Analysis and control of biped robot with variable stiffness ankle joints

Lin, Zhenkun; Zang, Xizhe; Zhang, Xuehe; Liu, Yubin; Heng, Shuai

doi:10.3233/THC-209046

Analysis and control of biped robot with variable stiffness ankle joints

Issue title: The 8th International Conference on Biomedical Engineering and Biotechnology

Guest editors: Severin P. Schwarzacher and Carlos Gómez

Article type: Research Article

Authors: Lin, Zhenkun | Zang, Xizhe^* | Zhang, Xuehe^* | Liu, Yubin^* | Heng, Shuai

Affiliations: State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, Heilongjiang, China

Correspondence: [*] Corresponding authors: Xizhe Zang, Xuehe Zhang and Yubin Liu, Technology Innovation Building, Harbin Institute of Technology, No.2 Yikuang Street, Nangang District, Harbin, Heilongjiang, China. Tel.: +86 451 86413392; Fax: +86 451 86414538; E-mails: [email protected], [email protected] and [email protected].

Keywords: Biped robot, variable stiffness ankle joint, foot-ground impact, locomotion control

DOI: 10.3233/THC-209046

Journal: Technology and Health Care, vol. 28, no. S1, pp. 453-462, 2020

Published: 4 June 2020

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Abstract

BACKGROUND:

Biped robot locomotion is an active topic of research, and the walking stability is one of the research objectives.

OBJECTIVE:

This paper discusses the variable stiffness ankle joint and the walking control of a biped robot.

METHODS:

A design is introduced that achieves the ankle joint variable stiffness by using a pneumatic unit. The actuation system of the biped robot is based on the hybrid use of electric and pneumatic. The locomotion control architecture has been proposed to exploit natural leg dynamics in order to improve the biped robot walking stability. We also present a dynamic simulation which matches the biped robot and experiments with the real biped robot.

RESULTS:

The simulation and experiments result that introducing the variable stiffness ankle joint and the controller achieve a significant improvement in foot-ground impact and walking stability of the biped robot.

CONCLUSION:

The biped robot with variable stiffness ankle joints has a better walking performance under the control method.

1.Introduction

A biped robot applied in the natural and unstructured environment should have the real human motion abilities. Many terrains in the real world are inaccessible by the wheeled vehicles [1], but humans can walk in almost all terrains with their own legs. Legged vehicles, similar to the natural creatures, are the best choice for field missions in the natural environment [2]. A key feature of the human system is the biomechanical compliance of the joint. And the compliance of the joint is the ability of the structure to deform in response to external stresses [3]. I In pervious researches, it was shown that the elastic ankle joint can use passive dynamics to assist the push away movement at the later stage of single support and reduce the dissipative collision losses during the contralateral heel strike [4, 5]. It has also been demonstrated that the ankle stiffness helps robot achieve dynamic biped walking and reduces energy expenditure [6, 7].

Actuation with variable stiffness has been introduced as a solution to make the robot joints compliant. Compliant actuators generally include springs in series with stiff actuators. These elements can include fixed intrinsic stiffness (Series Elastic Actuators, SEA), or variable stiffness (Variable Stiffness Actuators, VSA) [8]. For the example there are the Actuator with Mechanically Adjustable Series Compliance (AMASC) [9], the Actuator with Adjustable Stiffness-II (AwAS-II) [10], the Continuous-state Coupled Elastic Actuator (CCEA), and the Actuator with Variable Stiffness and Torque Threshold (AVASTT) [11].

However, these actuators are large in size and weight, relatively. Therefore, the pneumatic actuator for the ankle joints is proposed in this paper. We first study the mechanisms and the design for variable stiffness ankle joints. Then the variable ankle joints controller is introduced in the biped robot control system. We also show the data from simulations and experiments of the platform.

2.Design of the variable stiffness ankle joints

A pneumatic unit is introduced to the ankle joint, the mechanism is shown in Fig. 1a. The prototype of biped robot is shown in Fig. 1b. The abstract model of the ankle joint is shown in Fig. 1c.

Figure 1.

Design of the variable ankle joint. (a) Components used in the ankle joint. (b) Prototype of biped robot. (c) Abstract model of the ankle joint.

According to cosine theorem there are:

(1)

cos⁡(α0-αd)-cos⁡(α-αd)=x⁢(-2⁢l1+x)2⁢L⁢r2

The force F applied on pneumatic unit is shown as follows:

(2)

F=τ𝑎𝑛𝑘𝑙𝑒r2⁢sin⁡α=𝐾𝑠⁢(α-α0)2r2⁢sin⁡α

The stiffness can be derived from:

(3)

k⁢(x)=d⁢Fd⁢x=d⁢Fd⁢α⁢d⁢αd⁢x=-l1⁢𝐾𝑠⁢(α-α0)⁢(2⁢sin⁡α-(α-α0)⁢cos⁡α)r2⁢sin2⁡α⁢L2⁢r22-(l1⁢x+𝐿𝑟2⁢cos⁡α0)2

3.Control schemes

3.1Variable stiffness ankle joint controller

On the basis of linear inverted pendulum model, the foot and ankle joint of the support leg are added to establish the biped robot support leg model, as shown in Fig. 2.

