A structure-improved extended state observer based control with application to an omnidirectional mobile robot

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A structure-improved extended state observer based control with application to an omnidirectional mobile robot✩ Chao Ren a , Yutong Ding a , Shugen Ma a,b , a b

∗

School of Electrical and Information Engineering, Tianjin University, Tianjin, 300072, China Department of Robotics, Ritsumeikan University, Shiga 525-8577, Japan

article

info

Article history: Received 26 August 2019 Received in revised form 12 January 2020 Accepted 13 January 2020 Available online xxxx Keywords: Extended state observer Omnidirectional mobile robot Initial peaking phenomenon Trajectory tracking control

a b s t r a c t This paper presents a structure-improved extended state observer (SESO) based trajectory tracking control scheme with application to an omnidirectional mobile robot. To alleviate the initial peaking phenomenon of the traditional extended state observer (TESO), a SESO with reduced order is proposed by improving the structure of TESO. Moreover, the designed SESO can achieve superior estimation performances. The total disturbances are estimated by SESO and then compensated in the controller. Then a phase-based nonlinear proportional–differential controller with time-varying gains is applied for high trajectory tracking performance. The stability of SESO and the closed-loop system are analyzed, respectively. Finally, the effectiveness of the proposed control scheme is validated through simulations in both frequency domain and time domain as well as experimental tests. © 2020 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Omnidirectional mobile robots (OMRs) are a class of mobile robots which can perform both translational and rotational motion simultaneously and independently. High maneuverability and flexibility promote the wide applications of OMRs. Compared with non-holonomic mobile robots, OMRs are particularly suitable to narrow working spaces, such as warehouses, factories and hospitals. The trajectory tracking control of OMRs aims to control the robot to accurately track a reference trajectory in real time, which plays an important role in robot applications. In the literature, there have been a lot of works on the trajectory tracking control of OMRs, such as trajectory linearization control [1], adaptive sliding-mode control [2], model-predictive control [3], switching quasi-linear-parameter-varying control [4], active disturbance rejection control [5], and so on. Particularly, in the work [5], a linear extended state observer (ESO) based linear proportional–differential (LPD) control scheme with friction compensation was designed for a three-wheeled OMR. ESO was applied to estimate the unknown friction forces. The designed control scheme is practical from the perspective of implementation, since no friction model is required and the computation cost is quite low. ✩ This work was supported by the National Natural Science Foundation of China [grant numbers 61603270, 61573253]; and Tianjin Natural Science Foundation, China [grant number 18JCQNJC04600]. ∗ Corresponding author at: School of Electrical and Information Engineering, Tianjin University, Tianjin, 300072, China. E-mail addresses: [email protected] (C. Ren), [email protected] (Y. Ding), [email protected] (S. Ma).

However, in real applications, two main problems are in the control design of [5]. One problem is related with the initial peaking phenomenon of the linear traditional ESO (TESO). This is due to the large initial estimation error and high gains of the observer. It will seriously deteriorate the control performances and may cause the damage of motors or mechanisms. The other problem is that the simple LPD control with fixed gains was applied in [5], resulting in limited control performance. In practical applications, affected by complex and changeable operating conditions and environmental parameters, LPD control with fixed gains is usually difficult to meet the high-performance control requirements. Over the last decade, many works have been conducted to improve the performance of TESO by designing nonlinear ESO [6– 12]. In particular, the problem of initial peaking was involved only in a few works [13–16], in which the time-varying gain of nonlinear ESO is set small in the initial stage and then increases rapidly to a maximal value. However, the works mentioned above only focus on the gain design or the nonlinear error function design, while neglecting the structure improvement of TESO. Actually, one key problem lies in the structure of TESO, i.e., the achievement of estimating all states only depends on the estimation error of the first state. This will lead to high observer gains and limited estimation performance. In [17], the structure of TESO was improved instead of only using the estimation error of the first state. It greatly alleviates the initial peaking phenomenon, but the parameter tuning is time consuming due to the introduction of the nonlinear function fal(·) and a saturation function. On the other hand, in the literature, a number of nonlinear proportional–integral–differential (PID) control designs have

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Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

2

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been proposed [18–22], to overcome the performance limitations of linear PID control with fixed gains. The control gains of linear PID do not change with the operating conditions or control error. However, the gains of nonlinear PID control vary in real time with the control error, resulting in superior command tracking performances. Specifically, the nonlinear PID control enjoys several advantages, such as reduced rise time for rapid inputs, higher disturbance rejection performance and higher control accuracy. In this paper, a structure-improved ESO (SESO) based nonlinear PD (NPD) trajectory tracking scheme is proposed, with application to an OMR. Firstly, a linear SESO with reduced order is designed to alleviate the initial peaking phenomenon. The proposed SESO is used to estimate the total disturbances, i.e., modeling errors, parameter uncertainties and external disturbances. Compared with TESO, the main features of the proposed SESO are as follows: (1) alleviating the initial peaking phenomenon; (2) completely eliminating the initial peaking caused by initial posture estimation error; (3) improving the convergence rate and the estimation accuracy with much lower gains; (4) decreasing the phase lag. The nonlinear robot dynamics is decoupled into three double integral systems by SESO and then a phase-based NPD controller is applied to each channel. The gains of the NPD controller are time-varying according to the phase of tracking errors, with rapid convergence rate and high tracking accuracy. The stability behaviors of both the SESO and the closed-loop control system are then analyzed. Finally, simulations in frequency and time domain as well as experimental tests are carried out to validate the effectiveness of the designed control system. The main contributions of this paper are as follows. (1) A SESO is designed by improving the structure of TESO applied in [5], to alleviate the initial peaking phenomenon and to improve the estimation performances of TESO. (2) A trajectory tracking control scheme is proposed by combining the phase-based NPD controller and the SESO, and is applied to an OMR. (3) The theoretical stability analysis is given and the effectiveness of the proposed control design is verified by simulations and experiments. The remainder of this paper is organized as follows. Section 2 describes a nonlinear dynamic model with unknown disturbances for a three-wheeled OMR. The proposed control scheme is presented in Section 3, as well as the stability analysis. In Section 4, simulations in both frequency and time domain are displayed. Section 5 is devoted to present the experimental setup, results and discussions. Finally, the conclusions are drawn and the further research is described in Section 6.

