# Coordinated Motion Control and Event-based Obstacle-crossing for Four Wheel-
leg Independent Motor-driven Robotic System111This work was supported in part
by the National Key Research and Development Project of China under Grant
2019YFC1511401 and the National Natural Science Foundation of China under
Grant 51675041 and 61773060.

Dongchen Liu  Junzheng Wang<EMAIL_ADDRESS>Shoukun Wang  The National Key
Laboratory of Complex System Intelligent Control and Decision, School of
Automation, Beijing Institute of Technology, Beijing, 10081, China  The
Ministry of Industry and Information Technology Key Laboratory of Servo Motion
System Drive and Control, School of Automation, Beijing Institute of
Technology, Beijing, 10081, China

###### Abstract

This work investigates the coordinated motion control and obstacle-crossing
problem for a four wheel-leg independent motor-driven robotic system via a
model predictive control (MPC) approach based on an event-triggering
mechanism. The modeling of a wheel-leg robotic control system with a dynamic
supporting polygon is organized. The system dynamic model is 3 degrees of
freedom ignoring the pitch, roll, and vertical motions. The single wheel
dynamic is analyzed considering the characteristics of the motor-driven and
the Burckhardt nonlinear tire model. As a result, an over-actuated predictive
model is proposed with the motor torques as inputs and the system states as
outputs. As the supporting polygon is only adjusted at certain conditions, an
event-based triggering mechanism is designed to save hardware resources and
energy. The MPC controller is evaluated on a virtual prototype as well as a
physical prototype. The simulation results guide the parameter tuning for the
controller implementation in the physical prototype. The experimental results
on these two prototypes verify the efficiency of the proposed approach.

###### keywords:

Coordinated motion control, Obstacle-crossing, Model predictive control,
Event-based triggering mechanism, Dynamic supporting polygon.

††journal: Mechatronics

##  1 Introduction

Four-wheel independent motor-drive (4WIMD) vehicles play an important role in
wheeled intelligent systems, due to their advantages in driveability and
energy-saving [1, 2, 3, 4]. Coordinated motion control of these systems, which
is an essential point for satisfying handling and stability performance, has
been a popular topic in recent decades[5, 6, 7, 8]. Different achievements
about the coordinated motion control strategy have been achieved, e.g.,
proportional-integral-derivative control [9], sliding mode control [10],
adaptive control [11], robust control [12], and model predictive control (MPC)
[13]. In particular, the MPC control scheme, which is a closed-loop strategy
that adds feedback to typical open-loop optimal control solutions, suits
nonlinear systems with state quantities constraints [14]. In [15], a novel
visual servo-based model predictive control method is developed to steer a
wheeled mobile robot moving in a polar coordinate toward the desired target. A
tracking control system with an explicit nonlinear model predictive control
scheme is presented in [16] for electric-driven vehicles with in-wheel motors.
A key point of the MPC controller design is the predictive system modeling.

There exist various effective modeling approaches to meeting the Ackermann
steering mechanism. In [17], a synchronization control approach is proposed
with a two-wheeled-driven electric vehicle regarding two wheels as followers.
The synchronization action is achieved by using a general fictitious master,
and the Ackermann principle is used to compute an adaptive desired wheel
speed, resulting in a robust technique. As the existence of the nonlinear
coupling effects in the Ackermann steering mechanism, neural networks have
been applied to estimate the vehicle speed [18]. In [19], the design and
evaluation of electrical differential (ED) are discussed for over actuated
electric vehicles independently driven by in-wheel motors. From the comparison
of three patterns of ED, almost the same vehicle performances in terms of the
sideslip angle, yaw rate, and trajectory tracking have been achieved under
normal driving maneuvers. The stabilization of control architecture based on
the Ackermann steering mechanism is limited under adverse driving conditions
when the wheels are slipping or adjusting the relative position. Moreover,
another kind of widely used method targets the adhesion or slip ratio control
of each wheel. In [20], based on the measurements from wheel encoders, an
inertial measurement unit, and a global positioning system, the joint wheel-
slip, as well as the estimation about vehicle motion, are considered. As
automated vehicles and advanced driver-assistance systems benefit from vehicle
trajectory prediction to understand the traffic environment and perform tasks,
an integrated vehicle trajectory prediction method, which is based on
combining physics and maneuver approaches, is proposed in[21]. Actually, in
terms of the path-following and safety problem, a nonlinear coordinated motion
controller is proposed in [22] with the framework of the triple-step method
that potentially benefits the engineering-oriented implementation for
intelligent vehicles. However, the real-time observation of small slip rates
and the identification of optimal slip rates under complex driving conditions
are difficult to achieve. With the shortcomings of the methodologies
investigated above, the improvement of modeling for the 4WIMD systems is
significant for the MPC controller design.

In most of the studies cited above, the supporting polygon is changeless, and
the vehicles only can bypass obstacles from the side. Recently, as diverse
wheeled robots have realized the flexible adjustment of the wheels, the
systems can cross obstacles by adjusting the supporting polygon to shorten the
journey and strengthen the terrain adaptability [23, 24, 25]. The relative
position changes between wheels result in a nonlinear time-varying model for
the dynamics of the robotic system. The rapidly increasing number of
calculations place extra demands on hardware resources and battery life which
are limited on robotic systems. The MPC-based coordinated motion control data
for this kind of system are still scarce, and development efforts are still
necessary.

