NAVIGATION OF A MOBILE ROBOT IN OUTDOOR ENVIRONMENTS

Journal of the Chinese Institute of Engineers, Vol. 28, No. 6, pp. 915-924 (2005)915NAVIGATION OF A MOBILE ROBOT IN OUTDOORENVIRONMENTS†Chia-Ju Wu*, Ting-Li Chien, Tsong-Li Lee, and Li-Chun LaiABSTRACTImplementation of an outdoor navigation system for a mobile robot is described in this paper. In this system, two optical encoders are used to perform dead-reckon-ing for the mobile robot. Meanwhile, an electronic compass and a GPS receiver are used for self-localization of the robot. Fusing the sensory data from dead-reckoning and self-localization, position and orientation of the robot can be determined munication between the robot and the host computer is achieved through GSM modems. In controlling the right and left angular velocities of the robot, a PID con-trol law will be used. For illustration, computer simulation and a practical experi-ment are presented to show that the outdoor navigation of a mobile robot is feasible,in a convenient and cost-effective manner.Key Words: mobile robots, outdoor navigation, dead-reckoning, self-localization.†Based on an awarded paper presented at Automation 2005, the 8th international conference on automation technology, Taichung,Taiwan, R.O.C. during May 5-6, 2005.*Corresponding author. (Email: wucj@.tw)C. J. Wu is with the Department of Electrical Engineering,National Yunlin University of Science and Technology, Yunlin,Taiwan 640, R.O.C.T. L. Chien and L. C. Lai are with the Graduate School of Engi-neering Science and Technology, National Yunlin University of Science and Technology, Yunlin, Taiwan 640, R.O.C.T. L. Lee is with the Department of Automation Engineering,Nan Kai Institute of Technology, Nantou, Taiwan 542, R.O.C.I. INTRODUCTIONMobile robots is a field of great interest and rapid development. With related technologies chang-ing rapidly, different kinds of mobile robots have been widely used in many indoor/outdoor applications such as spot welding, loading/unloading of parcels, clean-ing or grass cutting in parks, spraying in agriculture,undersea inspection, nursing care, etc (Schraft and Schmierer, 2002).For indoor/outdoor navigation, a mobile robot must have the capability of self-positioning. In (Tsai,1998; Wu and Tsai, 2001), by fusing the information from an electronic compass, two optical encoders, and a set of ultrasonic sensors, position and orientation of a mobile robot can be determined accurately usingthe the extended Kalman filter and the least-square method, respectively. However, these two methods are applicable for indoor navigation only. In (Rintanen et al ., 1996; Abbott and Powell, 1999; Panzieri, Pascucci,and Ulivi, 2002), GPS is used for outdoor navigation of mobile robots. Though GPS has many successful applications, one disadvantage of GPS is that tall buildings, dense foliage, or terrain that stands between a GPS receiver and a GPS satellite will block the satellite’s signal. Since the GPS position may be un-available at times, many navigation systems utilize other navigation aids in conjunction with GPS posi-tioning to enhance overall system performance. These aids usually include some combination of sensors such as low-cost gyroscopes, compasses, odometers, incli-nometers and accelerometers.Selecting proper sensors in a navigation system is a tradeoff between cost and system performance.More accurate sensors usually mean more cost. In this paper, the mobile robot will be equipped with two low-cost optical encoders for dead-reckoning.These sensors are used since they have been proved to be very useful, economical external sensing sys-tems for mobile robot location. However, using optical encoders for dead-reckoning is subject to ac-cumulation errors caused by wheel slippage, mechani-cal tolerances, and surface roughness. In this way,the robot may fail to keep track of its true location916Journal of the Chinese Institute of Engineers, Vol. 28, No. 6 (2005) over a long distance. To compensate for the inaccu-racy of dead-reckoning, an electronic compass and aGPS receiver will be used for self-localization toeliminate the possible accumulation errors.Funded by the US Department of Defense, GPShas applications such as navigation, surveying, andpositioning. The overall GPS system includes 24navigational orbiting satellites, six earth controlstations, and many user-owned receivers. A GPS re-ceiver converts satellite signals into position, velocity,and time estimates. By combining signals from threedifferent satellites, the receiver can determine a 2Dposition of latitude and longitude. Civilian GPS re-ceivers