## 1 Introduction  
Since its invention by [PERSON] and [PERSON] [1], the six-degrees-of-freedom (six DOF) parallel platform has rapidly become popular because of its high stiffness, accuracy, and strong bearing capacity. Nowadays, the Stewart-Gough platform (SGP) is widely applied to motion simulation, optical precision machining, intelligent medical treatment, aerospace positioning, and space measurement. The forward kinematic problem (FKP) of the platform refers to predicting the posture in six degrees of freedom by the length of the six outriggers. The FKP is of great significance for posture detection, SGP calibrating, and high-precision positioning. However, the FKP mathematically means multivariate higher-order nonlinear equations, from which it is a big challenge to ascertain the unique solution. In order to obtain the high-precision forward solution and improve real-time applications, researchers have put forward many ideas. [PERSON] and [PERSON] found 40 forward solutions by algebraic method [2, 3]. [PERSON] et al. analyzed the polynomial solution of forward kinematics for the different parallel platform [4, 5]. [PERSON] et al. applied the algebraic method to choose one out of all possible postures [6, 7]. [PERSON] et al. proposed a kind of generic geometric-algebraic framework for FKP [8]. These above algebraic methods analyze the FKP as a math problem, which cannot be applied in engineering, therefore, numerical methods have attracted the attention of researchers. The Newton-Raphson (NR) method is popular for seeking the solutions of nonlinear equationsbecause of its simplicity and fast convergence. [PERSON] and [PERSON] put forward a global NR method to avoid the divergence caused by the selection of initial value [9; 10]. However, the NR method requires calculation of the Jacobi matrix or Hessian matrix, which may not exist in some situations, and the calculation of partial derivatives is not conducive to real-time application. [PERSON] et al. analyzed the number of extra sensors for the FKP of different SGP configurations [11]. [PERSON] et al. made use of two extra rotary sensors to help achieve the solution, and their method could reduce the computational burden [12]. By optimizing the sensor installation position, [PERSON] and [PERSON] proposed a method based on three redundant sensors, and this method is insensitive to the sensor position deviation and measurement error [13]. By adding additional sensors, the calculation can be simplified. However, the measurement system will increase the costs and the measurement error introduced is also a big challenge. With the improvement of computing power, many intelligent methods have been applied to the calculation of robot kinematics. [PERSON], [PERSON] and [PERSON] both trained a neural network to replace tedious kinematic calculation [14; 15]. [PERSON] et al. analyzed the SGP by using the principal component analysis [16]. In this way, the FKP was converted to a two-dimensional nonlinear minimization problem. [PERSON] et al. developed a popular machine learning method--support vector machines--for classification and regression, and they finally built a solution model for the FKP [17]. However, these intelligent algorithms require a large number of prior data, which are not scalable and are inconvenient to apply to practical engineering. In this paper, two vector spaces are built: one is composed of all possible length vectors, which is called space L, and the other, named space Q, is filled with posture vectors. The FKP now is regarded as mapping space L to space Q through the complex nonlinear relationship. Considering efficiency and accuracy, the PSO method combined with the SA process is raised to find the target posture from space Q. In Section 2, the geometric structure and kinematics of SGP are analyzed in detail. Section 3 redefines the FKP problem from the perspective of spatial mapping, and the SA-PSO method is introduced to deal with it.