In this work, we propose a thorough study of a coordinated motion control
problem for a four-wheel-leg robotic system. As the wheels can steer and
adjust position independently, the system has different dynamic
characteristics from 4WIMD vehicles. Motivated by the Ackermann steering
mechanism, the wheel steering angles are calculated according to the reference
trajectory obtained from the environment awareness system. The steering angle
for each wheel is along its own steering axis and the desired paths for all
wheels are around the same point, which meets the characteristics of our
independent driven robotic system. Furthermore, a predictive model for this
kind of system is proposed with the motor torques as inputs and the system
states as outputs. In practice, the adjustments only occur in certain
conditions, exemplified by deep pits or isolated obstacles. Therefore, the
event-based triggering mechanism is suitable. Pioneered by the works in [26,
27, 28], numerous results have been obtained in event-based control and
estimation [29, 30]. Li and Shi [31] combine the event-triggered theory with
the MPC strategy for continuous-time nonlinear systems with disturbances. In
[32], the event-triggered extended state observer (ET-ESO) is designed for a
continuous-time nonlinear system with system uncertainty and outside
disturbance. As for motion control of robotic systems, a behavior-based
switch-time MPC controller for mobile robots in [33] meets the characteristics
of the coordinated motion control of wheeled robots with a dynamic supporting
polygon. Admittedly, relevant literature has rarely been seen about the event-
based modeling for coordinated motion control of wheeled robotic systems. We
consider the coordinated motion control as different behaviors including
steering and adjusting supporting polygon with the switch time determined by
an event-based triggering mechanism. The main contributions of our work are as
follows:

  * 1.

A developed model considering both the Ackermann steering mechanism and the
wheel adhesion for the four wheel-legs independent motor-driven systems is
proposed. The model is 3 degrees of freedom and is established by the
characteristics of wheel motion with a dynamic supporting polygon.

  * 2.

An event-based triggering mechanism is designed based on the information from
the environment awareness system. The improved MPC controller is discussed in
theory and evaluated on a virtual prototype and a physical prototype. The
experimental results verify the effectiveness of the proposed approach.

The rest of this paper is organized as follows. In section II, the model of
four wheel-leg independent motor-driven robotic systems is proposed. In
section III, an event-based MPC coordinated motion controller is shown in
detail. Simulation and experiment results are presented in section IV, and the
paper concludes with a brief discussion in section V.

##  2 System Model and Problem Formulation

For the controller design of a four wheel-leg independent motor-driven robot,
a dynamic model is required.  With the reference trajectory of the system as
input, the wheel steering angles can be calculated by Ackermann steering
geometry [17, 18, 19], such that the desired paths for all wheels are around
the same point. The steering angle for each wheel is along its own steering
axis, which meets the characteristics of our independent driven robotic
system. However, the dynamics of the robotic system and each wheel are
required for this kinematic model under adverse driving conditions when the
wheels are slipping.

###  2.1 System Dynamic Model

Figure 1: Schematic diagram of the robotic system.

The schematic diagram of a four wheel-leg independent motor-driven robotic
system in the longitudinal, lateral, and yaw directions is shown in Fig. 1.
The wheels are numbered as $W_{i}$, where $i=1,2,3,4$. To express the ratio
between the wheelbase $l$ and track width $d$, $\delta$ is defined as

$\displaystyle\delta=\arctan\left(\frac{l}{d}\right).$ (1)

The tire forces in longitudinal, lateral and vertical directions are presented
by $[F_{x_{i}},F_{y_{i}},F_{z_{i}}]$. The origin $o$ of the dynamic coordinate
system $\Sigma_{b}$ is fixed on the system coinciding with the center of
gravity (COG), the $x-$axis is the longitudinal axis of the system (the
forward direction is positive), the $y-$axis is the lateral axis of the system
(the right-to-left direction is positive). The steering angle between the
longitudinal direction of wheel $W_{i}$ and $x-$axis is $\delta_{i}$. In the
geodetic coordinate system $\Sigma_{n}$, $X-$axis positive direction is due
east, and $Y-$axis positive direction is due north. This model is 3 degrees of
freedom (DOF) ignoring the pitch, roll, and vertical motions of the system. It
is assumed that the mechanical properties of the wheels are the same.

