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The geometric characterization of mouldboard plough surfaces by using splines
Soil & Tillage Research 112 (2011) 98–105
Contents lists available at ScienceDirect
Soil & Tillage Research journal homepage: www.elsevier.com/locate/still
Short communication
The geometric characterization of mouldboard plough surfaces by using splines ˜ oz-Piorno a, J.V. Gira´ldez b E. Gutie´rrez de Rave´ a,*, F.J. Jime´nez-Hornero a, J.M. Mun a b
Dept. of Engineering Graphics and Geomatics, University of Co´rdoba, Gregor Mendel Building, ctra. Madrid km 396, 14071 Co´rdoba, Spain Dept. of Agronomy, University of Co´rdoba and IAS CSIC, Leonardo da Vinci Building, ctra. Madrid km 396, 14071 Co´rdoba, Spain
A R T I C L E I N F O
A B S T R A C T
Article history: Received 23 July 2010 Received in revised form 24 November 2010 Accepted 26 November 2010 Available online 30 December 2010
For historic reasons, mouldboard ploughs have a wide variety of surfaces. The concentration of manufacturers has reduced the number of different types but there are still few complete geometric descriptions of the surfaces of a mouldboard plough. With the aim of describing and characterizing the three-dimensional shape of the plough, permitting the obtaining of a tool for an adequate preparation of the soil for cultivation, as well as reducing labour costs, in this work different approximation methods of three-dimensional surfaces have been explored. The results obtained allow us to conclude that the application of non uniform rational B-splines, combined with an optimization algorithm based on assigning different weights to the control points of the surface analyzed, is a method which succeeds in improving the quality of the fit with respect to other algorithms proposed by several authors in different previous works also based on the use of splines. The root mean square error of the fit of these algorithms to a mouldboard plough was the lowest 1.9 mm in the observed, and increases up to 14.4 mm using a simple Be´zier bicubic parametric surface. Using the efficiency index of Nash and Sutclife (1970), the highest value, 0.995, corresponded to the non uniform rational B-splines method. ß 2010 Elsevier B.V. All rights reserved.
Keywords: Mouldboard design Be´zier surface Computer graphic NURBS
1. Introduction The characterization of agricultural implements is currently a subject of great importance due to the revision of traditional cultivation methods for the new conservation tillage and precision agriculture techniques. The great variety of environmental, edaphic, agronomic, and even social conditions has given rise to the development of numerous types of agricultural implements, which require a simple characterization to permit tillage power optimization (Bentaher et al., 2008). The new types of conservation and precision agriculture methods need new tools which have to be characterized, since ploughs of very different characteristics are often found and it would be necessary to unify them (Horvat et al., 2008). The use of geometric algorithms describing the surfaces in a simple way will permit farmers and machinery manufacturers to carry out effective quality controls, taking advantage of the development of approximation methods in the motor car industry, especially since the proposal of Pierre Be´zier of the curve bearing his name in the 1950s (e.g. Bloomenthal, 1998; Rogers, 2001; Buss, 2003). Later, developments in industry and the universities have led to a wide range of techniques which are gradually being employed in the agricultural machinery field (Craciun and Leon, 1998). Their
* Corresponding author. Tel.: +34 957 218371; fax: +34 957 218455. E-mail address: [email protected] (E. Gutie´rrez de Rave´). 0167-1987/$ – see front matter ß 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.still.2010.11.009
use has been extended to solve the problem of defining the tools, among which the application of the Be´zier surfaces, as suggested by Richey et al. (1989), has been made. These authors proposed the use of a Be´zier biparametric surface, whose control points are determined by some optimization method. Ravonison and Destain (1994) simplified this method with a much easier algorithm to fix the control points. Although the results were not shown in their publication, these authors reported that the fit obtained was a good one. Ros et al. (1995) represented the tillage tool surfaces by a multiplicity of quadrilateral faces limited by user-selected bounding curves. Traditionally, plough design and manufacture have been based on empirical methods and experiments (Shrestha et al., 2001), depending on the type of soil in the different areas. Different types of surfaces are used in the plough, cylindrical, cylindroidal and helicoidal, the two latter ones being those most used (Craciun and Leon, 1998). It is important to describe and characterize the threedimensional shape of the plough, which permits the carrying out of studies and simulations with the aim of analyzing the effect of the soil interactions on the plough (Formato et al., 2005) and its lateral and draught forces (Godwin and O’Dogherty, 2007). Tillage with a mouldboard plough is the largest consumer of world agricultural energy (Craciun and Leon, 1998), its geometry being that which determines its consumption during the tillage operation. Shrestha et al. (2001) studied the optimal design of the parameters of the ploughs and their operations in relation to the power required.
[(Fig._1)TD$IG]
E. Gutie´rrez de Rave´ et al. / Soil & Tillage Research 112 (2011) 98–105
99
Fig. 1. Laser profilometer and data taken.
The geometrical characterization of the mouldboard plough enables a computer study of the behaviour of different designs and the effect produced by different soil types, also facilitating identical models in the manufacturing process. A good plough design is important for it to prepare the soil for the conservation of water. The effects of the design and the operational parameters like speed, incidence angle and curvature radius were studied by Gebresenbet (1995). The mouldboard plough surface has been traditionally approximated to conics such as hyperbolic paraboloids (White, 1917), circular arcs which move and rotate along a transversal line (Nichols
[(Fig._2)TD$IG]
and Kummer, 1932) and also some empirical methods have been used such as the studies made by Reed (1941) and So¨hne (1959) can be mentioned. More recently, computer graphic techniques of the parametric models are being used (Richey et al., 1989), which, in addition to representing them graphically, permit the use of computational models on the variations in their shape. Geometric modelling is a very useful tool in design and industrial manufacture, mainly the implicit and parametric representations (Dimas and Briassoulis, 1999). The employment of parametric modelling techniques like those developed by Be´zier, the B-splines and the Non-Uniform Rational B-Splines (NURBS)
Fig. 2. Real and simulated values obtained by using RDA and MRDA.
