## 1 Introduction  
Recently, the number of disasters, such as earthquakes and tsunamis, is increasing, and humanoid robots have commonly been expected to work in place of humans in such dangerous environments, in which the terrain can be uneven and unpredictable. Improvement of their locomotion abilities by addition of walking, crawling, climbing, fall protection, and jumping capabilities [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] is important to allow these robots to navigate across such adverse terrain. Unfortunately, most contemporary humanoid robots only have segmental functions. Some robots are designed simply for walking or crawling [2; 4; 7], while others are designed for jumping alone [9; 10]. The purpose of this paper is to add a jumping pattern to a versatile BHR6 robot, which is actuated using electric motors, and that already has walking, rolling, and fall protection capabilities [11; 12; 13]. Quick and harmonized coordination of the jumper's body segments is essential for jumping tasks. Humans take advantage of the elasticity of their tendons to store and use kinetic energy during jumping motions. Jumping using elastic elements has been studied intensively. [PERSON] and colleagues studied hopping robots [14] that were driven by pneumatic and hydraulic actuators to perform various actions, including somersaults [15]. Other researchers successfully demonstrated jumping motions using robots with artificial muscles [16; 17]. [PERSON] [9] and [PERSON] [18] proposed jumping mechanisms that use the potential energy of a spring. These research efforts are focused on development of a jumpingmechanism and its control. All of these robots have elastic mechanisms that are used to retrieve kinetic energy during dynamic movement cycles. Use of elastic elements has obvious advantages for storage and use of kinetic energy, but they may also affect the performance of the fundamental functions of the robots, e.g., walking, crawling, climbing, and carrying objects. Because our intention is to add jumping functionality to an already versatile robot BHR6 without any elastic mechanism, the robot is modeled as three rigid planer links. A jumping robot not only requires a high torque output to accelerate its COM (center of mass) at the beginning of the motion but also must realize a high velocity at the end of the motion. As the joints become increasingly extended, the transfer of the joint angular motion to produce the desired translation of the COM becomes less effective. When the joint is fully extended, the effect of this joint on the COM's translational motion in a specific direction is equal to zero. When these factors are taken into consideration, an appropriate reduction gear is required for the motor [19]. The prime criterion for the robot's obstacle negotiation ability is the required height of the jump, which is dependent on the vertical velocity of the robot's COM at the moment when its feet leave the ground. During the push-off phase, when the mechanical limitations are considered, a high-torque output is required to generate high angular acceleration. [PERSON] et al. proposed an adaptation of human jumping motion based on the ground force [8]. Using a similar algorithm, [PERSON] et al. designed a nonlinear gear ratio profile to maximize the ground force [19]. [PERSON] et al. used the base resonance frequency to realize high acceleration [18]. By addition of elastic devices, [PERSON] et al. realized jumping motion in the low-power humanoid robot Nao [20] that would not originally have had sufficient power to jump. All these methods planned jump patterns based on the trajectory of the leg length or the force of the ground. It leads to the lack of consideration of each joint state, e.g., the torque, velocity and acceleration. Consequently, the relevant motors could not work in the ultimate states required to supply higher kinetic energy and thus achieve a higher jump.