The motion of the system includes translation $v_{t}=[v_{x},v_{y}]$ and
rotation $\omega_{r}$. In this paper, the velocity vectors are uniformly
mapped to a velocity vector $v=[v_{t},\omega_{r}]^{T}$. Considering the
Ackermann steering mechanism, we have

$\displaystyle\delta_{1}=-\delta_{2}=\arctan\left(\frac{lK}{2-dKC_{d}}\right),$
(2)
$\displaystyle\delta_{3}=-\delta_{4}=\arctan\left(\frac{lK}{2+dKC_{d}}\right),$

where $K$ is the turning curvature which is inversely proportional to the
turning radius, and $C_{d}$ is the turning direction. As the reference
trajectory is described by
${x}^{\ast}=[P_{x}^{\ast}(t),P_{y}^{\ast}(t),P_{\theta}^{\ast}(t)]^{T}$, it
holds that

$\left\\{\begin{aligned}
&K=\frac{|\dot{P}_{x}^{\ast}(t)\ddot{P}_{y}^{\ast}(t)-\ddot{P}_{x}^{\ast}(t)\dot{P}_{y}^{\ast}(t)|}{\left[\dot{P}_{x}^{\ast}(t)^{2}+\dot{P}_{y}^{\ast}(t)^{2}\right]^{\frac{3}{2}}},\\\
&C_{d}=\left\\{\begin{array}[]{cc}\\!\\!1,&P_{\theta}^{\ast}(t)>0,\\\
\\!\\!0,&P_{\theta}^{\ast}(t)=0,\\\ \\!\\!-1,&P_{\theta}^{\ast}(t)<0.\\\
\end{array}\right.\end{aligned}\right.$ (3)

From dynamic analysis, considering the mass $m$ and inertia $I$ of the wheeled
system, we have

$\\!\\!\\!\left\\{\\!\\!\begin{aligned}
m\dot{v}_{x}\\!\\!=&\sum_{i=1}^{4}\left(F_{x_{i}}\cos\delta_{i}+F_{y_{i}}\sin\delta_{i}\right)+m\omega_{r}v_{y},\\\
m\dot{v}_{y}\\!\\!=&\sum_{i=1}^{4}\left(F_{x_{i}}\sin\delta_{i}+F_{y_{i}}\cos\delta_{i}\right)-m\omega_{r}v_{x},\\\
I\dot{\omega}_{r}\\!\\!=&\frac{\sqrt{l^{2}\\!\\!+\\!d^{2}}}{2}\\!\\!\sum_{i=1}^{4}\\!\\!\left(F_{\\!x_{i}}\\!\cos\left(\delta_{i}\\!\\!-\\!\delta\right)\\!+\\!F_{\\!y_{i}}\\!\sin\left(\delta_{i}\\!\\!-\\!\delta\right)\\!\right).\\\
\end{aligned}\right.$ (4)

The wheel sideslip angles $\alpha_{i}$ are calculated by

$\left\\{\begin{aligned}
\alpha_{1}&=\delta_{1}-\frac{2v_{y}+l\omega}{2v_{x}-d\omega},\\\
\alpha_{2}&=\delta_{2}-\frac{2v_{y}-l\omega}{2v_{x}-d\omega},\\\
\alpha_{3}&=\delta_{3}-\frac{2v_{y}+l\omega}{2v_{x}+d\omega},\\\
\alpha_{4}&=\delta_{4}-\frac{2v_{y}-l\omega}{2v_{x}+d\omega}.\\\
\end{aligned}\right.$ (5)

###  2.2 Single Wheel Dynamic Analysis

Figure 2: Single wheel dynamics

For a four wheel-leg independent motor-driven robotic system, each wheel is
driven by a motor. A transmission system is adopted to transfer the
electromagnetic driving torque $T_{e_{i}}(u_{i})$ from the motor to the
wheels. In practice, the transmission requires multiple components such as the
gearbox, torque shaft, and suspension mechanism, such that the resistance
$T_{s_{i}}$ exists. The dynamic of a single wheel is simplified as shown in
Fig. 2, and characterized by

$\displaystyle
T_{e_{i}}(u_{i})-T_{s_{i}}-F_{x_{i}}r=J_{\textit{w}}\dot{\omega}_{\textit{w}_{i}},$
(6)

where $\omega_{\textit{w}_{i}}$, $J_{\textit{w}}$ and $r$ are the rotation
velocity, inertia, and radius of the wheel respectively. Referring to [34],
$T_{e_{i}}(u_{i})$ is approximated by $k_{i}u_{i}$. However, the tire is
highly nonlinear and the Burckhardt tire model is suitable for controller
design to describe its nonlinearity [13]. The composition of the tire
longitudinal and lateral forces is calculated by

$\left\\{\begin{aligned}
&F_{t_{i}}=F_{z_{i}}c_{1}(1-e^{c_{2}s_{iRes}})-c_{3}s_{iRes},\\\
&F_{x_{i}}=\frac{\lambda_{i}}{s_{iRes}}F_{t_{i}},\\\
&F_{y_{i}}=\frac{\tan\alpha_{i}}{s_{iRes}}F_{t_{i}},\\\ \end{aligned}\right.$
(7)

where $F_{t_{i}}$ is the composition of the longitudinal and lateral tire
forces, $F_{z_{i}}$ is the vertical load of the wheels and
$[c_{1},c_{2},c_{3}]$ are parameters for the dry asphalt road. The combined
friction efficient satisfies

$\displaystyle s_{iRes}=\sqrt{\lambda_{i}^{2}+{\tan\alpha_{i}}^{2}}$ (8)

with the wheel longitudinal slip is calculated by

$\displaystyle\lambda_{i}=\frac{\omega_{\textit{w}_{i}}r-v_{x}}{\max(\omega_{\textit{w}_{i}}r,v_{x})}.$
(9)

