EDUMECH Mechatronic Instructional Systems. Ball on Beam System

Transcription

1 EDUMECH Mechatronic Instructional Systems Ball on Beam System Product of Shandor Motion Systems Written by Robert Hirsch Ph.D All Rights Reserved. 999 Shandor Motion Systems, Ball on Beam Instructional System

2 Table of Contents. Objective of Case Study Physical System Physical principles involved...3. System inputs and outputs Manufacturer s specifications of system components Physical Model Picture Description Problem statement Principles Assumptions Diagram of physical model: Mathematical Modeling Derivation of governing differential equations Analysis Analytical solutions... Numerical solutions Experimental Verification of Model Design experiments to determine parameter values...3 Conduct experiments Control-System Design Control-system planning Analog-control design Implementation Analog-control implementation Testing Experimental testing of closed-loop system and comparisons with predicted response Time-domain testing Evaluation of system performance vs. performance specifications Bibliography Shandor Motion Systems, Ball on Beam Instructional System

3 . Objective of Case Study Throughout the recent history of control theory there have been a number of classic problems that have been studied used to help explain overall concepts and small nuances. Classic systems include inverted pendulum, double inverted pendulum, and Magnetic Levitation. Another of these Classic systems is the Ball on Beam system (BOB) in which a ball is placed on a track and the track is dynamically angled so as to keep the ball in the center of the beam. A multitude of control schemes, both analog and digital has been tried on such a system. This case study described the design, analysis, simulation, and implementation of a ball on beam system.. Physical System The Ball on Beam system has five main parts to it. The driving force is supplied by a Canon CKT6-T5 DC motor and tachometer. The motor drives a gear train, which is attached to a piece of aluminum channel stock, which is the beam on which the ball rolls on. The power amplification to drive the motor is supplied by the power op-amp used in the EDUMECH DC motor system, which you should already be familiar with before starting this case study. The position of the rolling ball is resolved by an ultrasonic sensor, which sends out a sound wave and measures the time it takes for the echo to return. In doing this, the distance of the ball from the sensor head can be found. The electronics for accomplishing this is supplied on an independent board. The final component to the ball on beam system is the control circuitry. In fact two control loops are required. One for controlling the angle of the beam, the other for controlling the position of the ball. Figure. shows the layout of the Ball on Beam system. Figure. The Ball on Beam system. Physical principles involved Since the principles of a DC motor and power amplification for driving it are covered in another case study [] they will not be repeated here. Instead the focus will be on the dynamics of the ball on beam and on the basics of ultrasonic measurement... The Ball on Beam Mechanics The mechanics of the Ball on Beam system are rather straightforward. Figure. shows a schematic of the ball on beam system. Two quantities are measured: the position of the ball, x, and the angle of the beam, θ. 999 Shandor Motion Systems, Ball on Beam Instructional System 3

4 3. Picture Figure 3.: The Physical System 3. Description A controller to keep a ball that rolls on the beam at the center, modifies the angle of a beam. The beam is driven through a gear train by a Canon CKT6-T5 motor. The angle of the beam is measured though the use of a potentiometer. The ball position is measured with an ultrasonic sensor. Both angle and ball position signals are available for feedback control. The motor is driven with a high power op-amp circuit. 3.3 Problem statement The objective of this exercise is to keep the ball at the center position by modifying the beam angle. The beam angle should not exceed +/- 40 degrees. The ball position should settle to within % of its final state within 5 seconds. 3.4 Principles Derivation of dynamics of the entire Ball and Beam system comes from Newtons Laws. Newton s Second Law τ = J & θ and f = m& x The full derivation will be discussed later in this study. The results of these governing laws will be a set of differential equations that describe the dynamics of the motor-beam system and the Ball on Beam system. 3.5 Assumptions ) The inertia of the motor and tach rotors is insignificant relative to the inertia of the beam. ) The motor can be modeled simply as an amp to torque converter. 3) Other nonlinear characteristics of the motor and tachometer will be ignored 4) The beam will be considered a single lumped inertia. The gear train will be modeled as a torque proportionality constant. 5) There are only two degrees of motion, the ball moving linearly on the beam and the beam rotating about a pivot point. 6) All friction in the system is negligible. 999 Shandor Motion Systems, Ball on Beam Instructional System 7

5 7) Noise is unmodeled. 8) The motor-beam dynamics are much faster than the ball on beam dynamics. 9) The change in potential energy of the ball during while system is operational is negligible. 3.6 Diagram of physical model: mg sinθ θ mg x X xθ C (A) ϕ B R x +(xθ) (B) r θ A (C) (D) Figure 3. Schematic diagrams for the Ball on Beam system 4. Mathematical Modeling 4. Derivation of governing differential equations The construction and use of the DC motor is covered in another study []. For this reason we will simply move straight into the modeling and control of the beam angle and the ball position. The Ball and Beam dynamics are fully described by Shahian and Hassul []. The derivation supplied in that text is repeated here as follows. 4.. Ball Beam Dynamics Figure 3.4 shows the various schematics needed to understand and model the Ball on Beam system. A Lagrangian approach may be used to analyze the problem. Since this system stores no potential energy (according to the last assumption we made), the Lagrangian equation is the kinetic energy, as follows: U = mv + I b ω + I a θ& Eq Shandor Motion Systems, Ball on Beam Instructional System 8

