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University of Portland School of Engineering 5000 N. Willamette Blvd. Portland, OR 97203-5798 Phone 503 943 7314 Fax 503 943 7316 Theory of Operations Project Nuthatch: The Inverted Pendulum Contributors: Jason Boyce Jennifer Miller Approvals Name Dr. Albright UNIVERSITY OF PORTLAND Date Date Name Dr. Lillevik SCHOOL OF ENGINEERING Date Date CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE II Revision History Rev. 0.1 0.9 0.91 1.0 Date 02/02/03 02/06/03 02/09/03 02/14/03 UNIVERSITY OF PORTLAND Author J. Boyce, J. Miller J. Boyce, J. Miller J. Boyce, J. Miller J. Boyce, J. Miller Reason for Changes Began Initial Draft Completed Initial Draft Advisor Feedback Industry Rep Feedback SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE III Table of Contents Summary....................................................................................................................... 1 Introduction .................................................................................................................. 2 Background .................................................................................................................. 3 Architecture .................................................................................................................. 4 General Description ................................................................................................................................4 System Components and Interfacing .....................................................................................................4 Power Supply ...................................................................................................................................4 Control Circuit ...................................................................................................................................4 Motor .................................................................................................................................................4 Position and Velocity Sensor ...........................................................................................................5 Track .................................................................................................................................................5 Cart ...................................................................................................................................................5 Pendulum .........................................................................................................................................5 Pendulum Sensor ............................................................................................................................5 Cart-Track Architecture ...........................................................................................................................6 Tachometer ......................................................................................................................................6 Motor .................................................................................................................................................6 Pulleys and Belt................................................................................................................................6 Potentiometers .................................................................................................................................6 Cart and Track..................................................................................................................................6 Pendulum .........................................................................................................................................6 Feedback Diagram and Transfer Functions ..........................................................................................8 The Plant, G(s) .................................................................................................................................8 The Motor, M(s) ................................................................................................................................8 The Compensator, K(s) ...................................................................................................................8 UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE IV Input and Output...............................................................................................................................8 k’ ........................................................................................................................................................8 bs+c ..................................................................................................................................................8 Design Overview.......................................................................................................... 9 Circuit Schematics...................................................................................................................................9 Circuit Design ....................................................................................................................................... 10 Unity Gain Buffers ......................................................................................................................... 10 Summing Junctions....................................................................................................................... 10 Integrator........................................................................................................................................ 11 Amplifiers ....................................................................................................................................... 11 Power Transistor Block ................................................................................................................. 11 Motor and Tachometer ................................................................................................................. 