## 1 Introduction
[PERSON] showed in 1990 that the forward kinematics of a 6-3 SP can be solved by finding the roots of a 16 th-order polynomial, which [PERSON] [25] already presented in 1989. This is the case in the 3-3 SP (also in the 6-3 SP). As several papers about the forward kinematics of parallel manipulators have been published lately, they focus on the issues such as the use of machine learning [26], transforming the forward kinematics problem to a suitable inverse kinematics analysis [27], or improving and optimizing the weighting factor of a neural network for the forward kinematics of a Stewart platform [28] rather than proposing or developing new analytical or analytical-numerical combined methods. As is seen in the literature mentioned above, various methods have been proposed to solve forward kinematics problem, which can be separated into two main groups: closed-form solution (algebraic) methods and numerical methods. The details of the algebraic  
Figure 1: General view of the 3-3 Stewart platform. techniques proposed so far will not be given here because they are well-known in the relevant community and are beyond the scope of the paper. Regarding numerical methods, the Newton-Raphson method is the most common among these. In this method, once the user assigns the initial values to the unknowns of the problem, this method gives just one solution. The method is highly sensitive to initial values. Sometimes, it may converge toward an uninterested solution far from the expected, or even it might diverge. Finding only one solution at once is the weakness of this method. Although the forward kinematics problem of parallel manipulators has been intensively studied from various aspects, and the issue appears completed, it is still possible to develop new approaches specific to some manipulator architectures but extendable to more general assembly forms. Motivated by this fact, in the present paper, we propose two new and easy to implement methods for the forward kinematics of parallel manipulators of some specific types. Both methods can be used as a general solution procedure for 3-RRR planar manipulators, whereas they are applicable to the 3-3 or 6-3 type Stewart platforms only. The process in our both methods is the same as that in the present literature until kinematic constrain equations are obtained. The first of our methods is an analytical approach, while the second one is a combined analytical-numerical method. We call these methods \"the Solve-and-Substitute-Method\" (the SSM henceforth) and \"the Error-Function-Method\" (the EFM in short). However, our methods (especially the SSM) differ from those existing in the literature, in that they exploit the special structure of the constraint equations for the manipulators considered. In both the polynomial-based methods in the current literature and the method we propose here (the SSM), the problem of finding the position parameters (coordinates) is reduced to obtaining the roots of a polynomial that is a function of only one of these coordinates. However, our method, the SSM and the EFM differ from those existing in the literature, in that they exploit the special structure of the kinematic constraint equations for the manipulators considered. The steps of the SSM can be explained as follows. We have three constraint equations for both types of manipulators studied. These equations have three unknowns, say \(\lambda\), \(\mu\), and \(\
u\). Each of these equations includes just two of these unknowns. The power of these unknowns in the equation is at most two. In this case, two of these equations are considered, in which one unknown is common, say \(\lambda\). Hence, if this common unknown is considered as a parameter, these two equations can be viewed as polynomials of second order in terms of the remaining unknowns \(\mu\) and \(\
u\). The roots of these polynomials can be found in \(\lambda\). Afterwards, these roots are substituted in the third equation in which only \(\mu\) and \(\
u\) appear, and a polynomial of 32 nd order.