# Journal of Robotics, Networking and Artificial Life

Volume 8, Issue 4, March 2022, Pages 284 - 288

# Tracking Control of Mobile Robots based on Rhombic Input Constraints

Authors
Kai Gong, Yingmin Jia*, Yuxin Jia
The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang University (BUAA), Beijing 100191, China
*Corresponding author. Email: ymjia@buaa.edu.cn
Corresponding Author
Yingmin Jia
Received 26 November 2020, Accepted 15 September 2021, Available Online 29 December 2021.
DOI
10.2991/jrnal.k.211108.011How to use a DOI?
Keywords
Differential wheeled mobile robots; rhombic input constraints; trajectory tracking; vector analysis
Abstract

This paper focuses on the trajectory tracking control algorithm for Differential Wheeled Mobile Robots (DWMRs) based on rhombic input constraints. The kinematics and dynamics model of DWMRs are established, and vector analysis method is used to design the controller when the linear velocity and angular velocity of DWMRs were not mutually independently. Through the simulation of tracking 8-shaped curve, a good control performance is obtained.

Open Access

## 1. INTRODUCTION

Differential Wheeled Mobile Robots (DWMRs) are widely used in today’s society. There are many methods have been used in controller design for trajectory tracking. Sliding mode control [1], backstepping control [2], robust control [3], fuzzy control [4], active disturbance rejection control [5] etc. are used to solve tracking control problem. From a practical perspective, the input constraints must be considered when designing controller, but the current situation is that most of the research does not consider the mutual constraints relationship between the linear velocity v and the angular velocity ω of the mobile robot, they usually assume that the input constraints of the robots’ linear velocity v and angular velocity ω are mutually independently, that is, |v| ≤ m1, |ω| ≤ m2, where m1 and m2 are positive constants. The real situation is that input field of DWMRs is the rhombic area defined by |v/m| + |ωl/m| ≤ 1 as shown in Figure 1, and m represents the maximum velocity of two drive wheels, l represents half of the distance between the two drive wheels, the proof process will be given later. If a differential wheeled mobile robot uses the controller designed in Su and Zheng [6], the rectangular area where v and ω are independently needs to be obtained from the rhombic area mentioned above, it can be determined by |v| ≤ m/2 and |ω| ≤ m/2l. So we can see that the actual area where v and ω are mutually independently is only half of the hypothetical rectangular area. Also the mobility of DWMRs cannot be fully utilized. Rhombic input constraints are considered first time in Chen et al. [7], it proposed a geometric analysis method to design time-varying feedback parameters.

## 2.1. Rhombic Input Constraints

As shown in Figure 2, vl and vr respectively represent the velocity of robot’s driving wheels, and their maximum velocity is m, that is vlm and vrm. Usually v and ω of DWMRs are used as control inputs, and their relationship with the velocity of the driving wheel is

v=(vl+vr)2 (1)
ω=(vrvl)2l (2)

Thus v and ω are constrained by

{(m+v)/lω(m+v)/l,v[m,0)(mv)/lω(mv)/l,v[0,m] (3)

The above is collated into one expression:

Formula (3) can be sorted into one expression:

|v/m|+|ω l/m|1 (4)

Formula (4) can be expressed as the solid black rhombus in Figure 1. So far, the independent rhombic area of v and ω is obtained.

## 2.2. Tracking Control Based on Rhombic Input Constraints

The kinematics and dynamics equations of two-wheel differential mobile robots is

x˙=vcosθ,  y˙=vsinθ,  θ˙=ω (5)

(x, y) is the center point coordinates of DWMRs and θ is used to indicate its azimuth angle (see Figure 2).

### Assumption 1.

The input constraint of DWMRs is Equation (4), and its reference trajectory satisfies:

x˙r=vrcosθr,  y˙r=vrsinθr,  θ˙r=ωr (6)
and
|vr/m|+|ωrl/m|1lɛ/m (7)

Among them, (xr, yr, θr, vr, ωr) is the target values of (x, y, θ, v, ω), where ɛ is a constant satisfies 0 < ɛ < m/l.

### Remark 1.

We ensure the traceability of the trajectory by introducing a constant ɛ in formula (7).

Figure 2 shows that system errors of DWMRs are defined as:

[xeyeθe]=[  cosθ sinθ  0sinθ cosθ  0    0        0      1][xrxyryθrθ] (8)

The tracking errors system can be obtained by deriving the two sides of the above formula (8)

x˙e=vrcosθev+ωyey˙e=vrsinθeωxeθ˙e=vrωrω (9)

Now our task is how to design the controller with satisfying the input constraints to make errors tend to zero.

