Robot manipulators trends and development 2010 Part 10 doc

RobotManipulators,TrendsandDevelopment352 Fig. 16. Virtual impedance model for the mobile platform Fig. 17.Obstacle repulsion force The magnitude F obs is chosen to be (Borenstein & Koren, 1991):   2 F = a - b d(t) - d min obs obs obs where a obs and b obs are positive constants satisfying the condition 2 a = b (d - d ) max min obs obs , d max is the maximum distance between the robot and the detected obstacle that causes a nonzero repulsive force, d min represents the minimum distance accepted between the robot and the obstacle and d(t) is the distance measured between the robot and the obstacle d min < d(t) < d max ( Fig. 17). Note that the bound d max characterizes the repulsion zone. Which is inside the region where the repulsion force has a non-zero value. Desired interaction impedance is defined as the linear dynamic relationship Z d = B d s + K d where B d and K d are positive constants simulating the damping and the spring effects, respectively, involved in the robot obstacle interaction inside the repulsion zone. 3.4 Simulation results Simulations are conducted in order to show the performance of the proposed methodology. The numerical example supposes the lengths of the arm are such that a = 0.6, a = 0.4, a = 0.3 1 2 3 and the initial configuration of the mobile manipulator is such that: ξ = (0.1, 0.1, π/6) p and T /4]/2,/4,[ a q   . The end effector is supposed to track the following straight-line trajectory     T T ttttttt 1.01,1.0,1.0)(),(),()( * 3 * 2 * 1 *   ; Furthermore, we imposed the following additional tasks to the mobile platform * * * * ξ (t) =(x (t), y (t), (t)) = (t, t, π/4) p   . Fig. 18 shows the stance of the whole system when the end effector tracks the reference trajectory. The resulting trajectory of the end effector as well as that of the mobile plat form is depicted in Fig. 19. Figures 20, 21, 22 and 23describe the evolution of the angles of the arm and the orientation of the platform respectively. If the robot finds an obstacle at less than d = 1m max the impedance control is activated, and the collision is avoided as it can be seen in Fig. 24. -0.1 0 0.1 0.2 0. -0.4 -0.2 0 0.2 0.4 0 0.5 1 1.5 x y z Fig. 18. A 3D-view of the arm and the mobile platform evolutions in an obstacle free space. The resulted trajectories of the arm as well as of the mobile plat form appear in Fig. 26. The corresponding curves showing the evolution of the angles of the arm and the orientation of the platform are depicted in Figs. 27, 28, 29 and 30 respectively. TrajectoryGenerationforMobileManipulators 353 Fig. 16. Virtual impedance model for the mobile platform Fig. 17.Obstacle repulsion force The magnitude F obs is chosen to be (Borenstein & Koren, 1991):   2 F = a - b d(t) - d min obs obs obs where a obs and b obs are positive constants satisfying the condition 2 a = b (d - d ) max min obs obs , d max is the maximum distance between the robot and the detected obstacle that causes a nonzero repulsive force, d min represents the minimum distance accepted between the robot and the obstacle and d(t) is the distance measured between the robot and the obstacle d min < d(t) < d max ( Fig. 17). Note that the bound d max characterizes the repulsion zone. Which is inside the region where the repulsion force has a non-zero value. Desired interaction impedance is defined as the linear dynamic relationship Z d = B d s + K d where B d and K d are positive constants simulating the damping and the spring effects, respectively, involved in the robot obstacle interaction inside the repulsion zone. 3.4 Simulation results Simulations are conducted in order to show the performance of the proposed methodology. The numerical example supposes the lengths of the arm are such that a = 0.6, a = 0.4, a = 0.3 1 2 3 and the initial configuration of the mobile manipulator is such that: ξ = (0.1, 0.1, π/6) p and T /4]/2,/4,[ a q   . The end effector is supposed to track the following straight-line trajectory     T T ttttttt 1.01,1.0,1.0)(),(),()( * 3 * 2 * 1 *   ; Furthermore, we imposed the following additional tasks to the mobile platform * * * * ξ (t) =(x (t), y (t), (t)) = (t, t, π/4) p   . Fig. 18 shows the stance of the whole system when the end effector tracks the reference trajectory. The resulting trajectory of the end effector as well as that of the mobile plat form is depicted in Fig. 19. Figures 20, 21, 22 and 23describe the evolution of the angles of the arm and the orientation of the platform respectively. If the robot finds an obstacle at less than d = 1m max the impedance control is activated, and the collision is avoided as it can be seen in Fig. 24. -0.1 0 0.1 0.2 0. -0.4 -0.2 0 0.2 0.4 0 0.5 1 1.5 x y z Fig. 18. A 3D-view of the arm and the mobile platform evolutions in an obstacle free space. The resulted trajectories of the arm as well as of the mobile plat form appear in Fig. 26. The corresponding curves showing the evolution of the angles of the arm and the orientation of the platform are depicted in Figs. 27, 28, 29 and 30 respectively. RobotManipulators,TrendsandDevelopment354 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x-y plots of the end-effector and mobile platform x (m) y (m) end-effector mobile platform Fig. 19. End–effector and mobile platform trajectories in the x-y plane with no obstacles. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 The orientation qa1 of the arm time (sec) qa1 (rad) Fig. 20. Articulation q a1 curve 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 The orientation qa2 of the arm time (sec) qa2 (rad) Fig.21. Articulation q a2 curve 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 The orientation qa3 of the arm time (sec) qa3 (rad) Fig. 22. Articulation q a3 curve 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 The orientation  of platform time (sec)  (rad) Fig. 23. Articulation  curve -2 0 2 4 6 -1 0 1 2 3 4 5 6 0 0.5 1 1.5 x y obstacle Fig. 24. A 3D-View of the arm and the platform evolutions in presence of obstacles TrajectoryGenerationforMobileManipulators 355 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x-y plots of the end-effector and mobile platform x (m) y (m) end-effector mobile platform Fig. 19. End–effector and mobile platform trajectories in the x-y plane with no obstacles. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 The orientation qa1 of the arm time (sec) qa1 (rad) Fig. 20. Articulation q a1 curve 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 The orientation qa2 of the arm time (sec) qa2 (rad) Fig.21. Articulation q a2 curve 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 The orientation qa3 of the arm time (sec) qa3 (rad) Fig. 22. Articulation q a3 curve 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 The orientation  of platform time (sec)  (rad) Fig. 23. Articulation  curve -2 0 2 4 6 -1 0 1 2 3 4 5 6 0 0.5 1 1.5 x y obstacle Fig. 24. A 3D-View of the arm and the platform evolutions in presence of obstacles RobotManipulators,TrendsandDevelopment356 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x-y plots of the end-effector and mobile platform x (m) y (m) end-effector mibile platform Fig. 25. End–effector and mobile platform trajectories in the x-y plane in presence of obstacles 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -600 -500 -400 -300 -200 -100 0 100 The orientation qa1 of the arm time (sec) qa1 (rad) Fig. 26. Evolution curve of the joint 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 The orientation qa2 of the arm time (sec) qa2 (rad) Fig. 27. Evolution curve of the joint 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 The orientation qa3 of the arm time (sec) qa3 (rad) Fig. 28. Evolution curve of the joint 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 The orientation  of platform time (sec)  (rad) Fig. 29. Articulation  curve 4. Conclusion This work proposed two different methodologies to generating desired joint trajectories for both holonomic and non-holonomic mobile manipulators given prespecified operational tasks. The first part considers a non-holonomic platform where the generalized inverses in the resolution of a redundant system are used. The additional degrees of freedom are exploited to avoid unforeseen obstacles and joint limits. In the second part of the work