100
E. Gutie´rrez de Rave´ et al. / Soil & Tillage Research 112 (2011) 98–105
facilitate the design and analysis processes both of real and artificial models. It is advisable the use of NURBS because they offer a common mathematical form for their representation and design unified in a single database, invariability in affine transformations, ease in handling the parameters and simplicity of their geometric interpretation (Piegl, 1991). The purpose of this work is the exploration of the geometric algorithms based on the spline concept to improve the earlier traditional mouldboard plough design process. The exploration will compare different algorithms to characterize the threedimensional geometry of the plough comparing them on the basis of the quality of the results obtained.
2. Materials and methods
Table 1 Derivatives considered as null by Ravonison and Destain (1994).
x y z
@2 Pð0; 0Þ=@u@v
@2 Pð0; 1Þ=@u@v
@2 Pð1; 0Þ=@u@v
@2 Pð1; 1Þ=@u@v
196.0 702.0 929.7
486.0 208.8 1062.0
221.4 826.2 538.2
356.6 954.0 513.0
2.2.1. Be´zier bicubic parametric surface The Be´zier bicubic parametric surface is a tensor product patch based on the Be´zier curve, x(u), with u as a parameter in the range (0,1): xðuÞ ¼
n X P i Bni ðuÞ i¼0
2.1. Description of the mouldboard plough surface
[(Fig._3)TD$IG]
The plough studied had a complex surface, manufactured by stamping with a central body, close to a hyperbolic paraboloid, but not arriving at being a ruled surface (Fig. 1), having a larger curve in the upper end. It was limited by three spatial curves and a straight section in the lower part. It is not possible to describe this surface by analytical functions (Ravonison and Destain, 1994). However, it can be represented with NURBS, which have the potential to represent curves and surfaces of freely formed complex models (Dimas and Briassoulis, 1999). The data recording system of the plough’s points was based in the use a laser profilometer (Valera, 1997) that determined the coordinate z shown in Fig. 1. The laser was installed on an endless screw, which permitted its displacement. By using a multi-turn potentiometer, the x coordinates of some points were determined at 0.05 m whereas the y coordinates were recorded at 0.10 m, both distances referred to the displacement increments, on a surface taken as a reference level. Due to the plough features, at the two ends the displacement increment in the direction of the y axis was of 0.02 m, as indicated in Fig. 1. In this way, twenty profiles were measured to characterize the mouldboard shape. In order to refine the measurements of the edge of the plough, given the importance of these points in the mathematical models employed, the same plough was projected orthogonally on a lattice, measuring the height of each projected point. Both data sets were integrated into a system with a common reference. A total of 121 points were measured, 82 inner points and 39 points corresponding to the edge of the plough. The dimensions of the analyzed plough were of 415 mm 1102 mm 126 mm in x, y and z axes, respectively. 2.2. Spline surface models Mouldboards can be assumed as three-dimensional surfaces delimited by four curved edges in a warped quadrilateral. Although author such as Craciun and Leon (1998) have tried to describe these ploughs in a simple analytical way, it is not easy to do that. Ravonison and Destain (1994) defined the surface of the plough by means of Be´zier surfaces, by considering a simple approximation, the Be´zier cubic functions. Be´zier curves possess several properties, which made them very useful to design surfaces (e.g. Farin, 2002). In general, a Be´zier curve can be fitted to describe any curve with an unlimited number of control points, but, for reasons of simplicity, the degree of polynomials, k, between 2 and 5 is preferred. When interpolation techniques are employed, k = 2 or k = 3 is used (Piegl, 1991). The following sections include a description of the different interpolation methods used in this work to fit the mouldboard surface.
Fig. 3. Real and simulated values obtained with the ARG algorithm.
(1)
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Table 2 Root mean squared error (RMSE) and efficiency index (EI) obtained for different edge fitting methods. Algorithm
Method of tangents (7 points)
ARG fit with 7 points
RMSE (mm)
EI
RMSE (mm)
EI
RMSE (mm)
EI
RDA MRDA
26.71 6.83
0.082 0.929
12.74 3.51
0.750 0.981
12.75 3.27
0.750 0.984
The order of the curve is n, in this case n = 3. The Be´zier points, vertexes, or control points are Pi, and Bi3(u) represent the Bernstein polynomials: B3i ðuÞ ¼
3 ui ð1 uÞ3i i
(2)
The tensor product is then zðu; vÞ: zðu; vÞ ¼
3 X 3 X P i j B3i ðuÞB3j ðvÞ
(3)
i¼0 j¼0
The Ravonison and Destain algorithm (RDA) is based on a Be´zier bicubic parametric surface. The control points are determined on the edges of the warped quadrilateral which, approximately, defines the plough’s surface. According to Ravonison and Destain (1994), this algorithm starts from the ends of these curved edges that are the control points P0 and P3. The inner control points, P1 and P2, are determined by selecting proportional increments in distance over the tangents traced at the ends of these edges, in such a way that the resulting Be´zier curve passes through an intermediate point of known coordinates. The method proposed by Ravonison and Destain (1994) considered seven experimental points, three belonging to each of the ends of the edge and one central, which permitted the calculation of the inner control points defining the Be´zier curve. The selection of these seven experimental points influenced the fit obtained. The estimation of the coordinate of the interior points, Pij, i,j = {1,2}, is made through the cross derivatives of the z function, or
[(Fig._4)TD$IG]
ARG fit with 9 points
twists (Farin, 2002), at the corners of the quadrilateral surface. The Be´zier bicubic surface approximating the plough’s surface, Eq. (3), requires the four inner control points. With this aim, the cross derivatives:
@2 Sð0; 0Þ ¼ 9ðP 00 P 01 P 10 þ P 11 Þ @u@v @2 Sð0; 1Þ ¼ 9ðP30 P31 P 20 þ P 21 Þ @u@v @2 Sð1; 1Þ ¼ 9ðP 33 P 32 P 23 þ P22 Þ @u@v 2 @ Sð1; 0Þ ¼ 9ðP03 P02 P13 þ P 12 Þ @u@v
(4)
were considered null in the original RDA algorithm, which permitted the characterization of the mouldboard with only 24 experimental points, 20 of them located at the corners that link the four edges and the rest placed at an intermediate position in each edge, with no need to measure the inner points. The original RDA algorithm has been modified in the present work, MRDA, for the condition of null cross derivatives is not taken into account with respect to the parameters u and v at the inner control points. Thus, it is possible to estimate the optimal values of these derivatives in order to minimize the root mean squared error, RMSE, by using the Rosenbrock algorithm (e.g. Press et al., 1992). Proceeding in this way, the values of the derivatives shown in Table 1 were obtained, with the RMSE decreasing to a value of 8.94 103 m. Even so, the approximation was not very good so that it was necessary to select other interpolation functions.
