**5.1 Experiments of the ANPD controller**

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

parameters values parameters values parameters values parameters values *k* 508.7 <sup>21</sup> -16.4 <sup>12</sup> -18 23 *f* 500 *Bv*<sup>1</sup> 892.8 1 *d* -2.6 <sup>22</sup> 5.1 <sup>13</sup> -0.9 <sup>11</sup> *f* 1040.1 *Bv*<sup>2</sup> 1396.4 2 *d* -21.9 <sup>23</sup> 10 <sup>21</sup> *f* -268.0 12 *f* -309.0 *Bv*<sup>3</sup> 402.6 3 *d* 30

With the analysis of the identification of the nonlinear friction model, the corresponding work of the *Coulomb + viscous* friction model is much simpler. Substituting the *Coulomb + viscous* friction model (49) into (47), one can get a linear equation about the identified

*vvv c c c c c c a* **D K** *kf f f d f d f d f d f d f d f* **τ** (50)

2 2 2

*q u l*

0 0 0000

0 0 0000 0 0 00 00

3 3 3

*q u l*

1 1 1

*q ul*

*a*

For simplicity, parameter combinations *i ci d f* and *i ci d f* are viewed as identified parameters, and the coefficients *ui* and *li* of the parameters are determined by the following

There are 10 parameters to be identified in Eq. (50), but only three independent equations can be got for each sampling point. So a group of linear equations about the unknown parameters can be got with the sampling data of a continuous trajectory, then the Least

The identification experiment designed for the *Coulomb + viscous* friction model is the same with the nonlinear friction model discussed in section 4.2. Identification results of the

parameters values parameters values *k* 512.7 1 1*<sup>c</sup> d f* -261.7 *<sup>v</sup>*<sup>1</sup> *f* 1534.8 2 2*<sup>c</sup> d f* 212.6 *<sup>v</sup>*<sup>2</sup> *f* 1415.9 2 2*<sup>c</sup> d f* -256 *<sup>v</sup>*<sup>3</sup> *f* 1475.1 3 3*<sup>c</sup> d f* 179.2 1 1*<sup>c</sup> d f* 248.5 3 3*<sup>c</sup> d f* -129

As shown in Fig. 2, the actual experiment platform is a 2-DOF parallel manipulator with redundant actuation designed by Googol Tech. Ltd. in Shenzhen, China. It is equipped with

*T*

1 2 31 11 12 22 23 33 3

*a*

*a*

**K**

rules: 1, 0 *u l i i* when 0 *ai q* , and 0, 1 *u l i i* when 0 *ai q* .

Squares method is used to identify the unknown parameters.

Table 2. Identification results of the *Coulomb + viscous* friction model

*Coulomb + viscous* friction model are shown in Table 2.

<sup>11</sup> 0.7 22 *f* -61.8 13 *f* -867.7

Table 1. Identification results of the nonlinear friction model.

parameters as follows

**5. Experiments** 

where

The trajectory tracking control experiment is designed for the parallel manipulator to validate the ANPD controller. The desired trajectory of the end-effector is a straight line, the starting point is (0.22, 0.29) and the ending point is (0.37, 0.21), thus the motion distance is 0.17m. The profile of the desired velocity is an S-type curve (Cheng et.al., 2003). In the experiment, the low-speed and high-speed motions are both tested. For the low-speed motion, the max velocity is 0.2m/s, the max acceleration is 5m/s2, and the jerk is 200m/s3. For the high-speed motion, the max velocity is 0.5m/s, the max acceleration is 10m/s2, and the jerk is 400m/s3.

In order to implement the ANPD controller (17), the dynamic parameters in (18.a) and the friction parameters in (18.b) must be known. In the experiment, the nominal values of the dynamic parameters are used (Shang et.al., 2008). Then, with the known dynamic parameters, the friction parameters in the *Coulomb + viscous* friction model can be identified by the Least Squares method, as shown in Table 2. In fact, the control parameters in (18.c) are tuned and determined by the actual experiments. The procedures to tune the control parameters in (18.c) can be summarized as follows:


Nonlinear Dynamic Control and Friction Compensation of Parallel Manipulators 247

Fig. 3. Tracking errors of the end-effector at the low-speed: (a) X-direction; (b) Y-direction

