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In this paper, finding the maximum load carrying capacity of mobile manipulators for a given two-end-point task is formulated as an optimal control problem. The solution methods of this problem are broadly classified as indirect and direct. This work is based on the indirect solution which solves the optimization problem explicitly. In fixedbase manipulators, the maximum allowable load is limited mainly by their joint actuator capacity constraints. But when the manipulators are mounted on the mobile bases, the redundancy resolution and nonholonomic constraints are added to the problem. The concept of holonomic and nonholonomic constraints is described, and the extended Jacobian matrix and additional kinematic constraints are used to solve the extra DOFs of the system. Using the Pontryagin's minimum principle, optimality conditions for carrying the maximum payload in point-to-point motion are obtained which leads to the bang-bang control. There are some difficulties in satisfying the obtained optimality conditions, so an approach is presented to improve the formulation which leads to the two-point boundary value problem (TPBVP) solvable with available commands in different softwares. Then, an algorithm is developed to find the maximum payload and corresponding optimal path on the basis of the solution of TPBVP. One advantage of the proposed method is obtaining the maximum payload trajectory for every considered objective function. It means that other objectives can be achieved in addition to maximize the payload. For the sake of comparison with previous results in the literature, simulation tests are performed for a twolink wheeled mobile manipulator. The reasonable agreement is observed between the results, and the superiority of the method is illustrated. Then, simulations are performed for a PUMA arm mounted on a linear tracked base and the results are discussed. Finally, the effect of final time on the maximum payload is investigated, and it is shown that the approach presented is also able to solve the time-optimal control problem successfully.

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The aim of this paper is to determine the optimal trajectory and maximum payload of flexible link manipulators in point-to-point motion. The method starts with deriving the dynamic equations of flexible manipulators using combined Euler-Lagrange formulation and assumed modes method. Then the trajectory planning problem is defined as a general form of optimal control problem. The computational methods to solve this problem are classified as indirect and direct techniques. This work is based on the indirect solution of open-loop optimal control problem. Because of the offline nature of the method, many difficulties like system nonlinearities and all types of constraints can be catered for and implemented easily. By using the Pontryagin's minimum principle, the obtained optimality conditions lead to a standard form of a two-point boundary value problem solved by the available command in MATLAB R . In order to determine the optimal trajectory a computational algorithm is presented for a known payload and the other one is then developed to find the maximum payload trajectory. The optimal trajectory and corresponding input control obtained from this method can be used as a reference signal and feedforward command in control structure of flexible manipulators. In order to clarify the method, derivation of the equations for a planar two-link manipulator is presented in detail. A number of simulation tests are performed and optimal paths with minimum effort, minimum effort-speed, maximum payload, and minimum vibration are obtained. The obtained results illustrate the power and efficiency of the method to solve the different path planning problems and overcome the high nonlinearity nature of the problems.

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