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Zero-moment point trajectory modeling of a biped
walking robot using an adaptive neuro-fuzzy system
D. Kim, S.-J. Seo and G.-T. Park
Abstract: A bipedal architecture is highly suitable for a robot built to work in human environments
since such a robot will find avoiding obstacles a relatively easy task. However, the complex dynamics involved in the walking mechanism make the control of such a robot a challenging task.
The zero-moment point (ZMP) trajectory in the robot’s foot is a signi?cant criterion for the robot’s
stability during walking. If the ZMP could be measured on-line then it becomes possible to create
stable walking conditions for the robot and here also stably control the robot by using the measured ZMP, values. ZMP data is measured in real-time situations using a biped walking robot and this ZMP data is then modelled using an adaptive neuro-fuzzy system (ANFS). Natural walking motions on ?at level surfaces and up and down a 10° slope are measured. The modelling
performance of the ANFS is optimized by changing the membership functions and the consequent
part of the fuzzy rules. The excellent performance demonstrated by the ANFS means that it can not only be used to model robot movements but also to control actual robots.
1 Introduction
The bipedal structure is one of the most versatile setups for a walking robot. A biped, robot has almost the same movement mechanisms as a human and it able to operate in environments containing stairs, obstacles etc. However, the dynamics involved are highly nonlinear, complex and unstable. Thus, it is dif?cult to generate a human-like walking motion. The realisation of human-like walking robots is an area of considerable activity [1–4]. In contrast to industrial robot manipulators, the interaction between a walking robot and the ground is complex. The concept of a zero-moment point (ZMP) [2] has been shown to be useful in the control of this interaction. The trajectory of the ZMP beneath the robot foot during a walk is after taken to be an indication of the stability of the walk [1–6]. Using the ZMP we can synthesise the walking patterns of biped robots and demonstrate a walking motion with actual robots. Thus, the ZMP criterion dictates the dynamic stability of a biped robot. The ZMP represents the point at which the ground reaction force is taken to occur. The location of the ZMP can be calculated using a model of the robot. However, it is possible that there can be a large error between the actual ZMP value and the calculated value, due to deviations in the physical parameters between the mathematical model and the real machine. Thus, the actual ZMP should be measured especially if it is to be used in a to parameters a control method for stable walking.
In this work actual ZMP data taken throughout the whole walking cycle are obtained from a practical biped waling robot. The robot will be tested both on a ?at ?oor and also on 10 slopes. An adaptive neuro-fuzzy system (ANFS) will be used to model the ZMP trajectory data thereby allowing its use to control a complex real biped walking robot.
2 Biped walking robot
2.1 Design of the biped walking robot
We have designed and implemented the biped walking robot shown in Fig. 1. The robot has 19 joints. The key dimensions of the robot are also shown in Fig. 1.The height and the total weight are about 380mm and 1700 g including batteries, respectively. The weight of the robot is minimised by using aluminium in its construction. Each joint is driven by a RC servomotor that consists of a DC motor, gears and a simple controller. Each of the RC servomotors is mounted in a linked structure. This structure ensures that the robot is stable (i.e. will not fall down easily) and gives the robot a human-like appearance. A block diagram of our robot system is shown in Fig. 2.
Out robot is able to walk at a rate of one step (48mm) every 1.4 s on a ?at ?oor or an shallow slopes. The speci?cations of the robot are listed in Table 1.
The walkingmotions of the robot are shown in Figs. 3–6.- Figures 3 and 4 are show front and side views of the robot, respectively when the robot is on a ?at surface. Figure 5 is a snapshot of the robot walking down a slope whereas Fig. 6 is a snapshot of the robot walking up a slope.
The locations of the joints during motion are shown in Fig. 7. The measured ZMP trajectory is obtained from ten-degree-of-freedom (DOF) data as shown in Fig. 7. Two degrees of freedom are assigned to the hips and ankles and one DOF to each knee. Using these joint angles, a cyclic walking pattern has been realised. Our robot is able to walk continuously without falling down. The joint angles in the four-step motion of our robot are summarised in the Appendix.
