AP_07_6.vp 1 Introduction The problem of curve synthesis occurs frequently in me- chanical design [1, 2, 3, 4, 5, 6]. This problem arises whenever two flat – the case of a surface – or straight – the case of a line – segments of a machine element are to be joined either to close an orifice smoothly or to join that segment to the machine frame. The reason why a smooth transfer is required lies in the need to prevent stress concentrations. Mechanical designers over the years have designed ma- chines with rectangular bores using circular arcs to round the corners. This is done with two purposes: (i) to avoid stress con- centrations that would arise due to an inWnite curvature at the corner, and (ii) to ease the machining of the bore. The problem with circular arcs is that they provide only G1-conti- nuity. Second-order geometric continuity G2, on the other hand, requires that the two curvatures coincide at the blending point; however, the curvature of the straight segment is zero, while that of the circular arc is the reciprocal of the radius of the circle, which is a finite quantity different from zero. The outcome is that stress concentrations are not eliminated because of the curvature discontinuity at the point of blending [7]. In this paper, a methodology is proposed aiming at the production of G2-continuity at the blending of segments of two curves. The paper is based on the concept of curve synthe- sis, as first proposed in [1]. While the forgoing reference resorts to parametric cubic splines to synthesize geometric curves, we show in this paper that a geometric curve can also be synthesized using non-parametric splines, thereby stream- lining the procedure. 2 Problem statement First and foremost, a distinction is made between a curve representing the plot of a function and a geometric curve: while the slope of the former is bounded, the latter is not; moreover, a geometric curve can have cusps, can cross itself and need not be representable by analytic functions. The paper focuses on geometric curves, while resorting to non- -parametric cubic splines. Curve synthesis in this context is defined as: Given two curve segments �a and �b lying in the same plane, find a third curve segment �c that blends smoothly with both �a and �b at the blending points A and B, with G2-continuity. Notice that we stated the search of a segment, as opposed to that of the segment, to emphasize that the problem admits mul- tiple solutions. In fact, the problem admits infinitely many so- lutions. To pinpoint one particular solution, we must impose additional conditions. Many are possible, the one that we adopt here being that the segment sought �c be “as straight as possible.” What this means is that we want the segment to have the smallest possible curvature. This requirement makes sense in design engineering, since a straight segment is the simplest shape to fabricate and, in the realm of structural engineering, the least likely to offer high bending moments if, for example, �c were to be the neutral axis of a beam. The challenge here is to formulate the synthesis problem in a standard form, e.g., one that would lead to an optimization problem. To this end, we resort to a discretization of the curve. Many forms of curve discretization are available in the realm of geo- metric modeling, namely, Bézier curves, cubic splines, B-splines, non-uniform rational B-splines, or NURBS, and so on [8]. We limit the search to the simplest of these tools, namely, non-parametric cubic splines. 3 Problem formulation using cubic splines In discretizing the problem at hand by means of cubic splines, we define n � 2 supporting points � �Pk n 0 1� , the kth point having Cartesian coordinates Pk(xk, yk). Shown in Fig. 1 is a sketch of the blending segments �a and �b by means of �c, with P denoting a generic point of the latter. Alterna- tively, points � �Pk n 0 1� are defined by their polar coordinates Pk(�k, �k). Henceforth, we define P A0 � and P Bn � �1 . Now, let A(�A, �A) and B(�B, �B) be the polar coordinates of the blending points. Moreover, let � � � � � � � B A n 1 and define � � �k A k� � � , for k n� �1 2 1, , ,� , with � �n B� �1 , � � thus being the uniform increment over the po- 56 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 