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enough experiments to realize this on their own, but placement is vastly more important to performance than is routing. This is particularly true of FPGAs due to the coarse nature of a routing matrix (in an ASIC, there is more exibility to be creative with the routing). If you were to take a typical design and run hundreds of combinations of placement and routing effort levels and then plot your data, you would likely see something very similar to the graph shown in Figure 16.3. As can be seen from the curves in Figure 16.3, placement has a dominant (call it rst-order effect) on the performance of the design, and routing has a relatively minor effect.

Hence the connectivity of js and jt can be determined completely by studying Df , which is simpler than Jf because the dimensionality of Df is lower than that of Jf Furthermore, a path between two given points js = (j1s , j2s , l3s ), jt = (j1t , j2t , l3t ) Jf can be obtained by nding the path between the two points js = (j1s , j2s , l3s ), js = (j1t , j2t , l3t ) Df Because of the monotonicity property (Theorem 632), js and jt always exist and they can be respectively connected within Jf with js and jt via vertical line segments Hence the following statement: Corollary 632.

Near the back corner of the room, only about 20 feet from Craig s desk, a computer was set up where there hadn t been one the day before The desks to either side were empty, without computers or monitors or phones A note from Anderson told me that the desk was mine to use, that I should call the tech support guys if I needed to load software onto the machine, and that I should make myself at home Next to the note from Anderson was a note from someone named Alistair Martin It said: Harry: sweet trade on the Euro! Thanks for the idea! Who was Alistair Martin I looked up, toward the desk where I d sat the night before.

The problem of nding a path in the 3D subset Jf between points js , jt Jf can be reduced to one of nding a path in its 2D subset Df 635 Con guration Space and Its Retract Our motion planning problem has thus been reduced to one of moving a point in a 2D subset of J -space, J However, as pointed out in Section 631, the joint space J is not very representative when revolute joints are involved, because the mapping from J to workspace W is not unique Instead, we de ne con guration space C: De nition 6312 De ne an equivalence relation F in J as follows: for j = (j1 , j2 , jw ), j = (j1 , j2 , j3 ) J, j Fj if and only if (ji ji )%2 = 0, for i =.

Placement effort has a dominant effect on performance, whereas routing effort has a relatively minor effect.

1, 2, 3, where % is the modular operation The quotient space C = J/F is called the con guration space (C-space), with normal quotient space topology assigned, see Ref 57, A1 Let c = Fj represent an equivalence class; then the project f : J C is given by f (j ) = Fj Theorem 637 The con guration space C is compact and of nite volume (area) By de nition, J = J1 J2 J3 De ne equivalence relations Fi in Ji such that ji Fi ji if and only if (ji ji )%2 = 0 De ne Ci = Ji /Fi and the project fi : Ji Ci given by fi (j ) = Fi j Apparently, Ci S 1 with length vi = 2 if = Ji = , and Ci I 1 with length vi = 1 if Ji = I 1 .

Then I realized that I d left my Post-it note on the desk of a currency trader named Ellen Hansell, and that she had taken the trade, probably with Anderson s approval I immediately punched some keys on the computer (realizing that I had become very comfortable with a computer, without having even noticed it before) and pulled a chart to see where the Euro had traded that day (Figure 165) To say that I amazed myself is an understatement I shocked myself For five minutes, I just looked at the chart from several different angles Was I looking at it correctly Did I really plan that trade Had I actually analyzed the charts, picked an entry, a stop, and a profit target Without blowing up an account I wondered how much money Alistair Martin had made using my idea.

Because f is the product of = fi s, fi s are both open and closed, and the product topology and the quotient topology on C1 C2 C3 are the same (see Ref 57, Proposition A31); therefore, C C1 C2 C3 is of nite volume v1 v2 vn = For an RR arm, for example, C S 1 S 1 with area 2 2 = 4 2 ; for an = RRP arm, C S 1 S 1 I 1 with volume 2 2 1 = 4 2 = For c C, we de ne L(c) = L(j ), where j f 1 (c), to be the area the robot arm occupies in W when its joint vector is j De nition 6313.