Guosheng Xu, Guangming Lv and Nianli Lu

(School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China)

Abstract: An adaptive sliding mode control system for the excavator manipulator during straight-line digging operation was presented in this paper. The adaptive laws derived from Lyapunov stability theorem were used to update switching gains and compensate uncertainties. The dynamic model of the excavator manipulator considering link shape and weight was established and the straight-line trajectory of the bucket tooth tip of excavator was transformed into the desired joint angle trajectory for direct control. Finally, simulation results showed the superiority of the proposed control scheme compared with traditional variable structure control with sliding mode.

Keywords: dynamics; adaptive sliding mode control; excavator

1 Introduction

Automation of excavation work has been studied by many researchers. In order to overcome the parameter uncertainties of the excavator and the disturbances due to the interaction with soil and other nonlinearities, Hanh et al.[1]designed a fuzzy self tuning control system with neural network algorithm for the trajectory control of a mini excavator and the experimental results showed the better control performance compared with the conventional PID control. Some researchers[2-3]applied a time-delay control (TDC) with switching action using an integral sliding surface to the straight-line motion control of a hydraulic excavator and the achieved accuracy was mostly within 3 cm or 4 cm for task surfaces with various inclinations through experiments. A contour control algorithm (or cross-coupled pre-compensation algorithm) combined with PI control was applied to the control of a real excavator and was proved effective for improving tracking accuracy in Refs. [4-5]. Ha et al.[6]presented a robust sliding mode controller that implemented impedance control for a backhoe hydraulic excavator and obtained satisfactory results in autonomous excavation with the consideration of bucket-soil interaction. Haga et al.[7]developed the hydraulic excavator with an on-board depth control digging device for the purpose of improving the efficiency of the level digging work. Park et al.[8]employed impedance control and sliding mode control (SMC) in the rigid-body motion control of the excavator, and utilized the echo-state networks online learning method to control the hydraulic servo system. Chiang et al.[9]used model reference adaptive control in the control of excavator path tracking.

Excavators can be regarded as serial robotic manipulators which interact with environment. Many control techniques for robotic manipulators have been applied to the control of the excavators as mentioned above. Variable structure control with sliding mode has been widely used in motion control systems because it is immune to parameter changes and external disturbances[10]. It has also been applied to the control of the excavators in Refs. [3,6,8]. However, the main disadvantage of SMC is its high frequency chattering due to the discontinuous switching action used to handle the uncertainties. In order to attenuate or eliminate the chattering, one way is to use the saturation function other than the sign function and the tracking error is then determined by the width of the boundary layer[3,11]. A second way is the application of the adaptive sliding mode control (ASMC) in which switching gain is reduced by adaptive laws to a minimal admissible level defined by the conditions for the sliding mode to exist since the amplitude of the chattering phenomenon is proportional to the magnitude of the switching part[12-18]. Some other chattering suppression methods use neural network or fuzzy theory to approximate the sign function or tune the switching gain to adapt to the parameter uncertainties and load disturbances[19-25].

In this paper, an adaptive sliding mode controller is designed for the joint angle trajectory tracking control of the excavator manipulator. The adaptive laws derived from Lyapunov stability theorem are used to update the switching gains and compensate the model errors and uncertainties. The rest of the article is organized as follows. Section 2 presents the dynamic model of the excavator manipulator. Section 3 introduces the trajectory generation in joint space of the excavator system. Adaptive sliding mode controller is designed and presented in Section 4. Simulation results of the ASMC and the classical SMC for the trajectory tracking are given in Section 5 and Section 6 provides the brief conclusion.

2 Dynamic Model of the Excavator

The excavator can be seen as an open-chain manipulator. As shown in Fig.1, a Cartesian (rectangular and right-handed) coordinate systemO0x0y0z0was chosen as the fixed base frame.Oixiyizi(i=1,2,…,4) attached to the revolute axis was defined based on the D-H parameters methodology (Table 1).G1,G2,G3, andG4are the centers of gravity of the rotating platform with driver’s cab, boom, stick, and bucket, respectively.

Fig.1 Coordinate frames for the excavator manipulator

Since the joint angleθ1is not changed during the digging operation, the dynamic model for the digging motion of the excavator can be expressed as

(1)

Table 1 Kinematic parameters of the excavator

where

is the inertia matrix, which is symmetric and positive.

is the interactive torques between the excavator bucket and the ground.Ft,Fn,θb,δ, andθdgare presented in Fig.2.

