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.Automation in Construction 9 2000 421435 rlocaterautcon Impedance control of a hydraulically actuated robotic excavator Q.P. Ha), Q.H. Nguyen, D.C. Rye, H.F. Durrant-Whyte Australian Center for Field Robotics, The Uniersity of Sydney, J07, Sydney, 2006 NSW, Australia Abstract In robotic excavation, hybrid positionrforce control has been proposed for bucket digging trajectory following. In hybrid positionrforce control, the control mode is required to switch between position- and force-control depending on whether the bucket is in free space or in contact with the soil during the process. Alternatively, impedance control can be applied such that one control mode is employed in both free and constrained motion. This paper presents a robust sliding controller that implements impedance control for a backhoe excavator. The control law consists of three components: an equivalent control, a switching control and a tuning control. Given an excavation task in world space, inverse kinematic and dynamic models are used to convert the task into a desired digging trajectory in joint space. The proposed controller is applied to provide good tracking performance with attenuated vibration at bucketsoil contact points. From the control signals and the joint angles of the excavator, the piston position and ram force of each hydraulic cylinder for the axis control of the boom, arm, and bucket can be determined. The problem is then how to find the control voltage applied to each servovalve to achieve force and position tracking of each electrohydraulic system for the axis motion of the boom, arm, and bucket. With an observer-based compensation for disturbance force including hydraulic friction, tracking of the piston ram force and position is guaranteed using robust sliding control. High performance and strong robustness can be obtained as demonstrated by simulation and experiments performed on a hydraulically actuated robotic excavator. The results obtained suggest that the proposed control technique can provide robust performance when employed in autonomous excavation with soil contact considerations. q2000 Elsevier Science B.V. All rights reserved. Keywords: Robotic excavator; Hybrid positionrforce control; Sliding controller 1. Introduction The usual task of a backhoe excavator is to free and remove material from its original location and to transfer it to another location by lowering the bucket, digging by dragging the bucket through the soil, then ) Corresponding author. Department of Mechanics and Mecha- tronic Engineering, The University of Sydney, J07, 2006 NSW, Australia. Tel.: q 61-2-9351-3098; fax: q 61-2-9351-7474. .E-mail address: .au Q.P. Ha . lifting, slewing and dumping the bucket. In moving towards automatic excavation, there is a need for the development of a controller that is robust to uncer- w xtainties associated with these operations1 . For control purposes, kinematic and dynamic models of excavators that assume the hydraulic actuators act as infinitely powerful force sources are presented in wxRefs.24 . Position control with a conventional proportional and derivative controller is used in Refs. wx4,5for simulation of the digging process with limited soil interaction. Excavators are, however, subject to a wide varia- tion of soiltool interaction forces. When digging, 0926-5805r00r$ - see front matter q2000 Elsevier Science B.V. All rights reserved. .PII: S0926-5805 00 00056-X ()Q.P. Ha et al.rAutomation in Construction 9 2000 421435422 the bucket tip motion is effectively force-constrained by the nonlinear constitutive equations of the envi- ronment, and by the hydraulic forces. Compliance control approaches may therefore be considered to be more suitable than position control for the shap- ing of excavator dynamics. Compliant motion con- trol can generally be classified into two broad classes: hybrid positionrforce control and interactive control orimpedanceradmittancecontrol.Inhybrid forcerposition control, the Cartesian space of the end-effector co-ordinates is decomposed into a posi- tion sub-space and a force sub-space. Separate posi- tion- and force-trajectory tracking objectives are specified in each sub-space. Excessive force tran- sients may, however, occur at the instant of contact between the tool and the environment. Rather than tracking desired position and force trajectories, inter- active