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The study on Differential Steering Control of In-wheel Motor Vehicle Based on Double Closed Loop System
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Applied Energy Symposium andSymposium Forum 2018: carbon cities and urbancities energy CUE2018-Applied Energy andLow Forum 2018: Low carbon andsystems, Applied Energy Symposium and Forum 2018: Low carbon cities and urban energy systems, CUE2018, 5–7 June 2018, Shanghai, China urban energy systems, 5–7 June Shanghai, CUE2018, 5–7 June 2018,2018, Shanghai, ChinaChina
The study SteeringonControl of In-wheel Theon 15thDifferential International Symposium District Heating and CoolingMotor The study on Differential Steering Control of In-wheel Motor Vehicle Based on Double Closed Loop System Vehicle on Double Closed Loop System Assessing theBased feasibility of using the heat demand-outdoor
Zhichao Li,Junqiu Li*, Sen Yang temperature function for a Li,Junqiu long-term Zhichao Li*, district Sen Yangheat demand forecast National Engineering Laboratory of Electric Vehicles, Beijing Institute of Technology, Beijing 100081, China Nationala,b,c Engineering Laboratory Beijing Institute a of Electric Vehicles, a b of Technology, Beijing c100081, China
I. Andrić
*, A. Pina , P. Ferrão , J. Fournier ., B. Lacarrière , O. Le Correc
Abstract a IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal Abstract b Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, France c This paper describes a novel differential steering control system- for in-wheel4motor electric vehicle fourFrance axles. Basically, Département Systèmes Énergétiques et Environnement IMTan Atlantique, rue Alfred Kastler, 44300with Nantes, This paper describes novel and differential steering control in-wheel motor with axles. Basically, the vehicle dynamic amodel the differential steeringsystem modelfor areanestablished basedelectric on thevehicle platform of four MATLAB/Simulink the vehicle dynamic model and the differential steering model are to established based on yaw the platform of aMATLAB/Simulink respectively. A two-degree-of-freedom vehicle model is established get the referential rate. Next, double closed loop respectively. A two-degree-of-freedom vehicle model is established to get the referential yaw rate. Next, a double loop control system based on the sliding mode structure control is proposed to improve the handing stability. The yaw rate isclosed controlled control system based on the sliding mode structure control is compensation proposed to improve the for handing stability.wheel The yaw rateisisdesigned controlled Abstract through a differential torque controller. Meanwhile, a torque controller non-steering motors to through differential torque controller. a torque compensation controller non-steering wheel motors is designed to solve theaproblem of steering power loss.Meanwhile, Furthermore, the simulation for validation is for performed. The results show that the control solve the problem ofthe steering power loss. Furthermore, thethe simulation forasThe validation is performed. The results the control District heating networks are commonly addressed in one of the most solutions forthat decreasing the system can regulate vehicle states effectively to enhance theliterature stability. simulation also effective indicates that theshow control strategy can system can regulate the vehicle states effectively to enhance stability. The high simulation also indicates thatreturned the control strategy greenhouse gasperformance emissions from the building sector. These the systems require investments which are through the can heat provide a good even in complex driving conditions. provide a good in complex driving sales. Due to performance the changedeven climate conditions andconditions. building renovation policies, heat demand in the future could decrease, prolonging investment return Copyright © the 2018 Elsevier Ltd. Allperiod. rights reserved. Copyright © © 2018 2018 Elsevier Elsevier Ltd. Ltd. All rights reserved. reserved. Copyright rights The mainand scope of this paper isAll toresponsibility assess the feasibility of using the heat demand function for heat demand Selection peer-review under responsibility ofofthethe scientific committee of Applied Energytemperature Symposium and Forum 2018: Low Selection and peer-review under scientific committee of the– outdoor CUE2018-Applied Energy Symposium and Selection and peer-review under responsibility the scientific committee of Applied Energy Symposium and isForum 2018:ofLow forecast. The district of cities Alvalade, located inofLisbon (Portugal), was used as a case study. The district consisted 665 carbon cities and urban energy systems, CUE2018. Forum 2018: Low carbon and urban energy systems. carbon citiesthat andvary urban systems, CUE2018. buildings in energy both construction period and typology. Three weather scenarios (low, medium, high) and three district Keywords: Differential In-wheel motor, Electricintermediate, vehicle, Doubledeep). closedTo loop, Sliding mode controlobtained heat demand values were renovation scenariossteering, were developed (shallow, estimate the error, Keywords: steering, motor, vehicle, Double closed loop, Slidingand mode control by the authors. comparedDifferential with results from In-wheel a dynamic heatElectric demand model, previously developed validated The results showed that when only weather change is considered, the margin of error could be acceptable for some applications 1. Introduction error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation 1.(the Introduction scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered). The multi-axis vehicle with in-wheel motors an important parttoin8% theper large cargothat transportation. The value of slopeelectric coefficient increased on average withinplays the range of 3.8% up decade, corresponds toThe the The multi-axis electric vehicle the with in-wheel motors reduces plays an energy important part the large cargo transportation. The application structure, loss andinmakes accurate dynamic response, decrease in of the motors number simplifies of heating hoursvehicle of 22-139h during the heating season (depending on the combination of weather and application of motorsconsidered). simplifies the vehicle structure, energy loss for and makes accurate dynamic response, which also scenarios improves the controlOn flexibility [1-2]. Thereduces differential steering technology works through differential renovation the other hand, function intercept increased 7.8-12.7% per decade (depending on the which also improves the control flexibility [1-2]. The differential steering technology works through differential vertical steering generated different motorthe torques. This technique firstly proposed in the coupledmoments scenarios).ofThe valuesaxles suggested could by be used to modify function parameters for was the scenarios considered, and vertical of of steering axles generated by different motor torques. This technique washave firstly proposed in the improve the accuracy heat demand estimations. study of moments sliding steering system of wheeled military vehicles [3]. Since then, many researches been conducted in
study of sliding system of military vehicles [3].control Since then, many researches been conducted in this field. Some steering studies involve thewheeled differential steering motion and torque distributionhave of the in-wheel motor this field. Some studies involve the differential steering motion control and torque distribution of the in-wheel motor © 2017 The Authors. Published by Elsevier Ltd. vehicle [4-6]. Different control approaches are adopted to improve the steering performance [7-11]. The differential Peer-review under responsibility ofapproaches thedynamic Scientific Committee of The 15th International on steering District and and vehicle [4-6]. Different are adopted to improve the steeringSymposium performance [7-11]. Heating The differential steering technology hascontrol excellent characteristics, which is effective to improve capability Cooling.technology has excellent dynamic characteristics, which is effective to improve steering capability and steering handling stability. Owing to the obvious strengths of in-wheel motor vehicles in economy, the relevant study will be handlingfor stability. Owing to the of in-wheel motor vehicles economy,steering the relevant study will be helpful its popularization andobvious energy strengths saving. Large numbers of studies in in differential control have been Keywords: Heat demand; Forecast; Climate change helpful for its popularization and energy saving. Large numbers of studies in differential steering control have been * Corresponding author. Tel.: +861-362-123-9752 . * E-mail Corresponding Tel.: +861-362-123-9752 . address:author. [email protected] E-mail address: [email protected] 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. 1876-6102 Copyright © 2018 Elsevier All rights reserved.of The 15th International Symposium on District Heating and Cooling. Peer-review under responsibility of theLtd. Scientific Committee Copyright © 2018 2018 Elsevier Ltd. All Allof rights 1876-6102 Copyright © Elsevier Ltd. rights reserved. committee of the Applied Energy Symposium and Forum 2018: Low carbon cities Selection and peer-review under responsibility the scientific Selection andpeer-review peer-review under responsibility the scientific committee of the Energy CUE2018-Applied Energy and Forum Selection responsibility of theofscientific committee of the Applied Symposium and ForumSymposium 2018: Low carbon cities and urbanand energy systems, under CUE2018. 2018: Lowenergy carbon cities and urban energy systems. and urban systems, CUE2018. 10.1016/j.egypro.2018.09.215
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conducted around traditional four-wheel vehicle platform, while the researches on multi-axis vehicles need to be further deepened and improved. In this paper, a novel four-axle distributed driven vehicle is studied, which has two steering axles and eight independent in-wheel motors. A vehicle dynamic model using differential steering method is established and the differential torque calculation method is determined on MATLAB/Simulink platform. SMC strategy is developed to improve the steering stability. Meanwhile, the vehicle speed tracking controller is designed to solve the problem of power loss. Finally, the effectiveness of the above control strategy is verified by simulation. 2. Modeling of vehicle dynamic In order to reflect the vehicle actual working condition, vehicle model is set up to obtain the vehicle state parameters, a differential steering model is set up to obtain the differential torque and a 2 DoF model is set up to obtain the control reference. The model is shown in Fig. 1.
