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Fault-Tolerant Control of FWIA Electric Ground Vehicles with Differential Drive Assisted Steering
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Fault-Tolerant Fault-Tolerant Control Control of of FWIA FWIA Electric Electric Fault-Tolerant Control of FWIA Electric Ground Vehicles with Differential Ground Vehicles with Differential Drive Drive Ground Vehicles with Differential Drive Assisted Steering Assisted Assisted Steering Steering
∗∗ Chuan Hu ∗∗∗ Hui Jing ∗∗ Fengjun Yan ∗∗ ∗∗ Rongrong Wang ∗∗ ∗∗ Fengjun Yan ∗ Chuan Hui Cong Jing Rongrong Wang ∗∗∗ ∗∗ Chuan Hu Hu ∗ Hui Jing ∗∗ Rongrong Wang ∗∗ Fengjun Yan ∗ ∗∗∗ Nan Chen ∗∗ Chuan Hu Hui Cong Jing ∗∗LiRongrong Wang Fengjun Yan ∗∗ Nan Chen Cong Li Li ∗∗∗ Nan Chen Cong Li ∗∗∗ Nan Chen ∗∗ ∗ Department of Mechanical Engineering, McMaster University, ∗ ∗ Department of Mechanical Engineering, McMaster University, of Canada, Mechanical Engineering, McMaster University, ∗ Department Hamilton, (e-mail: [email protected], and Department of Canada, Mechanical Engineering, McMaster University, Hamilton, (e-mail: [email protected], and Hamilton, Canada, (e-mail: [email protected], and [email protected]). Hamilton, Canada, (e-mail: [email protected], and [email protected]). ∗∗ [email protected]). School of Mechanical Engineering, Southeast University, Nanjing, ∗∗ [email protected]). ∗∗ School of Mechanical Engineering, University, School (e-mail: of Mechanical Engineering, Southeast Southeast University, Nanjing, Nanjing, ∗∗China, [email protected], [email protected], and School of Mechanical Engineering, Southeast University, Nanjing, China, (e-mail: [email protected], [email protected], China, (e-mail: [email protected], [email protected], and and [email protected]). China, (e-mail: [email protected], [email protected], and [email protected]). ∗∗∗ [email protected]). Department of Mechanical Engineering, Guilin University of ∗∗∗ [email protected]). ∗∗∗ Department of Mechanical Engineering, Guilin Department of Mechanical Engineering, Guilin University University of of ∗∗∗ Aerospace Technology, Guilin, China, ([email protected]). Department of Mechanical Engineering, Guilin University of Aerospace Aerospace Technology, Technology, Guilin, Guilin, China, China, ([email protected]). ([email protected]). Aerospace Technology, Guilin, China, ([email protected]). Abstract: This paper investigates a novel fault-tolerant control approach against active steering Abstract: This paper novel fault-tolerant control approach against steering Abstract: This paper investigates adrive novelassisted fault-tolerant control approach against active active steering system failure based oninvestigates differential a steering (DDAS) on four-wheel independently Abstract: This paper investigates adrive novelassisted fault-tolerant control approach against active steering system failure based on differential steering (DDAS) on four-wheel independently system failure basedelectric on differential drive assisted steering (DDAS) on four-wheel independently actuated (FWIA) ground vehicles. The front-wheel steering angle can be generated system failure based on differential drive assisted steering (DDAS) on four-wheel independently actuated (FWIA) electric electric ground vehicles. Thewhen front-wheel steering angle can be generated generated actuated (FWIA) ground vehicles. The front-wheel steering angle be with the front-wheel differential driving torque the steering motor fails.can DDAS being the actuated (FWIA) electric ground vehicles. Thewhen front-wheel steering angle can be generated with the front-wheel differential driving torque the steering motor fails. DDAS being with the front-wheel differential driving torque when the steering motor fails. DDAS being the the only steering power source is a potential alternative to replace the regular steering system in with the front-wheel differential driving torque when the steering motor fails. DDAS being the only power source is a alternative to the steering system in only steering steering power On source isissue, a potential potential alternative toa replace replace the regular regular steering system in defective situation. this this paper deploys hierarchical control method to utilize only steering power source is a potential alternative to replace the regular steering system in defective situation. On this issue, this paper deploys a hierarchical control method to utilize defective situation. On this issue, this paper deploys a hierarchical control methodthe to desired utilize DDAS for automatic steering. The higher-level controller is dedicated to generate defective situation. On this issue, this paper deploys a hierarchical control method to utilize DDAS automatic higher-level controller dedicated to desired DDAS for for automatic steering. The higher-level controller isLQR dedicated to generate generate the desired control input includingsteering. the yawThe moments based on a robustis approach, and the the lower-level DDAS for automatic steering. The higher-level controller isLQR dedicated to generate the desired control input including the yaw moments based on a robust approach, and the lower-level control input including the yaw moments based on a robust LQR approach, and the lower-level controller is responsible for yaw control allocation out the approach, respectiveand desired tire forces control input including the moments basedand on giving agiving robust LQR the lower-level controller is responsible for control allocation and out the respective desired tire forces controller is responsible control allocation andongiving out the respective desired forces and the desired steering for angle. Simulation based a high-fidelity and full-car modeltire indicates controller is responsible for control allocation andongiving out the respective desired tire forces and the desired steering angle. Simulation based a high-fidelity and full-car model indicates and the desired steering angle. Simulation basedforonFWIA a high-fidelity and full-car modeltoindicates that DDAS-based fault-tolerant control design electric vehicle is effective achieve and the desired steering angle. Simulation based on a high-fidelity and full-car model indicates that DDAS-based DDAS-based fault-tolerant control design for for FWIA FWIA electric vehiclefailure. is effective effective to to achieve achieve that control design vehicle is equal handling andfault-tolerant maneuverability performance case ofelectric the steering that DDAS-based fault-tolerant control design forinFWIA electric vehicle is effective to achieve equal equal handling handling and and maneuverability maneuverability performance performance in in case case of of the the steering steering failure. failure. equal and maneuverability in caseHosting of the by steering © 2015,handling IFAC (International Federation ofperformance Automatic Control) Elsevierfailure. Ltd. All rights reserved. Keywords: four-wheel independently actuated (FWIA), electric ground vehicles, differential Keywords: four-wheel independently actuatedcontrol, (FWIA), electric ground vehicles, vehicles, differential differential Keywords: four-wheel actuated (FWIA), electric ground drive assisted steering independently (DDAS), hierarchical optimal control. Keywords: four-wheel actuatedcontrol, (FWIA), electric ground vehicles, differential drive assisted steering independently (DDAS), hierarchical hierarchical optimal control. drive assisted steering (DDAS), control, optimal control. drive assisted steering (DDAS), hierarchical control, optimal control. considerably enhance active safety and vehicle handling 1. INTRODUCTION 1. considerably safety and considerably enhance active safety Additionally, and vehicle vehicle handling handling 1. INTRODUCTION INTRODUCTION during severeenhance driving active maneuvers. another 1. INTRODUCTION considerably enhance active safety Additionally, and vehicle handling during severe driving maneuvers. another during severe driving maneuvers. Additionally, another advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal potential during severe driving maneuvers. Additionally, another potential advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal potential advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal drive assisted steering (DDAS), utilized to provide certain combustion engine vehicle, has showed its advantages in potential advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal drive assisted steering (DDAS), utilized to provide certain combustion engine vehicle, has its advantages in drive assisted steering (DDAS), utilized tocontrol, providehas certain combustion engine vehicle, has showed showed itsenergy advantages in degree of steering power in vehicle motion also fuel economy, emissions reductions, and security, drive assisted steering (DDAS), utilized to provide certain combustion engine vehicle, has showed itsenergy advantages in degree of steering power in vehicle motion control, has also fuel economy, emissions reductions, and security, degree of steering power in vehicle motion