control seeks to regulate the relationship be- tween the end-effector position and the interaction force. It is known that impedance control provides a unified approach to both unconstrained and con- w xstrained motion 6 . If hybrid positionrforce control is adopted, the control mode should be switched between position control and force control according to whether the excavator bucket is in free space or in contact with the soil during an excavation task. Impedance control is believed to be better suited to excavation tasks in the sense that it can be applied continuously to both free and constrained motions w x1 . An impedance controller has recently been re- w xported for an excavator arm 7 . This paper proposes a robust sliding mode control technique to imple- ment impedance control for an excavator using gen- eralised excavator dynamics. The bucket tip is con- trolled to track a desired digging trajectory in the presence of environment and system parameter un- certainties. In impedance control of hydraulic exca- vators, the piston position and the ram force of each hydraulic cylinder for the axis control of the boom, arm, and bucket can be determined. The problem is then how to find the control voltage applied to the servovalves to track these desired commands. Taking into account friction and nonlinearities, a discontinu- ous observer is developed for estimating both piston displacement velocity and disturbance including load force and friction. With an observer-based compen- sation for the disturbance force, robust tracking the piston ram force and position is guaranteed using robust sliding mode controllers for electrohydraulic systems. The validity of the proposed method is verified through simulation and filed tests performed on the Komatsu PC-05 mini-excavator. The remain- der of this paper is organised as follows. Section 2 is devoted to the derivation of the excavator dynamic model. The problem formulation and the develop- ment of impedance control for excavator dynamics are presented in Section 3. The control of electrohy- draulic systems is addressed in Section 4. The hard- ware organisation for the robotic excavator is de- scribed in Section 5 together with computer simula- tions and experimental results. Finally, conclusions are provided in Section 6. 2. Excavator dynamics The equations of motion for a generic excavator can be derived by applying the EulerLagrange equations to a Lagrangian energy function, or by writing the NewtonEuler equations successively for each link of the machine. In the latter approach, the dynamics of each link are described by equations that recurse through the link index. The driving joint torques of the boom, arm, and bucket are generated by the forces of the hydraulic ram actuators. The translational and rotational motions of these links are described by the dynamic model of the excavator system. Dynamic models of excavators are presented w xw xin Ref. 2 with refinements in Ref. 4 . Firstly, a ?4Cartesian co-ordinate frame O x y zfixed to the 0000 excavator body is chosen. Other Cartesian co-ordinate frames are systematically assigned by applying the wxDenavit and Hartenberg procedure as in Refs. 24 . ?4?4?4The framesO x y z ,O x y z ,O x y z , 111122223333 ?4andO x y zare respectively attached to the 4444 boom, arm, bucket, and bucket tip as seen in Fig. 1. Note that the movements of the excavator mecha- nism during digging usually occur in the vertical plane. It is therefore assumed that no boom-swing motion occurs during excavation, so that the boom .swing angleu is therefore held constantus0 11 during digging. The model equations can be written for each link of the excavator by considering the link as a rigid free body. By combining Newton and Euler equations for all links, the dynamical model for the excavator can be expressed concisely in a ()Q.P. Ha et al.rAutomation in Construction 9 2000 421435423 Fig. 1. Excavator joint variables. well-known form of manipulator equations of motion w x4 : Du uqCu,u uqBuqGu. sAuFyTF ,F,1. . Ltn wxTwhereusuuuis the vector of measured 234 shaft angles:ufor the boom joint,ufor the arm 23 joint, andufor the bucket joint; Trepresents the 4L load torques as functions of the tangential and nor- mal components, F and F , of the soil reaction force tn at the bucket, and F are the ram forces of the hydraulic actuators that produce the torques acting on the joint