- Email: [email protected]
Issues in geometric synthesis of mechanisms
Mechanism and Machine Theory 39 (2004) 1321–1330
Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Issues in geometric synthesis of mechanisms Dibakar Sen *, Vyankatesh Joshi Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India Accepted 25 May 2004
Abstract The paper presents a systematic non-iterative procedure for a priory design of link shapes for interference-free motion of a planar mechanism. We define the notions of unilayer and multi-layer links and show that generality of shapes that can be conceived depends upon the designerÕs choice about the physical implementation of the joints. We propose a matrix-based procedure for the determination of the nature of interaction among the links and joints during motion where each link can be multi-layer. Techniques for geometric synthesis as in the case of unilayer links could then be used by processing one layer at a time. The concept is demonstrated for the geometric synthesis of a six-bar Stephenson mechanism with one multi-layer link. A theory of simultaneous synthesis, using the concept of compliant motion, is also presented which does not have the limitations of order dependence inherently present in sweep-based methods. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Geometric synthesis of mechanisms involves determination of the spatial distribution of material associated with the links and joints under different design considerations. The design considerations can be non-interference of the links and joints of the mechanism itself or of the links and objects around the mechanism, material distribution for given inertial properties such as mass, centre of gravity, moment of inertia etc. The desired output of a geometric synthesis procedure *
Corresponding author. Tel.: +91 80 2293 3230; fax: +91 80 3600 648. E-mail address: [email protected] (D. Sen).
0094-114X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.05.018
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is an engineering description of the links and joints of the mechanism generated from the kinematic description of the mechanism. Work available in literature in this area is rather limited and they consider the question of interference of links alone. Most of the existing literature deals with relatively simple constructional elements namely, flat links of uniform thickness and revolute pairs in the pin-in-hole configuration [1–4]. For such systems even in the case of a four-bar mechanism, there are eight different stacking orders one can try to assemble the mechanism as shown in Fig. 1. The annotations shown in Fig. 1(a) are valid for all mechanisms in Figs. 1–3. The one shown in Fig. 1(h) is physically infeasible with flat links because it is same as the case in Fig. 1(f) except that near joint-4, link-1 is below link-4 and as shown in Fig. 2 it requires a bent link for assembling the links. Moreover, the cases in Fig. 1(b), (c), (e) and (g) cannot have full rotatability due to link or joint interferences, as shown in Fig. 3. Hence for geometric synthesis of an arbitrary mechanism one must systematically go about determining the possible stacking orders of links, determination of structural infeasibilities and design of interference-free link-shapes. Determination of feasible layer assignments has been dealt with in [2] and efficient determination of the interference-free link-shapes has been dealt with in [3].
Fig. 1. Stacking order of links in four-bar mechanism.
Fig. 2. Four-bar mechanism with a bent link: (a) link interference, (b) joint interference, (c) joint interference, (d) link interference.
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Fig. 3. Interference of links and joints: (a) link interference, (b) joint interference, (c) joint interference, (d) link interference.
The concept of layer is used to determine the combinatorial arrangements of links and the concept of sweep, as used in [1,5] is used to computationally determine the ‘‘prohibited areas’’ in the lamina associated with each link. The shape of a link obtained out of the feasible region is considered feasible. The configuration space based methods for the computer aided generation of geometric details of objects under given motion can also be found in [6]. This method is particularly useful for mechanisms with higher pairs, intermittent motions etc. Geometric synthesis for specified location of the centre of gravity has been dealt with in [7] which we envisage as a step after geometric synthesis for non-interference of links. The issue of non-interference arises in robotic path planning where the motion is synthesized for given part models and target location, which in a sense the inverse of the present problem where non-interfering geometric extent is to be determined for the given motion. The present paper deals with the issues related with more detailed a priori geometric design for interference-free motion in planar mechanisms. For that purpose we propose a more general model of the links and joints so as to get a wider variety of combinatorial arrangements of the links, which increases the feasibility of a design. We also present a theoretical investigation on the simultaneous synthesis of link geometry using the concept of compliant motion.