a holonomic platfrom is used. In this case, the trajectory is generated using a reactive approach based on virtual impedance and additional tasks. When the robot task is about a stationary point, the mobile manipulator showed a good tracking for the manipulator. As perspective an estimate procedure must be conducted in order to estimate the contact forces and the unknown holonomic mobile manipulator parameters driving the system Computer TrajectoryGenerationforMobileManipulators 357 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x-y plots of the end-effector and mobile platform x (m) y (m) end-effector mibile platform Fig. 25. End–effector and mobile platform trajectories in the x-y plane in presence of obstacles 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -600 -500 -400 -300 -200 -100 0 100 The orientation qa1 of the arm time (sec) qa1 (rad) Fig. 26. Evolution curve of the joint 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 The orientation qa2 of the arm time (sec) qa2 (rad) Fig. 27. Evolution curve of the joint 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 The orientation qa3 of the arm time (sec) qa3 (rad) Fig. 28. Evolution curve of the joint 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 The orientation  of platform time (sec)  (rad) Fig. 29. Articulation  curve 4. Conclusion This work proposed two different methodologies to generating desired joint trajectories for both holonomic and non-holonomic mobile manipulators given prespecified operational tasks. The first part considers a non-holonomic platform where the generalized inverses in the resolution of a redundant system are used. The additional degrees of freedom are exploited to avoid unforeseen obstacles and joint limits. In the second part of the work a holonomic platfrom is used. In this case, the trajectory is generated using a reactive approach based on virtual impedance and additional tasks. When the robot task is about a stationary point, the mobile manipulator showed a good tracking for the manipulator. As perspective an estimate procedure must be conducted in order to estimate the contact forces and the unknown holonomic mobile manipulator parameters driving the system Computer RobotManipulators,TrendsandDevelopment358 simulations have validated to show the effectiveness of the two approaches. The reference values obtained by the two methods can be used as inputs to controllers for real mtion. 5. References Khatib, O. (1986). Real-time obstacle avoidance for manipulators and mobile robots, International Journal of Robotics Research, 5(1):90{98. Sundar, S. & Shiller, Z. (1997). Optimal obstacle avoidance based on the Hamilton-Jacobi- Bellman equation, IEEE Trans.on Robotics and Automation, Vol. 13, pp. 305{310. Laumond, J. P., Jacobs, P. E., Taix, M., and Murray, R. M. (1994). A motion planner for nonholonomic mobile robots, IEEE Trans. on Robotics and Automation, Vol. 10, pp. 577{593. Reeds, J. A. and Shepp, R. A. (1990). Optimal paths for a car that goes both forward and backward, Pacific J. Math., vol. 145, pp. 367–393. Murray, R. M.; Li, Z. & Sastry, S. S. (1994). A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press. Tilbury, D.; Murray, R. M. & Sastry, S. S. (1995). Trajectory generation for the n-trailer problem using goursatnormal form, IEEE Trans. Automat. Contr., vol. 40, pp. 802– 819, May 1995. Abdessemed, F. Monacelli, E. & Benmahammed, K. (2008). Trajectory Generation In an Alternated and a Coordinated Motion Control Modes of a Mobile Manipulator, AMSE journal, Modelling, Measurements and Control B, Vol.77, No 1, pp 18-34. Djebrani, S. Benali, A. & Abdessemed, F. (2009). Force-position control of a holonomic mobile manipulator, 12 int. Conf. on Climbing & Walking Robotsand the support technologis for Mobile Machines Bogazaci Univ. Garanti Culture Center (North Campus). Qu, Z.; Wang, J. & Plaisted, C. E. (2004). A New Analytical Solution to Mobile Robot Trajectory Generation in the Presence of Moving Obstacles, IEEE Tran. on Robotics, Vol. 20, No. 6. Kant, K. & Zucker, S. W. (1988). Planning collision free trajectories in time varying environments: A two-level hierarchy, in