Research of the motion balance of spherical mobile robot based on fuzzy control

2.1. Selection of control algorithm

The large-size spherical robot in this paper is high nonlinear and non-holonomic, its motors are highly coupled, these properties result in hard control using classic control algorithm. The irregularity of robot has a great effect on the uncertainty of dynamical equation which result in the infeasibility of using dynamical equation merely to obtain suitable control method. In addition, spherical mobile robot is much sensitive to the disturbance of outside world including landform, so ordinary algorithm is easily unstable. For above problems, it is a significant issue to choose a suitable algorithm to control the deflection.

We have compared some kinds of common algorithms like PID, CMAC, BP and fuzzy control. Traditional PID algorithms are simple and mature, they can verify the algorithm by collecting PID parameters, but later we found that collecting PID parameters and delimiting dead zone again and again is still hard to keep the robot robust. So, classical PID algorithm will not be concerned.

As intelligence algorithm, the property of BP and CMAC is ability to study. BP neural network is widely applied on pattern recognition, function fitting and so on. CMAC is widely applied on robot control because of the excellent real-time performance and the simpleness of calculation. But neither of them are suitable for spherical mobile robot directly. First of all, considered the high nonlinearity of robot, the generalization ability of BP is doubtful, and its complicated calculation is inadaptable to real-time control. CMAC will occupy enormous room which will exceed the range of driving hardware. Second, the biggest problem of these two algorithms is the absence of exact dynamical model to simulate and online study. Given the lack of dynamical model and the motion characteristic of robot, we determined to adopt fuzzy control algorithm which includes the thought of experts.

The advantage of fuzzy control is that it can integrate the perceptual knowledge, intuitive feeling and rational analysis of people into calculation without exact dynamical model. For example, it was applied to inverted pendulum successfully. So, we took fuzzy control algorithm to solve the deflection of robot.

2.2. Design of angle controller based on fuzzy control

We took fuzzy algorithm to control the deflection of robot. The input of controller include declination angle, declination angle acceleration and declination angle of target, the output is torque of short shaft.

Fig. 3Fuzzy control diagram of declination angle

As shown in Fig. 3, ${\alpha }_t$ is target torque, $\alpha$ is declination angle about $x$-axis and $\alpha \text{'}$ is declination angular velocity about $x$-axis. $u_0$ is output of fuzzy controller, $k$ is parameter. $u$ is output current from short axis motor:

After uniform quantization of $\alpha$ and $\alpha \text{'}$ which collected by sensor, we can get fuzzy segmentation as shown in Fig. 4 which are divided into seven parts called NL, NM, NS, ZE, PS, PM, PL which represent –3, –2, –1, 0, 1, 2, 3 by number. The range of $\alpha$ is between –45 degrees and 45 degrees. The range of $\alpha \text{'}$ is between –450 degrees and 450 degrees, and we took trigonometric function as membership function.

The formula below is expression of membership function:

where ${\mu }_i$, ${\mu }_{i+1}$, ${\mu }_j$ and ${\mu }_{j+1}$ are memberships corresponding to $\alpha$ and $\alpha \text{'}$, $i$, $i$+1, $j$ and $j$+1 are fuzzy sets. Then we can establish a two-dimensional fuzzy rule table according to experimental formula as shown in Table 1.

Fig. 4Diagram of fuzzy segmentation

According to rules of fuzzy deduction, we can obtain premise credibility [6, 7]:

According to fuzzy rule base, our results are $R_{i,j}$, $R_{i+1,j}$, $R_{i,j+1}$ and $R_{i+1,j+1}$.

So the total credibility of fuzzy system is:

${\mu }_a=\mathit{max}\left\{\mathit{min}\left({\omega }_{i,j},R_{i,j}\right),\mathit{min}\left({\omega }_{i+1,j},R_{i+1,j}\right),\mathit{min}\left({\omega }_{i,j+1},R_{i,j+1}\right),\mathit{min}\left({\omega }_{i+1,j+1},R_{i+1,j+1}\right)\right\}.$

Finally, we made a defuzzification to above formula and took use of center of gravity, then we can obtain the output of fuzzy control, if the calculated results of ${\mu }_a$ included ${\omega }_1$, ${\omega }_2$,…, ${\omega }_n$, $R_1$, $R_2$,…, $R_n$, that $n$ represents 1, 2, …, then we can get the output:

where $u_0$ is the output of fuzzy control system. We can control the output torque of declination angle by making use of $u=k\bullet u_0$. We only considered the situation that the declination angle is zero as balance position, relevant fuzzy rule table need to be changed if the declination angle has a value.

3.1. Zero degree declination angle fuzzy controller

While using fuzzy algorithm to control robot, we found that the bigger the declination angle is, the better the performance is. But, if we tried to keep robot balance at zero degree position, problem was emerged. After our careful analysis, the major factor of difficult balance is the irregular shape of robot.

It is precisely because of the particular shape that if we tried to make robot keep balance at vertical position, robot could finish balance automatically between $\text{±}$5 degrees, but if external force made the declination angle bigger than $\text{±}$5 degrees, robot would tilt to one side immediately and finally keep balance between $\text{±}$10 degrees. It is because the shape of robot is similar to ellipse. So, the result stated that the fuzzy rule table which was made by experience should be improved.

