State feedback based on grey wolf optimizer controller for two-wheeled self-balancing robot

Wesam M. Jasim

doi:10.1515/jisys-2022-0035

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State feedback based on grey wolf optimizer controller for two-wheeled self-balancing robot

Wesam M. Jasim

Veröffentlicht/Copyright: 28. April 2022

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Aus der Zeitschrift Journal of Intelligent Systems Band 31 Heft 1

Abstract

The two-wheeled self-balancing robot (TWSBR) is based on the axletree and inverted pendulum. Its balancing problem requires a control action. To speed up the response of the robot and minimize the steady state error, in this article, a grey wolf optimizer (GWO) method is proposed for TWSBR control based on state space feedback control technique. The controller stabilizes the balancing robot and minimizes the overshoot value of the system. The dynamic model of the system is derived based on Euler formula and linearized to state space representation to enhance the control technique. Then, the GWO optimizes the state feedback controller parameters. Simulation results show that the system reaches the zero steady-state error with less than 2 ms, which proves the effectiveness of the proposed controller over the classical state feedback controller in terms of fast response, very small overall error, and minimum overshoot.

Keywords: TWSBR; grey wolf optimizer; state feedback controller; heuristics control technique; state space linearization

1 Introduction

The two-wheeled self-balancing robot (TWSBR) has a significant platform in control area as it is a high order and multivariable robot. It is an underactuated, nonlinear, and good coupling system [1]. Its structure is unstable, but it can be stabilized to behave like a human [2]. Selecting and designing a good controller for a robot control problem is a crucial task [3]. Several classical, optimal, and heuristic control techniques were designed to control the TWSBR.

In state of the art, the classical and optimal controllers were discussed; for example, Jiménez et al. [4] designed proportional-integral-derivative (PID) and linear quadratic regulator (LQR) controllers to control the vertical equilibrium position of the Two-Wheeled Automatic Balancing Robot with disturbance consideration. These two controllers were proposed in ref. [5]. The kinematics of the TWSBR was developed to separate the wheel and the frame for control purpose. The results show the effectiveness of the controllers in disturbance rejection. A comparison between the performance of the PID and the LQR controllers was established in ref. [6] when they were tested to control the TWSBR. The test shows the effectiveness of the LQR controller over the PID one. The LQR controller was designed in ref. [7] to control the TWSBR with different pitch and heading angles, weights, and disturbances. The experiments proved the ability of the controller to overcome the different situations. The PID controller was implemented to control the TWSBR in ref. [8]. The tilt angle was calculated using a combination of the accelerometer and the gyroscope. The ability of the PID controller was proven in the simulation results. The PID controller was used in ref. [9] to control the self-balancing mobile robot with good performance. Iwendi et al. [10] discussed the proportional-derivative-proportional-integral and proportional-integral-proportional-derivative based on Kalman filter controllers to control the TWSBR and to avoid the dynamic obstacles with disturbance consideration. The results show the ability of the proposed controller compared with that of adaptive fuzzy controller.

Another control approach was implemented such as the adaptive robust controller that was designed in ref. [11]. The controller was based on Moore–Penrose with system uncertainty consideration. The proposed controller was used to balance the TWSBR. The main results were analyzed based on Lyapunov function. In ref. [12], a hierarchical fast terminal sliding mode controller was proposed to balance a TWSBR with different traveling case consideration. The Lyapunov function was used to analyze the stability of the proposed controller. The results show a good balancing and shaking reduction. In ref. [13], two controllers were combined to control the TWSBR. The first controller was the PID controller; it was used in the inner loop to control the speed of the motor. However, the second controller was the active disturbance rejection controller controller; it was used to control the pitch angle based on extended state observer optimizer. The proposed heretical controller performs better than that of the PID in both loops even with external disturbance consideration. Another controller that was used to control the TWSBR was the model predictive controller (MPC) controller. It was implemented in ref. [14] for a heavy TWSBR. The results prove the stability of the proposed controller.