2. Layer based approach to geometric synthesis In planar mechanisms points on links move on parallel planes; hence the shape of the intersection of physical links with any plane parallel to these planes also do not change. A layer is defined as the region of maximum width within which the geometry of none of the links in the mechanism changes. A link is called unilayer if it spans only one layer. A link is called multi-layered if it spans more than one layer. In Fig. 2(d), link-4 is multi-layer (spans five layers) and all other links are unilayer. Physically a revolute pair can be implemented in many different ways. It can be the traditional ‘‘pin-in-hole’’ configuration where a circular cylinder passes through the links forming the pairs, which in turn puts the constraint that two unilayer links forming a revolute pair cannot lie in the same layer. For pairs not requiring full rotation, ‘‘in-plane hinges’’ can be designed. Implication of this on geometric synthesis will be discussed later. The ordinary revolute pairs can therefore be represented in the link–link adjacency matrix (L) defined as follows: Lij = 0 if link-i and link-j are not adjacent, Lij = +1 if link-i is above link-j and Lij = 1 if link-i is below link-j. Thus L is a n · n skew-symmetric where n is the number of links. Number of non-zero elements above the diagonal of L is equal to the number of joints (say J) in the mechanism. Arrangements represented by L
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and L are only mirror images of one another, they are considered to be the same. Thus it is easy to see that there are 2J1 distinct ways of assigning +1 or 1 to the non-zero elements of L. Each of these assignments leads to a specific stacking order of the links called a layer assignment. Hence L is called layer assignment matrix. A layer index of a link is nothing but an integer associated with each link in a mechanism indicating its height with respect to the link that is there at the bottom of the stack. A matrix-based method is presented in [2] for systematic enumeration of layer indices of links of any mechanism. The layer assignment matrix defined above can also be represented as a directed graph. It has been proved in [2] that for a layer assignment to be valid (structurally feasible), the directed graph must not contain any (oriented) cycle. In the matrix based method from the given link–joint adjacency matrix (M), link–link adjacency matrix (Lu) is derived which in turn gives different layer assignment matrices (L). From each L we can derive a link– layer matrix (B). The layer–joint matrix (A) is obtained as A = BTM. Suppose link-l with layer index-i is incident on joint-j, then Aij = 1 otherwise Aij = 0. At each joint at least two links are incident which need not be in consecutive layers. However, since a single straight pin is used to construct the pair, the pin will pass through all intermediate layers. Thus, the geometric designs of the links in the intermediate layers are affected by all the joints (pins) that pass through these layers, in addition to the design of other links in the same layer. This situation is not dealt with in [1]. To capture this in the matrix representation we modify A by replacing 0Õs bracketed by two 1Õs in each column of A with 2. This gives us the modified layer–joint matrix (A*). To obtain the modified link–joint matrix (M*), which indicates the joints that affect the geometric design of links is obtained by comparing BA* with M and elements with inconsistencies being replaced with 2. Fig. 4 shows the above matrices for a Stephenson chain. To design a link, we consult M* to identify the joints affecting it, indicated by columns containing 2 in its row. We also determine other links in the same layer by consulting B. The locus of these joints and links already designed in its layer determines the region (infeasible area), which the link in context cannot use. The infeasible areas are determined by either explicit sweep [1] or implicit sweep [3]. The link is designed using the remaining area in the domain. From M* it can be observed that the design of link-4 is affected by the joints 3, 6 and 7 and B indicates that it is also affected by the design of link-5. If link-4 is designed earlier, only the joint pins of link-5 are considered. This introduces order dependence in the design of links in a layer. If there are n links in a layer, then there are n! orders in which the links can be designed many of them are
Fig. 4. Matrices for determination of links–layers–joints interaction.
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Fig. 5. Geometric synthesis of unilayer links in a StephensonÕs chain: (a) mechanism with kinematic dimensions, (b) layer assignment as per Fig. 4, (c) links after geometric synthesis, (d) links after geometric synthesis.
geometrically infeasible due to the ad hoc decision of consideration of only joint pins for the links designed later and the preference given to convex shape of the links. The structural infeasibility (e.g. Fig. 1(h)) is taken care of by the use of multi-layer links and the order dependence can be taken care of by the simultaneous synthesis procedure discussed later in the paper. Neither multi-layer links nor simultaneous synthesis has been reported in literature. Designs obtained for the Stephenson chain for two different kinematic dimensions are illustrated in Fig. 5.
3. Multi-layer link synthesis The constraint of flat links of identical thickness and pin-in-hole type of joints limits the variety of designs, which are often found in existing machinery. For multi-layer links a different link model is required. This in turn necessitates a more general matrix based method for the synthesis. 3.1. Multi-layer link model For multi-layer links, joints on the links are allowed to be at different layers, since layer index for links does not make much sense here. Fig. 6 illustrates the different ways of modeling a multilayer binary link: (a) is a unilayer link, (b) is a multi-layer link with both the joints in the same layer, (c) and (d) have the joints in adjacent layers and (e) has two joints in non-adjacent layers. This example demonstrates that • multi-layer links span a range of layers; • minimum range of layer indices for a link is given by the indices of the layers in which the joints are there; • joints can be there in adjacent or non-adjacent layers.
Fig. 6. Link shapes for a multi-layer binary link.
