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1·The paper presents the feedback control based on Jacobi matrix.

2·Based on Jacobi matrix, a simple and useful method for the static transformation between coordinate systems of robot is developed.

3·The inverse Jacobi matrix of the 6-sps parallel manipulator is obtained from differential equations of the reverse displacement analysis.

4·This method based on modified nodal approach is easy to set up the fault diagnosis equations and has lower degree of nonlinearity and regular Jacobi matrix form.

5·Inverse Jacobi matrix of robot was obtained by means of the differentiation with respect to input and output equations for positions of 6-sps typed parallel robot.

6·The forward and inverse kinematics solutions of a 3-DOF parallel micro-nano manipulator were emphatically analyzed, and the Jacobi matrix of kinematics forward solution was derived.

7·New formulations are on the basis of the theory of the bifurcation and the Jacobi matrix eigenvalue structure analysis incorporating the margin and state index of the voltage stability.

8·The effects of posture changes on the accuracy of robot have been studied when the robot is in non-singular posture, namely the determinant of inverse Jacobi matrix is not close to zero.

9·Some measures, such as reasonable geological model, equivalent layer and equalization of Jacobi matrix elements, are taken to achieve less parameters, stable inversion j process and definite solution.

10·I analyse some conclusions of spectrum arbitrary and give two sign patterns. then I prove two classes sign pattern matrix that are spectrally arbitrary using Nilpotent-Jacobi method.