Dynamic+surface+control-backstepping+based+impedan.pdf

J. Cent. South Univ. Technol. 2010 17 807−815 DOI 10.1007/s11771−010−0560−y Dynamic surface control-backstepping based impedance control for 5-DOF flexible joint robots XIONG Gen-liang熊根良1, XIE Zong-wu谢宗武1, HUANG Jian-bin黄剑斌1, LIU Hong刘宏1, 2, JIANG Zai-nan蒋再男1, SUN Kui孙奎1 1. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China; 2. Institute of Robotics and Mechatronics, German Aerospace Center DLR, Wessling 82230, Germany Central South University Press and Springer-Verlag Berlin Heidelberg 2010 Abstract A new impedance controller based on the dynamic surface control-backstepping technique to actualize the anticipant dynamic relationship between the motion of end-effector and the external torques was presented. Comparing with the traditional backstepping that has “explosion of terms” problem, the new proposed control system is a combination of the dynamic surface control technique and the backstepping. The dynamic surface control DSC technique can resolve the “explosion of terms” problem that is caused by differential coefficient calculation in the model, and the problem can bring a complexity that will cause the backstepping hardly to be applied to the practical application, especially to the multi-joint robot. Finally, the validity of the was proved in the laboratory environment that was set up on the 5-DOF degree of freedom flexible joint robot. Tracking errors of DSC-backstepping impedance control that were 2.0 and 1.5 mm are better than those of backstepping impedance control which were 3.5 and 2.5 mm in directions X, Y in free space, respectively. And the anticipant Cartesian impedance behavior and compliant behavior were achieved successfully as depicted theoretically. Key words Cartesian impedance control; dynamic surface control; backstepping; PPSeCo; flexible joint robots 1 Introduction In contrast to traditional industrial robots, people in robotics domain transferred their interest into the service robot in the past several years, such as medical robots, mobile robots and exploration robots. These robots will work in the laboratory environments or practical environments, even in the space. The compliant behavior of the manipulator is predesigned whenever a robot is supposed to per some manipulation tasks such as picking and placing operation in practice. In order to achieve the compliant behavior by a control , an impedance control , a classical issue, which could provide a unified framework for achieving compliant behavior when robot contacted with an unknown environment, was brought up. Impedance control was theorized by HOGAN [1] and experimentally applied by KAZEROONI et al [2]. Based on a singular perturbation approach, flexible joint robot was controlled by impedance control [3], the feedback of the joint torques was therein considered as the control of a fast inner control loop that received its set point values from an outer impedance controller. HUANG et al [4] also proposed an impedance controller for the flexible joint robot based on the singular perturbation theory by DSP digital signal processing/ FPGA field programmable gate array hardware structure. However, the main drawback of the singular perturbation is lacking of theoretical justification for proving the stability due to the limitation of Tychonov’s theorem [5]. Therefore, ALIN and GERD [6] proposed Cartesian impedance control techniques for the torque control of light-weight flexible joint robots, using local stiffness control to enhance the impedance control. CHRISTIAN et al [7] developed decoupling based Cartesian impedance control of flexible joint robots and a al analyzed the stability of the proposed controller. CHRISTIAN et al [8−10] investigated the Cartesian impedance control of the light-weight flexible joint robots of DLR with torque feedback, gravity compensation and complete static states feedback, and proved asymptotical stability based on passivity theory. OZAWA and KOBAYASHI [11] proposed a new impedance control concept for elastic joint robots with programmable passive impedance devices in the transmission. The concept allows the user to use the same index for free motion and contacting task, but it Foundation item Project2006AA04Z228 supported by the National High-Tech Research and Development Program of China; ProjectPCSIRT supported by Program for Changjiang Scholars and Innovative Research Team in University Received date 2009−12−19; Accepted date 2010−04−01 Corresponding author XIONG Gen-liang, PhD; Tel 86−451−86412042; E-mail xgl.lijing J. Cent. South Univ. Technol. 