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The points above are to be taken as the necessary conditions of validity of layer-assignments for the links and layers. It can also be observed that a multi-layer link can be viewed as a stack of unilayer shapes glued over a common area on the interface between two successive layers. The presence of a finite common interfacial area between the two parts of a link in two consecutive layers is the condition of validity of a design obtained one layer at a time. 3.2. Alternate joint models The problem of structural feasibility is not there for mechanisms with multiple links. However, the conditions of validity mentioned above does not preclude the situation where two adjacent layers are assigned the same layer indices. This can be handled by either using the signed adjacency matrix discussed for unilayer links or by developing alternate models for the in-plane revolute pairs (Fig. 7). 3.3. Designing with a multi-layer link Relaxing the condition of flat links of uniform thickness (unilayer links) and the pin-in-hole type of revolute pairs makes the above matrix-based procedure need to be modified. A threedimensional matrix called Layer–Link–Joint (LLJ) matrix is used to represent the information of the multi-layer link–joint relationship. LLJijk = 1 implies Link-j is incident on Joint-k in Layer-i. From LLJ, the modified LLJ (MLLJ) is constructed as follows. Initially MLLJ = LLJ. Then, for a given k (joint) MLLJ is scanned over all i and j values; identify the two rows where there is a non-zero element, say they are i1 and i2; then for all the columns (j-values) in rows between and including i1 and i2 replace all 0Õs with 2Õs. Geometric synthesis is then carried out layerby-layer as in the case of unilayer links: in each layer find the joints affecting it; also find the links in the layer. For each of these links compute the swept areas of the joints not belonging to this link and those of links already designed in this layer. Compose geometry for the link out of the remaining area in the layer. Fig. 8 shows an example of a Stephenson chain with only one (ternary) link spanning layer-2 to layer-5.
4. A theory for simultaneous synthesis The shape of links one obtains using the concept of sweep, as described above, is dependent on the order in which the links are designed because the methods are inherently sequential. Although we would prefer to allocate maximal regions to each link for subsequent optimization of the mechanism, we have to remain conservative because, allocation of more regions to a link designed ear-
Fig. 7. Joint models for revolute pairs.
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Fig. 8. Geometric syntheses with one multi-layer link: (a) mechanism, (b) layer 2, (c) layer 3, (d) layer 4, (e) layer 5.
lier would produce larger prohibited areas for a link designed later which intern could render the design infeasible. Simultaneous synthesis refers to any procedure of designing link geometry in which the geometry of all the links in a layer is obtained simultaneously. In the following we describe a method of obtaining the geometry of two links simultaneously. 4.1. Condition for non-interference of two links Given two motions M1(t) and M2(t), we can identify two coordinate frames F1 and F2 (Fig. 9(a)) undergoing the respective motions with respect to a global coordinate frame, G. Then a point P in G can be identified in F1 and F2 as P1(t) and P2(t). Thus P represents a point common to both the moving bodies B1 and B2 associated with F1 and F2 at time t. This is physically possible only when P is situated on the boundaries of B1 and B2 (Fig. 9(b)). Let V1(t) and V2(t) be the velocities of P1 and P2, respectively; L be a line that locally separates B1 and B2; V t1 and V n1 be the components of V1 along and normal to L; V t2 and V n2 be the components of V2 along and normal to L; DV n ¼ V n1 V n2 . If V n1 and V n2 both pointing inside B2, DVn < 0 imply the two bodies are going to separate, DVn > 0 imply the two bodies are going to interfere, Vn = 0 imply the two bodies are going to slide past each other in the next instant. Thus the condition of non-interference of motion is DVn 6 0. This is similar to the law of conjugate profiles but without the constraint of pure rolling.
y2 M 2
y M1
o2
F2
x2 t
F1 P2
P1
V1
B1
V2
(a)
t V1
n
P P
o
V1
x
Fig. 9. Analysis of interference.
B2
t
V2
(b)
V2
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4.2. Locus of singular points When two bodies separate during motion (DVn < 0) it indicates that it is possible to enhance the extents of the bodies such that it still remains in contact. Thus for inclusion of maximum material in the moving bodies, Vn(t) = 0 should be satisfied for all t. Locus of the points satisfying this condition is called the locus of singular points (LSP). In [5] the concept of unsweep to enumerate the maximal area capable of undergoing a given motion within a constrained space is described. However, if more than one body lies in the constrained space, the method cannot be used. Following is a method proposed for tracing the LSP using the properties of instantaneous centres. Let L1, L2, . . . ,Ln be the lines joining the instantaneous centres of rotation (called lines of centroes) of F1 and F2 at discrete times t1, t2, . . . ,tn, respectively in sequence. Then, if P lies on Li, then V1(ti) and V2(ti) are parallel to each other and both perpendicular to Li which can be taken as the tangent at the point of contact between bodies. By tracing the corresponding positions of P on all these lines, locus of P can be obtained on G. This locus of singular point is a curve, C, orthogonal to the family of lines joining the corresponding points on the moving centroes of F1 and F2 with respect to G and it represents the locus of the point of contact of the two bodies during the compliant motion. For discrete time instances, intervals of C between two successive lines, say Li and Li+1 can be represented as an arc with centre at the point of intersection Xi between Li and Li+1. Thus, from the sequence of lines of centroes we can generate X1, X2, . . . , Xn1 from which piecewise circular approximation of C can be graphically constructed. Determination of Xi is difficult when Li and Li+1 are almost parallel. Let us define Qi (Fig. 10), the point on the circle, equidistant from Li and Li+1, the corresponding points on the lines being Pi and Pi+1. The sequence Pi Qi Pi+1 with i varying from 1 to n gives the sequence of triplets of points for the construction of the arcs. Using the notion of inverse motion, the inverse map of the curves could be traced on F1 and F2 to get the family of boundary curves of B1 and B2. For each point, P1, on a L1, we get an instance of B1 and B2. A one-dimensional optimisation problem then could be set up which would search the best combination of B1 and B2. The sequence of points is determined as per the following algorithm (derivation is omitted for brevity).
