徐 朋 趙東標(biāo) 應(yīng)明峰 程錦翔 李 奎

(南京航空航天大學(xué)機(jī)電學(xué)院, 南京 210016)

8自由度鋪絲機(jī)械手的自運(yùn)動(dòng)流形分析

徐 朋 趙東標(biāo) 應(yīng)明峰 程錦翔 李 奎

(南京航空航天大學(xué)機(jī)電學(xué)院, 南京 210016)

針對(duì)梯度投影算法所得冗余機(jī)械手關(guān)節(jié)逆解不一定包括最優(yōu)解的缺陷,提出一種分析8自由度冗余鋪絲機(jī)械手關(guān)節(jié)逆解的流形方法. 利用流形法所得逆解包含了冗余機(jī)械手的全部關(guān)節(jié)逆解,有利于實(shí)現(xiàn)自運(yùn)動(dòng)控制的全面優(yōu)化.根據(jù)逆解流形的空間多維特性,在8自由度鋪絲機(jī)械手的關(guān)節(jié)空間內(nèi)分別定義其位置關(guān)節(jié)子流形和姿態(tài)關(guān)節(jié)子流形,并分別得到三維空間中仿真映射曲面.結(jié)果表明,由于芯模自由度是在固定空間內(nèi)的運(yùn)動(dòng),相比于7自由度的鋪絲機(jī)械手,8自由度鋪絲機(jī)械手的仿真曲面流形在靈活性及避障礙能力方面均有較大提高.最后以飛機(jī)S形進(jìn)氣道為例進(jìn)行仿真,仿真結(jié)果表明,仿真軌跡與期望軌跡高度吻合,證明了所提方法的正確性.

流形;鋪絲;逆解;進(jìn)氣道

復(fù)合材料纖維鋪放成型技術(shù)[1-2](fiber placement technology)是一種復(fù)合材料成型技術(shù),在航空航天制造中擁有重要地位.隨著航空工業(yè)的發(fā)展,高強(qiáng)度、輕質(zhì)量的復(fù)合材料需求量進(jìn)一步加大,纖維鋪放成型技術(shù)逐漸成為了研究的熱點(diǎn).因此針對(duì)復(fù)合材料纖維鋪放工藝技術(shù)的成型裝備[3-4]——鋪絲機(jī)械手的研究已經(jīng)成為當(dāng)今先進(jìn)制造技術(shù)研究中的一個(gè)新方向.

鋪絲機(jī)械手一般都是冗余機(jī)械手[5],冗余機(jī)械手的逆運(yùn)動(dòng)學(xué)解具有自運(yùn)動(dòng)流形的結(jié)構(gòu)[6],自運(yùn)動(dòng)流形在空間反映了機(jī)械手的全部關(guān)節(jié)構(gòu)型.(2)Moll等[7]通過(guò)在關(guān)節(jié)空間尋找流形上的最小能量曲線方法,實(shí)現(xiàn)了末端運(yùn)動(dòng)軌跡的最優(yōu)化.Burdick[8]給出位置空間3R機(jī)構(gòu)和4R機(jī)構(gòu)的流形分析,并得出解流形的切空間同構(gòu)于Jacobian矩陣零空間的結(jié)論.Galicki[9]利用反自由控制算法詳細(xì)分析了冗余機(jī)械手的關(guān)節(jié)空間流形,得出逆解流形的最小和最大取值區(qū)間.趙建文等[10]得到了特殊結(jié)構(gòu)冗余機(jī)器人參數(shù)化的自運(yùn)動(dòng)流形,但沒(méi)有具體給出解空間和工作空間之間的流形映射關(guān)系.葛新鋒等[11]利用位姿分離方法分析了冗余鋪絲機(jī)械手關(guān)節(jié)逆解流形,7自由度鋪絲機(jī)械手只有一個(gè)冗余自由度,其在鋪絲工作過(guò)程中的靈活性及避障礙能力有一定局限性,不利于自運(yùn)動(dòng)控制的選擇.

