## 1 Introduction  
Control valves are commonly used devices for flow control in industries, with their use ranging from food industries to petrochemical industries. These control valves, especially the valves used in petrochemical industries, constitutes the major proportion of fugitive emissions from industries. According to studies by the European Sealing Association (ESA), valves account for up to 60% of fugitive emissions from a particular plant. Usually, valves with rising stems such as globe valves and gate valves tend to leak more often. A common reason for this leakage is the frequent operation of control valves that causes the stem and packing to wear quickly, thus allowing the emission paths to develop [1]. Regardless of how good the packing in the valve is, leakage still exists through the valve packing due to its design. Environment protection standards such as API 624 also consider leakage of up to 100 ppm as an acceptable emission [2]. In an attempt to solve this problem, a zero-leakage valve exists in industry known as a bellow valve [3]. Bellow valves use metal bellows for valve stem sealing, which removes the risk of emissions. Metal bellows provide the characteristic of no emission sealing by extending or compressing within the elastic limit while allowing the required motion to operate the valve. The structure of the metal bellow valve consists of a metal bellow attached to a reciprocating shaft, that allows disc motion, and the bonnet of the valve. One end of the bellow is attached to the bonnet of the valve whereas the other end is attached to the stem. The bellow extends or compresses with the axial movement of the stem, thus providing the perfect sealing. The disadvantage of bellow valve sealing is that to keep the extension or compression of the bellow within the elastic limit, a long length of bellow is required to produce the required displacement. This in turns increases the length of the stem, therefore, increasing the height of the valve. The valve height for a one-inch valve can go up to 17 inches. This unusual valve height makes the design of the valve aesthetically unsuitable for most applications. This is the reason for the very low usage of bellow valves in industry [4]. This study presents the synthesis and analysis of a novel extending mechanism that addresses the problem of requiring of long length of bellow. The synthesized linkage will convert the small displacement of a bellow to a large displacement at the valve, therefore reducing the length of the bellow required to generate the desired motion. Introducing this mechanism helps reduce the height of the valve exponentially, thus making the design of bellow valves more suitable for use in industrial applications. The bellow valve working principle is shown in Figure 1.  
Linkages are the most fundamental elements of mechanisms and machines. It is an assembly of a number of links and joints that generates the desired output motion in response to some specific input. The number of links possessed by a linkage gives the name to linkage, for example, a four-bar linkage consists of four links joined together to form a mechanism. A six-bar linkage, as the name suggests, is a single degree of freedom mechanism with six links and seven joints. A six-bar linkage can also be considered as a combination of two four bar linkages with two links shared. This assumption makes the synthesis and analysis of six-bar linkage less complicated by solving two four bar linkages [5]. There are several six-bar linkages that form the basis of various complicated mechanisms and can be considered as standard six-bar linkages. These standard six-bar linkages form a good starting point of linkage synthesis [6]. These include the Watt I six-bar linkage, Watt II six-bar linkage, Stephenson I six-bar linkage, Stephenson II six-bar linkage, and the Stephenson III six-bar linkage. Six-bar linkage also forms the basis of extending linkages used for magnifying displacements or forces.