Proc. IEEE Int. Conf. Robotics and Automation , Raleigh, NC, pp. 1644–1649. Murray, R. M. & Sastry, S. S. (1993). Nonholonomic motion planning: Steering using sinusoids, IEEE Trans. Automat. Contr., vol. 38, pp. 700–716. Abdessemed, F.; Benmahammed, K. & Eric Monacelli (2004). A Fuzzy Based Reactive Controller for Non-Holonomic Mobile Robot, Journal of Robotics and Autonomous Systms , 47 (2004) 31-46. Russell, S. & Norvig, P. (2000). Artificial Intelligence: A Modern Approach, Prentice Hall, New Jersey, 1995 A. Okabe, B. Boots, K. Sugihara and S.N. Chiu, Spatial Tessellations and Applications of Voronoi Diagrams, John Wiley & Sons, New York. Zhao, M.; Ansari, N. & Hou, E.S.H. (1994). Mobile manipulator path planning by a genetic algorithm, Journal of Robotic Systems, 11(3): 143-153. Pin, F. G. & Culioli, J. C. (1992). Optimal Positioning of Combined Mobile Platform- Manipulator systems for Material Handling Tasks, Journal of intelligent and Robotic Systems. 6: 165-182. Pin, F. G.; Morgansen, K. A.; Tulloch, F. A.; Hacker, C. J. & Gower, K. B. (1996). Motion Planning for Mobile Manipulators with a Non-Holonomic Constraint Using the FSP (Full Space Parameterization) Method, Journal of Robotic Systems 13(11), 723-736. Lee, J. K. & Cho, H. S. (1997). Mobile manipulator Motion Planning for Multiple Tasks Using Global Optimization Approach, Journal of Intelligent and Robotic Systems, 18: 169-190. Seraji, H. (1995) Configuration control of rover-mounted manipulators, IEEE Int. Conf. on Robotics and Automation, pp2261-2266. Campion, G.; Bastin, B. & D'Andrea-Novel. (1996). Structural proprieties and classifcation of kinematic and dynamic models of wheeled mobile robots. IEEE Trans. on Robotics and Automation , 2(1):47{62, February. Liegeois, A. (1997). Automatic supervisory control of the configuration and behavior of multibody mechanisms, IEEE Trans. Syst. Man Cybernet. 7, 842-868. Seraji, H. (1993). An on-line approach to coordinated mobility and manipulation, ICRA’93, pp. 28-35, May, 1993. Mourioux, G.; Novales, C.; Poisson, G. & Vieyres, P. (2006). Omni-directional robot with spherical orthogonal wheels: concepts and analyses, IEEE International Conference on Robotics and Automation, pp. 3374-3379. Seraji, H. (1998). A unified approach to motion control of mobile manipulators, The International Journal of Robotics Research , vol. 17, no. 2, pp. 107-118. Bayle, B.; Fourquet, J. Y.; Lamiraux, F. & Renaud, M. (2002). Kinematic control of wheeled mobile manipulators, IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1572-1577. Arai, T. & Ota, J. (1996). Motion planning of multiple mobile robots using virtual impedance, Journal of Robotics and Mechatronics, vol. 8, no. 1, pp. 67-74. Borenstein, J. & Koren, Y. (1991). The vector field histogram fast obstacle avoidance for mobile robots, IEEE Transactions on Robotics and Automation, vol. 7, no. 3, pp. 278- 288. TrajectoryGenerationforMobileManipulators 359 simulations have validated to show the effectiveness of the two approaches. The reference values obtained by the two methods can be used as inputs to controllers for real mtion. 5. References Khatib, O. (1986). Real-time obstacle avoidance for manipulators and mobile robots, International Journal of Robotics Research, 5(1):90{98. Sundar, S. & Shiller, Z. (1997). Optimal obstacle avoidance based on the Hamilton-Jacobi- Bellman equation, IEEE Trans.on Robotics and Automation, Vol. 13, pp. 305{310. Laumond, J. P., Jacobs, P. E., Taix, M., and Murray, R. M. (1994). A motion planner for nonholonomic mobile robots, IEEE Trans. on Robotics and Automation, Vol. 10, pp. 577{593. Reeds, J. A. and Shepp, R. A. (1990). Optimal paths for a car that goes both forward and backward, Pacific J. Math., vol. 145, pp. 367–393. Murray, R. M.; Li, Z. & Sastry, S. S. (1994). A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press. Tilbury, D.; Murray, R. M. & Sastry, S. S. (1995). Trajectory generation for the n-trailer problem using goursatnormal form, IEEE Trans. Automat. Contr., vol. 40, pp. 802– 819, May 1995. Abdessemed, F. Monacelli, E. & Benmahammed, K. (2008). Trajectory Generation In an Alternated and a Coordinated