(a) 13 DoF model of electric vehicle.
(b) Differential steering model.
(c) Linear 2 DoF vehicle model with 4 axles.
Fig. 1. Vehicle system model
In this paper, suffixes 𝑖𝑖 = 1,2,3,4 represent corresponding numbers of axles, suffixes 𝑗𝑗 = 1,2 represent the left and right wheel. 𝐿𝐿𝑖𝑖 is the distance between the axle and the vehicle center of gravity (C.G.). 𝐵𝐵 is the wheel track. 𝑣𝑣𝑥𝑥 、𝑣𝑣𝑦𝑦 and 𝑟𝑟 are the longitudinal speed, lateral speed, and yaw rate, respectively. 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 、𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 are the longitudinal and lateral force of the left or right wheels in the vehicle coordinate system. 𝐹𝐹𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 、𝐹𝐹𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 are the wheel longitudinal and lateral force in the tire coordinate system respectively. 𝛿𝛿𝑖𝑖𝑖𝑖 represents the steering angle of the corresponding wheel. 2.1. Vehicle Model
A nonlinear vehicle dynamics model including 13 degrees of freedom (DoF) is established, as is shown in Fig. 1(a). It has 3 degrees of freedom for longitudinal, lateral and yaw motion, 8 degrees of freedom for the rotation of 8 wheels and 2 degrees of freedom for steering system. The vehicle model is composed of the body dynamic model, wheel rotation model, wheel vertical force model, cornering and slip model as well as the magic tire model. 2.2. Differential steering Model Differential steering system is simplified and shown in Fig. 1(b). The first and second steering axles are connected with each other through the transmission mechanism. The differential steering torque is the vector superimposition of each wheel torque around the kingpin, which can be calculated as a a Tst ( Fx11 Fx 21 ) Fx12 Fx 22 a T11 T21 T12 T22 T , r r
where 𝐹𝐹𝑥𝑥11 、𝐹𝐹𝑥𝑥12 、𝐹𝐹𝑥𝑥21 、𝐹𝐹𝑥𝑥22 represent the longitudinal forces, and 𝑎𝑎 is the kingpin lateral offset. The angle
(1)
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3
acceleration is derived from the torque superimposition and described as 𝛼𝛼 = ∑ 𝑇𝑇 /𝐼𝐼𝑒𝑒𝑒𝑒 , where 𝐼𝐼𝑒𝑒𝑒𝑒 is equivalent moment of inertia. Therefore, the dynamic equation of the steering system can be expressed as 1
𝛿𝛿1̈ = [(𝐼𝐼
11 +𝐼𝐼12 )+𝑆𝑆𝑟𝑟 (𝐼𝐼21 +𝐼𝐼22 )]
(𝑆𝑆𝑔𝑔 𝐾𝐾𝑠𝑠𝑠𝑠 𝛿𝛿𝑠𝑠𝑠𝑠 + 𝑇𝑇𝑠𝑠𝑠𝑠 + 𝑀𝑀ℎ + 𝐸𝐸 − 𝐶𝐶𝑠𝑠𝑠𝑠 𝛿𝛿1̇ − 𝑆𝑆𝑔𝑔2 𝐾𝐾𝑠𝑠𝑠𝑠 𝛿𝛿1 ) − 𝑟𝑟̈ ,
(2)
where 𝐼𝐼11 、𝐼𝐼12 、𝐼𝐼21 、𝐼𝐼22 are the moments of inertia of the steering wheels around the kingpin, and 𝑆𝑆𝑔𝑔 is the angle ratio of the steering gear. 𝐾𝐾𝑠𝑠𝑠𝑠 is the angular stiffness around the kingpin, and 𝛿𝛿𝑠𝑠𝑠𝑠 is the steering wheel angle. 𝑀𝑀ℎ is the torque provided by the hydraulic system. E is the sum of the aligning torques around the kingpin. 𝐶𝐶𝑠𝑠𝑠𝑠 is the damping ratio of steering system and r is the vehicle yaw angle. 2.3. Application of linear 2 DoF reference model The linear 2 DoF vehicle model is shown in Fig. 1(c). The two degrees of freedom are the side slip angle and the yaw rate. The model can be expressed as
{
4
4
∑ 𝐶𝐶 ∑ (𝐶𝐶 𝐿𝐿 ) 𝛽𝛽̇ = − 1 𝑖𝑖 𝛽𝛽 − (1 + 1 𝑖𝑖2 𝑖𝑖 ) 𝑟𝑟 + ( 𝑚𝑚𝑣𝑣𝑥𝑥
𝑟𝑟̇ = −
∑4 1(𝐶𝐶𝑖𝑖 𝐿𝐿𝑖𝑖 ) 𝐽𝐽𝑧𝑧
𝛽𝛽 −
𝑚𝑚𝑣𝑣𝑥𝑥
2 ∑4 1(𝐶𝐶𝑖𝑖 𝐿𝐿𝑖𝑖 )
𝐽𝐽𝑧𝑧 𝑣𝑣𝑥𝑥
𝐶𝐶1 𝐿𝐿1
𝑟𝑟 + (
𝐽𝐽𝑧𝑧
𝐶𝐶1
𝑚𝑚𝑣𝑣𝑥𝑥
+
+
𝑁𝑁𝐶𝐶2
𝑚𝑚𝑣𝑣𝑥𝑥
𝑁𝑁𝐶𝐶2 𝐿𝐿2 𝐽𝐽𝑧𝑧
)𝛿𝛿
)𝛿𝛿
(3)
where 𝐶𝐶𝑖𝑖 represents the side slip angle stiffness of corresponding wheels, 𝑚𝑚 is the vehicle mass and 𝐽𝐽𝑧𝑧 is the moment 2𝐿𝐿 −𝐿𝐿 −𝐿𝐿 of inertia, 𝑁𝑁 represents the angle proportion and can be calculated as 𝑁𝑁 = 2 3 4. The desired relationship 2𝐿𝐿1 −𝐿𝐿3 −𝐿𝐿4
between the yaw rate and vehicle driving state is obtained from the model solution.
3. Double closed loop differential steering system based on the control of yaw rate and vehicle speed 3.1. Establishment of Double Closed Loop Control System The wheel output torque consists of the differential torque and the total demand torque. The differential torque is obtained according to the steering wheel angle and the vehicle speed. The total wheel demand torque is determined based on the accelerator pedal information. The final torques are calculated based on the overload limit control. The differential torque is evenly distributed both between wheels and between axles to ensure the steering smoothness and stability. The total demand torque is distributed between the steering axles according to the static axle load proportion for sufficient power and good adhesion. The final output torque is limited by the motor external characteristics.
Fig. 2. Double closed loop control system black diagram.
The differential steering improves the steering ability, which may also affect the stability due to the parameter changes. Therefore, the closed loop control is adopted to regulate the torque allocation and eliminate the potential bad influences, which will be reflected in two aspects. On the one hand, the differential torque is controlled based on the yaw rate feedback, on the other hand, the non-steering wheel torque is controlled based on the vehicle speed feedback.
