Mechatronic Motion System

Size: px

Start display at page:

Download "Mechatronic Motion System"

Sara Logan
5 years ago
Views:

1 echatronic otion System odeling, Identification, & Analysis Objectives Understand compliance, friction, & backlash as it relates to motion systems. Develop a control-oriented model of a motion system with appropriate simplifying assumptions. Apply system identification techniques for the parameters in the model. Analyze the model and run simulations using the model to demonstrate its effectiveness.

otor Rotor Inertia with Coulomb & Viscous Friction Gearbox with Backlash and Gear Inertias (neglect friction) Rigid oad oving with Belt by Coulomb Friction or Attached Rigid Shaft with Inertia

2 otor Rotor Inertia with Coulomb & Viscous Friction Gearbox with Backlash and Gear Inertias (neglect friction) Rigid oad oving with Belt by Coulomb Friction or Attached Rigid Shaft with Inertia Compliant Shaft with Inertia and Viscous Damping Complaint Shafts and Coupling with Inertias of Shafts & Coupling Rigid oad with Coulomb & Viscous Friction acting on it otion System Pulleys 1 & with Inertias and No Belt Slip or Backlash Belt with ength- Dependent Compliance and Damping; Belt Inertia may be included

4 Shaft has infinite stiffness (rigid). Shaft has a stiffness represented by a spring constant that leads to a resonance in the model. Shaft is represented by a PDE that leads to an infinite number of resonances.

5 Background on Compliance Key Questions Where do the resonances and anti-resonances come from and does the engineer have control over them? How do the physical system parameters affect the nature of the resonances and anti-resonances? Is compliance truly a parasitic effect, the result of highspeed behavior and lighter components, or an intentional part of the design that can be modeled and dealt with well? When is a flexible coupling needed in a motion system transmission? How stiff transverse or torsional does this coupling need to be? Is there a difference between static and dynamic stiffness of couplings?

6 A goal for mechatronic motion systems is high motion quality: high speed, high accuracy, high precision, high resolution robustness to system changes Systems have time-varying and position-varying characteristics: inertia, viscous friction, nonlinear dry friction which leads to stickslip behavior, compliances from flexible belts, couplings, and shafts, and backlash from gears Understanding and modeling these effects are essential for the mechatronic design of the system.

seen many times before, like the two-mass, three-spring

7 Resonance and anti-resonance are such mystifying phenomena that even when we see a laboratory demonstration we have seen many times before, like the two-mass, three-spring system, we are still amazed by what we see. Colocated Bode Plot

8 x 1 x F(t) K K K Frictionless Surface 3 1 ode Shapes undeflected 1 st Natural Frequency 1 K K K K nd Natural Frequency 3 3K K K K node fixed Anti-Resonance K K K K

9 Resonance, and its destructive potential, always brings to mind the Tacoma Narrows Bridge Disaster in During a 4 mphwind the bridge, designed for 10 mph winds, shed vortices which caused the bridge to oscillate.

10 Resonance in engineering mostly has a negative connotation something to be avoided. Of course, without resonance we wouldn t have radio, television, music, or swings on playgrounds, but mostly resonance brings to mind its dark side it can cause a bridge to collapse or a helicopter to fly apart. Resonance requires three conditions: a system with a natural frequency a forcing function applied at the natural frequency and in phase with velocity a lack of energy loss

11 Below Resonant Frequency Force in phase with Displacement Resonance Above Resonant Frequency Force 180 out of phase with Displacement B K +v F

12 At Resonant Frequency Force in phase with Velocity Resonance At Resonant Frequency Force 90 out of phase with Displacement B K +v F

13 Hand-held Barcode Scanner

14 Oscillating mirrors are used in hand-held barcode scanners to reflect laser light out & collect reflected light from the barcode. They use less power, have less moving mass, fit in a smaller space, & survive shocks and drops better than rotating polygon mirrors, which are used in fixed scanners. Reducing the energy required to oscillate mirrors at the required frequency & also producing wide oscillation angles are important. To accomplish these objectives, the system, essentially a torsional spring-mass-damper system, is driven into resonance by a solenoid. The solenoid has two coils: one to sense frequency of oscillation & another to drive the system. The system has no inherent failure mechanism, as there are no bearings and hence no friction. As the mechanical stresses in the flexing member of the system are kept below a threshold, fatigue failure is avoided. The result of this design is extreme reliability.

