Document Sample

Singularity Analysis of Geometric Constraint System Peng Xi aobo* Chen Li ping Zhou Fanli Zhou Ji CAD Center, Huazhong Univ. of Sci. & Tech., Wuhan, P.R.China 430074 Abstract Singularity analysis is an important subject of geometric constraint satisfact ion problem. In this paper, three kinds of singularity are described and corresponding identificat ion methods are presented for both under-constrained system and over-constrained system. Another special but common singularity for under-constrained geometric system, pseudo-singularity, is analy zed. Pseudo-singularity is caused by the variety of constraint matching for an under-constrained system and can be removed by improving constraint distribution. To avoid pseudo-singularity and decide redundant constraints adaptively, a d ifferentiation algorith m is proposed in the paper. Its correctness and efficiency have been validated through practical applications in a 2D/ 3D geometric constraint solver CBA. Keywords Geo metric Constraint Satisfaction, Paramet ric Des ign, Singularity, Redundant Constraint are co mprehensively analy zed and, further, three types of 1 Instruction singularity and corresponding judging methods are presented. Singularity analysis of constraint is a key technology for Geo met ric Constraint Satisfaction Problems(GCSP). 2 Ba sic concepts However, so far most researches and papers on GCSP have been focused on constraint solving planning[1,2,3,4,5,6,8], A geometric constraint system can be described with i.e. constraint matching, sorting, and decomposition , with a directed graph G[4], in wh ich a directed arc a uniquely only few papers involv ing singularity analysis . DCM stands for a constraint c, and c matches with the head 2D/ 3D is one of the best constraint solvers in the world. vertex of a. The process of creating a directed graph is in But according to our experiences, definition of constraints fact an iterat ive process of finding matching vertex for and dimensions in DCM is affected by operation order in every constraint. It is essentially the bipart ite matching of under-constrained system. It seems that DCM has not <c ,e>, where e is geo metric element. Fo llo wing are perfectly resolved the singularity problem for several important concepts used in the paper: under-constrained system. Light[1] tried to differentiate Compound Vertex: In directed graph G, a strongly the redundancy of a constraint according to the singularity connected sub-graph can be reduced to a single vertex of its Jacobian mat rix. Wang Bo xing[7] indicated that if called co mpound vertex. there is redundant constraint, the Jacobian matrix o f the Residual DOF(RDOF): The d ifference between the system must be singular. Degree of Freedo m(DOF) of a vertex and the number of Our research work is based on constraint directed its matching constraints is called its RDOF; RDOF of a graph. Generally, graph-based singularity analysis is fit for compound vertex is the sum of RDOF of all the vertices in well-constrained system and over-constrained system. In it. If the RDOF of a vertex is more than 0, it is free. this paper, singularity analysis method is extended to Propagati on Set and Predeterminati on Set: In a under-constrained system. The reasons leading to a DA G(d irected acyclic graph), the vertex set found by singular Jacobian matrix of a geo metric constraint system DFE(depth first ergodicity) fro m vertex v is called the propagation set of v. While the vertex set found by inverse *Corresponding author. DFE fro m v is called its predetermination set. E-mail: pxb_2001@mails1.hust.edu.cn (Peng Xiaobo); lpchen@mail.hust.edu.cn (Chen Liping); Transiti ve path: Constraints c 1 and c 2 correspond to c3 = PntOn Line(p 1 ,l3 ); c4 = PntOn Line(p 3 ,l3 ); an in-arc and an out-arc of vertex v1 respectively. The c5 = PntOn Line(p 2 , l2 ); c6 = PntOn Line(p 3 ,l2 ); explicit parameter sets in algebra expressions of c 1 and c 2 c7 = DistPP(p 1 ,p 2 ,d 1 ); c8 = DistPP(p 2 ,p 3 ,d 2 ); are P 1 and P 2 respectively. If P 1 ∩P 2 ≠Ф, then c 1 and c 2 