1. Introduction
Biped robots have attracted a lot of attention in recent years for their unique advantage of human-like structure, which allows them to easily work in a human-built environment [
1]. Up to now, a multitude of biped robot designs have been developed, such as Atlas, developed by Boston Dynamics [
2], Cassie, developed by the Oregon State University [
3], Asimo, developed by the Honda company [
4], and others [
5,
6]. Though developed by different research groups, what these biped robots have in common is the anthropomorphic leg structure, which is comprised of mechanical components, drive systems and transmission mechanisms, such as the thigh, the electric motor and the linkage mechanism.
Typically, biped robots are driven by electric motors [
7,
8], hydraulics [
9,
10] or pneumatics [
11,
12]. With respect to the publications of biped robots, most of them address the issues of bionic structure designs, gait panning, and control strategies [
13,
14,
15]. In these publications, the biped robots are usually taken as rigid bodies, which is not feasible in the real design, regardless of the driving approach. Hence, it is worth noting that the biped robots, in particular their legs, should be taken as flexible parts. Elastic deformation would occur when subjected to external forces during the biped robot’s locomotion process; consequently, the actual execution of the joint trajectory might deviate from the expected trajectory. For example, due to the elastic deformation of the drive chain of the knee joint motion, the execution angle position of the knee joint may deviate from its gait planning. Therefore, the stiffness of the drive chain would have an influence on the locomotion behavior of biped robots and is worth discussion.
However, few publications could be found on discussing the stiffness of the leg’s drive chain. Carbone et al. analyzed the static characteristics of the biped robot Wabian-RIV induced by stiffness. In their works, the robot was regarded as springs and the stiffness parameters were determined by measuring the displacement with respect to a certain static wrench being applied to the robot [
16,
17,
18]. However, the dynamic behavior of the robot was not accounted for. Kwon et al. analyzed the dynamic stiffness of mechanical components of the biped robot Mahru III for mass reduction using CAE methods [
19,
20]. However, only mechanical components were analyzed and the drive chain system was not accounted for. Lohmeier performed an elasto-dynamic analysis of the drive mechanisms of the biped robot, Lola utilizing two-mass systems as an equivalent model [
21,
22,
23]. However, the stiffness of the leg’s stiffness was not accounted for. ** control of the biped, while the stiffness of the drive chain was not discussed [
24,
25]. Moreover, it is worth noting that, in the above works, the legs of the biped robots were taken as two symmetrical legs. However, due to manufacturing errors, the two legs were not completely symmetrical; this would do harm to the locomotion behavior of the biped robot.
Research could be found on addressing the stiffness concerning other robotic systems, such as industrial robots, the lower limb exosuits, et al. [
26,
27]. Luca et al. analyzed the robot manipulator’s performance, taking the mechanical flexibility in to consideration, focusing on the stiffness and dam** properties [
28]. Klimchik et al. proposed a stiffness model for a serial robot, paying particular attention to the elastostatic parameters identification and calibration of the robot [
29]. Pashkevich et al. presented a methodology to enhance the stiffness analysis of serial manipulators, taking into account the loading influence on the manipulator configuration [
30]. Kim et al. analyzed the stiffness and optimized the design of an under-actuated tendon-driven robot [
31]. Geeroms et al. designed and analyzed a prosthetic knee joint actuator with a lockable parallel spring [
32]. However, in these works, the industrial robots were connected to fixed bases and lower limb exosuits usually played the role of assisting and interacting with humans; the biped robots are mobile robots that usually move on their own, with two legs that each take turns as the base during locomotion. Under this circumstance, the drive chain of the leg is of particular interest and discussion.
This work used the biped robot AIRO, shown in
Figure 1, as a platform. AIRO was designed and built by us in Zhejiang Lab. In this work, we conclude some lessons learned from designing and testing the biped robot AIRO and elucidate the effects of the stiffness of the leg’s drive chain on the locomotion behavior in biped robots using mathematical models. The model was experimentally validated using AIRO and the parameters of the leg’s drive chain of AIRO were further optimized.
The main contribution of this work are as follows: (1) a mass-spring model was established that allows analysis of the influence of the stiffness of the drive chain of the leg; (2) methods for determination of the parameters of the stiffness and the moment of inertia were introduced, including using ANSYS Workbench and the dynamic modelling of the biped robot; (3) guidance was given for the design of the biped robot’s legs, including the leg’s drive chain and the symmetrical design of the two legs.
The rest of this work is organized as follows.
