Geometric basis:a geometric solving cell for geometric computing

Geometric computing is an important tool in design and manufacturing and in arts. Conventionally, geometric computing is taken by algebraic computing. The vivid intuition of objects in visualization is lost in numeric functions, which is however very useful to human cognition as well as emotion. In this paper, we proposed a concept and theory of geometric basis （GB） as the solving cell for geometric computing. Each GB represents a basic geometric operation. GB works as both expressing and solving cell just like the concept of basis in linear algebra by which every element of the vector space can be expressed. For 3D problems, with a procedure of a projections reduction, the problem can be reduced to plane and the reduction function can be designed as a GB. A sequence of GB can construct a higher layer GB. Then, by the traversal of tree, a sequence of GB is got and this sequence is just the construction process and also the solution of this geometric problem.

Algebraic Solution for the Forward Displacement Analysis of the General 6-6 Stewart Mechanism

The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.