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The block diagram of the double closed loop control system is shown in Fig. 2. Based on the open loop torque calculation system, two closed loop feedback controllers are added. The outer ring is the yaw rate feedback loop, and the differential torque compensation control is performed. The yaw rate control reference is the given steady state response 𝑟𝑟0 of the 2 DoF vehicle system. The inner ring is the vehicle speed feedback loop, and the non-steering wheel torque compensation control is performed. The vehicle speed reference is the given vehicle speed 𝑢𝑢0 , and the vehicle speed is kept to compensate the vehicle power performance during steering process. Compared with the traditional steering control method, the use of dynamic reference will reflect the real vehicle state and match the stability requirement reliably, which can also bring more flexibility for control. The addition of the speed control part assists improving the reference model accuracy further. 3.2. System model for controller design The multi-axis drive vehicle system with differential steering has obvious nonlinear characteristics, and the traditional linear control methods are not robust enough to solve the above problems. Based on the vehicle model analysis, the state equation of the 13 DoF control object is simplified as 1/𝑀𝑀 0 0 𝐹𝐹𝑋𝑋 𝑣𝑣̇𝑥𝑥 𝑟𝑟𝑣𝑣𝑦𝑦 − (𝐶𝐶_𝐷𝐷 𝐴𝐴𝐴𝐴_𝑥𝑥^2)/21.15𝑀𝑀 ] + [0 1/𝑀𝑀 0] [ 𝐹𝐹𝑌𝑌 ], [𝑣𝑣̇𝑦𝑦 ] = [ 𝑟𝑟𝑣𝑣𝑥𝑥 0 0 1/𝐼𝐼 𝑀𝑀𝑍𝑍 0 𝑟𝑟̇
(4)
which can be abbreviated into the following form:
𝒙𝒙̇ = 𝑓𝑓(𝒙𝒙, 𝑡𝑡) + 𝑔𝑔(𝒙𝒙, 𝑡𝑡)𝒖𝒖 { (𝒙𝒙 ∈ 𝑅𝑅𝑛𝑛 , 𝒖𝒖 ∈ 𝑅𝑅𝑚𝑚 , 𝑡𝑡 ∈ 𝑅𝑅) 𝒚𝒚 = 𝒙𝒙
(5)
where the system state variable is 𝒙𝒙 = [𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 ]𝑇𝑇 = [𝑣𝑣𝑥𝑥 𝑣𝑣𝑦𝑦 𝑟𝑟 ]𝑇𝑇 , corresponding to the vehicle longitudinal speed, lateral speed and yaw rate; 𝒖𝒖 = [𝐹𝐹𝑋𝑋 𝐹𝐹𝑌𝑌 𝑀𝑀𝑍𝑍 ]𝑇𝑇 , representing the control input of vehicle's total longitudinal force, total lateral force, and yaw moment respectively; 𝒚𝒚 = [𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 ]𝑇𝑇 , is the system output vector of vehicle driving states. The control input variables are decoupled from each other, the state equation can be simplified to three single-input single-output systems for analysis, which can be described as 𝒙𝒙̇ 𝑖𝑖 = 𝑓𝑓𝑖𝑖 (𝒙𝒙, 𝑡𝑡) + 𝑔𝑔𝑖𝑖 (𝒙𝒙, 𝑡𝑡)𝒖𝒖𝒊𝒊
(6)
The yaw rate 𝑟𝑟 and the vertical speed 𝑣𝑣𝑥𝑥 are selected as the controlled states. From the above equation, the corresponding control input variables are 𝐹𝐹𝑋𝑋 and 𝑀𝑀𝑍𝑍 . Therefore, any specific parameters related to the two inputs can be considered as the direct objects of the control. According to the composition analysis of the input vector, 𝐹𝐹𝑋𝑋 、𝐹𝐹𝑌𝑌 and 𝑀𝑀𝑧𝑧 can be expressed together by the tire forces and 𝛿𝛿𝑖𝑖𝑖𝑖 . The tire forces and aligning moments can be expressed by the magic tire formula as the function of the tire slip ratio 𝜆𝜆𝑖𝑖𝑖𝑖 and the side slip angle 𝛼𝛼𝑖𝑖𝑖𝑖 . 𝜆𝜆𝑖𝑖𝑖𝑖 and 𝛼𝛼𝑖𝑖𝑖𝑖 are directly related to wheel rotation angular velocity 𝑤𝑤𝑖𝑖𝑖𝑖 . According to wheel motion differential equation, there is a functional relationship between 𝑤𝑤𝑖𝑖𝑖𝑖 and wheel torque 𝑇𝑇𝑖𝑖𝑖𝑖 . The above deduction shows that the control input variables can be described by the relationship with 𝑇𝑇𝑖𝑖𝑖𝑖 as 𝒖𝒖 = 𝑢𝑢(𝐹𝐹𝑋𝑋 (𝑇𝑇𝑖𝑖𝑖𝑖 ), 𝐹𝐹𝑌𝑌 (𝑇𝑇𝑖𝑖𝑖𝑖 ), 𝑀𝑀𝑍𝑍 (𝑇𝑇𝑖𝑖𝑖𝑖 ))
(7)
So the control input variables can be selected as 𝑇𝑇𝑖𝑖𝑖𝑖 , and the change of 𝒖𝒖 can be realized by the torques regulation.
3.3. The design of the SMC
Sliding Mode Structure Control (SMC) method has strong robustness among the common control methods, which is very suitable for the control of the nonlinear vehicle system.
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3.3.1. Determination of sliding surface ̃ Assuming that the actual value and its differential of the control variable are 𝑥𝑥1 and 𝑥𝑥2 , the system state variable 𝒙𝒙 should be the yaw rate and vehicle speed in this control system and can be written as 𝑟𝑟 𝑥𝑥1 ̃ = [𝑥𝑥 ]=[𝑣𝑣 ] 且𝒙𝒙 𝒙𝒙 ̃ ∈ 𝒙𝒙. 2
(8)
𝑥𝑥
For the vehicle system with strong nonlinearity, the linear switching function is developed for SMC design, and the switching function is shown as follows: s = c𝑒𝑒1 + 𝑒𝑒̇1 = c1 (𝑟𝑟 ∗ − 𝑟𝑟) + (𝑟𝑟̇ ∗ − 𝑟𝑟̇ ) , { 1 s2 = c𝑒𝑒2 + 𝑒𝑒̇2 = c2 (𝑣𝑣𝑥𝑥∗ − 𝑣𝑣𝑥𝑥 ) + (𝑣𝑣̇𝑥𝑥∗ − 𝑣𝑣̇𝑥𝑥 )
(9)
where c1 、c2 are constants that affect dynamic quality and satisfy the Hurwitz condition with c1 、c2 > 0. When the sliding mode is reached, the following equation is satisfied as 𝑠𝑠̇𝑖𝑖 = c𝑖𝑖 (𝑥𝑥𝑖𝑖∗ − 𝑥𝑥𝑖𝑖 ) + 𝑥𝑥̇ 𝑖𝑖∗ − 𝑥𝑥̇ 𝑖𝑖 =0
(10)
3.3.2. Selection of the sliding mode control law Since the exact equation of the vehicle system is not easy to determine, the proportional control law is used to obtain the control strategy, which is expressed as: 𝑢𝑢 = ∑𝑘𝑘𝑖𝑖=1 𝜓𝜓𝑖𝑖 𝑥𝑥𝑖𝑖
where
𝑘𝑘 < 𝑛𝑛,
𝛼𝛼 , 𝑥𝑥𝑖𝑖 𝑠𝑠 > 0 . 𝜓𝜓𝑖𝑖 = { 𝑖𝑖 𝛽𝛽𝑖𝑖 , 𝑥𝑥𝑖𝑖 𝑠𝑠 < 0
(11)
(12)
The differential torque and non-steering wheel torque are compensated and the control law can be described as {
𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐1 = (𝛼𝛼1 |𝑒𝑒1 | + 𝛽𝛽1 |𝑒𝑒̇1 |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠(𝑥𝑥)) = (𝛼𝛼1 |𝑟𝑟 ∗ − 𝑟𝑟| + 𝛽𝛽1 |𝑟𝑟̇ ∗ − 𝑟𝑟̇ |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠1 )
𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐2 = (𝛼𝛼2 |𝑒𝑒2 | + 𝛽𝛽2 |𝑒𝑒̇2 |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠(𝑥𝑥)) = (𝛼𝛼2 |𝑣𝑣𝑥𝑥∗ − 𝑣𝑣𝑥𝑥 | + 𝛽𝛽2 |𝑣𝑣̇𝑥𝑥∗ − 𝑣𝑣̇𝑥𝑥 |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠2 )
(13)
where 𝛼𝛼𝑖𝑖 and 𝛽𝛽𝑖𝑖 are constants greater than zero and can adjust the effect of the control law. The law should satisfy Lyapunov's second law to guarantee the existence, reachability and stability conditions of the sliding mode. 4. Simulation and analysis of double closed loop control Table 1. Vehicle basic parameters. Name
Symbol
Value
Vehicle rated mass
𝑀𝑀
20000kg
Vehicle moment of inertia Distance from axles to C.G. Wheel track Wheel rolling radius Frontal area of the vehicle Air drag coefficient Ratio of motor reducer
𝐼𝐼
(𝐿𝐿1 、𝐿𝐿2 、𝐿𝐿3 、𝐿𝐿4 )
B
R
A
𝐶𝐶𝐷𝐷 𝑖𝑖0
70748kg·m²
(2.23、0.81、 − 1.19、 − 2.61) m 2.6 m
0.59 m 4.8 m² 0.6
11.07
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In different types of steering driving conditions, the vehicle steering performance is simulated under different driving conditions. The simulation is based on the actual vehicle parameters, which are listed in Table 1. 4.1. Performance analysis in normal road At the medium speed and normal road, the accelerator pedal opening is kept 10% unchanged, and the initial speed is 39.24 km/h. The steering angle ramp input is used to express the sharp steering condition. At 0.5 seconds, the ramp angle signal with a slope of 1.67 rad/s is input, and after 1.5 seconds, it is kept unchanged at 2.5 rad. The simulation shows that the steering speed under closed-loop control has remained unchanged at 39.24 km/h, as shown in Fig. 3(a), which indicates that the speed can keep constant after torque compensation. Fig. 3(b) shows that the actual yaw rate always changes according to the reference value, which indicates yaw rate is effectively controlled. Fig. 3(c) shows that the rear wheel motor starts to provide torque under vehicle speed control, and the power output value increases with the steering process. The differential torque is compensated under the yaw rate control, as is shown in Fig. 3(d).