Fig. 4. Comparison between the results provided by the method of tangents and ARG procedure.
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Table 3 Root mean squared error (RMSE) and efficiency index (EI) obtained for the applied methods. Algorithm
RMSE (mm)
EI
RDA MRDA ARG EB NURBS
14.4 8.1 2.3 2.2 1.9
0.685 0.901 0.992 0.993 0.995
2.2.2. B-spline surfaces A B-spline surface is formed with piecewise polynomial curves such as (e.g. Rogers, 2001; Piegl and Tiller, 1997):
xðuÞ ¼
nþ1 X Q i Ni;k ðuÞumin u < umax
2knþ1
(5)
i¼1
where Qi are the de Boor or polygon control points and the Ni,k are the normalized basis splines, or B-splines, defined with the Cox–de
[(Fig._5)TD$IG]
Fig. 5. Control points considered for MRDA, EB, ARG and NURBS.
[(Fig._6)TD$IG]
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103
Fig. 6. Error distribution over the mouldboard surface obtained with MRDA, ARG, EB and NURBS.
Boor recursion formula: 1 ui u < uiþ1 N i;1 ðuÞ ¼ 0 otherwise ðu ui ÞNi;k1 ðuÞ ðuiþk uÞNiþ1;k1 ðuÞ N i;k ðuÞ ¼ þ uiþkþ1 ui uiþk uiþ1
modified version of the original Ravonison and Destain algorithm (ARG). The B-spline surface is zðu; vÞ: (6)
The B-spline curves have more flexibility than the Be´zier curves to adapting to any shape for the same polynomial order (Piegl and Tiller, 1995). The B-spline curves have been adapted in another
zðu; vÞ ¼
nþ1 m þ1 X X
Qi j Ni;k ðuÞN j;l ðvÞ
i¼1 j¼1
where k and l are the orders.
(7)
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As Rogers (2001) indicated, the matrix version of Eq. (7) is very convenient for the identification of the de Boor points. The approximation algorithm based on the B-splines method is EB. 2.2.3. Rational B-spline surfaces A rational B-spline surface is a Cartesian product of rational Bspline curves, non-uniform rational B-spline curves, or NURBS, introduced to represent simple forms like conics, each of them defined as (e.g. Rogers, 2001): Pnþ1 nþ1 T i hi N i;k ðuÞ X xðuÞ ¼ Pi¼1 T i Ri;k ðuÞ ¼ nþ1 i¼1 i¼1 hi N i;k ðuÞ
(8)
with Ti as the control points and the rational B-spline basis functions: hi N i;k ðuÞ Ri;k ðuÞ ¼ Pnþ1 i¼1 hi N i;k ðuÞ
(9)
The hi are weights of the interpolation. The rational B-spline surface is nþ1 m þ1 X X zðu; vÞ ¼ T i; j Si; j ðu; vÞ
(10)
i¼1 j¼1
where Si,j are the rational B-spline surface basis: hi; j Ni;k ðuÞM j;l ðvÞ Si; j ðu; vÞ ¼ Pnþ1 Pmþ1 i1¼1 j1¼1 hi1; j1 N i1;k ðuÞM j1;l ðvÞ
(11)
The optimal weights of the interpolation have been calculated by using an algorithm also called NURBS in this work. 3. Results and discussion In addition to the RMSE, as an indicator of the goodness of fit, the Nash–Sutcliffe efficiency index, EI (e.g. Beven, 2000) has been used, whose advantage is that it normalizes the RMSE with the variance s2 of the observed points. Thus, EI is defined as EI ¼ 1
RMSE 2
s
(12)
The index varies from 1 to 1. The closer the index is to unity the better the fit is. Fig. 2 compares the observed and generated values on the edge of the mouldboard according to the RDA and MRDA methods. The RDA algorithm fits apparently better the external contour of the mouldboard plow. Nevertheless, the RMSE for MRDA is 6.83 mm that is almost the fourth part of the analog value for RDA, Table 2. In addition, EI is closer to 1 for MRDA. In the same table the RDA and MRDA methods are compared when fitting the mouldboard surface when the control points defining its edges were calculated following method of tangents proposed by Ravonison and Destain (1994) or by the ARG procedure. As it can be checked, the RDA method yields better results when the control points needed to define the Bezier’s curve are determined by using the ARG method. Thus, the RMSE decreases from 26.71 mm to 12.74 mm and EI is closer to unity. However, no improvement was observed in the fit when the amount of points used for the ARG fit was increased for each edge. Removing the constrain of null twist in the surfaces of Ravonison and Destain (1994) and using B-splines the RMSE is reduced to almost one ninth (from 26.71 mm to 3.51–3.27 mm). Figs. 3 and 4 show the observed and simulated plough edge points for the ARG algorithm. The mouldboard edge BC is fitted by both methods by taking into account seven measured points. If the XY plane is considered, the obtained fits are similar and resemble the edge BC. However, when the ZY plane is taken into account, there
