機械外文文獻翻譯-15號規(guī)格自動挖掘機的線性、非線性和經(jīng)典的控制器【中文6000字】【PDF+中文WORD】,中文6000字,PDF+中文WORD,機械,外文,文獻,翻譯,15,規(guī)格,自動,挖掘機,線性,非線性,經(jīng)典,控制器,中文,6000,PDF,WORD 外文翻譯 中文題目：1/5號規(guī)格自動挖掘機的線性、非線性 和經(jīng)典的控制器 英文題目：Linear, Nonlinear and Classical Control of a 1/5th Scale Automated Excavator 【中文6000字】 1/5號規(guī)格自動挖掘機的線性、非線性和經(jīng)典的控制器 摘要： 這篇論文是研究機械手各種控制系統(tǒng)的論文，介紹了一種自動操作器的規(guī)格模式。機器人臂膀已經(jīng)在蘭開斯特大學(xué)機械研究教學(xué)里得到發(fā)展。論文考慮了經(jīng)典和現(xiàn)代兩種方法的應(yīng)用，包括：由傳統(tǒng)Ziegler-Nichol規(guī)則控制的比例積分(PI)控制;線性比例積分正(PIP)控制,它可以理解為對傳統(tǒng)PI方法的合理擴展;還有一個建立在一半線性模式結(jié)構(gòu)上的新型非線性PIP設(shè)計，其中參數(shù)作為一個變量改變的函數(shù)而變化.論文考慮到了在文章中必需的在設(shè)計和執(zhí)行的復(fù)雜性間的平衡，還有為提高閉環(huán)性的潛力。 1.介紹 建筑是對于許多工業(yè)部門的主要經(jīng)濟上的意義。激烈的競爭，熟練勞力的不足和科技的進步都是落后于建筑業(yè)快速變化的主要力量，也是自動化的一個動機[1]。基于操作上的挖掘機的例子包括一般的運土、挖掘和打板樁。一個小的規(guī)模，挖掘和打樁結(jié)構(gòu)正需要發(fā)展并限制挖掘機。全自動化或部分自動化能夠提供諸多好處，它可以降低對操作員技術(shù)的依靠性，而且會降低操作員的勞動負(fù)荷，這兩者都可能會在連續(xù)性和質(zhì)量的提高上作出不小的貢獻。 然而，對于開發(fā)者們一直持續(xù)的絆腳石就是在自動控制下如何作到足夠快地運作。這里，一個主要研究問題是去獲得在熟練人工操作員提高的基礎(chǔ)上的電腦控制反應(yīng)時間。這就為設(shè)計者提供了一個極大的挑戰(zhàn)，研究者們正在選擇使用眾多方法的一個很寬的范圍；可見例子[2，3，4]。 此論文考慮到了一個實驗機械手，是更廣泛知名的蘭開斯特大學(xué)計算機智能挖掘機（LUCIE）的1/5規(guī)格的代表，而它已經(jīng)發(fā)展為建筑地[4，5]上作為挖掘工具。不管它的小規(guī)格和輕重量，1/5模式與LUCIE擁有相似的運動學(xué)和動力學(xué)上的道具，因此也為新控制策略的發(fā)展提供了一個富有價值的實驗床。在這點上，本論文提出了經(jīng)典和現(xiàn)代兩種方法，包括：由傳統(tǒng)Ziegler-Nichol規(guī)則 控制的比例積分(PI)控制;線性比例積分正(PIP)控制, 它可以理解為對傳統(tǒng)PI方法[6，7]的合理擴展;還有一個建立在一半線性模式結(jié)構(gòu)上的新型非線性PIP設(shè)計，其中參數(shù)作為一個變量改變的函數(shù)而變化[8]。 對這個研究更進一步，文章簡要地介紹了機械手的應(yīng)用，作為一個工具用于在蘭開斯特大學(xué)有關(guān)機械的學(xué)習(xí)和教育。實際上，實驗室指示者為大學(xué)生和研究生都提供了眾多的學(xué)習(xí)機會和單獨的研究方案。完善挖掘系統(tǒng)的發(fā)展需要對技術(shù)領(lǐng)域有一個完備的知識認(rèn)識，包括傳感器、激發(fā)器、計算機硬件、電子學(xué)、水力學(xué)、機械學(xué)和智能控制。 這里，控制系統(tǒng)設(shè)計需要一個有關(guān)決定適當(dāng)末端受動器軌線的高級規(guī)則的分等級方法，以致挖掘一個指定尺寸的溝渠。在實踐中，控制器也應(yīng)該包括安全模數(shù)和在土壤中處理障礙物的工具[4]。最后，高級的運算法則外加每個聯(lián)接點的適當(dāng)?shù)图壙刂疲钱?dāng)前論文的著重點。 2.硬件 機械手與LUCIE[4，5]有一個相似的排列，除了用于采沙坑的挖掘和工作臺的實驗室1/5規(guī)格模式。與圖1.所示的一樣，臂由四個聯(lián)接點構(gòu)成，包括箍用角條、俯角條、回轉(zhuǎn)角和鏟角。它們中的三個是由水壓汽缸開動，而只有回轉(zhuǎn)鏈接是基于一個與簡約變速器有關(guān)的水壓旋轉(zhuǎn)激勵器：詳細(xì)資料見[9]。 聯(lián)接器的速度由實用電壓信號手段控制。因此，整個鉆探設(shè)備已經(jīng)由多重輸入輸出異步實時控制系統(tǒng)支持，它考慮到了通過在寫入Turbo C++中密碼有標(biāo)準(zhǔn)組件的多重任務(wù)處理的程序。計算機硬件有一個96MB RAM的AMDK6/PR2-166MHz的個人電腦。 聯(lián)接角直接由與每一個聯(lián)接樞軸同中心的裝備旋轉(zhuǎn)分壓計測量。每一個分壓器的輸出信號同地面線路來傳輸，伴隨有由于環(huán)境電子噪聲造成的最小信號畸變。 圖1：顯示有四個控制聯(lián)接的實驗挖掘機原理圖 這些信號發(fā)送給高級線性儀器中有在促進A/D轉(zhuǎn)換的訓(xùn)練卡里的放大器中。這里，輸入信號的范圍在訓(xùn)練之后就不會超過±5V。這個A/D轉(zhuǎn)換器是一個高性能的16通道多元的連續(xù)近似值轉(zhuǎn)換器，具有在小于25ms12位轉(zhuǎn)換的能力。目前僅有八個可利用的通道在被使用。因此將來，合并另外的傳感器進入系統(tǒng)將不成問題；例如一個探測障礙物的照相機，可以作為高級控制系統(tǒng)的一部分，或者壓力傳感器。 電子管標(biāo)度實際上提供了一個有意義的輸入值的臂聯(lián)接。這個標(biāo)度是基于滿足輸入要求的每一個聯(lián)接的正常輸入電壓，范圍是從最高可能向下速度的-1000到對于每一個聯(lián)接的最高可能向上速度的+1000。這里，零的輸入要求對應(yīng)于無運動。注明，沒有這種電子管標(biāo)度，臂會由于每一個聯(lián)接運載的載荷而逐漸松馳下去。 在開環(huán)模式中，臂是用手驅(qū)動去挖掘溝渠的，操作員運用兩個相似的操作桿，每個都有兩個自由度。第一個操作桿被用作去驅(qū)動箍用角鏈接和回轉(zhuǎn)鏈接而另一個用做移動浸漬聯(lián)接和鏟聯(lián)接。照這樣，一個熟練的操作員為完成任務(wù)同時地移動四個聯(lián)接。比較下，這里的目的是去設(shè)計一個無人工干涉的自動挖掘計算機控制系統(tǒng)。 3.運動學(xué) 運動學(xué)平衡的目的是考慮到在三維空間中鏟的空間位置和方向的控制。既 然這樣，工具端可能會跟隨計劃軌道編程，同時鏟角分別與控制和釋放沙子相諧調(diào)。在這點上，圖1顯示的是實驗挖掘機和它的尺寸，i.e.(聯(lián)接角)和li（鏈接長度），這里i=1，2，3，4分別為箍用角條、俯角條、回轉(zhuǎn)角和鏟角。任何一個操作員的運動學(xué)分析通常需要同類的臂的工具結(jié)構(gòu)變換矩陣的發(fā)展。這被用做找到有關(guān)同等系統(tǒng)的鏟的位置、方向、速度和加速度，在特定的聯(lián)合變換向量[10]。這種分析代表性地基于著名的Denavit-Hartenberg協(xié)定，而它主要用于由一個每一個接合點都有一個自由度的開環(huán)構(gòu)成的機械操作者，如此[9]。 3.2反轉(zhuǎn)運動學(xué) 在軌線計劃程序中指定﹛X,Y,Z﹜,如末端受動器的方位利用一個等同系統(tǒng)在工作臺上初始化,不有鏟的方向,隨之的反轉(zhuǎn)運動學(xué)運算法則 由于Shaban[9]而得出。這里的Ci和Si各自表示了和,且同時 3.2 軌道計劃 一個溝渠的挖掘機在挖掘操作過程中需要兩個“連續(xù)通道”（CP）和一個更簡單的“點-點”（PTP）運動，此時鏟子為了卸載而從溝渠中搬出。