4
Fig. 10. Construction of points on LSP.
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4.3. Algorithm for circular approximation of LSP 1. 2. 3. 4. 5. 6. 7. 8.
Do step-2 to step-8 for i = 1 to i = n 1. Choose any Pi on Li. Compute P aiþ1 where the perpendicular on Li at Pi meets Li+1. Compute ui the angle between Li and Li+1. Qi ¼ P i þ ðP aiþ1 P i Þ(cos ui)/(1 + cos ui). Compute P biþ1 the foot of the perpendicular from Pi on Li+1. P iþ1 ¼ P biþ1 þ ðP aiþ1 P biþ1 Þ(cos ui)/(1 + cos ui). Qi ¼ ðP i þ P iþ1 Þ=2 þ ðQi ðP i þ P iþ1 Þ=2Þðtanðui =4Þ= tanðui =2ÞÞ.
4.4. Issues in simultaneous synthesis The nature of LSP: By the above method we can construct circular arc approximation of the LSP. Taking a different location for Pi on Li we generate another sequence of circular arcs whose centres are also at Xi. Thus the families of curves that can be generated are merely offsets of each other. We do not know whether the actual (analytical) loci of singular points also are offsets to each other. Since we do not expect this to be so; because, instead of circular arc if we use an ArchimedesÕ spiral or a cubic curve, we can get a different curve still satisfying the orthogonality condition. The locus of point of contact of the relative centrodes (pitch curve) of F1 and F2 is also a LSP and it is unlikely to be a simple offset of any of the curves. Moreover, the boundary curves of the B1 and B2 must be simple which the above method does not guarantee; implications of a possibly self-intersecting LSP need to be looked into. The above method does not work F1 and F2 both have pure rotation motion. In that case we can use the theory of conjugate profiles for construction of B1 and B2 with the pitch point moving on the line of centres as dictated by the given angular velocity ratios of F1 and F2. Conditions of validity of a boundary curve are (a) B1 and B2 should lie on the two sides of the common tangent in a consistent manner (if B1 is to the left, it should always remain so and vice versa) and (b) B1 and B2 should connect all of their respective joint pins (for multi-layer links union of the shapes designed in all the layers should be considered). The pitch curve always passes through some of the joints in the link. Hence, this partitioning, even if it gives a pure rolling motion between F1 and F2, is not acceptable. Geometric synthesis is about the sharing of the material in a lamina by different links in the lamina (layer). When the material not associated with any link in a layer is minimized, we say that links are utilizing the maximal area in the lamina. This is the condition of optimality. In the present case, out of all the valid partitioning of a domain into B1 and B2 one that satisfies the above maximality criteria is said to be optimal. Since each C is associated with a point on Li, search for optimal geometry can be formulated as a one-dimensional optimization problem. Work on all the above aspects of simultaneous synthesis are under investigation by the authors. 5. Conclusions Geometric synthesis of mechanisms is essential for conversion of the results of kinematic synthesis into realizable artefacts. This paper discussed different issues related to the a priori
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geometric synthesis of planar mechanisms with revolute pairs. Using multi-layer links limitations from structural infeasibility of unilayer designs could be overcome. Potential of multi-layer links in generating much wider variety of link shapes have been discussed. A theory of simultaneous synthesis has been introduced for generation of the family of geometries, which would move without interference. Various issues related to multi-layer links and simultaneous synthesis have been discussed.
References [1] Z.K. Ling, T.R. Chase, A priori geometric design of an interference free complex planar mechanism, Mech. Mach. Theory 31 (4) (1996) 513–524. [2] S. Choudhury, Computer aided design of interference free planar mechanisms, ME Dissertation, Indian Institute of Science, Bangalore, 2001. [3] D. Sen, S. Choudhury, S.R. Pandey, Geometric Design of Interference-Free Planar Linkages, Mech. Mach. Theory 39 (7) (2004) 737–759. [4] K.J. Waldron, G.L. Kinzel, Kinematics Dynamics And Design of Machinery, John Wiley, New York, 1999, pp. 38– 42. [5] H. Ilies, V. Shapiro, On shaping with motion, Trans. ASME J. Mech. Des. 122 (4) (2000) 567–574, December. [6] M.-H. Kyung, E. Sacks, Parameter synthesis of higher kinematic pairs, CAD 35 (6) (2003) 567–575. [7] W.K. Chiu, S.T. Tan, An automatic method for controlling the centre of gravity of a model, CAD 34 (13) (2003) 997–1010.
Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Issues in geometric synthesis of mechanisms Dibakar Sen *, Vyankatesh Joshi Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India Accepted 25 May 2004
Abstract The paper presents a systematic non-iterative procedure for a priory design of link shapes for interference-free motion of a planar mechanism. We define the notions of unilayer and multi-layer links and show that generality of shapes that can be conceived depends upon the designerÕs choice about the physical implementation of the joints. We propose a matrix-based procedure for the determination of the nature of interaction among the links and joints during motion where each link can be multi-layer. Techniques for geometric synthesis as in the case of unilayer links could then be used by processing one layer at a time. The concept is demonstrated for the geometric synthesis of a six-bar Stephenson mechanism with one multi-layer link. A theory of simultaneous synthesis, using the concept of compliant motion, is also presented which does not have the limitations of order dependence inherently present in sweep-based methods. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Geometric synthesis of mechanisms involves determination of the spatial distribution of material associated with the links and joints under different design considerations. The design considerations can be non-interference of the links and joints of the mechanism itself or of the links and objects around the mechanism, material distribution for given inertial properties such as mass, centre of gravity, moment of inertia etc. The desired output of a geometric synthesis procedure *
Corresponding author. Tel.: +91 80 2293 3230; fax: +91 80 3600 648. E-mail address: [email protected] (D. Sen).
0094-114X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.05.018
1322
D. Sen, V. Joshi / Mechanism and Machine Theory 39 (2004) 1321–1330
is an engineering description of the links and joints of the mechanism generated from the kinematic description of the mechanism. Work available in literature in this area is rather limited and they consider the question of interference of links alone. Most of the existing literature deals with relatively simple constructional elements namely, flat links of uniform thickness and revolute pairs in the pin-in-hole configuration [1–4]. For such systems even in the case of a four-bar mechanism, there are eight different stacking orders one can try to assemble the mechanism as shown in Fig. 1. The annotations shown in Fig. 1(a) are valid for all mechanisms in Figs. 1–3. The one shown in Fig. 1(h) is physically infeasible with flat links because it is same as the case in Fig. 1(f) except that near joint-4, link-1 is below link-4 and as shown in Fig. 2 it requires a bent link for assembling the links. Moreover, the cases in Fig. 1(b), (c), (e) and (g) cannot have full rotatability due to link or joint interferences, as shown in Fig. 3. Hence for geometric synthesis of an arbitrary mechanism one must systematically go about determining the possible stacking orders of links, determination of structural infeasibilities and design of interference-free link-shapes. Determination of feasible layer assignments has been dealt with in [2] and efficient determination of the interference-free link-shapes has been dealt with in [3].
Fig. 1. Stacking order of links in four-bar mechanism.
Fig. 2. Four-bar mechanism with a bent link: (a) link interference, (b) joint interference, (c) joint interference, (d) link interference.
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Fig. 3. Interference of links and joints: (a) link interference, (b) joint interference, (c) joint interference, (d) link interference.
The concept of layer is used to determine the combinatorial arrangements of links and the concept of sweep, as used in [1,5] is used to computationally determine the ‘‘prohibited areas’’ in the lamina associated with each link. The shape of a link obtained out of the feasible region is considered feasible. The configuration space based methods for the computer aided generation of geometric details of objects under given motion can also be found in [6]. This method is particularly useful for mechanisms with higher pairs, intermittent motions etc. Geometric synthesis for specified location of the centre of gravity has been dealt with in [7] which we envisage as a step after geometric synthesis for non-interference of links. The issue of non-interference arises in robotic path planning where the motion is synthesized for given part models and target location, which in a sense the inverse of the present problem where non-interfering geometric extent is to be determined for the given motion. The present paper deals with the issues related with more detailed a priori geometric design for interference-free motion in planar mechanisms. For that purpose we propose a more general model of the links and joints so as to get a wider variety of combinatorial arrangements of the links, which increases the feasibility of a design. We also present a theoretical investigation on the simultaneous synthesis of link geometry using the concept of compliant motion.