are transitive at v1 and any change of either one will affect the c9 = DistPP(p 1 ,p 3 ,d 3 ) If add a new height constraint c 10 =Height(p 2 ,l3 ,h) other. If all the constraints along a path are transitive and between p 2 and l3 , c10 will cause saturated singularity. the end vertex is free, then the path is transitive. The remainder of this paper is organized as fo llo ws: Section 3 describes three types of singularity and analyzes pseudo-singularity for under-constrained system; Section 4 advances the symbol method and the numerical method for singularity differentiat ion. In the section, a singularity differentiat ion algorith m is presented and some examp les are illustrated; Section 5 gives a conclusion. (a) 3 Singularity type s p2{c7 c2 c10}:0 c5 l2{c5, A geometric constraint system can be expressed as a c6}:0 l1{c1, c2}:0 c6 set of non-linear equations. Based on the algorithm for the c8 c10 c7 c1 blocking triangular form of sparse matrix, many divid ing p3{c8 methods[1,3,4,5,8] were proposed to decompose p1{c9 c9 ,c4}:0 c4 }:1 c3 geometric constraint system so that a geometric system is l3{c3} :0 reduced to a set of sub-systems that can be solved orderly. In these methods, singularity analysis of constraints is a (b) key subject. Fig.1 Saturated singularity 3.1 Saturated singularity 3.2 Embranchment singularity Chen Liping[3] p resented the concept of Saturated Set: According to mechanism kinemat ics, if at the t ime of if under geometric constraint set C in which there is no t* there is a position q* for the system where the Jacobian redundant constraint, relative DOF between any two matrix q (q*,t*) is not singular, then according to the geometric entit ies in entity set E is zero, then E is called a Implicit Function Theorem, for δ>0, the system has a Saturated Set under C. A saturated set can be regarded as a stable unique solution q(t) near q* in the period of |t-t*|<δ. rig id body composed of its inner geometric entities. Solution 1 and Solution 2 in Fig.2(b ) are t wo examp les o f Any constraint c added into a saturated set will cause q*. On the other hand, if there exists locking or a singular Jacobian matrix and thus is called saturated embranch ment at the t ime of t*, then when t →t* there is redundant constraint. In this case, the b ipartite matching no solution and the motion of the system loses its certainty. operation of c, which is to find its matching vertex in the For examp le, at the position showed in Fig.2(a), the next directed graph, will fail. moving direct ion of the system can not be decided and the In Fig.1, when the lengths of the three sides of system becomes singular. In the research of mult i-body triangle P1 P2 P3 are decided, P1 P2 P3 is also decided. It can system, E.J Haug[9] and Hong Jiazhen[10] considered be regarded as a rigid body. The constraints are as follo ws: that this kind of singularity occurs only at isolated points, Geo metric entity set: E = {p 1 ,p 2 ,p 3 ,l1 ,l2 ,l3 }； e.g. position q* in Fig.2(a), and called it isolated Constraint set: C = {c i | i=1,9} singularity. c1 =PntOnLine(p 1 ,l1 ); c2 =PntOnLine(p 2 ,l1 ); This case also occurs in geometric constraint 2 satisfaction problems such as parametric design. Solution 1 解1 6 Considering that two curves are constrained by tangency Embranchment 6 分支点bb Point 5 constraint. The constraint system is non-linear and the 3 5 Jacobian matrix q is dependent on the orientation 3 ang parameter vector q of the system[12]. The system is 8.5 8.5 解2 singular. Obviously the singularity occurs only when Solution 2 tangency constraint exists or rather at the isolated points. (a) (b) That is to say, tangent point can be considered as the Fig.2 isolated point of the system. There is only one d ifference between embranch ment singularity of mechanism analysis and that of geometric constraint satisfaction problems: Kinemat ics analysis can be considered as geometric constraint problem based on time sequence. While for static GCSP like parametric design, when we say a system is of embranch ment Fig.3 Emb ranchment