Automata, languages, and machines by Samuel Eilenberg

By Samuel Eilenberg

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27) which is of fundamental importance in inertial navigation [Stevens and Lewis 1992]. Computing the Arm Jacobian Let us now return to the problem of defining and computing the arm Jacobian J(q). 2) does not afford an appropriate definition. 3) as the definition of J(q). 4). Then J(q) is a 6×n matrix. Let us consider a few examples showing how to compute the arm Jacobian to demonstrate that the procedure is not complicated. A methodical technique for computing J(q) will then be given. 1 to compute the Jacobian for the three-link cylindrical arm.

It says that Rk=k, or that the rotation axis is not rotated by R. , [IMSL]), which may be used to find k from R. See the problems to find an explicit formula for k in terms of (n, o, a) [Paul 1981]. The angle of rotation ␸ about the axis k may easily be found, for the trace of a rotation matrix is equal to 1+2 cos ␸. 16) where nx, for instance, denotes the x component of the normal vector n. , k and ␸) for orientation. Representing Cartesian Orientation Using Quaternions. The quaternion representation is a 4-vector which is often used to describe the end-effector frame orientation [Ickes 1970], [Sheppard 1978], [Yuan 1988], [Stevens and Lewis 1992].

5a where ᐉ is the fixed known length of the base link. 5b. The joint vector is (1) which corresponds to polar coordinates in the plane. 5: Two-link planar RP arm: (a) arm schematic; (b) D-H coordinate frames. By inspection, the A matrices are found to be Copyright © 2004 by Marcel Dekker, Inc. 5a. This may be interpreted as the forward kinematics solution for the arm. This example illustrates the freedom we have to select the base frame origin o0. We could have selected o0 coincident with o1. However, we have chosen to include the length ᐉ of link 0 by placing o0 at the bottom of the base link.

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