Motion Control of Four-Wheel Independently Actuated Electric Ground Vehicles considering Tire Force Saturations

Abstract

A vehicle stability control approach for four-wheel independently actuated (FWIA) electric vehicles is presented. The proposed control method consists of a higher-level controller and a lower-level controller. An adaptive control-based higher-level controller is designed to yield the vehicle virtual control efforts to track the desired vehicle motions due to the possible modeling inaccuracies and parametric uncertainties. The lower-level controller considering tire force saturation is given to allocate the required control efforts to the four in-wheel motors for providing the desired tire forces. An analytic method is given to distribute the high-level control efforts, without using the numerical-optimization-based control allocation algorithms. Simulations based on a high-fidelity, CarSim, and full-vehicle model show the effectiveness of the control approach.

1. Introduction

Electric vehicles typically achieve greater fuel economy, lower emissions, and increased energy security than conventional internal combustion engine vehicles [1]. Four-wheel independently actuated (FWIA) electric vehicles employ four in-wheel (or hub) motors to actuate the four wheels, and thus the torque of each wheel can be controlled independently. The actuation flexibility of the FWIA electric vehicles together with the fast and precise torque responses of electric motors enhances the vehicle control strategies, such as the traction control system and direct yaw-moment control [2–4].

The actuators in a FWIA electric vehicle are more than those in a conventional vehicle. This actuator redundancy makes the FWIA electric vehicle control problem more challenging but rewarding. This paper considers the motion control problems of FWIA electric vehicles. Both the vehicle longitudinal speed and yaw motion are controlled. Many studies have been carried out on the vehicle control methods for improving the vehicle stability and maneuverability. Most of them nevertheless are designed for the conventional vehicle architectures [5–7], not for the FWIA electric vehicles. Sakai et al. [8] proposed a direct yaw-moment control system for a FWIA electric vehicle, a half-vehicle model which is a linear approximation of vehicle dynamics was used in the controller design, and only the vehicle lateral motion was controlled. A braking control method for electric vehicle was proposed in [9], and the studied vehicle was driven by independent front and rear motors. The vehicle stability problem was not considered either in the paper. A stability control method for four-wheel driven hybrid electric vehicle was proposed in [10]. The studied vehicle in this paper was driven by a front and a rear motor, and the rear motor with an electrohydraulic brake was used to generate the required torque split for yaw motion control. As a FWIA electric vehicle is equipped with four in-wheel motors to independently actuate the four wheels, the control problem in [10] is thus different from the one considered in this study.

In this paper, the tracking control problem [11–14] of a FWIA electric vehicle is studied. The proposed control system consists of a higher-level controller and a lower-level controller. Due to the possible modeling inaccuracies and vehicle parametric uncertainties, an adaptive controller is designed as the higher-level control to give the required virtual total ground forces and the force split between the left and right sides of the vehicle. The vehicle longitudinal speed is controlled by the total ground forces while the yaw motion is compensated with the external yaw moment generated with the tire force difference between the two sides of the vehicle. The lower-level controller allocates the virtual ground forces from the higher-level controller to the four wheels. Control allocation algorithms are generally used to distribute the higher-level control signals to the lower-level actuators [15, 16]. As the control allocation algorithms usually require high computational costs, which may discourage their implementations in real time, an analytic solution of allocating the ground forces without using the numerical-optimization-based control allocation algorithms is given in this study. When the tire slip ratios become large and move into the unstable tire force region, the tire forces will be saturated and it may no longer be possible to fully transfer the desired tire force onto the road. What is more, locking/skidding wheels no longer provide any grip on the road and thus the cornering forces transferred to the ground will be limited. So the vehicle will be unsteerable if the tire slip ratios become too large. The constraints of the tire forces are also explicitly considered in the optimal solution.

