Coupling Control Strategy and Experiments for Motion Mode Switching of a Novel Electric Chassis

: A ﬂexible chassis (FC) is a type of electric vehicle driven by in-wheel motors that can be used in narrow conditions in agricultural facilities. The FC is composed primarily of four o ﬀ -center steering mechanisms (OSMs) that can be controlled independently. Various FC operation modes can be achieved including cross motion (CM), in-place rotation (IR), diagonal motion (DM), and steering motion (SM). However, it is di ﬃ cult to achieve satisfactory motion mode switching (MMS) results under traditional distribution control methodologies due to a lack of linkage relationships between the four OSMs. The goal of this study was to provide a coupling control method that can cope with this problem. First, dynamic MMS models were derived. Then, an MMS coupling error (CE) model was derived based on coupling control and Lyapunov stability theory. Second, a fuzzy proportional integral derivative (PID) controller with self-tuning parameters was designed to reduce the CE during MMS. A fuzzy PI controller was also employed to improve response times and decrease OSM tracking motion steady-state error. Finally, MATLAB / Simulink simulations were performed and experimentally validated on hard pavement. The results showed that the proposed methodology could e ﬀ ectively reduce CE and guarantee MMS control stability while substantially shortening response times. The proposed methodology is e ﬀ ective and feasible for FC MMS. stability theory. A fuzzy PID controller was designed to compensate for the coupling error and a fuzzy PI method was employed to reduce off-center steering mechanism tracking errors as well. The proposed control methodology was examined by simulation and validated experimentally on hard pavement. Compared to the conventional distribution control method, the proposed approach could effectively reduce coupling errors and guarantee MMS control stability while substantially shortening response times. The coupling control method proposed in this study demonstrated better effectiveness and feasibility than distribution control. The results of the study show that the DM and SM MMS performance with regard to coupling error was not as good as the CM and IR performance. Thus, we should pay attention to the reasons for this performance difference. It is important to devise a strategy to improve FC MMS control.

. However, due to the lack of a rigid constraint mechanism, it is difficult to coordinate the motion of each FC OSM during MMS in situ.
The objective of this paper is to provide a coordinated control strategy to reduce MMS angular velocity coupling errors (CEs) and guarantee smooth, steady MMS. To achieve this, a CE model is derived based on coupling control and Lyapunov stability theory. A fuzzy PID controller is then designed for CE control. To shorten the response time and reduce steady-state errors of OSM steering tracking motion, a fuzzy PI control method is employed for steering tracking control. The feasibility of the proposed control strategy is confirmed via MATLAB/Simulink simulations and then experimentally validated on hard pavement. This paper provides two main contributions to the engineering community: (i) a control algorithm that integrates coupling control, Lyapunov stability theory, fuzzy PID control, and fuzzy PI control is proposed for FC MMS in order to cope with the lack of a linkage relationship among the four OSMs, and (ii) the CE performance and MMS stability are improved under the proposed control methodology.
The rest of the paper is organized as follows. In Section 2, the overall structure of the FC and system modeling are described. In Section 3, a coupling control strategy for MMS of the FC is designed. In Section 4, we discuss the MATLAB/Simulink simulation. Section 5 is dedicated to experimental validation. Finally, our conclusions and proposals for future work are provided in Section 6.

Overall Structure
The overall structure of the FC drawn by Computer Aided Design (AutoCAD2007, Auto desk Company, San Rafael, CA, USA) is shown in Figure 1. The main components include four OSMs, an electronic control unit, and the control lines.   Each OSM consists of an electromagnetic friction lock (FBD-050, Taiwan Kaide, Taiwan), an offcenter steering shaft, an off-center arm, a suspension and an in-wheel motor (WX-WS4846, Fujitec, Tianjin, China. A bridge circuit made of four arms identified as R1, R2, R3, and R4 achieves steering tracking for the OSM. When the electromagnetic friction lock is in a release state, the vehicle is flexibly Tianjin, China. A bridge circuit made of four arms identified as R1, R2, R3, and R4 achieves steering tracking for the OSM. When the electromagnetic friction lock is in a release state, the vehicle is flexibly steered by rotating the OSM around the steering shaft. By controlling the electromagnetic friction locks and the in-wheel motors, the FC can achieve the various types of motion modes mentioned earlier.
Switching processes of all motion modes are performed in situ during MMS. In contrast, when the electromagnetic friction lock is in a locked state, the off-center arm is fixed on the FC frame and the FC can only move with fixed posture. This paper focuses on the MMS coupling control methodology.

Motion Models for Various Motion Modes
A schematic diagram of the FC motion mode models is presented in Figure 2. Different motion modes require different off-center arm steering angles. During MMS, the steering angles of the four off-center arms and the angular velocities of the four in-wheel motors must maintain fixed relationships. Each off-center arm has a specific target angle. During CM, the relationships are given by: where δ i is the steering angle of each off-center arm (I = fl, fr, rl, rr, which represent the left front, right front, left rear, and right rear wheel of the flexible chassis, respectively), ω i is the angular velocity of each off-center steering mechanism, and δ io denote the target angle of each off-center arm. For in-place rotation, the target angles and angular velocities have the following relationships: where L is the distance between the front and rear off-center shafts and B is the distance between the left and right off-center shafts. During the whole MMS process, the FC frame needs to keep its original posture. For diagonal motion, if CM is directly switched to DM, the moments at centroid from all wheels will be in the same direction, and the FC frame cannot keep its posture. To avoid this problem, DM is switched from IR in this study. This method can maintain symmetrical force on the FC. The target angle and angular velocity relationships are expressed by: Similarly, the relationships that govern steering motion are: where d is the off-center distance and R is the turning radius of the FC during steering motion.

