Journal of Robotics and Mechanical Engineering Research

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1 Journal of Robotics an Mechanical Engineering Research Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms Yang Zhang,2 an Vigen Arakelian,2* Mecaproce, INSA, 20 avenue es Buttes e oësmes, S 70839, Rennes eex 7, France 2 LS2N / EN, rue e la Noë, BP 920, 4432 Nantes eex 03, France Vol: 2, Issue: 3 *orresponing author: Vigen Arakelian, Mecaproce, INSA, 20 avenue es Buttes e oësmes, S 70839, Rennes eex 7, France ; Tel: +33(0) ; Fax : +33(0) ; E mail : vigen.arakelyan@insa-rennes.fr Article Type: Research, Submission Date: 27 July 208, Accepte Date: 3 August 208, Publishe Date: 22 August 208. itation: Yang Zhang an Vigen Arakelian (208) Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms. opyright: 208 Yang Zhang an Vigen Arakelian. This is an open-access article istribute uner the terms of the reative ommons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original author an source are creite. Abstract In the past ecaes, an extensive research has been focuse on legge walking robots. One of the most attractive trens in the esign of walking robots is the evelopment of bipe robots with reuce number of egrees of freeom. This paper eals with a new solution of a legge walking mechanism with four-bar mechanism an only one actuator. A Genetic algorithm base mechanism synthesis has been implemente in the esign of four-bar mechanism. The efficiency of the suggeste esign of the legge walking robot is illustrate by numerical simulations that carrie out via ADAMS, an the results show that this esign has the capability of walking both forwar an backwar. Keywors: Mechanism esign, Four-bar linkage, Walking robot, Mechanism synthesis, Genetic algorithm. Introuction It is well known that legge locomotion is more efficient, speey, an versatile than one by track an wheele vehicles when it operates on a rough terrain, steeps stairs or avoi obstacles. This research fiel has attracte great interest of many research groups an companies in the past few ecaes. The pioneering works in the fiel of legge robots were achieve aroun 970 by two famous researchers, Ichiro Kato [-3] an MiomirVukobratovic [4,5]. Both works were characterize by the esign of relevant experimental systems. In Japan, the first anthropomorphic robot, WABOT, was emonstrate in 973 at Wasea University. In Yugoslavia, Belgrae, at the MihailoPuppin Institute, MiomirVukobratovic an his team esigne the first active exoskeletons an several other evices. The en of the millennium was a perio of intense technological activities. Inustrial breakthroughs showe to the worl that builing true humanois was now possible. In Japan, the first humanoi robot P2 was exhibite by Hona in 996, followe by several more:asimo (Hona), QRIO (Sony), HRP (Kawaa), etc. To create bipe robots walking like a human is necessary to use a large number of actuators. Therefore, these robots are automorphic an flexible. However, there are several rawbacks: complexities of the esign an the control system, low energy efficiency ue to the masses of motors, as well as an overly high price complicating practical use. Hence, in orer to make a bipe robot more attractive, a ifferent methoology can be consiere such as constructing a bipe robot with reuce number of egrees of freeom [6-9]. In the last ecae, at the Laboratory of Robotics an Mechatronics in assino University, Marco eccarelli an his team have propose various solutions for Low-ost Easy-Operation leg esign [0]. The hebichev four-bar linkage has been successively use for generation a foot trajectory. In orer to amplify the prouce motion of the hebyshev linkage, a pantograph mechanism has been utilize. In other stuies [-3], five-bar mechanism, Klann Mechanism an Jansen Mechanism have been aopte for the esign of walking robots with reuce number of actuators. The four-bar mechanism is one of the simplest but practically wiesprea mechanisms. It has been use in various inustrial fiels, such as automatic oor-closing systems, train suspensions, front loaer mechanisms, etc. In orer to aopt the four-bar mechanism into ifferent engineering fiels, many researchers [4-7] have conucte researches on the graphical an analytical methos of the esign of four-bar mechanism. However, these methos are often with low precision an will increase the computational time an complexity when the prescribe points J Robot Mech Eng Resr Page

