A Stabilization Method Based on an Adaptive Feedforward Controller for the Underactuated Bipedal Walking with Variable Step-Length on Compliant Discontinuous Ground

Both compliance and discontinuity are the common characteristics of the real ground surface. -is paper proposes a stabilization method for the underactuated bipedal locomotion on the discontinuous compliant ground. Unlike a totally new control method, the method is actually a high-level control strategy developed based on an existing low-level controller meant for the continuous compliant ground. As a result, although the ground environment is more complex, the calculation cost for the robot walking control system is not increased. With the high-level control strategy, the robot is able to adjust its step-length and velocity simultaneously to stride over the discontinuous areas on the compliant ground surface.-e effectiveness of the developed method is validated with a numerical simulation and a physical experiment.


Introduction
Underactuated bipedal walking has attracted increasing attention due to the low energy-consumption characteristic [1][2][3][4]. In recent years, the research points mainly centralize on mechanism design [5], gait planning [6], motion control [7,8], and human-machine interaction [9]. However, a real ground surface is of compliance and discontinuity all the time. Because of the difficulty in stabilization control of a periodic system with multisource disturbance, the underactuated bipedal locomotion is still hard to act in real world.
In the early research studies, the ground is assumed to be rigid, and the robot-ground impact is modelled as the rigid body collisions of kinematic chains with an external surface [10,11]. With this assumption, a nominal gait should be preplanned by considering the actual unevenness of the terrain around the robot, and the bipedal locomotion can be stabilized by finding a control strategy to force each joint position always converging the gait. is method has been used prevalently to realize a stable bipedal locomotion on the rigid uneven ground [12,13]. However, with the heating up of studying bipedal locomotion on a real pavement, the rigid impact model was no longer capable to be used directly. For example, in a man-made city, to improve the walking comfort and safety, the roads are always paved with large quantities of compliant materials, such as the wood boards and semirigid polyvinyl chloride (PVC) mats which are used commonly in the hospitals, parks, and private houses. As a result, the rigid ground assumption is no longer valid, and the robot-ground impact model is no longer independent of the ground compliance as well [14][15][16][17][18]. In documents, the effect of ground compliance on the bipedal locomotion and the control strategies to cope with it have been studied [19][20][21][22][23][24][25][26]. e ground discontinuity is another obstruction to deploy the underactuated bipedal locomotion in practice. e compliant pavement being crushed and developing into potholes is common due to the inevitable fatigue and corrosion. Because of the difficulty in figuring out the exact conditions in the potholes standing in the way immediately, it is reasonable for the robot to stride over them with adjusted step length. However, the dramatic change of step length is bound to break the existing periodic stability of the bipedal locomotion system. And the stabilization of an underactuated bipedal locomotion with variable step length is a big challenge. In the literature, Nguyen utilizes a twostep periodic gait optimization technique to build a library of gaits for realizing a variable step-length bipedal walking [27]. Hu et al. used a feedback control method to adjust the robot's step length and walking speed to realize a planar bipedal walking on the uneven ground [28]. Yang et al. applied a reinforcement learning method to supervise the stride-frequency and find out the reasonable step-length online for bipedal walking [29]. However, since these methods are not designed for walking on the compliant ground, they are also incapable of coping with the effects of ground compliance and discontinuity simultaneously.
In this paper, a high-level control strategy is proposed to stabilize an underactuated bipedal walking on the discontinuous compliant ground. e main contribution of this work is the strategy being developed from the features of an adaptive feedforward controller (AFC) which was designed for the locomotion on a continuous compliant ground mentioned in [23] but not from a fresh start. As a result, although the ground environment is much more complex, the calculation cost for the walking system is not increased. e high-level control strategy is composed of a step-length control substrategy and a velocity control substrategy, both of which are developed based on the hysteresis and three monotonicities of the walking system controlled with AFC.
is paper is organized as follows: Section 2 summarizes some conclusions of the AFC method. Section 3 presents the control strategy to stabilize underactuated walking on the discontinuous compliant ground. Section 4 presents the validation of the proposed control strategy, and the conclusions are provided in Section 5.