Fig.2 Forces acting on the bucket due to environment and the bucket interaction

D23=D32=D33+Mbua3r4cos(θ4+α4)

D13=D31=D23+Mbua2r4cos(θ34+α4)

a2a3c3+a3r4cos(θ4+α4)]

Mbua3s3+Mbur4sin(θ34+α4)]-

Mbua3s3+Mbur4sin(θ34+α4)]-

a3sin(θ4+α4)]

a3sin(θ4+α4)]

α4)+Mstr3sin(θ3+α3)]-

a3sin(θ4+α4)]+

C33=0

G2=Mbug[a2c2+a3c23+r4cos(θ234+α4)]+

Mstg[a2c2+r3cos(θ23+α3)]+

Mbogr2cos(θ2+α2)

G3=Mbug[a3c23+r4cos(θ234+α4)]+

Mstgr3cos(θ23+α3)

G4=Mbugr4cos(θ234+α4)

whereMbo,Mst, andMbuare the masses of boom, stick, and bucket, respectively.Ibo,Ist, andIbuare the moments of inertia of boom, stick, and bucket, respectively.

r2=O2G2,r3=O3G3,r4=O4G4

α2=∠G2O2O3,α3=∠G3O3O4,α4=∠G4O4K

c3=cos(θ3),s3=sin(θ3),c23=cos(θ2+θ3)

θ34=θ3+θ4,θ234=θ2+θ3+θ4

3 Trajectory Generation of the Excavator

The bucket straight-line motion to scrape and flatten the ground is one of the most fundamental tasks of the excavator and an essential element for more complicated tasks[2]. Since the hydraulic actuators generate the desired torques acting on the joint shafts, the bucket straight-line motion should be transformed into the joint angle trajectory for direct control.

The attachments of the excavator working device, i.e., boom, stick, and bucket, can be seen as planar serial manipulators with 3 degree of freedom(3-DOF). However, the planned straight-line trajectory of the bucket provides only two constraints inO1x1direction andO1z1direction, respectively. As the joint angleθ1is held constant during digging operation, the trajectory is presented in coordinate systemO1x1y1z1. In order to get unique joint angle trajectory, the digging angleθdg(the angle between the bottom surface of bucket and the task surface) was kept constant[2]. As shown in Fig.3,θbis the angle between the bucket bottom and theO4x4direction,kis the slope of the task surface,φ=θ234=θ2+θ3+θ4is the attitude angle of the bucket inO1x1y1z1, and the additional constrain can be transformed as follows:

φ=arctan(k)+θb+θdg-π

(2)

From Eq. (2), it can be concluded that the attitude angleφis constant during straight-line digging operation and the transformed joint angle trajectory presented inO1x1y1z1is

θ3=-arccos([(x1-a1-a4cosφ)2+(z1-

(3a)

(3b)

β=arccos([(x1-a1-a4cosφ)2+

(3c)

θ2=α+β

(3d)

θ4=φ-θ2-θ3

(3e)

Once the slope of the task surface is obtained, the path of the bucket is determined. According to Refs. [2-3], the planned velocity trajectory is shown in Fig.4 and the straight-line motion is carried out from a stretched posture to a folded one with the direction tangential to the task surface. Then the planned straight-line displacement and the values ofx1andz1in Eq. (3) can be obtained by integrating the velocity trajectory, and the joint angle trajectory in Eq. (3) is thus calculated.

Fig.3 Position and orientation of the bucket

Fig.4 Planned velocity trajectory of the bucket

4 Design of Adaptive Sliding Mode Controller

4.1 Classical Sliding Mode Controller

D(θ)=D0(θ)+ED

G(θ)=G0(θ)+EG

τL=τL0+EL

To design the adaptive sliding mode controller and drive the joint angle positionθto the desiredθd, the sliding surfacesis defined as

wheree=θd-θis the tracking error vector andΛ=diag(λ11,λ22,λ33) is diagonal matrix in whichλnn(n=1,2,3) is a positive constant. Define the reference states as

Then the dynamic model of the excavator described by Eq. (1) can be written as

(4)

The control inputτis defined as

Substitute the control inputτinto Eq. (4), we can obtain

(5)

Define the Lyapunov function as

then the differential Lyapunov function is

-sTKps-Ks|s|+sTE

IfKsn≥|En| (n=1,2,3), then

SinceVis a Lyapunov function, so

However, the major disadvantage in designing the classical sliding mode controller is that the switching gainKsin Eq. (5) should be selected larger than the upper bound of the model uncertaintiesEto guarantee the stability of the control system. A sufficient large switching gain may aggravate the chattering phenomenon and induce a large amplitude jump to the continuous counterpart of the classical SMC.

4.2 Adaptive Sliding Mode Controller

In order to avoid the aforementioned problems in the classical SMC design, an adaptive sliding mode controller is proposed for the trajectory tracking control of the excavator manipulator.

