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Akademische Arbeit, 2017
21 Seiten
Abstract
1. Introduction
2. Gait cycle
3. Stability
3.1 Zero-Moment Point (ZMP)
3.2 Centroidal angular momentum
3.3 Footstep-based criteria
4. ZMP-based control
5. High-level control
5.1 Online reference walking patterns
5.2 Balance control
6. Low-level control
7. Conclusions
References
Researchers dream of developing autonomous humanoid robots which behave/walk like a human being. Biped robots, although complex, have the greatest potential for use in human-centered environments such as the home or office. Studying biped robots is also important for understanding human locomotion and improving control strategies for prosthetic and orthotic limbs. Control systems of humans walking in cluttered environments are complex, however, and may involve multiple local controllers and commands from the cerebellum. Although biped robots have been of interest over the last four decades, no unified stability/balance criterion adopted for stabilization of miscellaneous walking/running modes of biped robots has so far been available. The literature is scattered and it is difficult to construct a unified background for the balance strategies of biped motion. The zero-moment point (ZMP) criterion, however, is a conservative indicator of stabilized motion for a class of biped robots. Therefore, we offer a systematic presentation of multi-level balance controllers for stabilization and balance recovery of ZMP-based humanoid robots.
Keywords: Biped robot; Stability; Zero-moment point; Balance; Multi-level control
Industrial robots are often rigidly attached to the ground. In contrast, mobile robots are able to move in specific environments depending on the tasks they are required to perform. Much attention has been paid to the design and control of mobile robots because they have significant characteristics such as versatile mobility and can sense and reacting to specific environments 1. Mobile robots can be classified as legged robots, wheeled robots and tracks.
Legged robots offer significant advantages over wheeled robots and tracks in terms of workable environments, energy consumption (e.g. biped robots could be exploited to consume less energy than wheeled robots when walking on sand environments) and adaptability 2. Although wheeled robots are simple lightweight structures, they need regular terrains for motion and lack for instance the ability to climb stairs. Tracks help to overcome this drawback, but they consume a lot of energy because of the high amount of friction between the chains and the ground. In contrast, legged robots can move in regular and irregular terrains with versatile mobility. With configuration changes, they can easily adapt to irregular environments. The small contact areas of their feet mean legged robots can be efficiently operated 2.
The biped robots, a type of two-feet legged robots, are designed to imitate human-like locomotion and perform certain tasks such as activities in danger environments, assistance to the elderly and entertainment 3-11. The biped robot can consist of a trunk and legs with/without feet or even have an entirely human-like mechanism depending on the desired application.
Biped robots have advantages over multi-legged robots. They have considerably higher adaptability, enabling them to overcome obstacles like narrow paths or stairs with ease. This adaptability is particularly important when the robot is required to perform human-centred tasks. Because of the smaller ground contact area and fewer actuators used, their energy consumption can be lower than that of multi-legged robots. This is consistent with the assumption that two-legged animals have higher efficiency and adaptability than multi-legged animals 12. For a historical review of legged machines, we refer readers to references 3,13.
Some challenges encountered in the design of biped robots are as below:
- Biped robots have unstable structures because of the passive joint located at the unilateral foot-ground contact 14-17.
- Because of the unilateral foot-ground contact and the varying configurations throughout the gait cycle, their mechanical description is highly nonlinear. During the single-support phase, the robot is under-actuated, turning into an over-actuated system during the double-support phase, however 18. Consequently, the dynamic description and control laws change during transition from one phase to another 15. It is notable that the biped robot can be fully actuated during the swing phase, with some limitations. In fact, the dynamics of the robot depend on which legs make contact with the ground. Moreover, exchange of leg support is accompanied by an impact disturbing the robot’s motion 15.
- Biped robots have many degrees of freedom (DOFs). A humanoid robot may have more than 30 degrees of freedom, making its mechanical behaviour and control difficult 19-21. To avoid this difficulty most researchers have used simple models based on approximations and assumptions. A trade-off between simplicity and accuracy is necessary.
- Biped robots interact with different unknown environments. The surface could be elastic, sticky, soft or stiff. This requires robust algorithms for the generation of reference trajectories and control. These algorithms should be insensitive to possible disturbances and noises. Stabilization and online adaptive control schemes could solve this dilemma.
These challenges are associated with mechanics, control, electronics, artificial intelligence and human anatomy and so studying biped robots is an interdisciplinary exercise. Unified solutions are therefore difficult to develop. For further reading on difficulties encountered in biped design, we refer to references 22. In our previous paper 23, we concentrated on discussion of the different methods used for generation of biped walking patterns and the stability/balance criteria used for biped stabilization. In this paper, we have extended discussion of the stability/balance problem to include some other criteria used for balance recovery, and we have attempted to present systematically multi-level control systems of biped robots based on ZMP criterion.
