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notation

# notation - New Notation for Serial Kinematic Chains...

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New Notation for Serial Kinematic Chains Berthold K. P. Horn May 1987 Abstract: The description of a serial kinematic chain should be unique, unambiguous, simple to determine, easy to use and well-behaved when small changes are made in the arrangement of the elements of the chain. The notation currently in use, introduced by Denavit and Hartenberg, does not satisfy all of these criteria. It involves arbitrary choices, so that more than one description may apply to a given kinematic chain. More impor- tantly, the parameters relating the links in the chain can be very sensi- tive to small changes in the physical arrangement of the chain. This is particularly true of so-called ideal chains, ones that permit closed-form solution of the inverse kinematic problem, since these often involve ge- ometries where adjacent axes are parallel, perpendicular or intersect. A new notation is proposed here that does not suffer the above-mentioned short-comings. To demonstrate some of the advantages of the new nota- tion, it is applied to the problem of finger-printing a robot arm and to the solution of the inverse kinematic problem of near-ideal arms.

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1 Introduction Let us start right away by defining the new notation: Let the home position of a serial kinematic chain be an arbitrary posi- tion specified in terms of the joint variables. The home position may, for example, be chosen to be the one where all joint variables are zero. The base coordinate system is an arbitrary external coordinate system fixed with respect to the base of the kinematic chain. Erect a coordinate system in each link of the chain in such a way that the coordinate axes are paral- lel to those of the base coordinate system when the chain is in the home position. The kinematic chain is then fully specified when the following are given for the chain in the home position: The set of unit vectors { o } , parallel to the directions of motion of ˆ i the prismatic joints. The set of unit vectors { ω } , parallel to the axes of the revolute joints. ˆ i The set of offset vectors { d } determined recursively with respect to i the chosen base origin as follows: The first reference point is the origin of the given base coordinate system. Drop a perpendicular from this reference point to the first revolute joint axis. This defines the first offset vector as well as a reference point on the first revolute joint axis. Now drop a perpendicular from this new reference point onto the second revolute joint axis. This defines the second offset vector and a new reference point. Continue in this fashion to the last revolute joint. Connect the last reference point so found to a chosen tip reference point in the final link. This defines the last offset vector.
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