The heuristic approaches were also used in controlling the TWSBR. For example, the LQR based on particle swarm optimization (PSO) algorithm was proposed in ref. [15]. The obtained results show the improvement of the robot stability when the PSO algorithm was used to tune the LQR controller parameters over the classical controller. In this article, the grey wolf optimizer (GWO) algorithm is used to optimize the pole placement controller parameters to control the TWSBR. The pole placement controller is designed and implemented, and then the GWO algorithm is implemented to optimize the poles. To the best of my knowledge, it is for the first time the GWO was used to control this robot. This article contributes the following considerations:

Speed up the overall system response and minimize the steady-state error and the overshoot.
Overcoming the problem of finding the poles based on the classical method using the GWO method instead.
Overcoming the consideration of the effect of the external disturbances.

The remaining of the article is organized as follows: the mathematical model of the robot is driven in Section 2. The controller design is described in Section 3. Section 4 represents the simulation results and discussion of the proposed controller. The conclusions obtained from the control system performance are illustrated in Section 5.

2 The TWSBR mathematical model

The structure of the TWSBR is divided into two parts. The first part includes the two wheels with the main body, whereas the second part includes the coaxial two wheels combined with the motor. The motor is used to derive the two wheels as shown in Figures 1 and 2. The two wheel parameters such as moment of inertia and radius of the wheels are same and the center of gravity of the body is inverted above the balanced axletree [16].

Figure 1

Robot left wheel diagram.

Figure 2

Robot chassis diagram.

According to Figures 1 and 2, the dynamic model of the robot movement is derived in separate parts as follows [16]:

The left wheel:

(1) N r x ̈ r l = ( F d r l + F d r r ) + K T l − K l ,

(2) N r y ̈ r l = N r g + v T l − v l ,

(3) I r l ψ ̈ r l = c l − K T l r ,

(4) x ̇ r l = r ψ ̇ r l ,

(5) y ̇ b = − ψ ̇ b l sin ψ b ,

(6) x ̇ b = x r l + x r r 2 + ψ ̇ b l cos ψ b ,

(7) γ ̇ = x r l − x r r 2 F ,

where r and l are the right and left wheels, x , y , and z are the three position directions of the robot, as the robot moves in x direction, x ̇ , y ̇ , and z ̇ are its speed and x ̈ , y ̈ , and z ̈ are the related accelerations. The wheel is rotated with a pitch angle ψ and the whole robot rotated around the vertical axis with a yaw angle γ with its angular velocity ψ ̇ , γ ̇ and acceleration ψ ̈ , γ ̈ . I is the moment of inertia, N is the rotating mass related to both wheels, N b is the chassis mass, its moment of inertia with respect to the z axis is denoted by I b and with respect to the y axis is denoted by I γ . The distance between the center of gravity and the z axis is l and r is the wheel radius. The distance between the two wheels is d , the chassis shift position with respect to the x axis is x b , and y R is the shift position with respect to the y axis.

The chassis:

(8) N b x ̇ b = F D b + ( K r + K l ) ,

(9) N b y ̈ b = v l + v r − N b g + F c ψ ,

(10) ψ ̈ b I b = ( v r + v l ) l sin ψ b − ( K r + K l ) l cos ψ b − ( c l + c r ) ,

(11) γ ̈ I γ = ( K l − K r ) d 2 ,

where the forces affecting the free bodies are v T l , v T r , v l , v r , K T l , K T r , K l , K r . Linearizing the above equations around the operating point ( x r = 0 , γ = 0 , ψ b = 0 , sin ψ b = ψ b , cos ψ b = 1 ) and doing some modifications, the following state space equations are obtained [16]:

(12) x ̇ r x ̈ r ψ ̇ b ψ ̈ b γ ̇ γ ̈ = 0 1 0 0 0 0 0 0 a 23 0 0 0 0 0 0 1 0 0 0 0 a 43 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 x r x ̇ r ψ b ψ ̇ b γ γ ̇ + 0 0 b 21 b 22 0 0 b 41 b 42 0 0 b 61 b 62 c l c r ,

(13) x ̇ r x ̈ r ψ ̇ b ψ ̈ b = 0 1 0 0 0 0 a 23 0 0 0 0 1 0 0 a 43 0 x r x ̇ r ψ b ψ ̇ b + 0 b 2 0 b 4 c ψ ,

(14) γ ̇ γ ̈ = 0 1 0 0 γ γ ̇ + 0 b 6 c γ ,

with x = N b ( N b + 6 N r ) l 3 N b + 9 2 N r , y = N b N b + 3 2 N r r + 1 l , a 23 = g ( 1 − 4 3 l N b x ) , a 43 = g N b x b 2 = 4 l y 3 x − 1 N b l , b 4 = − y x , b 6 = 6 ( 9 N r + N b ) r d .

