Solution Geometry for Dyads and Triads

Transcription

1 Solution Geometry for Dyads and Triads by John R. Mlinar 3M Company St. Paul, MN and Arthur G. Erdman Department of Mechanical Engineering University of Minnesota

2 Abstract If a design angle is varied over all possible values when designing a dyad for three design positions, circle solution loci are formed. These loci are known as M- and K- circles. This paper presents the explanation of why regions without solutions, known as forbidden regions, exist for the case of path generation with prescribed timing and not for the case of motion generation. The extension of M-K circle theory to the linear solution of triads is also presented. The result of this work is circular solution curves and conditions for forbidden regions.

3 Introduction This paper begins with a brief introduction of the geometry of circles and the theory of M- and K-circles. Previous work with M- and K-circles showed there are regions without solutions. These forbidden regions exist for the case of path generation with prescribed timing and not motion generation. A geometric proof is presented that finally explains why this is so. A proof that the centers of K- circles for the case of path generation with prescribed timing is a hyperbola is also presented. Finally, M- and K- circle theory is expanding to triads. Using the direct linear method for triads, circular solution loci are presented along with the conditions for forbidden regions. 3

4 The Geometry of a System of Circles Let C and C represent two different circles. A system of circles is defined by letting C = λc + µc = 0 () where λ and µ are real and both are not zero. This system of circles is called a pencil of circles [3,4]. There are three types of pencils, elliptic, parabolic, and hyperbolic [3]. In the elliptic type, all the circles of the pencil go though two points as illustrated in Figure. Figure shows there is a single point common to all circles in a parabolic pencil. Finally, Figure 3 shows that with the hyperbolic type, there are no points in common with all the circles in the pencil. In the Figures through 3, there is a vertical line drawn near the center of each figure. This is the radical axis of the pencil. The radical axis is the case where λ µ = [4], or the radius of the circle is infinite. In the elliptic-pencil case, the radial line passes through the two fixed intersection points. For the parabolic pencil, the radical line proceeds through the single point of intersection. For a hyperbolic pencil, the radical line lies between the circles. 4

5 Review of M- and K-Circle Theory When designing a dyad for three design positions, the system of vector dyad-equations is linear []. ( e ( e iβ iβ3 ) ) ( e ( e iα iα 3 ) W δ = ) Z δ 3 () If one of the angles α, α 3, β, or β 3 varies from 0 to π, the solution points for the vectors W and Z lie on circles known as M- and K-circles respectively. The existence of solution circles (M- and K-circles) was first presented by Loerch [6]. For the case of motion generation, the α angles are constant and one of the β angles is used to generate the M- and K-circles (β varied from 0 to π). When β 3 is the trace or varied angle, all the M-circles intersect at the poles P 3 and P 3, and all the K-circles pass through the poles P 3 and P 3 [6]. When β is the trace angle, all the M-circles intersect at the poles P and P 3, and all the K-circles pass through the poles P and P 3. In the motion-generation case, the solutions for the M- and K-circles generate an elliptic pencil of circles. For the case of path generation with prescribed timing, the β angles are constant and one of the α angles is varied. When α changes, all the M-circles intersect at the pseudopoles S and S 3 [6]. Pseudopoles are defined as the poles obtained when the motion generation rotations, α, are set equal to the rotations of the grounded link, β []. When 5

6 the angle α 3 is varied, all the M-circles pass through the pseudopoles S 3 and S 3. For path generation with prescribed timing, the M-circles form an elliptic pencil of circles. The solutions for the K-circles have no stationary intersections; therefore, no pencil of circles is formed. 6

7 The Geometric Explanation of the Existence of Forbidden Regions in M- and K-Circle Theory When plotting all possible K-circles for path generation with prescribed timing, there are two circular regions without K-circles. These regions are called forbidden regions [6]. This section presents the geometric conditions that are necessary for the existence of these forbidden regions. Figure 4 is a sketch of an M- and K-circle and the first forbidden region. This region always surrounds the first design position, PP. The figure also shows one dyad solution, W and Z. This particular dyad has the minimum value for the length of the vector Z, Z min. For any particular K-circle, there is a single Z min and Z max. If one plots all possible K-circles (by varying α and α 3 from 0 to π), the forbidden region that surrounds PP is bounded by two circles. These circles, centered at PP, have radii equal to the largest Z min and the smallest Z min of all possible K-circles. This is illustrated in Figure 5. The explanation for the existence of the forbidden region begins with the question, what is required for Z min to equal zero? When Z = 0, only the vector W is required to move through the three design positions (see Figure 6) and the angles α and α 3 can take on any value. For Z = 0, the standard-form dyad equations become i ( e β ) = δ i ( e ) β 3 = δ 3 W (3) W (4) 7

