Certified Solving and Synthesis on Modeling of the Kinematics. Problems of Gough-Type Parallel Manipulators with an Exact Algebraic Method

The forward kinematics of the 6-6 parallel manipulator using an evolutionary algorithm based on generalized generation gap with parent-centric crossover

“…21 In numerical methods, specific geometric properties usually lead to computation simplifications and smaller mathematical model expression and in many instances, the methods cannot solve the general configurations. 29 For the sake of clarity and simplicity, OA | R f will be replaced by OA O and CB | Rm by CB O . This article shall only concentrate on the forward kinematics problem (FKP ), Fig.…”

“…The rotation matrix R can be expressed utilizing various orientation models with their specific rotation variable sets such as navigation angles (yaw, pitch and roll), Euler angles, quaternions or even taking the nine rotation matrix components as variables. 29 Implementing the equation 2 directly, various displacement based equation models can be derived depending on the selected orientation variables. 29 In our experience, several different evolutionary algorithms and optimization techniques could not isolate the orientation variables to acceptable accuracy levels less then 1 degree.…”

“…35 Homotopy Continuation methods might then be advocated, 26 but they are prone to miss solutions. 29 Resultant or Dialytic Elimination methods have also been successfully applied on a certain number of problems, 16,36 but they might add spurious solutions. 29 More recently, interval analysis was implemented that could find certified results.…”

“…29 Resultant or Dialytic Elimination methods have also been successfully applied on a certain number of problems, 16,36 but they might add spurious solutions. 29 More recently, interval analysis was implemented that could find certified results. 22 However, homotopy and interval analysis require one numerical method for solving an equation system and this is usually Newton's method, implying that, in certain conditions, some solutions may not be found.…”

A meta-heuristic paradigm for solving the forward kinematics of 6–6 general parallel manipulator

“…Resultant or dialytic elimination methods have been applied but they have the tendency to add spurious solutions. Moreover, homotopy methods might also be advocated but they are prone to Rohitash [9]. Therefore, the implementation of other methods which could find solutions numerically is justifiable.…”