因此,本文提出一種位置和姿態(tài)相耦合的8自由度鋪絲機(jī)械手模型.不僅增強(qiáng)了鋪絲機(jī)械手在整個(gè)工作過(guò)程中的靈活性及避障礙能力,而且有利于后續(xù)自運(yùn)動(dòng)控制的優(yōu)化選擇.8自由度鋪絲機(jī)械手關(guān)節(jié)空間擁有無(wú)數(shù)多個(gè)逆解,所有逆解一起構(gòu)成多維空間中的解流形.

1 8自由度鋪絲機(jī)械手模型

圖1為8自由度鋪絲機(jī)械手模型.由單自由度芯模、6自由度庫(kù)卡機(jī)械手及單自由度滑動(dòng)導(dǎo)軌組成. 相對(duì)于7自由度位姿分離式鋪絲機(jī)械手, 8自由度鋪絲機(jī)械手模型各個(gè)關(guān)節(jié)之間存在強(qiáng)耦合關(guān)系,可以適應(yīng)更復(fù)雜的工作環(huán)境及有助于逆解和自運(yùn)動(dòng)控制的進(jìn)一步優(yōu)化.

圖1 8自由度鋪絲機(jī)械手模型

2 8自由度鋪絲機(jī)械手關(guān)節(jié)逆解

本文基于旋量理論[12-13]通過(guò)設(shè)定雙冗余變量,計(jì)算分析8自由度鋪絲機(jī)械手的關(guān)節(jié)逆解,各連桿間的尺寸參數(shù)及初始運(yùn)動(dòng)狀態(tài)如圖2所示.圖中,a1～a6分別為機(jī)械手的尺寸參數(shù);d1為冗余關(guān)節(jié)變量值;r1～r5為各自軸線上的點(diǎn);ξ1～ξ8為相應(yīng)的單位運(yùn)動(dòng)旋量.

初始運(yùn)動(dòng)狀態(tài)下慣性坐標(biāo)系xsyszs與工具坐標(biāo)系xtytzt之間的齊次變換為

(1)

圖2 冗余鋪絲機(jī)械手初始運(yùn)動(dòng)狀態(tài)

一般工作狀態(tài)下慣性坐標(biāo)系與工具坐標(biāo)系之間的齊次變換為

(2)

式中,θ8為芯模旋轉(zhuǎn)角度;α為旋轉(zhuǎn)芯模軸線與鋪絲頭壓輥軸線之間的夾角;r為鋪絲運(yùn)動(dòng)軌跡點(diǎn)處的切線半徑.

將d1和θ8分別設(shè)定為冗余變量,根據(jù)旋量理論及各已知旋量子問(wèn)題,可得

θ2=arctan2(?(qx-d1),±qy)

(3)

(4)

(5)

(6)

(7)

θ7=arctan2((d1-q″x),(a2+a5-q″y))

(8)

(9)

(10)

(11)

(12)

(13)

式(3)～(8)中,當(dāng)冗余關(guān)節(jié)d1,θ8在固定范圍內(nèi)變化時(shí),本文計(jì)算所得關(guān)節(jié)逆解θ2,θ3,θ4,θ5,θ6,θ7均是以d1,θ8為變量的函數(shù).為了實(shí)際鋪絲工作中方便測(cè)量和計(jì)算,本文以d1,θ8作為冗余關(guān)節(jié)來(lái)計(jì)算逆解及流形分析,其中θ8包含于gst(θ).

3 仿真驗(yàn)證

定義關(guān)節(jié)構(gòu)型空間為

C=C1C2C3C4C5C6C7C8

(14)

式中,C1為移動(dòng)關(guān)節(jié)構(gòu)型空間;C2～C7為6自由度機(jī)械手各關(guān)節(jié)構(gòu)型空間;C8為芯模轉(zhuǎn)角構(gòu)型空間.

定義位置關(guān)節(jié)構(gòu)型空間為

Cwz=C1C2C3C4C8

(15)

定義位置關(guān)節(jié)工作空間流形為

(Cwl,Fwl)={(θwl,fwl(θwl))/fwl(θwl)=

low,θwl∈Cwz}

(16)

式中,low為從慣性坐標(biāo)系原點(diǎn)到機(jī)械手腕點(diǎn)的位置矢量;θwl={d1,θ2,θ3,θ4,θ8}T;Cwl為位置關(guān)節(jié)工作空間流形.