8 This is a system of two vector equations and one vector unknown, W. Unless the determinant of the augmented matrix is zero, this system of equations has no solution. For that reason, the angles β and β 3 may not be arbitrarily selected. When the values of β and β 3 are compatible, Z is zero for all values of α and α 3. The radii of the forbidden region and all K-circles is zero. This leads to the following theorem [8]. Theorem : When generating K-circles for path generation with prescribed timing, a forbidden region surrounding the first precision point, PP, will exist if Z min > 0 for all possible values of α, α 3, β, and β 3. When Z min equals zero, the values of β and β 3 are such that a single grounded vector passes through the three design positions. The angles α and α 3 may have any value. The radii of the forbidden region and all K-circles equal zero. In the case of path generation with prescribed timing, the M-circles intersect at the pseudopoles S 3 and S 3 for α specified and at S and S 3 for α 3 specified. When Z equals zero, these pseudopoles are coincident with the point where the vector W pivots (see Figure 6). The three coincident points define a circle with zero radius; consequently, the 8

9 radius of all the M-circles must be zero. This results in Theorem [8]. Theorem : When Z = 0 for the case of path generation with prescribed timing, the radius of all M- circles is equal to zero. For there to be no forbidden region, the angles β and β 3 must be compatible. This requirement is why the first forbidden region exists when plotting M- and K-circles for path generation with prescribed timing and not for motion generation. In the motion-generation case, the angles β and β 3 vary over all possible values. Thus, compatible values exist and there will be no forbidden region. In the pathgeneration case, there is single set of values for β and β 3. Unless the values are compatible, there will be a forbidden region surrounding the first design position. There is a second forbidden region when plotting K- circles for path generation with prescribed timing [6]. This region never includes the first precision position and does not intersect with the first forbidden region. Of all the possible K-circles for a path-generation-with-prescribedtiming problem, let us select the one that has the smallest Z max where Z max is the longest of all possible Z vectors for a given K-circle. The vector Z max originates on the K- circle and terminates at precision point (PP ). Since all K-circles are tangent to both of the forbidden regions [], the vector Z max passes through the center of the K-circle (see Figure 7). As the Z max point on the K-circle is the furthermost point from PP, there can be no solutions 9

10 further away along the vector Z max. Thus, the tail of the smallest of all possible Z max 's is the closest solution point to the second forbidden region. This leads to the following theorem [8]. Theorem 3: If there is a forbidden region around the first precision point, PP, there will also be a second forbidden region. The nearest point from PP to this second forbidden region will be equal to the smallest Z max for all possible values of β, β 3, α, and α 3. If Z = 0 leads to such significant results, what does the solution W = 0 lead to? The solution of W = 0 is similar to the result of Z = 0 except for the angles α and α 3 which must be compatible in order to obtain a solution. The solution W = 0 occurs where the M- and K-circles intersect. This solution is possible for all path generation and motion generation problems. 0

11 K-Circle Locus for Path Generation with Prescribed Timing K-circles for path generation with prescribed timing do not generate a pencil of circles. However, the K-circles for path generation with prescribed timing do form a pattern that fills the space between the two forbidden regions. The following discussion proves the supposition by Loerch [6] that the centers of all K-circles form a hyperbola. Figure 8 shows the two forbidden circles and one K- circle. The equations for the sides A and B of triangle ABC are A = r + (5) r k B = r + (6) r k where r is the radius of the first forbidden circle, r the radius of the second forbidden circle, and r k is the radius of the K-circle. The difference of these is ( r + rk ) = r = A B = r (7) + rk r Since r and r are each constant, is constant. The locus of the K-circle centers are those points whose difference from two fixed points (the centers of the forbidden regions) is a constant. This is the definition of a hyperbola [5]. Since we know that the centers of the K-circle lie on a hyperbola, the equation for the locus of the centers is readily determined. The equation of this locus is

12 where x a b y = (8) r + r a = (9) ( r ) k + r rk + rrk b = (0) 4 This locus is displayed in Figure 9.