are evident differences for both methods. Thus, the ARG procedure yields a good fit, especially in the central part of edge BC, in contrast to the method of tangents. Table 3 lists the RMSE and EI obtained for the applied methods to get the whole fit. The B-splines algorithms allow a better description of the plow since the RMSE are lower than those found with the Ravonison and Destain algorithm either the original model, RDA, or the modified, MRDA, and, conversely, the efficiency indexes are closer to unity. Fig. 5 shows the control points obtained for generating the mouldboard surface by applying the EB, ARG, MRDA and NURBS methods. The distribution of errors on plough surface can be seen in Fig. 6, where the algorithms of best results have been considered (ARG, EB, MRDA and NURBS). The best fit corresponds to the NURBS algorithm. This result is also appreciable in the data of Table 3. Nevertheless the differences between the B-splines surface (EB), and non uniform rational B-splines are rather small, 2.2 mm and 1.9 mm, respectively, as it is shown in Table 3. 4. Conclusions The manufacturing technique traditionally used for the mouldboard plough is stamping, starting from a highly irregularly shaped matrix made on the basis of the designer’s previous experience. Consequently, the three-dimensional surface of the resulting implement is far removed from the optimal one initially conceived, giving rise to an inadequate preparation of the soil, and, in some cases, to an additional energy expenditure due to the increase in resistance found by the plough as it moves along This circumstance makes it advisable to explore the use of different parametric approximation models with three-dimensional surfaces. Thus, among the different models studied, the best results were obtained by the application of NURBS, optimizing the weights of the control points, since they are capable of reproducing common surfaces like cylinders and revolution and extrusion cylinders, which are frequent on the mouldboard plough surface. Due to the computational power of current computers, the use of NURBS does not require any simplifications like those proposed by Ravonison and Destain (1994). For these authors, the derivatives shown in their Eq. (6) are assumed to be null. However, it has been demonstrated here that the root mean squared error decreases when this assumption is not considered in the proposed MRDA algorithm. Acknowledgements The authors gratefully acknowledge the technical assistance with the laser profile meter developed by him, and the fruitful discussions with Prof. Juan Agu¨era of the Dept. of Rural Engineering of the University of Cordoba. The support from the Spanish Ministry of Science and Innovation ERDF Project AGL2009-12936C03-02 is acknowledged. References Bentaher, H., Hamza, E., Kantchev, G., Maalej, A., Arnold, W., 2008. Three-point hitch-mechanism instrumentation for tillage power optimization. Biosyst. Eng. 100, 24–30. Beven, K.J., 2000. Rainfall-Runoff Modeling: The Primer. John Wiley, Colchester. Bloomenthal, J., 1998. Graphics remembrances. IEEE Ann. Hist. Comput. 20, 35–51. Buss, S.R., 2003. 3D Computer Graphics: A Mathematical Introduction with OpenGL. Cambridge University Press, Cambridge. Craciun, V., Leon, D., 1998. An analytical method for identifying and designing a moldboard plow surface. Trans. ASAE 41, 1589–1599. Dimas, E., Briassoulis, D., 1999. 3D geometric modelling bases on NURBS: a review. Adv. Eng. Softw. 30, 741–751. Farin, G., 2002. Curves and Surfaces for CAGD: A Practical Guide, 5th ed. MorganKaufmann, San Francisco.
E. Gutie´rrez de Rave´ et al. / Soil & Tillage Research 112 (2011) 98–105 Formato, A., Faugno, S., Paolillo, G., 2005. Numerical simulation of soil-plough mouldboard interaction. Biosyst. Eng. 92, 309–316. Gebresenbet, G., 1995. Optimization of animal drawn tillage implements. Part I. Performance of a curved tillage implement. J. Agric. Eng. Res. 62, 173–184. Godwin, R.J., O’Dogherty, M.J., 2007. Integrated soil tillage force prediction models. J. Terramech. 44, 3–14. Horvat, Z., Filipovic, D., Kosutic, S., Emert, R., 2008. Reduction of mouldboard plough share wear by a combination technique of hardfacing. Tribol. Int. 41, 778–782. Nichols, M.L., Kummer, T.H., 1932. The dynamic properties of soil. IV. A method of analysis of plow moldboard design based upon dynamic properties of soil. Agric. Eng. 13, 279–285. Piegl, L.A., 1991. On NURBS: a survey. IEEE Comput. Graph. Appl. 10, 55–71. Piegl, L.A., Tiller, W., 1995. Algorithm for degree reduction of B-spline curves. Comput. Aided Des. 27, 17–28. Piegl, L.A., Tiller, W., 1997. The NURBS Book, 2nd ed. Springer, New York. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in Fortran. The Art of Scientific Computing, 2nd ed. University Press, Cambridge. Ravonison, N.M., Destain, M.F., 1994. Parametric cubic equations for modelling mouldboard plough surfaces. Soil Tillage Res. 31, 363–373.