特別的是，每個挖掘周期都可以被分成四個明顯的階段，如下：布置鏟來穿透土層（PTP）；挖掘過程沿著指定的空間長度（CP）在一個水平垂直線上；掘起從空間中收集到的沙子到卸載的一邊（PTP）；卸下沙子（CP）。 圖2.實驗挖掘機的軌跡計劃 對于當(dāng)前的例子，CP軌線能夠以一個恒定速度來回移動。假設(shè)和分別表示末端受動器的最初和最后的位置，而此運動需在T秒內(nèi)得到實現(xiàn)。既然這樣，工具端統(tǒng)一直線軌跡是， 這里St是一個可變速分配函數(shù)，此處S0=0和ST=1。特別的是，速度剖面第一個斜面在進行到一個恒定速度之前以一個等加速度向上移動，最后以一個等減速度向下滑到零。在統(tǒng)一的直線運動的過程中，速度剖面是來成形。通過求積分，速度分配函數(shù)將是。 對于這個特別的應(yīng)用，實驗挖掘機有所約束，只允許挖掘鏟的長度和深度分別不會超過600mm和150mm，圖2.顯示了一個完整的挖掘過程，圖解出鏟的預(yù)定軌道。值得注意的是每一個挖掘軌道由掘地點（270，﹣150，0）跟隨，利用PTP移動有一個180度的方向。這個過程由另一個PTP運動跟隨，在卸載區(qū)以 坐標(biāo)（100，﹣100，﹣400）定位鏟。挖掘周期的最后一步是卸載過程，有一個﹣30度方位的坐標(biāo)（600，﹣100，﹣700）。 4.教育和學(xué)習(xí) 工程教育最重要的特征之一是將理論知識和實踐經(jīng)驗得到結(jié)合。因此，實驗室實驗在支持學(xué)生學(xué)習(xí)中扮有非常重要的角色。然而，還有幾個因素通常阻礙學(xué)生們與機械系統(tǒng)得到“從做中學(xué)”的機會。這包括它們的高費用、熟練技術(shù)支持的較弱而必需的供應(yīng)。雖然如此，機械的利用卻潛在地為許多不同的工程學(xué)科包括機械學(xué)、電子學(xué)、控制學(xué)和計算機工程提供了一個極好的基礎(chǔ)，見例[11，12，13，14]。自動機械為基礎(chǔ)工程問題的示范提供了一個很好的工具，而它們也促進了在創(chuàng)造力、集體諧做、工程設(shè)計、系統(tǒng)統(tǒng)合和問題解決等方面技術(shù)的發(fā)展。 在這點上，LUCIE1/5規(guī)格示例在蘭開斯特大學(xué)機械學(xué)上提供了研究和教學(xué)的支持。它是為信號處理和實時控制各種方法中的一個實驗臺；而且為大學(xué)生和研究生提供了許多學(xué)習(xí)機會和單獨的計劃。例如，因為在開環(huán)模式下僅有幾秒來收集實驗數(shù)據(jù)，機械臂為示范比較對于系統(tǒng)識別的機械的和數(shù)據(jù)庫兩種方法提供了一個很好的實驗例子。 關(guān)于控制系統(tǒng)設(shè)計，各種各樣經(jīng)典和現(xiàn)代的方法都是可行的。但是，當(dāng)前的作者們認(rèn)為PIP控制為學(xué)生們對一個對現(xiàn)代控制理論提供了一個具有深刻見解的介紹。這里，非最小狀態(tài)空間（NMSS）模式都用公式表示了，以便通過控制過程測定的輸入輸出信號使完整的變態(tài)數(shù)回饋控制得到實現(xiàn)，而不會對確定性狀態(tài)的重建器或一個隨機的Kalman濾波器采取設(shè)計和執(zhí)行[6，7]。的確，智能控制中MEng/MSc模式在學(xué)科中的教學(xué)占有很多領(lǐng)域，利用機械臂只是作為一個設(shè)計例子。 5.控制方法 每一個聯(lián)接點的基準(zhǔn)PID控制器建立在著名的Ziegler-Nichol方法上。系統(tǒng)被置于比例控制下而且通過漸多的增益有穩(wěn)定性的限制直至獲得永久的振動。在這種方式下獲得的“最終增益”隨后也用作確定控制增益。另一種方法是運用一個Nichol圖表來獲得指定的增益和狀態(tài)極限，由[15]描述。 線性PIP控制是一個與PID控制有相似結(jié)構(gòu)的模式基礎(chǔ)方法，同時另外的動態(tài)回饋和輸入補償在過程有第二個指令或更高動態(tài)或大于一個采樣間隔的單純時間托延時被引入。然而，與經(jīng)典的方法形成對比，PIP設(shè)計開拓了變態(tài)數(shù)回饋（SVF）方法的范圍，這里人工調(diào)諧的奇特之處由電極分配或線性二次（LQ）設(shè)計所替代[6，7]。 最后，許多最近的出版物都為非線性PIP控制描述了一個方法，它建立在對隨之的態(tài)獨立參量（SDP）模式[8]的識別上。 這里yk 和uk 分別是輸出和輸入變量，而a i﹛ xk ﹜（i=1,2,…,n）和bj ﹛ xk﹜（j=1, …,m）是態(tài)獨立參量。后者假定為一個非最小狀態(tài)向量的函數(shù)。對于SDP-PIP控制系統(tǒng)設(shè)計，它通常對限制模式（8）是足夠的，狀態(tài)如。（8）中的NMSS表示是 這里非最小狀態(tài)參量定義如下 而且是命令輸入和輸出之間的積分誤差。依靠這個積分誤差狀態(tài)，固有的模式1伺服機構(gòu)性能得到引入。為了簡短，這里被省略了但在例[9，16]中被定義。 狀態(tài)變量回饋控制運算法則隨后被定義為 這里在每一個采樣距離通過可電極分配或一個線性Quadratic余弦函數(shù)的最優(yōu)化得到的控制增益向量。關(guān)于后一種方法，最近的研究用一種“凍結(jié)參數(shù)”系統(tǒng)定義為一個NMSS模式系列的采樣， 去定義P矩陣[9]，離散時間代數(shù)Riccatti函數(shù)僅用于修正每一個采樣距離。最后，要注明的是當(dāng)NMSS/PIP線性控制環(huán)境由[6]得到發(fā)展,對非線性SDP系統(tǒng)的整個控制和穩(wěn)定結(jié)果的來歷是作者對這個課題進一步的研究。 6.控制設(shè)計 對于線性PIP設(shè)計，開環(huán)實驗是第一個為應(yīng)用電壓和最初環(huán)境范圍的操作，全部都建立在一個0.11秒的采樣率上。在此情況下，簡化精確的工具變數(shù)(SRIV)運算法則[17],提出了一個采樣時間延續(xù)的第一指令線性模式,見,yk=a1 yk-1+bT uK-T提供了一個對每一個聯(lián)接點的近似表示.這里yk是聯(lián)接角而uk是一個在±1000范圍之內(nèi)的規(guī)則電壓值,同時{a1,bT}是時間不變參量.注意的是臂本質(zhì)上是作為一個綜合者，因規(guī)格化的電壓已經(jīng)被校準(zhǔn)以使當(dāng)uk=0時無移動。事實上，a1=﹣1被確定為一個初值，以使僅僅分子參量是在實際中為線性PIP設(shè)計而估計的。 圖3.相對箍用角輸入要求的變化 就浸入角和鏟角聯(lián)接在控制利用線性PIP方法相關(guān)地表現(xiàn)出簡單化。既然這樣，運算法則處于PI結(jié)構(gòu)，因此執(zhí)行結(jié)果與利用經(jīng)典頻率方法調(diào)諧的PI運算法則是相似的。與將被預(yù)期的一樣，經(jīng)典的和PIP兩種方法間的不同是這些聯(lián)接點是定性的。這些不同僅涉及到為將運算法則去迎合規(guī)定的控制目標(biāo)而作的相關(guān)放松。 按照，有回轉(zhuǎn)和箍用聯(lián)接運用PIP方法得到更好地控制因為（見于許多早期出版物）后一種自動地運用增加的時間差[9]。當(dāng)然，一個對于這個問題可選擇的解決方案將會引入一個Smith預(yù)報器進入PI控制組織。研究者們目前正在調(diào)查與PIP方法相比較的一種方法的相關(guān)精力。 然而，對于開環(huán)數(shù)據(jù)的進一步分析在上面的線性模式中顯示出了局限性。尤其是，的值被10或更多的一個因素來改變，依靠于被應(yīng)用的電壓值，如果圖3表示出箍用的情形。這里，許多實驗都在一個范圍內(nèi)的應(yīng)用電壓值內(nèi)操作，在每種情況下，SRIV方法都用于估計線性模式，圖3圖示了對結(jié)構(gòu)化這些估計與階式信號輸入量是相反的（可靠的軌跡表現(xiàn)了一個簡單的多項式格式）。 實際上，SDP分析表現(xiàn)出一個對箍用角更合適的模式表現(xiàn)為公式8的形式， 這里 圖4：上部：線性PIP（細(xì)線），非線性SDP-PIP（粗線）和命令輸入（虛線）對應(yīng)于箍用角條，相對結(jié)構(gòu)的采樣號。