13 3 The Extension of M- and K-Circle Theory to Triads A triad is a chain of three vectors connected tail-totip (see Figure 0). The first vector in the chain (attached to ground) is W, the middle vector is V, and the final vector is Z. The angular position of these vectors at the design positions are αi, βi, and γi, respectively. The vector equation for a triad is developed in a similar fashion as for the dyad. Figure illustrates a triad in two design positions. This figure shows a more general case where the vector W is not grounded. By rearranging terms of the vector loop equation, the standard form triad equation is obtained. ( ) ( ) ( ) h e Z e V W e i i i = + + δ γ β α () Writing this equation for the positions 3 and 4 results in, = h h h Z V W e e e e e e e e e i i i i i i i i i δ δ δ γ β α γ β α γ β α () Equation is a linear system of three vector or six scalar equations. By selecting all the angles α j, β j, and γ j, the vectors W, V, and Z can be solved for directly. The first author calls this the direct linear method for triad synthesis [8]. Figure shows a Stephenson-3 which consists of one triad, W-V-Z, and two dyads, A-B, and, C-D. This linkage is synthesized using any of the sequences list in Table. The

14 first sequence is utilized for this example. The design positions are presented in Table. Table - Design Sequences for Stephenson-3 using Direct Linear Method Step Step Step 3 Dyad Solution Method C-D W-V-Z A-B motion generation W-V-Z A-B C-D motion generation W-V-Z C-D A-B motion generation Table - Design Positions for Stephenson-3 Synthesis Example. X Y q o o o The first step is to synthesize the dyad C-D. Using the standard form method [] with β, β 3, and β 4 equal to 0 o, 6.56 o, and 8.90 o, respectively, the vector C has the value of 3.546e i( ) and D is equal to 5.7e i(96.49). The next step is to solve for the triad W-V-Z. The displacements of the tail of the triad, h j, are equal to zero. The displacements of Z (δ j, γ j ) equal the design positions given in Table. The rest of the triad angles, α j and β j, are free choices for the designer. In this example the angles α j are 0 o, 35 o, and 65 o. The angles β j are 5 o, o, and 0 o. Solving Equation gives W =.96e i(-33.46), 4

15 V = 7.003e i(93.038), and Z = 0.04e i(-38.05). The final step in our example of a Stephenson-3 linkage is to solve for the dyad A-B. For this dyad the displacements, δ j, are equal to the motion of the tail of Z (the design positions minus Z). The motion-generation angles, α j, are equal to the angles of the vector V. Using the standard form method [] with β, β 3, and β 4 equal to 80 o, 87 o, and 85 o, respectively, results in the solution of A =.36e i(6.768) and B = 33.86e i(- 9.43). Figure 3 illustrates the completed Stephenson-3 linkage. Circular Solution Loci for Triads The locus of solutions for a three-precision-position dyad is a circle where one of the angles (α or β) varies from 0 to π. [6]. The locus of ground pivots is the M- circle and the locus of moving pivots is the K-circle. In the case of a triad with four design positions, the highest number of design positions that still allows for a linear solution, there are also circular solution loci [8]. Figure 4 illustrates the circular solution loci for the points M, V, and K. Figure 4 was generated using the data given in Table 3. The M points are the points for the tail of W, the V points are located at the tail of V, and the K points are found at the tail of Z. 5

16 Table 3 - Design Data for Triad Example(hi = 0) i di ai bi gi + i by i by i by 0-5 o 0 o 0 o 40 o 5 o 60 o If Equation is solved for the vector Z we obtain Z b c + b c 3 3 = (3) c 4 + b c where, b ( d h ) i = and, i i c a a 3 = a (4) a3 c a a = (5) 3 a a3 a aa3 a aa3 c = 3 a a (6) a3 a3 a3 a3 6

17 aa3 aa33 aa33 aa3 c = + 4 a a3 a3 a (7) a3 a3 a3 a3 a = e iα (8) 3 a = e iα (9) 4 a = e iα (0) 3 a = e iβ () 3 a = e iβ () 4 a = e iβ (3) 3 a = e iγ (4) 3 3 a = e iγ (5) 3 4 a = e iγ (6) 33 When α is the angle that varies, Equation 3 becomes Z Aθ = C θ + + B D (7) iα where θ = e and, b a b a a a a A + a = (8) 3 ( a ) a ( a ) 3 3 a 3 a a 3 b3aa B = b a b a + A a (9) 3 a3 c a a = (30) ( a ) a3 3 a a a 7

18 D = a a33 a3aa3 a3 + a3a C (3) a a 3 3 Equation 7 is a Mobius or bilinear transformation of θ into Z. This transformation maps circles into circles and lines into lines [3,4]. θ is a unit circle; therefore, the Mobius transformation takes the circle θ and transforms it into the circle Z. This proves that as α varies from 0 to π a circle is generated. A similar set of equations may be generated for V and W showing that the solution loci of V and W are circles. 8