rref
yaw rate (rad/s)
38
rreal
0.3
rerror
0.2
37.5
0.1
200 steering wheel torque (Nm)
0.4
rear-wheel compensation torque (Nm)
150
38.5 Vehicle speed (Km/h)
T11 T12 T21 T22
250
0.5
39
100
50
150 100 50
0
37
36.5
300
200
0.6
39.5
0
0
0
0.5
1
1.5 t(s)
2
2.5
3
-0.1
0
0.5
(a) Vehicle speed
1
1.5 t(s)
2
2.5
(b) Yaw rate
-50
3
0
0.5
1
1.5 t(s)
2
2.5
-50
3
0
0.5
(c) Compensation torque of non-steering wheel
1
1.5 t(s)
2
2.5
3
(d) Steering wheel torque
Fig. 3. Steering effect under double closed loop control with differential steering.
4.2. Performance analysis in complex road At the low speed and complex cross-country road, accelerometer pedal opening is kept 100% unchanged, and the initial speed is 26.35 km/h. The steering wheel triangulation waveform input is used to characterize vehicle lane change and hedging condition. At 0.5 seconds, a triangular angle signal is input with a slope of 3.14 rad/s, a period of 2 s and an amplitude of 1.57 rad. The simulation results show that the speed fluctuates slightly under the harsh steering conditions, as shown in Fig. 4(a). This is in accordance with the nonlinear control principle of SMC. The steering speed is constant with small fluctuation, and the speed loss is avoided. Fig. 4(b) shows that the yaw rate changes according to the law of the reference value, and the tracking error is almost 0. Meanwhile, Fig. 4(c) shows the compensation torque increases with steering angles increase. The steering wheel torque of one side is kept at a high level, while the torque on the other side decreases first and then increases, as is shown in Fig. 4(d). 0.2
26.2
1000
200
900
150
800
25.8
0 -0.05
25.7
-0.1
25.6
-0.15 -0.2
0
0.5
1
1.5 t(s)
2
(a) Vehicle speed
2.5
3
steering wheel torque (Nm)
rear-wheel compensation torque (Nm)
rerror
0.05 yaw rate (rad/s)
Vehicle speed (Km/h)
rreal
0.1
26 25.9
25.5
250
T11 T12 T21 T22
rref 0.15
26.1
100 50 0
0.5
1
1.5 t(s)
(b) Yaw rate
2
2.5
3
-150
600 500 400
-50
300
-100
0
700
0
0.5
1
1.5 t(s)
2
2.5
3
200
0
0.5
1
1.5 t(s)
2
2.5
(c) Compensation torque of non-steering wheel (d) Steering wheel torque
Fig. 4. Differential steering effect in complex road.
3
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5. Summary A differential steering vehicle dynamic model is established with the double front axles steering structure. Aiming at the stability problem caused by differential steering, a double closed loop joint control strategy is designed for torque distribution of hub motors. For this purpose, a 2 DoF vehicle model is constructed to provide a reference for vehicle stability control. The reference yaw rate is used for state tracking, the sliding mode variable structure control method is adopted to perform double closed loop control of the yaw rate and speed. The yaw rate control improves the vehicle's steering stability. At the same time, the vehicle speed control compensates the wheel torque and solves the problem of speed reduction in the steering process. Finally, through the simulation of the combined control strategy in different road conditions, the effectiveness and feasibility of the differential steering control strategy are verified. The research provides a reasonable theoretical reference for the further application of differential steering technology in multi-axle in-wheel motor vehicles. References [1] R. Wang, Y. Chen, D. Feng, X. Huang, J. Wang, Development and performance characterization of an electric ground vehicle with independently actuated in-wheel motors, J. Power Sources 196 (8) (2011) 3962–3971. [2] D. Li, Y. Song, D. Huang, H. Chen, Model-independent adaptive fault-tolerant output tracking control of 4WS4WD road vehicles, IEEE Trans. Intell. Transp. Syst. 14 (1) (2013) 169–179. [3] Francis B Hoogterp and William R Meldrum Jr. Differential torque steering for future combat vehicles[C]. SAE Paper no. 1999-01-3740, 1999. [4] Sakai S I, Sado H, HOd Y. Dynamic Driving/Braking Force Distribution In Electric Vehicles With Independently Driven Four Wheels[J]．Electrical Engineering in Japan, 2002, 138(1): 79-89. [5] Iwazaki Akihiro and Kunii RikiYa. Element technologies of a direct electromagnetic clutch[J]. Transaction of Society of Automotive Engineers of Japan, 2005, 36(3):133-138. [6] Yuichi Ushiroda, Kaoru Sawase, Takami Miura et al. Integrated vehicle dynamics control for high performance all wheel drive vehicle[C]. In Proceedings of the 9th International Symposium on Advanced Vehicle Control (AVEC08), Kobe, Japan, Oct., 2008:863-868. [7] He P, HOd Y, Kamachi M, et a1. Future Motion Sontrol To Be Realized By In-wheel Motored Electric Vehicle[C]. Proceedings of Industrial Electronics Conference, Raleigh, NC, United states, 2005:2632-2637. [8] Geng C, Mostefai L, Denai M,et al. Direct Yaw-moment Control Of an In-wheel-motored Electric Vehicle Based On Body Slip Angle Fuzzy Bbserver[J]. IEEE Transactions on Industrial Electronics, 2009, 56(5):1411- 1419. [9] B C Besselink. Computer controlled steering system for vehicles having two independently driven wheels[J]. Computers and Electronics in Agriculture, 2003, 39:209-226. [10] Junnian Wang, Qingnian Wang, Chuanxue Song, et al. Co-simulation and Test of Differential Drive Assist Steering Control System for Four-wheel Electric Vehicle. Transactions of the chinese society for angricultural machinery[J]. Transactions of the Chinese Society for Angricultural Machinery, 2010, 41(6):7-13. [11] Xiaolong Liu. Research on Driving Control System of Multi-wheel Independent Drive Electric Vehicles[D]. Zhejiang University, 2013.
