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Section-Map Stability Criterion for Biped Robots

The importance of stability for dynamical systems is well-known. Any real system, including biped robots, need to be working under all kinds of disturbances. Whether the biped robot can effectively keep the planned motion under these disturbances is a fundamental property, and that is the explanation of stability intuitively. Stability of biped walking is the key problem in the theoretical framework of biped robots. Roughly speaking, the research of biped robots can be classified as the following three aspects: stability criterion, walking pattern planning, and walking pattern control. The purpose of stability criterion is to give the condition that the robot can realize stable walking under some control strategy. The purpose of walking pattern planning is to generate a desired gait offline or online, and it plays the role of feed-forward (Huang et al., 2001). The purpose of walking pattern control is to modify the planning walking pattern based on sensory information, and it plays the role of feedback (Huang & Nakamura, 2005). Among the above three aspects, stability criterion is the most fundamental and important, and it is the basis of walking pattern planning and real-time control. Although some researchers have proposed several walking control methods which are not based on stability criterion (Raibert, 1986; Geng et al., 2006); however if these methods can not ensure walking stability from the aspect of theory, then it will need many trials on hardware before success, and it is difficult to generate them to other platforms. Presently, there are the following three stability criteria for biped walking. The first criterion is zero moment point (ZMP) criterion. The ZMP was originally defined as the point in the ground plane about which the net moments due to ground contacts become zero in the plane of ground (Vukobratovic & Juricic, 1969). As long as the ZMP lies strictly inside the support polygon of the foot, then the support foot will not rotate about its extremities, and the desired trajectories of the robot’s joints are dynamically feasible, just like a stationary manipulator. Takanishi et al. (1985) and Hirai et al. (1998) have proposed the methods of pattern synthesis based on ZMP offline. Recently, Kajita et al. (2001), Lim et al. (2002), and Nishiwaki et al. (2002) discussed the methods of online pattern generation. The ZMP criterion is not a necessary condition for stable walking. The ZMP criterion results in a flat-footed and short-step walking style which is less dynamic than human beings. During normal walking, human do not always obeys the ZMP requirement and the foot does not always remain flat on the ground. Humans, even with prosthetic legs, use foot rotation to decrease energy loss at impact (Kuo, 2002). Based on the ZMP criterion, the robot can only realize static walking or quasi-dynamic walking, as shown in Fig. 1(a) and (b). During the dynamic walking of human beings, the under-actuated degree-of-freedom (DOF) emerges between the support foot and the ground, as shown in Fig.1 (c).

In case (a), the robot's nominal trajectory has been planned so that the center of mass (COM) and ZMP are both within the interior of the footprint. In case (b), COM has moved out of the footprint while ZMP still keeps within the interior of the footprint. In both case (a) and (b), the foot will not rotate, and thus the foot is acting as a base, just like a normal robotic manipulator. In case (c), however, both COM and ZMP has moved out the interior of the footprint, allowing the foot to rotate The second stability criterion is Poincare return maps (Guckenheimer & Holmes, 1985), which is a technique for determining the existence of periodic orbits and their stability properties. With this method, the system is assumed to have a periodic limit cycle. Small deviations from the cycle follow the linear relation 1 nn where n X is the vector of deviations from the fixed point, K is a linear return map, and 1 n X + is the vector of deviations in the following cycle. If the eigenvalues of K have moduli less than one, then the limit cycle is stable. Hurmuzlu and Moskowitz (1993) first applied the Poincare map to the locomotion systems, McGeer (1990) and Goswami et al. (1996) used this technique to analyze stability issues of passive walking robots. Grizzle et al. (2001) developed an extension of Poincare method that reduces the stability calculation to a onedimensional map, and Westervelt et al. (2003a) used this method to design automated control for an under-actuated planar biped robot (Chevallereau et al., 2003). However, Using Poincare return maps as a stability criterion of biped walking has two serious limitations. Firstly, they are only applicable for periodic bipedal walking. There is nothing periodic about walking across unevenly spaces obstacles, or changing walking speed. Secondly, using eigenvalues of Poincare return maps is valid only for small deviations from a limit cycle. Large disruptions from a limit cycle, such as when being pushed, cannot be analyzed using this technique. Therefore, Poincare return maps are not necessary for analysing bipedal walking in general.

Section-Map Stability Criterion for Biped Robots 131
The third stability criterion is motivated by observation that human beings keep the relative small size of angular momentum about the center of mass (CoM) during walking. In the book Legged Robots that Balance, Raibert (1986) speculated that a control system that keeps angular momentum during stance could achieve higher efficiency and better performance. Popovic and Englehart (2004) have suggested that humanoid control systems should explicitly minimize global spin angular momentum during steady state forward walking. However, minimizing angular momentum is not a necessary condition for stable walking. Human can walk can walk while swinging his or her upper body which makes the global spin angular momentum larger than zero. Minimizing angular momentum is not a sufficient condition for stable walking, as a biped robot can fall down the ground while maintaining an angular zero momentum (Pratt & Tedrake, 2005). Therefore, angular momentum about the Center of Mass is not a good stability criterion for biped walking. In fact, the desirable characteristics of an ideal stability criterion for biped walking may include: 1. Universal. The ideal stability criterion should be applicable not only to static walking, but also to dynamic walking. The ideal stability criterion should be applicable not only to periodic walking, but also to non-periodic walking. 2. Sufficient and Necessary. If the stability margin is outside an acceptable threshold of values, the robot will fall down. If the stability margin is inside an acceptable threshold of values, the robot will walk stably. 3. Comparable and Measurable. Two walking patterns should be comparable for stability based on their relative stability margins. One should be able to measure the relevant state variables and estimate the stability margin on-line in order to use it for control purposes (Pratt & Tedrake, 2005). 4. Simple and Convenient. The ideal stability criterion should be easy to compute, and convenient to be used in analyzing and controlling robots. This chapter explores such a coherent stability criterion based on the description of biped walking from a global point of view. The organization of this chapter is organized as follows. Section II proposed an overall mathematical modeling method for biped walking is based on dimensional-variant hybrid automata. Section III presented a rigorous definition of biped walking stability by combining the character of biped locomotion with the notion of classical stability, and pointed out that the model in the task space is a length-varying and inertiavarying inverted pendulum. Section IV presented a stability criterion in task space of biped walking. Section V introduced application methods of the proposed criterion. Section VI provided the experimental results of a planar biped robot. Section VII concluded the chapter.

Overall mathematical model for dynamic biped walking 2.1 Assumption of dynamic walking
During biped locomotion, two legs alternately contact the ground. When only one leg contacts the ground, the robot is called in single support phase. When both legs contact the ground, the robot is called in double support phase. The overall biped walking consists of single support phase and double support phase. In the field of biped locomotion, there's still no accurate and rigorous definition for dynamic walking (Goswami & Kallem, 2004). For the purposes of this chapter, to eliminate complications, we assume that dynamic walking should satisfy the following two requirements: 132 1. The under-actuated DOF emerges between the support foot and the ground during dynamic walking. 2. The double support phase is instantaneous and can be modeled as a rigid contact. The robot is said to be in under-actuated phase when the robot is in the mode of toe or heel contact, and in fully-actuated phase when the robot is in the mode of full sole contact. A typical dynamic walking for biped robots with feet is shown in Fig.2 with state space :{ (;) | , } N TQ q q q Q q R =∈ ∈ .

Overall biped model based on dimension-variant hybrid automata
The overall biped model is hybrid and dimension-variant in nature, consisting of some continuous dynamics and re-initialization rules at the contact event. We propose an overall mathematical modeling method for biped walking based on dimension-variant hybrid automata. This method expresses the overall biped walking model as an 8-tuple

Stability definition of biped walking
The manuscript has to be submitted in MS Word (*.doc) and PDF format. If you use other word editors and can not transfer it in Word and PDF please contact us. The most intuitive definition of biped stability is likely that "the biped does not fall". This section will give a sequence of preliminary definitions leading to a rigorous mathematical definition of biped walking stability by combining the character of biped locomotion with the notion of classical stability from the view of hybrid automata. Since the main destination of biped walking is to avoid fall, following (Pratt & Tedrake, 2005), we define a fall in this chapter as follows.

Definition 4.1 (Fall) Let Fall
Q be a set of the robot's configuration in which a point on the biped, other than a point on the feet, touches the ground. There are three modes of fall for biped robots considering in this chapter as shown in Fig. 6. Let q and q denote the vector of generalized position and velocity respectively. Fall Q can be expressed as is a feasible orbit.

Definition 4.3 (Distance between a Point and an Orbit)
Given a norm || || ⋅ , the distance between a point ii ∈ xT Q and an orbit 0 () x Orb can be defined as can be defined as , then the biped walking is attractive. Definition 4.7 (Asymptotically Stable Walking) If a biped walking is both stable and attractive, then it is asymptotically stable.