19 Forbidden Regions for Triads One of the more interesting features of M- and K-circle theory is forbidden regions for the K-circles. These regions are two non-intersecting circles in the x-y plane and are found for the case of path generation with prescribed timing. When plotting the M-, V-, and K-circles for triads, forbidden regions also exist [8]. The conditions for these regions to exist are different for triads than for dyads. Figure 5 shows a plot of forbidden regions for the K- circles. It was generated using the data given in Table 4. Table 4 - Design Data for Triad (hi = 0) i di ai bi gi + i 0 o -5 o o by i 5 o 0 o o by i 0 o 5 o o by For there not to be a forbidden region around precision position (PP ), the vector Z must equal zero for some combination of the angles α, β, and γ. If the vector Z is zero, then only the dyad W-V is required to go though the four design positions. We know from Burmester theory, that for a dyad to go through four design positions, the angles α and β must be compatible. This leads to the following theorem. 9

20 Theorem 4: A forbidden region around the first design position will exist for the case of a triad when generating K-circles unless there is some combination of the angles α j and β j such that a dyad could be found to pass through the same four design positions. This theorem is applied to the following 3 cases. Case : If the γ and β angles are fixed and one of the α angles varies, then a forbidden region surrounding PP will exist when plotting the K-circles unless the remaining α and β angles allow a dyad solution for some value of the varied angle. In general, this is unlikely. If all the α angles vary, there will then be some combination of the α's that is compatible with the β's and the forbidden region will not exist. Case : If the γ and α angles are fixed and one of the β angles varies, a forbidden region surrounding PP will exist when plotting the K-circles unless the remaining β and the α angles allow a dyad solution for some value of the varied angle. If all the β angles are varied, there will then be some combination of the β's that is compatible with the α's and the forbidden region will not exist. Case 3: If the α and β angles are fixed and some of the γ angles change, a forbidden region is expected no matter which of the γ angles vary unless the α and β angles allow for a dyad solution. If the set of α and β angles is 0

21 compatible for a dyad solution, then the vector Z will be zero for all values of γ j. This leads to a K-circle with a zero radius. In addition, since there is only one dyad solution for a given set of α and β angles, the M- and V- circles will collapse to single point. Conclusion When a dyad is designed using three design positions and one of the design angles is varied over all possible values, circle solution loci are formed. These loci are known as M- and K-circles. This paper covered additions to the theory of M- and K-circles. The first was a geometric proof as to why there are forbidden regions when generating K-circles for the case of path generation with prescribed timing and no forbidden regions for the case of motion generation was presented. The second was a proof the centers of K-circles for the case of path generation with prescribed timing lie on a hyperbola. The expansion of M- and K-circle theory to the linear solution case for triads was also discussed. The result of circular solution loci for triads was presented along with the existence of forbidden regions for triads.

22 Bibliography Sandor, G. N., and Erdman, A. G., Advanced Mechanism Design: Analysis and Synthesis, Vol., Prentice-Hall, Englewood Cliffs, NJ, 984. Beyer, Rudolf, The Kinematic Synthesis of Mechanisms, English translation by Kuensel, H., Chapman & Hall Ltd, London, Schwerdtfeger, Hans, Geometry of Complex Numbers, Dover Publications, New York, Pedoe, Dan, Geometry, A Comprehensive Course, Dover Publications, New York, Salmon, George, A Treatise on Conic Sections, Sixth Edition, Chelesa Publishing, New York, pp Loerch, R. J., "Extensions of Revolute Dyad Synthesis for Three Finitely Displaced Positions", Master's thesis, University of Minnesota, Minneapolis, Minnesota, Hunt, K. H., Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, Mlinar, John R., An Examination of the Features of the Burmester Field and the Linear Solution Geometry of Dyads and Triads, Dissertation, University of Minnesota, 997

23 Figure - Elliptic Pencil of Circles 3

24 Figure - Parabolic Pencil of Circles 4

25 Figure 3 - Hyperbolic Pencil of Circles 5

26 Forbidden Region PP Z K-Circle W M-Circle Figure 4 - M- and K-Circles and the First Forbidden Region. 6

27 PP Forbidden Region Bounding Circle Figure 5 - Forbidden Circle Boundaries 7

28 d 3 d b 3 W b Figure 6 - Case of Z = 0 8

29 Figure 7 A K-Circle and the Second Forbidden Region 9

30 Forbidden Circle A Forbidden Circle C K-Circle B Figure 8 - K-Circle and Both Forbidden Regions 30

31 Figure 9 - Sketch of the Locus of K-circle Centers for Path Generation with Prescribed Timing 3

32 Z g i b i V W a i Figure 0 - Triad with Grounded First Vector. 3

33 Z e ig d V e ib V Z W e ia W h Figure - Triad in Two Design Positions. 33

34 V Z B D A C W Figure - Stephenson-3 Linkage 34

35 Figure 3 - Solution to Stephenson-3 Example 35

36 Figure 4 - Circular Solution Loci for Triads. 36

37 Figure 5 - Forbidden Regions for K-Circles 37