The robot is balanced by controlling its speed and direction (rotation angle). Then, the system can be formed based on pole placement theory. According to the GBOT1001 parameters shown in Table 1, the state space equations (13) and (14) can be rewritten as in equations (15) and (16), respectively [16].

(15) x ̇ r x ̈ r ψ ̇ b ψ ̈ b = 0 1 0 0 0 0 − 26.25 0 0 0 0 1 0 0 90.125 0 x r x ̇ r ψ b ψ ̇ b + 0 2.0296 0 − 5.4708 c ψ ,

(16) γ ̇ γ ̈ = 0 1 0 0 γ γ ̇ + 0 5.1917 c γ .

Table 1

The GBOT1001 robot parameters

Variable	Value	Unit
N b	21	kg
N r	0.42	kg
r	0.106	m
l	0.3	m
g	9.8	m/s²

3 Controller design

3.1 Full state–pole placement controller

Full state or pole placement is one of the most used control technique in feedback control system. Its theory is based on replacing the poles obtained from the closed loop of the system in the predetermining positions in the s-plane. The poles are located to match with the system eigenvalues and responsible for the control system performance. Therefore, the system should be fully controllable to use the pole placement technique for control purpose.

Implementing the pole placement technique indicates the choice of the feedback gain vector that locates the poles of the characteristic polynomial at the required locations. The system state space model expressed in equation (17) is controllable as the rank of its matrix is equal to the order of the matrix.

(17) X ̇ = A X + B U .

Its control law is described in the following equation:

(18) U = − K X .

Then, the closed-loop system is illustrated in the following equation:

(19) X ̇ = ( A − B K ) X = A c l X ,

where K = [ k 1 , k 2 , k 3 , ⋯ k n ] is the controller constant parameter vector. Then, finding the required poles is based on the following equation:

(20) ∣ S I − ( A − B K ) ∣ = ( s − p 1 ) ( s − p 2 ) ⋯ ( s − p n ) ,

where p 1 , p 2 , ⋯ p n are the required poles. Now, the appropriate gain K can be chosen such that the closed-loop poles are in the right position. Therefore, according to the system state space in equations (15) and (16), the required poles can be obtained (27).

3.2 GWO controller

The GWO technique is used in this article to find the optimal state feed controller matrix parameters that stabilize the robot in disturbance environment and then the results are compared with the classical one obtained in refs. [16,17]. The main reasons to select this algorithm are the algorithm mathematical derivation is free, the number of the adjustable parameters and operators is lower than the other algorithms, its iterative processes save the optimal solution, and therefore, it is easier than others in implementation [18].

GWO was introduced in 2014 by Mirjalili et al. [19]. It is a swarm metaheuristic algorithm inspired biologically by Grey Wolf family. It depends on the grey wolf hunting technique [20]. Its algorithm includes four groups of population: alpha, beta, gamma, and omega. The first three groups will guide the fourth group (Omega) to the best zones in the searching area. According to the hunting process (searching for prey, encircling prey, and attacking prey), the algorithm is started by initializing the population as a solution, then, it is improved to reach the optimal solutions [21,22].

Implementing the GWO algorithm to find the optimal controller parameters can be described as follows: first, the grey wolf population is created. In the second step, for each searching member, the fitness is calculated. In the third step, the optimal search member – for alpha, beta, and sigma – is chosen. Next step is to update the pace member location. Then, the fitness of the members is obtained. At the end, the optimal fitness for each member is updated [18,22]. These steps are illustrated as in the following equations; the locations of the three searching members are changed based on equations (21) and (22).

(21) P → = ∣ M → d ( i ) ⋅ S → − M → ( i ) ∣ ,

(22) M → ( i + 1 ) = ∣ M → d ( i ) − P → ⋅ W → ) ∣ ,

where the vector P → is the difference between the location of the prey and the computed predator, i represents the iterations, M → d represents the prey’s location vector, and M → is the vector of the grey wolf’s location. The following equations calculate the two vectors S → and W → , respectively:

(23) S → = r → 1 e ⋅ 2 e ,

(24) W → = r → 2 ⋅ 2 ,

where e is decreased from 2 to zero linearly while the two vectors r → 1 and r → 2 are of arbitrary values between zero and 1.