箍用角：相等于控制輸入。 圖5：上部：線性PIP（細(xì)線），非線性SDP-PIP（粗線）和命令輸入（虛線）對應(yīng)于俯角，相對結(jié)構(gòu)的采樣號。箍用角：相等于控制輸入。 這里增益f0,k,g1,k和kI,k在每一個采樣中立即以照一個預(yù)定的控制器的式樣。這種方法的完整詳細(xì)資料和相對的俯角、鏟角和回轉(zhuǎn)聯(lián)接SDP-PIP運算法則，由Shaban提出。 7.執(zhí)行 箍用臂的典型執(zhí)行結(jié)果在圖4.中表示出來，這里很明顯SDP-PIP運算法則比固定的增益有更加精確，線性PIP運算法則（可者相對的經(jīng)典PI控制器）在控制水平下有發(fā)展了很多。此外，非線性方法產(chǎn)生出了一個相當(dāng)平滑控制輸入信號。 指出的是線性和非線性控制器被設(shè)計去產(chǎn)生出一個在理論狀態(tài)下的響應(yīng)的相似速度，在圖4中可以看到的不同就歸于在b2（圖3）中的變化，此處僅考慮到在SDP-PIP狀態(tài)中。應(yīng)該指出，這個例子的反應(yīng)時間已經(jīng)被故意增加到對精確的線性PIP設(shè)計的實際限制，只是為了強調(diào)這些不同。 圖5是一個相似的實驗中俯臂的控制。盡管線性與非線性方法之間的不同通常在每一個聯(lián)接點在檢測時都相當(dāng)小，在空中運動都是孤立的，如圖5，這些不同在鏟角位置最后在采沙坑中撤離時會增加。在這點上，表1.比較了線性PIP和SDP-PIP兩種方法的反應(yīng)時間，表現(xiàn)出去完成三個完整溝渠的秒數(shù)，每個包括9個挖掘過程。這里，改良的聯(lián)接角控制考慮到了一個快速的SDP-PIP設(shè)計，典型地產(chǎn)生出一個在挖掘時間里有10％的提高 表1：完成一個溝渠的時間 最后，圖6.圖解了典型的SDP-PIP執(zhí)行結(jié)果，是鏟子的一個循環(huán)，顯示了一個末端受動器的3D縱坐標(biāo)結(jié)構(gòu)。這個圖顯示了鏟在第一低速下進入隨后才將沙子提取，移位，釋放好容易才完成。 圖6.在3-D空間中末端受動器的處理位置{X，Y，Z}，還有沿直線的移動點（mm） 8.結(jié)論 這篇論文已經(jīng)清楚地描述了實驗當(dāng)中機械斗桿的控制問題，代表了自動挖掘機的1/5號規(guī)格模式，為蘭開斯特大學(xué)的科研和教學(xué)提供了發(fā)展的機遇。與以前的研究項目相比，發(fā)表的這篇文章充分考慮了礦用自動挖掘機的完全控制系統(tǒng)的執(zhí)行。 古典和現(xiàn)代方法都采用了關(guān)聯(lián)控制，包括齊格勒-尼科爾森規(guī)則已協(xié)調(diào)的比例積分控制；線性比例積分控制和一個最近開始流行的基于國家統(tǒng)一規(guī)定參數(shù)模型鑒定的非線性比例積分控制。這里，線性比例積分運算法則已被有調(diào)整性的運用到古典和初期為了控制斗干和鏟斗運動角度而出現(xiàn)的現(xiàn)代比例積分控制當(dāng)中。反過來說，回轉(zhuǎn)和箍條連接是比可能會延遲裝卸時間的高度集中線性比例積分控制要好的一種控制方法。 然而，在這些所有聯(lián)接點的固有非線性有疑問地證實了當(dāng)反饋控制器與運動學(xué)方程相結(jié)合時，去控制末端受動器的位置，尤其是一旦鏟在沙子中移動。事實上，非線性SDP-PIP方法增加的復(fù)雜在這里表現(xiàn)出來了假定出改良的閉環(huán)性能。特別的是，完成整個挖掘過程的時間減少了大約10％。最后，利用實驗挖掘機獲得的經(jīng)驗最近已經(jīng)被開發(fā)，一個整個的振動層系統(tǒng)中被運用為在一個建筑地點的地面改善，見例[16]。 感謝 非常感謝工程和物理科學(xué)研究會（EPSRC）的支持。在本論文中使用過的統(tǒng)計工具已經(jīng)組合為一個CAPTAIN工具箱[18]，在MATLABTM軟件環(huán)境中，下載可得用網(wǎng)址：http://www.es.lancs.ac.uk/cres/captain/ 13 LINEAR,NONLINEAR AND CLASSICAL CONTROL OF A1/5TH SCALE AUTOMATED EXCAVATORE.Sidiropoulou,E.M.Shaban,C.J.Taylor,W.Tych,A.ChotaiEngineering Department,Lancaster University,Lancaster,UK,c.taylorlancaster.ac.ukEnvironmental Science Department,Lancaster University,Lancaster,UKKeywords:Identification;model-based control;proportional-integral-plus control;state dependent parameter model.AbstractThis paper investigatesvariouscontrol systems fora laboratoryrobot arm,representing a scale model of an autonomous exca-vator.The robot arm has been developed at Lancaster Univer-sity for research and teaching in mechatronics.The paper con-siders the application of both classical and modernapproaches,including:Proportional-Integral(PI)control tuned by conven-tional Ziegler-Nichols rules;linear Proportional-Integral-Plus(PIP)control,which can be interpreted as one logical exten-sion of the conventional PI approach;and a novel nonlinearPIP design based on a quasi-linear model structure,in whichthe parameters vary as a function of the state variables.Thepaper considers the pragmatic balance required in this context,between design and implementational complexity and the po-tential for improved closed-loop performance.1IntroductionConstruction is of prime economic significance to many indus-trial sectors.Intense competition,shortfall of skilled labourand technological advances are the forces behind rapid changein the construction industry and one motivation for automa-tion 1.Examples of excavationbased operationsinclude gen-eral earthmoving,digging and sheet-piling.On a smaller scale,trenching and footing formation require precisely controlledexcavation.Fullorpartialautomationcanprovidebenefitssuchas reduced dependence on operator skill and a lower operatorwork load,both of which are likely to contribute to