2. Layer based approach to geometric synthesis In planar mechanisms points on links move on parallel planes; hence the shape of the intersection of physical links with any plane parallel to these planes also do not change. A layer is defined as the region of maximum width within which the geometry of none of the links in the mechanism changes. A link is called unilayer if it spans only one layer. A link is called multi-layered if it spans more than one layer. In Fig. 2(d), link-4 is multi-layer (spans five layers) and all other links are unilayer. Physically a revolute pair can be implemented in many different ways. It can be the traditional ‘‘pin-in-hole’’ configuration where a circular cylinder passes through the links forming the pairs, which in turn puts the constraint that two unilayer links forming a revolute pair cannot lie in the same layer. For pairs not requiring full rotation, ‘‘in-plane hinges’’ can be designed. Implication of this on geometric synthesis will be discussed later. The ordinary revolute pairs can therefore be represented in the link–link adjacency matrix (L) defined as follows: Lij = 0 if link-i and link-j are not adjacent, Lij = +1 if link-i is above link-j and Lij = 1 if link-i is below link-j. Thus L is a n · n skew-symmetric where n is the number of links. Number of non-zero elements above the diagonal of L is equal to the number of joints (say J) in the mechanism. Arrangements represented by L
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and L are only mirror images of one another, they are considered to be the same. Thus it is easy to see that there are 2J1 distinct ways of assigning +1 or 1 to the non-zero elements of L. Each of these assignments leads to a specific stacking order of the links called a layer assignment. Hence L is called layer assignment matrix. A layer index of a link is nothing but an integer associated with each link in a mechanism indicating its height with respect to the link that is there at the bottom of the stack. A matrix-based method is presented in [2] for systematic enumeration of layer indices of links of any mechanism. The layer assignment matrix defined above can also be represented as a directed graph. It has been proved in [2] that for a layer assignment to be valid (structurally feasible), the directed graph must not contain any (oriented) cycle. In the matrix based method from the given link–joint adjacency matrix (M), link–link adjacency matrix (Lu) is derived which in turn gives different layer assignment matrices (L). From each L we can derive a link– layer matrix (B). The layer–joint matrix (A) is obtained as A = BTM. Suppose link-l with layer index-i is incident on joint-j, then Aij = 1 otherwise Aij = 0. At each joint at least two links are incident which need not be in consecutive layers. However, since a single straight pin is used to construct the pair, the pin will pass through all intermediate layers. Thus, the geometric designs of the links in the intermediate layers are affected by all the joints (pins) that pass through these layers, in addition to the design of other links in the same layer. This situation is not dealt with in [1]. To capture this in the matrix representation we modify A by replacing 0Õs bracketed by two 1Õs in each column of A with 2. This gives us the modified layer–joint matrix (A*). To obtain the modified link–joint matrix (M*), which indicates the joints that affect the geometric design of links is obtained by comparing BA* with M and elements with inconsistencies being replaced with 2. Fig. 4 shows the above matrices for a Stephenson chain. To design a link, we consult M* to identify the joints affecting it, indicated by columns containing 2 in its row. We also determine other links in the same layer by consulting B. The locus of these joints and links already designed in its layer determines the region (infeasible area), which the link in context cannot use. The infeasible areas are determined by either explicit sweep [1] or implicit sweep [3]. The link is designed using the remaining area in the domain. From M* it can be observed that the design of link-4 is affected by the joints 3, 6 and 7 and B indicates that it is also affected by the design of link-5. If link-4 is designed earlier, only the joint pins of link-5 are considered. This introduces order dependence in the design of links in a layer. If there are n links in a layer, then there are n! orders in which the links can be designed many of them are
Fig. 4. Matrices for determination of links–layers–joints interaction.
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Fig. 5. Geometric synthesis of unilayer links in a StephensonÕs chain: (a) mechanism with kinematic dimensions, (b) layer assignment as per Fig. 4, (c) links after geometric synthesis, (d) links after geometric synthesis.
geometrically infeasible due to the ad hoc decision of consideration of only joint pins for the links designed later and the preference given to convex shape of the links. The structural infeasibility (e.g. Fig. 1(h)) is taken care of by the use of multi-layer links and the order dependence can be taken care of by the simultaneous synthesis procedure discussed later in the paper. Neither multi-layer links nor simultaneous synthesis has been reported in literature. Designs obtained for the Stephenson chain for two different kinematic dimensions are illustrated in Fig. 5.
3. Multi-layer link synthesis The constraint of flat links of identical thickness and pin-in-hole type of joints limits the variety of designs, which are often found in existing machinery. For multi-layer links a different link model is required. This in turn necessitates a more general matrix based method for the synthesis. 3.1. Multi-layer link model For multi-layer links, joints on the links are allowed to be at different layers, since layer index for links does not make much sense here. Fig. 6 illustrates the different ways of modeling a multilayer binary link: (a) is a unilayer link, (b) is a multi-layer link with both the joints in the same layer, (c) and (d) have the joints in adjacent layers and (e) has two joints in non-adjacent layers. This example demonstrates that • multi-layer links span a range of layers; • minimum range of layer indices for a link is given by the indices of the layers in which the joints are there; • joints can be there in adjacent or non-adjacent layers.
Fig. 6. Link shapes for a multi-layer binary link.