Model of Electric Wheel
In MMS, the OSM driving force originates only from the electric wheels. To explore MMS control strategies, it is necessary to establish a theoretical model of the electric wheel. A force diagram of the electric wheel during acceleration assuming that tire deformation is small and can be ignored and the tire is rolling on a hard road is shown in Figure 3. The force diagram in Figure 3 shows that the relationship between the electric wheel angular velocity and the longitudinal tire force is as follows: where F x is the longitudinal tire force (N); r is the electric wheel radius (m); T e is the electromagnetic torque of the motor (N·m); B m is the viscous friction damping coefficient (N·m); ω m is the electric wheel angular velocity (rad·s −1 ); J is the moment of inertia (kg·m 2 ); M f is the rolling resistance moment (N·m); and t is time (s).  Figure 2. Schematic diagram of flexible chassis (FC) motion mode models. O is the turning center. R is the FC steering motion turning radius. Blue represents direct switching between two motion modes, and the gray arrow indicates that it cannot be switched directly.

Model of Electric Wheel
In MMS, the OSM driving force originates only from the electric wheels. To explore MMS control strategies, it is necessary to establish a theoretical model of the electric wheel. A force diagram of the electric wheel during acceleration assuming that tire deformation is small and can be ignored and the tire is rolling on a hard road is shown in Figure 3.
The force diagram in Figure 3 shows that the relationship between the electric wheel angular velocity and the longitudinal tire force is as follows: where Fx is the longitudinal tire force (N); r is the electric wheel radius (m); Te is the electromagnetic torque of the motor (N·m); Bm is the viscous friction damping coefficient (N·m); ωm is the electric wheel angular velocity (rad·s −1 ); J is the moment of inertia (kg·m²); Mf is the rolling resistance moment (N·m); and t is time (s). O is the turning center. R is the FC steering motion turning radius. Blue represents direct switching between two motion modes, and the gray arrow indicates that it cannot be switched directly.

Model of Electric Wheel
In MMS, the OSM driving force originates only from the electric wheels. To explore MMS control strategies, it is necessary to establish a theoretical model of the electric wheel. A force diagram of the electric wheel during acceleration assuming that tire deformation is small and can be ignored and the tire is rolling on a hard road is shown in Figure 3.
The force diagram in Figure 3 shows that the relationship between the electric wheel angular velocity and the longitudinal tire force is as follows: where Fx is the longitudinal tire force (N); r is the electric wheel radius (m); Te is the electromagnetic torque of the motor (N·m); Bm is the viscous friction damping coefficient (N·m); ωm is the electric wheel angular velocity (rad·s −1 ); J is the moment of inertia (kg·m²); Mf is the rolling resistance moment (N·m); and t is time (s). Figure 3. Electric wheel model during acceleration. ua is the forward speed of the wheel; W is the total weight of electric wheel and its load; N is the supporting force. Based on the OSM structure, the angular velocities of the off-center arm and the electric wheel have the following relationship: where K is a constant coefficient.

General Flexible Chassis Kinetic Model
A simplified chassis dynamics model ( Figure 4) is adopted to establish the FC MMS state equation. This study only considers the vehicle's longitudinal motion (x-direction), lateral motion (y-direction), and yaw motion (rotation around the z-axis). It is also assumed that there is no suspension system effect or air resistance. Changes in vehicle longitudinal speed are ignored. The coordinate system of the model takes the centroid (CG) of the FC as the origin. The x-axis is the longitudinal direction of the FC and the y-axis is the lateral direction.
Based on the OSM structure, the angular velocities of the off-center arm and the electric wheel have the following relationship: where K is a constant coefficient.

General Flexible Chassis Kinetic Model
A simplified chassis dynamics model ( Figure 4) is adopted to establish the FC MMS state equation. This study only considers the vehicle's longitudinal motion (x-direction), lateral motion (ydirection), and yaw motion (rotation around the z-axis). It is also assumed that there is no suspension system effect or air resistance. Changes in vehicle longitudinal speed are ignored. The coordinate system of the model takes the centroid (CG) of the FC as the origin. The x-axis is the longitudinal direction of the FC and the y-axis is the lateral direction. In the dynamic model, the kinetic equations for longitudinal, lateral, and yaw motion are expressed using Equations (7)-(9), respectively: ) cos sin ( ) ( where u is longitudinal speed, v is lateral speed, I is the moment of inertia of the FC, m is the mass of the FC, and i = 1(fl), 2(fr), 3(rl), 4(rr). According to the linear tire model [33], the steering angle, side-slip angle, longitudinal speed, and lateral speed have the following relationship: where αi denotes a side-slip angle, vj denotes longitudinal speed, and uj denotes lateral speed (j = CM, IM, DM, SM). The side-slip angle of each motion is calculated as: Figure 4. Simplified flexible chassis dynamics model. CG is the center of chassis gravity; l r and l f are the distances from the front and rear off-center shafts to CG, respectively; M Z represents the yaw moment generated by the four electric wheels; F xi represents the longitudinal tire force of each wheel; F yi is the lateral tire force of each wheel; and α yi is the slip angle of each wheel; γ is the yaw rate.
In the dynamic model, the kinetic equations for longitudinal, lateral, and yaw motion are expressed using Equations (7)-(9), respectively: where u is longitudinal speed, v is lateral speed, I is the moment of inertia of the FC, m is the mass of the FC, and i = 1(fl), 2(fr), 3(rl), 4(rr). According to the linear tire model [33], the steering angle, side-slip angle, longitudinal speed, and lateral speed have the following relationship: where α i denotes a side-slip angle, v j denotes longitudinal speed, and u j denotes lateral speed (j = CM, IM, DM, SM). The side-slip angle of each motion is calculated as: Appl. Sci. 2020, 10, 701 The tire side-slip angle generally is very small in normal driving. It can be assumed that there is a linear relationship between the tire side force and the side-slip angle of the FC. The relationship between tire side force F y , side-slip angle α, and tire cornering stiffness C α is then given by: Finally, the state equation of FC movement is derived as follows: Similarly, based on the above derivation, the state equations for CM, IR, DM, and SM can be derived according to Equation (13) and will not be derived here again.

