The kinematics and design for quasi-isotropy of 3U serial manipulators with reduced wrists
Introduction
Six-revolute (6R) serial manipulators are widely used in various applications, e.g., manufacturing, assembly, robotics, and space applications, among many others. Manipulators with reduced wrists refer to those whose last three axes do not intersect at one common point, but only the last two axes do, while manipulators with an offset wrist refer to a special subclass of the former, whose second to last axis intersects its two neighboring axes at different points [1], [2]. These manipulators offer higher load-carrying capacity and functional flexibility when compared to those with three axes intersecting at one common point, widely used in heavy-duty applications such as welding [2], [3], [4]. The reason why the latter, commonly referred to as “wrist-partitioned robots”, offer a lower load-carrying capacity than their “reduced-wrist” counterparts is that the former isolate the torque applied by the load from the three proximal links. Most reported research on manipulators with reduced/offset wrists focuses on the inverse displacement problem (IDP); since the robot does not have a decoupled architecture, normally a closed-form solution for its IDP is not available. Various numerical methods have been proposed for the IDP of general manipulators with reduced wrists [2], [3], [4]. A landmark work is that by Raghavan and Roth [5], who derived a procedure that yields the IDP of a general 6R serial manipulator as a univariate polynomial of 16th degree in one of its joint variables, which is minimal; moreover, it has been found that, under some special geometrical conditions, the number of postures of the IDP can be further reduced [6].
In this paper we study another subclass of 6R manipulators with reduced wrists, while focusing on their kinematics, singularity, and the design for isotropy, with extensions to what we term quasi-isotropy. Manipulators of this class bear six joint axes that intersect pairwise, as shown in Fig. 1, which can be alternatively termed 3U (three-universal-joint) manipulators. This special geometry yields some special features; the major motivation of this work is the study of the special properties brought about by this special layout, while targeting design applications. It will be shown that a concise symbolic expression exists for the inverse of the Jacobian matrix of this class of manipulators; moreover, a decomposition of this inverse is proposed here, based on which a clear geometric interpretation of the inverse of the Jacobian matrix is found. Applications to design and singularity-management are highlighted. Moreover, the concept of quasi-isotropy [7], introduced by the authors, is revisited, as a relaxation of the concept of isotropy; the design for both isotropy and quasi-isotropy is investigated for the 3U manipulators. The design for quasi-isotropy can also be extended to the design of a special type of architectures, under which the six columns of the Jacobian matrix are mutually orthogonal, the product being diagonal, thereby yielding decouplability. The results in this paper not only facilitate the utilization of this class of robots in heavy-duty applications, but can also be extended to the analysis and design for isotropy, quasi-isotropy and decouplability of other types of manipulators. Examples of the latter can be cited: a large number of six-dof parallel kinematics machines (PKMs) with three pairs of intersecting axes1, or of 7R manipulators bearing a 3U substructure, such as FREND, developed by NASA [10].
The kinematics relations include the linear mapping between the input joint rates and the end-effector (EE) twist, which is essential in analysis and control. The derivation is straightforward for serial manipulators. However, many applications call for the inverse of e.g., in numerical procedures for the IDP based on the Jacobian matrix, trajectory planning, simulation, and control. can be obtained numerically, which is not only computationally expensive but also sensitive to roundoff error, leading to large positioning errors in the presence of ill-conditioned postures [11]. The explicit use of is thus normally obviated by the LU-decomposition [12]. The latter, while in general reliable, is also sensitive to the presence of near-singular postures. If a symbolic expression for is available, it offers several advantages over numerical techniques: firstly, the calculation process is streamlined, with fewer floating-point operations (flops); moreover, a symbolic expression is robust to round-off error amplification, even when the robot is close to singular postures; furthermore, the symbolic nature can reveal more properties and characteristics of the mapping induced by which is difficult to capture by numerical procedures. These properties and characteristics are found to be helpful at the analysis and design stages.
In a previous work [7] the authors investigated the symbolic inverse of the Jacobian matrix of a class of six-dof PKMs with three pairs of intersecting axes; the six rows of the forward Jacobian matrix [13] of the said class are composed of the Plücker coordinates of six lines that intersect pairwise. Based on this result, we derive here the symbolic inverse of the Jacobian matrix for 3U serial manipulators. Indeed, the Jacobian matrix of this kind of manipulators is also composed of the Plücker coordinates of six lines, mathematically equivalent to the Jacobian of the class of PKMs whose actuated wrenches intersect pairwise. Hence, the symbolic inverse obtained therein also applies to the 3U serial manipulators. However, since the derivation in the previous work was purely based on algebraic manipulations, the expression obtained therein was cumbersome; its geometric meaning was not apparent either. In this paper, a concise decomposition of is found, the relevant terms bearing a clear geometric meaning; this allows us to further exploit the structure of simplifying the calculation process, while reducing the numerical error. Due to the clear geometric meaning and structure of its expression can also be helpful in design optimization.
