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boundaries, all possible ways to separate an area on the torus from the rest of its surface can be reduced to ve cases shown in Figure 5.8. The case in Figure 5.8a corresponds to a Type I obstacle; the four remaining cases correspond to Type II obstacles. The cases in Figure 5.8b and 5.8c are topologically equivalent; the cases in Figure 5.8d and 5.8e are equivalent as well. From the path planning standpoint, all ve cases are distinct and are treated in the algorithm separately.

Because the data path is often the critical path of the design (the throughput of the design will be related to the timing of the pipeline), it may be required that a oorplan is designed for this path to achieve maximum performance. The control logic, on the other hand, will often have slower timing requirements placed on it because it is not a part of the primary data path. For instance, it is very common to have a simple SPI (Serial Peripheral Interface) or I2C (Inter-IC) type bus to set up control registers in the design. If the pipeline is running at perhaps hundreds of megahertz, there will certainly be a large discrepancy between the two timing requirements. Thus, if a oorplan is required for the data path, the control logic can usually remain unconstrained (spatially) and scattered around the pipeline wherever it ts and as decided by the automatic place and route tool.

After dinner, and when the kids were in bed, I retreated to our small family room and I opened my wallet. Here is what I found: Regular identification Six credit cards I took out the cards one by one. I shifted them between my fingers. These would soon be history! My wife walked in and noticed what I was doing. I didn t mean to worry you about the rent, she said. I m not worried at all, I replied happily and put the cards away. As I got ready for bed, I started to calculate in my mind everything that we owed. The total came to $24,000, give or take a few hundred. Our monthly payments on the cards were low interest rates were falling so that wasn t so bad. Sometimes I paid our rent from one of the cards. But what was going to happen when the introductory low rates ended What was going to happen when we ran out of credit Without trading, we were going to be screwed. I couldn t believe how lucky I was to have found trading it had come along at just the right moment. My wife was trying to be as supportive as possible. But I knew that a big talk was coming between the two of us, and it was not going to be an easy one. I wanted to be optimistic. Ignoring the problems wasn t optimism, though. I had been lying to myself, living in a daze, expecting to ignore my problems. But now I didn t have to ignore them anymore. I could face them, three hundred dollars at a time!

q2 = 0

Figures 53 and 56 provide examples of the Type I case of Figure 58a An example for the Type II case in Figure 58b is shown in Figure 57 The Type II case shown in Figure 58c appears, for example, when an obstacle in W space presents a ring whose center is in the system origin and whose smaller radius is larger than l1 (see obstacle A, Figure 59) One might say the example is not excessively realistic This is true, except that with a bit more work one can come up with a rather realistic example that would still demonstrate the same phenomenon An example for the cases in Figures 58d and 58e appears in Figure 510 As these examples show, all ve cases of Figure 5.

Good design practice dictates that any given module should have only one type of clock and one type of reset. If, as is the case in many designs, there are multiple

8 are physically realizable, and therefore they should be accounted for in the algorithm Consider two counters, C1 and C2 , corresponding to the angles 1 and 2 of the arm joints, respectively When the arm travels in free space, the content of each counter is zero Once the arm hits an obstacle, both counters are turned on While the arm follows a closed curve of a virtual boundary, each counter integrates the corresponding angle, taking into account the sign As the arm completes a closed curve, the contents of each counter must be n 2 , |n| = 0, 1, 2, For a closed curve of some obstacle, the resulting values of the pair (C1 , C2 ) de ne its arm joints range (or, simply, range) An obstacle of Type I is de ned by the range of its single closed curve.