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Spatially-variable fertilizer and pesticide application with GPS and DGPS
Computers and electronics in agriculture ELSEVIER
Computers and Electronics in Agriculture 11 (1994) 69-83
Spatially-variable fertilizer and pesticide application with GPS and DGPS J o h n K. S c h u e l l e r * , M i n - W e n W a n g Department of Mechanical Engineering, University of Florida, Gainesville, FL 32611, USA
(Accepted 3 May 1994)
Abstract
Concepts of spatially-variable fertilizer and pesticide application are discussed. Commercial applicators are described which use Global Positioning Systems (GPS) or other location technologies to apply fertilizer and pesticide according to predetermined setpoint maps. The dynamics of the applicator are shown to have important performance effects. Command feedforward control can significantly improve the performance. Further research is needed on understanding the effects of the various error sources. Key words: Spatially-variable; Global Positioning System; Fertilizer; Pesticide; Applicator
1. I n t r o d u c t i o n
The basics of Global Positioning System (GPS), Differential GPS (DGPS) and G L O N A S S (all henceforth to be generically referred to GPS) are explained elsewhere in this Special Issue and do not need to be repeated here. Other articles explain how GPS can be used with sensors which monitor crop and soil properties to generate maps of these soil and crop properties. The generation of these maps will be very beneficial to the farm operator in that he can then know what sort of variabilities he has in his fields. But not only will he know the variabilities, he will be able to manage his fields spatially. T h e r e are several ways in which the farm manager can manage spatially. Not all of them require advanced technology. One way is for the fields to be reorganized according to the soil properties. Contour or strip farming is an example of this. However, there are problems with this in practice due to the constraints on field size and shape for efficient field operations, as well as the difficulty in assigning just a single set of field boundaries for all of the field variabilities. Another way is * Corresponding author. 0168-1699/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0168-1699(94)00016-J
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J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
for the farmer to manually operate his equipment in such a way that it responds properly to the spatial variabilities. However, in this case it is difficult for a human to successfully perform such control with reasonable accuracy. Operators without direct financial stake in the performance may also lack the motivation to perform the frequent, accurate control. Given the problems with the above two methods, it appears that automation may be a better approach. Field operations can be controlled automatically according to two major approaches: Automatic Control or Temporally Separate Control (Schueller, 1992). In Automatic Control, the piece of farm machinery performing the operation will have some sort of sensor and will control the operation based upon the sensor's output. In Temporally Separate Control, the control is decoupled from the sensing. The mapped data on spatial variability are gathered by any means, such as the yield and soil mapping discussed in other articles, and are used to generate a control setpoint map. This map is then used to guide the control operation. The Temporally Separate methodology relaxes the requirements on the sensor and permits more sophisticated management, but requires the development and use of the setpoint map and some method of locating the field equipment's position on the map. The location methods most often discussed are dead reckoning, electromagnetic (radio or microwave) trilateralation, or GPS. GPS can be used to determine position during such operations as landforming, terracing, and tillage. It could also be used to help determine the appropriate variety, population, or depth during planting operations. However, discussions with many agronomists and engineers indicate that they feel the greatest potential for spatially-variable control and the use of GPS lies in fertilizer and pesticide application. Pesticides by definition are only useful where there are pests. Fertilizers are only valuable where there are needs for the nutrients. At other places, pesticides and fertilizers only have the potential to create environmental problems. In North America there is a tendency to apply fertilizers and pesticides with the same equipment, in fact often at the same time. The emphasis in this article will be on fertilizer application, but most of the concepts and analyses are applicable also to pesticide application. The other articles show that soils and crops are spatially-variable. Soils have different nutrient levels, mobilities, and fixation abilities at different points within a field. Crops remove different amounts of nutrients and require deferent amounts of nutrients in a spatially-variable manner. Fertilizer is usually required to obtain the maximum economic yields (MEY). Generally, the yields of a crop increase with increasing fertilization, although the rate of increase tends to decrease. Eventually a point is reached where the yield increases do not justify the extra fertilizer costs. This is the MEY fertilization rate. This point will vary within fields, and is therefore the justification for spatially-variable fertilizer application. In addition, excess nutrients can cause quality problems in some crops. An example of these problems is nitrogen in sugar beets and potatoes. The excess nutrients will also be less likely to be utilized or immobilized and, therefore, may cause environmental problems. An example of the environmental concern is that the U.S. Environmental Protection Agency is sup-
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J.K. Schueller, M.-W. Wang~Computers and Electronics in Agriculture 11 (1994) 69-83
porting work to examine the benefits of spatially-variable fertilization on nitrogen pollution. 2. Commercial applicators
The potential of spatially-variable fertilizer and pesticide application has led to a number of attempts to commercialize the technology. The largest impact has been made based upon the U.S. patent of Ortlip (1986). This patent has broad claims which (assuming they are legally defendable and are defended) clearly patent application according to digital maps. The system patent is owned by Soil Teq, Inc., whose controlling interest (60%) is owned by Ag-Chem, Inc., also now the current owner of Lor-A1, the original licensee of the technology who manufactured spatially-variable applicators. The technology was first released as part of a dry fertilizer applicator. It has gone through a number of modifications and redesigns to improve performance and reliability. The latest redesign is not available, but an earlier version is shown in Fig. 1. The operating procedure is as follows. Fertilizer, chemical and micronutrient rates per acre per soil type are programmed into the 1 microprocessor. Sensing its location on the 2 field map the computer sends a signal to the 3 electro-hydraulic valves which control the rate at which product is dispensed from the 4 product bins by the 5 metering wheels. The fertilizer then drops onto a 6 stainless steel conveyor chain and is carried back to the 7 mixing auger where the chemicals (usually pesticides) from the 8 impregnation pump and micronutrients from the 9 micronutrient bin are also incorporated. The mixed product is then dropped into a short 10 horizontal auger that further mixes and force feeds the 11 main vertical auger. It is then distributed by a 12 double paddle, in the 13 distributor head, into twenty separate 14 pockets. The fertilizer mixture then drops into the 15 venturi