The location of the prey or the simplest three solutions during a GWO rule are perpetually expected to be alpha, beta, and delta, in this order, throughout optimization. The fourth is the wolves that are referred to as omegas; they will amend their locations toward alpha, beta, and delta. Then, the omega’s locations are updated based on equations (25)–(27), which calculate the distance between the location of the Wolfs of alpha, beta, and sigma and the presented solution.

(25) P → α = ∣ M → α ⋅ S → 1 − M → ∣ ,

(26) P → β = ∣ M → β ⋅ S → 2 − M → ∣ ,

(27) P → σ = ∣ M → σ ⋅ S → 3 − M → ∣ ,

where the three vectors M → α , M → β , and M → σ are the locations of the alpha, beta, and sigma wolf locations, respectively. M → represents the current position. S → 1 , S → 2 , and S → 3 are chosen randomly. The step sizes of omega to alpha, beta, and delta wolves are calculated in equations (25)–(27).

Next, based on the distances in equations (28)–(31), the final location of the current solution is obtained.

(28) M → 1 = M → α − P → α ⋅ W → 1 ,

(29) M → 2 = M → β − P → β ⋅ W → 2 ,

(30) M → 3 = M → σ − P → σ ⋅ W → 3 ,

(31) M → ( i + 1 ) = ( M → 1 + M → 2 + M → 3 ) 3 .

GWO algorithm was used in several applications such as text document clustering [23], time series classification [24], and optimizing the neural network weights [25].

4 Simulation results

The designed GWO based on full state feedback controller is implemented in MATLAB 2018b and tested to verify its effectiveness by comparing with the results of the classical full-state pole placement controller. The state variable initial conditions are set to x ̇ r = 0 , x ̈ r = 0.1 , ψ ̇ b = 0 , and ψ ̈ b = 0 . For comparison purpose, the pole placement controller parameters are obtained based on equations (15) and (16), where the required poles are p 1 , 2 = − 1 ± j and p 3 , 4 = − 2 ± 3 j and based on these poles, the full state feedback gain vector is K = [ − 0.4629 − 0.8176 110.7281 2.6640 ] .

These parameters are used to control the robot in the first step. In the second step, the GWO is used to optimize the fitness function and its results are compared with that of the pole placement controller. Its number of search agent is 30 and number of iterations is 100. It obtains required poles of p 1 , 2 = − 9.9993 ± 10 j and p 3 , 4 = − 10 ± 3.2930 j and the gain vector is K = [ − 394.6837 − 110.8947 483.3328 39.8296 ] .

The GWO algorithm showed its excellency over the classical pole placement one. Figure 3 shows the performance of the two controllers that were applied to the robot with disturbance consideration. While Figure 4 shows the test response of the system with sine wave input signal and disturbance consideration. It is obvious that the two controllers stabilize the robot. Moreover, the test of the proposed GWO controller performs better than the use of the pole placement controller in terms of the response speed and steady-state error. In spite of obtaining an acceptable result from the simulation test, the system suffers some limitations as the system is linearized based on an operating point to simplify the implementation of the system and finding the poles. There are reasons to linearize the system mathematical model instead of using the nonlinear one. It is to obtain a space around the operating point to validate the proposed linear controller.

Figure 3

System response with disturbance consideration.

Figure 4

System response with disturbance and sine wave input.

5 Conclusion

In this article, the full state feedback based on GWO controller was proposed to stabilize the TWSBR. The mathematical model of the robot was driven and linearized around the operating system and represented in state space form. The robot parameter used was of the GBOT1001. The robot was controlled using the classical pole placement control technique in the first phase. In the second phase, the GWO was used to optimize the controller parameters to stabilize the robot. The classical pole placement controller performed well, but the proposed full state feedback based on GWO controller performed better than the classical controller in terms of response speed and steady-state error. The next step toward this research direction is to use the hybrid controller and to consider the effect of the external disturbances.

Conflict of interest: Author states no conflict of interest.

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Received: 2022-01-14

Revised: 2022-02-13

Accepted: 2022-02-17

Published Online: 2022-04-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/jisys-2022-0035

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TWSBR; grey wolf optimizer; state feedback controller; heuristics control technique; state space linearization

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