The condition number of the Generalized Inertia Matrix (GIM) of a serial chain can be used to measure its ill-conditioning. However, computation of the condition number is computationally very expensive. Therefore, this paper investigates alternative means to estimate the condition number, in particular, for a very long serial-chain. For this, the diagonal elements of the GIM are examined. It is found that the ratio of the largest and smallest diagonal elements of the GIM, when scaled using an initial estimate of the condition number, closely resembles the condition number. This significantly simplifies the process of detecting ill-conditioning of the GIM, which may help to decide on stability of the system at hand.
The K4 graph and the inertia of the adjacency matrix for a connected planar graph. A substantial history exists about incorporating matrix analysis and graph theory into geography and the geospatial sciences. This study contributes to that literature, aiding in analyses of spatial relationships, especially in terms of spatial weights matrices. We focus on the n-by-n 0–1 binary adjacency matrix, whose rows and columns represent the nodes of a connected planar graph. The inertia of this matrix represents the number of positive (n+), negative (n−), and zero (n0) eigenvalues. Approximating the Jacobian term of spatial auto-normal models can benefit from calculating these matrix quantities. We establish restrictions for n- exploiting properties we uncover for the K4 graph.