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關(guān)于裝載適應(yīng)性神經(jīng)模糊系統(tǒng)的有兩足行走的機(jī)器人的零刻點(diǎn)彈道造型
D. Kim, S.-J. Seo and G.-T. Park
摘要：對(duì)于制造機(jī)器人來說兩足動(dòng)物的體系結(jié)構(gòu)高度適用于它們工作在人的環(huán)境里，因?yàn)檫@樣將使機(jī)器人避免障礙變成一項(xiàng)相對(duì)的容易的任務(wù)。 然而，在走動(dòng)的機(jī)制中介入復(fù)雜動(dòng)力學(xué)，這使得制作這樣的機(jī)器人的控制系統(tǒng)變成了一項(xiàng)富有挑戰(zhàn)性的任務(wù)。 機(jī)器人腳部的零刻點(diǎn)(ZMP)彈道是機(jī)器人行走時(shí)的穩(wěn)定性的重要保障。 如果ZMP可以在線測(cè)量那么就將使為機(jī)器人穩(wěn)定行走創(chuàng)造條件成為可能，而且通過運(yùn)用標(biāo)準(zhǔn)的ZMP還可以實(shí)現(xiàn)機(jī)器人的穩(wěn)定控制。ZMP數(shù)據(jù)是通過兩足行走機(jī)器人實(shí)時(shí)測(cè)量出來的，在這之后在通過一套適應(yīng)性神經(jīng)模糊系統(tǒng)（ANFS）將其造型。測(cè)量了在水平基準(zhǔn)面的自然行走和在帶有10度傾斜面的上下行走。通過改變模糊系統(tǒng)的成員作用和結(jié)果輸出部分的規(guī)則，使得ANFS造型的表現(xiàn)最優(yōu)化。由ANFS展示的優(yōu)秀表現(xiàn)意味著它不僅可以運(yùn)用于模型機(jī)器人的運(yùn)動(dòng)，還可以運(yùn)用于控制真正的機(jī)器人。
1 介紹
兩足動(dòng)物結(jié)構(gòu)是對(duì)走動(dòng)的機(jī)器人的最多才多藝的設(shè)定之一。兩足動(dòng)物結(jié)構(gòu)，使機(jī)器人即使在有臺(tái)階或障礙等的環(huán)境里也具備和人幾乎同樣的可支配的機(jī)械裝置。然而，介入的動(dòng)力學(xué)是高度非線性，復(fù)雜和不穩(wěn)定的。因此，它是引入模仿人體行走的最大的困難。模仿人體行走是一個(gè)可觀的研究領(lǐng)域（1）。與產(chǎn)業(yè)機(jī)器人的操作器相比，一個(gè)走動(dòng)的機(jī)器人和地面之間的相互作用是復(fù)雜的。在這種相互作用的控制上零刻點(diǎn)(ZMP) [2]概念被證明是有用的。在ZMP的彈道的幫助下機(jī)器人的腳在步行期間的行動(dòng)是受其穩(wěn)定性信息的誘導(dǎo)的。使用ZMP我們可以整合兩足的機(jī)器人的走的模式并用實(shí)際機(jī)器人示范行走行為。 因此，ZMP標(biāo)準(zhǔn)決定了一個(gè)兩足的機(jī)器人的動(dòng)態(tài)穩(wěn)定性。ZMP代表地面反作用力被采取發(fā)生的點(diǎn)。使用機(jī)器人的模型，ZMP的地點(diǎn)可以被計(jì)算。然而，ZMP價(jià)值指標(biāo)與計(jì)算值價(jià)值指標(biāo)之間有很大偏差也是有可能的，這是因?yàn)槲锢韰⒘康钠钤跀?shù)學(xué)模型和實(shí)際機(jī)器之間。 因此，實(shí)際ZMP是應(yīng)該測(cè)量的，尤其是在它作為穩(wěn)定行走的控制參數(shù)時(shí)。
在這項(xiàng)工作中，實(shí)際ZMP整周期走動(dòng)數(shù)據(jù)是通過一個(gè)實(shí)用兩足走動(dòng)機(jī)器人獲得的。機(jī)器人將在水平基準(zhǔn)面和10度傾斜面上被測(cè)試。一個(gè)適應(yīng)性神經(jīng)模糊系統(tǒng)(ANFS)將被用于控制一個(gè)復(fù)雜的真正的有兩足的走動(dòng)機(jī)器人，以便于ZMP的建模，使其能應(yīng)用與控制中。
2有兩足的走動(dòng)機(jī)器人
2.1有兩足的走動(dòng)機(jī)器人的設(shè)計(jì)
我們?cè)O(shè)計(jì)了并且制造了如圖1所示的有兩足的走動(dòng)機(jī)器人。 機(jī)器人有19聯(lián)接。 機(jī)器人的關(guān)鍵尺寸如圖1所示.高度308mm,總重量約為1700 g，包括個(gè)別電池。 通過使用鋁制結(jié)構(gòu)使機(jī)器人的重量減到了最小。每一個(gè)聯(lián)接都由一個(gè)遙控裝置控制，這個(gè)遙控裝置包括一個(gè)直流馬達(dá)、齒輪和一個(gè)簡(jiǎn)單的控制器。每一臺(tái)遙控裝置都安裝在聯(lián)接結(jié)構(gòu)上。 這個(gè)結(jié)構(gòu)保證機(jī)器人是穩(wěn)定的(即不會(huì)容易跌倒)并且給了機(jī)器人一個(gè)人類的外型。 我們的機(jī)器人系統(tǒng)結(jié)構(gòu)如圖2所示。
