Kinematic optimization of ball-screw transmission mechanisms.pdf

Kinematic optimization of ball-screw transmission mechanisms D. Mundo a,*, H.S. Yanb a Department of Mechanical Engineering, University of Calabria, 87036 Arcavacata di Rende (CS), Italy b Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC Received 20 June 2005; accepted 13 February 2006 Available online 4 April 2006 Abstract The paper proposes a method for the kinematic optimization of transmission mechanisms, where non-circular (NC) gears are used to perform a mechanical control on the output motion. The investigation presented here deals with the motion control of a ball-screw transmission mechanism. The objective is lowering the peak acceleration value of the screw, by designing a pair of variable radius gears as a driving mechanism. The kinematic characteristics of the ball-screw mechanism are analyzed by means of non-dimensional motion equations in order to formulate the optimization problem. A genetic algorithm (GA) is then implemented to optimize the objective function, and a penalty method is used to fulfi l the design rules. The kinematic analysis of the optimal mechanism revealed a 37% reduction of the peak acceleration of the screw in comparison with a constant pitch screw, operated at a constant speed. A kinematic simulation is used to validate the method. ? 2006 Elsevier Ltd. All rights reserved. Keywords: Ball-Screw transmission; Kinematic optimization; Genetic algorithm; NC gears 1. Introduction The pursuit of high productivity and high quality in industry urges researchers to investigate on eff ective mechanism design methods in order to improve the performances of automatic machines. The traditional methods of improving the output-motion characteristics assume the input-speed to be constant and propose to redesign and to manufacture a diff erent mechanism, with better kinematic or dynamic performances. An example is the optimal design of fl exible cam mechanisms proposed by Mills et al. 1. A diff erent approach to the problem is the active control of mechanism input-speed, by designing a variable input/output driving system. In 1956, Rothbart 2 proposed the use of a Withworth quick-return mechanism to provide a cam with a variable input-speed, thus reducing the cam dimensions and hence the pressure angle. Later, Tesar and Matthew 3 derived motion equations for the analysis of variable input-speed cam-follower mechanisms. 0094-114X/$ - see front matter ? 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.02.002 * Corresponding author. Tel.: +39 0984 494159; fax: +39 0984 494673. E-mail address: d.mundounical.it (D. Mundo). Mechanism and Machine Theory 42 (2007) 3447 Mechanism and Machine Theory The rapid development of servomotors and their control systems suggested researchers to design servo-inte- grated mechanisms, characterized by a computer controlled input-speed. In 1994, Chew and Plan 4 used a dc servomotor to minimize the residual vibrations in high-speed electromechanical bonding machines, while Yan et al. 5,6 demonstrated that the kinematic characteristics of followers are dependent on cams speed curve. Furthermore, they proposed the theory of Active Control of cam mechanisms 7, by developing a method to design the optimal computer-controlled input-speed function. In 1990, Kochev 8 proposed to actively balance shaking moments and torque fl uctuations in planar link- ages, while recently Yao et al. 9 studied the dynamics of variable-speed planar mechanisms. In spite of a wide literature about variable input-speed functions as a mean of motion optimization, few researches focus on the application of this technique to ball-screw transmissions. Such mechanisms, basically formed by a ball-screw linkage, driven by a slidercrank system, are used in several industrial applications. An example is the screw transmission mechanism used in textile machines 10. Because of their improved kine- matic behaviour, variable pitch screws are commonly used in commercial applications. In 1993, Yan and Liu 11 proposed a method to design and to manufacture variable pitch lead screws with cylindrical meshing elements. They further suggested a cubic polynomial relationship between the linear displacement of the slider and the rotation of the screw. Recently, Liu et al. 12 used a servomotor to actively control the input-speed of the slidercrank mechanism, in order to reduce the peak acceleration of the screw. The objective of the work presented in this paper is to optimize the output motion of ball-screw transmis- sions, by designing a driving mechanism, basically formed by a slidercrank system driven by a pair of non- circular gears. A combined mechanism is then proposed, where the input is the constant rotating speed of the driving NC gear, and the screw is forced to move according to an optimal law of motion. The pure mechanical control of the screw is based on the kinematic synthesis of variable-radius pitch lines, starting from the optimal input/output relationship 1315. Since a fl exible control strategy is not required in this application, com- puter-controlled servomotors can be replaced by a cheaper and eff ective pair of NC gears. In order to design an optimal driving mechanism, non-dimensional motion equations are derived. The objective function, in which the design constraints are inserted as penalty functions, is then defi ned, while the optimization problem is solved by using evolutionary theory 16. Genetic algorithms are widely used in problems involving global optimization. The main advantages of evolutionary techniques are their simplicity in implementing the numerical procedure and their low computational cost 17. Furthermore, a deep knowledge of the mathematical characteristics of searching space is not required. Once the optimal design of NC gears is performed, a virtual prototype of the combined mechanism and a kinematic simulation are used to validate the proposed control strategy. 