Coupling Error Control Model
In the previous study, the distribution control method was adopted for MMS of the FC. The command signal was directly assigned to each OSM according to the desired angle of each mode. The steering angle tracking control of each OSM was implemented based on bridge circuit, but the coupling motion of four OSMs was not considered. This methodology is simple and easy to implement when the four OSMs are steering independently without restraining each other. In this study, a coupling control strategy is proposed. The CE of an off-center arm is obtained by comparing the angular velocity between two adjacent OSMs. Then, the CE of each OSM is compensated using the coupling control algorithm. Any changes in the angular velocity of an OSM will provide feedback to an adjacent OSM. All adjacent steering mechanisms are coupled in pairs, eventually forming a coupling loop. This study observes the effect of the proposed coupling control by comparing it with the distribution control method.
In the MMS process, the expected angular velocity of an off-center arm, ω d , is the only input signal. However, the angular velocities of the off-center arms are different under different motion modes. In a motion mode, the proportional relationship between the angular velocities of all off-center arms is expressed as: where η p is proportionality coefficient (p = 1, 2, . . . , n).
The kth off-center arm with the worst control performance is selected as a reference, and its proportionality coefficient is η k . For convenience of synchronization error derivation, a normalized proportional coefficient η * p is introduced: In a control system with n OSMs, the angular velocity tracking error of OSM p is defined as where ω * p = η * p · ω d , and ω d represents the demand angular velocity.
Appl. Sci. 2020, 10, 701 The synchronization error between any two adjacent OSMs is expressed using Equation (17). Synchronization control of all OSMs is achieved when ε p = 0: The dynamic characteristics of the OSM are described using: .
where T s is the load torque (N·m); B o is the viscous friction damping coefficient of the OSM (N·m); and J o denotes the OSM moment of inertia (kg·m 2 ). Equation (18) can be simplified as: .
The CE of the pth OSM after compensation in coupling control is given as: where λ p represents the synchronization error coefficient, e p is the OSM tracking error, and ε p is the OSM coupling error. To achieve coupling control, E p must be maintained at zero. According to Lyapunov's direct method, a Lyapunov function can be constructed for a nonlinear differential equation to study its stabilization in a control system. To make E 1 zero, the Lyapunov function is used to judge the system stability. Similar to the studies of [34][35][36], taking the wheel 1 of the FC as an example, a Lyapunov function is constructed to judge the system stability, as shown by Equation (21): If Equation (22) exists and Equation (23) is true, the system is stable when E 1 approaches zero: . .
Combining Equations (17), (19), and (20) allows the following equation to be derived: . . . where and µ 1 is the control function of the left front off-center steering shaft.
If (27) is true: where For any positive real number c 1 , µ 1 can be constructed such that E 1 approaches zero. Therefore, we can construct Equation (28) to make E 1 →0. Similarly, µ i can be constructed to make E i →0. Based on the converse derivation of Equation (28), it also can be found that, if Equation (28) is established, then Equation (22) can be guaranteed. As one of the control parameters, c 1 will affect the regulation time and overshoot of the control system. Through pre-research simulation, it was found that, if c 1 is too small, the adjustment time of the system will increase, and, if c 1 is too large, the overshoot will increase.