6/2007 Shape Synthesis in Mechanical Design C. P. Teng, S. Bai, J. Angeles The shaping of structural elements in the area of mechanical design is a recurrent problem. The mechanical designer, as a rule, chooses what is believed to be the “simplest” shapes, such as the geometric primitives: lines, circles and, occasionally, conics. The use of higher-order curves is usually not even considered, not to speak of other curves than polynomials. However, the simplest geometric shapes are not necessarily the most suitable when the designed element must withstand loads that can lead to failure-prone stress concentrations. Indeed, as mechanical designers have known for a while, stress concentrations occur, first and foremost, by virtue of either dramatic changes in curvature or extremely high values thereof. As an alternative, we propose here the use of smooth curves that can be simply generated using standard concepts such as non-parametric cubic splines. These curves can be readily used to produce either extruded surfaces or surfaces of revolution. Keywords: Curve synthesis, cubic splines, optimum design, G2-continuity. lar coordinate �. In defining the polar coordinates of the blending curve �c, care must be taken to choose the origin conveniently, so as to avoid more than one point with the same �-coordinate, which would render the discretization adopted here invalid. Under these conditions, then we as- sume that a function � � �� ( ) exists in the interval [�a, �B]. Further, we define the n � 2-dimensional vectors, �, �� and ��� , namely, � � �� � � �0 1 1, , , ,� n n T (1) � � � � � � �� � � � �0 1 1, , , ,� n n T (2) �� � �� �� �� �� �� � � � �0 1 1, , , ,� n n T (3) Letting � �k( ) be the cubic polynomial between two consec- utive supporting points Pk k k( , )� � and Pk k k� � �1 1 1( , )� � , we have � � � � � � � � � � � k k k k k k k k k k A B C D( ) ( ) ( ) ( ) , , � � � � � � � � � �� 3 2 1 0 k n� . (4) By virtue of the G2-continuity of non-parametric cubic splines, a linear relationship between � and �� exists, namely, A C�� �� �6 , (5) where both A and C are n×(n � 1) matrices. Further, we re- call the expressions for the angle � made by the tangent to the curve with the radius vector, and the curvature, in polar coordinates: tan � � � � � , (6) � � � � �� � � � � � � �� � � 2 2 2 2 3 2 2( ) ( ) , (7) where � �� � � d d , (8) �� � � � � � d d . (9) Moreover, let tk and �k denote tan � and � at Pk, i.e., tk k k � � � � , (10) � � � � � � � � k k k k k k k � � � � �� � � 2 2 2 2 3 2 2( ) ( ) . (11) That is, t tk k k k� �( , )� � and � � � � �k k k k k� � ��( , , ). Let us now in- troduce two end-conditions: t tA A0 � � tan � , t tn B B� � �1 tan � , (12) where �A and �B are the known tangent angles at point A and B of �a and �b, respectively, matrices A and C then becoming square. A linear relation between � and �� also exist, namely, P Q� �� � , (13) where P and Q are ( ) ( )n n� � �1 2 matrices. Expressions for all four matrices A, C, P and Q are included in the Appendix. Hence, slope and curvature values at the unknown sup- porting points � �Pk n 1 of �c can all be expressed as functions of � ��k n 1 . In formulating the curve-synthesis problem within the realm of optimum design, we let x be the vector of design of variables, namely, x � [ ]� �1 � n T. (14) Notice that � �A � 0, � �B n� �1 , (15) which are known, and part of the data. We can now formulate the curve-synthesis problem at hand as an optimum-design problem: z n w kk k n x � ��1 2 1 min . (16) Where wk is the normal weight at the k th supporting point, whenever one needs to assign different importance to differ- ent points. We term the weights normal because they obey the relation wk n 1 1� � (17) the problem being constrained to obey the G2-continuity conditions at the two blending points. Specifically, these con- straints are, � �0( , , )� � �� �� � A , � �n B� � �� �1( , , )� � � , (18) where �A and �B are the curvature values of curve �a at point A and of curve �b at point B, respectively, thereby formulating an equality constrained optimum-design problem. Further- more, special cases may call for additional constraints. For ex- ample, if convexity is to be imposed, then we must add the inequality constraints �k � 0 , k n� 1 2, , ,� . (19) Once the optimization problem is formulated, a solution is possible with scientific code available in the market. We resort to our own Orthogonal Decomposition Algorithm, implemented in the ODA package [9], which is a library of C routines, especially suited to solve nonlinear least-square problems. