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We assume that Li has a nite volume; thus W is bounded. Arm links L1 and L2 can be of arbitrary shape and dimensions. Link L3 is assumed to present a generalized cylinder that is, a rigid body characterized by a straight-line axis coinciding with the corresponding joint axis, such that the link cross section in the plane perpendicular to the axis does not change along the axis. There are no restrictions on the shape of the cross section itself, except the physical-world assumption that it presents a simple closed curve it can be, for example, a circle (then the link is a common cylinder), a rectangle, an oval, or any nonconvex curve. We distinguish between the front end and rear end of link L3 . The front end coincides with the arm endpoint p (see Figure 6.13). The opposite end of link L3 is its rear end. Similarly, the front (rear) part of link L3 is the part of variable length between joint J3 and the front (rear) end of link L3 . Formally, we have the following de nition: De nition 6.3.2. At any given position j = (j1 , j2 , l3 ) J, the front part L3+ (j ) of link L3 is de ned as the point set L3+ (j ) = L3 (j ) L3 ((j1 , j2 , 0)) Similarly, the rear part L3 (j ) of link L3 is de ned as the point set L3 (j ) = L3 (j ) L3 ((j1 , j2 , 1))

The purpose of distinguishing between the front and rear parts of a prismatic (sliding) link as follows: When the front (respectively, rear) part of link L3 approaches an obstacle, the only reasonable local direction for maneuvering around the obstacle is by decreasing (respectively, increasing) the joint variable l3 . This makes it easy to decide on the direction of motion based on the local contact only. Since the point set is L3 ((j1 , j2 , 0)) L3 ((j1 , j2 , 1) L3 ((j1 , j2 , l3 )) for any l3 I , and it is independent of the value of joint variable l3 , the set is considered a part of L2 . These de nitions are easy to follow on the example of the PPP (Cartesian) arm that we considered in great detail in Section 6.2: See the arm s effective workspace in Figure 6.3a and its complete workspace in Figure 6.3b. The robot workspace may contain obstacles. We de ne an obstacle as a rigid body of an arbitrary shape. Each obstacle is of nite volume (in 2D of nite area), and its surface is of nite area. Since the arm workspace is of nite volume (area), these assumptions imply that the number of obstacles present in the workspace must be nite. Being rigid bodies, obstacles cannot intersect. Formally, we have the following de nition: De nition 6.3.3. In the 2D (3D) case, an obstacle Ok , k = 1, 2, . . . , is the interior of a connected and compact subset of 2 ( 3 ) satisfying Ok1 Ok2 = , k1 = k2 (6.2)

A good oorplan will allow various modules and groups of modules to be reused without a dramatic impact on the performance of those modules. In a sea of gates design, it is common to nd that changes to totally unrelated areas of the design can cause timing problems in other aspects of the design. This is due to the progressive nature of an unconstrained place and route algorithm. The chaos effect ensues when a small placement change on one side of the FPGA shifts logic resources over, forcing other local logic structures out of the way, and so on until the entire chip has been replaced with entirely new timing. For a constrained placement via a good oorplan, this is not an issue as the relative timing is xed for the critical-path modules, and any changes must operate around that oorplan. In Figure 15.11, the critical logic is constrained inside the oorplan region and will not be affected internally as placement changes around it.

We use notation O = M Oi to represent a general obstacle, where M is the k=1 number of obstacles in W. De nition 6.3.4. The free W-space is Wf = W O. Lemma 6.3.1 follows from De nition 6.3.1. Lemma 6.3.1. Wf is a closed set. The robot arm can simultaneously touch more than one obstacle in the workspace. In this case the obstacles being touched effectively present one obstacle for the arm. They will present a single obstacle in the joint space. De nition 6.3.5. An obstacle in J -space (J-obstacle) OJ J is de ned as OJ = {j J : L(j ) O = }. Theorem 6.3.1. OJ is an open set in J.

He then looked off into the distance and proceeded to tell me a longwinded story about how he had been out of college for six months, then was hired into a major bank, and how he did the working tour, like all the recent graduates, of the different parts of the bank At the end of his sixmonth tour, he started working at the equities trading desk Then he lost a bunch of money, almost got himself fired, and ended up with a story far more boring and far less informational than everything I had already learned When he was finished, he patted the desk in front of him Harry, if you d like to come down here after work and talk with some of the traders, that would be fine with me.