2.2 ZMP measurement system
The ZMP trajectory in a robot foot is a signi?cant criterion for the stability of the walk. In many studies, ZMP coordinates are computed using a model of the robot and information from the encoders on the joints. However, we employed a more direct approach which is to use data measured using sensors mounted on the robot’s feet.
The distribution of the ground is reaction force beneath the robot’s foot is complicated. However, at any point P on the sole of the foot to the reaction can be represented by a force N and moment M, as shown in Fig. 8. The ZMP is simply the centre of the pressure of the foot on the ground, and the moment applied by the ground about this point is zero. In other words, the point P on the ground is the point at which the net moment of the inertial and gravity forces has no component along the axes parallel to the ground [1, 7].
Figure 9 illustrates the used sensors and their placement on the sole of the robot’s foot. The type of force sensor used in our experiments is a FlexiForce A201 sensor [8]. They are attached to the four corners of the plate that constitutes the sole of the foot. Sensor signals are digitised by an ADC board, with a sampling time of 10ms. Measurements are carried out in real time.
The foot pressure is obtained by summing the force signals. Using the sensor data it is easy to calculate the actual ZMP values. The ZMPs in the local foot coordinate frame are computed using (1).
Where each fi is the force at a sensor ri is the sensor position which is a vector. These are de?ned in Fig. 10. In the ?gure, ‘O’ is the origin of the foot coordinate frame which is located at the lower-left-hand corner the left foot.
Experimental results are shown in Figs. 11–16. Figures 11, 13 and 15 show the x-coordinate and y-coordinate of the actual ZMP positions for the four-step motion of the robot walking on a ?at ?oor and also down and up a slope of 10 , respectively. Figures 12, 14 and 16 shown the ZMP trajectory of the one-step motion of the robot using the actual ZMP positions shown in Figs. 11, 13
and 15. As shown in the trajectories, the ZMPs exist in a rectangular domain shown by a solid line. Thus, the positions of the ZMPs are with in the robot’s foot and hence the robot is stable.
3 ZMP trajectory modelling
In many scienti?c problems an essential step towards their solution is to accomplish the modelling of the system under investigation. The important role of modelling is to establish empirical relationships between observed variables. The complex dynamics involved in making a robot walk
make the control of the robot control a challenging task. However, if the highly nonlinear and complex dynamics can be closely produced then this modelling can be used in the control of the robot. In addition, modelling, can even be used in robust intelligent control to minimise disturbances and noise.
3.1 ANFS
Fuzzy modelling techniques have become an active research area in recent years because of their successful application to complex, ill-de?ned and uncertain systems in which conventional mathematical models fail to give satisfactory results [9]. In this light we intend to use a system to model the ZMP trajectory.
The fuzzy inference system is a popular computing framework that is based on the concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. We will use the Sugeno fuzzy model in which since each rule has a crisp output, the overall output is obtained via a weighted average, thus avoiding the time-consuming process of defuzzi?cation. When we consider fuzzy rules in the fuzzy model, the consequent part can be expressed by either a constant or a linear polynomial. The different forms of polynomials that can be used in the fuzzy system are summarised in Table 2. The modelling performance depends on the type of consequent polynomial used in the modelling. Moreover, we can exploit various forms of membership functions (MFs), such as triangular and Gaussian, for the fuzzy set in the premise part of the fuzzy rules. These are another factor that contributes to the ?exibility of the proposed approach.
The types of the polynomial are as follows
A block diagram of the modelling system is shown in Fig. 17. The proposed method is ?rst used to model and then control a practical biped walking robot.
To obtain the fuzzy rules for the fuzzy modelling system we must notes that the nonlinear system to be identi?ed is a biped walking robot with ten input variables and each input variables has two fuzzy sets, respectively. For the fuzzy model, the if-then rules are as follows:
where Ai，Bi，,,, Ji in the premise part of the rules have linguistic values (such as ‘small’ or ‘big’) associated with the input variable, x1,x2,…,x10; respectively. Fj (x1, x2,…, x10); is the constant, or ?rst-order consequent polynomial function for the jth rule.