###  2.3 Problem Formulation

The supporting polygon changes when the robot crosses an isolated obstacle.
From this point, an event-based triggering mechanism is effective with the
triggering condition $\Gamma(t)$ proposed as

$\Gamma(t)=\left\\{\begin{aligned}
&1,~{}~{}\textrm{if}~{}d_{s}\in(d_{0},d_{0}+\Delta d_{\textrm{max}}),\\\
&0,~{}~{}\textrm{otherwise},\\\ \end{aligned}\right.$ (10)

where $d_{s}$ is the obstacle width which can be obtained by the environment
awareness system, $d_{0}$ is the initial lateral distance of wheels, and
$\Delta d_{\textrm{max}}$ is the maximum stretch width limited by the
workspace of the legs. Taking $x=v_{t}^{T}$ and
$u=[u_{1},u_{2},u_{3},u_{4}]^{T}$ as the states and control inputs of the
system model, the control system satisfies

$\displaystyle\dot{x}=g(x,u,\Gamma),$ (11)

where $g(x,u,\Gamma)$ is constructed from (4) to (9). This model includes the
coupling of the robotic system dynamics and single wheel nonlinearity. In this
work, we mainly focus on the coordinated motion control for the four-wheel
independent motor-driven robotic system with the following problems:

  * 1.

Considering the system control model in (11), can we propose an coordinated
motion controller based on MPC for a four wheel-leg independent motor-driven
robotic system with a dynamic supporting polygon?

  * 2.

As the complexities of the dynamic character, can we propose a event-based
framework to simplify the control problems? With the event-triggering
condition $\Gamma(t)$ come into concern, how do we achieve supporting polygon
switch with stable motion control?

##  3 Event-based MPC Controller of Coordinated Motion Control

In this section, the problems stated above are investigated. Consider the
event-based supporting polygon adjustment of the wheeled system and define the
behaviors as $\alpha(x(t),\Gamma(t))$. Then, the control system in (11) is
expressed as

$\displaystyle\dot{x}=f(x,u,\alpha(x,\Gamma)).$ (12)

As the number of the control inputs is more than the number of the model
states, the control problem is over-actuated. More specifically, it is
desirable to have the control systems execute a string of such behaviors, i.e.
$(\alpha_{0},t_{0}),(\alpha_{1},t_{1}),...,(\alpha_{n},t_{n})$, where
$t_{i}~{}(i=1,2,...,n)$ indicates the time when the system switch from
$\alpha_{i-1}$ to $\alpha_{i}$. Note that $\alpha_{0}$ is the original
behavior with $t_{0}=0$. For the robotic system,
$\alpha_{i}(x,\Gamma)|_{\Gamma=0}$ refers to the supporting polygon stay
original with the dynamic characteristics similar to 4WIMD vehicles, and
$\alpha_{i}(x,\Gamma)|_{\Gamma=1}$ means that the system adjusts the lateral
distance of wheels from $d_{0}$ to $d_{s}$. Utilizing this form of behaviors,
we can build upon a wealth of different control applications which all use
some form of parameterized control. Therefore, the system dynamics can be
simplified as

$\displaystyle\dot{x}=f_{i}(x,u,\Gamma)~{}\textrm{for}~{}t_{i}\leq t\leq
t_{i+1}.$ (13)

This model satisfies the system dynamics and the tire nonlinear effects in
behavior $(\alpha_{i},t_{i})$. Based on the references $x^{\ast}$, the
tracking error $e=x-x^{\ast}$ is defined to describe the control performance.
To find the optimal control input $u$ for each behavior, the cost function is
presented as

$\displaystyle J_{i}=$
$\displaystyle\int_{t_{j}}^{t_{j+1}}\left(e(t)^{T}Qe(t)+u(t)^{T}Ru(t)\right)dt$
(14) $\displaystyle+x(t_{j+1})^{T}Sx(t_{j+1}),$

where $Q$, $R$, and $S$ are positive definite weight matrices. The current
time is represented by $t_{j}$, and $t_{j+1}$ is related to the prediction
horizon $\Delta{t}$ and triggering condition $\Gamma(t)$ with the following
form

$t_{j+1}=\left\\{\begin{aligned} &t_{i},~{}~{}\textrm{if}~{}t_{j}<t_{i}\leq
t_{j}+\Delta{t},\\\
&t_{j}+\Delta{t},~{}\textrm{otherwise}.\end{aligned}\right.$ (15)

$e(t)^{T}Qe(t)$ is related to the tracking error and the velocity cost of the
control system. $u(t)^{T}Ru(t)$ stands for the future value of the deviations
of the model response raised by the controlled variables at the end of the
prediction horizon. As the existence of $x(t_{j+1})$ for the cost function
$J$, the controller drives the system state $x(t)$ at the end of the behavior
$(\alpha_{i},t_{i})$ approaches $0$. In such condition, the stability in the
switch between behaviors is ensured. The optimization problem of the
controller is summarized as

$\displaystyle J$ $\displaystyle=\min_{u(t)}\sum_{i=0}^{N_{p}}J_{i},$ (16)
$\displaystyle\textrm{s.t.}~{}\dot{x}$
$\displaystyle=f_{i}(x,u,\Gamma)~{}\textrm{for}~{}t_{i}\leq t\leq t_{i+1},$
$\displaystyle x$ $\displaystyle\in\chi,$ $\displaystyle u$ $\displaystyle\in
U,$

where $\chi$ is limited by the wheeled robotic system, $U$ is obtained by the
actuator saturation of the electric motors, and $N_{p}$ is the prediction
horizon. The function (16) corresponds to the tracking ability and control
volume optimization of the system.