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Applied Energy Symposium andSymposium Forum 2018: carbon cities and urbancities energy CUE2018-Applied Energy andLow Forum 2018: Low carbon andsystems, Applied Energy Symposium and Forum 2018: Low carbon cities and urban energy systems, CUE2018, 5–7 June 2018, Shanghai, China urban energy systems, 5–7 June Shanghai, CUE2018, 5–7 June 2018,2018, Shanghai, ChinaChina
The study SteeringonControl of In-wheel Theon 15thDifferential International Symposium District Heating and CoolingMotor The study on Differential Steering Control of In-wheel Motor Vehicle Based on Double Closed Loop System Vehicle on Double Closed Loop System Assessing theBased feasibility of using the heat demand-outdoor
Zhichao Li,Junqiu Li*, Sen Yang temperature function for a Li,Junqiu long-term Zhichao Li*, district Sen Yangheat demand forecast National Engineering Laboratory of Electric Vehicles, Beijing Institute of Technology, Beijing 100081, China Nationala,b,c Engineering Laboratory Beijing Institute a of Electric Vehicles, a b of Technology, Beijing c100081, China
I. Andrić
*, A. Pina , P. Ferrão , J. Fournier ., B. Lacarrière , O. Le Correc
Abstract a IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal Abstract b Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, France c This paper describes a novel differential steering control system- for in-wheel4motor electric vehicle fourFrance axles. Basically, Département Systèmes Énergétiques et Environnement IMTan Atlantique, rue Alfred Kastler, 44300with Nantes, This paper describes novel and differential steering control in-wheel motor with axles. Basically, the vehicle dynamic amodel the differential steeringsystem modelfor areanestablished basedelectric on thevehicle platform of four MATLAB/Simulink the vehicle dynamic model and the differential steering model are to established based on yaw the platform of aMATLAB/Simulink respectively. A two-degree-of-freedom vehicle model is established get the referential rate. Next, double closed loop respectively. A two-degree-of-freedom vehicle model is established to get the referential yaw rate. Next, a double loop control system based on the sliding mode structure control is proposed to improve the handing stability. The yaw rate isclosed controlled control system based on the sliding mode structure control is compensation proposed to improve the for handing stability.wheel The yaw rateisisdesigned controlled Abstract through a differential torque controller. Meanwhile, a torque controller non-steering motors to through differential torque controller. a torque compensation controller non-steering wheel motors is designed to solve theaproblem of steering power loss.Meanwhile, Furthermore, the simulation for validation is for performed. The results show that the control solve the problem ofthe steering power loss. Furthermore, thethe simulation forasThe validation is performed. The results the control District heating networks are commonly addressed in one of the most solutions forthat decreasing the system can regulate vehicle states effectively to enhance theliterature stability. simulation also effective indicates that theshow control strategy can system can regulate the vehicle states effectively to enhance stability. The high simulation also indicates thatreturned the control strategy greenhouse gasperformance emissions from the building sector. These the systems require investments which are through the can heat provide a good even in complex driving conditions. provide a good in complex driving sales. Due to performance the changedeven climate conditions andconditions. building renovation policies, heat demand in the future could decrease, prolonging investment return Copyright © the 2018 Elsevier Ltd. Allperiod. rights reserved. Copyright © © 2018 2018 Elsevier Elsevier Ltd. Ltd. All rights reserved. reserved. Copyright rights The mainand scope of this paper isAll toresponsibility assess the feasibility of using the heat demand function for heat demand Selection peer-review under responsibility ofofthethe scientific committee of Applied Energytemperature Symposium and Forum 2018: Low Selection and peer-review under scientific committee of the– outdoor CUE2018-Applied Energy Symposium and Selection and peer-review under responsibility the scientific committee of Applied Energy Symposium and isForum 2018:ofLow forecast. The district of cities Alvalade, located inofLisbon (Portugal), was used as a case study. The district consisted 665 carbon cities and urban energy systems, CUE2018. Forum 2018: Low carbon and urban energy systems. carbon citiesthat andvary urban systems, CUE2018. buildings in energy both construction period and typology. Three weather scenarios (low, medium, high) and three district Keywords: Differential In-wheel motor, Electricintermediate, vehicle, Doubledeep). closedTo loop, Sliding mode controlobtained heat demand values were renovation scenariossteering, were developed (shallow, estimate the error, Keywords: steering, motor, vehicle, Double closed loop, Slidingand mode control by the authors. comparedDifferential with results from In-wheel a dynamic heatElectric demand model, previously developed validated The results showed that when only weather change is considered, the margin of error could be acceptable for some applications 1. Introduction error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation 1.(the Introduction scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered). The multi-axis vehicle with in-wheel motors an important parttoin8% theper large cargothat transportation. The value of slopeelectric coefficient increased on average withinplays the range of 3.8% up decade, corresponds toThe the The multi-axis electric vehicle the with in-wheel motors reduces plays an energy important part the large cargo transportation. The application structure, loss andinmakes accurate dynamic response, decrease in of the motors number simplifies of heating hoursvehicle of 22-139h during the heating season (depending on the combination of weather and application of motorsconsidered). simplifies the vehicle structure, energy loss for and makes accurate dynamic response, which also scenarios improves the controlOn flexibility [1-2]. Thereduces differential steering technology works through differential renovation the other hand, function intercept increased 7.8-12.7% per decade (depending on the which also improves the control flexibility [1-2]. The differential steering technology works through differential vertical steering generated different motorthe torques. This technique firstly proposed in the coupledmoments scenarios).ofThe valuesaxles suggested could by be used to modify function parameters for was the scenarios considered, and vertical of of steering axles generated by different motor torques. This technique washave firstly proposed in the improve the accuracy heat demand estimations. study of moments sliding steering system of wheeled military vehicles [3]. Since then, many researches been conducted in
study of sliding system of military vehicles [3].control Since then, many researches been conducted in this field. Some steering studies involve thewheeled differential steering motion and torque distributionhave of the in-wheel motor this field. Some studies involve the differential steering motion control and torque distribution of the in-wheel motor © 2017 The Authors. Published by Elsevier Ltd. vehicle [4-6]. Different control approaches are adopted to improve the steering performance [7-11]. The differential Peer-review under responsibility ofapproaches thedynamic Scientific Committee of The 15th International on steering District and and vehicle [4-6]. Different are adopted to improve the steeringSymposium performance [7-11]. Heating The differential steering technology hascontrol excellent characteristics, which is effective to improve capability Cooling.technology has excellent dynamic characteristics, which is effective to improve steering capability and steering handling stability. Owing to the obvious strengths of in-wheel motor vehicles in economy, the relevant study will be handlingfor stability. Owing to the of in-wheel motor vehicles economy,steering the relevant study will be helpful its popularization andobvious energy strengths saving. Large numbers of studies in in differential control have been Keywords: Heat demand; Forecast; Climate change helpful for its popularization and energy saving. Large numbers of studies in differential steering control have been * Corresponding author. Tel.: +861-362-123-9752 . * E-mail Corresponding Tel.: +861-362-123-9752 . address:author. [email protected] E-mail address: [email protected] 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. 1876-6102 Copyright © 2018 Elsevier All rights reserved.of The 15th International Symposium on District Heating and Cooling. Peer-review under responsibility of theLtd. Scientific Committee Copyright © 2018 2018 Elsevier Ltd. All Allof rights 1876-6102 Copyright © Elsevier Ltd. rights reserved. committee of the Applied Energy Symposium and Forum 2018: Low carbon cities Selection and peer-review under responsibility the scientific Selection andpeer-review peer-review under responsibility the scientific committee of the Energy CUE2018-Applied Energy and Forum Selection responsibility of theofscientific committee of the Applied Symposium and ForumSymposium 2018: Low carbon cities and urbanand energy systems, under CUE2018. 2018: Lowenergy carbon cities and urban energy systems. and urban systems, CUE2018. 10.1016/j.egypro.2018.09.215
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conducted around traditional four-wheel vehicle platform, while the researches on multi-axis vehicles need to be further deepened and improved. In this paper, a novel four-axle distributed driven vehicle is studied, which has two steering axles and eight independent in-wheel motors. A vehicle dynamic model using differential steering method is established and the differential torque calculation method is determined on MATLAB/Simulink platform. SMC strategy is developed to improve the steering stability. Meanwhile, the vehicle speed tracking controller is designed to solve the problem of power loss. Finally, the effectiveness of the above control strategy is verified by simulation. 2. Modeling of vehicle dynamic In order to reflect the vehicle actual working condition, vehicle model is set up to obtain the vehicle state parameters, a differential steering model is set up to obtain the differential torque and a 2 DoF model is set up to obtain the control reference. The model is shown in Fig. 1.
