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本科畢業(yè)設(shè)計(jì)(論文)
外文翻譯（附外文原文）
系 ( 院 )：機(jī)械與控制工程學(xué)院
課題名稱：乒乓球發(fā)球器的結(jié)構(gòu)設(shè)計(jì)
專業(yè)(方向)：機(jī)械設(shè)計(jì)制造及其自動(dòng)化
（機(jī)械裝備設(shè)計(jì)與制造 ）
班 級： 機(jī)械11-2
學(xué) 生： 蔡書斌
指導(dǎo)教師： 張聲嵐
日 期： 2015年3月12日
第29屆中國控制會議論文集
7月29日至31日，2010年，北京，中國
基本姿勢5自由度混合機(jī)械臂控制算法適合乒乓球機(jī)器人
ZHENG Kuijing1, CUI Pei 1, MAO Haixia2
1.機(jī)械工程學(xué)院，燕山大學(xué)，秦皇島，066004中華人民共和國電子郵件：kjzheng@ysu.edu.cn
2. E＆啊科學(xué)河北科技師范學(xué)院的學(xué)院，秦皇島，郵編：066004
摘要：發(fā)展和乒乓球機(jī)器人的組成進(jìn)行了介紹?；谄古仪?，一種3-RPUR+ RP5自由度混合機(jī)械臂提出，它可以執(zhí)行三個(gè)平移自由度和兩個(gè)旋轉(zhuǎn)自由度的運(yùn)動(dòng)特性。通過使用DH參數(shù)法和XYZ歐拉角，混合動(dòng)力車機(jī)械臂的運(yùn)動(dòng)學(xué)逆溶液進(jìn)行分析，球拍的姿勢被方便地描述。姿態(tài)控制方程被推導(dǎo)，這可變換球拍構(gòu)成在工作空間到關(guān)節(jié)空間的驅(qū)動(dòng)軸的參數(shù)。通過ADAMS軟件，將運(yùn)動(dòng)仿真被執(zhí)行，從而有效地證明了理論分析?；舅惴ǖ於顺晒Φ?軸聯(lián)動(dòng)控制的乒乓球機(jī)器人的理論基礎(chǔ)。
關(guān)鍵詞：乒乓球機(jī)器人，自由的五度，混合機(jī)械手臂和姿態(tài)的反解
1引言
作為一個(gè)服務(wù)機(jī)器人，乒乓球機(jī)器人可用于不僅為專業(yè)運(yùn)動(dòng)員作為試馬針對性的訓(xùn)練，而且在行使對業(yè)余運(yùn)動(dòng)員。因此，乒乓球機(jī)器人吸引了來自學(xué)術(shù)界和工業(yè)界國內(nèi)外越來越多的關(guān)注。許多大學(xué)和公司都在打乒乓球的機(jī)器人了深入的研究，并開發(fā)了多種乒乓球的機(jī)器人在不同結(jié)構(gòu)和類型的自1980年以來初步乒乓球機(jī)器人具有比服務(wù)多樣化球的功力沒有其他的話，機(jī)械臂進(jìn)行開發(fā)反擊即將到來的球。 1983年，約翰·比林斯利[1]從英國樸茨茅斯理工大學(xué)約占乒乓球機(jī)器人法規(guī)。羅素L.Andersson [2]，宮崎文雄[3]等開發(fā)的乒乓球機(jī)器人一個(gè)接一個(gè)。建昌元[4]從西安理工大學(xué)，德許[5]從北京自動(dòng)化研究所和魏巍[6]浙江大學(xué)還研究了乒乓球的機(jī)器人。的詳細(xì)介紹可以在參考進(jìn)行檢查[7]。