Fig. 1. Coordinate frames. Table 1 Nomenclature. Parameters

θ

Robot’s position and orientation angle in the frame {W }

y˙

θ˙

]T

Robot’s linear velocity and angular velocity in the frame {W }

y¨

θ¨

]T

Robot’s linear acceleration and angular acceleration in the frame {W }

q=

x

y

q˙ =

[

x˙

q¨ =

[

x¨

n

Gear reduction ratio

r

Wheel radius

m

Robot mass

I0

Combined moment of inertia of wheel, gear train and motor referred to the motor shaft

kt

Motor torque constant

kb

Motor back electromotive force constant

Iv

Robot moment of inertia around the mass center of the robot

L0

Average contact radius

b0

Combined viscous friction coefficient of the gear, wheel shaft and motor

Ra

Motor armature resistance

represents the control input vector, and its elements correspond to the supplied voltage of three motors; and

⎡3 2

p0 + m

M =⎣

0 In the robot dynamic modeling, assume that no slippage exists between the wheel and the moving ground. Two coordinate frames ∑ are defined, as shown in Fig. 1: XM OM YM : the moving coordinate frame fixed on the robot geometric center (also regarded as the center of gravity), expressed ∑ as {M }; XW OW YW : the world coordinate frame fixed on the ground, expressed as {W }. Table 1 shows the robot parameters. A nonlinear dynamic model with unknown disturbances is given as follows [23]: M q¨ + C q˙ + d = Bu,

[

(1)

]T

dx dy dθ where d = is the unknown disturbances, including the unmodeled dynamics (e.g., friction forces), external disturbances and parameter uncertainties. u = [u1 u2 u3 ]T

⎡ C = ⎣−

⎢

3 p 2 1 3 p 2 0

0

0

⎤

+m

0

⎥ ⎦,

3 p 2 0

0

⎢

2. Dynamic modeling

Definition

]T

[

3p0 L0 2 + Iv

0

θ˙

θ˙

3 p 2 0 3 p 2 1

0

⎤

0

0

3p1 L0 2

⎥ ⎦,

0

√ ⎡ 1 − 2 cos θ − 23 sin θ √ ⎢ B = p2 ⎣− 1 sin θ + 3 cos θ 2 2

− 12 cos θ + − 12 cos θ −

L0 where p0 =

n2 I0 , r2

p1 =

√ 3 2

sin θ

cos θ

3 2

sin θ

sin θ ⎦,

√

L0 n2 (b0 r2

+

kt kb ), Ra

and p2 =

⎤ ⎥

L0 nkt rRa

.

3. Control design In this section, the proposed SESO-based NPD trajectory tracking control scheme is presented in detail. The SESO-based NPD

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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3

To solve this problem, a linear reduced-order ESO (RESO) is introduced without estimating the robot posture. Unlike the posture-estimation-error-driven TESO, RESO is velocityestimation-error-driven, and therefore the problem mentioned above can be avoided. Define system state variables as x1 = q˙ , x2 = −M −1 F and their estimations as xˆ 1 , xˆ 2 , respectively. The estimation errors are defined as x˜ i = xi − xˆ i , i = 1, 2. Transform the robot dynamic model (2) into the following state space form: Fig. 2. The block diagram of the proposed SESO-based NPD control design.

{ control design mainly consists of two parts: the design of a linear second-order SESO and the design of a phase-based NPD controller with disturbance compensation. The block diagram is shown in Fig. 2.

x˙ 2 = −M −1 F˙

M q¨ + F = τ.

(2)

In the linear TESO design [5], the state variables are defined as x01 = q, x02 = q˙ , x03 = −M −1 F . Then the linear TESO is constructed as follows:

⎧ ˙ ⎪ ⎨ xˆ 01 = xˆ 02 + β01 x˜ 01 x˙ˆ 02 = xˆ 03 + M −1 τ + β02 x˜ 01 , ⎪ ⎩ ˙ xˆ 03 = β03 x˜ 01

(3)

where xˆ 0i (i = 1, 2, 3) are the estimations of x0i , and the estimation errors are defined as x˜ 0i = x0i − xˆ 0i . Three diagonal positive definite matrices β0i ∈ R3×3 (i = 1, 2, 3) are the gains of TESO. All the poles of TESO in three channels are usually configured at the same location −ωo for simplicity. To this end, a common setting of the observer gains is:

0

⎡ β03

ωo

0 3ωo

β01 = ⎣ 0

0 3

(4)

x˙ˆ 1 = xˆ 2 + M −1 τ + β11 x˜ 1 x˙ˆ 2 = β12 x˜ 1

The proposed SESO design consists of two steps: reducing the order [and improving the structure of linear TESO. In this ]T part, F = fx fy fθ represents the total disturbances, including the coupling term C q˙ and the unknown disturbances d in the dynamic model (1), i.e., F = C q˙ + d. Then the robot dynamic model (1) is transformed to:

3ωo

.

