工程科学与技术   2021, Vol. 53 Issue (1): 155-161
Stephenson-Ⅲ型平面六杆机构轨迹综合的代数求解

1. 北京邮电大学 自动化学院，北京 100876;
2. 华北理工大学 机械工程学院，河北 唐山 063210

Algebraic Solution for Path Synthesis of Planar Stephenson-Ⅲ Six-bar Linkage
LI Xuegang1,2, ZHANG Lijuan2, WEI Shimin1, LI Heqing2
1. School of Automation, Beijing Univ. of Posts and Telecommunications, Beijing 100876, China;
2. College of Mechanical Eng., North China Univ. of Sci. and Technol., Tangshan 063210, China
Key words: planar Stephenson-Ⅲ six-bar linkage    path synthesis    Fourier series    analytical approach

1 连杆曲线的傅氏级数表示

 图1 Stephenson-Ⅲ六杆机构轨迹生成图 Fig. 1 Illustration of a planner Stephenson-Ⅲ six-bar linkage

Stephenson-Ⅲ平面六杆机构为多环组合机构，其可以看作在四杆机构ABCD上串联了一个二杆组EFG。其浮动杆Ⅰ上的E点产生轨迹曲线即为四杆机构ABCD的连杆曲线，浮动杆Ⅱ上的P点可以产生更为复杂的连杆曲线。

 ${r_E}(t) = {x_{\rm{1}}}(t) + {\rm{i}}{y_{\rm{1}}}(t) = \sum\limits_{ - \infty }^{ + \infty } {{c_n}{{\rm{e}}^{{\rm{i}}\varphi }}}$ (1-1)
 ${r_{{P}}}(t) = {x_{\rm{2}}}(t) + {\rm{i}}{y_{\rm{2}}}(t) = \sum\limits_{ - \infty }^{ + \infty } {{b_n}{{\rm{e}}^{{\rm{i}}\varphi }}}$ (1-2)

 ${\;\;\;\;\;\;\;\;\;\;\;\;{\rm{e}}^{{\rm{i}}{\theta _1}}}(t) = x_{\rm{1}}'(t) + {\rm{i}}y_1'(t) = \sum\limits_{ - \infty }^{ + \infty } {{{c}}_n'{{\rm{e}}^{{\rm{i}}\varphi }}}$ (2-1)
 ${\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{e}}^{{\rm{i}}\rho }}(t) = x_{\rm{2}}'(t) + {\rm{i}}y_2'(t) = \sum\limits_{ - \infty }^{ + \infty } {b_n'{{\rm{e}}^{{\rm{i}}\varphi }}}$ (2-2)

 ${c_n} =\frac{1}{T} \int_0^T {({x_{\rm{1}}}(t) + {\rm{i}}{y_{\rm{1}}}(t)){{\rm{e}}^{{\rm{i}}n\omega t}}{\rm{d}}t}$ (3-1)
 ${b_n} =\frac{1}{T} \int_0^T {({x_{\rm{2}}}(t) + {\rm{i}}{y_{\rm{2}}}(t)){{\rm{e}}^{{\rm{i}}n\omega t}}{\rm{d}}t}$ (3-2)
 $c_n' =\frac{1}{T} \int_0^T {(x_{\rm{1}}'(t) + {\rm{i}}y_1'(t)){{\rm{e}}^{{\rm{i}}n\omega t}}{\rm{d}}t}$ (3-3)
 $b_n' =\frac{1}{T} \int_0^T {(x_{\rm{2}}'(t) + {\rm{i}}y_2'(t)){{\rm{e}}^{{\rm{i}}n\omega t}}{\rm{d}}t}$ (3-4)

 ${c_n} = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {({x_{1m}}(t) + {\rm{i}}{y_{1m}}(t))[\cos (n m \omega ) + {\rm{i}}\sin(n m \omega )]}$ (4-1)
 ${c_n'} = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {(x_{1m}'(t) + {\rm{i}}y_{1m}'(t))[\cos(n m \omega ) + {\rm{i}}\sin(n m \omega )]}$ (4-2)
 ${b_n} = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {({x_{2m}}(t) + {\rm{i}}{y_{2m}}(t))[\cos (n m \omega ) + {\rm{i}}\sin (n m \omega )]}$ (4-3)
 ${b_n'} = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {(x_{2m}'(t) + {\rm{i}}y_{2m}'(t))[\cos (n m \omega ) + {\rm{i}}\sin (n m \omega )]}$ (4-4)

