Synthesis of Two Four-bar in Series for Body Guidance

Using planar two-degree-of-freedom mechanism for the task of body guidance is evaluated and presented. This mechanism consists of two four-bar in series. For some applications of simple planar movement, planar robot with three actuators can be replaced by this mechanism to save cost and reduce complexity. This work reveals that at most seventeen precision positions can be prescribed theoretically. Based on synthetic consideration, each four-bar is proposed to be in charge of some prescribed positions and thus can be designed separately. Decomposing the prescribed precision positions into the assignment of body guidance for each constituent four-bar is developed. Both four-bar are then assembled to successfully execute the task of at most nine precision positions. Numerical example is given to illustrate the synthesis process. Problems of order defect, branch defect and ratio of link lengths are all considered to ensure the performance and applicability of the mechanism.


Introduction
Body guidance is one of major applications for various low-cost linkages. The well-known planar four-bar can be synthesized by prescribing at most five positions theoretically [1]- [2]. The six-bar linkages [3] and a special cam-linkage [4] mechanism were synthesized and introduced for similar tasks. Eleven given positions are expected to be visited by using a complicated eight-bar linkage [5]. The concepts of multi-phase and adjustable link length are also proposed to improve the capability of planar four-bar [6]- [8].
The RRRRR five-link is a two-degree-of-freedom linkage, and the joints at both ends are fixed on the frame and driven by two actuators [9]- [10]. The type of some joints can be prismatic [11]- [12], and the PRRRP parallel mechanism actuated by linear actuators was also proposed [13]. For the purpose of improving performances, such as payload capability and stiffness, the mechanism with seven links including a redundant dyad was suggested [14]- [16]. However, the feature of these linkages is that the middle moving pivot can reach any points within the workspace, and path generation is thus the only application.
An expensive planar robot usually has three DOF (degrees of freedom) and is equipped with three actuators [17]. Hence, its end-effector can be moved to any positions and orientations. The present work focuses on the evaluation and synthesis of mechanisms with two DOF. Body guidance is the major task and the precision positions that can be prescribed should be more than five. Moreover, the synthesized mechanism should have no branch defect and order defect. The parameter i C that is the coordinate of point i C with respect to the base frame can thus be derived as

Description of mechanism
The moving frame Therefore, there are k 4 constraints like Eq. 4. The maximum value of k can be calculated from the equality The value of k is 16, and this implies that at most 17 positions can be prescribed by using this mechanism.

Synthesis of mechanism
This mechanism can be synthesized by theoretically solving 16 nonlinear equations after seventeen positions are prescribed, but this task is definitely cumbersome. Even though the solutions can be obtained, the linkage might have order defect and branch defect and is thus useless.
An alternate strategy for applying or synthesizing this mechanism is to let each four-bar be in charge of some prescribed positions separately. For example, if the j th prescribed position is under the charge of four-bar o o ABB A , only A A o is actuated when the guided body is moved from the (j-1) th position to the j th position. Consequently, each four-bar can be synthesized independently as explained in the following.
The coordinates of point E at the 1 st and i th positions can be related by a displacement matrix T i 1 , and i f m T can be derived successively by using Eq. 12, Eq. 14, and Eq. 15 or Eq. 16, Eq. 17, and Eq. 20. Both four-bar are synthesized separately and then assembled with

Order defect and branch defect
If a linkage is used to execute body guidance or path generation, problems of branch defect and order defect should be considered and avoided. A four-bar has two configurations for a given input as shown in Fig. 2 Eq. 21 is a three dimensional vector operation and k is a unit vector normal to the plane. The linkage synthesized should be branch defect-free and associate with the same configuration for all prescribed positions. In other words, the signs of all i Δ must be the same. On the other hand, the order defect-free implies the satisfaction of the following inequality where 1j θ represents the value of input angle 1 θ corresponding to the j th position, and j is between i and k. Example: A mechanism as shown in Fig. 1 is to be designed to execute body guidance. There are eleven prescribed positions as listed in Table 1. Two points 0) 2, ( : E 1 and 0) 1, ( : F 1 on the coupler link CD are used to verify the movement of body guidance. The precribed movements of points E and F or coupler link CD are shown in Fig. 3.
The data of prescribed positions were proposed in [5], and a linkage with eight links and one degree of freedom was synthesized. However, neither order defect nor branch defect was considered.  guides through the other positions. Therefore, each four-bar has to match six prescribed positions 2. Although a four-bar can match precisely at most five prescribed positions based on Burmester theory, the linkages obtained usually have branch defect or even order defect. Therefore, only four of six prescribed positions are chosen to derive the circle point curve, and the others become approximate positions. A linear spaced array covering a specified interval is schemed, and the elements are used for the x coordinate of pivot 1 A or 1 B . For each x coordinate, at most three solutions are obtained for pivot 1 A or 1 B . Numerous four-bar are then synthesized by combining any 1 A and 1 B . Both branch defect and ratio of link length are checked to discard unacceptable linkages. The others are compared by evaluating the deviations corresponding to both approximate positions. For the i th approximate position, the ideal or desired coordinate of pivot A or B can be derived as . The deviations are defined by the following functions where a is the length of input link  Table 1 are used to derive circle point curve. The data of the best linkage, after comparing the deviations or index, are listed in Table 2. . The data like those in Table 1 are obtained and listed in Table 3.
By choosing the 6 th , 7 th , 9 th , and 11 th as the precision positions, this four-bar is synthesized as in Step 2. The coordinates of four pivots with respect to the moving frame m m y x are listed in Table 4.

Conclusion
A mechanism with two four-bar in series and with two DOF is proposed for the task of body guidance. Its application and synthesis is presented for the first time. This mechanism can be synthesized by theoretically solving numerous nonlinear equations and at most seventeen positions can be prescribed.
An example illustrates the effective design of this mechanism by mainly actuating each four-bar in sequence.
The well-developed techniques for synthesizing four-bar can thus be used. Although the mechanism can guide the body through nine positions precisely, there is few choices left to match other requirements. Therefore, the mechanism synthesized in this example passes through only seven positions precisely. The problems of order defect, branch defect and ratio of link lengths are all considered, and the errors for approximate positions are trivial. The techniques used in this example can certainly be integrated with optimization search so that more positions can be prescribed and more consideration can be included. Evidently, this mechanism can replace complicated and expensive planar robots in some ways.