# A New Model for Human Running Micro-Doppler FMCW Radar Features
## 2 Design of the Human Running Model  
The human running model is based on the Thalmann model, which is a model of human walking motion. Through extensive experiments, the Thalmann model was proposed by [PERSON], [PERSON], and [PERSON]. The [PERSON] human model with 62 degrees of freedom and 32 joints is simplified to a model consisting of 12 regular rigid bodies and 16 reference points. The 12 rigid body parts of the human body are the head, the torso, the left upper arm, the right upper arm, the left lower arm, the right upper arm, the left thigh, the right thigh, the left lower leg, and the right lower leg. The 16 reference points are the head, the base point of the spine, the left shoulder, the right shoulder, the left elbow, the right elbow, the left hand, the right hand, the left hip, the right hip, the left knee, the right knee, the left ankle joint, the right ankle joint, the left toe, and the right toe. The distribution of these 16 reference points is shown on the left side of Figure 1, and the reference points are all located in the x-y-z coordinate system. This human motion model uses three simple geometries: sphere, ellipsoid, and cylinder. These three geometric shapes construct the human head, hands, limbs, and neck. The specific construction process is shown in Figure 1. The positive x direction is forward, the positive y direction is to the right, the positive z direction is upward, and the base point of the spine is the origin. The continuous changes in the spatial coordinates of various body parts are recorded and analyzed during running. By tracking these spatial coordinate change trajectories, the model extracts 12 key degrees of freedom. These degrees of freedom include three translation degrees of freedom, three rotation axis angular velocity degrees of freedom, and six main joint rotation degrees of freedom, which are closely related to the characteristic information of running kinematics. The coordinated motion between these 12 degrees of freedom describes the motion state and control strategy in a running gait cycle. The range of motion and the rate of change are determined by speed and body proportions, such as height, leg length and arm length ratio. Different running speeds and human body structures will lead to different motion patterns and gait characteristics in these 12 degrees of freedom. The 12 motion trajectories of the human body refer to the motion paths or posture changes of various parts, which can be described by 12 kinematic degrees of freedom. The changes of 12 kinematic degrees of freedom can be precisely defined and described in the form of functions. The form is divided into three types. The first one is described by sine function. When the trajectory exhibits periodic oscillation characteristics, it can be described by a sine function. The second one is described by the piecewise sine function. If different segments of the motion trajectory show different motion characteristics, it needs to be described by a piecewise function. The third is described by the interpolation function, which uses the positions of several key points on the trajectory as the endpoints of the interpolation function. These endpoints are connected through functions to express the motion trajectory. These three forms of description methods are universal. For brevity, typical examples of each of the three motion trajectories are given for explanation. 1. Vertical translation: The up and down translation of the spine center along the vertical direction, which can be described by a sine function: \[T_{v}(t^{*})=-a_{v}+a_{v}sin[2\pi(2t_{R}-0.35)],\] (1) if \(V_{R}\) is the relative walking speed, \(a_{v}=0.020V_{R}\). 2. Left and right rotation: The flexing way of the human body's pelvis to swing the legs, which can be described by a piecewise function: \[R_{LIR}=