**7. Conclusion**

The presented approaches for on-line obstacle avoidance for redundant manipulators are a) based on redundancy resolution at the velocity level or b) considering also the dynamics of the manipulator. The primary task is determined by the end-effector trajectories and for the obstacle avoidance the internal motion of the manipulator is used. The goal is to assign each point on the body of the manipulator, which is close to the obstacle, a motion component in a direction that is away from the obstacle.

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In the case of kinematic control (a) this is a velocity component. We have shown that it reasonable to define the avoiding motion in a one-dimensional operational space. In this way some singularity problems can be avoided when not enough "redundancy" is available locally. Additionally, the calculation of the pseudoinverse of the Jacobian matrix **J***<sup>o</sup>* is simpler as it includes scalar division instead of a matrix inversion. Using an approximate calculation of the avoiding velocities has its advantages computationally and it makes it easier to consider more obstacles simultaneously.

The second group of control algorithms (b) used in case b) is based on real or virtual forces. We compare three approaches regarding the sensors used to detect the obstacles: proximity sensors or vision, tactile sensors and no sensors. When proximity sensors are used we propose virtual forces strategy, where a virtual force component in a direction away from the obstacle is assigned to each point on the body of the manipulator, which is close to an obstacle. The algorithm based on the virtual forces avoids the problem of singular configurations and can be also easily applied when many obstacles are present. Additionally, the proposed control scheme enables us to use the null-space velocity controller for additional subtasks like the optimization of a performance criterion. Next, we have shown that under certain conditions obstacle avoidance can also be done without any information about the position and the size of the obstacles. This can be achieved by using a strategy that utilizes the self-motion caused by the contact forces to avoid an obstacle after the collision. Of course, an obstacle can be avoided only after a contact. The necessary prerequisite for this strategy to be effective is that the manipulator is backdrivable. As an alternative for the stiff systems we propose the use of tactile sensors. Here, a tactile sensor detects an obstacle and the controller generates the avoiding motion. The drawback of the last two control approaches is that they do not prevent the collision with the obstacle. Hence, they can only be applied if the collision occurs at a low speed so that the impact forces are not too high.

For the tasks where end-effector tracking is not essential for performing a given task we proposed a modified prioritized task control at the velocity level. The proposed approach enables a soft continuous transition between two different tasks. The obstacle-avoidance task only takes place when the desired movement approaches a given threshold, and then smoothly switches the priority of the tasks. The usefulness of this approach was shown on a two Kuka LWR robot to prevent a collision between them.

The computational efficiency of the proposed algorithms allows real-time application in an unstructured or time-varying environment. The simulations of highly redundant planar manipulators and the experiments on a four-link planar manipulator confirm that the proposed control algorithms assure an effective obstacle avoidance in an unstructured environment.

#### **8. References**


26 Will-be-set-by-IN-TECH

In the case of kinematic control (a) this is a velocity component. We have shown that it reasonable to define the avoiding motion in a one-dimensional operational space. In this way some singularity problems can be avoided when not enough "redundancy" is available locally. Additionally, the calculation of the pseudoinverse of the Jacobian matrix **J***<sup>o</sup>* is simpler as it includes scalar division instead of a matrix inversion. Using an approximate calculation of the avoiding velocities has its advantages computationally and it makes it easier to consider

The second group of control algorithms (b) used in case b) is based on real or virtual forces. We compare three approaches regarding the sensors used to detect the obstacles: proximity sensors or vision, tactile sensors and no sensors. When proximity sensors are used we propose virtual forces strategy, where a virtual force component in a direction away from the obstacle is assigned to each point on the body of the manipulator, which is close to an obstacle. The algorithm based on the virtual forces avoids the problem of singular configurations and can be also easily applied when many obstacles are present. Additionally, the proposed control scheme enables us to use the null-space velocity controller for additional subtasks like the optimization of a performance criterion. Next, we have shown that under certain conditions obstacle avoidance can also be done without any information about the position and the size of the obstacles. This can be achieved by using a strategy that utilizes the self-motion caused by the contact forces to avoid an obstacle after the collision. Of course, an obstacle can be avoided only after a contact. The necessary prerequisite for this strategy to be effective is that the manipulator is backdrivable. As an alternative for the stiff systems we propose the use of tactile sensors. Here, a tactile sensor detects an obstacle and the controller generates the avoiding motion. The drawback of the last two control approaches is that they do not prevent the collision with the obstacle. Hence, they can only be applied if the collision occurs at a low

For the tasks where end-effector tracking is not essential for performing a given task we proposed a modified prioritized task control at the velocity level. The proposed approach enables a soft continuous transition between two different tasks. The obstacle-avoidance task only takes place when the desired movement approaches a given threshold, and then smoothly switches the priority of the tasks. The usefulness of this approach was shown on a

The computational efficiency of the proposed algorithms allows real-time application in an unstructured or time-varying environment. The simulations of highly redundant planar manipulators and the experiments on a four-link planar manipulator confirm that the proposed control algorithms assure an effective obstacle avoidance in an unstructured

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more obstacles simultaneously.

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**8. References**

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13(3): 398 – 410.

two Kuka LWR robot to prevent a collision between them.


**12** 

*P.R. China* 

**Nonlinear Dynamic Control and Friction** 

**Compensation of Parallel Manipulators** 

Comparing with the serial ones, parallel manipulators have potential advantages in terms of high stiffness, accuracy and speed (Merlet, 2001). Especially the high accuracy and speed performances make the parallel manipulators widely applied to the following fields, like the pick-and-place operation in food, medicine, electronic industry and so on. At present, the key issues are the ways to meet the demand of high accuracy in moving process under the condition of high speed. In order to realize the high speed and accuracy motion, it's very

In literatures, there are two basic control strategies for parallel manipulators (Zhang et.al., 2007): kinematic control strategies and dynamic control strategies. In the kinematic control strategies, parallel manipulators are decoupled into a group of single axis control systems, so they can be controlled by a group of individual controllers. Proportional-derivative (PD) control(Ghorbel et.al., 2000; Wu et.al., 2002), nonlinear PD (NPD) control (Ouyang et.al., 2002; Su et.al., 2004), and fuzzy control (Su et.al., 2005) all belong to this type of control strategies. These controllers do not always produce high control performance, and there is no guarantee of stability at the high speed. Unlike the kinematic control strategies, full dynamic model of parallel manipulators is taken into account in the dynamic control strategies. So the nonlinear dynamics of parallel manipulators can be compensated and

The traditional dynamic control strategies of parallel manipulators are the augmented PD (APD) control and the computed-torque (CT) control (Li & Wu, 2004; Cheng et.al., 2003; Paccot et.al., 2009). In the APD controller (Cheng et.al., 2003), the control law contains the tracking control term and the feed-forward compensation term. The tracking control term is realized by the PD control algorithm. The feed-forward compensation term contains the dynamic compensation calculated by the desired velocity and desired acceleration on the basis of the dynamic model. Compared with the simple PD controller, the APD controller is a tracking control method. However, the feed-forward compensation can not restrain the trajectory disturbance effectively, thus the tracking accuracy of the APD controller will be decreased. In order to solve this problem, the CT controller including the velocity feed-back is proposed based on the PD controller (Paccot et.al., 2009). The CT control method yields a controller that suppresses disturbance and tracks desired trajectories uniformly in all configurations of the manipulators. Both the APD controller and the CT controller contain two parts including the PD control term and the dynamic compensation term. For the

important to design efficient control strategies for parallel manipulators.

better performance can be achieved with the dynamic strategies.

**1. Introduction** 

Weiwei Shang and Shuang Cong

*University of Science and Technology of China* 