2 itation: Yang Zhang an Vigen Arakelian (208) Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms. increase [8]. In the last three ecaes, with the significantly evelopment of the computer inustry, the computational ability of computer has been greatly increase which allow researchers via numerical methos to solve sophisticate optimization problems. Among these numerical methos, genetic algorithm (GA) [9-2]has been wiely use among researches from various fiels as one of the most effective approaches to solve non-linear optimization problems. Recently, several researchers have implement GA in the esign of four-bar mechanism. J.A abrera et al. from University of Malaga applie GA to esign the four-bar mechanism [22]. In their research, they use the ifference between esire points an actual points as the goal function to be minimize an several constraints ha been set such as Grashof conition an sequence conition. Base on this research, N. Nariman-Zaeh et al. from University of Guilan introuce the Pareto optimum synthesis, an the influences of transmission angle an maximum angular velocity ratio of fourbar mechanism were concerne [23]. This paper is organize as follows. In the first section the optimization metho is iscusse. Then, optimal synthesis of four-bar mechanism is consiere. After, the esign of oneegree of freeom walking robot is shown. Finally, the results of numerical simulations are presente. Optimization metho Genetic algorithm (GA) is a kin of mathematical algorithm which is inspire by the process of natural selection. Nowaays, GA is wiely an commonly use for solving multi-parameter optimization problems. In GA, a population of caniate solutions (iniviuals) to an optimization problem is evolve towar better solutions. Like the creature, each iniviual has a set of properties (chromosomes) which can be mutate an altere. Normally, the GA starts from a population of iniviuals which are ranomly generate, an the population in each iteration is calle a generation. In each generation, there is an objective function to evaluate the fitness of each iniviual in the population. The more fit iniviuals are stochastically selecte from the current population, an each iniviual s genome is moifie (recombine an mutate) to form a new generation. The new generation of caniate solutions is then use in the next iteration of the algorithm. ommonly, when the maximum number of generation is or a satisfactory fitness level is reache, the GA will be terminate. The general proceure of GA is shown in Figure. The selection proceure chooses parents for the next generation base on their fitness values. During each successive generation, a portion of the existing population is selecte to bree a new generation. ertain selection methos rate the fitness of each solution an preferentially select the best solutions. Other methos rate only a ranom sample of the population, as the former process may be very time-consuming. In the present stuy, we use the stochastic uniform metho in the selection proceure. It lays out a line in which each parent correspons to a section of the line of length proportional to its expectation. The algorithm moves along the line in steps of equal size, one step for each parent. At each step, the algorithm allocates a parent from the section it lans on. The first step is a uniform ranom number less than the step size. Start Generate initial population alculate fitness Selection rossover Mutation Juge termination criteria Yes Optimal solution Figure : General proceure of GA In GA, crossover is a genetic operator use to vary the programming of a chromosome or chromosomes from one generation to the next. It is analogous to reprouction an biological crossover, upon which genetic algorithms are base. ross over is a process of taking more than one parent solutions an proucing a chil solution from them. There are methos for selection of the chromosomes (i.e. Scattere, Single point, Intermeiate). Scattere has been use in the rossover process of the present stuy. Scattere creates a ranom binary vector. It then selects the genes where the vector is a from the first parent, an the genes where the vector is a 0 from the secon parent, an combines the genes to form the chil. This metho can be shown as p = [ abcef gh] p2 = [ ] ranom crossover vector = [ ] chil = [ a b 3 4 e 6 7 8] () Mutation is a genetic operator use to maintain genetic iversity from one generation to the next. It is analogous to biological mutation. Mutation alters one or more gene values in a chromosome from its initial state. In mutation, the solution may No J Robot Mech Eng Resr 2(3) Page 2