Figure 2.

Biped robot support leg model.

The equilibrium equation of the model can be expressed as:

(4)

{M⁢x¨=f⁢xr𝑀𝑔=f⁢zrτ+𝑀𝑔𝑥=M⁢x¨⁢z

Where f is ground reaction force, x is the center of mass x direction displacement, z is the center of mass z direction displacement, r is the length between center of mass and ankle joint, τ is ankle joint torque, x¨ is center of mass acceleration. As noticed in Fig. 2 ZMP is at the intersection of the force f reverse extension line and the sole of the foot, so the position of ZMP in the x direction is:

(5)

p=-rhz⁢x-τ⁢(z+rh)𝑧𝑀𝑔

Where rh is the ankle joint height from the sole of the foot, p is the position of ZMP in the x direction. Because rh is much smaller than z, p can be simplified to p≈-τ/𝑀𝑔. When ZMP exceeds the stability area, a torque can be applied at the ankle joint to move ZMP back into the stability area to ensure the walking stability of the robot. τmax is maximum ankle joint torque, so the maximum ZMP position error that can be adjusted by this control method is Δ⁢emax=τmax/𝑀𝑔.

Due to the flexibility of the variable stiffness ankle joint, the real angle θ would lag behind the desired angle μ. Utilizing the Lagrangian function to establish the linear equation between the energy of the system and the joint torque and represent as a state space expression:

(6)

[x˙1x˙2]=[01K+𝑚𝑔𝑟𝑚𝑟20]⁢[x1x2]+[01]⁢μ

(7)

p=[K2M2⁢r2⁢g⁢ 0]⁢[x1x2]+K𝑀𝑔⁢μ

Where K is the stiffness of ankle joint, x1=-𝑚𝑟2⁢θ/K and x2=x˙1 are state space variables. Discrete the equations as follows:

(8)

x⁢(k+1)=𝐴𝑥⁢(k)+𝐵𝑢⁢(k)

(9)

pk⁢(k)=𝐶𝑥⁢(k)+𝐷𝑢⁢(k)

Design the evaluation function of tracking error J as follows:

(10)

J=∑j=1∞{Q⁢(pj𝑟𝑒𝑓-pj)2+𝑅𝑢j2}

Where R and Q are weighted coefficients, pk𝑟𝑒𝑓-pk is tracking error. Utilizing pk𝑟𝑒𝑓 to minimize the evaluation function J to improve the tracking accuracy of the system. Based on the ZMP preview control theory design the ankle joint angle controller as follows:

(11)

u⁢(k)=-𝑆𝑥⁢(k)+[f1,f2,…,fn]⁢[p𝑟𝑒𝑓⁢(k+1)⋮p𝑟𝑒𝑓⁢(k+N)]

Where

(12)

{S≡(R+BT⁢𝑃𝐵)-1⁢BT⁢𝑃𝐴fi≡(R+BT⁢𝑃𝐵)-1⁢BT⁢(A-𝐵𝑆)T⋅(i-1)⁢CT⁢Q

3.2Overall control structure

Based on the position control, the hip and knee joint controllers are designed as:

(13)

u⁢(k⁢Δ⁢t)=kP⁢e⁢(k⁢Δ⁢t)+KP⁢Δ⁢tTI⁢∑j=0ke⁢(j⁢Δ⁢t)+KP⁢TDΔ⁢t⁢{e⁢(k⁢Δ⁢t)-e⁢[(k-1)⁢Δ⁢t]}

Where KP, TI and TD are the scale factors of controller, e⁢(k) is the input of controller, Δ⁢t is the sample time. In single support phase the swing leg ankle joint stiffness is constant, so during this period the ankle joint controller is designed in the same manner as Eqs (3)–(10).

After the swing leg touches the ground, the stiffness of the ankle joint is changed and the motor output angle is not equal to the ankle joint real angle:

(14)

θ=τK+θm⁢Kj

Figure 3.

Overall control structure of the biped robot.

Where θ is the real angle of ankle joint, θm is the motor output angle, K is stiffness of ankle joint, Kj is reciprocal of the motor reduction ratio. Combining dynamics of the biped robot with energy equation of the variable stiffness ankle joint, design the stiffness controller of the ankle joint. Combining these controllers the overall control system of the biped robot can be designed as shown in Fig. 3.

4.Simulations

4.1Simulation settings

A simplified virtual prototype model of the biped robot is established to verify the gait planning and control strategy of the biped robot by using Adams (a multibody dynamics simulation solution software). The basic parameters of the robot model is set as shown in Table 1. The dynamic simulation of the biped robot walking is carried out several times, and the optimal stiffness of the ankle joint under the current gait planning is determined as KS= 0.48 N⋅m/deg. The dynamic simulation process of the biped robot under this stiffness is shown in Fig. 4.

Table 1

Basic parameters of the robot model

Robot	Length (mm)	Weight (kg)
Torso	200	3
Thigh	279	2.5
Shank	298	1
Foot	160	0.75
Total	560	11.5

Figure 4.

Dynamic simulation process of the biped robot.

