Abstract— The two main important factors to be considered while designing an efficient biped robot are human like gait cycle and energy efficiency. The researchers have sought the solution through passive dynamic walking which can provide highly energy efficient model for biped locomotion exhibiting human like gait under gravity. By adding actuation at some joints, the passive dynamic walking robot can walk stably on level ground. In this paper, we present numerical simulation of a simple robotic dynamic walker using Matlab. We derived the equation of motion using Euler-Lagrange method. We first demonstrate that a simple passive biped walker, vaguely resembling human legs, can walk down a shallow slope without any actuation and control. Later, we extended the passivity property to active dynamic walkers on level ground. Our model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a point- mass at the hip, and point-masses at the feet. After nondimensionalizing the governing equations, the model has only one free design parameter, the ramp slope γ. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. Keywords—Passive walkers, Biped Robots, Dynamics, Passive dynamic walking I. INTRODUCTION In the past few decades, robotics research has made huge progress in the area of biped locomotion for various reasons, running from prosthesis development [1], rehabilitation [2], transportation underwater locomotion [3] and entertainment industries [4]. Based on their actuation system, biped walkers are categorized to fully-actuated, under-actuated and passive walkers. Fig. 1 shows an example of each category. Fully actuated biped walkers are widely illustrated in literature [5, 6], this type suffers from the high costs of both energy consumption and control [8]. In order to reduce these costs, the under-actuated walkers were presented (see, Fig. 1, middle). Adding actuation to passive dynamic walkers result in highly efficient robotic walkers. Such walkers can be Irfan Hussain is corresponding author (Irfan.hussain@kustar.ac.ae) 1 Authors are with the Khalifa University Robotics Institute, Khalifa University of Science Technology and Research, Abu Dhabi, United Arab Emirates 2Authors are with the Dept of Mechanical Engineering, Khalifa University of Science Technology and Research, Abu Dhabi, United Arab Emirates implemented at lower mass and use less energy because they walk effectively with only a couple of motors. Several designs Figure. 1: From left to right, examples of passive walker, underactuated and fully actuated powered biped robots are shown. and control laws were developed for this type of biped walkers [7 -9]. A solution for energy efficiency is the exploitation of the ‘natural dynamics’ of the multi-body system, or by incorporating actuators that are energy dissipative, energy- redirecting, or energy storing to the system [10]. Recently, some other solutions for exploiting the passive compliance in the underactuated robotic devices have also been introduced [11-13]. In 1989 McGeer [14] introduced the idea of ‘passive dynamic walking’, inspired by the research of Mochon and McMahon [12]. They showed that in human locomotion the motion of the swing leg is merely a result of gravity acting on an unactuated double pendulum. McGeer [16] extended the idea and showed that a completely unactuated and therefore uncontrolled robot can perform a stable walk. Garcia et al [17] have researched several stability and efficiency issues of those passive dynamic walkers, and showed that in the limiting case, energy consumption can even be reduced to zero, like an ideal rolling wheel. The walking 3Department of Mechanical Engineering, KU Leuven, 3000 Leuven, Belgium. Dynamic Modeling and Numerical Simulations of a Passive Robotic Walker Using Euler- Lagrange Method Irfan Hussain 1 *, IEEE Member, PhD, Mohammad I. Awad 1 , IEEE Student Member, Ali Bin Junaid 3 Federico Renda 1,2 , IEEE Member, PhD, Lakmal Seneviratne 1 , IEEE Member, PhD, Dongming Gan 1 , PhD