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 57 Acta Polytechnica Vol. 47 No. 6/2007 Fig. 1: Blending of two curve segments with a third one 4 Examples Several shape-synthesis problems have been solved using cubic-splines. Two examples belonging to structural optimi- zation are outlined in the balance of this paper. 4.1 Synthesis of the neutral axis of a curved robotic link The first case arose during the design of a novel spherical parallel robot, the “Agile Wrist” [10], as shown in Fig. 2. As a module of an 11-degree-of-freedom (dof) long-reach robot operating on fragile objects such as the fuselage of aircraft, the wrist requires a high positioning accuracy since any error in the positioning of the tool could lead to expensive damage. The kinematic chain of the “Agile Wrist” was borrowed from the design of the “Agile Eye”, developed at Laval Uni- versity in Quebec City [11]. Many factors can affect robot accuracy, such as manufacturing and assembly errors, cali- bration errors, etc. Our concern is the flexibility of the manipulator links, which significantly reduces the positioning accuracy of the robot due to external and inertial loading. For the most part, the geometric shape and the cross-section dimensions of robot links of manipulators are designed arbitrarily, with simple shapes and constant cross sections. Current structural optimization technology and the use of highly accurate CNC machine tools enable designers to pro- duce optimum, if less obvious, shapes; an efficient selection of the design parameters should improve link stiffness without making the link heavier. A general trend in the design of lighter and stiffer structures has led to the use of sophisti- cated materials, like carbon fibre-reinforced composites, with the consequent slendering of structural components. These components should ensure that no failure can occur under a given range of loads, which has motivated intensive research in the area of structural optimization, a very active field since Schmidt set forth an approach for coupling finite element analysis and nonlinear programming [12]. In our work, struc- tural optimization was conducted through the optimum de- sign of the links of the Agile Wrist in order to enhance the load-carrying capacity, while minimizing weight. We focus here on the shaping of the links, rather than on material selection. The material of choice, as decided on earlier, was aluminum AL2014. For optimization purposes, the problem at hand was decomposed into two main steps: The first one consists in defining the neutral axis of the links, i.e., the mid-curve; the second consists in defining the cross section at each point of the mid-curve. The idea is rather simple: once the mid-curve of the link is defined, the link shape is obtained by sweeping and blending simultaneously a cross-section of a given shape (rectangle, circle, etc.), whose dimensions are variable along the curve, and lie in a plane normal to the mid-curve. Since the second step can be completed with CAE software, we focus on the synthesis of the mid-curve. To this end, the mid-curve � of the proximal link, shown in Fig. 3, is defined so as to min- imize the root-mean square value of the curvature throughout � and to ensure a blending of the linear segments with the curved segment as smooth as possible. Notice that the two ends lie at different distances from the center of the wrist, in order to increase the workspace of the Agile Wrist, besides reducing the likelihood of collisions among moving links and the tool installed on top of the upper platform. Twenty spline supporting points Pi, for i � 0 19, ,� , are used, whose angles �i in polar coordinates are given by: � � � � i i n � � � � � � �� � ��1 2 ( ) (20) with � � � � � � � �arctan l a v and � � � � � � �arctan l b h . With the aid of ODA, the optimum values of the design variables were found. The objective function to minimize is the rms value of the array of curvature