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J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
where it is mixed with the air stream. The air is generated by a 16 hydraulic blower and is channeled to an 17 air manifold where the pressure is increased. The air stream is accelerated and the fertilizer is then carried out each 18 stainless steel boom pipe where it hits the 19 stainless steel nozzle deflector and is deflected to the ground in a fan-shaped pattern. (Based on Lor-Al, 1990.) Early versions of the applicator appear to have been sold with either one or six main nutrient bins. The latest redesign uses four bins which extend across the width of the truck. It also halved the dynamic response delay time discussed below. An applicator for liquid fertilizer and pesticides has also been recently released for commercial sale. The applicator utilizes two tanks with one of the following approximate splits: 1300/4500, 1500/5300, or 1900/5700 1 (350/1200, 400/1400, or 500/1500 U.S. gal). The smaller tank system uses what in the U.S. is termed a 3 x 3 centrifugal pump while the larger tank system uses a 4 x 4 or 3 x 5. Both fertilizer pumps are driven by hydraulic motors whose speeds are controlled by flow-control valves. The two systems are totally independent except for a common hydraulic reservoir. The fertilizer flow from each system is kept separate and is sprayed through separate nozzle systems mounted on a common boom structure. At each nozzle spray point there are three nozzles which can be individually switched on and off. If the nozzles are selected in a 1 : 2 : 4 ratio, the flow at a given pressure can be varied to 0, 1, 2, 3, 4, 5, 6 and 7 x the smallest nozzle flowrate. The use of essentially two separate systems adds weight and cost to the applicator design. However, it avoids dynamic interactions between the two systems and establishes familiarity and redundancy to the potential purchaser. The applicators spread according to zones on a digital map stored in an E P R O M or on a floppy disk. The location of the applicator in the field has variously been determined by dead reckoning, trilateralation, or GPS. From the applicator's viewpoint, it does not matter much what source of location is provided. However, the location should be accurate. The Soil Teq system accounts for the dynamics of the applicator with what can be classified as a form of command feedforward control (Bollinger and Duffle, 1988). Based upon the current velocity of the applicator, the position of the applicator corresponding to the applicator dynamics modelled as a pure transportation delay (Ogata, 1970) is predicted and the appropriate mixture and rate of fertilizer is commanded. The 4-s delay on early dry fertilizer applicators has been halved and is the subject of further design investigations. The liquid applicator is said to respond much quicker. Tyler, a competitor of Ag-Chem, has introduced an applicator based upon the organic matter sensor developed by Larry Gaultney (Gaultney et al., 1988; Shonk et al., 1991). The Soil Doctor system marketed by Crop Technology, Inc., varies side-dress nitrogen according to a nitrate sensor based upon technology developed from the work of John Colburn (1991). A system developed by Felton and others is available which sprays herbicides based upon the detection of green spots in fallow land (ASAE, 1992). These systems fit the classification of Automatic Control and since they are not Temporally Separate Control, they do not use any location device, such as GPS. There are also other systems of both classifications in various stages of development and marketing.
J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
73
3. Public sector research
After a slow start, there has been a sizeable amount of research effort to develop and evaluate systems for spatially-variable application of fertilizer and pesticide. These attempts are documented in the literature in such sources as the Proceedings of the Automated Agriculture for the 21st Century Conference (ASAE, 1992), Schueller (1992), and Robert et al. (1993). Particular notice should be paid to the activities at the following U.S. universities: Florida, Idaho State, Illinois, Minnesota, Missouri, Montana State, South Dakota State, and Texas A&M. European centers of activity include Freising-Weihenstephan, Silsoe, Leuven, and Braunschweig. The public sector activities have had a wide range of focuses. 4. GPS fertilization systems
It seems that a GPS fertilizer or pesticide application system needs to be constructed of at least four subsystems (location, setpoint map, controller, actuator). The location subsystem must provide the current location. The setpoint map must provide the application rates as a function of location. The controller must determine the desired application rates and adjust the applicator (actuator) accordingly. By following some path across the field x(t), y(t), where x ( t ) is one location coordinate, y (t) another orthogonal co-ordinate, and t the time, a desired application map r(x, y) results in an actual application, c(x, y). A well-designed applicator should generate an actual application which is close to the desired application. And hopefully the desired application map was the correct map which would produce the MEY. In designing an applicator there are a number of design choices which have to be made. These mainly involve tradeoffs between improved accuracy on one hand and extra sophistication and costs on the other. Some of the decisions involve type of control (open-loop/closed-loop/feedforward), type of applicator (variable rate only/variable rate and mixture), distribution dynamics (shortest path/equi-temporal path), and mixing point (near storage/near nozzle). Generally the latter options in each of these decision areas improve performance at a cost penalty, and maybe unwanted side effects. Open-loop control in which an application rate is just commanded is the simplest. But, as with any open-loop control system, parameter variations and disturbances may cause inaccuracies. A closed-loop control system is probably needed in which metering speed or opening is fed back on dry material applicators and flow (or more likely pressure for known nozzles) is fed back on liquid applicators. A variable rate only applicator does not allow the mixture of nutrients or pesticides to be varied. In some situations, especially in pest control, that is adequate. However, to change mixtures, such as nitrogen/phosphorous/potassium ratios, a more expensive and complex variable rate and mixture applicator is required. Such an applicator requires pumping or metering systems for each mixture component. If the mixture occurs near the storage location, there then is a substantial transportation delay which must be compensated for, probably by
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J.K. Schueller, M.-W Wang/Computersand Electronicsin Agriculture11 (1994) 69-83