Section 2 describes the structural design of the biped robot AIRO, especially its leg structure and the drive chain.
Section 3 brings the mathematical model for analyzing the stiffness of the drive chain.
Section 4 describes the simulation results and related discussions on the influence of the stiffness.
Section 5 shows the experiment setup and related experimental results. After that,
Section 6 presents the conclusions and the future work.
2. Description of the Biped Robot AIRO
Figure 2 presents the configuration of AIRO, a biped robot built in Zhejiang Lab. The robot was approximately 30 kg in weight, 1.4 m in height and 0.625 m in width. AIRO’s leg weighed 8 kg and its length was 600 mm, including 300 mm for the thigh and 300 mm for the shank. AIRO had 20 degrees of freedom (DOFs), including 2 for the head, 3 for each arm, and 6 for each leg. Regarding the leg, it consisted of three working parts: the hip joint with 3 DOFs, the knee joint with 1 DOF and the ankle joint with 2 DOFs. It was expected that AIRO could achieve omnidirectional walking with a maximum step size of 488 mm for forward walking and 328 mm for lateral walking.
The main design concept for AIRO, especially the leg structure, is based on the inverted pendulum model, which is characterized by a mass mainly concentrated above the leg and by the leg being lightweight. To satisfy this, as shown in
Figure 3a, the motors and the joint motions they drive are connected via linkages such that the motors can be placed closer to the hip, reducing the weight of the distal end of the leg. Specifically, the motor for driving motion of the knee is placed in the thigh; the motors for driving the motion of the ankle are placed in the shank.
AIRO is expected to be able to travel with only position control of the motor, without measuring and feeding back the actual trajectory of the joint. Specifically, the motors with encoders integrated within would receive commands from the walking gait generator and then execute the corresponding rotation motion. For a simpler leg structure and a lower cost, no extra encoders were added on the joints. By rotating the motors, the joints are driven to rotate an angle via the linkages, i.e., the drive chain. The motors are seen to rotate the same angles with the commands generated by the walking gaits; it is hoped that the angles at which the joints rotate should be the same as the angles at which the motors rotate.
However, during the period of the preliminary testing of the robot walking, two unfavorable walking phenomena of AIRO could be observed: (1) AIRO was observed to be unable to lift his legs into the air and then touch the ground alternately; (2) A relatively large upper body vibration could be observed; the upper body vibrated with different magnitudes in different directions, despite the symmetrical design of the two legs.
As mentioned above, we firstly examined the feedback data of the motors, and confirmed that the output trajectories of the motors matched well with those produced by the gait planning, proving that the output speed and torque of the motor were adequate.
After a thorough test of the robot walking, it was found out that the knee pitch motion trajectory deviated from the trajectory generated by the walking gaits, leading to the two unfavorable walking phenomena described above.
The reason behind these phenomena lies in the stiffness of the leg’s drive chain. To be specific, as shown in
Figure 3a, the knee pitch motion was driven via a linkage between the hip pitch motor and the knee joint, where a parallelogram was formed between the output of the motor and the knee pitch joint. As shown in
Figure 3b, the red lines denote the initial position of linkages. If the parallelogram was a rigid body, the rotation angle of the knee pitch motion would be the same as the output of the motor, as shown by the black lines, which can be expressed as
φ1 =
φ2. However, due to the elastic deflection of the linkages, the rotation angle of the knee pitch motion could not catch up with the output of the motor, as shown by the blue lines, which meant that
φ2 was no longer equal to
φ1.
In addition to this, for the biped robot’s walking, some parts would be driven by the knee pitch motion, as shown in
Figure 3a for the shank. If those parts were rigid bodies, they would be driven by the same rotation angle by the knee pitch motion, as shown by the black line in
Figure 3b, which can be expressed as
φ2 =
φ3. However, these parts would also undergo elastic deformation, leading to a different value of
φ3 to that of
φ2, as shown by the blue lines.
Hence, judging from the analysis above, by not measuring and feeding back the actual trajectory of the joint during walking, the actual rotation angle of the parts driven by the knee pitch motion φ3 would deviate from the angle φ1 generated by gait planning, which would inevitably lead to a negative influence on the locomotion behavior of the robot. Manufacturing and assembly errors lead to differences in the stiffness of the drive chain in the two legs, which, in turn, caused the upper body of AIRO to vibrate in different directions at different degrees.
A further discussion on the relations between the angles of φ1, φ2 and φ3 needs to be addressed to provide guidance for optimizing the structure of the leg and to improve the walking performance.
