Both the linear leg spring model and the two-segment leg model with constant spring stiffness have been broadly used as template models to investigate bouncing gaits for legged robots with compliant legs. In addition to these two models, the other stiffness leg spring models developed using inspiration from biological characteristic have the potential to improve high-speed running capacity of spring-legged robots. In this paper, we investigate the effects of “J”-curve spring stiffness inspired by biological materials on running speeds of segmented legs during high-speed locomotion. Mathematical formulation of the relationship between the virtual leg force and the virtual leg compression is established. When the SLIP model and the two-segment leg model with constant spring stiffness and with “J”-curve spring stiffness have the same dimensionless reference stiffness, the two-segment leg model with “J”-curve spring stiffness reveals that (^{−1} both the tolerated range of landing angle and the stability region are the largest. It is suggested that the two-segment leg model with “J”-curve spring stiffness is more advantageous for high-speed running compared with the SLIP model and with constant spring stiffness.

Owing to the elastic elements (muscles, tendons, ligaments, and other soft tissues) of legged systems, in fast animal locomotion spring-like leg behavior is discovered to represent bouncing gaits like running, hopping, and trotting [

According to spring-like leg behavior of bouncy gaits, biomechanists [

However, biological limbs are not the telescopic linear leg model, rather, they are made up of multiple joints; their compliance is situated at the joint level [

It is well known that the SLIP model and the two-segment leg model with constant spring stiffness can be seen as an effective tool to research on bouncing gaits for legged robots. In addition to these two models, Karssen and Wisse [^{−1} is implemented on MABEL by adjusting its effective leg stiffness [

Although there are many running robots and running models at present, we focus only on the two-segment leg model in this study. This is because it is the most reduced leg configuration; in spite of the low complexity of this two-segment leg model, it is still suitable to solve the question of how leg segmentation and joint stiffness influence the stability of running at different speeds [

In this study, we not only hope that the two-segment leg model with “J”-curve spring stiffness will show the largest range of running speed for self-stable high-speed running in all three models but also expect that results of our work will be regarded as a promising concept for the design of bioinspired high-speed robots.

As shown in Figure

SLIP model during a running cycle.

Model parameters

Locomotion phases and transition conditions

The following section describes the configuration of the two-segment leg model and its dynamics of running with spring-like legs. As illustrated in Figure

The configuration of the two-segment leg model.

Figure

Two-segment leg model during a running period.

As can be seen in Figures

In this section we illustrate a nonlinear “J”-curve spring force-elongation relationship of the proposed model and its dynamics of running. A schematic diagram of a joint of large mammals, presented in [

Five points representing “J” curve.

Elongation | 0% | 25% | 50% | 75% | 100% |

Force | 0% | 4% | 17% | 48% | 100% |

The virtual leg force vector

Seeing that the maximum compression of the virtual leg rarely exceeds 30% of the rest virtual leg length among running animals [

The steps-to-fall analysis and the apex return map, which are reported in [

The second approach, the apex return map, can identify the fixed point. Furthermore, the stability of the fixed point

For the purpose of facilitating the comparison of all three models, simulation parameters can be defined as follows. (^{−1}, respectively. Thus, two speed ranges (5 to 92 m s^{−1} and 5 to 40 m s^{−1}) are utilized to investigate the advantages and disadvantages of our model compared with the other two models with respect to high-speed running capacity, respectively. Here, the speed of 5 m s^{−1} is the minimum speed of stable running in our model at

In this section, we not only analyze normalized force-compression relationships of the two-segment leg model with “J”-curve spring stiffness but also investigate the effects of different nominal joint angles

Normalized force-compression relationships of the SLIP model and the two-segment leg model. Solid curves are the proposed model and dashed ones denote the two-segment leg model with constant spring stiffness. The same nominal joint angle