shafts. The tangential component, F, t which is parallel to the digging direction, represents the resistance to the digging of ground by excavators bucket teeth. This resistance is considered as the sum of soils resistance to cutting, the friction between the bucket and the ground, and the resistance to movement of the prism of soil and soil movement in the bucket. The tangential component can be calcu- w xlated according to Ref. 8 as F sk bh,2 . t1 w y2x where kis the specific digging force N m, and 1 h and b are respectively the thickness and width of w xthe cut slice of soil m . The normal component, F , n is calculated as F sCF ,3 . nt .wherecs 0.10.45 is a factor depending on the digging angle, digging conditions, and the wear and w xtear of the cutting edge 8 . The determination of the .matrices of inertia, Du, of Coriolis and centripetal . . effects, Cu,u u, of gravity forces, Gu, and of . functions of the moment arms, Au, is comprehen- wxsively described in Refs. 24 . All entries in these w xmatrices are given in Ref. 4 . The 3=1-matrix of . viscous friction Buis treated in this paper as a source of uncertainty. .In the digging plane, the Jacobian Judefined as xsJu u,4. . w xwcan be obtained from Ref.3 , where xs xz 44 xTurepresents the Cartesian co-ordinates and orien- O4 .tation of the bucket tipOwith respect to 4 ?4 .O x y z . Assuming that the Jacobian matrix J e 0000 .is non-singular, Eq. 1 in joint space can be rewrit- ten in Cartesian space as: H x xqCx,x xqBx qGx. xxx yT sJAFyF ,5 . e where HsJyTDJy1,C sJyTCyDJy1J Jy1,. x G sJyTG,B sJyTB,6 . xx and F sJITTdenotes the generalised forces of eL .interaction between the end-effector bucket tip and .the environment soil . They consist of digging forces acting on the bucket with force entries for the co- .ordinatesx ,zand a torque entry around y . 444 The determination of the forward and inverse . y1 . kinematic relationships, xsLuandusLx w x .is detailed in Ref. 3 . As Eq. 5 has the form of the generalised robotic manipulator dynamics where x is a vector of the co-ordinates of the contact point of the manipulator with the environment, below and in the next section, we will consider in general xgRn and ugRn. We assume A1:HsHqAH,C sC qDC , xxx G sG qDG ,AsAqDA,F sF qDF , xxxeee 7 . where matrices H, C , G , and A are known, F is to xxe be measured by force sensors such as load pins, and DH, DC , DG , DA, and DF are uncertainties. xxe Denoting friction and uncertainties by Df x,x,x sDHxqDC xqDG qB x. xxx yT qDF yJDAF,8 . e ()Q.P. Ha et al.rAutomation in Construction 9 2000 421435424 .Eq. 5 can be rewritten as H x xqCx,x xqGx. xx suyF yDf x,x,x ,9. . e where yT usJAF10. is the control input. .Remark 1. As Duis a 3=3-symmetric positive- definite matrix satisfying the skew symmetric prop- w x.erty 9 , for the nominal dynamics Df x,x,x s0 w .x of the excavator, H x -2Cx,xis also a skew- x symmetric matrix, i.e. T xH x y2Cx,xxs0,;x.11. x 3. Excavator dynamics impedance control 3.1. Problem formulation One of the excavating task elements is penetration of the soil by an excavator bucket to follow a pre-planned digging trajectory. During digging, three main tangential resistance forces arise: the resistance to soil cutting, the frictional force acting on the bucket surface in contact with the soil, and the resistance to movement of the prism of soil ahead of and in the bucket. The magnitude of the digging resistance forces depends on many factors such as the digging angle, volume of the soil prism, volume of material ripped into the bucket, and the specific resistance to cutting. These factors are generally variable and unavailable. Moreover, due to soil plas- ticity, spatial variation in soil properties, and poten- tial severe inhomogeneity of material under excava- tion, it is impossible to exactly define the force needed for certain digging conditions. The objective of impedance control is to establish a desired dynamical relationship between the end-ef- .fectorbucket tipposition and the contact force. This dynamical relationship is referred to as the .target impedance. Let xt be the desired trajectory r of the end-effector. Typically, the target impedance is chosen as a linear second-order system to mimic mass-spring-damper dynamics: Zs e s M s2qB sqKe . tPtttP sM e qB e qK e se ,12. tPtPtPF where s is the derivative operator and the constant positive-definite n=n-matrices M , B and Kare ttt