Selection of Servomotors and Reducer Units for a 2 DoF PKM

Transcription

1 Selection of Servomotors and Reducer Units for a 2 DoF PKM Hermes GIBERTI, Simone CINQUEMANI Mechanical Engineering Department, Politecnico di Milano, Campus Bovisa Sud, via La Masa 34, 20156, Milano, Italy ABSTRACT The paper describes the problem of selecting the motor and reducer units for 2 degrees of freedom parallel kinematic machine. Parallel kinematic machines attract researchers and companies, because they are claimed to offer several advantages over their serial counterparts, like high structural rigidity and high dynamic performance. The need to increase production capacity, while maintaining the quality standards, required the implementation of automatic machines performance ever higher. In this context, it is of strategic importance in the machine design phase the correct selection of the motor-reducer unit. Unfortunately, the choice of the electric motor required to handle a dynamic load, is closely related to the transmission choice. The selection of suitable motors and transmissions is carried on introducing some parameters that describe the performance of the motor, the power required by the system and the influence of the transmission mechanical characteristics on the machine performance. 1. INTRODUCTION The evolution of electronics in recent years has led to a wide diffusion of electric drives and their control systems. The ready availability and low cost of electronic devices has allowed rapid diffusion of mechatronic applications, highlighting the need for appropriate methods for selecting a motor-reducer unit. These procedures must be at the same time accurate and easy to use, they should be able to identify the available alternatives, compare them and help the designer in choosing the most appropriate one for his needs. The choice of the electric motor required to handle a dynamic load is closely related to the choice of transmission. This operation, in fact, is bound by the limitations imposed by the motor s working range and is subjected to several constraints that depend indirectly on the motor (through its inertia J M ) and on the reducer (through its transmission ratio τ), whose selection is the subject of this paper. A methodology for choosing the gear motor in order to ensure maximum acceleration of the system and reduce execution time for a particular law of motion is presented in [1]. This article introduces the so-called problem of inertia matching, showing how best performance can be reached when the inertia of the load, referred to the motor shaft, coincides with the inertia of the motor itself. In [2] a procedure for the selection of an AC synchronous motor with permanent magnets and its reducer, for a generic load, is shown. The authors use normalized torques, velocities, and transmission ratios to separate the load from the motor characteristics. By virtue of this normalization, the simulations for one standard motor (J M = 1 [kg m 2 ]) are applicable to other motors. In [3], the same procedure is extended to all types of servomotors. This methodology produces a chart representing all the usable motors and their corresponding normalized transmission ratios range, but not the available and actually usable commercial transmissions. In [5] the choice of the motor-reducer unit is analyzed with regard to the dependence on the law of motion used to operate a generic load, while [6] evaluates the gain in motor torque as a consequence of the optimization of trajectories and highlights the effect that a variable transmission ratio has on the performance of the machine. An in-depth discussion on the problem of motor reducer coupling can be found in [7]. Although this work is very accurate, it is extremely hard to use in a real industrial situation. On the other hand, [8] proposes a simpler approach, that consists in creating a database including the commercial motors and reducers and then trying all possible combinations. In [9], [10] the choice of motor and reducer is made by comparing two parameters respectively related to motor features and to load demands. The relationships between motor and transmission are investigated by introducing some easily calculated factors useful for comparing all the available motor-reducer couplings and selecting the best solution. The procedure is carried on with the use of graphs that allow allow showing all the possible alternatives. Following this approach the paper shows how it is possible to select the motor-reducer units for a 2 d.o.f. parallel kinematic manipulator. The paper is structured as follows. Section 2 gives a brief description of the manipulator with its main subsystems. Section 3 recalls the conditions to select a servomotor and a speed reducer and the corresponding checkouts. Section 4 introduces the multibody model of the system, developed to calculate the inverse dynamic. Section 5 shows the selection of the motor reducer units and the corresponding checkouts. Finally conclusions are drawn

