Kinematics optimization of a mechanical scissor system of tipping using a genetic algorithm

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1 Kinematics optimization of a mechanical scissor system of tipping using a genetic algorithm R. Figueredo a, P. Sansen a a. ESIEE-Amiens, 14 Quai de la Somme, BP 10100, Amiens Cedex 2 Résumé : Le compas hydraulique de bennage est un mécanisme très pratique équipant une large gamme de bennes, et donc nécessite une étude approfondie afin de l optimiser au mieux par rapport à son utilisation. Actuellement, les méthodes d optimisation ont donné des résultats qui restent encore trop approximatives. Présentée dans cette étude, une nouvelle approche d optimisation multi-objective plus robuste s applique entièrement à la cinématique du compas, où l un des algorithmes génétiques les plus efficaces ainsi que le formalisme Lagrangien sont utilisés respectivement en tant qu optimiseur et analyseur du système. Cherchant à obtenir un compas optimal à la fois plus résistant et plus léger, la méthode est appliquée à deux cinématiques : la première, classique, étant celle principalement adoptée dans l industrie; et la deuxième, version modifiée du classique, étant une innovation brevetée conçue pour réduire les efforts. Tous les résultats optimisés sont comparés aux originaux pour démontrer l efficacité de la méthodologie proposée. Abstract : Mechanical scissor systems of tipping are a very useful mechanism for a wide range of tipper vehicles. Currently in industry, the methods of optimization gave results which remain still too approximate. In this paper, we investigate the kinematics optimization of the mechanism using a new numeric approach in which an efficient genetic algorithm and the Lagrangian formalism in mechanics are used respectively as optimizer and analyzer. Thus, multiobjective functions are considered and based on the reaction force of the hydraulic cylinder and the size of the whole mechanical system to obtain a final optimum one, more resistant and lighter. Two kind of kinematics are optimized : a classical one which is usually adopted in industry; and a modified version which is a patented innovation conceived to reduce efforts. All optimized results are compared to the original one to demonstrate the effectiveness of the proposed methodology. Mots clefs : Mechanical scissor system of tipping; Genetic algorithm; Multiobjective optimization 1 Introduction The aim of this work is to present an efficient approach based on a genetic algorithms (GA) able to optimize the kinematics of the mechanical scissor system of tipping. Designed for light vehicles, very heavy tractor-trailers, agricultural and civil engineering equipment, it can raise important loads ranging from 1.5 to 60 Tons. Considered of rigid body, the classical kinematics is entirely characterized by the position of the links whose their coordinates will evolve thanks to a GA as optimizer. Indeed, GA can easily be applied to multiobjective functions and are able to find solutions close to the global optimum. In order to complete the mathematical formulation of the optimization problem, considered specific constraints constraints are described and written using the optimized variables. Finally, the implementation of the approach can be also extended to more complex cases. Here, a modified version of kinematics is studied and optimized. 1

2 2 Description of the mechanical scissor systems of tipping 2.1 Schematization of the two kinematics "pivots" M 4 F b M 6 1 Classical kinematics : y M 5 M 3 0 x M 1,M 2 "chassis" "pivots" M 4 F b M 6 2 Modified kinematics : M 3 M 7 y 0 x M 2 M 1 M 5 "chassis" "fixed support" "pivot with locking" (α 0 = 90 ) Figure 1 Kinematics at the initial position (t = 0) The classical scissor system is composed of three main parts (see Figure1-1 ). The arm [M 2 M 3 ] is the green part attached to the vehicle chassis (M 1 ) and the box beam (M 3 ). The box beam [M 3 M 4 M 5 ] is the red superior part attached to the arm (M 3 ) and the dumpster (M 5 ). And finally the blue one is the hydraulic cylinder [M 2 M 4 ] which is composed of a cylinder body attached to the arm (M 2 ) and a rod attached to the box beam (M 4 ). The rod moves versus the cylinder body in the direction M 2 M 4, inducing the tipping mechanism. A force F b is applied to the point M 5 modeling the load of the dumpster and its contents. As for the point M 6, it sets the dumpster to the chassis. As regards the modified version (see Figure1-2 ), the small rod [M 1 M 2 ] is the lower black part which is attached to the vehicule chassis (M 1 ) and to the arm (M 2 ). Two pivots are localized at the point M 2 : the first one, which is quite usual, links the arm to the hydraulic cylinder; the second one links the small rod to the arm and has a locking angular which the threshold α 0 is equal to 90. Therefore the tipping takes place in two stages : as long as the angular threshold α 0 is not reached, the arm slides on the fixed support and the hydraulic cylinder straightens up without maximum effort (see Figure2-1 ); when α 0 is reached, the locking allows the overall mechanical system to lift until the end of tipping (see Figure2-2 ). 1 Before the locking 2 After the locking Figure 2 The two stages of the tipping of the modified version 2