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The points above are to be taken as the necessary conditions of validity of layer-assignments for the links and layers. It can also be observed that a multi-layer link can be viewed as a stack of unilayer shapes glued over a common area on the interface between two successive layers. The presence of a finite common interfacial area between the two parts of a link in two consecutive layers is the condition of validity of a design obtained one layer at a time. 3.2. Alternate joint models The problem of structural feasibility is not there for mechanisms with multiple links. However, the conditions of validity mentioned above does not preclude the situation where two adjacent layers are assigned the same layer indices. This can be handled by either using the signed adjacency matrix discussed for unilayer links or by developing alternate models for the in-plane revolute pairs (Fig. 7). 3.3. Designing with a multi-layer link Relaxing the condition of flat links of uniform thickness (unilayer links) and the pin-in-hole type of revolute pairs makes the above matrix-based procedure need to be modified. A threedimensional matrix called Layer–Link–Joint (LLJ) matrix is used to represent the information of the multi-layer link–joint relationship. LLJijk = 1 implies Link-j is incident on Joint-k in Layer-i. From LLJ, the modified LLJ (MLLJ) is constructed as follows. Initially MLLJ = LLJ. Then, for a given k (joint) MLLJ is scanned over all i and j values; identify the two rows where there is a non-zero element, say they are i1 and i2; then for all the columns (j-values) in rows between and including i1 and i2 replace all 0Õs with 2Õs. Geometric synthesis is then carried out layerby-layer as in the case of unilayer links: in each layer find the joints affecting it; also find the links in the layer. For each of these links compute the swept areas of the joints not belonging to this link and those of links already designed in this layer. Compose geometry for the link out of the remaining area in the layer. Fig. 8 shows an example of a Stephenson chain with only one (ternary) link spanning layer-2 to layer-5.
4. A theory for simultaneous synthesis The shape of links one obtains using the concept of sweep, as described above, is dependent on the order in which the links are designed because the methods are inherently sequential. Although we would prefer to allocate maximal regions to each link for subsequent optimization of the mechanism, we have to remain conservative because, allocation of more regions to a link designed ear-
Fig. 7. Joint models for revolute pairs.
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Fig. 8. Geometric syntheses with one multi-layer link: (a) mechanism, (b) layer 2, (c) layer 3, (d) layer 4, (e) layer 5.
lier would produce larger prohibited areas for a link designed later which intern could render the design infeasible. Simultaneous synthesis refers to any procedure of designing link geometry in which the geometry of all the links in a layer is obtained simultaneously. In the following we describe a method of obtaining the geometry of two links simultaneously. 4.1. Condition for non-interference of two links Given two motions M1(t) and M2(t), we can identify two coordinate frames F1 and F2 (Fig. 9(a)) undergoing the respective motions with respect to a global coordinate frame, G. Then a point P in G can be identified in F1 and F2 as P1(t) and P2(t). Thus P represents a point common to both the moving bodies B1 and B2 associated with F1 and F2 at time t. This is physically possible only when P is situated on the boundaries of B1 and B2 (Fig. 9(b)). Let V1(t) and V2(t) be the velocities of P1 and P2, respectively; L be a line that locally separates B1 and B2; V t1 and V n1 be the components of V1 along and normal to L; V t2 and V n2 be the components of V2 along and normal to L; DV n ¼ V n1 V n2 . If V n1 and V n2 both pointing inside B2, DVn < 0 imply the two bodies are going to separate, DVn > 0 imply the two bodies are going to interfere, Vn = 0 imply the two bodies are going to slide past each other in the next instant. Thus the condition of non-interference of motion is DVn 6 0. This is similar to the law of conjugate profiles but without the constraint of pure rolling.
y2 M 2
y M1
o2
F2
x2 t
F1 P2
P1
V1
B1
V2
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t V1
n
P P
o
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x
Fig. 9. Analysis of interference.
B2
t
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4.2. Locus of singular points When two bodies separate during motion (DVn < 0) it indicates that it is possible to enhance the extents of the bodies such that it still remains in contact. Thus for inclusion of maximum material in the moving bodies, Vn(t) = 0 should be satisfied for all t. Locus of the points satisfying this condition is called the locus of singular points (LSP). In [5] the concept of unsweep to enumerate the maximal area capable of undergoing a given motion within a constrained space is described. However, if more than one body lies in the constrained space, the method cannot be used. Following is a method proposed for tracing the LSP using the properties of instantaneous centres. Let L1, L2, . . . ,Ln be the lines joining the instantaneous centres of rotation (called lines of centroes) of F1 and F2 at discrete times t1, t2, . . . ,tn, respectively in sequence. Then, if P lies on Li, then V1(ti) and V2(ti) are parallel to each other and both perpendicular to Li which can be taken as the tangent at the point of contact between bodies. By tracing the corresponding positions of P on all these lines, locus of P can be obtained on G. This locus of singular point is a curve, C, orthogonal to the family of lines joining the corresponding points on the moving centroes of F1 and F2 with respect to G and it represents the locus of the point of contact of the two bodies during the compliant motion. For discrete time instances, intervals of C between two successive lines, say Li and Li+1 can be represented as an arc with centre at the point of intersection Xi between Li and Li+1. Thus, from the sequence of lines of centroes we can generate X1, X2, . . . , Xn1 from which piecewise circular approximation of C can be graphically constructed. Determination of Xi is difficult when Li and Li+1 are almost parallel. Let us define Qi (Fig. 10), the point on the circle, equidistant from Li and Li+1, the corresponding points on the lines being Pi and Pi+1. The sequence Pi Qi Pi+1 with i varying from 1 to n gives the sequence of triplets of points for the construction of the arcs. Using the notion of inverse motion, the inverse map of the curves could be traced on F1 and F2 to get the family of boundary curves of B1 and B2. For each point, P1, on a L1, we get an instance of B1 and B2. A one-dimensional optimisation problem then could be set up which would search the best combination of B1 and B2. The sequence of points is determined as per the following algorithm (derivation is omitted for brevity).