(a) 13 DoF model of electric vehicle.
(b) Differential steering model.
(c) Linear 2 DoF vehicle model with 4 axles.
Fig. 1. Vehicle system model
In this paper, suffixes 𝑖𝑖 = 1,2,3,4 represent corresponding numbers of axles, suffixes 𝑗𝑗 = 1,2 represent the left and right wheel. 𝐿𝐿𝑖𝑖 is the distance between the axle and the vehicle center of gravity (C.G.). 𝐵𝐵 is the wheel track. 𝑣𝑣𝑥𝑥 、𝑣𝑣𝑦𝑦 and 𝑟𝑟 are the longitudinal speed, lateral speed, and yaw rate, respectively. 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 、𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 are the longitudinal and lateral force of the left or right wheels in the vehicle coordinate system. 𝐹𝐹𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 、𝐹𝐹𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 are the wheel longitudinal and lateral force in the tire coordinate system respectively. 𝛿𝛿𝑖𝑖𝑖𝑖 represents the steering angle of the corresponding wheel. 2.1. Vehicle Model
A nonlinear vehicle dynamics model including 13 degrees of freedom (DoF) is established, as is shown in Fig. 1(a). It has 3 degrees of freedom for longitudinal, lateral and yaw motion, 8 degrees of freedom for the rotation of 8 wheels and 2 degrees of freedom for steering system. The vehicle model is composed of the body dynamic model, wheel rotation model, wheel vertical force model, cornering and slip model as well as the magic tire model. 2.2. Differential steering Model Differential steering system is simplified and shown in Fig. 1(b). The first and second steering axles are connected with each other through the transmission mechanism. The differential steering torque is the vector superimposition of each wheel torque around the kingpin, which can be calculated as a a Tst ( Fx11 Fx 21 ) Fx12 Fx 22 a T11 T21 T12 T22 T , r r
where 𝐹𝐹𝑥𝑥11 、𝐹𝐹𝑥𝑥12 、𝐹𝐹𝑥𝑥21 、𝐹𝐹𝑥𝑥22 represent the longitudinal forces, and 𝑎𝑎 is the kingpin lateral offset. The angle
(1)
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3
acceleration is derived from the torque superimposition and described as 𝛼𝛼 = ∑ 𝑇𝑇 /𝐼𝐼𝑒𝑒𝑒𝑒 , where 𝐼𝐼𝑒𝑒𝑒𝑒 is equivalent moment of inertia. Therefore, the dynamic equation of the steering system can be expressed as 1
𝛿𝛿1̈ = [(𝐼𝐼
11 +𝐼𝐼12 )+𝑆𝑆𝑟𝑟 (𝐼𝐼21 +𝐼𝐼22 )]
(𝑆𝑆𝑔𝑔 𝐾𝐾𝑠𝑠𝑠𝑠 𝛿𝛿𝑠𝑠𝑠𝑠 + 𝑇𝑇𝑠𝑠𝑠𝑠 + 𝑀𝑀ℎ + 𝐸𝐸 − 𝐶𝐶𝑠𝑠𝑠𝑠 𝛿𝛿1̇ − 𝑆𝑆𝑔𝑔2 𝐾𝐾𝑠𝑠𝑠𝑠 𝛿𝛿1 ) − 𝑟𝑟̈ ,
(2)
where 𝐼𝐼11 、𝐼𝐼12 、𝐼𝐼21 、𝐼𝐼22 are the moments of inertia of the steering wheels around the kingpin, and 𝑆𝑆𝑔𝑔 is the angle ratio of the steering gear. 𝐾𝐾𝑠𝑠𝑠𝑠 is the angular stiffness around the kingpin, and 𝛿𝛿𝑠𝑠𝑠𝑠 is the steering wheel angle. 𝑀𝑀ℎ is the torque provided by the hydraulic system. E is the sum of the aligning torques around the kingpin. 𝐶𝐶𝑠𝑠𝑠𝑠 is the damping ratio of steering system and r is the vehicle yaw angle. 2.3. Application of linear 2 DoF reference model The linear 2 DoF vehicle model is shown in Fig. 1(c). The two degrees of freedom are the side slip angle and the yaw rate. The model can be expressed as
{
4
4
∑ 𝐶𝐶 ∑ (𝐶𝐶 𝐿𝐿 ) 𝛽𝛽̇ = − 1 𝑖𝑖 𝛽𝛽 − (1 + 1 𝑖𝑖2 𝑖𝑖 ) 𝑟𝑟 + ( 𝑚𝑚𝑣𝑣𝑥𝑥
𝑟𝑟̇ = −
∑4 1(𝐶𝐶𝑖𝑖 𝐿𝐿𝑖𝑖 ) 𝐽𝐽𝑧𝑧
𝛽𝛽 −
𝑚𝑚𝑣𝑣𝑥𝑥
2 ∑4 1(𝐶𝐶𝑖𝑖 𝐿𝐿𝑖𝑖 )
𝐽𝐽𝑧𝑧 𝑣𝑣𝑥𝑥
𝐶𝐶1 𝐿𝐿1
𝑟𝑟 + (
𝐽𝐽𝑧𝑧
𝐶𝐶1
𝑚𝑚𝑣𝑣𝑥𝑥
+
+
𝑁𝑁𝐶𝐶2
𝑚𝑚𝑣𝑣𝑥𝑥
𝑁𝑁𝐶𝐶2 𝐿𝐿2 𝐽𝐽𝑧𝑧
)𝛿𝛿
)𝛿𝛿
(3)
where 𝐶𝐶𝑖𝑖 represents the side slip angle stiffness of corresponding wheels, 𝑚𝑚 is the vehicle mass and 𝐽𝐽𝑧𝑧 is the moment 2𝐿𝐿 −𝐿𝐿 −𝐿𝐿 of inertia, 𝑁𝑁 represents the angle proportion and can be calculated as 𝑁𝑁 = 2 3 4. The desired relationship 2𝐿𝐿1 −𝐿𝐿3 −𝐿𝐿4
between the yaw rate and vehicle driving state is obtained from the model solution.
3. Double closed loop differential steering system based on the control of yaw rate and vehicle speed 3.1. Establishment of Double Closed Loop Control System The wheel output torque consists of the differential torque and the total demand torque. The differential torque is obtained according to the steering wheel angle and the vehicle speed. The total wheel demand torque is determined based on the accelerator pedal information. The final torques are calculated based on the overload limit control. The differential torque is evenly distributed both between wheels and between axles to ensure the steering smoothness and stability. The total demand torque is distributed between the steering axles according to the static axle load proportion for sufficient power and good adhesion. The final output torque is limited by the motor external characteristics.
Fig. 2. Double closed loop control system black diagram.
The differential steering improves the steering ability, which may also affect the stability due to the parameter changes. Therefore, the closed loop control is adopted to regulate the torque allocation and eliminate the potential bad influences, which will be reflected in two aspects. On the one hand, the differential torque is controlled based on the yaw rate feedback, on the other hand, the non-steering wheel torque is controlled based on the vehicle speed feedback.