定義姿態(tài)關(guān)節(jié)構(gòu)型空間為

Czt=C2C3C4C5C6C7C8

(17)

定義姿態(tài)關(guān)節(jié)工作空間流形為

(Czl,Fzl)= {(θzl,fzl)/fzl(θzl)=

(low,xt,zt),θzl∈Czt}

(18)

式中,xt,zt為工具坐標(biāo)系的姿態(tài)矢量;θzl={θ2,θ3,θ4,θ5,θ6,θ7,θ8}T;Czl為姿態(tài)工作空間流形.

8自由度鋪絲機(jī)械手本體結(jié)構(gòu)參數(shù)定義如下:

(19)

(20)

(a) 右上臂形

(b) 右下臂形

(c) 左上臂形

(d) 左下臂形

由于在實(shí)際工作過(guò)程中鋪絲機(jī)械手一般處于左上臂形和右上臂形2種情形.本文在鋪絲機(jī)械手正常工作過(guò)程中針對(duì)芯模鋪絲軌跡,在位置關(guān)節(jié)空間和姿態(tài)關(guān)節(jié)空間作了相應(yīng)的自運(yùn)動(dòng)流形仿真.其中式(16)展示的關(guān)節(jié)位置空間是五維的自運(yùn)動(dòng)流形,受空間維數(shù)的限制,本文僅將位置流形映射到θ2,θ3,θ4組成的位置關(guān)節(jié)空間(見(jiàn)圖4),將式(18)展示的七維姿態(tài)流形映射到θ5,θ6,θ7組成的腕關(guān)節(jié)空間(見(jiàn)圖5和6).

(a) 左上臂形

(b) 右上臂形

(a) 翻腕

(b) 不翻腕

鑒于鋪絲機(jī)械手關(guān)節(jié)空間存在自運(yùn)動(dòng)特性,所以也可以將姿態(tài)空間流形和位置空間流形分別映射到其他關(guān)節(jié)定義的三維空間.這樣得到的自運(yùn)動(dòng)流形是不同的,但是不同自運(yùn)動(dòng)流形之間的關(guān)節(jié)運(yùn)動(dòng)構(gòu)型是一致的.

(a) 翻腕

(b) 不翻腕

對(duì)于7自由度鋪絲機(jī)械手,以位置關(guān)節(jié)為例得出其在右上臂形和左上臂形標(biāo)志下的仿真流形,圖7分別為空間中的曲線流形.由于芯模自由度是在固定空間內(nèi)的運(yùn)動(dòng),因此當(dāng)芯模上的鋪絲軌跡點(diǎn)運(yùn)動(dòng)完全確定時(shí),芯模自由度就會(huì)被約束.這樣7自由度鋪絲機(jī)械手就會(huì)失去冗余特性,其逆運(yùn)動(dòng)學(xué)解將是空間中的有限離散點(diǎn),靈活性和避障礙能力將會(huì)變差,也不利于后續(xù)最優(yōu)自運(yùn)動(dòng)控制的實(shí)現(xiàn),而8自由度鋪絲機(jī)械手可以很好地解決這一問(wèn)題.

(a) 左上臂形

(b) 右上臂形

下面通過(guò)飛機(jī)S形進(jìn)氣道鋪絲軌跡點(diǎn)為例來(lái)驗(yàn)證本文所得逆解流形的正確性.

圖8 飛機(jī)S形進(jìn)氣道仿真軌跡

由圖8可見(jiàn),期望軌跡和仿真軌跡高度吻合,由此證明了本文所用自運(yùn)動(dòng)流形法的正確性.

