The slider-crank mechanism is one of the most fundamental elements in industrial machinery。 It converts translational motion into rotational motion on the one hand。 A widely known application is in internal combustion engines。 On the other hand, conversion from rotation to translation is also realized by this mechanism。 Typical examples are water pumps and compressors。 Historically speaking, slidercrank mechanism is one of the oldest machines ever used to satisfy human needs。 According to Nolle [1], slider-crank mechanism was utilized in pumping machinery as early as AD 970。 However, recent archaeological findings in Western Anatolia indicate that founding of ancient cities 2000 years ago was made possible by marble cutting machines based on slider-crank mechanisms [2]。 Despite its historical significance, full understanding and control of motion in his mechanism, which is essential for its effective use, has materialized only within the last century。 Some part of this knowledge has well been established in classical books like [3–7]。 Slider-crank mechanism is still continuing to capture

the unpided attention of researchers, which in some cases lead to creation of new devices such as fitness and rehabilitation equipment [8], intermittent motion generator [9], a new type of parallel manipulator [10]。 In fact, the motivation for the current article has come out of a need to design an economical, effective and efficient cutting system, in which case the control of the cutting tool position and velocity has been important。 It turned out that slider-crank mechanism was the most economical solution to satisfy the cutting requirements。 Figure 1 shows an example of a cutting application。 

Fig。 1 A Slider-crank based cutting system

Investigation and control of motion in the slider-crank mechanism has been the subject of many works from many different perspectives。 For instance, Andrews [11] considered the relative positions of the slider and crank points to describe the functional role of a muscle in the extremities of a human body。 Freudenstein [12] investigated the velocity fluctuations in the mechanism algebraically。 Coordination

of the crank and coupler’s acceleration pole-positions was studied in [13]。 The solution to the problem of determining crank positions corresponding to the maximum slider velocity in the centric slider-crank mechanism was given in closed form by Zhang [14]。

There are other researchers which incorporate into studies some dynamic elements involving elastic deflections as well。 For instance, Liu [15] exhibited a work on the analysis and synthesis of a cam-actuated flexible slider-crank mechanism to produce a desired output motion at a given design speed and damping ratio。 Position control and vibration suppression of a flexible slider-crank mechanism was studied in [16]。 Rotational speed control of the crank in the mechanism was the objective of two papers with two different methods [17, 18]。 Translational position control of the slider was investigated by means of a new approach in [19]。

The basic disadvantage of the works [15–19] is that they require extra elements such as cam mechanism in [15] or controllers of several types in [16–19], adding cost to the system at hand。 Alternatively, geometric potential of the slider-crank mechanism appears as a good basis to provide

desirable positions and velocities for the slider。 Thus, based on this geometric potential, one way to generate desirable positions and velocities in the slider-crank mechanism is to require that the motion transformation between the slider and crank of the mechanism take place according to a suitable

function。 Conclusively, geometric design of the slidercrank mechanism for the mentioned purpose leads to the classical problem known as function generation [20]。 Approaches to the design of slider-crank mechanism as a function generator have mainly taken two paths: one graphical, the other analytical。 The problem of co-ordinating two and three positions of the crank with the corresponding configurations of the slider has first been treated geometrically to come up with single construction techniques such as the one called relative-pole method [5, 6, 20] or as the variations of the kinematic inversion method [3, 7]。 The latter method is further extended to four positions of the slider and crank,