improve-ments in consistency and quality.However,a persistent stumbling block for developers is theachievement of adequate fast movement under automatic con-trol.Here,a key research problem is to obtain a computercontrolled response time that improves on that of a skilled hu-man operator.This presents the designer with a difficult chal-lenge,which researchers are addressing using a wide range ofapproaches;see e.g.2,3,4.This paper considers a laboratory robot arm,a 1/5th scale rep-resentation of the more widely known Lancaster UniversityComputerised Intelligent Excavator(LUCIE),which has beendeveloped to dig trenches on a construction site 4,5.De-spite its smaller size and light weight,the 1/5th model has sim-ilar kinematic and dynamic properties to LUCIE and so pro-vides a valuable test bed for the development of new controlstrategies.In this regard,the paper considers both classicaland modern approaches,including:Proportional-Integral(PI)control tuned by conventional Ziegler-Nichols rules;linearProportional-Integral-Plus(PIP)control,which can be inter-preted as one logical extension of the conventional PI ap-proach 6,7;and a novel nonlinear PIP design based on aquasi-linear model structure in which the parameters vary asa function of the state variables 8.Further to this research,the paper briefly considers utilisationof the robot arm as a tool for learning and teaching in mecha-tronics at Lancaster University.In fact,the laboratory demon-strator provides numerous learning opportunities and individ-ual research projects,for both undergraduate and postgraduatestudents.Development of the complete trench digging systemrequiresa thoroughknowledgeofa widerangeoftechnologies,including sensors,actuators,computing hardware,electronics,hydraulics,mechanics and intelligent control.Here,control system design requires a hierarchical approachwith high-level rules for determining the appropriate end-effector trajectory,so as to dig a trench of specified dimen-sions.In practice,the controller should also include modulesfor safety and for handling obstructions in the soil 4.Finally,the high-level algorithm is coupled with appropriate low-levelcontrol of each joint,which is the focus of the present paper.2HardwareThe robot arm has a similar arrangement to LUCIE 4,5,ex-ceptthatthislaboratory1/5thscale modelisattachedtoawork-bench and the bucket digs in a sandpit.As illustrated in Fig.1,thearmconsistsoffourjoints,includingtheboom,dipper,slewand bucket angles.Three of these are actuated by hydrauliccylinders,with just the slew joint based on a hydraulic rotaryactuator with a reduction gearbox:see 9 for details.The velocity of the joints is controlled by means of the appliedvoltage signal.Therefore,the whole rig has been supportedby multiple I/O asynchronousreal-time control systems,whichallow for multitasking processes via modularisation of codewritten in Turbo C+R?.The computer hardware is an AMD-K6/PR2-166 MHz personal computer with 96 MB RAM.The joint angles are measured directly by mounting rotary po-tentiometers concentric with each joint pivot.The output sig-nal from each potentiometer is transmitted with an earth lineto minimise signal distortion due to ambient electrical noise.Figure 1:Schematic diagram of the laboratory excavator showing the four controlled joints.These signals are routed to high linearity instrumentation am-plifiers within the card rack for conditioningbefore forwardingto the A/D converter.Here,the rangeof the input signal just af-ter conditioning does not exceed 5 volts.This A/D converteris a high performance 16 channel multiplexed successive ap-proximation