# Dynamic Modeling of the Dissipative Contact and Friction Forces of a Passive Biped-Walking Robot
## 1 Introduction  
The passive movement is a new concept of walking. Researchers have been working on this area with both theoretical and experimental analyses since the work of [PERSON] [1]. Over the decades, many authors have shown that completely unactuated and uncontrolled machines could walk stably downhill on a gentle slope, powered only by gravity, both in numerical simulations and physical experiments [2].  
In fact, different research groups have developed robots based on passive walking techniques [3], namely the one-meter length Robot Ranger of Cornell University [4]. Robot Toddlers from MIT University [5] is a small robot that has only a single passive pin joint at the hip, and the 3D movement is achieved by means of the feet surface design. The model developed at the Nagoya Institute of Technology has two legs, includes a stability mechanism and is able to move about 4000 steps (more than 30 minutes) without any power supply [6, 7]. [PERSON] present a running biped with elastic elements [8].  
In addition, some updated bipeds have been built following the passive philosophy [3; 9]. The biped \"PASIBOT\" of Universidad Carlos III de Madrid is able to walk in a steady mode with only one actuator/drive [10; 11]. The robot can walk in a similar way to humans, by means of the balance and the dynamics of the natural swinging, in order to consume a minimum energy to walk. This proves that biped robots based on passive walking have a good energy efficiency and can perform more natural gaits [12]. It seems that the mechanical parameters of these walkers work better than the complicated control system of the conventional robots in generating natural looking gaits [13; 14].  
Thus, this research has focused on the contact/impact of a walking robot based on the formulation of Multibody Systems Dynamics. For this, the model has been considered as a plane. The contact/impact has been defined between the feet and the ground, the feet being spherical, and the ground being flat, smooth and rigid. These approaches have been made in simpler models, such as those of [PERSON] [15], [PERSON] [16] or [PERSON] [17]. None of these works develop realistic behaviors with the friction force and with a smooth contact/impact.  
In contrast to other models that study the contact while taking as a reference the foot, either by its shape [18], with spring elements [19] or by analyzing the model as an inverted pendulum [20], the dynamics equations of our model are smooth (work for simple and double support phases).  
A method has been developed to obtain the impact/contact between the floor and the feet. This impact/contact detection is a critical issue for the correct functioning of the model, as well as for its computational efficiency [21]. The normal contact forces developed during the dynamic walking of the robot are evaluated using several models: Hertz, Kelvin-Voight, Hunt and Crossley, Lankarani and Nikravesh, and Flores. [22]. This will allow for the determination of which normal force works best for this model.  
In turn, the friction forces are computed with different models with the purpose to appraise the most relevant and appropriate options [23]. In the sequel of this process, several parameters associated with the friction force models utilized here are considered in order to get an appropriate physical and realistic behavior of the model [24].  
The model equations have been implemented on Matlab software [25]. This program is able to perform Multibody Dynamics Systems simulations [26]. The numerical results reveal that the stable periodic solutions are sufficiently robust for a large range of the parameter space.