Figure 3: Block diagram of the hierarchical control system.

Based on the dynamics of the system, a control sequence in the next $N_{p}$
steps is carried out by minimizing the defined cost $J$. Then, the control
sequence is used for the next $N_{c}$ steps, which indicates $N_{p}\geq
N_{c}$. Therefore, in theory, an increasing $N_{p}$ leads to a rising cost
(e.g., hardware resource demands or time consuming) of computing the
optimization solution, and a decreasing $N_{c}$ improves the real-time
performance and robustness. The MPC-based coordinated motion control combining
the event-based triggering mechanism is designed as shown in Fig. 3. The
simultaneous localization and mapping (SLAM) are implied by the information
from the environment awareness system and the integrated navigation system.
The obstacle information $d_{s}$ is sent to the event-triggered mechanism and
supporting polygon adjustment to realize behaviors switch. The robotic system
reference $x^{\ast}$ is calculated by the path planning results and the system
state $x$.

##  4 Application and Result Analysis in Simulation and Experiment

In this section, the integrating approach is employed on a robotic system.
Both the virtual prototype and the physical prototype are presented in detail.

###  4.1 Control System Design

Figure 4: The physical prototype BITNAZA. Table 1: Structure Parameters of the Robotic System Parameter | Value
---|---
Robot Length*Width*Height (m) | 1.5*1.5*1.2
Robot Weight (kg) | 278.0
Tire Diameter (m) | 0.2
Battery Life (h) | 2.5
Carrying Capacity (kg) | 300.0
Reducer Reduction Ratio | 40:1
Maximum Angle of Climb (∘) | 27.0
Maximum Velocity (km/h) | 10.0

We have established a four wheel-leg independent motor-driven robotic system
with a dynamic supporting polygon. The physical prototype named BITNAZA is
shown in Fig. 4, and the structure parameters of the robotic system in listed
in Table 1. The robotic system consists of four Stewart structured wheel-legs,
the control system, the environment awareness system, the integrated
navigation system, and the power system. The wheel-leg is presented in detail
inside the green box. A transmission link and a T-type reducer make up the
transmission system of the wheel-leg. The hook joints and the spring dampers
connect the other components. Driven by 6 electric cylinders and a servo
motor, the wheels can realize translations and rotations in 6 degrees of
freedom. As a result, the BITNAZA can not only move and steer like a 4WIMD
system but also adjust the supporting polygon meeting the dynamic model in
Fig. 1. The environment awareness system is introduced in the red box,
including a lidar, a charge-coupled device (CCD) camera, and an infrared
camera. Equipped with a compact, portable, and tightly coupled synchronized
position attitude navigation (SPAN-CPT), the integrated navigation system
bring together two different, but complementary technologies: global
navigation satellite system (GNSS) positioning and inertial navigation.
Therefore, the robotic system achieves all-weather path planning and obstacle
detection.

To improve the safety and efficiency of designing and testing, a co-simulation
platform is built as shown in Fig. 5. The proposed controller is implied by
Matlab/Simulink and drives the virtual prototype in Adams, while the state
data of the prototype are fed back to the controller in real-time. The control
inputs of the virtual prototype are $F_{x}$ and the elongations of electric
cylinders, while the wheel dynamics and nonlinear tire model are concerned in
the Matlab/Simulink. Note that the virtual prototype ignores some factors such
as the friction in the electric cylinders and the hook joints.

Figure 5: The co-simulation platform.

###  4.2 Simulation Results

To evaluate the coordinated control performance of this robotic system, a
series of comparative simulation tests are carried out. Test 1 and Test 2
stands for the tracking performance of the robotic system controlled by the
proposed algorithm with different parameters. For controller No.1, the control
horizon $N_{c}=30$ and prediction horizon $N_{p}=60$ are implied. The
decreased $N_{c}=5$ and $N_{p}=20$ are set in controller No.2. The reference
Line 1 is a single lane change that is widely used to test the tracking
ability for wheeled systems. The road is flat without any obstacle so that the
supporting polygon of the robotic system stays static.  In Test 3 and Test 4,
the controllers are the same. However, there exist two obstacles on the road
in Test 4, while no obstacle is placed on the road in Test 3. For clarity and
intuition, Table 2 is organized.

