BALL ON PLATE BALANCING SYSTEM Proposal for ECSE-4962 Control Systems Design

Transcription

1 BALL ON PLATE BALANCING SYSTEM Proposal for ECSE-4962 Control Systems Design Greg Andrews Chris Colasuonno Aaron Herrmann February 18, 2004 Rensselaer Polytechnic Institute

2 Abstract This report describes the proposed design and development strategy for implementing a control system to balance a ball on a plate. A pan-tilt device is placed on its side so as to create a tilt-tilt mechanism capable of moving a ball within an X-Y plane. A resistive touch pad is placed on the plane to allow the measurement of the location of said ball. Dynamic modeling of this system allows the creation of a digital controller capable of placing the ball at certain locations or following a preset path. The project goal is to create a system capable of moving the ball at a rapid rate of speed in any of several predefined complex paths with precision and accuracy.

3 Contents 1 Introduction 4 2 Objective 5 3 Design Strategy Model Development Performance Specifications vs. Available Components Parameter Identification Simulation Controller Design and Tuning Design Alternatives Subsystem Development Touch pad Inclinometer Verification Testing Procedure Tolerance Analysis Cost and Schedule Cost Analysis Phases Schedule Statement of Contribution 20 A Touch pad data sheet 22 B Coordinate System - Body A 24 C Coordinate System - Body B 25 D CAD Model 26 1

4 List of Figures 3.1 System States System States with Initial Conditions Simulation Diagram for Ball Dynamics Ball States

5 List of Tables 3.1 Arbitrary Friction Values Touchpad Pinout List of parts List of raw materials Labor costs Schedule

6 Chapter 1 Introduction The goal of this project is to develop a ball-on-plate balancing system, capable of controlling the position of a ball on a plate for both static positions as well as smooth paths. We intend that the initially horizontal plate will be tilted along each of two horizontal axes in order to control the position of the ball. Each tilting axis will be operated on by an electric motor. Each motor will be controlled using software, with a minimum of position feedback for control. After an extensive search, few systems of similar scale were found. The ball-on-beam system, a 1-dimensional similarity to the ball-on-plate, however, is a classic control problem, and has been studied in great detail, and solved a great many ways; PID control, optimal control, fuzzy-logic controllers, etc. The fuzzy-logic controllers would seem to be the current state of the art, however classical approaches using discrete approximations are certainly adequate, if not preferred for their relative ease of implementation. Two ball-on-plate systems were uncovered during the search: one, developed at Rensselaer Polytechnic Institute by Professor Kevin Craig [3] using a similar method to that which we intend, and another at the University of Newcastle, Australia, which was developed using image processing techniques in conjunction with a textbook by Professors Graham Goodwin, Stefan Graebe and Mario Salgado [4]. While this would seem to be a short list, the ball and beam system seems to be a more popular and less structurally complex system to implement. The aim of this project will be to create a ball-plate system using a resistive touch sensor to allow the movement of a ball by means of automatic control. The system should have accuracy of < 2% in the placement of the ball on the screen, as well as the ability to move the ball from one end of the plate to the other in less than 2 seconds (long side). Overshoot should be minimal, to reduce the chances of losing the ball off of the plate or incurring damage to the touch element due to striking the physical structure of the system. 4

7 Chapter 2 Objective The goal of the ball-plate system will be to initially create a system that can hold a ball in a static position on the plate. From there, the goal will then be to be able to move the ball around the plate in varying defined motion paths. The system should also be able to compensate for disturbances in the intended motion path of the ball, as well as disturbances to the physical support system. Initially, the desire of the team was to design a control system that could traverse a maze using the ball-plate system, using image processing to view the maze and plot a solution. However, given the processing power and estimation inherent to image processing techniques, as well as the team s lack of experience with the theory behind these techniques, this was abandoned. The traversal of a maze might still be possible, however, the traversal would be based on a known set of moves, as opposed to an on-the-fly calculation. Several challenges in the design and construction of this system can be foreseen. In order to construct the physical system, it will be necessary to machine several metal parts. A method of keeping the ball on the plate rolling when in motion rather than sliding is also necessary. A rubber membrane is on order to cover the touch pad with a frictional surface. However, the thickness of this membrane will also affect the sensitivity of the touch pad and therefore the necessary mass of the steel ball will need to be large, mandating a ball of large proportions. Gathering position data from the touch pad will also provide an added challenge. A serial interface controller is included with the kit, however additional precision is needed for our application, so we will have the added chore of developing a system to drive and sample the touch pad in order to generate the X-Y coordinates required. Finally, the control problem itself will be a significant challenge. Currently we intend to design the system as two uncoupled links, yet there may in truth be greater than negligible nonlinearities and coupling effects between the links. In addition, designing the system using a full state-feedback control system or a linearquadratic state-feedback regulator (LQR) will be far more complicated than simple PID control. 5

