Research paperAn evaluation approach for motion-force interaction performance of parallel manipulators with closed-loop passive limbs
Introduction
As the counterparts of serial mechanisms, parallel manipulators (PMs) are able to accomplish tasks performing high precision and high speed under heavy loads due to their multi-closed-loop structure. After the Gough-Stewart platform being built and introduced [1,2], a new era of parallel mechanisms or manipulators was opened. To produce a manipulator that would cater only to translations of the payload, i.e., operations that require three degrees of freedom (DOF), the Delta high-speed PM with closed-loop parallelogram mechanism was proposed [3]. As one of the most typical applications, PMs with closed-loop passive limbs play significant roles in the field of high-speed pick-and-place operations. These PMs have been widely used to implement high-speed manipulations of sorting and packaging in the food, pharmaceuticals, and electronics industries. Examples of such manipulators are the Par4 [4], Heli4 [5,6], Gantry-Tau [7], SCARA-Tau [8] and its variants [9], [10], [11], etc. Compared with the normal passive limbs, the closed-loop ones have more structural parameters. How to evaluate the influence of structural parameters inside closed-loop passive limbs on the performance of PMs is still a problem need to be solved.
The working mechanism of an n-DOF lower-mobility PM is illustrated in Fig. 1. In this PM, n-DOF permitted motions are allowed, and (6-n)-DOF restricted motions are limited. Motions of the mobile platform in the allowed/limited subspace are transmitted/constrained by the actuation/constraint wrenches from the allowed/limited joint subspace. When general external forces are applied on the mobile platform, the forces belonging to the allowed subspace and limited subspace are balanced by the actuation wrenches and constraint wrenches from limbs, respectively. In an n-DOF lower-mobility PM, the relationships between motions and forces are coupled and of great significance. How to evaluate the motion/force transmissibility and constrainability is a critical issue.
Throughout the literature, in the allowed and limited subspaces, many efforts have been put into the motion/force transmission and constraint performance of PMs. For the motion/force transmissibility, the initial and milestone work can be dated back to the transmission angle proposed by Alt [12]. After this, based on the virtual coefficient, several indices were proposed for spatial mechanisms with one DOF [13], [14], [15], [16], [17]. To evaluate n-DOF fully PMs, Takeda and Funabashi [18,19] proposed a transmission index by extracting n-DOF PMs into n single-DOF mechanisms. Inspired by these works, Wang et al. [20] proposed the input transmission index (ITI), output transmission index (OTI), and minimum value of both as the local transmission index (LTI) to evaluate the transmission performance of non-redundant PMs. Then, the concept was extended for redundant PMs [21], [22], [23], [24], [25], [26]. For the motion/force constrainability, a phenomenon of unexpected additional DOF appeared when the actuated joints were locked in the 3-UPU (U, universal joint; P, actuated prismatic joint) PM [27]. Then the constraint singularities were identified by Zlatanov et al. to explain the above case [28]. Thereafter, the constraint transmission index (CTI) was proposed and further refined by Liu et al. [29], [30], [31]. As a result, novel approaches for singularity analysis and closeness measurement to singularities of non-redundant PMs were established.
The above works divided the motion/force transmission and constraint performance into two parts. However, in PMs with closed-loop passive limbs, the transmission and constraint wrenches are always strongly coupled. How to separate them is still an open problem. To develop a better understanding of this issue, a 2P6R planar PM and a spatial 3-[PP]S PM ([PP], 2-DOF actuated prismatic joints; R, revolute joint; S, spherical joint), illustrated in Fig. 2, with closed-loop passive limbs are presented as examples. In the 2P6R PM, the closed-loop passive limb consists of two passive chains (bar B1,1C1,1 and bar B1,2C1,2 with two revolute joints, respectively). These two passive chains are connected between the mobile platform and the actuated joint. Here, a closed loop (B1,1-B1,2-C1,2-C1,1-B1,1) exists in the 1st limb, as shown in Fig. 2(a). In the spatial 3-[PP]S PM, the closed-loop passive limb consists of two passive chains (bar Bi Ci,1 and bar Bi Ci,2 with two spherical joints, respectively). There is also a closed loop (Bi-Ci,1-Ci,2-Bi) in each limb, as shown in Fig. 2(b). In these two cases, wrenches which strictly belong to the concept of actuation or constraint are ambiguous according to the screw-based identification method [32]. So far, there seems to be no uniform approach to distinguishing actuation and constraint wrenches of these PMs. However, two wrenches are physically available inside the closed-loop passive limb, and these wrenches lie along two passive chains due to the two-force rod. In this paper, the physically available wrenches are adopted to carry out the evaluation of the coupled motion/force transmissibility and constrainability, that is, the motion-force interaction performance, of PMs with closed-loop passive limbs.