The rest of the paper is organized as follows. System modeling is presented in Section 2. The proposed higher-level controllers are designed in Section 3. Tire force distribution design considering the tire force constraints is described in Section 4. Simulation results based on a high-fidelity, CarSim, full-vehicle model are provided in Section 5 followed by conclusive remarks.

2. System Modelling

Vehicle yaw control enhances the vehicle handling performance and maintains vehicle stability in cornering maneuvers [17]. When the vehicle yaw rate can be controlled to track the reference, the vehicle lateral speed and slip angle will be small [18]. A schematic diagram of a vehicle model is shown in Figure 1. If the vehicle longitudinal speed and yaw rate are controlled at the same time, the vehicle model can be expressed as 𝑣̇_𝑥=𝑣_𝑦𝐶𝑟−^𝑎𝑚𝑣²_𝑥+1𝑚²_𝑖=1𝐹_𝑦𝑖(𝛿)+𝐹sin^𝐷𝑚,̇1𝑟=𝐼_𝑧²_𝑖=1𝐹_𝑦𝑖𝑙_𝑓cos(𝛿)+(−1)^𝑖𝑙_𝑠sin(𝛿)−⁴_𝑖=3𝑙_𝑟𝐹_𝑦𝑖+1𝐼_𝑧Δ𝑀_𝑧, (1) where 𝑣_𝑥 and 𝑣_𝑦 are the vehicle longitudinal speed and lateral speed, respectively, 𝑟 is the yaw rate, 𝑚 is the mass of the vehicle, 𝐼_𝑧 is the yaw inertia, and 𝐶_𝑎 is the aerodynamic drag term. 𝐹_𝐷 is the total driving/braking forces in vehicle longitudinal direction, Δ𝑀_𝑧 is the external yaw moment generated with the longitudinal tire force difference between the left and right side wheels, 𝐹_𝐷 and Δ𝑀_𝑧 can be written as 𝐹_𝐷=²_𝑖=1𝐹_𝑥𝑖cos(𝛿)+⁴₃₌₁𝐹_𝑥𝑖,Δ𝑀_𝑧=²_𝑖=1𝐹_𝑥𝑖(−1)^𝑖cos(𝛿)𝑙_𝑠+sin(𝛿)𝑙_𝑓+⁴_𝑖=3(−1)^𝑖𝑙_𝑠𝐹_𝑥𝑖. (2)

Figure 1 

Schematic diagram of a vehicle planar motion model.

The wheel slip angle is a function of the vehicle states and can be calculated as 𝛽_1,2=−𝛿+tan⁻¹𝑣_𝑦+𝑙_𝑓𝑟𝑣_𝑥∓𝑙_𝑠𝑟,𝛽_3,4=tan⁻¹𝑣_𝑦−𝑙_𝑟𝑟𝑣_𝑥∓𝑙_𝑠𝑟. (3) The tire lateral forces are functions of the tire slip angles which can be calculated with the vehicle states, which indicates that the tire lateral forces in 𝐹_𝑦𝑖 are also functions of the vehicle states. Denoting 𝑥=[𝑣_𝑥𝑣_𝑦𝑟]^𝑇, the vehicle model can be rewritten as 𝑥̇₁=𝑓₁(𝑥)+1𝑚𝑢₁,𝑥̇₂=𝑓₂(𝑥)+1𝐼_𝑧𝑢₂, (4) where 𝑢₁=𝐹_𝐷 and 𝑢₂=Δ𝑀_𝑧 are the control signals, 𝑓₁(𝑥)=𝑣_𝑦𝑟−(𝐶_𝑎/𝑚)𝑣²_𝑥+(1/𝑚)∑²_𝑖=1𝐹_𝑦𝑖sin(𝛿), and 𝑓₂(𝑥)=(1/𝐼_𝑧)(∑²_𝑖=1𝐹_𝑦𝑖(𝑙_𝑓cos(𝛿)+(−1)^𝑖𝑙_𝑠sin(𝛿))−∑⁴_𝑖=3𝑙_𝑟𝐹_𝑦𝑖).