Fig. 4. Tracking errors of the end-effector at the high-speed: (a) X-direction; (b) Y-direction. where ( ) *<sup>d</sup> x j* and ( ) *<sup>d</sup> y j* represent the X-direction and Y-direction position coordinates at the *j*th sampling point of the desired trajectory respectively; *x j* ( ) and *y*( )*j* represent the Xdirection and Y-direction position coordinates of the *j*th sampling point of the actual

The RSME results of the trajectory tracking experiment of the ANPD and APD controller are shown in Table 3. From the data of the RPE (reduced percentage of error) in Table 3, the ANPD controller can increase the position accuracy of the end-effector above 30%,

> APD 1.04×10-4 4.55×10-4 ANPD 7.22×10-5 2.78×10-4 RPE 30.6% 38.9%

In order to validate the NCT controller further, the trajectory tracking control experiment is designed for the parallel manipulator. Both the linear and circular trajectories in the



at slow-speed(m) at high-speed(m)

tracking error(m)

0 0.2 0.4 0.6 0.8 1 time(s)

> ANPD APD

0 0.1 0.2 0.3 0.4 0.5 time(s)

ANPD APD


0

tracking error(m)

1

2

x 10-4




trajectory respectively.

tracking error(m)

0

tracking error(m)

1

ANPD APD

0 0.2 0.4 0.6 0.8 1

ANPD APD

0 0.1 0.2 0.3 0.4 0.5 time(s)

compared with the conventional APD controller.

Table 3. RSME of the APD and ANPD controller

**5.2 Experiments of the NCT controller** 

time(s)

(a) (b)

(a) (b)

2 x 10-4

value of 1 is tuned to the half value of the maximum error, and the value of 2 is tuned to the half of the maximum error rate. This choice has good control performance and it's easy to implement.


Using the above procedures, the ANPD controller parameters are tuned as follow:

$$k\_p = 4500 \text{ , } k\_d = 470 \text{ , } \delta\_1 = 3 \times 10^{-4} \text{ , } \delta\_2 = 3 \times 10^{-3} \text{ , } \alpha\_1 = 0.7 \text{ , } \alpha\_2 = 1.1 \text{ , }$$

In order to make a comparison between the ANPD controller and the APD controller, the same tracking experiments are implemented on the parallel manipulator. We choose the APD controller is because it has nonlinear dynamics compensation and friction compensation. In the APD controller, the control input vector of the three actuated joints can be calculated as (Shang et.al., 2009)

$$\mathbf{r}\_a = (\mathbf{S}^T)^+ \left( \mathbf{M}\_c \ddot{\mathbf{q}}\_c^d + \mathbf{C}\_c \dot{\mathbf{q}}\_c^d + \mathbf{K}\_{lp} \mathbf{e} + \mathbf{K}\_{ld} \dot{\mathbf{e}} \right) + \mathbf{f}\_a \tag{51}$$

where **K***lp* and **K***ld* are both symmetric, positive definite matrices of constant gains. In the APD controller (51), **M***e* and **C***e* can be calculated with the nominal dynamic parameters, and *a***f** can be calculated with the values of the friction parameters shown in Table 2. The procedures of tuning parameters **K***lp* and **K***ld* in APD controller are similar to the procedures of tuning parameters *<sup>p</sup> k* and *<sup>d</sup> k* in ANPD controller. Thus, the tuning procedures (1) to (3) can be used to tune the parameters **K***lp* and **K***ld* .

The experiment results of the APD and ANPD controller are shown in Fig. 3-4. Fig. 3a and Fig. 3b are the tracking errors of the end-effector at the low-speed on the X-direction and Ydirection respectively. From the experimental curves, one can see that the ANPD controller can decrease the tracking errors during the whole motion process obviously, and the maximum error in the motion is smaller. Fig. 4a and Fig. 4b are the tracking errors of the end-effector at the high-speed on the X-direction and Y-direction respectively. One can find that the tracking errors are much smaller with the ANPD controller than with the APD controller, especially at the acceleration process. And one can conclude that, by the ANPD controller, the performance improvement of trajectory tracking accuracy at the high-speed is more obvious than at the low-speed.