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The block diagram of the double closed loop control system is shown in Fig. 2. Based on the open loop torque calculation system, two closed loop feedback controllers are added. The outer ring is the yaw rate feedback loop, and the differential torque compensation control is performed. The yaw rate control reference is the given steady state response 𝑟𝑟0 of the 2 DoF vehicle system. The inner ring is the vehicle speed feedback loop, and the non-steering wheel torque compensation control is performed. The vehicle speed reference is the given vehicle speed 𝑢𝑢0 , and the vehicle speed is kept to compensate the vehicle power performance during steering process. Compared with the traditional steering control method, the use of dynamic reference will reflect the real vehicle state and match the stability requirement reliably, which can also bring more flexibility for control. The addition of the speed control part assists improving the reference model accuracy further. 3.2. System model for controller design The multi-axis drive vehicle system with differential steering has obvious nonlinear characteristics, and the traditional linear control methods are not robust enough to solve the above problems. Based on the vehicle model analysis, the state equation of the 13 DoF control object is simplified as 1/𝑀𝑀 0 0 𝐹𝐹𝑋𝑋 𝑣𝑣̇𝑥𝑥 𝑟𝑟𝑣𝑣𝑦𝑦 − (𝐶𝐶_𝐷𝐷 𝐴𝐴𝐴𝐴_𝑥𝑥^2)/21.15𝑀𝑀 ] + [0 1/𝑀𝑀 0] [ 𝐹𝐹𝑌𝑌 ], [𝑣𝑣̇𝑦𝑦 ] = [ 𝑟𝑟𝑣𝑣𝑥𝑥 0 0 1/𝐼𝐼 𝑀𝑀𝑍𝑍 0 𝑟𝑟̇
(4)
which can be abbreviated into the following form:
𝒙𝒙̇ = 𝑓𝑓(𝒙𝒙, 𝑡𝑡) + 𝑔𝑔(𝒙𝒙, 𝑡𝑡)𝒖𝒖 { (𝒙𝒙 ∈ 𝑅𝑅𝑛𝑛 , 𝒖𝒖 ∈ 𝑅𝑅𝑚𝑚 , 𝑡𝑡 ∈ 𝑅𝑅) 𝒚𝒚 = 𝒙𝒙
(5)
where the system state variable is 𝒙𝒙 = [𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 ]𝑇𝑇 = [𝑣𝑣𝑥𝑥 𝑣𝑣𝑦𝑦 𝑟𝑟 ]𝑇𝑇 , corresponding to the vehicle longitudinal speed, lateral speed and yaw rate; 𝒖𝒖 = [𝐹𝐹𝑋𝑋 𝐹𝐹𝑌𝑌 𝑀𝑀𝑍𝑍 ]𝑇𝑇 , representing the control input of vehicle's total longitudinal force, total lateral force, and yaw moment respectively; 𝒚𝒚 = [𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 ]𝑇𝑇 , is the system output vector of vehicle driving states. The control input variables are decoupled from each other, the state equation can be simplified to three single-input single-output systems for analysis, which can be described as 𝒙𝒙̇ 𝑖𝑖 = 𝑓𝑓𝑖𝑖 (𝒙𝒙, 𝑡𝑡) + 𝑔𝑔𝑖𝑖 (𝒙𝒙, 𝑡𝑡)𝒖𝒖𝒊𝒊
(6)
The yaw rate 𝑟𝑟 and the vertical speed 𝑣𝑣𝑥𝑥 are selected as the controlled states. From the above equation, the corresponding control input variables are 𝐹𝐹𝑋𝑋 and 𝑀𝑀𝑍𝑍 . Therefore, any specific parameters related to the two inputs can be considered as the direct objects of the control. According to the composition analysis of the input vector, 𝐹𝐹𝑋𝑋 、𝐹𝐹𝑌𝑌 and 𝑀𝑀𝑧𝑧 can be expressed together by the tire forces and 𝛿𝛿𝑖𝑖𝑖𝑖 . The tire forces and aligning moments can be expressed by the magic tire formula as the function of the tire slip ratio 𝜆𝜆𝑖𝑖𝑖𝑖 and the side slip angle 𝛼𝛼𝑖𝑖𝑖𝑖 . 𝜆𝜆𝑖𝑖𝑖𝑖 and 𝛼𝛼𝑖𝑖𝑖𝑖 are directly related to wheel rotation angular velocity 𝑤𝑤𝑖𝑖𝑖𝑖 . According to wheel motion differential equation, there is a functional relationship between 𝑤𝑤𝑖𝑖𝑖𝑖 and wheel torque 𝑇𝑇𝑖𝑖𝑖𝑖 . The above deduction shows that the control input variables can be described by the relationship with 𝑇𝑇𝑖𝑖𝑖𝑖 as 𝒖𝒖 = 𝑢𝑢(𝐹𝐹𝑋𝑋 (𝑇𝑇𝑖𝑖𝑖𝑖 ), 𝐹𝐹𝑌𝑌 (𝑇𝑇𝑖𝑖𝑖𝑖 ), 𝑀𝑀𝑍𝑍 (𝑇𝑇𝑖𝑖𝑖𝑖 ))
(7)
So the control input variables can be selected as 𝑇𝑇𝑖𝑖𝑖𝑖 , and the change of 𝒖𝒖 can be realized by the torques regulation.
3.3. The design of the SMC
Sliding Mode Structure Control (SMC) method has strong robustness among the common control methods, which is very suitable for the control of the nonlinear vehicle system.
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3.3.1. Determination of sliding surface ̃ Assuming that the actual value and its differential of the control variable are 𝑥𝑥1 and 𝑥𝑥2 , the system state variable 𝒙𝒙 should be the yaw rate and vehicle speed in this control system and can be written as 𝑟𝑟 𝑥𝑥1 ̃ = [𝑥𝑥 ]=[𝑣𝑣 ] 且𝒙𝒙 𝒙𝒙 ̃ ∈ 𝒙𝒙. 2
(8)
𝑥𝑥
For the vehicle system with strong nonlinearity, the linear switching function is developed for SMC design, and the switching function is shown as follows: s = c𝑒𝑒1 + 𝑒𝑒̇1 = c1 (𝑟𝑟 ∗ − 𝑟𝑟) + (𝑟𝑟̇ ∗ − 𝑟𝑟̇ ) , { 1 s2 = c𝑒𝑒2 + 𝑒𝑒̇2 = c2 (𝑣𝑣𝑥𝑥∗ − 𝑣𝑣𝑥𝑥 ) + (𝑣𝑣̇𝑥𝑥∗ − 𝑣𝑣̇𝑥𝑥 )
(9)
where c1 、c2 are constants that affect dynamic quality and satisfy the Hurwitz condition with c1 、c2 > 0. When the sliding mode is reached, the following equation is satisfied as 𝑠𝑠̇𝑖𝑖 = c𝑖𝑖 (𝑥𝑥𝑖𝑖∗ − 𝑥𝑥𝑖𝑖 ) + 𝑥𝑥̇ 𝑖𝑖∗ − 𝑥𝑥̇ 𝑖𝑖 =0
(10)
3.3.2. Selection of the sliding mode control law Since the exact equation of the vehicle system is not easy to determine, the proportional control law is used to obtain the control strategy, which is expressed as: 𝑢𝑢 = ∑𝑘𝑘𝑖𝑖=1 𝜓𝜓𝑖𝑖 𝑥𝑥𝑖𝑖
where
𝑘𝑘 < 𝑛𝑛,
𝛼𝛼 , 𝑥𝑥𝑖𝑖 𝑠𝑠 > 0 . 𝜓𝜓𝑖𝑖 = { 𝑖𝑖 𝛽𝛽𝑖𝑖 , 𝑥𝑥𝑖𝑖 𝑠𝑠 < 0
(11)
(12)
The differential torque and non-steering wheel torque are compensated and the control law can be described as {
𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐1 = (𝛼𝛼1 |𝑒𝑒1 | + 𝛽𝛽1 |𝑒𝑒̇1 |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠(𝑥𝑥)) = (𝛼𝛼1 |𝑟𝑟 ∗ − 𝑟𝑟| + 𝛽𝛽1 |𝑟𝑟̇ ∗ − 𝑟𝑟̇ |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠1 )
𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐2 = (𝛼𝛼2 |𝑒𝑒2 | + 𝛽𝛽2 |𝑒𝑒̇2 |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠(𝑥𝑥)) = (𝛼𝛼2 |𝑣𝑣𝑥𝑥∗ − 𝑣𝑣𝑥𝑥 | + 𝛽𝛽2 |𝑣𝑣̇𝑥𝑥∗ − 𝑣𝑣̇𝑥𝑥 |)𝑠𝑠𝑠𝑠𝑠𝑠(𝑠𝑠2 )
(13)
where 𝛼𝛼𝑖𝑖 and 𝛽𝛽𝑖𝑖 are constants greater than zero and can adjust the effect of the control law. The law should satisfy Lyapunov's second law to guarantee the existence, reachability and stability conditions of the sliding mode. 4. Simulation and analysis of double closed loop control Table 1. Vehicle basic parameters. Name
Symbol
Value
Vehicle rated mass
𝑀𝑀
20000kg
Vehicle moment of inertia Distance from axles to C.G. Wheel track Wheel rolling radius Frontal area of the vehicle Air drag coefficient Ratio of motor reducer
𝐼𝐼
(𝐿𝐿1 、𝐿𝐿2 、𝐿𝐿3 、𝐿𝐿4 )
B
R
A
𝐶𝐶𝐷𝐷 𝑖𝑖0
70748kg·m²
(2.23、0.81、 − 1.19、 − 2.61) m 2.6 m
0.59 m 4.8 m² 0.6
11.07
6
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In different types of steering driving conditions, the vehicle steering performance is simulated under different driving conditions. The simulation is based on the actual vehicle parameters, which are listed in Table 1. 4.1. Performance analysis in normal road At the medium speed and normal road, the accelerator pedal opening is kept 10% unchanged, and the initial speed is 39.24 km/h. The steering angle ramp input is used to express the sharp steering condition. At 0.5 seconds, the ramp angle signal with a slope of 1.67 rad/s is input, and after 1.5 seconds, it is kept unchanged at 2.5 rad. The simulation shows that the steering speed under closed-loop control has remained unchanged at 39.24 km/h, as shown in Fig. 3(a), which indicates that the speed can keep constant after torque compensation. Fig. 3(b) shows that the actual yaw rate always changes according to the reference value, which indicates yaw rate is effectively controlled. Fig. 3(c) shows that the rear wheel motor starts to provide torque under vehicle speed control, and the power output value increases with the steering process. The differential torque is compensated under the yaw rate control, as is shown in Fig. 3(d).