乒乓球機(jī)器人由機(jī)械系統(tǒng)，視覺系統(tǒng)和控制系統(tǒng)。作為手眼協(xié)調(diào)系統(tǒng)，三個(gè)子系統(tǒng)必須彼此協(xié)調(diào)。機(jī)械系統(tǒng)，類似于人類的手臂，直接進(jìn)行打乒乓球的功能。視覺系統(tǒng)，類似于人的眼睛，監(jiān)視乒乓球運(yùn)動(dòng)，并預(yù)測其運(yùn)動(dòng)軌跡。控制系統(tǒng)，類似于人類的大腦，控制所述機(jī)器人臂以敏捷擺動(dòng)球拍根據(jù)移動(dòng)軌跡乒乓去的規(guī)劃位置和方向，并實(shí)現(xiàn)了精確的命中。
A排序五自由度混合機(jī)械臂包括并行機(jī)制和串行機(jī)制提出，它可以執(zhí)行三個(gè)平移自由度和兩個(gè)旋轉(zhuǎn)自由度。混合機(jī)械臂的運(yùn)動(dòng)學(xué)反解進(jìn)行了深入分析。的位置和方向的控制方程推導(dǎo)。在前述的算法仿真，通過ADAMS軟件的方式進(jìn)行驗(yàn)證。該算法還規(guī)定，為人們控制機(jī)器人手臂的姿勢的理論基礎(chǔ)。
2方案乒乓球機(jī)器人
基于5自由度混合機(jī)械臂柔性雙眼視乒乓球機(jī)器人的方案示于圖1的混合式機(jī)器人臂裝置串行機(jī)制連接到并行平臺。它包括三個(gè)RPRU（回轉(zhuǎn)-棱柱回轉(zhuǎn)通用型），四肢和RP（回轉(zhuǎn)-棱鏡）肢。球拍安裝在機(jī)器人臂的末端。在該并聯(lián)機(jī)構(gòu)中三個(gè)平移對和在串行機(jī)制兩個(gè)旋轉(zhuǎn)對被用作驅(qū)動(dòng)軸達(dá)到5軸同步控制。球拍能夠擺動(dòng)到達(dá)需要的位置，方向和速度。兩個(gè)2自由度搖籃頭上面的機(jī)械臂安裝和CCD照相機(jī)被安裝在每個(gè)托架的頭。兩款相機(jī)都可以進(jìn)行旋轉(zhuǎn)2自由度，形成靈活的雙眼視覺。
圖1：乒乓球機(jī)器人計(jì)劃
乒乓球的機(jī)器人是一個(gè)手眼協(xié)調(diào)系統(tǒng)與快速的眼睛和靈巧的手。該機(jī)器人可以擺動(dòng)它的球拍敏捷，靈活，精確打擊乒乓球和避免現(xiàn)有的人類擊球的正反手問題。
3說明5自由度混合機(jī)械臂
乒乓球具有速度快，各種墜落點(diǎn)，廣泛和強(qiáng)烈的旋轉(zhuǎn)等特點(diǎn)。因此，機(jī)械臂必須滿足順序執(zhí)行這些要求，以適合打回乒乓球。一方面，它必須是多自由度來實(shí)現(xiàn)的各種位置和方向和擺動(dòng)球拍去的規(guī)劃點(diǎn)。另一方面，它需要有足夠的工作空間到蓋體內(nèi)部并在表外部更大的面積和回?fù)舾鱾€(gè)到來乒乓球。此外，速度快，精度高，還需要快速，準(zhǔn)確地回?fù)袅似古仪颉?
