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河北建筑工程學院 畢業(yè)設計計算書 指導老師：孫有亮 設計題目：蛇形管彎管機 設計者：劉海洋 設計項目 計算與說明 結果 第4章 新型彎管機的主要結構及性能 4.1 新型彎管機的主要結構 4.1.1 主傳動 4.1.2 夾緊機構 4.1.3 導向壓料機構 4.1.4 液壓系統(tǒng) 1. 液壓系統(tǒng)及電磁鐵動作 2. 液壓系統(tǒng)設計特點 4.1.5 中頻加熱系統(tǒng) 4.1.6 控制系統(tǒng) 1.系統(tǒng)的硬件構成 2.系統(tǒng)的軟件設計特點 4.2新型彎管機的性能特點 第4章 新型彎管機的主要結構及性能 4.1 新型彎管機的主要結構 4.1.1 主傳動 如圖4－1所示主傳動主要由床身、主軸、主驅(qū)動油缸、雙排鏈條及鏈輪等組成。鏈輪與主軸之間用雙平鍵聯(lián)接。驅(qū)動油缸的活塞桿兩頭與鏈條固接在一起。彎管時液壓缸左腔進油而右腔回油推動活塞右移，從而帶動鏈條右移，鏈條通過鏈輪帶動主軸順時針旋轉(zhuǎn)，使管坯在彎管模上纏繞彎曲。彎管結束后，液壓缸右腔進油而左腔回油，使主軸及彎管模復位。通過液壓油量的調(diào)節(jié)可以使彎管機平穩(wěn)工作快速復位，提高工作效率。 4.1.2 夾緊機構 如下圖4－2所示，夾緊機構主要由滑座、連桿、液壓缸、夾緊塊、齒形定位塊和絲杠微調(diào)副組成。因為在彎管的整個過程中夾緊塊都必須與彎管模一起夾緊管坯，故夾緊塊及整套夾緊機構都要與彎管模同步旋轉(zhuǎn)，彎管結束后也要在一起復位。因此宏觀調(diào)整通過夾緊機構和導向壓料機構上的齒形定位塊與定位齒條的嚙合傳動實現(xiàn)，精確定位則通過絲杠螺母副實現(xiàn)。 夾緊管坯時，通過液壓油驅(qū)動四連桿機構動作致使滑座沿著導軌上升并同時推進，直至夾緊塊與彎管模在同一平面內(nèi)并將管子夾緊。夾緊塊是通過夾緊滑塊、齒形定位塊和定位齒條固定聯(lián)接在滑座上的。通過齒形的定位和絲杠的調(diào)節(jié)，夾緊塊能在滑座上移動。松開管坯時，也是由四連桿機構使滑座和夾緊塊后退并下降，直至整個夾緊機構藏到轉(zhuǎn)臂內(nèi)腔為止。 為了適應管材規(guī)格及彎曲工藝參數(shù)的變化，夾緊模的行程和壓料模的行程都能進行宏觀調(diào)整和精確微調(diào)。 4.1.3 導向壓料機構 導向壓料機構與夾緊機構基本相似，主要由滑座、連桿、液壓缸、導板、導向滑塊、齒形定位塊和絲杠微調(diào)副組成。導板很長，內(nèi)壁為圓弧形，起導向作用。導向力由油缸通過連桿增力機構帶動導向滑塊來實現(xiàn)。導向滑塊在垂直于管坯軸線方向上的行程也是通過齒形定位塊和絲杠來調(diào)節(jié)。彎制管件時，導板在導向力的作用下十周壓著管坯，并隨著管坯的前進而向前移動。 4.1.4 液壓系統(tǒng) 1. 液壓系統(tǒng)及電磁鐵動作 圖4－3所示為新型彎管機的液壓系統(tǒng)原理圖。該系統(tǒng)為普通開式液壓系統(tǒng)，主要由一個PVL12－26雙聯(lián)葉片泵、一個Y80－315－180M－4三相異步電動機、一個雙作用雙活塞桿油缸、四個YG系列標準油缸、五個三位四通電磁換向閥、一個三位兩通電磁換向閥、四個電磁換向閥、一個電磁溢流閥、兩個單向閥、兩個遠程調(diào)壓閥以及四個液控單向閥等元件所組成。 操作人員裝好管坯后，通過點動按鈕控制電磁鐵11DT得電，從而使抽芯油缸活塞動作以調(diào)整芯棒的位置。芯棒的位置調(diào)整好后，點動另一按鈕控制7DT得電，調(diào)整支撐托輥的位置。接著按下啟動按鈕，在可編程控制器（PLC）的控制下依次進行以下動作： 1） 磁鐵11DT得電，卡盤2夾緊管坯 2） 服電機2得電，管坯自動進給 3） 電磁鐵7DT得電，卡盤1夾緊管坯 4） 電磁鐵12DT得電，卡盤2松開管坯 5） 步進電機2得電，絲杠復位 如果進給量＞1m,重復步驟1）--5） 6）電磁鐵6DT得電，壓料模壓緊管坯 7）電磁鐵4DT得電，夾緊模夾緊管坯 8）電磁鐵9DT得電，主驅(qū)動油缸帶動鏈輪使主軸旋轉(zhuǎn)，從而帶動彎管模和加緊模一起轉(zhuǎn)動，對管坯進行彎曲加工 9） 彎管結束后電磁鐵10DT 得電，彎管模復位 10） 磁鐵3DT得電，夾緊模退 11） 電磁鐵5DT得電，壓料模退 12） 電磁鐵11得電，卡盤2夾緊管坯 13） 電磁鐵8DT得電，卡盤1松開管坯 14） 伺服電機2得電，絲杠進給 15） 電磁鐵7DT得電，卡盤1夾緊管坯 16） 電磁鐵12DT得電，卡盤2松開管坯 17） 伺服電機2得電，絲杠復位 18） 伺服電機1得電，管坯旋轉(zhuǎn)空間角 重復步驟6－18操作，直到彎管結束 系統(tǒng)中電磁鐵的動作順序見表4-1 2. 液壓系統(tǒng)設計特點 1) 系統(tǒng)回路為集成塊回路，其連接方式采用疊加式，從而降低了成本，減少了連接管路和液壓油的泄漏與污染，使得系統(tǒng)集成度高，便于維護，并保證了工作的可靠性 2) 四個油缸回路都采用液控單向閥的鎖定回路，減少了工作中的壓力損失，并保證油缸活塞在任意位置都能夠保持，從而使彎管機的夾緊、壓料、支撐等動作可靠 3) 主驅(qū)動油缸采用了回油節(jié)流調(diào)速回路，使得系統(tǒng)速度可調(diào)，工作速度平穩(wěn)、無沖擊，速度負載特性好 4) 僅用一個電磁溢流閥形成卸荷回路，系統(tǒng)結構簡單，能量利用合理，功耗小，節(jié)能效果明顯 5) 采用兩個遠程調(diào)壓閥、兩個電磁換向閥荷一個電磁溢流閥做成調(diào)壓單元，可以實現(xiàn)三級系統(tǒng)壓力（14MPa、8MPa和5MPa）以適應不同直徑和彎曲半徑的管坯 6) 采用組合泵供油，通過開關控制電磁閥1DT、2DT得、失電來實現(xiàn)容積調(diào)速，以達到平穩(wěn)彎管、快速復位的目的，工作效率高，結構簡單。 4.1.5中頻加熱系統(tǒng) 中頻彎管技術是利用中頻電流對管件局部感應加熱，使管件在彎曲力矩的作用下連續(xù)彎曲成型。利用該技術可以得到高質(zhì)量的彎管，被廣泛應用于電力、石化、造船、鍋爐等行業(yè)，尤其能滿足電力建設中高質(zhì)量彎管的要求。如果在電站管系中采用小曲率半徑的彎管，則可使之結構緊湊，減少管系占用空間、運輸和安裝費用，并能通過優(yōu)化設計，提高管道系統(tǒng)的運行質(zhì)量。 實驗表明，平面假設在一直到破壞的大變形條件下仍可近似的運用，而且當彎曲時甚至是在大變形下，橫剖面的形狀畸變也可略去，感應加熱小半徑彎管是一種塑性大變形彎曲，在 800-1000℃ 的彎管溫度下，可以近似看作純塑性彎曲。 4.1.6控制系統(tǒng) 1.系統(tǒng)的硬件構成 系統(tǒng)的硬件為SIMATIC S7—300系列主要由CPU312IFM可編程控制器（PLC）、擴展模塊 SM321、SM322 、SM332各一個、E6CQ增量旋轉(zhuǎn)編碼器、MPT001－微型可編程終端、各種主令電器（按鈕和行程開關）及各種執(zhí)行機構（電動機、伺服電機和電磁閥）構成。其中，PLC的最大程序存儲能量為6KB，可處理52個開關量（輸入26點，輸出26點）。它采用編程器或STEP 7 Basis支持軟件和PG720適配器進行梯形圖編程，完全滿足新型彎管機的控制需要。 另外本系統(tǒng)設有急停按鈕，一旦出現(xiàn)緊急事故，可以通過按下急停按鈕停止系統(tǒng)的一切工作，確保人身和設備安全。 2. 系統(tǒng)的軟件設計特點 可編程控制器的軟件設計就是設計系統(tǒng)的梯形圖。根據(jù)加工工藝要求，梯形圖程序主要包括： 1) 油泵電機的啟動及主控; 2) 工作循環(huán)及整個加工過程控制； 3) 發(fā)生誤操作時系統(tǒng)自鎖控制； 4) 故障報警控制； 5) 急停控制 4.2新型彎管機的性能特點 與原有彎管機相比，新型彎管機的整體性能有顯著提高，主要有以下幾點： 1. 