singularity caused by tangency singularity, we mean that the system is at a position where there are constraints leading to isolated singular points for l1{c3, c{c12, c12 the system. For examp le, if the influence of time is not c3 c5}:0 c6,c13 c7 }:0 c13 considered in Fig.2(a), when some constraints make the p1{c1, c11 p3{c7, c2}:0 c11}:0 c8 quadrilateral to be a triangle, the system is of isolated c4 l2{c8, c9}:0 c6 singularity. Similarly, in Fig.3, p 1 is fixed, l1 and l2 are c10 horizontal, c is tangent with l1 and l2 . The constraints can p2{c4, c10}:0 be written as follows: c1 =FixX(p 1 ); c2 =FixY(p 1 ); c3 =PntOnLine(p 1 ,l1 ); Fig. 4 A directed graph of embranchment singularity c4 =PntOnLine(p 2 ,l1 ); c5 =Horizontal(l1 ); c6 = PntOnCircle(p 2 ,c); c7 =PntOnCircle(p 3 ,c); c8 =PntOnLine(p 3 ,l2 ); c9 =Horizontal(l2 ); c10 =DistPP(p 1 ,p 2 ,d1 ); c11 =DistPP(p 1 ,p 3 ,d 2 ); c12 =TanLC(l1 ,c); c13 =TanLC(l2 ,c); Fig.4 is the d irected graph o f Fig.3. The strongly connected sub-graph SC={p 3 ,l2 ,c} corresponds to a 7×7 Jacobian matrix q . q is singular and rank(q )=5. Its rank deficit is 2, being equal to the number of tangent l2{c2, c9 } c6 p0{c4, points. Obviously, there is no redundant constraint here l1{c1} :0 :1 c6}:0 and the singularity comes fro m the tangencies between c c9 c7 c5 c4 and l1 , l2 . c3 c{c8, p1{c3} c8 c7,c5} :1 :3 3.3 Redundancy singularity (c) There is no essential difference between redundancy Fig. 5 Redundancy singularity singularity and saturated singularity, wh ich are both singularity. Redundancy singularity often produced by the caused by redundant constraints. When a redundant existence of equivalent geometric constraints. constraint can be decided by a symbol method, it is called Unlike saturated singularity, this type of singularity is saturated singularity; Otherwise, it is called redundancy common when system is under-constrained, especially 3 A D during the initial phase of design. For an under-constrained l4 vertex, symbol method can not be used to judge its l1 l3 singularity by simply calculat ing its RDOF, because in this 60° B l2 C case, singularity might still exist even if the vertex is free. For instance, when a vertex of DOF 3 is constrained by (a) two geometrically equal constraints, the vertex is free and c6 l4 D the directed graph is normal, but singularity occurs. c7 c5 Considering the constrained geometry in Fig.5(a): A l3 c1 =Horizontal(l1 ); c2 =Vert ical(l2 ); c0 c11 c9 c4 c3 =PntOnLine(p 1 ,l1 ); c4 =Cent_X(p 0 ,c); l1 c10 C c5 =Cent_Y(p 0 ,c); c6 =PntOnLine(p 0 ,l2 ); c0 8 c7 =Tangent(l1 ,c); c8 =PntOnCircle(p 1 ,c); c1 c3 B l2 c9 =PntOnLine(p 1 ,l2 ); c2 Fig.5(c) is the directed graph of Fig.5(a). It can be (b) seen that circle c and its center point p 0 constitute a Fig.6 Redundancy singularity strongly connected component SC. A lthough the RDOF of Here it is also impossible to judge the singularity by SC seems to be zero in the graph, when the radius of c calculating the DOF of ve rtices. In fact, constraints changes, all the constraints can still be satisfied(Fig.5(b)). AB//DC and AD//BC are equivalent to |AB|=|CD|, which That is to say, SC is not fixed and the system is singular. leads to the singularity. This singularity is just caused by the existence of equivalent geometric constraints. This kind of singularity 3.4 Pesudo-singularity can not be decided by symbol method and numerical method should be adopted. For an under-constrained system, the matching Fig.6(a) is another example of redundancy singularity: modes between constraints and entities are various . AB//DC, A D//BC, |A B|=|CD|, ∠ B=60 °, |A B|=5. The Different constraint matching modes produce different constraint set is: solving sequences. Part of these solving sequences may c0 =PntOnLine(A,l1); c1 =PntOnLine(B,l2); contain singular units and are called ill-conditioned c2 =PntOnLine(B,l2); c3 =PntOnLine(C,l2); solving sequence. However, this kind of singularity can be c4 =PntOnLine(C,l3); c5 =PntOnLine(D,l3); c6 =PntOnLine(A,l4); c7 =PntOnLine(D,l4); Absolute constraint: l2 c8 =AngBtLine(l1,l2,60); c9 =ParalLL(l1,l3); A l1 is horizontal and fixed; p c10 =ParalLL(l2,l4); c11 =DistPP(A,B,5); l1 c1=OnLine(p,l1); c12 =EqualLen((A,B),(C,D)); c2=OnLine(p,l2); ABCD is a parallelogram. Fig.6(b) is the directed c1 c1 v(l1) v(p) v(l1) v(p) graph of fig.6(a). Three dashed arcs in Fig.6(b) denote the c2 c2 four-element constraint c 12 =EqualLen((A,B),(C,D)), which v(l2) v(l2) is different to binary constraints. Three arcs of c 12 mean that c12 involves 4 vertices. But the three arcs work as a (a-1) (b-1) single one and reduce the DOF o f the vertex D by 1. It can c1 c1 v(l1) v(p) v(l1) v(p) be seen that the total RDOF of the system is 3 and there is c3 c2 c3 c2 no local over-constraint in the graph. The system seems to v(l2) v(l2) be well-constrained. Ho wever, it is obvious that CD can move along AD. That is to say, the system is not (a-2) (b-2) under-constrained. So in the system there exists singularity. Fig.7 An examp le of pseudo-singularity 4 removed by improving the constraint distribution. If a kind constraint directed graph. There must be a matching vertex of singularity can be eliminated by adjusting constraint for a normal constraint. Otherwise, the directed graph can distribution, it is called pseudo-singularity. not be created correctly. In the examp le of Fig.7, l1 is horizontal and fixed. p Property 2: If constraint c is redundant, then the is the intersection point of l1 and l2 . A is the angle between Jacobian matrix q o f the matching vertex of c is singular. l1 and l2 . This under-constrained system has two possible That is, in our research, q has row rank deficit. directed graphs(Fig.7(a-1) and Fig.7(b-1)). No w add an Any redundant constraint belongs to an interrelated angle constraint c3 =AngLL(l1 ,l2 ,180) between l1 and l 2 . constraint set. In our research, each row vector of the Then (a-1) and (b-1) develop to (a-2) and (b-2) Jacobian matrix o f the interrelated constraint set is respectively. In the case of (b-2), intersection point p can composed of the partial derivatives of orientation not be solved from two superposed lines l1 and l2 . The parameters of one constraint equation in the set. Because system is singular. Ho wever, for the same system, the of the relativity of the constraint set, the row vectors of its solving sequence in (a-2) is normal. This situation is call Jacobian matrix are also relat ive and there must be deficit pseudo-singularity. To avoid the pseudo-singularity in of row rank of the Jacobian matrix. (b-2), ad just constraint c2 and let it be matched with v(l2 ), Property 3 : Assume that vs is a compound vertex and then the directed graph is optimized to (a-2) and the there is no unary constraint that describes absolute position solving sequence v(l 1 )v(p)v(l 2 ) becomes normal. in vs , if (i) the predetermination set of vs is empty and the So we can conclude that: (1) For under-constrained RDOF of vs is less than that of a rigid body or (ii) the system, singularity of a constraint can not be simply predetermination set of vs contains only one vertex P, the determined by the singularity of a single vertex; (2) For RDOF of vs is zero and nu mber of the out-arcs fro m P to under-constrained system, if a constraint has at least a vs is mo re than the DOF of P, then there must be saturated normal solving sequence, then it is normal. singularity. We adopt the steps as follows trying to eliminate Generally, there are t wo methods to position vs : one pseudo-singularity: Once an ill-conditioned vertex v ss is is using unary constraints that describe absolute position obtained, constraint matching adjustment[8] will be such as fixed position, horizontal, vertical and so on. This implemented on the non-propagation set of v ss to search situation is excluded by property 3. The other method is to for a inverse transitive path P. Reverse the arcs along P to l2{c5, move the constraint matched with v ss so that the c6}:0 l1{c1, c5 propagation set of v ss is enlarged and a new solving c2}:0 c2 p2{c7 c6 c10}:0 sequence is obtained. To ensure the new solving