機(jī)器人能在平面或小斜度面以1.4s一步，每步48mm的速度行走。機(jī)器人的配置如表一所示。
機(jī)器人的行走動(dòng)作如圖3–6所示。圖3、4分別為機(jī)器人在平面行走時(shí)正視圖和側(cè)視圖。圖5是機(jī)器人沿著傾斜面向下步行的快照，而圖6是機(jī)器人沿著傾斜面向上步行的快照。
行動(dòng)時(shí)聯(lián)接的位置如圖7.所示。 被測(cè)量的ZMP彈道是從這十個(gè)自由(DOF)（如圖7.所示）的數(shù)據(jù)得到的。 二個(gè)自由度被分配到臀部和腳腕，每個(gè)膝蓋分配一個(gè)自由度。 使用這些連接角，一個(gè)循環(huán)走的樣式就會(huì)體現(xiàn)出來。 我們的機(jī)器人能連續(xù)地走，無需跌倒。 在附錄里總結(jié)了我們的機(jī)器人的四步行動(dòng)的連接角。
2.2 ZMP測(cè)量系統(tǒng)
在一個(gè)機(jī)器人腳部的ZMP彈道是步行的穩(wěn)定的一個(gè)重要標(biāo)準(zhǔn)。 在許多研究中， ZMP坐標(biāo)是通過使用機(jī)器人模型和連接處的編碼器傳出的信息用計(jì)算機(jī)計(jì)算出來的。然而，我們使用更直接的方法，使用了機(jī)器人腳部上的傳感器測(cè)量的數(shù)據(jù)。
在機(jī)器人腳部的作用之下地面的反作用力的分布是復(fù)雜的。 然而，如圖8.所示，在腳的腳底的任意點(diǎn)P點(diǎn)的反作用力都可以用力量N和M時(shí)刻之前在任意時(shí)候代的力表。 ZMP是在地面上的腳的壓力的中心，并且關(guān)于這點(diǎn)的地面運(yùn)用的片刻是零。 換句話說，在地面上的點(diǎn)P是慣性和重力在0刻沒有沿軸的組分，平行與地面的點(diǎn)[1， 7]。
圖9說明了使用的傳感器和他們的在機(jī)器人腳的腳底的安置情況。 用于我們的實(shí)驗(yàn)的力量傳感器的種類是Flexi Force A201傳感器[8]。 他們附在構(gòu)成腳的腳底板材的四個(gè)角落。 傳感器信號(hào)由一個(gè)ADC板數(shù)字化，與10ms的采樣時(shí)光。 測(cè)量在實(shí)時(shí)被執(zhí)行。
腳壓力通過求和力量信號(hào)得到。 使用傳感器數(shù)據(jù)計(jì)算實(shí)際ZMP價(jià)值是容易的。 使用(1)，計(jì)算位置腳坐標(biāo)框架的ZMP。
式中每fi在傳感器ri的力量是傳感媒介的傳感器位置。 這些是在圖10.的詳細(xì)說明。 在圖形中， ‘O’是位于低左手角落左腳坐標(biāo)框架的起源。
實(shí)驗(yàn)性結(jié)果如圖11–16所示。 圖11，13和15顯示的是走動(dòng)機(jī)器人在平面和10度傾斜面的四步走動(dòng)的x坐標(biāo)和y坐標(biāo)轉(zhuǎn)化的實(shí)際ZMP位置。圖12，14和16顯示了機(jī)器人運(yùn)用圖11，13和15 的準(zhǔn)確ZMP坐標(biāo)的單步行走情況。如彈道所顯示，ZMP存在于實(shí)線顯示的一個(gè)長(zhǎng)方形領(lǐng)域。因此，ZMP的位置是與機(jī)器人腳部相關(guān)的，因此機(jī)器人是穩(wěn)定的。
3 ZMP彈道建模
在許多科學(xué)問題中，通往他們答案的實(shí)質(zhì)性的一步就是在他們的實(shí)驗(yàn)下建立（數(shù)學(xué)）模型。 建模的重要性體現(xiàn)在是建立被觀察物和可變物之間的經(jīng)驗(yàn)性的關(guān)系。 機(jī)器人步行介入的復(fù)雜動(dòng)力學(xué)使做機(jī)器人控制系統(tǒng)變?yōu)橐豁?xiàng)富挑戰(zhàn)性的任務(wù)。 然而，如果高度非線性和復(fù)雜動(dòng)力學(xué)可以被嚴(yán)密地建模，之后他的模型可以用于機(jī)器人的控制。 另外，建模，甚至能用于機(jī)器智能控制與干擾、噪聲的最小化處理。
3.1 ANFS
模糊建模技術(shù)近些年已經(jīng)成為一項(xiàng)活躍的研究領(lǐng)域，因?yàn)樗趶?fù)雜的，不清楚的，不明確的系統(tǒng)中依然能有出色的表現(xiàn)，而這些時(shí)候常規(guī)的數(shù)學(xué)建模很難給出讓人滿意的答案[9]。就此而論我們打算使用此系統(tǒng)為ZMP彈道建模。
模糊推理系統(tǒng)是以模糊集合理論的概念、模糊的if-then 語句和模糊推理為基礎(chǔ)的一個(gè)普遍的計(jì)算的框架。 我們將使用Sugeno 模糊模型，因?yàn)樵谶@個(gè)系統(tǒng)中，每一個(gè)規(guī)則都有明顯的輸出，總體的輸出將通過加權(quán)平均值給出。這樣就避免了計(jì)算的費(fèi)時(shí)過程。當(dāng)我們考慮在模糊建模時(shí)的模糊規(guī)則時(shí)發(fā)現(xiàn)，結(jié)果部分可以由一個(gè)恒定或一個(gè)線性的多項(xiàng)式表達(dá)。 可以用于模糊系統(tǒng)的多項(xiàng)式的不同的形式如表2.所示。