Then the RESO is designed as follows:

{

3.1. SESO design

⎡

x˙ 1 = x2 + M −1 τ

0

⎢ =⎣ 0

ωo

0

0

0

3ωo 2

⎡

⎢ 0 ⎦, β02 = ⎣ 0

3ωo 0

3

⎤

0

0 3ωo 2 0

0

⎥

3ωo 2

⎤

⎥ 0 ⎦, ωo 3

where ωo is the observer bandwidth. The estimation of the total disturbances F is Fˆ = −M xˆ 03 . It is obvious that the TESO is posture-estimation-error-driven, i.e., only the estimation error of robot posture (i.e., x˜ 01 ) is used in estimating all the state variables. As a consequence, the TESO (3) completes the tracking of x01 first, then x02 and x03 successively. In this case, when x˜ 01 becomes small, high gains are necessary for xˆ 02 and xˆ 03 to achieve good estimation, i.e., β02 and β03 should be selected large. In real applications, the initial state estimations of TESO are usually zero. In other words, the initial estimation of the robot posture xˆ 01 is zero at the initial moment. On the contrary, the robot initial posture q is usually not zero. Therefore, the initial posture estimation error x˜ 01 is usually large. This causes the initial peaking phenomenon of TESO with high gains. It will seriously deteriorate the control performances and may cause the damage of motors or robot mechanism.

(5)

where β1i ∈ R3×3 (i = 1, 2) are gain matrices of RESO and are positive definite; and 2ωo

0

⎡

0

2ωo

β11 = ⎣ 0 0

⎤

0 ⎦, β12 2ωo

0

⎡ 2 ωo ⎢ =⎣ 0 0

0

0

⎤

0 ⎦.

ωo 2

⎥

ωo 2

0

Comparing RESO (5) with TESO (3), the order is reduced to avoid the initial posture estimation error and the observer gains are decreased. Therefore, the initial peaking phenomenon caused by initial posture estimation error is avoided. However, the initial peaking phenomenon caused by the initial velocity estimation error exists in RESO due to its similar structure to TESO, i.e., both TESO and RESO are driven by the estimation error of the first state. In order to alleviate the initial peaking phenomenon caused by the initial velocity estimation error and improve the estimation performances of RESO, the structure of RESO is improved in a similar manner of [17]. The disturbance estimation error x˜ 2 is used instead of x˜ 1 in the second equation of RESO (5). From (5), the following equation can be obtained: xˆ 2 = x˙ˆ 1 − M −1 τ − β11 x˜ 1 = x˙ 1 − x˙˜ 1 − M −1 τ − β11 x˜ 1

= x2 − (x˙˜ 1 + β11 x˜ 1 ),

⎤

0 ⎦,

,

(6)

that is, x˜ 2 = x˙˜ 1 + β11 x˜ 1 . The SESO is designed as follows:

{

x˙ˆ 1 = xˆ 2 + M −1 τ + β21 x˜ 1 x˙ˆ 2 = β22 x˜ 2

,

(7)

where

β21 =

[ ωo 0 0

0

ωo 0

0 0

ωo

]

[ , β22 =

ωo 0 0

0

ωo 0

0 0

ωo

] .

Note that the estimation of the total disturbances F is Fˆ = −M xˆ 2 . Remark 1. It can be observed that the only difference between SESO (7) and RESO (5) is the second equation, i.e., the estimation of x2 . The estimation of the total disturbance in SESO is based on the estimation error itself (i.e., x˜ 2 ), instead of the estimation error of the first state (i.e., x˜ 1 ). Therefore, the problem mentioned in TESO and RESO that the estimation of disturbance can accomplish only after the estimation of the first state is solved. It is worth noting that the gains of SESO are significantly lower than those of TESO. Remark 2. SESO not only completely eliminates the initial peaking phenomenon caused by initial posture estimation error, but

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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areas in Fig. 3), the control gains K p and K d become higher, so as to accelerate the error convergence. On the contrary, when the error is moving towards zero, i.e., {e < 0, e˙ > 0} or {e > 0, e˙ < 0}, the control gains become lower. To show the effect of the parameter αp , Fig. 4 shows the 3D-views of the nonlinear proportional control gain kpi with different αp . As shown in Fig. 4, the value of αp affects the transition process of the gain kpi when the phase changes, and the high value implies the quick transition. Similarly, the transition process of the gain kdi depends on αd . Besides, βp and βd determines whether the values of kpi and kdi at the setpoint (i.e., the point when the phase changes) are closer to their maximum (smaller βp and βd ) or minimum (larger βp and βd ) according to [31], respectively. They are set to 1.5 in Fig. 4. Finally, by combining (8) and (9), the real control input is derived as: Fig. 3. Illustration of NPD control: the control gains become higher at the shaded areas.

also greatly alleviates the initial peaking phenomenon caused by initial velocity estimation error. Besides, compared with TESO applied in [5,24–29], simulations in both frequency domain and time domain demonstrate that SESO can greatly reduce the phase lag and improve the convergence rate, as well as the estimation accuracy. It will be presented in detail in Section 4.

u = B−1 (τ 1 + τ 2 ) = B−1 M (q¨ d − K p e − K d e˙ − xˆ 2 ).

