ORIGINAL ARTICLE Maximum payload for flexible joint manipulators in point-to-point task using optimal control approach M. H. Korayem & A. Nikoobin Received: 21 October 2006 / Accepted: 19 June 2007 / Published online: 25 July 2007 # Springer-Verlag London Limited 2007 Abstract In this paper, finding the maximum dynamic load carrying capacity of flexible joint manipulators in point- to-point motion is formulated as an optimal control prob- lem. The computational methods are classified as indirect and direct methods. This work is based on the indirect solution method of optimal control problem. The applica- tion of Pontryagin’ s minimum principle to this problem results in a two-point boundary value problem (TPBVP) solved numerically. Two algorithms are developed on the basis of the solution of the TPBVP. The first one is used to plan the optimal path for a given payload, and the other one is exploited to find the maximum payload and corresponding optimal path. The main advantage of the proposed method is that the various optimal trajectories are obtained with different characteristics and different maximum payload. Therefore, the designer can select a suitable path among the numerous optimal paths. In order to illustrate the power and efficiency of the proposed method in the presence of flexibility in joints, a number of simulation tests are performed for a two-link manipulator. Then, the effect of flexibility on the maximum payload value is investigated and compared with rigid one. Finally, for the sake of com- parison with previous results in the literature, simulation is performed for a rigid-joint three-link manipulator, and a reasonable agreement is observed between the results. Keywords Robot . Flexible joint . Maximum payload trajectory . Optimal control Nomenclature n Manipulator degree of freedom θ 2i-1 Angular position of ith link θ 2i Angular position of ith motor q 1 Vector of generalized cordinates associted with links q 2 Vector of generalized cordinates associted with motors D(q 1 ) Inertia matrix of the manipulator Cq 1 ; q : 1   Vector of damping, Coriolis and centrifu- gal forces G(q 1 ) Vector of gravity forces K Diagonal joint stiffness matrix J Diagonal motor inertia matrix U Vector of generalized force inserted into actuators X State variables vector X : State derivatives vector m p ,m pmax Payload mass, maximum payload (kg) J o (U) Objective criterion or performance measure W 1 ,W 2 ,W e , W p ,R Weighting matrices U Vector of admissible control torque H Hamiltonian function = Costate vector τ s Stall torque of the motor ω m Maximum no load speed of the motor U + ,U - Extrimal bound of motor torque ɛ Desired accuracy in TPBVP solution e Desired accuracy in maximum payload calculation L i ,m i ,I i Length, mass and moment of inertia of ith Int J Adv Manuf Technol (2008) 38:1045–1060 DOI 10.1007/s00170-007-1137-2 M. H. Korayem (*) : A. Nikoobin Robotic Research Laboratory, College of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran e-mail: hkorayem@iust.ac.ir