建模的表現(xiàn)形式取決于用于建模的表示結(jié)果的多項(xiàng)式的種類。 而且，我們可以為模糊規(guī)則的前期部分的模糊嵌入拓展各種各樣單元作用(MFs)，例如三角和高斯。 這些是為算式貢獻(xiàn)可行方法另一個(gè)因素。
多項(xiàng)式的種類如下是
建模系統(tǒng)的結(jié)構(gòu)圖如圖17所示。 提出的方法首先用于建模，而后用于控制一個(gè)實(shí)際的兩足結(jié)構(gòu)行走機(jī)器人。為了得到模糊建模系統(tǒng)的模糊規(guī)則，我們必須記錄一個(gè)非線性系統(tǒng)，這個(gè)系統(tǒng)是通過兩足行走機(jī)器人的十個(gè)輸入變量產(chǎn)生的模糊坐標(biāo)建立的，每個(gè)輸入變量會(huì)產(chǎn)生兩個(gè)模糊坐標(biāo)。
模糊建模的if-then法規(guī)如下：
在式中Ai，Bi，…J1，在規(guī)則的假設(shè)部分中起到語言上判斷的作用，分別結(jié)合輸入變量x1， x2， …， x10。 fj (x1、x2、…， x10); 是常數(shù),或者jth規(guī)則的已知結(jié)果多項(xiàng)式函數(shù)。
如圖18所示， 檢定了MFs的二種類型。 一個(gè)是三角式，另一個(gè)是高斯式。
圖19是適應(yīng)性神經(jīng)模糊系統(tǒng)體系結(jié)構(gòu)，考慮到讓它等同于十輸入模糊模型。在這個(gè)系統(tǒng)中假設(shè)每個(gè)輸入有兩個(gè)模糊值與它對(duì)應(yīng)，如圖18所示。標(biāo)記P的值給出的是所有輸入信號(hào)的乘積，，而這些標(biāo)記的N的值計(jì)算的是某一確定的反作用力與總反作用力之和的比。關(guān)于如何使ANFIS參量變化，我們使用梯度下降算法或一種遞歸最小平方的估計(jì)算法重復(fù)調(diào)整前提和結(jié)果參量。 然而，我們不使用復(fù)雜雜種學(xué)習(xí)算法，反而使用一般最小平方的估計(jì)算法并且只確定結(jié)果多項(xiàng)式函數(shù)的趨勢(shì)。
3.2模仿結(jié)果
使用ANFS，模型大致建成了。 然后準(zhǔn)確性在中間領(lǐng)域誤差（MSE）中被量化了。ANFS系統(tǒng)被申請(qǐng)為兩足走動(dòng)機(jī)器人的ZMP彈道建模，通過運(yùn)用機(jī)器人測(cè)量傳出的數(shù)據(jù)。ANFS的表現(xiàn)取決于MF的機(jī)警性和模糊規(guī)則的結(jié)果輸出。從我們的機(jī)器人輸出的ZMP彈道數(shù)據(jù)（如附錄的圖32－41所示）將用于過程參量。
當(dāng)三角和高斯MFs用于前提部分或用于結(jié)果部分的不變參數(shù)，那么相應(yīng)的MSE值列在表3中。我們?cè)趫D20－25中繪出了我們的結(jié)果。由ANFS產(chǎn)生的ZMP彈道圖如圖20，22，24所示分別為水平基準(zhǔn)面的行走圖，10度傾斜面下行圖和10度傾斜面上行圖。在圖21，23，25，我們可以看見由ANFS產(chǎn)生的相應(yīng)的ZMP彈道。
簡(jiǎn)而言之，兩個(gè)膝蓋的過程參數(shù)可以被忽略。 作為結(jié)果，我們可以減少模糊規(guī)則的維度和從而降低計(jì)算負(fù)擔(dān)。 在這種情況下ANFS的仿真條件和它對(duì)應(yīng)的MSE（均方的誤差）價(jià)值在表4列出。
從給出的模仿結(jié)果的圖和表中，我們能看到從模糊系統(tǒng)得到的ZMP彈道非常類似于我們的行走機(jī)器人所測(cè)量出的實(shí)際ZMP彈道（如圖11－16所示）。ANFS被展示的高準(zhǔn)確性能力，意味著ANFS可以有效地被用于建模和控制一個(gè)實(shí)際的兩足結(jié)構(gòu)走動(dòng)機(jī)器人。
3.3比較
我們現(xiàn)在把ANFS的表現(xiàn)與三種統(tǒng)計(jì)回歸模型的數(shù)學(xué)模型相比較。對(duì)于每個(gè)統(tǒng)計(jì)回歸模型，四個(gè)不同案件類型被修建了。它們?cè)趦煞N輸入下的一般表達(dá)式如下：
這里ci是回歸常數(shù)。
對(duì)應(yīng)的MSE值在表5–7里被給出。它測(cè)量第二類型給x和Y坐標(biāo)的最佳的結(jié)果所有被考慮的走的條件的。產(chǎn)生的ZMP彈道和相應(yīng)的產(chǎn)生它們的第二類型回歸模型如圖26－31所示。我們可以認(rèn)為， ANFS比統(tǒng)計(jì)回歸模型展示了一條相當(dāng)?shù)馗玫腪MP彈道。
4個(gè)結(jié)論
