Quadruped robots are machines designed to imitate the movement of four-legged animals. These robots have four extremities, each of them with their own sensors and actuators, that enable a movement similar to quadruped animals, such as dogs, cats or horses [1–4]. Quadruped robots have been receiving increasing attention in the robotics area because they can have many real applications, such as search and rescue, exploration, agriculture and entertainment [5–7].
Quadruped robots are designed with the goal of imitating the real movement of quadruped animals and require appropriate control algorithms for achieving stability and equilibrium while moving [8–10]. There are several kinds of quadruped robots in the literature, but we selected one that has not been previously considered with fuzzy logic. This robot is called Robot Dog Kit FNK0050 from Freenove, for which only a proportional-integral-derivative (PID) control existed in the previous literature. This is an existing research gap in the state of the art that we decided to consider as our research work.
The contribution of this paper is the proposed design of a fuzzy proportional derivative (PD) controller for the goal following problem, which can be viewed as an enhancement to the existing PID controller. The main idea is that fuzzy logic enables having a nonlinear control model, which can provide better results for complex problems. In this way, fuzzy PD control surpasses the traditional linear PID control in the goal following task.
The rest of the document is structured as follows: Section 2 explains the quadruped robot utilized in this work. Section 3 outlines the existing PID controller for this robot and then the proposal for a fuzzy PD controller. Section 4 summarizes the experimental results, and Section 5 offers the conclusions and future work.
In this section, we describe the particular quadruped robot utilized in this research work. The Robot Dog Kit FNK0050 is one of the models from Freenove, which is a quadruped design with open code that is compatible with Raspberry Pi. It has an acrylic light structure with several sensors, such as a camera, an ultrasound sensor, a gyroscope and an accelerometer. The robot has a total of 12 degrees of freedom, which makes it an ideal system for low-cost robotic applications. Figure 1 shows a side-view of the robot dog after assembly. Figure 2 depicts a front view of the robot.
Robot Dog side view
Robot Dog front view
Controlling the stability of quadruped robots is very important for several reasons: stability will contribute to reduced likelihood of falling, robustness against external perturbations, energy efficiency and improved performance. In addition, controlling the motion of the robot in following a trajectory to achieve a particular goal is very important. In this paper, we are addressing this last problem by providing the robot with a controller that will make the robot autonomously move to achieve a goal. Figure 3 illustrates the control problem, which basically consists of reaching the value of an input command starting from an initial point. In Figure 4, we illustrate how disturbances can affect the behavior of the controller in the closed feedback loop.
Control problem definition
Effect of disturbances on the controller
In Figure 5 we show the actual implementation of the controllers in the server to control the physical robot.
Controlling the robot with the server
For the test scenario, we created two configurations: the first one with a distance of 1 meter with the objective on the side and an irregular terrain, and the second one with a distance of 80 centimeters with the goal in front and a plain terrain. These configurations could generate trajectories like the ones illustrated in Figure 6.
Possible robot trajectories
In this section, we describe the implementation of the two fuzzy PD controllers, one for controlling the direction and velocity on the x axis (to maintain the red ball centered in the robot view), and the other to control the direction and velocity on the z axis (to position the robot in the front of the objective at a desired distance.
The fuzzy PD controller for the x axis is structured as illustrated in Figure 7, containing two inputs and two outputs. The first input is the error, and the second input is the change of the error called delta_error. The first output is the velocity of turning, and the second output is used to control the direction. The fuzzy system is of type-1 Mamdani form.
Structure of the fuzzy PD controller (x axis)
The fuzzy rules that were used in the fuzzy PD controller to position the goal in the center of the x axis are listed below.
- 1)
If error is left and delta_error is low then velocity is medium, direction is right
- 2)
If error is left and delta_error is medium then velocity is low, direction is right
- 3)
If error is left and delta_error is high then velocity is low, direction is right
- 4)
If error is center and delta_error is low then velocity is medium, direction is center
- 5)
If error is center and delta_error is medium then velocity is low, direction is center
- 6)
If error is center and delta_error is high then velocity is low, direction is center
- 7)
If error is right and delta_error is low then velocity is medium, direction is left
- 8)
If error is right and delta_error is medium then velocity is low, direction is left
- 9)
If error is right and delta_error is high then velocity is low, direction is left
In Figure 8 we illustrate the nonlinear surface of the fuzzy PD controller for the x axis.
