^{1}

^{*}

^{2}

^{*}

In this paper, we modeled a simple planer passive dynamic biped robot without knee with point feet. This model has a stable, efficient and natural periodic gait which depends on the values of parameters like slope angle of inclined ramp, mass ratio and length ratio. The described model actually is an impulse differential equation. Its corresponding poincare map is discrete case. With the analysis of the bifurcation properties of poincare map, we can effectively understand some feature of impulse model. The ideas and methods to cope with this impulse model are common. But, the process of analysis is rigorous. Numerical simulations are reliable.

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 [

Section 3 explains the basic definitions for the solution of discrete system which developed in Section 2 [

Section 4 describes the bifurcation route to chaos as an influence of slope angle [

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 [

The angular coordinates

cated in

where

The effect of centrifugal and coriolis forces on the walk is very low so it can be ignored to find the walking pattern of a periodic motion of a biped robot. Neglecting the above forces, the non-linear Equation (1) formed into liearized equation,

where

where the state space variable

where

For the biped with equal length and without a knee, the identification of the point of contact of the swing leg with the walking surface would seem to be physically ambiguous because the swing leg must scuff along the ramp when it passes through the stance leg. It is assumed that the walker is prevented from the scuffing and the impact of the swing leg with the walking surface has no rebound and no slipping and stance leg naturally lifting from the ground without integration [

The conservation laws of the angular momentum gives to the following compressed equation between the pre- and post-impact angular velocities after the impact [

where

After the impact the swing leg will be the new stance and the stance leg will be the new swing leg for the next subsequent step, it can be expressed by the following equation:

The state space form of the impact model can be written as follows:

where

The complete model of a biped passive dynamic robot is a hybrid of the motion equation and impact model which described as:

where

Regarding this model, the evolution process of periodic motion of a walking robot is described in the following steps: first, a trajectory of the robot is evaluated by the solution of motion equation until an impact occurs. The impact occurs when attains the impact condition and generates a contact point on the walking ramp. As the result of impact, there is a very fast change in the velocity components of the state vector of the contact point which is done by the impact model. This instantaneous change in the velocity components is resulting in a discontinuity of the velocities which make the biped system discontinuous. The last result of the impact model is a new initial state of a robot for the evolution of next step until the next impact. Due to the instantaneous change at the time of impact, the robot have two different state space positions at the same time which not to be obliged, so the impact event is explained with two notation: first, the state vector “

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).

If we assume that the function

tem (5). (i) If

The analytic solution of linearized Equation (5) of motion is _{ }is the initial condition of

walking. To get the stable walking, the initial condition should be proper otherwise random selection of initial conditions leads to walking falling forward or backward. To find the proper initial condition, considered the Poincare map of linearized model which is given by:

The eigenvector y corresponding to unity eigenvalue of the matrix of Poincare map for some values of

This y could be the proper choice of initial conditions for some range of slope angle [

The state variable y, the half inter leg angle

Analyzing

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

riables repeat themselves after every heel-stride i.e.

closed limit cycle which is identical to both legs, shown in

successive heel-strike:

each one is associated to one leg, shown in

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.

n-Periodic | Range of slope angle |
---|---|

1-Periodic | |

2-Periodic | |

4-Periodic | |

8-Periodic | |

Chaotic |

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