feedforward control. If the mixing is near the nozzles, the delay is minimized, but care must be taken to insure uniform mixing. Mixing near storage may cause non-uniform delays (which cannot be compensated for) unless the transportation time to each nozzle is equi-temporal. 5. Applicator accuracy Conventional applicators are designed to provide a uniform application rate, both with respect to time and with respect to nozzle position on the boom. Although controllers have been added to many conventional applicators which vary the application rate based upon applicator ground speed, the design of applicators has not been affected by this development because most of the time the applicators are operated at a nearly constant speed and with a constant desired areal application rate. With spatially-variable applicators, the desired application rate will change often and significantly. Dynamic performance becomes important. There has been an on-going effort to study the dynamics of applicators at the University of Florida. Although only liquid applicators were studied in the experiments and simulations, some of the concepts are applicable with modification to dry material applicators. The dynamics of a liquid applicator appear to depend to a large degree on the total fluid volume from the pump to nozzles and the effective bulk modulus of the fluid-enclosure system in the small applicators which were studied at the University of Florida. Wang (1991) derived a simplified and linearized dynamic model for an electric motor-driven diaphragm dual-pump, dual-tank sprayer system. The models for the flow rate changes of the pumps with reference to the input voltage changes to the first pump are: Ql(s) KlvnKlnq(Cll + KlpeKlvnKlnq) - - KlvnKlnq Zs(s) E1 (s) and K l~n K lnq (C2I + K2pe K2vn K2nq)
O2(s) E1 (s)
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where
v
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Zs = -~s + Cll + C2t + 24--'--~m + K lpeK lvnKlnq + K2peK2vnK2nq and in which s = the Laplace operator, E1 = voltage input to pump motor 1, Q1 = flowrate out of pump 1, Qe = flowrate out of pump 2, and other terms are system and component constants. Studies by Xu (1991) found slightly more complicated, but related, results for a hydraulically-driven centrifugal pump of the type used on larger applicators. The models have parameters and coefficients which are difficult to obtain in practice. However, when step changes in desired application rate are commanded, the response of the system can reasonably be approximated as a first order system with a transportation delay. That is, the transfer function can be represented as:
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83 C(s)
e -sD
R(s)
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75
where C ( s ) = actual applicator output, R ( s ) = commanded applicator output, e = natural antilogarithm, D = transportation delay, and r = time constant. This simple model probably models the system well enough considering the other sources of error and inaccuracies. The transportation delay is determined by the volume of fluid from the point of interest to the nozzles divided by the flowrate. For example, if a pesticide is being mixed at the nozzle there will be a very short transportation lag. However, if the mixing occurs far from the nozzle, there will be a substantial delay until the pesticide is applied. Transportation lag is therefore affected by the location of the point of interest (usually the mixing point) in the fluid circuit, the flowrate of material, and the length and area of the hoses from the point of interest to the nozzles. Branches of the fluid circuit which are not equi-temporal (having the same volume/flowrate ratio) will have different delays, making accurate feedforward compensation impossible. The problems with transportation delay is one of the reasons the Soil Teq liquid applicator has two separate liquid circuits. Modern computer hardware and software are fast enough compared to applicator dynamics that it is unlikely that they will significantly contribute to transportation delays. The time constant of the system represents the inability of the system to immediately respond to a commanded change. The types of systems tested by Wang (1991) did not seem to have their time constants determined by the response of the controller, amplifier, drive motor, or pump. Instead, the time constant appeared to be primarily caused by the effective bulk modulus effects. The desired flow was soon put into the hose system, but a time constant of about 0.5-1.0 s was seen for the nozzle flow. Higher flowrates or stiffer hoses would reduce the time constant. Design guidelines based upon these studies have been published (Schueller, 1991). 6. Effects of accuracy
The dynamics of the applicator can affect the applicator's performance in applying the correct amount of fertilizer or pesticide in the correct locations. A study was undertaken to determine the effects of applicator dynamics and command feedforward compensation. A liquid applicator was modeled in which the transportation delay was set at 5 s and the time constant was selected to be 1 s. The time constant was selected from the above studies of Wang (1991) and the example transportation delay was hypothesized based upon the geometries of a large conventional applicator and some assumed operating conditions. This model applicator was then used to study the effects of applicator dynamics in example field conditions through simulation. The procedure for this study was to: - generate a map of desired application rate in the field, - select a path for the applicator to follow in the field, - determine the time history of desired rate while following the path,
76
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
find the applicator's output to the desired rate time history, and study the difference between the desired output and the simulated output. In this way the effect of applicator dynamics was simulated. The simulations were conducted in C initially. Later, it was found easier to complete the simulations in a Quattro Pro spreadsheet where easy-to-use tools are available for graphical presentation and summary calculations of the results. -
-
7. Test
case
The map of desired P205 fertilization shown in Fig. 2 was used as the basis for primary test case. The map was generated by Karl Wild of the Technical University of Munich-Weihenstephan with our assistance. It is based on soil type, previous crop yield, and soil test P205 data obtained from the Institut ffir Pflanzenbau and the Institut ffir Landtechnik (results from FAM special research project Scheyern). The desired fertilization was determined based upon the 1985 Bavarian publication Die Dfingung von Acker und Grfinland nach Ergebnissen der Bodenuntersuchung (roughly The Fertilizing of Crop Fields and Pasture According to the Results of Soil Survey). The field was broken up into 40 m x 40 m cells with the desired P205 application rates, shown in each cell in Fig. 2. The applicator path chosen for the test case is shown in Fig. 3. If that path is followed at a speed of 5 m/s (18 km/h), the first 200 s of desired application rate time history are as shown in Fig. 4. It can be seen that the desired P205 application rate varies quite widely and rather rapidly. An applicator with a 5-s transportation delay and a 1-s time constant will response to the desired time rate history as shown in Fig. 5. It can be seen that the response significantly lags the desired application rate and causes poor performance. One method of improving the performance is to use command feedforward control. One such feedforward technique is to command the rate corresponding to
90
Fig. 2. Desired P205 fertilization rates (kg/ha).
J.K. Schueller, M.-W. Wang / Computers and Electronics in Agriculture 11 (1994) 69-83
77
Fig. 3. Applicator path in field.
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applicator location some time in the future. A 5-s precommand would eliminate the transportation delay. The time constant cannot be eliminated by feedforward control. However, its effects can be lessened. Both the average error and the root-mean-square error in application rate would be minimized by a feedforward c o m m a n d c o m p o n e n t equal to 69.3% of the time constant. Therefore, the overall feedforward precommand was chosen to be 5.7 s.
