## 1 Introduction  
Parallel manipulators, either planar or spatial, have been used for the last three decades due to their higher rigidity, accuracy, and payload capacity. However, they have smaller workspaces than serial manipulators. The Stewart platform (SP henceforth) is a well-known type of spatial manipulator. These manipulators have one fixed and one moving platform connected to each other by actuators, as seen in Figure 1. For this type of manipulator, various assembly modes are available, such as 6-6, 6-3, 3-3, etc. [1]. These numbers are attributed to the number of joints on the fixed and moving platforms, respectively. Contrary to serial manipulators, the forward kinematics of parallel manipulators are more complex than the inverse kinematics, hence many authors have focused on the forward kinematics of the SP, along with other parallel manipulators [2]. In the forward (direct) kinematics, loop equations and rigidity conditions are utilized to transition from actuators' kinematic parameters to those of the moving platform, i.e., to the pose, velocity, and acceleration of the moving platform. Vast literature exists on parallel manipulators, especially on the Stewart platform. However, here some early published and remarkable papers will be mentioned, as fundamental methods are developed and proposed in these works. For detailed literature on the Stewart platform, the paper by [PERSON] et al. can be referred to [2]. [PERSON] et al. studied a 3-dof spatial parallel platform in which linear actuators are connected to the fixed plate by revolute joints [3]. Inspired by this work, [PERSON] et al. developed a solution procedure for the 6-3 and 3-3 Stewart platforms [4]. [PERSON] demonstrated that a 6-dof Stewart platform with a triangle-shaped moving platform leads to a 16 th-degree polynomial [5]. In another paper, [PERSON] used symbolic computation to determine the solution polynomial of minimal degree and singular configurations of parallel manipulators [6]. [PERSON] proposed a method to count the real roots of the solution polynomial [7]. W forward kinematics of a general Stewart platform using Soma coordinates [8]. These papers differ from each other in the methods used or proposed. Parallel planar manipulators (PPM) usually have three dof (degrees-of-freedom) unless redundant freedom is desired. Several design variants are encountered in the relevant literature, such as 3 RPR or 3 RRR. [PERSON] et al. [9] proposed a new architecture for the 3-dof planar manipulator. [PERSON] studied the forward kinematics of different types of planar parallel manipulators [10]. [PERSON] and [PERSON] obtained the maximum number of solutions for the forward kinematics of planar parallel manipulators using [PERSON]'s theorem [11]. They showed that even though the proposed polynomial has six roots, only four solutions exist because the two left solutions are not feasible. [PERSON] et al. performed inverse kinematic model of a 3 RRR parallel robot and did workspace analysis [12]. Kinematic and dynamic analysis of a 3-RRR parallel planar robot using four different neural network algorithms was carried out [13]. [PERSON] [14] dealt with the kinematic and singularity-free workspace analysis of a 3-dof asymmetrical planar parallel robot (3-RPR) mechanism. For both spatial and planar parallel manipulators, the forward kinematics analysis leads to a set of non-linear equations based on satisfying kinematic constraint equations. These equations can be converted into algebraic equations by appropriate transformations. Researchers have tried to solve these equations using different methods, such as [PERSON]'s method [15; 16], the Groebner bases method [17; 18; 19], the interval analysis method [20; 21], the continuation approach [8], algebraic elimination method [22], and numerical methods such as the Newton-Raphson method [16; 23], or using the conformal geometric algebra framework [24]. However, in some special cases, this set of equations can be solved with simpler techniques [4].