are accurate to within 100 meters. However,the accuracy has been improved significantly sincethe White House removed the intentional degradationof GPS signals.In addition to the sensor-fusion of dead-reck-oning and self-localization, another feature in this paperis the use of GSM, which is the cellular standard thatprevails throughout Europe and much of the rest ofthe world. Basic GSM usually provides data services in two ways, the circuit-switched mode or short mes-sage service. GSM is used in this paper since it is inexpensive and widely used in many countries. Control commands to the robot and the status of the robot will be transformed into short messages before they are transmitted. In this manner, control and monitoring of the robot can be performed in a convenient and cost-effective way from a very remote site.In controlling the DC motors on the two wheels, due to the desire for simplicity, from both the design and the parameter tuning points of view, the well-known PID control law will be used. To implement such a controller, the proportional gains, the integral gains, and the derivative gains must be determined. To date, great effort has been devoted to develop methods to reduce the time spent on optimizing the choice of controller parameters. Among the existing gain tuning techniques, unarguably the Ziegler-Nichols formula (Ziegler and Nichols, 1942) is one of the most well-known and popular methods. For a wide range of practical processes, this gain tuning approach works quite well. However, sometimes it is laborious and time-consuming, particularly for pro-cesses with serious nonlinearities. Therefore, this method usually needs retuning before being used to control an industrial process (Gawthrop and Nomikos, 1990).In addition to the aforementioned disadvantages, another limitation of the Ziegler-Nichols formula is that it can be applied to single-input single-output (SISO) systems only. To overcome this difficulty, an extension of the Ziegler-Nichols formula to mul-tivariable (multi-input or multi-output) systems was developed (Niederlinski, 1997). This extension method works well for linear, interacting multivari-able systems. However, it still cannot be applied to nonlinear multivariable systems.Recently, to enhance the capabilities of tradi-tional PID tuning techniques, new methodologies such as neural networks (Kawafuku, Sasaki, and Kato, 1998; Tang et al., 1998), fuzzy logic controllers (Tzafestas and Papanikolopoulos, 1990), and genetic algorithms (GAs) (Porter and Jones, 1992; Wang and Kwok, 1994; Mitsukura, Yamamoto, and Kaneda, 1999) have been applied to tune the parameters of PID controllers. The results of these studies indicate that better control performance can be achieved than by use of the Ziegler-Nichols method. However, it should be pointed out that the methods (Tzafestas and Papanikolopoulos, 1990; Zhao, Tomizuka, and Isaka, 1993; Kawafuku, Sasaki, and Kato, 1998; Tang et al., 1998) are applicable to SISO systems only. For PID parameter tuning of nonlinear multivariable systems, only GAs can be used. Therefore, genetic PID pa-rameter tuning will be adopted in this paper.The remaining sections of this paper are organized as follows. Section II shows the system configuration of the mobile robot. PID control for navigation of the robot is introduced in Section III. Genetic PID tuning is described in Section IV. Measurements of position and orientation of the robot are shown in Section V. Simulation and experiment are performed in Section VI to confirm the feasibility of the proposed system. Conclusions are given in Section VII.II. THE MOBILE ROBOT SYSTEMThe appearance and the block diagram of themobile robot are shown in Fig. 1 and Fig. 2, respectively.Fig. 1 Appearance of the mobile robot systemC. J. Wu et al.: Navigation of a Mobile Robot in Outdoor Environments917The dead-reckoning module in Fig. 2 consists of two high-resolution optical encoders mounted on the DC motors. The optical encoders provide a measure of the angular velocities of the right and left wheels, respectively. Integrating the angular velocities over one sampling period, a dead-reckoning procedure is executed to determine the position and the orienta-tion of the mobile robot.As described above, using optical encoders for dead-reckoning is subject to accumulative errors. Therefore, an electronic compass and a GPS receiver are used for self-localization to eliminate possible accumulative errors. The compass is a flux compass with one-degree resolution and two-degree accuracy, which provides a measure of the orientation