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Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 13

Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 15.3 Examples 355 sinful Jo Mi 72Ỡ sinMi qd2 Jo ds Ớ1 O HMM ds 72Ỡ sinMi qd2 Jo 1 W2 q ds Ớ2 0 We describe next the laboratory experimental results. The initial conditions corresponding to the positions and velocities are chosen as 71 0 0 72 0 0 71 0 0 72 0 0 . The desired joint positions are chosen as .-dr -1- r -. 1 qdi 7T 10 7d2 70 30 rad . In terms of the state vector of the closed-loop equation the initial state is ộ 0 l 1Ao -0.3141cl _q 0 . 0 . 0 . l 0 . 0 . rad . Figure 15.1. Graphs of position errors 71 and 72 Figures 15.1 and 15.2 present the experimental results. In particular Figure 15.1 shows that the components of the position error ặ í tend asymptotically to zero in spite of the non-modeled friction phenomenon. The evolution in time of the adaptive parameters is shown 356 15 PD Control with Adaptive Desired Gravity Compensation Figure 15.2. Graphs of adaptive parameters 01 and 02 in Figure 15.2 where we appreciate that both parameters tend to values which are relatively near the unknown values of Ớ and ớ2 i.e. lim I rp Mt to 3.2902 0.1648 m2 _ 2.0458 m2 c2 0.047 Ớ1 Ớ2 As mentioned in Chapter 14 the latter phenomenon i.e. that 0 t -0 east-0 00 s called parametric convergence and the proof of this property rehes on a property called persistency of excitation. Verifying this property in applications is in general a difficult task and as a matter of fret often incemplex nonhnear adapliec control systems it may be expected that parameters do not converge to their true val . D. Similarlyas for PID coni rol itmay bepppreciatedfrom Figure 15.1 that the temporal evolution of the position errors is slow. Note that the timescale spans50s. Hence as for the erne of PID control the CranslenSresponte heheis alowtethaahhetunder I l hni ml with gravity compensation see Figure 7.3 or PD control with desired gravity compensation see Fi2ure 8.4 . As before if instead ef limiting the value of o we use thr same gains as for lh latter controhers the performance is improved