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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. Mohamedc, Atef A. Atad
a Department of Mechanical Engineering, Alexandria University, Alexandria 21544, Egypt
b 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
Received 5 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.
© 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/).
1. 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
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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
[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-to-point 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.
Figure 1 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 (Fmin-con) 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.
2. Problem formulation
In our investigation, a 3R planar redundant manipulator shown in Fig. 1 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 m, length moment of inertia about its center of rotation It 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:
dt{dq) (öl) Q';
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. Q refers to the generalized forces acting on the system under investigation. The torques applied to the manipulator joints represent these generalized forces.
The kinetic and potential energies of the manipulator are given by [35]:
K.E. = 2 (h + /«)6? + ^ /2(01 + 02) + ^ Ih2022 + ^ m1V\
1 • • • 2 1 1
+1 /3(01 + 02 + 03) +1 mEVl + 2 /h302
P. E. = 1 m1gl1 sin 01 + m2g(l2 sin 01 +1l2 sin 02
+ m3g ( l1 sin 01 + l2 sin 02 +113 sin 03
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:
M11 M12 M13" M21 M22 M23 M31 M32 M33 R11 R12 R13 + R21 R22 R23
R31 R32 R33
C11 C12 C13 021
C21 C22 C23 02
C31 C32 C33 023
01 02 "G„"
01 03 + G22
.02 03 . G33 _
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.
3. 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]:
1. The cost or fitness function used to resolve the redundancy has no requirements for continuity in derivatives. So any cost function can be selected.
2. There is no need to compute the Jacobian matrix.
3. The GA solution needs only the forward kinematics equation, which is easy to obtain.
4. 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.
5. 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:
0,(t) = a,t4 + b.t3 + c,t2 + d,t + e, (5)
where a,-, b,, c,-, d,, and e,- are coefficients to be determined based on the end-conditions as well as the cost function to be minimized. The constant e, always represents the initial angle while the constant d, 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 d, is always zero. The other initial conditions are as follows:
01 (tf) = p rad, e2(tf) = 0, and 03(t/) = 0,
0,(0) = 0 rad, 0,(0) = 0, and 0,(t/) = 0
where , = 1, 2, and 3 and / 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]:
Penalty Function = X IK'T-)2! = X I^T')2! (6)
where , refers to the joint number which varies between 1 and 3.
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.
4. 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
Figure 2 The program flowchart and the pseudo code of the combined GA-PS and GA-Fmincon optimization techniques.
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. 2 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 — 1 kg, L1 — 0.2 m Ih1 — 5 x I1 m2 — 2 kg, L2 — 0.4 m Ih2 — 5 x I2 m3 — 1 kg, L3 — 0.2 m Ih3 — 5 x I3
The simulation results have been conducted using the MATLAB 7.1 on a PC 8 GB ram and 3.1 GHz processor of 6 MB cache. The parameters of optimization techniques defined in the MATLAB programs are listed in Table 1.
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 1 s and 20 s. For each optimization technique, the joints' displacements of the manipulator are not constrained as they can be any value between 0 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, 01, 02, and 03 of the first trajectory are (p, 0,0 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. 3.
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 8 and 20 s, the optimum trajectory time is considered equal 8 s for compromising between the fastness of mission and the power curtailing. Table 2 contains the values of the consumed powers for both trajectories and both optimization techniques at two trajectory periods, 8 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 3. As mentioned before, according to the values of the joints' displacement and velocities at the end points of the trajectories, the values of e,'s and diS coefficients are zeros. For each technique, the program was run with constraints on the joints' angles (—p/2 6 ¿i 6 ft/2). 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 3. The listed values of the
Table 1 The parameters used with the optimization
techniques.
Population size 400
Initial range of population values [0;0.3]
Generation 100
The fraction of genes to be swapped Cross over
between individuals heuristic = 0.4
Hybrid function Fmincon or PS
Compute fitness functions of a Always
population in parallel
Reproduction Elite count = 2,
crossover rate =1.2
Mutation Gaussian (scale = 1,
shrink =1)
Crossover Scattered function
Migration Forward (fraction 0.2,
interval = 20)
Figure 3 Total consumed powers during using Hybrid GA-Fmincon and Hybrid GA-PS techniques (a) the first mission and (b) the second mission.
Table 2 The minimum power at tf = 8 s and tf = 20 s using Ga-Fmincon and Ga-PS techniques.
First trajectory (GA-Fmincon) First trajectory (GA-PS)
tf = 8 s Power = 6.2798 N m/s tf = 8 s Power = 6.2798 N m/s
tf = 20 s Power = 1.0048 N m/s tf = 20 s Power = 1.0202 N m/s
Second trajectory (GA-Fmincon) Second trajectory (GA-PS)
tf = 8 s Power = 5.8575 N m/s tf = 8 s Power = 4.4847 N m/s
tf = 20 s Power = 0.7176 N m/s tf = 20 s Power = 0.7176 N m/s
Table 3 The coefficients of the angular displacements polynomials using two hybrid optimization techniques.