Considering that in [

In this section, we concentrate on the effects of a running speed and the dimensionless reference stiffness on each other. Again, to gain better insights into advantages and disadvantages of our model compared with the other two models regarding high-speed running capacity, we also analyze the effects of these speeds on angle of attack and the stability region in all three models, respectively. Here, for ease of understanding, some typical examples, which are the regions of stable running at different running velocities (^{−1}, ^{−1}, and ^{−1}) and nominal joint angles (^{−1}, which is the maximum running speed of the cheetah, is the highest running speed recorded from land animal [

In all three models, the proposed model has the tolerated maximum range of the dimensionless reference stiffness at a running speed from 7 to 40/92 m s^{−1}; the smaller the nominal joint angles, the larger the stability regions for given combinations of ^{−1}. However, at the same dimensionless reference stiffness, the tolerated speed range is from 5 to 27 m s^{−1} in the SLIP model; worse still, in the two-segment leg model with constant spring stiffness this range is only from 5 to 19 m s^{−1}. In addition, a tolerated minimum ^{−1}; the smaller the nominal joint angle, the smaller the rate of increment in this tolerated minimum

Properties of regions of stable running for given combinations of

Regions of stable running for given combinations of

At fast running speed form 25 to 40/92 m s^{−1}, our model has the maximum range in angle of attack (^{−1}), ^{−1} and

At fast running speed form 17 to 40/92 m s^{−1}, regions of stable running of the proposed model are larger than those of the other two models, as illustrated in Figure ^{−1} the stability region consists of the 247 equidistant grids in our model (^{−1}, our model demonstrates the minimum stability region in all three models. Again, in the two-segment leg model, an increase in nominal joint angle leads to a decrease of the stability region during high-speed locomotion. For instance, at high running speed (36 m s^{−1}), the stability region of our model is decreased from the 205 at

In the following section we analyze stability of the proposed model by using of this single apex return map. For a given total energy ^{−1}.

Return maps of the apex height

Return maps function

Return maps function

Figure

We can obtain that the lower the running speeds, the higher the values of fixed points, and the high running velocities (31 m s^{−1}) result in the small basin of attraction containing all apex heights.

In this paper, we discuss the effects of the two-segment leg with “J”-curve spring stiffness on running speeds during high-speed running. Two methods, the steps-to-fall analysis and the apex return map [

Compared with the other two models, during fast running it reveals that (

These characteristics mentioned above are due to the two-segment leg structural configuration and “J”-curve spring stiffness properties, resulting in the nonlinear force-compression relationships depicted in Figure

Additionally, in the two-segment leg model with “J”-curve spring stiffness, the running is simulated across smooth and level terrain. Yet, the terrain is not the same in the real world, where the ground surface irregularities must be taken into account for running robots and running models. Therefore, in order to achieve stable running of the proposed model on uneven terrain, it is necessary to adjust apex height with adequate ground clearance in response to disturbances in ground height. Currently, for a given total system energy, the return map of the apex height ^{−1},

Return maps function

^{−1}), the tolerated minimum dimensionless reference stiffness in our model at

Practically, several researchers have designed their robotic legs similar to our model configuration and leg stiffness. For example, the robotic leg proposed by Schmiedeler and Waldron [

Virtual leg force-compression relationships of KOLT quadruped robot.

In this paper, we have presented the two-segment leg model with “J”-curve spring stiffness and have analyzed the effects of “J”-curve spring stiffness on running speeds of the proposed model during high-speed running. According to simulation results of all three models, for a given dimensionless reference stiffness, we have demonstrated that (^{−1} our model has not only the largest range of angle of attack but the largest region of stable running. It has already been successfully applied to quadruped robot, such as KOLT, for high-speed running.

Next, we will investigate the effects of a broad ratio of leg length and nominal joint angle range on high-speed running performance, respectively. Again, we will analyze the whole stability zone of segmented leg in order to gain better insights into advantages and disadvantages of our model at fast running.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This project is supported by National Natural Science Foundation of China (Grant no. 51475373).