convertercapable of 12 bit conversion in less than25 micro seconds.At present only eight available channels arebeing used.In the future,therefore,there would be no prob-lem for incorporating additional sensors into the system;e.g.acamera for detecting obstacles,to be used as part of the higherlevel control system,or force sensors.Valve calibration is essential to provide the arm joints withmeaningful input values.This calibration is based on normal-izing the input voltage of each joint into input demands,whichrange from-1000 for the highest possible downward velocityto+1000for the highest possible upwardvelocity of each joint.Here,an input demand of zero corresponds to no movement.Note that,without such valve calibration,the arm will gradu-ally slack down because of the payload carried by each joint.In open-loop mode,the arm is manually driven to dig thetrench,with the operator using two analogue joysticks,eachwith two-degrees of freedom.The first joystick is used to drivethe boom and slew joints while the other is used to move thedipper and bucket joints.In this manner,a skillful operatormoves the four joints simultaneously to perform the task.Bycontrast,the objective here is to design a computer controlledsystem to automatically dig without human intervention.3KinematicsThe objective of the kinematic equations is to allow for con-trol of both the position and orientation of the bucket in 3-dimensional space.In this case,the tool-tip can be pro-grammed to follow the planned trajectory,whilst the bucketangleis separatelyadjustedtocollectorreleasesand.Inthisre-gard,Fig.1 shows the laboratoryexcavator and its dimensions,i.e.i(joint angles)and li(link lengths),where i=1,2,3,4for the boom,dipper,bucket and slew respectively.Kinematic analysis of any manipulator usually requires devel-opment of the homogeneous transformation matrix mappingthe tool configuration of the arm.This is used to find the po-sition,orientation,velocity and acceleration of the bucket withrespect to the reference coordinate system,given the joint vari-able vectors 10.Such analysis is typically based on the well-known Denavit-Hartenberg convention,which is mainly usedfor robot manipulators consisting of an open chain,in whicheach joint has one-degree of freedom,as is the case here 9.3.1Inverse kinematicsGiven X,Y,Z from the trajectory planning routine,i.e.theposition of the end effector using a coordinate system origi-nating at the workbench,together with the orientation of thebucket =1+2+3,the following inversekinematic algo-rithm is derived by Shaban 9.Here Ciand Sidenotes cos(i)and sin(i)respectively,whilst C123=cos(1+2+3).X=X l4C4C4 l3C123(1)Y=Y l3S123(2)1=arctan?(l1+l2C2)Y l2S2X(l1+l2C2)X l2S2Y?(3)2=arccos?X2+Y2 l21 l222l1l2?(4)3=1 2(5)4=arctan?ZX?(6)3.2Trajectory planningExcavation of a trench requires both continuous path(CP)motion during the digging operation and a more primitivepoint-to-point(PTP)motion when the bucket is moved outof the trench for discharging.In particular,each digging cyclecan be divided into four distinct stages,as follows:positioningthe bucket to penetrate the soil(PTP);the digging process ina horizontal straight line along the specified void length(CP);Figure 2:Trajectory planning for the laboratory excavator.picking up the collected sand from the void to the dischargeside(PTP);discharging the sand(CP).For the present example,the CP trajectory can be traversed ata constant speed.Suppose v0and vfdenote,respectively,theinitial and final position vector for the end-effectorand that themovement is required to be carried out in T seconds.In thiscase,the uniform straight-line trajectory