The tire longitudinal slip ratio 𝑠_𝑖 is defined as the relative difference between the tire center speed and tire circumferential speed, and the tire slip ratio can be written as 𝑠_𝑖=𝜔_𝑖𝑅−𝑣_𝑥𝑖max𝑣_𝑥𝑖,𝜔_𝑖𝑅, (5) where 𝜔_𝑖 is the tire longitudinal rotational speed of the 𝑖th wheel, 𝑅 is the tire effective rolling radius, and 𝑣_𝑥𝑖 are the speeds at the centers of the wheels and are given as 𝑣_𝑥1,2=𝑣_𝑥∓𝑙_𝑠𝑟cos𝛿+𝑣_𝑦+𝑙_𝑓𝑟𝑣sin𝛿,_𝑥3,4=𝑣_𝑥∓𝑙_𝑠𝑟. (6) The vehicle states can be measured with the global positioning system (GPS) and inertia measurement unit (IMU) [19, 20]. The wheel speeds can be measured with wheel speed sensors. Thus, in this study, we assume all of the required signals to be known.

3. Higher-Level Controller Design

The proposed control system consists of a higher-level controller and a lower-level controller. The vehicle longitudinal speed is controlled by the total ground forces while the yaw rate is compensated with the external yaw moment. For the first channel of (4), the following controller 𝑢^∗₁=𝑚−𝑓₁(𝑥)+𝐾₁𝑒_𝑥+𝑣̇_𝑥𝑟, (7) with 𝑣_𝑥𝑟 being the reference vehicle longitudinal speed and 𝐾₁ being a positive constant, can make the longitudinal speed tracking error, 𝑒_𝑥=𝑣_𝑥𝑟−𝑣_𝑥, converge to 0. However, 𝑓₁(𝑥) and the vehicle mass 𝑚 may not be accurately obtained due to the modeling error and parameter uncertainties, and an adaptive controller is thus designed to yield the control signal 𝑢₁. The controller for the first channel is thus modified as 𝑢₁=𝑚−𝑓₁(𝑥)+𝐾₁𝑒_𝑥+𝑣̇_𝑥𝑟, (8) where 𝑓₁ and 𝑚 are the estimated values of 𝑓₁(𝑥) and 𝑚, respectively. If the following control law are used: ̇𝑓₁=−𝛾₁𝑒_𝑥,𝑚̇=𝑒_𝑥𝛾₂𝑢₁, (9) with 𝛾₁ and 𝛾₂ being positive constants, the longitudinal speed tracking error 𝑒_𝑥 can be bounded as 𝑒_𝑥=𝑓_1max−𝑓_1min𝑓̇_1max𝐾₁𝛾₁, (10) where |𝑓̇₁|_max is the upper boundary of |𝑓̇₁| and 𝑓_1max and 𝑓_1min are the upper and lower boundaries of 𝑓₁(𝑥), respectively. Proof can be found in Appendix A. Note that by choosing sufficiently large 𝛾₁ and 𝐾₁, the longitudinal speed tracking error 𝑒_𝑥 can be arbitrarily small.

The adaption law for 𝑓₁ and 𝑚 in (9) may cause the control signals grow out of the boundary. Thus the following control law modifications are introduced: 𝑢₁=𝑚−𝑓₁(𝑥)+𝐾₁𝑒_𝑥+𝑣̇_𝑥𝑟,̇𝑓₁=−𝛾₁𝑒_𝑥−𝜅₁𝑓₁−𝑓₁,𝑚̇=𝑒_𝑥𝛾₂𝑢₁−𝜅₂(𝑚−𝑚), (11) where 𝜅₁ and 𝜅₂ are positive constants, 𝑓₁=𝑓₁if𝑓_1min≤𝑓₁≤𝑓_1max𝑓_1minif𝑓₁<𝑓_1min𝑓_1maxif𝑓₁>𝑓_1max,𝑚=𝑚if𝑚_min≤𝑚≤𝑚_max𝑚_minif𝑚<𝑚_min𝑚_maxif𝑚>𝑚_max, (12) with 𝑚_max and 𝑚_min being the upper and lower bounds of 𝑚. The modified control law (11) for the first channel can still make the tracking error 𝑒_𝑥 be arbitrarily small. Proof can be found in Appendix B.