基于上述分析，一種3- RPUR+ RP 5自由度混合機(jī)械臂提出。如圖2所示，混合機(jī)構(gòu)由穩(wěn)定的平臺，移動(dòng)平臺，其與移動(dòng)平臺，旋轉(zhuǎn)對和平移一對串聯(lián)在移動(dòng)平臺和安裝在端球拍連接穩(wěn)定的平臺四肢機(jī)器人手臂。其在特征在于：在穩(wěn)定的平臺和移動(dòng)平臺都具有相同的連接
3 RPUR（旋轉(zhuǎn)，平移萬能旋轉(zhuǎn)）駕駛的肢體。通過控制P對三個(gè)RPUR驅(qū)動(dòng)四肢的位置和移動(dòng)平臺的取向的運(yùn)動(dòng)可以被改變以實(shí)現(xiàn)兩維轉(zhuǎn)動(dòng)和一維平移。旋轉(zhuǎn)一對R 4與移動(dòng)臺連接的，使周圍的移動(dòng)平臺的中心軸的擺動(dòng)桿L4轉(zhuǎn)動(dòng)。在擺桿L4平移對P5使得沿?cái)[桿軸方向的球拍P數(shù)據(jù)搬移。
a）機(jī)器人臂模型 b）該坐標(biāo)機(jī)械臂系統(tǒng)
圖2：3 RPUR+ RP5自由度混合機(jī)械臂
通用對和旋轉(zhuǎn)對的軸的兩個(gè)軸在一個(gè)點(diǎn)上相交的3- RPUR并聯(lián)機(jī)構(gòu)，其等于球體對，即3-RPS機(jī)構(gòu)。 3-RPS+ BP混合機(jī)械臂自由度可以計(jì)算通過使用Kutzbach Grubler的公式如下：
M == 6×﹙10－11-1﹚＋17 =5
這樣的3-RPS+ BP混合機(jī)器人手臂的自由度是5。
該混合機(jī)械臂結(jié)合高剛性，速度快，慣性小，誤差小，高負(fù)荷和簡單的敏捷和串行機(jī)制，寬大的空間足夠并聯(lián)機(jī)構(gòu)的結(jié)構(gòu)。的慣性和累積誤差被降低。的剛性提高。的運(yùn)動(dòng)精度和運(yùn)動(dòng)速度提高。的位置和取向以及動(dòng)態(tài)屬性的敏捷性被有效地提高。該混合機(jī)械臂能夠進(jìn)行球拍的運(yùn)動(dòng)計(jì)劃更迅速，敏捷，準(zhǔn)確地在不同的速度，落點(diǎn)，角度和各種未來乒乓球的條款。
4. 混合機(jī)械臂的運(yùn)動(dòng)學(xué)逆解
4.1轉(zhuǎn)發(fā)和RP肢體位置分析反解
用DH法[8]，該坐標(biāo)系上的旋轉(zhuǎn)對R4，平移對P5，哪些是在移動(dòng)的平臺上鏈接的乒乓球拍分別成立。如圖2 b）的移動(dòng)坐標(biāo)系統(tǒng){B}是基準(zhǔn)坐標(biāo)系中的{0}，坐標(biāo)系{4}對應(yīng)于R 4，坐標(biāo)系{5}對應(yīng)于P5的坐標(biāo)系統(tǒng){P}對應(yīng)的乒乓球拍。表1示出了相應(yīng)的D-H參數(shù)。 Q4和d5的是變量，A1，A2，d1和d3的是常數(shù)。90度A190°A2，D1lB4，D3 l5P。LB4是坐標(biāo)系統(tǒng){B}和坐標(biāo)系統(tǒng){P}的原點(diǎn)的原點(diǎn)之間的距離。
表1：D-H RP肢的參數(shù)
根據(jù)表1中的參數(shù)，變換矩陣（_B^ P）的坐標(biāo)系統(tǒng){P}相對于坐標(biāo)系統(tǒng){B}的T被給出如下：
在公式（1），S是sin，c是COS。
從等式（1），相對于{P}的原點(diǎn)的位置，可以在{B}表示：
等式（2）是反相肢體的位置的正解，所以逆溶液給出如下：
4.23-RPS肢體位置分析的反解
如圖2，3-RPS并聯(lián)機(jī)構(gòu)的移動(dòng)平臺是正三角形S1S2S3。移動(dòng)坐標(biāo)系{B}是建立在移動(dòng)平臺上。產(chǎn)地OB位于動(dòng)平臺的幾何中心。 Axis_XB恰逢載體OBS1。穩(wěn)定的平臺也是一個(gè)正三角形R1R2R3。穩(wěn)定的坐標(biāo)系{A}是建立在穩(wěn)定的平臺。產(chǎn)地OA位于穩(wěn)定的平臺的幾何中心。 Axis_XA恰逢載體OAR1。