一個(gè)實(shí)用的裝載模糊神經(jīng)系統(tǒng)的零彈道兩足結(jié)構(gòu)走動(dòng)機(jī)器人被展示出來。ZMP彈道是確保機(jī)器人行走穩(wěn)定性的重要保障。但是地面復(fù)雜的反作用力使控制變得困難。
我們?cè)噲D建立過程參數(shù)之間的經(jīng)驗(yàn)的關(guān)系，并且通過將其運(yùn)用于一個(gè)兩足結(jié)構(gòu)走動(dòng)機(jī)器人來解釋經(jīng)驗(yàn)規(guī)律。整個(gè)走動(dòng)過程的ZMP數(shù)據(jù)通過讓一個(gè)實(shí)際兩足結(jié)構(gòu)機(jī)器人在水平基準(zhǔn)面和斜面行走而獲得。ANFS的適用性取決于使用的MF和模糊的規(guī)則的結(jié)果部分。 使用ANFS產(chǎn)生的ZMP彈道嚴(yán)密地匹配于被測(cè)量的ZMP彈道。 然后模仿結(jié)果也表示，使用ANFS引起的ZMP可以改善兩足結(jié)構(gòu)走動(dòng)機(jī)器人的穩(wěn)定性并且ANFS不僅可以有效地用于建模，而且可以用于控制實(shí)際兩足結(jié)構(gòu)走動(dòng)機(jī)器人。如圖32－41所示。
5鳴謝
這項(xiàng)工作由韓國(guó)科學(xué)和工程學(xué)基金會(huì)的基礎(chǔ)性研究計(jì)劃的第R01-2005-000-11-44-0支持。
6參考文獻(xiàn)
1 Erbatur、F.、Okazaki、A.、Obiya、K.、Takahashi、T.和Kawamura， A. ：“一項(xiàng)關(guān)于兩足結(jié)構(gòu)走動(dòng)機(jī)器人的零刻點(diǎn)測(cè)量的研究”。 Proc.7th Int。 關(guān)于先進(jìn)的運(yùn)動(dòng)控制2002年，第 431–436頁。
2 Vukobratovic、M.、Brovac、B.、Surla、D.和Stokic， D. ： ‘運(yùn)動(dòng)機(jī)器人’ (Springer-Veriag1990)
3 Takanishi、A.、Ishida、M.、Yamazaki、Y.和Kato， I. ： “動(dòng)態(tài)走的機(jī)器人WL-10RD的認(rèn)識(shí)”。 Proc。 Int. Conf。 先進(jìn)機(jī)器人， 1985年， 第. 459–466頁。
4 Hirai、K.、Hirose、M.、Haikawa、Y.和Takenaka， T. ： “本田類人機(jī)器人的”。 Proc。國(guó)際電氣電子工程師協(xié)會(huì)。 Conf。 在機(jī)器人技術(shù)和自動(dòng)控制， 1998年，第 1321–1326頁。
5 Park,、J.H.和Rhee， Y.K. ： ‘減少兩足結(jié)構(gòu)走動(dòng)機(jī)器人的干線行動(dòng)的ZMP彈道世代。 Proc。國(guó)際電氣電子工程師協(xié)會(huì)。 Conf。 在智能機(jī)器人和系統(tǒng)， 1998年，第 90–95頁。
6Park、J.H.和Cho， H.C. ： “提高兩足結(jié)構(gòu)走動(dòng)機(jī)器人的基本聯(lián)接的在線ZMP彈道測(cè)量’。 Proc。國(guó)際電氣電子工程師協(xié)會(huì)。 Conf。 在機(jī)器人技術(shù)和自動(dòng)控制， 2000年， 第. 3353–3358頁。
7 Tak、S.、Song、O.和Ko， H.S. ： ‘行動(dòng)平衡過濾’。 Proc。 歐洲制圖，第19卷，第3日2000年。
8 FlexiForce A201傳感器模型， http://www.tekscan.com/ ?exiforce/?exiforce.html， (訪問2004 4月)。
9 Takagi、T.和Sugeno， M. ： ‘神經(jīng)模糊系統(tǒng)和它的建模和控制’， 國(guó)際電氣電子工程師協(xié)會(huì)，傳感器.， 1985年， S-15，第116–132頁。
10 Jang, J.S.: ‘適應(yīng)性網(wǎng)絡(luò)神經(jīng)模糊系統(tǒng): Adaptive-Networks-Based Fuzzy Inference Sys- tem’, 國(guó)際電氣電子工程師協(xié)會(huì)，傳感器., 1993, 23, (3), 第 665–685頁。
7附錄
這個(gè)附錄總結(jié)了我們兩足結(jié)構(gòu)走動(dòng)機(jī)器人的四步行動(dòng)的連接角。 這些連接角如下。
圖1兩足結(jié)構(gòu)走動(dòng)的機(jī)器人(所有尺寸單位為毫米)
圖2機(jī)器人系統(tǒng)的結(jié)構(gòu)圖 圖3機(jī)器人在水平基準(zhǔn)面行走的正視圖
圖4與圖3對(duì)應(yīng)的機(jī)器人的 圖5機(jī)器人沿帶有10度斜度 圖6機(jī)器人沿帶有10度斜