of the robot by sensing the strength of the earth’s magnetic field. The GPS receiver used in this paper is the GARMIN GPS25-HVS. This receiver tracks up to twelve satellites at a time while providing one sec-ond sampling time and low-power consumption. The receiver is designed to withstand heavy operating conditions and is completely water resistant. Inter-nal memory backup allows the GPS to retain critical data such as satellite orbital parameters, last position, date and time. The sensor can be connected to a PC via an RS-232 interface.The host PC communicates with the GPS receiver using National Marine Electronics Association output. Using NMEA 0183 format, which is a standard for connection of GPS receivers to other devices, the GPS receiver periodically outputs specific data from its serial port. In this paper, the latitude and longitude of the robot are taken from the NMEA GPGGA data.To transfer data between the robot and the host computer, two GSM modems and AT commands will be used. AT is a contraction of attention, a command used to program modems from Hayes Microcomputer Products. AT commands program a variety of modem hardware settings and are adopted by other modem manufacturers who want to market their wares with the coveted phrase Hayes-compatible. Many websites con-tain the technical information of AT commands. For more details of the AT commands, one can visit the website http:/cellular.co.za/hayesat.htm.The industrial PC (IPC) in Fig. 2 includes a Pentium III CPU, an IDE Flash Disk (IFD 180), and an A/D-D/A card (PCI-1202H). During the operation of the robot, the Flash Disk will replace an HDD. This is because the robot is designed for outdoor applications. Therefore, no HDD will be used since the vibration caused by rough terrain may damage a mechanical disk drive.The PCI-1202H card offers the following functions: A/D conversion (12 bits, 40KHz), D/A conversion, digital input, digital output, and timer. Since the outputs of sensors are usually analog, func-tions of A/D conversion and digital input will be used when the sensory information is sent to the CPU. On the other hand, when the digital control commands generated by the CPU are sent to the DC motors, func-tions of digital output and D/A conversion are needed. The driver of each DC motor is an H-bridge circuit, which uses Power MOSFET IRFP-250 and the DC/ DC converter MAU225, and has the advantages of space savings and high power density.III. DESIGN OF PID CONTROLLER FORNAVIGATIONThe navigation problem of the mobile robot is to control the angular velocities of the robot, which will be denoted by ωR(t) and ωL(t), respectively, such that the robot can trace a desired trajectory. PID con-trol law will be used in this paper since it is popular in industries and easy to implement in practice. For the PID controller, the output variables are ωR(t) and ωL(t), and the input variables are e P(t) and eθ(t), which denote the errors in position and heading angle, and are defined as follows:e p(t)≡(1)eθ(t) ≡θ(t) – θd(t)(2) where x(t), y(t), and θ(t) denote the position and the heading angle of the robot, and x d(t ), y d(t), and θd(t) are the desired trajectory of x(t), y(t), and θ(t), respectively.With the above definitions for error signals, the input-output relation of the PID controller is described asDead-reckoning module(two optical encoders)Host computer(remote site)IPCInterface busDriver circuitRight wheelRadio linkLeft wheelGSMmodem GSMmodemTrajectory planningalgorithmPIDcontroller Self-localization module (an electronic compass and a GPS receiver)Fig. 2 Block diagram of the mobile robot system918Journal of the Chinese Institute of Engineers, Vol. 28, No. 6 (2005)ωR(t)=K P11⋅e p(t)+K I11⋅e p(t)dt+K D11de p(t) dt+K P21⋅eθ(t)+K I21⋅eθ(t)dt+K D21deθ(t) dt (3)ωL(t)=K P12⋅e p(t)+K I12⋅e p(t)dt+K D12de p(t) dt+K P22⋅eθ(t)+K I22⋅eθ(t)dt+K D22deθ(t) dt (4)where K P11, K I11, K D11, K P21, K I21, K D21,K P12, K I12, K D12, K P22, K I22, and K D22 denote the PID gains to be determined.In (3) and (4), one can find that there are 12 PID gains to be determined. Since there are no system-atic approaches for parameter tuning of a multivari-able system, it is obvious that the design of a PID controller is not an easy task. To reduce the diffi-culty of this design task and for convenience of prac-tical implementation, the following relation will be used (Tsai, 1998).