Furthermore, to evaluate the performances of the two controllers, the root-square mean error (RSME) of the end-effector position is selected as the performance index

$$\begin{aligned} \text{RMSE} &= \sqrt{\frac{1}{N} \sum\_{j=1}^{N} \left( e\_x^2(j) + e\_y^2(j) \right)} \\ &= \sqrt{\frac{1}{N} \sum\_{j=1}^{N} \left( \left( \mathbf{x}^d(j) - \mathbf{x}(j) \right)^2 + \left( \mathbf{y}^d(j) - \mathbf{y}(j) \right)^2 \right)} \end{aligned} \tag{52}$$

6. For the parameters 1 1 and 2 1 , the proportional gain ( ) *<sup>p</sup> <sup>i</sup> k e* is a constant of *<sup>p</sup> k* , and the derivative gains ( ) *d i k e* is a constant of *<sup>d</sup> k* . Thus the NPD algorithm can be considered as the linear PD algorithm. So decrease the value of <sup>1</sup> ( <sup>1</sup> 0.5 1 ), and decrease the value of *<sup>p</sup> k* at the same time to improve the error curve further, and make tradeoffs between the two values. Using this step, one can get the nonlinear

7. Increase the value of <sup>2</sup> ( <sup>2</sup> 1 1.5 ), and decrease the value of *<sup>d</sup> k* at the same time to improve the error rate curve further, then make tradeoffs between the two values.

<sup>1</sup> 3 10 , <sup>3</sup>

In order to make a comparison between the ANPD controller and the APD controller, the same tracking experiments are implemented on the parallel manipulator. We choose the APD controller is because it has nonlinear dynamics compensation and friction compensation. In the APD controller, the control input vector of the three actuated joints can

> ( ) *T dd a ee ee lp ld a*

where **K***lp* and **K***ld* are both symmetric, positive definite matrices of constant gains. In the APD controller (51), **M***e* and **C***e* can be calculated with the nominal dynamic parameters, and *a***f** can be calculated with the values of the friction parameters shown in Table 2. The procedures of tuning parameters **K***lp* and **K***ld* in APD controller are similar to the procedures of tuning parameters *<sup>p</sup> k* and *<sup>d</sup> k* in ANPD controller. Thus, the tuning

The experiment results of the APD and ANPD controller are shown in Fig. 3-4. Fig. 3a and Fig. 3b are the tracking errors of the end-effector at the low-speed on the X-direction and Ydirection respectively. From the experimental curves, one can see that the ANPD controller can decrease the tracking errors during the whole motion process obviously, and the maximum error in the motion is smaller. Fig. 4a and Fig. 4b are the tracking errors of the end-effector at the high-speed on the X-direction and Y-direction respectively. One can find that the tracking errors are much smaller with the ANPD controller than with the APD controller, especially at the acceleration process. And one can conclude that, by the ANPD controller, the performance improvement of trajectory tracking accuracy at the high-speed is

Furthermore, to evaluate the performances of the two controllers, the root-square mean

2 2

*x y*

<sup>1</sup> () ()

<sup>1</sup> () () () ()

*d d*

*x j xj y j yj <sup>N</sup>*

2 2

(52)

error (RSME) of the end-effector position is selected as the performance index

1

*j N*

*RSME e j e j <sup>N</sup>*

*N*

1

*j*

procedures (1) to (3) can be used to tune the parameters **K***lp* and **K***ld* .

<sup>2</sup> 3 10 , 1 0.7 , 2 1.1

**τ S Mq Cq K e K e f** (51)

Using the above procedures, the ANPD controller parameters are tuned as follow:

and it's easy to implement.

be calculated as (Shang et.al., 2009)

more obvious than at the low-speed.

proportional gain of the ANPD controller.

4500 *<sup>p</sup> k* , <sup>470</sup> *<sup>d</sup> <sup>k</sup>* , <sup>4</sup>

value of 1 is tuned to the half value of the maximum error, and the value of 2 is tuned to the half of the maximum error rate. This choice has good control performance

Fig. 3. Tracking errors of the end-effector at the low-speed: (a) X-direction; (b) Y-direction

Fig. 4. Tracking errors of the end-effector at the high-speed: (a) X-direction; (b) Y-direction.

where ( ) *<sup>d</sup> x j* and ( ) *<sup>d</sup> y j* represent the X-direction and Y-direction position coordinates at the *j*th sampling point of the desired trajectory respectively; *x j* ( ) and *y*( )*j* represent the Xdirection and Y-direction position coordinates of the *j*th sampling point of the actual trajectory respectively.

The RSME results of the trajectory tracking experiment of the ANPD and APD controller are shown in Table 3. From the data of the RPE (reduced percentage of error) in Table 3, the ANPD controller can increase the position accuracy of the end-effector above 30%, compared with the conventional APD controller.