2010 17 807−815 808 was applied to one-DOF elastic joint robots. GIANNI et al [12] presented an impedance controller for elastic joint industrial manipulators. Special attention was paid to all aspects that qualify an industrial robot, including decentralized proportional integral derivative position control, torsional flexibility, and friction at the joints, etc. CHIEN and HUANG [13] proposed a regressor-free adaptive impedance controller for an n-link flexible joint robot, the function approximation technique FAT was employed to trans the time-varying uncertainties into finite combinations of orthogonal basis functions. HUANG et al [14] proposed an adaptive impedance controller for flexible joint robot based on the friction model. The friction model includes viscous friction, payload and motor position based friction. The closed loop stability was investigated. LIU et al [15] investigated the Cartesian impedance control and nonlinear compensation for a harmonic drive robot based on joint torque sensors, and the imperfect Cubic model for harmonic drive friction was detected according to friction identification experiments. A Cartesian impedance control law was introduced by virtual decomposition [16] to realize the compliance control which incorporated with three means to make the flexible manipulator come into compliant contact with the objects. Compared with the above singular perturbations, passivity theories and decoupling s, the backstepping technique was barely applied to the flexible joint robot control except the bcakstepping on Cartesian impedance control of flexible joint manipulators [17] even though it represented a complete solution of impedance control problem for flexible joint robot model including the tracking case and inertia shaping. On the contrary, the backstepping technique was widely used to design output or state feedback controller for flexible joint robots in the tracking case [18−26]. Because of these issues, a new impedance controller based on dynamic surface control-backstepping to actualize the anticipated dynamical relationship between the motion of end-effector and the external torques was presented. The dynamic surface control technique was able to resolve the “explosion of terms” problem in backstepping controller. The Lyapunov function guaranteed global stability of the proposed controller. Moreover, a DSP/FPGA-FPGA hardware structure was established to support the proposed impedance controller. 2 Dynamics of flexible joint robots Generally, the reduced dynamic model of an n-link flexible joint robot refers to robot dynamics and actuator dynamics, which could be described as the following proposed by SPONG [27] ext , M q qC q q qg q , n n ∈M qR , ∈C q q n ∈Rτ τ denotes the joint torque; ext n ∈Rτ τdenotes the external torque that shown by the manipulator’s environment; m n ∈Rτ τ denotes the motor torque served as the control ; the constant positive definite diagonal matrices n n ∈KR and n n ∈BR represent the joint stiffness and the actuator inertia, respectively. Moreover, two well-known properties of the robot model utilized in the following sections are described as. Property 1 Link inertia matrix Mq is symmetric and positive definite TT , M qM qy M q y＞0, , 0 n ∀≠∈q yR 4 Property 2 Matrix 2 , −M qC q q and x2d, x3d and x4d denote the stabilizing functions for the subsystem consisting of dynamic surfaces S2, S3 and S4, respectively. According to Eq.15, the derivatives of Eq.16 could be expressed as follows 111d22d 1T 222d1112 T 3ext2d 333d43d 1 444dm34d [, ] −− − − ⎧−− ⎪ −−−⎪ ⎪ − ⎨ ⎪ −− ⎪ ⎪ −−−− ⎩ Sxxxx Sxxxx x xJ qg q J qxτx Sxxxx SxxKBBqxx ei denotes the boundary layer error; ˆ iiii −x xxx denotes the observation error; and i x ˆ denotes the the estimation value of xi. Lyapunov function 22 was differentiated with respect to time, and Si, ei, , i xtriangle inequality and property P2 were utilized. The derivative could be obtained as follows T 121112 T1T 21322212 TT 34333444424 3 TTTT 1121233 11 , , , [ ] iii i ii ii k k Vα α αα −− − − −− − ∑ extSeSx x J qSeSx SeSxSx e ex xxx xx x t SS S SS ∆, βi and ci represent positive constants; and λ, V represent design function. The derivative implied that , , , iii VextS mi represents the joint quality; Bi represents the joint damp; and Ki represents the joint stiffness. 