Biped model in the task space
The definition of biped stability is established in high-DOF space; however it is difficult to study the stability in this high-DOF space directly. Although biped robots are typically high DOF mechanisms, the task of biped walking is inherent a low DOF task. Considering planar biped walking, the task space is only 1-DOF problem, as shown in Fig where i I plays the role of an inertia, and i J plays the role of a net moment around the pivot.

Basic definitions for the biped model in the task space
donate the vector field in task space. We assume the following conditions are satisfied: R is open and connected; is right continuous on t, and depends continuously on the initial condition i z ; H4) Transition section is designed as H6) 0 i σ > during normal forward walking.

Definition 4.9 (Time Function)
The meaning of time function is the time to the transition section at the first time.
The meaning of distance function is the maximum distance between By H3), H5), and Lemma 2, is continuous. In the same way, any item in the right side of the equation (14) Fig. 9. Section sequence and section map is stable (resp., asymptotically stable, or exponentially stable), orbit * 0 () z Orb is stable (resp., asymptotically stable, or exponentially stable). Proof: The process can be summed up into the following three parts: Take (17) into (16), and yield    (21) and (23) The proof of asymptotical stability can be finished by taking (25) into (24).

Section map and its analytical form Definition 5.2 (Section Map)
The meaning of section map is the map between two contiguous section sequences, as shown in Fig. 9.

Remark 1: It should be noted that section map does not need the system is periodic. Since
is a one-dimensional restriction; therefore, section map and section sequence are both one-dimensional essentially. By H6),

Section-map stability criterion
When there exists an initial perturbation  {} ii ξ ∞ = is exponential stable. ♠ Remark 2: Theorem 2 can be intuitively explained as: If the error arising from an initial perturbation can be shrinked at each impact, then the stability of the orbit can be achieved, vice versa, as shown in Fig. 10.

Fig. 10. Intuitive explaination of section-map stability criterion
Remark 3: Section-map Stability Criterion is an extension of Poincare return maps, and it is applicable to non-periodic walking which Poincare return map criterion can not solve. In Section I, we have asserted that a desirable stability criterion for biped walking should satisfy four characteristics. In followings, we will give an explanation for section-map stability criterion about the above four characteristics.

Universal. Section-map stability criterion is established on a rigorous definition of biped
walking; therefore it can not only be applicable to static walking, but also to dynamic walking, and can not only be used to study periodic walking, but also to non-periodic walking. 2. Necessary and Sufficient. Section-map stability criterion is a sufficient condition for biped walking, is a necessary condition for periodic walking, and is a quasi-necessary condition for non-periodic walking. The quasi-necessary condition means that the condition that all section-map factors are less than one is not necessary for non-periodic walking, but the number of section-map factors which is less than one should be larger than the number of section-map factors which is more than one. 3. Comparable and Measurable. By comparing the section-map factors of two walking patterns, one can determine which pattern is more stable. The lower δ , the faster the convergence toward the reference trajectory after perturbation. One can measure the relevant state variables and calculate or estimate the stability margin on-line in order to use it for control purposes. 4. Simple and Convenient. Comparing with ZMP criterion, it is not necessary to calculate all points of trajectories, and only transition points need to be calculated. Comparing with Poincare methods, the proposed criterion study biped walking in low-dimension task space and has a concise form; therefore section-map stability criterion is easy to compute, and convenient to be used in analyzing and controlling biped walking.