(11)

Remark 3. The phase-based NPD control with time-varying gains, can overcome the performance limitations of LPD control with fixed gains in [5,32–35]. The NPD control can achieve higher accuracy and faster convergence rate. The comparison of control gains and tracking performances between LPD and NPD control will be presented in the experimental tests. 3.3. Stability analysis

3.2. Controller design The controller applied in this paper is a phase-based NPD controller with disturbance compensation, in which the output of SESO (i.e., the estimation of the system’s total disturbance) is required. In the controller design, one part of the virtual control efforts τ is designed for the disturbance compensation as follows:

τ 1 = −M xˆ 2 .

(8)

The other part is used to achieve trajectory tracking. In this part, a phase-based NPD controller proposed in [30] is applied. Define the control law as:

τ 2 = M (q¨ d − K p e − K d e˙ ), (9) ]T [ where qd (t) = xd yd θd is the desired trajectory, e = q − qd is the posture tracking error, e = [e1 e2 e3 ]T , and e˙ is the time derivative of e, representing the velocity error. Nonlinear control gains K p and K d are third-order diagonal positive definite matrices: kp1

⎡

Kp = ⎣ 0 0

0

0

kp2 0

⎤

⎡

kd1

0 ⎦ , Kd = ⎣ 0 kp3

0

0 kd2 0

0

kpi = kdi =

1 + βp exp(−αp ei e˙ i ) d1 1 + βd exp(−αd ei e˙ i )

{

x˙˜ 1 = x˜ 2 − β21 x˜ 1 x˙˜ 2 = −β22 x˜ 2 − M −1 F˙

.

(12)

Theorem 1. Assuming the derivative of the total disturbances F˙ is bounded, ∀ωo > 0, there exist positive ⏐constants σij > 0 such that ⏐ the estimation errors x˜ i finally satisfy ⏐x˜ ij ⏐ ≤ σij (i = 1, 2; j = 1, 2, 3), where σij and x˜ ij represent the jth element of σ i and x˜ i respectively. Proof. Define y 1 = x˜ 1 , y 2 = x˜ 2 − β21 x˜ 1 , then (12) is transformed to:

{

y˙ 1 = y 2 y˙ 2 = −(β21 + β22 )y 2 − β21 β22 y 1 − M −1 F˙

.

0 ⎦;

(13)

Let g = β21 + β22 , h = β21 β22 , then the characteristic polynomial of system (13) is obtained as follows:

ρ(λ) = λ2 I 3 + g λI 3 + h.

⎤

(14)

As mentioned before, β21 = β22 = diag(ωo , ωo , ωo ), i.e., all of the poles of SESO are configured at −ωo :

kd3

ρ(λ) = (λ + ωo )2 I 3 .

and p1

3.3.1. Convergence of SESO The observer error dynamics in state space is derived as:

+ p0 ,

(15)

The system (13) can be rewritten into the following form: (10)

+ d0 ,

y¨ 1 + g y˙ 1 − hy 1 = M −1 F˙ ,

(16)

−1 ˙

where

where M F is the perturbation term. Then the nominal system of (16) is obtained as follows:

p0 = ωc 2 , d0 = 2ωc , p1 = 10p0 , d1 = 0.01d0 ,

y¨ 1 + g y˙ 1 − hy 1 = 0.

where ωc is the controller bandwidth when the control gains are minimum, and αp , αd , βp , βd > 0 are controller parameters. The principle of the phase-based NPD controller is shown in Fig. 3. When the tracking error is moving away from zero, i.e., {e < 0, e˙ < 0} or {e > 0, e˙ > 0} (see the shaded

According to Hurwitz criterion, the origin (y 1 = 0, y 2 = 0) is a globally asymptotically stable equilibrium point of the nominal system (17) if the disturbance F is constant. ⏐ ⏐ ⏐ ⏐ f Consider that F˙ is bounded and ⏐F˙j ⏐ ≤ fhj , i.e., ⏐(M −1 F˙ )j ⏐ ≤ Mhj j (j = 1, 2, 3), where Mj is the jth diagonal element of M . It can be

(17)

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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5

Fig. 4. 3-D views of nonlinear proportional control gain kpi (i = 1, 2, 3) function with different αp values: (a) αp = 2; (b) αp = 10; (c) αp = 100.

concluded that the SESO is bounded-input bounded-output stable and the estimation errors are bounded by:

⎧⏐ ⏐ fhj fhj ⎪ ⎪ ⎨ ⏐x˜ 1j ⏐ ≤ M β β = M ω 2 = σ1j j 21j 22j j o , j = 1, 2, 3, ⏐ ⏐ f f ⎪ hj hj ⎪ ⏐x˜ 2j ⏐ ≤ ⎩ = = σ2j Mj β22j Mj ωo

(18)

where β21j and β22j represent the jth diagonal element of β21 and β22 , respectively.