實現(xiàn)了彎管過程的自動化，工作效率高 用原有彎管機加工管坯時，整個彎管過程都需要工作人員手動操作，工作效率低下。新型彎管機采用可編程控制器控制，使管坯從進給、自動加熱、壓料到彎管都可按預定程序自動進行，通過容積調(diào)速使彎管模平穩(wěn)彎管而快速復位，大大提高了加工效率，并減輕了工人的勞動強度。而且由于雙彎管模彎管系統(tǒng)的采用減少了蛇形管旋轉(zhuǎn)機構，在節(jié)約成本的同時大大提高了生產(chǎn)率。 2. 彎管工藝精度高 原有彎管機全憑工作人員通過自制量角模片和通過卷尺測量及目測判斷，進行工作，難以保證彎管的加工精度，致使成品率較低。新型彎管機采用可編程控制器對彎管角度自動準確控制，彎管精度達到±1°，完全滿足彎管產(chǎn)品精度要求。 3. 機械效率高，能耗低 原有彎管機采用單泵供油，用溢流閥控制系統(tǒng)壓力，系統(tǒng)壓力一旦調(diào)定即不可再變，這樣會造成能源浪費。新型彎管機采用雙聯(lián)定量泵組合供油，通過調(diào)壓單元可實現(xiàn)三級系統(tǒng)壓力，從而降低功耗，提高總的機械效率。 4. 安全可靠，操作方便 由于原有彎管機的電氣控制完全由繼電接觸器組成，在切換過程中時常出現(xiàn)繼電器接觸不實、觸點粘連甚至燒毀等事故，造成器件損毀，影響生產(chǎn)，系統(tǒng)可靠度低。新型彎管機的控制系統(tǒng)采用可編程控制器，大大提高了系統(tǒng)的可靠性及設備的自動化程度，減輕了工作人員的勞動強度。另外，系統(tǒng)自鎖功能和故障報警功能從根本上保證了操作者和設備的安全。 61 Control Engineering Practice 10 (2002) 697–711 Control of a heavy-duty robotic excavator using time delay control with integral sliding surface Sung-Uk Lee*, Pyung Hun Chang Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Science Town, 373-1 Koosung-dong, Yusung-ku, Taejon 305-701, South Korea Received 22 March 2001; accepted 14 December 2001 Abstract The control of a robotic excavator is dif?cult from the standpoint of the following problems: parameter variations in mechanical structures, various nonlinearities in hydraulic actuators and disturbance due to the contact with the ground. In addition, the more the size of robotic excavators increase, the more the length and mass of excavator’s links; the more the parameters of a heavy-duty excavator vary. A time-delay control with switching action (TDCSA) using an integral sliding surface isproposed in thispaper for the control of a 21-ton robotic excavator. Through analysis and experiments, we show that using an integral sliding surface for the switching action of TDCSA is better than using a PD-type sliding surface. The proposed controller is applied to straight-line motions of a 21-ton robotic excavator with a speed level at which skillful operators work. Experiments, which were designed for surfaces with various inclinations and over broad ranges of joint motions, show that the proposed controller exhibits good performance. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Time-delay control; Robust control; Switching action; Robotic excavator; Trajectory control 1. Introduction A hydraulic excavator isa multi-functional construc- tion machine. Workers in the construction industry use it for tasks such as excavating, dumping, ?nishing, lifting work, etc. However, operatorswho control hydraulic excavatorsmust be trained for many years to do such work quickly and skillfully. A hydraulic excavator hasthree links: boom, arm and bucket; and the operator hastwo arms. Thus, it isnot easy for beginnersto execute elaborate work that manipulates three links at the same time. Moreover, because the operatorshave to run work in variousdangerousand dirty environments, the number of skillful operators is ever decreasing. For that reason, studying the automa- tion of hydraulic excavators is necessary for improving productivity, ef?ciency, and safety. The automation of hydraulic excavatorshasbeen studied by several researchers (Singh, 1997). Among the several tasks to be automated, Bradley and Seward (1998) developed the Lancaster University computerized intelligent excavator (LUCIE) and used it to automate the digging work. Stentz, Bares, Singh, and Rowe (1998) developed a complete