15 Flexural stiffness Flexural stiffness Two coils: (1) electromagnet to oscillate mirror & () coil to sense frequency agnetic plunger

16 Nonlinear Resonance kr F mg cos m t r r mg sin m r r Spring-Pendulum Dynamic System l + r k θ m r

17 Then we have anti-resonance, an even more intriguing phenomenon, which brings to mind tuned mass-damper systems that quiet lively buildings excited by the wind or earthquakes. Taipei 101 s 730-Ton Tuned ass Damper ay 005

18 This is all very interesting, but what does this have to do with mechatronic motion systems? There are no free lunches in design; there is always a tradeoff. The best path to good design is to become aware of these tradeoffs, assess the effects of these tradeoffs through modeling and analysis, and then make an intelligent choice based on what you need. Compliance is always present in real systems. It can be parasitic and degrade motion, but it also can be used to significantly enhance motion quality. The difficulty arises when it is not modeled effectively or simply ignored.

19 A goal for mechatronic motion systems is high motion quality high resolution, precision, accuracy, and speed as well as robustness to system changes. In an ideal world, machine components would be rigid, machining and assembly imperfections or tolerances would be non-existent, and there would be no friction or backlash to overcome. Incorporating compliance into a system design can significantly enhance motion quality and it can do so in three ways.

To eliminate friction and backlash in a load-bearing situation, a designer might use a magnetic bearing or an air bearing, where there is no contact.

20 To eliminate friction and backlash in a load-bearing situation, a designer might use a magnetic bearing or an air bearing, where there is no contact. Both are very complicated, high-maintenance systems. A flexure bearing provides both load bearing and motion guidance, albeit small motion, while eliminating friction and backlash. It is designed to have an optimal distribution of rigidity and compliance. Example of a twoaxis flexure bearing in a motion system Dr. Shorya Awtar U. ichigan

21 In motion transmission, flexible couplings are used to accommodate misalignments inherent to the design or due to manufacturing and assembly tolerances, while eliminating friction and backlash.

22 Stiffness is a function of the misalignment in two ways: For a bearing selection, more misalignment requires more transverse coupling compliance. ore transverse compliance means more torsional compliance. An ideal coupling has infinite torsional stiffness and zero transverse stiffness. A universal joint has this, but with friction and backlash!

23 When fixing two components together, flexible clamps provide similar benefits. Flexure Clamps Flexible Clamps, the correct substitute for set-screw-based shaft connections In-Plane Clamping echanism

24 Designs in nature often exploit compliance, while manmade designs often avoid compliance. As long as the compliance in the system design is captured in the model, high-quality motion and robustness can be achieved with the aid of compliance a friend! Compliance is a foe when it is not understood and accounted for in the system design. Ignoring inherent compliance or avoiding using compliance to one s advantage makes compliance a foe!

25 Some Comments Accurate modeling of the dynamic behavior of a mechanical system will result in a dynamic system of higher order than you probably would want to use for the design model. For example, consider a shaft that connects a drive motor to a load. Possibilities include: Shaft has infinite stiffness (rigid) Shaft has a stiffness represented by a spring constant that leads to a resonance in the model Shaft is represented by a Partial Differential Equation that leads to an infinite number of resonances

26 Shaft has infinite stiffness (rigid). Shaft has a stiffness represented by a spring constant that leads to a resonance in the model. Shaft is represented by a PDE that leads to an infinite number of resonances.

27 In most situations, the frequencies of these resonances will be orders of magnitude above the operating bandwidth of the control system and there will be enough natural damping present in the system to prevent any trouble. In applications that require the system to have a bandwidth that approaches the lowest resonance frequency, difficulties can arise. A control system based on a design model that does not account for the resonance may not provide enough loop attenuation to prevent oscillation and possible instability at or near the frequency of the resonance.

28 If the precise nature of the resonances are known, they can be modeled and included in the design model. However, in many applications the frequencies of the poles (and neighboring zeros) of the resonances are not known with precision or may shift during the operation of the system. A small error in a resonance frequency, damping, or distance between the pole and zero might result in a compensator design that is even worse than a compensator that ignores the resonance phenomenon.

29 echanical resonance is a pervasive problem in servo systems usually caused by compliance of power-transmission components. This compliance often reduces stability margins, forcing gains down and reducing machine performance. Servo performance is enhanced when control-law gains are high; however, instability results when a high-gain control law is applied to a compliantlycoupled motor and load.

