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Docstoc, Inc. Elaina Griffo Table of Contents Task 1: Become Familiar with ModelCenter ................................................................................................. 2 Task 2: Identify and Model the Uncertainty in the Design Analysis Model .................................................. 5 Task 3: Elicit a Detailed CDF for the Most Significant Uncertain Variables ................................................ 12 Task 4: Determine the Distribution of the Output of the Model................................................................ 14 Task 5: Lessons Learned .............................................................................................................................. 20 SCARA Robot Design for Wafer Transport 1 Elaina Griffo Task 1: Become Familiar with ModelCenter The goal of this project is to design a SCARA style robot for the handling of a semiconductor wafer in an ion implanter. The design engineer must select the belt size and pretension along with the controller gains to minimize move time and positional error. The minimization of positional error combined with the maximization of implanter throughput result in the maximization of company profit, the fundamental design objective. Prior to wrapping the Modelica model for use in Model Center, some modifications were made in order to generate outputs which clearly measure robot move time and positional accuracy. A system path planning block was created to combine the path planning blocks from both the rotary and translational joints. Within this system path planning block a condition was added to terminate the simulation at a time equal or greater than the sum of the path time and an input parameter, swing time. The path time is the total move time of the generated path, which is not necessarily equivalent to the time required to move the robot dynamics into position. To account for this difference, the swing time variable was added. This variable represents the desired settling time. The simulation will automatically be terminated at the path run time plus the settling time, so settling error can be observed. It is important to note that with this definition of simulation time, we expect no changes in simulation time without a change in the path or swing time variables. In later simulations this variable will be observed as a sanity check, to verify that the system is responding as expected. SCARA Robot Design for Wafer Transport 2 Elaina Griffo Figure 1. The system path planning block with terminate simulation condition Once the path planning modification was completed, a loop was added to compute the integrated positional error at each joint. SCARA Robot Design for Wafer Transport 3 Elaina Griffo Figure 2. The addition of the integral of positional error for each of the four joints The error calculation blocks calculate the difference between the commanded and actual positions. The absolute value of the error is integrated over the course of the move to provide a metric for the total positional following error. Note that the summation block is modified to integrate the inputs. This method is applicable when the position is critical throughout the course of the move. If only the final position is important, the integration method is not appropriate. The error block and modified code are shown in the figure below. SCARA Robot Design for Wafer Transport 4 Elaina Griffo Figure 3. The error calculation block integrates the absolute value of the positional error The error calculation block was tested by inputting a unit step into the reference input and a constant zero value at the actual position input. The output was verified to be a ramp with unit slope. The SCARA Robot model with simulation termination and error measurement features is used in the following analyses. Task 2: Identify and Model the Uncertainty in the Design Analysis Model Selected uncertain input variables were analyzed to determine their effects on the robot move time and integrated joint error. Initially, the input variables selected were the proportional, speed and integral gain terms for each of the four axes, along with the belt stiffness, damping and tension. However, a central composite experiment with such a large number of variables is not feasible within the time constraints of this course. For the purposes of this experiment, the integral gain terms were ignored as preliminary experimentation in Dymola indicated that the effects of these terms were relatively small. The triangular distribution of values for each variable are shown in the table below. SCARA Robot Design for Wafer Transport 5 Elaina Griffo Table 1. Triangular distributions for uncertain variables of interest Variable 5th Percentile Center 95th Percentile Controller Gains Prismatic Joint kp 20 168 600 ks 2 12 75 First Rotational Joint kp 30 250 500 ks 5 25 55 Second Rotational Joint kp 30 280 800 ks 5 30 100 Third Rotational Joint kp 10 150 500 ks 5 30 100 Belt Parameters Spring Stiffness (Nm/rad) 100 997.98 1800 Damping (Nms/rad) 20 99 110 