11 Potentiometers .............................................................................................................................. 11 Determination of Values....................................................................................................................... 11 Stability.................................................................................................................................................. 14 Conclusions ...............................................................................................................15 Research Appendix ...................................................................................................16 UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE V List of Figures Figure 1. System Block Diagram… …………………………………………. ………………….. ….. 4 Figure 2. Cart-Track Schematic .................................................................................................................5 Figure 3. Classic Feedback Diagram ........................................................................................................7 Figure 4. Transfer Functions ......................................................................................................................7 Figure 5. Control Circuit Schematic .................................................................................................... 9 Figure 6. Color-Coded Circuit Schematic ............................................................................................. 10 Figure 7. Matlab Plot: Impulse Response at the Output of M(s)........................................................ 14 Figure 8. Matlab Plot: Impulse Response at ...................................................................................... 14 UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH UNIVERSITY OF PORTLAND REV. 0.9 SCHOOL OF ENGINEERING PAGE VI CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH Chapter REV. 0.9 PAGE 1 Summary 1 This document contains an introduction, background, architecture description, design overview, and conclusion. The introduction and background define the scope of this document and provide a review of accomplishments prior to this document. Project Nuthatch is a solution to the classic inverted pendulum problem. A cart with an inverted pendulum on top is situated on a track. The cart will move back and forth in an attempt to balance the pendulum. The design has been completed and a first cut prototype is currently being implemented. The architecture section begins with a high-level system block diagram of Project Nuthatch and proceeds to explain each system block with diagrams and descriptions. The high level system blocks include the power supply, control circuit, motor, cart position and velocity sensors, pendulum sensor, pendulum, cart, and track. The diagrams and descriptions in the architecture section explain each block and the interactions between blocks. One of these diagrams is a slightly lower level cart-track schematic, and the classic feedback diagram for Project Nuthatch is also provided. The design overview is where the design of our control circuit can be found, along with an explanation of the design process and each circuit block. The control circuit implements the classic feedback diagram for our system that is shown in the architecture section. Matlab plots that demonstrate system stability are also included. After our conclusion, the research appendix section documents the research we have used to help us in our design. UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 Chapter PAGE 2 Introduction 2 The purpose of the Theory of Operations document is not only to aid the reader in understanding the technical design, but also to assist in future projects or maintenance. If the project develops any problems in the future, this document will be helpful in understanding the layout of our system so that the problems are fixed quickly. Future pendulum designs may be more complex, such as using a hinged pendulum, but our basic design scheme may be used and modified. That information will be contained in this document. The document will also help advisors, industry representatives, students, and other interested engineers to understand how Project Nuthatch works. The Theory of Operations will provide high level models such as major mechanical blocks and their interfacing and feedback diagrams for the control system. The bulk of the document will show and explain the lower level circuit implementation. The document begins with a background section to set the scene and review product features. Then we will move to the architecture where we will see the foundation of the design including architectural block diagrams and explanations of major components. The key pieces and how they fit together are made clear in the architecture section. In the design overview section, which is the bulk of the document, we will include system block diagrams at a lower level including circuit schematics. Finally, the conclusion will recap UNIVERSITY OF PORTLAND key points and make recommendations SCHOOL OF ENGINEERING for future work. CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH Chapter REV. 0.9 PAGE 3 Background 3 Project Nuthatch is a control system demonstration. The intended use is purely instructive and educational. The only users of this system will be students and other members of the University community. It is also an important academic exercise for the designers because it will reinforce prior knowledge of control systems and greatly expand upon that knowledge. Nuthatch is a classic control system known as the “inverted pendulum.” The inverted pendulum is composed of three main visible elements: a cart, track, and inverted pendulum. The car is situated on a short, straight track. It may only move along the track. A hinged inverted pendulum is located on the top of the car. Only one degree of freedom exists for both the pendulum and the car. The car may only move to the left or the right, and this also holds true for the pendulum. The ultimate goal is to balance the pendulum by moving the car either left or right on the track, as necessary to maintain equilibrium. This can be equated to tasks that the average person may be familiar