30 echanical resonance needs only two inertias coupled by compliant components to manifest itself. achine designers specify transmission components (e.g., couplings, gearboxes) to be rigid in an effort to minimize mechanical compliance. Some compliance is unavoidable. Also, cost and weight limitations force designers to choose lighter-weight components than would otherwise be desirable, leading to low transmission rigidity.

31 Curing Resonance: echanical Cures Stiffen the Transmission This usually improves resonance problems. The key to stiffening a transmission is to improve the loosest components in the transmission in an effort to raise the total spring constant K S. This has the effect of raising both the resonant and anti-resonant frequencies and moving them away from the frequencies where they cause harm. Some Suggestions: Use multiple belts, wide belts, or reinforce belts 4 Shorten shafts; JG ( d / 3)G K use large-diameter shafts

32 Use stiffer gear boxes Use larger lead screws and stiffer ball nuts Use idlers to support belts that run long distances Reinforce the frame of a machine Oversize coupling components; be cautious here as this will also add inertia, which may slow acceleration When stiffening a machine, start with the loosest components, as a single loose component can single-handedly reduce the overall spring constant significantly. Add Damping K In practice, it is difficult to add damping between the motor and load. aterials with large inherent damping do not normally make good transmission components. total K K K coupling gear box n

33 Steps used to stiffen a machine can actually make the machine perform more poorly because they also reduce damping. Sometimes the unexpected loss of damping can cause resonance problems. Reduce oad-to-otor Inertia Ratio Reducing the load-to-motor inertia ratio will improve resonance problems. The smaller the J /J ratio, the less compliance will affect the system. At low frequency, the system appears to have a noncompliant inertia, J T = J + J. At high frequency, the load inertia is disconnected; the system sees only J, the motor inertia.

34 agnitude (db) Phase (deg) Compliantly-Coupled otor + oad J J s 1 AR anti-resonance Rigidly-Coupled otor + oad Bode Diagram resonance R always R AR 1 J s Frequency (rad/sec)

35 In a sense, compliance gives the system an apparent inertia that varies with frequency. The smaller the J /J ratio, the less variation in apparent inertia as would be indicated by a smaller distance between the two parallel lines representing the low-frequency and high-frequency amplitude ratio vs. frequency plots. Reducing the load inertia is the best way to reduce the ratio J /J ; reduce the mass of the load or change its dimensions. The reflected inertia (load inertia felt by the motor) can also be reduced by increasing the gear ratio. 1 N Jeffective J J Rigid Coupling N

36 Unfortunately, increasing N can reduce the top speed of the application. Similar effects are realized by changing lead screw pitch or pulley diameter ratios. Any steps taken to reduce the load inertia will usually help the resonance problem; most machine designers work hard to minimize load inertia for non-servo reasons, e.g., cost, peak acceleration, weight, structural stress. Increasing J does help the resonance problem. Unfortunately, raising motor inertia increases the total inertia, which reduces total acceleration or requires more torque and power from the drive to maintain the acceleration. Increasing motor inertia therefore increases the cost of both motor and drive. Despite the increase in size and cost by increasing J, it is commonly used because it so effectively improves resonance problems.

37 Common isconception: J /J ratio is optimized when the ratio is 1 or inertias are equal or matched. Based on a fixed J and J, the gear ratio N that maximizes the power transferred from motor to load is the ratio that forces the reflected load inertia J /N to be equal to the rotor inertia J. T J NT N J J T eq eq J J N J eq T T NT eq d dn For T 0 N J J 0

38 This fact has little bearing on how motors and gear ratios are selected in practice because the assumption that the motor inertia is fixed is usually invalid; each time the gear ratio increases, the required torque from the motor decreases, allowing the use of a smaller motor. The primary reason that J and J should be matched is to reduce resonance problems; actually, this is an oversimplification. arger J improves resonance but increases cost. The more responsive the control system, the smaller the J /J ratio. J /J ratios of 3 to 5 are common in typical servo applications. Highest bandwidth applications require that the load inertia be no larger than about 70% of the motor inertia. The J /J ratio also depends on the compliance of the machine; stiffer machines will forgive large load inertias.

