Some imbalance and balance postures of a passive quadruped robot with a simplified mathematical model are studied. Through analyzing the influence of the touchdown angle of the rear leg on the posture of the trunk during the flight phase, the stability criterion is concluded: the closer are the two moments which are the zero time of the pitching angle and the peak time of the center of mass, the better is the stability of the trunk posture during the flight phase. Additionally, the validity of the stability criterion is verified for the cat, greyhound, lion, racehorse, basset hound, and giraffe. Furthermore, the stability criterion is also applicable when the center of the mass of body is shifted. Based on the stability criterion, the necessary and sufficient condition of the galloping stability for the quadruped robot is proposed to attain a controlled thrust. The control strategy is designed by an optimization dichotomy algorithm for seeking the zero point of the balance condition. Through the control results, it is demonstrated that the imbalance posture of the trunk could be stabilized by adjusting the stiffness of four legs.
In recent years, some models were presented to study the trotting, bounding, and galloping gait of quadruped robot. The spring loaded inverted pendulum (SLIP) model was suitable to study the hopping of one-legged robot [
The stability was an important property of locomotion, since the stability could ensure the quadruped robot form a successive locomotion. Most of the authors studied the motion stability of quadruped robot by employing the Poincaré return map (PRM) based on the Newton-Raphson method [
Meanwhile, other authors also focused on achieving the stable bounding or galloping gait by control strategies. Marhefka et al. [
As stated in previous paragraph, the stability indicators in the sagittal plane are the pitching angle and the height of COM. Meanwhile, the pitching angular velocity of the trunk during the flight phase also plays an important role in maintaining stable locomotion of the robot. In this paper, the instable state is defined as the tumble of robot; that is, the bottom or the head falls down on the ground while the quadruped robot is galloping. Through the BSL model, the imbalance postures of the robot with improper pitching angular velocity are studied in this paper. Then, the stability criterion is achieved. Next, the controlled thrust is designed by adjusting the leg stiffness on the basis of the stability criterion. Finally, through some simulations, it is demonstrated that the control method is feasible to stabilize the trunk posture under the instable initial parameter while the quadruped robot is galloping.
In Methods, the simplified model is introduced and the controlled thrust is proposed. In Results, by analyzing the impacts of the pitching angle and angular velocity on the trunk posture, the stability criterion is proposed. Meanwhile, the controlled thrusts are obtained to stabilize the locomotion by adjusting the leg stiffness. In Discussions, the validity of the stability criterion is verified for various animals. Finally, in Conclusions, some summaries are given.
In the paper, a quadruped robot is used for the purpose of analyzing the stability of the galloping gait and is simplified as a BSL model in sagittal plane [
The simplified quadruped model of a cheetah.
The dynamic model is derived as
Through the dynamic model Equation (
System parameters and initial conditions.
Variable | Value |
---|---|
|
640 |
|
62200 |
|
1.5 |
|
0.75 |
|
240 |
|
11 |
|
6 |
|
1.6 |
|
0 |
|
0 |
|
0 |
|
0.6 |
|
0.6 |
|
0.3540 |
|
0.2122 |
|
0 |
|
0 |
The galloping process of a cheetah.
From Figure
However, practically, the trajectories of the galloping gait for the quadruped robot cannot be always a single kind of symmetrical locomotion. When the cheetah suffers from a disturbance, such as stepping on a loose stone, catching a small animal, or jumping over an obstacle, the cheetah then must keep the balance of the trunk for running forward. The trunk stability is dependent on the pitching angle
Based on the passive dynamic model [
Stance phase of the rear tailing leg.
In addition, the maximum elastic potential energy
Maximum elastic potential energy stored in the rear legs.
From (
During locomotion of the quadruped robot, the stiffness of legs has not been fully used to support the trunk. The leg stiffness is not constant but it can be adjusted by itself. When the robot wants to accelerate forward, the leg stiffness will decrease and when the robot wants to decelerate forward, the leg stiffness will increase. Therefore, in the paper, the stiffness compensation is thought as a passive control method [
For the passive quadruped robot, both the imbalance and balance postures depend on the initial mechanical parameters. When the touchdown angle
Pitching angular velocities at different touchdown angles.
Pitching angular velocities during the flight phase.
Figure
Imbalance posture of trunk with
Furthermore, when the initial touchdown angle
Balance posture of trunk with
When the initial touchdown angle
Imbalance posture of trunk with
By analyzing the balance and imbalance postures of the trunk Figures
The zero time of pitching angle and the peak time of COM.