Fuzzy PID Coupling Error Control Strategy
PID controllers are frequently used for vehicle control. However, satisfactory results are difficult to achieve when working conditions and control parameters change. In contrast, a fuzzy logic controller (FLC) does not require a precise mathematical model of the system. This FLC property guarantees stable system operation even if the control parameters and conditions undergo dynamic change [37]. The fuzzy method has the advantages of keeping a simple control structure and avoiding costly sensor use [28,38]. In this paper, the angular velocity CE is adjusted via control of the in-wheel motor speed. The speed of the in-wheel motor is controlled according to the CE and its change rate. Additional control of the in-wheel motor speed is needed to avoid the interference of multiple random factors. Therefore, developing a controller based on fuzzy logic is an attractive choice, and a fuzzy PID controller was designed for CE control of MMS.
The CE and its change rate are the two key parameters in the process of synchronous motion adjustment. Therefore, the CE and its change rate are used as fuzzy controller inputs E c and ∆E c , and the two-dimensional fuzzy controller is the most suitable type. The fuzzy PID controller structure is shown in Figure 5. Seven fuzzy linguistic terms are adopted for both inputs and outputs: negative big (NB), negative medium (NM), negative small (NS), zero (Z), positive small (PS), positive medium (PM), and positive big (PB). Some basic fuzzy PID control processes are described as follows. When E c and ∆E c are small, larger K P and K I and proper K D should be adopted compared with current parameters, in order to stabilize the steering system. When E c and ∆E c are medium-sized, smaller K P and proper K I and K D are used to reduce steering angle overshoot. When E c and ∆E c are large, larger K P and smaller K I and K D are used to avoid excessive overshoot and expedite the steering response. The membership functions were fine-tuned experimentally based on human experiences of vehicle operation until the system performed acceptably [39]. Figure 6 shows the membership function curves. The fuzzy rules are shown in Table 1. Exact changes in the PID parameters are calculated using the center-of-area method. front off-center steering shaft.
, then Equation (27) is true: For any positive real number c1, μ1 can be constructed such that E1 approaches zero. Therefore, we can construct Equation (28) to make E1→0. Similarly, μi can be constructed to make Ei→0. Based on the converse derivation of Equation (28), it also can be found that, if Equation (28) is established, then Equation (22) can be guaranteed. As one of the control parameters, c1 will affect the regulation time and overshoot of the control system. Through pre-research simulation, it was found that, if c1 is too small, the adjustment time of the system will increase, and, if c1 is too large, the overshoot will increase.

Fuzzy PID Coupling Error Control Strategy
PID controllers are frequently used for vehicle control. However, satisfactory results are difficult to achieve when working conditions and control parameters change. In contrast, a fuzzy logic controller (FLC) does not require a precise mathematical model of the system. This FLC property guarantees stable system operation even if the control parameters and conditions undergo dynamic change [37]. The fuzzy method has the advantages of keeping a simple control structure and avoiding costly sensor use [28,38]. In this paper, the angular velocity CE is adjusted via control of the in-wheel motor speed. The speed of the in-wheel motor is controlled according to the CE and its change rate. Additional control of the in-wheel motor speed is needed to avoid the interference of multiple random factors. Therefore, developing a controller based on fuzzy logic is an attractive choice, and a fuzzy PID controller was designed for CE control of MMS.
The CE and its change rate are the two key parameters in the process of synchronous motion adjustment. Therefore, the CE and its change rate are used as fuzzy controller inputs Ec and ΔEc, and the two-dimensional fuzzy controller is the most suitable type. The fuzzy PID controller structure is shown in Figure 5. Seven fuzzy linguistic terms are adopted for both inputs and outputs: negative big (NB), negative medium (NM), negative small (NS), zero (Z), positive small (PS), positive medium (PM), and positive big (PB). Some basic fuzzy PID control processes are described as follows. When Ec and ΔEc are small, larger KP and KI and proper KD should be adopted compared with current parameters, in order to stabilize the steering system. When Ec and ΔEc are medium-sized, smaller KP and proper KI and KD are used to reduce steering angle overshoot. When Ec and ΔEc are large, larger KP and smaller KI and KD are used to avoid excessive overshoot and expedite the steering response. The membership functions were fine-tuned experimentally based on human experiences of vehicle operation until the system performed acceptably [39]. Figure 6 shows the membership function curves. The fuzzy rules are shown in Table 1. Exact changes in the PID parameters are calculated using the center-of-area method.

Off-Center Arm Steering Angle Tracking Error Control Strategy
In previous research, bridge circuits [40] were used for off-center arm angle tracking control. However, the bridge circuit-based control system exhibits large steady-state errors. To solve this problem, the fuzzy PI control method is employed. It is also possible to use PID for steering angle tracking control, but its parameter adjustment is more complex than PI control [41,42]. From the previous test, PI control was sufficient for steering angle tracking control. Therefore, a fuzzy PI controller was employed to improve processing efficiency. Fast steering response can be achieved and angle error can be reduced via the auxiliary PI algorithm and self-tuning of the proportional and integral coefficients. The fuzzy method still uses a two-dimensional fuzzy controller and the inputs are the errors of the steering angles and the error rate (Et and ΔEt). The outputs are the KP and KI corrections for PI control. In this controller, the seven fuzzy linguistic terms are introduced to describe the input and output variable values:

Off-Center Arm Steering Angle Tracking Error Control Strategy
In previous research, bridge circuits [40] were used for off-center arm angle tracking control. However, the bridge circuit-based control system exhibits large steady-state errors. To solve this problem, the fuzzy PI control method is employed. It is also possible to use PID for steering angle tracking control, but its parameter adjustment is more complex than PI control [41,42]. From the previous test, PI control was sufficient for steering angle tracking control. Therefore, a fuzzy PI controller was employed to improve processing efficiency. Fast steering response can be achieved and angle error can be reduced via the auxiliary PI algorithm and self-tuning of the proportional and integral coefficients. The fuzzy method still uses a two-dimensional fuzzy controller and the inputs are the errors of the steering angles and the error rate (E t and ∆E t ). The outputs are the K P and K I corrections for PI control. In this controller, the seven fuzzy linguistic terms are introduced to describe the input and output variable values: input variable 1:  Figure 7. The membership functions were also fine-tuned experimentally. Figure 8 shows a surface plot of the input and the output fuzzy logic variables of ∆K P and ∆K I .