Figure 6: Tracking curves and error curves in X-axis for Test 1 and Test 2. Figure 7: Tracking curves and error curves in Y-axis for Test 1 and Test 2. Figure 8: Tracking curves and error curves of Yaw for Test 1 and Test 2. Figure 9: The time consuming of the controllers for Test 1 and Test 2. Table 2: Evaluation Procedures Test | Trajectory | Obstacle | Controller Number
---|---|---|---
Test 1 | Line 1 | N | No.1
Test 2 | Line 1 | N | No.2
Test 3 | Line 2 | N | No.1
Test 4 | Line 2 | Y | No.1
Figure 10: The result recording of Test 4. Figure 11: Tracking curves for Test
3 and Test 4.

Considering the parameters in Table 1 and the system model, the weight matrix
are chosen as $Q=\textrm{diag}(1.0,10.0)$ and
$R=\textrm{diag}(1.0,1.0,1.0,1.0)$. The constraint of the input $u$ is

$U=\left[\begin{array}[]{cc}-10.0&10.0\\\ -10.0&10.0\\\ -10.0&10.0\\\
-10.0&10.0\\\ \end{array}\right].$ (17)

The simulation results of Test 1 and Test 2 are shown as follows. In Fig. 6, X
indicates the position of the robot centroid in the X-axis and Xe implies the
tracking error in X-axis, which are similarly defined in Fig. 7 and Fig. 8.
The maximum tracking errors in X-axis, Y-axis, and Yaw are 0.042 m, 0.043 m,
and $1.78^{\circ}$ respectively for Test 1, while the same indicators are
0.095 m, 0.086 m, and $2.13^{\circ}$ for Test 2. Therefore, with the
increments of $N_{p}$ and $N_{c}$, tracking accuracy is improved for the
proposed algorithm.

Moreover, as presented in Fig. 9, the time consuming for each cycle of the
controller is measured. Controller No.1 needs 30 ms for each cycle of
calculation on average, while controller No.2 only needs 7.5 ms approximately.
Therefore, with the increments of $N_{p}$ and $N_{c}$, the time consumption is
also increased, which may limit the application of the proposed algorithm for
real-time motion control systems. This feature indicates that the $N_{p}$ and
$N_{c}$ must be reasonably chosen for our robotic system in practice.

Since the hardware of the simulation system is sufficient, controller No.1 is
adopted in the following simulation tests. Test 4 is recorded in Fig. 10,
where there are two different obstacles. The first obstacle in green is 1.6 m
in width and 0.4 m in height. The second obstacle in black is 0.5 m in width
and 1.45 m in height. Considering the structure parameters in Table 1, the
robotic system can adjust the lateral wheel-track or the body height to
overcome the obstacles. From the analysis in Fig. 10, the system dynamic model
has the same characteristics with different body heights. As shown in (10), it
holds $\Gamma(t)=1$ in the process of the lateral wheel-track adjustment.

The tracking curves of the robotic system in Test 3 and Test 4 are as shown in
Fig. 11. Note that the small black box is set when $\Gamma(t)=1$. It is
pointed out there exists a peak of up to -0.074 m of the tracking error in the
second process of the lateral wheel-track adjustment. However, the tracking
error of the system with controller No.1 converges quickly. On the other hand,
the small black box occurs only in the process of supporting polygon
adjustment. The event-triggering mechanism efficiently splits the control
model into piecewise functions by behaviors. The system function is linear
when the robot tracking the desired path with $\Gamma(t)=0$. In such a
situation, the proposed controller helps to save hardware resources and
energy.

The simulation results presented above prove that the proposed control
algorithm is stable and effective. On the other hand, the different
performances in Test 1 to Test 4 guide the parameter tuning of the controller
in practical applications.

###  4.3 Experiment Results

To estimate the coordinated motion control performance, the naturalistic
driving data are collected in $39.970^{\circ}$ E and $116.316^{\circ}$ N. Some
cases that include straight, turning, passing intersections, bypassing
pedestrians, and crossing obstacles are set at markers $A$ to $F$ to analyze
the efficiency of the event-based MPC controller. The route is shown in Fig.
12, covering a distance of approximately 134 m. In the experiment, the SLAM is
carried out by the environment awareness system and the integrated navigation
system as shown in Fig. 4. A large number of experiments are conducted with
various obstacles. One case from the naturalistic driving database is
presented and analyzed in Fig. 13. The processes at the marked points are
recorded in detail.

Figure 12: Experimental environment and the reference trajectory on map.
Figure 13: Tracking curve for the experiment.

The experiment starts at marker $A$ followed by a short straight road and a
left turn. At marker $B$, a white obstacle, which is 1.5 m in width, is set. A
black and yellow obstacle appears at marker $C$, which is 1.3 m in height. The
triggering condition $\Gamma(t)=1$ holds while the robot is crossing the
obstacle at marker $B$ and adjusts the supporting polygon, while $\Gamma(t)=0$
holds at marker $C$ with the robot overcoming the obstacle. A pedestrian and a
slope are placed at markers $D$ and $E$. As the pedestrian is 1.65 m in height
which is more than the maximum height, the robotic system avoids the
pedestrian by passing from the side. The angle of the slope is 16.5∘ which is
smaller than the maximum angle of climb, the robotic system climbs up the
slope. The robot smoothly passes the different terrains tracking the desired
path from the SLAM. The experiment ends at the marker $F$ and the results of
the physical prototype trajectory are shown in Fig. 13. The experiment takes
about 2.5 minutes entirely, such that the average cruise speed approximates
0.9 m/s.

