By Ned Chapin

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1(r) a nal yticall y or numerically becomes much easier. This approach is developed in the following subsections. 2 Link Parameters Consider a manipulator consisting of n links co n nected serially by n joints, with one degree of freedom each, which may be either revolute or pri smatic. As is shown in figure 2. 1 5, the links and the joints are numbered 1 , ... 16 Joint axes, joint variables, and link parameters: (a) when joint i is revolute; (b) when joint i is prismatic. Kinematics 33 starting from the base side.

Alan2 (x. x! 14 Solutions of inverse kinematics problem for two-link manipulator. Joint 2 ~. 15 Link and j oint numbers for n-link manipulator. 9 shows , it is rather easy to solve the direct kinematics problem and the inverse kinematics problem for manipulators with around two degrees of freedom. However, both problems become increasingly diffi cul t as the number of degrees of freedom increases. One way to cope with this diffic ul ty is to as si g n an ap p ropria te coordinate frame to each link and to describe the relation among the links by the relation among these frames.

First, 'T , � r� �l l ° ° 1 ° ° 1 ° and ° ° ° ° 1 ° ° ° 1 Ig ° ° 1 . 33, the corresponding position vector by '2 '3 '4 '5 ' 6 = Py = = = = pz + R 2 3 ig , + la + R33 1g, atan2(R 2 3 , R 1 3 ), atan2 ( { JR 1 3 2 + R 2 / , R 3 3 )' 2 + R2 / atan2(R 3 2 , - R 3 1 ) if R 1 3 atan2 ( R 2 1 , R 2 2 ) - R 3 3 '4 if R 1 3 2 + Second, when q = q * we have =F ° 0. ). One approach to this problem is to find a closed-form solution by using algebra or geometry. 6 Although the former approach is generally more desirable in applying the solution to real-time control of robots, it is not always possible to obtain closed-form solutions for manipu lators with an arbitrary mechanism.