3 itation: Yang Zhang an Vigen Arakelian (208) Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms. change entirely from the previous solution. Hence GA can come to better solution by using mutation. Mutation occurs uring evolution accoring to a user-efinable mutation probability. This probability shoul be set low. If it is set too high, the search will turn into a primitive ranom search. There are several algorithms for Mutation in GA, for example, Gaussian, Uniform, an Aaptive feasible. In the present stuy, the Aaptive feasible has been use in the Mutation proceure. It ranomly generates irections that are aaptive with respect to the last successful or unsuccessful generation. A step length is chosen along each irection so that linear constraints an bouns are satisfie. Leg esign via synthesis of four-bar mechanism with prescribe path trajectory To aapt the GA to the path-generation base optimal synthesis of four-bar mechanism (Figure 2), the goal function in the GA must be efine at first. Figure 2: A planar four-bar mechanism with esign parameters The ultimate goal of the path-generation base optimal synthesis is to optimize a four-bar mechanism which can generation a path (the trajectory of point in Figure 2) as close as possible to the prescribe points (also known as esign points). Hence, the first part of the goal function is the ifference between actual points an esign points, which can be expresse by: N i i 2 i i 2 ( X X ) ( Y Y ) (2) i= where N is the number of esign points, i =[ i X, i Y]=[ X (θ i 2), Y (θ i 2)]is the coorinates of actual path points of the optimize four-bar mechanism, i =[ix, iy] is the coorinates of the esign points. In the optimal synthesis, the esign variables are r, r 2, r 3, r 4, r cx, r cy, θ 0, x 0, y 0, an input angle θ 2 as shown in Figure 2. However, to calculate the coorinates of actual points i, the value of θ 3 must be solve first. In orer to calculate θ 3, the close equation of four-bar mechanism has been introuce: r2sinθ2 + r3sinθ3 = r4sinθ4 r2cos θ2 + r3cos θ3 = r+ r4cos θ4 (3) It can be rewritten as: cos θ4 = ( r2 cosθ2 + r3cos θ3 r) / r4 sin θ4 = ( r2sinθ2 + r3sin θ3) / r4 (4) By squaring both the equations of Eq. 4 an aing them together, we can obtain a quaratic equation with respect to cosθ 3. An the solution can be represente as flowing: AB AB B cosθ3 = ± D D D (5) where: A=2r 3 (r 2 cosθ 2 -r ) B=r2 4-r2 -r2 2-r2 3+2r r 2 cosθ 2 =2r 2 r 3 sinθ 2 D=(A ) /2 thenθ 3 an θ 4 can be easily calculate: / AB AB B θ3 = arccos ± D D D θ4 = arccos [( r2 cosθ2 + r3cos θ3 r) / r4] (6) Thus, the position of point in the local reference O 2 X r Y r, which has been efine in Figure 2, is: Xr = r2cos θ2 + rcx cosθ3 rcy sinθ3 Yr = r2sinθ2 + rcx sinθ3+ rcy cosθ3 (7) an it can be transferre to the global coorinate in reference OXY: X cosθ0 sinθ0 Xr x0 = Y sinθ0 cosθ + 0 Yr y 0 (8) The first part of goal function can be calculate from Eq. 2. In orer to obtain a esirable four-bar mechanism, there are several constraints which have to be complie. For example, the Grashof conition which allows at least one link has the entire rotation, sequence conition of input angle θ 2 which guarantee the continuity of input angles, an the limitation of esign variables. Therefore, in the secon part of the goal function, these constraints have been introuce. Now, the synthesis conitions can be escribe as follows: N i i 2 i i 2 min ( ( X) ( X)) + ( ( X) ( X)) i= X X Y Y subject to: (a) Grashof conition: 2(r max +r min )<(r +r 2 +r 3 +r 4 ) (b) Sequence conition: θ j 2<θmo(j+,N) 2< <θmo(j+n,n) 2 l u (c) Limitation of esign variables: x [ L, L ] x X /2 i i i i where r max is the length of the longest linkage, r min is the length of the shortest linkage, θ i 2 is the value of θ 2 in its ith position, θ j 2=min{θ i 2}, X is the set of all esign variables, Ll i an Lu i are the lower an upper limitation of ith esign variable. J Robot Mech Eng Resr 2(3) Page 3

4 itation: Yang Zhang an Vigen Arakelian (208) Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms. In GA, the limitation of esign variables can be prescribe. With the aition of the penalty of the constraints, the entire goal function can be written as: N i i 2 i i 2 min ( X ( X) X ( X)) + ( Y ( X) Y ( X)) + Pf( X) + P2f2( X) i= wheref (X)=0 when Grashof conition is satisfie f (X)= when Grashof conition is not satisfie f 2 (X)=0 when Sequence conition is satisfie f 2 (X)= when Sequence conition is not satisfie an P an P 2 are the penalty coefficients of Grash of an Sequence conition, which are usually set to a high value in orer to have suitable results. Design of the single-actuate walking robot The typical walk consists of a repeate gait cycle. The cycle itself contains two phases: a propelling phase an non-propelling phase [0]. In Figure 3, the thicker line represents the supporting leg (right leg) in propelling phase an the thin line represents the swing leg (left leg) in non-propelling phase. It is known that for esign a one-egree of freeom leg mechanism with back-forth an up-own motion capability, the foot point shoul generate an ovoi