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Reed, I.F., 1941. Tests of tillage tools. III. Effect of shape on the draft of 14-inch moldboard plow bottoms. Agric. Eng. 22, 101–104. Richey, S.B., Srivastava, A.K., Segerlind, L.J., 1989. The use of three dimensional computer graphics to design mouldboard plough surfaces. J. Agric. Eng. Res. 43, 245–258. Rogers, D.F., 2001. An Introduction to NURBS, 1st ed. Morgan-Kaufmann, New York. Ros, V., Smith, R.J., Marley, S.J., Erbach, D.C., 1995. Mathematical modelling and computer-aided design of passive tillage tools. Trans. ASAE 38, 675–683. Shrestha, D.S., Singh, G., Gebresenbet, G., 2001. Optimizing design parameters of a mouldboard plough. J. Agric. Eng. Res. 78, 377–389. So¨hne, W., 1959. Untersuchungen u¨ber die Form von Pflugko¨rpern bei erho¨hter Fahrgeschwindigkeit, Investigations on the shape of plough bodies for high speed. Grundl. Landtech. 11, 22–39. Valera, D.L., 1997. Medida sin contacto y ana´lisis del microrrelieve de los suelos ˜ os provocados por el agrı´colas: aplicacio´n al estudio experimental de los dan paso de tractores sobre los campos. Ph.D. diss. Dept of Rural Engineering, University of Cordoba, Spain (in Spanish). White, E.A., 1917. A study of the plow bottom and its action on the furrow slice. Ph.D. Diss. Dept. of Biological Engineering. Cornell University. available at (http:// ecommons.cornell.edu/bitstream/1813/7641/1/Study%20of%20the%20Plow% 20Bottom%20online.pdf).
Contents lists available at ScienceDirect
Soil & Tillage Research journal homepage: www.elsevier.com/locate/still
Short communication
The geometric characterization of mouldboard plough surfaces by using splines ˜ oz-Piorno a, J.V. Gira´ldez b E. Gutie´rrez de Rave´ a,*, F.J. Jime´nez-Hornero a, J.M. Mun a b
Dept. of Engineering Graphics and Geomatics, University of Co´rdoba, Gregor Mendel Building, ctra. Madrid km 396, 14071 Co´rdoba, Spain Dept. of Agronomy, University of Co´rdoba and IAS CSIC, Leonardo da Vinci Building, ctra. Madrid km 396, 14071 Co´rdoba, Spain
A R T I C L E I N F O
A B S T R A C T
Article history: Received 23 July 2010 Received in revised form 24 November 2010 Accepted 26 November 2010 Available online 30 December 2010
For historic reasons, mouldboard ploughs have a wide variety of surfaces. The concentration of manufacturers has reduced the number of different types but there are still few complete geometric descriptions of the surfaces of a mouldboard plough. With the aim of describing and characterizing the three-dimensional shape of the plough, permitting the obtaining of a tool for an adequate preparation of the soil for cultivation, as well as reducing labour costs, in this work different approximation methods of three-dimensional surfaces have been explored. The results obtained allow us to conclude that the application of non uniform rational B-splines, combined with an optimization algorithm based on assigning different weights to the control points of the surface analyzed, is a method which succeeds in improving the quality of the fit with respect to other algorithms proposed by several authors in different previous works also based on the use of splines. The root mean square error of the fit of these algorithms to a mouldboard plough was the lowest 1.9 mm in the observed, and increases up to 14.4 mm using a simple Be´zier bicubic parametric surface. Using the efficiency index of Nash and Sutclife (1970), the highest value, 0.995, corresponded to the non uniform rational B-splines method. ß 2010 Elsevier B.V. All rights reserved.
Keywords: Mouldboard design Be´zier surface Computer graphic NURBS
1. Introduction The characterization of agricultural implements is currently a subject of great importance due to the revision of traditional cultivation methods for the new conservation tillage and precision agriculture techniques. The great variety of environmental, edaphic, agronomic, and even social conditions has given rise to the development of numerous types of agricultural implements, which require a simple characterization to permit tillage power optimization (Bentaher et al., 2008). The new types of conservation and precision agriculture methods need new tools which have to be characterized, since ploughs of very different characteristics are often found and it would be necessary to unify them (Horvat et al., 2008). The use of geometric algorithms describing the surfaces in a simple way will permit farmers and machinery manufacturers to carry out effective quality controls, taking advantage of the development of approximation methods in the motor car industry, especially since the proposal of Pierre Be´zier of the curve bearing his name in the 1950s (e.g. Bloomenthal, 1998; Rogers, 2001; Buss, 2003). Later, developments in industry and the universities have led to a wide range of techniques which are gradually being employed in the agricultural machinery field (Craciun and Leon, 1998). Their
* Corresponding author. Tel.: +34 957 218371; fax: +34 957 218455. E-mail address: [email protected] (E. Gutie´rrez de Rave´). 0167-1987/$ – see front matter ß 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.still.2010.11.009
use has been extended to solve the problem of defining the tools, among which the application of the Be´zier surfaces, as suggested by Richey et al. (1989), has been made. These authors proposed the use of a Be´zier biparametric surface, whose control points are determined by some optimization method. Ravonison and Destain (1994) simplified this method with a much easier algorithm to fix the control points. Although the results were not shown in their publication, these authors reported that the fit obtained was a good one. Ros et al. (1995) represented the tillage tool surfaces by a multiplicity of quadrilateral faces limited by user-selected bounding curves. Traditionally, plough design and manufacture have been based on empirical methods and experiments (Shrestha et al., 2001), depending on the type of soil in the different areas. Different types of surfaces are used in the plough, cylindrical, cylindroidal and helicoidal, the two latter ones being those most used (Craciun and Leon, 1998). It is important to describe and characterize the threedimensional shape of the plough, which permits the carrying out of studies and simulations with the aim of analyzing the effect of the soil interactions on the plough (Formato et al., 2005) and its lateral and draught forces (Godwin and O’Dogherty, 2007). Tillage with a mouldboard plough is the largest consumer of world agricultural energy (Craciun and Leon, 1998), its geometry being that which determines its consumption during the tillage operation. Shrestha et al. (2001) studied the optimal design of the parameters of the ploughs and their operations in relation to the power required.