Adaptive Feedforward Controller.
e AFC strategy, inspired by the man's gait characteristic that the walking speed increases when the man's body leans forward and decreases when the body leans back, was proposed to stabilize the underactuated bipedal walking on the compliant ground in our preview work [23]. For brevity, only the principle of AFC is described in this paper.
Firstly, the bipedal locomotion under the effect of ground compliance is modelled as a parameterized single-inputsingle-output (SISO) system, where the output u f and the input x f describe the horizontal velocity and the relative horizontal position of the robot's CoM at the end of each cycle, respectively. And then, a parameter, named as the dissipation ratio and denoted with λ, is used to describe the combined effect of ground compliance and robot's gait on the variation of u f . When the AFC is working, supposing u fd is the desired value of u f , λ should be identified online based on the variations of u f and x f during the last two walking cycles. And a desired x fc for the current cycle is calculated to control u f always converging to u fd and thereby keep the robot walking continuous. e schematic of AFC is shown in Figure 1, and only the key equations of [23] are introduced as follows.
Based on the SISO model and identified λ, the theoretical value of u f can be calculated and expressed as where u and v denote the horizontal and vertical velocities of rod CoM, respectively; x and y are the horizontal and vertical displacements of CoM toward the contact point at the initial time of impact, respectively; and subscripts "i" and "f " denote the initial and final value, respectively. e calculation of λ is transformed to solving a transcendental equation. en, by applying the method of 1 storder linearization, λ should be formulated as e controlled input x fc can be obtained as where L S denotes the step length and subscripts "c" and "d" denote the controlled and desired value, respectively.

Model of the Ground Discontinuity.
Based on the model of the compliant ground established in [23], a discontinuous compliant ground is constructed, as shown in Figure 2.
Although the pothole positioning the step-length planning online is already available for a real humanoid robot [30], for simplicity, the position of each pothole is described with the number of steps relative to the origin in this paper. And the width of the largest pothole is not up to 1.5 times longer than the step length of nominal gait.

Stabilization Control for Walking on the Discontinuous Compliant Ground
In this section, two features of AFC are studied, and the high-level strategy for stabilizing the walking with variable step length is developed.

Control cycle
Step: (n -1)-th n-th (n + 1)-th (n + 2)-th  Hysteresis is an inherent characteristic for an underactuated bipedal locomotion system controlled with the AFC. According to Section 2.1, the output of AFC is a periodically stable gait meant for a ground with certain compliance. eoretically, with the gait, periodically stable walking should be realized immediately. However, in practice, the initial state of the robot of each cycle is determined only by the end state of the last cycle. In other words, before a periodically stable state being arrived, the walking system is aperiodic. For example, during this stage, x fc n always equals to x f n but defers from x f n−1 . As a result, the ideal effect of AFC is weakened, and the variation of u f n lags behind the variation of u fd in a few cycles. Since the underactuated bipedal locomotion system is an inherent unstable system, this hysteresis would lead to system instability if without any measure. Fortunately, with the using of λ, the effect of hysteresis is suppressed. And the walking system can be stabilized with AFC even on a ground with variable compliance, as shown in [22]. us, to study the hysteresis effect suppression, the relationship between the aperiodicity and the value of λ is studied at first. For brevity, the derivation is detailed in Appendix A, and only conclusions are given as follows.
If u fn ≡ u fd would be realized on a ground with invariant compliance, the following conditions must be satisfied: where Δx � x i n − x id n is the difference between the actual and theoretical initial configurations of the robot of the n-th cycle. According to the inequalities, if the robot's CoM at the initial state is leaning back compared with the previous cycle, the total dissipated kinetic energy of the walking system through this cycle is high than the previous cycle and the walking speed tends to decline and vice versa. With this conclusion, λ not only quantifies the influence from ground compliance but also reflects the effect of the difference between the actual and theoretical initial configurations of the robot at each cycle. It suppresses the effect of hysteresis of a controlled underactuated bipedal locomotion. is feature is useful to stabilize the walking with variable step length.

Monotonicity.
According to the relationship between the three equations introduced in Section 2.1, three monotonicities of AFC are studied and obtained: (1) x i is inversely proportional to u f cal ; (2) when x i is constant, the change of L S is directly proportional to u f cal ; and (3) to meet u f cal � u fd , the change of x i is directly proportional to the change of L S . e derivations of the three monotonicities are detailed in Appendix B.

Deductions for Walking on the Compliance Ground with AFC.
When a biped robot, controlled with AFC, walks on a ground with an identical compliance: L s n < L s n−1 , u fc n < u fd must be satisfied.
In summary, the relationship between robot gait and walking speed is shown in Table 1, where "↑" means up, and "↓" means down.