Surface of the controller for the velocity
The fuzzy PD controller for the z axis is structured as illustrated in Figure 9, containing two inputs and two outputs. The first input is the error, and the second input is the change of the error called delta_error. The first output is the velocity of motion in the z axis, and the second output is used to control the direction. The fuzzy system is of type-1 Mamdani form.
Input and outputs of the fuzzy PD controller (z axis)
The fuzzy rules that were used in the fuzzy PD controller to position the goal in the center of the z axis are listed below
- 1)
If error is left and delta_error is low then velocity is medium, direction is backward
- 2)
If error is left and delta_error is medium then velocity is low, direction is backward
- 3)
If error is left and delta_error is high then velocity is low, direction is backward
- 4)
If error is center and delta_error is low then velocity is medium, direction is stay
- 5)
If error is center and delta_error is medium then velocity is low, direction is stay
- 6)
If error is center and delta_error is high then velocity is low, direction is stay
- 7)
If error is right and delta_error is low then velocity is medium, direction is forward
- 8)
If error is right and delta_error is medium then velocity is low, direction is forward
- 9)
If error is right and delta_error is high then velocity is low, direction is forward
In the same form as with the fuzzy PD controller for the x axis, the rules for the z axis were obtained by physically experimenting with the FNK0050 robot to avoid slipping and collision during the movement on the trajectory.
In Figure 10, we illustrate the nonlinear surface of the fuzzy PD controller for the z axis.
Control surface for the velocity for the z axis
In this section, we summarize the results that were obtained with the controllers mentioned in Section 3. These results were achieved by performing physical tests with the FNK0050 robot platform at distances of 80 cm and 100 cm, respectively. In Figure 11, we illustrate the test scenario for experimenting with the controllers.
Test scenario for the experiments
The PID controller was tested with two different scenarios, the first one is to find the objective in a straight line with a distance of 80 cm on a plain surface, and the second one is to find the objective that is positioned on the sides at a distance of 1 m on a rough surface.
In the first case, we performed 30 tests with a distance of80 cm and the objective positioned in front of the robot, and in all cases, the robot reached the objective with an average time of 24.2056 seconds. It is worth mentioning that this controller (by default) gives priority to the distance to the objective before centering the objective on the x axis.
In the second case, we performed 40 tests with a distance of 1 m and the objective positioned on the left side of the robot. In this situation, the robot was not able to reach the objective in all the tests, which we believe is because the default controller gives priority to distance over centering the objective.
The fuzzy PD controllers were tested with the same scenarios as for the default controller of the robot: the first one with the straight-line trajectory of 80 cm on a plain surface, and the second one with a lateral trajectory on a rough surface.
We performed 30 tests with a distance of 80 cm and the objective positioned in front of the robot, and in all cases, the robot reached the objective with an average time of 13.4796 seconds. The acceleration oscillated between 2 and 5, and the velocity between 2 and 3. The main advantages of the fuzzy PID controller were the dynamic adjustment of velocities and the adjustment to the position of the objective in the x axis, which enabled better movement of the robot, as shown in Figure 12.
Fuzzy PD controller adjusting the position with respect to the ball in x axis
The fuzzy PD controller was used in 40 tests on a rough surface, which could randomly add noise to the motion of the robot by affecting the leg movement of the robot. Even with the noise, the controller was able to reach the objective in all cases with an average time of 29.5325 seconds. In Figure 13, we see the result of one of the tests.
Fuzzy PD controller performance in the 1 m distance case
Tables 1 and 2 show the results of each test for the above-mentioned controllers, respectively. The results with underline and bold are the minimal times in reaching the objective, and in bold are the maximum times.