相對于三個(gè)旋轉(zhuǎn)對R1，R2和R3的三個(gè)軸是相切的穩(wěn)定的平臺的外接圓。外接圓半徑為A R。的三個(gè)關(guān)節(jié)的R1，R2和R3可以在坐標(biāo)系{A}中被表示為如下的位置：
正三角形S1S2S3的外接圓的半徑為Rb。三關(guān)節(jié)S1，S2和S3的位置可以被表示在坐標(biāo)系統(tǒng){B}如下：
然后變換{B}相對于{A}可以表示如下的矩陣：
在公式（4），OB原產(chǎn)地在{A}的位置。
表示旋轉(zhuǎn)矩陣和（A，B，g）為{B}相對于{A}的取向的歐拉角。
關(guān)節(jié)的Si（ⅰ=1，2,3）的位置的坐標(biāo)系{A}中可被表示為如下：
然后，在{A}的驅(qū)動(dòng)軸的長度矢量可以舉出：
從等式（6），所有的驅(qū)動(dòng)軸的長度可以計(jì)算：
在3-RPS并聯(lián)機(jī)構(gòu)三個(gè)約束方程給出[9]：
.
從等式（8），下面的方程可以給出：
從方程（9），在移動(dòng)平臺的位置和取向的6個(gè)參數(shù)，以及是獨(dú)立的。一個(gè)，并且可以通過上述三個(gè)約束方程來解決。然后將三個(gè)驅(qū)動(dòng)軸的長度可以表示為如下：
4.3姿態(tài)混合機(jī)械臂的逆解
從等式（1）和（4），{P}相對于{A}的變換矩陣給出如下：
在公式（11），是相對于{P}的{A}原點(diǎn)OP的位置。它可以表示為如下：
在歐拉角表示的旋轉(zhuǎn)矩陣為{P}相對于{A}。是{P}相對的歐拉角{A}的歐拉角。
4.4姿勢3-RPS+ RP混合機(jī)械手臂的控制方程
的球拍可以通過所描述的位置和方向表示的球拍中{A}的中央點(diǎn)的位置，并且表示相對于球拍的取向的方向角。通過使用方向余弦的Z{P}之間以及在三個(gè)坐標(biāo)中{A}軸分別描述。
從變換方程，規(guī)劃的參數(shù)所構(gòu)成的球拍，移動(dòng)平臺和（D5，Q4），反相肢的DH參數(shù)的姿勢的參數(shù)由等式（12）中描述的。
通過使用等式（12），的位置和移動(dòng)平臺的取向的數(shù)據(jù)可以根據(jù)相對于球拍的位置和方向的輸入數(shù)據(jù)來計(jì)算。然后，利用位置反式（10），該規(guī)劃姿勢在工作空間的球拍可以被翻譯成大約在關(guān)節(jié)空間的旋轉(zhuǎn)軸的驅(qū)動(dòng)軸和角度的長度。機(jī)器人手臂的運(yùn)動(dòng)控制可以通過等式（13）的方式來實(shí)現(xiàn)。機(jī)器人臂可以被控制以擺動(dòng)其火箭到達(dá)規(guī)劃姿勢很快以便回?fù)粑磥砬驕?zhǔn)確。
5仿真
圖3：模擬的流程圖
3 RPUR+ RP5自由度混合機(jī)械臂可以在SolidWorks中構(gòu)建。然后，該實(shí)體模型是通過一種數(shù)據(jù)轉(zhuǎn)換格式命名的Parasolid導(dǎo)入ADAMS?，F(xiàn)有在SolidWorks中裝配和約束關(guān)系成為unvalid當(dāng)他們在ADAMS。因此，有必要定義約束模型中的所有部分。首先，成立了工作狀態(tài)。然后定義運(yùn)動(dòng)副的約束，包括固定對，對平移和旋轉(zhuǎn)對。運(yùn)動(dòng)關(guān)系可以通過運(yùn)動(dòng)副裝載驅(qū)動(dòng)運(yùn)動(dòng)構(gòu)造，其中四對平移一轉(zhuǎn)動(dòng)對[10-11]。最后，仿真可以通過導(dǎo)入的駕駛數(shù)據(jù)來獲得。的流程圖被示出為圖3.
