تجزیه و تحلیل حساسیت حرکتی از مکانیزم های موازی 3-UPU

This paper addresses the kinematic sensitivity of the three degree-of-freedom 3-UPU parallel mechanism, a mechanism consisting of a fixed base and a moving platform connected by three serial UPU chains. Although a mathematical mobility analysis confirms that the mechanism has three degrees of freedom, hardware prototypes exhibit unexpected large motions of the platform even when the three prismatic joints are locked at arbitrary configurations. Existing mathematical classifications of kinematic singularities also fail to explain the gross motions of the 3-UPU. This paper resolves this apparent paradox. We show that the 3-UPU is highly sensitive to certain minute clearances in the universal joint, and that a careful kinematic sensitivity analysis of the 3-UPU augmented with virtual joints satisfactorily explains the gross motions. Observations with a hardware experimental prototype confirm the results of our sensitivity analysis.

Despite their many advantages vis-à-vis serial mechanisms, classical 6-6 parallel mechanisms such as the Stewart platform suffer from a smaller workspace, complex mechanical design, and more difficult motion generation and control due to their complex kinematic analysis. In an attempt to overcome these and other limitations of 6-6 parallel mechanisms, many researchers have investigated various three and six degree-of-freedom 3-3 parallel mechanism designs as an alternative (e.g., [1], [3] and [7]). One of the more fascinating 3-3 designs is the three degree-of-freedom 3-UPU parallel mechanism; the basic structure of the mechanism is shown in Fig. 1. It consists of a fixed base and moving platform connected by three serial chains, with each chain having a universal–prismatic universal joint arranged in sequence. The universal joints are passive; only the three prismatic joints are actuated. In contrast to other 3-3 mechanisms, because the 3-UPU mechanism consists of only universal and prismatic joints, it is quite attractive from the manufacturing point of view. More interestingly, as first pointed out by Tsai [8], the universal joints can be attached in such a way that the moving platform only undergoes pure translational motion. Motivated by these results, Di Gregorio et al. [2] and [5] explore the conditions under which the more general 3-RRPRR mechanism (which includes the 3-UPU mechanism as a special case) can be arranged to undergo strictly translational motion. Full-size image (9 K) Fig. 1. The 3-UPU parallel mechanism: (a) the general 3-UPU mechanism; (b) the translational 3-UPU. Figure options Analysis of the kinematic constraint equations for both the general 3-UPU mechanism and the more general 3-RRPRR mechanism confirms that both have three degrees of freedom. Experiments with hardware prototypes of two representative 3-UPU designs, however, reveal an unexpected set of additional degrees of freedom––regardless of the platform configuration, when the prismatic joints are locked, the mechanism behaves as if it has additional degrees of freedom, rather than being a rigid structure as predicted by kinematic mobility analysis (see Fig. 2). Full-size image (86 K) Fig. 2. Hardware prototype of the SNU 3-UPU mechanism: (a) initial configuration, (b) redundant self-motion. Figure options In this paper we first show that existing classifications of kinematic singularities fail to explain these redundant self-motions of the 3-UPU. We then show that this unexpected behavior can in fact be traced to minute clearances and manufacturing tolerances in each UPU assembly. Specifically, clearances in the bearing and shaft of each UPU assembly admit small torsional rotations about the leg axes, which in turn cause the gross motions of the moving platform. To show this we develop a more complete model that accounts for all possible infinitesimal motions of the mechanism resulting from manufacturing errors or tolerances––each of the universal joints is augmented with a virtual revolute joint, effectively modeling the universal joint as a spherical joint. A first-order sensitivity analysis is then performed with the more complete kinematic model that resolves the apparent paradox. The importance of identifying kinematic singularities of parallel mechanisms has long been recognized in the literature. Our study emphasizes the importance of kinematic sensitivity analysis, particularly when designing new parallel mechanisms. In this context careful attention must be paid to ensure that the kinematic models used for the kinematic sensitivity analysis reflect the complete range of design variations, clearances, and manufacturing tolerances that may occur; for parallel mechanisms containing universal joints it is particularly important to take into account torsional clearances of the type mentioned above. From this perspective the 3-UPU mechanism can be regarded as inherently more unstable than other three degree-of-freedom parallel mechanism architectures.

This paper has examined the causes of the gross self-motions observed in hardware prototypes of the 3-UPU parallel mechanism. After eliminating kinematic singularities as a primary cause of these motions, we show that the mechanism is extremely sensitive to small torsional rotations about the legs arising from clearances and manufacturing errors in the bearing-shaft assembly. A kinematic sensitivity analysis confirms that the suspected infinitesimal torsional rotations about the legs are indeed the source of the redundant self-motions. One of the lessons to be drawn from this work is the importance of kinematic sensitivity analysis when designing parallel mechanisms. Our results indicate that some parallel mechanism designs are inherently more robust than others, and that certain designs are unstable. It is particularly important to verify that all possible design parameter variations, clearances, manufacturing and other assembly errors are accounted for in the kinematic model used for sensitivity analysis. Particularly in the case of universal joints, they should be augmented with additional virtual revolute joints to model torsional clearances.