Similar to the controller design for the first channel, the control law for the second channel can be designed as 𝑢₂=𝐼_𝑧̂𝑙_𝑠−𝑓₂+𝐾₂𝑒₂+𝑟̇_ref,̇𝑓₂=−𝛾₃𝑒_𝑥−𝜅₃𝑓₂−𝑓₂,𝐼̇_𝑧=𝑒₂𝛾₄𝑙_𝑠𝑢₂−𝜅₄(𝑚−𝑚), (13) where 𝑒₂=𝑟_ref−𝑟 is the yaw rate tracking error and 𝛾₃, 𝛾₄, 𝜅₃, and 𝜅₄ are positive constants, 𝑓₂=𝑓₂if𝑓_1min≤𝑓₂≤𝑓_2max𝑓_2minif𝑓₂<𝑓_2min𝑓_2maxif𝑓₂>𝑓_2max,𝐼_𝑧̂=𝐼_𝑧if𝐼_𝑧min≤𝐼_𝑧≤𝐼_𝑧max𝐼_𝑧minif𝐼_𝑧<𝐼_𝑧min𝐼_𝑧maxif𝐼_𝑧>𝐼_𝑧max. (14) Here, 𝑓_2max and 𝑓_2min are the upper and lower bounds of 𝑓₂, respectively. 𝐼_𝑧max and 𝐼_𝑧min are the upper and lower bounds of 𝐼_𝑧, respectively.

4. Lower-Level Controller Design

When the higher-level controller signals are obtained, the lower-level controller operates the four in-wheel motors such that the control requirements from the higher-level controller can be satisfied. An analytic solution is given to distribute the higher-level control efforts without using the numerical-optimization-based control allocation algorithms.

The cost function for allocating the four tire forces can be defined as 1𝐽=2𝐹^𝑇_𝑥𝑊𝐹_𝑥+𝐵𝐹_𝑥−𝑢^𝑇𝑄𝐵𝐹_𝑥−𝑢, (15) where 𝑄=diag[𝑞₁𝑞₂],𝑊=diag[𝑤₁𝑤₂𝑤₃𝑤₄], 𝐹_𝑥=[𝐹_𝑥1𝐹_𝑥2𝐹_𝑥3𝐹_𝑥4]^𝑇, 𝑢=[𝑢₁,𝑢₂]^𝑇 is the desired control effort given by the higher-level controller, and 𝐵 is the control effectiveness matrix and can be written according to (2) (𝛿)(𝛿)(𝛿)𝑙𝐵=coscos11−cos_𝑠+sin(𝛿)𝑙_𝑓cos(𝛿)𝑙_𝑠+sin(𝛿)𝑙_𝑓−𝑙_𝑠𝑙_𝑠. (16) Based on (15), we have 𝜕𝐽𝜕𝐹_𝑥=𝐹^𝑇_𝑥𝑊+𝐹^𝑇_𝑥𝐵^𝑇𝑄𝐵−𝐵^𝑇𝑄𝑢=𝐹^𝑇_𝑥𝑊+𝐵^𝑇𝑄𝐵−𝐵^𝑇𝜕𝑄𝑢,𝐻=²𝐽𝜕²𝐹_𝑥=𝑊+𝐵^𝑇𝑄𝐵. (17)(18) As 𝑊>0 and 𝐵^𝑇𝑄𝐵≥0, one can claim 𝐻>0, which implies that the objective function 𝐽 has a global minimum with the minimizing 𝐹_𝑥0 given by 𝐹_𝑥0=𝑊+𝐵^𝑇𝑄𝐵⁻¹𝐵^𝑇𝑄𝑢. (19)