control, has also fuel economy, emissions reductions, and energy security, been extensively researched. Jang et al. (2004), Nozaki. and thus become a worldwide focal point (Wang et al. degree of steering researched. power in vehicle motion control, has also fuel economy, emissions reductions, and energy security, been extensively Jang et al. (2004), Nozaki. and thus become a worldwide focal point (Wang et al. been extensively researched. Jang et al. (2004), Nozaki. and thusAmong becomeelectric a worldwide focal point (Wang et al. (2005) and Wu et al. (2008)Jang proposed similar Nozaki. idea to (2011)). vehicles,focal the four-wheel indepenbeen extensively researched. et al. (2004), and thus become a worldwide point (Wang et al. (2005) and Wu et al. (2008) proposed similar idea (2011)). Among electric the Wu etassist al. (2008) proposed similar idea to to (2011)).actuated Among (FWIA) electric vehicles, vehicles, the four-wheel four-wheel indepengenerateand steering force by differential driving of dently electric vehicle, mountedindepenwith in- (2005) (2005) and Wu et al. (2008) proposed similar idea to (2011)). Among electric vehicles, the four-wheel indepengenerate steering assist force by differential driving of dently actuated (FWIA) electric vehicle, mounted with ingenerate steering assist force by differential driving of dently actuated (FWIA) electric vehicle, mounted with intwo front wheels of electric vehicle, which has electrical wheel motors (or(FWIA) hub motors), isvehicle, an emerging and supegenerate steering assist force by differential driving of dently actuated electric mounted with intwo front wheels of electric vehicle, which has electrical wheel motors (or hub motors), is an emerging and supetwo front wheels of electric vehicle, which has electrical wheel motors (or hub motors), is an emerging and supe- steering system according to the steering geometry. Wang rior vehicle architecture, owing to its actuation flexibility two front wheels of electric vehicle, which has electrical wheel motors (or hub motors), is an emerging and supesteering system according to the the steering steering geometry. Wang rior to flexibility system according to geometry. Wang rior vehicle vehicle architecture, architecture, owing to its its actuation flexibility et al. (2011) proposed an indirect power steering measure combined the electricowing motors’ fastactuation and precise torque steering steering system according to the steering geometry. Wang rior vehiclewith architecture, owing to its actuation flexibility et al. (2011) proposed an indirect power steering measure combined with the electric motors’ fast and precise torque et al. (2011) proposed an indirect power steering measure combined with the electric motors’ fast and precise torque named as differential torquepower assisted steering, and response (2012),motors’ Chen etfast al. (2012), Shuaitorque et al. et al. (2011) proposed drive an indirect steering measure combined(Nam with et theal. electric and precise named as differential drive torque assisted steering, response (Nam et al. (2012), Chen et al. (2012), Shuai et al. named as its differential torque steering, assisted steering, and response Especially, (Nam et al. the (2012), Chen et al. (2012), Shuai et al. validated feasibilitydrive of assisting capabilityand of (2014)). actuator redundancies in FWIA named as differential drive torque assisted steering, and response (Nam et al. (2012), Chen et al. (2012), Shuai et al. validated its feasibility feasibility of effect assisting steering, capability of (2014)). Especially, the actuator in FWIA its of assisting steering, capability of (2014)). vehicles Especially, the actuator redundancies redundancies in FWIA validated road feel keeping and the of the torque distribution electric can considerably enhance the handling its feasibility of effect assisting steering, capability of (2014)). Especially, the actuator redundancies in FWIA validated road feel keeping and the of the torque distribution electric vehicles can considerably enhance the handling road feel keeping and the effect of the torque distribution electric vehicles can considerably enhance the handling system. However, none ofofthe thetorque abovedistribution literatures and maneuverability, and thus the enhance stability and safety in control road feel keeping and the effect electric vehicles can considerably the handling control system. However, none of the above literatures and and stability safety in system. of the above literatures and maneuverability, maneuverability, and thus thus the stability and safety in control the However, case that none the differential drive acts as defective driving conditions (Li the et al. (2013),and Djeziri et al. control system. However, none of the above literatures and maneuverability, and thus the stability and safety in investigated investigated the case that the differential drive acts defective driving conditions (Li et al. (2013), Djeziri et al. investigated the case that the differential drive acts as defective driving conditions (Li et al. (2013), Djeziri et al. the only power source of the steering, which can be as a (2013), Wang et al. (2014)). (Li et al. (2013), Djeziri et al. investigated the source case that the steering, differential drivecan acts as defective driving conditions the only power of the which be (2013), Wang et al. (2014)). the only power source of the steering, which can be a a (2013), Wang et al. (2014)). potential fault-tolerant control approach against active only power source of the steering, which canactive be a (2013), yaw Wang et al. (2014)). potential fault-tolerant control approach against Direct moment control (DYC) (Geng et al. (2009)) the potential fault-tolerant control approach against active steering system failure. Previous researchersagainst mostlyactive dealt Direct yaw moment (Geng fault-tolerant control approach Direct yaw moment control (DYC) (Geng et et al. al. (2009)) (2009)) steering system failure. researchers mostly dealt has been proved as ancontrol effective(DYC) and promising for potential steering system failure.ofPrevious Previous researchers mostly dealt Direct yaw moment control (DYC) (Geng etapproach al. (2009)) with the technology DDAS as an assisting steering has been proved as an effective and promising approach for steering system failure. Previous researchers mostly dealt has been proved ascontrol, an effective and promising approach for with the technology of DDAS as an assisting steering lateral dynamics with the advantage that it can with thefor technology ofdriver, DDAS asaddressed an assisting steering has been proved as an effective and promising approach for strategy easing the few DDAS being lateral the technology of DDAS as an assisting steering lateral dynamics dynamics control, control, with with the the advantage advantage that that it it can can with strategy for easing easing thepower driver,infew few addressed DDAS DDAS being for the driver, addressed being lateral dynamics control, with the advantage that it can strategy the exclusive steering faulted-steering situation. ⋆ strategy for easing thepower driver,infew addressed DDAS being This research is partly sponsored by the Foundation of Educathe exclusive steering faulted-steering situation. the exclusive steering power in faulted-steering situation. ⋆ ⋆ This This research is partly partly sponsored by the the(Grant Foundation of EducaEducaresearch is sponsored by Foundation of the exclusivedrive steering power in used faulted-steering situation. tion Office of Guangxi Province of China No. 2013YB272, ⋆ Differential is commonly in the lateral motion This research is partly sponsored by the(Grant Foundation ofthe Education Office of of No. 2013YB272, tion Office of Guangxi Guangxi Province of China China (Grant No. for 2013YB272, KY2015YB101) and theProvince Fundamental Research Funds CenDifferential drivevehicles, is commonly commonly used in the the lateral lateral motion Differential drive is used in motion control of FWIA with or without combination tion Office of Guangxi Province of China (Grant No. for 2013YB272, KY2015YB101) and Fundamental Research Funds the Differential drivevehicles, is commonly used in thethe lateral motion KY2015YB101) and the Fundamental Research Funds for the CenCentral Universities andthe Jiangsu Postgraduate Innovation Programm control of FWIA with or without the combination control of FWIA vehicles, with or without the combination KY2015YB101) and the Fundamental Research Funds for the Central and of activeof front-wheel steering (Shuai al. (2014), tral Universities Universities and Jiangsu Jiangsu Postgraduate Postgraduate Innovation Innovation Programm Programm (Grant No. KYLX-0102). control FWIA vehicles, with(AFS) or without theet combination of active front-wheel steering (AFS) (Shuai et al. (2014), tral Universities and Jiangsu Postgraduate Innovation Programm of active front-wheel steering (AFS) (Shuai et al. (2014), (Grant (Grant No. No. KYLX-0102). KYLX-0102). of active front-wheel steering (AFS) (Shuai et al. (2014), (Grant No. KYLX-0102).
