THE use of an artificial potential field (APF) to describe the

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1 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 4, AUGUST Fuzzy-GA-Based Trajectory Planner for Robot Manipulators Sharing a Common Workspace Emmanuel A. Merchán-Cruz and Alan S. Morris Abstract This paper presents a novel fuzzy genetic algorithm (GA) approach to tackling the problem of trajectory planning of two collaborative robot manipulators sharing a common workspace, where the manipulators have to consider each other as a moving obstacle whose trajectory or behaviour is unknown and unpredictable, as each manipulator has individual goals and where both have the same priority. The goals are not restricted to a given set of joint values, but are specified in the workspace as coordinates at which it is desired to place the end-effector of the manipulator. By not constraining the goal to the joint space, the number of possible solutions that satisfies the goal increases according to the number of degrees of freedom of the manipulators. A simple GA planner is used to produce an initial estimation of the movements of the robots articulations and collision free motion is obtained by the corrective action of the collision-avoidance fuzzy units. Index Terms Collision avoidance, fuzzy logic, genetic algorithms (GAs), multiple manipulators, trajectory planning. I. INTRODUCTION THE use of an artificial potential field (APF) to describe the workspace of a manipulator is naturally appealing for the implementation of a fuzzy collision-avoidance strategy, since the value of the potential field increases or decreases accordingly to the nearness of obstacles or the goal. In contrast to many methods, robot motion planning based on an APF is able to consider the problems of collision avoidance and trajectory planning simultaneously, as this approach can drive the robot to its desired goal while keeping it from colliding with static or dynamic obstacles present in the workspace. However, the use of this method is often associated with local minima problems which cause the planning algorithms to get stuck in a suboptimal solution. While some previous research has been done in constructing potential fields free from local minima, other research has focused on the development of strategies to identify and move away from local minima within the planning algorithm. Following the original approach proposed by Khatib in [1], several authors have proposed different ways to construct APFs to overcome the local-minima problem in the Cartesian space Manuscript received April 21, 2005; revised October 11, This paper was recommended for publication by Associate Editor H. Zhuang and Editors I. Walker and H. Arai upon evaluation of the reviewers comments. This work was supported in part by the CONACYT-Mexico and the Instituto Politécnico Nacional-Mexico. E. A. Merchán-Cruz was with the University of Sheffield, Sheffield, South Yorkshire S1 3JD, U.K. He is now with the Instituto Politécnico Nacional at the Escuela Superior de Ingeniería Mecánica y Eléctrica, Azcapotzalco, México D.F., C.P , México ( eamerchan@ipn.mx). A. S. Morris is with the Automatic Control and Systems Engineering Department, University of Sheffield, Sheffield, South Yorkshire S1 3JD, U.K. ( a.morris@shef.ac.uk). Digital Object Identifier /TRO [2], as well as in the configuration space (C-Space) representation [3] [7]. More recently, a different alternative considering evolutionary algorithms for the derivation of local minima free AFPs was presented in [8], but its application is restricted to static workspaces due to the computations involved if a dynamic obstacle is considered. With regard to the algorithms for motion planning of robot manipulators using fuzzy logic, very little work can be referred to in comparison to the research done for mobile robots. The first reported approach of a fuzzy navigator for robot manipulators is that reported in [9] and further discussed in [10] [12]. In this approach, the goal is specified as a particular arm configuration to which the manipulator is required to reach, that is, the goal is specified in the joint space of the robot. A fuzzy navigator is presented consisting of individual fuzzy units that govern the motions of each link individually and determine the distance to an obstacle by constantly scanning a predetermined rectangular area surrounding each link. The application of this approach is presented for two-degree-of-freedom (DOF) and 3-DOF planar manipulators in static environments. The paper by Althoefer [10] presents a brief discussion of the application of this approach where a 3-DOF planar manipulator deals with a point obstacle whose trajectory is known. In [13], a similar approach is presented, where an harmonic potential field is used to drive a 2-DOF planar manipulator to its desired goal in the joint space. The