[(Fig._1)TD$IG]
E. Gutie´rrez de Rave´ et al. / Soil & Tillage Research 112 (2011) 98–105
99
Fig. 1. Laser profilometer and data taken.
The geometrical characterization of the mouldboard plough enables a computer study of the behaviour of different designs and the effect produced by different soil types, also facilitating identical models in the manufacturing process. A good plough design is important for it to prepare the soil for the conservation of water. The effects of the design and the operational parameters like speed, incidence angle and curvature radius were studied by Gebresenbet (1995). The mouldboard plough surface has been traditionally approximated to conics such as hyperbolic paraboloids (White, 1917), circular arcs which move and rotate along a transversal line (Nichols
[(Fig._2)TD$IG]
and Kummer, 1932) and also some empirical methods have been used such as the studies made by Reed (1941) and So¨hne (1959) can be mentioned. More recently, computer graphic techniques of the parametric models are being used (Richey et al., 1989), which, in addition to representing them graphically, permit the use of computational models on the variations in their shape. Geometric modelling is a very useful tool in design and industrial manufacture, mainly the implicit and parametric representations (Dimas and Briassoulis, 1999). The employment of parametric modelling techniques like those developed by Be´zier, the B-splines and the Non-Uniform Rational B-Splines (NURBS)
Fig. 2. Real and simulated values obtained by using RDA and MRDA.
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facilitate the design and analysis processes both of real and artificial models. It is advisable the use of NURBS because they offer a common mathematical form for their representation and design unified in a single database, invariability in affine transformations, ease in handling the parameters and simplicity of their geometric interpretation (Piegl, 1991). The purpose of this work is the exploration of the geometric algorithms based on the spline concept to improve the earlier traditional mouldboard plough design process. The exploration will compare different algorithms to characterize the threedimensional geometry of the plough comparing them on the basis of the quality of the results obtained.
2. Materials and methods
Table 1 Derivatives considered as null by Ravonison and Destain (1994).
x y z
@2 Pð0; 0Þ=@u@v
@2 Pð0; 1Þ=@u@v
@2 Pð1; 0Þ=@u@v
@2 Pð1; 1Þ=@u@v
196.0 702.0 929.7
486.0 208.8 1062.0
221.4 826.2 538.2
356.6 954.0 513.0
2.2.1. Be´zier bicubic parametric surface The Be´zier bicubic parametric surface is a tensor product patch based on the Be´zier curve, x(u), with u as a parameter in the range (0,1): xðuÞ ¼
n X P i Bni ðuÞ i¼0
2.1. Description of the mouldboard plough surface
[(Fig._3)TD$IG]
The plough studied had a complex surface, manufactured by stamping with a central body, close to a hyperbolic paraboloid, but not arriving at being a ruled surface (Fig. 1), having a larger curve in the upper end. It was limited by three spatial curves and a straight section in the lower part. It is not possible to describe this surface by analytical functions (Ravonison and Destain, 1994). However, it can be represented with NURBS, which have the potential to represent curves and surfaces of freely formed complex models (Dimas and Briassoulis, 1999). The data recording system of the plough’s points was based in the use a laser profilometer (Valera, 1997) that determined the coordinate z shown in Fig. 1. The laser was installed on an endless screw, which permitted its displacement. By using a multi-turn potentiometer, the x coordinates of some points were determined at 0.05 m whereas the y coordinates were recorded at 0.10 m, both distances referred to the displacement increments, on a surface taken as a reference level. Due to the plough features, at the two ends the displacement increment in the direction of the y axis was of 0.02 m, as indicated in Fig. 1. In this way, twenty profiles were measured to characterize the mouldboard shape. In order to refine the measurements of the edge of the plough, given the importance of these points in the mathematical models employed, the same plough was projected orthogonally on a lattice, measuring the height of each projected point. Both data sets were integrated into a system with a common reference. A total of 121 points were measured, 82 inner points and 39 points corresponding to the edge of the plough. The dimensions of the analyzed plough were of 415 mm 1102 mm 126 mm in x, y and z axes, respectively. 2.2. Spline surface models Mouldboards can be assumed as three-dimensional surfaces delimited by four curved edges in a warped quadrilateral. Although author such as Craciun and Leon (1998) have tried to describe these ploughs in a simple analytical way, it is not easy to do that. Ravonison and Destain (1994) defined the surface of the plough by means of Be´zier surfaces, by considering a simple approximation, the Be´zier cubic functions. Be´zier curves possess several properties, which made them very useful to design surfaces (e.g. Farin, 2002). In general, a Be´zier curve can be fitted to describe any curve with an unlimited number of control points, but, for reasons of simplicity, the degree of polynomials, k, between 2 and 5 is preferred. When interpolation techniques are employed, k = 2 or k = 3 is used (Piegl, 1991). The following sections include a description of the different interpolation methods used in this work to fit the mouldboard surface.
Fig. 3. Real and simulated values obtained with the ARG algorithm.