3 2.2 Boundary conditions In this context, the Lagrangien formalism from analytical mechanics is the most suitable to describe the kinematics problem. It considers the entire studied system directly from its kinematic energy, in which the maximal reaction force F of the hydraulic cylinder can be evaluated. As boundary conditions, a constant velocity v 0 is applied to the rod of the hydraulic cylinder and is equal to 100mm/s, i.e. t [0,t max ] M 2 M 4 (t) = M 2 M 4 (0)+v 0.t (1) with t max = (M 2 M 3 +M 3 M 4 M 2 M 4 (0))/v 0 (2) The final time t max is by default the moment where the hydraulic cylinder is aligned with the arm, i.e. M 2 M 4 (t max ) = M 2 M 3 + M 3 M 4. In the following we consider the stroke, the opening ǫ of the hydraulic cylinder and the angle of the tipping θ by : = M 2 M 4 (t) M 2 M 4 (0) ǫ = /M 2 M 4 (0) (3) θ = M 2 M 6 M 5 (t) θ 0 with θ 0 = M 2 M 6 M 5 (0) (4) Finally, we will assume that the applied force F b is a continuous function and mostly decreasing of θ in order to reproduce a progressive emptying of the dumpster until a maximum angle of tipping defined by θ m : F b (θ) = STEP(θ,0,0,θ s,f b )+STEP(θ,θ s,0,θ m, F b ) (5) where : STEP(x,x 0,h 0,x 1,h 1 ) = h 0 x x 0 h 0 +AB 2 (3 2B) x 0 x x 1 h 1 x x 1 A = h 1 h 0 B = (x x 0 )/(x 1 x 0 ) (6) The value of the parameter θ s is usually quite small (θ s << 1) to avoid some useless numerical perturbations from dynamic computations, and F b is the maximum load applied to the mechanical scissor system (i.e. when the dumpster is full). 3 Numerical optimization An interesting optimization of the scissor system is to find the best kinematics in which the thrust of the hydraulic cylinder and the global size of the system are minimized. Indeed, the maximum reaction force F should be lowered to improve the mechanical performance of the system, and also to increase its life span. As for the global size, it is decided that the height H of the kinematics at the initial position should be also minimized, in order to reduce the material quantity, and also to lighten the scissor system of tipping. Thus, our optimization problem is postulated multiobjective with two conflicting objective functions (F and H) and the problem resolution gives rise to a set of compromised solutions, i.e. the Pareto-optimal solutions. In this context, the particular NSGA-II for Non-dominated Sorting Genetic Algorithm - II is considered [1, 2] to resolve problem. It uses an elitist approach i.e. it allows saving the best solutions found in previous iterations, and specifically, its fast non-dominanted sorting procedure makes it more efficient in space research (getting many local optimums) and faster than other classical genetic algorithms, such as Pareto Archived Evolution Strategy (PAES) and Strength Pareto EA (SPEA) [3, 4]. Indeed, authors prove that the proposed NSGA-II is able to find much better spread of solutions and better convergence near the true Pareto-optimal front with O(MN 2 ) computational complexity (M and N being respectively the number of objectives and the population size). In order to find as many Paretooptimal solutions as possible, an initial population of 100 individuals [x i,y i ] (coordinates set of the points M i ) is randomly generated and the maximum number of iterations is 200. It means that iterations will be made during the optimization process, and each of them represent no more than computational complexity according to the NSGA-II optimizer. 3

4 3.1 Geometry constraints The previous problem is subjected to geometry constraints in which the goal is to adjust the optimization process to satisfied solutions. Therefore, constraints must be formulated relative to the optimized parameters [x i,y i ]. Concisely, three kind of constraints are considered : Compatibility constraints : In order to make the scissor system easily interchangeable with the new optimized one, the three points M 1, M 5 and M 6 will be constant during the optimization process. Especially for the modified version, M 2 must be always located on the segment [M 1 M 3 ] at the initial position. The fixed support, which M 7 is the contact point with the arm, will be always located at the middle of [M 3 M 2 ]. Then : x 2 [x 3,x 1 ] y 2 = (y 3 y 1 )x 2 +y 1 x 3 y 3 x 1 x 7 = x 2 +x 3 x 3 x 1 2 y 7 = (y 3 y 2 )x 7 +y 2 x 3 y 3 x 2 x 3 x 2 (7) Hydraulic cylinder constraint : A special hydraulic cylinder has been designed for the future optimized scissor system whose its initial size M 2 M 4 (0) and its maximum stroke m cannot be modified and are respectively equal to 458mm and 261mm. In that case, the parameters y 4 will be evaluated by : x 4 [ r+x 2 ;r +x 2 ] y 4 = y 2 + r 2 (x 4 x 2 ) 2 with r = M 2 M 4 (0) (8) And the maximum opening ǫ m of the hydraulic cylinder will be fixed with a value of : ǫ m = m /r 57%. The associated time t m can be determined and its value should be always smaller than t max. From the equations (1) and (3) the expression between t m and ǫ m is given by : t m = min(ǫ m.r/v 0,t max ) (9) Angle constraint : A goodtippingshouldbedoneifthemaximal angleθ m ofthecontainer, reached at the moment t m, is greater than or equal to 55. From some mathematical arrangements relative to previous equations, we deduce that : ( M2 M6 2 θ m = arccos +M 5M6 2 M 2M4 2(t m) M 4 M5 2 +2M ) 2M 4 (t m )M 4 M 5 cosα 245 θ 0 (10) 2M 2 M 6.M 5 M 6 with : ( M3 M4 2 α 245 = arccos +M 4M5 2 M 3M5 2 2M 3 M 4.M 4 M 5 ) ( M3 M4 2 arccos +M 2M 4 (t m ) 2 M 2 M3 2 ) 2M 3 M 4.M 2 M 4 (t m ) (11) 4 Results and comparisons In this section we optimize a classical 3T5 scissor system of tipping from industry which can raise up to 3.5Tons; hence its name. Its classical kinematics is firstly optimized whose results are shown in Table1. Here, two far solutions of x are retained : the one in which the F is minimized, noted Minimal F, and the other one in which H is minimized, noted Minimal H. Individual x 3 [mm] x 4 [mm] y 3 [mm] F [N] H [mm] θ m [degree] 3T Minimal F E Minimal H Table 1 Minimal F and Minimal H from the classical kinematics optimization From now on, Minimal F is noted Opti3T5 and is the most desirable solution. It improves the 3T5 one with a fall in force of 11% approximately. Then, the Opti3T5 is optimized with the new modified 4

