4, one can notice that the minimum torques are applied at the mid-third of the trajectory and the absolute maximum values of the torques are applied at the beginning of the mission to overcome the friction at the beginning of motion On the other hand, the torque applied to the third joint is always smaller than the torques applied to the first two joints during the mission In Fig 5, the maximum value of the third joint displacement and the minimum value of the second joint displacement are taking place at the mid- Figure The torques applied to the joints during the second trajectory based on GA-PS optimization technique Figure The angular displacements of the joints during the second trajectory based on GA-PS optimization technique time of the mission In the same figure, it is noticed that the first joint displacement is gently increasing until it reaches its maximum value at the end of the mission In Fig 6, it is noticed that the second and the third joints come to complete stop at the mid-time of the mission Notice also that the maximum speed of the first joint is taking place a bit before the mid-time of the mission In the same figure, it is also noticed that the first and the third joints accelerate at the first half of the mission and decelerate at the second half, while the second joint is behaving inversely In Fig 7, it is also noticed that the end effector is following a sinusoidal trajectory around the straight line extended between the first and last points of the mission In the simulations of the second trajectory, it is noticed that the joints torques are behaving in the same mode of the torques applied in the first trajectory In Fig 8, both of the displacements of the first and second joints are increasing steeply between the end points of the mission In the same figure, Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Figure 10 The angular velocities of the joints during the second trajectory based on GA-PS optimization technique M.M.Y.B Elshabasy et al ulation cases, the larger the mission period, the smaller the optimum consumed power during the mission The optimum consumed power is abruptly decreasing between the mission period of s and that of s Between the mission time of s and s, the power decreases with a moderate rate Starting from a mission period of s, the variation in the power is very trivial Consequently, it is not recommended to decrease the mission period below s as a dramatic increase in the consumed power is taking place The GA-PS shows a superior performance over the GA-Fmincon Because of the inertias of the manipulator components and other parameters, there should be limits on the mission periods for optimum consumed power Applying the same hybrid GA-based approaches for the soft motion trajectory to investigate their applicability rather than the normal polynomial of Linear Segment with Parabolic Blends LSPB trajectories will be considered in our future investigations Besides, other Evolutionary Algorithms such as Ant Colony and Particle Swarm optimization will be considered The comparisons of these state-of-the-art optimization techniques with the classical optimization, GA, and GA-hybrid techniques are also possible Appendix A Elements of equation of motion matrices À Á À M11 ẳ Iact1 ỵ l21 m31 ỵ m2 ỵ m3 ỵ l22 m32 ỵ m3 ỵl23 m33 þ l1 l2 C2 ðm2 þ 2m3 Þ þ m3 l2 l3 C3 ỵ l1 l3 C23 ị: À Á À Á M12 ¼ M21 ¼ l22 m32 þ m3 þ l23 m33 þ l1 l2 C2 m22 þ m3 þ m23 ð2l2 l3 C3 þ l1 l3 C23 ị: M13 ẳ M31 ẳ l23 m33 þ m23 ðl2 l3 C3 þ l1 l3 C23 Þ: M22 ẳ Iact2 ỵ l22 m32 þ m3 þ l23 m33 þ m3 l2 l3 C3 : M23 ẳ M32 ẳ l32 m33 ỵ m3 l22l3 C3 : M33 ẳ Iact3 ỵ l32 m33 : Figure 11 The second trajectory and the manipulator configuration based on GA-PS optimization technique notice that displacement of joint reaches its maximum value a bit after the mid-time of the mission before it decreases to satisfy its final value at the end of the trajectory In Fig 9, the angular velocities of the first two joints are almost the same, while the velocity curve of the third joint has the same sinusoidal shape of