橢圓軌跡直擺凸輪組合機(jī)構(gòu)的設(shè)計(jì)【說(shuō)明書(shū)+CAD+PROE+仿真】

橢圓軌跡直擺凸輪組合機(jī)構(gòu)的設(shè)計(jì)【說(shuō)明書(shū)+CAD+PROE+仿真】,說(shuō)明書(shū)+CAD+PROE+仿真,橢圓軌跡直擺凸輪組合機(jī)構(gòu)的設(shè)計(jì)【說(shuō)明書(shū)+CAD+PROE+仿真】,橢圓,軌跡,凸輪,組合,機(jī)構(gòu),設(shè)計(jì),說(shuō)明書(shū),仿單,cad,proe,仿真 歡迎各位老師指導(dǎo)歡迎各位老師指導(dǎo)橢圓軌跡直擺凸輪組合機(jī)構(gòu)的設(shè)計(jì)專(zhuān)專(zhuān) 業(yè)：機(jī)械設(shè)計(jì)制造及其自動(dòng)化業(yè)：機(jī)械設(shè)計(jì)制造及其自動(dòng)化班班 級(jí)：機(jī)制級(jí)：機(jī)制051班班導(dǎo)導(dǎo) 師：姚明印師：姚明印學(xué)學(xué) 生：胡鵬法生：胡鵬法目目 錄錄第一章第一章 緒論緒論第二章第二章 機(jī)構(gòu)的理論設(shè)計(jì)機(jī)構(gòu)的理論設(shè)計(jì)第三章第三章 機(jī)構(gòu)的計(jì)算機(jī)輔助設(shè)計(jì)機(jī)構(gòu)的計(jì)算機(jī)輔助設(shè)計(jì)第四章第四章 機(jī)構(gòu)的運(yùn)動(dòng)仿真機(jī)構(gòu)的運(yùn)動(dòng)仿真致致 謝謝第一章第一章 緒論緒論在現(xiàn)代生產(chǎn)領(lǐng)域中，很多都要求其機(jī)器設(shè)備在現(xiàn)代生產(chǎn)領(lǐng)域中，很多都要求其機(jī)器設(shè)備能實(shí)現(xiàn)某種預(yù)期的軌跡，比如在饅頭生產(chǎn)線能實(shí)現(xiàn)某種預(yù)期的軌跡，比如在饅頭生產(chǎn)線上，其饅頭的堆放要走如下圖所示的軌跡上，其饅頭的堆放要走如下圖所示的軌跡 這樣就需要該堆放機(jī)構(gòu)能實(shí)現(xiàn)這樣這樣就需要該堆放機(jī)構(gòu)能實(shí)現(xiàn)這樣的軌跡。由此看來(lái)，對(duì)這類(lèi)機(jī)構(gòu)的的軌跡。由此看來(lái)，對(duì)這類(lèi)機(jī)構(gòu)的研究是有現(xiàn)實(shí)意義的。研究是有現(xiàn)實(shí)意義的。本課題的設(shè)計(jì)內(nèi)容本課題的設(shè)計(jì)內(nèi)容 1.機(jī)構(gòu)的理論設(shè)計(jì)機(jī)構(gòu)的理論設(shè)計(jì) 2.機(jī)構(gòu)的實(shí)體設(shè)計(jì)機(jī)構(gòu)的實(shí)體設(shè)計(jì) 3.機(jī)構(gòu)實(shí)體的運(yùn)動(dòng)仿真機(jī)構(gòu)實(shí)體的運(yùn)動(dòng)仿真第二章第二章 機(jī)構(gòu)的理論設(shè)計(jì)機(jī)構(gòu)的理論設(shè)計(jì)已知參數(shù)已知參數(shù) 直動(dòng)從動(dòng)件凸輪的基圓半徑為直動(dòng)從動(dòng)件凸輪的基圓半徑為60；直動(dòng)從動(dòng)件凸輪的偏心距為直動(dòng)從動(dòng)件凸輪的偏心距為20；擺動(dòng)從動(dòng)件凸輪的基圓半徑為擺動(dòng)從動(dòng)件凸輪的基圓半徑為50；擺桿長(zhǎng)度為擺桿長(zhǎng)度為400；擺桿上段長(zhǎng)度為擺桿上段長(zhǎng)度為200。頂桿位移與擺桿轉(zhuǎn)角的變化規(guī)律頂桿位移與擺桿轉(zhuǎn)角的變化規(guī)律 如圖所示，依據(jù)機(jī)構(gòu)運(yùn)動(dòng)時(shí)所滿足的幾何如圖所示，依據(jù)機(jī)構(gòu)運(yùn)動(dòng)時(shí)所滿足的幾何關(guān)系，可得出頂桿位移與擺桿轉(zhuǎn)角的變化關(guān)系，可得出頂桿位移與擺桿轉(zhuǎn)角的變化規(guī)律：規(guī)律：擺桿轉(zhuǎn)角擺桿轉(zhuǎn)角 其中其中 i=0,1,2，n 頂桿位移頂桿位移 其中其中 i=0,1,2，n直動(dòng)從動(dòng)件凸輪理論廓線的參數(shù)計(jì)算公式直動(dòng)從動(dòng)件凸輪理論廓線的參數(shù)計(jì)算公式 參看下圖計(jì)算得直動(dòng)從動(dòng)件凸輪理論廓線的向參看下圖計(jì)算得直動(dòng)從動(dòng)件凸輪理論廓線的向徑以徑以 及向徑極角及向徑極角 的計(jì)算公式如下的計(jì)算公式如下：在整條軌跡在整條軌跡上取上取36個(gè)等個(gè)等分點(diǎn)，也即分點(diǎn)，也即i=36。計(jì)算。計(jì)算得直動(dòng)從動(dòng)得直動(dòng)從動(dòng)件凸輪理論件凸輪理論廓線上廓線上36個(gè)個(gè)離散點(diǎn)的參離散點(diǎn)的參數(shù)如表數(shù)如表 序號(hào)向徑向徑 極角極角 序號(hào)向徑向徑 極角極角011902200321042205230624072508260927102811291230133114321533163417351836直動(dòng)從動(dòng)件的理論廓線直動(dòng)從動(dòng)件的理論廓線 在在CAD下，用描點(diǎn)法可以畫(huà)出直動(dòng)從動(dòng)件下，用描點(diǎn)法可以畫(huà)出直動(dòng)從動(dòng)件的理論廓線，如圖：的理論廓線，如圖：同理，可以得到擺動(dòng)從動(dòng)件的理論廓線，同理，可以得到擺動(dòng)從動(dòng)件的理論廓線，如圖所示：如圖所示：第三章第三章 機(jī)構(gòu)的計(jì)算機(jī)輔助設(shè)計(jì)機(jī)構(gòu)的計(jì)算機(jī)輔助設(shè)計(jì)機(jī)構(gòu)實(shí)體的裝配圖機(jī)構(gòu)實(shí)體的裝配圖 裝配體的爆炸圖裝配體的爆炸圖凸輪的實(shí)體設(shè)計(jì)凸輪的實(shí)體設(shè)計(jì) 選擇凸輪的從動(dòng)件為滾子從動(dòng)件。嚴(yán)格按選擇凸輪的從動(dòng)件為滾子從動(dòng)件。嚴(yán)格按照滾子半徑的選擇原則，選取滾子的半徑照滾子半徑的選擇原則，選取滾子的半徑為為10。則在。則在CAD下可以畫(huà)出凸輪的實(shí)際廓下可以畫(huà)出凸輪的實(shí)際廓線。線。直動(dòng)從動(dòng)件凸輪的實(shí)際廓線如下圖：直動(dòng)從動(dòng)件凸輪的實(shí)際廓線如下圖：將在將在CAD下得到的凸輪理論廓線導(dǎo)入下得到的凸輪理論廓線導(dǎo)入Proe中，生成凸輪的實(shí)體。設(shè)計(jì)結(jié)果如圖中，生成凸輪的實(shí)體。設(shè)計(jì)結(jié)果如圖：擺動(dòng)從動(dòng)件凸輪的實(shí)體設(shè)計(jì)擺動(dòng)從動(dòng)件凸輪的實(shí)體設(shè)計(jì) 同理，可以生成擺動(dòng)從動(dòng)件凸輪的實(shí)體。同理，可以生成擺動(dòng)從動(dòng)件凸輪的實(shí)體。結(jié)構(gòu)如圖：結(jié)構(gòu)如圖：頂桿的實(shí)體設(shè)計(jì)頂桿的實(shí)體設(shè)計(jì) 頂桿需在滑道上滑動(dòng)，因此其結(jié)構(gòu)形式應(yīng)頂桿需在滑道上滑動(dòng)，因此其結(jié)構(gòu)形式應(yīng)與滑到對(duì)應(yīng)，并且其上還需安裝銷(xiāo)軸和滾與滑到對(duì)應(yīng)，并且其上還需安裝銷(xiāo)軸和滾子。設(shè)計(jì)結(jié)果如圖：子。設(shè)計(jì)結(jié)果如圖：擺桿的設(shè)計(jì)擺桿的設(shè)計(jì) 當(dāng)擺桿采用直桿的形式時(shí)，機(jī)構(gòu)運(yùn)動(dòng)將發(fā)當(dāng)擺桿采用直桿的形式時(shí)，機(jī)構(gòu)運(yùn)動(dòng)將發(fā)生擺桿與擺動(dòng)從動(dòng)件凸輪的運(yùn)動(dòng)干涉。為生擺桿與擺動(dòng)從動(dòng)件凸輪的運(yùn)動(dòng)干涉。為了避免該情況，將擺動(dòng)桿設(shè)計(jì)成非直桿形了避免該情況，將擺動(dòng)桿設(shè)計(jì)成非直桿形式，其實(shí)體如圖：式，其實(shí)體如圖：座板的實(shí)體設(shè)計(jì)座板的實(shí)體設(shè)計(jì) 座板是支撐整個(gè)機(jī)構(gòu)的零件，其上需要安座板是支撐整個(gè)機(jī)構(gòu)的零件，其上需要安裝軸承和后支架，它們之間的聯(lián)接均為螺裝軸承和后支架，它們之間的聯(lián)接均為螺栓聯(lián)接。設(shè)計(jì)結(jié)果如圖：栓聯(lián)接。設(shè)計(jì)結(jié)果如圖：后支撐板的實(shí)體設(shè)計(jì)后支撐板的實(shí)體設(shè)計(jì) 后支撐板是安裝導(dǎo)軌的零件，同時(shí)還需用后支撐板是安裝導(dǎo)軌的零件，同時(shí)還需用螺栓固定于座板上。設(shè)計(jì)結(jié)果如圖：螺栓固定于座板上。設(shè)計(jì)結(jié)果如圖：滑道的設(shè)計(jì)滑道的設(shè)計(jì) 導(dǎo)軌采用簡(jiǎn)單的滑道截面為方形的形式，導(dǎo)軌采用簡(jiǎn)單的滑道截面為方形的形式，并且由兩滑道附件和后支撐板組合而成。并且由兩滑道附件和后支撐板組合而成?；栏郊脑O(shè)計(jì)結(jié)果如圖滑道附件的設(shè)計(jì)結(jié)果如圖：第三章第三章 機(jī)構(gòu)實(shí)體的運(yùn)動(dòng)仿真機(jī)構(gòu)實(shí)體的運(yùn)動(dòng)仿真為了驗(yàn)證設(shè)計(jì)的正確性，對(duì)該機(jī)構(gòu)的實(shí)體進(jìn)行運(yùn)動(dòng)仿真。