8 Chapter 3 Design Strategy 3.1 Model Development Due to the complexity of this system, a highly accurate, non-linear model must be developed. In order to consider a Lagrange-Euler dynamic model, the kinetic and potential energies of the system must be found. The kinetic energy is comprised of the energy due to both the linear and angular motion of the system, and can be represented as an inertia tensor. Gravity, friction, and velocity coupling terms must also be considered to represent the full non-linear dynamic model. Professor Wen s pantilt.m script was used to define the symbolic equations of motion for our system. In this file, the gravity vector had to be changed to point in the negative-x direction to account for our system s orientation change. This script returned symbolic values for the inertia tensor, velocity coupling matrix, gravity loading vector, and total energy. As expected, the gravity loading vector contained a term for the pan axis due to our orientation. Based on a Solidworks model of our system, the inertia tensors for bodies A and B with respect to the defined output coordinate systems as shown in Figures B and C were found to be: I a = I b = (3.1) Solidworks also provided the total mass of the bodies. Since the equations of motion were left in symbolic form, values for inertia, mass, and other parameters can be easily changed. In addition to these equations of motion for bodies for bodies A and B, a model must be developed for the ball itself, body C. In Professor Craig s previous work in the Mechatronics department here at RPI, equations of motion for the ball based on the platform angles were developed. Based on the small angle assumption made, Eqs. (3) and (4) in [3], should hold true for our system. This provides us with equations of motion for the ball in non-linear form. For now, the non-linear model will be considered, however Professor Craig s 6

9 system was linearized to decouple the two modes of motion. See Eq. (5) in [3]. The inertia of the ball was found for the equation of inertia for a solid sphere[5]. I c = 2 5 m cr 2 b 3.2 Performance Specifications vs. Available Components Beginning with the physical specifications of the desired system, quickly we see that we require several simple parts that are unavailable from the set of components with which we have been provided. In order to construct the system, several metal parts need to be machined in order to accommodate the 10.4 inch touch-screen platform. These parts include a new yoke, motor mount plate, and specialized shaft. The yoke and mount plate are much larger than those originally provided, and material has been removed from a larger diameter shaft in order to position the platform at the center of the axis of rotation. The overall weight of the final system will be approximately 1.2 kg. These parts can all be seen in the cad model in Figure D. High torque, high speed motors will be employed in the system. They are a necessary result of the combination of several factors. The system will be fairly heavy and a large gravity loading will be placed upon each motor, especially the base tilt axis which will support the entire system. The plate will tilt over a ±35 range in one second and thus fairly fasts speeds will be required. These speeds will be necessary to accurately control the ball s path with any amount of reasonable speed. It is expected that the motors will be required to rotate at speeds of at least 1.35 radians/sec. High resolution optical encoders will also be employed in the system. In order to control the motion of the ball with precision, a decent value for velocity feedback will be required. However, since the touch pad and encoders will only provide position feedback, it will be necessary to integrate the position to find velocity. Making use of optical encoders with a high resolution will help ensure greater accuracy and less noise in this integration. The encoders have a resolution of 2048 levels, with quadrature and our A/D converters we can achieve an effective resolution of 4096 levels. A rubber membrane of 1/16 thickness will be applied to the contact surface of the touch-screen. This membrane will serve to provide the friction necessary for the ball to roll rather than slide on the surface while remaining thin enough as to not greatly increase the weight of the ball which will be necessary to read position information from the touch-screen. A steel ball bearing of 1.25 diameter and weighing in at approximately 130 grams will serve as the ball in the system. This should provide sufficient weight to cause a response in the touch-screen setup. If after experimentation it can be determined that this is excessive, a smaller ball may be substituted. The touch-screen itself that will be used to detect the position of the ball on the surface is a 10.4 diagonal Dynapro wire resistive touch pad. The screen s total outside dimensions can be seen on the attached cad drawing. Rather than using the supplied serial encoder, we intend to interface directly with the screen, performing A/D conversions on the read lines in order to determine the position of the ball on the surface. The screen will also have a resolution of 4096 levels, due to the 12-bit A/D converters on the interface computer. 7

