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Still, understanding the Bug family algorithms from 3 will help one grasp the algorithms for robot arms that we are about to develop We can now sketch the idea behind a motion planning algorithm for a planar robot arm manipulator It is easier to describe the operation in C-space; the actual operation in W -space proceeds accordingly As one will notice, the sketch sounds much like the algorithm Bug2; deviations and complexities will be added later At the beginning, the C-space arm image point moves along a simple M-line, which is a desired path from point S to point T , an equivalent of the straight-line M-line for the mobile robot (Section 33) During this motion, when (in W -space) some point of the arm body meets an obstacle, in C-space this corresponds to the image of M-line intersecting the obstacle s virtual boundary.

The point of intersection is said to de ne a hit point, Hj , where j is the running index enumerating such points We will show below that the virtual boundary is a simple curve, a curve with no self-intersections or double points This being so, at the hit point the arm has a simple choice: to walk along the virtual boundary in one or the opposite direction along the curve Since no information is available beforehand as to which of the two directions is better, one direction, called the local direction, will be chosen once and for all While following the obstacle virtual boundary, the arm may meet the M-line again.

12.2 Traps if(ctrl[0]) else if(ctrl[1]) else if(ctrl[2]) else if(ctrl[3]) end endmodule out[0] out[1] out[2] out[3] = = = = in[0]; in[1]; in[2]; in[3];

If it does, and if this occurs at a distance (measured appropriately along the M-line) from point T shorter than the distance from the latest hit point Hj to T , the arm will de ne a leave point, Lj Hit and leave points will play an important role in the path planning procedure We will see below that these points come in pairs, (Hj , Lj ), j = 1, 2, For convenience, denote Lo = S, Start, with no corresponding Ho The motion planning algorithm proper, the proof of its convergence, and the test for target reachability will emerge from our analysis of the described scheme and of C-space properties Similar to the mobile robot case ( 3), under our scheme the arm will need no beforehand information about the obstacles in order to move properly.

arvey knew his way around the SoHo/Chinatown border, and led me through some alleys and backstreets until we emerged on Mulberry Street, which was just about all that was left of a neighborhood that once had been home to tens of thousands of Italians. From Mulberry, we turned left onto Grand Street, where we found Di Palo Fine Foods. This is where, he said, we could find the very best soft drink in the world. Harvey bought us a half-pound of mixed olives, some prosciutto (Italian ham), and two cans of Aranciata, which he believed to be the nectar of the gods and that was essentially a lightly sweetened orange soda. And yes, it was the very best soft drink in the world. As we ate the olives and the ham, Harvey told me the story of George Sisler s hedge fund. George managed a group of traders for a major European bank about 10 years ago. He had been a rising star in the commodities trading desk, and they promoted him to run a separate, outside-the-office proprietary trading operation in New York. It was a team of currency traders, just seven or eight, and they were essentially running a little hedge fund inside the bank, with the bank s money. It wasn t really a hedge fund, but it operated like one. They made their own trading decisions, the traders were paid a portion of their gains, and their entire mission was to produce spectacular returns on the bank s own money. How much money did they have At least five hundred million. It was a good deal of money at the time and George was young. He knew how to trade but he really didn t know

The C-space presentation is used primarily for the analysis, the algorithm development, and the proof of convergence No explicit mapping of any kind from W -space to C-space and no explicit calculation of C-space will ever take place before or during the actual arm motion..

Because the last assignment is the one that sticks, the above modi cation sets an initial value for all output bits and then only changes one output as necessary. Although this code will synthesize to the same logic structure as the more complex nonblocking structure, race conditions may appear in simulation. Although it is less intuitive, nonblocking assignments can be used to accomplish the same thing with a similar coding style as shown below.