非常感謝各位老師和同學(xué)在百忙之中聽(tīng)取我的答辯橢圓軌跡直擺凸輪組合機(jī)構(gòu)的設(shè)計(jì) 1 緒 論 本課題要求設(shè)計(jì)一直擺凸輪組合機(jī)構(gòu)，使給定在擺桿上的某個(gè)點(diǎn)實(shí)現(xiàn)預(yù)期橢圓軌跡，并在此基礎(chǔ)上進(jìn)一步設(shè)計(jì)出整個(gè)機(jī)構(gòu)所需的所有零件的實(shí)體模型，然后將其裝配組合，并進(jìn)行運(yùn)動(dòng)仿真。機(jī)構(gòu)示意圖如圖1－1： 圖1－1 直擺組合凸輪機(jī)構(gòu)示意圖 眾所周知，人類(lèi)創(chuàng)造發(fā)明機(jī)構(gòu)和機(jī)器的歷史十分悠久，并且隨著人們對(duì)不同機(jī)器和機(jī)構(gòu)的需求的日益增多，對(duì)它們的研究也在不斷的深入，特別是在近代，科學(xué)技術(shù)的飛速發(fā)展使得機(jī)構(gòu)和機(jī)器的種類(lèi)和它們所能完成的功能得到了極大的豐富。也正因?yàn)槿绱?，機(jī)構(gòu)和機(jī)器理論已經(jīng)發(fā)展成為一門(mén)重要的技術(shù)基礎(chǔ)學(xué)科。在這一學(xué)科中，進(jìn)一步完善傳統(tǒng)典型機(jī)構(gòu)的分析與綜合方法，例如實(shí)現(xiàn)預(yù)期軌跡的機(jī)構(gòu)的類(lèi)型和設(shè)計(jì)方法的創(chuàng)新，仍是值得研究的課題。在這一方面，對(duì)本課題的研究就有著重要的意義。 現(xiàn)代化的生產(chǎn)，許多都要求設(shè)備能實(shí)現(xiàn)某種預(yù)期軌跡來(lái)更好的生產(chǎn)，比如在食品加工機(jī)械中的饅頭自動(dòng)化生產(chǎn)線上，其饅頭堆放機(jī)構(gòu)就是一個(gè)利用組合機(jī)構(gòu)來(lái)完成預(yù)期的饅頭堆放軌跡的。在實(shí)現(xiàn)預(yù)期軌跡的組合機(jī)構(gòu)中，直擺凸輪組合機(jī)構(gòu)是一種非常實(shí)用的機(jī)構(gòu)，通過(guò)不同輪廓的直動(dòng)凸輪和擺動(dòng)通論驅(qū)動(dòng)連桿配合運(yùn)動(dòng)，既能實(shí)現(xiàn)連續(xù)性預(yù)期軌跡，如星形線、內(nèi)擺線、旋輪線、漸開(kāi)線、正態(tài)曲線等；又能實(shí)現(xiàn)離散化預(yù)期軌跡，如人頭像、金魚(yú)、黑桃、三菱商標(biāo)等。所涉及到的工業(yè)生產(chǎn)：如專(zhuān)用線切割機(jī)床、專(zhuān)用電火花加工機(jī)床、專(zhuān)用焊接焊切機(jī)械手、專(zhuān)用幾何測(cè)量?jī)x器、行程控制機(jī)構(gòu)及各類(lèi)輕工機(jī)械等?？梢詫?shí)現(xiàn)圖案加工、電火花刻線等等。因此，研究本課題不僅有其理論意義，也有著其現(xiàn)實(shí)意義。 該機(jī)構(gòu)是由直動(dòng)從動(dòng)件凸輪機(jī)構(gòu)與擺動(dòng)從動(dòng)件凸輪機(jī)構(gòu)組成的聯(lián)動(dòng)凸輪機(jī)構(gòu)（圖－1），該機(jī)構(gòu)具有3個(gè)活動(dòng)構(gòu)件（n=3），3個(gè)低副(Pl=3)，2個(gè)高副(Ph=2)，由平面機(jī)構(gòu)自由度計(jì)算公式[1] 故其機(jī)構(gòu)自由度η為：該機(jī)構(gòu)原動(dòng)件數(shù)目為1,與其機(jī)構(gòu)自由度相等,故該機(jī)構(gòu)成立。 通過(guò)建立直、擺組合凸輪機(jī)構(gòu)的設(shè)計(jì)公式，從而得出該機(jī)構(gòu)各構(gòu)件位置、大小及形狀尺寸、凸輪實(shí)際廓線、理論廓線。在此基礎(chǔ)上，再合理設(shè)計(jì)出機(jī)構(gòu)所需的每個(gè)零部件的結(jié)構(gòu)，之后將它們裝配組合，并進(jìn)行運(yùn)動(dòng)彷真，驗(yàn)證設(shè)計(jì)的正確性。 此機(jī)構(gòu)的設(shè)計(jì)可以分為如下幾個(gè)部分：直動(dòng)從動(dòng)件凸輪和擺動(dòng)從動(dòng)件凸輪的設(shè)計(jì)，直動(dòng)桿和擺動(dòng)桿的設(shè)計(jì)，直動(dòng)導(dǎo)軌的設(shè)計(jì)，軸系零部件的設(shè)計(jì)和機(jī)架的設(shè)計(jì)。其中最為關(guān)鍵也最為困難的是直動(dòng)從動(dòng)件凸輪和擺動(dòng)從動(dòng)件凸輪的設(shè)計(jì)，而采用何種方法進(jìn)行設(shè)計(jì)又是首先需要考慮的問(wèn)題。因此在設(shè)計(jì)過(guò)程中應(yīng)該先確定所要采用的凸輪設(shè)計(jì)方法。 在以上部分設(shè)計(jì)完成后，機(jī)構(gòu)的運(yùn)動(dòng)仿真，包括機(jī)構(gòu)各個(gè)部件的裝配和裝配后的動(dòng)態(tài)仿真。在這一階段需仔細(xì)計(jì)劃各個(gè)部件的安裝位置和安裝順序，將每一個(gè)部件都正確安裝到位。其中值得注意的是直動(dòng)凸輪與擺動(dòng)凸輪的安裝滯后角，這一角度需嚴(yán)格控制，稍微的誤差可能就直接影響預(yù)期的曲線。 本課題所用到的硬件主要是計(jì)算機(jī)。用到的軟件有：AutoCAD 2004，Proe Wildfire3.0，Word2000，Powerpoint2000。 2 橢圓軌跡直擺組合凸輪機(jī)構(gòu)理論設(shè)計(jì) 由于該組合機(jī)構(gòu)綜合了單一的直動(dòng)凸輪和擺動(dòng)凸輪兩種機(jī)構(gòu)，其運(yùn)動(dòng)的復(fù)雜性，靠單純的傳統(tǒng)的方法求凸輪廓線，非常復(fù)雜，本課題采用一種準(zhǔn)確、快捷，簡(jiǎn)便的離散化方法 [2]。 2.1 直、擺組合凸輪機(jī)構(gòu)設(shè)計(jì)基本思想 圖2-1 直、擺組合凸輪機(jī)構(gòu)參數(shù)的幾何關(guān)系 設(shè) 為預(yù)期曲線上n + 1 個(gè)坐標(biāo)點(diǎn)，它們與下列數(shù)值一一對(duì)應(yīng)[3]，如圖2—1 ——頂桿位移； ——擺桿轉(zhuǎn)角； ——直動(dòng)從動(dòng)件凸輪向徑與極角； ——擺動(dòng)從動(dòng)件凸輪向徑與極角； e ——直動(dòng)凸輪偏心距； a,b——預(yù)期曲線起始點(diǎn)坐標(biāo)； R , R 1 , R 2 ——擺桿長(zhǎng)度，擺桿上端長(zhǎng)度，頂桿長(zhǎng)度。 依據(jù)預(yù)期曲線上的點(diǎn) 與頂桿位移 、擺桿轉(zhuǎn)角之間的幾何關(guān)系，求出它們的變化規(guī)律 ，，再分別設(shè)計(jì)直動(dòng)從動(dòng)件凸輪廓形與擺動(dòng)從動(dòng)件凸輪廓形。 2.2直、擺組合凸輪機(jī)構(gòu)設(shè)計(jì)步驟 2.2.1在預(yù)期曲線L上求取坐標(biāo)點(diǎn) 預(yù)期曲線可以是由一條或若干條平面曲線組成的封閉曲線,首先寫(xiě)出它的參數(shù)方程表達(dá)式,并且要求參數(shù)方程表示的曲線位于第Ⅰ、第Ⅳ象限,初定其起始點(diǎn)為坐標(biāo)原點(diǎn)。曲線方程為： (2-1) 積分求弧長(zhǎng)，得 (2-2) 其中,t0,tn分別表示曲線的起始參數(shù)與終了參數(shù)。 再按照設(shè)計(jì)要求將曲線分成若干段 ,其中任意一段定一位置，則有，且 , 令k 0 = 0。 下面采用勻速運(yùn)動(dòng)規(guī)律將預(yù)期曲線分段，k i求解公式為： 式中，i=0,1,2……n。 如果將預(yù)期曲線L對(duì)應(yīng)的凸輪轉(zhuǎn)角都分成n等份，使之與 :相對(duì)應(yīng)，那么當(dāng)凸輪軸勻速轉(zhuǎn)動(dòng)時(shí)，通過(guò)組合凸輪機(jī)構(gòu)，將使從動(dòng)點(diǎn)以預(yù)期的勻速運(yùn)動(dòng)規(guī)律沿預(yù)期曲線運(yùn)動(dòng)。 2.2.2機(jī)構(gòu)初始位置參數(shù)確定 參看圖2-2，直、擺組合凸輪機(jī)構(gòu)的結(jié)構(gòu)參數(shù)為：直動(dòng)凸輪基圓半徑，擺動(dòng)凸輪基圓半徑 ，偏心距e以及擺桿長(zhǎng)度R及R1 ，頂桿長(zhǎng)度R2等。由這些機(jī)構(gòu)參數(shù)可得到如下機(jī)構(gòu)初始位置參數(shù)（初始位置）： ① 擺桿與頂桿在初始位置的夾角[4] （2-3） 式中，， ② 從動(dòng)點(diǎn)起始位置坐標(biāo) （2-4） （2-5） 圖2-2 直、擺組合凸輪機(jī)構(gòu)初始位置參數(shù) 考慮機(jī)構(gòu)的初始位置,應(yīng)該將上節(jié)求到的坐標(biāo)點(diǎn)平移到從(a,b)為初始點(diǎn)的位置上來(lái)，于是有： . 平移后的坐標(biāo)點(diǎn)仍記作。 2.2.3 確定頂桿位移與擺桿轉(zhuǎn)角的變化規(guī)律 分析圖2-1,可以得到以下關(guān)系式： ①擺桿轉(zhuǎn)角 ， i=0,1,2，…，n （2-6） ②頂桿位移 ， i=0,1,2，…，n。 （2-7） 從而得到與對(duì)應(yīng)的和。式中,i=0，1，2，…，n。 2.2.4 凸輪廓形設(shè)計(jì) （1）擺動(dòng)從動(dòng)件凸輪輪廓設(shè)計(jì) 首先，設(shè)計(jì)擺動(dòng)從動(dòng)件凸輪廓形，參見(jiàn)圖2-3，分析△AOB，應(yīng)用余弦定理，則擺動(dòng)從動(dòng)件凸輪理論廓線上任意一點(diǎn)的向徑： 圖2-3 確定擺動(dòng)從動(dòng)件凸輪的向徑及向徑極角 . （2-8） 式中， ； . 其向徑極角 . （2-9） 式中，為凸輪累加轉(zhuǎn)角： 以上各式中，i=0，1，2，…，n，由此可以得到擺動(dòng)從動(dòng)件凸輪的向徑與極角。