Copyright © 2015, 2015 IFAC 1180Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 1180 Copyright ©under 2015 responsibility IFAC 1180Control. Peer review of International Federation of Automatic Copyright © 2015 IFAC 1180 10.1016/j.ifacol.2015.09.686
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Yang et al. (2009)). However, when the steering motor breaks down, the front wheels of vehicles with regular steering systems (e.g., steering motor, or hydraulic power steering) will be liberated from the manipulation of the driver, the vehicle will consequently be out of control and vulnerable. In that case, if the DDAS in FWIA electric vehicles can be used to regulate the front wheel steering angle with the actuation generated by the differential drive torque of the front axle through the steering system mechanism (Wang et al. (2011)), the front wheel can still steer even though there is no steering power from the steering motor. This principle can be utilized for automatic steering without external steering power in faulted-steering condition, but few researchers had noticed that benefit. To this issue, this paper studies the failure condition that the steering motor breaks down, and presents a novel fault-tolerant control approach using the DDAS on an FWIA electric vehicle, to realize normal driving desire but guarantee the equal handling and maneuverability when the active steering system is in complete failure. This paper deploys a hierarchical control method, where higher-level controller is dedicated to generate the desired control input (the yaw moments) based on a robust LQR approach, the lower-level controller is responsible for control allocation and giving out the respective desired tire forces and the desired steering angle. The rest of this paper is organized as following. The vehicle model and the DDAS model are described in Section 2. The optimal control law design based on DDAS is presented in Section 3. Simulation based on CarSim platform using a high-fidelity and full-vehicle model is conducted in Section 4. Followed is the conclusion in Section 5. 2. MODELING OF VEHICLE DYNAMICS 2.1 Vehicle Model The vehicle model is shown in Fig. 1, which has three planar degrees of freedom for longitudinal, lateral and yaw motion. The vehicle pitch, roll and vertical motions are neglected, and the four tires are assumed to be completely consistent in the dynamics properties. The vehicle has the mass m, the moment of inertia Iz through the CG (center of gravity) about the yaw axis. The front and rear wheel axles are located at distances lf and lr from CG, respectively. vx and vy represent the longitudinal and lateral velocities of the vehicle, respectively. β and r stand for the sideslip angle and the yaw rate, respectively. Fxi and Fyi represent the longitudinal and lateral force of the ith tire, respectively, where i = f l, f r, rl, rr = 1, 2, 3, 4 indicates the specific wheel. Here the steering wheels are the front wheels, and we assume the steering angles of the front wheels are the same and sufficiently small, represented by δf . ls is half of the wheel-base. Here we assume that the lateral velocity is sufficiently small and bounded, then the yaw dynamics equation for the 2 degree-of-freedom (DoF) model of vehicle in the yaw plane can be given as follows Mz r˙ = , (1) Iz
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Fig. 1. 2 DoF model of vehicle in the presence of sliding effects
where the generalized yaw moment Mz are given by Mz = M z + lr (−Fyrl − Fyrr ),
(2)
with M z = ls (−Fxf l cos δf + Fyf l sin δf − Fxrl ) +ls (Fxf r cos δf − Fyf r sin δf + Fxrr ) (3) +lf (Fxf l sin δf + Fyf l cos δf + Fxf r sin δf + Fyf r cos δf ) . Based on the assumption about small tire-sliding effects, tire lateral forces Fyi (i = rl, rr) can be given by Fyi = cri αr , (4) where cri represents the cornering stiffness of the rear wheel, and here we assume the cornering stiffnesses of the rear wheels are the same, that is crrl = crrr = cr . αr is the rear wheel slip angle, where we assume the equality of the slip angles of the rear wheels. αr can be given by lr r − vy αr = . (5) vx From (1)- (5), (1) can be transformed into r˙ = −
Mz 2lr2 cr r+ + d1 , Iz vx Iz
(6)
r cr where d1 = 2l Iz vx vy is regarded as a disturbance, and assumed small and bounded. The explanation for that practice is that vy is pretty small and hard to be measured.
Remark 1 : Note that in the above description of vehicle lateral dynamics, we only derive the lateral forces of the rear wheels in the linear region of tires operation as shown in (4). The explanation is that in this paper, we address the differentially steered vehicle, and the front wheel steering angle is not a control input anymore, but a vehicle state, which is determined by the front-axle external yaw moment. Then a tricky problem occurs that the desired steering angle is unknown. So here we choose to use hierarchical control to render the real front wheel to steer tracking the desired steering angle, which is fixed later. Generally, the control objective for vehicle lateral motion control is to make the sideslip angle and yaw rate track their respective desired values.
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2.2 Differential Drive Assisted Steering Model In this paper we deal with the fault-tolerant control of FWIA electric vehicles, the failure condition is set that the steering motor breaks down completely, i.e., the steering motor torque becomes zero, as a consequence, the steering system is out of the driver’s control. To this end, here we employ differential drive assisted steering (DDAS), shown in Fig. 2, to achieve safety tolerances in case of active steering fault. The differential steering is generated by the differential drive torque of the front axle, through the steering system mechanism. The steering angle thus becomes a vehicle state, which is utilized to make the real lateral tire force of the front wheels to track their desired values. The detailed design method will be presented in the controller design procedure.
αf ≤ αst αf > αst
(10)
where αsl = arctan (1/θy ) is the sliding slip angle, θy = 2cp l2 / (3µFz ) , σy = tan αf , l is the half of the tire contact length, µ is the tire-road adhesion coefficient, Fz is the normal force. cp is the stiffness coefficient of tire tread in unit length, and cp = cf /(2l), where cf is the cornering stiffness of the front tires, and we assume the cornering stiffnesses of the front tires are the same. αf is the slip angle of the front wheel, where we assume the slip angles of the front wheels are the same. Assuming the tire slip angle is small, from (10), we have τa ≈ κ1 cf αf , (11) where κ1 = l2 /3 is a constant, and αf is given by lf r + vy . (12) αf = δf − vx τds can be given by τds = (Ftf r − Ftf l ) rσ , (13) where rσ stands for the scrub radius as shown in Fig. 2. The steering force Ftf j is given by Ftf j = Fxf j / cos δf , (j = 1, 2) , (14) which derives τds = κ2 ∆Mzf . (15) rσ where κ2 is given by κ2 = ls cos δf . Since δf is assumed small enough, so we have κ2 ≈ rσ /ls .
µFz lθy σy [1 − 3 |θy σy | + � 2 3 τa = 3(θy σy ) − |θy σy | , 0,
Fig. 2. Differential drive assisted steering model When the regular steering system (e.g., steering-by-wire (SBW), active front-wheel steering (AFS)) of the vehicle works normally, the electronic power steering system is a simple inertia-damping system (Ahn et al. modeled as (2013)) Jeff δ¨f + beff δ˙f = τa + τm − τf , (7) where Jeff and beff are the effective moment of inertia of the steering system, respectively. and effective damping τa represents the tire self-aligning moment, τm is the steering motor torque, and τf is the friction torque. τm is related with the driver’s steering wheel torque τs through a steering wheel sensor, which measures the angle and the torque of the steering wheel. When the regular steering system of the vehicle fails totally, which means τm = 0, the differential steering system is activated, described by the following model Jeff δ¨f + beff δ˙f = τa + τds − τf , (8)
From (10)- (15), (8) can be transformed into κ 1 cf l f κ1 cf κ2 (16) δ˙f = − r+ δf + ∆Mzf + d2 , beff vx beff beff � � κ c 1 −τf − Jeff δ¨f , and can be where d2 = − beff1 vfx vy + beff assumed as a bounded disturbance. The explanation for that practice is that Jeff and δ¨f are sufficiently small, regular treatment of them will increase control complexity unworthily. Define the following state variables: x1 = r, x2 = δf , T and denote x = [x1 , x2 ] , the state space form of the dynamic model integrating vehicle lateral dynamics (6) and differential steering system (16) is given by x˙ = Ax + B (u + d) , (17) where
where τds is the differential steering moment generated by the front-axle external yaw moment ∆Mzf , which is generated with the longitudinal tire force difference between the two sides of the front wheels, and can be written as ∆Mzf = (Fxf r − Fxf l ) ls cos δf . (9) From the brush model, the self-aligning moment are given by 1182
2lr2 cr − 0 Iz vx A= , κ 1 cf l f κ 1 cf − beff vx beff B = diag (1/Iz , κ2 /beff ) , �T � u = M z , ∆Mzf ,
d = diag (d1 ,
d2 ) ,
(18)
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κ c where d1 = Iz d¯1 = 2lvrxcr vy and d2 = bκeff2 d¯2 = − κ21vfx vy + � � 1 ¨ are assumed to be bounded disturκ2 −τf − Jeff δ f bances.
3. OPTIMAL CONTROL BASED ON DIFFERENTIAL STEERING SYSTEM 3.1 Higher-Level Controller Design Here we deploy a robust and adaptive LQR controller (Liu et al. (2014)) to minimize the errors of the yaw rate r and the front wheel steering δf with their respective desired values, and simultaneously, restrict the control inputs M z and ∆Mzf in reasonable regions. The advantage of the proposed LQR method lies on that the effect of the matched disturbance can be eliminated by an add-on nonlinear controller. Denote xd = [rd , δf d ]T , where rd and δf d mean the respective desired values of the yaw rate and the steering angle. rd is generally related with the driver’s steering command δ0 , while δf d is proposed in the lower-level controller design procedure. Hence the states errors are given by ε = x − xd = [r − rd , δf − δf d ]T , (19) and the integrated performance index of the optimal control is defined as: �∞ � 1 � T J1 = ε Qε + uT Ru dt, (20) 2 0
where Q = diag [Qz , Qs ], R = diag [Rz , Rs ]. Here we utilize a theorem to design the robust LQR controller, one can refer to (Liu et al. (2014)) for the detailed proof of Theorem 1. Theorem 1 Assume d (x, t) is norm-bounded. The control input vector u (t) can be designed as: u (t) = ul (t) + unl (t) , (21) where the linear item ul (t) is (22) ul = −R−1 B T P (x − xd ) , in which the symmetric positive definite P is the solution of the Algebraic Riccati Equation P A + AT P + Q − P BR−1 B T P = 0. (23) And the nonlinear unl (t) item is BT P ε −Λ T , B T P ε �= 0 unl (t) = (24) �B P ε�2 0, BT P ε = 0 where the gain of nonlinear item Λ should satisfy Λ > W > �d (x, t)�2 . (25) If the controller is designed as shown in (21), system (17) will possess globally asymptotic stability. T
ε Remark 2 : Even if ε = 0, since �BBT PPε� is a relative 2 small value compared with ul , so the effect of the control switch to the vehicle stability is rather limited and can be neglected.