with, like attempting to balance an object, such as a baseball bat, on the palm of the hand. The hand must be moved in the direction that the bat begins to fall in order to keep the bat balanced. The project is somewhat simplified from this example by only allowing the car and pendulum to move in one fixed plane of motion, as described above. Balance will be kept using a control system with negative feedback circuitry rather than by physical human means. UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 Chapter PAGE 4 Architecture 4 General Description Pendulum Stoppers Pendulum Power Supply Pendulum Sensor Control Circuit Position and Velocity Sensors Cart Motor Track Cart and Pendulum Movement Figure 1. System Block Diagram System Components and Interfacing Power Supply Will supply +15 to –15 volts DC to power the control circuit, pendulum sensor, position sensor, and motor. Control Circuit An analog feedback circuit to control the cart using the motor. Motor A DC motor. UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 5 Position and Velocity Sensor Components which sense the position of the cart on the track, as well as the velocity of the cart. Track A straight track for the cart to move along. Cart The cart that moves along the track. Pendulum The inverted pendulum is attached to the top of the cart by a hinge. Pendulum Sensor A mechanism for sensing the position of the pendulum. Cart-Track Schematic Pendulum (t) V(t) X(t) Hinge Motor Motor Shaft Pulley Cart Track Belt 15 15 15 15 Potentiometer Voltage proportional to (t) Ground Multi-turn potentiometer Voltage proportional to x(t) Ground From William McC. Siebert, Circuits, Signals, and Systems, MIT Press, Cambridge, Mass., 1986 UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 6 Figure 2. Cart-Track Schematic Cart-Track Architecture Tachometer The tachometer attaches to the motor shaft to convert the rotational velocity of the motor shaft into a voltage. This is used for the velocity feedback. Motor A 24-volt DC motor with one end of the shaft attached to the voltage tachometer and the other end of the shaft turning a pulley and a multi-turn potentiometer. Pulleys and Belt One pulley is connected to the motor shaft and the other is free-ended at the other end of the track. A belt wraps around the two pulleys and is attached to the cart. As the motor shaft turns, the belt pulls the cart back and forth on the track. Potentiometers A multi-turn potentiometer is attached to the shaft of the motor to sense the position of the cart. This potentiometer adds variable resistance and is connected between +15 and –15 volts DC. Another potentiometer is located on the hinge of the pendulum and is used to measure . Cart and Track The metal cart rests on a single, straight metal rail allowing motion only along the rail. Friction is kept to a minimum using ball bearings, which are the only interface between the cart and the track. Pendulum The lightweight metal inverted pendulum is attached to the center of the top of the cart by a hinge. It is also attached to a potentiometer to sense its position. UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 7 Classic Feedback Diagram bs + c Qc + + -1 - Q k´ M(s) - G(s) K(s) Figure 3. Classic Feedback Diagram X ( s) kM V ( s ) s( M s 1) s 1 K ( s) K Ks M ( s) G( s ) s2 / g ( L s 1)( L s 1) L l g Figure 4. Transfer Functions UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 8 Feedback Diagram and Transfer Functions The Plant, G(s) This transfer function describes the cart-pendulum system and is derived using Newtonian Physics, LaPlace transforms, and a lot of research. The Motor, M(s) This transfer function describes the motor and is derived using the circuit model for a DC motor and control theory. The Compensator, K(s) This transfer function describes the compensation used to obtain stability by moving poles and zeros into the left half of the plane. Input and Output The input, C, is the reference position at 90 relative to the track, or fully erect. The output, , is the angular position of the pendulum relative to C. k’ This is simply an adjustable gain block. bs+c This block implements velocity feedback, where b and c are gain blocks and s is a differentiation block representing the tachometer. UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 Chapter PAGE 9 Design Overview 4 Circuit Schematics +15 V R2 10k C2 9.3 uF +15V 10k 10k + + + _ 10k + 33 _ _ 100 10k _ + 10k Rotary Pot 1 _ 10k -15V -15V +15V 10M 10k Multi-turn Potentiometer + _ 1k + 24 V DC MOTOR _ _ + -15 V Figure 5. Control Circuit Schematic UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 +15 V R2 PAGE 10 C2 10k +15V 9.3 uF 10K Rotary Pot 1 100 10k 10k + + + _ 10k + 33 _ _ _ + 10k _ 10k -15V -15V +15V 10M 10K Multi-turn potentiometer + _ 1k + 24 V DC MOTOR _ _ + -15 V Figure 6. Color-Coded Circuit Schematic Circuit Design In the following descriptions, we will be referring to the colored circuit schematic in Figure 6. Unity Gain Buffers The orange-colored blocks are unity gain buffers. Unity gain buffers are used to isolate resistance values. For example, we do not want the resistance of the potentiometers in series with the resistance of our amplifiers because the behavior of the amplifier would change. Summing Junctions The red-colored blocks represent our summing junctions. The summing junction located at the bottom of our schematic is used to sum the two velocity feedback gains, bs and c. The upper summing junction is used to add the negative feedback of the compensator, K(s), to the velocity feedback portion of the circuit. UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 11 Integrator The blue-colored block is our compensator, which is just a simple integrator. represents our K(s) block in the feedback diagram. This Amplifiers The purple-colored blocks are basic amplifier blocks, which represent k, b, and c. Power Transistor Block The green-colored block is our power transistor block. This is used to generate enough current to power the DC motor. Motor and Tachometer The 24V DC motor is connected to the tachometer by the shaft. The tachometer serves as a voltage generator that is controlled by the velocity of the motor shaft. Potentiometers The multi-turn