ωR(t) + ωL(t) = 2 .v(t)(5) where v(t) denotes the linear velocity of the robot.With the relation in (5), the PID controller will contain only 6 PID gains which are expressed asωR(t)=K P1⋅e p(t)+K I1⋅e p(t)dt+K D1de p(t) dt+K P2⋅eθ(t)+K I2⋅eθ(t)dt+K D2deθ(t) dt(6)ωL(t)=2⋅v d(t)–ωR(t)=2⋅v d(t)–K P1⋅e p(t)–K I1⋅e p(t)dt–K D1de p(t)dt–K P2⋅eθ(t)–K12⋅eθ(t)dt–K D2deθ(t)dt(7)where v d(t) denotes the desired linear velocity of the robot, and K P1, K I1, K D1, K P2, K I2, and K D2 denote the new PID gains to be determined.Comparing (3) and (4) with (6) and (7), one can find that the number of PID gains is reduced from 12 to 6. Therefore, the design of a PID controller will be simplified significantly.IV. GENETIC PID PARAMETER TUNINGThe theoretical basis of GAs is that chromosomes better suited to the environment will have greater chance of survival and better chance of producing offspring (Janikow and Michalewicz, 1990; Man, Tang, and Kwong, 1996; Gen and Cheng, 1997). In this paper, in terms of the error signals e p(t) and eθ(t), the fitness function is defined asfitness(8)After encoding possible solutions into corre-sponding chromosomes, the evolutionary process is based on two primary operators: mutation and crossover. The crossover combines the features of two parent structures to form two similar offspring. The mutation operation arbitrarily alters one or more components of a selected structure, which increases the variability of the population.1. Chromosome RepresentationThe way to encode a solution of the problem into a chromosome is a key question in GAs. In the past, binary encoding has been the most popular approach used in GA research since it is simple and easy to implement. However, binary coding is difficult to apply directly in many GA applications because it is not a natural coding. For problems from the indus-trial engineering world, such as the design of PID controllers, the parameters to be determined are usu-ally all real. Therefore, real number encoding will be used in this paper.Once the real-coded chromosomes are used, the next step is to determine the number of genes in a chromosome. Since there are 6 PID gains to be determined, a chromosome will contain 6 genes, which will be denoted by x = [x1, x2, ..., x6].2. Crossover and Mutation OperationsArithmetical crossover and nonuniform mutation in (Gen and Cheng, 1997) will be used in this paper since they are applicable to real-coded chromosomes. For two real-coded chromosomes x1 and x2, the op-eration of arithmetical crossover is defined as follows:x1, offspring = λx1 + (1 – λ)x2(9)x2, offspring = λx2 + (1 – λ)x1(10)where λ ∈ (0, 1).For a given parent x, if a gene x k of it is selected for mutation, then the resulting offspring will beC. J. Wu et al.: Navigation of a Mobile Robot in Outdoor Environments 919randomly selected from one of the following two choices:x k ,offspring =x k +(x k U –x k )⋅r ⋅(1–g G)b (11)x k ,offspring =x k –(x k –x kL )⋅r ⋅(1–g G)b(12)where x k U and x kL are the upper and lower bounds for x k , r is a random number from [0, 1], g is the genera-tion number, G is the maximal generation number, and b is a parameter determining the degree of nonuniformity.In addition to arithmetical crossover and non-uniform mutation, dynamic crossover and mutation probability rates (Sheble and Britting, 1995) will also be used for fast convergence, in which the crossover and mutation rates are defined ascrossover rate =exp(–g G)(13)mutation rate =exp(g4G)–1(14)3. Enlarged Sampling SpaceIn GAs, if the selection methods to generate off-spring are developed based on a regular sampling space (Man, Tang, and Kwong, 1996; Gen and Cheng, 1997),then parents are replaced by their offspring soon af-ter they give birth. In this manner, some fitter chro-mosomes will be lost in the evolutionary process and it is possible that the offspring will have lower fit-ness values than their parents. To cope with this problem,the selection method used in this paper will be per-formed in enlarged sampling space, in which parents have an equal chance to compete with the offspring.In this manner, there will be more possibility to gen-erate fitter chromosomes in the next generation.4. Ranking MechanismIf the selection probability of a chromosome is proportional to its fitness, then some undesirable properties will exhibit. For example, a few super chromosomes will dominate the selection process in early generations. Moreover, competition among chromosomes will be less strong and a random search behavior will emerge in later generations. To miti-gate these problems, a ranking mechanism is used in this paper, in which the chromosomes are selected proportionate to their ranks rather than actual fitness values. This means that the fitness values will be integer numbers from 1 to N p , where N p is the popu-lation size. The best chromosome will have a fitness value equal to N p and the worst one will have a fit-ness value equal to 1.The details of the GA-based PID parametertuning procedure can be summarized as follows:Algorithm A (Genetic PID parameter tuning)Step 1.Define the fitness function as shown in (8).Step 2.Determine the population size and the maxi-mal generation number.Step 3.Produce an initial generation in a randomway.Step 4.Evaluate the fitness for each member of theinitial generation.Step 5.With the crossover rate in (13), generate off-spring according to (9) and (10), in which the ranking mechanism is used for selection of chromosomes.Step 6.With mutation rate in (14), generate offspringaccording to (11) and (12).Step 7.Select the members of the new generationfrom the parents in the old generation and the offspring in Step 5 and Step 6 according to their fitness values.Step 8.Repeat the procedure in Step 5 through Step7 until the number of generations reaches a prescribed value.V. MEASUREMENTS OF POSITION ANDORIENTATION Accurate measurements of x (t ), y (t ), and θ(t )play an important role in the navigation procedure.In the dead-reckoning procedure, the values of ωR (t )and ωL (t ) can be determined by counting the pulses generated in the two optical encoders. Then by nu-merically integrating over a sampling period, the value of θ(t ) can be calculated by (Tsai, 1998)θ(k )=θ(k –1)+∆T ⋅ωR (k –1)–ωL (k –1)0.6(15)where ∆T is the length of the sampling period and θ(k ) is the value of θ(t ) at the sampling instant t = k . ∆T .The values of x (t ) and y (t ) at the sampling in-stant t = k . ∆T are then determined byx (k )=x (k –1)+∆T ⋅ωR (k –1)+ωL (k –1)2⋅cos θ(k –1)(16)andy (k )=y (k –1)+∆T ⋅ωR (k –1)+ωL (k –1)2⋅sin θ(k –1)(17)When using optical encoders to measure the values of x (k ), y (k ), and θ(k ), due to wheel slippage or surface roughness, cumulative error may occur.920Journal of the Chinese Institute of Engineers, Vol. 28, No. 6 (2005) Therefore, an electronic compass and a GPS receiverare used in this paper for self-localization. When thevalues of x(k), y(k), and θ(k) measured by the opticalencoders according to (15) through (17) are differentfrom the ones measured by the electronic compassand the GPS receiver, fusion of different bits of sen-sory information will be needed.Since the accuracy of the flux compass used inthis paper is two degrees, the value of the heading anglewill be determined in a sensor-fusion manner as follows:θ(k)=θe(k)forθe(k)–θc(k)≤2°θc(k)otherwise(18)where θe(k) and θc(k) denote the heading angle of therobot measured by the encoders and the compass,respectively.Similarly, accumulative error due to wheel slip-page or surface roughness may also affect the accu-racy of the measurements of x(k) and y(k). To copewith this problem, a GPS receiver is used. When theGPS receiver is used in the differential mode, its ac-curacy is within 5 meters RMS. However, if the ex-tended Kalman filter (Tsai, 1998) or the least-squaredmethod (Wu and Tsai, 2001) is used, then the accu-racy of the GPS receiver can be within 3 meters.Therefore, the values of x(k) and y(k) will be deter-mined bywhere (x e(k), y e(k)) and (x g(k), y g(k)) are the position of the robot measured by the encoders and the GPS receiver, respectively.In the above sensor-fusion routine, details of the measurements of position and orientation of the mo-bile robot are summarized as follows:Algorithm B (Measurements of x(t), y(t), and θ(t)): Step 1:Use the GPS receiver and the compass to measure the initial configuration of therobot. This means that (x(0), y(0), θ(0)) =(x g(0), y g(0), θc(0)).Step 2:Set k = 0.Step 3:Set k = k + 1.Step 4:Within the time interval [(k – 1) .∆T, k.∆T], use the PID control law in (6) and (7)to navigate the mobile robot.Step 5:According to the number of pulses generated in both optical encoders, determine the val-ues of ωR(k – 1) and ωL(k – 1).Step 6:Perform the integration in (15) over one sam-pling period to obtain θe(k) = θ(k – 1) + ∆T.ωR(k–1)–ωL(k–1)0.6.Step 7:If |θe(k) – θc(k)| ≤ 2°, then set θ(k) = θe(k);Otherwise, set θ(k) = θc(k).Step 8:According to (16) and (17), determine the values of x e(k) = x(k – 1) + ∆T.ωR(k–1)+ωL(k–1)2. cosθ(k –1) and y e(k) = y(k – 1) + ∆T.