Off-Center Arm Steering Angle Tracking Error Control Strategy
In previous research, bridge circuits [40] were used for off-center arm angle tracking control. However, the bridge circuit-based control system exhibits large steady-state errors. To solve this problem, the fuzzy PI control method is employed. It is also possible to use PID for steering angle tracking control, but its parameter adjustment is more complex than PI control [41,42]. From the previous test, PI control was sufficient for steering angle tracking control. Therefore, a fuzzy PI controller was employed to improve processing efficiency. Fast steering response can be achieved and angle error can be reduced via the auxiliary PI algorithm and self-tuning of the proportional and integral coefficients. The fuzzy method still uses a two-dimensional fuzzy controller and the inputs are the errors of the steering angles and the error rate (Et and ΔEt). The outputs are the KP and KI corrections for PI control. In this controller, the seven fuzzy linguistic terms are introduced to describe the input and output variable values: input variable 1:

Simulation Parameters
To assess the feasibility of the proposed control strategy, MATLAB/Simulink (R2014a, MathWorks Company, Natick, MA, USA, 2014) simulations of coupling and distribution control were performed. The simulation model is composed primarily of an angle distribution planner, a fuzzy PID controller, and a fuzzy PI controller for each wheel. The main FC parameters used in the simulations are shown in Table 2. The CE is the key index that characterize the coupling control performance of the four OSMs. Longitudinal and lateral acceleration are the main indicators used to determine stability during MMS. Changes in the steering angles and angular velocities of the four OSMs also must be detected. Thus, the simulations included three parts: the steering responses of four off-center arms, the CE performance, and the longitudinal and lateral acceleration changes under distribution and coupling control. The feasibility of the proposed method can be determined by comparing these two control methods.
According to the analysis in Section 2.2, the target steering angle in any off-center arm is 90° during CM switching and 63° during IR switching. During DM switching, the target steering angle of off-center arms 1 and 3 is set to 45°, while off-center arms 2 and 4 are set to 135°. During SM switching, the target steering angles of off-center arms 1 and 3 is 30°, while arms 2 and 4 use 39° based on Ackermann steering geometry.

Simulation Results
MMS steering angle simulation results with distribution and coupling control are shown in Figure 9. In CM, IR, DM, and SM, the switching time in coupling control is shorter than that used in distribution control. The mode switching times for CM, IR, DM, and SM are respectively 4.9 s, 3.4 s,

Simulation Parameters
To assess the feasibility of the proposed control strategy, MATLAB/Simulink (R2014a, MathWorks Company, Natick, MA, USA, 2014) simulations of coupling and distribution control were performed. The simulation model is composed primarily of an angle distribution planner, a fuzzy PID controller, and a fuzzy PI controller for each wheel. The main FC parameters used in the simulations are shown in Table 2. The CE is the key index that characterize the coupling control performance of the four OSMs. Longitudinal and lateral acceleration are the main indicators used to determine stability during MMS. Changes in the steering angles and angular velocities of the four OSMs also must be detected. Thus, the simulations included three parts: the steering responses of four off-center arms, the CE performance, and the longitudinal and lateral acceleration changes under distribution and coupling control. The feasibility of the proposed method can be determined by comparing these two control methods.
According to the analysis in Section 2.2, the target steering angle in any off-center arm is 90 • during CM switching and 63 • during IR switching. During DM switching, the target steering angle of off-center arms 1 and 3 is set to 45 • , while off-center arms 2 and 4 are set to 135 • . During SM switching, the target steering angles of off-center arms 1 and 3 is 30 • , while arms 2 and 4 use 39 • based on Ackermann steering geometry. Figure 9. In CM, IR, DM, and SM, the switching time in coupling control is shorter than that used in distribution control. The mode switching times for CM, IR, DM, and SM are respectively 4.9 s, 3.4 s, 3.3 s, and 4.2 s, respectively, under coupling control. Under distribution control, the mode switching times are 6.1 s, 4.8 s, 5.1 s, and 5.5 s, respectively. Of these four motions, the IR mode switching time is the shortest whether coupling or distribution control is used. This is primarily due to the small, symmetrical rotation angles of the four OSMs. The average steering angle errors of the four OSMs are 0. In addition, Figure 9d shows that the relationships between the two front and two rear wheels are better maintained under coupling control than under distribution control. The results demonstrate that there is good steering angle symmetry and uniformity among the four OSMs during coupling control mode switching. Overall, the four OSM steering angles exhibit better synchronization under the proposed control methodology than under distribution control.

