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624 Connectivity of C A space or manifold is connected relative to two points in it if there is a path that connects both points and that lies fully in the space (manifold) For a given path l, the corresponding trajectory l(t) de nes this path as a function of a scalar variable t; for example, t may indicate time Denote the 2D Cartesian space formed by joint values l1 , l2 as Cp , Cp = [0, l1 max ] [0, l2 max ] We intend to show here that for the 3D Cartesian arm the connectivity in C can be deduced from the connectivity in Cp .

Such a relationship will mean that the problem of path planning for the 3D Cartesian arm can be reduced to that for a point automaton in the plane, and hence the planar strategies of 3 can be utilized here, likely with some modi cations De ne the conventional projection Pc (E) of a set of points E = {(l1 , l2 , l3 )} C onto space Cp as Pc (E) = {(l1 , l2 ) | l3 , (l1 , l2 , l3 ) E} Thus, Pc (S), Pc (T ), Pc (M-line), and Pc ({O}) are, respectively, the projections of points S and T , the M-line, and C-obstacles onto Cp See, for example, projections Pc of three obstacles, O1 , O2 , O3 (Figure 612) It is easy to see that Pc (O1 O2 ) = Pc (O1 ) Pc (O2 ).

In Figure 14.22, the timing failures may still need to feed back to the physical synthesis tool. Even if timing failed due to poor routing, either the placement or the fundamental logic structures will need to be reoptimized. The key here is that because synthesis and placement are tightly coupled, the number of iterations before achieving timing closure will be far fewer. The degree to which this process can be automated is directly related to algorithms used in the physical synthesis for placement estimates as discussed in the next section.

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De ne the minimal projection Pm (E) of a set of points E = {(l1 , l2 , l3 )} C onto space Cp as Pm (E) = {(l1 , l2 ) | l3 , (l1 , l2 , l3 ) E} Thus, if a C-obstacle O stretches over the whole range of l3 [0, l3 max ], and E contains all the points in O, then Pm (E) is the intersection between the (l1 , l2 )-space and the maximum cylinder that can be inscribed into O and whose axis is parallel to l3 Note that if a set E is a cylinder whose axis is parallel to the l3 axis, then Pc (E) = Pm (E) Type I and Type I I obstacles present such cylinders In general, Pm (S) = Pm (T ) = ..

One of the keys to high-performance physical synthesis is to create accurate estimates of routing congestion and delay. Synplicity has developed a technology for FPGA physical synthesis called graph-based physical synthesis. The idea is to

Existence of Collision-Free Paths. We will now consider the relationship between a path in C and its projection in Cp . The following statement comes directly from the de nition of Pc and Pm :

Lemma 6.2.1. For any C-obstacle O in C and any set Ep in Cp , if Ep Pc (O) = , then P 1 (Ep ) O = . m If the hypothesis is not true, then P 1 (Ep ) O = . We have Pc (P 1 (Ep ) m m O) = Pc (P 1 (Ep )) Pc (O) = Ep Pc (O) = . Thus a contradiction. m The next statement provides a suf cient condition for the existence of a path in C-space: Lemma 6.2.2. Given a set of obstacles {O} in C and the corresponding projections Pc ({O}), if there exists a path between Pc (S) and Pc (T ) in Cp , then there must exist a path between S and T in C. Let lp (t) = {l1 (t), l2 (t)} be a trajectory of Pc (C-point) between Pc (S) and Pc (T ) in Cp . From Lemma 6.2.1, P 1 (lp (t)) {O} = in C. Hence, for example, m the path l(t) = {(lp (t), (1 t)l3S + t l3T )} P 1 (lp (t)) connects S and T in C. c