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2 Outline 1.Introduction 2.Modeling 3.Simulation 4.Implementation 5.Demo 6.Conclusion

3 Outline 1.Introduction 2.Modeling 3.Simulation 4.Implementation 5.Demo 6.Conclusion

4 Introduction Rotating arm and inverted pendulum. Rotating arm is actuated by a DC motor. The angular disturbance will be sensed by the potentiometer. l1l1 length from the center of rotating arm to the pendulum. l2l2 length of the inverted pendulum. m1m1 mass of the rotating arm. m2m2 mass of the inverted pendulum. α The angular displacement of the rotating arm rotated. θ The angular displacement of the inverted pendulum. linear velocity of the mass center of the inverted pendulum.

5 Introduction The system is controlled by a PID control circuit. Two equilibrium points existed. Use a cut-off device to protect the system.

6 Outline 1.Introduction 2.Modeling Find the transfer function of input voltage and the angle of inversed pendulum. –Equation of motion. –Linearization –Laplace transform –Transfer function 3.Simulationment 4.Implementation 5.Demo 6.Conclusion

7 Modeling -Equation of motion Step 1 : Find the equation of motion by Lagrange equation

8 Modeling -Equation of motion

9 Step 2 : Linearization –To do the linearization, we have to find the equilibrium points first. –Find the position where the extreme value of the potential energy exist. Modeling -Linearization

10 In this case, we set the equilibrium point at θ=0° Expand the nonlinear terms in Taylor series. Modeling -Linearization

11 System modeling -Linearization If the angle of disturbance is 5°, the max. error between linear and nonlinear model is 0.046°, less then 1%.

12 Step 3 : Laplace transform of the motion equations System modeling -Laplace transform

13 System modeling -Transfer function Step 4 : Find the transfer function of a DC motor According to Kirchhoff’s voltage law (KVL) Where is the voltage of coil is the induced voltage of the motor is the torque generate by motor Equivalent circuit of a DC motor

14 System modeling -Transfer function Step 5 : Transfer function of the system

15 SymbolValueUnit 0.10m 0.32m Kg 0.046Kg 9.81m/s e-5Kgm e-4Kgm e-4Kgm 2 Modeling -Transfer function Set the values we need SymbolValueUnit 1Ω 0.03 Assume the values we need but we don’t know Ref. : Stephen J. Chapman “Electric Machinery Fundamentals” Chap. 9 McGraw. Hill

16 Modeling -Transfer function Transfer function.

17 Modeling -Transfer function Unit step command test

18 Modeling -Transfer function Command unit step and disturbance is zero to check transfer function.

19 Modeling –Routh-Hurwitz Stability Using Routh-Hurwitz stability to find the stable range of the gain of PID or PD controller.

20 Modeling -Reference S. Awtar, N. king, T. Allen, I. Bang, M, Hagan, D.Skidmore, K. Craig, “Inverted pendulum systems: rotary and arm-driven- a mechatronic system design case study.” Mechatronic 12 (2002) Y. Yavin, “Control of a Rotary Inverted Pendulum.” Applied Mathematics Letters 12 (1999)

21 Outline 1.Introduction 2.Modeling 3.Simulation –Open loop –PD controller –PI controller –PID controller 4.Implementation 5.Demo 6.Conclusion

22 Simulation Use SimMechanics to build a nonlinear system model

23 Simulation Use Simulink to build a nonlinear system model

24 Simulation Use Simulink to build a linear system model

25 。 Simulation –open loop (angular V)

26 Simulation -PD controller

27 Simulation -PD controller

28 Simulation -PD controller Response simulation. (PD controller) Absolute error between the simulation of SimMechanics and Simulink.

29 Simulation -PI controller

30 Simulation -PI controller

31 Simulation -PI controller Response simulation. (PI controller) Absolute error between the simulation of SimMechanics and Simulink.

32 Simulation -PID controller

33 Simulation -PID controller

34 Simulation -PID controller Response simulation. (PID controller) Absolute error between the simulation of SimMechanics and Simulink.

35 Outline System introduction System modeling Simulation Implementation –Inversed pendulum –Control circuit Demo Conclusion

36 Implementation System block diagram

37 The length and mass of pendulum: 32 cm and 28.41g The length and mass of rotating arm: 10 cm and 46 g Gear ratio: 5 Implementation - Inversed pendulum

38 Implementation -Control circuit Circuit block diagram

39 Implementation -Control circuit PID controller Power amplifier Cut-off circuit Power supply II On/Off Sensor Signal light Limit switch Motor Circuit board Power supply I

40 Implementation -Potentiometer Use a variable resistor as a potentiometer. Inverted pendulum Potentiometer

41 Implementation - Potentiometer How does it work?

42 Implementation -PID controller Use operational amplifier Modes switch Elements shiftable PID controller

43 Implementation -PID controller

44 Implementation -Cut-off circuit, signal light NPN transistor Relay 5V 2 Form C Contact 500 ohm resistances Resistance with signal light 7404 NOT 7408 AND

45 Implementation -Cut-off circuit, signal light

46 Implementation -Power amplifier NPN TIP41 NPN TIP107 Diode

47 Implementation Why we use two power supply? The DC motor turns on, the voltage of power supply drops. Input ： triangular ±200mV;2Hz Output ： DC power supply +15V port The DC motor use the power from +15V port normal

48 Outline 1.Introduction 2.Modeling 3.Simulation 4.Implementation 5.Demo 6.Conclusion

49 Demo -PD controller Steady state error exist

50 Steady state error is zero Demo -PID controller

51 Outline 1.Introduction 2.Modeling 3.Simulation 4.Accomplishment 5.Demo 6.Conclusion

52 Conclusion We use different ways to model the system by MATLAB. For a small disturbance, linearized model is reliable. The rotary inverted pendulum can be controlled by a PID controller. I controller can eliminate the steady state error.