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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