The Inertia Projection Method

The Structurally Recursive Method is then expanded, and a second, non-recursive algorithm fw the manipulator inertia matrix is derived from it A finite summation, which is a function of individual link inertia matrices and columns of the propriate Jacobian matrices, is defined fw each element of the joint space inertia matrix in the Inertia Projection Method. Further manipulation of this expression and application of the composite-rigid-body inertia concept [42] are used to obtain two additional algwithms, the Modified Composite-Rigid-Body Method and the Spatial Composite-Rigid-Body Method, also in the fourth section. These algorithms do make use of recursive expressions and are more computationally efficient. 

Four algorithms for computing the joint space inertia matrix of a manipulator are presented in this section. We begin with the most physically intuitive algorithm the Structurally Recursive Method. Development of the remaining three methods, namely, the Inertia Projection Method, the Modified Composite-Rigid-Body Method, arid the Spatial Composite-Rigid-Body Method, follows directly from the results of this tot intuitive derivation. 

Table 3.2 Algorithm for the Inertia Projection Method (Method II)...

In the next analysis, we will examine the components of successive inotia matrices as defined by the algorithm given in Table 3.1. First, the expansion of the equations for the Structurally Recursive Method leads to an exjnession for H,j, the Tii X itj submatrix of H, in the form of a summation. Its terms involve projections of individual link inertias onto the preceding joint axis vectors, which... 

In Equation 12.6 p, is the permanent dipole moment, h is Planck s constant, I the moment of inertia, j the angular momentum quantum number, and M and K the projection of the angular momentum on the electric field vector or axis of symmetry of the molecule, respectively. Obviously if the electric field strength is known, and the j state is reliably identified (this can be done using the Stark shift itself) it is possible to determine the dipole moment precisely. The high sensitivity of the method enables one to measure differences in dipole moments between isotopes and/or between ground and excited vibrational states (and in favorable cases dipole differences between rotational states). Dipole measurements precise to 0.001 D, or better, for moments in the range 0.5-2D are typical (Table 12.1). 

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