Figure 5.

Vertical force on foot-ground interface of biped robot with rigid ankle joint.

4.2Simulation results

Figures 5 and 6 show the simulation results of the vertical force on foot-ground interface. Figure 5 shows the force simulation result of biped robot with rigid ankle joint. It can be seen form the figure that the force on foot-ground interface has obvious abrupt change at the time of foot-ground collision, take the left foot as example, the force increases steeply to about 206 N at the heel collision time and increase from about 70 N to 121 N at the toe collision time. Meanwhile, the force of right foot drops from 108 N to 0 N steeply. After introducing the variable stiffness ankle joint controller, as shown in Fig. 6, the left foot force on foot-ground interface is about 100 N at the time of heel collision, but quickly decreased to about 20 N, and then the force increases steadily until toe collision. Meanwhile, the right foot force on foot-ground interface decreases steadily. Obviously, the simulation results show that introducing the variable stiffness ankle joint can absorb the impact effectively and achieve smooth switching between the swinging leg and the supporting leg in the double support phase.

Figure 6.

Vertical force on foot-ground interface of biped robot with variable stiffness ankle joint.

Figure 7 shows the desired and simulation ZMP trajectories. Obviously, the biped robot with variable stiffness ankle joint generates trajectories with a better ZMP curve than the biped robot with rigid ankle joint. It demonstrates that introducing the variable stiffness ankle joint controller can improve the stability of the biped robot walking.

Figure 7.

Desired and simulation ZMP trajectories. (a) Trajectories of biped robot with rigid ankle joint. (b) Trajectories of biped robot with variable ankle joint.

5.Experiments

5.1Experimental setup

The proposed control scheme for walking control is evaluated on a biped robot system, as shown in Fig. 8. Its front width is 320 mm and hip joint is 612 mm from ground. The robot weighs 10.92 kg with the control system hardware. Control system of the robot is set as shown in Fig. 9. Fourteen pressure sensors (Flexiforce A201 Pressure Sensor) are installed in the robot foot to measure the force on foot-ground interface. Six drivers (Elmo-G-Whistle) are used to guarantee the servo-control of drive motor. Six magnetic encoders are installed in each joint to measure the rotary displacements which can be used to control the leg posture. The host computer controls the proportional valves and the solenoid valves via the data acquisition card to realize the control of ankle joints pneumatic units.

Figure 8.

Biped robot prototype platform.

Figure 9.

Control system of the biped robot.

5.2Walking experiments

The experiments for verifications are set in two ways: the walking experiment with rigid ankle joints and the walking experiment with variable stiffness ankle joints. For the walking experiment with rigid ankle joint the pneumatic variable stiffness unit pressure is set as 0.6 MPa, which is approximately regarded as rigid joint. For the walking experiment with variable stiffness ankle joint the pneumatic variable stiffness unit is controlled as mentioned before to realize the variation stiffness of the ankle joint. As we can see from Figs 10 and 11, the biped robot can walk steadily in both conditions, in particular, achieve the rotation of ankle joints during the starting and landing of the gait, which validates the feasibility of the gait planning and control strategy.

Figure 10.

Walking experiment of biped robot with rigid ankle joint.

Figure 11.

Walking experiment of biped robot with variable stiffness ankle joint.

Figure 12.

Desired and real ZMP trajectories. (a) Trajectories of biped robot with rigid ankle joint. (b) Trajectories of biped robot with variable ankle joint.

Figure 12 shows the desired and real ZMP trajectories. Obviously, the walking experiment with variable stiffness ankle joints generates trajectories with a better ZMP curve than the walking experiment with rigid ankle joints. The real ZMP curve with rigid ankle joints has abruptly changes in the opposite direction when the swing leg touches the ground, as depicted in Fig. 12a, indicating that the biped robot has suffered a large impact at this moment. At the end of the double support phase, the support foot is lifting off the ground and the real ZMP curve has abruptly changes in the opposite direction too, indicating that the robot has a tendency to fall over backwards. As shown in Fig. 12b, the real ZMP curve with variable stiffness ankle joints has better tracking the desired ZMP. Due to the compliance of the ankle joints, these oscillations are small when the feet touch and leave the ground. It demonstrates that the ankle joints with a variable stiffness control strategy reduce the foot-ground impact and improve the stability of the bipedal walking.

6.Conclusion

In this paper, variable stiffness ankle joint for designing, analyzing and controlling robotic systems has been investigated. A design is introduced that achieves the ankle joint variable stiffness by using a pneumatic unit, and a control architecture is proposed to make the biped robot walk stability. The simulated and experimental results have shown that the variable stiffness ankle joint and the controller in the biped robot system provide an improvement in foot-ground impact and walking stability of biped robot. We will study the effects of variable stiffness knee joints and variable stiffness hip joints on the walking stability of biped robots in future researches. And we will study the performance of the variable damping and the control strategies for best utilizing the variable dynamics of the biped robot.

Acknowledgments

The work was funded by the National Key R&D Program of China (no. 2018YFF01012304), and the National Natural Science Foundation of China (NSFC) (no. 51675116).

Conflict of interest

None to report.

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