cases of static obstacles and that of a trajectory-known point obstacle are also considered. A neurofuzzy implementation of this approach is used to solve the same proposed cases as presented in [14]. This paper presents a novel fuzzy genetic algorithm (GA) approach to tackle the problem of trajectory planning of two manipulators sharing a common workspace, where the manipulators have to consider each other as a moving obstacle whose trajectory or behavior is unknown and unpredictable and which, under certain circumstances, can act as a static obstacle (i.e., when one manipulator has reached its goal and does not need to move any further). The start conditions of the system are indicated as the current manipulator configurations, while the goal is not restricted to a given set of joint values, but is specified in the workspace as coordinates at which it is desired to place the end-effector of the manipulator. By not constraining the goal to the joint space, the number of possible solutions that satisfy the goal increases according to the number of DOFs of the manipulators. II. FUZZY NAVIGATION The application of fuzzy logic to generate useful information to drive a manipulator from a start configuration to a given goal configuration was first investigated in [9]. This approach, where /$ IEEE

2 614 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 4, AUGUST 2006 Fig. 1. (a), (b): Althoefer approach applied successfully. (c), (d): Althoefer approach fails to reach the desired goal. the problem is formulated and solved entirely in the C-Space obtaining a suitable trajectory in the workspace of the manipulator, has been used as a basis for the development of other similar approaches [10] [15]. However, its application has been limited to one manipulator with two or three DOFs moving in static environments, or, in the case of the 3-DOF planar manipulator, considering a mobile point obstacle moving along a linear trajectory. Although this approach has been successfully applied for the aforementioned cases, it has been restricted to simply reach prespecified angular positions for the manipulator s articulations and it is not suitable for application to redundant manipulators. This is because when a large number of DOFs are considered, some of the joints reach their angular goals before others. This generates an effect comparable to that of search algorithms based on potential fields getting stuck in a locus that the manipulator cannot pass through, because while the fuzzy units for some articulations are compensating their outputs due to the presence of an obstacle, others do not change, as they have already reached their individual goals. To illustrate this limitation, Fig. 1 shows two 7-DOF planar manipulators where one, manipulator B, is acting as a static obstacle. The starting configuration of manipulator A is (90, 0, 0, 0, 0, 0, 0), and the following goals have been considered: From these examples, it can be appreciated that specifying the goal as a set of joint values for the manipulator that describe the

3 MERCHÁN-CRUZ AND MORRIS: FUZZY-GA-BASED TRAJECTORY PLANNER 615 desired arm configuration is easily solvable when the final configuration is located some distance away from the obstacle, as illustrated in Fig. 1 (a) and (b). However, when the desired final configuration happens to be close to an obstacle, the approach fails by getting stuck, as shown in Fig. 1 (c) and (d), since the fuzzy units do not provide an output large enough to overcome the obstacle, as some of the articulations have already reached their individual goals and, at the same time, are away from the zone of influence of the obstacle. In practical terms, the end-effector of a manipulator is often required to be positioned somewhere within the workspace of the manipulator, expressed in ( ) coordinates, where the base of the manipulator is the origin of such a reference system. This would indicate the need for solving the inverse kinematics of the manipulator to find the set of angular positions of the robot s articulations that comply with the desired goal. III. FUZZY-GA TRAJECTORY PLANNER The fuzzy-ga-based trajectory planner (FuGABTP) approach presented in this section employs a simple GA-based trajectory planner (simple GABTP) to initially drive the manipulators towards their goals. Individual fuzzy units provide a correction in the displacement of each articulation in the event that a link is approaching an obstacle. The goals are specified in workspace coordinates to take full advantage of the dexterity of the redundant manipulators. The manipulators are modeled as obstacles in the workspace using the APF method. Since this is a local approach, the APF is only calculated for the near vicinity of the links of the manipulator to save time in the modelling process. Because each manipulator is considered by the other as a mobile obstacle where