The remainder of the paper is organized as follows. The gait cycle of biped locomotion is presented in Section 2. Section 3 discusses the stability/balance criteria of the biped mechanism, and Sections 4, 5 and 6 introduce multi-level control of biped locomotion based on ZMP criterion. Section 7 concludes.
Remark 1. Despite numerous 3D applications have been reported in the literature, this paper tries to capture the fundamental characteristics of biped walking by limiting our study to this super-abstractive structure.
The complete gait cycle of human walking consists largely of two successive phases: the double-support phase (DSP) and the single-support phase (SSP) with intermediate sub-phases 24, 25. The DSP arises when both feet contact the ground, resulting in a closed-chain mechanism, and the SSP starts when the rear foot is not supported by the ground and the front foot is flat on the ground. It should be noted that the DSP accounts for about 20% of the time taken for one stride of the gait cycle and the SSP about 80% 16, 25.
Because of the complexity of biped mechanisms, most researchers have simplified the gait cycle to explore their kinematics, biomechanics and control schemes. For details on different walking patterns of biped robot, see 23 and the references therein.
The biped mechanism is unstable during the SSP. One of the challenges in the design and control of biped robots is to maintain their balance while walking in different kinds of environment. The reason for the instability is under-actuation owed to the passive joint of the foot-ground contact. This means that control of the feet is dependent on control of the mechanism above the feet 14, 29. Common stability theories such as analysis of eigenvalues, gain and phase margins, and Lyapunov stability can be applied to particular modes of biped robot gait but cannot guarantee biped stability for all modes of motion 30.
In general, there are two types of stability criteria that the trajectories of a biped mechanism depend on: static stability and dynamic stability. Static stability restricts the vertical projection of the centre of the mass of the biped to the inside of the support polygon. The support polygon is defined as the area represented by the stance foot during the SSP and the bounded area between the supported feet during the DSP 18. This type of stability leads to slow gait and biped robots with large feet 31. Thus, the position of the centre of mass, , can be calculated as
Abbildung in dieser Leseprobe nicht enthalten
where is the number of biped links, is the mass of link (i) and is the position of the centre of mass of link (i). The ground projection of can be found easily by determining its components.
Dynamic stability provides more freedom than static stability since the projected centre of mass of the biped may leave the support polygon and thus allow for faster gait 31. There are four specific techniques for analysing dynamic stability 30, 32-34:
(1) Zero-moment point (ZMP), (2) Centroidal angular momentum, (3) Footstep-based criteria (4) Periodicity-based gait.
The first three criteria will be described in details due to their association with humanoid biped locomotion. The periodicity–based gait is not locally stable such that the biped cannot stop, turn etc; therefore, it will not be considered in this paper.
The notion of the ZMP was proposed by Vukobratovic and colleagues 207,35, who exploited the passive joint of foot-ground contact. It is applied to the biped mechanism in the design of walking patterns and control schemes. The ZMP is the point on the ground at which the net moment vector of the inertial and gravitational forces of the entire body has zero components in the horizontal planes 14, 29. In brief, if ZMP is located inside the support polygon, then the system is stable and the centre of pressure (COP) of the foot coincides with the ZMP. If the ZMP is outside the support polygon, the system is unstable and the ZMP will be outside the stability margin comprising the fictitious ZMP (FZMP) 14, as shown in Fig. 1. The point P in the mentioned figure represents the location of the zero components of the net moments affecting the foot on the horizontal planes. It is clear that in the stable case one can determine the position of the ZMP by calculating the position of the COP. This is done by using force sensors at the sole plate of the foot. In effect, the theoretical calculation of the ZMP can be performed by the following two formulations:
Abbildung in dieser Leseprobe nicht enthalten
Fig. 1 Relationship between the ZMP, FZMP and COP. (a) Dynamically stable (b) Dynamically unstable 14
(a) Formulation 1 (more computational)
By knowing the inertial and gravitational forces, one can find the ZMP coordinate as in Eqs. (2) and (3) 36
Abbildung in dieser Leseprobe nicht enthalten
Abbildung in dieser Leseprobe nicht enthalten
where x and y axes are parallel to the horizontal plane for the biped motion, z axis is pointing upwards, is the inertial vector of link (i), and are the angular acceleration of link (i) about x and y respectively, g is the gravitational acceleration, ( is the coordinate of the ZMP and ( is the coordinate of the mass centre of link (i). The above equations can be used to investigate the stability of the biped mechanism during the SSP and DSP.