Motion/force transmissibility was used to carry out optimal design of two novel parallel robots with articulated travelling plate [33,34]. Recently, the concept of the pressure angle was extended to the transmission index for lower-mobility parallel manipulators such as the 3-PRS [35,36], 3-RPS [37] and Delta Robot [38]. These works attempted to evaluate the motion/force transmission and constraint performance with one index. Although some studies have focused on the performance evaluation of PMs with multi-loop mechanisms or closed-loop subchains [39,40], only analyses of planar subchains have been performed.
As an extension of our previous works [20,29,35], by drawing on the physically available wrenches inside the passive limbs and their corresponding twists on the mobile platform and actuated joints, this paper intends to develop the motion-force interaction performance indices for PMs with closed-loop passive limbs. Furthermore, the indices are expected to reveal the effect of the structural parameters inside closed-loop passive limbs on the motion-force interaction performance. This will be very helpful for optimization of such structural parameters to give full play to the potentials of the investigated robots. Having outlined in Section 1 the significance of evaluating motion/force transmissibility and constrainability and its challenges in PMs with closed-loop passive limbs, the rest of this paper is organized as follows. In Section 2, a new blocking-and-actuating strategy is proposed as a physical foundation. Accompanied by the evaluation strategy, the distal wrenches and proximal wrenches are identified. The distal interaction index (DII) and proximal interaction index (PII) are then proposed to evaluate the coupled motion/force transmissibility and constrainability of the investigated PMs. In Section 3, the motion-force interaction performance evaluation of the 3-P PM with closed-loop passive limbs is described to demonstrate the generality and validity of the proposed approach. Conclusions are summarized in Section 4.
Section snippets
Performance evaluation strategy and indices
A PM consists of a mobile platform and a fixed base linked together by at least two independent limbs. This structure brings challenges to research, such as output of multi-loop and multi-input coupling, interaction among limbs, and nonlinear and high sensitivity of performance to the change of parameters. To analyze the motion/force transmissibility of PMs, a blocking strategy was proposed in [18,19] and adopted in [20,29]. This strategy was then extended in [23,24] to convert an n-DOF PM to
Application
In this section, the motion-force interaction performance evaluation of a spatial 3-PS PM is carried out to illustrate the effectiveness of the approach. As is well known, the zero-torsion parallel mechanism, which has two orientational DOF and one translational DOF, has attracted much attention due to the increasing demand of A/B-axis tool heads in the manufacture of structural aircraft parts with thin walls [43]. The 3-PRS parallel mechanism is a very typical zero-torsion parallel
Conclusion
The core function of a parallel mechanism is to transmit or restrain motion and force from the actuated joints to the mobile platform. The motion-force interaction performance is one of the essential attributes for PMs. In this paper, an approach is proposed to evaluate the motion-force interaction performance of PMs with closed-loop passive limbs via screw theory. In the proposed approach, a blocking-and-actuating strategy is provided to divide the motion-force interaction performance into two
Declaration of Competing Interest
No conflict of interest exits in the submission of this manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants 91748205, 51922057 and 51675290. The first author, Qizhi Meng, thanks Xiaoqing Li for her invaluable support over the years.
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