側(cè)視圖 的斜坡向下步行的快照 度的斜坡向上步行
圖7由連接角的表示法構(gòu)成的 圖8 ZMP的概念 圖9力量傳感器和他們的安置
十個(gè)自由程度 a力量傳感器
b安置在構(gòu)成機(jī)器人腳部板材下面的四個(gè)角落
圖10傳感器位置和左右腳的應(yīng)用力
圖11在機(jī)器人的四步行動(dòng)的實(shí)際ZMP位置在基準(zhǔn)水平面的
a x坐標(biāo)的by坐標(biāo)
圖12一步行動(dòng)的ZMP彈道與圖11相對(duì)應(yīng)圖14 一步行動(dòng)的ZMP彈道與圖13相對(duì)應(yīng)
圖13沿著一個(gè)10度傾斜的面向下步行的機(jī)器人的四步行動(dòng)的實(shí)際ZMP位置的
a x坐標(biāo) b y坐標(biāo)
圖15沿著一個(gè)10度傾斜的面向上步行的機(jī)器人的四步行動(dòng)的實(shí)際ZMP位置的
a x坐標(biāo)b y坐標(biāo)
圖16一步行動(dòng)的ZMP彈道與圖15相應(yīng) 圖17塑造方法的ANFS的結(jié)構(gòu)圖
圖18在與二個(gè)模糊的標(biāo)簽的模糊的模型的三角和高斯MFs用于輸入變數(shù)
a三角MF b高斯MF
圖19與ANFIS是等效的能適應(yīng)的神經(jīng)模糊的結(jié)構(gòu)
圖20引起了使用ANFS的四步行動(dòng)的ZMP位置與被測(cè)量的數(shù)據(jù)（機(jī)器人在水平基準(zhǔn)面行走）的比較a x坐標(biāo) b y坐標(biāo)
圖21一步行動(dòng)的引起的ZMP彈道與圖20相對(duì)應(yīng) 圖23一步行動(dòng)的引起的ZMP彈道與圖
22對(duì)應(yīng)
圖22引起了使用ANFS的四步行動(dòng)的ZMP位置與被測(cè)量的數(shù)據(jù)（機(jī)器人在一個(gè)10度斜面向下行走）的比較a x坐標(biāo) b y坐標(biāo)
24引起了使用ANFS的四步行動(dòng)的ZMP位置與被測(cè)量的數(shù)據(jù)（機(jī)器人在一個(gè)10度斜面向上行走）的比較a x坐標(biāo)b y坐標(biāo)
圖25一步行動(dòng)的引起的ZMP彈道與圖24相應(yīng) 圖27一步行動(dòng)的引起的ZMP彈道與圖26相對(duì)應(yīng)
圖26引起了四步行動(dòng)的ZMP位置使用一個(gè)統(tǒng)計(jì)回歸模型與被測(cè)量的數(shù)據(jù)比較為案件
機(jī)器人在水平基準(zhǔn)面上走的a x坐標(biāo) b y坐標(biāo)
圖28引起了四步行動(dòng)的ZMP位置使用統(tǒng)計(jì)回歸模型與被測(cè)量的數(shù)據(jù)比較為案件
機(jī)器人步行沿著向下10傾斜的a x坐標(biāo) b y坐標(biāo)
圖29一步行動(dòng)的引起的ZMP彈道與圖28相應(yīng) 圖31一步行動(dòng)的引起的ZMP彈道與圖30相對(duì)應(yīng)
圖30引起了四步行動(dòng)的ZMP位置使用統(tǒng)計(jì)回歸模型與被測(cè)量的數(shù)據(jù)比較為案件
機(jī)器人向上走10傾斜的面a x坐標(biāo) b y坐標(biāo)
圖32我們的機(jī)器人的四步行動(dòng)的連接角1 圖33在我們的機(jī)器人的四步行動(dòng)的連接角2
圖34在我們的機(jī)器人的四步行動(dòng)的連接角3 圖35在我們的機(jī)器人的四步行動(dòng)的連接角4
圖36在我們的機(jī)器人的四步行動(dòng)的連接角5 圖37在我們的機(jī)器人的四步行動(dòng)的連接角6
圖38在我們的機(jī)器人的四步行動(dòng)的連接角7 圖39在我們的機(jī)器人的四步行動(dòng)的連接角8
圖40在我們的機(jī)器人的四步行動(dòng)的連接角9圖41在我們的機(jī)器人的四步行動(dòng)的連接角10
表1機(jī)器人規(guī)格
尺寸
高：300mm, 寬；225mm
重
1.7kg
CPU
S3C3410X
驅(qū)動(dòng)
RC電機(jī)（11kg,4.8V）
自由度
19
動(dòng)力源
AA號(hào)鎳鎘電池（2100MA）
行走速度
48mm/1.4s
表2神經(jīng)模糊系統(tǒng)運(yùn)用的不同形式的多項(xiàng)式
輸入
多項(xiàng)式
1
2
3
0－命令
不變
不變
不變
1－命令
直線的
雙線性的
三線性的
表3我們兩足結(jié)構(gòu)走動(dòng)機(jī)器人在仿真條件的下和相應(yīng)的實(shí)際的四部走動(dòng)的ZMP值
行走條件
度
樂觀因素
前提的MF
結(jié)果類型
MSE mm
X 坐標(biāo)
Y 坐標(biāo)
0
三角
常量
4.325
4.615
－10
3.571
7.008
＋10
8.125
5.579
0
高斯
常量
4.249
4.59
