# Mechatronics Engineering. Li Wen

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1 Mechatronics Engineering Li Wen

2 Bio-inspired robot-dc motor drive Unstable system Mirko Kovac,EPFL

3 Modeling and simulation of the control system

4 Problems 1. Why we establish mathematical model of the control system? 2. Modeling methods and procedures? 3. How to create a mathematical model of the DC servo motor? 4. The simulation tool of the control system?

5 Contents 1 Significance of control system simulation analysis 2 Modeling methods and procedures 3 DC motor modeling examples 5 4 Analysis and correction for linear motion unit closed-loop simulation Introduction to MATLAB /SIMULINK

6 Contents 1 Significance of control system simulation analysis 2 Modeling methods and procedures 3 DC motor modeling examples 5 4 Analysis and correction for linear motion unit closed-loop simulation Introduction to MATLAB /SIMULINK

7 Establish the significance of the mathematical model By the specific physical problems, from a qualitative understanding of engineering problems to rise to the precise quantitative understanding of the key. Research and to analyze a mechanical control system, not only to qualitatively understand the working principle and characteristics of the system, but also quantitatively describe the system dynamic performance.

8 Basic concepts of mathematical models Mathematical description of the dynamic characteristics of the system: Because during the transition process, the system variable you want to change over time, thus describing the system appears not only in mathematical model of dynamic characteristics of the variable itself, but also contain all order derivatives of these variables, so the system of dynamic equations are differential equations, it is the most basic form representing mathematical model of the system.

9 Bio-inspired soft robot _ slow response

10 Control system modeling approach First, analysis, starting from the physical or chemical laws, establishing mathematical model and experimental verification Two is an experimental method, by adding a certain forms of input signals to the system or component, evaluating output response for system or component, building mathematical models. This lesson uses analysis

11 Principle of establishing mathematical models Inexact: Theoretically none can be absolutely accurate mathematical expressions to describe a system because, in theory, any system is nonlinear, time and distribution parameters change, the random factors are present, the more complex the system, the situation is more complicated. Simplification: Ignore secondary factors, seize the main problem for modeling, quantitative analysis.

12 Mathematical model is divided into: Time Domain Model Complex domain model Frequency domain model Time domain model: Mathematical Model Types Advantages: it is describes the control system in the time domain, and has the advantage of an intuitive, accurate, all of the responses and can provide the system time information. Disadvantages: complex; hard to find system of structure parameters on performance of control system of general rules, cannot find improvement program is not easy on the system analysis and design.

13 Mathematical Model Types Complex domain models: It includes the transfer function and structure of the system. It demonstrates its characteristic of the system and of the input signal;it not only characterize the dynamic performance of the system, but can also affect the structure or the study of changes in system parameters on system performance Frequency domain model: Describes the frequency characteristics of the system, with a clear physical meaning, experimental methods are available to determine.

14 Mathematical Model Types Relationship between the three commonly used mathematical models Linear Systems Transfer function Rumsfeld Transform Differential Equations Fourier Transform Frequency Characteristics

15 Modeling steps 1. A linear system of equations: 1 determine the input and output of the system 2 The system is divided into several areas, from the input start signal is transmitted in the order, according to the laws of physics followed each variable (Newton's law, Kirchhoff's current and voltage law), etc., lists various aspects of linearization original equation; 2. For the establishment of differential equations, Laplace transform one by one, eliminating the intermediate variables, get the system transfer function model

16 Biomimetic robot-dc drive Self-stable system Mirko Kovac,EPFL

17 Biomimetic robot-dc drive Self-stable system with steering Mirko Kovac,EPFL

18 Contents 1 Significance of control system simulation analysis 2 Modeling methods and procedures 3 DC motor modeling examples 5 4 Analysis and correction for linear motion unit closed-loop simulation Introduction to MATLAB /SIMULINK

19 DC modeling analysis Solution: armature controlled DC motor is essentially the work of the input electrical energy into mechanical energy, which is the Input of the armature voltage U a (t) generated armature current I a (t) in the armature circuit, and then by the current I a (t) and the excitation flux generated by the interaction of electromagnetic torque M m (t), to drag the load movement. Therefore, the equation of motion of the DC motor by the following three components. Armature circuit voltage balance equation Electromagnetic torque equation Turn the motor shaft from the balance equation 19

20 DC modeling analysis (1) According to Kirchhoff's voltage law, the armature winding voltage balance equation u i R L di E dt a a a a a a (1) Where, L a and R a were inductance (Henry) and the resistance of the armature windings (Ohm)

21 DC modeling analysis (2) When the rotation of the DC motor armature, the armature windings produce anti potential, it is generally proportional to the motor speed, i.e., d m Ea Ke dt (2) Where, E a is the back EMF (V), K e is a scaling factor (V.rad / s)

22 DC modeling analysis (3) the interaction between the armature current and the magnetic field to produce an electromagnetic torque. General electromagnetic torque is proportional to the armature current, namely: M m K i m a (3) Where M m is the electromagnetic torque (Nm), I a is the armature current (A), K m for the moment coefficient (Nm / A)