取直、擺組合凸輪機(jī)構(gòu)的結(jié)構(gòu)參數(shù)為：直動(dòng)凸輪基圓半徑=60，擺動(dòng)凸輪基圓半徑=50 ，偏心距e=20，擺桿長(zhǎng)度R=400及R1 =200。 其計(jì)算結(jié)果如表2—1： 表2—1 擺動(dòng)凸輪輪廓參數(shù) 序號(hào) 擺動(dòng)凸輪向徑 擺動(dòng)凸輪極角 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 94.3 -77.75 89.97 -67.16 85.6 -56.49 81.3 -45.76 77.19 -34.98 73.35 -24.18 69.87 -13.37 66.83 -2.59 64.28 8.13 62.28 18.73 60.88 29.18 60.13 39.43 60.05 49.46 60.67 59.25 62 68.82 64.06 78.19 70.19 96.55 74.15 105.65 78.55 114.75 83.28 123.9 88.15 133.11 93 142.42 97.65 151.82 101.9 161.32 105.6 170.92 108.6 -179.39 110.8 -169.6 112.13 -159.73 112.54 -139.72 112.05 -139.72 110.69 -129.59 108.55 -119.38 105.71 -109.09 102.3 -98.73 100.45 -91.23 98.45 -88.28 94.3 -77.75 采用描點(diǎn)法可得其理論輪廓如圖2-4： 圖2-4 擺動(dòng)凸輪理論輪廓 凸輪從動(dòng)件采用滾子從動(dòng)件，滾子半徑的選擇原理參見(jiàn)[1]。選取滾子半徑為5，則可得凸輪實(shí)際廓線如圖2-5： 圖2-5 擺動(dòng)凸輪實(shí)際輪廓 （2）直動(dòng)從動(dòng)件凸輪輪廓設(shè)計(jì) 設(shè)計(jì)直動(dòng)從動(dòng)件圖輪廓形，參見(jiàn)圖2-6，直動(dòng)從動(dòng)件凸輪理論廓線上任意一點(diǎn) 圖2-6 確定直動(dòng)從動(dòng)件凸輪的向徑及向徑極角 B的向徑： （2-10） 式中： 表示 中的最小值 其向徑極角： （2-11） 以上各式中，i=0，1，2，…，n，由此可以得到直動(dòng)從動(dòng)件凸輪的向徑與極角。 取直、擺組合凸輪機(jī)構(gòu)的結(jié)構(gòu)參數(shù)為：直動(dòng)凸輪基圓半徑=60，擺動(dòng)凸輪基圓半徑=50 ，偏心距e=20，擺桿長(zhǎng)度R=400及R1 =200。 其計(jì)算結(jié)果如表2—1： 表2—2 直動(dòng)凸輪輪廓參數(shù) 序號(hào) 直動(dòng)凸輪向徑 直動(dòng)凸輪極角 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 50 0 50.3 15.01 51.56 29.81 53.7 44.05 56.54 57.53 59.9 70.18 63.58 82.04 67.41 93.16 71.23 103.64 74.92 113.57 78.36 123.03 81.47 132.1 84.18 140.82 86.43 149.23 88.19 157.37 89.45 165.27 90.21 172.95 90.49 180.43 90.31 187.74 89.72 194.93 88.75 202.02 87.43 209.07 85.81 216.14 83.91 223.3 81.73 230.6 79.29 238.15 76.61 246 73.69 254.26 70.56 263 67.27 272.32 63.9 282.32 60.56 293.11 57.79 304.77 55.02 317.35 52.25 330.86 51.13 345.16 50 360 采用描點(diǎn)法得到直動(dòng)從動(dòng)件凸輪的理論輪廓如圖2-7： 圖2-7 直動(dòng)凸輪輪廓圖 與直動(dòng)從動(dòng)件凸輪的滾子半徑選擇原理一樣，選取滾子半徑為5，則凸輪實(shí)際輪廓如圖 2.2.5計(jì)算直動(dòng)從動(dòng)件凸輪的頂桿長(zhǎng)度及安裝滯后角 圖2-8表示的是直、擺組合凸輪機(jī)構(gòu)的初始位置（初始位置），其直動(dòng)從動(dòng)件凸輪機(jī)構(gòu)的頂桿長(zhǎng)度： . （2-12） 式中，為機(jī)構(gòu)初始位置時(shí)，直動(dòng)從動(dòng)件凸輪B點(diǎn)的向徑，其極角記為。 代入數(shù)據(jù)計(jì)算得：R2=107.58 圖2-8 頂桿位移與擺桿轉(zhuǎn)角的變化 如圖2-8所示，定義機(jī)構(gòu)初始位置時(shí)，擺動(dòng)從動(dòng)件凸輪向徑（即其基圓半徑）與使頂桿處于最低位置時(shí)直動(dòng)從動(dòng)件凸輪向徑（即其基圓半徑）之間的夾角為安裝滯后角,則安裝滯后角： 。 （2-13） 代入各數(shù)據(jù)計(jì)算得：=78.45 2.3直動(dòng)凸輪和擺動(dòng)凸輪的厚度 確定直動(dòng)從動(dòng)件凸輪和擺動(dòng)從動(dòng)件凸輪的厚度為20mm.。 3 機(jī)構(gòu)的計(jì)算機(jī)輔助設(shè)計(jì) 3.1 直動(dòng)桿和擺動(dòng)桿的設(shè)計(jì) 由于該機(jī)構(gòu)所傳遞的功率比較小，因此各個(gè)零件的設(shè)計(jì)所考慮的重點(diǎn)不在于零件的力學(xué)性能，而在于設(shè)計(jì)的簡(jiǎn)易度，零件的外形，零件的加工工藝性能，和整體機(jī)構(gòu)的緊湊程度等方面，即機(jī)構(gòu)的結(jié)構(gòu)設(shè)計(jì)。 3.1.1直動(dòng)桿部分的設(shè)計(jì) 直動(dòng)桿的兩端分別聯(lián)接擺動(dòng)桿和直動(dòng)凸輪，其中和擺動(dòng)桿的聯(lián)接運(yùn)動(dòng)副為轉(zhuǎn)動(dòng)副，與凸輪的聯(lián)接為通過(guò)滾子的高副。由2的設(shè)計(jì)計(jì)算結(jié)果：直動(dòng)桿的理論長(zhǎng)度為107.58mm，也即直動(dòng)桿與擺動(dòng)桿的聯(lián)接轉(zhuǎn)動(dòng)副的中心軸和滾子的中心軸的距離為107.58mm，這一長(zhǎng)度必須嚴(yán)格控制，否則將會(huì)直接影響到指定點(diǎn)的運(yùn)動(dòng)軌跡。 綜合考慮桿的理論長(zhǎng)度，與其他零部件的聯(lián)接形式，自身的加工工藝性能，零件的重量和機(jī)構(gòu)的緊湊性后，其設(shè)計(jì)結(jié)果如圖3—1： 圖3—1 直動(dòng)桿主要尺寸 其詳細(xì)尺寸見(jiàn)所附圖紙。 其Proe效果圖如圖3—2： 圖3—2 頂桿效果圖 3.1.2 擺動(dòng)桿的設(shè)計(jì) 擺動(dòng)桿的兩端分別聯(lián)接直動(dòng)動(dòng)桿和擺動(dòng)凸輪，其中和直動(dòng)桿的聯(lián)接運(yùn)動(dòng)副為轉(zhuǎn)動(dòng)副，與凸輪的聯(lián)接為通過(guò)滾子的高副。由第2章的設(shè)計(jì)計(jì)算結(jié)果：擺動(dòng)桿的理論長(zhǎng)度為400mm，擺動(dòng)桿上段長(zhǎng)度為200mm，也即直動(dòng)桿與擺動(dòng)桿的聯(lián)接轉(zhuǎn)動(dòng)副的中心軸和滾子的中心軸的距離為200mm，直動(dòng)桿與擺動(dòng)桿的聯(lián)接轉(zhuǎn)動(dòng)副的中心軸與指定點(diǎn)間的距離為400mm。這幾個(gè)尺寸必須嚴(yán)格控制，否則將會(huì)直接影響到指定點(diǎn)的運(yùn)動(dòng)軌跡。 綜合考慮桿的理論長(zhǎng)度，與其他零部件的聯(lián)接形式，自身的加工工藝性能，零件的重量和機(jī)構(gòu)的緊湊性后，其設(shè)計(jì)結(jié)果如圖3—3： 圖3—3 擺動(dòng)桿的主要尺寸 其詳細(xì)尺寸見(jiàn)所附圖紙。 其Proe實(shí)體效果圖如圖3—4： 圖3—4 擺動(dòng)桿的效果圖 3.2 直動(dòng)導(dǎo)軌的設(shè)計(jì) 導(dǎo)軌的形式多種多樣，根據(jù)以上直動(dòng)桿的設(shè)計(jì)結(jié)果，選用橫截面為長(zhǎng)方形的組合導(dǎo)軌，組合形式為兩個(gè)前扣（導(dǎo)軌附件）和后支撐板。 由第2章的計(jì)算可得直動(dòng)凸輪從動(dòng)件的行程為小于55。綜合其行程以及直動(dòng)桿滑塊部分的高度，選取導(dǎo)軌長(zhǎng)度為80。設(shè)計(jì)結(jié)果如圖3-5： 圖3-5 導(dǎo)軌的主要尺寸 其詳細(xì)的結(jié)構(gòu)尺寸見(jiàn)所附圖紙。 其Proe實(shí)體效果圖如圖3-6： 圖3-6 導(dǎo)軌附件實(shí)體效果圖 3.3 機(jī)架的設(shè)計(jì) 機(jī)架支撐著導(dǎo)軌，軸承座和另外的不是直接連接在機(jī)架上的與機(jī)架具有相對(duì)運(yùn)動(dòng)關(guān)系的零件。對(duì)機(jī)架的設(shè)計(jì)應(yīng)該力求結(jié)構(gòu)簡(jiǎn)單，緊湊，輕便，安裝方便，并且外觀好看。該機(jī)構(gòu)的機(jī)架分為后支撐板和座板兩部分，其設(shè)計(jì)如下： 3.3.1 后支撐板的設(shè)計(jì) 由所選用的導(dǎo)軌形式，選取機(jī)架為方形，板狀。為了減少導(dǎo)軌部分的加工面，在導(dǎo)軌部分設(shè)計(jì)一厚度為3mm的凸臺(tái)，并按照第4章所設(shè)計(jì)的前扣的結(jié)構(gòu)尺寸設(shè)計(jì)導(dǎo)軌部分的結(jié)構(gòu)尺寸。