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
78
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If a 5.7 precommand is issued, the response of the applicator model is as shown in Fig. 6. It can be seen that the performance is much improved and that the errors in application are reduced. Table 1 presents a comparison of the different application methods. The three columns represent conventional constant rate application, spatially-variable application without precommand, and spatiallyvariable application with 5.7 s precommand. All methods approximately achieve the desired application rate of 56 kg/ha. However, the dynamic errors in application vary widely. Two measures of dynamic error are included in Table 1: the mean absolute error and the room-mean-square error. It can be seen in this case that spatially-variable control with precommand (the last column) significantly reduces
J.K. Schueller, M.- W. Wang / Computers and Electronics in Agriculture 1! (I 994) 69-83
79
Table 1 Comparison of different application methods (kg/ha) Constant rate
No precommand
Precommand(5.7 s)
Desired average application rate Applied average application rate
56 56
56 55
56 55.9
Mean absolute application error Root-mean-square application error
25.3 35.4
24.7 41
2.9 7.7
the errors in application. Without the precommand, the use of spatially-variable control does not appreciably reduce application errors. It must be remembered that the above example is just that - - an example. If the application rate changed in a slower manner, the spatially-variable control without precommand would perform better, as it also would if the applicator traveled at a lower speed or had a lower transportation delay or time constant. The advantages of precommand increase with: increasing applicator speed, increasing transportation delays and time constants, and increasing rate of desired application rate change. It must be remembered that there were two assumptions which improved the performance of the precommand. The first was that the path of the applicator was known. The second was that the speed of the applicator remained constant. Violation of these assumptions would degrade the performance of the precommand spatially-variable control. However, this example still illustrates how precommand is necessary to achieve good performance in certain situations. Other fields will have different application rate time histories. Figs. 7 and 8 illustrate other fields in which the desired application rate varies. Care should be taken in using these figures because the desired application rates are not based upon proper algorithms, but rather just estimates based upon yield maps. Their corresponding desired application rate time histories, shown in Figs. 9 and 10, are therefore not absolutely correct, but do present illustrations of relative variabilities. -
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J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
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8. Error source B e s i d e s the applicator dynamics, there are other potential error sources in spatially-variable fertilizer and pesticide application. T h e s e can be classified into: - locator errors, - m a p errors, and - applicator transverse errors. T h e locator errors are the errors due to inaccurate position determination. T h e s e s errors are discussed in articles on GPS and other locator technologies. Based on the n e e d to keep these errors within bounds, it is anticipated that m o s t GPS units o n fertilizer and pesticide applicators will use D G P S .
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
81
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The map errors include all the errors that result from the generation of the desired application rate map. These include the errors in determining the sensed qualities, the errors in calculating the desired application rate, and the errors in storing the desired rate in the map. Portions of these errors result from quantization, interpolation, and aggregation. In order to achieve large field capabilities, applicators usually cover a wide swath. This swath was set to 20 m in the test case above. These need to be a uniform application rate (with uniform dynamics) across the width of the swath. However, since it will generally take longer for any changes to reach the nozzles at the end of the boom than the nozzles at the center, uneven transportation delays can result. In addition, greatly decreasing hose size to reduce transportation delays may cause excessive pressure drops in the hoses which would result in uneven flows through the nozzles. The various errors need to be combined to find the integrated system errors. If expressions are developed for errors and reasonably accurate typical numerical values obtained, it will be possible to determine where tradeoffs should be made between accuracy versus complexity and cost. This is an area of needed research. One of the biggest factors affecting the errors and perceptions of errors is the cell size chosen. The Soil Teq system establishes boundaries of irregularly-shaped areas. However, many of the other investigations use square or rectangular cells. The selection of cell size is an important activity. Han et al. (1992) have proposed a mean correlation distance (MCD) as a criterion to determine the upper limit of cell size. This is based upon the variation in the soil and crop properties. They suggest that the upper limit of cell size be the minimum MCD of all spatial variables. For example, if the MCD of phosphorous is 40 m and the MCD of nitrogen is 36 m, then the upper limit of the cell size would be 36 m 2. Based upon these upper limits and the computer capabilities to handle a large number of cells, there may be a tendency to reduce cell sizes. However, the cells
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J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
should not be m a d e too small. Applicator dynamics, location accuracy, and swath width all provide lower bounds on cell size. Current research at the University of Florida is attempting to remove the swath width constraint by using pinch valves. These valves are currently pneumatically actuated to turn individual nozzles on and off. This research will investigate the feasibility of controlling pressure (and hence flow) at individual nozzles by accurately controlling the actuation pressure of the pinch valve.
9. Summary and conclusions This article discussed some of the issues involved in spatially-variable control of fertilizer and pesticide application. Some general concepts can be stated: (1) Commercial spatially-variable applicators are available which control the application of fertilizers and pesticides. They may use GPS or other location technologies. (2) T h e r e is active ongoing research in spatially-variable crop production in public sector institutions in North America and Western Europe. (3) T h e dynamics of the spatially-variable applicator affect its performance. (4) C o m m a n d feedforward control (also known as p r e c o m m a n d ) improves the p e r f o r m a n c e of spatially-variable applicators. (5) The various error sources can degrade the performance of spatially-variable applicators. These effects, especially in combination, should be further studied. (6) Besides upper bounds on cell size due to the agronomics, there are also lower bounds due to engineering concerns.
Acknowledgements Portions of this work were financially supported by the Fluid Fertilizer Foundation. Mr. Karl Wild helped with the test case and Mr. Liangji Xu with the applicator modeling.
References ASAE (1992) Proceedings of the Conference on Automated Agriculture for the 21st Century. American Society of Agricultural Engineers, St. Joseph, Mich., ASAE-PubI., 11-91, 540 pp. Bollinger, J.G. and Duffle, N.A. (1988) Computer Control of Machines and Processes. Addison-Wesley, Reading, Mass., 613 pp. Colburn, J. (1991) Soil chemical sensor and precision agricultural chemical delivery system and method. U.S. Patent 5,033,397. Gaultney, L.D., Schueller, J.K., Shonk, J.L. and Yu, Z. (1988) Automatic soil organic matter mapping. A S A E Paper 88-1607. Han, S., Hummel, J.W., Goering, C.E. and Chan, C.D. (1992) Selection of cell size for site-specific crop management. ASAE Paper 92-7008. Lor-Al (1990) Lor-Al Soilection: Farming by the Foot. Advertising brochure, Lor-Al, Benson, Minn. Ogata, K. (1970) Modern Control Engineering. Prentice-Hall, Englewood Cliffs, N.J., 836 pp. Ortlip, E.W. (1986) Method and apparatus for spreading fertilizer. U.S. Patent 4,630,774. Robert, P.C., Rust, R.H. and Larson, W.E. (1993) Proceedings of the Soil Specific Crop Management:
J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
83
Workshop on Research and Development Issues, 14-16 April 1992, ASA/CSSA/SSSA, Minneapolis, Minn., 395 pp. Schueller, J.K. (1991) Design for dynamic response of sprayer applicators. J. Fert. Issues, 8(3): 69-73. Schueller, J.K. (1992) A review and integrating analysis of spatially-variable control of crop production. Fert. Res., 33: 1-34. Shonk, J.L., Gaultney, L.D., Schulze, D.G. and Van Scoyoc, G.E. (1991) Spectroscopic sensing of soil organic matter. Trans. ASAE, 34: 1978-1984. Wang, M.W. (1991) Dynamic model of a multi-pump sprayer applicator. Master thesis, University of Florida Mechanical Engineering, Gainesville, Fla. Xu, U (1991) Dynamic and static analysis of a servovalve controlled hydraulic motor driven centrifugal pump. Master thesis, University of Florida Mechanical Engineering, Gainesville, Fla.