Based on the discussion of Figure
In a word, (
Figure
Imbalance posture of the quadruped robot without control.
Therefore, the controlled thrusts of the hind limb are required to stabilize the pitching motion and the height of COM which are considered to be the indicators of stability in sagittal plane. The designed thrusts are given as follows based on (
The stiffness variation as the control input is achieved by solving the dynamic equation (
The control result is presented in Figure
Balance posture of the quadruped robot with control.
Controlled thrusts.
As shown in Section
Parameters of animals and the initial values of simulation [
Variable | Cat | Greyhound | Lion | Racehorse | Basset hound | Giraffe |
---|---|---|---|---|---|---|
|
3.7 | 35 | 167 | 900 | 25 | 1200 |
|
3200 | 7000 | 62200 | 62200 | 7000 | 62200 |
|
0.285 | 0.6 | 0.755 | 1.4 | 0.22 | 2 |
|
0.16 | 0.26 | 0.45 | 0.65 | 0.3 | 0.5 |
|
5 | 1 | 9 | 8 | 1 | 3 |
|
12 | 12 | 6 | 7 | 12 | 9 |
|
0.3 | 0.7 | 0.8 | 1.5 | 0.3 | 2.1 |
|
0 | 0 | 0 | 0 | 0 | 0 |
|
0 | 0 | 0 | 0 | 0 | 0 |
|
0 | 0 | 0 | 0 | 0 | 0 |
As shown in Figure
Imbalance and balance postures of cat. (a) The imbalance posture when
The imbalance and balance postures of greyhound, lion, and racehorse are shown in Figures
Imbalance and balance postures of greyhound. (a) The imbalance posture when
Imbalance and balance postures of lion. (a) The imbalance posture when
Imbalance and balance postures of racehorse. (a) The imbalance posture when
The basset hound and the giraffe are two special animals for the length proportion between leg and torso. The legs of basset hound are much shorter than the torso, while the legs of giraffe are much longer than the torso. The lengths of leg and torso are listed in Table
The imbalance and balance postures of basset hound are shown in Figure
Imbalance and balance postures of basset hound. (a) the imbalance posture when
The imbalance and balance postures of giraffe are shown in Figure
Imbalance and balance postures of giraffe. (a) The imbalance posture when
The COM of the carnivore will be shifted when it runs with a prey in its mouth. The running dynamics will be changed with the displacement of COM. So, we analyze the influence of the COM shift on the stability and dynamics of carnivore. The additional mass is in the mouth of the carnivore and varies from 0 to 75 kg. The largest additional mass is twelve percent of the carnivore mass. The ratio between additional weight and carnivore weight can mimic the case that the carnivores hold a prey in the mouth. The simplified force diagram is shown in Figure
The diagram of force analysis when the carnivores hold a prey in the mouth.
All of the simulations comply with the stability criterion equation (
Zero time of pitching angle and peak time of COM. The dark triangles indicate different intersections between the zero time curve of the pitching angle and the peak time curve of COM with different weights of prey.
The structural parameters of a robot have been fixed when the dynamic model is built; in this case, the stable running could be achieved by adjusting the stiffness and the touchdown angle of leg when the initial conditions are given [
In the paper, a stability criterion is presented based on the passive dynamic galloping. First, the dynamical model of the quadruped robot is constructed on the basis of the energy conservation law. Through the mathematical model, the imbalance and balance postures of a passive quadruped robot are analyzed. Then, a necessary and sufficient balance condition that is the stability criterion is attained. In addition, the validity of the stability criterion is verified against cat, greyhound, lion, and racehorse with the body weight varying from small to large. Besides, the stability criterion is applicable to the basset hound and giraffe, whose legs are much shorter or much longer than the torso. Furthermore, the applicability of the stability criterion is verified when the carnivore holds a prey in its mouth. The results support the conclusion that the influence of the COM shift on the stability criterion is almost negligible. Next, the controlled thrust is obtained with an optimization dichotomy algorithm for seeking the zero point of the stability criterion when the animal exposes imbalance postures. Finally, the imbalance postures of a galloping gait are stabilized with the controlled thrust by adjusting the stiffness of the legs. The validity of the stability criterion and the control method are verified by the controlled results.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project is supported by the National Natural Science Foundation of China (Grant no. 51375303).