10 It will be necessary to keep noise at a minimum by making clean connections and perhaps shielding wires. The touch-screen operates in an analog fashion over only on a 0-3v range. Any significant noise could lead to inaccurate reading of the position of the ball which would severely limit the accuracy and precision of the final system. 3.3 Parameter Identification Once the system has been constructed, it will be necessary to identify physical parameters of the system such as friction and gravity loading. Friction will occur within the system at various locations such as motor, gear, and joint bearings as well as on the belts and pulleys, however if properly identified this friction can be canceled. Coulomb friction, which is constant when an object is moving, and viscous friction, which is proportional to velocity, will be lumped together as one friction term and determined experimentally. A Simulink diagram capable of outputting constant voltages and measuring the encoders will be used to repeatedly estimate joint velocity. A MATLAB script is being written to automate this process so that multiple trials may be easily run and averaged to get the best approximation of the friction parameter. It should be noted however that this measurement will be limited by the fact that velocity is being estimated. Any variance from the actual friction measurement will have to be considered a disturbance. 3.4 Simulation Based on Ben Potsaid s Laser Pointer Demo, a simulation has been developed to test the validity of our developed model. The simulation is initialized with our numerical values with the pantiltinit.m script attached to this proposal. The first step in simulation was to check conservation of energy. After disabling gravity, friction, and torque input, the simulation was run, and energy was plotted vs. time. The energy was calculated based on the equations returned from Professor Wen s pantilt.m script for our system. Energy is indeed conserved (remains at a constant 0 value for all time), as would be expected. The next step was to enable gravity and friction, and run the simulation. Chosen values for friction are shown in Table 3.1. As Table 3.1: Arbitrary Friction Values Joint Viscous Coulomb Pan Tilt expected, the pan axis swings down, and settles to a final value because of the gravity loading as seen in Figure 3.1. In addition to this, a small movement is noticed in the tilt axis presumably due to the velocity coupling between the axes. When initial conditions for the pan axis are 2.1rad, the pan axis swings in the other direction, finally reaching the original settling point. This is as expected since at this point, the motor crosses vertical plane formed by the pan axis. See Figure 3.2. Using the equations of motion from Professor Craig s [3] paper, the ball dynamics have been implemented 8

11 Figure 3.1: System States Figure 3.2: System States with Initial Conditions 9

12 into the simulation. From the equations, we can solve for ẍ and ÿ in terms of the state variables: [ ] u = θ1 θ2 θ 1 θ 2 θ 1 θ 2 ẋ ẏ x y (3.2) To do this, initial conditions are given to ẋ, ẏ, x, and y. Then, ẍ and ÿ are calculated and integrated to find x and y. A simulation diagram is given in Figure 3.3. More simulation needs to be done to test the validity of this model, however some simple simulations have been run. To test, gravity was removed from the base system, but kept on the ball. The plate was given an initial condition in one direction of 0.1rad. As expected, the ball accelerates in one direction, as seen in Figure 3.4. Figure 3.3: Simulation Diagram for Ball Dynamics Figure 3.4: Ball States 10