sequence c1 c8 p3{c8 is normal, the algorith m needs to check all the vertices in c7 c10 ,c4}:0 p1{c9 c9 the new solving sequence. Readers please refer to }:1 c4 c3 reference[8] for details. l3{c3} :0 (a) 4 Singularity differentiation l2{c5, c6}:0 l1{c1, c2}:0 c5 4.1 Graph di fferentiati on method c2 p2{c7 c6 c10}:0 c1 c8 p3{c8 A geometric constraint system has fo llo wing three ,c4}:0 c7 c9 c1 properties: 0 c4 p1{c9 Property 1: If constraint c is not redundant, then ,c3}:1 c3 l3{}:2 there must be a geo metric entity e in the system, (b) DOF(e)>0 and c can be matched with e(<c,e>|e ∈E ). This property comes fro m the definit ion of geometric Fig.8 Graph character of saturated singularity 5 position by its predetermination set P. If P is empty or perturbation direction; F is the constraint equation. Then there is only one geometric entity whose DOF is less than compute with the iteration formu la that of vs , obviously it is impossible fo r vs to be positioned and there must be saturated singularity among the q j 1 q j j Fj , q constraints of vs . Where q is the orientation parameter vector of v i , is qj Property 1 and 2 are two necessary conditions for a the general inverse of q . Because the perturbation is normal constraint. Property 3 indicates the graph character small, after few steps of iteration can we determine if there of saturated singularity and becomes an efficient symbol is a solution or not. If there is a solution for either differentiat ion method for this kind of singularity. For perturbation direction for ci and row ran k deficit of q instance, Fig.8(a) is the strongly connected graph of decreases, the singularity caused by ci is embranch ment, or Fig.1(b), in which the only strongly connected sub-graph it is redundancy. By this means, the types of singularity of SC has 4 entities and its RDOF is 2×4-6 = 2. After the optimal matching adjustment[8], wh ich can ensure the all the constraints in Cr can be decided. RDOF of any compound vertex v i Dof(v i )→0, SC contains In Fig.5, there is redundancy among the constraints 3 entit ies Vs={p 1 ,p 2 ,p 3 } and RDOF=2×3-6 = 0(Fig.8 (b)). between compound vertex Vs and c6 ,c7 ,c8 . When Cr={c6 }, The predetermination set of Vs is P={l3 }. The DOF of l3 is the perturbation is: 2. However, it can be seen that there are three arcs from l3 a2 x0 + b 2 y 0 + c2 = ， to Vs, corresponding to constraints c 3 , c4 , c10 , which means where a2 ,b 2 ,c2 are parameters of l2 . A fter perturbation, l2 rig id body Vs whose DOF is 3 is fu lly determined by l3 will have a slight translation and then the intersection whose DOF is 2. It is obviously impossible. According to point of l1 and l2 can not be on circle c(Fig.9(a)). There is Property 3, there must be saturated singularity in SC. no solution for this perturbation. If Cr={c7 }, the perturbation is: 4.2 Constraint Resi due Perturbation Method a1 xc + b 1 y c + c1 r = , where a1 ,b 1 ,c1 are parameters of l1 . It means that l1 moves For redundancy singularity and emb ranchment singularity, nu merical judging method should be applied. l1 l1 l1 p1 p1 p1 To decide singular points, E.J.Haug[9] presented a variable perturbation method. This method is efficient. Ho wever, it p0 c p0 c p0 c applies perturbation to all the entit ies in the system l2 l2 l2 indiscriminately and the scope of perturbation does not Perturbation l1 p1 vary for different entities, which increases the expense of solution No solution No solution 无解 time of the method. 