是機(jī)器人手臂的結(jié)構(gòu)參數(shù)如下：
rA=300mm, rB=220mm, lB4=132m, l5P=40mm
是在驅(qū)動(dòng)軸的初始參數(shù)如下：
l1=l2=l3=679.72mm, q4 =0 , d5=500mm
機(jī)器人手臂的規(guī)劃動(dòng)作如下：（500，0，847，90°，90°，0°）→（0,400,1147,110°，108.75°，27.99 O）→（-200，0，847，90°，90°，0°）→（0，-400，1147,70°，71.253°，27.99°）→（500，0，847，90°，90°，0°）
表2：姿勢火箭數(shù)據(jù)
表3：機(jī)器人手臂的駕駛數(shù)據(jù)
圖4：混合機(jī)械臂的運(yùn)動(dòng)模擬圖
通過使用所構(gòu)成的機(jī)器人手臂的控制方程，所述規(guī)劃的位置和方向的數(shù)據(jù)（參照表2）的火箭，可以計(jì)算以獲取控制數(shù)據(jù)（參照表3）然后，控制數(shù)據(jù)可以在每個(gè)驅(qū)動(dòng)軸被裝載在ADMAS軟件。和火箭的運(yùn)動(dòng)軌跡可以生產(chǎn)。如圖4的模擬結(jié)果一致的規(guī)劃軌跡。
6結(jié)論
3 RPUR+ RP5自由度混合機(jī)械臂可以執(zhí)行三個(gè)平移自由度和兩個(gè)旋轉(zhuǎn)自由度。通過使用XYZ歐拉角表示了火箭的位置和取向，所述機(jī)器人的運(yùn)動(dòng)學(xué)逆溶液簡明解決。關(guān)于火箭的姿態(tài)控制式成立。通過ADAMS軟件，模擬執(zhí)行，從而有效地證明了理論分析。基本算法將為5軸同步控制的乒乓球機(jī)器人的理論基礎(chǔ)。
參考
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Proceedings of the 29th Chinese Control Conference
July 29-31, 2010, Beijing, China
Basic Pose Control Algorithm of 5-DOF Hybrid Robotic Arm Suitable for Table Tennis Robot
ZHENG Kuijing1, CUI Pei 1, MAO Haixia2
1. Mechanical Engineering College, Yanshan University, Qinhuangdao, 066004, P.R.China E-mail: kjzheng@ysu.edu.cn
2. E&A College of Hebei Normal University of Science & Technology, Qinhuangdao, 066004, P.R.China
Abstract: The development and the composition of table tennis robot are introduced. Based on the moving characteristic of table tennis, a sort of 3-RPUR+RP 5-DOF hybrid robotic arm is put forward, which can perform three translational DOFS and two rotational DOFS. By using D-H parameter method and XYZ Euler angle, the kinematic inverse solution of the hybrid robotic arm is analyzed and the pose of the racket is described conveniently. The pose control equation is deduced, which can transform the racket pose in working space into the parameters of the driving axis in joint space. By using ADAMS software, the motion simulation is performed so as to prove the theoretical analysis effectively. The basic algorithm lays the theoretical foundation for the successful 5-axis simultaneous control of the table tennis robot.