4 結(jié)語(yǔ)

利用自運(yùn)動(dòng)流形方法得到了8自由度鋪絲機(jī)械手的全部關(guān)節(jié)逆解,由于流形中肯定含有最優(yōu)逆解,所以有效避免了梯度投影算法所得逆解不一定是最優(yōu)逆解的缺陷,為后續(xù)的自運(yùn)動(dòng)最優(yōu)控制奠定了基礎(chǔ).根據(jù)關(guān)節(jié)逆解的自運(yùn)動(dòng)流形特點(diǎn),可以利用微分流形理論在曲面上研究鋪絲機(jī)械手的逆解,最后通過(guò)飛機(jī)S形進(jìn)氣道驗(yàn)證了本文所得結(jié)論的正確性.

References)

[1]Bruyneel M, Zein S. A modified fast marching method for defining fiber placement trajectories over meshes[J].ComputersandStructures, 2013, 125: 45-52. DOI:10.1016/j.compstruc.2013.04.015.

[2]方宜武, 王顯峰, 顧善群, 等. 自動(dòng)鋪絲過(guò)程中預(yù)浸料的側(cè)向彎曲[J]. 材料工程, 2015, 43(4): 47-52.DOI:10.11868/j.issn.1001-4381.2015.04.009. Fang Yiwu, Wang Xianfeng, Gu Shanqun, et al. Lateral bending of prepreg during automated fiber placement[J].JournalofMaterialsEngineering, 2015, 43(4): 47-52.DOI:10.11868/j.issn.1001-4381.2015.04.009.(in Chinese)

[3]段玉崗, 董肖偉, 葛衍明, 等. 基于CATIA生成數(shù)控加工路徑的機(jī)器人纖維鋪放軌跡規(guī)劃[J]. 航空學(xué)報(bào), 2014, 35(9): 2632-2640 DOI:10.7527/S1000-6893.2013.0471. Duan Yugang, Dong Xiaowei, Ge Yanming, et al. Robotic fiber placement trajectory planning based on CATIA CNC machining path[J].ActaAeronauticaetAstronauticaSinica, 2014, 35(9): 2632-2640. DOI:10.7527/S1000-6893.2013.0471. (in Chinese)

[4]Schmidt C, Schultz C, Weber P, et al. Evaluation of eddy current testing for quality assurance and process monitoring of automated fiber placement[J].CompositesPartB:Engineering, 2014, 56(17): 109-116. DOI:10.1016/j.compositesb.2013.08.061.

[5]郭迪, 孫富春, 劉華平, 等. 基于Kinect的冗余機(jī)械臂直線推移操作控制[J]. 東南大學(xué)學(xué)報(bào)(自然科學(xué)版), 2013, 43(I): 143-147. DOI:10.3969/j.issn.1001-0505.2013.S1.030. Guo Di, Sun Fuchun, Liu Huaping, et al. Line pushing control of redundant manipulator based on Kinnect[J].JournalofSoutheastUniversity(NaturalScienceEdition), 2013, 43(Ⅰ): 143-147.DOI:10.3969/j.issn.1001-0505.2013.S1.030.(in Chinese)

[6]An H H, Clement W I, Reed B. Analytical inverse kinematic solution with self-motion constraint for the 7-DOF restore robot arm[C]//2014IEEE/ASMEInterna-tionalConferenceonAdvancedIntelligentMechatronics. Besancon, France, 2014: 1325-1330. DOI:10.1109/aim.2014.6878266.

[7]Moll M, Kavraki L E. Path planning for minimal energy curves of constant length[C]//Proceedingsofthe20 04IEEEInternationalConferenceonRoboticsandAuto-mation. New Orleans, USA, 2004: 2826-2831. DOI:10.1109/robot.2004.1307489.

[8]Burdick J W. On the inverse kinematics of redundant manipulators: Characterization of the self-motion mani-folds[C]//Proceedingsofthe1989IEEEInternationalConferenceonRoboticsandAutomation. Scottsdale, USA, 1989: 264-270. DOI:10.1109/robot.1989.99999.

[9]Galicki M. Inverse-free control of a robotic manipulator in a task space[J].RoboticsandAutonomousSystems, 2014, 62(2): 131-141. DOI:10.1016/j.robot.2013.11.005.

