Optimal Trajectory of Underwater Manipulator Using Adjoint Variable Method for Reducing Drag

Author(s)
Kazunori Shinohara

ABSTRACT

In order to decrease the fluid drag on an underwater robot manipulator, an optimal trajectory method based on the variational method is presented. By introducing the adjoint variables, which are Lagrange multipliers, we formulate a Lagrange function under certain constraints related to the target angle, target angular velocity, and dynamic equation of the robot manipulator. The state equation (the partial differentiation of the Lagrange function with respect to the state variables), adjoint equation (the partial differentiation of the Lagrange function with respect to the adjoint variables), and sensitivity equation (the partial differentiation of the Lagrange function with respect to torques) can be derived from the stationary conditions of the Lagrange function. Using the state equation, we can calculate the state variables (angles, angular velocities, and angular acceleration) at every time step in the forward time direction. These state variables are stored as data at every time step. Next, by using the adjoint equation, we can calculate the adjoint variables by using these state variables at every time step in the backward time direction. These adjoint variables are stored as data at every time step. Third, the sensitivity equation is calculated by using both the state variables and the adjoint variables. Finally, the optimal trajectory of the manipulator is obtained using the sensitivities. The proposed method is applied to the problem of two-link manipulators. It can obtain the optimal drag reduction trajectory of the manipulator under the constraints mentioned above.

In order to decrease the fluid drag on an underwater robot manipulator, an optimal trajectory method based on the variational method is presented. By introducing the adjoint variables, which are Lagrange multipliers, we formulate a Lagrange function under certain constraints related to the target angle, target angular velocity, and dynamic equation of the robot manipulator. The state equation (the partial differentiation of the Lagrange function with respect to the state variables), adjoint equation (the partial differentiation of the Lagrange function with respect to the adjoint variables), and sensitivity equation (the partial differentiation of the Lagrange function with respect to torques) can be derived from the stationary conditions of the Lagrange function. Using the state equation, we can calculate the state variables (angles, angular velocities, and angular acceleration) at every time step in the forward time direction. These state variables are stored as data at every time step. Next, by using the adjoint equation, we can calculate the adjoint variables by using these state variables at every time step in the backward time direction. These adjoint variables are stored as data at every time step. Third, the sensitivity equation is calculated by using both the state variables and the adjoint variables. Finally, the optimal trajectory of the manipulator is obtained using the sensitivities. The proposed method is applied to the problem of two-link manipulators. It can obtain the optimal drag reduction trajectory of the manipulator under the constraints mentioned above.

KEYWORDS

Robot Manipulator Dynamics, Optimal Trajectory, Adjoint Variable Method, Euler-Lagrange Equation, Fluid Drag Force, Calculus of Variations

Robot Manipulator Dynamics, Optimal Trajectory, Adjoint Variable Method, Euler-Lagrange Equation, Fluid Drag Force, Calculus of Variations

Cite this paper

nullK. Shinohara, "Optimal Trajectory of Underwater Manipulator Using Adjoint Variable Method for Reducing Drag,"*Open Journal of Discrete Mathematics*, Vol. 1 No. 3, 2011, pp. 139-152. doi: 10.4236/ojdm.2011.13018.

nullK. Shinohara, "Optimal Trajectory of Underwater Manipulator Using Adjoint Variable Method for Reducing Drag,"

References

[1] T. L. McLain, S. M. Rock, and M. J. Lee, “Experiments in the Coordinated Control of an Underwater Arm/Vehicle System,” Autonomous Robots, Vol. 3, No. 2-3, 1996, pp. 213-232. doi:10.1007/BF00141156

[2] K. N. Leabourne and S. M. Rock, “Model Development of an Underwater Manipulator for Coordinated Arm-Vehicle Control,” Proceedings of the OCEANS 98 Conference, Nice France, No. 2, 1998, pp. 941-946.

[3] J. Yuh, S. Zhao and P. M. Lee, “Application of Adaptive Disturbance Observer Control to an Underwater Manipulator,” International Conference on Robotics and Automation, Vol. 4, 2001, pp. 3244-3249.

[4] S. Sagara, T. Tanikawa, M. Tamura and R. Katoh, “Experiments on a Floating Underwater Robot with a Two-Link Manipulator,” Artificial Life and Robotics, Vol. 5, No. 4, 2001, pp. 215-219. doi:10.1007/BF02481505

[5] G. R. Vossoughi, A. Meghdari and H. Borhan, “Dynamic Modeling and Robust Control of an Underwater ROV Equipped with a Robotic Manipulator Arm,” 2004 Japan USA Symposium on Flexible Automation, Denver USA, 2004.

[6] K. Ioi and K. Itoh, “Modelling and Simulation of an Underwater Manipulator,” Advanced Robotics, Vol. 4, No. 4, 1989, pp. 303-317. doi:10.1163/156855390X00152

[7] M. L. Nagurka and V. Yen “Optimal Design of Robotic Manipulator Trajectories: A Nonlinear Programming Approach,” Technical Report CMU-RI-TR-87-12, the Robotics Institute, Carnegie Mellon University, April, 1987.

[8] M. Zefran, “Review of the Literature on Time-Optimal Control of Robotic Manipulators,” Technical Report MS- CIS-94-30, University of Pennsylvania, Philadelphia, 1994.

[9] M. E. Kahn and B. Roth, “The Near Minimum-Time Control of Open-Loop Articulated Kinematic Chains,” Journal of Dynamic Systems, Measurement and Control, Vol. 93, No. 3, 1971, pp. 164-172. doi:10.1115/1.3426492

[10] M. Vukobratovi? and M. Kir?anski, “A Method for Optimal Synthesis of Manipulation Robot Trajectories,” Journal of Dynamic Systems, Measurement, and Control, Vol. 104, No. 2, 1982, pp. 188-193. doi:10.1115/1.3139695

[11] M. A. Townsend, “Optimal Trajectories and Controls for Systems of Coupled Rigid Bodies with Application of Biped Locomotion,” Thesis (Ph.D.), University of Wisconsin Madison, Madison, 1971.