the position and orientation of their links are constantly changing, there would be a major computational problem if the APF for the whole of the shared workspace were to be calculated each time one of the manipulators has moved as would be necessary in a global approach. The simple GABTP algorithm determines an initial set of configurations that best reduce the cartesian error between the current end-effector position of the manipulator and the goal. The fuzzy units provide a correction based on the magnitude of the AFP that modifies the original displacement of the articulations given by the GABTP. This correction ranges from slowing the velocity of the articulation to completely change the direction of the original motion. The fuzzy units determine the best direction to correct the original motion of an articulation by determining the direction of the motion that implies an increment in the magnitude of the APF at the particular instant considered by the planner. This results in a signed correction to affect the initial estimation given by the GABTP. The direct output from the FuGABTP algorithm is given by (1) Fig. 2. General outline of fuzzy-ga trajectory planner. The diagram shown in Fig. 2 illustrates the general outline of the trajectory planner. The simple GABTP algorithm carries out an initial estimation of the motion of the manipulators without considering the presence of any obstacles. In order to favor fully extended motion of the manipulator to minimize unnecessary displacements of the links when moving in a space free from obstacles, the fitness function for the GA search considers the degree of extension of the manipulator (2) where Error ; ; manipulator s base coordinates; goal coordinates; DKM ; DKM direct kinematic model as a function of s. The fuzzy correction is achieved by individual Mamdami-type fuzzy units that provide a correction value per articulation in order to avoid collision. Each of the units, as illustrated in Fig. 3, has three inputs: obstacle direction, APF magnitude and error, and a single output named correction. The obstacle direction can either be right if the magnitude of the APF increases by moving the considered link around its axis of movement in a positive direction, or left if the magnitude of the APF increases in the opposite direction. (2)

4 616 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 4, AUGUST 2006 Fig. 3. Fuzzy correction unit. Fig. 5. Fuzzy sets for magnitude APF. Fig. 4. Fuzzy sets for obstacle direction. The magnitude of the APF indicates how close a link is from a nearby obstacle, and the error is the cartesian error between the actual position of the end-effector and the goal. Finally, the correction output indicates the positive or negative correction needed to ensure that no collision takes place. The range of each variable, or universe of disclosure, is partitioned into fuzzy sets. Each of the sets, represents a mapping by which the degree of membership of within a fuzzy set is associated with a number in the interval [0,1]. The functions employed to represent the fuzzy sets have been chosen to be asymmetrical triangular functions as they are easily computed. The degree of membership of each input is calculated by where are the coordinates of the left and right zero crossing, and is the coordinate where the fuzzy set becomes 1. For the values that lie either at the left, (4), or right, (5), of each input range, the triangular functions are continued as constant values of magnitude 1 as and if (5) if Figs. 4 7 illustrate the partition of the fuzzy sets that describe each of the fuzzy variables. The defuzzification of the data into a crisp output is accomplished by combining the results of the inference process and then computing the fuzzy centroid of the area. The weighted strengths of each output member function if if if if (3) (4) Fig. 6. Fuzzy sets for error. Fig. 7. Fuzzy sets for output correction. are multiplied by their respective output membership function center points and summed. Finally, this area is divided by the sum of the weighted member function strengths, and the result is taken as the crisp output. The correction output of each fuzzy unit is given as the product of the intersection of the following fuzzy rules. 1) If (obstacle is left) and (APF magnitude is small), then (correction is SN). 2) If (obstacle is left) and (APF magnitude is med), then (correction is MN). 3) If (obstacle is left) and (APF magnitude is medbig), then (correction is MBN). 4) If (obstacle is left) and (APF magnitude is big), then (correction is BN). 5) If (obstacle is left) and (APF magnitude is zero), then (correction is zero). 6) If (obstacle is right) and (APF magnitude is small), then (correction is SP). 7) If (obstacle is right) and (APF magnitude is med), then (correction is MP). 8) If (obstacle is right) and (APF magnitude is medbig), then (correction is MBP).

