The Study of Fractional Order Controller with SLAM in the Humanoid Robot

Abstract

We present a fractional order PI controller (FOPI) with SLAM method, and the proposed method is used in the simulation of navigation of NAO humanoid robot from Aldebaran. We can discretize the transfer function by the Al-Alaoui generating function and then get the FOPI controller by Power Series Expansion (PSE). FOPI can be used as a correction part to reduce the accumulated error of SLAM. In the FOPI controller, the parameters (𝐾_𝑝,𝐾_𝑖,and𝛼) need to be tuned to obtain the best performance. Finally, we compare the results of position without controller and with PI controller, FOPI controller. The simulations show that the FOPI controller can reduce the error between the real position and estimated position. The proposed method is efficient and reliable for NAO navigation.

1. Introduction

More than 95% of the control loops are of PI/PID type in process control, but only a few of control loops work well [1]. There are some problems in PI/PID control, such as bad tuning, incorrect implementation techniques, some restrictions and limitations, and actuator and sensor problems.

Using the notion of fractional order, it may be a step closer to the real world life, because many real physical systems are well characterized by fractional order differential equations [2]. So a considerable amount of research in fractional order dynamic systems and controllers was published in scientific and engineering literature in the last few years. Modern examples of applications are bioengineering [3], physics [4], chaos theory [5], viscoelasticity [6], and many others (see, e.g., [7]). Fractional order controllers can be expressed by fractional order differential equations. Fractional order controllers have been used in industrial applications [8] and various fields such as power electronics [9], system identification [10], robotic manipulators, irrigation canal control [11], mechatronics systems [12], and heat diffusion systems [13]. In addition, the stability is an essential property of systems, for example, for the stability of conventional systems of integer order and for fractional systems. So the stability issues for fractional order have been solved [2, 14–19].

In this paper, we will study fractional order PI (FOPI) controller combined with SLAM in the navigation of humanoid robot because of the limitation of PI controller. In the FOPI controller, the parameters (𝐾_𝑝, 𝐾_𝑖, and 𝛼) need to be tuned based on some design specifications. The results will be compared without controller and with PI controller, fractional order PI controller.

The rest of this paper is organized as follows: the FOPI controller used in the NAO robot navigation control is presented in Section 2, the background knowledge of SLAM and the NAO robot are described in Section 3, and the simulation results and conclusion are given in Sections 4 and 5, respectively.

2. Fractional Order PI Controller

2.1. Fractional Calculus

The fractional order 𝑃𝐼^𝜆𝐷^𝑢 was proposed as a generalization of the PID controller [20]. A general fundamental operator is denoted as follows: _𝑎𝐷^𝛼_𝑡=𝑑^𝛼𝑑𝑡^𝛼(𝛼)(𝛼)Re>01Re=0_𝑡𝑎(𝑑𝜏)^−𝛼,Re(𝛼)<0, (1) where 𝑎 and 𝑡 are the limit of the operation, 𝛼 is the fractional order, which can be any complex number, and 𝜏 is the variable of integration.

Commonly, there are three methods to define the fractional calculus [21].

2.1.1. Grünwald-Letnikov Definition

Consider _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=lim_ℎ→01ℎ^{𝛼[(𝑡−𝑎)/ℎ]}_𝑗=0(−1)^𝑗𝛼𝑗𝑓𝑡−𝑗ℎ, (2) where ^𝛼_𝑗=Γ(𝛼+1)/Γ(𝑗+1)Γ(𝛼−𝑗+1)=𝛼!/𝑗!(𝛼−𝑗)! is the binomial coefficient, [⋅] is the integer part, ℎ is the calculation of step length, and Γ(•) is gamma function.

2.1.2. Riemann-Liouville Definition

Consider _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡_𝑎𝑓(𝜏)(𝑡−𝜏)^{𝛼−𝑛+1}().𝑑𝜏,𝑛−1<𝛼<𝑛 (3)

2.1.3. Caputo Definition

Consider _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑓^(𝑛)(𝜏)(𝑡−𝜏)^{𝛼−𝑛+1}𝑑𝜏,(𝑛−1<𝛼<𝑛). (4)

For zero initial conditions and lower limit 𝑎=0, the Laplace transform of fractional derivatives is 𝐿₀𝐷^𝛼_𝑡𝑓(𝑡)=𝑠^𝛼𝐹(𝑠). (5)

2.2. Fractional Order Controller

From the definition, it is shown that the fractional order differential is different from the integer order differential. It is not to obtain the limit values at the point, but it is related to the function values from the initial time to this point time. So the fractional order PI has memory.