system for loading trucks fully autonomously on a 25-ton robotic excavator. Chang and Lee (2002) automated straight-line motions on a 13- ton robotic excavator under working speed conditions. Here, the straight-line motion represents the important task of scraping or ?attening the ground and serves as a fundamental element used as a basis for developing more complicated tasks. As illustrated in Fig. 1, the end- effector of the manipulator needsto be controlled to track a linear path on the task surface. An operator should manipulate three links simultaneously to execute it. Though an operator isskillful, performing the straight-line motions for a long time results in the fatigue of an operator and decreases productivity. The control of robotic excavator isdif?cult from the standpoint of the following problems: parameter varia- tionsin mechanical structures, variousnonlinearitiesin hydraulic actuators, and disturbance due to the contact with the ground. In mechanical structures, the inertial *Corresponding author. Tel.: +82-42-869-3266; fax: +82-42-869- 5226. E-mail address: s sulee123@cais.kaist.ac.kr (S.-U. Lee). 0967-0661/02/$-see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0967-0661(02)00027-8 force and gravitational force varieslargely with joint motions. Hydraulic actuators, massively coupled and complexly connected, have variousnonlinear compo- nents. For such reasons, various dif?culties exist in controlling a robotic excavator. To solve these problems, several research works have been performed, which may be categorized aseither simulation studies or experimental studies. In terms of simulation studies, for instance, Chiba and Takeda (1982) applied an optimal control scheme to the control of the manipulator of an excavator. Morita and Sakawa (1986) used PID control with feedforward control based on inverse dynamics. Medanic, Yuan, and Medanic (1997) proposed a polar controller-based variable structure control. Song and Koivo (1995) used a feedforward multiplayer neural network and a PID controller over a wide range of parameter variations. As for experimental studies, Bradley and Seward (1998) used a high-level controller that was based on rules obtained by observation of skilled operators, and a PID low-level motion controller that moved the end-effector in response to a demand from the high-level controller. Lee (1993) used P control together with a fuzzy control technique that used response error and its derivative on the phase plane. Sepehri, Lawrence, Sassani, and Frenette (1994) analyzed the phenomenon of coupling in the hydraulic actuator, and proposed a feedforward scheme that compensates coupling and load variation by using a simple valve model and measured pressure. Yokota, Sasao, and Ichiryu (1996) used disturbance observer and PI control, and applied it to a mini excavator. Chang and Lee (2002) used time-delay control (TDC) and compensators based on the dy- namicsof the excavator and applied it to straight-line motionsof a 13-ton excavator with a bucket speed of 0:5m=s; a speed level at which skillful operators work. However, almost all the research works above tend to be limited to experimentson a mini excavator under relatively lower speed conditions. Among the experi- mental research works above, only that of Chang and Lee (2002), which wasperformed on the control of a heavy-duty 13-ton robotic excavator, wasperformed under working speed conditions. The