Computers and Electronics in Agriculture 11 (1994) 69-83
Spatially-variable fertilizer and pesticide application with GPS and DGPS J o h n K. S c h u e l l e r * , M i n - W e n W a n g Department of Mechanical Engineering, University of Florida, Gainesville, FL 32611, USA
(Accepted 3 May 1994)
Abstract
Concepts of spatially-variable fertilizer and pesticide application are discussed. Commercial applicators are described which use Global Positioning Systems (GPS) or other location technologies to apply fertilizer and pesticide according to predetermined setpoint maps. The dynamics of the applicator are shown to have important performance effects. Command feedforward control can significantly improve the performance. Further research is needed on understanding the effects of the various error sources. Key words: Spatially-variable; Global Positioning System; Fertilizer; Pesticide; Applicator
1. I n t r o d u c t i o n
The basics of Global Positioning System (GPS), Differential GPS (DGPS) and G L O N A S S (all henceforth to be generically referred to GPS) are explained elsewhere in this Special Issue and do not need to be repeated here. Other articles explain how GPS can be used with sensors which monitor crop and soil properties to generate maps of these soil and crop properties. The generation of these maps will be very beneficial to the farm operator in that he can then know what sort of variabilities he has in his fields. But not only will he know the variabilities, he will be able to manage his fields spatially. T h e r e are several ways in which the farm manager can manage spatially. Not all of them require advanced technology. One way is for the fields to be reorganized according to the soil properties. Contour or strip farming is an example of this. However, there are problems with this in practice due to the constraints on field size and shape for efficient field operations, as well as the difficulty in assigning just a single set of field boundaries for all of the field variabilities. Another way is * Corresponding author. 0168-1699/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0168-1699(94)00016-J
70
J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
for the farmer to manually operate his equipment in such a way that it responds properly to the spatial variabilities. However, in this case it is difficult for a human to successfully perform such control with reasonable accuracy. Operators without direct financial stake in the performance may also lack the motivation to perform the frequent, accurate control. Given the problems with the above two methods, it appears that automation may be a better approach. Field operations can be controlled automatically according to two major approaches: Automatic Control or Temporally Separate Control (Schueller, 1992). In Automatic Control, the piece of farm machinery performing the operation will have some sort of sensor and will control the operation based upon the sensor's output. In Temporally Separate Control, the control is decoupled from the sensing. The mapped data on spatial variability are gathered by any means, such as the yield and soil mapping discussed in other articles, and are used to generate a control setpoint map. This map is then used to guide the control operation. The Temporally Separate methodology relaxes the requirements on the sensor and permits more sophisticated management, but requires the development and use of the setpoint map and some method of locating the field equipment's position on the map. The location methods most often discussed are dead reckoning, electromagnetic (radio or microwave) trilateralation, or GPS. GPS can be used to determine position during such operations as landforming, terracing, and tillage. It could also be used to help determine the appropriate variety, population, or depth during planting operations. However, discussions with many agronomists and engineers indicate that they feel the greatest potential for spatially-variable control and the use of GPS lies in fertilizer and pesticide application. Pesticides by definition are only useful where there are pests. Fertilizers are only valuable where there are needs for the nutrients. At other places, pesticides and fertilizers only have the potential to create environmental problems. In North America there is a tendency to apply fertilizers and pesticides with the same equipment, in fact often at the same time. The emphasis in this article will be on fertilizer application, but most of the concepts and analyses are applicable also to pesticide application. The other articles show that soils and crops are spatially-variable. Soils have different nutrient levels, mobilities, and fixation abilities at different points within a field. Crops remove different amounts of nutrients and require deferent amounts of nutrients in a spatially-variable manner. Fertilizer is usually required to obtain the maximum economic yields (MEY). Generally, the yields of a crop increase with increasing fertilization, although the rate of increase tends to decrease. Eventually a point is reached where the yield increases do not justify the extra fertilizer costs. This is the MEY fertilization rate. This point will vary within fields, and is therefore the justification for spatially-variable fertilizer application. In addition, excess nutrients can cause quality problems in some crops. An example of these problems is nitrogen in sugar beets and potatoes. The excess nutrients will also be less likely to be utilized or immobilized and, therefore, may cause environmental problems. An example of the environmental concern is that the U.S. Environmental Protection Agency is sup-
71
J.K. Schueller, M.-W. Wang~Computers and Electronics in Agriculture 11 (1994) 69-83
porting work to examine the benefits of spatially-variable fertilization on nitrogen pollution. 2. Commercial applicators
The potential of spatially-variable fertilizer and pesticide application has led to a number of attempts to commercialize the technology. The largest impact has been made based upon the U.S. patent of Ortlip (1986). This patent has broad claims which (assuming they are legally defendable and are defended) clearly patent application according to digital maps. The system patent is owned by Soil Teq, Inc., whose controlling interest (60%) is owned by Ag-Chem, Inc., also now the current owner of Lor-A1, the original licensee of the technology who manufactured spatially-variable applicators. The technology was first released as part of a dry fertilizer applicator. It has gone through a number of modifications and redesigns to improve performance and reliability. The latest redesign is not available, but an earlier version is shown in Fig. 1. The operating procedure is as follows. Fertilizer, chemical and micronutrient rates per acre per soil type are programmed into the 1 microprocessor. Sensing its location on the 2 field map the computer sends a signal to the 3 electro-hydraulic valves which control the rate at which product is dispensed from the 4 product bins by the 5 metering wheels. The fertilizer then drops onto a 6 stainless steel conveyor chain and is carried back to the 7 mixing auger where the chemicals (usually pesticides) from the 8 impregnation pump and micronutrients from the 9 micronutrient bin are also incorporated. The mixed product is then dropped into a short 10 horizontal auger that further mixes and force feeds the 11 main vertical auger. It is then distributed by a 12 double paddle, in the 13 distributor head, into twenty separate 14 pockets. The fertilizer mixture then drops into the 15 venturi