Strategy for Walking with Variant
Step Length. e bipedal locomotion falling when striding over the pothole shall be attributed to the lack of stall control when only one leg of the biped robot has stepped over the pothole and the other is trying to be recovered, as shown in Table 1. us, there are two subobjectives of the control strategy for striding over the potholes and walking stably on the compliant ground: (1) In the cycle of only one leg stepping over the pothole, u fc should keep the law, the higher the better. (2) In the cycle after both the legs stepping over the pothole, the decline of u fc must be suppressed.
To meet these objectives, by observing the man's movement when striding over the pothole, a regulatory step with a step length smaller than the nominal should be executed before the robot striding over the pothole to increase u fc sharply, and the desired velocity of the robot's CoM after the both legs have been step over the pothole will change adaptively to suppress the dramatic decreasing of u fc . e control strategy is formulated with two substrategies: steplength control strategy and desired velocity control strategy, as shown in Figure 3.
Step: Figure 2: Models of the rigid robot and the compliant ground structure.

Complexity
In the step-length control strategy, the designed step length is L Sd , ground discontinuity span is W g , and the whole walking process includes N cycles. e robot crosses the ground discontinuous area at step i, step i is the span cycle, step i − 1 is the adjustment cycle, and step i + 1 is the recovery cycle. Here, the step length of (i − 1)-th cycle is the difference between two times of the designed step length and the span of ground discontinuity. erefore, the step-length control strategy is given as follows: In the desired velocity control strategy, the designed velocity of robot's CoM is u fd , and the change of CoM velocity before and after the adjustment cycle is given as follows: u f1 � · · · � u fd2 � · · · � u fdi � u fdi+3 · · · � u fdn , In order to prove the effectiveness of the control strategy, theoretical evaluation is given in the Appendix.

Experimental Setup.
A planar point-feet biped robot prototype, UABOT, is shown in Figure 4. UABOT's lateral stabilization is ensured by a directional wheeled cage, and thus, only 2D motion in the sagittal plane is considered. To perform anthropomorphic gaits, UABOT had to have at least a hip and two knees, giving a minimum of four links, where the three joints are actuated. For this external stabilization device not to limit the motion of the robot, including falling down, the cage is only attached to UABOT via a slide-revolute joint system, where the revolute joint is aligned with the axes of the hip, and the sliding joint is along with the normal direction of the ground. In the upright position, with both legs together and straight, the hip is 0.6 m above the ground. UABOT's total mass is 6.5 kg and the cage's 5.8 kg.
To prevent the numerical experiment being diverse from reality, the coefficient of stiffness and damping of real compliant materials to be used in the physical experiment is calculated at first, and the coefficients applied in the numerical experiment should be obtained accordingly.
In the physical experiment, the compliant ground on which UABOT walks is concrete covered with 4 mm poplar particle boards and semirigid PVC mats. According to the elastic modular of the two materials, their equivalent coefficients of stiffness and damping are obtained experimentally, as shown in Table 2. en, the sets of coefficients of stiffness and damping of the compliant ground to be used in the numerical experiment are determined and shown in Table 3 and Table 4, respectively.

Effectiveness Evaluation of the Control Strategy for Walking with Variable
Step-Length. To evaluate the effectiveness of the control strategy developed for walking with variable step-length, the variations of controlled and uncontrolled u f when the robot is striding over a pothole are collected. Subject to the geometric constraint of the principle of inverse-kinematics proposed for the gait generation online, the width of the pothole is not larger than 1.5 times the step-length of nominal gait and thus ranges from 0.19 m to 0.27 m. Meanwhile, to suppressing of the effect of ground compliance on the experiment results, a compliant ground with high stiffness and low damping is set, where the coefficients of stiffness and damping are 48 × 10 4 N/m and 0.01 × 10 4 N · s/m, respectively. Furthermore, to examine the effectiveness of the whole strategy and the two substrategies, the results with both substrategies simultaneously, only with the step-length control substrategy and velocity control substrategy independently, are shown in Figure 5.
With the step-length control substrategy merely, when W g > 0.22 m, after one leg steps over the pothole, u f decreases significantly which causes the falling of the walking process. With the velocity control substrategy merely, only when W g < 0.27 m, after one leg steps over the pothole, the other leg is recovered easily and the walking can continue. With the whole strategy, stable walking is realized all the time.
From the above, the necessity of the control strategy is proved which is also in accordance with the result derived in the second half of 2 in Appendix. On the contrary, with the control strategy, u f is boosted after one leg steps over the pothole which is also in accordance with the proof result of sufficiency discussed in the first half of 2 in Appendix.