Table 3. RSME of the APD and ANPD controller

#### **5.2 Experiments of the NCT controller**

In order to validate the NCT controller further, the trajectory tracking control experiment is designed for the parallel manipulator. Both the linear and circular trajectories in the

Nonlinear Dynamic Control and Friction Compensation of Parallel Manipulators 249

Fig. 5. Linear trajectory tracking errors of the end-effector: (a) X-direction; (b) Y-direction

Fig. 6. Circular trajectory tracking errors of the end-effector: (a) X-direction; (b) Y-direction.

In order to compare with the compensation performances of the nonlinear friction model and the Coulomb + viscous friction model, the trajectory tracking experiments are implemented on the parallel manipulator. In the actual experiment, the augmented PD (APD) controller is designed in the task space for the parallel manipulator (Shang et.al., 2009). In the APD

> ( ) *T dd a ee ee lp ld a*

where the term *a***f** is the friction compensation calculated by the nonlinear friction model (46) with the parameter values in Table 1. Moreover, *a***f** can be calculated by the Coulomb + viscous friction model (49) with the parameter values in Table 2. If the term *a***f** is neglected in the APD controller (54), it means the friction compensation is not considered in the

Both the straight line and the circle in the task space are selected as the desired trajectory to study the friction compensation. For the straight line, the starting point is (0.22, 0.29) and the ending point is (0.37, 0.21), thus the motion distance is 0.17m. The profile of the desired velocity is trapezoidal curve. In the experiment, both the low-speed and high-speed motions are implemented. For the low-speed motion, the maximum velocity is 0.2m/s and the acceleration is 5m/s2. For the high-speed motion, the maximum velocity is 0.5m/s and the



**τ S Mq Cq K e K e f** (54)


0

tracking error(m)

0.5

1 x 10 -3

0 0.1 0.2 0.3 0.4

NCT CT

> NCT CT

time(s)

0 0.2 0.4 0.6 0.8

time(s)

0

tracking error(m)

5 x 10 -4

NCT CT

(a) (b)

NCT CT

(a) (b)

controller, the control input vector of the three active joints can be calculated as

controller and the friction is ignored in the parallel manipulator.




0

tracking error(m)

0.5 1 x 10 -3

0 0.1 0.2 0.3 0.4 time(s)

0 0.2 0.4 0.6 0.8

time(s)

**5.3 Experiments of nonlinear friction compensation** 


0

tracking error(m)

5 x 10 -4

workspace are selected as the desired trajectory. For the linear trajectory, the starting point is (0.22, 0.19) and the ending point is (0.35, 0.29), thus the motion distance is 0.164m. The velocity profile of the linear trajectory is an S-type curve (Cheng et.al., 2003), the max velocity is 0.5m/s, the max acceleration is 10m/s2, and the jerk is 400m/s3. For the circular trajectory with the constant speed of 0.5m/s, the center is (0.29, 0.25) and the starting point is (0.29, 0.31), thus the radius is 0.06m.

The actual implement of the NCT controller is similar to the ANPD controller, and the dynamic parameters in (33.a) and the friction parameters in (33.b) must be known. In the experiment, the nominal values are selected as the values of the actual dynamic parameters (Shang et.al., 2008). Then, with the known dynamic parameters, the friction parameters can be identified by the Least Squares method (Shang et.al., 2008). And the values of the control parameters in (33.c) are tuned and determined by the actual experiments. The tuning procedures for the ANPD controller can be used to tune the NCT controller. Using those procedures, the NCT controller parameters are tuned as follows: <sup>2400</sup> *<sup>p</sup> <sup>k</sup>* , 240 *<sup>d</sup> <sup>k</sup>* , 4 <sup>1</sup> 3 10 , <sup>3</sup> <sup>2</sup> 3 10 , 1 0.7 , 2 1.1 . Moreover, to demonstrate that the NCT controller can improve the tracking accuracy of the end-effector, experiments using the CT controller are carried out as comparison (Shang & Cong, 2009). The CT controller is chosen because it has friction compensation and feedback dynamics compensation. In the CT controller, the control input vector of the three active joints can be calculated as

$$\mathbf{r}\_a = \left(\mathbf{S}^T\right)^+ \left(\mathbf{M}\_e \ddot{\mathbf{q}}\_e^d + \mathbf{C}\_e \dot{\mathbf{q}}\_e + \mathbf{M}\_e \left(\mathbf{K}\_{l\eta} e + \mathbf{K}\_{ld} \dot{e}\right)\right) + \mathbf{f}\_a \tag{53}$$

where **K***lp* and **K***ld* are both symmetric, positive definite matrices of constant gains.