2 in Section THE SYSTEM The parallel kinematic machine under study is a 5R-2 d.o.f. manipulator consisting on 4 links (5 considering the ground) connected by five revolutionary joints (R) two of which are located on the ground and driven by motors (Fig.1). The manipulator can reach a place inside its workspace of coordinates x = [x e ; y e ] T which is as function of the actuated joints coordinates q = [θ 1 ; θ 2 ] T. It is constituted by 4 main elements: 1. the support, which is fixed and connected to the ground; 2. the driving system, constituted by 2 brushless motors, each actuating a joint. Main features of each motor are resumed in Tab.1 motor transmission manipulator support Table 1: Motor main features Figure 1: The manipulator Symbol T M J M T M,N TM,max T H T M,max ω M ω M,max Description motor torque motor moment of inertia motor nominal torque motor theoretical maximum torque servo-motor maximum torque motor angular speed maximum speed achievable by the motor ω M motor angular acceleration α accelerating factor [9] 3. the transmission, which changes the torque and the speed supplied by the motor to the ones requested at joints. Main features of transmission are resumed in Tab.2 Symbol τ η J T Table 2: Transmission main features Description transmission ratio transmission mechanical efficiency transmission inertia pick and place operation; execution of a job along a defined trajectory. The power supplied by the motor depends on the external load applied (T L ) and on the inertia acting on the system (J L ω L ). Since different patterns of speed (ω L ) and acceleration ( ω L ) generate different loads, the choice of a proper law of motion is the first project parameter that should be taken into account when sizing the motorreducer unit. Otherwise, it may be that the law of motion has been already defined and therefore represents a problem datum, and not a project variable. Once the law of motion is defined, all the characteristics of the load are known (Tab.3). 3. CONDITIONS OF SELECTION AND CHECKOUTS As it is well known [2], the selection of the motor-reducer unit means checking the following conditions: rated motor torque: 1 T M,rms = t a ta 0 T 2 M dt T M,N ; (1) maximum motor speed: 4. the manipulator (coloured with light blue), machine consisting in 4 connected by five revolutionary joints The task of the robot is defined by the user and generally can consists on: maximum servo-motor torque: ω M ω M,max ; (2) T M (ω M ) T M,max (ω M ). (3)

3 Symbol T L J L TL Table 3: Load main features Description load torque load moment of inertia generalized load torque (TL = T L + J L ω L ) TL,rms generalized load root mean square torque T L,max load maximum torque ω L load angular speed ω L load angular acceleration ω L,rms load root mean square acceleration ω L,max maximum speed achieved by the load t a cycle time β load factor [9] Once the available motor-reducer units have been identified it is necessary to check some further conditions: The maximum torque supplied by the servo-motor for each angular velocity achieved: max ( τt L η + JM + J T τ T M,max (ω M ) + J ) Lτ ω L ω. η (4) the effect of the transmission mechanical efficiency (η) and its moment of inertia (J T ) on the root mean square torque: = ta 0 T 2 M,N T 2 M,rms = 1 t a ta TM 2 ( (J M + J T ) ω L τ + τt L η 0 dt = t a ) 2 dt (5) the resistance of the transmission as supplied by the manufacturer. Conditions expressed by inequalities (1), (2) and (3) are well known in the literature and represent the starting point of all the procedures for motor and reducer selection. This paper follows the procedure of selection described in [9], [10] where these conditions have been rewritten by introducing certain parameters related to motor and load features. Most important are the accelerating factor α: and the load factor β: α = T 2 M,N J M, (6) β = 2 [ ω L,rms T L,rms + ( ω L T L) mean ] (7) It is important to highlight that all the parameters used have a physical meaning and are easily obtained. In this way the designer will have a clear idea of the needs and of the steps to follow. The preliminary choice of motor is made by comparing only the values α and β; these values are easily calculated if we know the mechanical properties of the motor and the load features. A motor must be rejected if α < β, while if α β the motor can have enough rated torque if τ is chosen properly. Once these parameters are calculated, conditions (1), (2), can be easily expressed, for each considered motor, as functions of τ. = JM 2T L,rms τ min, τ max = [ α β + 4 ω L,rms TL,rms ± ] α β. (8) τ M,lim = ω L,max ω M,max (9) The result is a range of acceptable transmission ratios for each motor. τ max τ max (τ min ; τ M,lim ). (10) However, since it is difficult to express the constraint imposed on the servo-motor maximum torque (3) as a function of τ, this condition will be checked after the motor and its transmission have been chosen. 4. MULTIBODY MODEL To evaluate the power required to the motors when the manipulator performs the desired task, a multibody model has been developed.the model allows to define the trajectory the robot has to follow, the end effector motion law and the external forces applied to the manipulator. Once the task has been defined, simulations allow to calculate: - torques required to the motor-reducer unit as a function of time (T M (t)). From this result it is possible to calculate the root mean square torques (T M,rms ) and the maximum torque (T M,max ) required during the task; - the motor angular velocity as a function of time (ω M (t)); - reaction forces on joints to properly size the structure. For example it is possible to define a linear trajectory (Fig.2), with trapezoidal motion law ( 1 3, 1 3, 1 3 ) (Fig.3), with a cycle time t a = 0.8s. The corresponding motion laws of joints and the required motor torques are depicted in Fig.4,5.