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72
J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
where it is mixed with the air stream. The air is generated by a 16 hydraulic blower and is channeled to an 17 air manifold where the pressure is increased. The air stream is accelerated and the fertilizer is then carried out each 18 stainless steel boom pipe where it hits the 19 stainless steel nozzle deflector and is deflected to the ground in a fan-shaped pattern. (Based on Lor-Al, 1990.) Early versions of the applicator appear to have been sold with either one or six main nutrient bins. The latest redesign uses four bins which extend across the width of the truck. It also halved the dynamic response delay time discussed below. An applicator for liquid fertilizer and pesticides has also been recently released for commercial sale. The applicator utilizes two tanks with one of the following approximate splits: 1300/4500, 1500/5300, or 1900/5700 1 (350/1200, 400/1400, or 500/1500 U.S. gal). The smaller tank system uses what in the U.S. is termed a 3 x 3 centrifugal pump while the larger tank system uses a 4 x 4 or 3 x 5. Both fertilizer pumps are driven by hydraulic motors whose speeds are controlled by flow-control valves. The two systems are totally independent except for a common hydraulic reservoir. The fertilizer flow from each system is kept separate and is sprayed through separate nozzle systems mounted on a common boom structure. At each nozzle spray point there are three nozzles which can be individually switched on and off. If the nozzles are selected in a 1 : 2 : 4 ratio, the flow at a given pressure can be varied to 0, 1, 2, 3, 4, 5, 6 and 7 x the smallest nozzle flowrate. The use of essentially two separate systems adds weight and cost to the applicator design. However, it avoids dynamic interactions between the two systems and establishes familiarity and redundancy to the potential purchaser. The applicators spread according to zones on a digital map stored in an E P R O M or on a floppy disk. The location of the applicator in the field has variously been determined by dead reckoning, trilateralation, or GPS. From the applicator's viewpoint, it does not matter much what source of location is provided. However, the location should be accurate. The Soil Teq system accounts for the dynamics of the applicator with what can be classified as a form of command feedforward control (Bollinger and Duffle, 1988). Based upon the current velocity of the applicator, the position of the applicator corresponding to the applicator dynamics modelled as a pure transportation delay (Ogata, 1970) is predicted and the appropriate mixture and rate of fertilizer is commanded. The 4-s delay on early dry fertilizer applicators has been halved and is the subject of further design investigations. The liquid applicator is said to respond much quicker. Tyler, a competitor of Ag-Chem, has introduced an applicator based upon the organic matter sensor developed by Larry Gaultney (Gaultney et al., 1988; Shonk et al., 1991). The Soil Doctor system marketed by Crop Technology, Inc., varies side-dress nitrogen according to a nitrate sensor based upon technology developed from the work of John Colburn (1991). A system developed by Felton and others is available which sprays herbicides based upon the detection of green spots in fallow land (ASAE, 1992). These systems fit the classification of Automatic Control and since they are not Temporally Separate Control, they do not use any location device, such as GPS. There are also other systems of both classifications in various stages of development and marketing.
J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
73
3. Public sector research
After a slow start, there has been a sizeable amount of research effort to develop and evaluate systems for spatially-variable application of fertilizer and pesticide. These attempts are documented in the literature in such sources as the Proceedings of the Automated Agriculture for the 21st Century Conference (ASAE, 1992), Schueller (1992), and Robert et al. (1993). Particular notice should be paid to the activities at the following U.S. universities: Florida, Idaho State, Illinois, Minnesota, Missouri, Montana State, South Dakota State, and Texas A&M. European centers of activity include Freising-Weihenstephan, Silsoe, Leuven, and Braunschweig. The public sector activities have had a wide range of focuses. 4. GPS fertilization systems
It seems that a GPS fertilizer or pesticide application system needs to be constructed of at least four subsystems (location, setpoint map, controller, actuator). The location subsystem must provide the current location. The setpoint map must provide the application rates as a function of location. The controller must determine the desired application rates and adjust the applicator (actuator) accordingly. By following some path across the field x(t), y(t), where x ( t ) is one location coordinate, y (t) another orthogonal co-ordinate, and t the time, a desired application map r(x, y) results in an actual application, c(x, y). A well-designed applicator should generate an actual application which is close to the desired application. And hopefully the desired application map was the correct map which would produce the MEY. In designing an applicator there are a number of design choices which have to be made. These mainly involve tradeoffs between improved accuracy on one hand and extra sophistication and costs on the other. Some of the decisions involve type of control (open-loop/closed-loop/feedforward), type of applicator (variable rate only/variable rate and mixture), distribution dynamics (shortest path/equi-temporal path), and mixing point (near storage/near nozzle). Generally the latter options in each of these decision areas improve performance at a cost penalty, and maybe unwanted side effects. Open-loop control in which an application rate is just commanded is the simplest. But, as with any open-loop control system, parameter variations and disturbances may cause inaccuracies. A closed-loop control system is probably needed in which metering speed or opening is fed back on dry material applicators and flow (or more likely pressure for known nozzles) is fed back on liquid applicators. A variable rate only applicator does not allow the mixture of nutrients or pesticides to be varied. In some situations, especially in pest control, that is adequate. However, to change mixtures, such as nitrogen/phosphorous/potassium ratios, a more expensive and complex variable rate and mixture applicator is required. Such an applicator requires pumping or metering systems for each mixture component. If the mixture occurs near the storage location, there then is a substantial transportation delay which must be compensated for, probably by
74
J.K. Schueller, M.-W Wang/Computersand Electronicsin Agriculture11 (1994) 69-83