无解 p0 c Considering that singularity is caused by redundant 扰动解 l2 constraints, we advance the Constraint Residue ( a) (a) (b) (c) Perturbation Method(CRPM) which only involves the Fig. 9 Redundancy singularity redundant constraint set: If Jacobian matrix q of a solving unit v i is singular, then redundant constrain set C r C can be found easily Perturbation using Gaussian Eliminat ion Method. Here C is the l l l soloution 扰动解 constraint set of v i . Set a perturbing value for each Perturbation 扰动解 p p p constraint ci ∈ Cr and then adopt Newton-Raphson soloution iteration. For c c c ci ∈Cr Fi = Fi , (a) (b) (c) where > ， is the iteration precision; means Fig.10 Embranchment singularity 6 along l2 (Fig.9(b)). Obviously, the perturbing result also singularity) then c is redundant and return makes p 1 not on circle c, which is inconsistent to c8 and FALSE; there is also no solution. When Cr={c8 }, perturbation is else if (there is v∈S and v is of redundancy singularity) then {search for reverse transitive ( x c x1 ) 2 ( y c y1 ) 2 r . paths P on the non-propagation set of v;} It means that the radius of circle c changes for the else {print(“c is normal constraint”) output the perturbation. Fro m Fig.9(c), it can be seen that after solving sequence; return TRUE;} perturbation, all the constraints are still satisfied and step5 If (P is empty) c is redundant; return FALSE; perturbing solution can be easily obtained. The Jacobian else{apply reverse operation on P; reset the matrix q of Vs is: solving sequence S; goto Step2;} x0 y0 xc yc r c4 : 1 0 1 0 0 In Algorith m 1, step 4 needs to judge the singularity c5 0 1 0 1 0 type of vertex v. A lgorith m 2 gives this judging method: c6 a2 b2 0 0 0 Algorithm 2: Judging the singularity type of vertex v c7 0 0 a1 b1 1 step1 if (v satisfies Property 3) c8 0 0 ( xc x1 ) / R ( y c y1 ) / R 1 return “v is of saturated singularity”; where E { p 0 , c} , C {c4 , c5 , c6 , c7 , c8 , } , step2 Co mpute the Jacobian matrix q of v; step3 If (q is normal) return “v is normal” q x0 , y0 , xc , yc , r , R ( x c x1 ) 2 ( y c y1 ) 2 . T step4 Co mpute dependant constraint set Cr in v by Because p 0 and p 1 are always on l2 before and after Gaussian Elimination Method. perturbation and the values of (xc-x1 )/ R and (y c-y 1 )/R do Step5 i = 1, r is the number of constraints in Cr. not change, the row rank deficit of q is unaltered. Step6 ci∈Cr, r0=rank(q ); According to CRPM, c8 causes redundancy singularity. Step7 Add a perturbation δ on ci; Similarly, in Fig.10(b), line l is tangent with circle c . Step8 Solve v; The system is singular. After either perturbation shown in Step9 If (there is no solution) Fig.10(a) or Fig.10(c), the row ran k deficit of the Jacobian return “v is of redundancy singularity”; matrix of the system decreases to 0, which shows the else if(r1=rank(q)≤r0) system has embranchment singularity. return “v is of embranch ment singularity”; The algorith m of singularity d ifferentiation can be step10 If (i<r) {i+1, go to step5}; summed up as follows: step11 return “v is normal”; Algorithm 1: Globally judging the redundancy of constraint c in an under-constrained system Algorith m 1 adaptively enlarges the propagation set step1 If the bipartite matching of c fails, c is of constraint c according to the singularity types of the redundant; return FALSE; vertices in the solving sequence SC and eliminates the step2 Initialize directed graph G; Get the propagation singularity. Because the repeat of Step2 is limited, the set PS of c. complexity of the algorith m is approximately equal to that step3 Adopt optimal adjustment algorith m[8] to of DFE: O(n+e)[11]. The algorithm works well for obtain the strongly connected sub-graphs on PS under-constrained systems. Ho wever, it is also efficient for and make G a d irected non-loop graph. Then the well-constrained and over-constrained systems. solving sequence S={vi} fro m c can be obtained along the our-arcs of c. step4 if (there is v ∈ S and v is of saturated 7 CbaDigraph CBA classes CbaObject CbaSolveUnit CbaConstraint CbaModel CbaBody CbaModel2D CbaBody2D CbaDim2D CbaConstraint2D CbaModel3D CbaBody3D CbaDim3D CbaConstraint3D CbaGeomEntity CbaSolveUnit2D CbaPoint2D CbaPoint3D CbaRelation CbaSolveUnit3D CbaLine2D CbaLine3D CbaDivertex CbaDigraph2D CbaCircle2D CbaPlane3D CbaDigraph3D CbaEllipse2D .......... CbaDiEdge Fig.11 the hiberarchy of CBA Fig.12 An examp le of constraint identification 1 1 3 3 2 2 (a) (b) Fig.13 Assembling a t ripod constraints in these constraints. Finally, 2n+1 horizontal 4.3 Some examples constraints, 2 vertical constraints, 4+3(n-1)=3n+1 tangent constraints and 3 point-on-line