more the size of excavators increase, the more the length and mass of excavator’slinksincrease, and the more the parameters of a heavy-duty excavator vary. Therefore, the control of a heavy-duty excavator becomesmore dif?cult than the control of a mini excavator. The control of a heavy- duty excavator (a 21-ton robotic excavator (Fig. 2) used in thispaper) requiresa robust controller. In thispaper, we apply time-delay control with switching action (TDCSA) using an integral sliding surface (ISS) to the control of a 21-ton robotic excavator and validate the proposed control algorithm through experimentson a straight-line motion tracking control. In addition, we show the advantage of the TDCSA using an ISS. TDCSA, which wasproposed by Chang and Park (1998), consists of a TDC and a switching action. The switching action based on sliding mode control (SMC) compensates for the error of the time-delay estimation (TDE) and makes the TDC more robust. Chang and Park (1998) used a PD-type sliding surface (PDSS) for the switching action and applied the TDCSA using a PDSS to a pneumatic system for compensating the stick-slip, but we use an ISS for the switching action to improve the control performance in thispaper (Slotine Utkin only the boom, arm and bucket are considered. The mathematical model that isneeded for designing a controller isdescribed in Appendix A. Since a robotic excavator consists of a manipulator and actuators, the characteristic of these two parts will be described brie?y. 2.1. Manipulator In the dynamic equation (Eq. (A.1)), the inertial forcesand gravitational forcesaswell asthe centrifugal and Coriolisforcesvary nonlinearly with the change of anglesof the links, and have coupling elementsbetween links. Among these terms, the centrifugal and Coriolis forceshave a smaller effect on the control performance, since the velocity of each link is not that great. In comparison, the inertial forces and gravitational forces vary largely, since the total weight of boom, arm and bucket used in this research is 2:67 ton and the range of joint anglesare broad. The size and variation in each of the inertial and the gravitational forcesin a straight-line motion with incline of 01 are shown in Fig. 3. We can observe that the inertial forces and gravitational forces vary largely. 2.2. Hydraulic actuators The hydraulic actuator of the robotic excavator used in thispaper hasat least three kindsof nonlinearitiesas follows: valve characteristics, dead zone and time lag. 2.2.1. Valve characteristics Hydraulic valvesare devicesthat transfer the ?ow from the pump to cylinder. From the general valve ?ow equation eQ ? c d A ??????? DP p T; the ?ow that istransferred from the pump to cylinder isdetermined by ?ow coef?cient, the area of the valve and pressure difference. The area of a spool valve has a nonlinear shape as shown in Fig. 4. Therefore, valves have nonlinear characteristics according to the nonlinear area of the valve and ??????? DP p : 2.2.2. Dead zone The geometry of the spool valve used in a Ro- bex210LC-3 excavator is an overlapped shape as shown in Fig. 4 and causes the dead zone nonlinearity. The overlapped region isdesigned for the convenience of an operator. Therefore, when the spool is displaced in the overlapped region, the valve becomes closed: this causes 0 2 4 6 8 _ 3000 _ 2000 _ 1000 0 1000 2000 3000 time[sec] force[Newton] (a) Inertial force boom arm 0 2 4 6 8 _ 5 0 5 10 15 20 x 10 4 time[sec] force[Newton] (b) Gravitational force boom arm Fig. 3. Inertial forcesand gravitational forcesof boom and arm. S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697–711 699 the dead zone nonlinearity. The overlapped region is about 30 percent of the whole spool displacement. 