Key Words: Table Tennis Robot, Five Degrees of Freedom, Hybrid Robotic Arm and Inverse Solution of Pose
1 INTRODUCTION
As a service robot, table tennis robot can be used not only in pertinent training for professional athletes as a trial horse, but also in exercising for amateur athletes. Therefore table tennis robot attracts increasing concern from academic and industrial community home and abroad. Many universities and companies have researched deeply in table tennis robot and developed a variety of table tennis robots in different structure and type since 1980. The initial table tennis robots had no other than the skill of serving diverse balls, then the robotic arm were developed to hit back the coming balls. In 1983, John Billingsley[1] from Portsmouth Polytechnic University of Britain constituted regulations about table tennis robots. Russel L.Andersson[2], Fumio Miyazaki[3] and so on developed table tennis robots one by one. Jianchang Yuan[4] from Xi'an Polytechnic University, De Xu[5] from Beijing Research Institute of Automation and Wei Wei[6] from Zhejiang University have also researched on table tennis robot. The detailed presentation can be checked in reference [7].
Table tennis robot consists of mechanical system, vision system and control system. As a hand-eye coordinating system, the three subsystems must coordinate with each other. Mechanical system, similar to human arm, performs the function of hitting table tennis directly. Vision system, similar to human eye, monitors the movement of the table tennis and predicts its moving track. Control system, similar to human brain, controls the robotic arm to swing the racket agilely according to the moving track of the table tennis to get to the planning position and orientation and realize the accurate hit.
A sort of 5-DOF hybrid robotic arm including parallel mechanism and serial mechanism is put forward, which can perform three translational DOFS and two rotational DOFS. The kinematic inverse solution of the hybrid robotic arm is analyzed deeply. The control equations of position and orientation are deduced. The forenamed algorithm is simulated and verified by means of ADAMS software. The algorithm also lays the theoretical foundation for people to control the pose of the robotic arm.
2 SCHEME of TABLE TENNIS ROBOT
The scheme of table tennis robot based on a 5-DOF hybrid robotic arm with flexible binocular vision is shown in Figure 1. The hybrid robotic arm means connecting the serial mechanism onto the parallel platform. It includes three RPRU(Revolute-Prismatic-Revolute-Universal) limbs and a RP(Revolute-Prismatic) limb. The racket is installed at the end of the robotic arm. Three translational pairs in the parallel mechanism and two rotational pairs in the serial mechanism are used as driving axes to achieve 5-axis simultaneous control. The racket can be swung to get to the required position, orientation and velocity. Two 2-DOF cradle heads are installed eudipleurally above the robotic arm and a CCD camera is installed in each cradle head. Each camera can perform 2 rotational DOFS to form agile binocular vision.
Fig.1: Scheme of table tennis robot
The table tennis robot is a hand-eye coordinating system with quick of eye and deft of hand. The robot can swing its racket agilely and flexibly to hit table tennis precisely and avoid the forehand and backhand problems existing in human hitting.
3 DESCRIPTION of THE 5-DOF HYBRID ROBOTIC ARM
The table tennis has the characteristics of fast speed, various falling points, wide range and strong spin and so on. Therefore, the robotic arm must satisfy these requirements in order to be suitable for hitting back table tennis. On the one hand, it is required to be multi-degrees of freedom to realize the various position and orientation and swing the racket to get to the planning point. On the other hand, it is required to have adequate work space to cover more area inside and outside the table and hit back the various coming table tennis. In addition, fast speed and high precision are also required to hit back the table tennis quickly and accurately.