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3.2 Lower-Level Controller Design In this section, the required input is allocated to the four tyres. Since ∆Mzf is only related with the longitudinal forces of the front wheels, we shall preferentially allocate ∆Mzf , and then deal with M z . Make Fx1 = −Fx2 as the allocation rule, and follow the definition of ∆Mzf Fx1 (−ls cos δf ) + Fx2 (ls cos δf ) = ∆Mzf , (26) we have ∆Mzf ∆Mzf Fx1 = − , Fx2 = . (27) 2ls cos δf 2ls cos δf Note that there exist Fx1 and Fx2 items in the expression of M z as shown in (3), we need to eliminate them before the subsequent allocation. Denote M ze as the allocation object after the elimination as M ze = M z − ls (−Fx1 cos δf + Fx2 cos δf ) −lf (Fx1 sin δf + Fx2 sin δf ) ,
(28)
where at this moment M z comes from the higher-level controller, and Fx1 and Fx2 are obtained from (27). Then an analytic optimization control allocation algorithm (Wang et al. (2013)) is adopted to distribute the control efforts without using the numerical method. The cost function for allocating M ze to the four tire forces can be defined as � �T � � 1 � GF − M ze Q GF − M ze + J2 = 2 (29) � T F WF , where W = diag [w1 , w2 , w3 , w4 ], Q = q1 , F = T [Fx3 , Fx4 , Fy1 , Fy2 ] , where we assume Fyf = Fy1 = Fy2 . G is the control effectiveness matrix and can be written according to (3) G = [−ls , ls , (ls sin δf + lf cos δf ) , (30) (−ls sin δf + lf cos δf )] , Based on (29), we have ∂J2 = W F +GT Q[GF − M ze ] ∂F T =� W F +[GT QGF � − G QM ze ] T = W + G QG F − GT QM ze , and
(31)
∂ 2 J2
(32) = W + GT QG, ∂2F As W > 0 and GT QG ≥ 0, we can claim that H > 0, which indicates that the objective function J2 has a global minimum when F is chosen as the minimizing value F ∗ given by � �−1 F ∗ = W + GT QG GT QM ze , (33) H=
where we can choose W = I4×4 , Q = 1 in the following simulation. In order to realize the actuation of the desired lateral tire ∗ ∗ forces Fy1 and Fy2 of the front wheels, the following control law (Wang et al. (2015)) is used to generate the desired steering angle � ∗ � δf d = Kp Fyf − Fyf , (34)
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6
2
r (deg/s)
0
0
δ (deg)
1
−1 −2 0
Controlled Reference
4 2 0 −2 −4 1
2 3 Time (sec)
4
5
−6 0
Fig. 3. Driver’s steering angle command
1
2 3 Time (sec)
4
5
4
5
4
5
Fig. 4. Reference and real yaw rate
where Kp is a positive constant, δf d is the desired value of ∗ the steering angle. Fyf is the desired lateral tire force of the front wheel given by lower-level controller (33), where ∗ ∗ ∗ = Fy1 = Fy2 . Fyf is the real lateral it is assumed that Fyf tire force of the front wheel which can be measured by sensors or estimated with the vehicle model (Doumiati et al. (2011), Cho et al. (2010)). It can be known that in practice δf d is bounded, hence if Kp is chosen as an ∗ enough big value, we will eventually have Fyf → Fyf . It should be noted that δf d which is given by (34) needs to be substituted into (19) to obtain the states error. 4. SIMULATION
1
α (deg)
0.5
f
0
−0.5 −1 0
1
2 3 Time (sec)
Fig. 5. Front tire slip angle
5. CONCLUSION This paper addresses the fault-tolerant control issue against active steering system failure for FWIA electric
δf (deg)
0.5
0
−0.5 0
1
2 3 Time (sec)
Fig. 6. Real steering angle of the front wheel 4000 Yaw moment (Nm)
In this section, we present a single-lane change simulation case, where the vehicle is controlled to make a single-lane change on high adhesion (µ = 0.85) road with a high speed (vx = 30 m/s). The control objective is to make the vehicle yaw rate follow the desired value with only the DDAS. The desired yaw rate can be generated from the driver’s steering angle and vehicle longitudinal speed as (Du et al. (2010)) vx rd = δ0 , (35) L (1 + kus vx2 ) where L = lf + lr is the distance between the front and rear axles, kus is the stability factor, δ0 is the driver’s steering angle command which is shown in Fig. 3. The yaw control result is shown in Fig. 4, from which we can see the tracking objective of the real yaw rate towards the reference value is achieved. Fig. 5 presents the slip angle of the front wheel. Since the slip angles of the front wheels differ pretty small based on the previous assumption of the equality of them, so here we only show their average value. We can see that the variation tendencies of the front wheel slip angle and the yaw rate are roughly the same, which validates the effectiveness of the proposed control law. From Fig. 5 we can also infer that the vehicle is stable in the simulation. The real front wheel steering angle is presented in Fig. 6, the control inputs M z and ∆Mzf are shown in Fig. 7, we can see that they are all maintained in reasonable regions. Note that the signs of M z and ∆Mzf are with the same sign, which means the tracking control of the yaw rate and steering angle is complementary.
¯z M ∆Mzf
2000 0 −2000 −4000 0
1
2 3 Time (sec)
4
5
Fig. 7. The control inputs vehicles, and proposes a novel fault-tolerant control approach based on the DDAS mechanism. When the regular steering system breaks down, the differential steering can be generated by the front-wheel differential driving torque. To guarantee the equal handling and maneuverability as the normal status, this paper adopts a hierarchical con-
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trol method to utilize DDAS for automatic steering. The higher-level controller generates the desired control inputs, i.e., the yaw moments, based on a robust LQR strategy, while the lower-level controller arranges the control allocation and gives out the respective desired tire forces and the desired steering angle. CarSim-Simulink simulation of a sing-lane change case based on a high-fidelity and full-car model verifies the availability and the effectiveness of the proposed controller for fault-tolerant control on steering failure. REFERENCES R. Wang, Y. Chen, D. Feng, X. Huang and J. Wang. Development and Performance Characterization of an Electric Ground Vehicle with Independently Actuated In-Wheel Motors. Journal of Power Sources, volume 196, issue 8, pages 3962–3971, April 2011. K. Nam, H. Fujimoto, and Y. Hori. Lateral Stability Control of In-Wheel-Motor-Driven Electric Vehicles Based on Sideslip Angle Estimation Using Lateral Tire Force Sensors. IEEE Transactions on Vehicular Technology, volume 61, issue 5, pages 1972–1985, June 2012. Y. Chen and J. Wang. Fast and Global Optimal EnergyEfficient Control Allocation With Applications to OverActuated Electric Ground Vehicles. IEEE Transactions on Control Systems Technology, volume 20, issue 5, pages 1202-1211, September 2012. Z. Shuai, H. Zhang, J. Wang, J. Li, and M. Ouyang. Combined AFS and DYC Control of Four-WheelIndependent-Drive Electric Vehicles over CAN Network with Time-Varying Delays. IEEE Transactions on Vehicular Technology, volume 63, issue 2, pages 591–602, February 2014. D. Y. Li, Y. D. Song, D. Huang, and H. N.Chen. Modelindependent adaptive fault-tolerant output tracking control of 4WS4WD road vehicles. IEEE Transactions on Intelligent Transportation Systems, volume 14, issue 1, pages 169-179, March 2013. M. A. Djeziri, R. Merzouki, B. O. Bouamama, and M. Ouladsine. Fault Diagnosis and Fault-Tolerant Control of an Electric Vehicle Overactuated. IEEE Transactions on Vehicular Technology, volume 62, issue 3, pages 986– 994, January 2013. R. Wang and J. Wang. Actuator-Redundancy-Based Fault Diagnosis for Four-Wheel Independently Actuated Electric Vehicles. IEEE Transactions on Intelligent Transportation Systems, volume 15, issue 1, pages 239– 249, February 2014. C. Geng, L. Mostefai, M. Denai, and Y. Hori. Direct YawMoment Control of an In-Wheel-Motored Electric Vehicle Based on Body Slip Angle Fuzzy Observer. IEEE Transactions on Industrial Electronics, volume 56, issue 5, pages 1411–1419, May 2009. B.C. Besselink. Computer controlled steering system for vehicles having two independently driven wheels. Computers and Electronics in Agriculture, volume 39, issue 3, pages 209–226, August 2003. B. C. Jang, Y. H. Yun, S. C. Lee. Simulation of vehicle steering control through differential braking. International Journal of the Korean Society of Precision Engineering, volume 5, issue 3, pages 26–34, July 2004. H. Nozaki. Effect of differential steering assist on drift running performance. SAE technical paper, 2005-01-
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Fault-Tolerant Fault-Tolerant Control Control of of FWIA FWIA Electric Electric Fault-Tolerant Control of FWIA Electric Ground Vehicles with Differential Ground Vehicles with Differential Drive Drive Ground Vehicles with Differential Drive Assisted Steering Assisted Assisted Steering Steering