Belt Tension (Nm) 50 200 300 It should be noted that these ranges represent the feasible ranges of parameters that may exist for this system. The belt stiffness, for example, represents the range of stiffnesses for belts that may be used in a system of this size, not the uncertainty in the stiffness of a single belt. Through this process the design engineer is able to quantify the effects of a variety of belt and control conditions prior to choosing the alternative which will maximize profits for the company. The above distributions were input into a Central Composite Experiment in ModelCenter. The resulting effect of the input variables on simulation run time is illustrated in the figure below. SCARA Robot Design for Wafer Transport 6 Elaina Griffo Figure 4. The effects of parameter variation on the simulation time of the SCARA Robot model As mentioned previously, the simulation run time is expected to remain constant throughout this experiment as no path or swing time parameters are being modified. However, the results above clearly show the belt parameters and certain gain values having an effect. This inconsistency may be caused by errors in the sampling process. Throughout the sampling, but particularly toward the end, errors were observed in which the simulation was unable to solve the model. In addition to simulation time, the cumulative errors at each of the four joints were evaluated to determine which inputs had the largest effect on positional error. SCARA Robot Design for Wafer Transport 7 Elaina Griffo Figure 5. The effects of parameter variation on the cumulative error of the prismatic joint of the SCARA Robot model The figure above shows the effects of each input variable on the cumulative position error of the prismatic joint. The three largest factors are the velocity gains of the prismatic and first rotational joints, along with the positional gain of the prismatic joint. It is intuitive that the gains for the prismatic joint will affect its positional error. A mechanism with a larger proportional gain, kp, value tends to track a command position more accurately, resulting in a smaller cumulative error. However, if the proportional gain is too large, the system can become unstable. This instability will result in oscillation which will increase the error. A mechanism with a larger velocity gain term, ks, will tend to oscillate less (similar to an overdamped system) however, the settling time will increase. In this simulation it is likely that a large ks value would result in a positional error at the time of simulation termination, since the simulation terminates based on the planned path not the actual robot dynamics. A small ks value may result in system oscillation (similar to an underdamped system), thus increasing cumulative positional error. It is more difficult to explain the strong interaction of the first rotational joint velocity gain with the prismatic positional error. While cross-talk between the rotational joints is expected since they are moving in the same plane, I expected any interaction between the rotary joints and the prismatic joint to be secondary. The magnitude of the interaction may have been affected by the errors in the simulation. SCARA Robot Design for Wafer Transport 8 Elaina Griffo Figure 6. The effects of parameter variation on the cumulative error of the first rotary joint of the SCARA Robot model The above figure illustrates the effects of input parameter variation on the magnitude of the cumulative positional error of the first rotational joint. The four parameters which contribute the largest variation to the response are the position and velocity gains of the third and first rotary joints. It is intuitive that the gains for the controller of the first rotary joint would have a large effect on the cumulative error of that joint. Less intuitive is the more significant relationship between the gains of the third rotational joint and the cumulative error of the first joint. It is possible that this relationship does not really exist and that its presence in this data stems from a number of errored samples which occurred in this simulation when the system was unable to be solved. The second possibility, which is based on observation of system performance in Dymola, is that the third link tends to be less stable, so variations in the controller gains of the third rotary joint cause an oscillation in the third link which is significant enough to affect the performance on the first joint. SCARA Robot Design for Wafer Transport 9 Elaina Griffo Figure 7. The effects of parameter variation on the cumulative error of the second rotary joint of the SCARA Robot model The figure above illustrates the effects of input parameter variation on the cumulative error of the second rotary joint. Immediately it is apparent that the controller gains for the third joint do not contribute significantly to the variation in the error of the second rotary joint. This would appear to indicate that the phenomenon observed in the first rotary joint (the cumulative error behavior being dominated by the third