## 3. CONTROLLER DESIGN BASED ON RHOMBIC INPUT CONSTRAINTS

To design the controller, we need to use the following two lemmas:

### Lemma 1 [7].

f:[0,∞) → R is first-order continuous differentiable and limtf(t) is a finite value, if f˙(t),t[0,) is uniformly continuous, then limtf˙(t)=0.

### Lemma 2 [7].

There is a scalar function ρ(x), x ∈ [0, ∞], which satisfies the following properties:

1. (1)

ρ(x) is a continuous and non-decreasing function;

2. (2)

ρ(0) = 0, and 0 < ρ(x) ≤ 1 for x ∈ [0, ∞];

3. (3)

limx0+ρ(x)=ρ0, which ρ0 is a positive constant.

Define Ψ(x) as

ψ(x)={ρ(x)/x    x(0,)     ρ0       x=0 (10)

Then, for ∀σ ∈ (0, ∞), there always exist α and β, such that α < Ψ(x) ≤ β holds for x ∈ [0, σ], where both α and β are positive constants.

ρ(x) = tanh(x) is a function that satisfies the above conditions.

In this paper, we refer to the controller designed in Blažič [8] as follows:

v=vrcosθe+kxxeω=ωr+kyvryesinθeθe+kθθe (11)
where kx, ky and kθ are positive constants. If the errors are too large, then v and ω are more likely to break through the range of the rhombic input area through analysis formula (11). In this way, the control commands cannot be executed well.

### Lemma 3 [7].

For controller (11), if following conditions are met:

1. (1)

k_xkxk¯x,k_ykyk¯y,k_θkθk¯θ

2. (2)

ky is differentiable and k˙y0.

where k_x,k¯x,k_y,k¯y,k_θ,k¯θ are positive constant values. Then, trajectory tracking errors of DWMRs will converge to zero, that is xe, ye, θe will converge to zero.

To use the vector method to design controller (11), we first need to define the controller v and ω as a vector

OD=[vω]
then by defining other vectors as:
OA=[vrcosθeωr],AB=[0kyvryesinθeθe]BC=[kxxe0],CD=[0kθθe] (12)
the controller can be represented by a combination of several vectors:
OD=OA+AB+BC+CD (13)

It is necessary to design each vector in turn, so that controller can finally meet the rhombic input constraints.

|vrcosθem|+|ωrlm||vrm|+|ωrlm|1lɛm (14)

From formula (14) we know that OA satisfies the rhombic input constraints, without loss of generality, we represent OA as shown in Figure 3, and because the length of AB is proportional to ky, we can definitely find a ky to make AB within the rhombic input constraints. Similarly, we can also find suitable kx and kθ, so BC and CD can meet the rhombic constraints respectively. Obviously, the controller OD will definitely meet the rhombic input constraints.

Since the requirement for ky is k˙y0, we intuitively thought of designing if from Lyapunov function V(t).

V(t)=12(xe2+ye2+θe2ky) (15)

Let ky be

ky=λɛm2V(t)+μ2ky=mθe2+m2θe4+4λ2ɛ2(xe2+ye2+μ2)2m(xe2+ye2+μ2) (16)
where λ and μ are constants, 0 < λ < 1 and μ > 0.

According to Equations (15) and (16), we can get

ky=mθe2+m2θe4+4λ2ɛ2(xe2+ye2+μ2)2m(xe2+ye2+μ2)>0 (17)
k˙y=2kxky2xe2+2kθkyθe22ky(xe2+ye2+μ2)+θe2 (18)

If kx > 0 and kθ > 0, then according to formula (17) and (18), k˙y<0 can be derived, and further from formula (15) we can get

k_y=λɛm2V(0)+μ2kyλɛmμ=k¯y (19)

In this way, the vector OB can be expressed as:

OB=OA+AB=(vrcosθe,ωr+kyvryesinθeθe)T (20)

We can verify that OB satisfies the rhombic input constraints through formulas (14)(16). Because of kx, kθ > 0, so the directions of the vectors BC and CD are determined by the signs of xe and θe. We want to occupy the entire area as much as possible under the premise that the controller meets the rhombic input constraints. First, we need to determine the triangle area ΔBEF where the points of C and D are located, as shown in Figure 3, when xe < 0 and θe > 0, we take the constraint segment ② to determine the reference triangle ΔBE2F2, similarly, when xe > 0 and θe > 0, we take the constraint segment ① to determine the reference triangle ΔBE1F1, when xe < 0 and θe < 0, we get the reference triangle ΔBE3F3, and when xe < 0 and θe < 0, we get the reference triangle ΔBE4F4. Through the formulas of the four constraint lines and the coordinates of point B, we can easily obtain the coordinates of point E as:

E:(sgn(xe)(msgn(θe)(ωr+kyvryesinθeθe)l),       wr+kyvryesinθeθe) (21)