constraints are obtained by Using O-O method we presented a united modeling CBA2D. In this examp le, all the types of singularity method for both 2D and 3D geo metric constraints and mentioned above occur. Particu larly, after the constraint developed a 2D/3D geo metric constraint solver identification, labeling some d imensions (e.g. Fig.12(b )) CBA(Constraint Broadcasting Automation). Fig.11 will lead to sophisticated embranchment singularity. illustrates the hiberarchy of CBA. CBA2D also can deal with such kind of problems . The module o f constraint identificat ion in CBA2D In another 3D examp le in Fig.13(a), pole 2 mates transforms general engineering graphs to parametric ones with pole 1 and pole 3 by two co-axis relat ions. Each and identifies geo metric constraints according to relative co-axis relation includes 4 constraints. If add another positions of geometric entit ies. Fig.12(a) is a keyway that co-axis relation between pole 1 and pole 3, CBA3D will has n loops. By geometric detection, 2n+1 horizontal close the tripod and find out two redundant constraints, 2 vertical constraints, 4n tangent constraints constraints(Fig.13(b)). (Readers please refer to and 4n+4 point-on-line constraints are made out. The reference[12] fo r our research of geometric constraint algorith m identifies successfully that there are n-1 expression and decomposition). Practice has proven the redundant tangency and 4n+1 redundant point-on-line differentiat ion method presented in this paper fast and efficient. perturbation method are advanced to differentiate efficiently singular constraints. All the algorith ms and 5 Conclusi on methods proposed in this paper can also be applied to 3D geometric constraint system and has proved efficient and Singularity is an important factor that affects directly fast in CBA 3D. the solving ability of geometric constraint solver. In this paper, three types of singularity are presented and 6 Acknowledgement pseudo-singularity, which is very common in under-constrained systems, is analyzed. As one part of the This work was supported by the National “863” kernel of our geo metric constraint solver CBA, the graph Project under grant 9842-003 in the field of automation. differentiat ion method and the constraint residue References 1 Light , R, Lin V Gossard D. Variational geometry in CAD. key technologies of geometric constraint driven function in Computer Graphics, 1981,15(3)171-177 traditional drafting systems. Chinese Computer Research & 2 Kramer G.A. A geometric constraint engine, Artificial Development, Vol.35, No.10, pp935-940,1998. intelligence, 1992, 58: 327-360 8 Chen Liping, Wang Boxing, An Optimal method of Bipartite 3 Chen liping, Zhou Ji. Research of geometric constraint Graph M atching for Underconstrained Geometry Solving. system reasoning. Journal of Huazhong Univ. of Sci & ol. Chinese Journal of Computers V 23, No. 5, M ay 2000: Tech，1995, 6:70-74. pp523-530. 4 Dong jinxiang. A new thought of constraint solving in 9 Haug E.J, Computer Aided Kinematics and Dynamics of variable drawing system. Journal of computer-aided design ol. M echanical Systems, V I Basic M ethod, 1989. & computer graphics, 1997, 9(6):513-519. 10 Hong Jiazhen, Computational Dynamics of M ultibody 5 Jae Yael Lee, Kwangsoo Kim. A 2-D geometric constraint System. Advanced Education Press, China, 1996(4):70. solver using DOF-based graph reduction. Computer-Aided 11 Yan Weimin, Wu Weiming, Data Structure, Tsinghua Design, 1998,30(11):883-896 University Press, China, 1992. 6 Xiao-Shan Gao,Shang-Ching Chou, Solving geometric 12 Chen Liping, Peng Xiaobo, Proceedings of ASM E constraint systems. A symbolic approach and decision of DETC’00, Baltimore, M aryland, September 10-13, 2000. Rc-constructibility. Computer-Aided Design Vol.30,No.20. pp115-122,1998 7 Wang Boxing, Chen Liping, Zhou Ji, Study and practice of 9

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 6 |

posted: | 11/15/2010 |

language: | English |

pages: | 9 |

OTHER DOCS BY hcj

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.