 $c_n' = \frac{{{c_n}{{\rm{e}}^{ - {\rm{i}}(\beta + {\alpha _2})}}}}{{{l_{21}}}};n \ne {\rm{0}}{\text{,}}{\rm{1}}$ (5-3)
 $c_0' = \frac{{({c_0} - r{{\rm{e}}^{{\rm{i}}\mu }}){{\rm{e}}^{ - {\rm{i}}(\beta + {\alpha _2})}}}}{{{l_{21}}}}$ (5-1)
 $c_1' = \frac{{({c_1} - {l_1}{{\rm{e}}^{{\rm{i}}{\varphi _{\rm{0}}}}}){{\rm{e}}^{ - {\rm{i}}(\beta + {\alpha _2})}}}}{{{l_{21}}}}$ (5-2)
2 平面六杆机构综合设计方程建立

2.1 左侧四杆机构设计变量求解

 \begin{aligned}[b] &\;\;\;\;\;r = \pm \sqrt {ls} ,\mu = - {\rm{i}}\ln \frac{s}{r},{l_1} = \pm \sqrt {uv}, {l_2} = \pm \sqrt {xy} ,{l_{21}} = w,\\ &{l_4} = {\textit{z}},{l_{61}} = \pm \sqrt {fg} ,{\varphi _{\rm{0}}} = - {\rm{i}}\ln \frac{\mu }{{{l_1}}} ,{\alpha _{\rm{2}}} = - {\rm{i}}\ln \frac{x}{{{l_2}}},\beta = - {\rm{i}}\ln \frac{f}{{{l_{61}}}}{\text{。}} \\ \end{aligned}

2.2 右侧二杆组设计变量求解

 ${{{l}}_1} + {{{l}}_{21}} + {{{l}}_3} = {{{l}}_6} + {{{l}}_5}$ (6)

 ${\;\;\;\;\;\;\;l_1}{{\rm{e}}^{{\rm{i}}(\varphi + {\varphi _{\rm{0}}} + \beta )}} + {l_{21}}{{\rm{e}}^{{\rm{i}}({\alpha _2} + {\theta _{\rm{1}}} + \beta )}} + {l_3}{{\rm{e}}^{{\rm{i}}{\theta _{\rm{2}}}}} - {l_6}{{\rm{e}}^{{\rm{i}}({\alpha _1} + \beta )}} \\ = {l_5}{{\rm{e}}^{{\rm{i}}{\phi _{\rm{2}}}}}$ (7)

 ${l_1}{{\rm{e}}^{ - {\rm{i}}(\varphi + {\varphi _{\rm{0}}} + \beta )}} + {l_{21}}{{\rm{e}}^{ - {\rm{i}}({\alpha _2} + {\theta _{\rm{1}}} + \beta )}} + {l_3}{{\rm{e}}^{ - {\rm{i}}{\theta _{\rm{2}}}}} - {l_6}{{\rm{e}}^{ - {\rm{i}}({\alpha _1} + \beta )}} \\ = {l_5}{{\rm{e}}^{ - {\rm{i}}{\phi _{\rm{2}}}}}$ (8)

 ${{\rm{e}}^{{\rm{i}}{\theta _{\rm{2}}}}} = {{\rm{e}}^{ - {\rm{i}}{\alpha _3}}}{{\rm{e}}^{{\rm{i}}\rho }}$ (9)

 \begin{aligned}[b] {\;\;\;\;uv} +& hk + cd + ab - {o^2} + {{\rm{e}}^{{\rm{i}}\varphi }}{{\rm{e}}^{ - {\rm{i}}\rho }}uc + {{\rm{e}}^{ - {\rm{i}}\varphi }}{{\rm{e}}^{{\rm{i}}\rho }}vd- \\ & {{\rm{e}}^{{\rm{i}}\varphi }}ud - {{\rm{e}}^{ - {\rm{i}}\varphi }}va - {{\rm{e}}^{{\rm{i}}\rho }}bd - {{\rm{e}}^{ - {\rm{i}}\rho }}ac + {{\rm{e}}^{{\rm{i}}\varphi }}{{\rm{e}}^{ - {\rm{i}}{\theta _{\rm{1}}}}}uk+ \\ & {{\rm{e}}^{ - {\rm{i}}\varphi }}{{\rm{e}}^{{\rm{i}}{\theta _{\rm{1}}}}}vh + {{\rm{e}}^{{\rm{i}}\rho }}{{\rm{e}}^{ - {\rm{i}}{\theta _{\rm{1}}}}}dk + {{\rm{e}}^{ - {\rm{i}}\rho }}{{\rm{e}}^{{\rm{i}}{\theta _{\rm{1}}}}}ch - {{\rm{e}}^{ - {\rm{i}}{\theta _{\rm{1}}}}}ak- \\ & {{\rm{e}}^{{\rm{i}}{\theta _{\rm{1}}}}}bh = 0 \end{aligned} (10)

