This paper proposes four geometric invariants to be used to eliminate joint variables in closure equations for a 6R manipulator instead of the Gaussian elimination used in other studies. The geometric invariants are determined by the structure of any three consecutive joints in space. They have specific geometric meanings such as angle, length, area, volume. For a general 6R manipulator, its four basic closure equations containing only three angular variables can be directly constructed in a few minutes from the geometric invariants using Maple on a general personal computer. These resulting equations have no extraneous roots and are algebraically independent. Because the basic closure equations are obtained from the geometric invariants, they have the most simple forms and provide a very good chance to solve the input-output equation for the inverse kinematics problem. As a result, we use the set of basic closure equations to derive the symbolic 16th degree input-output equation and compute the input-output equation for a special case with 16 different real solutions. All position and orientation coordinates of the end-effector may be symbolic parameters. The definition of the geometric invariants is independent of joint types and can be applied to manipulators with any serial geometry.

Dixon's method for computing multivariate resultants by simultaneously eliminating many variables is reviewed. The method is found to be quite restrictive because often the Dixon matrix is singular, and the Dixon resultant vanished identically yielding no information about solutions for many algebraic and geometry problems. We extend Dixon's method for the case when the Dixon matrix is singular, but satisfies a condition. An efficient algorithm is developed based on the proposed extension for extracting conditions for the existence of affine solutions of a finite set of polynomials. Using this algorithm, numerous geometric and algebraic identities are derived for examples which appear intractable with other techniques of triangulation such as the successive resultant method, the Gro¨bner basis method, Macaulay resultants and Characteristic set method. Experimental results suggest that the resultant of a set of polynomials which are symmetric in the variables is relatively easier to compute using the extended Dixon's method.

This paper is to present an algorithm to decide whether a real-time system satisfies a set of invariants which are constructed from linear inequalities of integrated durations of system states. Real-time systems in the paper are taken to be real-time automata which set up for each of state transitions a lower time bound and an upper time bound. The satisfaction problem can be translated into a family of linear programming problems. The algorithm is, according to the invariants, to reduce the infinite family of linear programming problems to an equivalent one with only finite members, and then to solve each of linear programming problems efficiently. The algorithm is so simple that no prerequisite of linear programming theory is assumed.