Similarly, we can get the coordinates of F as:

F:(vrcosθe,sgn(θe)(msgn(xe)vrcosθe)l) (22)
where sgn(⋅); is sign function
sgn(x){|x|/x   x00          x=0 (23)

Further we can get the expressions of BE and BF as

BE=OEOB=(sgn(xe)(msgn(θe)(ωr+kyvryesinθeθe)l)vrcosθe,0)T (24)
BF=OFOB=(0,sgn(θe)(msgn(xe)vrcosθe)lωrkyvryesinθeθe)T (25)

To design kx and kθ, let

BC=ρ(|xe|)2BECD=ρ(|θe|)2BF (26)

Then, we get from (12), (24), (26), that

kx=ψ(|xe|)2(msgn(θe)(ωr+kyvryesinθeθe)lsgn(xe)vrcosθe)kθ=ψ(|θe|)2(msgn(xe)vrcosθelsgn(θe)(ωr+kyvryesinθeθe)) (27)

By formulas (15), (16), (20) and Lemma 2 we can easily get

k_xα(1λ)ɛl2kxβmk¯xk_θα(1λ)ɛ2kθβm2k¯θ (28)

At this point, the kx, ky and kθ meet the two conditions in Lemma 3, so the system error will converge to zero. And because of our vector method design the parameters ensure that the control variables v and ω meet the rhombic input constraints too.

## 4. SIMULATION RESULTS

In this section, we verify the performance of the controller through simulation and compare it with the controller in Chen et al. [7]. Before the simulation starts, some parameters are set as follows:

The maximum velocity of the drive wheels is set to m = 0.4 m/s, the wheel spacing is set to l = 0.16 m, for setting some parameters of the controller, we choose ρ(x) = tanh(x), ɛ = 0.1, λ = 0.99, μ = 0.01.

Figure 4 shows a robot gradually tracks on the reference trajectory, where the blue line represents the expected trajectory, and the red line represents the actual trajectory. Figure 5 shows the tracking errors xe, ye, θe are each gradually converge to zero, also we can guarantee the control variables v and ω satisfy the rhombic input constraints through Figure 6, and sometimes v can basically reach the bounds of rhombic input constraints. Figure 7a is the tracking errors diagram under the controller in Chen et al. [7], Figure 7b is the tracking errors diagram under our controller, it can be seen that our controller can make the errors converge faster, and the oscillation is smaller.

## 5. CONCLUSION

The tracking control problem of DWMRs with rhombic input constraints is solved in this paper. Compared with existing methods, we have improved the design of controller parameters and achieved better performance. Also our method can better exert the robots’ mobility and makes the tracking errors converge faster. The controller simultaneously solves the tracking problem and stability problem, its effectiveness can be confirmed by simulation results. Future work will focus on the controller design with uncertainty based on a more complex application environment.

## CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

## ACKNOWLEDGMENTS

This work was supported by the NSFC (62133001, 61520106010) and the National Basic Research Program of China (973 Program: 2012CB 821200, 2012CB821201).

## AUTHORS INTRODUCTION

Mr. Kai Gong

He received the B.S. degree in measurement & control technology and instrument from Harbin Engineering University, Harbin, China, in 2016. He is currently pursuing the PhD degree with the Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang University. His research interests include robotics and motion control.

Prof. Yingmin Jia

He received the B.S. degree in control theory from Shandong University, China, in 1982, and the M.S. and PhD degrees both in control theory and applications from Beihang University, China, in 1990 and 1993, respectively. Then, he joined the Seventh Research Division at Beihang University where he is currently Professor of automatic control. His research interests include robust control, intelligent control and their applications in robot systems.

Mr. Yuxin Jia

He received the B.S. degree in Automation from Chongqing University, Chongqing, China, in 2017. He is currently pursuing the PhD degree with the Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang University. His research interests include gravity compensation control and intelligent control of mobile robots.

Journal
Journal of Robotics, Networking and Artificial Life
Volume-Issue
8 - 4
Pages
284 - 288
Publication Date
2021/12/29
ISSN (Online)
2352-6386
ISSN (Print)
2405-9021
DOI
10.2991/jrnal.k.211108.011How to use a DOI?
Open Access

TY  - JOUR
AU  - Kai Gong
AU  - Yingmin Jia
AU  - Yuxin Jia
PY  - 2021
DA  - 2021/12/29
TI  - Tracking Control of Mobile Robots based on Rhombic Input Constraints
JO  - Journal of Robotics, Networking and Artificial Life
SP  - 284
EP  - 288
VL  - 8
IS  - 4
SN  - 2352-6386
UR  - https://doi.org/10.2991/jrnal.k.211108.011
DO  - 10.2991/jrnal.k.211108.011
ID  - Gong2021
ER  -