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Trajectory Optimization of a Mobile Robot with Flexible Links Using Pontryagin’s Method

Arian Almasi , Mostafa Ghayour, and Mohammad Jafar Sadigh13
Isfahan University of Technology, Isfahan 84156-83111, Iran

Abstract—In this study, the open-loop optimal control method is used to optimize the trajectory of a mobile robot with flexible links. Equations of motion for the mobile robot are initially obtained via the Lagrangian method. The assumed modes method is then used to obtain a model with limited degrees of freedom. The relevant kinematic model is established based on the standard frame transformation matrices including rigid rotations and elastic displacements assuming that the values for these parameters are small. The Book modified method is also used to obtain the kinematic elastic links. Elastic manipulator links are modeled as Euler-Bernoulli beams with Clamped-Mass (CM) boundary conditions. Nonholonomic constraints and additional kinematic constraints are considered in order to specify the base motion. A performance criterion is defined that involves the square of the angular velocity and joint torque. The torque and velocity are determined such that the performance criterion is minimized. The optimal control problem is converted into a two-point boundary value problem using the calculus of variations, the Hamiltonian function, and the Pontryagin’s minimum principle. By considering the different weight coefficients for angular velocity joints and the torques on the joints of the robot in the performance criterion, the effects of weight coefficients are investigated on the solution. In another stage of the study, a flexible arm with a mobile base is simulated to illustrate the capability of the proposed method and the relevant equations are solved by MATLAB. Finally, the results are compared with those reported in previous studies to evaluate the dynamic model.

Index Terms—Mobile robot, Optimal control, Flexible link, Trajectory planning

Cite: Arian Almasi, Mostafa Ghayour and Mohammad Jafar Sadigh, "Trajectory Optimization of a Mobile Robot with Flexible Links Using Pontryagin’s Method," International Journal of Mechanical Engineering and Robotics Research, Vol. 3, No. 3, pp. 149-164, July 2014.