∗∗ Chuan Hu ∗∗∗ Hui Jing ∗∗ Fengjun Yan ∗∗ ∗∗ Rongrong Wang ∗∗ ∗∗ Fengjun Yan ∗ Chuan Hui Cong Jing Rongrong Wang ∗∗∗ ∗∗ Chuan Hu Hu ∗ Hui Jing ∗∗ Rongrong Wang ∗∗ Fengjun Yan ∗ ∗∗∗ Nan Chen ∗∗ Chuan Hu Hui Cong Jing ∗∗LiRongrong Wang Fengjun Yan ∗∗ Nan Chen Cong Li Li ∗∗∗ Nan Chen Cong Li ∗∗∗ Nan Chen ∗∗ ∗ Department of Mechanical Engineering, McMaster University, ∗ ∗ Department of Mechanical Engineering, McMaster University, of Canada, Mechanical Engineering, McMaster University, ∗ Department Hamilton, (e-mail: [email protected], and Department of Canada, Mechanical Engineering, McMaster University, Hamilton, (e-mail: [email protected], and Hamilton, Canada, (e-mail: [email protected], and [email protected]). Hamilton, Canada, (e-mail: [email protected], and [email protected]). ∗∗ [email protected]). School of Mechanical Engineering, Southeast University, Nanjing, ∗∗ [email protected]). ∗∗ School of Mechanical Engineering, University, School (e-mail: of Mechanical Engineering, Southeast Southeast University, Nanjing, Nanjing, ∗∗China, [email protected], [email protected], and School of Mechanical Engineering, Southeast University, Nanjing, China, (e-mail: [email protected], [email protected], China, (e-mail: [email protected], [email protected], and and [email protected]). China, (e-mail: [email protected], [email protected], and [email protected]). ∗∗∗ [email protected]). Department of Mechanical Engineering, Guilin University of ∗∗∗ [email protected]). ∗∗∗ Department of Mechanical Engineering, Guilin Department of Mechanical Engineering, Guilin University University of of ∗∗∗ Aerospace Technology, Guilin, China, ([email protected]). Department of Mechanical Engineering, Guilin University of Aerospace Aerospace Technology, Technology, Guilin, Guilin, China, China, ([email protected]). ([email protected]). Aerospace Technology, Guilin, China, ([email protected]). Abstract: This paper investigates a novel fault-tolerant control approach against active steering Abstract: This paper novel fault-tolerant control approach against steering Abstract: This paper investigates adrive novelassisted fault-tolerant control approach against active active steering system failure based oninvestigates differential a steering (DDAS) on four-wheel independently Abstract: This paper investigates adrive novelassisted fault-tolerant control approach against active steering system failure based on differential steering (DDAS) on four-wheel independently system failure basedelectric on differential drive assisted steering (DDAS) on four-wheel independently actuated (FWIA) ground vehicles. The front-wheel steering angle can be generated system failure based on differential drive assisted steering (DDAS) on four-wheel independently actuated (FWIA) electric electric ground vehicles. Thewhen front-wheel steering angle can be generated generated actuated (FWIA) ground vehicles. The front-wheel steering angle be with the front-wheel differential driving torque the steering motor fails.can DDAS being the actuated (FWIA) electric ground vehicles. Thewhen front-wheel steering angle can be generated with the front-wheel differential driving torque the steering motor fails. DDAS being with the front-wheel differential driving torque when the steering motor fails. DDAS being the the only steering power source is a potential alternative to replace the regular steering system in with the front-wheel differential driving torque when the steering motor fails. DDAS being the only power source is a alternative to the steering system in only steering steering power On source isissue, a potential potential alternative toa replace replace the regular regular steering system in defective situation. this this paper deploys hierarchical control method to utilize only steering power source is a potential alternative to replace the regular steering system in defective situation. On this issue, this paper deploys a hierarchical control method to utilize defective situation. On this issue, this paper deploys a hierarchical control methodthe to desired utilize DDAS for automatic steering. The higher-level controller is dedicated to generate defective situation. On this issue, this paper deploys a hierarchical control method to utilize DDAS automatic higher-level controller dedicated to desired DDAS for for automatic steering. The higher-level controller isLQR dedicated to generate generate the desired control input includingsteering. the yawThe moments based on a robustis approach, and the the lower-level DDAS for automatic steering. The higher-level controller isLQR dedicated to generate the desired control input including the yaw moments based on a robust approach, and the lower-level control input including the yaw moments based on a robust LQR approach, and the lower-level controller is responsible for yaw control allocation out the approach, respectiveand desired tire forces control input including the moments basedand on giving agiving robust LQR the lower-level controller is responsible for control allocation and out the respective desired tire forces controller is responsible control allocation andongiving out the respective desired forces and the desired steering for angle. Simulation based a high-fidelity and full-car modeltire indicates controller is responsible for control allocation andongiving out the respective desired tire forces and the desired steering angle. Simulation based a high-fidelity and full-car model indicates and the desired steering angle. Simulation basedforonFWIA a high-fidelity and full-car modeltoindicates that DDAS-based fault-tolerant control design electric vehicle is effective achieve and the desired steering angle. Simulation based on a high-fidelity and full-car model indicates that DDAS-based DDAS-based fault-tolerant control design for for FWIA FWIA electric vehiclefailure. is effective effective to to achieve achieve that control design vehicle is equal handling andfault-tolerant maneuverability performance case ofelectric the steering that DDAS-based fault-tolerant control design forinFWIA electric vehicle is effective to achieve equal equal handling handling and and maneuverability maneuverability performance performance in in case case of of the the steering steering failure. failure. equal and maneuverability in caseHosting of the by steering © 2015,handling IFAC (International Federation ofperformance Automatic Control) Elsevierfailure. Ltd. All rights reserved. Keywords: four-wheel independently actuated (FWIA), electric ground vehicles, differential Keywords: four-wheel independently actuatedcontrol, (FWIA), electric ground vehicles, vehicles, differential differential Keywords: four-wheel actuated (FWIA), electric ground drive assisted steering independently (DDAS), hierarchical optimal control. Keywords: four-wheel actuatedcontrol, (FWIA), electric ground vehicles, differential drive assisted steering independently (DDAS), hierarchical hierarchical optimal control. drive assisted steering (DDAS), control, optimal control. drive assisted steering (DDAS), hierarchical control, optimal control. considerably enhance active safety and vehicle handling 1. INTRODUCTION 1. considerably safety and considerably enhance active safety Additionally, and vehicle vehicle handling handling 1. INTRODUCTION INTRODUCTION during severeenhance driving active maneuvers. another 1. INTRODUCTION considerably enhance active safety Additionally, and vehicle handling during severe driving maneuvers. another during severe driving maneuvers. Additionally, another advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal potential during severe driving maneuvers. Additionally, another potential advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal potential advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal drive assisted steering (DDAS), utilized to provide certain combustion engine vehicle, has showed its advantages in potential advantage of DYC in FWIA vehicles, differential Electric ground vehicle, compared to conventional internal drive assisted steering (DDAS), utilized to provide certain combustion engine vehicle, has its advantages in drive assisted steering (DDAS), utilized tocontrol, providehas certain combustion engine vehicle, has showed showed itsenergy advantages in degree of steering power in vehicle motion also fuel economy, emissions reductions, and security, drive assisted steering (DDAS), utilized to provide certain combustion engine vehicle, has showed itsenergy advantages in degree of steering power in vehicle motion control, has also fuel economy, emissions reductions, and security, degree of steering power in vehicle motion control, has also fuel economy, emissions reductions, and energy security, been extensively researched. Jang et al. (2004), Nozaki. and thus become a worldwide focal point (Wang et al. degree of steering researched. power in vehicle motion control, has also fuel economy, emissions reductions, and