(b) Formulation 2 (less computational)
Equations (2) and (3) are highly nonlinear; therefore, most researchers use Eqs. (4) and (5) alternately during the SSP, using the static equations of the fixed stance foot 15.
Abbildung in dieser Leseprobe nicht enthalten
where and are the ankle joint torques about the referred axes and is the normal component of the ground reaction force.
Because the ZMP coincides with the COP, the ZMP coordinate can be found during the DSP according to Eq. (6) 38
Abbildung in dieser Leseprobe nicht enthalten
where represent the centre of pressure for the biped robot during the DSP, the front foot COP and the rear foot COP respectively, and and are the normal components of the ground reaction forces for the front and rear foot respectively.
In general, the ZMP-based biped mechanisms are characterized by the following.
- The stance foot of the biped should remain in full contact all the time 39.
- All the joints of the biped mechanism are actuated and often rigidly controlled to track predetermined trajectories which can simplify the control task during the SSP.
- They do not exploit natural dynamics, so their locomotion usually tends to be less natural than human-like motion (often bent-knee walking) and with high energy consumption.
- Most conventional humanoid biped robots are driven by electric motors via gears and their walking patterns are based on the linear inverted pendulum mode.
Several algorithms have been used to generate a stable reference trajectory for the biped mechanism which satisfies the ZMP constraint. This implies that the solution of Eqs. (2) and (3) to find a relationship between the centre of mass (COM) of the biped and the ZMP trajectory requires a lot of computational effort. Therefore these algorithms can be applied offline 42, 43. Alternatively, most researchers use simple models to generate the desired biped walking patterns such as the linear inverted pendulum for online implementation. Biped robot stability approaches based on the ZMP still lack efficiency, robustness, easy handling and natural motion 39, 44. For further reading, refer to references 27, 45-54.
According to the principle of dynamics, the rate of change of linear/angular momentum of a rigid body about its centre of gravity G is equal to the resultant external forces/moments. As a result the linear and angular momentums are zero for zero external disturbances (forces and moments). Exploiting this property, the angular momentum can be used as an index for rotational posture control of biped robots and even for legged systems 55. Sano and Furusho 56, 57 proposed using the angular momentum of the biped system as a controlled variable. It has been shown that the angular momentum during the SSP about the ankle joint of the stance foot is a function of the gravity effect and the ankle torque. Therefore, the ankle torque can be considered as an angular momentum control input. In their method, the torque control rather than the position control have been used for the ankle joints during the SSP and one of the hip joints during the DSP with specific selection of the desired angular momentum function. Goswami and Kallem 55 suggested an important criterion called the zero rate of change of angular momentum (ZRAM) point. They observe that the rate of change of centroidal angular momentum is a useful index/criterion for the analysis and control of posture control of biped robots in different interactive environments. Figure 2 shows the detail of their method.
They proposed three control strategies for balance control, manipulating the value of the rate of change of the angular momentum as follows.
Abbildung in dieser Leseprobe nicht enthalten
with notations shown in Fig. 2.
These strategies can be applied through the following:
1. Enlarging the support polygon.
2. Moving G relative to point P; this coincides with strategy of COG position-based compensation discussed in Table III.
3. Changing the direction of ground reaction forces by changing the centroidal (spin) acceleration of the biped mechanism; this coincides with the strategy of COG acceleration-based compensation discussed in Table III.
Remark 2. Although some researchers have not directly considered the angular momentum as a balance criterion, the latter two strategies have been used successfully to compensate for modelling errors and external disturbances, as we will see later. In addition, sudden forward movements of the body and arms make use of the angular momentum to maintain balance 32. Therefore, all the methods described in Section 5.2 try to regulate the centroidal angular momentum.
Popov and colleagues 54, 58-61 proposed the centroidal moment pivot CMP as a stability criterion for measuring the unbalance of the biped mechanism. It is exactly the same criterion proposed by 55. It is the point where a line parallel to the ground reaction force, passing through the centre of mass, intersects with the external contact surface 54, as shown in Fig. 2.
Abbildung in dieser Leseprobe nicht enthalten
Fig. 2 The relationship between ZMP, CMP and ZRAM points. (a) Dynamically stable biped (b) Dynamically unstable biped 55,54
It can be written in terms of the centre of mass and ground reaction forces as in (8) and (9) and in terms of the ZMP location, the vertical ground reaction force, and the moment about the centre of mass as in (10) and (11).