5 MERCHÁN-CRUZ AND MORRIS: FUZZY-GA-BASED TRAJECTORY PLANNER 617 Fig. 8. FuGABTP approach, schematic representation. 9) If (obstacle is right) and (APF magnitude is big), then (correction is BP). 10) If (obstacle is right) and (APF magnitude is zero), then (correction is zero). 11) If (obstacle is left) and (error is zero), then (correction is zero). 12) If (obstacle is right) and (error is zero), then (correction is zero). The final fuzzy correction for each joint is also affected by the correction of the distal links, since the motion of the distal links is not only given by the action of a single articulation. Fig. 8 illustrates the overall approach for a 3-DOF manipulator, where (1) shows the manipulator at the current configuration, (2) indicates the initial GA estimation that simply reduces the Cartesian error between the current position of the end-effector and its goal, and finally, (3) shows the final position of the manipulator after the fuzzy units have corrected the final s for the manipulator. Since the s are obtained by the planner, the velocities and accelerations necessary to calculate the required torque of each articulation are easily calculated as a function of time. IV. SIMULATIONS FOR TWO MANIPULATORS SHARING A COMMON WORKSPACE In this section, the approach described is applied to solve the motion-planning problem for two-manipulator systems, with each having (a) three DOFs and (b) seven DOFs. Initially, the problem of static obstacles is dealt with by considering manipulator B as a stationary obstacle. Following this, the case is covered where both manipulators are required to move to reach independent goals while considering the other manipulator as a mobile obstacle with unknown trajectory. For the study of the static obstacle case, manipulator A is required to reach a goal in the workspace from a given starting configuration, while manipulator B remains in a stationary position, where it behaves as a static obstacle. The following cases are presented, as they test the performance of the algorithm in different circumstances and are considered to be representative of the common task that the manipulator performs in its workspace. Table I summarizes these cases for the 3-DOF and 7-DOF manipulators. The example cases for the 3-DOF system confirm the effectiveness of the proposed approach for a planar manipulator with low redundancy. For the 7-DOF manipulator, Cases I IV correspond to the cases (a, b, c, d) of Section II solved using Althoefer s approach and are presented to prove that the proposed algorithm successfully solves those cases where the other failed. Table II summarizes the results for the application of the fuzzy-ga trajectory planner considering a static obstacle. Table II also compares the execution times obtained and number of trajectory segments with those obtained with the GABTP algorithm with adaptive population size as described in [16]. It can be immediately observed that the execution time necessary to solve each case is considerably lower than that of the GABTP. However, the number of segments that describe the trajectory of the manipulator returned by the FuGABTP is higher than those returned by the GABTP, as a result of the constant correction in the trajectory made by the fuzzy units in order to avoid a collision. Fig. 9 illustrates the described path and trajectory profiles for Case III of the 3-DOF system. The figure shows how the 3-DOF manipulator moves from the given start configuration to the given end-effector goal in the workspace. Collision with the static manipulator is successfully avoided. The trajectory produced by running the FuGABTP algorithm under MATLAB consisted of 50 segments and was obtained in 2 s on an IBM Workstation with a 3.8-GHz Pentium IV processor. The average time to compute each iteration was 0.04 s.

6 618 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 4, AUGUST 2006 TABLE I REPRESENTATIVE CASES FOR THE TWO-MANIPULATOR SYSTEM TABLE II SUMMARY OF RESULTS FOR THE TWO-MANIPULATOR SYSTEM, CONSIDERING ONE AS A STATIC OBSTACLE The combination of the simple GABTP algorithm with the fuzzy correction drives the manipulator towards the desired goal in the workspace while keeping it away from the obstacle by moving the link away from it through the correction action determined by the fuzzy units. For the purpose of comparing two different solutions to Case III, Fig. 10 shows the described path and trajectory profiles obtained with the GABTP algorithm with fuzzy adaptive population [16]. It can be appreciated that while with the FuGABTP algorithm the manipulator avoids colliding with the obstacle by moving the element at risk of collision away from the obstacle, the GABTP algorithm with fuzzy adaptive population drives the manipulator around the obstacle as a result of moving the end-effector under the influence of the APF. As a result of the different path followed by the manipulator s elements, the obtained trajectory profiles vary, reflecting the changes between one trajectory and the other. The manipulator is successfully driven to its goal with both approaches and, although the number of segments returned by the FuGABTP is greater than the one from the GABTP, the total execution time is considerable lower. Regarding the 7-DOF manipulator, Cases I IV corresponding to the four cases illustrated in Section II are used to illustrate the disadvantages of constraining the solution of a redundant system to a single goal configuration, since this results in the manipulator not reaching its goal configuration in Cases III and IV. For its solution with the FuGABTP, the goals have been expressed in their corresponding coordinates in the workspace (Table I), in order to take full advantage of the dexterity of a redundant