 \begin{aligned}[b] {H_{{\rm{ - 4}}}}{{\rm{e}}^{ - {\rm{4}}{\rm{i}}\varphi }} + &{H_{{\rm{ - 3}}}}{{\rm{e}}^{ - {\rm{3}}{\rm{i}}\varphi }} + {H_{{\rm{ - 2}}}}{{\rm{e}}^{ - {\rm{2}}{\rm{i}}\varphi }} + {H_{{\rm{ - 1}}}}{{\rm{e}}^{ - {\rm{i}}\varphi }} + {H_{\rm{0}}} + \\ {H_{\rm{1}}}{{\rm{e}}^{{\rm{i}}\varphi }} +& {H_{\rm{2}}}{{\rm{e}}^{2{\rm{i}}\varphi }} + {H_{\rm{3}}}{{\rm{e}}^{3{\rm{i}}\varphi }} + {H_{\rm{4}}}{{\rm{e}}^{4{\rm{i}}\varphi }} = 0 \end{aligned} (11)

 \begin{aligned}[b] - & ad{J_5} + cu{J_6} - ah{J_7} + ku{J_9} + dv{J_{11}} - bc{J_{12}}+ \\ & hv{J_{13}} + hc{J_{14}} + dk{J_{15}} - bk{J_{21}}{\rm{ = 0}} \end{aligned} (12-1)
 \begin{aligned}[b] -& bc{{\overline J_5}} + vd{{\overline J_6}} - bk{{\overline J_7}} + hv{{\overline J_9}} + cu{{\overline J_{11}}} - ad{{\overline J_{12}}}+ \\ & ku{{\overline J_{13}}} + kd{{\overline J_{14}}} + hc{{\overline J_{15}}} - ah{{\overline J_{21}}} {\rm{ = 0}} \end{aligned} (12-2)
 \begin{aligned}[b] -& bv - ad{J_{11}} + cu{J_{12}} - ah{J_{13}} + vd{J_{16}} - bc{J_{17}}+ \\ & hv{J_{18}} + hc{J_{19}} - bk{J_{20}} + ku{J_{21}} + kd{J_{22}}{\rm{ = 0}} \end{aligned} (12-3)
 \begin{aligned}[b] -& au - bc{{\overline J_{11}}} + vd{{\overline J_{12}}} - bk{{\overline J_{13}}} + cu{{\overline J_{16}}} - ad{{\overline J_{17}}} + \\ & ku{{\overline J_{18}}} + kd{{\overline J_{19}}} - ah{{\overline J_{20}}} + hv{{\overline J_{21}}} + hc{{\overline J_{22}}} {\rm{ = 0}} \end{aligned} (12-4)
 \begin{aligned}[b] & ab + hk + uv + cd - {o^2} - ad{J_{16}} - bc{{\overline J_{16}}}+ \\ & vd{{\overline J_{17}}} + cu{J_{17}} - ah{J_{18}} - bk{{\overline J_{18}}} + hv{{\overline J_{20}}} +\\ & ku{J_{20}} + hc{J_{23}} + kd{{\overline J_{23}}} {\rm{ = 0}} \end{aligned} (12-5)

 ${\;\;\;\;\;\;\;\;\;\;\;\;k_{\rm{4}}}{a^4} + {k_{\rm{3}}}{a^3} + {k_{\rm{2}}}{a^2} + {k_{\rm{1}}}a + {k_{\rm{0}}} = 0$ (13)