that in the first mission The second trajectory shown in Fig 10 is different from the first trajectory Conclusion Two hybrid optimization algorithms with constrained ranges of the variables during the search have been presented and compared to design an optimal trajectory for planar redundant manipulator based on minimum power to be consumed These techniques are GA-Fmincon and the GA-PS The manipulator is following two trajectories which are distinguished by the final values of the joints (p/2, 0, 0) and (1, 1, 1) For the two sim- where Iact1, Iact2, and Iact3 represent the mass moment of inertias of the actuators 1, 2, and respectively In the current investigation, the inertias of the actuators are neglected C11 ¼ C22 ¼ C33 ¼ C12 ¼ ÀC21 ¼ À 12 ðm2 l1 l2 S2 ỵ m3 2l1 l2 S2 ỵ l1 l3 S23 ị C13 ẳ C31 ẳ 12 m3 l2 l3 S3 ỵ l1 l3 S23 ị C23 ẳ ÀC32 ¼ À 12 m3 l2 l3 S3 R11 R12 R13 R21 ẳ m2 l1 l2 S2 ỵ m3 2l1 l2 S2 ỵ l1 l3 S23 ịị ẳ m3 l2 l3 S3 ỵ l1 l3 S23 ị ẳ 12 m3 l1 l2 S23 À m3 l2 l3 S3 À 12 m3 l1 l3 S23 ¼0 R22 ¼ R23 ¼ Àm3 l2 l3 S3 R31 ¼ m3 l2 l3 S3 R32 ¼ R33 ¼ À Á À Á À G11 ẳ gl1 C1 m21 ỵ m2 ỵ m3 þ gl2 C12 m22 þ m3 þ gl3 C123 m23 m m3 G22 ẳ gl2 C12 22 ỵ m3 ỵ gl3 C123 G33 ẳ gl3 C123 m23 where S23 ẳ sinh2 ỵ h3 ị, C123 ẳ cosh1 ỵ h2 ỵ h3 ị Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Power optimization of planar redundant manipulator References [1] H Hanafusa, T Yoshikawa, Y Nakamura, Analysis and control of articulated robot arms with redundancy, in: 8th Triennial World Congress, IFAC, XIV, Kyoto, Japan, August 1981, vol 14, pp 78–83 [2] T Yoshikawa, Analysis and control of robot manipulators with redundancy, in: Robotics Research: the First International Symposium, MIT Press, Cambridge, MA, 1984, pp 735–747 [3] G Field, Y Stepanenko, Iterative dynamic programming: an approach to minimum energy trajectory planning for robotic manipulators, in: Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, Minnesota, pp 2755–2760 [4] A.R Hirakawa, A Kawamura, Trajectory planning of redundant manipulators for minimum energy consumption without matrix inversion, in: Proceedings of the 1997 IEEE International Conference on Robotics and Automation, ICRA Part (of 4) – Albuquerque, New Mexico, pp 24152420 [5] C Hansen, J Oăltjen, D Meike, T Ortmaier, Enhanced approach for energy-efficient trajectory generation of industrial robots, in: 8th IEEE International Conference on Automation Science and Engineering, Seoul, Korea, 2012, pp 1–7 [6] T Chettibi, P Lemoine, Generation of point to point trajectories for robotic manipulators under electro-mechanical constraints, Int Rev Mech Eng., IREME (2) (2009) 131–145 [7] G Schreiber, M Otter, G Hirzinger, Solving the singularity problem of non-redundant manipulators by constraint optimization, Proc IEEE/RSJ Int Conf (1999) 1482–1488 [8] K Kazerounian, Z Wang, Global versus local optimization in redundancy resolution of robotic manipulators, Int J Robot Res (5) (1988) 3–12 [9] R Kurtz, H Vincent, Multiple-goal kinematic optimization of a parallel spherical mechanism with actuator redundancy, IEEE Trans Robot Autom (5) (1992) 644–651 [10] Y Davidor, Genetic algorithms and robotics: a heuristic strategy for optimization, in: World Scientific Series in Robotics and Automated Systems, vol 1, Teaneck, NJ, USA, 1991 [11] S Yue, D Henrich, Manipulating deformable linear objects attachable adjustment-motions for vibration reduction, J Robot Syst 18 (7) (2001) 375–389 [12] T Huang, M Li, Z Li, D Chetwynd, D Whitehouse, Optimal kinematic design of 2-DOF parallel manipulators with wellshaped workspace bounded by a specified conditioning index, IEEE Trans Robot Autom 20 (3) (2004) 538–543 [13] A Nearchou, Path planning of a mobile robot using genetic heuristics, Robotica 16 (1998) 575–588 [14] D Garg, M Kumar, Optimization techniques applied to multiple manipulators for path planning and torque minimization, Eng Appl Artif Intell 15 (3–4) (2002) 241–252 [15] X Lei, Y Du, A linear domain system identification for small unmanned aerial rotorcraft based on adaptive genetic algorithm, J Bionic Eng (2) (2010) 142–149 [16] E Shintaku, Minimum energy trajectory for an underwater manipulator and its simple planning method by using a genetic algorithm, Adv Robot 13 (2) (1998) 115–138 [17] W.M Yun, Y.G Xi, Optimum motion planning in joint space for robots using genetic algorithms, Robot Auton Syst 18 (4) (1996) 373–393 [18] Y.D Lee, B.H Lee, Genetic trajectory planner for a manipulator with acceleration parametrization, J Univers Comput Sci (9) (1997) 1056–1073 [19] L Tian, C Collins, An effective robot trajectory planning method using genetic algorithm, J Mechatron 14 (2004) 455– 470 [20] A Khadwilard, P Pongcharoen, Application of genetic algorithm for trajectory planning of two degrees of freedom robot arm with two dimensions, Thammasat Int J Sci Technol 12 (2) (2007) 40–47 [21] A Elshamli, H.A Abdullah, S Areibi, Genetic algorithm for dynamic path planning, in: Proceedings of the Canadian Conference on Electrical and Computer Engineering, Ontario, Canada, May 2004, pp 677–680 [22] E.S Pires, J.T Machado, P.D.M Oliveira, An evolutionary approach to robot structure and trajectory optimization, in: Transactions of the 10th IEEE Conference on Advanced Robotics, Budapest, Hungary, 2001, pp 333–338 [23] B Kazem, A Mahdi, A Oudah, Motion planning for a robot arm by using genetic algorithm, Jordan J Mech Ind Eng (3) (2008) 131–136 [24] C Hansen, J Kotlarski, T Ortmaier, Experimental validation of advanced minimum energy robot trajectory optimization, in: Advanced Robotics (ICAR), 16th International Conference, Montevideo, 2013, pp 1–8 [25] A Ruiz, J Fontes, M Da Silva, The influence of kinematic redundancies in the energy efficiency of planar parallel manipulators, in: Proceedings of the ASME 2015 International Mechanical Engineering Congress and Exposition, IMECE2015, Houston, Texas, pp 1–10 [26] A Ruiz, J Fontes, M da Silva, Energy consumption of redundant planar parallel manipulators: a comprehensive study, in: 23rd ABCM International Congress of Mechanical Engineering, (COBEM 2015), Rio de Janeiro, Brazil, 2015, pp 1–8 [27] A.A Ata, T.R Myo, Optimal point-to-point trajectory tracking of redundant manipulators using generalized pattern search, Int J Adv Robot Syst (3) (2005) 239–244 [28] A.A Ata, T.R Myo, Collision-free trajectory planning for manipulators using generalized pattern search, Int J Simul Model (4) (2006) 145–154 [29] A.A Ata, Optimal trajectory planning of manipulators: a review, J Eng Sci Technol (1) (2007) 32–54 [30] R Chandra, L Rolland, On solving the forward kinematics of 3RPR planar parallel manipulator using hybrid metaheuristics, Appl Math Comput 217 (22) (2011) 8997–9008 [31] R Chandra, L Rolland, Global-local population memetic algorithm for solving the forward kinematics of parallel manipulators, Connect Sci 27 (1) (2015) 22–39 [32] L Rolland, R Chandra, The forward kinematics of the 6–6 parallel manipulator using an evolutionary algorithm based on generalized generation gap with parent-centric crossover, Robotica 34 (1) (2016) 1–22 [33] D.E Goldberg, Genetic Algorithm in Search Optimization, and Machine Learning, Addison Wesley, 1989 [34] J Parker, A Khoogar, D.E Goldberg, Inverse kinematics of redundant robots using genetic algorithms, IEEE Int Conf Robot Autom (1989) (271-176) [35] R.N Jazar, Theory of applied Robotics: Kinematics, Dynamics, and Control, second ed., Springer, New York, USA, 2010 Please cite this article in press as: M.M.Y.B Elshabasy et al., Power optimization of planar redundant manipulator moving along constrained-end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 ... al., Power optimization of planar redundant manipulator moving along constrained- end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Power. .. al., Power optimization of planar redundant manipulator moving along constrained- end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Power. .. al., Power optimization of planar redundant manipulator moving along constrained- end trajectory using hybrid techniques, Alexandria Eng J (2017), http://dx.doi.org/10.1016/j.aej.2017.01.040 Power