I . INTRODUCTIONLinear oscillatory actuators (LOAs) have been used in a wide range of applications because they have a lot of advantages: high efficiency, simple structure, easy control, etc . [1] . Small-sized LOAs have a large potential to be used as handheld haptic devices in the near future . Haptic devices that generate a force vector using asymmetric acceleration can create the illusion of a pulling sensation without needing a grounded structure [2] . However, these devices are not suitable for mobile applications because of their size . Furthermore it can only produce a one-DOF force, and also generates unnecessary torque due to the rotation of the connecting rod . We have proposed a 2-DOF oscillatory actuator, shown in Fig . 1, and succeeded in creating an acceleration waveform that is similar to the one generated by the slider-crank mechanism [3] . However, it was then realized that the slider-crank waveform synthesized by our 2-DOF actuator is unique and specific to the mechanical structure of the slider-crank mechanism . Therefore it is actually not necessary to use it for our 2-DOF actuator, which can synthesize a wide range of acceleration waveforms . In a previous paper, we proposed a new acceleration waveform that consumed less power [4] . However, the asymmetric acceleration waveform shown in [4] is not necessarily the best waveform since it has not been optimized . In this paper, first, how the properties of the asymmetric acceleration properties are relate to haptic perception is investigated by conducting experiments on volunteers . Then, by applying a genetic algorithm to the optimization method, an optimized asymmetric acceleration waveform is created . Finally, We verify the effectiveness of the optimized acceleration waveform is verified through 3-D finite element analysis and also experiments on volunteers . II . REQUIRED ACCELERATION CHARACTERISTICS In this section, the effects of the properties of the asymmetric acceleration on haptic perception were determined through experiments on volunteers . Two effects on haptic perception were studied using the acceleration waveform shown in [3] . Fig . 2 shows the relationship between the size of the negative acceleration peak and the percentage of correct answers, with the ratio of the positive to the negative peaks being constant . As the peak value increases, the percentage of correct answers also increases . The percentage of correct answers seems to saturate when the peak value is about 4 .0 m/s 2 . Therefore we determined that a safe negative peak value to obtain a high percentage of correct answers would be 4 .5m/s 2 . Fig . 3 shows the relationship between the ratio of the positive to the negative peaks and the percentage of correct answers . The percentage of correct answers increased linearly with the ratio . From these results, we determined that the ratio of the negative to the positive peaks needs to be as large as possible and the negative acceleration peak should be more than 4 .5m/s 2 to achieve a high percentage of correct answer...