curve, which is compose of a straight-line segment an a curve segment (see Figure ). The straight-line segment is relate to the propelling phase when the corresponing leg touches the groun an coul guarantee stable propelling of the boy. The curve segment is relate to the non-propelling phase, which is prouce by leg when it swings from back to forth. Figure 3: Two-stage step cycle for human walking gait However, the use of multiple actuators increases not only the whole weight of the robot which makes the robot very bulky, but also the complexity of the control system. In orer to tackle this rawback, many researches [9-3,24-26] have been conucte focus on the esign of walking robot with reuce number of egrees of freeom. In the present paper, the four-bar linkage is use as a leg of a walking robot. The four-bar mechanism has the capability to generate ifferent types of trajectory. However, to esign a fourbar mechanism which can generate humanoi walking step is (9) often intractable because in human step, there has both straight segment an curve segment. But with the optimal synthesis which has been iscusse in the thir section, the four-bar mechanism allows one to generate the straight segment an curve segment of the mentione trajectory. In the present stuy, Matlab an its GA toolbox have been use for the implementation of the metho escribe in Section 3. The general setting of the problem is showe by Table. During the optimization, the best fitness values with respect to the generation are shown in Figure 4. The optimization satisfie the stopping criteria at 50 th generation with a final error of an the best optimize parameters are: r = ; r 2 = ; r 3 = ; r 4 = ;r cx = ; r cy = ; θ 0 = ; x 0 = ; y 0 = ; θ 2= ; θ2 2=6.2089; θ3 2=0.2207; θ4 2=0.5238; θ5 2= ; θ6 2= Figure 4: Best fitness values with respect to the generation Figure 5 shows the comparison of the trajectory generate by the esigne mechanism an prescribe points. As we can see, this is a very esirable trajectory for esign a walking robot. Figure 6 shows the change transmission angle of the optimal mechanism with the rotation of input bar. It can be seen that the optimal mechanism is a proper esign sine the transmission angles are between 35 an 80. After obtaining the parameters of the optimal four-bar mechanism, the esign the walking robot can be carrie out. Figure 7 shows the relationship between input bar motion an leg trajectory, the thick an thin soli lines below represent the leg trajectory in stan an swing phase respectively; the input link motion with respect to the stan an swing phase is represente by the above thick an thin ot lines respectively. It can be seen that the input link motion with respect to the stan an swing phase are equally ivie an the ifference of the angles of input bar between two legs is always 80, hence the left an right leg can be riven by a single actuator. The AD moel of the suggeste walking robot was built by using ynamics simulation software ADAMS (Figure 8). It shoul be note that two feets are connecte at the output points of the four-bar mechanisms by torsion springs an ampers which maintain the balance of the robot uring walking. J Robot Mech Eng Resr 2(3) Page 4

5 itation: Yang Zhang an Vigen Arakelian (208) Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms. 80 A B Figure 5: omparison of the optimal mechanism s trajectory an esign points 80 Figure 7: Leg trajectory with respect to the motion of input link Input Bar Angle θ 2 (eg) Figure 6: hange of transmission angle with respect to the input bar angle Simulation results To verify whether the robot which esigne in previous section has the capability of walking, a simulation was conucte by using ADAMS. In orer to simulate the propose esign in both ifferent spees an ifferent walking irections, there ifferent rotational spees of actuator have been set, which are 60 /s, 20 /s, -60 /s respectively. The total simulation time of each simulation is 30s. Figure 8: AD moel of the walking robot Figure 9 an Figure 0 show the snapshots of part of the motion of the walking robot uring simulation when it s walking forwar an backwar with the spees of actuator of 60 /s an -60 /s respectively. The simulation verifie that the robot has the ability of both walking forwar an backwar. The isplacements of the mass centre of robot s moving frame along walking irection with ifferent actuator spees are shown in Figure. As we can see from these two figures, the robot was walking at a constant spee with both low an high spees an ifferent walking irections. J Robot Mech Eng Resr 2(3) Page 5

