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譯文題目: 基于使用DTFC最優(yōu)滑動控制的
新型混合防抱死制動系統(tǒng)電動汽車
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文獻出處：國際科學26.1(Jan-Mar 2014): 197-203.
基于使用DTFC最優(yōu)滑動控制的新型混合防抱死制動系統(tǒng)電動汽車
Sharifian, Mohammad Bagher Bannae;?Yousefi, Babak;?Ebadpour, Mohsen
（伊朗大不里士大學，電氣和計算機工程學院）
摘要：一輛汽車的制動性能是汽車安全的一個重要因素。一個成功設計的制動系統(tǒng)的車輛必須滿足不同的需求，迅速減少車速和維護方向穩(wěn)定性。事實證明，轉(zhuǎn)速和車速在一定的比例范圍內(nèi)，制動力最大。當代，制動系統(tǒng)的主要功能就是防抱死制動系統(tǒng)(ABS)。首先是阻止車輪鎖止，第二處于理想滑動。在本文中該方法結合的原則應用于ABS、電氣制動系統(tǒng)和由兩個機械和電氣部分組成。模擬轉(zhuǎn)矩控制和滑移控制系統(tǒng)設計利用MATLAB / Simulink軟件已經(jīng)完成。為了證明在可變化性和比較小附著系數(shù)的道路上，提出的制動系統(tǒng)的高性能，傳統(tǒng)的剎車所面臨的問題。該方法的可靠性已經(jīng)得到證實。
關鍵詞：混合動力電動汽車(HEVs)；防抱死制動系統(tǒng)(ABS)；直接轉(zhuǎn)矩和磁通控制(DTFC)；滑動控制
引言
近幾十年來，相關的研究和開發(fā)活動交通一直強調(diào)發(fā)展高效、清潔、安全的交通工具。通常建議電動車((EVs)、混合動力電動汽車((HEVs)和燃料電池汽車在不久的將來取代傳統(tǒng)汽車。人們對混合動力電動汽車日益增長興趣是由于這一事實：在低速電動汽車和高速柴油機都可以高性能工作。除了高效的電機由于發(fā)電機這些機器的功能,他們節(jié)約電力的能力 [1-3]。在電動汽車或混合動力電動汽車，都可以使用電機制動。車輪制動過程中，鎖止是對車輛的穩(wěn)定性造成負面影響。另一方面，它還危害乘客的生命。那就是為什么今天是很常見的使用汽車防抱死系統(tǒng)[4]。
安裝防抱死制動系統(tǒng)(ABS)的車輛的目的是保持在一定范圍內(nèi)，以確保車輪滑移最大制動力，減小制動距離。此外，它直接增加制動力和側(cè)向摩擦力，幫助維持車輛穩(wěn)定[5]。在每個輪子上通過應用傳感器和使用四個調(diào)節(jié)器石油龍頭來調(diào)整制動力將在ABS獲得最好的制動狀態(tài)。在[6]，提出了一種方法來優(yōu)化石油閥門的開啟和關閉來計算合適的機油壓力在每個輪子上的最少時間。有另一種類型的電氣制動只適合普通事故，稱為再生制動。再生制動過程中部分動能轉(zhuǎn)化為電能。這種制動器使用低可靠性的傳統(tǒng)的制動器 [7-9]。在防抱死制動系統(tǒng)(ABS)中，由于各種各樣的壓力，制動管、變附著系數(shù)的道路及其依賴速度，道路彎曲不同輪滑移值和其他參數(shù)，滑移的計算方法非常重要。在[10]，通過使用一個反饋，控制循環(huán)已經(jīng)形成衡量動態(tài)滑動率。通過李雅普諾夫理論策略開啟和關閉閥門的油壓控制提出了控制滑移?；谥悄芗夹g可以使用算法。
在[11]，ABS系統(tǒng)提供檢測路況和滑移控制。為了估計摩擦和道路外形，LuGre模型已經(jīng)使用。然后，應用一種基于模糊神經(jīng)網(wǎng)絡的估計量。在一些報紙中，估計滑移率是根據(jù)灰色模型和由滑?？刂破?SMC)證實了它們的準確性。在這些方法中，滑移估計值要結合最優(yōu)滑移調(diào)節(jié)器[12]。有一些方法，以提高車輛穩(wěn)定性原理類似于ABS的原則。在[13]，基于模糊邏輯的方法來控制偏航和橫向滑動。在這種方法中，在高速車輛的后方馬達通過安裝控制器來控制橫向穩(wěn)定性。
在本文中，獲得最大利用機械制動器制動力，在同一時間利用恒力和電氣制動的可能性來調(diào)整滑移是目的?？紤]到制動力依賴于道路狀況，提高車輪制動力矩導致車輪抱死，從而導致制動力降低大約30%和100%的側(cè)向穩(wěn)定性。在提出的方法中,通過從車輪滑動得到反饋，它的數(shù)量是固定在最優(yōu)數(shù)量(通常0.2)獲得最大制動力。此外，一些其他問題，ABS系統(tǒng)在結冰或塵土飛揚的道路上，一直有影響。
一、制動力的原則
當車輛勻速前進，速度正比于車輪速度并且車輪沒有滑動。但當司機按下制動踏板降低速度時，車輪速度逐漸減少，車身的比例也被毀壞。應該指出，在這種情況下由于慣性力，車身傾向于移動。因此，在車輪與路面之間形成一個小坡度。車輪速度和車速之間的差異表示滑移量?；坡视嬎闳缦拢?