5 21e me Congre s Franc ais de Me canique Individual Opti3T5 Minimal F Minimal H x2 [mm] x3 [mm] Bordeaux, 26 au 30 aou t 2013 x4 [mm] y3 [mm] -7.8E F [N] H [mm] θm [degree] 55, Table 2 Minimal F and Minimal H from the modified version optimization version kinematics whose results are shown in Table 2. Minimal F is still chosen to be the best solution, and henceforth noted Opti3T5Rod. Comparing the results between Opti3T5 and Opti3T5Rod, we see a fall in force of 27.64% which proves that the modified kinematics assumption is very satisfying. In the end, we can also see that the Opti3T5Rod improves the original 3T5 scissor system with a gap of 37.6% in terms of force. In Figure 3 all results are plotted in the (F, H)-space. For both optimizations, the set of good compromised solutions is depicted as the Pareto-optimal front (blue points) between the Minimal F point and the Minimal H point. 5 2 x 10 5 x T Opti3T5 1.4 Reaction force F [N] Reaction force F [N] {Pareto optimal Solutions} Minimal H 1.1 Minimal H {Pareto optimal Solutions} Minimal F (Opti3T5) Minimal F (Opti3T5Rod) Height H [mm] Height H [mm] Figure 3 Pareto-optimal Front in the (F,H)-space An interesting comparison is also made in Figure 4, where three curves of the hydraulic cylinder reaction force associated to the three cases (3T5, Opti3T5 and Opti3T5Rod) are plotted. On the red curve (Opti3T5Rod case), we clearly characterize the two stages of tipping previously described in Figure 2. Before the locking, the small rod plays a significant role in terms of force because the hydraulic cylinder doesn t make as much effort as both of the classical 3T5 and Opti3T5 cases. Figure 4 Reaction forces of the hydraulic cylinder 5

6 5 Conclusion Mostly, manufacturers who are facing a kinematics problem are tempted to optimize each member of the mechanical system according to general mechanical theories. Here, the proposed method based on the NSGA-II optimization techniques allows to find good solutions from a large map of classical kinematics of mechanical scissor system. In order to describe the overall kinematics movement and adequate mechanical measures (reaction forces, opening and stroke of the hydraulic cylinder), the suitable Lagrangian formalism is employed as analyzer. Applied to a multiobjective problem with geometric constraints, in which hydraulic cylinder thrust and the overall size of the mechanical system are minimized, the optimized solution choiced from the Pareto-optimal front seems satisfying with a fall in force/in size about 11%. As well, a modified kinematics has been proposed in which a small rod has just been added between the arm and the chassis with a locking angular system, in order to allow the hydraulic cylinder straightening up without maximum effort. Applied to the same optimization method, these results are very satisfying because the thrust of the hydraulic cylinder has now greatly reduced, until 37.6% of decrease from the original scissor system. Acknowledgements On behalf of ESIEE-Amiens, the authors would like to express their gratitude to the European Fund for Regional Development FEDER and the Picardy Regional Council, France, for the support to this project. We also express our gratitude to our Picardy industrial partners. Références [1] Beb, K. and Pratap, A. and Agarwal, S. and Meyarivan, T A fast and elitist multiobjective genetic algorithm : NSGA-II. IEEE Transactions on Evolutionary Computation. vol. 6 num. 2 [2] Srinivas, N. and Deb, K Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms. Evolutionary Computation. vol.2 pp [3] Knowles, J. and Corne, D The Pareto archived evolution strategy : a new baseline algorithm for Pareto multiobjective optimisation. Proceedings of the 1999 Congress on Evolutionary Computation. [4] Zitzler, E Evolutionary algorithms for multiobjective optimization : Methods and applications. Swiss Federal Institute of Technology Zurich., 6