5. Experiments
Some experiments were conducted using the biped robot AIRO for validating the influence of the stiffness of the leg’s drive chain. Two types of linkages were manufactured with different stiffness, as shown in
Figure 14. The black linkages are those used during the preliminary tests and have lower stiffness, while the white linkages are optimized ones with greater stiffness. As shown in
Figure 15, using the white linkages, AIRO could alternately lift his legs and touch the ground, successfully achieving omnidirectional walking.
The pitch data of the IMU installed at the hip of the robot was used to analyze the influence of the stiffness of the leg’s drive chain, since no encoders were added specifically to measure the execution trajectories of the knee pitch motion. The IMU of AIRO was Xsens MTI-630, which could provide the pitch angle data for analyzing the locomotion dynamics of the robot. Three different cases were analyzed, as depicted in
Figure 16: (1) both are black linkages; (2) both are white linkages; (3) one is black; the other is white. During the experiment, the gaiting planning for both legs were completely symmetrical with a step frequency of 2.0 Hz; the rest of the robot was kept constant. The difference in the IMU’s pitch angle data could be taken as caused by the linkages.
As depicted in
Figure 16, for cases of both black and both white, peaks could be observed around 2.0 Hz, 6.0 Hz and 10.0 Hz. The peaks were larger for white at 2.0 Hz, the same for both cases at 6.0 Hz, and larger for black at 10.0 Hz. As analysis in
Section 4, the peak at 2.0 Hz indicates the peak-to-peak value, while the high frequencies (here 6.0 Hz and 10.0 Hz) indicate the wave envelope as well as the phase error. This indicates that, for the same gait command, the white linkage obtained a relatively larger peak-to-peak value, a relatively smaller wave envelope, and a smaller phase error, suggesting a better execution of the planned trajectory. The black linkage had a smaller peak-to-peak value, smaller wave envelope, and phase error, suggesting a poorer execution of the planned trajectory.
It is also worth mentioning that the upper body vibrating in different directions with different degrees could, however, still be observed with both white linkages. Moreover, peaks could be observed between 0 and 2.0 Hz, especially at 1.0 Hz. This was mainly due to the asymmetry of the robot’s legs. Although the robot was designed to be symmetrical, due to manufacturing and assembly errors, the structure of the legs was not completely identical, which led to a peak at 1.0 Hz. Under this circumstance, the behavior of the left leg stance was repeated only at the next left leg stance, while the behavior of the right leg stance was repeated only at the next right leg stance. Therefore, the frequency was half of the step frequency, i.e., 1.0 Hz.
This phenomenon was particularly evident in the case of one black and one white. Using one black and one white, the stiffness of the two legs was set asymmetry deliberately. The IMU’s pitch angle data exhibited a larger peak at 1.0 Hz, and many significant peaks at other frequencies between 0 and 2.0 Hz. These experimental results demonstrate that asymmetry in the stiffness, not the upper body vibrating in many different directions with many different degrees, which could also be observed during the robot walking. It suggests that attention should be paid to the tolerances of the leg structure during the design process, in particular, those that would influence of the drive chain. Reducing the manufacturing assembly errors could improve the symmetry of the legs of biped robots, which would in turn lead to an improvement in the locomotion behavior of biped robots, such as walking smoother with lower vibration of the upper body.
6. Conclusions and Future Work
This paper investigates the influence of the stiffness of the leg’s drive chain on the locomotion behavior of the biped robots. A mass-spring model was proposed based on the biped robot AIRO built in Zhejiang Lab, and methods for determination of the parameters in the mass-spring model were introduced, including determining the stiffness parameters using ANSYS Workbench and the inertia parameters through dynamic modelling of the biped robot.
Simulation results suggest that special attention should be paid to the stiffness of the drive train of the leg when designing a biped robot to ensure the walking capability of the robot. If the stiffness is not large enough, the motion of the joint will deviate from the gait planning, especially in the stance phase. In addition, manufacturing tolerances should be paid attention for symmetry of the legs of the biped robots. Asymmetry in stiffness of the legs would lead to greater vibration of the robot’s body. The simulation analysis was validated via experiments conducted on AIRO.
Upon the analysis of the preliminary testing of AIRO, this paper mainly focused on the knee pitch motion and the related stiffness that would influence the knee pitch motion. As is known, the locomotion of the biped robot is a whole-body dynamic problem and the stiffness of all the joints should be accounted for. Future work will be focused on the influence of the stiffness of the whole body based on the whole-body-control strategy of the biped robot to better understand the biped robot’s locomotion behavior.