The typical 𝑛-term linear fractional order differential equation as is follows: 𝑎_𝑛𝐷^𝛽𝑛𝑦(𝑡)+𝑎_𝑛−1𝐷^𝛽𝑛−1𝑦(𝑡)+⋯𝑎₁𝐷^𝛽1𝑦(𝑡)+𝑎₀𝐷^𝛽0𝑦(𝑡)=𝑏_𝑛𝐷^𝛼𝑛𝑢(𝑡)+𝑏_𝑛−1𝐷^𝛼𝑛−1𝑢(𝑡)+⋯𝑏₁𝐷^𝛼1𝑢(𝑡)+𝑏₀𝐷^𝛼0𝑢(𝑡), (6) where 𝛼_𝑘 and 𝛽_𝑘 are arbitrary real numbers. Assuming 𝛼_𝑘>𝛼_𝑘−1>⋯𝛼₀>0, 𝛽_𝑘>𝛽_𝑘−1>⋯𝛽₀>0 because of no loss of generality.

Assume (6) meets the zero initial condition; we can get the continuous fractional order transfer function by using Laplace transform technique: 𝐺(𝑠)=𝑌(𝑠)𝑈(𝑠)=𝑏_𝑚𝑠^𝛽𝑚+𝑏_𝑚−1𝑠^𝛽𝑚−1+⋯+𝑏₀𝑠^𝛽0𝑎_𝑛𝑠^𝛼𝑛+𝑎_𝑛−1𝑠^𝛼𝑛−1+⋯+𝑎₀𝑠^𝛼0. (7) For the discrete fractional order control system, the equation 𝑧 transform is available: 𝐺(𝑧)=𝑌(𝑧)𝑈(𝑧)=𝑏_𝑚𝑤𝑧^−1𝛽𝑚+𝑏_𝑚−1𝑤𝑧^{−1𝛽𝑚−1}+⋯+𝑏₀𝑤𝑧^−1𝛽0×𝑎_𝑛𝑤𝑧^−1𝛼𝑛+𝑎_𝑛−1𝑤𝑧^{−1𝛼𝑛−1}+⋯+𝑎₀𝑤𝑧^{−1𝛼0−1}, (8) where 𝑤(𝑧⁻¹) is the transform function from 𝑠 domain to 𝑧 domain.

In general, the discretization of continuous fractional order differentiator/integrator 𝑠^±𝛾 (𝛾∈𝑅) can be expressed as 𝑠^±𝛾≈(𝑤(𝑧⁻¹))^±𝛾. So the generating function and its expansion determine both the form of the approximation and the coefficients [22]. We can get different generating function because of different calculation method. As a generating function, 𝑤(𝑧⁻¹), the following formula can be used in general: 𝑤𝑧⁻¹=1𝜇𝑇1−𝑧⁻¹𝛾+1−𝛾𝑧⁻¹, (9) where 𝜇 and 𝛾 denote the gain and phase tuning parameters, respectively, and 𝑇 is the sampling period. For example, when 𝜇=1 and 𝛾={0,1/2,7/8,1,3/2}, the generating function (9) becomes the forward Euler, the Tustin, the Al-Alaoui, the backward Euler, and the implicit Adams rules, respectively [23]. We will consider the Al-Alaoui generating function of the form 𝑤𝑧^−1±𝛾=87𝑇1−𝑧⁻¹1+𝑧⁻¹/7^±𝛾. (10)

The discretization formal for GL formula (2) is given after performing the PSE (Power Series Expansion) of (1−𝑧⁻¹)^±𝛾. By using the short memory principle [24], the discrete equivalent of the fractional order integrodifferential operator (𝑤(𝑧⁻¹))^±𝛾 is given by 𝑤𝑧^−1±𝛾=𝑇^∓𝛾𝑧^{−[𝐿/𝑇][𝐿/𝑇]}_𝑗=0(−1)^𝑗𝑗𝑧±𝛾_{[𝐿/𝑇]−𝑗}, (11) where 𝑇 is the sampling period, 𝐿 is the memory length, and (−1)^𝑗_𝑗^±𝛾 are binomial coefficients 𝑐_𝑗^(𝑟) (𝑗=0,1,…), 𝑐₀^(𝑟)=1, and 𝑐_𝑗^(𝑟)=(1−(1+(±𝛾))/𝑗)𝑐^(𝛾)_𝑗−1.

