## 1 Introduction  
Robots can be used in place of humans to perform remote tasks such as remote meetings/interactions [1; 2], telesurgery [3; 4; 5; 6], rehabilitation [7], and search and rescue [8]. However, the ability to control a humanoid robot with a high degree of freedom using a traditional controller or through preprogramming based on human experience strongly depends on the operation skills of the users, the programming skills of the developers, and the availability of extensive experience. As an alternative, imitation is an efficient social learning approach that endows a robot with the ability to transmit and reproduce human postures, actions, behaviors, etc., as a human does. The questions of how to achieve accurate, stable, and complete whole-body imitation and how to evaluate the imitation similarity are two basic issues facing this approach. Generally, robotic imitation consists of three phases, i.e., observation, representation and reproduction [9]. The observation process usually relies on either marker-based or markerless visual capture systems or dynamic capture systems to acquire human motions. In a marker-based visual capture system, markers must be placed on the demonstrators to indicate the positions of their body parts [10; 11; 12; 13; 14], whereas a markerless visual capture system can directly extract 2D or 3D information from video frames [15; 16; 17; 18; 19; 20]. The imitation system presented in [21] obtained motion data using the Xsens MVN dynamic capture system (Enschede, The Netherlands), which offers high precision but is inconvenient and expensive. Our real-time imitation system uses the Microsoft Kinect II ([PERSON], Washington, USA), which costs much less and is more user-friendly, to capture 3D skeleton points and track the faces of the demonstrators. In the representation process, the observed actions are mapped to the internal representation of the robot. The traditional approaches for posture mapping are numerical methods of inverse kinematic (IK) computations [10; 16; 17; 21; 22; 23; 24]. The numerical IK method is effective for manipulating the end effectors of robots in the work space, but it is a relatively indirect and imprecise means of producing motions in joint space, and it incurs a rather high computational cost. Because the joint angles determine the posture of the robot, a precise mapping in joint space will be beneficial for the imitation similarity. To enable the generation of more human-like actions with less computation, new methods will be necessary. To this end, several analytical methods have been exploited [13; 14; 19; 20; 25; 26; 27; 28]. In addition to the positions of the end effectors, other information such as the positions of the joints is obtained in these methods. An inherent advantage of analytical methods with constraints is that the solutions generated using such analytical methods are more direct, accurate and efficient. Consequently, it is easier to obtain accurate joint angles for mapping that correspond to an observed posture and thus achieve a high imitation similarity. However, most existing geometrical analytical mapping methods used in imitation learning cannot obtain the specific orientations of each \"body link\" in a demonstrated human posture [13; 20; 25; 26; 27; 28]. In addition, the details of the robot's joint structures, such as mother and child joint features, are not fully considered when calculating the joint angles for mapping, and the imitation similarity of the generated postures suffers as a result. Some researchers have directly used the rotation matrices provided by the Kinect software development kit (SDK) or the Open NI library, which have been proven to be unstable and imprecise [19; 25]. In this paper, a novel method for motion mapping, GA-LVVJ, is proposed to solve the above problems. In the existing robot imitation work, to the best of our knowledge, no similar analytical motion mapping method is systematically documented in detail.