4
Fig. 10. Construction of points on LSP.
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4.3. Algorithm for circular approximation of LSP 1. 2. 3. 4. 5. 6. 7. 8.
Do step-2 to step-8 for i = 1 to i = n 1. Choose any Pi on Li. Compute P aiþ1 where the perpendicular on Li at Pi meets Li+1. Compute ui the angle between Li and Li+1. Qi ¼ P i þ ðP aiþ1 P i Þ(cos ui)/(1 + cos ui). Compute P biþ1 the foot of the perpendicular from Pi on Li+1. P iþ1 ¼ P biþ1 þ ðP aiþ1 P biþ1 Þ(cos ui)/(1 + cos ui). Qi ¼ ðP i þ P iþ1 Þ=2 þ ðQi ðP i þ P iþ1 Þ=2Þðtanðui =4Þ= tanðui =2ÞÞ.
4.4. Issues in simultaneous synthesis The nature of LSP: By the above method we can construct circular arc approximation of the LSP. Taking a different location for Pi on Li we generate another sequence of circular arcs whose centres are also at Xi. Thus the families of curves that can be generated are merely offsets of each other. We do not know whether the actual (analytical) loci of singular points also are offsets to each other. Since we do not expect this to be so; because, instead of circular arc if we use an ArchimedesÕ spiral or a cubic curve, we can get a different curve still satisfying the orthogonality condition. The locus of point of contact of the relative centrodes (pitch curve) of F1 and F2 is also a LSP and it is unlikely to be a simple offset of any of the curves. Moreover, the boundary curves of the B1 and B2 must be simple which the above method does not guarantee; implications of a possibly self-intersecting LSP need to be looked into. The above method does not work F1 and F2 both have pure rotation motion. In that case we can use the theory of conjugate profiles for construction of B1 and B2 with the pitch point moving on the line of centres as dictated by the given angular velocity ratios of F1 and F2. Conditions of validity of a boundary curve are (a) B1 and B2 should lie on the two sides of the common tangent in a consistent manner (if B1 is to the left, it should always remain so and vice versa) and (b) B1 and B2 should connect all of their respective joint pins (for multi-layer links union of the shapes designed in all the layers should be considered). The pitch curve always passes through some of the joints in the link. Hence, this partitioning, even if it gives a pure rolling motion between F1 and F2, is not acceptable. Geometric synthesis is about the sharing of the material in a lamina by different links in the lamina (layer). When the material not associated with any link in a layer is minimized, we say that links are utilizing the maximal area in the lamina. This is the condition of optimality. In the present case, out of all the valid partitioning of a domain into B1 and B2 one that satisfies the above maximality criteria is said to be optimal. Since each C is associated with a point on Li, search for optimal geometry can be formulated as a one-dimensional optimization problem. Work on all the above aspects of simultaneous synthesis are under investigation by the authors. 5. Conclusions Geometric synthesis of mechanisms is essential for conversion of the results of kinematic synthesis into realizable artefacts. This paper discussed different issues related to the a priori
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geometric synthesis of planar mechanisms with revolute pairs. Using multi-layer links limitations from structural infeasibility of unilayer designs could be overcome. Potential of multi-layer links in generating much wider variety of link shapes have been discussed. A theory of simultaneous synthesis has been introduced for generation of the family of geometries, which would move without interference. Various issues related to multi-layer links and simultaneous synthesis have been discussed.
References [1] Z.K. Ling, T.R. Chase, A priori geometric design of an interference free complex planar mechanism, Mech. Mach. Theory 31 (4) (1996) 513–524. [2] S. Choudhury, Computer aided design of interference free planar mechanisms, ME Dissertation, Indian Institute of Science, Bangalore, 2001. [3] D. Sen, S. Choudhury, S.R. Pandey, Geometric Design of Interference-Free Planar Linkages, Mech. Mach. Theory 39 (7) (2004) 737–759. [4] K.J. Waldron, G.L. Kinzel, Kinematics Dynamics And Design of Machinery, John Wiley, New York, 1999, pp. 38– 42. [5] H. Ilies, V. Shapiro, On shaping with motion, Trans. ASME J. Mech. Des. 122 (4) (2000) 567–574, December. [6] M.-H. Kyung, E. Sacks, Parameter synthesis of higher kinematic pairs, CAD 35 (6) (2003) 567–575. [7] W.K. Chiu, S.T. Tan, An automatic method for controlling the centre of gravity of a model, CAD 34 (13) (2003) 997–1010.