rref
yaw rate (rad/s)
38
rreal
0.3
rerror
0.2
37.5
0.1
200 steering wheel torque (Nm)
0.4
rear-wheel compensation torque (Nm)
150
38.5 Vehicle speed (Km/h)
T11 T12 T21 T22
250
0.5
39
100
50
150 100 50
0
37
36.5
300
200
0.6
39.5
0
0
0
0.5
1
1.5 t(s)
2
2.5
3
-0.1
0
0.5
(a) Vehicle speed
1
1.5 t(s)
2
2.5
(b) Yaw rate
-50
3
0
0.5
1
1.5 t(s)
2
2.5
-50
3
0
0.5
(c) Compensation torque of non-steering wheel
1
1.5 t(s)
2
2.5
3
(d) Steering wheel torque
Fig. 3. Steering effect under double closed loop control with differential steering.
4.2. Performance analysis in complex road At the low speed and complex cross-country road, accelerometer pedal opening is kept 100% unchanged, and the initial speed is 26.35 km/h. The steering wheel triangulation waveform input is used to characterize vehicle lane change and hedging condition. At 0.5 seconds, a triangular angle signal is input with a slope of 3.14 rad/s, a period of 2 s and an amplitude of 1.57 rad. The simulation results show that the speed fluctuates slightly under the harsh steering conditions, as shown in Fig. 4(a). This is in accordance with the nonlinear control principle of SMC. The steering speed is constant with small fluctuation, and the speed loss is avoided. Fig. 4(b) shows that the yaw rate changes according to the law of the reference value, and the tracking error is almost 0. Meanwhile, Fig. 4(c) shows the compensation torque increases with steering angles increase. The steering wheel torque of one side is kept at a high level, while the torque on the other side decreases first and then increases, as is shown in Fig. 4(d). 0.2
26.2
1000
200
900
150
800
25.8
0 -0.05
25.7
-0.1
25.6
-0.15 -0.2
0
0.5
1
1.5 t(s)
2
(a) Vehicle speed
2.5
3
steering wheel torque (Nm)
rear-wheel compensation torque (Nm)
rerror
0.05 yaw rate (rad/s)
Vehicle speed (Km/h)
rreal
0.1
26 25.9
25.5
250
T11 T12 T21 T22
rref 0.15
26.1
100 50 0
0.5
1
1.5 t(s)
(b) Yaw rate
2
2.5
3
-150
600 500 400
-50
300
-100
0
700
0
0.5
1
1.5 t(s)
2
2.5
3
200
0
0.5
1
1.5 t(s)
2
2.5
(c) Compensation torque of non-steering wheel (d) Steering wheel torque
Fig. 4. Differential steering effect in complex road.
3
592
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5. Summary A differential steering vehicle dynamic model is established with the double front axles steering structure. Aiming at the stability problem caused by differential steering, a double closed loop joint control strategy is designed for torque distribution of hub motors. For this purpose, a 2 DoF vehicle model is constructed to provide a reference for vehicle stability control. The reference yaw rate is used for state tracking, the sliding mode variable structure control method is adopted to perform double closed loop control of the yaw rate and speed. The yaw rate control improves the vehicle's steering stability. At the same time, the vehicle speed control compensates the wheel torque and solves the problem of speed reduction in the steering process. Finally, through the simulation of the combined control strategy in different road conditions, the effectiveness and feasibility of the differential steering control strategy are verified. The research provides a reasonable theoretical reference for the further application of differential steering technology in multi-axle in-wheel motor vehicles. References [1] R. Wang, Y. Chen, D. Feng, X. Huang, J. Wang, Development and performance characterization of an electric ground vehicle with independently actuated in-wheel motors, J. Power Sources 196 (8) (2011) 3962–3971. [2] D. Li, Y. Song, D. Huang, H. Chen, Model-independent adaptive fault-tolerant output tracking control of 4WS4WD road vehicles, IEEE Trans. Intell. Transp. Syst. 14 (1) (2013) 169–179. [3] Francis B Hoogterp and William R Meldrum Jr. Differential torque steering for future combat vehicles[C]. SAE Paper no. 1999-01-3740, 1999. [4] Sakai S I, Sado H, HOd Y. Dynamic Driving/Braking Force Distribution In Electric Vehicles With Independently Driven Four Wheels[J]．Electrical Engineering in Japan, 2002, 138(1): 79-89. [5] Iwazaki Akihiro and Kunii RikiYa. Element technologies of a direct electromagnetic clutch[J]. Transaction of Society of Automotive Engineers of Japan, 2005, 36(3):133-138. [6] Yuichi Ushiroda, Kaoru Sawase, Takami Miura et al. Integrated vehicle dynamics control for high performance all wheel drive vehicle[C]. In Proceedings of the 9th International Symposium on Advanced Vehicle Control (AVEC08), Kobe, Japan, Oct., 2008:863-868. [7] He P, HOd Y, Kamachi M, et a1. Future Motion Sontrol To Be Realized By In-wheel Motored Electric Vehicle[C]. Proceedings of Industrial Electronics Conference, Raleigh, NC, United states, 2005:2632-2637. [8] Geng C, Mostefai L, Denai M,et al. Direct Yaw-moment Control Of an In-wheel-motored Electric Vehicle Based On Body Slip Angle Fuzzy Bbserver[J]. IEEE Transactions on Industrial Electronics, 2009, 56(5):1411- 1419. [9] B C Besselink. Computer controlled steering system for vehicles having two independently driven wheels[J]. Computers and Electronics in Agriculture, 2003, 39:209-226. [10] Junnian Wang, Qingnian Wang, Chuanxue Song, et al. Co-simulation and Test of Differential Drive Assist Steering Control System for Four-wheel Electric Vehicle. Transactions of the chinese society for angricultural machinery[J]. Transactions of the Chinese Society for Angricultural Machinery, 2010, 41(6):7-13. [11] Xiaolong Liu. Research on Driving Control System of Multi-wheel Independent Drive Electric Vehicles[D]. Zhejiang University, 2013.