MMS steering angle simulation results with distribution and coupling control are shown in
Angular velocity distribution and coupling control MMS simulation results are shown in Figure  10. In all motion modes, the angular velocity first increases and then decreases smoothly. All modes exhibit maximum angular velocity. Under distribution control, the angular velocity responses are slower than under coupling control. The maximum angular velocities are all smaller as well. The maximum angular velocities of the four OSMs do not maintain good symmetry under distribution control. For example, in CM, the maximum angular velocities are all nearly 0.74 rad·s −1 but only 0.63 rad·s −1 , 0.61 rad·s −1 , 0.60 rad·s −1 , and 0.62 rad·s −1 for the left front, right front, left rear, and right rear OSMs, respectively, under distribution control. Results from the other three motion modes are similar to those from CM. The CM, IR, and SM time differences are all above 1 s. The biggest mode switching In addition, Figure 9d shows that the relationships between the two front and two rear wheels are better maintained under coupling control than under distribution control. The results demonstrate that there is good steering angle symmetry and uniformity among the four OSMs during coupling control mode switching. Overall, the four OSM steering angles exhibit better synchronization under the proposed control methodology than under distribution control.
Angular velocity distribution and coupling control MMS simulation results are shown in Figure 10. In all motion modes, the angular velocity first increases and then decreases smoothly. All modes exhibit maximum angular velocity. Under distribution control, the angular velocity responses are slower than under coupling control. The maximum angular velocities are all smaller as well. The maximum angular velocities of the four OSMs do not maintain good symmetry under distribution control. For example, in CM, the maximum angular velocities are all nearly 0.74 rad· s −1 but only 0.63 rad· s −1 , 0.61 rad· s −1 , 0.60 rad· s −1 , and 0.62 rad· s −1 for the left front, right front, left rear, and right rear OSMs, respectively, under distribution control. Results from the other three motion modes are similar to those from CM. The CM, IR, and SM time differences are all above 1 s. The biggest mode switching time difference between coupling and distribution control occurs during DM. The time difference for this motion can reach approximately 2 s. One can speculate that the angular velocities of the four OSMs under coupling control show better synchronicity than those under the proposed control methodology. The OSM CE curves for all motion modes are shown in Figure 11. We take the CE between wheel 1 and wheel 2 as an example for analysis. Obviously, the CEs are greater under distribution control than under coupling control. The respective absolute maximum angular velocity CEs during CM, IR, DM, and SM are 0.008 rad·s −1 , 0.006 rad·s −1 , 0.011 rad·s −1 , and 0.007 rad·s −1 under distribution control. The respective average CEs are 0.002 rad·s −1 , 0.001 rad·s −1 , 0.003 rad·s −1 , and 0.003 rad·s −1 for these four motions. In coupling tests, the maximum CEs are 0.004 rad·s −1 , 0.002 rad·s −1 , 0.006 rad·s −1 , and 0.003 rad·s −1 , respectively, and the average CEs are 0.0008 rad·s −1 , 0.0004 rad·s −1 , 0.001 rad·s −1 , and 0.0009 rad·s −1 , respectively. Clearly, the CEs change randomly throughout MMS during distribution control tests. In coupling control tests, the initial errors are somewhat large but gradually approach zero in later stages. This indicates that the coupling algorithm plays a role in MMS. The CE is greatest during DM regardless of which control method is used. This may be caused by the asymmetric kinematic characteristics and wide steering angles used by off-center arms 2 and 4 during this motion. Overall, the CE performance improves substantially under the proposed control methodology. The OSM CE curves for all motion modes are shown in Figure 11. We take the CE between wheel 1 and wheel 2 as an example for analysis. Obviously, the CEs are greater under distribution control than under coupling control. The respective absolute maximum angular velocity CEs during CM, IR, DM, and SM are 0.008 rad· s −1 , 0.006 rad· s −1 , 0.011 rad· s −1 , and 0.007 rad· s −1 under distribution control. The respective average CEs are 0.002 rad· s −1 , 0.001 rad· s −1 , 0.003 rad· s −1 , and 0.003 rad· s −1 for these four motions. In coupling tests, the maximum CEs are 0.004 rad· s −1 , 0.002 rad· s −1 , 0.006 rad· s −1 , and 0.003 rad· s −1 , respectively, and the average CEs are 0.0008 rad· s −1 , 0.0004 rad· s −1 , 0.001 rad· s −1 , and 0.0009 rad· s −1 , respectively. Clearly, the CEs change randomly throughout MMS during distribution control tests. In coupling control tests, the initial errors are somewhat large but gradually approach zero in later stages. This indicates that the coupling algorithm plays a role in MMS. The CE is greatest during DM regardless of which control method is used. This may be caused by the asymmetric kinematic characteristics and wide steering angles used by off-center arms 2 and 4 during this motion. Overall, the CE performance improves substantially under the proposed control methodology.  Figure 12 shows simulated mode switching longitudinal and lateral acceleration changes. In all motion modes, acceleration values are greater than simulation values and fluctuations occur. Meanwhile, acceleration is smaller under coupling control than under distribution control. In CM mode switching, the maximum absolute longitudinal acceleration rates under coupling control and distribution control are −0.033 m·s −2 and −0.011 m·s −2 , respectively. In contrast, the maximum absolute lateral acceleration rates under coupling and distribution control are 0.031 m·s −2 and 0.010 m·s −2 , respectively. The four values in the above order are 0.028 m·s −2 , −0.008 m·s −2 , −0.023 m·s −2 , and −0.006 m·s −2 during IR mode switching, 0.029 m·s −2 , 0.008 m·s −2 , −0.035 m·s −2 , and 0.007 m·s −2 during DM mode switching, and −0.028 m·s −2 , 0.004 m·s −2 , 0.028 m·s −2 , and −0.006 m·s −2 during SM mode switching. From these results, we can see that the longitudinal and lateral disturbances are smaller under coupling control than under conventional distribution control. The maximum CE reduction reaches 50%. Therefore, MMS occurs with better stability under coupling control. This further demonstrates that the method proposed in this study is feasible and effective.