feedforward control. If the mixing is near the nozzles, the delay is minimized, but care must be taken to insure uniform mixing. Mixing near storage may cause non-uniform delays (which cannot be compensated for) unless the transportation time to each nozzle is equi-temporal. 5. Applicator accuracy Conventional applicators are designed to provide a uniform application rate, both with respect to time and with respect to nozzle position on the boom. Although controllers have been added to many conventional applicators which vary the application rate based upon applicator ground speed, the design of applicators has not been affected by this development because most of the time the applicators are operated at a nearly constant speed and with a constant desired areal application rate. With spatially-variable applicators, the desired application rate will change often and significantly. Dynamic performance becomes important. There has been an on-going effort to study the dynamics of applicators at the University of Florida. Although only liquid applicators were studied in the experiments and simulations, some of the concepts are applicable with modification to dry material applicators. The dynamics of a liquid applicator appear to depend to a large degree on the total fluid volume from the pump to nozzles and the effective bulk modulus of the fluid-enclosure system in the small applicators which were studied at the University of Florida. Wang (1991) derived a simplified and linearized dynamic model for an electric motor-driven diaphragm dual-pump, dual-tank sprayer system. The models for the flow rate changes of the pumps with reference to the input voltage changes to the first pump are: Ql(s) KlvnKlnq(Cll + KlpeKlvnKlnq) - - KlvnKlnq Zs(s) E1 (s) and K l~n K lnq (C2I + K2pe K2vn K2nq)
O2(s) E1 (s)
Z~(s)
where
v
cd
Zs = -~s + Cll + C2t + 24--'--~m + K lpeK lvnKlnq + K2peK2vnK2nq and in which s = the Laplace operator, E1 = voltage input to pump motor 1, Q1 = flowrate out of pump 1, Qe = flowrate out of pump 2, and other terms are system and component constants. Studies by Xu (1991) found slightly more complicated, but related, results for a hydraulically-driven centrifugal pump of the type used on larger applicators. The models have parameters and coefficients which are difficult to obtain in practice. However, when step changes in desired application rate are commanded, the response of the system can reasonably be approximated as a first order system with a transportation delay. That is, the transfer function can be represented as:
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83 C(s)
e -sD
R(s)
rs + 1
75
where C ( s ) = actual applicator output, R ( s ) = commanded applicator output, e = natural antilogarithm, D = transportation delay, and r = time constant. This simple model probably models the system well enough considering the other sources of error and inaccuracies. The transportation delay is determined by the volume of fluid from the point of interest to the nozzles divided by the flowrate. For example, if a pesticide is being mixed at the nozzle there will be a very short transportation lag. However, if the mixing occurs far from the nozzle, there will be a substantial delay until the pesticide is applied. Transportation lag is therefore affected by the location of the point of interest (usually the mixing point) in the fluid circuit, the flowrate of material, and the length and area of the hoses from the point of interest to the nozzles. Branches of the fluid circuit which are not equi-temporal (having the same volume/flowrate ratio) will have different delays, making accurate feedforward compensation impossible. The problems with transportation delay is one of the reasons the Soil Teq liquid applicator has two separate liquid circuits. Modern computer hardware and software are fast enough compared to applicator dynamics that it is unlikely that they will significantly contribute to transportation delays. The time constant of the system represents the inability of the system to immediately respond to a commanded change. The types of systems tested by Wang (1991) did not seem to have their time constants determined by the response of the controller, amplifier, drive motor, or pump. Instead, the time constant appeared to be primarily caused by the effective bulk modulus effects. The desired flow was soon put into the hose system, but a time constant of about 0.5-1.0 s was seen for the nozzle flow. Higher flowrates or stiffer hoses would reduce the time constant. Design guidelines based upon these studies have been published (Schueller, 1991). 6. Effects of accuracy
The dynamics of the applicator can affect the applicator's performance in applying the correct amount of fertilizer or pesticide in the correct locations. A study was undertaken to determine the effects of applicator dynamics and command feedforward compensation. A liquid applicator was modeled in which the transportation delay was set at 5 s and the time constant was selected to be 1 s. The time constant was selected from the above studies of Wang (1991) and the example transportation delay was hypothesized based upon the geometries of a large conventional applicator and some assumed operating conditions. This model applicator was then used to study the effects of applicator dynamics in example field conditions through simulation. The procedure for this study was to: - generate a map of desired application rate in the field, - select a path for the applicator to follow in the field, - determine the time history of desired rate while following the path,
76
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
find the applicator's output to the desired rate time history, and study the difference between the desired output and the simulated output. In this way the effect of applicator dynamics was simulated. The simulations were conducted in C initially. Later, it was found easier to complete the simulations in a Quattro Pro spreadsheet where easy-to-use tools are available for graphical presentation and summary calculations of the results. -
-
7. Test
case
The map of desired P205 fertilization shown in Fig. 2 was used as the basis for primary test case. The map was generated by Karl Wild of the Technical University of Munich-Weihenstephan with our assistance. It is based on soil type, previous crop yield, and soil test P205 data obtained from the Institut ffir Pflanzenbau and the Institut ffir Landtechnik (results from FAM special research project Scheyern). The desired fertilization was determined based upon the 1985 Bavarian publication Die Dfingung von Acker und Grfinland nach Ergebnissen der Bodenuntersuchung (roughly The Fertilizing of Crop Fields and Pasture According to the Results of Soil Survey). The field was broken up into 40 m x 40 m cells with the desired P205 application rates, shown in each cell in Fig. 2. The applicator path chosen for the test case is shown in Fig. 3. If that path is followed at a speed of 5 m/s (18 km/h), the first 200 s of desired application rate time history are as shown in Fig. 4. It can be seen that the desired P205 application rate varies quite widely and rather rapidly. An applicator with a 5-s transportation delay and a 1-s time constant will response to the desired time rate history as shown in Fig. 5. It can be seen that the response significantly lags the desired application rate and causes poor performance. One method of improving the performance is to use command feedforward control. One such feedforward technique is to command the rate corresponding to
90
Fig. 2. Desired P205 fertilization rates (kg/ha).