－10
3.567
7.225
＋10
7.943
5.797
表4我們兩足結(jié)構(gòu)走動(dòng)機(jī)器人在仿真條件的下和相應(yīng)的實(shí)際的四部走動(dòng)的ZMP值
行走條件
度
樂觀因素
前提的MF
結(jié)果類型
MSE mm
X 坐標(biāo)
Y 坐標(biāo)
0
三角
常量
6.716
10.928
－10
6.092
13.446
＋10
11.031
12.252
0
1－命令
4.539
6.985
－10
4.114
7.648
＋10
8.862
6.443
0
高斯
常量
6.404
10.823
－10
5.670
12.207
＋10
10.966
11.179
0
1－命令
4.164
4.763
－10
3.879
9.928
＋10
8.552
5.011
表5我們兩足結(jié)構(gòu)走動(dòng)機(jī)器人在仿真條件的下和相應(yīng)的實(shí)際的四部走動(dòng)的ZMP值
行走條件
度
統(tǒng)計(jì)的
回歸模型
MSE mm
X 坐標(biāo)
Y 坐標(biāo)
0
一型
32.175
48.793
二型
7.780
13.558
三型
8.126
15.353
四型
13.018
21.420
表6我們兩足結(jié)構(gòu)走動(dòng)機(jī)器人在仿真條件的下和相應(yīng)的實(shí)際的四部走動(dòng)的ZMP值
行走條件
度
統(tǒng)計(jì)的
回歸模型
MSE mm
X 坐標(biāo)
Y 坐標(biāo)
－10
一型
34.564
46.773
二型
7.734
16.743
三型
8.193
19.377
四型
11.606
25.290
表7我們兩足結(jié)構(gòu)走動(dòng)機(jī)器人在仿真條件的下和相應(yīng)的實(shí)際的四部走動(dòng)的ZMP值
行走條件
度
統(tǒng)計(jì)的
回歸模型
MSE mm
X 坐標(biāo)
Y 坐標(biāo)
＋10
一型
34.421
50.216
二型
13.661
15.560
三型
14.409
17.436
四型
17.543
24.889
Zero-moment point trajectory modeling of a biped
walking robot using an adaptive neuro-fuzzy system
D. Kim, S.-J. Seo and G.-T. Park
Abstract: A bipedal architecture is highly suitable for a robot built to work in human environments
since such a robot will find avoiding obstacles a relatively easy task. However, the complex dynamics involved in the walking mechanism make the control of such a robot a challenging task.
The zero-moment point (ZMP) trajectory in the robot’s foot is a signi?cant criterion for the robot’s
stability during walking. If the ZMP could be measured on-line then it becomes possible to create
stable walking conditions for the robot and here also stably control the robot by using the measured ZMP, values. ZMP data is measured in real-time situations using a biped walking robot and this ZMP data is then modelled using an adaptive neuro-fuzzy system (ANFS). Natural walking motions on ?at level surfaces and up and down a 10° slope are measured. The modelling
performance of the ANFS is optimized by changing the membership functions and the consequent
part of the fuzzy rules. The excellent performance demonstrated by the ANFS means that it can not only be used to model robot movements but also to control actual robots.
