Matrix Jacobean

As seen from the above equations, the (mxp) matrix (Ve1)1 is the Jacobean matrix, J, of the vector function e, and the (mxp) matrix(VfT)T is the Jacobean matrix, G, of the vector function f(x,k). The srth element of the Jacobean matrix J. is given by... 

In other words, the observation matrix C from the case of a linear output relationship is substituted with the Jacobean matrix (dhT/dx)T in setting up matrix A and vector b. 

The partial derivative of fp is called the Jacobean matrix Jr. The index R that this Jacobean is for the specific robot studied. Using the Jacobean designation, Eq. 6 can be written as ... 

The stability of x t) is determined by whether the solution wiU return back to Xg, yg following a perturbation. This will occur if the solutions x are stable, and this requires that the real part of A. be negative. This can also be regarded as a problem of finding the eigenvalues of the matrix [ j] from the Jacobean of the original linear equations. Solutions of x t) can be as shown in Figure 6-5. This pair of equations is stable if [(a 8) + 4 y] > 0 (A real). When we insert coefficients a, defined,... 

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