7 MERCHÁN-CRUZ AND MORRIS: FUZZY-GA-BASED TRAJECTORY PLANNER 619 Fig. 9. Described path and trajectory profiles for the 3-DOF manipulator case considering a static obstacle obtained with the FuGABTP algorithm. Fig. 10. Described path and trajectory profiles for the 3-DOF manipulator case considering a static obstacle obtained with the GABTP algorithm. manipulator to reach its goal. Fig. 11 shows the described path and trajectory profiles for Case III. Fig. 12 presents a detailed motion sequence for Case IV. The effect of the fuzzy units to correct the trajectory to avoid collision with the static manipulator can be clearly appreciated. Dynamic Case: In this section, the FuGABTP algorithm is applied to solve the trajectories for two manipulators sharing a common workspace where both manipulators have the same priority when moving towards individual goals. As in the previous section, the examples presented here are just a selection to illustrate the applicability of the algorithm when one manipulator has to consider the other as a mobile obstacle of unknown trajectory. Table III summarizes these representative cases where each one tests the algorithm under different circumstances. When individual goals have been specified and these require that both manipulators move, the algorithm starts solving the trajectory for both manipulators in parallel as described in the algorithm outline (Fig. 4) by considering the current individual configuration of each manipulator. The simple GABTP built in the algorithm provides an initial approximation which is corrected by the fuzzy units in the event that this proposed configuration lays within the vicinity of an obstacle, in this case, the links of the other manipulator. The components of the algorithm remain the same as the static obstacle case, with the only difference that the manipulator B is now required to move. Fig. 13 illustrates the motion of the 3-DOF manipulators for the example Case II. It can be observed that both manipulators adapt their trajectories and successfully avoid collision. The FuGABTP took 1.25 s to return the 39 pairs of segments that describe the trajectories of the manipulators. The case of two 7-DOF planar manipulators is shown in Fig. 14, where the manipulators are required to reach the goals from the starting configuration according to Case I defined in Table III. In this example, the applicability of the present approach for a highly redundant system is proven. The FuGABTP took s to return 71 pairs of segments that described the trajectories of the manipulators, taking an average of 0.06 s per iteration. The trajectory profiles for the two 7-DOF manipulators are shown in Figs The FuGABTP maintains the manipulators moving towards their goals as a result of the simple GATP which optimizes the configuration of the manipulators to reduce the error to the de-

8 620 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 4, AUGUST 2006 Fig. 11. Described path and trajectory profiles for the 7-DOF manipulator case considering a static obstacle obtained with the FuGABTP algorithm. Fig. 12. Detailed sequence of motion for the 7-DOF manipulator considering a static obstacle (Case IV). sired goals. Solving in parallel for both manipulators, as described in Fig. 2, produces individual trajectories that do not require the further scheduling required by traditional decoupled methods. Given that both manipulators have the same priority, both adapt their trajectories when necessary to avoid collision. This avoids over compensating the motion of a single given manipulator as happens with a conventional hierarchical approach. The example cases have also been solved by the GABTP algorithm with fuzzy adaptive population [17] to establish a point of comparison between the two approaches. Table IV summarizes the execution times for the 3-DOF and 7-DOF systems. V. CONCLUSION This paper has presented a novel approach based on a combination of fuzzy logic and genetic algorithms to solve in parallel the trajectory planning of two manipulators sharing a common workspace where both manipulators have to adjust their trajectories in the event of a possible collision. The proposed planner combines a simple GABTP with fuzzy correction units. The simple GABTP solves the trajectory planning problem as an initial estimation without considering the influence of any obstacles. Once this stage is reached, fuzzy units assigned to each articulation verify that the proposed s do not take the manipulator to a possible collision based on the magnitude of the APF exerted by the manipulators when considered as obstacles. If any of the original s takes a link or links of the manipulator near the influence of the modelled APF, the fuzzy units evaluate a correction value for the that corresponds to the link and articulation involved in the possible collision and modifies not only the for that articulation, but