 \left\{ \begin{aligned}{l_6} =& \pm \sqrt {ab} ;\\ {l_3} = & \pm \sqrt {cd}; \\ {l_5} = & o; \\ {\alpha _{\rm{1}}} = & - {\rm{i}}\ln \frac{a}{{{l_8}}};\\ {\alpha _3} = & - {\rm{i}}\ln \frac{c}{{{l_3}}} - \beta \end{aligned} \right. (28)

3 综合步骤

1）将Stephenson-Ⅲ型平面六杆机构拆分为四杆机构和二级杆组，根据E点的轨迹生成任务，利用式（5）得到连杆曲线的谐波参数 ${c_n}$ ，将其代入平面四杆机构轨迹综合的设计参数计算通用公式，得到左侧四杆机构的设计参数 $r{\text{、}}\mu {\text{、}}{l_1}{\text{、}}{l_2}{\text{、}}{l_{21}}{\text{、}}{l_4}{\text{、}}{l_{61}}{\text{、}}{\varphi _{\rm{0}}}{\text{、}}{\alpha _2}{\text{、}}\beta$

2）由E点和P点轨迹坐标，计算其对应点的转角 ${\;\rho _i}$ ，根据 ${\;\rho _i}$ 利用式（3）计算得到连杆转角函数 ${{\rm{e}}^{{\rm{i}}\rho }}$ 的谐波参数 $b_n'$ ，利用式（13）～（15）计算得到 $c_n'$ ，将 $b_n'$ $c_n'$ 代入右侧二杆组设计变量的计算通式，求解得到右侧二杆组设计参数 ${l_6}{\text{、}}{l_3}{\text{、}}{l_5}{\text{、}}{\alpha _{\rm{1}}}{\text{、}}{\alpha _3}$

3）对所得48组综合结果进行运动仿真，检验其是否存在曲柄，有无分支问题、顺序问题，并依据综合误差，最终得到满足设计要求的Stephenson-Ⅲ型平面六杆机构。

4 综合实例

1）根据步骤2），得到平面四杆机构连杆曲线的谐波参数 ${c_n}$ ，如表2所示，将上述所得的 ${c_n}$ 的值代入四杆机构轨迹综合设计变量的计算通式，可求得设计参数 $r{\text{、}}\mu{\text{、}}{l_1}{\text{、}}{l_2}{\text{、}}{l_{21}}{\text{、}}{l_4}$ ${l_{61}}{\text{、}}{\varphi _0}{\text{、}}{\alpha _2}{\text{、}}\beta$

2）利用式（5）计算得到 $c{}_n'$ ，计算转角 ${\rho _i}$ ，进而得到连杆转角函数 ${{\rm{e}}^{{\rm{i}}\rho }}$ 的谐波参数 $b_n'$ ，如表2所示，将所得的 $c_n'$ $b_n'$ 的值代入右侧二杆组设计变量的计算通式，可求得设计参数 ${l_6}{\text{、}}{l_{\rm{3}}}{\text{、}}{l_5}{\text{、}}{\alpha _1}{\text{、}}{\alpha _3}$

3）将左侧四杆机构与右侧二杆组的设计参数进行组合得到48组解。应用仿真程序对所得综合结果进行运动分析和检验，得到满足设计要求的一组机构参数为： $r = 5,{l_1} = 75,{l_{21}} = 110,\mu = 0.349,{\varphi _0} = 0.175, {l_2} = 139.982,\; {l_4} = 162.027,\;{l_{61}} = 160.008,\;{\alpha _2} = 0.349,\; \beta$ = $0.174, {l_6} = 263.{\rm{886}}, {l_3} = 201.{\rm{871}},{l_5} = 16{\rm{1}}{\rm{.421}},{\alpha _1} = 0.175$ ${\alpha _3} = 0.875$

 图2 生成轨迹与目标轨迹点的比较 Fig. 2 Comparison between prescribed points and the corresponding generated path