As depicted in Fig. 18, two types of MFs were examined. One is the triangular and the other is Gaussian.
Figure 19 is an adaptive neuro-fuzzy inference system [10] architecture that is equivalent to the ten-input fuzzy model considered here, in which each input is assumed to have one of the twoMFs shown in Fig. 18. Nodes labelled P give the product of all the incoming signals and these labelled N calculate the ratio of a certain rule’s ?ring strength to the sum of all the rule’s ?ring strengths. Parameter variation in ANFIS is occured using either a gradient descent algorithm or a recursive least-squares estimation algorithm to adjust both the premise and consequent parameters iteratively. However, we do not use the complex hybrid learning algorithm but instead use the general least-squares estimation algorithm and only determine the coef?cients in the consequent polynomial function.
3.2 Simulation results
Approximately models were constructed using the ANFS. Then accuracy was quanti?ed in terms of there mean- squared error (MSE), values.
The ANFS was applied to model the ZMP trajectory of a biped walking robot using data measured from out robot. The performance of the ANFS was optimised by warying the MF and consequent type in the fuzzy rule. The measured ZMP trajectory data from our robot (shown in Figs. 32–41A in the Appendix) are used as the process parameters.
When triangular and Gaussian MFs are used in the premise part and a constant in the consequent part then, the corresponding MSE values are listed in Table 3. We have platted our results in Figs. 20–25. The generated ZMP positions from the ANFS are shown in Figs. 20, 22 and 24 for a ?at level ?oor, walking down a 10 slope and walking up a 10 slope, respectively. In Figs. 21, 23 and 25, we can see the corresponding ZMP trajectories which are generated from the ANFS.
For simplicity, the process parameter of both knees can be ignored. As a result, we can reduce the dimension of the fuzzy rules and thereby lower the computational burden. In this case the simulation conditions of the ANFS and its corresponding MSE values are given in Table 4.
From the Figures and Tables that present the simulation results, we can see that the generated ZMP trajectory from the fuzzy system is very similar to actual ZMP trajectory of measured for our walking robot shown in Figs. 11–16. The demonstrated high performance ability of the ANFS, means that ANFS can be effectively used to model and control a practical biped walking robot.
3.3 Comparisons
We now compare the performance of ANFS with numerical methods including three types of statistical regression models. For each statistical regression model, four different case types were constructed. Their general forms in the case of two inputs are given as:
where the ci are the regression coef?cients.
The corresponding MSE values are given in Tables 5–7 which reveals that type 2 gives the best results for the x and y coordinates for all the considered walking conditions. The generated ZMP positions and their corresponding trajectons generated using the type 2 regression model are shown in Figs. 26–31. We can conclude that the ANFS demonstrated a considerably better ZMP trajectory than the statistical regression models.
4 Conclusions
The ANFS modelling at the ZMP trajectory of a practical biped walking robot has been presented. The trajectory of the ZMP is an important criterion for the balance of a IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005 walking robot but the complex dynamics involved make robot control dif?cult.
We have attempted to establish empirical relationships between process parameters and to explain empirical laws by incorporating them into a biped walking robot. Actual ZMP data throughout the whole walking phase was obtained from a real biped walking robot both on a ?at level ?oor and
on slopes. The applicability of the ANFS depends on the MF used and the consequent part of the fuzzy rule. The generated ZMP trajectory using ANFS closely matches the measured ZMP trajectory. Then simulation results also show that the ZMP generated using the ANFS can improve
the stability of the biped walking robot and therefore ANFS can be effectively used to not only to model but also control practical biped walking robots. Figs. 32–41A
5 Acknowledgments
This work was supported by grant no.R01-2005-000-11-44-0 from the Basic Research Program of the Korea Science & Engineering Foundation.
6 References
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7 Appendix
This Appendix summarise the joint angles in the four-step motion of our biped walking robot. These joint angles are as follows.