It is known that the tire force may become saturated if a sufficiently large motor control signal is applied in some extreme cases such as hard brake on a low-𝜇 road. Once the tire longitudinal force reaches its maximal value, further increasing of slip makes the tire work in the unstable range and the tire force will decrease quickly. So the constraints of the tire forces should be explicitly considered in the tire force allocation design. Note that a bigger weighting factor 𝑤_𝑖 in (15) for a wheel means that a smaller portion of the total torque is required from this wheel. So one can control a certain wheel to provide a larger or less portion of the total torque by selecting a smaller or bigger weighting factor for this wheel. The constraint violations of the tire forces can be discouraged by defining the weighting factors 𝑤_𝑖 in (15) as 𝑤_𝑖=𝑤₀1−𝑠^∗_𝑖^𝜅, (20) where 𝑤₀ is a positive constant, 𝜅>0 is a constant which is used to determine the shape of the weighting factor, and 𝑠^∗_𝑖=𝑠_𝑖/𝑠^peak_𝑖 is the normalized tire slip ratio with 𝑠^peak_𝑖 corresponding to the maximum longitudinal forces of the 𝑖th tire. The curves of the weighting factor at different 𝜅>0 are plotted in Figure 2. When the slip ratio of a certain wheel reaches the 𝑠^peak_𝑖 which corresponds to the maximum longitudinal tire force, the weighting factor for this wheel will become large according to (20), which means that the required tire force from this wheel will become small; in this way the tire slip ratio can always be limited within the stable region. Based on the desired tire force calculated from (19), motor control signals can be generated such that the desired tire forces can be provided [21].

Figure 2 

Weighting factor curves at different 𝜅 (𝑤0=1).

5. Simulation Results

Two simulation cases based on a high-fidelity, full-vehicle model constructed in CarSim were conducted. The vehicle parameters in the simulations were taken from an actual FWIA electric vehicle with in-wheel motors developed at the Ohio State University [21]. The desired vehicle yaw rate and speed can be generated from the drivers steering angle, accelerator/brake pedal positions. The vehicle reference model can be found in the literatures such as [22, 23].

5.1. J-Turn Simulation

In this simulation, the vehicle ran at a low-speed range. A counter-clockwise turn was introduced with the front wheel steering angle shown in Figure 3. We aim at controlling the vehicle such that the actual vehicle states can follow the references. The nominal vehicle mass was set to 800 kg in the simulation. As the designed controller does not depend on the actual vehicle parameters, we set the vehicle mass in the controller as 600 kg, which is different than the actual vehicle mass.

Figure 3 

Front wheel steering angle in the J-turn simulation.

The generated external yaw moment with the tire force difference between two sides of the vehicle is shown in Figure 4. The vehicle yaw rate and vehicle speed are plotted in Figures 5 and 6, respectively. One can see from these two figures that both vehicle yaw rate and speed could be well controlled. Note that, in this simulation, a big steering angle was applied to the vehicle, and this steering angle could make the vehicle yaw rate reach up to 30 deg/s, which indicates that the proposed control method can control the vehicle well at extreme conditions.

Figure 4 

The generated external yaw moment in the J-turn simulation.

Figure 5 

Vehicle yaw rates in the J-turn simulation.

Figure 6 

Vehicle speeds in the J-turn simulation.

5.2. Single-Lane Change

In the above simulation, we investigated the performance of the proposed controller in the cases where the vehicle runs on a high-𝜇 road. In this simulation, the vehicle was controlled to make a single-lane change on a low-𝜇 road. The tire-road friction coefficient was set as 0.2 and a big steering which would make the vehicle loss of stability was introduced at 2 s. The desired speed increased from 24.5 m/s to around 27.8 m/s in 6 seconds. The vehicle mass in the controller was set to 1000 kg, which is bigger than the actual vehicle mass.