2. Motion equations A ball-screw transmission is basically formed by two combined mechanisms: a ball-screw linkage, driven by a crankslider mechanism. Crank rotation is the input of this mechanical system, while the reciprocating rotation of the screw is the output. In Fig. 1 the mechanism is schematically represented as consisting of fi ve members: link 1 is the frame, links 2, 3 and 4 are, respectively, the crank, the connecting rod and the slider of the driving mechanism, link 5 is the screw. Dimensionless motion equations can be derived by considering the Fig. 1. Schematic representation of a ball-screw transmission mechanism. D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344735 two basic mechanisms separately, as shown in Fig. 2. By referring to Fig. 2(a), the position equations of the slidercrank mechanism are the following: r3cosh3 r2cosh2 r4;1 r3sinh3 r2sinh2;2 being r2and r3the lengths of the crank and of the connecting rod, h2and h3the angles the links form with the negative X-axis, r4the position of the slider. By combining Eqs. (1) and (2), the displacement s of the slider can be determined as s r2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi r3 r2 ? ?2 ? sin2h2 s ? cosh2? r3 r2 ? 1 ? 2 4 3 5. 3 While the crank rotates an angle p in a time s equal to half a period of the mechanism motion, the slider completes a stroke of 2r2 . Dimensionless motion equations can be then derived by defi ning the following non- dimensional parameters: T t s ;4 S s 2r2 ;5 H h2 p ;6 where t 2 0,s, s 2 0,2r2, h 2 0,p, while T, S and H vary between 0 and 1. By substituting Eqs. (4)(6) in Eq. (3), the non-dimensional slider displacement is obtained: S 1 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 2R32? sin2pH q ? cospH ? 2R3? 1 ? GH;7 where the non-dimensional length R3 is defi ned as r3/2r2. Fig. 2. Schematic representation of the crankslider mechanism (a) and of the ball-screw linkage (b). 36D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447 By diff erentiating Eq. (7) with respect to T, the dimensionless slider velocity (V) and then acceleration (A) and jerk (J) are obtained: V dS dT G0H ? _ H;8 A d2S dT2 G00H ? _ H2 G0H ? _ H ? H;9 J d3S dT 3 G000H ? _ H3 3G00H ? _ H ? H G0H v H.10 By referring to Fig. 2(b) and assuming the pitch p of the screw to be constant, the output rotation / is given by / 2p p s;11 from where the dimensionless screw rotation r can be derived as U / /max S.12 Therefore, since the rotation of a constant-pitch screw is linearly dependent on the slider displacement, the screw and the slider dimensionless motion equations are the same. The screw non-dimensional velocity, accel- eration and jerk can be then computed by means of Eqs. (8)(10). Dimensionless motion equations show that the kinematics of constant-pitch screw-ball transmissions can be improved by forcing the crank to rotate according to an optimal law of motion H(T). In the following sec- tions an optimization strategy will be implemented in order to design an optimal control function. Once non-dimensional motion characteristics of the optimal screw are determined, the actual kinematic curves can be determined by the following relationships, deriving from Eqs. (8)(10): / 2p p ? 2r2? S;13 v d/ dt 2p p ? 2r2? dS dt 2p p ? 2r2? dS dT ? dT dt 2p p ? 2r2 s ? V ;14 a dv dt 2p p ? 2r2 s ? dV dt 2p p ? 2r2 s2 ? A;15 j da dt 2p p ? 2r2 s2 ? dA dt 2p p ? 2r2 s3 ? J;16 where /, v, a and j are the actual angular displacement, velocity, acceleration and jerk of the screw. 3. Optimal control strategy The optimal control of the screw motion requires that the objective function and a set of design rules are defi ned. Once the optimization problem is formulated, a genetic algorithm will be used to minimize the cost function. The penalty method is employed to ensure the optimal solution fulfi ls the design rules. 3.1. Formulation of the optimization problem Main objective of the optimization problem is to design a rotating-speed function of the crank in order to actively control the screw kinematics and to minimize its peak acceleration during the forward stroke of the slider, so that inertial loading problems can be reduced during the working period. Moreover, both the screw and the crank motion characteristics should fulfi l a set of kinematic requirements and general design rules. The screw velocity and the acceleration curves must be continuous, while fi nite values of the jerk are required. Therefore, it is known from Eqs. (7)(10) that the crank rotating speed X(T) must be at least a second order D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344737 diff erentiable function. If the crank rotation H(T) is selected as a control function, a fourth- or higher-order polynomial expression can be then defi ned in the following form: HT a0 X N i1 aiT i; 17 where the design variables (a0, ., aN) must be determined so that the screw rotates according to the optimal law of motion and the following boundary conditions, deriving from Eq. (6), are satisfi ed: H0 0;18 H1 1.19 From Eqs. (17)(19), the following conditions are derived: a0 0;20 1 ? X N i1 ai 0.21 To complete the formulation of the optimization problem, the cost function must be defi ned. The main objective is to reduce the peak acceleration of the screw. However, the optimal control function must fulfi l the following design rules: 1. A typical crank continuously rotates without changing direction. Therefore, the time-derivative of the con- trol function cannot change in sign. Without loss of generality, in this work the crank speed will be kept positive. 