also modifies the s for the more anterior articulations in the case where a distal articulation is involved. The approach presented here has shown a robust and stable performance throughout the different simulated cases. It produced suitable trajectories that are easily obtained since the direct output from the FuGABTP algorithm is a set of s after each iteration. Since the planning is done in parallel for both manipulators, no further planning or scheduling of their motions are necessary as is required with decoupled approaches, since the trajectories are free from any collision. The implementation of the simple GABTP in the algorithm solves the problem of over-compensation associated with the fuzzy units by maintaining the direction of the manipulators towards the desired goal at all times, keeping the links of the manipulator from over-reacting when approaching an obstacle. Finally, the proposed algorithm was applied to solve the trajectory planning problem of two manipulators sharing a common workspace with individual goals. In this case, each manipulator considers the other as a mobile obstacle of unknown trajectory. The two considered cases, 3-DOF and 7-DOF manipulator systems, were successfully solved, obtaining suitable trajectories for both manipulators. Referring to the average time per trajectory segment that the algorithm took to solve for 3-DOF and 7-DOF manipulators in both, the static and the dynamic scenarios, ranging from 0.03 to 0.06 s to produce a trajectory update, it can be concluded that

9 MERCHÁN-CRUZ AND MORRIS: FUZZY-GA-BASED TRAJECTORY PLANNER 621 TABLE III REPRESENTATIVE CASES FOR THE TWO-MANIPULATOR SYSTEM WITH INDIVIDUAL GOALS Fig. 13. Detailed motion sequence for the two 3-DOF manipulators with individual goals Case II. Fig. 15. Joint displacements for Manipulator A Case I. Fig. 14. Detailed motion sequence for the two 7-DOF manipulators with individual goals Case I. the approach discussed in this paper is suitable for its consideration on real-time applications, since the compilation of the algorithms that conform the suggested approach into executable files and as faster computer power is available, the iteration time could be reduced even more. APPENDIX I SIMPLE GA-BASED TRAJECTORY PLANNER The motion planning of robot manipulators is more complex than that of a single point moving in space. The trajectory described by the tip of the manipulator is kinematically constrained by the links and joints of the manipulator which define its kinematic structure. The simplest problem in trajectory planning is that of considering a workspace free of obstacles, where the manipulator is required to move from a known Fig. 16. Joint velocities for Manipulator A Case I. starting configuration to a desired goal expressed in coordinates in the workspace of the manipulator. The DKM of the manipulators is used to relate the error given from the initial/current configuration of the manipulator to its desired goal. This transformation from the joint space to the workspace is particularly useful in the implementation of a GA for local path planning, since the planner has to calculate the next position

10 622 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 4, AUGUST 2006 Fig. 17. Joint accelerations for Manipulator A Case I. Fig. 20. Joint velocities for Manipulator B Case I. Fig. 21. Joint accelerations for Manipulator B Case I. Fig. 18. Joint torques for Manipulator A Case I. Fig. 22. Joint torques for Manipulator B Case I. Fig. 19. Joint displacements for Manipulator B Case I. that best minimizes the error. This calculation is performed by finding the best set of joint displacements within a pre-established range of s per unit of time until the goal is reached, without solving the inverse kinematics of the manipulator or dealing with singularities in the case of redundant manipulators. Each chromosome of the population considered by the GA is composed of substrings of equal length corresponding to a particular joint variation. Fig. 23 shows how these strings are decoded from the binary representation into joint increments/ decrements which are then applied to the DKM of the manipulators to calculate the new error due to these variations. In this representation, determines the minimum increment, and is the number of joints in the manipulator. The algorithm returns the set of s that best reduce the error, updates the configuration of the manipulator, and continues until the goals are reached. The algorithm performs a search for the best set of joint values that minimizes the error to the goal within the prespecified range of possible values for each joint. The trajectory is then updated and the resulting position is evaluated. If the goal has been reached, the planner terminates its execution;

11 MERCHÁN-CRUZ AND MORRIS: FUZZY-GA-BASED TRAJECTORY PLANNER 623 TABLE IV SUMMARY OF RESULTS FOR THE TWO-MANIPULATOR SYSTEM WITH INDIVIDUAL GOALS Fig. 23. Chromosome structure. Fig. 24. General outline for the planner. otherwise, the process is repeated starting from the new updated configuration of the manipulator. Fig. 24 outlines the algorithm. The GA search, illustrated in Fig. 25, is based on a simple GA structure where the maximum number of generations is a set value, a new population is generated and its fitness function evaluated. If a previous backup of a fittest chromosome exists, it is copied into the new population replacing the weakest chromo- Fig. 25. Outline for the GA search process. some. A new population is generated and its fitness is computed. This process is repeated until the average fitness of the population has become stable, or the value of the fittest chromosome has not improved in a space of ten generations. At this stage,

