Stability Analysis of Bipedal Robots Using the Concept of Lyapunov Exponents

Abstract

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 second method 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.

1. 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.

2. Lyapunov Exponents

Consider the two following equations 𝑥_𝑛+1=𝑓𝑥_𝑛,𝑦_𝑛+1=𝑓𝑦_𝑛. (1)

Suppose that the error of initial value is |𝑥₀−𝑦₀|, through the first iteration by 𝑥₁−𝑦₁=𝑓𝑥₀−𝑓𝑦₀=𝑓𝑥₀−𝑓𝑦₀𝑥₀−𝑦₀𝑥₀−𝑦₀≈𝑑𝑓𝑑𝑥_𝑥0𝑥₀−𝑦₀, (2) where 𝑑𝑓𝑑𝑥_𝑥0=lim_{𝑥0→𝑦0}𝑓𝑥₀−𝑓𝑦₀𝑥₀−𝑦₀. (3)

After the second iteration, we have 𝑥₂−𝑦₂≈𝑑𝑓𝑑𝑥_𝑥1𝑥₁−𝑦₁≈𝑑𝑓𝑑𝑥_𝑥1𝑑𝑓𝑑𝑥_𝑥0𝑥₀−𝑦₀. (4)

And through the 𝑛 time iteration 𝑥_𝑛−𝑦_𝑛≈^𝑛−1_𝑛=0𝑑𝑓𝑥_𝑛,𝜇𝑑𝑥_𝑥𝑛𝑥₀−𝑦₀. (5)

From the above, the sensitivity of two systems to initial disturbance is effected by the value of derivative |𝑑𝑓/𝑑𝑥| at 𝑥₀.

The sensitivity of overall system mapping to initial value is achieved depending on the average of all initial conditions that needs the 𝑛th time iteration; the value of every deviation is ^𝑛−1_𝑛=0𝑑𝑓𝑑𝑥_𝑥𝑛^1/𝑛. (6)

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: 1𝜆=𝑛ln^𝑛−1_𝑛=0𝑑𝑓(𝑥_𝑛,𝜇)𝑑𝑥_𝑥𝑛, (7) where 𝑥_𝑛 is the value of 𝑛 times iteration. The computational formula of Lyapunov exponent (8) is received with 𝑛 tending to infinity 𝜆=lim_𝑛→∞1𝑛^𝑛−1_𝑛=0𝑥ln𝑑𝑓_𝑛,𝜇.𝑑𝑥 (8)

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 𝑚 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 𝜆_𝑖 can be achieved by the following form: 𝜆_𝑖=lim_𝑛→∞1𝑛𝜀_𝑖(𝑛)𝜀₀(𝑛),(𝑖=1,2,…). (9)

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 𝜆₁≥𝜆₂≥𝜆₃≥⋯≥𝜆_𝑚.

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.

3. Lyapunov Exponent of Continuous System

An 𝑛-dimensional differential equation of continuous smooth dynamics system is defined as the following form: 𝑥̇=𝐹(𝑥), (10) where 𝑥̇=𝑑𝑥/𝑑𝑡, 𝑥(𝑡)∈𝑅^𝑛 is the state vector. Usually, suppose two close points 𝑥₀, 𝑥₀+𝑢₀, and the initial point 𝑥₀ located in the basin of attraction; 𝑢₀ is the disturbance of initial 𝑥₀ (Figure 1). After a period of time t, the disturbance 𝑢_𝑡 is as the following form: 𝑢_𝑡≡𝑓^𝑡𝑥₀+𝑢₀−𝑓^𝑡𝑥₀=𝐷_𝑥0𝑓^𝑡𝑥₀⋅𝑢₀, (11) where 𝑢_𝑡 is tangent vector in (11), which satisfies vary equation in following form [10]: Φ̇_𝑡𝑥₀=𝐷_𝑥𝐹𝑓^𝑡𝑥₀⋅Φ_𝑡𝑥₀,Φ₀𝑥₀=𝐼, (12) where Φ_𝑡(𝑥₀) is the derivative at 𝑥₀, that is, Φ_𝑡(𝑥₀)=𝐷_𝑥0𝑓^𝑡(𝑥₀). In order to calculate the trajectory, (13) needs to be integrated. Consider the following: 𝑥̇Φ̇=𝐹(𝑥)𝐷_𝑥𝐹(𝑥),𝑥𝑡⋅ΦΦ𝑡⁰₀=𝑥₀𝐼. (13)

Figure 1 

Distance evolution of the adjacent trajectory.

Then, the average exponent of two trajectories’ divergence or convergence is defined as 𝜆𝑥₀,𝑢₀=lim_𝑡→∞1𝑡𝑢_𝑡𝑢₀=lim_𝑡→∞1𝑡ln𝐷_𝑥0𝑓^𝑡𝑥₀⋅𝑢₀, (14) where ‖𝑢_𝑡‖ is the vector of length. If 𝜆>0, the exponent will be divergent nearby track. For 𝑥₀∈𝑀, (14) can calculate the maximum Lyapunov exponent in a very weak smoothness conditions.

4. 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 𝑚, 𝐿, and 𝐼. The two legs are connect with a passive joint. The distance between the center of mass and hip joint is 𝐿₀=𝐿/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.

Figure 2 

The simplified biped model.

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: 𝑀₁₁𝑀₁₂𝑀₂₁𝑀₂₂{𝑢̇}(𝑢),𝑓(𝑢)=[𝜃]𝑞=𝑓𝑔𝐿𝑚cossin₁(𝑡)+𝑚𝑔𝐿−𝐿₀cos[𝜃]sin𝑞₁(𝑡)+𝑚𝐿𝐿₀sin𝑞₁(𝑡)−𝑞₂(𝑡)𝑢²₂−𝑚𝑔𝐿₀cos[𝜃]sin𝑞₂(𝑡+)𝑚𝐿²₀+𝑚𝐿₀𝐿₀−𝐿sin𝑞₁(𝑡)−𝑞₂(𝑡)𝑢²₁,𝑀₁₁=−𝐼₀+𝑚𝐿𝐿−𝐿₀−𝑚𝐿₀𝐿+𝑚𝐿−𝐿₀²,𝑀₁₂=𝑚𝐿₀𝐿cos𝑞₁(𝑡)−𝑞₂(𝑡,𝑀₂₁)=𝑚𝐿₀𝐿−𝐿₀+𝑚𝐿²₀cos𝑞₁(𝑡)−𝑞₂(𝑡,𝑀₂₂)=−𝐼₀−𝑚𝐿²₀. (15)

5. Simulation

Supposing 𝐿=0.62 m, 𝜃=31^∘, and 𝑚=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.

Figure 3 

Lyapunov exponent spectrum.

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.

Figure 4 

Time series of the variable.

6. Conclusion

It is significant to optimize 𝑡 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Natural Science Foundation of Jiangsu Province (BK20130999), the Natural Science Foundation of Colleges and Universities in Jiangsu Provincel (13KJB460012), the Postdoctoral Science Foundation of China (230210235), the Open Project (KDX1102) of Jiangsu Key Laboratory of Meteorological Observation and Information Processing, and Nanjing University of Information Science and Technology.