4 T 1 T 2 Figure 5: Required motor torques Figure 2: Linear trajectory (diametrical) limited space and weight; displacement high positioning accuracy and repeatability; high torque availability; velocity low wear; high torsional stiffness; acceleration Figure 3: Motion law along the desired trajectory displacement velocity acceleration Figure 4: Motion law q 1 (t), q 2 (t) q 1 q 2 5. MOTOR REDUCER UNIT SELECTION The choice of the motor reducer unit should satisfy some design requirements: high transmission ratio. For these reasons DC brushless motors and Hermonic Drive speed reducers have been considered. Available motors are models: F HA 8C, F HA 11C, F HA 14C, while speed reducers have transmisison ratios: 1/30, 1/50, 1/100. Once the task the robot has to perform (trajectory, mnotion law, external forces applied), numerical simulatuions on multibody model allows to calculate all the terms related to the load as, generalized resistant toque (T L ) applied to motor shafts, velocities ( (q) 1, (q) 2 ) and acceletions ( (q) 1, (q) 2 ). using these values it is possible to calculate the load factor β that can be compared with the accelerating factors α calculated for each considered motor using information on manufacturers catalogues. Consider now the motor task described in Section 4. Figure 6 shows the values of α and β, highlighting that all the considered motors can be profitably used to perform the required task since the condition α β is verified. For each motor it is possible to calculate the range of suitable transmission ratios using eq.(10). Figure 7 highlights that transmission ratios 1/50 e 1/100 can not be selected becaouse of the limit on the maximum achievable speed. On the other side it results that motor FHA-11C, coupled with the transmission ratio 1/30 is a good combination to perform the task. However, it may be simplistic to choose the motor only on a specific task, since the robot should be suitable for many movements. It is also desirable to optimize the choice of the motor-reducer unit, investigating which solution can

5 Radial Diametrical Circular Figure 6: α and β [s] Figure 8: Comparison between α values and β as a function of the cycle time Radial Diametrical Circular Figure 7: Ranges of suitable transmission ratios τ offer the best performance, ie which allows smaller cycle times with the same law of motion. Considering three simplified trajectories (radial, diametrical and circular movement) and maintaining a trapezoidal law of motion, one can gradually decrease the cycle time to obtain the limit transmission ratio. Figure 8 shows the trend of the load factor β when the cycle time decreases. This value can be compared with the accelerating factors α calculated for each motor. The graph gives an overview to easily understand when a motor can be no longer suitable to perform the task along a certain trajectory. Figure 9 shows the maximum speed required by the task for each trajectory. It can be compared with the maximum speed achievable by each speed reducer allowing to identify which is the most suitable transmission ratio for the purposes. Looking at Fig.8 and Fig.9 it is evident that, for every considered machine task, motor F HA 11C and transmission ratio 1/30 represent alway the best solution to perform the task. Once the motor reducer unit has been selected, checkout described in Sec.3 have to be performed. In particular, Figures from 10 to 12 shows the limits of the selected motor reducer unit in terms of maximum torque and nominal torque exertable. By reducing the cycle time Figure 9: Comparison between ω Lmax values and τ as a function of the cycle time the points (T rms,ω rms ) and (T max,ω max ), move to the border of the motor working zone. The closest point represent a limit working condition. 6. CONCLUSION The paper presents a the selection of the most suitable motor and speed reducer pairing to drive a parallel kinematic machine, once its task has been defined. The methodology is based on information achievable on manufacturers catalogues and it is carried out through a graphical representation of the characteristics of the machine and of available motors and speed reducers. The designer has then a useful procedure to compare all the feasible solutions and to choose the best one.

6 Dynamic working zone Dynamic working zone Max Continuous working zone Continuous working zone Max Rms Rms Figure 10: Checkout for a task with radial trajectory Figure 12: Checkout for a task with circular trajectory Continuous working zone Rms Dynamic working zone Max Figure 11: Checkout for a task with diametrical trajectory REFERENCES [1] Pasch K.A.,Seering W.P. (1984) On the drive systems for High-performance Machines, Transactions of ASME 106: [2] Van de Straete H.J, Degezelle P., de Shutter J., Belmans R.(1998) Servo Motor Selection Criterion for Mechatronic Application, IEEE/ASME Transaction on mechatronics 3:43-50 [3] Van de Straete H.J, de Shutter J., Belmans R. (1999) An Efficient Procedure for Checking Performance Limits in Servo Drive Selection and Optimization, IEEE/ASME Transaction on mechatronics 4: loads, Mechanism and Machine Theory 38, (2003) [5] Cusimano G. (2003) A procedure for a suitable selection of laws of motion and electric drive systems under inertial loads, Mechanism and Machine Theory 38: [6] Van de Straete, H.J, de Shutter, J., Leuven, K.U., Optimal Variable Transmission Ratio and Trajectory for an Inertial Load With Respect to Servo Motor Size, Transaction of the ASME 121, (1999) [7] Cusimano G. (2007) Optimization of the choice of the system electric drive-device-transmission for mechatronic applications, Mechanism and Machine Theory 42:48-65 [8] Roos F., Johansson H., Wikander J. (2006) Optimal selection of motor and gearhead in mechatronic application, Mechatronics 16:63-72 [9] H. Giberti, S. Cinquemani, G. Legnani (2011) A Practical Approach for the Selection of the Motor-Reducer Unit in Electric Drive Systems, Mechanics Based Design of Structures and Machines (in press) [10] H. Giberti, S. Cinquemani, G. Legnani (2010) Effects of the mechanical characteristics of the transmission on the choice of motor-reducers, Mechatronics 20(5): [4] Cusimano, G., A procedure for a suitable selection of laws of motion and electric drive systems under inertial