rotational joint controller gains) is an artifact of simulation error rather than some sort of coupled oscillatory behavior, otherwise a similar behavior would likely be observed in the second rotary joint. The input parameters which dominate the error behavior of the second rotary joint are: the position and velocity gains of the first rotary joint and the belt stiffness. It is expected that the gains of the first rotary joint are relevant; since the second joint is mounted to the first link, any oscillatory behavior in the first link will be transmitted directly into the belt drive. Similarly, the belt stiffness directly affects the performance of the second rotary joint. When the belt stiffness is low, the positional tracking will be less accurate, leading to larger cumulative error values since the error can occur over the entirety of the move. SCARA Robot Design for Wafer Transport 10 Elaina Griffo Figure 8. The effects of parameter variation on the cumulative error of the third rotary joint of the SCARA Robot The figure above shows the effects of input parameter variation on the third rotational joint. The parameters with the largest effect on the error of the third joint are: the velocity gain for the second rotary joint, the velocity and positional gains of the first rotary joint, and the belt stiffness. The effect of the second joint velocity gain is logical, as the third joint mounts directly to the second link. Thus any instability in the second link is transmitted directly to the third joint. Similarly, the interaction between belt stiffness and the cumulative positional error of the third joint is expected because a lack of belt stiffness can lead to system vibration which would increase error in the third joint. The interaction between the gains of the first rotational joint also makes sense. This relationship was also observed in the second rotational joint, indicating that oscillation in the first rotational joint is transmitted through the first link through the second and third joints. From this uncertainty analysis it is difficult to say which variables make the largest contribution to the cumulative error of the system. The controller gains for the first rotational joint made larger contributions to all four cumulative positional errors. The belt parameters were also consistently in the top few sources of uncertainty. Based on this uncertainty analysis and my own interest in the effects of a drive belt on robot dynamics, I have decided to model the distribution of belt parameters in more detail in the following section. SCARA Robot Design for Wafer Transport 11 Elaina Griffo Task 3: Elicit a Detailed CDF for the Most Significant Uncertain Variables For each of the three belt parameters, a cumulative distribution function (CDF) was defined based on my beliefs. These curves were created through a series of elicitation questions of the following form: “How much would I be willing to pay for a bet which pays $1 if a parameter is equal or less than a specified value and $0 if the parameter is larger than the specified value?” In order for this elicitation process to be successful, the bet must be fair, so at the price I select I must be willing to take either side of the bet. The table below shows the list of parameter and payment values determined through this process. Table 2. Price for willingness to bet based on a specified parameter value Belt Stiffness (Nm/rad) Belt Damping (Nms/rad) Belt Tension (Nm) Parameter Bet Price Parameter Bet Price Value ($) Parameter Value Bet Price ($) Value ($) 0 0.00 0 0.00 0 0.00 100 0.01 20 0.08 20 0.01 450 0.12 60 0.35 125 0.15 998 0.60 99 0.95 200 0.50 1400 0.80 110 0.98 250 0.85 1800 0.90 200 1.00 300 0.95 3000 1.00 NA NA 1000 1.00 From the above table the CDFs for each belt parameter were created in Excel. It is worth mentioning that the prices which were initially elicited did not make sense once the PDF was viewed. A decrease in value in the probability density function (PDF) is indicative of a decrease in the likelihood of an even occurring. Through the combination of the values I selected and the cubic spline function, there were a number of undesirable dips in the PDF which were not representative of any real phenomenon. With minor modifications to the prices used in the definition of the CDF, a smoother PDF was generated. I believe this process of elicitation and modification is acceptable since my initially elicited prices did not create a PDF which represented my beliefs about the uncertainty of the variable in its entirety (rather than at a single point). Additionally, within a certain margin willingness to bet is equivalent. If a bet is $0.62 as compared to $0.60 it is still similarly appealing. The figures below show the CDFs and PDFs for the belt stiffness, tension and damping. SCARA Robot Design for Wafer Transport 12 Elaina Griffo Figure 9. Cumulative distribution and probability density functions for belt stiffness The figure above shows the CDF and PDF for the belt stiffness distribution. Notice that while there are inflection points in the PDF there are no