## 1 Introduction  
The dimensional synthesis of a mechanism determines the dimensions of the individual links that generate a desired motion. Since the development of an analytical approach to the design of four-bar linkages introduced by [PERSON] [1], numerous studies have been conducted on the dimensional synthesis, which can be classified into two categories. One is the exact synthesis method, known as the precision point approach, in which a finite number of exact positions are to be satisfied. The other is the approximate synthesis, which seeks the dimensions that approximately satisfy prescribed positions. In general, dimensional synthesis problems are formulated as a system of nonlinear equations and require heavy computation. For this reason, various approximate synthesis approaches using numerical methods [2, 3], optimization methods such as least square techniques [4, 5, 6], genetic algorithms [7, 8, 9, 10], and evolutionary algorithms [11, 12, 13, 14, 15] have been used widely in recent times. However, the approximate synthesis methods require appropriate initial values that are close to the actual solutions of the synthesis equations. When unsuitable initial values are assumed, the solutions may not be found. In addition, the methods may not find all the solutions of the synthesis equations. In contrast, since the exact synthesis method can find all the solutions without estimating an initial value, it yields diverse mechanism dimensions that satisfy a desired motion. Among the studies on the precision point approach, [PERSON] et al. [16] and [PERSON] et al. [17] used elimination methods to perform the dimensional synthesis of four-bar path generators for five precision points and slider-crank function generators for four precision points, respectively. [PERSON] and [PERSON] [18] and [PERSON] et al. [19] carried out the dimensional synthesis of four-bar path generators for five precision points and nine precision points, respectively. For the dimensional synthesis of six-bar linkages with revolute joints only, [PERSON] et al. [20] and [PERSON] and [PERSON] [21, 22, 23] applied the homotopy continuation method for the synthesis of Watt II, Stephenson II, and Stephenson III six-link function generators, proposing design methods for eight, nine, and eleven precision points, respectively. It is worth noting that the continuation method is developed from the parameter perturbation procedure introduced by [PERSON] and [PERSON] [24] for solving a system of nonlinear equations starting from a known solution of a similar system. Most studies on the precision point approach deal with four-bar linkages, four-link slider-crank mechanisms, and six-bar linkages with revolute joints only, but the exact synthesis of six-link slider-crank mechanisms with six revolute joints and one prismatic joint are extremely rare. Since the six-bar slider-crank linkage has more design parameters, it can satisfy more precision points than its four-link counterpart. Different types of six-bar slider-crank linkages are widely used in various applications such as ore crushers, mechanical presses [25], punching mechanisms [26], and quick return mechanisms [27] in machinery. These precision mechanisms are required to satisfy precise motion between input and output links. Furthermore, by adding one link and one joint, the six-link slider-crank can be modified to a two-degree-of-freedom seven-link adjustable mechanism that can be used in controllable presses for punching and metal forming [28; 29], variable pumps [30], and variable compression ratio engines [31; 32; 33; 34]. For the design of this machinery, the dimensional synthesis of the six-link slider-crank is essential. This paper proposes a systematic exact dimensional synthesis method to find all Watt II and Stephenson III six-bar slider-crank function generators that satisfy nine precision points of the output slider with respect to prescribed input crank angles.