Since p0 < kpi < p1 + p0 , d0 < kdi < d1 + d0 , let kpmin = p0 , kpmax = p1 + p0 and kdmin = d0 , kdmax = d1 + d0 . K pmin , K dmin and Aemax are the matrices when kpi = kpmin and kdi = kdmin , and the matrices K pmax , K dmax and Aemin are the opposite, i = 1, 2, 3. The in Eq. (23) can be further derived as:

ϕ(t) ≤ e

t

∫

Aemax t

1 1 −Aemin t e−τ Aemin dτ Γ = eAemax t (−A− + A− emin e emin )Γ , 0

(24)

⏐( ) ⏐ ⏐( 1 Aemax t ) ⏐ 1 (Aemax −Aemin )t |ϕi (t)| ≤ ⏐ −A− Γ i ⏐ + ⏐ A− Γ i⏐ , emin e emin e

3.3.2. Convergence of the closed-loop system The error dynamics of the closed-loop system can be derived by combining (2) and (11) as:

where i = 1, 2, · · · , 6. In addition, since

M e¨ + MK d e˙ + MK p e − M x˜ 2 = 0.

Aemax − Aemin =

(19)

[

e˙ T ]T , then the following equation can be

Define E = [eT obtained:

O3

−K pmin + K pmax

−K dmin + K dmax

(20)

⏐( (A −A )t ) ⏐ ⏐ e emax emin Γ ⏐ =

{

where O3 Ae = −K p

]

]

[

O I3 , Ax = 3 , I3 −K d

where O3 , I 3 ∈ R3×3 represent the third-order zero matrix and the identity matrix respectively.

Proof. Solving Eq. (20), E(t) can be expressed as follows:

where

E(t) =

t

e−

E(0) +

∫τ

0 Ae (ξ )dξ

Ax x˜ 2 (τ )dτ

)

∫t

e

0 Ae (τ )dτ

.

(21)

0

It can be obtained from the convergence conclusion of SESO that:

= 0, i = 1, 2, 3 ⏐( ⏐ )⏐ ⏐ , ⏐ Ax x˜ 2 ⏐ = ⏐x˜ 2(i−3) ⏐ ≤ σ2(i−3) , i = 4, 5, 6 Ax x˜ 2

)

i

where Ax x˜ 2 ∈ 6×1 , and the ith element of which is (Ax x˜ 2 )i (i = 1, 2, · · · , 6). x˜ 2j and σ2j represent the jth element of x˜ 2 and σ 2 respectively. ∫t ∫ t ∫τ Define ϕ(t) = e 0 Ae (τ )dτ 0 e− 0 Ae (ξ )dξ Ax x˜ 2 (τ )dτ , Γ =

]T

0 0 0 λ1 λ2 λ3 , where λj = σ2j (j = 1, 2, 3). The numerical comparison of the matrices below represents the numerical comparison between corresponding elements. Then it has

[

ϕ(t) ≤ e

∫t

0 Ae (τ )dτ

t

∫

e− 0

∫τ

0 Ae (ξ )dξ

Γ dτ .

(23)

.

, i = 4, 5, 6

(26)

(27)

where λmax = max {λ1 , λ2 , λ3 }. The inverse of matrix Aemin is

[

Ae1 I3

Ae2 O3

]

,

(28)

⎡

− kkdmax pmax

0

0

⎤

⎢

0

dmax − kkpmax

0

⎥ ⎥, ⎦

Ae1 = ⎢ ⎣

0

0

− kkdmax pmax

⎡

1 − kpmax

0

0

⎤

⎢

0

1 − kpmax

0

0

0

1 − kpmax

⎥ ⎥. ⎦

(22)

i

i = 1, 2, 3 −kd min +kd max

⏐( A t ) ⏐ λmax ⏐ e emax Γ ⏐ ≤ (i = 1, 2, · · · , 6), i ωc6

1 A− emin =

∫

,

Since Ae is Hurwitz, the following inequation is established after a finite time T when the estimation errors of SESO are bounded:

Theorem 2. Assuming the derivative of the total disturbances F˙ is bounded, ∀ωc > 0, there exists a positive constant vector ρ = [ ]T ρ1 ρ2 · · · ρ6 such that every element Ei of E finally satisfies |Ei | ≤ ρi (i = 1, 2, · · · , 6).

(

0,

λi−3 e

i

[

]

it can be obtained that

E˙ = Ae E + Ax x˜ 2 ,

{(

O3

(25)

Ae2 = ⎢ ⎣

Combining (25)–(28), the closed-loop system satisfies:

|ϕi (t)| ≤

⎧ ⎨

λi e−kdmin +kdmax

⎩

λmax , ωc6

kpmax

+

λmax (kdmax +1) , kpmax ωc6

i = 1, 2, 3

i = 4, 5, 6

.

