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job; this can be some kind of a gripper, a screwdriver, a paint or welding gun, and so on. The arm end effector may have its own DOF. To use tools, the end effector needs to be put not only in a particular position but also at a particular orientation: For example, not only the screwdriver s blade has to be in the screw s slot, but its axis has to be perpendicular to the surface into which a screw is driven. In the human arm, it is of course the hand that handles the tool orientation, a relatively compact device, compared with the rest of the arm, with its own (many) DOF. A similar organization of kinematics is common to robot arm manipulators: The robot s DOF are divided into the major linkage, relatively long links whose function is to bring the end effector to the vicinity of the job to be done, and the more compact minor linkage (called the wrist or hand), which is the end effector proper. Although from the theoretical standpoint such division is not necessary, it is useful from the design considerations. Often a real arm s major linkage consists of three DOF, and its minor linkage presents a 3-DOF hand. The values of all robot arm DOF de ne its coordinates in physical space. A given position plus orientation can be described via two coordinate (reference) systems: Cartesian coordinates and joint (joint space) coordinates. The latter are also called con guration space coordinates. As long as both systems describe the robot con guration that is, position plus orientation in a unique way, they are equivalent. For the 3-link 3-DOF arm manipulator shown in Figure 1.2b, its

Example Third-party administrator who works directly with payroll provider to receive contributions

1' (a)

11 T

Key role Invest assets in accordance with fiduciaries /trustees instructions Responsibilities Provide prudent investment management, including diversifying assets in order to reduce risk of substantial loss Example Investment management arm of financial institution (e.g., mutual fund)

3 10 B 4

1 6 5 A

The following roles, given their specific requirements, are typically always outsourced to the appropriate professional. It is extremely rare for any of these roles to be performed in house. These professionals must comply with their own industry professional standards.

9 8 1

7 5 6 10

Key role Determining the liabilities of a defined benefit plan (not applicable to defined contribution plans)

q1 = 90

q1 = 0

+ q1 (b)

Responsibilities Calculate cash contribution and accounting expense requirements according to regulations Assist plan sponsor in administration issues

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Figure 5.24 Revolute prismatic (RP) parallel two-link arm. (a) W -space, with obstacles A and B. Shadows (shaded) result from interaction between the rear end of link l2 with obstacle A and the front end of l2 with obstacle B. (b) C-space images of the corresponding virtual obstacles. Line (S, T ) is the image of the straight line (S, T ) in W -space.

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Note also that these same obstacles A and B create another two shadows, which are not of importance in our current task but might show up with some other M-line: the arm endpoint will not be able to reach any point within the gures described by points in the left part of Figure 524a, (4, 5, 5 , 1 , 1, 2, 3 , 4) and (0, 10 , 6 , 0) An obstacle that extends into the arm s dead zone forms only one shadow, which extends from the obstacle all the way to the W -space outer periphery An obstacle X that is in the shadow of another obstacle Y will never interact with the arm If obstacle X partially intersects the shadow of obstacle Y, it will be treated by the arm as a part of Y.

Key role Assist other parties (employer, fiduciaries, plan administrator, etc.) with complying with legal requirements

Therefore, with the exception of obstacles that extend into the arm s dead zone, any virtual obstacle in the workspace of this RP arm includes the actual obstacle itself plus two shadow components: one front shadow and one rear shadow The front shadow extends from the obstacle to the outer boundary of W -space [see the curve (6, 7, 8, 9, 10, 11, 12, 6) in Figure 524a]; the rear shadow extends from the arm origin O into the periphery, but in general not to the boundary, of W -space [see the curve (1, 0, 5, 4, 3, 2, 1) in Figure 524a] Unlike the more complicated situation with the RR arm (Section 52), in the case of our RP arm the virtual lines of obstacles are always simple curves.

In a special case when the M-line crosses the dead zone, the latter can be treated simply as an obstacle interfering with the rear end of link l2 Two independent variables, one angular and the other linear displacement, can be represented by the surface of a cylinder The C-space of our arm is therefore a cylinder whose at sides, called base circles, correspond to the rst joint value, 1 , and whose height corresponds to the second joint value, l2 The C-space of the example in Figure 524a is shown in Figure 524b Shown in the gure are the images of an M-line and of the shadow components of virtual obstacles A and B that interfere with the arm motion.

Key role Audit the trust Responsibilities Provide assurance that financial statement of plan s trust is accurate Sample professional designation Certified public accountant

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