39 odeling of Compliance T K S J J B B B Compliantly-Coupled otor and oad J = rotor inertia of a motor J = driven-load inertia K S = elasticity of coupling B = viscous damping of coupling B = viscous damping between ground and motor rotor B = viscous damping between ground and load inertia T = electromagnetic torque applied to motor rotor

40 Comments K S, the elasticity of the coupling; it is often neglected in low-power systems; modeling it in high-power systems is essential. B, the viscous damping of the coupling; it is usually small, as transmission materials provide little damping. B and B can be neglected in the following analysis, as they have a small effect on resonance. They are included here for completeness. Coulomb friction has been neglected. The fixed value of Coulomb friction has little impact on stability when the motor is moving. At rest, the impact of stiction on resonance is more complex. Sometimes stiction is thought of as increasing the load inertia when the motor is at rest. This accounts for the tendency of systems to change resonance behavior when the motion stops.

41 T K S J J B B B Equations of otion T B B ( ) K ( ) J S B B ( ) K ( ) J S

42 Equations of otion T B B ( ) K ( ) J S B B ( ) K ( ) J S aplace Transform of the Equations of otion J s B B s KS s Bs KS s T s J s B B s KS s Bs KS s J s B B s K B s K Ts s S S Bs KS Js B B s K S s 0

43 atab / Simulink Block Diagram (B = 0 and B = 0) T B ( ) K ( ) J S B ( ) K ( ) J S

44 Transfer Functions T s J s B B s K S D s T s Bs K D s S 4 3 D s J J s J J B J B J B s J J K B B B B B s B B K s S S

45 Transfer Functions (B = 0 and B = 0) 1 J s B s K s T J J s S JJ s Bs K S J J 1 B s K s T J J s S JJ s Bs K S J J As K S or as s0 1 s s Rigid-Body otion J J s

46 Transfer Functions in Standard Form (B = 0 and B = 0) s AR K 1 AR AR s T s Rs s 1 R R s K s 1 s T s Rs s 1 R R Natural frequency of load connected to ground through the compliance 1 K J J R R AR AR B K S K J J S B S J J K J J J J KS J B K J S

47 agnitude (db) Phase (deg) Bode Diagram Frequency (rad/sec) 316 rad/s 50.3 Hz AR 447 rad/s 71. Hz R AR R T s agnitude (db) Phase (deg) Sample Values: Compliantly-Coupled otor + oad J J s 1 Rigidly-Coupled otor + oad AR anti-resonance B Bode Diagram J J K S resonance R Frequency (rad/sec) 0.00 kg-m 0.00 kg-m 00 N-m/rad 0.01 N-m-s/rad always R AR 1 J s

48 T s 1 J s B s K s T J J s agnitude (db) Phase (deg) S JJ s Bs K S J J Collocated System J J s 1 Rigidly-Coupled otor + oad AR anti-resonance Bode Diagram resonance R always R AR 1 J s Frequency (rad/sec)

49 1 B s K s T J J s S JJ s Bs K S J J Non-Collocated System Bode Diagram 100 T s agnitude (db) Phase (deg) R resonance Frequency (rad/sec)

50 100 T s T s Bode Diagram Bode Plot Comparison agnitude (db) Phase (deg) º additional phase lag Frequency (rad/sec)

51 s s ARs 1 AR AR s1 50 Bode Diagram s agnitude (db) Phase (deg) in phase out of phase Frequency (rad/sec) Note: There is no anti-resonant frequency in this transfer function. Also, there is 90º more phase lag at high frequency.

52 B J K Effect of J / J Ratio on Resonance and Anti-Resonance J J J J 1 5 J 1 J 5 S T 0.00 kg-m 00 N-m/rad 0.01 N-m-s/rad agnitude (db) s Phase (deg) ω R lower limit = 316 rad/sec ω AR lower limit = 0 Bode Diagram Frequency (rad/sec)

53 Effect of Varying J J Bode Diagram -0 T agnitude (db) s Phase (deg) Frequency (rad/sec)

54 J J 100, 10, 1, 0. 1 Bode Diagram ω R lower limit = 316 rad/sec T s agnitude (db) Phase (deg) rad/sec Frequency (rad/sec)

55 J J T 100, 1000 agnitude (db) s Phase (deg) Bode Diagram ω R lower limit = 316 rad/sec Frequency (rad/sec)

56 Effect of K S on Resonance and Anti-Resonance Bode Diagram J J 1 K K K S S S N-m 0 rad N-m 00 rad N-m 000 rad agnitude (db) KS 0 KS 00 KS T s Phase (deg) Frequency (rad/sec)

57 imiting Behavior: R AR K J J K J S S J J 1 J s B s K s T J J s S JJ s Bs K S J J 1 J J s 1 J s as as 0 As J, 0 and AR R K J S

58 Observations For J > 0, the anti-resonance frequency always occurs before the resonance frequency. R AR K J J K J S S J J At a low J /J ratio, the resonance and antiresonance frequencies are close to each other at a high frequency.