Coefficient Mean value Standard Coefficient Mean value Standard Coefficient Mean value Standard
deviation deviation deviation
First trajectory (GA-Fmincon)
a1 2.0614 * 10 2.0156e—04 b1 -9.434* 10-3 0.0032 c1 8.683 * 10-2 0.0129
a2 -5.1 * 10-4 4.9842e-04 b2 8.162* 10-3 0.008 c2 -3.266 * 10-2 0.0319
a3 67.709 * 10-4 0.0066 b3 -108.3 * 10-3 0.1057 c3 43.334 * 10-2 0.4230
First trajectory (GA-PS) -9.436 * 10-3 8.683 * 10-2
al 2.0625 * 10-4 2.2082e-04 b1 0.0035 c1 0.0141
a2 -5.1 * 10-4 5.4604e-04 b2 8.165 * 10-3 0.0087 c2 -3.266 * 10-2 0.0349
a3 67.710 * 10-4 0.0072 b3 -108.3 * 10-3 0.1158 c3 43.334 * 10-2 0.4633
Second trajectory (GA-Fmincon)
ai 2.394 * 10-4 2.5479e-04 b1 -7.736 * 10-3 0.0041 c1 6.2195 * 10-2 0.0163
a2 4.193 * 10-4 2.4816e—04 b2 -10.62* 10-3 0.004 c2 7.371 * 10-2 0.0159
a3 -69.41 * 10-4 0.006 b3 107.1 * 10-3 0.0957 c3 -39.74* 10-2 0.383
Second trajectory (GA-PS)
a1 2.827 * 10-4 2.7911e—04 b1 0.618 * 10-3 0.0045 c1 2.8777 * 10-2 0.0179
a2 -0.893 * 10-4 2.7185e—04 b2 -2.477 * 10-3 0.0043 c2 4.1159 * 10-2 0.0174
a3 53.222 * 10-4 0.0066 b3 -89.06 * 10-3 0.1049 c3 38.750 * 10-2 0.4195
Figure 4 The torques applied to the joints during the first trajectory based on GA-Fmincon optimization technique.
Figure 5 The angular displacements of the joints during the first trajectory based on GA-Fmincon optimization technique.
coefficients are the mean values of 7 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-
sented in Figs. 4, 5, 6, and 7. Fig. 4 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. 6. The first trajectory and the manipulator configurations at each selected end-
■e -o.»l-.-.-.-.-.-.-.-1
^012345678
Time of End Effector Trajectory (s)
Figure 6 The angular velocities of the joints during the first trajectory based on GA-Fmincon optimization technique.
0.8 | i 07 \
\ Coordlantc of the End Effector (ni)
Figure 7 The first trajectory and the manipulator configuration based on GA-Fmincon optimization technique.
effector location are shown in Fig. 7. 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 2. Thus, the simulation results of the GA-PS method are adopted and presented in Figs. 8-11. Fig. 8 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-
-0.15--1-1-1-1-1-'-1-1
012345678
Time of End Effector Trajectory (s)
Figure 8 The torques applied to the joints during the second trajectory based on GA-PS optimization technique.
-------,-,-,-,-.-,-,
012349678 Time of End Effector Trajectory (s)
Figure 9 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,
Figure 10 The angular velocities of the joints during the second trajectory based on GA-PS optimization technique.
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 1 s. and that of 4 s. Between the mission time of 4 s and 8 s, the power decreases with a moderate rate. Starting from a mission period of 8 s, the variation in the power is very trivial. Consequently, it is not recommended to decrease the mission period below 8 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.
Figure 11 The second trajectory and the manipulator configuration based on GA-PS optimization technique.
notice that displacement of joint 3 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.
5. 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-
Appendix A
Elements of equation of motion matrices
Mu — Iact1 + l2{f + m2 + m3) + l2(m3L + m3) +l2 (m3) + I1I2C2K + 2m3) + m3)hhC3 + (1I3C23).
M12 = M21 = l2 (f + m3) + l2 (f) + hhC2 (f + m3)
+ f (2kl3C3 + hhC23).
M13 = M31 = l2 (231) + m (hhC3 +1113 C23).
M22 = Iact2 + + mi) + l2 (f) + mil2l3C3.
M23 — M32 — l32 (f) + . M33 = Iaca + l32(f).
where Iact1, Iact2, and Iact3 represent the mass moment of inertias of the actuators 1, 2, and 3 respectively. In the current investigation, the inertias of the actuators are neglected.
C11 — C22 — C33 — 0
C12 — -C21 — -5 (m2hhS2 + m3{2kkS2 + hhSis) C13 — -C31 — -1 m3(l2l3S3 + hhS23) C23 — — C32 — — 1 m3l2 l3S3
R11 — -(m2hhS2 + m3(2l1l2S2 + ^23))
R12 — -m3(l2l3S3 + hhS23)
R13 — -1m3l1l2S23 - m34^3 -1 m3khS23
R21 — 0
R22 — R23 — - mi^hSi R31 — m3l2l3S3 R32 — R33 — 0
G11 — gl1 C1 (m + m2 + mi) + gl( Cn(f + mi) + gl3 C123 (f) G22 — gl2C12 (m + mi) + gliC123 (f) G33 — gl3 C123 (m3)
where S23 — sin(h2 + h3), C123 — cos(h1 + h2 + h3).
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