dips in probability density. The peak of the probability density function is around 700Nm/rad, with the distribution skewed to the right. This skew is fitting as most belt stiffnesses are below 1500Nm/rad; however, there are a limited number of specialty belts designed and manufactured for high-stiffness applications so the tail of the distribution extends up to 3000Nm/rad. The belt I used in my SCARA model, with a stiffness of 997.98Nm/rad, falls above the peak probability density, but it was one of the stiffest steel-banded polyurethane belts. Fully metal belts would likely have larger stiffness values. Figure 10. Cumulative distribution and probability density functions for belt tension The figure above shows the CDF and PDF for the belt tension. While manufacturer recommended tension is ultimately a function of pulley diameter, spacing and span, the constraints on the design in the form of link length, mass and payload constrain the tension to a fairly limited range of values. However, since belt width and speed are also relevant in this calculation, and since these values may vary between belt and pulley choices, there is an extended tail skewing the distribution to the right. From the PDF above it can be observed that the most probable belt tension values are between 125Nm and 300Nm. SCARA Robot Design for Wafer Transport 13 Elaina Griffo The selected value, 200Nm, falls within this range. Figure 11. Cumulative distribution and probability density functions for belt damping In the figure above, the CDF and PDF for the belt damping value are shown. The PDF is again shifted to the right. Mechanisms without the addition of damping elements are almost always underdamped. The PDF increases steadily from 0Nms/rad, peaking near 80Nms/rad. The value used in the SCARA Robot model, 99Nms/rad, represents critical damping in the second rotary joint, so the peak of the PDF is in the underdamped range of damping values. The curve is not as skewed to the right as one might expect (damping in metal links is only around 5%) due to the inherent damping in the belt system in the form of the interaction between the belt and pulley teeth, and as increased friction in the bearings due to larger radial loads from the belt. Task 4: Determine the Distribution of the Output of the Model The CDFs generated in the Excel files described in the previous section included an inverse CDF calculator which calculates the parameter value corresponding to an input probability. A probabilistic analysis was set up with triangular distributions (described above) for all variables except belt stiffness, tension and damping. For belt stiffness, tension and damping a uniform distribution was applied from 0 to 1. The outputs of the uniform distribution were input into the Excel inverse CDF, and the parameter value calculated from Excel was input into the SCARA Robot model. This setup is depicted in the figure below. SCARA Robot Design for Wafer Transport 14 Elaina Griffo Figure 12. Setup of probabilistic analysis with derived belt parameter distributions A Monte Carlo analysis with 1000 samples was performed, and the results are displayed in the following figures. The figure below shows the resulting distribution for the simulation time. As expected, there is no spread, the simulation time is only a single value since belt parameters and gain values have no effect on the simulation time as it is defined in this model. SCARA Robot Design for Wafer Transport 15 Elaina Griffo Figure 13. Monte Carlo histogram for simulation time based on 1000 samples The figure below shows the distribution of the cumulative error in the prismatic joint. This distribution is skewed to the right with a mean of 0.0136m and a standard deviation of 0.0222m. The skewness makes sense, as we expect the error to typically be close to zero. However, even an error of 13mm is quite large for the precision motion required from a SCARA Robot. It should be noted that the median is 0.0085m, which is often a better descriptor for a skewed distribution as it is unaffected by outliers. From this distribution it is possible to conclude that the design of the belt parameters and the selection of the controller gains are not trivial, as the certain values within the current distribution lead to significant positional error. It should also be mentioned that in Dymola simulations this axis did appear to oscillate more than the rotational joints. This oscillation would also lead to an increased cumulative error. Such oscillation can be decreased by increasing the system stiffness (raising the natural frequency) or by adding a damping element. SCARA Robot Design for Wafer Transport 16 Elaina Griffo Figure 14. Monte Carlo histogram for cumulative positional error of the prismatic joint based on 1000 samples The histogram below shows the resulting distribution of cumulative error values for the first rotary joint. The distribution is once more skewed to the right with a mean of 0.0171rad, a standard deviation of 0.0281rad, and a median of 0.0069rad. While the mean and standard deviation are larger in magnitude