# Dynamic Modeling of the Dissipative Contact and Friction Forces of a Passive Biped-Walking Robot
## 2 Biped Model
### Geometric Considerations of the Model  
Derived from the simplest walking model [15; 17], the following geometric model is employed and investigated in this study. The geometric considerations are proposed on relative coordinates in order to use the Multibody System Dynamics formulation. The positions are defined in a global coordinate system using the Euler parameters: p = {cos \(\theta\)/2,0,sin \(\theta\)/2,0}T. The model we considered was based on the planar model with a straight leg and round feet. The general view of the model is shown in Figure 1. The biped walking model consists of two legs linked by a spherical joint. The spherical joint is a kinematic holonomic constraint defined by the condition that the point Pi on body i coincides with the point Pj of body i. This condition can be written in a scalar form as Equation (3), using generalized coordinates:  
\[\phi(sph)=r_{j}^{p}-r_{i}^{p}=r_{j}+s_{j}^{p}-r_{i}-s_{i}^{p}=0 \tag{3}\]  
where \(r_{j}^{p}\) and \(r_{i}^{p}\) are the position of the point Pi on body i and Pj on body j with respect to the global system. Note that \(r_{j}\) and \(r_{i}\) are the position of the point Pi on body I and Pj on body j with respect to the local system, and \(s_{j}^{p}\) and \(s_{i}^{p}\) define the locations of the origin of the local coordinate system.  
The first-time derivative of Equation (3) results in the velocity constraint equations for a spherical joint, and it gives the contribution to the Jacobian matrix. The first-time derivative of Equation (3) yields the velocity constraints that provide the relation between the velocity variables, and can be expressed as:  
\[\dot{\phi}=Dv=0 \tag{4}\]  
where D denotes the Jacobian matrix, and v contains the velocity term. In a similar manner, the acceleration constrains equations of the spherical joint can be obtained by again taking the time  
Figure 1: The planar model of the passive dynamic of the biped.  
derivative, and it gives the contribution to the right-hand side of the acceleration of the spherical joint constraint. [25] The second derivative of Equation (3) results in:  
\[\ddot{\phi}=D\dot{v}+\dot{D}v=0 \tag{5}\]  
where \(\dot{v}\) denotes the acceleration terms, and the term \(-\dot{D}v\) is referred to as the right-hand side of the kinematic acceleration equation, \(\varphi\).  
The constraint equations represented before are non-linear and can be solved by any usual method. In short, the kinematic analysis of this multibody system for a specific instant can be carried out by solving the set of Equations (1) to (5) together. Then, the position, velocities and acceleration of the initial time can be obtained. Note that this is an iterative method; afterwards, the forces are obtained and everything is recalculated for the next time step. That allows us to solve this multibody model by forward dynamics.  
Every leg has round feet that can act with the ground. The ground is a massive rigid body, with the surface being flat and smooth. Note that the model is symmetrical.  
The initial conditions of the biped have been set from the literature [17; 27]. Thanks to these initial conditions, it is possible to verify the results of this paper with real experimental results. The masses, momentum of inertia and length of each leg are m = 1 kg, J = 0.01 kg\(\cdot\)m\({}^{2}\) and l = 0.40 m. The radius of the feet is: r = 0.08 m, and the distance between the spherical joint S and the center of mass of the leg is d = 0.10 m. The initial angles and rotational velocity are: \(\theta_{1}\)= \(-\)0.2479 rad; \(\omega_{1}\)= \(-\)0.0052 rad/s; \(\theta_{2}\)= 0.1655 rad; \(\omega_{2}\)= \(-\)1.2565 rad/s.