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may not be appropriate for another crank angle. Therefore, in order to obtain a proper solution quickly, the code used for the
solution ismodified so that initial guesses are obtained fromthe previous cycle's set of solutions. In Fig. 4, θ13, θ14, s15 and T12 vs. θ12
are shown respectively. It is observed that link 4 remains approximately at the same angle during the reverse and forward strokes.
During the work stroke, θ14=115° and is almost constant.
3. Generalization of the variable stroke mechanism
In this section, a procedure to obtain design charts for the generalization of the compliant variable stroke mechanism is
introduced. In Eqs. (7), (8), (15), and (16), there are seven structure parameters, namely a2, a3, a4, k34, k45, c34, and c45. To obtain a
simple design chart, the number of free parameters must be reduced. Since this is an underactuated mechanism, the output-
loading condition is also a function of the kinematics of the mechanism.
First of all, all related equations can be pided or multiplied with a2 or kij to obtain dimensionless link proportions. Also, the
force and spring constants can be simplified. As can be seen in Eqs. (15) and (16), T12 and F15 are linearly proportional with the
spring constant kij, and hence increasing or decreasing both spring constants and output force linearly yields exactly the same
kinematics. Therefore, the spring constant and output force can be considered as a single design parameter. Dividing Eqs. (15) and
(16) by k34, more useful F15a2/k34 and T12/k34 ratios can be introduced. Also, for simplification the spring initial position constants
can be the same (c34=c45=c) during the design procedure. In order to generalize the approach, the unit length can be taken into
consideration. Therefore, the unit of spring constants becomes Nunit/rad.Now, in Eqs. (17), (18), (19), and (20), there are five free parameters: four design parameters (a3/a2, a4/a2, c, and k45/k34) and
force ratios Fa2/k34 or T/k34 (whichever is given).
4. Design charts of the variable stroke mechanism according to a given output loading
The procedure of obtaining a design chart for a given output-loading constitution is as follows: for a constant spring ratio (k45/
k34=1, k=k45=k34) and heuristically chosen range of a4/a2 and a3/a2 (or c)with small increments, Eqs. (17), (18), (19), and (20)
are solved numerically for a whole crank rotation. After full-rotation of the crank, the maximum and minimum values of the ratio
s15/a2 (which yields to the stroke Δs/a2) are determined. This procedure is repeated for the interval of the pre-determined design
parameters. Finally, by interpolation of the data (e.g., cubic spline interpolation), the design charts are obtained. Thismethodology
is shown in Fig. 5. By employing this procedure, two design charts are obtained as displayed in Fig. 6. The chart shown on Fig. 6a
represents the variation of the stroke (Δs/a2) of themechanismwith respect to a4/a2 and a3/a2,while keeping c at a constant value.
It can be easily observed from the chart that a3/a2 has a very minor effect on the stroke (a similar effect is also observed by
reconstructing the chart for several different values of the spring initial constant c). Therefore by keeping a3/a2 constant with a 摘要：在本文中，欠驱动柔性变行程机构的设计与分析是采用的伪刚体模型（模型）。在设计过程中考虑两种情况：规定输出负荷和恒定的输入扭矩。结果表明，这一机制是适用于可变行程相对应的可变输出负载的场合。还能够观察到的是，该机构能够在一个较宽的输入范围内提供的力几乎是恒定的。在本研究中，介绍的分析和设计方法也适用于其它类似的欠驱动柔性机构。
关键词：欠驱动多自由度机构，伪刚体模型
1. 导言
柔性机构已被许多研究人员调查研究，因为它们相比刚性连接的机制，价格上不太昂贵，质量上更轻，而且更容易被加工生产[1]。特别是近年来，对柔性机构领域的兴趣正在逐步增长[2–9]。柔性机构的一个缺点是他们的分析和设计的复杂性。为了简化分析，伪刚体模型技术（PRBM）已经在文献中被经常使用[1]。如果一个柔性机构的小弯曲支点的数量增加，那么将会使得自由度超过1，由此产生的机构的设计会变得相当具有挑战性的。近年来，对多自由度的柔性机构已经有了一些研究[10–13]。而且，如果输入机构的数目是小于其自由度，那么多自由度柔性机构将成为欠驱动。这种情况介绍了这种系统设计的另一个额外的复杂性。欠驱动多自由度机构的分析不是和一个自由度的机构一样简单。对于这样的机构，必须进行运动分析和受力分析。因此，这些机构的设计与分析可以指定一个特定的输出负载条件。欠动机构的运动学关系到他们的加载条件以及链接的比例。有几项研究欠驱动机构：一个多端口杆机械逻辑元件已通过Söylemez和Freudenstein的调查研究[14]。Laliberte和Gosselin [15]在引进欠驱动的原则使得机械手的形状自适应和提供现有的欠驱动机械手设计的研究进展上，进行了欠驱动机械手的仿真和设计。Tanık和Söylemez[16] 推导出一个合成过程用于使用两种不同类型的七连杆欠驱动机制的变结构。Mahindakar，Rao，和Banavar[17]对欠驱动机械臂的控制的方面进行了研究。最后，Cheng，Carbone和Ceccarelli[18] 研究了欠驱动手指操作运转。变行程机构还具有力恒定的特性。一些作者在具有恒定的力传递特性的柔性机构的方方面作出了重大贡献[19,20]。