for the tool-tip is,v=(1 St)v0+Stvf0 t T(7)Here,Stis a differentiable speed distribution function,whereS0=0 and ST=1.Typically,the speed profileStfirst rampsup at a constant acceleration,before proceeding at a constantspeed and finally ramping down to zero at a constant deceler-ation.In the case of uniform straight-line motion,the speedprofile will take the formSt=1/T.By integrating,the speeddistribution function will be St=t/T.For this particular application,the kinematic constraints of thelaboratoryexcavatorallow for digging a trench with length anddepth not exceeding 600 mm and 150 mm,respectively.Fig.2shows one complete digging cycle,illustrating the proposedpath for the bucket.Note that each digging path is followed bypicking up the soil to the point(270,150,0)with an orien-tation of 180 degrees using PTP motion.This step is followedby another PTP motion to position the bucket inside the dis-charging area at coordinate(100,100,400).The last stepin the digging cycle is the discharging process which finishesat(600,100,700)with an orientation of-30 degrees.4Teaching and learningOne of the most important features of engineering education isthe combination of theoretical knowledge and practical expe-rience.Laboratory experiments,therefore,play an importantrole in supporting student learning.However,there are severalfactors that often prevent students from having access to suchlearning-by-doinginteractionwith roboticsystems.Thesein-clude their high cost,fragility and the necessary provision ofskilled technical support.Nonetheless,the utilization of robotspotentially offers an excellent basis for teaching in a number ofdifferent engineering disciplines,including mechanical,elec-trical,control and computer engineering;e.g.11,12,13,14.Robotsprovideafascinatingtoolforthedemonstrationofbasicengineering problems and they also facilitate the developmentof skills in creativity,teamwork,engineering design,systemsintegration and problem solving.In this regard,the 1/5th scale representation of LUCIE pro-vides for the support of research and teaching in mechatronicsat Lancaster University.It is a test bed for various approachesto signal processingandreal-time control;and providesnumer-ous learning opportunities and individual projects for both un-dergraduate and postgraduate research students.For example,since only a few minutes are needed to collect experimentaldata in open-loop mode,the robot arm provides a good labo-ratory example for demonstrating contrasting mechanistic anddata-based approaches to system identification.With regards to control system design,various classical andmodern approaches are feasible.However,the present au-thors believe that PIP control offers an insightful introductionto modern control theory for students.Here,non-minimal statespace(NMSS)models are formulated so that full state vari-able feedback control can be implemented directly from themeasured input and output signals of the controlled process,without resort to the design and implementation of a determin-istic state reconstructor or a stochastic Kalman Filter 6,7.Indeed,a MEng/MSc module in Intelligent Control taught inthe Department covers all these areas,utilising the robot armas a design example.5Control methodologyThe benchmark PID controller for each joint is based onthe well known Ziegler-Nichols methodology.The system isplaced under proportional control and taken to the limit of sta-bility by increasing the gain until permanent oscillations areachieved.The ultimate gain obtained in this manner is subse-quently used to determine the control gains.An alternative ap-proach using a Nichols