During the control process, the motor torques are distributed according to the
driving conditions of the control system. There is no collision with obstacles
or slippage. From the experiment results, it can be concluded that the
proposed controller has significant performance in coordinated motion control
for the four-wheel-leg independent motor-driven robotic systems.

##  5 Discussion

This paper studies a coordinated motion control and obstacle-crossing problem
for the four wheel-leg independent motor-driven robotic systems via the model
predictive control (MPC) approach based on an event-triggering mechanism. The
wheel-leg robotic system with a dynamic supporting polygon is analyzed and
modeled. The system model are proposed in detail, and the single wheel
dynamics considering the characteristics of motor-driven and the Burckhardt
nonlinear tire model. The relative position changes between wheels result in a
nonlinear time-varying model for the control system, which brings the main
challenge for MPC controller design. As a matter of the fact that the
supporting polygon is only adjusted at certain conditions, an event-based
triggering mechanism is designed to split the control model into piecewise
functions. When the robot requires no more supporting polygon adjustment, the
system function is time-invariant, such that the optimal solution of the MPC
controller is simplified to save hardware resources and energy. Based on the
developed cost function, the controller drives the system state at the end of
the behavior approaches $0$, which guarantees the switching stability between
behaviors. The simulated and experimental results verify the efficiency of the
proposed controller.