[12] Y. D. Lee and B. H. Lee, “Genetic Trajectory Planner for a Manipulator with Acceleration Parameterization,” Journal of Universal Computer Science, Vol. 3, No. 9, 1997, pp. 1056-1073.

[13] D. Constantinescu and E. A. Croft, “Smooth and Time-Optimal Trajectory Planning for Industrial Manipulators Along Specified Paths,” Journal of Robotic Systems, Vol. 17, No. 5, 2000, pp. 233-249. doi:10.1002/(SICI)1097-4563(200005)17:5<233::AID-ROB1>3.0.CO;2-Y

[14] E. Shintaku, “Minimum Energy Trajectory for an Underwater Manipulator and Its Simple Planning Method by Using a Genetic Algorithm,” Advanced Robotics, Vol. 13, No. 6-13, 1999, pp. 115-138.

[15] Y. Uno, M. Kawato and R. Suzuki, “Formation and Control of Optimal Trajectory in Human Multijoint Arm Movement,” Biological cybernetics, Vol. 61, No. 2, 1989, pp. 89-101. doi:10.1007/BF00204593

[16] J. Saleh, “Fluid Flow Handbook,” McGraw Hill, New York, 2002.

[17] H. Kobayashi et al., “The Robot Control Actually,” The Society of Instrument and Control Engineers, Tokyo, 1997.

[1] T. L. McLain, S. M. Rock, and M. J. Lee, “Experiments in the Coordinated Control of an Underwater Arm/Vehicle System,” Autonomous Robots, Vol. 3, No. 2-3, 1996, pp. 213-232. doi:10.1007/BF00141156

[2] K. N. Leabourne and S. M. Rock, “Model Development of an Underwater Manipulator for Coordinated Arm-Vehicle Control,” Proceedings of the OCEANS 98 Conference, Nice France, No. 2, 1998, pp. 941-946.

[3] J. Yuh, S. Zhao and P. M. Lee, “Application of Adaptive Disturbance Observer Control to an Underwater Manipulator,” International Conference on Robotics and Automation, Vol. 4, 2001, pp. 3244-3249.

[4] S. Sagara, T. Tanikawa, M. Tamura and R. Katoh, “Experiments on a Floating Underwater Robot with a Two-Link Manipulator,” Artificial Life and Robotics, Vol. 5, No. 4, 2001, pp. 215-219. doi:10.1007/BF02481505

[5] G. R. Vossoughi, A. Meghdari and H. Borhan, “Dynamic Modeling and Robust Control of an Underwater ROV Equipped with a Robotic Manipulator Arm,” 2004 Japan USA Symposium on Flexible Automation, Denver USA, 2004.

[6] K. Ioi and K. Itoh, “Modelling and Simulation of an Underwater Manipulator,” Advanced Robotics, Vol. 4, No. 4, 1989, pp. 303-317. doi:10.1163/156855390X00152

[7] M. L. Nagurka and V. Yen “Optimal Design of Robotic Manipulator Trajectories: A Nonlinear Programming Approach,” Technical Report CMU-RI-TR-87-12, the Robotics Institute, Carnegie Mellon University, April, 1987.

[8] M. Zefran, “Review of the Literature on Time-Optimal Control of Robotic Manipulators,” Technical Report MS- CIS-94-30, University of Pennsylvania, Philadelphia, 1994.

[9] M. E. Kahn and B. Roth, “The Near Minimum-Time Control of Open-Loop Articulated Kinematic Chains,” Journal of Dynamic Systems, Measurement and Control, Vol. 93, No. 3, 1971, pp. 164-172. doi:10.1115/1.3426492

[10] M. Vukobratovi? and M. Kir?anski, “A Method for Optimal Synthesis of Manipulation Robot Trajectories,” Journal of Dynamic Systems, Measurement, and Control, Vol. 104, No. 2, 1982, pp. 188-193. doi:10.1115/1.3139695

[11] M. A. Townsend, “Optimal Trajectories and Controls for Systems of Coupled Rigid Bodies with Application of Biped Locomotion,” Thesis (Ph.D.), University of Wisconsin Madison, Madison, 1971.

[12] Y. D. Lee and B. H. Lee, “Genetic Trajectory Planner for a Manipulator with Acceleration Parameterization,” Journal of Universal Computer Science, Vol. 3, No. 9, 1997, pp. 1056-1073.

[13] D. Constantinescu and E. A. Croft, “Smooth and Time-Optimal Trajectory Planning for Industrial Manipulators Along Specified Paths,” Journal of Robotic Systems, Vol. 17, No. 5, 2000, pp. 233-249. doi:10.1002/(SICI)1097-4563(200005)17:5<233::AID-ROB1>3.0.CO;2-Y

[14] E. Shintaku, “Minimum Energy Trajectory for an Underwater Manipulator and Its Simple Planning Method by Using a Genetic Algorithm,” Advanced Robotics, Vol. 13, No. 6-13, 1999, pp. 115-138.

[15] Y. Uno, M. Kawato and R. Suzuki, “Formation and Control of Optimal Trajectory in Human Multijoint Arm Movement,” Biological cybernetics, Vol. 61, No. 2, 1989, pp. 89-101. doi:10.1007/BF00204593

[16] J. Saleh, “Fluid Flow Handbook,” McGraw Hill, New York, 2002.

[17] H. Kobayashi et al., “The Robot Control Actually,” The Society of Instrument and Control Engineers, Tokyo, 1997.