1 Introduction
The bipedal structure is one of the most versatile setups for a walking robot. A biped, robot has almost the same movement mechanisms as a human and it able to operate in environments containing stairs, obstacles etc. However, the dynamics involved are highly nonlinear, complex and unstable. Thus, it is dif?cult to generate a human-like walking motion. The realisation of human-like walking robots is an area of considerable activity [1–4]. In contrast to industrial robot manipulators, the interaction between a walking robot and the ground is complex. The concept of a zero-moment point (ZMP) [2] has been shown to be useful in the control of this interaction. The trajectory of the ZMP beneath the robot foot during a walk is after taken to be an indication of the stability of the walk [1–6]. Using the ZMP we can synthesise the walking patterns of biped robots and demonstrate a walking motion with actual robots. Thus, the ZMP criterion dictates the dynamic stability of a biped robot. The ZMP represents the point at which the ground reaction force is taken to occur. The location of the ZMP can be calculated using a model of the robot. However, it is possible that there can be a large error between the actual ZMP value and the calculated value, due to deviations in the physical parameters between the mathematical model and the real machine. Thus, the actual ZMP should be measured especially if it is to be used in a to parameters a control method for stable walking.
In this work actual ZMP data taken throughout the whole walking cycle are obtained from a practical biped waling robot. The robot will be tested both on a ?at ?oor and also on 10 slopes. An adaptive neuro-fuzzy system (ANFS) will be used to model the ZMP trajectory data thereby allowing its use to control a complex real biped walking robot.
2 Biped walking robot
2.1 Design of the biped walking robot
We have designed and implemented the biped walking robot shown in Fig. 1. The robot has 19 joints. The key dimensions of the robot are also shown in Fig. 1.The height and the total weight are about 380mm and 1700 g including batteries, respectively. The weight of the robot is minimised by using aluminium in its construction. Each joint is driven by a RC servomotor that consists of a DC motor, gears and a simple controller. Each of the RC servomotors is mounted in a linked structure. This structure ensures that the robot is stable (i.e. will not fall down easily) and gives the robot a human-like appearance. A block diagram of our robot system is shown in Fig. 2.
Out robot is able to walk at a rate of one step (48mm) every 1.4 s on a ?at ?oor or an shallow slopes. The speci?cations of the robot are listed in Table 1.
The walkingmotions of the robot are shown in Figs. 3–6.- Figures 3 and 4 are show front and side views of the robot, respectively when the robot is on a ?at surface. Figure 5 is a snapshot of the robot walking down a slope whereas Fig. 6 is a snapshot of the robot walking up a slope.
The locations of the joints during motion are shown in Fig. 7. The measured ZMP trajectory is obtained from ten-degree-of-freedom (DOF) data as shown in Fig. 7. Two degrees of freedom are assigned to the hips and ankles and one DOF to each knee. Using these joint angles, a cyclic walking pattern has been realised. Our robot is able to walk continuously without falling down. The joint angles in the four-step motion of our robot are summarised in the Appendix.
2.2 ZMP measurement system
The ZMP trajectory in a robot foot is a signi?cant criterion for the stability of the walk. In many studies, ZMP coordinates are computed using a model of the robot and information from the encoders on the joints. However, we employed a more direct approach which is to use data measured using sensors mounted on the robot’s feet.
The distribution of the ground is reaction force beneath the robot’s foot is complicated. However, at any point P on the sole of the foot to the reaction can be represented by a force N and moment M, as shown in Fig. 8. The ZMP is simply the centre of the pressure of the foot on the ground, and the moment applied by the ground about this point is zero. In other words, the point P on the ground is the point at which the net moment of the inertial and gravity forces has no component along the axes parallel to the ground [1, 7].
Figure 9 illustrates the used sensors and their placement on the sole of the robot’s foot. The type of force sensor used in our experiments is a FlexiForce A201 sensor [8]. They are attached to the four corners of the plate that constitutes the sole of the foot. Sensor signals are digitised by an ADC board, with a sampling time of 10ms. Measurements are carried out in real time.
The foot pressure is obtained by summing the force signals. Using the sensor data it is easy to calculate the actual ZMP values. The ZMPs in the local foot coordinate frame are computed using (1).
Where each fi is the force at a sensor ri is the sensor position which is a vector. These are de?ned in Fig. 10. In the ?gure, ‘O’ is the origin of the foot coordinate frame which is located at the lower-left-hand corner the left foot.
Experimental results are shown in Figs. 11–16. Figures 11, 13 and 15 show the x-coordinate and y-coordinate of the actual ZMP positions for the four-step motion of the robot walking on a ?at ?oor and also down and up a slope of 10 , respectively. Figures 12, 14 and 16 shown the ZMP trajectory of the one-step motion of the robot using the actual ZMP positions shown in Figs. 11, 13
and 15. As shown in the trajectories, the ZMPs exist in a rectangular domain shown by a solid line. Thus, the positions of the ZMPs are with in the robot’s foot and hence the robot is stable.
3 ZMP trajectory modelling
In many scienti?c problems an essential step towards their solution is to accomplish the modelling of the system under investigation. The important role of modelling is to establish empirical relationships between observed variables. The complex dynamics involved in making a robot walk
make the control of the robot control a challenging task. However, if the highly nonlinear and complex dynamics can be closely produced then this modelling can be used in the control of the robot. In addition, modelling, can even be used in robust intelligent control to minimise disturbances and noise.
3.1 ANFS
Fuzzy modelling techniques have become an active research area in recent years because of their successful application to complex, ill-de?ned and uncertain systems in which conventional mathematical models fail to give satisfactory results [9]. In this light we intend to use a system to model the ZMP trajectory.
