Stability Analysis of Bipedal Robots Using the Concept of Lyapunov Exponents

The dynamics and stability of passive bipedal robot have an important impact on the mass distribution, leg length, and the angle of inclination. Lyapunov’s secondmethod is difficult to be used in highly nonlinear multibody systems, due to the lack of constructive methods for deriving Lyapunov fuction.The dynamics equation is established by Kane method, the relationship between the mass, length of leg, angle of inclination, and stability of passive bipedal robot by the largest Lyapunov exponent. And the Lyapunov exponents of continuous dynamical systems are estimated by numerical methods, which are simple and easy to be applied to the system stability simulation analysis, provide the design basis for passive bipedal robot prototype, and improve design efficiency.


Introduction
The walking gait of active bipedal robot is achieved by control and drive system tracking the joint angle trajectories.The huge energy constrains the development of bipedal robot [1].The passive bipedal robot can walk naturally under the drive of gravity without the outside force.It is important to study the bipedal robot walking in low energy from the bionics [2,3]; some research by the universities of Cornell, Mit, and Delft was published in the Science [4].
However, the stability of the passive bipedal robot has high sensitivity to its structure parameters; any change of structure parameter will lead to its gait characteristics being fluctuated evidently [5].Therefore, it is very important to optimize the configuration for the stability of the system [6,7].The effect of structure parameters to the stable fixed point and speed of periodic motion was analyzed under the given initial conditions in literature [5].Asymptotical stability of passive bipedal robot in three dimensional space was researched by the method that combined extended virtual constraints and hybrid zero dynamics [8].
The dynamics equation of passive bipedal robot is modeled using the Kane method in this paper, and the method of Lyapunov exponent is applied to analyze the relationship between the stability of the system and mass, length of leg, and slope angle.This method is simple and reliable for the optimization design of passive bipedal robot prototype and is easy to program.

Lyapunov Exponents
Consider the two following equations Suppose that the error of initial value is | 0 −  0 |, through the first iteration by And through the  time iteration From the above, the sensitivity of two systems to initial disturbance is effected by the value of derivative |/| at  0 .
The sensitivity of overall system mapping to initial value is achieved depending on the average of all initial conditions that needs the th time iteration; the value of every deviation is If there are small deviations from the initial value in the two systems, the result will be divergent along with the time (or times of iteration).The deviation is measured by Lyapunov exponent, the logarithmic of geometric average in following form: where   is the value of  times iteration.The computational formula of Lyapunov exponent ( 8) is received with  tending to infinity The stability of system state is related to the divergence or convergence of two adjacent trajectories through the time evolution, which can be measured by Lyapunov exponent.
In phase space, the initial conditions of the system are defined as an infinitesimal  dimensional ball.The ball will naturally deform as a super circle ellipsoid due to the dynamics effects.All the main axes of ellipsoid are arranged according to their length, and the Lyapunov exponent   can be achieved by the following form: The Lyapunov exponent is related to the divergence or convergence of system.The trajectory is convergent in the direction of the value of Lyapunov exponent of less than 0, and the system is stable with no sensitivity to the initial conditions [9].The trajectory is divergent in the direction of the value of positive Lyapunov exponent, and the system is unstable with sensitivity to the initial conditions.Usually, Lyapunov exponent is arranged according to Where the convergence and divergence of two adjacent trajectories in phase space are quantitative description by maximum Lyapunov exponent; the motion is stable when the maximum Lyapunov exponent of system is less than zero.

Lyapunov Exponent of Continuous System
An -dimensional differential equation of continuous smooth dynamics system is defined as the following form: where ẋ = /, () ∈   is the state vector.Usually, suppose two close points  0 ,  0 +  0 , and the initial point  0 located in the basin of attraction;  0 is the disturbance of initial  0 (Figure 1).After a period of time t, the disturbance   is as the following form: where   is tangent vector in (11), which satisfies vary equation in following form [10]: where Φ  ( 0 ) is the derivative at  0 , that is, Φ  ( 0 ) =   0   ( 0 ).In order to calculate the trajectory, (13) needs to be integrated.Consider the following: Then, the average exponent of two trajectories' divergence or convergence is defined as where ‖  ‖ is the vector of length.If  > 0, the exponent will be divergent nearby track.For  0 ∈ , (14) can calculate the maximum Lyapunov exponent in a very weak smoothness conditions.

Dynamics Model of Passive Bipedal Robot
The simplified biped model is showed in Figure 2. The mass, length, moment of inertia are, respectively, represented with , , and .The two legs are connect with a passive joint.
The distance between the center of mass and hip joint is  0 = /2.The body of leg is rigid.The collision between foot and ground is completely inelastic contact with no friction and slippage.The ground is also rigid.The robot will walk along the slope face automatically under its own gravity and inertia with the leg given an initial speed.
The process of robot's motion is divided into parts.(1) The leg 1 will swing around the hip joint after it left off the ground.The total procedure is only in gravity acting, so the total mechanical energy is conserved.(2) When the swing leg contacts with the ground, it will exchange role with the supporting leg.The collision between the foot and the ground is instantaneous, and there is no sliding during the process of collision.The angular momentum of system is conservation.
The dynamics equations are modeled by Kane method in following form:

Simulation
Supposing  = 0.62 m,  = 31 ∘ , and  = 14 kg, the Lyapunov exponent spectrum will be received by ( 14) (showed in  Figure 3).The whole calculation is used by software Mathematica.From the Lyapunov exponent spectrum of Figure 3, we can find one of the values of Lyapunov exponent above 0, so the passive biped robot system is unstable.It is greatly important for optimization of the structure parameter and stability of control to analyze the dynamic characteristic of the passive biped robot.The time series of all variables is showed in Figure 4.The dynamics characteristic of system can be observed by changing one of the structure parameters.From lots of results of simulation, it is concluded that small changes in mass cannot affect the dynamic characteristics of the whole system, but the length of leg and slope angle's effect are obvious.

Conclusion
It is significant to optimize  mass distribution, leg length, slope angle, and other parameters of passive bipedal robot for improving the stability of the system.The dynamics equation of passive bipedal robot is modeled by Kane method in this paper, and the method of Lyapunov exponent is applied to analyze the relationship between the stability of the system and mass, length of leg, and slope angle.This method is simple and reliable for the optimization design of passive bipedal robot prototype.

Figure 1 :
Figure 1: Distance evolution of the adjacent trajectory.

Figure 4 :
Figure 4: Time series of the variable.