Motion System Classes. Motion System Classes K. Craig 1

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1 Motion System Classes Motion System Classes K. Craig 1

2 Mechatronic System Design Integration and Assessment Early in the Design Process TIMING BELT MOTOR SPINDLE CARRIAGE ELECTRONICS FRAME PIPETTE Fast Component Mounter Placement Module Motion System Classes K. Craig

3 Introduction During conceptual design of controlled electromechanical motion systems, one has to obtain feasible technical design specifications for the: path generator the control system the electromechanical plant with appropriate sensor locations And this must be done in an integrated way! Let s investigate how this might be done for the task of positioning an end effector at a certain location within a limited period. Motion System Classes K. Craig 3

4 During the conceptual design, the aim is not to complete a final design, but rather to identify the performance limiting factors of the design proposals and to choose satisfactory specifications for these factors. Experience has shown that the following factors dominantly determine system performance: Task specification: motion distance, motion time, required positional accuracy after motion time Path Generator: smoothness of the path Controller: proportional and differential gains Plant: total mass to be moved, lowest eigenfrequency, location of the position and velocity sensors Motion System Classes K. Craig 4

5 The dominant plant factors motivate the use of simple 4 th - order models which take only the rigid-body mode and the lowest mode of vibration into account. These models have the following characteristics: Simple and of low order Have a small number of parameters Completely describe the performance-limiting factor Are a good basis to provide reliable estimates of the dominant dynamic behavior and the attainable closedloop bandwidth A Mechatronic Approach to Design allows for the assessment of the influence of these design factors on the system performance. Let s first begin by classifying plant dynamics. Motion System Classes K. Craig 5

6 Basic Open-Loop Transfer Function Types Consider the plant transfer function P(s) from the input force u to a measured position y. The mechanical damping in the plant is neglected, as damping does not in general dominate the dynamics of a mechanism and it unnecessarily complicates the mathematics. The influence of friction is also not explicitly considered, as mechanical friction is difficult to estimate and highly nonlinear; it should be preferable to minimize it by proper mechanical design. In any way, it would be impossible to anticipate friction characteristics of a plant during the conceptual design stage. Motion System Classes K. Craig 6

7 The denominator polynomial of P(s) is always the same for a particular dynamic system, but the numerator polynomial depends on the locations of the actuator and the position sensor. The zero pair of the transfer function P(s) moves along the imaginary axis for different locations of the sensor. This is called migration of zeros and can be used to characterize five different types of plant transfer functions at an abstract level. We will consider the location of the complex conjugate zero pair in the s plane with respect to the complex conjugate pole pair. We will refer to these pairs as the anti-resonance frequency ω ar and the resonance frequency ω r of a plant transfer function. Motion System Classes K. Craig 7

8 Basic Plant Transfer Function Types 1 ms ar s s ar r 1 1 r ar 1 ms r 1 ms ar s s ar r 1 1 r 1 1 ms s 1 r 1 ms s s ar r 1 1 Type AR Type D Type RA Type R Type N Motion System Classes K. Craig 8

9 In the previous diagram: Type AR: anti-resonance resonance Type D: double integrator Type RA: resonance anti-resonance Type R: resonance Type N: non-minimum phase These basic transfer functions describe the dynamic behavior of four classes of electromechanical motion systems. These classes are characterized by the mechanical subsystem that contains the dominant stiffness. They are typically obtained after simplification and reduction of more extensive plant models. Motion System Classes K. Craig 9

10 Classes of Electromechanical Motion Flexible Mechanism Systems Dominant stiffness is located in the mechanism. X m1s K t e G s s F m s m s K X1 K G1 s s F m s m s K ar K m 1 K t e r me Motion System Classes K. Craig 10 AR R m m m t 1 mm 1 me m 1 m F m m 1 actuator x x 1 K end effector

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13 Flexible Frame Dominant stiffness is located in the supporting frame. The mass of the frame is m at position x. The mass m 1 is a rigid body containing the mass of the actuator and end effector. Its position x 1 is the position to be controlled. This position can be measured with respect to the frame or with respect to the ground, resulting in different types of transfer functions. x x 1 K m F m 1 Motion System Classes K. Craig 13

14 The transfer function from the input force F to the position measurement with respect to the frame (y = x 1 x ) is of type AR, while it is type D when the position measurement is with respect to the ground (y = x 1 ). X1 1 s F m s 1 X1 X s m1m s K F m s m s K 1 ar m K m 1 K x x 1 r K m m F m 1 Motion System Classes K. Craig 14

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17 Flexible Actuator Suspension The figure represents a system consisting of a rotating actuator with transmission that is contained in a flexible linear suspension. Motion System Classes K. Craig 17

18 Linear movements of the end effector m e are a combination of movements due to actuator rotations and suspension vibrations. i = x a / θ, where x a is the end-effector translation due only to actuator rotation. The transfer function from the input force (u = T/i) to the position of the actuator (y = θ) is of type AR. When the position of the end effector is measured (y = x e ), a type RA transfer function is obtained. x a i T u i Motion System Classes K. Craig 18 J i me k r J me mf i memf k k ar m m m ar e f f y y x e

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27 A physical interpretation of transfer function zeros for simple control systems with mechanical flexibilities is as follows. The poles of the transfer function are the resonances of a flexible structure, while the zeros are the resonances of a constrained substructure. In the case of the flexible mechanism, flexible frame, and flexible actuator suspension, the antiresonance can be looked upon as the resonance frequency of the system in case the actuator is blocked, i.e., constrained. Motion System Classes K. Craig 7

28 Flexible Guidance Due to an input force (u = F) the mass m will move in the x direction. Additionally, F will excite a rocking mode around the center of mass due to flexibilities K. F φ x a f a x m, J C.M. ar J Kb ma a f x K K r Kb J 1.00 b 1.00 b Motion System Classes K. Craig 8

29 The type of transfer function from the input force (u = F) to the measured position (y = x) depends on geometrical properties: the distances a f and a x. Five situations can be distinguished: When F and x are on the same side of the center of mass (a x > 0 and a f > 0 or a x < 0 and a f < 0) we obtain a type AR transfer function because ω ar < ω r. When either F or x are exactly located at the center of mass (a x = 0 or a f = 0) we obtain a type D transfer function, as ω ar = ω r. Note that a x = 0 or a f = 0 are two different situations, where the plant is unobservable and uncontrollable, respectively. Motion System Classes K. Craig 9

30 When F and x are on different sides of the center of mass (a x > 0 and a f < 0 or a x < 0 and a f > 0) we obtain a type RA transfer function, under the condition that J + ma x a f > 0. When F and x are on different sides of the center of mass and J + ma x a f 0, we obtain a R type transfer function, as ω ar is located at infinity. When F and x are on different sides of the center of mass and J + ma x a f < 0, we obtain a N type transfer function, as ω ar is complex. Motion System Classes K. Craig 30

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