chart to obtain specified gain and phasemargins is described by 15.Linear PIP control is a model-based approach with a similarstructure to PID control,with additionaldynamic feedbackandinput compensators introduced when the process has secondorder or higher dynamics,or pure time delays greater than onesample interval.In contrast to classical methods,however,PIPdesign exploits the power of State Variable Feedback(SVF)methods,where the vagaries of manual tuning are replaced bypole assignment or Linear Quadratic(LQ)design 6,7.Finally,a number of recent publications describe an approachfor nonlinear PIP control based on the identification of the fol-lowing state dependent parameter(SDP)model 8,yk=wTkpk(8)where,wTk=?yk1yknuk1ukm?pk=?p1,kp2,k?Tp1,k=?a1kank?p2,k=?b1kbmk?Here ykand ukare the output and input variables respectively,while aik(i=1,2,.,n)and bjk(j=1,.,m)are state dependent parameters.The latter are assumed to befunctions of a non-minimalstate vector Tk.For SDP-PIP con-trol system design,it is usually sufficient to limit the model(8)to the case that Tk=wTk.The NMSS representation of(8)is,xk+1=Fkxk+gkuk+dyd,k(9)yk=hxkwhere the non-minimal state vector is defined,xk=?ykykn+1uk1ukm+1zk?Tand zk=zk1+yd,k yk is the integral-of-error betweenthe command input yd,kand the output yk.Inherent type 1servomechanism performance is introduced by means of thisintegral-of-error state.For brevity,Fk,gk,d,h are omittedhere but are defined by e.g.9,16.The state variable feedback control algorithm uk=lkxkissubsequently defined by,lk=?f0,k.fn1,kg1,k.gm1,kkI,k?where lkis the control gain vector obtained at each samplinginstant by either pole assignment or optimisation of a LinearQuadratic(LQ)cost function.With regard to the latter ap-proach,the present research uses a frozen-parameter systemdefined as a sample member of the family of NMSS modelsFk,gk,d,h to define the P matrix 9,with the discrete-timealgebraic Riccatti equation only used to update lkat each sam-plinginstant.Finally,notethatwhiletheNMSS/PIP linearcon-trollability conditions are developed by 6,derivation of thecomplete controllability and stability results for the nonlinearSDP system is the subject on-going research by the authors.6Control designFor linear PIP design,open-loop experiments are first con-ducted for a range of applied voltages and initial conditions,allbased on a sampling rate of 0.11seconds.In this case,the Sim-plified Refined Instrumental Variable(SRIV)algorithm 17,suggests that a first order linear model with samples timedelay,i.e.yk=a1yk1+buk,provides an approximaterepresentation of each joint.Here ykis the joint angle and ukis a scaled voltage in the range 1000,while a1,b are timeinvariant parameters.Note that the arm essentially acts as anintegrator,since the normalised voltage has been calibrated sothat there is no movement when uk=0.In fact,a1=11000800600400200020040060080010000.0050.010.0150.020.0250.030.0350.04ParameterScaled voltageFigure 3:Variation of bagainst input demand for the boom.is fixed a priori,so that only the numerator parameter bisestimated in practice for linear PIP design.With =1,the dipper and bucket joints appear relativelystraightforward to control using linear PIP methods.In thiscase,the algorithm reduces to a PI structure 6,hence the im-plementationresults are similar to the PI algorithm tuned usingclassical frequencymethods.As would be expected,the differ-ence between the classical and PIP methods for these joints isqualitative.Such differences relate only to the relative ease oftuning the algorithm to meet the stated control objectives.By contrast,with =2,the slew and boom joints