energy security, been extensively Jang et al. (2004), Nozaki. and thus become a worldwide focal point (Wang et al. been extensively researched. Jang et al. (2004), Nozaki. and thusAmong becomeelectric a worldwide focal point (Wang et al. (2005) and Wu et al. (2008)Jang proposed similar Nozaki. idea to (2011)). vehicles,focal the four-wheel indepenbeen extensively researched. et al. (2004), and thus become a worldwide point (Wang et al. (2005) and Wu et al. (2008) proposed similar idea (2011)). Among electric the Wu etassist al. (2008) proposed similar idea to to (2011)).actuated Among (FWIA) electric vehicles, vehicles, the four-wheel four-wheel indepengenerateand steering force by differential driving of dently electric vehicle, mountedindepenwith in- (2005) (2005) and Wu et al. (2008) proposed similar idea to (2011)). Among electric vehicles, the four-wheel indepengenerate steering assist force by differential driving of dently actuated (FWIA) electric vehicle, mounted with ingenerate steering assist force by differential driving of dently actuated (FWIA) electric vehicle, mounted with intwo front wheels of electric vehicle, which has electrical wheel motors (or(FWIA) hub motors), isvehicle, an emerging and supegenerate steering assist force by differential driving of dently actuated electric mounted with intwo front wheels of electric vehicle, which has electrical wheel motors (or hub motors), is an emerging and supetwo front wheels of electric vehicle, which has electrical wheel motors (or hub motors), is an emerging and supe- steering system according to the steering geometry. Wang rior vehicle architecture, owing to its actuation flexibility two front wheels of electric vehicle, which has electrical wheel motors (or hub motors), is an emerging and supesteering system according to the the steering steering geometry. Wang rior to flexibility system according to geometry. Wang rior vehicle vehicle architecture, architecture, owing to its its actuation flexibility et al. (2011) proposed an indirect power steering measure combined the electricowing motors’ fastactuation and precise torque steering steering system according to the steering geometry. Wang rior vehiclewith architecture, owing to its actuation flexibility et al. (2011) proposed an indirect power steering measure combined with the electric motors’ fast and precise torque et al. (2011) proposed an indirect power steering measure combined with the electric motors’ fast and precise torque named as differential torquepower assisted steering, and response (2012),motors’ Chen etfast al. (2012), Shuaitorque et al. et al. (2011) proposed drive an indirect steering measure combined(Nam with et theal. electric and precise named as differential drive torque assisted steering, response (Nam et al. (2012), Chen et al. (2012), Shuai et al. named as its differential torque steering, assisted steering, and response Especially, (Nam et al. the (2012), Chen et al. (2012), Shuai et al. validated feasibilitydrive of assisting capabilityand of (2014)). actuator redundancies in FWIA named as differential drive torque assisted steering, and response (Nam et al. (2012), Chen et al. (2012), Shuai et al. validated its feasibility feasibility of effect assisting steering, capability of (2014)). Especially, the actuator in FWIA its of assisting steering, capability of (2014)). vehicles Especially, the actuator redundancies redundancies in FWIA validated road feel keeping and the of the torque distribution electric can considerably enhance the handling its feasibility of effect assisting steering, capability of (2014)). Especially, the actuator redundancies in FWIA validated road feel keeping and the of the torque distribution electric vehicles can considerably enhance the handling road feel keeping and the effect of the torque distribution electric vehicles can considerably enhance the handling system. However, none ofofthe thetorque abovedistribution literatures and maneuverability, and thus the enhance stability and safety in control road feel keeping and the effect electric vehicles can considerably the handling control system. However, none of the above literatures and and stability safety in system. of the above literatures and maneuverability, maneuverability, and thus thus the stability and safety in control the However, case that none the differential drive acts as defective driving conditions (Li the et al. (2013),and Djeziri et al. control system. However, none of the above literatures and maneuverability, and thus the stability and safety in investigated investigated the case that the differential drive acts defective driving conditions (Li et al. (2013), Djeziri et al. investigated the case that the differential drive acts as defective driving conditions (Li et al. (2013), Djeziri et al. the only power source of the steering, which can be as a (2013), Wang et al. (2014)). (Li et al. (2013), Djeziri et al. investigated the source case that the steering, differential drivecan acts as defective driving conditions the only power of the which be (2013), Wang et al. (2014)). the only power source of the steering, which can be a a (2013), Wang et al. (2014)). potential fault-tolerant control approach against active only power source of the steering, which canactive be a (2013), yaw Wang et al. (2014)). potential fault-tolerant control approach against Direct moment control (DYC) (Geng et al. (2009)) the potential fault-tolerant control approach against active steering system failure. Previous researchersagainst mostlyactive dealt Direct yaw moment (Geng fault-tolerant control approach Direct yaw moment control (DYC) (Geng et et al. al. (2009)) (2009)) steering system failure. researchers mostly dealt has been proved as ancontrol effective(DYC) and promising for potential steering system failure.ofPrevious Previous researchers mostly dealt Direct yaw moment control (DYC) (Geng etapproach al. (2009)) with the technology DDAS as an assisting steering has been proved as an effective and promising approach for steering system failure. Previous researchers mostly dealt has been proved ascontrol, an effective and promising approach for with the technology of DDAS as an assisting steering lateral dynamics with the advantage that it can with thefor technology ofdriver, DDAS asaddressed an assisting steering has been proved as an effective and promising approach for strategy easing the few DDAS being lateral the technology of DDAS as an assisting steering lateral dynamics dynamics control, control, with with the the advantage advantage that that it it can can with strategy for easing easing thepower driver,infew few addressed DDAS DDAS being for the driver, addressed being lateral dynamics control, with the advantage that it can strategy the exclusive steering faulted-steering situation. ⋆ strategy for easing thepower driver,infew addressed DDAS being This research is partly sponsored by the Foundation of Educathe exclusive steering faulted-steering situation. the exclusive steering power in faulted-steering situation. ⋆ ⋆ This This research is partly partly sponsored by the the(Grant Foundation of EducaEducaresearch is sponsored by Foundation of the exclusivedrive steering power in used faulted-steering situation. tion Office of Guangxi Province of China No. 2013YB272, ⋆ Differential is commonly in the lateral motion This research is partly sponsored by the(Grant Foundation ofthe Education Office of of No. 2013YB272, tion Office of Guangxi Guangxi Province of China China (Grant No. for 2013YB272, KY2015YB101) and theProvince Fundamental Research Funds CenDifferential drivevehicles, is commonly commonly used in the the lateral lateral motion Differential drive is used in motion control of FWIA with or without combination tion Office of Guangxi Province of China (Grant No. for 2013YB272, KY2015YB101) and Fundamental Research Funds the Differential drivevehicles, is commonly used in thethe lateral motion KY2015YB101) and the Fundamental Research Funds for the CenCentral Universities andthe Jiangsu Postgraduate Innovation Programm control of FWIA with or without the combination control of FWIA vehicles, with or without the combination KY2015YB101) and the Fundamental Research Funds for the Central and of activeof front-wheel steering (Shuai al. (2014), tral Universities Universities and Jiangsu Jiangsu Postgraduate Postgraduate Innovation Innovation Programm Programm (Grant No. KYLX-0102). control FWIA vehicles, with(AFS) or without theet combination of active front-wheel steering (AFS) (Shuai et al. (2014), tral Universities and Jiangsu Postgraduate Innovation Programm of active front-wheel steering (AFS) (Shuai et al. (2014), (Grant (Grant No. No. KYLX-0102). KYLX-0102). of active front-wheel steering (AFS) (Shuai et al. (2014), (Grant No. KYLX-0102).