We use the FOPI controller, and its transfer function can be expressed as 𝐺(𝑠)=𝑈(𝑠)𝐸(𝑠)=𝐾_𝑝+𝐾_𝑖𝑠^𝛼=𝐾_𝑝𝑠^𝛼+𝐾_𝑖𝑠^𝛼. (12)

With the Al-Alaoui operation, we can obtain 𝐺(𝑠)=𝑈(𝑠)𝐸(𝑠)=𝐾_𝑝(8/7𝑇)^𝛼𝑃_𝑝/𝑄_𝑞+𝐾_𝑖(8/7𝑇)^𝛼𝑃_𝑝/𝑄_𝑞=𝐾_𝑝(8/7𝑇)^𝛼𝑃_𝑝+𝐾_𝑖𝑄_𝑞(8/7𝑇)^𝛼𝑃_𝑝. (13)

The third order expansion by using PSE method is 𝐺(𝑠)=𝑈(𝑠)𝐸(𝑠)=𝐾_𝑝𝐾^𝛼𝑝₃𝑧⁻³+𝑝₂𝑧⁻²+𝑝₁𝑧⁻¹+1+𝐾_𝑖𝑞₃𝑧⁻³+𝑞₂𝑧⁻²+𝑞₁𝑧⁻¹×𝐾+1_𝛼𝑝₃𝑧⁻³+𝑝₂𝑧⁻²+𝑝₁𝑧⁻¹+1⁻¹, (14) where 𝑝₁=−𝛼,𝑝₂=𝛼²−𝛼2,𝑝₃=−𝛼³+3𝛼²−2𝛼6,𝑞₁=𝛼7,𝑞₂=𝛼²−𝛼98,𝑞₃=𝛼³−3𝛼²+2𝛼,𝐾2058_𝛼=87𝑇^𝛼. (15) Equation (14) can be given by 𝐺(𝑠)=𝐾_𝑝𝐾^𝛼𝑝₃+𝐾_𝑖𝑞₃𝑧⁻³+𝐾_𝑝𝐾^𝛼𝑝₂+𝐾_𝑖𝑞₂𝑧⁻²+𝐾_𝑝𝐾^𝛼𝑝₁+𝐾_𝑖𝑞₁𝑧⁻¹+𝐾_𝑖+𝐾_𝑝𝐾^𝛼×𝐾^𝛼𝑝₃𝑧⁻³+𝑝₂𝑧⁻²+𝑝₁𝑧⁻¹+1⁻¹. (16) And it can be expressed as 𝑈(𝑧)𝐾^𝛼𝑝₃𝑧⁻³+𝑝₂𝑧⁻²+𝑝₁𝑧⁻¹(𝑧𝐾+1=𝐸)_𝑝𝐾^𝛼𝑝₃+𝐾_𝑖𝑞₃𝑧⁻³+𝐾_𝑝𝐾^𝛼𝑝₂+𝐾_𝑖𝑞₂𝑧⁻²+𝐾_𝑝𝐾^𝛼𝑝₁+𝐾_𝑖𝑞₁𝑧⁻¹+𝐾_𝑖+𝐾_𝑝𝐾^𝛼. (17) So we can get 𝑢(𝑡)=𝐾_𝑝𝑝₃+𝐾_𝑖𝑞₃𝐾^𝛼𝑒()+𝐾𝑡−3_𝑝𝑝₂+𝐾_𝑖𝑞₂𝐾^𝛼𝑒()+𝐾𝑡−2_𝑝𝑝₁+𝐾_𝑖𝑞₁𝐾^𝛼𝑒()+𝐾𝑡−1_𝑝+𝐾_𝑖𝐾^𝛼𝑒(𝑡)−𝑝₁𝑢(𝑡−1)−𝑝₂𝑢(𝑡−2)−𝑝₃𝑢(𝑡−3). (18)

3. Background of SLAM and NAO Humanoid Robot

The simultaneous localization and mapping (SLAM) is a process by which a robot can build a map of an environment and at the same time use this map to deduce its location. In SLAM, both the trajectory of the platform and the location of all landmarks are estimated online without the need for any a priori knowledge of location [25, 26]. EKF-SLAM is very well known to navigation problems. The main steps of SLAM include robot motion prediction, new landmarks initialization, and known landmarks correction.

The observation model is 𝑦=ℎ(𝑥)+𝑣, (19) where 𝑥 is robot state vector and landmark states, ℎ(𝑥) is the observation function, and 𝑣 is the measurement noise.