J.K. Schueller, M.-W. Wang / Computers and Electronics in Agriculture 11 (1994) 69-83
77
Fig. 3. Applicator path in field.
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applicator location some time in the future. A 5-s precommand would eliminate the transportation delay. The time constant cannot be eliminated by feedforward control. However, its effects can be lessened. Both the average error and the root-mean-square error in application rate would be minimized by a feedforward c o m m a n d c o m p o n e n t equal to 69.3% of the time constant. Therefore, the overall feedforward precommand was chosen to be 5.7 s.
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
78
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If a 5.7 precommand is issued, the response of the applicator model is as shown in Fig. 6. It can be seen that the performance is much improved and that the errors in application are reduced. Table 1 presents a comparison of the different application methods. The three columns represent conventional constant rate application, spatially-variable application without precommand, and spatiallyvariable application with 5.7 s precommand. All methods approximately achieve the desired application rate of 56 kg/ha. However, the dynamic errors in application vary widely. Two measures of dynamic error are included in Table 1: the mean absolute error and the room-mean-square error. It can be seen in this case that spatially-variable control with precommand (the last column) significantly reduces
J.K. Schueller, M.- W. Wang / Computers and Electronics in Agriculture 1! (I 994) 69-83
79
Table 1 Comparison of different application methods (kg/ha) Constant rate
No precommand
Precommand(5.7 s)
Desired average application rate Applied average application rate
56 56
56 55
56 55.9
Mean absolute application error Root-mean-square application error
25.3 35.4
24.7 41
2.9 7.7
the errors in application. Without the precommand, the use of spatially-variable control does not appreciably reduce application errors. It must be remembered that the above example is just that - - an example. If the application rate changed in a slower manner, the spatially-variable control without precommand would perform better, as it also would if the applicator traveled at a lower speed or had a lower transportation delay or time constant. The advantages of precommand increase with: increasing applicator speed, increasing transportation delays and time constants, and increasing rate of desired application rate change. It must be remembered that there were two assumptions which improved the performance of the precommand. The first was that the path of the applicator was known. The second was that the speed of the applicator remained constant. Violation of these assumptions would degrade the performance of the precommand spatially-variable control. However, this example still illustrates how precommand is necessary to achieve good performance in certain situations. Other fields will have different application rate time histories. Figs. 7 and 8 illustrate other fields in which the desired application rate varies. Care should be taken in using these figures because the desired application rates are not based upon proper algorithms, but rather just estimates based upon yield maps. Their corresponding desired application rate time histories, shown in Figs. 9 and 10, are therefore not absolutely correct, but do present illustrations of relative variabilities. -
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J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
p---1
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8. Error source B e s i d e s the applicator dynamics, there are other potential error sources in spatially-variable fertilizer and pesticide application. T h e s e can be classified into: - locator errors, - m a p errors, and - applicator transverse errors. T h e locator errors are the errors due to inaccurate position determination. T h e s e s errors are discussed in articles on GPS and other locator technologies. Based on the n e e d to keep these errors within bounds, it is anticipated that m o s t GPS units o n fertilizer and pesticide applicators will use D G P S .
J.K. Schueller, M.-W. Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
81
~00 3.80 1GO
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The map errors include all the errors that result from the generation of the desired application rate map. These include the errors in determining the sensed qualities, the errors in calculating the desired application rate, and the errors in storing the desired rate in the map. Portions of these errors result from quantization, interpolation, and aggregation. In order to achieve large field capabilities, applicators usually cover a wide swath. This swath was set to 20 m in the test case above. These need to be a uniform application rate (with uniform dynamics) across the width of the swath. However, since it will generally take longer for any changes to reach the nozzles at the end of the boom than the nozzles at the center, uneven transportation delays can result. In addition, greatly decreasing hose size to reduce transportation delays may cause excessive pressure drops in the hoses which would result in uneven flows through the nozzles. The various errors need to be combined to find the integrated system errors. If expressions are developed for errors and reasonably accurate typical numerical values obtained, it will be possible to determine where tradeoffs should be made between accuracy versus complexity and cost. This is an area of needed research. One of the biggest factors affecting the errors and perceptions of errors is the cell size chosen. The Soil Teq system establishes boundaries of irregularly-shaped areas. However, many of the other investigations use square or rectangular cells. The selection of cell size is an important activity. Han et al. (1992) have proposed a mean correlation distance (MCD) as a criterion to determine the upper limit of cell size. This is based upon the variation in the soil and crop properties. They suggest that the upper limit of cell size be the minimum MCD of all spatial variables. For example, if the MCD of phosphorous is 40 m and the MCD of nitrogen is 36 m, then the upper limit of the cell size would be 36 m 2. Based upon these upper limits and the computer capabilities to handle a large number of cells, there may be a tendency to reduce cell sizes. However, the cells
82
J.K. Schueller, M.-W Wang/Computers and Electronics in Agriculture 11 (1994) 69-83
should not be m a d e too small. Applicator dynamics, location accuracy, and swath width all provide lower bounds on cell size. Current research at the University of Florida is attempting to remove the swath width constraint by using pinch valves. These valves are currently pneumatically actuated to turn individual nozzles on and off. This research will investigate the feasibility of controlling pressure (and hence flow) at individual nozzles by accurately controlling the actuation pressure of the pinch valve.
9. Summary and conclusions This article discussed some of the issues involved in spatially-variable control of fertilizer and pesticide application. Some general concepts can be stated: (1) Commercial spatially-variable applicators are available which control the application of fertilizers and pesticides. They may use GPS or other location technologies. (2) T h e r e is active ongoing research in spatially-variable crop production in public sector institutions in North America and Western Europe. (3) T h e dynamics of the spatially-variable applicator affect its performance. (4) C o m m a n d feedforward control (also known as p r e c o m m a n d ) improves the p e r f o r m a n c e of spatially-variable applicators. (5) The various error sources can degrade the performance of spatially-variable applicators. These effects, especially in combination, should be further studied. (6) Besides upper bounds on cell size due to the agronomics, there are also lower bounds due to engineering concerns.
Acknowledgements Portions of this work were financially supported by the Fluid Fertilizer Foundation. Mr. Karl Wild helped with the test case and Mr. Liangji Xu with the applicator modeling.
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