The front wheel steering angle is shown in Figure 7. The generated yaw moment which regulated the vehicle yaw rate was plotted in Figure 8. The yaw rate control results are shown in Figure 9. To better show the effectiveness of the proposed control method, the yaw rate of an uncontrolled vehicle which ran on the same low-𝜇 road was compared. One can see from Figure 9 that the yaw rate of the controlled vehicle could always follow the reference, while the yaw rate of the uncontrolled vehicle deviated from the reference when the front wheel steering angle became large. The yaw rate control results indicate that stability of the controlled vehicle was ensured. The vehicle longitudinal speeds are plotted in Figure 10. One can see again that the vehicle speed could be well controlled as well.

Figure 7 

Front wheel steering angle in the single-lane change simulation.

Figure 8 

The generated external yaw moment in the single-lane change simulation.

Figure 9 

Vehicle yaw rates in the single-lane change simulation.

Figure 10 

Vehicle longitudinal speeds in the single-lane change simulation.

6. Conclusion

A vehicle stability control system for an FWIA electric vehicle is presented. The proposed adaptive control-based higher-level controller does not need the accurate vehicle parameters or tire force models but can still yield the desired control signals. An analytic solution considering tire force constraints is designed to allocate the required control efforts from the higher-lever controller to the four wheels. Simulations under various driving scenarios are carried out with a high-fidelity, CarSim, and full-vehicle model. Simulation results show the effectiveness of the proposed control approach.

Appendices

A.

By defining 𝑚=𝑚−𝑚 and 𝑓₁=𝑓₁−𝑓₁, the dynamics of the vehicle speed can be written as 𝑣̇_𝑥=𝑓₁+1𝑚𝑢₁=𝑓₁−𝑓₁+1𝑚(𝑚+𝑚)−𝑓₁+𝐾₁𝑒₁𝑣̇_𝑟𝑥=𝑓₁−𝑓₁+−𝑓₁+𝐾₁𝑒₁+𝑣̇_𝑟𝑥+𝑚𝑚𝑢₁=−𝑓₁+𝐾₁𝑒₁+𝑣̇_𝑟𝑥+𝑚𝑚𝑢₁, (A.1) which means that the error dynamics of the first channel can be written as 𝑒₁̇=𝑣̇_𝑟𝑥−𝑣̇_𝑥=−𝐾₁𝑒₁+𝑓₁−𝑚𝑚𝑢₁. (A.2) Define the Lyapunov function candidate for this channel as 𝑉₁=12𝑒²₁+𝑓²₁𝛾₁+𝑚²𝛾₂𝑚, (A.3) whose time derivative can be expressed as 𝑉̇₁=𝑒₁𝑒₁̇−𝑓₁̇𝑓₁−𝑓̇₁𝛾₁+̇𝑚𝑚̇𝛾₂𝑚=𝑒₁−𝐾₁𝑒₁+𝑓₁−𝑚𝑚𝑢₁−𝑓₁̇𝑓₁−𝑓̇₁𝛾₁+𝑚𝑚̇𝛾₂𝑚=−𝐾₁𝑒²₁+𝑒_𝑥𝑓₁+𝑓₁̇𝑓₁𝛾₁−𝑒_𝑥𝑚𝑢₁𝑚−𝑚𝑚̇𝛾₂𝑚−𝑓₁𝑓̇₁𝛾₁. (A.4) Assuming that the upper bound of 𝑓₁ is the same as that for 𝑓₁, the update law (11) can make the time derivative of 𝑉₁ to be written as 𝑉̇₁=−𝐾₁𝑒²₁−𝑓₁𝑓̇₁𝛾₁≤−𝐾₁𝑒²₁−𝑓₁−𝑓₁𝑓̇₁𝛾₁≤−𝐾₁𝑒²₁−𝑓_1max−𝑓_1min𝑓̇_1max𝛾₁, (A.5) where |𝑓̇₁|_max is the upper boundary of |𝑓̇₁| and 𝑓_1min and 𝑓_1max are the upper and lower boundaries of 𝑓₁, respectively. The error can be bounded according to (10). The proof is completed.