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Yang et al. (2009)). However, when the steering motor breaks down, the front wheels of vehicles with regular steering systems (e.g., steering motor, or hydraulic power steering) will be liberated from the manipulation of the driver, the vehicle will consequently be out of control and vulnerable. In that case, if the DDAS in FWIA electric vehicles can be used to regulate the front wheel steering angle with the actuation generated by the differential drive torque of the front axle through the steering system mechanism (Wang et al. (2011)), the front wheel can still steer even though there is no steering power from the steering motor. This principle can be utilized for automatic steering without external steering power in faulted-steering condition, but few researchers had noticed that benefit. To this issue, this paper studies the failure condition that the steering motor breaks down, and presents a novel fault-tolerant control approach using the DDAS on an FWIA electric vehicle, to realize normal driving desire but guarantee the equal handling and maneuverability when the active steering system is in complete failure. This paper deploys a hierarchical control method, where higher-level controller is dedicated to generate the desired control input (the yaw moments) based on a robust LQR approach, the lower-level controller is responsible for control allocation and giving out the respective desired tire forces and the desired steering angle. The rest of this paper is organized as following. The vehicle model and the DDAS model are described in Section 2. The optimal control law design based on DDAS is presented in Section 3. Simulation based on CarSim platform using a high-fidelity and full-vehicle model is conducted in Section 4. Followed is the conclusion in Section 5. 2. MODELING OF VEHICLE DYNAMICS 2.1 Vehicle Model The vehicle model is shown in Fig. 1, which has three planar degrees of freedom for longitudinal, lateral and yaw motion. The vehicle pitch, roll and vertical motions are neglected, and the four tires are assumed to be completely consistent in the dynamics properties. The vehicle has the mass m, the moment of inertia Iz through the CG (center of gravity) about the yaw axis. The front and rear wheel axles are located at distances lf and lr from CG, respectively. vx and vy represent the longitudinal and lateral velocities of the vehicle, respectively. β and r stand for the sideslip angle and the yaw rate, respectively. Fxi and Fyi represent the longitudinal and lateral force of the ith tire, respectively, where i = f l, f r, rl, rr = 1, 2, 3, 4 indicates the specific wheel. Here the steering wheels are the front wheels, and we assume the steering angles of the front wheels are the same and sufficiently small, represented by δf . ls is half of the wheel-base. Here we assume that the lateral velocity is sufficiently small and bounded, then the yaw dynamics equation for the 2 degree-of-freedom (DoF) model of vehicle in the yaw plane can be given as follows Mz r˙ = , (1) Iz
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Fig. 1. 2 DoF model of vehicle in the presence of sliding effects
where the generalized yaw moment Mz are given by Mz = M z + lr (−Fyrl − Fyrr ),
(2)
with M z = ls (−Fxf l cos δf + Fyf l sin δf − Fxrl ) +ls (Fxf r cos δf − Fyf r sin δf + Fxrr ) (3) +lf (Fxf l sin δf + Fyf l cos δf + Fxf r sin δf + Fyf r cos δf ) . Based on the assumption about small tire-sliding effects, tire lateral forces Fyi (i = rl, rr) can be given by Fyi = cri αr , (4) where cri represents the cornering stiffness of the rear wheel, and here we assume the cornering stiffnesses of the rear wheels are the same, that is crrl = crrr = cr . αr is the rear wheel slip angle, where we assume the equality of the slip angles of the rear wheels. αr can be given by lr r − vy αr = . (5) vx From (1)- (5), (1) can be transformed into r˙ = −
Mz 2lr2 cr r+ + d1 , Iz vx Iz
(6)
r cr where d1 = 2l Iz vx vy is regarded as a disturbance, and assumed small and bounded. The explanation for that practice is that vy is pretty small and hard to be measured.
Remark 1 : Note that in the above description of vehicle lateral dynamics, we only derive the lateral forces of the rear wheels in the linear region of tires operation as shown in (4). The explanation is that in this paper, we address the differentially steered vehicle, and the front wheel steering angle is not a control input anymore, but a vehicle state, which is determined by the front-axle external yaw moment. Then a tricky problem occurs that the desired steering angle is unknown. So here we choose to use hierarchical control to render the real front wheel to steer tracking the desired steering angle, which is fixed later. Generally, the control objective for vehicle lateral motion control is to make the sideslip angle and yaw rate track their respective desired values.
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2.2 Differential Drive Assisted Steering Model In this paper we deal with the fault-tolerant control of FWIA electric vehicles, the failure condition is set that the steering motor breaks down completely, i.e., the steering motor torque becomes zero, as a consequence, the steering system is out of the driver’s control. To this end, here we employ differential drive assisted steering (DDAS), shown in Fig. 2, to achieve safety tolerances in case of active steering fault. The differential steering is generated by the differential drive torque of the front axle, through the steering system mechanism. The steering angle thus becomes a vehicle state, which is utilized to make the real lateral tire force of the front wheels to track their desired values. The detailed design method will be presented in the controller design procedure.
αf ≤ αst αf > αst
(10)
where αsl = arctan (1/θy ) is the sliding slip angle, θy = 2cp l2 / (3µFz ) , σy = tan αf , l is the half of the tire contact length, µ is the tire-road adhesion coefficient, Fz is the normal force. cp is the stiffness coefficient of tire tread in unit length, and cp = cf /(2l), where cf is the cornering stiffness of the front tires, and we assume the cornering stiffnesses of the front tires are the same. αf is the slip angle of the front wheel, where we assume the slip angles of the front wheels are the same. Assuming the tire slip angle is small, from (10), we have τa ≈ κ1 cf αf , (11) where κ1 = l2 /3 is a constant, and αf is given by lf r + vy . (12) αf = δf − vx τds can be given by τds = (Ftf r − Ftf l ) rσ , (13) where rσ stands for the scrub radius as shown in Fig. 2. The steering force Ftf j is given by Ftf j = Fxf j / cos δf , (j = 1, 2) , (14) which derives τds = κ2 ∆Mzf . (15) rσ where κ2 is given by κ2 = ls cos δf . Since δf is assumed small enough, so we have κ2 ≈ rσ /ls .
µFz lθy σy [1 − 3 |θy σy | + � 2 3 τa = 3(θy σy ) − |θy σy | , 0,
Fig. 2. Differential drive assisted steering model When the regular steering system (e.g., steering-by-wire (SBW), active front-wheel steering (AFS)) of the vehicle works normally, the electronic power steering system is a simple inertia-damping system (Ahn et al. modeled as (2013)) Jeff δ¨f + beff δ˙f = τa + τm − τf , (7) where Jeff and beff are the effective moment of inertia of the steering system, respectively. and effective damping τa represents the tire self-aligning moment, τm is the steering motor torque, and τf is the friction torque. τm is related with the driver’s steering wheel torque τs through a steering wheel sensor, which measures the angle and the torque of the steering wheel. When the regular steering system of the vehicle fails totally, which means τm = 0, the differential steering system is activated, described by the following model Jeff δ¨f + beff δ˙f = τa + τds − τf , (8)
From (10)- (15), (8) can be transformed into κ 1 cf l f κ1 cf κ2 (16) δ˙f = − r+ δf + ∆Mzf + d2 , beff vx beff beff � � κ c 1 −τf − Jeff δ¨f , and can be where d2 = − beff1 vfx vy + beff assumed as a bounded disturbance. The explanation for that practice is that Jeff and δ¨f are sufficiently small, regular treatment of them will increase control complexity unworthily. Define the following state variables: x1 = r, x2 = δf , T and denote x = [x1 , x2 ] , the state space form of the dynamic model integrating vehicle lateral dynamics (6) and differential steering system (16) is given by x˙ = Ax + B (u + d) , (17) where
where τds is the differential steering moment generated by the front-axle external yaw moment ∆Mzf , which is generated with the longitudinal tire force difference between the two sides of the front wheels, and can be written as ∆Mzf = (Fxf r − Fxf l ) ls cos δf . (9) From the brush model, the self-aligning moment are given by 1182
2lr2 cr − 0 Iz vx A= , κ 1 cf l f κ 1 cf − beff vx beff B = diag (1/Iz , κ2 /beff ) , �T � u = M z , ∆Mzf ,
d = diag (d1 ,
d2 ) ,
(18)
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κ c where d1 = Iz d¯1 = 2lvrxcr vy and d2 = bκeff2 d¯2 = − κ21vfx vy + � � 1 ¨ are assumed to be bounded disturκ2 −τf − Jeff δ f bances.
3. OPTIMAL CONTROL BASED ON DIFFERENTIAL STEERING SYSTEM 3.1 Higher-Level Controller Design Here we deploy a robust and adaptive LQR controller (Liu et al. (2014)) to minimize the errors of the yaw rate r and the front wheel steering δf with their respective desired values, and simultaneously, restrict the control inputs M z and ∆Mzf in reasonable regions. The advantage of the proposed LQR method lies on that the effect of the matched disturbance can be eliminated by an add-on nonlinear controller. Denote xd = [rd , δf d ]T , where rd and δf d mean the respective desired values of the yaw rate and the steering angle. rd is generally related with the driver’s steering command δ0 , while δf d is proposed in the lower-level controller design procedure. Hence the states errors are given by ε = x − xd = [r − rd , δf − δf d ]T , (19) and the integrated performance index of the optimal control is defined as: �∞ � 1 � T J1 = ε Qε + uT Ru dt, (20) 2 0
where Q = diag [Qz , Qs ], R = diag [Rz , Rs ]. Here we utilize a theorem to design the robust LQR controller, one can refer to (Liu et al. (2014)) for the detailed proof of Theorem 1. Theorem 1 Assume d (x, t) is norm-bounded. The control input vector u (t) can be designed as: u (t) = ul (t) + unl (t) , (21) where the linear item ul (t) is (22) ul = −R−1 B T P (x − xd ) , in which the symmetric positive definite P is the solution of the Algebraic Riccati Equation P A + AT P + Q − P BR−1 B T P = 0. (23) And the nonlinear unl (t) item is BT P ε −Λ T , B T P ε �= 0 unl (t) = (24) �B P ε�2 0, BT P ε = 0 where the gain of nonlinear item Λ should satisfy Λ > W > �d (x, t)�2 . (25) If the controller is designed as shown in (21), system (17) will possess globally asymptotic stability. T
ε Remark 2 : Even if ε = 0, since �BBT PPε� is a relative 2 small value compared with ul , so the effect of the control switch to the vehicle stability is rather limited and can be neglected.