values at the support- ing points, while respecting the tangency and zero-curvature conditions at points P0 and Pn � 1. A convexity condition was added, so as to ensure that no changes in curvature-sign oc- 58 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 6/2007 Fig. 2: Prototype of the Agile Wrist Fig. 3: The mid-curve of the proximal link cur. These twenty points, expressed first in polar coordinates, were then expressed in Cartesian coordinates as required by CAE software. The radius �i and the Cartesian coor- dinates (xi, yi) obtained for a � 89 5. mm, b � 75 5. mm and l lv h� � 12.0 mm, are listed in Table 1. The rms value of the curvature distribution over the synthesized curve is 0.0144 mm�1. The curve is plotted in Fig. 4a, its curvature distribution being shown in Fig. 4b for verification of the cur- vature continuity at the two blending points. Based on this mid-curve, structural optimization was completed by consid- ering the cross-section at each point of the curve. The overall stress analysis on an optimized link with two circular cross- -sections at its ends is graphically displayed in Fig. 5, where no stress concentrations are observed. It is noted that the link fabricated, shown in Fig. 2, exhibits a rectangular, uniform cross-section, which is cheaper to machine, and we decided to adopt to stay within budget. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 59 Acta Polytechnica Vol. 47 No. 6/2007 i �i [mm] xi [mm] yi [mm] i �i [mm] xi [mm] yi [mm] 0 90.3009 89.5000 12.0000 10 87.7878 60.7257 63.3962 1 91.2159 89.3859 18.1795 11 86.3873 55.4222 66.2658 2 91.9843 88.7004 24.3585 12 84.9587 49.9954 68.6910 3 92.4646 87.3132 30.4319 13 83.5441 44.5046 70.7032 4 92.6339 85.2225 36.3065 14 82.1737 38.9943 72.3323 5 92.4845 82.4522 41.8929 15 80.8669 33.4957 73.6036 6 92.0253 79.0508 47.1128 16 79.6338 28.0313 74.5372 7 91.2811 75.0882 51.9039 17 78.4782 22.6175 75.1484 8 90.2919 70.6501 56.2245 18 77.4004 17.2674 75.4497 9 89.1084 65.8309 60.0549 19 76.4477 12.0000 75.5000 Table 1: Numerical results for the optimum mid-curve of the proximal link (a) (b) Fig. 4: Mid-curve synthesized for the Agile Wrist: (a) the curve, (b) its curvature distribution Fig. 5: Von Mises stress distribution along the optimized proxi- mal link (thickest cross-section is located closest to the motor) 4.2 The design of a wrist mechanism housing The second example pertains to an innovative mecha- nism, a gearless pitch-roll wrist (PRW) [13], namely, a robotic device intended to produce two-degree-of-freedom rotations of a robotic gripper about a fixed point, the wrist center. Conventional means of producing such motions rely on a bevel-gear differential train, similar to those found in auto- motive driving-wheel axes. Moreover, such trains, in robotic wrists, invariably bear straight-tooth bevel gears, which are the source of noise and significant power losses. In an attempt to overcome these drawbacks, a PRW is being designed at the Robotic Mechanical Systems Laboratory, McGill Univer- sity, Montreal, as depicted in Fig. 6a, with gears replaced by cam-roller pairs. The key component of the mechanism is an array of spherical Stephenson mechanisms (SSMs) used to transmit the power from the two independent cam rotations to the gripper, as illustrated in Fig. 6b. The two cams are driven by the motion of the two roller-carrying disks of Fig. 6a, which are rigidly coupled to their respective motors. When the two face-to-face motors turn at opposite angular velocities of identical absolute values, the whole array turns about the common axis of the two cams, as a single rigid body (pitch); when the two motors turn at identical angular veloci- ties, the plane containing the four spherical-linkage centers remains stationary, but the gripper turns about its axis (roll). The array must be supported by a housing that doubles as a protection means to isolate the spherical linkages from the environment dust and dirt. One approach in designing the housing is to make use of Lamé curves [14], which are given by the implicit function x a y b p p � � 1 , (21) where p is an integer and a and b are real numbers that deter- mine the dimensions, 2 2a b� , of the box circumscribing the curve. A PRW housing design [15] is shown