The covariance matrix is 𝑃=𝐹𝑃𝐹^′+𝑄, (20) where 𝐹 is Jacobian matrix and 𝑄 is the Gaussian noise.

The EKF correction step is written as 𝑧=𝑦−ℎ𝑥^′,𝑍=𝐻𝑃𝑥^𝑇+𝑅, (21) where 𝑧 is the innovation’s mean, 𝑥^′ is the observation position of the robot, 𝐻 is Jacobian matrix, and 𝑅 is the covariance matrix of the measurement noise.

Kalman gain 𝐾 is 𝐾=𝑃𝐻^𝑇𝑍.

The update position of the robot is 𝑥^′=𝑥^′+𝐾𝑧. (22)

The correction covariance matrix is 𝑃=𝑃−𝐾𝑍𝐾^𝑇. (23)

NAO is a new biped robot recently developed by French company Aldebaran-Robotics. This humanoid robot had been designed purposely to look approachable. With a height of 0.57 meters and weight of about 4.5 kilogram, NAO has the appealing appearance of a human toddler [27]. Based on NAO robot, the FOPI controller as the correction part is used in the SLAM simulation of NAO navigation in this paper.

4. Simulation Result

4.1. Control System Model

When NAO robot moves in 2D space, the motion is mainly influenced by three variables, namely, the robot current position, the control variable, and the disturbance variable. Therefore, it is very important to reduce the accumulated error for robot and effectively overcome the negative influence of disturbance variable.

NAO will deviate from a set target because there is no correction part to compensate noise of the sensors. When the control U is a fixed value, the real position and estimated position of the NAO will be largely different. So we use controller as the correction part to reduce the error. The correction part is based on the return error of the real position and estimated position of NAO. The outline of this correction part is presented in Figure 1.

Figure 1 

The correction part, where 𝑥𝑅 is the real position and 𝑥𝑒𝑠 is the EKF-estimated position.

4.2. Algorithm Performance Index

Because the NAO in SLAM mainly produces two kinds of errors, so the average error index of robot position, the average error index of landmarks, and the average variance index of robot position are proposed to test the performance of the controller. The average error index of robot position is the average different value between the real position and the estimated position. When NAO moves in 𝑠 steps, the definition 𝑒 is expressed by ∑𝑒=^𝑠𝑖=1𝑒𝑠, (24) where 𝑒 is the error of 𝑥_𝑅 and 𝑥_𝑒𝑠 and 𝑠 is the number of steps.

If the number of landmarks is 𝑛 and 𝑙𝑒 is the error of the real landmarks and estimated landmarks, the average error index of landmarks 𝑙𝑒 is defined as follows: ∑𝑙𝑒=^𝑛𝑖=1𝑙𝑒𝑛. (25)

According to (24) and (25), we can see that the smaller the average error, the better the control effect. But it could not judge the stability of the control system from the single average error index. So we define the average variance that shows the distribution degree of error. The average variance 𝜎 is defined as follows: ∑𝜎=^𝑠𝑖=1(𝑒−𝑒)²𝑠. (26) The smaller the average variance, that is, every step error of NAO is close to the average error, the better the stability of the control.

4.3. Control Algorithm Comparison

Figures 2 and 3 show the simulation result of SLAM with the fixed U and PI controller, respectively. The three stars are the landmarks. The red points and green points are the real position and the estimated position, respectively. The circles on the green points are the covariance of the robot. The circles become smaller when the robot is near the landmark. The green circles are the estimated landmarks.

Figure 2 

The simulation motion result of NAO in SLAM with the fixed U.

Figure 3 

The simulation motion result of NAO in SLAM with PI controller.

In this paper, if not specified, abscissa represents steps, and ordinate represents the error. Figure 4 shows the error of the different 𝐾_𝑝 with PI controller in EKF-SLAM. When 𝐾_𝑝<0.7, the PI controller cannot effectively adjust the control variable according to the error. And when 0.7<𝐾_𝑝<1, the control variable generated from PI controller can basically make the estimated position trace to the real position. But when 𝐾_𝑝 is too large, the PI control variable cannot keep stable, so we cannot follow the estimated position of robot.

Figure 4 

The error of the different 𝐾𝑝 with PI controller.