13 3.5 Controller Design and Tuning Based on the simulation that we are currently developing, a control system can be developed. As noted in Chapter 5, different controllers are planed to be developed depending on our success in each phase. Initially, we aim simply to balance the ball on the plate. This can be achieved with a traditional PID controller through root-locus and frequency-response analysis. Upon successful completion of this phase, a more sophisticated controller will be designed to allow us to accurately control the position of the ball on the plate and reject disturbances. To do this, we plan on implementing a full state-feedback controller. For our system, the state vector would be of the form: [ x := θ 1 θ 2 θ 1 θ 2 x b y b x b ] T y b (3.3) Since our model is in a non-linear form, the system must be linearized about an operating point [6] (θ, θ) = (θ d, 0). Another more advanced technique we may pursue upon completion of the former controller is to implement a linear-quadratic state-feedback regulator (LQR). To obtain higher accuracy, the system will be linearized around a number of operating points corresponding to different variations of θ and θ. Gains will be calculated off-line for the LQR at each operating point, and stored in a look-up-table. A simple interpolation function will determine the gains for intermediate states. This will allow the controller to respond quickly since all gains are calculated off-line, and state variables should be optimally controlled. An additional approach may be to implement an observer design. Since velocity cannot be directly measured, finite difference and/or washout filter methods would normally be used to estimate velocity from position measurements. However, these methods often produce unwanted noise. A possible solution to this would be to implement an observer to estimate the velocity: ˆx(k + 1) = Aˆx(k) + Bu(k) + L(y(k) ŷ(k)) (3.4) ŷ(k) + C ˆx(k) where L in Eq. 3.4 represents a column vector designed to cause the observer error e(k) := ˆx(k) x(k) to converge to zero [2]. 3.6 Design Alternatives Several control design alternatives are currently being contemplated in an attempt to achieve optimum performance from the system. Initially, PID control will be utilized using a linearized model in order to achieve an initial working system with which to further develop other systems. However, due to the nonlinear nature of the system, its inherent unbalanced state, and the complexity of the feedback system, it is expected that in order to achieve the desired performance specifications and a high degree of accuracy, a state-feedback controller will be required. Several designs will need to be developed and compared, however. Working with MATLAB Simulink, a script will be created to run each control design through a specified set of maneuvers: simple plate motions, static ball balancing, and ball path following. After the completion of the runs, graphs of the actual motions and states of the system can be compared to analyze overshoot, response time, and overall accuracy and performance. 11

14 3.7 Subsystem Development Touch pad The touch pad is the key piece to achieving our objective of balancing a ball on a plate. The touch pad we will use is a resistive element, returning an analog voltage corresponding to the location of the pressure applied to it. The pad is an 8-wire panel, but it needs to be pulsed in order to retrieve location information. Each axis needs to be pulsed separately, and the analog voltage read off a single pin. This pin diagram is show in Table 3.2 [1]. The touch pad as received came with a serial (RS-232) interface controller card capable of controlling the touch pad and retrieving data points from it. Our concern, however, is the accuracy of the controller, as well as the difficulty in using it with the MATLAB xpc target software. The controller uses a Texas Instruments 10-bit A/D converter, and the concern is that the resolution of 1024x1024 capable with this converter will be insufficient for our application. To achieve a higher resolution, a software and hardware interface will be created to connect the touch pad directly into the A/D hardware of the computer system being employed to develop the control system. The 12-bit A/D converters included will increase the possible resolution from the screen to 4096x4096, a 16-fold increase in resolution. This system will use MATLAB code in the Simulink model to pulse the excite pins of the touch pad, and simultaneously retrieve A/D samples from the output pin. The code will do this for each axis, back and forth, to create (X,Y) coordinate pairs for use in the control system code. We believe that using this level of integration, we can easily best the provided controller s sampling rate of samples per second. Table 3.2: Touchpad Pinout Axis Xe + Xe Y e + Y e Xs + Xs Y s + Y s X 5v GND NC READ Ref+ Ref- NC NC Y NC READ 5v GND NC NC Ref+ Ref Inclinometer The ball-on-plate system is based on the concept of balance. However, in order to be balanced, the system has to know what it means to be balanced. This means having both axes exactly parallel to ground, perpendicular to gravity. To accomplish this, inclinometers will be installed on each of the two tilt axes. This will make certain that during initial start-up and calibration, the system starts in a position that it knows to be zero. The inclinometers will be attached to the tilt axes, and their output signal interfaced directly to the A/D converters of the computer system. These signals will then be read from the MATLAB code for use in the system calibration during start-up. 12

15 Chapter 4 Verification 4.1 Testing Procedure Though little testing will be necessary for the newly machined parts, it should be noted that they must be accurately made. It is especially important that the axle be straight and the holes in the yoke be of the proper size and exact placement in the yoke so that the platform will properly spin about the tilt axis. This can be tested by simple measurement and observation. The test of the motors ability to perform their task will be quite simply whether they are able to rotate this heavy system at the speeds desired in order to control the ball. This will be testable with a Simulink/MATLAB setup. Various inputs and the resulting position graphs read from the encoder can easily be used to test the limits of the motor s ability. The optical encoder s performance also can easily be tested with MATLAB. Especially of interest will be resulting velocity graphs that will be a result of integrating the position feedback provided by the encoder. Testing of the touch-screen subsystem will be necessary to ensure accurate position information will be available for feedback in the control algorithm. In addition, the touch-screen sensitivity to the ball weight and the effectiveness of the rubber membrane covering the screen, will require testing to ensure that all components of this subsystem are effective but in no way excessive. To test touch-screen feedback, once the subsystem has been implemented the ball will have to be placed on the screen at several locations and the position read by the subsystem compared to the actual location of the ball. Precise measuring will be necessary here. It may also be desirable to check for precision of position feedback for the ball in motion, however it will be nearly impossible to measure this accurately enough for comparison in the physical world. At minimum however, a plot of the balls path could be observed. The smoother this resulting plot, the better. During the testing of the position feedback, the effect of the ball s weight on the touch-screen must also be noted. If the ball is not always successful in exciting the touch-screen, it will be necessary to increase the size and weight of the ball. However if the ball is performing perfectly, this will be an indication that a smaller ball of less weight could be used. This is desirable as there is finite amount of space on the platform and the 13

