This paper investigates two situations in which the forward kinematics of planar 3-RPR parallel manipulators degenerates. These situations have not been addressed before. The first degeneracy arises when the three input joint variables r1, r2 and r3 satisfy a certain relationship. This degeneracy yields a double root of the characteristic polynomial in t, which could be erroneously interpreted as two coalesce assembly modes. But, unlike what arises in non-degenerate cases, this double root yields two sets of solutions for the position coordinates (x, y) of the platform. In the second situation, we show that the forward kinematics degenerates over the whole joint space if the base and platform triangles are congruent and the platform triangle is rotated by 180 deg about one of its sides. For these "degenerate" manipulators, which are defined here for the first time, the forward kinematics is reduced to the solution of a 3rd-degree polynomial and a quadratics in sequence. Such manipulators constitute, in turn, a new family of analytic planar manipulators that would be more suitable for industrial applications.
Deep Dive into Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators.
This paper investigates two situations in which the forward kinematics of planar 3-RPR parallel manipulators degenerates. These situations have not been addressed before. The first degeneracy arises when the three input joint variables r1, r2 and r3 satisfy a certain relationship. This degeneracy yields a double root of the characteristic polynomial in t, which could be erroneously interpreted as two coalesce assembly modes. But, unlike what arises in non-degenerate cases, this double root yields two sets of solutions for the position coordinates (x, y) of the platform. In the second situation, we show that the forward kinematics degenerates over the whole joint space if the base and platform triangles are congruent and the platform triangle is rotated by 180 deg about one of its sides. For these “degenerate” manipulators, which are defined here for the first time, the forward kinematics is reduced to the solution of a 3rd-degree polynomial and a quadratics in sequence. Such manipulato
Solving the forward kinematic problem of a parallel manipulator often leads to complex equations and non analytic solutions, even when considering planar 3-DOF parallel manipulators [1]. For these planar manipulators, Hunt showed that the forward kinematics admits at most 6 solutions [2] and several authors [3,4] have shown independently that their forward kinematics can be reduced as the solution of a characteristic polynomial of degree 6.
In [3], a set of two linear equations in the position coordinates (x, y) of the moving platform is first established, which makes it possible to write x and y as function of the sine and cosine of the orientation angle ϕ of the moving platform. Substituting these expressions of x and y into one of the constraint equations of the manipulator and using the tan-half angle substitution leads to a 6 th -degree polynomial in tan( / 2) t ϕ =
. Conditions under which the degree of this characteristic polynomial decreases were investigated in [5,6]. Four distinct cases were found, namely, (i) manipulators for which two of the joints coincide (ii) manipulators with similar aligned platforms (iii) manipulators with nonsimilar aligned platforms and, (iv) manipulators with similar triangular platforms. For cases (i), (ii) and (iv) the forward kinematics was shown to reduce to the solution of two quadratics in cascade while in case (iii) it was shown to reduce to a 3 rd -degree polynomial and a quadratic in sequence. To the best of the author’s knowledge, no other degenerate cases have been identified yet. In this paper, we show that the forward kinematics of planar 3-RPR 1 manipulators degenerates under conditions that have not been identified before. More precisely, the system of linear equations in x and y that needs be established prior to the derivation of the characteristic polynomial becomes singular under certain conditions. Moreover, we show that the forward kinematics may degenerate over the whole joint space. 1 The underlined letter refers to the actuated joint Next section recalls the kinematic equations and points out the singularity that may occur when solving the system of linear equations. Section 3 derives the first degeneracy condition, which arises when the input joint coordinates satisfy a certain relationship. The second degeneracy condition is set in section 4. This condition pertains to the geometry of the manipulator and is shown to define new analytic manipulators. These manipulators have a characteristic polynomial of degree 3 instead of 6, and feature more simple singularities. Last section concludes this paper.
Figure 1 shows a general 3-RPR manipulator, constructed by connecting a triangular moving platform to a base with three RPR legs. The actuated joint variables are the three link lengths ρ 1 , ρ 2 and ρ 3 . The output variables are the position coordinates ( , )
x y of the operation point P chosen as the attachment point of link 1 to the platform, and the orientation ϕ of the platform. A reference frame is centred at A 1 with the x-axis passing through A 2 . Notation used to define the geometric parameters of the manipulator is shown in Fig. 1. x y
( )
A system of two linear equations in x and y is first derived by subtracting Eq. (1) from Eqs. ( 2) and (3), thus obtaining
where, As pointed out in [7], x and y can be solved only if the determinant RV-SU is different from zero. If it is so, the 6 thdegree characteristic polynomial is obtained upon substituting the expressions of x and y into Eq. ( 1). The general expression of this characteristic polynomial is not reported here but can be found in [8]. Otherwise, the system degenerates and the forward kinematics cannot be solved this way. To the best of the authors’ knowledge, however, the degenerate case RV-SU =0 has never been examined. In the following sections, we will investigate the conditions under which the determinant of the linear system vanishes and will derive the forward kinematics equations associated with these conditions.
Since RV-SU depends only on the geometric parameters of the manipulator and the orientation ϕ of the platform, RV-SU =0 yields a condition on ϕ for the linear system to degenerate. This condition is
Resorting to the tan-half angle substitution provides a quadratic in tan( / 2) t ϕ = , which has the following form
and may define two orientation angles of the platform.
In order to be able to check the degeneracy condition while solving the forward kinematics, it is useful to set it in terms of ρ 1 , ρ 2 and ρ 3 . When the determinant of the linear system of Eqs. (4,5) vanishes, the Cramer’s rule tells us that an additional condition must be satisfied for a solution to exist, that is 0
)
Let t 1 and t 2 define the two solutions of the quadratic defined by Eq. ( 7). Substituting t 1 and t 2 into Eq. ( 8) yields two degeneracy conditions D 1 and D 2 that depend only on ρ 1 , ρ 2 and ρ 3 and on the geometric parameters. Since Eq. ( 8) is a polynomial of degree
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