The fuzzy inference system is a popular computing framework that is based on the concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. We will use the Sugeno fuzzy model in which since each rule has a crisp output, the overall output is obtained via a weighted average, thus avoiding the time-consuming process of defuzzi?cation. When we consider fuzzy rules in the fuzzy model, the consequent part can be expressed by either a constant or a linear polynomial. The different forms of polynomials that can be used in the fuzzy system are summarised in Table 2. The modelling performance depends on the type of consequent polynomial used in the modelling. Moreover, we can exploit various forms of membership functions (MFs), such as triangular and Gaussian, for the fuzzy set in the premise part of the fuzzy rules. These are another factor that contributes to the ?exibility of the proposed approach.
The types of the polynomial are as follows
A block diagram of the modelling system is shown in Fig. 17. The proposed method is ?rst used to model and then control a practical biped walking robot.
To obtain the fuzzy rules for the fuzzy modelling system we must notes that the nonlinear system to be identi?ed is a biped walking robot with ten input variables and each input variables has two fuzzy sets, respectively. For the fuzzy model, the if-then rules are as follows:
where Ai，Bi，,,, Ji in the premise part of the rules have linguistic values (such as ‘small’ or ‘big’) associated with the input variable, x1,x2,…,x10; respectively. Fj (x1, x2,…, x10); is the constant, or ?rst-order consequent polynomial function for the jth rule.
As depicted in Fig. 18, two types of MFs were examined. One is the triangular and the other is Gaussian.
Figure 19 is an adaptive neuro-fuzzy inference system [10] architecture that is equivalent to the ten-input fuzzy model considered here, in which each input is assumed to have one of the twoMFs shown in Fig. 18. Nodes labelled P give the product of all the incoming signals and these labelled N calculate the ratio of a certain rule’s ?ring strength to the sum of all the rule’s ?ring strengths. Parameter variation in ANFIS is occured using either a gradient descent algorithm or a recursive least-squares estimation algorithm to adjust both the premise and consequent parameters iteratively. However, we do not use the complex hybrid learning algorithm but instead use the general least-squares estimation algorithm and only determine the coef?cients in the consequent polynomial function.
3.2 Simulation results
Approximately models were constructed using the ANFS. Then accuracy was quanti?ed in terms of there mean- squared error (MSE), values.
The ANFS was applied to model the ZMP trajectory of a biped walking robot using data measured from out robot. The performance of the ANFS was optimised by warying the MF and consequent type in the fuzzy rule. The measured ZMP trajectory data from our robot (shown in Figs. 32–41A in the Appendix) are used as the process parameters.
When triangular and Gaussian MFs are used in the premise part and a constant in the consequent part then, the corresponding MSE values are listed in Table 3. We have platted our results in Figs. 20–25. The generated ZMP positions from the ANFS are shown in Figs. 20, 22 and 24 for a ?at level ?oor, walking down a 10 slope and walking up a 10 slope, respectively. In Figs. 21, 23 and 25, we can see the corresponding ZMP trajectories which are generated from the ANFS.
For simplicity, the process parameter of both knees can be ignored. As a result, we can reduce the dimension of the fuzzy rules and thereby lower the computational burden. In this case the simulation conditions of the ANFS and its corresponding MSE values are given in Table 4.
From the Figures and Tables that present the simulation results, we can see that the generated ZMP trajectory from the fuzzy system is very similar to actual ZMP trajectory of measured for our walking robot shown in Figs. 11–16. The demonstrated high performance ability of the ANFS, means that ANFS can be effectively used to model and control a practical biped walking robot.
3.3 Comparisons
We now compare the performance of ANFS with numerical methods including three types of statistical regression models. For each statistical regression model, four different case types were constructed. Their general forms in the case of two inputs are given as:
where the ci are the regression coef?cients.
The corresponding MSE values are given in Tables 5–7 which reveals that type 2 gives the best results for the x and y coordinates for all the considered walking conditions. The generated ZMP positions and their corresponding trajectons generated using the type 2 regression model are shown in Figs. 26–31. We can conclude that the ANFS demonstrated a considerably better ZMP trajectory than the statistical regression models.
4 Conclusions
The ANFS modelling at the ZMP trajectory of a practical biped walking robot has been presented. The trajectory of the ZMP is an important criterion for the balance of a IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005 walking robot but the complex dynamics involved make robot control dif?cult.
We have attempted to establish empirical relationships between process parameters and to explain empirical laws by incorporating them into a biped walking robot. Actual ZMP data throughout the whole walking phase was obtained from a real biped walking robot both on a ?at level ?oor and
on slopes. The applicability of the ANFS depends on the MF used and the consequent part of the fuzzy rule. The generated ZMP trajectory using ANFS closely matches the measured ZMP trajectory. Then simulation results also show that the ZMP generated using the ANFS can improve
the stability of the biped walking robot and therefore ANFS can be effectively used to not only to model but also control practical biped walking robots. Figs. 32–41A
5 Acknowledgments
This work was supported by grant no.R01-2005-000-11-44-0 from the Basic Research Program of the Korea Science & Engineering Foundation.
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7 Appendix
This Appendix summarise the joint angles in the four-step motion of our biped walking robot. These joint angles are as follows.