6 itation: Yang Zhang an Vigen Arakelian (208) Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms. Figure 0: Snapshots of the simulation when robot moving backwar (-60 /s) Displacement (mm) Actuator Spee: 60 o /s Actuator Spee: 20 o /s Actuator Spee: -60 o /s Time (s) Figure : Displacements of the mass centre of robot s moving frame along walking irection with ifferent walking spees References. Kato I. The hyraulically powere bipe walking machine with a high carrying capacity. In: Proc. of the 4th Int. Symposium on External ontrol of Human Extremities, Dubrovnik; 972. p Takanishi A, Ishia M, Yamazaki Y,Kato I. The realization of ynamic walking by the bipe walking robot wl-0 r. Journal of the Robotics Society of Japan. 985; 3(4): Takanishi A, Tochizawa M, Takeya T, Karaki H,Kato I. Realization of ynamic bipe walking stabilize with trunk motion uner known external force. In:Avance Robotics.Berlin, Heielberg Springer;989. p Vukobratovic M, Juricic D. ontribution to the synthesis of bipe gait. IEEE Transactions on Biomeical Engineering. 969; : Vukobratović M. Legge locomotion robots an anthropomorphic mechanisms. Mihailo Pupin Institute; Song SM, Walron KJ. Machines that walk: the aaptive suspension vehicle. MIT press; arbone G, eccarelli M. Legge robotic systems. In utting Ege Robotics. InTech; onclusion This paper eals with the esign of a walking robot with reuce egree of freeom. A four-bar mechanism has been use as the leg of this walking robot. In the first part of the stuy, a mechanism synthesis base on the Genetic Algorithm has been evelope. It propose geometric synthesis of the four-bar linkage allows one to fin the lengths of links that ensure the reprouction of prescribe points of the given trajectory. The suggeste walking robot consist of a moving frame connecte with two synthesize four-bar linkages. The actuator of the walking robot is mounte at the centre of its moving frame an moves both legs. To valiate the performance of the walking robot, a simulation has been carrie out via ADAMS. The obtaine results showe that the robot is capable to walk steaily both forwar an backwar at ifferent constant spees. 8. Funabashi H, Horie M, Tachiya H,Tanio S. A synthesis of robotic pantograph mechanisms base on working spaces an static characteristic charts. JSME international journal. Ser. 3, Vibration, control engineering, engineering for inustry. 99; 34(2): Shieh WB, Tsai LW, Azarm S. Design an optimization of a one egree of freeom six bar leg mechanism for a walking machine. Journal of Fiel Robotics. 997; 4(2): Liang, eccarelli M, Takea Y. Operation analysis of a one-dof pantograph leg mechanisms. Proceeings of the RAAD. 2008; 8:- 0.. Lokhane NG, Emche VB. Mechanical spier by using Klann mechanism. IJMA. 203; (5): Kim SW, Kim DH. Design of leg length for a legge walking robot base on Theo Jansen using PSO. Journal of Korean Institute of Intelligent Systems. 20; 2(5): Na B, Kong K. Design of a One Degree-of-Freeom Quarupe Robot Base on a Mechanical Link System: heetaroi-ii. IFA- PapersOnLine. 206; 49(2): Hrones JA. Analysis of the Four-bar Linkage. Publishe Jointly by the technology Press of the Massachusetts Institute of Technology an Wiley. 95. J Robot Mech Eng Resr 2(3) Page 6

7 itation: Yang Zhang an Vigen Arakelian (208) Design of a Single Actuator Walking Robot via Mechanism Synthesis Base on Genetic Algorithms. 5. Hartenberg RS, Denavit J. Kinematic synthesis of linkages. McGraw-Hill Freuenstein F. Approximate synthesis of four-bar linkases. Trans. ASME. 995; 77: Artobolovski I.Theoric es Mecanismeset es Machines, Dituon MIR, Moscou; Sanor GN,Erman AG. Avance Mechanism Design: Analysis an Synthesis.Vol. 2. Prentice Hall, Englewoo liffs, NJ; Hollan JH. Genetic algorithms an the optimal allocation of trials. SIAM Journal on omputing. 973; 2(2): Hollan JH. Aaptation in Natural an Artificial Systems. Univ. of Mich. Press. Ann Arbor; Golberg DE. Genetic algorithms in search, optimization, an machine learning, 989. Reaing: Aison-Wesley; abrera JA, Simon A, Prao M. Optimal synthesis of mechanisms with genetic algorithms. Mechanism an Machine theory. 2002; 37(0): Nariman-Zaeh N, Felezi M, Jamali A, Ganji M.Pareto optimal synthesis of four-bar mechanisms for path generation. Mechanism an Machine Theory. 2009; 44(): Feng S, Amur SAY, Sun Z. Bipe walking on level groun with torso using only one actuator. Science hina Information Sciences. 203; 56(): Jo HS, Mir-Nasiri N, Jayamani E. Design an Trajectory Planning of Bipeal Walking Robot with Minimum Sufficient Actuation System. In: WASET Int. onf. on ontrol, Automation, Robotics an Vision Engineering; p Jiang Z, Luo Y, Jin Y. New cable-riven continuun robot with only one actuator. In: ybernetics an Intelligent Systems (IS) an IEEE onference on Robotics, Automation an Mechatronics (RAM); 207. p J Robot Mech Eng Resr 2(3) Page 7