Based on the above analysis, a sort of 3-RPUR+RP 5-DOF hybrid robotic arm is put forward. As shown in Figure 2, the hybrid mechanism consists of the stable platform, the moving platform, the limbs which connect the stable platform with the moving platform, the rotational pair and translational pair in series with the moving platform and the racket installed at the end of robotic arm. Its characteristic lies in: the stable platform and the moving platform are connected with the same
three RPUR (Rotational-Translational-Universal-Rotational) driving limbs. By controlling the motion of P pair of the three RPUR driving limbs, the position and orientation of the moving platform can be changed to realize two-dimension rotation and one-dimension translation. The rotational pair R4 linked with the moving platform makes the swing rod L4 rotate around the central axis of the moving platform. The translational pair P5 on the swing rod L4 makes the racket P move along axial direction of the swing rod.
a) robotic arm model b) the coordinate systems of the robotic arm
Fig.2: 3-RPUR+RP 5-DOF hybrid robotic arm
The two axes of the Universal pair and the axis of the Rotational pair intersect at one point in the 3-RPUR parallel mechanism, which is equal to a sphere pair, namely 3-RPS mechanism. The degrees of freedom of the 3-RPS+RP hybrid robotic arm can be calculated by using the following equation of Kutzbach Grubler:
M == 6×﹙10－11-1﹚＋17 =5
So the degrees of freedom of the 3-RPS+RP hybrid robotic arm are 5.
The hybrid robotic arm combines high rigidity, fast speed, small inertia, small error, high load and simple structure of parallel mechanism with agility and large work space of serial mechanism sufficiently. The inertia and the accumulative error are reduced. The rigidity is enhanced. The kinematic accuracy and the kinematic velocity are improved. The agility of the position and orientation and dynamic properties are improved efficiently. The hybrid robotic arm is able to carry out the planning movement of the racket more quickly, agilely and accurately in terms of different speed, falling points, angles and variety of the coming table tennis.
4 KINEMATIC INVERSE SOLUTION of THE HYBRID ROBOTIC ARM
4.1 Forward and inverse solution of position analysis of RP limb
By using D-H method[8], the coordinate systems are established respectively on rotational pair R4, translational pair P5 and the table tennis racket which are linked in the moving platform. As shown in Figure 2 b): the moving coordinate system {B} is the basic coordinate system {0}, the coordinate system {4} corresponds to R4 , the coordinate system {5} corresponds to P5, the coordinate system {P} corresponds to table tennis racket. Table 1 shows the corresponding D-H parameters. q4 and d5 are variables, a1 ,a2 , d1 and d3 are constants. 90o a1 = - , 90o a2 = , d1 = lB4 , d3 = l5P . lB4 is the distance between the origin of the coordinate system {B} and the origin of the coordinate system {P}.
Tab.1: D-H parameters of RP limb
According to the parameters in Table 1, the transform matrix BPT of the coordinate system {P} relative to the coordinate system {B} is given as follows:
In equation (1), s is sin and c is cos.
From equation (1), the position with respect to the origin of {P} can be represented in {B}:
Equation (2) is the forward solution of the position of RP limb, so the inverse solution is given as follows:
4.2 Inverse solution of the position analysis of 3-RPS limb
As shown in Figure 2, the moving platform of 3-RPS parallel mechanism is a regular triangle S1S2S3 . The moving coordinate system {B} is established on the moving platform. Origin OB is located in the geometric centre of the moving platform. Axis_XB coincides with vector OBS1 . The stable platform is also a regular triangle R1R2R3. The stable coordinate system {A} is established on the stable platform. Origin OA is located in the geometric centre of the stable platform. Axis_XA coincides with vector OAR1 .
The three axes relative to the three rotational pairs R1,R2 and R3 are tangent to the circumcircle of the stable platform. The radius of the circumcircle is A r . The position of the three joints R1, R2 and R3 can be represented in the coordinate system {A} as follows:
The radius of the circumcircle of the regular triangle S1S2S3 is rB . The position of the three joints S1, S2 and S3 can be represented in coordinate system {B} as follows:
Then the transform matrix of {B} relative to {A} can be represented as follows:
In equation (4), the position of origin OB in {A}.
represents the rotation matrix and (a,b ,g ) is Euler angle of orientation of {B} relative to {A}.
