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For example, if the arm endpoint follows a sharp corner on an obstacle, or if the arm contacts some obstacle while passing around another obstacle [as in the link position (a8 , b8 ), Figure 54], the virtual line may form sharp turns Nor is a virtual line necessarily a non-self-intersecting curve (see virtual boundary of obstacle B, Figure 53), differing in this respect from the boundaries of physical two-dimensional objects We will discuss this issue later, when analyzing the arm C-space properties Points of contact on the arm may undergo a discontinuous jump when passing around obstacles This can happen because of the shapes of obstacles and arm links involved, or because of the arm obstacle interaction In Figure 5.

Combinatorial loops are logic structures that contain feedback without any intermediate synchronous elements. As can be seen from Figure 12.11, a combinatorial loop occurs when the output of a cloud of combinatorial logic feeds back to itself with no intermediate registers. This type of behavior is rarely desirable and typically indicates an error in the design or implementation. In 18, we will discuss how to handle the timing analysis of such a structure, but for this

4, for example, during link l2 motion through positions (a1 , b1 ), (a2 , b2 ), and so on, an instant before position (a8 , b8 ) link l2 is in contact with obstacle A; an instant after position (a8 , b8 ) the link is in contact with obstacle B Accordingly, in this short period the contact point on the arm jumps from a point of contact on one side of link l2 to a completely different point on the link s other side Note, however, that even in such cases there will be no discontinuity in the virtual curve2 For example, in the area of point b8 , which corresponds to the jump of the contact point mentioned above (Figure 54), the virtual line remains continuous There will be more on the virtual line continuity in our analysis of the arm C-space.

Observe also that some distinct pieces of the virtual line may be associated with the same physical curve Such is, for example, a part of the virtual line (b14 , b8 ) (Figure 53), which is a part of obstacle A boundary When trying to do a complete rotation by the arm around A, the arm endpoint will follow the curve segment (b14 , b8 ) twice, once in each of the two directions The requirement of continuous contact while passing around the obstacle is equivalent to adding a constraint on the arm motion In general, the arm s position relative to obstacles is described by one of these three situations: 1 No contact with obstacles takes place; the motion is unconstrained, and all points in the vicinity of the arm endpoint are available for its next position..

discussion we will discuss traps that may create such a structure and how to avoid them. Consider the following code segment.

He was on the sales side of trading, and was in the business of providing liquidity for funds and for some midsize manufacturing companies I was helping.

Given the physics of the underlying phenomenon, this is not surprising: Physical motion is continuous, so the arm endpoint must be moving through a continuous curve.

// BAD CODING STYLE module combfeedback( output out, input a); reg b; // BAD CODING STYLE: this will feed b back to b assign out = b; // BAD CODING STYLE: incomplete sensitivity list always @(a) b = out ^ a; endmodule

2. One degree of freedom of the system (not necessarily one arm link) is constrained by an obstacle boundary; then only points along the virtual line that is, a one-dimensional curve are available for the next positions of the arm endpoint. 3. Two degrees of freedom of the system are constrained: No motion is possible. Because of our model s assumption that some motion is always possible, case 3 is impossible. Case 2 thus includes all cases of interaction between the arm and obstacles.

Obstacles in C-Space. Con guration space (C-space) of our RR arm manipulator is presented as the surface of a common two-dimensional torus de ned by two independent angular variables, 1 and 2 [57]. Values of these variables are the arm joint values, respectively. An arm position P with coordinates (joint valp p ues) 1 and 2 in W -space corresponds to a point P with the same coordinates on the surface of the C-space torus. Continuity is preserved in this mapping: A small change in the position of arm links in W -space translates into a small displacement of the corresponding image point in C-space. A closed curve in W -space has its closed curve counterpart in C-space [105]. For an M-line in W -space, there is an M-line image in C-space (Figure 5.5).