respectively the matrices of inertia, damping and stiffness. The position error, e , and the force error, P e , are defined as F e sx yx,e sF y yF,13. PrFre .where F t sM x qB x qK xis the force rtrtrtr set-point. The control problem is to asymptotically drive the .system state to implement the target impedance 12 even in the presence of uncertainty. If the position error eapproaches zero, the force error ealso PF approaches zero and vice versa, according to a speci- fied dynamical relationship defined by the numerical .values of the matrices M , B and K in Eq. 12 . In ttt some contact tasks, the force set-point, F , will be r specified to be constant rather than time-varying. During free-space motion where there is no contact with the environment, F syF s0, so etends to reP . 2 zero since Z s sM s qB sqKis stable. The tttt choice of the matrices M , B and K will determine ttt the shape of the desired transient response of the system. When the end-effector contacts the environ- ment, the interaction is characterised by the target .impedance 12 , which results in a compromise be- tween the position error and the force error. If the end-effector position tracks the desired trajectory .xxthen the contact force follows the force r .set-point yF F . er 3.2. Controller deelopment Consider a manipulator dynamical model of the . .form 5 with uncertainty satisfying condition 7 . It wxis well known 10 that robustness is the most distin- guished feature of variable structure control with sliding mode. In this section, a robust sliding mode controller will be developed for the manipulator . .dynamics5 . As Eq.5represents a 2n-dimen- ()Q.P. Ha et al.rAutomation in Construction 9 2000 421435425 sional system with an n-dimensionalcontrol input, a sliding surface in the state space will be a manifold wxwof dimension 2nynsn 10 . Let us define ss s1 . . .xTx , sx ,., sx, the sliding functions, as 2n wxfollows 11 : ssye yMy1B e yMy1Ke dtH PttPttP qMy1e dtsxyx ,14.H tFs where x sx qMy1B e qMy1Ke dtyMy1e dt.HH srttPttPtF 15. The existence of a sliding mode, ss0, requires that ssye yMy1B e yMy1K e qMy1e s0. PttPttPtF 16. It can be seen that once the system state is in the .sliding mode associated with Eq. 14 , condition 16 .guarantees that the target impedance 12 is reached. .Thus, in the sliding mode s x s0, is1,2,.,n , i .the force error tends to zero. The magnitudes s x i .is1,2,.,nrepresent then the deviations of the system state from the sliding surface. We assume further that: .A2: Each entry of the uncertainty Df x,x,xis bounded: D fx,x,xFb, . ii ; x,x,xis1,2,.,n .17. Let us now define the control input usuyQsgn s ,18 . where usHx qC x qG qF ,19. sxsxe T Qs Q sgn s,.,Q sgn s,. 11nn Q )bis1,2,.,n .20. ii .Remark 2. The control law18consists of two .components. The component19 , calculated with the nominal system dynamic model, is called the equialent control. The other component, with the .discontinuous gain given in Eq.20 , is called the switching control. ( )Theorem 1. Consider the system of Eq. 5 associ- ()ated with sliding functions14and the target ()impedance 12 . If the assumptions A1 and A2 are ()satisfied, and the control law 18 is employed, then ()the impedance error 14 asymptotically conerges to zero 9 . Implementation implies that sufficiently large switching gains, Q , are available. Large values of i Q will, however, tend to excite chattering. To accel- i erate the reaching phase and to reduce chattering, the .control law 18 is added by a tuning component: usuyQsgn s yKs,21 . where Ksdiag Ksis1,2,.,n .22. ii Employing the fuzzy tuning technique proposed in wx.Ref. 12 , the expressions for Ks )0 are chosen ii as: K sK1yexp y s rd . ii maxii is1,2,.,n ,23. where Kanddare some positive constants. i maxi ( )Theorem 2. Consider the system of Eq. 5 associ- ()ated with the sliding functions 14 and the target ()impedance 12 . If the assumptions A1 and A2 are ()satisfied, and the control law 21 is employed then ()the impedance error 14 asymptotically conerges to zero 9 . Remark 3. The switching component is for ensuring robust stability only. It can be omitted in practice. Note that from the geometry of the excavator, there exists a trigonometric mapping between each joint angleuand the corresponding linear displace- i ment y of each hydraulic cylinder piston, is2,3,4 i wx.13 . Using this relationship and Eq. 10 , the cylin- wxTder positions ys yyyand the ram forces