Experiment Equipment and Method
To further verify the effectiveness of the control strategy, the chassis system model was loaded from MATLAB into the MicroAutoBox real-time system. The embedded ECUs (Electronic Control Units) were implemented on STM32 units in order to perform tests on hard pavement. The control panel is shown in Figure 13a. The controller mainly included single-chip modules (STM32F103ZET6, STMicroelectronics, Geneva, Switzerland), stepper motors (YH42BYGH47-401A, Microstep, Bratislava, Slovakia) and their drivers, bridge circuit modules, and accessory circuits. Figure 13b shows the detailed configuration of the FC used for test. A photoelectric encoder (GTS06-OC-RA1000B-2M, pulse: 1000; Kasei Electronics Ltd., Tokyo, Japan) was used to acquire the speeds of inwheel motors. The off-center arm steering angles were measured using a multi-turn potentiometer (22HP-10, 0-5 kΩ; Sakae Company, Tokyo, Japan). Acceleration sensors (WT61C232, Wit Technology Company, Dublin, Ireland) were used to detect the longitudinal and lateral FC acceleration rates. The duration of MMS was calculated by the clock integrated into the data acquisition equipment, which included a data acquisition card (USB7648B, Beijing Zhongtai Research Ltd., Beijing, China) and an industrial personal computer (610H, Advantech Technology Corporation, Beijing, China).

Experiment Equipment and Method
To further verify the effectiveness of the control strategy, the chassis system model was loaded from MATLAB into the MicroAutoBox real-time system. The embedded ECUs (Electronic Control Units) were implemented on STM32 units in order to perform tests on hard pavement. The control panel is shown in Figure 13a. The controller mainly included single-chip modules (STM32F103ZET6, STMicroelectronics, Geneva, Switzerland), stepper motors (YH42BYGH47-401A, Microstep, Bratislava, Slovakia) and their drivers, bridge circuit modules, and accessory circuits. Figure 13b shows the detailed configuration of the FC used for test. A photoelectric encoder (GTS06-OC-RA1000B-2M, pulse: 1000; Kasei Electronics Ltd., Tokyo, Japan) was used to acquire the speeds of in-wheel motors. The off-center arm steering angles were measured using a multi-turn potentiometer (22HP-10, 0-5 kΩ; Sakae Company, Tokyo, Japan). Acceleration sensors (WT61C232, Wit Technology Company, Dublin, Ireland) were used to detect the longitudinal and lateral FC acceleration rates. The duration of MMS was calculated by the clock integrated into the data acquisition equipment, which included a data acquisition card (USB7648B, Beijing Zhongtai Research Ltd., Beijing, China) and an industrial personal computer (610H, Advantech Technology Corporation, Beijing, China).
The FC MMS tests were performed as follows. First, we confirmed that all parts of the FC worked correctly. This included the mechanical connections, control lines, power supply lines, data acquisition system, etc. Second, we activated the data acquisition system and started CM mode switching under distribution control. After MMS was complete, we stopped saving data and restored the FC to its original state. Similar tests were performed using the coupling control method proposed in this study. Next, we used the method described above to conduct IR, DM, and SM MMS tests. Finally, we deactivated the controller, electric wheel power supply, and overall power supply in turn, and put the FC in standby mode. The tests were conducted on hard pavement on the north campus of Northwest A&F University. Images from various MMS tests are shown in Figure 14.
To further observe the effect of the proposed controller, we also tested the trajectory of the FC centroid. In this test, a soft marker pen fixed on the frame was used to obtain the trajectory of the FC (Figure 15a). The test was conducted when the FC moved with fixed posture after MMS was completed at the initial position ( Figure 15b). The soft marker pen has a slight contact with the ground and the driving resistance of the FC was not affected by this pen. We created a coordinate system on the ground (Figure 15b). Through measuring the trajectory coordinate, the path behavior of the FC can be obtained.

