Abstract

Both compliance and discontinuity are the common characteristics of the real ground surface. This 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. The effectiveness of the developed method is validated with a numerical simulation and a physical experiment.

1. Introduction

Underactuated bipedal walking has attracted increasing attention due to the low energy-consumption characteristic [1–4]. 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. This 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 As a result, the rigid ground assumption is no longer valid, and the robot-ground impact model is no longer independent of the ground compliance as well [14–18]. In documents, the effect of ground compliance on the bipedal locomotion and the control strategies to cope with it have been studied [19–26].

The ground discontinuity is another obstruction to deploy the underactuated bipedal locomotion in practice. The 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 two-step 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. The 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. The 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.

This 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.

2. Background

2.1. Adaptive Feedforward Controller

The 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-input-single-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. The schematic of AFC is shown in Figure 1, and only the key equations of [23] are introduced as follows.

Figure 1 

Schematic of AFC.

Based on the SISO model and identified λ, the theoretical value of u_f can be calculated and expressed aswhere u and 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.

The calculation of λ is transformed to solving a transcendental equation. Then, by applying the method of 1^st-order linearization, λ should be formulated as

The controlled input can be obtained aswhere denotes the step length and subscripts “c” and “d” denote the controlled and desired value, respectively.

2.2. 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.

Figure 2 

Models of the rigid robot and the compliant ground structure.

3. 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.

3.1. Essentials of the Stabilization Method Development Based on AFC

3.1.1. Hysteresis

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. Theoretically, 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, always equals to but defers from . As a result, the ideal effect of AFC is weakened, and the variation of lags behind the variation of 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]. Thus, 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 would be realized on a ground with invariant compliance, the following conditions must be satisfied:where 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. This feature is useful to stabilize the walking with variable step length.

3.1.2. Monotonicity

According to the relationship between the three equations introduced in Section 2.1, three monotonicities of AFC are studied and obtained: (1) is inversely proportional to ; (2) when is constant, the change of is directly proportional to ; and (3) to meet , the change of is directly proportional to the change of . The derivations of the three monotonicities are detailed in Appendix B.

3.1.3. Deductions for Walking on the Compliance Ground with AFC

When a biped robot, controlled with AFC, walks on a ground with an identical compliance:(1)λ is reciprocally correlated with L_s. For example, if must be satisfied.(2)x_i is positively correlated with L_s. For example, if must be satisfied.(3)u_f is positively correlated with L_s. For example, if must be satisfied.

In summary, the relationship between robot gait and walking speed is shown in Table 1, where “↑” means up, and “↓” means down.

3.2. Strategy for Walking with Variant Step Length

The 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. Thus, 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, should keep the law, the higher the better.(2)In the cycle after both the legs stepping over the pothole, the decline of 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 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 . The control strategy is formulated with two substrategies: step-length control strategy and desired velocity control strategy, as shown in Figure 3.

Figure 3 

The variable step-length control strategy.

In the step-length control strategy, the designed step length is L_Sd, ground discontinuity span is , and the whole walking process includes N cycles. The 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. Therefore, the step-length control strategy is given as follows:

In the desired velocity control strategy, the designed velocity of robot’s CoM is , and the change of CoM velocity before and after the adjustment cycle is given as follows:

In order to prove the effectiveness of the control strategy, theoretical evaluation is given in the Appendix.

4. Experiment and Discussion

4.1. 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.

Figure 4 

The prototype UABOT.

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. Then, 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.

4.2. Numerical Experiment

4.2.1. 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 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 and , 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.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 5 

Walking results with different controllers. (a) Step-length control substrategy. (b) Velocity control substrategy. (c) The whole strategy.

With the step-length control substrategy merely, when  > 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  < 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, 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.

4.2.2. Adaptability Evaluation of the Control Strategy for Walking with Variable Step-Length

To evaluate the adaptability of the control strategy, the variation of when the robot walks on the ground with nonidentical compliance and strides over a series of potholes with different widths is collected. The 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, is always higher than zero, and then, a long-lived bipedal walking is realized even when the ground compliance and the potholes width varying simultaneously. From the above, the adaptability of the control strategy in copy with a composite disturbance of the ground environment is demonstrated.

(a)

(b)

(a)
(b)

Figure 6 

Walking on the discontinuous compliant ground. (a) The relative position changes of the robot CoM. (b) The horizontal velocity changes of the robot CoM.