2. The second time-derivative of the control function must be moderate, since non-circular gears will be used to provide the crank with the variable-speed function and sudden changes in _ HT would determine irreg- ular pitch lines. On the basis of the design rules, the cost function can be defi ned as Ca0;.;aN w1maxabsA w2maxabsH P;22 where the weighing factors w1and w2 can be adjusted according to diff erent optimal strategies, while the term P is used to penalize the control function when one or more changes in the sign of _ H occur. The penalty method assures that the design constraints are fulfi lled, since any infeasible set of design parameters will have a greater value of the cost function than an admissible solution. Obviously, if _ HT 0 T 2 0,1, the penalty term in Eq. (22) is set to zero. 3.2. Optimization method In order to solve the optimization problem as formulated in the previous section, the strategy of evolution- ary methods is used. A modifi ed genetic algorithm, schematically represented in Fig. 3, is employed in this paper 18. The fi rst step is the generation of the starting population, formed by NPindividuals (chromo- somes). Each individual consists of a set of design-variable admissible values (genes). Therefore, the generic individual is a possible solution of the optimization problem and can be represented by a vector of real num- bers in the form Xi x1x2?xn?;i 1;.;NP;23 where n is the number of independent design variables. The evolutionary optimization strategy is based on the survival of the fi ttest individuals. These individuals undergo a set of genetic operations (reproduction) in order to promote the population evolution. This process is known as natural selection. The fi rst step of reproduction is the selection of NPcouples of individuals (par- ents), whose genetic information will be combined to generate one individual (child) for the next population. 38D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447 Therefore the population size is kept constant. Selection of individual for reproduction can be based on dif- ferent probability distributions, including uniform distribution. In the algorithm used in this work, the method of normalized geometric ranking selection is used. The probability for an individual to be selected is based on its performance (fi tness), according to the following expression: Pr Pb 1 ? 1 ? PbNP 1 ? Pbr?1;24 where Pbis a constant and is proportional to the probability of selecting the best individual, and r is the rank of the individual. The rank is 1 for of the best individual, Np for the individual with the lowest fi tness value. Therefore, before selecting individuals for reproduction, the cost function of each chromosome must be evaluated in order to establish a fi tness-based order inside the population. Once two individuals X1and X2are selected for reproduction, a genetic operation (crossover) is necessary to generate a new individual X? 1, whose genes are derived from parents chromosomes. In this work the new individual is created by means of a heuristic crossover operation, as follows: X? 1 X1 rnX1? X2; 25 where rn is a real number randomly selected in the range 0,1. According to an elitist strategy, the new individual will enter the next population only if it has a greater fi tness than its parents, otherwise X? 1will be rejected and the best chromosome between X1and X2is retained. Fig. 3. Scheme of the genetic algorithm. D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344739 The last step of reproduction is mutation, an operator that changes a piece of genetic information of the new individual. Mutation is necessary in order to prevent the algorithm from converging towards a local min- imum condition, and it is carried out with a probability PM2 0,1. In this work non-uniform mutation is employed. A gene of the mutating individual is randomly changed according to the following scheme: xi? xi ui? xifGif ri 0:5; xi? xi? lifGif ri6 0:5; ? 26 where riis a random number in the range 0,1; uiand liare the upper- and lower-bound of the ith gene; f(G) is a function defi ned as follows: fG rn21 ? G Gmax ?b ;27 being rn2a random number in the range 0,1, b the shape parameter of mutation, G the current generation, Gmaxthe maximum number of iterations. The sequence of fi tness evaluation, selection of individuals for reproduction, crossover and mutation is iterated, according to the scheme of Fig. 3, until the maximum number of generations is reached or a cost-function low value is achieved. 3.3. Optimal control function Before starting the iterative procedure as described in the previous section, GA parameters must be initi- alized. The size of starting population should be established on the basis of the number of design variables. In the application presented here a seventh order polynomial function is chosen as a control function. There- fore, according to Eqs. (17), (20) and (21), six design variables must be determined by means of evolutionary theory. A starting population of Np= 60 individuals is then generated. By setting Pb= 0.5, PM= 0.1, Gmax = 100 and b = 0.85, the best individual of the fi nal population was found to be the following: XBEST 1:1685?0:25711:2e?52:3984?5:90515:0918?.28 The optimal control function, that is the dimensionless crank rotation as defi ned by Eqs. (17)(21) and by the genes of the best individual, is the following: HOPTT 1:168 ? T ? 0:257 ? T 2 2:398 ? T 4 ? 5:905 ? T 5 5:091 ? T 6 ? 1:495 ? T 7. 29 Fig. 4. Optimal kinematic curves of the crank. 40D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447 In Fig. 4 the curves of dimensionless displacement, velocity, acceleration and jerk of the crank are shown, as deriving from the optimization procedure. By substituting HOPT(T) and its time-derivatives in Eqs. (7)(10), the optimal curves of dimensionless angu- lar displacement, velocity, acceleration and jerk o