in Fig. 7, which is based on Lamé curves, with p � 4 for the inner surface, and p � 6 for its outer counterpart. In most common applications, Lamé curves are needed only in the first quadrant, at which we can dispense with the absolute-value signs, and rewrite eq. (21) as x a y b p p � � � � � � � � � � � � � � 1 , 0 1� �x , 0 1� �y . (22) In any event, moreover, the curvature of these curves at the points of intersection with the coordinate axes vanishes, thereby allowing for G2-continuity at the points of blending with two line segments at 90°. While Lamé curves, in the foregoing case, provide a thick- er cross section at the points of maximum curvature, a plus for the curves, the additional material, placed at a distant loca- tion from the cam axis, adds significantly to the moment of inertia of the whole device. An alternative design is thus desirable to replace the cur- rent housing shape by means of a new shape that is free of the drawbacks of Lamé curves. The new housing is to have both uniform thickness and zero curvature at the blending points with the straight bearing housings. The profile of a new design is displayed in Fig. 8a, which was generated based on non-parametric cubic splines. The numerical results, namely, the values of the �-coordinate at the unknown supporting points, are listed in Table 2. In this case, a � 198 mm, b � 98 mm, the half-length of the horizontal line being equal to 100 mm and the half-length of its vertical counterpart being equal to 20 mm. A Lamé curve with p � 4 is displayed in Fig. 8a together with the synthesized curve for comparison. The curvature distributions of the two curves are shown in Fig. 8b, which indicates that the synthesized curve has a lower maximum curvature value, implying a higher allowable bending moment. The rms value of the curvature of the synthesized curve is 0.0119 mm�1, as compared with 0.0142 mm�1 of the Lamé curve with p � 4. A new housing is currently under design. 60 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 6/2007 (a) (b) Fig. 6: An innovative PRW with cam-roller pairs: (a) the mecha- nism; (b) an array of two SSMs and their mirror-images Fig. 7: The Stephenson-linkage housing made up of two identical covers 5 Conclusions A methodology of curve synthesis with G2-continuity was proposed, based on non-parametric cubic splines. Two syn- thesis examples are provided to demonstrate the effectiveness of the methodology. The proposed methodology was applied to planar curves. It can also be used in spatial curves, if these curves are synthe- sized via their projections on the orthogonal planes [16]. For shape optimization, the procedure described herein can be integrated with CAE software. The procedure is also expected to find applications in areas such as trajectory generation and path planning for manipulators and mobile robots. References [1] Angeles, J.: Synthesis of Plane Curves with Prescribed Geometric Properties Using Periodic Splines. Computer- -Aided Design, Vol. 15 (1983), No. 3, p. 147–155. [2] Böhm, W., Farin, G., Kahmann, J.: A Survey of Curve and Surface Methods in CAGD. Computer Aided Geometric Design, Vol. 1 (1984), No. 1, p. 1–60. [3] Hosaka, M.: Modeling of Curves and Surfaces in CAD/CAM. Springer-Verlag, 1996. [4] Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. Springer-Verlag, 1993. [5] Lin, P. D., Lin, M. F.: Geometric Modelling of an Elliptic Ball-End Mill. Proc. IMechE – Part B, Vol. 219 (2005), No. 1, p. 87–97. [6] Pottmann, H.: Industrial Geometry: Recent Advances and Applications in CAD. Computer Aided Design, Vol. 37 (2005), No. 7, p. 751–766. [7] Neuber, H.: Theory of Notch Stresses: Principles for Exact Calculation of Strength with Reference to Structural Form and Material. U.S. Dept. of Commerce, Office of Technical Services, Washington, 1961. [8] Pottmann, H., Wallner, J.: Computational Line Geometry. Springer-Verlag, Heidelberg, Berlin, 2001. [9] Teng, C. P., Angeles, J.: A Sequential-Quadratic-Pro- gramming Algorithm Using Orthogonal Decomposition with Gerschgorin Stabilization. ASME J. of Mechanical De- sign, Vol. 123 (2001), No. 4, p. 501–509. [10] Bidault, F., Teng, C. P., Angeles, J.: Structural Optimiza- tion of a Spherical Parallel Manipulator Using a Two- -Level Approach. Proc. of ASME 2001 Design Engineering Technical Conferences, 2001, #DETC 2001-21030. [11] Gosselin, C.M., Hamel, J.