(29)

On the other hand,

⏐( ∫ t ) ⏐ ⏐( ) ⏐ Emax (0) ⏐ 0 Ae (τ )dτ ⏐ E(0) ⏐ ≤ ⏐ eAemax t E(0) i ⏐ ≤ , ⏐ e i ωc6

(30)

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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Similarly, the transfer functions between the estimated disturbance and the real disturbance of TESO and RESO are obtained as follows: TESO :

−Mj Xˆ 03j (s) Fj (s)

β03j

=

s3 + β01j s2 + β02j s + β03j

=

RESO :

Fig. 5. Bode diagram of TESO, RESO and SESO.

where Emax (0) = max{Ei (0)}(i = 1, 2, · · · , 6). Finally, the following conclusion can be obtained:

|Ei (t)| ≤ ρi ,

(31)

where

ρi =

⎧ ⎨ ⎩

Emax (0)

ωc6 Emax (0)

ωc6

+ +

λi e−kd min +kd max kpmax

+

λmax (kdmax +1) ,i kpmax ωc6

λmax , ωc6

= 1, 2, 3

i = 4, 5, 6

In this part, simulations in both frequency domain and time domain are conducted in MATLAB. For the sake of frequencydomain simulation, the transfer functions between the estimated disturbance and the real disturbance of TESO, RESO and SESO are derived for each channel. Then, bode diagrams are plotted. On the other hand, time-domain simulations for estimating different disturbances are carried out to verify the estimation performances of SESO.

s , ⎩ sX˜ 2j (s) = − Fj (s) − β22j X˜ 2j (s)

(32)

Mj

where j = 1, 2, 3, represent x, y, θ channel, respectively. The Laplace transform of (4) for each channel is as follows:

.

(33)

Mj

Remind that X˜ i (s) = Xi (s) − Xˆ i (s), i = 1, 2. Then, combining (32) and (33), the transfer function between the estimated disturbance and the real disturbance of SESO is derived; that is

−Mj Xˆ 2j (s) Fj (s)

=

Mj X˜ 2j (s) Fj (s)

(36)

[ ωo = 5 rad/s, K p =

Kd =

18 0 0

0 18 0

0 0 18

81 0 0

0 81 0

−

Mj X2j (s) Fj (s)

=

β22j s + β22j

=

ωo s + ωo

.

(34)

0 0 81

] ,

] .

[

⎧ ⎨ sX˜ 1j (s) = X˜ 2j (s) − β21j X˜ 1j (s)

SESO :

β12j ωo 2 = 2 . + β11j s + β12j s + 2ωo s + ωo 2

Since the real total disturbances in experiments cannot be measured, simulations are necessary to verify the estimation performances of SESO, such as the initial peaking phenomenon, the estimation accuracy and the response speed. The robot physical parameters used in simulations are as follows: m = 19.1 kg, Iv = 0.65 kg m2 , r = 0.05 m, L0 = 0.25 m, I0 = 1.47×10−5 kg m2 , b0 = 1.0 × 10−8 Nms/rad, kb = 0.02076 V s/rad, kt = 0.0259 N m/A, Ra = 1.53 , n = 71. For the sake of comparison, the LPD controller is employed in the three closed-loop systems with TESO, RESO and SESO, respectively. The control parameters are set as follows:

[

The Laplace transform of the error dynamics of SESO (12) for each channel is as follows:

Mj

s2

The observer bandwidth ωo is set to 5 rad/s. Bode diagrams of TESO, RESO and SESO are shown in Fig. 5. In the top graph of Fig. 5, it can be seen that SESO has the lowest attenuation in the high frequency range, resulting in the fastest response speed. In other words, SESO to a great extent improves the rapidity of the system. On the other hand, the lowest attenuation in the high frequency range leads to a poor noise-tolerant performance, which is the only drawback of SESO. As seen from the bottom graph of Fig. 5, the SESO significantly reduces the phase lag. Specifically, compared with TESO, SESO reduces the phase lag by more than 60% in the high frequency range when the phase lag is stable.

4.1. Simulations in frequency domain

s ⎪ ⎪ ⎩ sX2j (s) = − Fj (s)

Fj (s)

=

(35)

+ 3ωo

s2

4.2. Simulations in time domain

.

4. Simulations

⎧ 1 ⎪ ⎪ ⎨ sX1j (s) = X2j (s) + τj (s)

−Mj Xˆ 2j (s)

ωo 3 . + 3ωo 2 s + ωo 3

s3

]T

As mentioned before, F = fx fy fθ is the total disturbances, including C q˙ and the unknown disturbances d of the robot. The estimation error of F is defined as ef = Fˆ − F , and ef = [ ]T efx efy ef θ . 4.2.1. Initial peaking phenomenon Take the x channel as an example, and a step disturbance signal is introduced into the system at the initial moment as fx [N ] = 10. In this part, two cases are respectively considered, with different initial estimation conditions of TESO, RESO and SESO. In the first case, only the initial posture estimation error exists. In the second case, both the initial posture estimation error and the initial velocity estimation error are considered. In the first case, the initial posture and velocity estimation errors are set as 0.2 m and 0 m/s, respectively. The total disturbance and its estimations estimated by TESO, RESO and SESO are presented in Fig. 6(a). Fig. 6(b) displays the estimation errors. The results confirm that RESO and SESO can completely eliminate the initial peaking phenomenon caused by initial posture estimation

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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Fig. 6. Simulation results of initial peaking phenomenon only with initial posture estimation error: (a) Total disturbance and estimated disturbances in x channel. (b) Estimation errors of total disturbance in x channel.

Fig. 7. Simulation results of initial peaking phenomenon with initial posture and velocity estimation errors: (a) Total disturbance and estimated disturbances in x channel. (b) Estimation errors of total disturbance in x channel.

