**6. Conclusions**

250 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

acceleration is 10m/s2. In the circle motion with constant speed, the center coordinates of the circle are (0.29,0.25) and radius is 0.04, also both the low-speed motion of 0.2m/s, and the

Linear trajectory tracking errors of the end-effector at the slow-speed and the high-speed are shown in Fig.7. From the curves one can see, the tracking errors are much smaller with the friction compensation methods based on the *Coulomb + viscous* model or the nonlinear model, compared with the without friction compensation method which means the term *a***f** is neglected in the APD controller (54). Especially, the maximum error at the acceleration process is decreased greatly with the friction compensation methods. Also one can see that, compared with the friction compensation based on the *Coulomb + viscous* model, the tracking accuracy has improved further at the low-speed, and the improvement at the high-speed is even more apparent by using the friction compensation based on the nonlinear model. Circular trajectory tracking errors of the end-effector at the slow-speed and the high-speed are shown in Fig.8. From the curves one can see, the tracking accuracy is improved obviously using the two friction compensation methods, compared with the without friction compensation method. Also one can see that, with the friction compensation based on the nonlinear model, the tracking error is decreased at the low-speed, and the improvement at the high-speed is more obvious than the friction compensation based on the *Coulomb + viscous*.

0 0.1 0.2 0.3 0.4

nonlinear Coulomb+viscous without compensation

time(s)

0 0.2 0.4 0.6

nonlinear Coulomb+viscous without compensation

time(s)

0

0

0.5

error(m)

1 x 10 -3

0.5

error(m)

1

1.5 x 10 -3

high-speed motion of 0.5m/s are implemented.

0 0.2 0.4 0.6 0.8 1.0

nonlinear Coulomb+viscous without compensation

(a) (b)

(a) (b)

nonlinear Coulomb+viscous without compensation

Fig. 7. Linear trajectory tracking error of the end-effector: (a) at the low-speed; (b) at the

Fig. 8. Circular trajectory tracking error of the end-effector: (a) at the low-speed; (b) at the

time(s)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

time(s)

0

0

high-speed

2

4

error(m)

6 x 10 -4

high-speed.

2

4

error(m)

6

8 x 10 -4 In order to realize the high-speed and high-accuracy motion control of parallel manipulator, nonlinear control method is used to improve the traditional dynamic controllers such as the APD controller and the CT controller. The common feature of the two controllers is eliminating tracking error by linear PD control, and the friction compensation is realized by using the Coulomb + viscous friction model. However, the linear PD control is not robust against the uncertain factors such as modeling error and external disturbance. To overcome this problem, the NPD control is combined with the conventional control strategies and two nonlinear dynamic controllers are developed. Moreover, a nonlinear model is used to construct the friction of the parallel manipulator, and the nonlinear friction can be compensated effectively. Our theory analysis implies that, the proposed controllers can guarantee asymptotic convergence to zero of both tracking error and error rate. And for its simple structures and design, the proposed controllers are easy to be realized for the industry applications of parallel manipulators. Our experiment results show that, the position error of the end-effector decrease obviously with the proposed controllers and the nonlinear friction compensation method, especially at the high-speed. So the nonlinear dynamic controller and nonlinear friction compensation can realize high-speed and highaccuracy trajectory tracking of the parallel manipulator in practice. Also these new methods can be used to other manipulators, such as serial ones, or parallel manipulator without redundant actuation to realize high-speed and high accuracy motion.