are bettercontrolledusing PIP methodssince(as shownin numerousear-lier publications)the latter automatically handles the increasedtime delay 9.Of course,an alternative solution to this prob-lem would be to introduce a Smith Predictor into the PI controlstructure.The authors are presently investigating the relativerobustness of such an approach in comparison to PIP methods.However,further analysis of the open-loop data reveals limi-tations in the linear model above.In particular,the value ofbchanges by a factor of 10 or more,depending on the ap-plied voltage used,as illustrated in Fig.3 for the case of theboom.Here,numerous experiments are conducted for a rangeof applied voltages and,in each case,SRIV methods used toestimate linear models.Fig.3 illustrates these estimates of bplotted against the magnitude of the step input(the solid tracerepresents a straightforward polynomial fit).In fact,SDP analysis suggests that a more appropriate modelfor the boom takes the form of equation(8)with,wTk=?yk1uk1uk2?pk=?a1k0b2k?T(10)where,a1k=0.238 106u2k2 1b2k=5.8459 106uk2+0.0189880901001101201301401502002040608090100110120130140150100050005001000Figure 4:Top:linear PIP(thin trace),nonlinear SDP-PIP(thick)and command input(dashed)for the boom angle,plot-ted against sample number.Bottom:equivalent control inputs.The associated SDP-PIP control algorithm takes the form,uk=?f0,kg1,kkI,k?ykuk1zk?T(11)where the gains f0,k,g1,kand kI,kare updated at each sam-pling instant in the manner of a scheduled controller.Full de-tails of this approach and the equivalent SDP-PIP algorithmsfor the dipper,bucket and slew joints are given by Shaban 9.7ImplementationTypical implementation results for the boom arm are illus-trated in Fig.4,where it is clear that the SDP-PIP algorithmis more robust than the fixed gain,linear PIP algorithm(orequivalentclassical PIcontroller)tolargestepsinthecommandlevel.Furthermore,the nonlinear approach yields a consider-ably smoother control input signal.Note that the linear and nonlinear controllers are designed toyield a similar speed of response in the theoretical case,i.e.thedifferences seen in Fig.4 are due to the variation in b2(Fig.3)which is only taken account of in the SDP-PIP case.It shouldpointed out that the response time for this example has beendeliberately increased to the practical limit of robust linear PIPdesign,in order to emphasis these differences.Fig.5 shows controlof the dipperarm fora similar experiment.Although the differences between the linear and nonlinear ap-proachesare oftenrelativelysmall when each joint is examinedin isolation for movement in air,as in Fig.5,such differencesare multiplied up when the bucket position is finally resolvedin the sandpit.In this regard,Table 1 compares the responsetime of the linear PIP and SDP-PIP approaches,representedby the number of seconds taken to complete three completetrenches,each consisting of 9 digging cycles.Here,the im-proved joint angle control allows for a faster SDP-PIP design,typically yielding a 10%improvement in the digging time.204060801001201401601802001201008060402040608010012014016018020020015010050050100Figure 5:Top:linear PIP(thin trace),nonlinear SDP-PIP(thick)and command input(dashed)for the dipper angle,plot-ted against sample number.Bottom:equivalent control inputs.Table 1:Time taken to complete one trench.TrenchLinear PIPSDP-PIP1338.46s369.01s2334.39s370.43s3336.13s372.86sFinally,Fig.6 illustrates typical SDP-PIP implementation re-sults for one cycle of the bucket showing a 3D co-ordinate plotof the end-effector.This graph shows the bucket being firstlowered into and subsequentlybei