2.2.3. Time lag A phenomenon similar to a dead zone occurs because of the time taken for the pump output pressure to reach the pressure level that is suf?cient to move the link. Note that thisphenomenon issomewhat different from the pure time delay often found in transmission lines. Fig. 5 illustrates this phenomenon with the experimental results. In the presence of maximum control input, the boom doesnot move until the time is0 :13 s; when the pump pressure begins to exceed the pressure of the boom cylinder plus the offset pressure, as shown in Fig. 5. Thisphenomenon occursonly when the boom link beginsto move and it doesnot exist any more once the pump output pressure reaches the pressure level suf?cient to move the link. Moreover, this phenomenon can be compensated by the compensator which will be proposed in Section 3.2. 3. Controller design A robotic excavator hasthe following nonlinearities: variationsin the inertial and gravitational forcesin the manipulator; and nonlinear valve characteristics, dead zone and time lag in the hydraulic actuator. To overcome these aforementioned nonlinearities, Chang and Lee (2002) suggested the TDC and compensators and used these to control a 13-ton robotic excavator, but we need a more robust controller to control a 21-ton robotic excavator effectively. The greatest difference between the 21-ton robotic excavator used in this paper and the 13-ton robotic excavator used in Chang and Lee (2002) exists in the length and mass of excavator’s links. The linksof the former are one and half timesthe length and weight of those of the latter. Therefore, the parameter variationsof a 21-ton excavator are more serious than those of a 13-ton excavator. A more robust controller than TDC isrequired to control the straight- line motion of a 21-ton robotic excavator. For controlling a 21-ton robotic excavator, we have considered TDCSA, which is more robust than TDC. Spool displacement Vavle Area overlap region Fig. 4. Rough shapes for areas of spool valve. 00.10.20.30.40.50.60.70.80.9 1 0 50 100 150 200 (a) Pressures of pump and head side of boom cylinder time[sec] pressure[bar] pump head side of boom cylinder 00.10.20.30.40.50.60.70.80.9 1 94 96 98 100 102 104 106 (b) Boom response time[sec] angle[deg] Fig. 5. Illustration of the nonlinearity due to time lag. S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697–711700 Then, instead of a PDSS used by Chang and Park (1998), we use an ISS in this paper for improving the control performance (Slotine Utkin whereas % HetT consists of terms representing uncertainties and time-varying factors, which are expressed as % HetT?HetTteM K etTC0 % MT . letT: e2T Now we de?ne the desired dynamics of the closed-loop system with the following error dynamic: .eetTtK v ’eetTtK p eetT?0; e3T where eetT?l d etTC0letT denotesthe position error vector with l d etT denoting the vector of desired piston displace- ments, K v the derivative gain matrix, and K p the proportional gain matrix. The TDC law that meetsthe requirement isobtained as u tdc etT? % M? . l d etTtK v ’eetTtK p eetTC138 t # HetT; e4T where # HetT denotesan