In the CT controller (53), the dynamic parameters in **M***e* and **C***<sup>e</sup>* , and the friction parameters in *a***f** are the same with these of the NCT controller. The procedures of tuning parameters of **K***lp* and **K***ld* in the CT controller are similar to the procedures of tuning parameters of *<sup>p</sup> k* and *<sup>d</sup> k* in the NCT controller. Thus, the tuning procedures (1), (2), and (3) can be used to tune the parameters of **K***lp* and **K***ld* . Using the above methods, the CT controller parameters are tuned as follows: **K***lp diag*20000, 20000 , **K***ld diag*150, 150 .

The tracking error curves of the end-effector controlled by the CT and NCT controller are shown in Fig. 5-6. Fig. 5 is the linear trajectory tracking errors of the end-effector on the Xdirection and Y-direction. From the experiment curves, one can see that the NCT controller can decrease the tracking errors during the whole motion process obviously, and the maximum error in the motion is smaller. Fig. 6 is the circular trajectory tracking errors of the end-effector on the X-direction and Y-direction. From the curves one can see, the tracking accuracy is improved obviously using the NCT controller, compared with the CT controller. The RSME results of the trajectory tracking experiment of the NCT and CT controller are shown in Table 4. From the data of the RPE in Table 4, the NCT controller can increase the position accuracy of the end-effector above 35%, compared with the conventional CT controller.


Table 4. RSME of the CT and NCT controller

workspace are selected as the desired trajectory. For the linear trajectory, the starting point is (0.22, 0.19) and the ending point is (0.35, 0.29), thus the motion distance is 0.164m. The velocity profile of the linear trajectory is an S-type curve (Cheng et.al., 2003), the max velocity is 0.5m/s, the max acceleration is 10m/s2, and the jerk is 400m/s3. For the circular trajectory with the constant speed of 0.5m/s, the center is (0.29, 0.25) and the starting point

The actual implement of the NCT controller is similar to the ANPD controller, and the dynamic parameters in (33.a) and the friction parameters in (33.b) must be known. In the experiment, the nominal values are selected as the values of the actual dynamic parameters (Shang et.al., 2008). Then, with the known dynamic parameters, the friction parameters can be identified by the Least Squares method (Shang et.al., 2008). And the values of the control parameters in (33.c) are tuned and determined by the actual experiments. The tuning procedures for the ANPD controller can be used to tune the NCT controller. Using those procedures, the NCT controller parameters are tuned as follows: <sup>2400</sup> *<sup>p</sup> <sup>k</sup>* , 240 *<sup>d</sup> <sup>k</sup>* , 4

controller can improve the tracking accuracy of the end-effector, experiments using the CT controller are carried out as comparison (Shang & Cong, 2009). The CT controller is chosen because it has friction compensation and feedback dynamics compensation. In the CT

> *T d <sup>a</sup> ee ee e lp ld a e e*

In the CT controller (53), the dynamic parameters in **M***e* and **C***<sup>e</sup>* , and the friction parameters in *a***f** are the same with these of the NCT controller. The procedures of tuning parameters of **K***lp* and **K***ld* in the CT controller are similar to the procedures of tuning parameters of *<sup>p</sup> k* and *<sup>d</sup> k* in the NCT controller. Thus, the tuning procedures (1), (2), and (3) can be used to tune the parameters of **K***lp* and **K***ld* . Using the above methods, the CT controller parameters are tuned as follows: **K***lp diag*20000, 20000 ,

The tracking error curves of the end-effector controlled by the CT and NCT controller are shown in Fig. 5-6. Fig. 5 is the linear trajectory tracking errors of the end-effector on the Xdirection and Y-direction. From the experiment curves, one can see that the NCT controller can decrease the tracking errors during the whole motion process obviously, and the maximum error in the motion is smaller. Fig. 6 is the circular trajectory tracking errors of the end-effector on the X-direction and Y-direction. From the curves one can see, the tracking accuracy is improved obviously using the NCT controller, compared with the CT controller. The RSME results of the trajectory tracking experiment of the NCT and CT controller are shown in Table 4. From the data of the RPE in Table 4, the NCT controller can increase the position accuracy of

> Line (m) Circle (m) CT 4.77×10-4 4.41×10-4 NCT 3.08×10-4 2.59×10-4 RPE 35.4% 41.3%

the end-effector above 35%, compared with the conventional CT controller.