Results for the experiments with 80 cm
| Track of 80 cm | ||
|---|---|---|
| Tests | PID | Fuzzy PD |
| 1 | 26.1 | 12.45 |
| 2 | 28.39 | 12.26 |
| 3 | 24.61 | 11.54 |
| 4 | 22.43 | 11.51 |
| 5 | 26.71 | |
| 6 | 26.43 | 12.72 |
| 7 | 22.73 | 10.63 |
| 8 | 21.04 | 11.16 |
| 9 | 20.05 | 13.87 |
| 10 | 15.84 | |
| 11 | 22.71 | 13.52 |
| 12 | 22.91 | 12.70 |
| 13 | 20.76 | 14.55 |
| 14 | 21.36 | 12.66 |
| 15 | 30.03 | 13.39 |
| 16 | 25.47 | 14.07 |
| 17 | 23.00 | 17.08 |
| 18 | 21.43 | 16.62 |
| 19 | 24.01 | 12.71 |
| 20 | 26.51 | 15.32 |
| 21 | 27.09 | 13.42 |
| 22 | 20.82 | 15.71 |
| 23 | 23.97 | 14.47 |
| 24 | 26.47 | 13.63 |
| 25 | 24.78 | 14.84 |
| 26 | 24.56 | 12.77 |
| 27 | 23.11 | 14.16 |
| 28 | 27.17 | 13.89 |
| 29 | 29.47 | 14.69 |
| 30 | 22.63 | 11.90 |
| s | 24.2056 | 13.4796 |
| 2.8009 | 1.6871 | |
Results for the experiments with 1 m
| Track of 1 m | ||
|---|---|---|
| Tests | Fuzzy PD | PID |
| 1 | 26.28 | NAO |
| 2 | 28.12 | NAO |
| 3 | NAO | |
| 4 | 36.12 | NAO |
| 5 | 26.50 | NAO |
| 6 | 33.47 | NAO |
| 7 | 30.47 | NAO |
| 8 | 27.02 | NAO |
| 9 | 32.56 | NAO |
| 10 | 35.92 | NAO |
| 11 | 22.87 | NAO |
| 12 | 31.83 | NAO |
| 13 | 33.00 | NAO |
| 14 | 31.26 | NAO |
| 15 | 33.33 | NAO |
| 16 | 27.30 | NAO |
| 17 | 29.89 | NAO |
| 18 | 20.31 | NAO |
| 19 | 23.18 | NAO |
| 20 | 33.55 | NAO |
| 21 | 25.78 | NAO |
| 22 | 36.65 | NAO |
| 23 | 25.83 | NAO |
| 24 | 31.04 | NAO |
| 25 | 23.51 | NAO |
| 26 | 38.31 | NAO |
| 27 | 32.30 | NAO |
| 28 | 26.93 | NAO |
| 29 | 36.39 | NAO |
| 30 | 33.30 | NAO |
| 31 | 26.70 | NAO |
| 32 | 32.59 | NAO |
| 33 | 28.74 | NAO |
| 34 | 31.02 | NAO |
| 35 | 33.36 | NAO |
| 36 | 27.88 | NAO |
| 37 | 28.07 | NAO |
| 38 | 23.83 | NAO |
| 39 | 29.32 | NAO |
| 40 | 28.15 | NAO |
| 29.5325 | NAO | |
| s | 4.6036 | NAO |
The results obtained in the experiments with a scenario of 80 cm show a clear advantage in the average times of the fuzzy PD controller with respect to the PID controller. This statement is true despite this scenario being ideal for the PID controller. Note that the PD controller was better in all cases as well as on average. The reason for the superiority of the fuzzy PD controller is that it dynamically changes the velocity values according to the situation. A statistical test comparing the averages produces a t value of 17.96 and p value of 3.82 × 10−23, which are evidence of the superiority of the proposed fuzzy controller.
In Table 2, the acronym NAO (not able objective) is used to represent that the controller is not able to reach the objective. The results from Table 2 show that the PID controller is not able to reach the objective in all cases, which is due to the limited capability that it has to adapt to noisy situations (terrain with perturbations). On the other hand, the fuzzy PD controller is able to reach the objective in all cases, although it takes more time than in the previous table, and it is able to adapt to noisy situations.
In this paper, we presented fuzzy PD control for the problem of a quadruped robot. A fuzzy PD controller was designed and implemented with fuzzy logic to enhance its performance with respect to traditional linear controllers. The problem is very important to achieve efficient movement of the robot, as well as to minimize energy usage. A comparison of the designed fuzzy PD controller with respect to the PID controller was presented to verify the superiority of the proposal.
The experimentation was performed with a robot called FNK0050 for which only a PID controller was previously used, so the contribution is in the enhancement of the control using fuzzy logic. Future works include optimizing the design of the fuzzy PID controller, as well as elevating its design to type-2 [12–15] and possibly type-3 fuzzy logic [16–18] with the goal of handling higher levels of uncertainty in the control process. Finally, we envision optimizing the fuzzy PID design with metaheuristics, as in [19–21].