5 结　论

 [1] 梁崇高,陈海宗.平面连杆机构的计算设计[M].广州:广东教育出版社,1993. [2] Sun Jianwei,Wu Xin,Wang Jun,et al. Research on straight-lines mechanism path generation by using numerical atlas method[J]. Mechanical Science and Technology for Aerospace Engineering, 2005(6): 693-695. [孙建伟,吴鑫,王军,等. 利用数值图谱法进行多杆直线导向机构的轨迹综合[J]. 机械科学与技术, 2005(6): 693-695. DOI:10.13433/j.cnki.1003-8728.2005.06.018] [3] 褚金奎.连杆机构尺度综合的谐波特征参数法[M].北京:科学出版社,2010. [4] Bulatović R R,Đorðević S R. Optimal synthesis of a path generator six-bar linkage[J]. Journal of Mechanical Science and Technology, 2012, 26(12): 4027-4040. DOI:10.1007/s12206-012-0906-5 [5] Guo G,Zhang J,Gruver W A. Optimal design of a six-bar linkage with one degree of freedom for an anthropomorphic three-jointed finger mechanism[J]. Proceedings of the Institution of Mechanical Engineers(Part H:Journal of Engineering in Medicine), 1993, 207(3): 185-190. DOI:10.1243/PIME_PROC_1993_207_291_02 [6] Tang Dunbing,Yang Jun,Dai Min. Object-oriented multiple poles and multi-objective constraint optimization design method[J]. Machine Building & Automation, 2016, 45(4): 1-4. [唐敦兵,杨俊,戴敏. 面向对象的多杆机构多目标多约束优化设计方法[J]. 机械制造与自动化, 2016, 45(4): 1-4. DOI:10.19344/j.cnki.issn1671-5276.2016.04.001] [7] Li Hongzhong. Dimension synthesis of planar multi-bar linkages[J]. China Science and Technology Information, 2005(11): 126. [李洪忠. 平面多杆机构的尺度优化综合[J]. 中国科技信息, 2005(11): 126. DOI:10.3969/j.issn.1001-8972.2005.11.115] [8] Wei Feng,Wei Shimin,Zhang Ying,et al. The algebraic solution for five precision points path synthesis of stephenson-Ⅲ planar six-bar linkage[J]. Journal of Beijing University of Posts and Telecommunications, 2015, 38(5): 104-108. [魏锋,魏世民,张英,等. Stephenson-Ⅲ型平面六杆机构五精确点轨迹综合代数求解[J]. 北京邮电大学学报, 2015, 38(5): 104-108. DOI:10.13190/j.jbupt.2015.05.020] [9] Ma Jiayi.Singularity analysis of six degrees of freedom of parallel manipulators using geometric algebra[D].Hangzhou:Zhejiang Sci−Tech University,2016,97:112–126. 马嘉熠.基于几何代数的六自由度并联机构奇异分析[D].杭州:浙江理工大学,2017. [10] Plecnik M M,McCarthy J M. Kinematic synthesis of Stephenson Ⅲ six-bar function generators[J]. Mechanism and Machine Theory, 2016, 97: 112-126. DOI:10.1016/j.me-chmachtheory.2015.10.004 [11] Nafees K,Mohammad A. Dimensional synthesis of six-bar Stephenson Ⅲ mechanism for 12 precision points path generation[J]. International Journal of Mechanisms and Robotic Systems, 2016, 3(1): 80-90. DOI:10.1504/IJMRS.2016.077037 [12] Li Xuegang,Wei Shimin,Liao Qizheng,et al. The algebraic solution for path synthesis of planar five-bar linkages with timing[J]. Journal of Beijing University of Posts and Telecommunications, 2017, 40(1): 23-27. [李学刚,魏世民,廖启征,等. 平面五杆机构计时轨迹综合的代数求解[J]. 北京邮电大学学报, 2017, 40(1): 23-27. DOI:10.13190/j.jbupt.2017.01.003] [13] Li X G,Wei S M,Liao Q Z,et al. A novel analytical method for function generation synthesis of planar four-bar linkages[J]. Mechanism and Machine Theory, 2016, 101: 222-235. DOI:10.1016/j.mechmachtheory.2016.03.013 [14] Li X G,Wei S M,Liao Q Z,et al. A novel analytical method for four-bar path generation synthesis based on Fourier series[J]. Mechanism and Machine Theory, 2020, 144: 103671. DOI:10.1016/j.mechmachtheory.2019.103671 [15] Li X G,Wu J,Ge Q J. A Fourier descriptor-based approach to design space decomposition for planar motion approximation[J]. Journal of Mechanisms and Robotics, 2016, 8(6): 064501. DOI:10.1115/1.4033528 [16] McGarva J,Mullineux G. Harmonic representation of closed curves[J]. Applied Mathematical Modelling, 1993, 17(4): 213-218. DOI:10.1016/0307-904X(93)90109-T