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... by Nielsen and Roth, who gave an algorithm to derive the Dixon resultant of any planar linkage [14]; the work by Wampler, which improves on Nielsen and Roth’s by applying a complex plane formulation =-=[15]-=-; the work by Celaya et al. [16], which provides an interval propagation algorithm; and finally, the work by Porta et al. [17], which attacks the problem via linear relaxations. An examination of thes...

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... mapped isomorphically onto the set of real points of the image. A special case of this occurs in planar mechanism theory, in which isotropic coordinates can be used to represent vectors in the plane =-=[2, 26]-=-. These are pairs (z, �z) ∈ C 2 that are “real” when z and �z are complex conjugates of each other, that is, real points are the stationary points of the generalized conjugation operation κ : (z, �z) ...

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...nd Roth also gave an elimination-based method that uses Dixon’s resultant to derive the lowest degree polynomial of the algebraic system under study [17]. This technique was later improved by Wampler =-=[25]-=-, who used a complex-plane formulation to reduce the size of the final eigenvalue problem by half. The problem can also be tackled using general continuation-based solvers like [23], that start with a...

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...ethod to compute the reachable workspace of a planar mobility-three double-butterfly linkage (see Fig. 6). This mechanism has been used to compare the performance of general position analysis methods =-=[39]-=-–[41], but no complete method has been given to compute the boundaries and barriers of its reachable workspace yet, as far as we know. For this example, it will be assumed that the end-effector is the...

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... [41] (Vector method with Sylvester resultant), Dhingra et. al. [38] (Vector method with Sylvester resultant), Dhingra et. al. [42] (Vector method with Gröbner basis and Sylvester resultant), Wampler =-=[43]-=- (Complex number method with Dixon resultant), Rojas and Thomas [13] (Bilateration method). Numerical: Hang et. al. [8] (Vector method with homotopy continuation), Shen et. al. [44] (Single-opened-cha...

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...s the positions [B4, j], j = 1, . . . ,5. Similarly, we compute the RR chain G2W2 using (7) with [D1 j] = [B2,1][B2, j]−1[B4, j][B4,1]−1 to constrain B2 and B4. The solutions are listed in Table 2, and 3. A schematic of the resulting planar six bar linkage at each of the five positions is shown by the image sequence labelled 1-5 in Figure 5. ANALYSIS OF PLANAR SIX-BAR LINKAGES: THE STEPHENSON IIA In this section, we formulate the configuration analysis of a planar six-bar linkage to simulate its movement using complex number coordinates and the Dixon determinant as presented by Wampler (2001) [15]. Our focus is on the Stephenson IIa planar six-bar, but the approach can be applied to all of the mechanically constrained PRR chains. Consider the general Stephenson IIa linkage shown in Figure 6. Introduced a coordinate frame for this analysis such that the prismatic slide axes of the PRR chain is parallel to and C1 is along the y-axis, its x-axis is directed toward G1, and with joint parameters θi as shown in Figure 6. The seat of the wheelchair FIGURE 5. The image sequence for the Stephenson IIa linkage reaching a set of five task position. is denoted by the link C2W1. The Complex Loop Eq...

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...sm at the specific input. The traditional solution method for a direct kinematic problem is completed by using tangent-half-angle substitutions for the output variables [2]. More efficiently, Wampler =-=[3, 4]-=- presents techniques for using isotropic coordinates to generate polynomial equations that describe the position of a general single-dof linkage. Solution of the complex, polynomial systems can be rea...

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... nine have a ground-connected input and match the nine identified by Manolescu. Linkage synthesis solves for the specific dimensions of the links using one of the enumerated topologies. Soh and McCarthy [10] published a methodology specific to the 8-bar family for synthesizing linkages that can be constructed from a pair of constrained 3-R chains. Linkage synthesis approaches are published for a variety of selected kinematic chain topologies [11–13]. Linkage configuration analysis solves for the angles of all of the output links. Approaches are typically shown for specific topologies. Wampler [14] referencing Dixon [15] analyzed a double butterfly 8-bar linkage using the Dixon determinant in a complex plane formulation. The same linkage was also evaluated in rational formulation by Nielsen and Roth [16]. Various other methods for solving the configuration of a linkage have been published such as the Gr€obner–Sylvester method by Dhingra et al. [17] and the linear relaxation method by Porta et al. [18]. To determine if a particular assembly configuration moves smoothly through a range of input angles, the singular configurations must be avoided. Linkages that encounter a singularity with...