初步定軸的軸線位于后支撐板的中心面，由直動(dòng)從動(dòng)件凸輪的偏心距20mm可得導(dǎo)軌中心面與后支撐板的中心面的距離為20mm。 再綜合考慮支撐板與座板的安裝，凸輪的運(yùn)動(dòng)范圍，其設(shè)計(jì)結(jié)果和主要尺寸如圖3-7： 圖3-7 支撐板主要尺寸 其詳細(xì)的結(jié)構(gòu)尺寸見(jiàn)所附圖紙。 其Proe實(shí)體效果圖如圖3-8: 圖3-8 支撐板實(shí)體效果圖 3.3.2 座板的設(shè)計(jì) 座板連接了后支撐板，并且要安裝上兩個(gè)軸承座，其設(shè)計(jì)的主要部分在于各個(gè)螺栓孔的布置。 根據(jù)后支撐板的結(jié)構(gòu)尺寸和所初步選取的軸承的結(jié)構(gòu)尺寸，其Proe實(shí)體圖如圖3-9： 圖3-9 座板實(shí)體效果圖 3.4 軸系零部件的設(shè)計(jì) 由于該機(jī)構(gòu)所傳遞的功率很小，因此這里的軸的設(shè)計(jì)就是軸的結(jié)構(gòu)設(shè)計(jì)[5]。 3.4.1 擬定軸上零件的裝配方案 需安裝到軸上的零件有：軸承，直動(dòng)從動(dòng)件凸輪和擺動(dòng)從動(dòng)件凸輪。裝配方案采用圖3-10形式： 圖3-10 軸上零件的裝配方案 3.4.2 確定軸的各段直徑和長(zhǎng)度 對(duì)照?qǐng)D3-11，初步選擇滾動(dòng)軸承為內(nèi)徑25的深溝球軸承，其軸向定位為軸肩定位，并取軸肩高度為2.5mm，則4段的直徑為30mm。查表，取軸承厚度為15mm，則取安裝軸承的軸頸部分長(zhǎng)度為18mm（分別為圖中的3，5段） 再計(jì)算6段長(zhǎng)度和直徑，其長(zhǎng)度的計(jì)算照看圖3-11： 其直徑為20mm。 圖3-11 6段長(zhǎng)度計(jì)算示意圖 則計(jì)算得其長(zhǎng)度為48mm。 計(jì)算軸頭部分的長(zhǎng)度和直徑：取直徑為15mm，長(zhǎng)度為兩凸輪厚40mm。 3.4.3 軸上零件的周向定位 兩凸輪與軸的周向定位采用平鍵連接。按照軸頭直徑15mm選平鍵的截面為b*h=5*5。按照軸段的寬度選其l=30mm. 軸的設(shè)計(jì)結(jié)果見(jiàn)所附圖紙。 軸的Proe實(shí)體效果如圖3-12： 圖3-12 軸的實(shí)體效果圖 3.5 標(biāo)準(zhǔn)件 以上的設(shè)計(jì)所需選定的標(biāo)準(zhǔn)件整理如下[6]： 聯(lián)接后支撐板與座板用的螺栓：GB5780-86 M6×30 數(shù)量為4 ； 與其配合的螺母：GB41-86 M6 數(shù)量為4； 組合導(dǎo)軌用的螺栓：GB5780 M6×30 數(shù)量為4； 與其配合的螺母：GB41-86 M6 數(shù)量為4； 聯(lián)接軸承座與座板用的螺栓：GB5780 M12×45 數(shù)量為4； 其配合的螺母：GB41-86 M12 數(shù)量為4； 凸輪的軸向定位用的軸端C型外擋圈：GB894.1-86 15 數(shù)量為1； 擺桿的軸向定位用的軸端擋圈：GB895.1-86 7 數(shù)量為1； 凸輪與軸的周向定位用的A型普通平鍵：截面尺寸 5×5 數(shù)量為1； 安裝在軸上的軸承：GB276-89 6205； 與其配合的軸承座：GB7813-87 SN103。 4 機(jī)構(gòu)實(shí)體的運(yùn)動(dòng)仿真 為了驗(yàn)證本課題所設(shè)計(jì)的直白組合凸輪機(jī)構(gòu)能否使擺桿的末端點(diǎn)實(shí)現(xiàn)預(yù)期的橢圓軌跡，現(xiàn)對(duì)該機(jī)構(gòu)的實(shí)體進(jìn)行運(yùn)動(dòng)仿真。 所有零件的實(shí)體模型構(gòu)建完成后，在Proe下將它們裝配[7]組合，效果如圖4-1： 圖4-1 裝配效果圖 其分解視圖如圖4-2： 圖4-2 分解視圖 參 考 文 獻(xiàn) [1] 鄭文緯，吳克堅(jiān).機(jī)械原理（第七版）.高等教育出版社 [2] 周全申，郭建生.直、擺組合凸輪機(jī)構(gòu)設(shè)計(jì). 1992年， 第9卷，第1期 [3] 鄒慧君，董師予.凸輪機(jī)構(gòu)的現(xiàn)代設(shè)計(jì). 1991年，上海交通大學(xué)出版社 [4] 趙韓.凸輪機(jī)構(gòu)運(yùn)動(dòng)幾何學(xué)的通用解析公式. 1995年第31卷第3期 [5] 楊明忠，朱家誠(chéng).機(jī)械設(shè)計(jì).武漢理工大學(xué)出版社 [6] 蔡春源.新篇機(jī)械設(shè)計(jì)手冊(cè).遼寧科學(xué)技術(shù)出版社 [7] 邵立新，夏素民，孫江宏.Pro/ENGINEER Wildfire3.0標(biāo)準(zhǔn)教程.清華大學(xué)出版社 22 致 謝 經(jīng)過(guò)一個(gè)多月的忙碌和工作，本次畢業(yè)設(shè)計(jì)已經(jīng)接近尾聲，作為一個(gè)本科生的畢業(yè)設(shè)計(jì)，由于經(jīng)驗(yàn)的匱乏，難免有許多考慮不周全的地方，如果沒(méi)有導(dǎo)師的督促指導(dǎo)，以及一起做設(shè)計(jì)的同學(xué)們的支持，想要完成這個(gè)設(shè)計(jì)是難以想象的。 在這里首先要感謝我的導(dǎo)師姚明印。她平日里工作繁多，但在我做畢業(yè)設(shè)計(jì)的每個(gè)階段，從給我們下放課題，設(shè)計(jì)前期的引導(dǎo)，中期檢查，后期詳細(xì)設(shè)計(jì)，裝配草圖等整個(gè)過(guò)程中都給予了我悉心的指導(dǎo)。我的設(shè)計(jì)一直都做得不太順利，但是姚老師仍然細(xì)心地糾正我設(shè)計(jì)中的錯(cuò)誤，不厭其煩的給我悉心教導(dǎo)。她的治學(xué)嚴(yán)謹(jǐn)和科學(xué)研究的精神是我永遠(yuǎn)學(xué)習(xí)的榜樣，并將積極影響我今后的學(xué)習(xí)和工作。 其次要感謝和我一起作畢業(yè)設(shè)計(jì)的董仁財(cái)同學(xué)，他在本次設(shè)計(jì)中勤奮工作，克服了許多困難來(lái)完成此次畢業(yè)設(shè)計(jì)。如果沒(méi)有他的努力工作，此次設(shè)計(jì)的完成將變得非常困難。 然后還要感謝大學(xué)四年來(lái)所有的老師，為我們打下機(jī)械專(zhuān)業(yè)知識(shí)的基礎(chǔ)；同時(shí)還要感謝所有的同學(xué)們，正是因?yàn)橛辛四銈兊闹С趾凸膭?lì)。此次畢業(yè)設(shè)計(jì)才會(huì)順利完成。 最后感謝江西農(nóng)業(yè)大學(xué)四年來(lái)對(duì)我的大力栽培。 and as rol freedom final states of the unactuated DOFs are viewed as an indirect consequence of the profile of the policies. Dynamical systems are used as acceleration policies, providing the actuated system with convenient 1. Introduction are tasks. verse types viors. compe tball. a viors est posture transitions. Moreover, cyclic and composed tasks may be Procedures that synthesize behaviors rely on the availability of ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier Neurocomputin 15630-C02-01). Neurocomputing 72 (2009) 36243630 offering kinematic solutions to the problem of achieving the upc15838 (C. Angulo). specifications in terms of the workspace, which may include a complete path or simply the initial and final positions of the controlled element. These specifications may be derived from a direct human imposition or by path-planning methods, often 0925-2312/$-see front matter meanwhile, the resulting .com/locate/neucom g ARTICLE IN PRESS desired motor behavior. Subsequently, robot control methods are required to compute torque-level actions that drive the mechan- ism according to the given specifications. As an inheritance from experience in the control of robot manipulators, where the element that defines the workspace is the end-effector of a joint chain with a limited motion domain, most of the approaches to robot control consider a workspace reference as input data, computed off-line or generated within the methodology of the relevant approach, starting from pure kinematic information about the solution to the