Adaptability Evaluation of the Control Strategy for Walking with Variable
Step-Length. To evaluate the adaptability of the control strategy, the variation of u f when Table 1: e relationship among x i , x fc , L S , u f , and x ic .
Complexity the robot walks on the ground with nonidentical compliance and strides over a series of potholes with different widths is collected. e model of discontinuous compliant ground is built, where the coefficients of stiffness and damping of each compliant unit are selected randomly from Tables 5 and 6, respectively, and the width of each pothole, ranging from 0.19 m to 0.25 m, is also generated randomly, as shown in Tables 5-9. In Figure 6, the results are shown without and with the controller. With the controller, u f is always higher than zero, and then, a long-lived bipedal walking is realized even when the ground compliance and the potholes width varying    Figure 3: e variable step-length control strategy.

Rubber ground
Particle board Concrete ground    6 Complexity for the type of ground material, GC is 1 or 2 representing poplar particle board and semirigid, respectively. v sci and u fci represent the corresponding value in the i-th experiment.
With the controller, v sc is always higher than zero, and a long-lived bipedal locomotion is realized. e relationship between the three variables, u fc , x fc , and L S , is in accordance with the principle of stabilization control strategy developed       above.
e physical experiment results demonstrate the availability and effectiveness of the control strategy for realizing a bipedal locomotion with variable step length.

Conclusions
In this study, a variable step-length control strategy based on an AFC is proposed to stabilize underactuated bipedal walking on the discontinuous compliant ground.
(1) e control strategy achieves a good adaptability to the underactuated bipedal walking system diversity.
Since only the robot's CoM state under the ground effect is considered in the controlled input calculation, regardless of the specific ground compliance, robot's structure, and initial gait, the walking process can be stabilized; meanwhile, a desired walking speed can be obtained as well.
(2) e control strategy is low cost and high error tolerance. Since the controlled input is deduced with a polynomial with definite number of degrees, the algorithm is fast and stable, which is beneficial to real-time control. Furthermore, since the joint trajectories' tracking error will be counted in the total effect on the robot's CM motion in the real walking system, the control system has good error tolerance. (3) e control strategy has broad application prospects.
Since the control strategy is essentially originated from the human's gait, it can be integrated into a more advanced bionic control system for bipedal robots to realize a human-like walking in more complicated road environment.
Future work should primarily focus on two issues: (1) how to achieve underactuated bipedal walking on uneven terrains and (2) how to realize 3D bipedal walking.

A. Relationship between the Aperiodicity and the Value of λ
Set x i n and x id n which are the actual and ideal initial configurations of the robot at the very beginning of the n-th equivalent cycle, respectively. x i n and x id n must meet the following conditions: then, According to (2), by replacing x i n with Δx + x id n , λ n is formulated as 10 Complexity For β(Δx) in (A.4), if β(Δx) ∈ R, the following inequation will be satisfied: en, with the consideration that the robot's CoM is behind the swing foot at the horizontal direction at the very beginning of a DSP, the following inequation will be satisfied: and then, by differentiating σ with respect to Δx, the effective domain of σ and the trend of it over Δx can be obtained: the following inequation is satisfied: e trending of λ(β(Δx)) over Δx can be described qualitatively by using a diagram, where only the relative magnitudes of Num(β) and Den(β) are considered, as shown in Figure 11. With the consideration of the sign of λ n−1 , the effective domain of β(Δx) consists of two parts. When λ n−1 > 0, the effective domain of β(Δx) is described as When λ n−1 ≤ 0, the effective domain of β(Δx) is described as (A.14) According to the above, with the increase of Δx, β(Δx) is decreased and λ n−1 is increased. In other words, if u f n ≡ u fd is to be realized on a certain compliant ground with invariant compliance, the following conditions must be satisfied: λ n > λ n−1 , Δx > 0, λ n ≤ λ n−1 , Δx ≤ 0.
(A. 15) In summary, the AFC can effectively suppress the effect of hysteresis of online identification and the influence of high frequency variation of ground compliance.

(B.15)
On the contrary, if η < 2, the following inequality must be satisfied: is satisfied. en, if ε ≤ 1, the following inequality must be satisfied: