## 1 Introduction  
There are millions of people in the world who have lost their limbs for various reasons, such as accidents, vascular disease including diabetes, peripheral arterial diseases, trauma, and cancer [1]. This situation negatively affects the quality of life and psychology of amputees. Prostheses are of great importance for people with limb losses. A prosthesis may provide both good prosthetic positioning and natural appearance. Current lower-limb prostheses are mainly grouped into three categories: (i) passive prostheses, (ii) semi-active prostheses [2; 3] and (iii) active prostheses [4; 5; 6; 7]. Passive prostheses have only passive mechanical structures; therefore, amputees cannot walk comfortably and aesthetically [4]. These cause high metabolic energy consumption and cannot provide sufficient gait symmetry. An amputee with passive prosthesis spends 60% more metabolic energy compared to that used by a healthy person [8]. At the same time, they are not suitable for long-term usage since they cause irritation on the stump. Passive prostheses are preferred by the amputees due to their lower costs compared to semi-active and active prostheses. One of the main issues in prosthesis design is the need to ensure acceptable joint stiffness for maintaining motion during the gait phase. In semi-active prosthesis, joint stiffness of the prosthesis is controlled by a microprocessor. This control system is powered by a small-size battery that supplies the sensor and microcontroller only. Semi-active prostheses have better joint stiffness characteristics compared to passive prostheses. At the same time, the metabolic energy consumption of semi-active prostheses decreases compared to that of passive prostheses. However, semiactive prostheses are heavier and more expensive than passive prostheses. Active prostheses are the new generation of prostheses, designed considering the principles of robotic systems [9]. These types of prostheses control joint angles; thus, they require less metabolic energy than passive and semi-active prostheses. They also provide a natural walking ability to the amputees on an uneven terrain [10]. Each joint of active prostheses is actuated by an independent DC motor and controlled dynamically by a microprocessor. The whole prosthesis mechanism is powered by a battery. Although this active joint mechanism provides a more aesthetic walking ability to the amputees, DC motors particularly increase the weight of the prostheses. This situation both increases energy consumption in the battery and makes amputees consume more metabolic energy. At this stage, lighter active prosthesis with acceptable joint stiffness is the main issue for designers [7]. There are several studies which have been performed in the literature for minimizing the energy consumption of active- or robotic-type prostheses. [PERSON] et al. [11] minimized energy consumption of an ankle prosthesis (only one-axis prosthesis driven by a DC motor). They focused on optimizing the drive-train of the prosthesis and spring stiffness values. [PERSON] et al. [12] reduced the torque and power requirements of the ankle prosthesis during the natural walking. The researchers in studies [10; 11] used springs in their prostheses. They analyzed motor peak power and energy values. They compared these values with those of conventional actuators. Unlike the single-axis active prostheses mentioned above, a new robotic-type three-axes above-knee prosthesis has been designed. Its kinematics and dynamic equations are derived symbolically. Energy consumption in robotic-type active above-knee prosthesis is mainly derived from the required torque at each joint [7; 11; 13; 14; 15]. The dominant part of the torque equation is composed of the link masses. Considering this fact, in this study the link masses of new above-knee prosthesis are optimized for minimizing the electrical energy drawn from the battery subject to the kinematic and dynamic constraints.