ωR(k–1)+ωL(k–1)2. sinθ(k –1).Step 9:If ≤ 3 me eset (x(k), y(k)) = (x g(k), y g(k)).Step 10:Repeat Step 3 through Step 9 until the navi-gation procedure is completed.VI. SIMULATION AND EXPERIMENTALEXAMPLESThough GAs can be used to tune PID param-eters of multivariable systems, it should be noted thatthere still exist several problems. The major one isthat the PID gains are adjusted almost randomly inthe genetic tuning procedure. Hence the control plantmay be damaged if on-line tuning is performed. Thisexplains the fact that though many successful appli-cations of GAs in PID tuning of nonlinear systemshave been reported, most of them are computer simu-lation results based on a mathematical model of theplant only, rather than practical experimental results.Therefore, in the following two examples, it will beshown how to tune the PID gains in an off-line man-ner first by performing computer simulation. Afterthe PID gains are determined by computer simulation,the designed PID controller is then applied for prac-tical experiments in the second example.Example 1:In this simulation example, Algorithm A will beapplied to determine the PID gains in (6) and (7) tomaximize the fitness functionC. J. Wu et al.: Navigation of a Mobile Robot in Outdoor Environments921fitness(20)while using the PID controller in (6) and (7) to navi-gate the robot moving along a square that has the sizeof 2 m × 2 m. A square trajectory is used here sinceit is a very common test-bench for checking the track-ing control of mobile robots. When applying Algo-rithm A, referring to the work of (Grefenstette, 1998),the population size and the maximal generation num-ber are chosen to be 100, and 2000, respectively. Al-gorithm B will not be used since this is a simulationexample only.With the above genetic parameters, the PID gainsdetermined by Algorithm A are found to be K P1 = -3.26,K I1 = -1.39, K D1 = -1.26, K P2 = 4.82, K I2 = 2.38, and K D2= 2.24. Corresponding to this PID controller, the move-ment of the robot is shown in Fig. 3.Example 2:After determining the PID gains, the design task of producing a PID controller is completed. In this example, the PID controller designed in Example 1 will be used to perform a practical experiment of tra-jectory-tracking control. The desired trajectory is a rectangle that has the size of 32 m × 22 m. During the experiment, the values of x(t), y(t), and θ(t) are measured by the optical encoders, the electronic compass, and the GPS receiver according to the pro-cedure described in Algorithm B. The length of the sampling period ∆T is chosen to be 0.1s. Communi-cation between the host computer and the mobile robot is achieved through GSM modems and an in-terface program written in C++ language will be used to display the status of the mobile robot on the screen of a remote host computer. Performing the proposed PID navigation procedure, the robot will take about 4 minutes to finish the task of trajectory-tracking. The movement of the robot and the plot of e p(t) and eθ(t) are shown in Fig. 4 and Fig. 5, respectively.VII. CONCLUSIONSA navigation system for outdoor navigation of a mobile robot is implemented. In this navigation system, two low-cost optical encoders are used for dead-reckoning, and an electronic compass and GPS are used for self-localizaiton. Data between the host computer and the mobile robot are transferred via two GSM modems, which provide an economic and reliable way for communication. Fusing the infor-mation from different sensors, it is found from the experimental results that the error in position is less than 3 m and the error in heading angle is less than 2degrees. Such kind of accuracy will meet the require-ments of many outdoor applications of mobile robots.Compared with previous methods, the proposed method has three note worthy features. The first one is the integration of dead-reckoning and self-local-ization to eliminate possible accumulative errors. The second one is the use of GSM such that the control and monitoring of the robot can be performed in a convenient and cost-effective way from a very remote site. Finally, the last one is the development of a GA-based method for PID parameter tuning of non-linear multivariable systems. Combining these features, the proposed method provides a simple and effective way to navigate a robot along a prescribed trajectory in outdoor environments.ACKNOWLEDGMENTSThis work was supported in part by the National Science Council, Taiwan, R.O.C., under grants NSC93-2213-E-224-001 and NSC93-2218-E-224-020.NOMENCLATUREb degree of nonuniformitye p(t), eθ(t)errors in position and headingangleg generation numberG maximal generation numberK P1, K I1, K D1,PID gainsK P2, K I2, K D2N p population sizev(t)linear velocity of the robotv d(t)desired linear velocity of therobot.-150-10015010050-50-100-150-500X (cm)Y(cm)50100150 Fig. 3 Movement of the mobile robot in the simulation example。