(1)
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Table 2 Root mean squared error (RMSE) and efficiency index (EI) obtained for different edge fitting methods. Algorithm
Method of tangents (7 points)
ARG fit with 7 points
RMSE (mm)
EI
RMSE (mm)
EI
RMSE (mm)
EI
RDA MRDA
26.71 6.83
0.082 0.929
12.74 3.51
0.750 0.981
12.75 3.27
0.750 0.984
The order of the curve is n, in this case n = 3. The Be´zier points, vertexes, or control points are Pi, and Bi3(u) represent the Bernstein polynomials: B3i ðuÞ ¼
3 ui ð1 uÞ3i i
(2)
The tensor product is then zðu; vÞ: zðu; vÞ ¼
3 X 3 X P i j B3i ðuÞB3j ðvÞ
(3)
i¼0 j¼0
The Ravonison and Destain algorithm (RDA) is based on a Be´zier bicubic parametric surface. The control points are determined on the edges of the warped quadrilateral which, approximately, defines the plough’s surface. According to Ravonison and Destain (1994), this algorithm starts from the ends of these curved edges that are the control points P0 and P3. The inner control points, P1 and P2, are determined by selecting proportional increments in distance over the tangents traced at the ends of these edges, in such a way that the resulting Be´zier curve passes through an intermediate point of known coordinates. The method proposed by Ravonison and Destain (1994) considered seven experimental points, three belonging to each of the ends of the edge and one central, which permitted the calculation of the inner control points defining the Be´zier curve. The selection of these seven experimental points influenced the fit obtained. The estimation of the coordinate of the interior points, Pij, i,j = {1,2}, is made through the cross derivatives of the z function, or
[(Fig._4)TD$IG]
ARG fit with 9 points
twists (Farin, 2002), at the corners of the quadrilateral surface. The Be´zier bicubic surface approximating the plough’s surface, Eq. (3), requires the four inner control points. With this aim, the cross derivatives:
@2 Sð0; 0Þ ¼ 9ðP 00 P 01 P 10 þ P 11 Þ @u@v @2 Sð0; 1Þ ¼ 9ðP30 P31 P 20 þ P 21 Þ @u@v @2 Sð1; 1Þ ¼ 9ðP 33 P 32 P 23 þ P22 Þ @u@v 2 @ Sð1; 0Þ ¼ 9ðP03 P02 P13 þ P 12 Þ @u@v
(4)
were considered null in the original RDA algorithm, which permitted the characterization of the mouldboard with only 24 experimental points, 20 of them located at the corners that link the four edges and the rest placed at an intermediate position in each edge, with no need to measure the inner points. The original RDA algorithm has been modified in the present work, MRDA, for the condition of null cross derivatives is not taken into account with respect to the parameters u and v at the inner control points. Thus, it is possible to estimate the optimal values of these derivatives in order to minimize the root mean squared error, RMSE, by using the Rosenbrock algorithm (e.g. Press et al., 1992). Proceeding in this way, the values of the derivatives shown in Table 1 were obtained, with the RMSE decreasing to a value of 8.94 103 m. Even so, the approximation was not very good so that it was necessary to select other interpolation functions.
Fig. 4. Comparison between the results provided by the method of tangents and ARG procedure.
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Table 3 Root mean squared error (RMSE) and efficiency index (EI) obtained for the applied methods. Algorithm
RMSE (mm)
EI
RDA MRDA ARG EB NURBS
14.4 8.1 2.3 2.2 1.9
0.685 0.901 0.992 0.993 0.995
2.2.2. B-spline surfaces A B-spline surface is formed with piecewise polynomial curves such as (e.g. Rogers, 2001; Piegl and Tiller, 1997):
xðuÞ ¼
nþ1 X Q i Ni;k ðuÞumin u < umax
2knþ1
(5)
i¼1
where Qi are the de Boor or polygon control points and the Ni,k are the normalized basis splines, or B-splines, defined with the Cox–de
[(Fig._5)TD$IG]
Fig. 5. Control points considered for MRDA, EB, ARG and NURBS.
[(Fig._6)TD$IG]
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103
Fig. 6. Error distribution over the mouldboard surface obtained with MRDA, ARG, EB and NURBS.
Boor recursion formula: 1 ui u < uiþ1 N i;1 ðuÞ ¼ 0 otherwise ðu ui ÞNi;k1 ðuÞ ðuiþk uÞNiþ1;k1 ðuÞ N i;k ðuÞ ¼ þ uiþkþ1 ui uiþk uiþ1
modified version of the original Ravonison and Destain algorithm (ARG). The B-spline surface is zðu; vÞ: (6)
The B-spline curves have more flexibility than the Be´zier curves to adapting to any shape for the same polynomial order (Piegl and Tiller, 1995). The B-spline curves have been adapted in another
zðu; vÞ ¼
nþ1 m þ1 X X
Qi j Ni;k ðuÞN j;l ðvÞ
i¼1 j¼1
where k and l are the orders.