23 DC modeling analysis (4) for driving the electromagnetic torque to overcome the friction and load torque, assuming only consider the viscous friction is proportional to the speed, the DC motor torque balance equation 2 d m d m M m J m B () 2 m M c t dt dt The formula: J m The total moment of inertia of the motor shaft (Including the moment of inertia of the rotor and the load) 牛米.. 秒 2 m B m M () c t The angular displacement of the motor shaft (rad); As a viscous friction coefficient of the motor shaft The role of the applied load on the motor shaft torque 牛. 米 / 弧度 / 秒

24 DC modeling analysis m To find the angular velocity and load control model of the motor armature voltage U, namely the transfer function. We assume zero initial conditions in these kinds of Laplace Transform, respectively U s L I s s R I s E s a a a a a a E s K s a e m m m a M s K I s M s ( J S B ) s M s m m m m c

25 DC modeling analysis Erasing the armature current ia, and then take the armature voltage Ua is input, the angular velocity of the motor output shaft, i.e., m m s s. U s Whereby the DC motor can be controlled in the model, i.e., the transfer function is: a s 2 m Km (s) U s L J s ( L B J R ) s R B K K a a m a m m a a m e m

26 DC modeling analysis Created in MATLAB using the Simulink simulation model DC servo motor Structure

27 Li Wen et al, Beihang University

28 DC modeling analysis

29 Open Simulink

30 Modeling Overview of Linear Motion Units Linear motion units has flexibility and mechanical friction, etc., so it is virtually impossible to establish a precise mathematical model. We usually use approximate model, assuming driver and transmission is ideal rigid, and there is no elastic deformation

31 New Simulink model

32 The system block diagram of the various modules and drag it to the model file Pull in

33 Modeling of linear motion unit control system (2) Linear motion unit components Coupling Ball screw Reducer DC motor

34 The system block diagram of the various modules and drag it to the model file

35 The system block diagram of all the modules file and drag it to the model and adjust the layout and orientation

36 Connection

37 Variable and label

38 DC modeling analysis Substituting parameters L a,r a,j m,b m,k m,k e L a = 0.001Hery;R a = 1.2oum;J m = 1e-5 kg.m 2 ;B m = 5e-4;K m = 0.08N.m/A;K e = 0.08V.S/rad; Input signal: Amplitude of Input voltage U a is 1V, the frequency of square wave is 1Hz Amplitude of the interference torque Mf 0.01Nm, frequency of sinusoidal signal 1Hz Simulink block diagram to obtain arguments:

39 matlab m files for variable assignment

40 Generate input and load (or disturbance torque) signal generator Click on Run Amplitude of Input voltage is Ua 1V, the frequency of square wave is 1Hz Amplitude of the interference torque Mf 0.01Nm, frequency sinusoidal signal of 1Hz For example: motor Slip ring friction Produce

41 DC modeling analysis Judging from the simulation curve, the response curve is a cycle curve, is in response to a step input and Input of the linear superposition of the load cycle, from the curve to see the system is still stable, which can be from the poles and zeros of the transfer function are located to the left half plane verify get.

42 DC modeling analysis Necessary and sufficient conditions for stability of the system is necessary All the roots of the characteristic equation must be negative real part, that is all the roots in the complex plane of the left half-plane Root system characteristics 0.08 G(s) = e-08 s^ e-05 s Continuous-time transfer function i i 2 s 2 m Cm U s L J s ( L B J R ) s R B C C a a m a m m a a m e m

43 DC modeling analysis J m Impact on system performance Overshoot Jm=10-5 Jm=10-4 Jm=10-3 In three Jm, the system is stable, but smaller overshoot Jm more powerful; Jm greater the longer the rise time of the system.

44 DC modeling analysis B m impact on performance(b m bigger (output / input) becomes smaller, shorter adjustment time) Bm =1X10-4 Bm =5X10-4 Bm =2X10-3 The greater the damping coefficient, the smaller the value of the unit step response (speed / voltage value becomes smaller), the rise time becomes longer, but the time to reach steady state becomes shorter.

45 DC modeling analysis Load impact on performance Mc = 1Nm Mc = 2Nm Mc = 3Nm Load increase reduce the system response (moving speed), larger changes in the steady-state error of the system (open-loop steady-state error is large), the adjustment time becomes longer, the rise time becomes long.

46 Contents 1 Significance of control system simulation analysis 2 Modeling methods and procedures 3 DC motor modeling examples 5 4 Analysis and correction for linear motion unit closed-loop simulation Introduction to MATLAB /SIMULINK

47 Modeling Overview of linear motion units Model building Specify the slider velocity (unit: mm/s) as the input, and the slider actual speed (mm/s) as the output, establish a mathematic model for the linear motion unit speed control system.