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3.2 Lower-Level Controller Design In this section, the required input is allocated to the four tyres. Since ∆Mzf is only related with the longitudinal forces of the front wheels, we shall preferentially allocate ∆Mzf , and then deal with M z . Make Fx1 = −Fx2 as the allocation rule, and follow the definition of ∆Mzf Fx1 (−ls cos δf ) + Fx2 (ls cos δf ) = ∆Mzf , (26) we have ∆Mzf ∆Mzf Fx1 = − , Fx2 = . (27) 2ls cos δf 2ls cos δf Note that there exist Fx1 and Fx2 items in the expression of M z as shown in (3), we need to eliminate them before the subsequent allocation. Denote M ze as the allocation object after the elimination as M ze = M z − ls (−Fx1 cos δf + Fx2 cos δf ) −lf (Fx1 sin δf + Fx2 sin δf ) ,
(28)
where at this moment M z comes from the higher-level controller, and Fx1 and Fx2 are obtained from (27). Then an analytic optimization control allocation algorithm (Wang et al. (2013)) is adopted to distribute the control efforts without using the numerical method. The cost function for allocating M ze to the four tire forces can be defined as � �T � � 1 � GF − M ze Q GF − M ze + J2 = 2 (29) � T F WF , where W = diag [w1 , w2 , w3 , w4 ], Q = q1 , F = T [Fx3 , Fx4 , Fy1 , Fy2 ] , where we assume Fyf = Fy1 = Fy2 . G is the control effectiveness matrix and can be written according to (3) G = [−ls , ls , (ls sin δf + lf cos δf ) , (30) (−ls sin δf + lf cos δf )] , Based on (29), we have ∂J2 = W F +GT Q[GF − M ze ] ∂F T =� W F +[GT QGF � − G QM ze ] T = W + G QG F − GT QM ze , and
(31)
∂ 2 J2
(32) = W + GT QG, ∂2F As W > 0 and GT QG ≥ 0, we can claim that H > 0, which indicates that the objective function J2 has a global minimum when F is chosen as the minimizing value F ∗ given by � �−1 F ∗ = W + GT QG GT QM ze , (33) H=
where we can choose W = I4×4 , Q = 1 in the following simulation. In order to realize the actuation of the desired lateral tire ∗ ∗ forces Fy1 and Fy2 of the front wheels, the following control law (Wang et al. (2015)) is used to generate the desired steering angle � ∗ � δf d = Kp Fyf − Fyf , (34)
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6
2
r (deg/s)
0
0
δ (deg)
1
−1 −2 0
Controlled Reference
4 2 0 −2 −4 1
2 3 Time (sec)
4
5
−6 0
Fig. 3. Driver’s steering angle command
1
2 3 Time (sec)
4
5
4
5
4
5
Fig. 4. Reference and real yaw rate
where Kp is a positive constant, δf d is the desired value of ∗ the steering angle. Fyf is the desired lateral tire force of the front wheel given by lower-level controller (33), where ∗ ∗ ∗ = Fy1 = Fy2 . Fyf is the real lateral it is assumed that Fyf tire force of the front wheel which can be measured by sensors or estimated with the vehicle model (Doumiati et al. (2011), Cho et al. (2010)). It can be known that in practice δf d is bounded, hence if Kp is chosen as an ∗ enough big value, we will eventually have Fyf → Fyf . It should be noted that δf d which is given by (34) needs to be substituted into (19) to obtain the states error. 4. SIMULATION
1
α (deg)
0.5
f
0
−0.5 −1 0
1
2 3 Time (sec)
Fig. 5. Front tire slip angle
5. CONCLUSION This paper addresses the fault-tolerant control issue against active steering system failure for FWIA electric
δf (deg)
0.5
0
−0.5 0
1
2 3 Time (sec)
Fig. 6. Real steering angle of the front wheel 4000 Yaw moment (Nm)
In this section, we present a single-lane change simulation case, where the vehicle is controlled to make a single-lane change on high adhesion (µ = 0.85) road with a high speed (vx = 30 m/s). The control objective is to make the vehicle yaw rate follow the desired value with only the DDAS. The desired yaw rate can be generated from the driver’s steering angle and vehicle longitudinal speed as (Du et al. (2010)) vx rd = δ0 , (35) L (1 + kus vx2 ) where L = lf + lr is the distance between the front and rear axles, kus is the stability factor, δ0 is the driver’s steering angle command which is shown in Fig. 3. The yaw control result is shown in Fig. 4, from which we can see the tracking objective of the real yaw rate towards the reference value is achieved. Fig. 5 presents the slip angle of the front wheel. Since the slip angles of the front wheels differ pretty small based on the previous assumption of the equality of them, so here we only show their average value. We can see that the variation tendencies of the front wheel slip angle and the yaw rate are roughly the same, which validates the effectiveness of the proposed control law. From Fig. 5 we can also infer that the vehicle is stable in the simulation. The real front wheel steering angle is presented in Fig. 6, the control inputs M z and ∆Mzf are shown in Fig. 7, we can see that they are all maintained in reasonable regions. Note that the signs of M z and ∆Mzf are with the same sign, which means the tracking control of the yaw rate and steering angle is complementary.
¯z M ∆Mzf
2000 0 −2000 −4000 0
1
2 3 Time (sec)
4
5
Fig. 7. The control inputs vehicles, and proposes a novel fault-tolerant control approach based on the DDAS mechanism. When the regular steering system breaks down, the differential steering can be generated by the front-wheel differential driving torque. To guarantee the equal handling and maneuverability as the normal status, this paper adopts a hierarchical con-
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trol method to utilize DDAS for automatic steering. The higher-level controller generates the desired control inputs, i.e., the yaw moments, based on a robust LQR strategy, while the lower-level controller arranges the control allocation and gives out the respective desired tire forces and the desired steering angle. CarSim-Simulink simulation of a sing-lane change case based on a high-fidelity and full-car model verifies the availability and the effectiveness of the proposed controller for fault-tolerant control on steering failure. REFERENCES R. Wang, Y. Chen, D. Feng, X. Huang and J. Wang. Development and Performance Characterization of an Electric Ground Vehicle with Independently Actuated In-Wheel Motors. Journal of Power Sources, volume 196, issue 8, pages 3962–3971, April 2011. K. Nam, H. Fujimoto, and Y. Hori. Lateral Stability Control of In-Wheel-Motor-Driven Electric Vehicles Based on Sideslip Angle Estimation Using Lateral Tire Force Sensors. IEEE Transactions on Vehicular Technology, volume 61, issue 5, pages 1972–1985, June 2012. Y. Chen and J. Wang. Fast and Global Optimal EnergyEfficient Control Allocation With Applications to OverActuated Electric Ground Vehicles. IEEE Transactions on Control Systems Technology, volume 20, issue 5, pages 1202-1211, September 2012. Z. Shuai, H. Zhang, J. Wang, J. Li, and M. Ouyang. Combined AFS and DYC Control of Four-WheelIndependent-Drive Electric Vehicles over CAN Network with Time-Varying Delays. IEEE Transactions on Vehicular Technology, volume 63, issue 2, pages 591–602, February 2014. D. Y. Li, Y. D. Song, D. Huang, and H. N.Chen. Modelindependent adaptive fault-tolerant output tracking control of 4WS4WD road vehicles. IEEE Transactions on Intelligent Transportation Systems, volume 14, issue 1, pages 169-179, March 2013. M. A. Djeziri, R. Merzouki, B. O. Bouamama, and M. Ouladsine. Fault Diagnosis and Fault-Tolerant Control of an Electric Vehicle Overactuated. IEEE Transactions on Vehicular Technology, volume 62, issue 3, pages 986– 994, January 2013. R. Wang and J. Wang. Actuator-Redundancy-Based Fault Diagnosis for Four-Wheel Independently Actuated Electric Vehicles. IEEE Transactions on Intelligent Transportation Systems, volume 15, issue 1, pages 239– 249, February 2014. C. Geng, L. Mostefai, M. Denai, and Y. Hori. Direct YawMoment Control of an In-Wheel-Motored Electric Vehicle Based on Body Slip Angle Fuzzy Observer. IEEE Transactions on Industrial Electronics, volume 56, issue 5, pages 1411–1419, May 2009. B.C. Besselink. Computer controlled steering system for vehicles having two independently driven wheels. Computers and Electronics in Agriculture, volume 39, issue 3, pages 209–226, August 2003. B. C. Jang, Y. H. Yun, S. C. Lee. Simulation of vehicle steering control through differential braking. International Journal of the Korean Society of Precision Engineering, volume 5, issue 3, pages 26–34, July 2004. H. Nozaki. Effect of differential steering assist on drift running performance. SAE technical paper, 2005-01-
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