Abbildung in dieser Leseprobe nicht enthalten
where ( is the position of COM, and other notations are shown in Fig. 2
It is clear from (10) and (11) that if the centroidal moments of the biped are zero, the CMP will coincide with the ZMP and the biped is stable (balanced). This now means that the centroidal moments should be zero for a balanced biped; corrective moments for minimizing the non-zero spin angular momentum should be applied in the case of unbalanced bipeds. As a result, the ZMP may not coincide with the CMP in some cases. One of the important problems inherent in the notion of ZMP which motivates researchers to look for a general powerful stability index represented by angular momentum is that of the point-foot biped robot. In that case the ZMP is constrained at the contact point and cannot be repositioned. The only way to stabilize such a system is to produce a non-zero moment about the COM of the biped. Kajita and colleagues 62, 63 used the ZMP to generate reasonable walking patterns and integrated the notion of specific angular momentum as a high-level control to maintain the balance of the biped mechanism.
Remark 3. Most common humanoid robots depend on the ZMP in generating stable walking patterns. Motion planning is applied with the ZMP and the balance strategy can be performed by regulating angular momentum and the ZMP stabilizer. In other words, as proved in 63, the ZMP can be maintained inside the support polygon by selection of suitable values of the total linear and angular momentum and the position of the COM.
Remark 4. The foot rotation indicator (FRI) 29 and FZMP are equivalent terms which can be used to measure the unbalance of the biped mechanism without explaining their relationships with the linear/angular momentums, except in the work of 63. The ZRAM point and CMP are exactly equivalent in terms of regulating the angular momentum in relation to the COG. Even now there is no general criterion that can describe all stability cases of biped locomotion with different gaits 55.
Pratt and Tedrake 32 proposed five subtasks that the biped should perform to generate balanced gait: (1) maintaining the body (trunk) orientation, (2) maintaining virtual leg length resulting from simplified linear inverted pendulum plus flywheel body, (3) swinging the swing leg, (4) transition of different phases, and (5) regulating the velocity of the centre of mass. They called them velocity-based stability margins. Point 5 plays an important role in maintaining balance. It has been shown that the first three points are fully controllable and can be implemented by means of traditional high gain joint position control techniques, provided the requirements of joint torques are satisfied to avoid slipping. The fourth point can be dealt with by low impedance force control 64, or a shift transition function that guarantees smooth transition of the ground reaction forces from the rear foot to the front one 65.The most challenging problem is point 5 such that the COM velocity, beyond a certain point, is not controllable once the projection of the COM moves away from the support foot. The only possible strategy in this case is to take a step forward to maintain balance. Therefore, capture points have been proposed 33 to make a step for balance maintenance. A capture point is a point on the ground that can be stepped on by the biped robot in order to stop. Pratt and colleagues 33, 66 extended the linear inverted pendulum to include a flywheel body at its COM to explicitly model angular momentum about the COM. A closed form solution of the capture region was devised. Their model is approximate, however, and its application to complex humanoid robots is questionable! Wight et. al. 34 proposed a dynamic measure of balance to estimate the footstep to restore balance. The ZMP criteria can be augmented with this dynamic measure to broaden the stability conditions of the biped as it walks.
Comparison between the above mentioned stability criteria are described in Table I.
Table I Comparison of different stability criteria 32, 33
Abbildung in dieser Leseprobe nicht enthalten
Human locomotion on level terrain with constant speed walking is controlled by a local pattern generator on the spinal cord without the use of brain commands. The control system of human walking in a cluttered environment is more complex because this may involve multiple local controllers and commands from the cerebellum 3. Maintaining human balance while walking, standing, etc. is a complex process associated with different levels of the central nervous system (CNS) 67. In general, balance depends on the task characteristics and environment context as detailed in 68. The control system which is responsible for human walking can be described by hierarchical controls ranging from the highest level of action planning (motion planning) to the low-level control (reflex/local control) 3.
In robotics, multiple control loops have been used efficiently for generating feasible walking for biped mechanisms. As mentioned earlier, most ZMP-based humanoid robots need to use multi-level control to adapt to changes in the environment and internal perturbations. In contrast, the periodicity-based biped does not afford robust walking and cannot stop or stand etc; therefore, using a balance control is challenging. Figure 3 shows the classification of control levels of general biped mechanisms.
Remark 5. Offline walking patterns do not need multi-level controllers, because they have been prepared in advance to satisfy the dynamic/kinematic constraints 65, 69, 70,102. It needs only low-level control to track the desired angular joint trajectories.
Abbildung in dieser Leseprobe nicht enthalten
Fig. 3 General classification of multi-level control of biped mechanism according to its stability criterion.
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