task. When workspace references are followed, the torque control actions are computed with the The key attribute in both approaches was the direct connection between the desired behavior and the torque commands, i.e. the workspace requirements were almost null, leaving the system free to be modulated in order to fulfill the behavior, i.e. lift a defined weight. Both approaches use optimization as the main route; nevertheless, the analytical solution in 19 implies a detailed formulation of the problem and its restrictions, which is perfectly viable for manipulators in structured environments, but this is not the case for AMRs. On the other hand, the solution given in 15 is not analytical but numerical; it searches in the solution space using a learning algorithm, i.e. a numerical optimization of policy D. Pardo et al. / Neurocomputing 72 (2009) 36243630 3625 purpose of achieving desired joint accelerations, which implies compensation of rigid-body dynamics, canceling the effects of gravity, and other nonlinearities. Some of the approaches described in 9 consider the kinematics and rigid-body dynamics of the robot in the genera- tion of the workspace reference, but this is usually done to establish sufficient conditions for the accomplishment of the behavior, rather than to take advantage of the kinematics and rigid-body dynamics. An example of this is evidenced in the gait control system used in the ASIMO humanoid (which has one of the best humanoid gaits developed so far), where control forces are computed to maintain balance stability during gait execution, i.e. the effects of gravity are canceled while suitable accelerations are imposed to accomplish the motion; consequently, the energy consumption is more than 15 times (scaled) the amount required during human gait 18. However, it has been demonstrated that during human gait, not only are the dynamic effects of gravity not always canceled but also they are actually employed 14. It seems that the current strategies to carry out a given motor behavior are well-suited to obtaining a particular solution of the problem. Thus, the space of behavior solutions is narrowed by the approach used rather than by the capacities of the robot. However, some results using new perspectives showevidence of alternative solutions, ones that favor the execution of the motion and expand the capacities of the robot. For instance, results in 19 show that the given factory-maximum payload of an industrial manipulator (a 6-DOF PUMA-762) can be greatly increased by exploring new zones of the solution space with suitable control policies. The approach used was the formulation of a parameter- ized optimal control problem, where body dynamics and time ranges were stated as restrictions. Torque-level actions were found such that the payload lifted by the manipulator was much more (six times) than the load reachable by the default aggregation of path planning, workspace reference and torque control. Surpris- ingly, contrary to standard procedures, the resulting trajectories included singularities, letting the robot rest the payload against its structure on its way to the goal. Along the same lines, a similar result was later presented in 15, where a simple manipulator (2D, 3-DOF) accomplished a weightlifting behavior, avoiding workspace restrictions in the formulation. Besides maximizing the payload lifted, the results included quite different workspace trajectories that accomplished the same behavior. Fig. 1. (a) Initial state of the robot for the standing-up behavior. (b) Desired final configur parameters by means of iterative evaluation of experiences. Nevertheless, its control framework, based on the coordination of lower-level PID controllers, cannot be directly extrapolated to more complex problems. Recently, the attention given to the use of learning as a paradigm to exploit the capacity of robots has being growing. The latest publications on learning of motions by robots 7 revolve around early results on imitation 6, where the initial solution in the workspace is directly guided by a human, and afterwards the robot joints are controlled by parameterized policies that are intended to accomplish the behavior. The type of the functions used as control policies is that of dynamical systems (DSs). The optimal parameters of the policy are found using reinforcement learning (RL) 17 algorithms. Extensive work on RL algorithms suitable for computing robot control policies has been presented in 13. In the methodology presented in this paper, we assume the availability of kinematic information equivalent