[10]趙建文, 杜志江, 孫立寧. 7自由度冗余手臂的自運(yùn)動(dòng)流形[J]. 機(jī)械工程學(xué)報(bào), 2007, 43(9): 132-137.DOI:10.3321/j.issn:0577-6686.2007.09.027. Zhao Jianwen, Du Zhijiang, Sun Lining. Self-motion manifolds of a 7-DOF redundant manipulator[J].JournalofMechanicalEngineering, 2007, 43(9): 132-137.DOI:10.3321/j.issn:0577-6686.2007.09.027.(in Chinese)

[11]葛新鋒, 趙東標(biāo). 7自由度自動(dòng)鋪絲機(jī)器人參數(shù)化的自運(yùn)動(dòng)流形[J]. 機(jī)械工程學(xué)報(bào), 2012, 48(13): 27-31. DOI:10.3901/JME.2012.13.027. Ge Xinfeng, Zhao Dongbiao. Parameterized self-motion manifold of 7-DOF automatic fiber placement robotic manipulator[J].JournalofMechanicalEngineering, 2012, 48(13): 27-31. DOI:10.3901/JME.2012.13.027. (in Chinese)

[12]戴建生. 機(jī)構(gòu)學(xué)與旋量理論的歷史淵源以及有限位移旋量的發(fā)展[J]. 機(jī)械工程學(xué)報(bào), 2015, 51(13): 13-26. DOI:10.3901/JME.2015.13.013. Ddi Jiansheng. Historical relation between mechanisms and screw theory and the development of finite displace-ment screws[J].JournalofMechanicalEngineering, 2015, 51(13): 13-26.DOI:10.3901/JME.2015.13.013.(in Chinese)

[13]Zheng F Y, Hua L, Han X H. The mathematical model and mechanical properties of variable center distance gears based on screw theory[J].MechanismandMachineTheory, 2016, 101: 116-139. DOI:10.1016/j.mechmachtheory.2016.03.005.

Self-motion manifold analysis on 8-DOF fiber placement manipulator

Xu Peng Zhao Dongbiao Ying Mingfeng Cheng Jinxiang Li Kui

(College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)

In consideration of the disadvantage that the inverse solutions of redundant manipulator’sjoints based on the gradient projection method were not always optimal, a new method using manifolds to analyze the inverse solutions of an 8-DOF fiber placement manipulator’s joints was proposed in this study. The self-motion manifolds obtained by this method contain all of the inverse kinematic solution of fiber placement manipulator that helps to optimize the self-motion control roundly. According to the multi-dimensional characteristics of self-motion manifolds, the position joints space sub-manifolds and posture joints space sub-manifolds were respectively defined in working space of the 8-DOF fiber placement manipulator joint space, and the corresponding triaxial simulation curves of the manifolds were obtained respectively. The results show that the 8-DOF fiber placement manipulator, compared with the 7-DOF fiber placement manipulator, improved a lot in its flexibility and obstacle avoidance abilities because the motion corresponding to the mandrel’s degree of freedom was in the fixed space. In the last, the method was verified by using the S-shaped inlet simulation. The result shows that the simulation trajectory was highly consistent with the desired trajectory which proved validity of the method proposed by the paper.

manifolds; fiber placement; inverse solutions; inlet

10.3969/j.issn.1001-0505.2017.02.010

2016-08-21. 作者簡(jiǎn)介: 徐朋(1982—)，男，博士生；趙東標(biāo)(聯(lián)系人)，男，博士，教授，博士生導(dǎo)師，zdbme@nuaa.edu.cn.

國(guó)家自然科學(xué)基金資助項(xiàng)目(51175261)、國(guó)家重點(diǎn)基礎(chǔ)研究發(fā)展計(jì)劃(973計(jì)劃)資助項(xiàng)目(2014CB046501)、高等學(xué)校博士學(xué)科點(diǎn)專項(xiàng)科研基金資助項(xiàng)目(20123218110020).

徐朋,趙東標(biāo),應(yīng)明峰,等.8自由度鋪絲機(jī)械手的自運(yùn)動(dòng)流形分析[J].東南大學(xué)學(xué)報(bào)(自然科學(xué)版),2017,47(2):254-258.

10.3969/j.issn.1001-0505.2017.02.010.

TP242.2

A

1001-0505(2017)02-0254-05