16 more space that is available, the greater the range of movement the ball will have on the plate. One less complex, but still important test of this subsystem remains. The ball must be placed on the surface when tilted at its steepest angle and observed. If the ball rolls, then the rubber membrane is providing sufficient friction. However if any slipping at all occurs, it will be necessary to upgrade to a thicker membrane. This however could also affect the necessary weight of the ball and perhaps position feedback, so this simple test of the touch-screen subsystem will be performed first. 4.2 Tolerance Analysis The physical machine parts require great accuracy. While this should not be overly difficult to implement and test, any deviations from the correct measurements could lead to unrecoverable faults in the performance of the system. Care will have to be taken in their construction and the final test of their performance will be the smoothness of rotation of the tilt axis which they implement. It also is important to the success of the system that the motors and encoders perform well. However, in the event that they are sub-par some correction other than replacement will need to be implemented if the system is still to be a success. There simply are no better motors or encoders available to us. Fortunately, they should be satisfactory As stated earlier, it is necessary for the touch-screen subsystem to be highly accurate. If the position feedback for the ball is not accurate, the overall ability to control the system will be greatly reduced. The system will be tested as described in the prior section and any possible calibration or noise cancellation will be implemented. The final test will of course be the overall ability of the system. Barring some unforeseen problem with the machined parts, motors, or encoders, it is likely that the greatest barrier for perfection in the performance of the system will be the accuracy of the touch-screen subsystem. If this subsystem does not provide an accurate error signal, an ideal control system will never be achieved. 14

17 Chapter 5 Cost and Schedule 5.1 Cost Analysis The cost for developing the system can be broken down into the cost for parts, the cost for raw materials, and the cost for labor. Table 5.1: List of parts Item Qty Cost Total Source 1 1/4 diam. 440C stainless ball 1 $9.17 $9.17 McMaster-Carr Dynapro wire resistive touch pad 1 $39 $39 Ebay Pittman motor GM9234S017 (pan) 1 $97.59 $97.59 Supplied Pittman motor GM9234S017 (tilt) 1 $97.59 $97.59 Supplied Pan gear A 1 $9.97 $9.97 Supplied Pan gear B 1 $22.02 $22.02 Supplied Tilt gear A 1 $7.95 $7.95 Supplied Tile gear B 1 $22.02 $22.02 Supplied Pan belt 1 $3.92 $3.92 Supplied Tilt belt 1 $4.00 $4.00 Supplied Total $ Table 5.2: List of raw materials Item Qty Cost Total Source 1/2 aluminum stock 5 lb $4/lb $20 RPI Machine Shop 1 1/4 diam. aluminum round stock 4 lb $4/lb $16 RPI Machine Shop 1/16 latex rubber membrane 1 $9.38 (12 x12 ) $9.38 McMaster-Carr Total $