to the initial and final states of the desired behavior. In contrast to the imitation approach, a reference in the workspace is not specified. Our control framework, based on local control policies at the joint acceleration level, attracts actuated DOFs to the desired final configuration; meanwhile, the resulting final states of the unactuated DOFs are viewed as a consequence of the actuated acceleration profiles. DSs are used as acceleration controllers, providing the systemwith these attractor properties. Additionally, the control policies are parameterized around imposed simple primitives, which may by deformed by means of changes in the parameters in order to obtain complex accelerations. Subsequently, we present an example that provides a qualita- tive description of the type of problems that this paper addresses. The standing-up behavior illustrates those motor behaviors of underactuated systems in which dynamic balance is compro- mised. Fig. 1 shows the initial and final states for this behavior. Note that the behavior is enclosed by a motion where the initial and final velocities are equal to zero. The robot starts in a lying- down posture and should stand up, ending up as shown in Fig.1b. However, gravity and other nonlinearities can influence the behavior in such a way that the robot ends up in a different state (see Fig. 1c). The achievement of desired values for the actuated DOFs is not enough for the desired behavior to be the result. ation. (c) Undesired final configuration, where motor behavior has failed. ARTICLE IN PRESS u Mq q N _ q;q4 D. Pardo et al. / Neurocomputing 72 (2009) 362436303626 where u 2 R n represents the vector of joint torques, M 2 R nC2n is the mass matrix that describes the inertial properties of the robot, and N 2 R n contains nonlinear terms such as Coriolis forces and gravity. The accelerations of the actuated DOFs q b 2 R b and the unactuated DOFs q c 2 R c , may be considered separately without We present the basic definitions and a formal formulation of the problem in Section 2, and then the methodology for computing controllers is described in Section 3. A demonstration motor behavior in a simulated humanoid is synthesized by applying this methodology in Section 4. Finally, the conclusions are gathered together in Section 5. 2. Controlling robot motor behaviors The configuration of a robot specifies the location of all parts of the mechanismwith respect to a reference frame, and is given bya vector of independent position variables (generalized coordinates) q 2 R n ; the number of generalized coordinates defines the number of degrees of freedom of the mechanism. The joint space is the set of possible values for the joints of the robotH2 R b ; brn. The state of the robot is given by the set formed by the positions and velocities of the generalized coordinates, i.e. z q _ qC1382R 2n . An element controlled by a robot also has a configuration that determines its position and orientation. This configuration defines the workspace, denoted by x 2 R m . The geometrical relation that maps the generalized coordinates to the configuration of the controlled element is known as the forward kinematics x f kinem q1 In those cases where the number of DOFs of a robot is greater than that of the controlled element, i.e. n4m, the system is called redundant owing to the existence of more DOFs than are required for the control of the operational space. The relation between the velocities and accelerations in the operational space and those in the configuration space is obtained from the derivative and second derivative of (1), leading to _ x Jq _ q x Jq q _ Jq _ q 2 where Jq is the Jacobian matrix of f kinem q. Now that the elements that describe the motion of a robot have been established, we now focus on the forces that generate the motion. The relation between the control vector (the applied forces) u 2 R b and the resulting accelerations q 2 R n is given by the robot dynamics, and may be written in the constrained form q f 1 q; _ q; tf 2 q; _ q;tu 3 The system is called underactuated when the configuration is not able to command an instantaneous acceleration in an arbitrary direction of q. In the case of an AMR, the assumption that all joints are actuated is coherent, but, because they are not secured to the ground, the AMR can move in directions other than those allowed by its actuators, and therefore it has more DOFs than joints, i.e. rankf 2 C138bon. The nonactuated DOF may be represented as virtual joints that correspond to the motion of a reference frame at the robot base with respect