## References

  * [1] X. J. Jin, G. Yin, and N. Chen, “Gain-scheduled robust control for lateral stability of four-wheel-independent-drive electric vehicles via linear parameter-varying technique,” _Mechatronics_ , vol. 30, pp. 286–296, 2015\.
  * [2] H. Zhou, Z. Liu, and X. Yang, “Motor torque fault diagnosis for four wheel independent motor-drive vehicle based on unscented kalman filter,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 3, pp. 1969–1976, March 2018.
  * [3] W. Xu, H. Chen, H. Zhao, and B. Ren, “Torque optimization control for electric vehicles with four in-wheel motors equipped with regenerative braking system,” _Mechatronics_ , vol. 57, pp. 95–108, 2019.
  * [4] C. Liu, W. Sun, and J. Zhang, “Adaptive sliding mode control for 4–wheel sbw system with ackerman geometry,” _ISA Transactions_ , vol. 96, pp. 103–115, 2020.
  * [5] D. Lee, K. Yi, S. Chang, B. Lee, and B. Jang, “Robust steering-assist torque control of electric-power-assisted-steering systems for target steering wheel torque tracking,” _Mechatronics_ , vol. 49, pp. 157–167, 2018.
  * [6] J. Guo, Y. Luo, and K. Li, “An adaptive hierarchical trajectory following control approach of autonomous four-wheel independent drive electric vehicles,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 19, no. 8, pp. 2482–2492, 2018.
  * [7] M. A. Kamel, X. Yu, and Y. Zhang, “Fault-tolerant cooperative control design of multiple wheeled mobile robots,” _IEEE Transactions on Control Systems Technology_ , vol. 26, no. 2, pp. 756–764, March 2018.
  * [8] H. Marzbani, H. Khayyam, C. N. TO, . V. Quoc, and R. N. Jazar, “Autonomous vehicles: Autodriver algorithm and vehicle dynamics,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 4, pp. 3201–3211, April 2019.
  * [9] Z. Zhang, L. Wang, J. Zhang, and R. Ma, “Study on requirements for load emulation of the vehicle with an electric braking system,” _IEEE Transactions on Vehicular Technology_ , vol. 66, no. 11, pp. 9638–9653, Nov 2017\.
  * [10] H. Du, J. Shi, J. Chen, Z. Zhang, and X. Feng, “High-gain observer-based integral sliding mode tracking control for heavy vehicle electro-hydraulic servo steering systems,” _Mechatronics_ , vol. 74, p. 102484, 2021.
  * [11] J. Huang and Q.-G. Wang, “Decentralized adaptive control of interconnected nonlinear systems with unknown control directions,” _ISA Transactions_ , vol. 74, pp. 60–66, 2018.
  * [12] B. Li, X. Zhang, Y. Fang, and W. Shi, “Visual servo regulation of wheeled mobile robots with simultaneous depth identification,” _IEEE Transactions on Industrial Electronics_ , vol. 65, no. 1, pp. 460–469, Jan 2018\.
  * [13] H. Zhou, F. Jia, H. Jing, Z. Liu, and L. G$\ddot{\textrm{u}}$ven, “Coordinated longitudinal and lateral motion control for four wheel independent motor-drive electric vehicle,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 5, pp. 3782–3790, May 2018.
  * [14] N. Guo, B. Lenzo, X. Zhang, Y. Zou, R. Zhai, and T. Zhang, “A real-time nonlinear model predictive controller for yaw motion optimization of distributed drive electric vehicles,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 5, pp. 4935–4946, 2020.
  * [15] Z. Li, C. Yang, C. Su, J. Deng, and W. Zhang, “Vision-based model predictive control for steering of a nonholonomic mobile robot,” _IEEE Transactions on Control Systems Technology_ , vol. 24, no. 2, pp. 553–564, March 2016.
  * [16] D. Tavernini, M. Metzler, P. Gruber, and A. Sorniotti, “Explicit nonlinear model predictive control for electric vehicle traction control,” _IEEE Transactions on Control Systems Technology_ , vol. 27, no. 4, pp. 1438–1451, July 2019.
  * [17] F. J. Perez-Pinal, I. Cervantes, and A. Emadi, “Stability of an electric differential for traction applications,” _IEEE Transactions on Vehicular Technology_ , vol. 58, no. 7, pp. 3224–3233, Sep. 2009.
  * [18] A. Haddoun, M. E. H. Benbouzid, D. Diallo, R. Abdessemed, J. Ghouili, and K. Srairi, “Modeling, analysis, and neural network control of an ev electrical differential,” _IEEE Transactions on Industrial Electronics_ , vol. 55, no. 6, pp. 2286–2294, June 2008.
  * [19] Y. Chen and J. Wang, “Design and evaluation on electric differentials for overactuated electric ground vehicles with four independent in-wheel motors,” _IEEE Transactions on Vehicular Technology_ , vol. 61, no. 4, pp. 1534–1542, May 2012.
  * [20] K. Berntorp, “Joint wheel-slip and vehicle-motion estimation based on inertial, gps, and wheel-speed sensors,” _IEEE Transactions on Control Systems Technology_ , vol. 24, no. 3, pp. 1020–1027, May 2016.
  * [21] G. Xie, H. Gao, L. Qian, B. Huang, K. Li, and J. Wang, “Vehicle trajectory prediction by integrating physics- and maneuver-based approaches using interactive multiple models,” _IEEE Transactions on Industrial Electronics_ , vol. 65, no. 7, pp. 5999–6008, July 2018.
  * [22] F. Wang, H. Chen, and D. Cao, “Nonlinear coordinated motion control of road vehicles after a tire blowout,” _IEEE Transactions on Control Systems Technology_ , vol. 24, no. 3, pp. 956–970, May 2016.
  * [23] H. Peng, J. Wang, S. Wang, W. Shen, D. Shi, and D. Liu, “Coordinated motion control for a wheel-leg robot with speed consensus strategy,” _IEEE/ASME Transactions on Mechatronics_ , vol. 25, no. 3, pp. 1366–1376, 2020.
  * [24] X. Li, H. Zhou, S. Zhang, H. Feng, and Y. Fu, “Wlr-ii, a hose-less hydraulic wheel-legged robot,” in _2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2019, pp. 4339–4346.
  * [25] K. Xu, S. Wang, B. Yue, J. Wang, H. Peng, D. Liu, Z. Chen, and M. Shi, “Adaptive impedance control with variable target stiffness for wheel-legged robot on complex unknown terrain,” _Mechatronics_ , vol. 69, p. 102388, 2020\.
  * [26] K. E. Årzen, “A simple event-based pid controller,” _IFAC World Congress_ , vol. 32, pp. 8687–8692, Dec. 1999.
  * [27] K. J. Åstrom and B. Bernhardsson, “Comparison of periodic and event based sampling for first order stochastic systems,” _IFAC World Congress_ , vol. 32, pp. 5006–5011, 1999.
  * [28] W. P. M. H. Heemels, R. J. A. Gorter, A. V. Zijl, P. P. J. V. D. Bosch, S. Weiland, W. H. A. Hendrix, and M. R. Vonder, “Asynchronous measurement and control: a case study on motor synchronization,” _Control Engineering Practice_ , vol. 7, no. 12, pp. 1467–1482, 1999.
  * [29] D. Shi, L. Shi, and T. Chen, _Event-Based State Estimation_ , 2016.
  * [30] Y. Huang, J. Wang, F. Wang, and B. He, “Event-triggered adaptive finite-time tracking control for full state constraints nonlinear systems with parameter uncertainties and given transient performance,” _ISA Transactions_ , vol. 108, pp. 131–143, 2021.
  * [31] H. Li and Y. Shi, “Event-triggered robust model predictive control of continuous-time nonlinear systems,” _Automatica_ , vol. 50, no. 5, pp. 1507–1513, 2014.
  * [32] Y. Huang, J. Wang, D. Shi, and L. Shi, “Toward event-triggered extended state observer,” _IEEE Transactions on Automatic Control_ , vol. 63, no. 6, pp. 1842–1849, 2018.
  * [33] G. Droge, P. Kingston, and M. Egerstedt, “Behavior-based switch-time mpc for mobile robots,” in _2012 IEEE/RSJ International Conference on Intelligent Robots and Systems_ , Oct 2012, pp. 408–413.
  * [34] D. Shi, J. Xue, L. Zhao, J. Wang, and Y. Huang, “Event-triggered active disturbance rejection control of dc torque motors,” _IEEE/ASME Transactions on Mechatronics_ , vol. 22, no. 5, pp. 2277–2287, 2017.