18 Table 5.3: Labor costs Description Hours Cost Total Andrews, Greg (engineer) 300 $35/hr $10,500 Colasuonno, Chris (engineer) 300 $35/hr $10,500 Herrmann, Aaron (engineer) 300 $35/hr $10,500 Caskey, Ryan (machinist) 10 $35/hr $350 Total 910 $31, Phases The development of the ball-on-plate system can be generalized into several phases to overall project completion. Basic system completion entails the ability to balance a ball in a static position on the sensor plate. The target advanced system builds on the basic system and allows for simple as well as complex trajectory following of the ball. 1. Initial development System modeling Using physical parameters for the individual system pieces and a CAD model, the inertia and mass matrices can be generated. These can then be used to generate the coriolis/centrifugal matrices, which can be used to generate the dynamics of the system. Additional parts construction Additional parts are needed for the central plate yoke, which need to be machined in the machine shop. 2. Identification Parameter identification Using a MATLAB script, the physical system can be run through several tests to determine the parameters describing the full model, including friction terms. Validation Once these parameters are defined, the physical system response can be compared to the projected response, and the std.dev of this response can be used to further define the system parameters. 3. Controller Control design Once the system is defined, the controller can be developed. Several different designs are planned, including a PID controller, a State-feedback observer controller, and an Optimal controller. Sensors The sensor development and integration will need to be finalized by this point, including the testing of the inclinometers, touch pad, and encoders. 4. Integration Integration Move the sensors from testing into the physical system. Also, integrate the control design into the Simulink system for usage with MATLAB xpc target. Testing With the control system now operating, test to make sure the system can be calibrated, and maintains position to within a small percentage of the desired position. Also, check for response time and disturbance rejection in the joint angles. 16

19 5. Basic System Static ball balancing Move system from basic controller to balancing system. This includes feedback from the touchpad in order to monitor ball position and dynamics in an attempt to maintain static ball position. Disturbance rejection With the ball static, attempt to nudge the ball from its current position, to make certain that the control system can compensate for unknown random forces. 6. Advanced System Line trajectory tracking (slow) With static ball balancing working, work on moving the ball along a line trajectory (e.g. y = x), such that it can traverse from one corner of the work surface to another. Line trajectory tracking (fast) Increase the speed of the traverse until the system can move the ball rapidly, but under control. Complex trajectory tracking Now that a line path can be rapidly tracked, attempt to make figureeight, circular or more complex paths for the ball to follow. MATLAB GUI interface to tracking (optional) Develop a MATLAB gui to allow for simplified control of different tracking examples. This system could possibly include a grid on which a user could pick points, and have the ball trace out a simple path. 5.3 Schedule A proposed overall development schedule is shown in Table 5.4. While preliminary, the schedule is realistic and aims for advanced system completion before the final demonstration day. The schedule also includes information on task breakdown between members; however, the foreseen complexity of the project and of each task suggests that each task might likely be completed by the group at large, with the designated team member merely acting as lead. 17

20 Table 5.4: Schedule Week Task Member Week 4 Model development Greg Research sensing hardware Aaron Find machinist to make metal parts Chris Week 5 Model development Greg Test sensing hardware Aaron Work on machining parts Chris Work on project proposal Team Week 6 Model development Greg Research sensing interface options Aaron Friction ID script Chris Work on project proposal Team Week 7 Friction ID Team Week 8 Model validation Chris Develop sensor interface Aaron Preliminary control design Greg Week 9 Control design Greg, Chris Finish sensor interface Aaron Week 10 Integration of sensors Aaron Control system testing Greg Sensor testing Chris Week 11 Static ball balancing Team Week 12 Complex path following Team Week 13 Work on final report Team Final demonstration Team Week 14 Work in final report Team Final presentation Team Week 15 Final report Team 18

21 Bibliography [1] 3M Touch Systems. SC4 Touch Screen Controller: User s Guide, 2nd edition, [2] Dr. Murat Arcak. Discrete time systems - lecture notes [3] S. Awtar, C. Bernard, N. Boklund, A. Master, D. Ueda, and K. Craig. Mechatronic design of a ball-onplate balancing system. Technical report, Rensselaer Polytechnic Institute, [4] Graham Goodwin, Stefan Graebe, and Mario Salgado. Control system design - ball-on-plate tutorial. Available WWW: sim.html, [5] Eric W. Weisstein. Moment of inertia - sphere. Available WWW: physics/momentofinertiasphere.html. [6] Dr. John Wen. Control system design - lecture notes

22 Chapter 6 Statement of Contribution For the project proposal document: Greg completed the following sections: Model development Controller design and tuning Simulation CAD model Chris completed the following sections: Performance specs vs. available components Parameter identification Testing procedures Tolerance analysis Aaron completed the following sections: Abstract Introduction Objectives Design alternatives Subsystem development Cost analysis Phases Schedule 20

23 Greg Andrews Chris Colasuonno Aaron Herrmann 21

24 Appendix A Touch pad data sheet 22

25 23

26 Appendix B Coordinate System - Body A 24

27 Appendix C Coordinate System - Body B 25

28 Appendix D CAD Model 26