Fig. 8. Simulation results of estimation accuracy: (a) Total disturbance and estimated disturbances in x channel. (b) Estimation errors of total disturbance in x channel.

error. This is because the order of TESO is reduced and then the posture estimation error has no effect on RESO and SESO. In the second case, the initial posture estimation error and the initial velocity estimation error are set to 0.2 m and 0.2 m/s, respectively. Simulation results are shown in Fig. 7. It is obvious that once the initial velocity estimation error exists, the initial peaking phenomenon appears in RESO. However, SESO can achieve the disturbance estimation with almost no peaking, due to the structure improvement and much lower gains. More importantly, it is observed from Figs. 6 and 7 that the convergence rate is greatly improved and the settling time is greatly shortened by SESO. It needs to be emphasized that the

superior estimation performance of SESO is achieved with much lower gains. 4.2.2. Estimation accuracy Take the x channel as an example, and consider a sinusoidal disturbance signal fx [N ] = 2sin(t). The initial estimation errors are set to zero. The estimation performances are plotted in Fig. 8. It can be observed in Fig. 8 that the maximum estimation error of SESO is reduced by more than 50% compared with TESO. As a result, the high estimation accuracy is guaranteed by SESO. The results in time-domain agree well with the conclusion about the phase lag in the frequency-domain analysis.

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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4.2.3. Estimation performances for stochastic disturbances Take the x channel as an example, and set the reference trajectory to xd (t) [m] = 0 and the initial estimation errors to zero. Then consider a stochastic unknown disturbance signal dx [N ] = 5 + 2r(t), where r(t) is a function generating a set of stochastic normally distributed time series. This is achieved by using the ‘‘randn’’ function in MATLAB. The estimation results and the tracking errors are shown in Fig. 9. The estimation results in Fig. 9(a) show the much better estimation performances of SESO compared with TESO. The disturbance estimation of SESO is more consistent with the overall trend of the total disturbance. Hence, the SESO can achieve high estimation accuracy in the presence of stochastic disturbances. On the other hand, in the enlarged figure, the estimation of TESO is smooth while the estimation of SESO oscillates with the stochastic disturbance. This verifies that the response speed of SESO is greatly improved. It should be emphasized that for fair comparison, the controllers applied in both closed-loop systems are LPD controllers. Therefore, the different tracking errors in Fig. 9(b) only depend on the different estimation performances between SESO and TESO. Then, it can be concluded that the excellent estimation performances of SESO result in the high tracking accuracy of the closed-loop control system using SESO.

Table 2 Evaluation index calculation results of the circle trajectory test. Control design

IAExy [m]

IAEθ [rad]

MAExy [m]

MAEθ [rad]

TESO + LPD SESO + NPD

1.1194 0.2669

0.4794 0.2556

0.0961 0.0471

0.0932 0.0558

Table 3 Evaluation index calculation results of the square trajectory test. Control design

IAExy [m]

IAEθ [rad]

MAExy [m]

MAEθ [rad]

TESO + LPD SESO + NPD

1.0624 0.3110

1.7185 0.3957

0.0623 0.0306

0.2057 0.0705

(1) IAE (integral of the absolute error):

{

IAExy =

∫T (

IAEθ =

∫

0 T 0

⏐ ⏐) |ex | + ⏐ey ⏐ dt [m],

|eθ |dt [rad].

(2) MAE (maximum absolute error):

{

MAExy = max

{

max |ex | ,

⏐ ⏐ }

max ⏐ey ⏐

[m],

MAEθ = max |eθ | [rad].

5. Experiments 5.2. Experiment results In this section, to validate the effectiveness of the proposed SESO-based NPD control design, extensive experimental tests are carried out. First, implementation details are introduced, and experimental results are then presented and analyzed. For comparison purposes, experimental tests of TESO-based LPD control scheme are carried out. 5.1. Implementation Fig. 10(a) shows the experimental setup. MATLAB is used to run the control law on a personal computer (Intel(R) Core(TM) i7-4770 [email protected] GHz). The WIFI wireless communication is employed to transmit the control signal to the omnidirectional mobile robot (OMR). The robot position is obtained using the OptiTrack motion capture system, which consists of eight capture cameras (part NO. Prime 41) and can realize the positioning accuracy within 2 mm. The robot velocity is then obtained using a second-order low-pass filter. The robot prototype is shown in Fig. 10(b). Three DC motors are used to actuate the three wheels. A motor microcontroller (STM32F103VET6) is used to generate the PWM signal to three motor drivers. Besides, the rated voltage of each motor is 24 V. Three asymmetric markers are placed on the robot for OptiTrack to track the OMR. The robot physical parameters used in experiments are the same as those in simulation. The control parameters of the proposed SESO-based NPD control design are set as follows:

ωo = 5 rad/s, ωc = 2.5 rad/s, αp = αd = 100, βp = βd = 1.5. The control parameters of the TESO-based LPD control scheme are set as follows:

[ ωo = 5 rad/s, K p =

[ Kd =

5.02 0 0

0 5.02 0

32 0 0 0 0 5.02

0 32 0

0 0 32

] ,

] .