potentiometer connected to the motor shaft is used to measure the position of the cart with a zero voltage equivalent to the cart being positioned in the center of the track. The other rotary potentiometer is used to measure the angle of the pendulum with zero voltage referenced to a fully erect pendulum. Determination of Values The inverted pendulum with no control circuit is an inherently unstable fourth-order system. Our goal was to design an analog control system to stabilize this system. Using Newtonian physics, we derived a mathematical model and transfer function for the inverted pendulum. This portion of the system is known as the plant, G(s), which is clearly labeled in our feedback diagram. The plant is what we want to control. This is intrinsic to the system, therefore is not part of the circuit design, but is represented in our feedback diagram for the purpose of mathematical understanding. We have also included the well-known motor transfer function, which is represented by M(s) in our feedback diagram. In implementation, this is simply the DC motor that will be controlled by our UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH circuit. REV. 0.9 PAGE 12 Through research, we found the system could be made stable by adding compensation (Lundberg). This compensator is represented in the feedback diagram by K(s) and is implemented in circuitry by an integrator. Through further research, we found that adding velocity feedback contributes additional stability and reduces the undesired drift of the cart (Siebert). The velocity feedback is implemented by the bs+c block on our feedback diagram. The bs portion is implemented in circuitry by a tachometer and an operational amplifier and the c portion is simply another operational amplifier. The bs portion of the velocity feedback is used to stabilize the position of the pendulum more quickly, while the c portion is used to prevent the cart from drifting and eventually moving to the end of the track. The success of this project is contingent on designing the correct compensator for the particular traits of the pendulum and the motor. This was accomplished through research and the help of Matlab. Research suggested that the time constant of our compensator transfer function , K(s), should be between the time constant of our motor transfer function , M(s), and the plant transfer function, G(s) (Siebert). Please refer to Figure 4 for the transfer functions. The plant time constant is simply a function of the pendulum length. The time constant of the compensator is the product of the capacitor value C2 and the resistor value R2. The motor time constant was determined using the data sheet for our DC motor. After deciding on the pendulum length and finding the motor constants, we selected a value for the compensator time constant that was directly between the motor and plant time constants. From our motor data sheet, we determined the following constants: k M 18.559 M 12.4ms N *m A The plant constants are: UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 13 l 0.43m L l 0.2094 s g Therefore, we needed to pick a compensator time constant in the range .01245s K .2094s We then picked a compensator time constant of K 0.093s R2 C 2 R2 10k C 2 9.3F With these determined values, we could construct the following transfer functions for Matlab simulations. 18.559 s (0.0124s 1) 0.093s 1 K ( s) 0.093s 2 s 9.81 G ( s) (0.2094s 1)(0.2094s 1) M (s) Once all the values of the transfer function were obtained, it was time to prove that the compensator stabilizes the inverted pendulum. A Matlab script was developed to show the impulse response of our system at both the output (s) and at the output M(s). The output of M(s) can be realized as the position of the cart as a function of frequency. The impulse response of the system at (s) represents the stabilization of the pendulum and the impulse response at M(s) shows the stability of the position of the cart. Please refer to Figures 7 and 8 to see the Matlab simulations. As seen clearly from the Matlab plots, both plots depict a stable system. It should be noted that these two plots show the impulse response of our system with no velocity feedback included. It is not seen in Figure 7, but if we zoom out of this plot, it is apparent that the position of the cart will drift. UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 14 This drift is very subtle and may not be a problem. If it is a problem, we will use velocity feedback to compensate. Stability Figure 7. Matlab Plot: Impulse Response at the Output of M(s) UNIVERSITY OF PORTLAND SCHOOL OF ENGINEERING CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH REV. 0.9 PAGE 15 Figure 8. Matlab Plot: Impulse Response at Conclusions 5 This document outlined the implementation of Project Nuthatch. The problem was revisited, and our solution to the problem was to design a control circuit that will stabilize the cart-pendulum system. The design of the control circuit was presented and the derivation of the circuit was thoroughly explained. As shown in the Matlab plots of the impulse response, the system is stable. The layout of the project has been determined as shown in our system block diagrams, and the circuit design has been completed. The construction of a first cut prototype is in process. The first time we construct the circuit we will leave out the velocity feedback portion. If the cart drift is a problem then velocity feedback will be implemented during the debugging process. In the future it would be interesting to try to solve this problem using a mixed digital and analog solution or by using computer control. Another fascinating approach would be to use a pendulum that is double-hinged, further complicating the problem. If our project and research were used as the starting point, this may be a realistic project for a larger team UNIVERSITY OF PORTLAND of SCHOOL OF ENGINEERING students. CONTACT: JASON BOYCE THEORY OF OPERATIONS PROJECT NUTHATCH Chapter REV. 0.9 PAGE 16 Research Appendix 6 References Kent Lundberg. http://web.mit.edu/klund/www/pendulum.pdf William McC. Siebert. Massachusetts, 1986. UNIVERSITY OF PORTLAND Circuits, Signals, and Systems. SCHOOL OF ENGINEERING MIT Press, Cambridge, CONTACT: JASON BOYCE