With countless modification of fuzzy rule table, the fuzzy controller could keep balance at zero degree position which ensured the smooth use of standstill or linear motion.

After many times of experiments and observation, we obtained fuzzy rule table as shown in Fig. 3, the output would be high when the declination angle is NS or PS. The reason is that the declination angle could easily come to balance point when it was between $\text{±}$10°, that is to say, only the short axis motor provided bigger torque can make the robot turn to zero degree angle continually. So, in order to make robot generate high torque, the output at NS and PS of fuzzy rule table should be high too.

Then we can make use of the same method of section 2.2, and we can get the output current u of short axis motor after fuzzy quantization, fuzzy inference and center of mass. But this fuzzy algorithm still cannot well performance the property of flat joint of shell. The output close to zero position was influenced by the high output at NS and PS, so the shaking and bad stability was emerged inevitably, we developed dead area rule for this:

So far, the fuzzy controller can control the robot return to its zero degree position on the flat ground. And it made a preparation for the control of linear motion.

3.2. Self-adaption declination angle control based reference model

We have illustrated the fuzzy control of declination angle of SMR before. But in practice, because the fuzzy rule table is mainly rely on the subjective perception, the control effect is not ideal enough. The stability needs to be improved, and there is also some place to advance in the adaptability to environment due to the structure of robot. It should be optimized. For that, we introduced a kind of self-adapted algorithm to improve the fuzzy controller what we already have.

The self-adaption ability defined that robot should adjust parameter in real time. To this algorithm, the parameters include fuzzy rule table, fuzzy quantization coefficient and output coefficient $k$ (as shown in Fig. 4), we made $k$ adapt itself here to advance the performance of system for the convenience. In order to improve the algorithm, the shell is almost smooth except the joint and can obtain good control property by the simple fuzzy controller. So we can adjust coefficient $k$ to adapt fuzzy controller and optimize controller finally.

The writer have already done dynamical model despite there is a little error with actual model. The fuzzy controller in section 2.2 can well control the dynamical model, so we chose it as reference model to make the dynamical performance of robot tend to reference model by adjusting $k$. So we can achieve great control effect through fuzzy controller.

The diagram of self-adaption algorithm is shown in Fig. 5, and we took gratitude reduction as study algorithm. In order to make dynamical property of robot tend to reference model at any position, the value of $k$ should be relative to declination angle of shell. If we took similar method, which selected several value of $k$ to represent function $k$, the resolution is too small to meet the requirement. But if the resolution is too large, it would not meet the requirement because the lack of generalization.

We decided to adopt the method of fuzzy $k$ to meet control requirement. First, we made fuzzy quantization to $\alpha$ as shown in Fig. 2, the fuzzy rule table of $R_k$ is shown in Fig. 4.

Fig. 5Control diagram of self-adaption algorithm

We could make the output of $k$ obtain generalization and convenient of study. From gradient study algorithm, we can get:

where $\mu$ is membership, $\gamma$ is study efficiency which is 100 here, ${\alpha }_r$ is the output of reference model. In conclusion, we can use the algorithm to adjust $k$ by adjust fuzzy rule table in real time. But for reference model, fuzzy controller can obtain favorable effect, and own the resistance to the disturbance of outside world. The self-adaption algorithm can greatly achieve large-size SMR control which has been proven by our experiment.

4.1. The experimental analysis of the declination angle extension

In order to verify that the algorithm can improve the performance of robot, we adopted a standstill control method that robot stand still which is relatively easy. The main purpose of this experiment is verifying that robot can keep balance by the effect of balance torque at the time when $\mathrm{\alpha }$ is bigger than $\text{±25}$ degrees. The parameter of robot is shown in Table 5.

In the experiment, we took 0 degree as original position, took 34 degrees as target position. According to Table 2, the output current of motor at balance position is 2500 mA, the experimental result is shown in Figs. 6 and 7.

Fig. 6The declination angle

Fig. 7The motor output

According to the experimental results, with the adjustment of balance torque, the output of fuzzy control algorithm can achieve great control effect at any declination angle.

4.2. Experiment analysis of robot’s optimized algorithm

In this experiment, we mainly tested the balance property of robot at zero degree before and after optimize. The parameter of robot is shown in Table 6.

Fig. 8The declination angle

Fig. 9Output current

The angle of original position was set as 30°, and the angle of target position was set as 0°. The reason that we took 0° as target position is that robot’s position is regular out of $\text{±}$5° and basically abide the dynamical model of robot, the difference is also minor compared to reference model, so the experiment benefited us much in verifying the effectiveness of this optimized method.

The experimental result is shown in Figs. 8 and 9. First, we compelled robot deflecting several times to make robot study optimized algorithm, then we released it from original position, when robot returned to its zero degree, we obtained the data. Figs. 8 and 9 is the experimental results of declination angle and output current.

After analyzed the data, we can get that the optimized algorithm resolved the insufficiency of output current which caused by the irregular shell, and this problem is hard to be resolved by changing fuzzy rule table. It’s because the joint of robot is also irregular, so the performance of control is bad, but self-adaption can guarantee the controllability of robot in a certain extent.