to the inertial frame 16, here this virtual joints are denoted as q c 2 R c , where c n C0 b. Provided that articulated robots are rigid bodies, their dynamics can be described as second-order systems where torque commands (control actions) interact with the rest of the forces acting on the mechanism. A well-known model describing this interaction is given by loss of generality 2. We can expand the dynamics of the system in (4) as u b 0 C20C21 M bb M cb M bc M cc # q b q c # N b N c # 5 where u b 2 R b is the vector of actuation torques, and N b ;N c C138 T gather together the projections of other forces (gravitational forces, Coriolis forces and ground reactions) in the corresponding subspace. The inertia matrix M is divided into four matrices (M bb ;M cb ;M bc ;M cc ), consequently relating causes and effects for actuated and unactuated accelerations. We denote an element of the given robot posture set byP, i.e. P2fsitting; standing; arm raised; etc:g. Let us assume that a certain function P maps robot configurations q 2 R n to the corresponding robot postures, i.e. P : q-P. Let us denote by Q P the set of all configurations that represent the posture P, i.e. Q P fqjPqPg. We define as motor behavior the process of taking the robot from an initial robot posture Q 0 to a final posture Q F within a delimited amount of time 0otot f .Itis assumed that representative elements of the initial and final posture sets are known, i.e. q 0 2Q 0 ;q f 2Q F , including actuated and unactuated DOFs, i.e. q 0 q 0b ;q 0c , q f q fb ;q fc .Itis assumed that all joints are actuated, i.e. q b H. We control actuated joints at the acceleration level using dynamical systems as policies, which allows on-line modification of acceleration commands. Provided the DSs have attractor properties, the policies are designed to attract each joint to its corresponding final state q fb H g , while it follows a smooth trajectory. In order to select an appropriate policy, we first define the following term as a local primitive: p i ta i C0 _ y i tb i y g i C0y i tC138; i 2f1; .; bg6 This represents the state of the velocity and position error of joint i at instant t. The parameters a i ;b i locally encode the properties of the primitive. We define as a local control policy the combination of the local primitives involved q d b;i t X b j1 o ij C1 p j t7 where the o ij are scaling factors for the influence of the primitives j on the desired accelerations q d b;i of individual joints. In this paper, we assume that there exists a lower-level controller that converts the desired accelerations into suitable torque commands (e.g. by the computed torque method). Therefore we assume that the actual accelerations of the actuated DOFs are given by (7). Before continuing, and for clarificationpurposes, we define as a basic policy the case where the scaling factors are given by o i;j 0; iaj 1; i j ( 8 When a basic policy is used, the position and velocity errors of the joints behave as a simple damped harmonic system with an attractor in (y g i ;0). The dynamics of this system are modulated when the nondiagonal weighting factors are not zero. This allows the generation of complex forms in the actual profiles followed by the joints. The whole-body policy, generically denoted by q b hq; _ q, may be explicitly presented, dropping the time dependency notation, in a single expression q b WC1 A _ q b q b # 9 where the matrix A 2 R bC22b is formed by the set of a;b-factors from the local primitives represented in (6), and the matrix W 2 bC2b R gathers together the scaling factors of the policy. ARTICLE IN PRESS (13). Given that the objective of the task is to take the robots states from an initial to a final value, the euclidian distance from by the expected value of a special term G APP W r E b 2 C1 RW r mn j m C1n j C20C21 17 where G APP W r represents the estimated value of the gradient in the point W r , whereas n j is a uniformly random unitary vector in the search space and m is a size factor; the term b 2 is the size of the vector of parameters. It has been demonstrated in 4 that with a suitable selection of these parameters, the gradient approximation leads to convergence at least to a local minimum. In order to solve (17), attempts, also known as roll-outs, to perform the task must take place (with 0ojrJ). The resulting performance of a single roll-out is obtained using an acceleration policy with perturbed parameters, i.e. RW r mn j . In practice, a single roll-out may be enough for an unbiased gradient estimate 4. Here we present the corresponding algorithm A