Motion Control Modes of a Mobile Manipulator, AMSE journal, Modelling, Measurements and Control B, Vol.77, No 1, pp 18-34. Djebrani, S. Benali, A. & Abdessemed, F. (2009). Force-position control of a holonomic mobile manipulator, 12 int. Conf. on Climbing & Walking Robotsand the support technologis for Mobile Machines Bogazaci Univ. Garanti Culture Center (North Campus). Qu, Z.; Wang, J. & Plaisted, C. E. (2004). A New Analytical Solution to Mobile Robot Trajectory Generation in the Presence of Moving Obstacles, IEEE Tran. on Robotics, Vol. 20, No. 6. Kant, K. & Zucker, S. W. (1988). Planning collision free trajectories in time varying environments: A two-level hierarchy, in Proc. IEEE Int. Conf. Robotics and Automation , Raleigh, NC, pp. 1644–1649. Murray, R. M. & Sastry, S. S. (1993). Nonholonomic motion planning: Steering using sinusoids, IEEE Trans. Automat. Contr., vol. 38, pp. 700–716. Abdessemed, F.; Benmahammed, K. & Eric Monacelli (2004). A Fuzzy Based Reactive Controller for Non-Holonomic Mobile Robot, Journal of Robotics and Autonomous Systms , 47 (2004) 31-46. Russell, S. & Norvig, P. (2000). Artificial Intelligence: A Modern Approach, Prentice Hall, New Jersey, 1995 A. Okabe, B. Boots, K. Sugihara and S.N. Chiu, Spatial Tessellations and Applications of Voronoi Diagrams, John Wiley & Sons, New York. Zhao, M.; Ansari, N. & Hou, E.S.H. (1994). Mobile manipulator path planning by a genetic algorithm, Journal of Robotic Systems, 11(3): 143-153. Pin, F. G. & Culioli, J. C. (1992). Optimal Positioning of Combined Mobile Platform- Manipulator systems for Material Handling Tasks, Journal of intelligent and Robotic Systems. 6: 165-182. Pin, F. G.; Morgansen, K. A.; Tulloch, F. A.; Hacker, C. J. & Gower, K. B. (1996). Motion Planning for Mobile Manipulators with a Non-Holonomic Constraint Using the FSP (Full Space Parameterization) Method, Journal of Robotic Systems 13(11), 723-736. Lee, J. K. & Cho, H. S. (1997). Mobile manipulator Motion Planning for Multiple Tasks Using Global Optimization Approach, Journal of Intelligent and Robotic Systems, 18: 169-190. Seraji, H. (1995) Configuration control of rover-mounted manipulators, IEEE Int. Conf. on Robotics and Automation, pp2261-2266. Campion, G.; Bastin, B. & D'Andrea-Novel. (1996). Structural proprieties and classifcation of kinematic and dynamic models of wheeled mobile robots. IEEE Trans. on Robotics and Automation , 2(1):47{62, February. Liegeois, A. (1997). Automatic supervisory control of the configuration and behavior of multibody mechanisms, IEEE Trans. Syst. Man Cybernet. 7, 842-868. Seraji, H. (1993). An on-line approach to coordinated mobility and manipulation, ICRA’93, pp. 28-35, May, 1993. Mourioux, G.; Novales, C.; Poisson, G. & Vieyres, P. (2006). Omni-directional robot with spherical orthogonal wheels: concepts and analyses, IEEE International Conference on Robotics and Automation, pp. 3374-3379. Seraji, H. (1998). A unified approach to motion control of mobile manipulators, The International Journal of Robotics Research , vol. 17, no. 2, pp. 107-118. Bayle, B.; Fourquet, J. Y.; Lamiraux, F. & Renaud, M. (2002). Kinematic control of wheeled mobile manipulators, IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1572-1577. Arai, T. & Ota, J. (1996). Motion planning of multiple mobile robots using virtual impedance, Journal of Robotics and Mechatronics, vol. 8, no. 1, pp. 67-74. Borenstein, J. & Koren, Y. (1991). The vector field histogram fast obstacle avoidance for mobile robots, IEEE Transactions on Robotics and Automation, vol. 7, no. 3, pp. 278- 288. RobotManipulators,TrendsandDevelopment360 [...]