The position of joints Si (i =1, 2,3) in the coordinate system {A} can be represented as follows:
Then, the length vector of the driving axes in {A} can be given:
From equation (6), the length of all driving axes can be calculated:
Three constraint equations in 3-RPS parallel mechanism are given[9]:
From equation (8), the following equation can be given:
From equation (9), in the six parameters of the position and orientation of the moving platform, , and are independent. a , and can be solved by the three constraint equations above. Then the length of the three driving axes can be represented as follows:
4.3 Pose inverse solution of hybrid robotic arm
From equation (1) and (4), the transform matrix of {P} relative to {A} is given as follows:
In equation (11), is the position with respect to the origin OP of {P} in {A}. It can be represented as follows:
is the rotation matrix represented in Euler angle and is the Euler angle of {P} relative to {A}.is the Euler angle of {P} relative to {A}.
4.4 Pose control equation of 3-RPS+RP hybrid robotic arm
The position and orientation of the racket can be described by represents the position of the central point of the racket in {A} and represents the direction angle with respect to the orientation of the racket. is described by using the direction cosine between axis_Z of {P} and the three coordinate axes in {A} respectively.
The transform equations from , parameters of the planning pose of the racket to , parameters of the pose of the moving platform and (d5,q4 ) , D-H parameters of RP limb are described by equation (12).
By using equation (12), the data of position and orientation of the moving platform can be calculated according to the input data with respect to the position and orientation of the racket. Then, making use of inverse position equation (10), the planning pose of the racket in work space can be translated into lengths of the driving axes and the angles about the rotation axes in joint space. The motion control of the robotic arm can be implemented by means of equation (13). The robotic arm can be controlled to swing its rocket to get to the planning pose quickly so as to hit back the coming ball accurately.
5 SIMULATION
Fig.3: The flow chart of simulation
The 3-RPUR+RP 5-DOF hybrid robotic arm can be constructed in Solidworks. Then the entity model is imported into ADAMS through a sort of data conversion format named parasolid. The assemblage and constraint relationship existing in Solidworks become unvalid when they are in ADAMS. Therefore, it is necessary to define constraints for all the parts in the model. First, set up the working condition. Then define kinematic pairs constraints, including fixed pair, translational pair and rotational pair. The motion relation can be constructed through loading driving motion on kinematic pairs, including four translational pairs and one rotational pair [10-11]. Finally, the simulation can be gained through importing driving data. The flow chart is shown as Figure3.
The structural parameters of the robotic arm are as follows:
rA=300mm, rB=220mm, lB4=132m, l5P=40mm
The initial parameters of the driving axes are as follows:
l1=l2=l3=679.72mm, q4 =0 , d5=500mm
The planning motion of the robotic arm is as follows: (500, 0, 847, 90 ° , 90° , 0 ° )→(0,400,1147,110 °,108.75 °,27.99 o )→(-200, 0, 847, 90° , 90 ° , 0 °)→(0, -400, 1147,70 °, 71.253° , 27.99 ° )→(500, 0, 847, 90° , 90 ° , 0° )
Tab.2: Pose data of the rocket
Tab.3: Driving data of the robotic arm
Fig.4: Graph of motion simulation of the hybrid robotic arm
By using pose control equation of the robotic arm, the planning position and orientation data (refer to table 2) of the rocket can be calculated to acquire the control data (refer to table 3) Then the control data can be loaded in each driving axis in ADMAS software. And the motion track of the rocket can be produced. As shown in Figure 4,The simulation result coincides with the planning track.
6 CONCLUSION
3-RPUR+RP 5-DOF hybrid robotic arm can perform three translational DOFS and two rotational DOFS. By using XYZ Euler angle to represent the position and orientation of the rocket, the kinematic inverse solution of the robotic is solved concisely. The pose control equation about the rocket is established. By using ADAMS software, the simulation is perform