Singularity affects the robot performance significantly, whose characterization and verification is quite important. According to Merlet [14], singularity analysis can generally be conducted via four different methods, based on: screw theory [15]; Grassmann geometry [16]; differential geometry [17]; and the rank-deficiency condition of the Jacobian matrix at hand [18]. In this paper, a singularity index is established based on a term that arises during the calculation of whose geometric meaning is revealed here; moreover, this index is found to be roughly proportional to the condition number of the Jacobian matrix, especially when the manipulator is near a singular posture. Hence, the singularity evaluation is simultaneously conducted during the calculation of . A concise symbolic singularity expression is also derived, while incorporating simplifications due to the special layout of the class of manipulators of interest to the paper.
In terms of optimum design, different performance indices have been proposed to characterize different aspects of performance requirements, e.g., workspace volume, dexterity and accuracy—References [19], [20] include a comprehensive summary of these indices. Among these indices, isotropy is an important design objective, which refers to manipulators that can achieve postures under which the condition number of the corresponding normalized Jacobian matrix, denoted as is unity [21]. Here, we recall [21] that the condition number of a 6 × 6 Jacobian matrix is meaningless because of the disparate units of its entries2; as a result, a normalization is needed beforehand. Several techniques have been proposed to handle the inconsistency of units, such as those resorting to the characteristic length [21], weighting factors [22], and the “point-based methods” [23], the “speed” of the EE in the foregoing reference being represented in terms of the velocity of several points attached to the EE [23], [24]. The isotropy condition not only guarantees the orthogonality of the different columns of the normalized Jacobian matrix, but also that all these columns bear the same Euclidean norm, thus yielding six identical diagonal values of thereby guaranteeing a homogeneous motion capability along all six directions in the twist space. This leads to a higher positioning accuracy [19], [20], [25]. Moreover, the concept of quasi-isotropy, proposed by the authors [7] to refer to those manipulators that can attain postures under which all the columns of the normalized Jacobian matrix are orthogonal, besides bearing similar Euclidian-norm values, is recalled. More specifically, under quasi-isotropy, the product is diagonal, with only three distinct singular values, the condition number of being close to 1.0, e.g., below 2.0, as imposed by the designer, according to the design specifications. Due to these features, quasi-isotropic designs offer a performance similar to their isotropic counterparts. Indeed, the concept of quasi-isotropy can be regarded as a generalization of isotropy, the former offering a significantly larger number of design possibilities than the latter, i.e., more design alternatives. This is especially important when isotropy is not possible under a given set of design conditions. The same procedure can be extended to the design of a special class of manipulators, under which the six columns of are mutually orthogonal, but with different Euclidian norms, the product becoming diagonal. Moreover, the diagonal entries can be chosen based on the design specifications at hand.
The paper is organized as follows: in Section 2, the derivation of a simple form of the inverse of the Jacobian matrix is described; in Section 3, the singularity analysis is discussed, including a case study for the use of in kinematics and singularity analyses; the design for isotropy and quasi isotropy is conducted in Section 4. Section 5 includes the concluding remarks.
Section snippets
The kinematics relations and the symbolic inverse of the Jacobian matrix
A sketch of a 3U manipulator is shown in Fig. 1. Now we derive the kinematics relation, which bears the formwith representing the six-dimensional twist of the end-effector (EE), the six-dimensional array of joint rates, and the kinematics Jacobian matrix. Moreover, is composed of six columns, each representing the Plücker coordinates [21] of the corresponding axis, namely,where and represent the unit vectors parallel to
Singularity analysis
In this section we discuss briefly the singularity of the class of manipulators of interest. As stated in the previous section, the Jacobian matrix bears a pattern similar to those of a class of six-dof PKMs with three pairs of intersecting axes. The singularity of this class has been investigated intensively via different approaches [8], [9], [30], [31], [32], which has led to a clear geometric interpretation, namely, singularity occurs iff the four planes—Πi, for and O2O4O6, the
Design for isotropy
Isotropy, an important design objective, not only implies the uniformity of the transformation between the joint rates and the EE twist, but also guarantees the kinetostatic accuracy of the robot [21]. An isotropic posture can be characterized by where σ is a positive scalar, and represents the normalized Jacobian matrix. In this paper, we resort to the robot characteristic length [21] for the normalization, thus obtained upon multiplying the first three—non-dimensional—rows
Conclusions
A subclass of 6R manipulators with reduced wrists is investigated, whose six joint axes intersect pairwise. Manipulators with reduced wrists offer higher load-carrying capacity and maneuverability, as compared to those with three axes intersecting at one common point, which makes the subclass quite suitable for heavy-duty applications. The special layout of the class of interest leads to some special features, such as the existence of a concise symbolic expression for , the inverse of the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The first and third authors acknowledge the support from NSFC (National Natural Science Foundation of China) through grant no. 51927809. The second author acknowledges the support received from NSERC (Canada’s Natural Sciences and Engineering Research Council) through grant no. 2015-03864.
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