... Control for Robotic Manipulators, Automatica, Vol.38, pp.235-242 Clark, C M.; Mills, J K (2000) Robotic System Sensitivity to Neural Network Learning Rate: Theory, Simulation, and Experiments, The International Journal of Robotics Research, Vol.19,No .10, pp.955-968 376 Robot Manipulators, Trends and Development Performance Evaluation of Autonomous Contour Following Algorithms for Industrial Robot 377... few points such as approach, insert and depart points, contour following for painting, arc welding and sealing application requires a large number of points recorded and at the best location After all the best program and process 378 Robot Manipulators, Trends and Development parameters are achieved for one sample part, the same quality is expected for the subsequent parts in a batch This expectation... for the third joint can be designed independently and easily Fig 6 AdeptOne robot manipulator 370 Robot Manipulators, Trends and Development Focusing on control of the most complex part of the robot, we do not take the third joint into consideration in the control design The fourth joint is extremely light-weight designed comparing with other joints and its link length is zero Fourth joint does not... Simulation results: planned joint trajectories and tracking results Trajectory Control of Robot Manipulators Using a Neural Network Controller Fig 8 Simulation results: planned joint velocity trajectories and tracking results Fig 9 Simulation results: control inputs of joint 1 Fig 10 Simulation results: control inputs of joint 2 371 372 Robot Manipulators, Trends and Development 6.3 Trajectory tracking control... main role in generating actuation voltages for the robot On the other hand, in the experimental 374 Robot Manipulators, Trends and Development results Fig.13 and Fig.14 “the roles changing” seems not as evident as in their simulation counterparts The reason lies on a fact that we had added the dead-zone compensating inputs into PD control inputs Standing on the dynamics point of view, with the same... model errors and uncertain parameters, and a great number of research results have been reported, for example, (Hsia, 1989), (Kou, and Wang, 1989), (Slotine and Li, 1989), ( Spong, 1992), and (Cheah, Liu and Slotine, 2006) However, almost parts of results associate with complicated control system design approaches and difficulties in the control system implementation for industrial robot manipulators. .. effectiveness and usefulness of the proposed control system 2 Dynamic models of robot manipulators 2.1 Torque-based dynamic model A torque-based dynamic model of robot manipulator describes relationship between motion and joint torque of the robot without concerning what generates the torque and how This class of dynamic formulation is most popular and widely used in the control design and simulation of the robot. .. given as follows u  ul  un u l is control input of the PID controller, and can be simply described as below (8) 366 Robot Manipulators, Trends and Development t    u l   k v (θ  θ d )  k p (θ  θd )  k i (θ  θ d ) dt (9) 0  where θ d and θ d are planned trajectories of joint displacements and velocities, k v , k p , and k i are gain matrices u n is the control input of the neural network... for Robots with Uncertainties in Kinematic, Dynamic and Actuator Models, IEEE Transactions on Automatic Control, Vol.51 No.6, pp .102 4 -102 9 Kwan, C.; Lewis, F L (2000) Robust backstepping control of nonlinear systems using neural networks, IEEE Transactions on Systems, Man, and Cybernetics Part A:Systems and Humans., Vol 30, No 6, pp.753-766 Sanger, T D (1994) Neural network learning control of robot manipulators. .. present a simple control system consisting of a traditional controller and a neural network controller with parallel structure for trajectory tracking control of 362 Robot Manipulators, Trends and Development industrial robot manipulators First, a PD controller is designed Second, a neural network with three layers is designed and added to the control system in the parallel way to the PD controller . avoidance for mobile robots, IEEE Transactions on Robotics and Automation, vol. 7, no. 3, pp. 278- 288. Robot Manipulators, Trends and Development3 60 TrajectoryControlof Robot Manipulators UsingaNeuralNetworkController. nonlinear and complicated systems such as robot manipulators (Sanger, 1994), (Kim and Lewis, 1999), (Kwan and Lewis, 2000), (Jung and Yim, 2001) (Yu and Wang, 2001). A new field in robot control. control of 16 Robot Manipulators, Trends and Development3 62 industrial robot manipulators. First, a PD controller is designed. Second, a neural network with three layers is designed and added