than those of the prismatic joint, the median is lower. This pattern indicates a large cluster of error values closer to zero with a wider spread of outlying values. Again, this distribution makes sense as we expect stable combinations to have low cumulative errors. The outliers represent unstable parameter combinations. The oscillation caused by instability leads to an increase in error. SCARA Robot Design for Wafer Transport 17 Elaina Griffo Figure 15. Monte Carlo histogram for cumulative positional error of the first rotary joint based on 1000 samples The figure below depicts the cumulative error distribution for the second rotary joint for a sample size of 1000. The distribution is once again skewed to the right, and the average and standard deviation of the cumulative error of the joint are significantly larger in magnitude than the prismatic or first rotary joints. With an average of 0.1389rad, a standard deviation of 0.3935rad and a median of 0.0125rad, this distribution has quite a large spread. Of all four joints, it is expected that the second rotational joint would contain the samples with the largest error magnitudes. This is because the second rotational joint is belt driven. The belt parameters directly affect the cumulative error of this joint, and when the belt stiffness or tension are low the second joint may have a following error through the entire move—when integrated this error becomes quite large. Additionally, as damping is decreased oscillation of the joint response is expected, which will similarly increase the cumulative error. This theory is further supported with the observation that the median joint error is only slightly larger in magnitude than the first rotational joint. SCARA Robot Design for Wafer Transport 18 Elaina Griffo Figure 16. Monte Carlo histogram for cumulative positional error of the second rotary joint based on 1000 samples The final figure below shows the cumulative positional error distribution of the third rotational joint. This distribution is difficult to explain as it does not seem to make sense. The distribution is again skewed to the right; however, some of the error values are negative. Given that the model takes the absolute value of the error prior to integration, I would expect to only have non-negative error values. Since this distribution spans zero, I believe the mean, standard deviation, and median are not accurate descriptors of the true error distribution for this joint. My only explanation is that it is an error in the ModelCenter simulation setup. Unlike in the central composite experiment, no errors were noted in the trial calculations for this simulation, so it does not appear this error stems from the Monte Carlo analysis itself. SCARA Robot Design for Wafer Transport 19 Elaina Griffo Figure 17. Monte Carlo histogram for cumulative positional error of the third rotary joint based on 1000 samples Task 5: Lessons Learned I had a number of difficulties with this project. First, there are a number of default parameters in Windows 7 which interfere with ModelCenter and the file wrapping process. Since file extensions are not shown, changing the name of the dsin file to .template creates a text file (full name dsin.txt.template.txt). I also saw multiple file access/write errors since the wrapper files are saved in the program files directory. Once I successfully wrapped my model and brought it into ModelCenter, my Central Composite experiment would not run past the third set of trial values. I believe this was related to the ranges I had selected for my variables. For each input, I ran a parametric study over my desired input range. When the parametric study failed, I adjusted the range until the study ran to completion. I mentioned this issue to Professor Paredis, and he suspects it is because the certain input parameters make the system stiff. The same values, run in Dymola, caused the output to become wildly unstable, and the simulation had to be stopped as it was unable to solve in a reasonable time period. In the creation of the CDF, I struggled with the different forms of preference elicitation. Initially I was comparing two bets: a bet with a percentage probability of winning $1 versus winning a dollar if the variable of interest was less than or equal to a specified value. When these two bets are equally appealing, the probability of winning is used to describe the CDF of the variable of interest at the specified value. However, after reviewing the course notes, I decided to use a method of elicitation SCARA Robot Design for Wafer Transport 20 Elaina Griffo which asks how much the decision maker would pay to participate in a bet which pays one dollar when the parameter value is less than or equal to a certain value. I am still a bit uncertain about the appropriate use for each of these two methods, as they seem quite similar and I got the same results from both approaches. The problem-solving approach outlined in this assignment worked well, and I would use it again. SCARA Robot Design for Wafer Transport 21

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