59 As J /J increases, both the anti-resonance and resonance frequency decrease, with the antiresonance frequency decreasing at a faster rate. J J 1 1 K J J J J K K S R S S JJ J J S AR J K 1 KS J At J = J, ω AR = 0.707ω R. As J, 0 and AR R K J S

60 For a given J, to increase the resonance frequency, either increase the shaft stiffness or decrease the motor inertia. J J 1 1 K J J J J K K S R S S JJ J J As K S increases, both ω R and ω AR increase. A general guideline to avoid instability problems is to keep the desired closed-loop bandwidth well below the resonance frequency and the ratio J / J less than 5.

61 Heavy-oad Approximation: J > 5J J J J J J J J J J J 1 1 J s B s K s T J J s J S JJ s Bs K S J J 1 Js Bs KS Js Js Bs K S

62 Heavy-oad Approximation agnitude (db) approximate Bode Diagram exact J J Phase (deg) Frequency (rad/sec)

63 Belt-Driven oad and otor Rigid Belt Case: R R R J J T R

64 J B KR BR T KR R BR R J B KR BR KR R BR R T

65 Equations of otion J B KR BR T KR R BR R J B KR BR KR R BR R T aplace Transform of the Equations of otion (T = 0) Js B BR s KR s BR R s KR R s T s Js B BR s KR s BR R s KR R s Js B BR s KR BR R s KR R s Ts s 0 BR R s KR R Js B BR s KR

66 Transfer Functions T s J s B BR s KR D s T s BR R s KR R D s 4 3 D s JJ s B JR JR JB JB s K J R J R B B B B R B R s K BR BR s

67 Transfer Functions (B = 0 and B = 0) J s BR s KR T J J s B J R J R s K J R J R s s 4 3 BR R s KR R s T J J s B J R J R s K J R J R s 4 3

68 Transfer Functions (B = 0 and B = 0) 1 Js BR s KR s T JR JR s JJ s Bs K JR JR 1 BR R s KR R s T JR JR s JJ s Bs K JR JR

69 Transfer Functions in Standard Form (B = 0 and B = 0) s AR K1 1 AR AR s T s Rs s 1 R R s K s 1 s T s Rs s 1 R R K K JR JR 1 R R AR AR R R R B J R J R K K J R JR B J J KJ J J R KR J BR KJ J R

70 odeling of Backlash In everyday language, the word backlash sounds as undesirable as its meaning, i.e., a strong adverse reaction or a violent backward movement. In engineering, the situation is no different. Backlash, the excessive play between machine parts, as often occurs in gears and flexible couplings, is highly undesirable and usually exists with compliance. It gives rise to inaccuracies in the position and velocity of a machine, as well as to delays and oscillations. The model that is most beneficial for control design is the least complex model that still retains sufficient accuracy to capture the gross dynamic behavior of the system.

71 The diagram shows the physical system under investigation with the accompanying assumptions. In addition, we assume that collisions due to backlash are sufficiently plastic to avoid bouncing.

72 It is critical that the model capture the fact that the output from the backlash element is a torque on the load inertia, not a displacement of the load inertia. The model presented here also captures the situation where the assumed-massless compliant element has damping. The importance of this can be demonstrated as follows. Imagine that you are compressing with your hand a massless spring that possesses no internal damping. If you were to suddenly move your hand away, the spring would stay in contact as its response is instantaneous, since, being pure, it has no mass or damping. But if the spring has damping and you repeat the experiment, the spring s response would not be instantaneous and it would start to lose contact. The model shown in the block diagram, developed from the system equations of motion, captures these essential attributes and fosters insight.

73 J B T T S J B T T S TS B shaft K Define : c b shaft d TS TS T S (Bd K d) (Bb K b) Equations of otion Shaft Torque K max d d b b 0, ( ) if (TS 0) B K b d ( d b ) if b (T S 0) B K min 0, d ( d b) if b (TS 0) B

Acceleration Feedback

Acceleration Feedback Mechanical Engineer Modeling & Simulation Electro- Mechanics Electrical- Electronics Engineer Sensors Actuators Computer Systems Engineer Embedded Control Controls Engineer Mechatronic