6 Then equation 4.9 reduces to: X ( g G( = = Θ( R + 5 r s Eq. 4. It is interesting to note that the ball on beam system is independent of the ball s mass, but is in fact dependent on the radius of the ball. However, the maximum mass of the ball is limited by the maximum output torque of the motor. The entire derivation of the Ball and Beam plant assumes an angular input. In fact we know that applying power to a motor will not supply an angle but instead will supply a torque. Therefore, a second control loop must be developed to control the angular position of the beam. 4.. Motor Beam Dynamics Since the inertia of the beam is rather large we would expect that a simple proportional controller might not work. The transfer function of the beam alone is simply: Θ( M ( = = Eq. 4. T( J s Since the beam inertia is large and the motor is geared down, we ignore the inertia of the motor and tach. The following SIMULINK block diagram shows the model of the beam. It takes into account the torque constant of the motor and the gear train. b Amps Oz-In. Kt Kg Oz-In Radians Torque constant Gear Ratio Transfer Fcn (with inital output 0.468s Figure 4. Open Loop Beam Model One possibility for the motor beam system would be to design a controller for it, thereby creating a voltage to angle converter, then assume that the bandwidth of that system is far larger than the bandwidth of the Ball on Beam system. If this approximation were made, then the open loop transfer function of the Ball on beam system would become: X ( αk b g G( = = Θ( R + 5 r s Eq. 4. Where K b is the ball position gain and α is the Voltage to angle conversion constant. Later this idea will be further explored. 5. Analysis Analysis is the process of solving the mathematical model for a specific set of parameters to predict the response of the physical system. The prediction is a simplified representation of the response of the real system, but we hope that the model adequately represents the important dynamics of the physical system. It is only through the analysis process, and subsequent comparison to experimental measurements, that we are able to verify the predictive capability of the model. 999 Shandor Motion Systems, Ball on Beam Instructional System 0

7 6. Design experiments to determine parameter values In section 5, a method of controlling the beam angle was presented and a model was proposed and simulated. The response of the motor-beam system will be shown to be quite adequate for the entire ball on beam system if the accuracy of the model is shown to be good. Once the PD controller had been implemented (implementation will be covered later in this text), two experiments may be used to determine the step and frequency responses. These responses can be used to compare both the characteristic performance of the system and the accuracy of the model. The experimental results will be compared with the analytical model. Once the model of the beam dynamics has been verified, we can design a controller for the Ball on Beam system. In the first experiment, the transient response to a step input will be determined. A square wave with a DC bias will be applied to the angle command input of the PD beam angle controller discussed in the previous chapter. The beam angle will be measured (in fact, a potentiometer will measure the beam angle and supply a voltage that represents the angle i.e. K a ). The response we be recorded and compared with the step response found the chapter 5. In the second experiment a sinusoidal voltage will be applied to the angle command input of the motorbeam system. The frequency of this signal will be increased from 0.Hz to about 0 Hz since the simulation shows a significant roll of at about 5 Hz. 6. Conduct experiments 6.. Time-response experiments The time response to a step input is shown in Figure 6.. In addition to the experimental results, the step response of the model has been replotted for comparison. Clearly the model is quite accurate. There is some steady state error most likely due to some unmodeled friction in the system. The settling time is about 50 ms. This should be fast enough to supply any command that the Ball on Beam controller should require..5 Step response of the motor-beam system.5 V ol t s 0.5 Command signal Simulated repsonse Actual Response Time (sec) Figure 6. Step response of the motor-beam system 999 Shandor Motion Systems, Ball on Beam Instructional System 3

8 Figure 7.3 Root locus of the Lead compensated Ball on beam with motor-beam dynamics modeled as a constant Since the models are available for analysis in MATLAB, various zeros and poles were experimented with. The system appears to be sensitive to the location of the controller zero and less sensitive to the location of the controller pole. The zero should be located such that the locus of the systems with unmodeled motorbeam dynamics and the locus of the full model both have stable regions. This happens when the following controller is chosen. As we will see later, implementation also plays a role in choosing the pole and zero locations. K( s +.4) K c ( = Eq. 7. ( s + 4) Where K is the controller gain. A gain of.4 was chosen to give acceptable results. This may have to be reduced to as low as.5 to keep the more complex model stable. Figure 7. 3 Ball on Beam dynamics with lead compensator and Motor Beam Dynamics included. 999 Shandor Motion Systems, Ball on Beam Instructional System 6