Figure 5 shows the comparison of the errors with fixed U and PI controller. It can be seen that the error with PI controller is far less than the one with the fixed U. However, when the steps of the robot motion increase, the error generated from PI controller also gradually increases. So PI controller cannot efficiently reduce the accumulated error of the robot with the increasing steps. Therefore, we use FOPI controller to decrease the error in the motion process. There are three parameters (𝐾_𝑝, 𝐾_𝑖, and 𝛼) to be tuned.

Figure 5 

The comparison of the error with fixed U and PI controller.

From Figure 6, we can know that the scope of 𝐾_𝑝 is 0.7–0.9, 𝐾_𝑖 is about 0.2, and 𝛼 is the value of about 0.1. In Figure 6(c), if the parameter 𝛼 increases, the result of controller will become bad. But when 𝛼 is too small, the effect of integral is not obvious, so the value of 𝛼 is about 0.1. In the permitted range of proportional coefficient 𝐾_𝑝 and integral coefficient 𝐾_𝑖, the FOPI error influenced by the changing of 𝐾_𝑝 or 𝐾_𝑖 will not be a linear increasing trend, and its curve will move up or down with the increasing 𝐾_𝑝 or 𝐾_𝑖. There is a fundamental difference between FOPI controller and PI controller. The error in FOPI controller has oscillation trend, which reflects that FOPI has a better control performance.

(a) The error of different 𝐾𝑝

(b) The error of different 𝐾𝑖

(c) The error of different 𝛼

Figure 6 

The effect of three parameters (𝐾𝑝, 𝐾𝑖, and 𝛼) of FOPI controller.

Based on the analysis of the parameters, this paper selects the optimal parameters (𝐾_𝑝=0.7, 𝐾_𝑖=0.2, and 𝛼=0.1) of FOPI controller. The simulation of motion result of EKF-SLAM with FOPI controller is given in Figure 7. We can conclude that FOPI control can keep stable and get a better dynamics performance. The NAO robot’s estimated position has almost completely been close to the real position.

Figure 7 

The simulation motion result of SLAM with FOPI controller.

Figure 8 is the comparison between the optimal parameters of FOPI controller (𝐾_𝑝=0.7, 𝐾_𝑖=0.2, and 𝛼=0.1) and PI controller (𝐾_𝑝=0.9, 𝐾_𝑖=0.02). We compare them with the fixed U. The results verify the validity of FOPI controller. The average error and average variance are listed in Table 1. When NAO moves 0.5 each step, we can get that the error rate has got to 14% with PI controller and the error rate to approximately 3% with FOPI controller from Table 1. According to the index, it is shown that FOPI controller has better stability and higher accuracy. In addition, the average errors of the real landmarks and estimated landmarks with PI and FOPI controller are given in Table 2. We get the errors between the real landmarks and estimated landmarks based on the coordinates [1, 2, 5, 8, 11, 14]. In order to ensure the accuracy of simulation, we do the simulation 30 times, respectively, and get the 𝑙𝑒 in Table 2. We can see that the errors between the real landmarks and estimated landmarks based on FOPI controller with EKF-SLAM are much smaller than PI controller.

Table 1 

The average error index and variance index of robot position with PI/FOPI controller.

Table 2 

The average error of the real landmarks and estimated landmarks with PI/FOPI controller.

Figure 8 

The comparison of the error with fixed U/PI/FOPI controller.

From the above figures and tables we can see that the pros and cons of three control methods are very obvious. FOPI controller is superior to PI controller, and PI controller is better than the fixed U control. The error of the robot movement with FOPI controller gradually tends to zero with the increasing steps. Moreover, the error of FOPI controller will not gradually increase and remain oscillation trend within a certain number of steps. It is shown that FOPI has a better performance than PI.

5. Conclusion

In this paper, FOPI controller with EKF-SLAM is proposed. FOPI can be applied as a correction part. FOPI controller with EKF-SLAM can effectively reduce the accumulated error. The simulation shows that the FOPI controller is better than PI controller in NAO robot. The feasibility of FOPI is valid for the error of robot position and the error of landmarks. The feasibility of the proposed algorithm is validated by simulation results. In the future work, we can apply FOPI controller with SLAM to NAO platform and do the experiment in order to finish the navigation indoors environment. In addition, it is worth studying how to generalize the two-dimensional EKF-SLAM problem to the multidimensional EKF-SLAM problem.

Conflict of Interests

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

Acknowledgments

The work was partly supported by the National Natural Science Foundation of China (NSFC) under the Project Grant no. 51275439 and the 973 Plan under the Project no. 2013CB733003, and the Natural Science Foundation of Hebei Province of China under the project no. F2014203095.