estimate of % HetT: The estimated # HetT can be obtained by using both Eq. (1) and the fact that % HetT isusually a continuous function. More speci?cally, when L issmall enough, then # HetTE % Het C0 LT?uet C0 LTC0 % M . let C0 LT: e5T Combining Eq. (5) with Eq. (4), the TDC law is obtained asfollows: u tdc etT? % M? . l d etTtK v ’eetTtK p eetTC138 t u tdc et C0 LTC0 % M . let C0 LT: e6T More details about the stability condition and the design of TDC can be found in Youcef-Toumi and Ito (1990) and Hsia and Gao (1990). L should be suf?ciently small for TDC to meet the desired error dynamics of Eq. (3). The valve used for L; however, is set to be that of the sampling time, when TDC isimplemented in a real-time controller. The variation of system nonlinearities and disturbances, occurred during the time delay eLT; caused TDE error asfollows: % HetTC0 # HetT? % HetTC0 % Het C0 LT?DHetT: e7T More speci?cally, the friction dynamics cause large TDE error. Because of the TDE error, TDC does not have the desired error dynamics of Eq. (3), but the following error dynamics: .eetTtK v ’eetTtK p eetT? % M C01 DHetT; e8T where the right term e % M C01 DHetTT denotesthe effect of the TDE error. The TDCSA is proposed by adding the switching action based on the sliding mode control to TDC, as follows: u tdcsa etT? % M? . l d etTtK v ’eetTtK p eetTC138 t u tdcsa et C0 LT C0 % M . let C0 LTtK w sgnesT; e9T where s represents the sliding surface and K w isa switching gain matrix. The TDCSA has the following error dynamic: .eetTtK v ’eetTtK p eetT? % M C01 DHetTC0 % M C01 K w sgnesT: e10T In Eq. (10), we see that the switching action can reduce the TDE error. In order to match the desired error dynamics (Eq. (4)) with the sliding surface (s), we use the integral sliding surface as follows: setT?’eetTtK v eetTtK p Z t 0 eetT dtC0 ’ee0TC0K v ee0T; e11T where the sliding surface (s) hasthe initial value of zero and itsderivative isequal to desired error dynamics (Eq. (3)). The necessity and advantage of using an integral sliding surface will be shown in Section 3.1.4. 3.1.2. Stability analysis of TDCSA using an integral sliding surface For the stability analysis of the overall system, we use the second method of Lyapunov. If the Lyapunov function isselected as V ? 1 2 s T s; itstime derivative isas follows: ’ V ?s T ’s ? s T ?.e tK v ’e tK p eC138 ?s T ? . l d C0 % M C01 u t % M C01 % H tK v ’e tK p eC138 ?s T f . l d C0 % M C01 ? % Me . l d tK v ’e t K p eTt # H t K w sgnesTC138 t % M C01 % H t K v ’e tK p eg ?s T ?C0 % M C01 # H t % M C01 % H C0 % M C01 K w sgnesTC138 ?s T ? % M C01 DH C0 % M C01 K w sgnesTC138: e12T Therefore, the following condition isneeded so that the time derivative of the Lyapunov function should be negative de?nite: eK w T ii > jeDHT i j for i ? 1;y;3: e13T S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697–711 701 In other words, the magnitude of the switching gain eK w T must be larger than that of the term due to the TD estimation error. 