Instead of sign function, shifted sigmoid functionΦ(s)=2/(1+e-s)-1 is used in Eq. (5) to eliminate the chattering and the switching gainKsis updated by the following adaptive law:

(6)

The differential Lyapunov function is calculated as

-sTKps+sT[E-KdΦ(s)]+

-sTKps+sT[E-KdΦ(s)]<0

Hence,s→0 andKs→Kdin finite time. The tracking error will converge to zero asymptotically and the control system is obviously stable. Then the control input of ASMC is defined as

The switching gainKsis updated by the adaptive law in Eq. (6). Fig.5 shows the block diagram of the proposed controller.

Fig.5 Block diagram of the ASMC

5 Simulation Results

Parameters of the excavator in Refs. [26-27] were used for simulation, where

a1=0.05 m,a2=5.16 m,a3=2.59 m,a4=1.33 m

r2=2.71 m,r3=0.64 m,r4=0.65 m

α2=0.256 6 rad,α3=0.331 6 rad

α4=0.394 4 rad,Mbo=1 566 kg

Mst=735 kg,Mbu=432 kg

Ibo=14 250.6 kg·m2，Ist=727.7 kg·m2

Ibu=224.6 kg·m2,g=9.8 N/kg

Fig.6 Angle trajectories for task surface with the slope k=0

Fifth-order polynomials were used for approximating the boom, stick, and bucket joint angle trajectories[28]. The approximation error curves are shown in Fig.7. According to the error curves, the maximum approximation error was less than 0.015 rad, which can be neglected.

Fig.7 Approximation errors of the joint angle trajectories using fifth-order polynomials

The interactive force between the environment and the bucket is expressed as (Fig.8)

F=1.275 2×10-2t5-6.215 9×10-1t4+

7.088 1t3-30.507 7t2+46.316 1t-

1.805 2 kN(0≤t≤6 s)

δ=0.1 rad, and thenFn=FsinδandFt=Fcosδ. The nominal parameters were chosen as

D0(θ)=0.9D(θ)

G0(θ)=0.9G(θ),τL0=0.95τL

(0.1rad0rad/s-0.6rad0rad/s-1.2rad0rad/s)

Λ=diag(5,5,5)

Kp=diag(100,100,100)

Ki=diag(100,100,100)

The switching gain

Ks=diag(20 000,1 500,1 200)

is for the classical SMC. The design parameter of the adaptive law for the switching gain of the proposed ASMC is

Γ=diag(1 500 000,100 000,100 000)

Fig.8 The force F acting on the bucket

The simulation results are presented in Figs.9-14. Fig.9 shows the joint angle trajectories tracking performance of the excavator using classical SMC and ASMC respectively and Fig.10 shows the angle tracking errors. Results indicate that the proposed ASMC and the classical SMC can both have good control results in trajectory tracking of the excavator. However, the chattering phenomenon in ASMC was eliminated according to the comparison between the control input torque of the classical SMC and that of the ASMC as shown in Fig.11.

According to Eq. (4) and the parameters set for simulation, the model errors or parameter uncertaintiesEis

Fig.9 Tracking performance of classical SMC and ASMC

Fig.10 Angle tracking errors of classical SMC and ASMC

Fig.11 Control input torque of classical SMC and ASMC

Figs.12-14 show the compensation for the actual model errors or uncertaintiesEusing classical SMC and ASMC. For the classical SMC,Kssign(s) is used to compensate the model errorsEwith the premise that the boundary of model errors or uncertainties is known. SinceKsis unchanged and usually chosen bigger than the model errors, it may intensify the chattering problem on the sliding surface. Compared with the classical SMC, for the ASMC presented in this paper,KsΦ(s) withKsupdated by the adaptive law in Eq.(6) could successfully track the time-variant model errors or uncertaintiesEwithout any prior knowledge of the excavator uncertainties.

6 Conclusions

In this paper, an adaptive sliding mode controller is designed for the straight-line motion control of the excavator manipulator. The dynamic model considering the link shape and weight was presented, and the bucket straight-line motion was transformed into the desired joint angle trajectory for direct control. In general, the nominal parameters or known parts of the dynamic model can be obtained from the design drawings or technical documents of the excavator. The model errors or parameter uncertainties were estimated by the adaptive switching part of the proposed ASMC in this research successfully. Compared with the classical SMC, the prior knowledge of the boundary of the uncertainty is not required in the proposed ASMC, and the chattering phenomenon can be eliminated. Simulation results show the superiority of the adaptive sliding mode controller in the joint trajectory tracking control of the excavator.