B.

Redefine the Lyapunov function candidate (A.3) as 𝑉₁=12𝑒²₁+𝑓₁−𝑓₁²−𝑓₁−𝑓₁²𝛾₁+(𝑚−𝑚)²−(𝑚−𝑚)²𝑚𝛾₂. (B.1) Based on (12), the time derivative of the above Lyapunov function candidate is 𝑉̇₁=𝑒₁−𝐾₁𝑒₁+𝑓₁−𝑚𝑚𝑢₁−𝑓₁−𝑓₁̇𝑓₁−𝑓̇₁−𝑓₁−𝑓₁̇𝑓₁−̇𝑓₁𝛾₁−()̇𝑚̇−(𝑚)̇̇𝑚−𝑚𝑚−𝑚−𝑚−𝑚𝛾₂𝑚=−𝐾₁𝑒²₁+𝛾₁𝑒₁𝑓₁−𝑓₁−𝑓₁𝑓̇₁−𝑓₁̇‌𝑓₁+𝑓₁𝑓̇₁+𝑓₁̇𝑓₁+𝑓₁̇‌𝑓₁−𝑓₁̇𝑓₁×𝛾₁⁻¹+𝛾₂𝑒₂(𝑚−𝑚)𝑢₁−𝑚𝑚̇−𝑚𝑚̇+𝑚𝑚̇−𝑚𝑚̇𝛾₂𝑚=−𝐾₁𝑒²₁+𝑓₁−𝑓₁𝑓̇₁𝛾₁+𝛾₁𝑒₁+̇𝑓₁𝑓₁−𝑓₁+𝑓₁−𝑓₁̇𝑓₁𝛾₁+𝑚̇−𝑒₁𝛾₂𝑢₁(𝑚−𝑚)+(𝑚−𝑚)𝑚̇𝛾₂𝑚. (B.2) If 𝑓_1min≤𝑓₁≤𝑓_1max, the following holds 𝑓₁=𝑓₁,̇𝑓₁=𝛾₁𝑒₁. (B.3) And if 𝑓₁<𝑓_1min or 𝑓₁>𝑓_1max, we have ̇𝑓₁𝛾=0,₁𝑒₁+̇𝑓₁𝑓₁−𝑓₁=−𝜅₁𝑓₁−𝑓₁𝑓₁−𝑓₁≤0, (B.4) which means the following always holds: 𝛾₁𝑒₁+̇𝑓₁𝑓₁−𝑓₁+𝑓₁−𝑓₁̇𝑓₁𝛾₁≤0. (B.5)

Similarly, one has 𝑚̇−𝑒₁𝛾₂𝑢₁(𝑚−𝑚)+(𝑚−𝑚)𝑚̇𝛾₂𝑚≤0, (B.6) so (B.2) can be rewritten as 𝑉̇₁≤−𝐾₁𝑒²₁+𝑓₁−𝑓₁𝑓̇₁𝛾₁. (B.7) Based on (20), one can see that if 𝑓₁ tends to move out of its boundary [𝑓_1min,𝑓_1max], the feedback term −𝜅₁(𝑓₁−𝑓₁) will pull 𝑓₁ back to close to 𝑓₁; thus 𝑓₁ is also bounded. Thus, the modified control law given by (11) and (12) can also make 𝑒₁ arbitrarily small if sufficiently large 𝛾₁ and 𝐾₁ are taken.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was supported by National Science Foundation of China (51105074, 51205058, and 51375086), Foundation of State Key Laboratory of Mechanical Transmission (SKLMT-KFKT-201206), and by the Polish-Norwegian Research Programme operated by the National Centre for Research and 24 Development under the Norwegian Financial Mechanism 2009–2014 in the frame of Project Contract no. Pol-Nor/200957/47/2013.