12 624 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 4, AUGUST 2006 the algorithm is considered to have found the best solution. If the maximum number of generations is reached before an optimum solution is found, the algorithm returns the best solution derived up to that point. Since the error between the current/initial and a desired position of the tip of the manipulator is to be minimized, the fitness function can be expressed as where Error ; goal coordinates; DKM ; DKM direct kinematic model as a function of s. This simple fitness function drives the simple GABTP from its initial configuration to a goal given in coordinates of the workspace of the robot manipulator. This approach can be used either for planar 2-D manipulators or 3-D spatial configurations as this condition is contained within the kinematic description of the manipulator. REFERENCES [1] O. Khatib, Real-time obstacle avoidance for manipulators and mobile robots, Int. J. Robot. Res., vol. 5, pp , [2] R. Volpe and P. K. Khosla, Manipulator control with superquadric artificial potential functions: theory and experiments, IEEE Trans. Syst., Man, Cybern., vol. 20, no. 6, pp , Nov./Dec [3] E. Rimon and D. E. Koditschek, Exact robot navigation using artificial potential functions, IEEE Trans. Robot. Autom., vol. 8, no. 5, pp , Oct [4] D. E. Koditschek, The control of natural motion in mechanical systems, J. Dyn., Syst., Meas. Control, vol. 113, pp , [5], Some applications of natural motion, J. Dyn., Syst., Meas. Control, vol. 113, pp , [6] C. I. Connolly, Harmonic functions as a basis for motor control and planning, Dept. Comput. Sci., Univ. Mass., Amherst, [7] J. Kim and P. K. Kohsla, Real time obstacle avoidance using harmonic potential functions, IEEE Trans. Robot. Autom., vol. 8, no. 3, pp , Jun [8] P. Vadakkepat, C. T. Kay, and W. Ming-Liang, Evolutionary artificial potential fields and their application in real time robot path planning, in Proc. Congr. Evol. Comput., 2000, vol. 1, pp [9] K. Althoefer and D. A. Fraser, Fuzzy obstacle avoidance for robotic manipulators, Neural Netw. World, vol. 6, pp , (6) [10] K. Althoefer, Neuro-fuzzy motion planning for robotic manipulators, Dept. Electron. Elect. Eng., King s College London, Univ. London. London, U.K., 1997, p [11] K. Althoefer, B. Krekelberg, D. Husmeier, and L. D. Seneviratne, Reinforcement learning in a rule-based navigator for robotic manipulators, Neurocomputing, vol. 37, pp , [12] K. Althoefer, L. D. Seneviratne, P. Zavlangas, and B. Krekelberg, Fuzzy navigation for robotic manipulators, Int. J. Uncertainty, Fuzziness, Knowledge-Based Syst., vol. 6, pp , [13] J. B. Mbede, X. Huang, and M. Wang, Fuzzy motion planning among dynamic obstacles using artificial potential field for robot manipulators, Robot. Auton. Syst., vol. 32, pp , [14] J. B. Mbede and W. Wei, Fuzzy and recurrent neural network motion control among dynamic obstacles for robot manipulators, J. Intell. Robot. Syst., vol. 30, pp , [15] P. Zavlangas and S. G. Tzafestas, Industrial robot navigation and obstacle avoidance using fuzzy logic, J. Intell. Robot. Syst., vol. 27, pp , [16] E. A. Merchán-Cruz, Soft-computing techniques in the trajectory planning of robot manipulators sharing a common workspace, Ph.D. dissertation, Dept. Autom. Control Syst. Eng., Univ. Sheffield, Sheffield, U.K., [17] E. A. Merchán-Cruz and A. S. Morris, GA trajectory planner for robot manipulators sharing a common workspace, in Proc. IASTED Int. Conf. Appl. Simul. Modeling, 2004, pp Emmanuel A. Merchán-Cruz received the B.Eng. degree in industrial robotics engineering and the M.Sc. degree in mechanical engineering, specializing in robotics, from the National Polytechnic Institute (IPN), Mexico City, México, in 1998 and 2000, respectively, and the Ph.D. degree from the University of Sheffield, Sheffield, U.K., in 2005, after carrying out research on trajectory planning for multirobot manipulator systems. Currently, he is an Assistant Professor and Head of the Robotics Lab, Postgraduate Studies and Research Department, Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Azcapotzalco, México. His main research interests are automatic systems, robotics, biomechanics, and process optimization. Alan S. Morris was born and educated in the U.K. He received the B.Eng. degree in electrical and electronic engineering and the Ph.D. degree from the University of Sheffield, Sheffield, U.K., in 1969 and 1978, respectively. Following five years of employment with British Steel as a Research and Development Engineer in control system applications, he returned to the University of Sheffield to carry out research in electric arc furnace control. He was a Lecturer, and, more recently, a Senior Lecturer, with the University of Sheffield, where he now holds the position of B.Eng. Course Leader. His main research interests lie in robotics, and he is now the author of over 140 refereed research papers.