Table 4. RSME of the CT and NCT controller

controller, the control input vector of the three active joints can be calculated as

where **K***lp* and **K***ld* are both symmetric, positive definite matrices of constant gains.

<sup>2</sup> 3 10 , 1 0.7 , 2 1.1 . Moreover, to demonstrate that the NCT

**τ S Mq Cq M K K f** (53)

is (0.29, 0.31), thus the radius is 0.06m.

<sup>1</sup> 3 10 , <sup>3</sup>

**K***ld diag*150, 150 .

Fig. 5. Linear trajectory tracking errors of the end-effector: (a) X-direction; (b) Y-direction

Fig. 6. Circular trajectory tracking errors of the end-effector: (a) X-direction; (b) Y-direction.

#### **5.3 Experiments of nonlinear friction compensation**

In order to compare with the compensation performances of the nonlinear friction model and the Coulomb + viscous friction model, the trajectory tracking experiments are implemented on the parallel manipulator. In the actual experiment, the augmented PD (APD) controller is designed in the task space for the parallel manipulator (Shang et.al., 2009). In the APD controller, the control input vector of the three active joints can be calculated as

$$\mathbf{r}\_a = (\mathbf{S}^T)^+ \left( \mathbf{M}\_c \ddot{\mathbf{q}}\_c^d + \mathbf{C}\_c \dot{\mathbf{q}}\_c^d + \mathbf{K}\_{lp} \mathbf{e} + \mathbf{K}\_{ld} \dot{\mathbf{e}} \right) + \mathbf{f}\_a \tag{54}$$

where the term *a***f** is the friction compensation calculated by the nonlinear friction model (46) with the parameter values in Table 1. Moreover, *a***f** can be calculated by the Coulomb + viscous friction model (49) with the parameter values in Table 2. If the term *a***f** is neglected in the APD controller (54), it means the friction compensation is not considered in the controller and the friction is ignored in the parallel manipulator.

Both the straight line and the circle in the task space are selected as the desired trajectory to study the friction compensation. For the straight line, the starting point is (0.22, 0.29) and the ending point is (0.37, 0.21), thus the motion distance is 0.17m. The profile of the desired velocity is trapezoidal curve. In the experiment, both the low-speed and high-speed motions are implemented. For the low-speed motion, the maximum velocity is 0.2m/s and the acceleration is 5m/s2. For the high-speed motion, the maximum velocity is 0.5m/s and the

Nonlinear Dynamic Control and Friction Compensation of Parallel Manipulators 251

Furthermore, the RSMEs of the trajectory tracking experiment are shown in Table 5. From the data in the table one can see, by using the two friction compensation methods, the RSMEs are much smaller than the method ignoring friction compensation. And the RSMEs of the friction compensation based on the nonlinear model are smaller than the friction

Without compensation 3.83×10-4 7.07×10-4 4.00×10-4 7.45×10-4 *Coulomb + viscous* model 1.03×10-4 4.56×10-4 1.29×10-4 6.68×10-4 Nonlinear model 8.88×10-5 2.71×10-4 9.53×10-5 4.49×10-4

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

This work was supported by the National Natural Science Foundation of China with Grant No. 50905172, the Anhui Provincial Natural Science Foundation with Grant No.090412040,

Cheng H., Yiu Y.K., Li Z.X. (2003) Dynamics and control of redundantly actuated parallel

Straight line motion (m) Circle motion (m) 0.2m/s 0.5m/s 0.2m/s 0.5m/s

compensation with the *Coulomb + viscous* model, especially when the speed is higher.

Friction compensation method

**6. Conclusions** 

**7. Acknowledgments** 

**8. References** 

Table 5. RSME of the trajectory tracking experiments

redundant actuation to realize high-speed and high accuracy motion.

and the Fundamental Research Funds for the Central Universities.

manipulators. *IEEE Trans. Mechatronics*, 8(4): 483-491

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

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*.

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

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

Furthermore, the RSMEs of the trajectory tracking experiment are shown in Table 5. From the data in the table one can see, by using the two friction compensation methods, the RSMEs are much smaller than the method ignoring friction compensation. And the RSMEs of the friction compensation based on the nonlinear model are smaller than the friction compensation with the *Coulomb + viscous* model, especially when the speed is higher.


Table 5. RSME of the trajectory tracking experiments