48 Modeling of linear motion unit control system (1) Linear motion unit system components and parameters Rated voltage 24V Back-EMF constant (Ke) v*s/rad Reduction ratio (i) 29:1 Amplifier (Ka) 2.4 Motor resistor (Ra) 21.6 欧 Torque constant (Km) N*m/A Motor inductance (La) Screw lead (p) 1.97mH 2mm Rotor moment of inertia (Jx) Equivalent damping (Bm) kg. m Screw diameter (d) 11.5mm Speed gain (Ka) v*s/rad Screw Length (L) Workbench mass (m) 540mm 0.315kg Equivalent moment of inertia of the motor shaft d ( ) 2 p d l m ( ) J e -7 kg m 2 m J x i 2

49 Modeling of linear motion unit control system 1. The relationship between the motor and the screw speed The actual relationship between the rotational speed of the motor shaft and the screw speed: ( i is reduction ratio and the value is 29) m o t t m Motor shaft speed t i t Actural speed of screw o

50 Modeling of linear motion unit control system 2. Modeling of DC servo motor

51 Modeling of linear motion unit control system 2. Modeling of DC servo motor Potential balance equation of armature windings: Relations between the counter-electromotive force and speed The relationship between the armature current and the armature torque is: Torque balance equation is:

52 Modeling of linear motion unit control system 3. Angular velocity feedback To constitute the load shaft speed control system, there must be speed feedback of load shaft, the error voltage can be obtained by velocity error: n t t n u t k t t a a n m is the input shaft speed of motor; k a u a k a is the speed feedback gain t t m

53 Modeling of linear motion unit control system Laplace transform of the above formula : m t i t o s i s M K i ( N m) m m a a u k () t L di R i dt a e m a a a d m(t) M m(t) J m Bm m (t) Mc(t) dt m o m m a M s K I s U s L I s s R I s k s a a a a a e (s) M s J s s B s M m m m m m c u t k t t a a n m u s k s s a a n m This equation describes the relationship between the input control voltage U and the rotational angular velocity of the drive shaft.

54 Modeling of linear motion unit control system Specify the slider velocity (unit: mm/s) as the input, and the slider actual speed (mm/s) as the output, establish a mathematic model for the linear motion unit speed control system. The above notation is expressed as the angular velocity, as the screw lead P is 2mm, we can Build relationships between line speed and angular velocity. o s V o P 2 Vo 2 o s P o t is input speed Vo is slider speed

55 Modeling of linear motion unit control system Build the system model in Simulink:

56 After determining the mathematical model of the system, you can use several different methods to analyze the dynamic performance and steady-state performance of the control system. Method: Simulation analysis of control systems Time domain analysis Frequency domain analysis.

57 Simulation analysis of control systems Dynamic performance and steady-state performance

58 Simulation analysis of control systems Input step signal (amplitude is 1) to analyze the time-domain response From the results, we can get that rise time, peak time and settling time are relatively small, although there is a certain system overshoot, but eventually stabilized, but there is an error between the input and output. Therefore, system stability and accuracy are not very well.

59 Simulation analysis of control systems Analyze the system frequency response, draw bode plot

60 Simulation analysis of control systems Analyze the system frequency response, draw Nyquist plot Conclusion: The system gain margin is infinite, phase margin is 71.8, system stability is very good.

61 Impacts on system when change parameters Impact on the performance of the screw lead 2mm 6mm 8mm In the open-loop control, when lead increases, changes in steady-state error is large and the system response increases, rise time get lower.

62 Impacts on system when change parameters Impact on the performance of the screw lead 2mm 6mm 8mm In the open-loop control, changes in lead will not change the dynamic characteristic.

63 Impacts on system when change parameters Impact on the performance of the reduction ratio In the open-loop control, when the reduction ratio increases, system response time decreases

64 Impacts on system when change parameters Impact on the performance of the reduction ratio In the open-loop control, changes in reduction ratio will not change the dynamic characteristic.

65 Impacts on system when change parameters Changing the system structure parameters, and analyze its impact on system performance

66 Impacts on system when change parameters Bode plot

67 Open Simulink

68 Contents 1 Significance of control system simulation analysis 2 Modeling methods and procedures 3 DC motor modeling examples 5 4 Analysis and correction for linear motion unit closed-loop simulation Introduction to MATLAB /SIMULINK

69 biorobotics_fast response X0.1 Active control Rob Wood lab, Harvard University

70 New Simulink model

71 According to the system block diagram, and drag various modules to the model file Drag

72 According to the system block diagram, and drag various modules to the model file

73

74 Connection

75 Brought variables and mark

76 DC modeling analysis Brought parameters: L a,r a,j m,b m,k m,k e L a = 0.001H; R a = 1.2Ω; J m = 1e-5 kg.m 2 ;B m = 5e-4;K m = 0.08N.m/A; K e = 0.08V.S/rad; Input signal is: Square wave: Input voltage U a is 1V,frequency is 1Hz; Sin signal: Disturbance torque M f is 0.01N.m, frequency is 1Hz; Simulink block diagram:

77 Variable assignment in matlab m file

78 Generate input and load (or disturbance torque) signal using generator Run Input voltage U a : 1V, frequency 1Hz, of square wave. Disturbance torque M f is 0.01N.m, frequency 1Hz, of sin signal.

79 DC modeling analysis Judging from the simulation curve, the response curve is a cycle curve, which is a linear superposition response to a step input and periodic load input. From the curve to see the system is still stable, which can be verified from the transfer function poles and zeros are in the left half plane.

80

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