Human walking can be considered as a complex mechanical process controlled by Newton’s law of motion which inspired to model the biped robot. Furthermore, the biped robot is more capable to work in the hazardous atmosphere than the wheeled vehicles. The fundamental problem of biped robot is to achieve the stable gait but it is not a trivial task. So, most of the researchers have focused on locomotion while working on bipedal robots till today. We will mainly focus on the walking pattern of robots. This paper will discuss about the effect of slope on the behavior of robot while walking on the inclined ramp.

This paper organized in several steps. Section 2 presents mathematical model of passive dynamic knee-less biped robot, powered only by gravity while walking on the inclined ramp. The knee-less biped robot is built of two rods where rods are considered as the legs of biped while joint of rods is considered as a hip. The gait of biped is dividing into two phase: swing phase and impact phase. The motion of the swing phase is equivalent to the motion of double inverted pendulum [1] . The model has two basic parts: one is a differential equation for the swing motion and second is the impact function for the heel-strike when foot of swing leg touching the ground. The model build here is a representative of many biped models developed in the literature [2] - [7] .

Section 3 explains the basic definitions for the solution of discrete system which developed in Section 2 [8] - [10] . It also describes the method of poincare section to the system with consideration of impact effect [11] - [14] . The discrete-time map defines from the plane of Poincare section to itself, that is, physically define from one heel-strike to next successive heel-strike which is called poincare return map [15] - [17] . The fixed point analysis of the Poincare return map will generate the periodic motion of the system. One of difficult task to generate the periodic motion is the selection of initial condition which described in this section.

Section 4 describes the bifurcation route to chaos as an influence of slope angle [18] . The steps convergent sequences and phase space diagrams simply explain bifurcation theory. Section 5 gives the conclusion and scope of future work.

This section presents the dynamic model of simplest passive biped robots, consisting two equal length rigid legs pivoted at the hip with no ankles and no knees. It is a structure of three point-masses: one at the hip and other two at the centre of mass of legs. These point masses are not independent due to the distance constraints imposed by the stance and the swing legs. So, the system has four degrees of freedom. The walk of robot is considered only in the sagittal plane and on a level surface. Furthermore, the a periodic motion consists of successive swing phases, means one leg (swing leg) is free for moving and other leg (stance leg) is touching the ground, and transition phase, means the transition from one leg to another taking place in an infinitesimal length of time [3] [8] . With this consideration, the model of the biped robot made up of two parts: the differential equations which describe the dynamics of the biped robot during the swing phase and the impact model which describe the contact event and transition phase. Such models are very common in the field of biped locomotion. The linearized form of this model will be used for further analysis. It is assumed that the biped has no foot scuffing and no bounce during the walking.

In this section, we define the poincare map for discontinuous system and describe the stability in sense of a fixed point of poincare map and describe the analytic solution of the impulse model (5).

The state variable y, the half inter leg angle, the step period are focused to examine the influence of slope angle on the periodic orbit. The bifurcation diagram is the graphical evolutions of such variables as the function of slope angle. We analyzed that the one periodic gaits turns periodic when slope angle is increased and further it directed to chaos. This research found that deg is the bifurcation point where the bifurcation occurs. The continuous spreading of points in the bifurcation diagrams represent the chaotic gaits, shown in Figures 4-6. We observed that the behavior of variables is monotonic during the period doubling flow. The following table details the behavior of the biped robot during a period doubling flow as the influenced of slope angle α.

Analyzing Table 1, the intervals of slope angle are gradually decreasing for periodic orbit. 8-periodic orbits are very sensitive so it difficult to observed, shown in Figure 10 and after that number of bifurcations are occurred for small change in the value of slope angle but those bifurcations are difficult to separate individually.

The phase plane diagram can be used to show the period doubling flow. The 1 periodic gait occurs when the sequence of impact time converses to one point after some finite number of steps with respect to the initial condition, shown in Figure 7(a). In other words, after some finite steps from the initial conditions, the state va-

riables repeat themselves after every heel-stride i.e., so the phase plane diagram has only one

closed limit cycle which is identical to both legs, shown in Figure 7(b) [3] . For the 2-periodic gaits, the sequence of impact time has two convergent points after some finite number of steps with respect to the initial condition, shown in Figure 8(a), it means, after some finite steps, the state variables repeat themselves after two

successive heel-strike:. As a result, the phase plane diagram generates two closed limit cycles

each one is associated to one leg, shown in Figure 8(b) [3] . In case of 4-periodic gaits, the sequence of impact time has four convergent points, shown in Figure 9(a) and the state variables repeat themselves after 4 steps. Consequently, the phase plane has four closed limit cycles where each leg is associated two closed limit cycles, shown in Figure 9(b) [3] . Also in case of 8-periodic gaits, the sequence of impact time has eight convergent points Figure 10(a) and the phase space plane has eight closed limit cycles where four for each leg, shown in Figure 10(b). In general, for -periodic gaits, each leg is associated to closed limit cycles in the phase plane [3] . In case of chaotic gait, difficult to find number of convergent points for the sequence of impact time, shown in Figure 11(a), so hard to separates the closed limit cycle associate to legs in the phase plane, shown in Figure 11(b) [3] .

In this work, we analyzed the effect of slope angle on the stability of walking of passive dynamic biped robot on linearized model. From the results of bifurcation diagram, the stable walking range is obtained with respect to slope angle. Also we observed that the stability of walking of passive biped robot is deceased as the slope is increased. This linear model can be helpful for guessing the initial states of the robot for the stable walking of its nonlinear model. These results can be helpful for finding effect of slope angle on non-linearized model and also for designing a biped robot.

This research is partially funded by DST-FIST2014 file number: MS1-097.