4.3. Physical Experiment

In Figure 7, UABOT is walking on a compliant pavement which is constructed with particle board and semi-PVC mats and striding over a virtual pothole, where . Among them, (a)∼(c) are the robot’s normal gait, (d)∼(f) are the adjustment period, (g)∼(i) are the span period, (j)∼(k) are the recovery period, and (l) is the follow normal gait.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)

Figure 7 

Proceeding of UABOT striding over a pothole where and the compliance of ground is nonidentical (“↓↑”: direction of foot movement, “−”: sole being in contact with the ground surface, “⟶⟵”: direction of hip movement, “NG”: nominal gait, SG: gait with short step-length, LG: gait with long step-length, and RG: gait for back leg recovery). (a) Initial stage of NG. (b) SSP of NG. (c) Final state of NG. (d) Initial state of SG. (e) SSP of SG. (f) Final state of SG. (g) Initial state of LG. (h) SSP of LG. (i) Final state of LG. (j) SSP of RG. (k) Final state of RG. (l) SSP of NG.

The variations of , , and the configuration of the robot’s CoM and step length in the three-time experiments are depicted in Figures 8∼10, respectively. GC is representing for the type of ground material, GC is 1 or 2 representing poplar particle board and semirigid, respectively. and represent the corresponding value in the i-th experiment.

Figure 8 

Average speed of each cycle.

Figure 9 

Horizontal velocity of robot’s CoM at the end of each DSP.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 10 

UABOT’s configuration of each cycle in the physical experiment. (a) Variations of and L_s in the 1^st experiment. (b) Variations of and L_s in the 2^nd experiment. (c) Variations of and L_s in the 3^rd experiment.

With the controller, is always higher than zero, and a long-lived bipedal locomotion is realized. The relationship between the three variables, , , and L_S, is in accordance with the principle of stabilization control strategy developed above. The physical experiment results demonstrate the availability and effectiveness of the control strategy for realizing a bipedal locomotion with variable step length.

5. 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)The 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)The 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)The 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.

Appendix

A. Relationship between the Aperiodicity and the Value of λ

Set and which are the actual and ideal initial configurations of the robot at the very beginning of the n-th equivalent cycle, respectively. and must meet the following conditions:then,

According to (2), by replacing with , λ_n is formulated aswhere

For in (A.4), if , the following inequation will be satisfied:

Then, 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 with respect to ,the effective domain of and the trend of it over can be obtained:

Set

If , wherethe following inequation is satisfied:

The trending of over can be described qualitatively by using a diagram, where only the relative magnitudes of and are considered, as shown in Figure 11. With the consideration of the sign of , the effective domain of consists of two parts. When , the effective domain of is described as

Figure 11 

Changes of Num and Den with β.

When , the effective domain of is described aswhere

According to the above, with the increase of , is decreased and is increased. In other words, if is to be realized on a certain compliant ground with invariant compliance, the following conditions must be satisfied:

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. Monotonicity of AFC

Let , , and satisfy the following relationship:where and are two variables. Then, the calculated walking speed can be rewritten aswhere

Then,where

By expending and further differentiating it with respect to , and are obtained:

Then,

By taking (B.15) and (B.16) into (B.7) and (B.8), the following results are obtained:

With the consideration of the definition of , the domain of it is

Thus, if , the following inequality must be satisfied:

While if , the following inequality must be satisfied:

On the contrary, if , the following inequality must be satisfied:

Furthermore, if (B.1) is solvable in real number field, the following inequality must be satisfied:

Thus, if ,is satisfied. Then, if , the following inequality must be satisfied:

While, if , the following inequality must be satisfied:

From the above discussion, two conclusions are obtained at first: (1) is positively correlated with but reciprocally correlated with ; (2) L_s is positively correlated with .

Furthermore, with the AFC, the relationship between the three variables, , , and , can be described as

Therefore, when , the following equationsshould be satisfied. In other words, to meet , must be always positively correlated with .

C. Theoretical Evaluation of the Control Strategy

1. Effectiveness Evaluation of Step-Length Control Strategy

Suppose the robot will meet a pothole in the n-th step. According to step-length control strategy and the model of ground discontinuity, the sequence of robot’s step length around the pothole can be given aswhere .

According to subobjective I, should be increased sharply compared with the nominal value of preplanned gait, . Thus, to evaluate the effectiveness of the step-length control strategy on the increasing of , the walking system is assumed to have been arrived a periodically stable state before the (n − 2)-th cycle. Some parameters of this stable state can be defined as

It must be mentioned that if the bipedal locomotion with invariant step length is periodically stable, the following equation must be satisfied: . For simplicity, (B.10)–(B.12) are expressed as

In the (n − 1)-th walking cycle, if only the step length varying is considered, the theoretical value of should be calculated:

Then, to realize , the ideal gait for the (n − 1)-th cycle will be obtained:

Therefore, the total increment of the robot’s CoM kinematic energy through the SSP of the (n − 1)-th can be expressed as

The actual value of should be calculated theoretically aswhereand the kinematic energy of the robot’s CoM at the end the DSP of the (n − 1)-th walking cycle is calculated as