Analysis of Results
The CE test results are presented in Figure 16. The CE between wheel 1 and wheel 2 is still analyzed as an example. All of the experimental CEs are greater than those that were simulated CEs. The maximum experimental CE absolute values are 0.012 rad· s −1 , 0.01 rad· s −1 , 0.017 rad· s −1 , and 0.015 rad· s −1 for CM, IR, DM, and CM mode switching, respectively. Although the experimental and simulated values are different, the CE change trends are consistent. In particular, large CE fluctuations occur as the MMS tests start. These gradually decay until they approach zero. Obviously, the proposed controller plays a role in MMS and the CEs are well controlled. These results prove that the proposed control method is effective.
The maximum and average acceleration rates are the main values used to evaluate longitudinal and lateral motion trends. Therefore, the results also focus on these two indices. Table 3 shows the longitudinal and lateral acceleration test results. As with the CEs, the experimental maximum absolute acceleration rates all exceed their simulated values, but are all in an acceptable range. The average absolute values are all quite small at approximately 9 to 12% of the maximum absolute value in most tests. This indicates that the changes in longitudinal and lateral FC directions are not notable. This shows that the proposed methodology can effectively guarantee FC stability during MMS. Figure 17 shows the centroid trajectory under distribution control and coupling control when the FC is moved in a fixed posture. Under the coupling control, the trajectory of the FC is more consistent with the expected trajectory. For the CM and DM process, the trajectory deviates early under distribution control. As the displacement increases, the trajectory deviation becomes more serious. In the coupling control, the trajectory does not shift slightly until the FC moves a large distance. In the IR process, the trajectory under distribution control is farther away from the desired center than coupling control. During the SM process, the trajectory deviates most from the expected path under distribution control. These results indicate that, under coupled control, the MMS effect is better, and the mode switching accuracy is higher than the distribution control. The FC MMS tests were performed as follows. First, we confirmed that all parts of the FC worked correctly. This included the mechanical connections, control lines, power supply lines, data acquisition system, etc. Second, we activated the data acquisition system and started CM mode switching under distribution control. After MMS was complete, we stopped saving data and restored the FC to its original state. Similar tests were performed using the coupling control method proposed switching under distribution control. After MMS was complete, we stopped saving data and restored the FC to its original state. Similar tests were performed using the coupling control method proposed in this study. Next, we used the method described above to conduct IR, DM, and SM MMS tests. Finally, we deactivated the controller, electric wheel power supply, and overall power supply in turn, and put the FC in standby mode. The tests were conducted on hard pavement on the north campus of Northwest A&F University. Images from various MMS tests are shown in Figure 14. To further observe the effect of the proposed controller, we also tested the trajectory of the FC centroid. In this test, a soft marker pen fixed on the frame was used to obtain the trajectory of the FC (Figure 15a). The test was conducted when the FC moved with fixed posture after MMS was completed at the initial position ( Figure 15b). The soft marker pen has a slight contact with the ground and the driving resistance of the FC was not affected by this pen. We created a coordinate system on the ground (Figure 15b). Through measuring the trajectory coordinate, the path behavior of the FC can be obtained.

Analysis of Results
The CE test results are presented in Figure 16. The CE between wheel 1 and wheel 2 is still analyzed as an example. All of the experimental CEs are greater than those that were simulated CEs. The maximum experimental CE absolute values are 0.012 rad·s −1 , 0.01 rad·s −1 , 0.017 rad·s −1 , and 0.015 rad·s −1 for CM, IR, DM, and CM mode switching, respectively. Although the experimental and simulated values are different, the CE change trends are consistent. In particular, large CE fluctuations occur as the MMS tests start. These gradually decay until they approach zero. Obviously, the proposed controller plays a role in MMS and the CEs are well controlled. These results prove that the proposed control method is effective.
The maximum and average acceleration rates are the main values used to evaluate longitudinal and lateral motion trends. Therefore, the results also focus on these two indices. Table 3 shows the longitudinal and lateral acceleration test results. As with the CEs, the experimental maximum absolute acceleration rates all exceed their simulated values, but are all in an acceptable range. The average absolute values are all quite small at approximately 9 to 12% of the maximum absolute value in most tests. This indicates that the changes in longitudinal and lateral FC directions are not notable. This shows that the proposed methodology can effectively guarantee FC stability during MMS.

Conclusions
This study presents a flexible chassis coupling control methodology designed to reduce coupling errors and improve motion mode switching stability and handling performance. A coupling error model was established for linkage control of an off-center steering mechanism based coupling control, and Lyapunov stability theory. A fuzzy PID controller was designed to compensate for the coupling error and a fuzzy PI method was employed to reduce off-center steering mechanism tracking errors as well. The proposed control methodology was examined by simulation and validated experimentally on hard pavement. Compared to the conventional distribution control method, the proposed approach could effectively reduce coupling errors and guarantee MMS control stability while substantially shortening response times. The coupling control method proposed in this study demonstrated better effectiveness and feasibility than distribution control. The results of the study show that the DM and SM MMS performance with regard to coupling error was not as good as the CM and IR performance. Thus, we should pay attention to the reasons for this performance difference. It is important to devise a strategy to improve FC MMS control.
Author Contributions: J.Q. and K.G. proposed the original ideas and wrote the manuscript. J.Q. developed the algorithm for differential steering coordinated control. J.Q., Y.L., and S.S. ran the simulations. J.Q. developed the test. K.G. and Z.Z. directed the overall project. All authors have read and agreed to the published version of the manuscript.

Conclusions
This study presents a flexible chassis coupling control methodology designed to reduce coupling errors and improve motion mode switching stability and handling performance. A coupling error model was established for linkage control of an off-center steering mechanism based coupling control, and Lyapunov stability theory. A fuzzy PID controller was designed to compensate for the coupling error and a fuzzy PI method was employed to reduce off-center steering mechanism tracking errors as well. The proposed control methodology was examined by simulation and validated experimentally on hard pavement. Compared to the conventional distribution control method, the proposed approach could effectively reduce coupling errors and guarantee MMS control stability while substantially shortening response times. The coupling control method proposed in this study demonstrated better effectiveness and feasibility than distribution control. The results of the study show that the DM and SM MMS performance with regard to coupling error was not as good as the CM and IR performance. Thus, we should pay attention to the reasons for this performance difference. It is important to devise a strategy to improve FC MMS control.
Author Contributions: J.Q. and K.G. proposed the original ideas and wrote the manuscript. J.Q. developed the algorithm for differential steering coordinated control. J.Q., Y.L., and S.S. ran the simulations. J.Q. developed the test. K.G. and Z.Z. directed the overall project. All authors have read and agreed to the published version of the manuscript.
Funding: This research was supported by the General Program of the National Natural Science Foundation of China (51375401).

Conflicts of Interest:
The authors declare no conflict of interest.