(7)
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104
As Rogers (2001) indicated, the matrix version of Eq. (7) is very convenient for the identification of the de Boor points. The approximation algorithm based on the B-splines method is EB. 2.2.3. Rational B-spline surfaces A rational B-spline surface is a Cartesian product of rational Bspline curves, non-uniform rational B-spline curves, or NURBS, introduced to represent simple forms like conics, each of them defined as (e.g. Rogers, 2001): Pnþ1 nþ1 T i hi N i;k ðuÞ X xðuÞ ¼ Pi¼1 T i Ri;k ðuÞ ¼ nþ1 i¼1 i¼1 hi N i;k ðuÞ
(8)
with Ti as the control points and the rational B-spline basis functions: hi N i;k ðuÞ Ri;k ðuÞ ¼ Pnþ1 i¼1 hi N i;k ðuÞ
(9)
The hi are weights of the interpolation. The rational B-spline surface is nþ1 m þ1 X X zðu; vÞ ¼ T i; j Si; j ðu; vÞ
(10)
i¼1 j¼1
where Si,j are the rational B-spline surface basis: hi; j Ni;k ðuÞM j;l ðvÞ Si; j ðu; vÞ ¼ Pnþ1 Pmþ1 i1¼1 j1¼1 hi1; j1 N i1;k ðuÞM j1;l ðvÞ
(11)
The optimal weights of the interpolation have been calculated by using an algorithm also called NURBS in this work. 3. Results and discussion In addition to the RMSE, as an indicator of the goodness of fit, the Nash–Sutcliffe efficiency index, EI (e.g. Beven, 2000) has been used, whose advantage is that it normalizes the RMSE with the variance s2 of the observed points. Thus, EI is defined as EI ¼ 1
RMSE 2
s
(12)
The index varies from 1 to 1. The closer the index is to unity the better the fit is. Fig. 2 compares the observed and generated values on the edge of the mouldboard according to the RDA and MRDA methods. The RDA algorithm fits apparently better the external contour of the mouldboard plow. Nevertheless, the RMSE for MRDA is 6.83 mm that is almost the fourth part of the analog value for RDA, Table 2. In addition, EI is closer to 1 for MRDA. In the same table the RDA and MRDA methods are compared when fitting the mouldboard surface when the control points defining its edges were calculated following method of tangents proposed by Ravonison and Destain (1994) or by the ARG procedure. As it can be checked, the RDA method yields better results when the control points needed to define the Bezier’s curve are determined by using the ARG method. Thus, the RMSE decreases from 26.71 mm to 12.74 mm and EI is closer to unity. However, no improvement was observed in the fit when the amount of points used for the ARG fit was increased for each edge. Removing the constrain of null twist in the surfaces of Ravonison and Destain (1994) and using B-splines the RMSE is reduced to almost one ninth (from 26.71 mm to 3.51–3.27 mm). Figs. 3 and 4 show the observed and simulated plough edge points for the ARG algorithm. The mouldboard edge BC is fitted by both methods by taking into account seven measured points. If the XY plane is considered, the obtained fits are similar and resemble the edge BC. However, when the ZY plane is taken into account, there
are evident differences for both methods. Thus, the ARG procedure yields a good fit, especially in the central part of edge BC, in contrast to the method of tangents. Table 3 lists the RMSE and EI obtained for the applied methods to get the whole fit. The B-splines algorithms allow a better description of the plow since the RMSE are lower than those found with the Ravonison and Destain algorithm either the original model, RDA, or the modified, MRDA, and, conversely, the efficiency indexes are closer to unity. Fig. 5 shows the control points obtained for generating the mouldboard surface by applying the EB, ARG, MRDA and NURBS methods. The distribution of errors on plough surface can be seen in Fig. 6, where the algorithms of best results have been considered (ARG, EB, MRDA and NURBS). The best fit corresponds to the NURBS algorithm. This result is also appreciable in the data of Table 3. Nevertheless the differences between the B-splines surface (EB), and non uniform rational B-splines are rather small, 2.2 mm and 1.9 mm, respectively, as it is shown in Table 3. 4. Conclusions The manufacturing technique traditionally used for the mouldboard plough is stamping, starting from a highly irregularly shaped matrix made on the basis of the designer’s previous experience. Consequently, the three-dimensional surface of the resulting implement is far removed from the optimal one initially conceived, giving rise to an inadequate preparation of the soil, and, in some cases, to an additional energy expenditure due to the increase in resistance found by the plough as it moves along This circumstance makes it advisable to explore the use of different parametric approximation models with three-dimensional surfaces. Thus, among the different models studied, the best results were obtained by the application of NURBS, optimizing the weights of the control points, since they are capable of reproducing common surfaces like cylinders and revolution and extrusion cylinders, which are frequent on the mouldboard plough surface. Due to the computational power of current computers, the use of NURBS does not require any simplifications like those proposed by Ravonison and Destain (1994). For these authors, the derivatives shown in their Eq. (6) are assumed to be null. However, it has been demonstrated here that the root mean squared error decreases when this assumption is not considered in the proposed MRDA algorithm. Acknowledgements The authors gratefully acknowledge the technical assistance with the laser profile meter developed by him, and the fruitful discussions with Prof. Juan Agu¨era of the Dept. of Rural Engineering of the University of Cordoba. The support from the Spanish Ministry of Science and Innovation ERDF Project AGL2009-12936C03-02 is acknowledged. References Bentaher, H., Hamza, E., Kantchev, G., Maalej, A., Arnold, W., 2008. Three-point hitch-mechanism instrumentation for tillage power optimization. Biosyst. Eng. 100, 24–30. Beven, K.J., 2000. Rainfall-Runoff Modeling: The Primer. John Wiley, Colchester. Bloomenthal, J., 1998. Graphics remembrances. IEEE Ann. Hist. Comput. 20, 35–51. Buss, S.R., 2003. 3D Computer Graphics: A Mathematical Introduction with OpenGL. Cambridge University Press, Cambridge. Craciun, V., Leon, D., 1998. An analytical method for identifying and designing a moldboard plow surface. Trans. ASAE 41, 1589–1599. Dimas, E., Briassoulis, D., 1999. 3D geometric modelling bases on NURBS: a review. Adv. Eng. Softw. 30, 741–751. Farin, G., 2002. Curves and Surfaces for CAGD: A Practical Guide, 5th ed. MorganKaufmann, San Francisco.
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