20 Modeling and Measurement of a Pedestrian’s Center-of-Mass Trajectory 161 20.2 Background 20.2.1 Model of Pedestrian’s Motion The purpose of this paper is to present a simple closed-form description of the motion of the pedestrian and verify it against experimental data. In this paper, a pedestrian walking on a rigid surface. The analytical description is intended for use in Human Structure Interaction (HSI) modeling, as proposed earlier in [3]. The focus is therefore on simplicity. Since one may roughly assume that the subtleties of motion of the human body, like hand movement, do not have a significant impact on structural vibrations during HSI, we disregard them. We shall only consider the “first order” characteristics of human walking, which can be represented by the trajectory of the human’s CoM. This trajectory describes the position of the CoM at a given time instant with respect to the surface, on which the human walks. The main assumptions about/features of the motion can be summarized as: 1. For the purpose of an HSI analysis, a human can be adequately represented as a mass point, placed at the human’s CoM and having three translational degrees of freedom. 2. During constant-speed walking, the CoM’s trajectory is a periodic function of time (except in the longitudinal direction). 3. The vertical, z.t/, and lateral, y.t/, components of the trajectory, as well as the longitudinal velocity v.t/ D Px.t/, have the same phase. 4. The frequency of z.t/ and v.t/ are equal, and is twice the frequency of y.t/. Remark. During walking, the ground reaction force (GRF) and its placement result from the balance of forces acting on the mass centered at the CoM. If one assumes, for simplicity, that the double-support phase of walking is negligibly short, then: (1) GRF is equal to the sum of the forces of gravity and inertia acting at the CoM during its movement along the prescribed trajectory. (2) The position of the supporting foot’s center of pressure is the spot on the ground from which the GRF points towards the CoM. In the remainder of the paper, we will use the below—example—form of the trajectory [3], for comparison with experimental results. Its free parameters are: step length ax, lateral sway amplitude ay, vertical sway amplitude az, mean vertical positionz0, maximum longitudinal velocityv0, and amplitude of longitudinal velocityav. The parameters should be adjusted to fit experimental data. 1. Longitudinal velocity of the CoM: Px.t/ Dv.t/ Dv0 jav sin.ft/j; (20.1) where f D.v0 2av/=ax. 2. Longitudinal position of the CoM, obtained by integrating Px between 0 and t: x.t/ Dv0t av f 2 ft C1 cos.ft mod / : (20.2) 3. Lateral and vertical positions of the CoM: y.t/ Day sin ax x.t/ ; (20.3) z.t/ Dz0 az cos 2 ax x.t/ : (20.4) 20.2.2 Bayesian Model Updating The trajectory of the pedestrian can be idealized following the model presented in the previous section. The trajectory depends on six parameters (ax, ay, az, av, v0 andz0). In this paper, Bayesian inference is used to update the parameters of the model based on experimental data [16, 17]. Bayesian inference is based on the Bayes theorem:
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