5.2 DSP/FPGA-FPGA based hardware system The hardware system based on DSP/FPGA-FPGA as shown in Fig.3 was given to realize the proposed controller. In order to minimize cabling and weight of the 5-DOF flexible joint manipulator, a fully mechatronic design ology was introduced to develop the hardware system. All the analog signals were converted into proper digital signals and serially transmitted into joint FPGA board and further to PCI peripheral component interconnect-based central processor. The hardware system consisted of PCI-based DSP/FPGA board configured as a Cartesian level and joint FPGA board for five-joint control configured as a joint level. The control algorithm is illustrated in Fig.4. Joint’s FPGA board Slave took charge of the joint level controller, and a PCI-based DSP/FPGA board Master cuted as Cartesian level. In the joint control level, the FPGA technology was chosen to achieve a more flexible implementation of the joint controller with a high control rate and a small sized joint electronics. To implement real time control of the robot, the Table 1 Manipulator parameters Frame ai/mm αi/˚ di/mm θi/˚ mi/kg Bi/kgm2 Ki/Nmrad−1 {L1} 0 90 110.76 0 0.800 0.85 82.175 {L2} 530 0 0 0 1.090 0.85 68.236 {L3} 470 0 0 0 1.069 0.85 68.236 {L4} 0 −90 135.66 0 0.700 0.85 68.236 {L5} 0 90 75.26 0 0.700 0.85 68.236 J. Cent. South Univ. Technol. 2010 17 807−815 812 Fig.3 Hardware system based on DSP/FPGA-FPGA Fig.4 Block diagram of controller Cartesian level needed the feedback ination of positions, velocities and torques of the joints and calculated the required torques swiftly. At the same time, the joint level should update the data in time especially for the transient state. Therefore, a high speed data bus of point-to-point serial communication PPSeCo was designed for this requirement, in which the cycle time is less than 200 s and communication rate is up to 25 Mb/s. The communication and other control programs for FPGA were written in VHDL and run in FPGA. 5.3 Experiments To illustrate the validity of the proposed s, the following three experiments on manipulator were carried out. The first experiment was to use the 5-DOF flexible joint robot to track sine curve in X- and Y-directions, and remain static in Z-direction for the case of free motion Fext0 by backstepping impedance control and DSC-backstepping impedance control in Cartesian space. Fig.5 shows Cartesian coordinate positions and tracking errors in different directions under the s of backstepping impedance control BIC and DSC- backstepping impedance control DSC-BIC. From Figs.5b and d, it can be seen that the tracking errors were 3.5 and 2.5 in X- and Y-directions by backstepping impedance control, while the tracking errors were 2.0 and 1.5 mm in X- and Y-directions by DSC-backstepping J. Cent. South Univ. Technol. 2010 17 807−815 813 Fig.5 Cartesian coordinate positions and tracking errors in different directions a Positions in X-direction; b Errors in X-direction; c Positions in Y-direction; d Errors in Y-direction impedance control, respectively. The control parameters of two controllers are listed in Tables 2 and 3. According to Fig.5, the accuracy of the tracking errors was improved evidently by using DSC-backstepping impedance control. In addition, the trajectory tracking accuracy of impedance control was lower than that of position control because the impedance control sacrificed some Table 2 Parameters of backstepping impedance control Joint No. KP,i KD,i 1 33.955 3.480 2 27.354 2.804 3 27.354 2.257 4 27.354 2.325 5 27.354 2.325 Table 3 Parameters of DSC-backstepping impedance control Filter No. Positive design parameter αi ti/s 1 14.0 − 2 0.8 0.01 3 1.5 0.01 4 16.0 0.01 tracking accuracy. Moreover, the more flexible the joint, the lower the accuracy of track, even not meeting the requirements. The tracking accuracy would be improved when the joint with little flexibility was regarded as a rigid joint, but the impedance perance would be down. Therefore, the above experiment compromised the impedance perance and the tracking accuracy. And that the phenomenon of the error was asymmetric about X-axis in Fig.5 was caused by the imperfect gravity and friction compensation. The second experiment was Cartesian impedance experiment made on a 5-DOF flexible joint manipulator. Firstly, the robot was placed on a virtual equilibrium position CD[0, 0, 0]. The anticipant stiffness Ki and damping Di in Table 4 were used. Then, the robot was pulled in different directions shown in Fig.6. Finally, the robot overcame the gravity and returned to the CD as soon as the force was released. Fig.6 shows the corresponding Cartesian forces along with the Cartesian position. It could be concluded that the theoretic Cartesian impedance behavior was achieved successfully. In Cartesian impedance control, the desired Cartesian impedance stiffness could be set a small value J. Cent. South Univ. Technol. 2010 17 807−815 814 Table 4 Impedance parameters in workspace Coordinate Ki/Nm−1 Di/Nsm−1 X-axis 2 000 482.6 Y-axis 2 000 482.6 Z-axis 2 000 78.3 Fig.6 Variation of Cartesian coordinates in DSC-backstepping Cartesian impedance control a Position; b Force smaller than 20 N/m. Under this condition, the robot could be pushed freely using a small force, and when the force was removed, the robot could stay at the last position stably, which was called zero-force control. The third experiment, which was based on DSC-