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This technical note addresses a tradeoff in the tree based trajectory optimization algorithms for open-loop optimal control problem of rigid body system. In this technical note, linear prediction based uniform state sampling method (LPUSS), which relaxes the tradeoff between solutions' quality and computational efficiency, is proposed on the basis of the local linearity of motion equations. LPUSS has a lower calculation order than randomized kinodynamic planning (RKP) and rapid semi-optimal motion planning (RASMO). In the validation using multi degree-of-freedom (DOF) under actuated manipulator models, the solutions' quality and the computational efficiency of LPUSS were better than those of RKP and RASMO. LPUSS finished the optimization for the 6 DOF model within 40 minutes. This was the world's first success of the optimization of the swing up motion of a 6 DOF under actuated manipulator model. [ABSTRACT FROM PUBLISHER]