3.1.3. Saturation function TDCSA uses a switching action for compensating the TDE error, but the switching action in TDCSA causes a chattering problem. Therefore, we use a saturation function to reduce the chattering problem (Slotine fT? e setT f T if jsetTjof; sgnesetTT otherwise; 8 jDH i j; the minimum tracking guarantee is je ss i jo % M C01 i DH i etT K p i t % M C01 i eK w i l i =f i T : e22T For a constant right-hand side of Eq. (21), however, the steady-state solution of Eq. (21) is e i etT-0 and R t 0 e i etT dt-0: Therefore, TDCSA using an ISS can drive the tracking errorsresulting from biasin uncertainties (such as constant and slowly varying parametric errors) to zero. From Eqs. (20) and (21), the relationship in the Laplace domain between the TDE error and position error eeetTT isasfollows: where p isthe Laplace operator. Eq. (23) isthat of the TDCSA using a PDSS and Eq. (24) is that of the TDCSA using an ISS. The bode plot of Eqs. (23) and (24) is shown in Fig. 6. The TDCSA using an ISS has the same high-frequency behavior as the TDCSA using a PDSS and TDC. In the low-frequency range, however, TDCSA using an ISS has a lower gain than the other controller. Thus, the TDCSA using an ISS reduces effectively the position error, which is caused by TDE error in the low-frequency range, and then the TDCSA using an ISS is more robust than the other controller against the disturbances and variation of parameters which occur in the low-frequency range. 3.2. Design of compensators Compensators are designed to overcome the dead zone and the time lag. Since the size of the dead zone coming from the overlapped area of the spool valve is constant, we add the size of the dead zone to TDCSA input as follows: u ? u tdcsa t u comp1 ; e25T E i epT DH i epT ? % M C01 i p 2 teK v i t % M C01 i eK w i =f i TTp teK p i t % M C01 i eK w i l i =f i TT ; e23T E i epT DH i epT ? % M C01 i p p 3 teK v i t % M C01 i eK w i =f i TTp 2 teK p i t % M C01 i eK w i K v i =f i TTp t % M C01 i eK w i K p i =f i T ? % M C01 i p ep t % M C01 i eK w i =f i TTep 2 t K v i p t K p i T ; e24T 10 _ 2 10 _ 1 10 0 10 1 10 2 10 3 10 _ 6 10 _ 5 10 _ 4 10 _ 3 10 _ 2 10 _ 1 |E i (s)|/|delH i (s)| Frequency[Hz] Magnitude TDC TDCSA using a integral sliding surface TDCSA using a PD type sliding surface Fig. 6. Bode diagram of closed-loop error dynamics. S.-U. Lee, P. Hun Chang / Control Engineering Practice 10 (2002) 697–711 703 where u denotesthe overall control input and u comp1 the 3 C2 1 vector whose elements are constants equivalent to the dead zone of each link. To compensate for the time lag, Chang and Lee (2002) designed the compensator using the pressure difference of the pump and cylinder. Thiscompensator increases the pump pressure to cylinder pressure quickly and works until pump pressure begins to exceed the pressure of the cylinder plus the offset pressure, but it needs pressure sensors. In this paper, as shown in Fig. 7, we add the constant value eu comp2 T to the control law until the pump pressure is increased to the pressure level suf?cient to move the link, and then decrease the value slowly to zero once the link begins to move. The whole control input, which now consists of the TDCSA input and the compensation inputs, is obtained asfollows: u ? u tdcsa t u comp1 tu comp2 : e26T Note that u comp2 isused for the control inputsof boom and arm. 4. Experiment To evaluate TDCSA using an ISS in real circum- stance, we have experimented the method in a heavy- duty excavator carrying out realistic tasks. The task of concern is primarily a straight-line motion in free spaces; yet, we have applied the straight-line motion to scraping the ground with the bucket in contact with the ground. The excavator used is a Hyundai Robex210LC-3, which hasthe following speci?cations: it weights21 ton ; the total length of the manipulator is10 :06 m and the t