Planar biped robot THR-I
To test the validation of the proposed criterion, a planar biped robot called THR-I has been developed, and this robot has five links which are connected by revolute joints. To constrain motions in the frontal plane, THR-I was constructed with a boom attached at the hip joint, as shown in Fig. 11. The boom constrains the sagittal plane to be tangent to a sphere centered at the universal joint, and still allows the robot to freely trip or fall forward or backward. The material of the boom is made of carbon fiber which is rather light, and the length of the boom is more than 5 times leg length of THR-I; therefore, the influence of the boom on THR-I' dynamics in the sagittal plane is very small. Encoders are located between the boom and hip joint, and binary contact switches are located at the tip of the leg to detect whether or not a leg is in contact with the walking surface. There is no actuation at the stance leg tip. Hence, the robot is underactuated. It is assumed that walking consists of two successive phase: a single support phase and an instantaneous impact phase. Although this robot is simple, it captures the main difficulties: hybrid, static instability, and under-actuation. This model was also adopted in Chevallereau et al., 2003). To describe the shape of the biped, let 1234 (, ,,) c qq q q q ′ = denote the configuration coordinates, and 5 q denote the absolute coordinate of the torso with respect to the coordinate frame as shown in Fig.12. The vector of the generalized coordinates of the biped robot is defined as 12345 (, ,, ,) q qqqqq′ = . Let (,) com com xy denote the Cartesian coordinates of the center of mass. Torques i u , 1 i = to 4, are applied between each connection of two links.
Let σ denote the biped angular momentum around the pivot point of the stance leg. For the above choice of the coordinates in the support phase, σ has the following form (Chevallereau, 2004): Since only the gravity affects the angular momentum around the pivot point, the angular momentum dynamics can therefore be written as Let q − , q + , q − , and q + denote the pre-and post-impact generalized positions and generalized velocities, respectively. The superscript "-" and "+" will denote quantities immediately before and after impact thereinafter.
We assumed that the impact is instantaneous and inelastic. After impact the former stance leg is lifted off immediately, and the legs swap roles, which can be written as the following transformation equation: q Rq where 01000 10000 00010 00100 00001 is a circular matrix describing the exchange of the support leg.
According to the conservation of momentum about impact point and no rebound nor slip at impact of swing leg tip, the map from q − and q + can be obtained respectively by where ( ) q q − Δ can be found in (Westervelt et al., 2003a).

Synthesizing periodic walking patterns based on section-map factor
Consider the following output function (Westervelt et al., 2003a): where 0 () hqspecifies the four actuated joints that are to be controlled and () d hq θ specifies the desired evolution of these joints as a function of the monotonic quantity θ(q), as shown in Fig. 13. Driving y to zero will force 0 () hq to track () Since one of the basic aspects of biped locomotion is to maintain a constant erect torso, we specify q5 = 0 in the whole walking cycle, as shown in Fig. 13. To shape the impact posture, we define two normalized non-dimensional parameters: Since we assume that the robot maintain a constant erect posture during the whole walking cycle, the following two conditions must be satisfied: Observing from human walking, we find that human beings appear to hold his support knee joint and relative angle between two thighs intendedly just before impact, so we get the following equation: Let σ − denote the angular momentum just before impact. According to (33), one can obtain According to equation (44) to (48), the generalized velocities q − can be expressed as According to (52), the section-map factor only depends on q at impact. Since there are only two parameters s k and h k , we can easily obtain a small section-map factor by exhaustive search computation (Fu et al., 2006).

Stable walking transition and its stability analysis
According to section-map stability criterion, the robot can achieve stable walking provided that all section-map factors is less than one. Fig. 14 shows the property of angular momentum during one-step transition.
To realize a stable one-step transition, the following two conditions must be satisfied: The domain of attraction of walking pattern a can be steered into the domain of attraction of walking pattern b under transition map (Westervelt, 2003b), that is, The walking with perturbation should be in the intersection set of domains: Since one-step transition map (55) is a monotonic increasing function, as shown in Fig. 15 Fig. 16 shows the property of angular momentum during multi-step transition, and we assume the domains before and after transition are respectively In the first experiment, THR-I was controlled to walk on a flat floor with a section-map factor 0.89 δ = . The experiment lasted more than 120s and THR-I took approximately 600 steps, which indicates the walking period is 0.2s per step. Fig. 17 gives video frames of THR-I taking four steps for a typical walking motion. Fig. 18 is the real joint angles versus time during walking. Fig. 19 are the section-map factors calculated from encoders during walking. . The robot is walking at 0.20 s per step

Unstable walking experiment
The second experiment demonstrated the walking result with a section-map factor 1.20 δ = , which indicates the corresponding biped walking is unstable stable. Fig. 20 shows the desired and real values of holonomic constraints, from which we can observe that the walking pattern can not be imposed on the robot. Fig. 21 is the corresponding snapshot of the walking experiment, from which one can see the robot falls forward finally. For periodic forward walking, the minimum of the angular momentum around the pivot point during a walking cycle should be positive; otherwise the robot has no enough energy to achieve a step and will fall backward. Fig. 22 is the desired and real values of holonomic constraints during walking on level ground with the section factor min 0 σ < , which indicates that the robot will fall backward finally. Fig. 23 is the snapshot of walking experiment.

Stable walking transition experiment
The fourth experiment demonstrated the walking transition. Fig. 24 is simulation results of phase portraits during one-step transition with a 5% error from limit cycles before transition. The section-map factor before transition is 0.86, and after transition is 0.89; therefore, the walking is stable. Fig. 25 gives video frames of THR-I walking from 0.2s/step to 0.3s/step.