In order to quantitatively evaluate the control performance, the following two evaluation indicators are used in this paper:

In the first experimental scenario, consider the following circle trajectory as the reference trajectory:

⎧ (π ) ⎪ ⎨ xd [m] = 0.8 cos ( 15 t) π . yd [m] = 0.8 sin 15 t ⎪ ⎩ π θd [rad] = 20 t

(37)

In this [ experiment, the initial ]T posture of the robot is set as 0.8 m 0 m 0 rad q(0) = . Since the initial posture estimation is zero in TESO, an initial estimation error of 0.8 m exists in the x direction. Experimental results are shown in Fig. 11. IAE and MAE of circle trajectory test are listed in Table 2. In Fig. 11(a) and (b), it is observed that the tracking errors of the proposed control design are much smaller than those of the TESO-based LPD control scheme. Specially, the superior tracking and estimation performances of the proposed design are reflected in the initial stage, such as the higher response speed and convergence rate. Note that, the outperformance of the proposed control design is not only due to the SESO, but also the phase-based NPD controller. Fig. 11(e) and (f) show the control gains K p and K d of NPD controller and LPD controller, respectively. The control gains of NPD controller are time-varying according to the phase of tracking errors, resulting in fast convergence performance. In addition, as verified in Table 2, both IAE and MAE of the proposed control design are much smaller than those of the TESO-based LPD control scheme. It should be noted that the initial posture estimation error exists in the x direction. As seen from Fig. 11(c), an evident initial peaking (see x-channel) appears in the disturbance estimation of TESO. However, it is observed that in the x-channel, no initial peaking appears in the disturbance estimation of SESO. This is due to the fact that the initial peaking problem resulting from the initial posture estimation error has been avoided in SESO. In the second experimental scenario, consider a square trajectory as the reference trajectory. Note that, the square trajectory has several features, such as sudden changes of the reference velocity at each corner. The reference square trajectory is selected

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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Fig. 9. Simulation results of estimation performances for stochastic disturbances: (a) The total disturbances and the estimated disturbances in x channel. (b) Tracking errors employing SESO and TESO in x channel.

Fig. 10. The experimental setup and the robot prototype: (a) Experimental setup. (b) Robot prototype.

as follows:

⎧ 0.1t , 0 s ≤ t < 10 s ⎪ ⎪ ⎨ 1, 10 s ≤ t < 20 s xd [m] = ⎪ 1 − 0.1(t − 20), 20 s ≤ t < 30 s ⎪ ⎩ 0, 30 s ≤ t ≤ 40 s ⎧ 0 , 0 s ≤ t < 10 s ⎪ ⎪ ⎨ 0.1(t − 10), 10 s ≤ t < 20 s yd [m] = ⎪ 1 , 20 s ≤ t < 30 s ⎪ ⎩ 1 − 0.1(t − 30), 30 s ≤ t ≤ 40 s ⎧ 0, 0 s ≤ t < 10 s ⎪ ⎪ ⎪ ⎨ π (t − 10), 10 s ≤ t < 20 s 10 θd [rad] = ⎪ π , 20 s ≤ t < 30 s ⎪ ⎪ ⎩ π (t − 30) + π, 30 s ≤ t ≤ 40 s 10

(38)

Moreover, compared with the LPD with fixed-gains, the control gains K p and K d of the phased-based NPD controller have sharp variations when the tracking errors tend to be large, as shown in Fig. 12(e) and (f). Finally, IAE and MAE in Table 3 illustrate that the proposed control design achieves much better tracking performance. 6. Conclusions

(39)

(40)

The initial posture of the robot [ ]T is set as q(0) = 0 m 0 m 0 rad . Experimental results are shown in Fig. 12. IAE and MAE of the square trajectory test are listed in Table 3. As seen in Fig. 12(a) and (b), the experimental results confirm that the tracking performances of the proposed control design significantly outperform the TESO-based LPD control scheme. Attention should be paid to the corners of the square trajectory in Fig. 12(a) and (b). When the reference velocity has sudden changes, the proposed control system can rapidly converge to steady state with much smaller overshoot and much shorter settling time, compared with the TESO-based LPD control system.

In this paper, a SESO-based NPD trajectory tracking control scheme has been proposed and applied to an OMR. A SESO has been designed by reducing the order and improving the structure of TESO, to alleviate the initial peaking phenomenon and improve the estimation performances of TESO. Besides, the designed SESO enjoys several advantages while requiring much lower gains: (1) accelerating the response speed; (2) improving the convergence rate; (3) improving the estimation accuracy; and (4) decreasing the phase lag in frequency domain. The robot dynamics is decoupled into three double integral systems by SESO and then a phase-based NPD controller is applied to each channel, in which the control gains can change based on the phase of the tracking error in real time. Stability analysis has shown that both SESO and the closed-loop control system are bounded-input boundedoutput stable. Both simulations and experiments have confirmed the advantages of the designed SESO. Experimental results have verified that the proposed control design can achieve much better tracking performance than TESO-based LPD control. Finally, the proposed control scheme can be extended to other robotic systems.

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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Fig. 11. Results of the first experimental scenario, i.e., the circle trajectory tracking control: (a) Comparison between the reference circle trajectory and the tracking results of two control systems in the xy-plane. (b) Tracking errors of two control systems in three directions. (c) Estimated disturbances by TESO and SESO. (d) Control inputs u(t). (e) Control gains K p of LPD and NPD. (f) Control gains K d of LPD and NPD.

Fig. 12. Results of the second experimental scenario, i.e., the square trajectory tracking control: (a) Comparison between the reference square trajectory and the tracking results of two control systems in the xy-plane. (b) Tracking errors of two control systems in three directions. (c) Estimated disturbances by TESO and SESO. (d) Control inputs u(t). (e) Control gains K p of LPD and NPD. (f) Control gains K d of LPD and NPD.

Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.

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Please cite this article as: C. Ren, Y. Ding and S. Ma, A structure-improved extended state observer based control with application to an omnidirectional mobile robot. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.01.024.