-F.: The Agile Eye: A High- -Performance Three-Degree-of-Freedom Camera Ori- enting Device. IEEE Int. Conf. on Robotics and Automation, 1994, p. 781–786. [12] Schmidt, L. A.: Structural Design by Systemastic Synthe- sis. Proc. 2nd ASCE Conf. Electronic Computation, Pitts- burgh, Pennsylvania, USA, 1960, p. 1105–1122. [13] Hernandez, S., Bai, S., Angeles, J.: The Design of a Chain of Stephenson Mechanisms for a Gearless Pitch- -Roll Wrist. Proc. ASME Design Engineering Technical Con- ferences (DETC’04), Salt Lake City, Utah, USA, 2004, #DETC 2004-57424. [14] Gardner, M.: The Superellipse: a Curve that Lies Be- tween the Ellipse and the Rectangle. Scientific American, Vol. 21 (1965), p. 222–234. [15] Hernandez, S.: Optimum Design of Epicyclic Trains of Spherical Cam-Roller Pair. Master’s thesis, McGill Univer- sity, Montreal, Canada, 2004. [16] Angeles, J., Akhras, R., Liu, Z.: The Synthesis of Smooth Trajectories for Pick-and-Place Operations of Spherical Wrists. Mechanism and Machine Theory, Vol. 28 (1992), No. 2, p. 261–269. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 61 Acta Polytechnica Vol. 47 No. 6/2007 (a) (b) Fig. 8: (a) New housing profile and (b) curvature distribution over the synthesized curve; shown dashed is the Lamé curve i �i [mm] �i i �i [mm] �i 0 199.0075 0.0000 10 189.4187 0.0155 1 199.7687 0.0094 11 185.4056 0.0146 2 200.3707 0.0109 12 180.7433 0.0133 3 200.6720 0.0122 13 175.4735 0.0117 4 200.6039 0.0135 14 169.6913 0.0098 5 200.1041 0.0145 15 163.5592 0.0076 6 199.1170 0.0153 16 157.2998 0.0054 7 197.5937 0.0159 17 151.1588 0.0033 8 195.4919 0.0161 18 145.3498 0.0014 9 192.7760 0.0160 19 140.0142 0.0000 Table 2: Numerical results for the housing design Chin-Pun Teng Department of Technical R&D The Original Cakerie Ltd, B.C., Canada Shaoping Bai e-mail: shb@ime.aau.dk Department of Mechanical Engineering Aalborg University, Denmark Jorge Angeles e-mail: angeles@cim.mcgill.ca Department of Mechanical Engineering & Centre for Intelligent Machines McGill University, Montreal, Canada 62 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 6/2007 Appendix The linear relationship between � and ��� is expressed as A C�� �� �6 where A � � � 2 1 0 0 0 0 1 4 1 0 0 0 0 1 4 1 0 0 0 0 1 4 1 0 0 0 0 � � � � � � � � � � � � � � � � � � � 1 4 1 0 0 0 0 1 2� � � � � � � � � � � � � � � � � � � � � � � � � � � , C � � � � 1 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 11 � � c � � � � � � � � � � � � � � � � � � 1 0 0 0 0 1 2 1 0 0 0 0 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� ��cn n , (23a) in which �� � �n n 2 and c tA 11 1� � � �� , c tn n B �� �� � � �1 �� . (23b) Similarly, the linear relationship between � and ��� is P Q� �� � where P � � � � �1 0 0 0 0 0 1 4 1 0 0 0 0 1 4 0 0 0 0 0 1 4 1 0 0 0 � � � � � � � � � � � � � � � � � � 0 1 4 1 0 0 0 0 0 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � , Q � � � � 1 0 0 0 0 0 3 0 3 0 0 0 0 3 0 3 0 0 0 0 3 0 3 0 0 tA � � � � � � � � � � � � � � � � � � 0 0 3 0 3 0 0 0 0 0 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � �tB (23c) It is noteworthy that when either �A or �B is equal to � 2, Q becomes singular. In our procedure, however, we need not invert Q, for all we compute is � � �� �P Q1 . Table of Contents Signal Separation in Ultrasonic Non-Destructive Testing 3 V. Matz, M. Kreidl, R. Šmíd Design and Validation of a Probe for Spatially and Temporally Resolved Measurements of Vorticity and Strain Rates in Compressible Turbulence Interactions 10 S. Xanthos, M. Gong, Y. Andreopoulos Dynamics of the Flow Pattern in a Baffled Mixing Vessel with an Axial Impeller 17 O. Brùha, T. Brùha, I. Foøt, M. Jahoda Dimension of Fracture Surfaces 27 T. Ficker Statistics of Electron Avalanches and Streamers 31 T. Ficker Influence of Oxidation Media on THE Transport Properties of Thin Oxide Layers of Zirconium Alloys 36 H. Frank EU Emissions Trading Scheme as Implemented in the Czech Republic 43 T. Chmelík, P. Zámyslický Simulation Environment for Artificial Creatures 46 K. Kohout, P. Nahodil Finite Automata Implementations Considering CPU Cache 51 J. Holub Shape Synthesis in Mechanical Design 56 C. P. Teng, S. Bai, J. Angeles