Classification and complete solution of the kinetostatics of a compliant Stewart–Gough platform

This paper studies a compliant Stewart–Gough platform whose six flexible links balance external loads, including force and moment, applied to a general point at the platform. Soma coordinates are used to express the location of the moving plate with respect to the base plate. The mathematical modeling involves both the kinematics and the statics. Seven kinematic and six static constraints are obtained. Vector dot- and cross-products are also cast in quaternion form which reduces the degree of the resulting constraints. By classifying the parameters to the knowns and the unknowns, five major problem types are recognized. Four of them are mathematically decoupled and the remaining one is coupled for which the obtained 13 polynomials are of the total degree of 5,971,968 which to the authorsʹ best of knowledge is the largest kinematic problem ever investigated. After solving the equations with the numerical polynomial solver Bertini, we conclude that the upper bound to the number of nonsingular solutions is 29,272. In addition, for practical problems a parameter continuation is devised to recompute for only 29,272 generically nonsingular solutions. At last a numerical example is provided to demonstrate the solution process.