simp ed in ding int wh al confi The humanoid ankle, ever n D. Pardo et al. / Neurocomputing 72 (2009) 36243630 3627 the resulting final states qt f to the desired states q f is a valid representation of the quality of the controllers R 1 Wq f C0 qt f C138 T q f C0 qt f C138 1=2 14 Moreover, as the unactuated DOFs are a virtual concept used for representation purposes, an equivalent measure of the distance function is obtained if the actual and desired values of the workspace vector are measured RWx f C0 xt f C138 T x f C0 xt f C138 1=2 15 This implies that the workspace variables x must be observable. If we aim to minimize this function, the computation of the optimal parameters may be performed by iteratively calculating the steepest descent for the distance function, i.e. W r1 W r gr W RC138 WWr 16 where r denotes the iteration number, and 0ogo1 is a learning rate. The gradient of R with respect to W evaluated in W r is written as r W RC138. However, because an analytical computation of this gradient is not viable in the current framework, we approximate the gradient based on data collected from the robots experience, i.e. we use an instance of the Policy Gradient Reinforcement Learning (PGRL) algorithms (see e.g. 13). The With the intention of analyzing the consequences of the joint policies for the whole-body behavior, using (5) we write the resulting effects on the unactuated DOFs, generically denoted by q c g q b , explicitly as q c M C01 cc C0M bc q b C0N c C13810 Note that the external forces N c interact with the forces related to q b and together modify the motor behavior, i.e. they affect (2) and generate workspace trajectories from the internal dynamics of the robot. The problem of completing a motor behavior can be solved if a function h C3 is found such that q C3 c g3h C3 q; _ q11 accomplishing Z t f 0 Z t f 0 g3h C3 q; _ qdtdt C25 q fc C18 12 The parameters w i;j in (7) must be such that above condition is satisfied. An optimization framework is presented in the following section for the establishment of a numerical methodology to find the optimal parameters of the acceleration policies and therefore solve for the motor behavior. 3. Learning control policies The problem of finding the parameters of the acceleration policies that fulfill the motor behavior has the necessary elements for it to be formulated as a parametric optimization problem. We evaluate the quality of a given set of parameters as the objective function, by measuring the performance of the robot attempting to accomplish the task RWHq b ; _ q b ; _ q c ;q c ; q b 13 where R 2 R is a scalar that represents how the state and actions of the robot proceed for some particular parameters in the policy. Without loss of generality, we may specify a type of relation for stochastic approximation of the gradient proposed in 4 is given Fig. 2. Fujitsu Hoap-2 simulated humanoid used as a test bed for motion behavior synthesis. absolute displacement) is q 2 R , n 7. As mentioned in the humanoid, needed the size of the vector of generalized coordinates to determine its posture and orientation (assuming no How , owing to the dynamic balance condition of the right hip and knee, i.e. b 4. (see Fig. 3 for details). uses four joints during the motion: left ankle, fin gurations of the task are shown in Fig. 2. po ere the humanoid stands using justone leg. The initial and stan -up position, the goal of this task is to reach an equilibrium test le motor behavior, named humanoid equilibrium, was a simulated humanoid (Fujitsu Hoap-2). Starting from a where /C1S denotes the average of the gradient estimators. 4. Results input: m; b;x f 1 repeat 2 select n j 2 R b 2 , Jn j J 1;n j C24U0;1 3 perform roll-out with perturbed parameters W mn j 4 sense controlled element configuration xt f 5 compute the performance RWmn j x f C0 xt f C138 T x f C0 xt f C138 1 2 6 estimate the gradient vector G APP W r b 2 C1 RWrmn j m C1n j DE 7 until gradient estimate G APP W r converged. return: gradient estimate G APP W r (a) Initial posture; and (b) final posture. ARTICLE IN PRESS description of the methodology in Section 2, a goal is manually established for each actuated DOF q f;b . Parameters for the acceleration primitives of joint i, i.e. a i ;b i , are imposed to generate standard exponential trajectories, such as the one shown (dotted line) in Fig. 4. Using the basic policy defined in (8) to generate accelerations, despite the fact that the joints arrive at their desired positions, the unactuated DOFs finish in a completely different state, i.e. the robot falls down. Using th