Λ=(Vv-Vw)/Vv (1)
在等式(1)中，參數(shù)Vv和Vw分別表示車速和車輪速度。0%的滑移率顯示了輪自由滑動，沒有任何障礙，100%的滑移率顯示模型在路上完全鎖止和輪下滑。通過增加車輪速度和車速之間的區(qū)別，車輪和路面之間的滑移太高了。這引起摩擦和制動力，進而車速降低。制動力和滑移率關系圖1所示。制動力并不總是與滑動率有關。然而，當10 - 30%之間的滑移率，獲得的最大制動力[14]。
圖1 制動力和滑移率關系圖
二、最優(yōu)制動力的分配
按下制動踏板時，制動盤上創(chuàng)建一個制動轉(zhuǎn)矩。這種制動力矩在車輪和接觸的地面之間產(chǎn)生一個力。如果這個力大于最大制動力矩，它將使車輛停止。制動力及其最大值可以得到如下：
Fb=Tb/rd (2)
Fbmax=Ub*w (3)
在方程(2)和(3),，Tb、rd、參考轉(zhuǎn)矩、有效半徑分別為輪子的速度。Hb是道路和車輪的附著系數(shù)和滑移率變化?？色@得最大數(shù)值(1520)的比例下滑。制動力隨著制動力矩的增加而增加。如圖2所示，當制動功率達到最大可容忍的路輪，其值仍然幾乎是不變的[14]。在道路和輪子之間最大可實現(xiàn)的制動力是依賴/ ib和車重。
圖2 a)制動時車輪受力圖 b)轉(zhuǎn)矩和制動力關系圖
三、制動力之間的前后軸
在平坦道路上車力如圖3所示。轉(zhuǎn)向盤和氣動抗性被忽視，是由于他們與制動力相比其值小。負加速度的車輛，制動模式定義為j可以容易得到：
j=(Fbf+Fbr)/Mv (4)
Fbr和Fbf分別表示制動力作用于前后車輛的輪軸。最大制動力被限制是由于輪路間的粘附系數(shù)和每個輪子的機械負載荷。因此，從制動力矩獲得的制動力應正比前后輪軸的負載。結果，前后輪軸同時實現(xiàn)他們的最大制動力。忽視了在制動期間從后軸到前軸的整體移動，輪子的重量在前后輪軸接觸點圖3 A和B可以計算(5)和(6)[4]。
Wf=Mv*g*(Lb+hg*j/g)/L (5) Wr=Mv*g*﹙La-hg*j/g﹚/L (6)
圖3 前后輪軸受力分布圖
此外，在前、后軸制動力的值應該等于重量的名義值。所以：
Fbf/Fbr=Wf/Wr=(Lb+hg*j/g)/(La-hg*j/g) (7)
(7)(4)相比，在理想的制動，前后輪軸的制動力圖4所示，j是車輛在道路上的最大負加速度。
圖4 理想制動力分配圖（I-curve）
根據(jù)圖4，理想制動力分配曲線命名I-curve是非線性雙曲線。同時鎖定前輪和后輪，前后輪軸的制動力應遵循I-curve。在實際設計中，這些力和他們的比率被認為是線性和定義前軸的制動力與總制動力的比例。根據(jù)(8)。
β=Fbf/Fbr (8)
圖5顯示了理想和實際制動曲線[14]。圖5表明，這些曲線相交與一點，在這一點上前輪和后輪同時被鎖止。這一點似乎為一定值的附著系數(shù)Ho，可以從以下公式計算：
Fbf/Fbr=β/(1-β) (9)
β/(1-β)=(Lb+hg*j/g)/(La-hg*j/g) (10)
u0=(L*β-Lb)/hg (11)
制動時，ia小于Ho(一個地區(qū)?曲線在I-curve之下)，前輪比后輪提前鎖止，反之亦然。當后輪首先鎖止時，車輛失去方向穩(wěn)定性和后輪的橫向穩(wěn)定性降低為零。在這種情況下，一個小側(cè)向力像風力或離心力等將會導致側(cè)向不穩(wěn)定。車輛將旋轉(zhuǎn)90，然后180度脫離道路。另一方面，制動時前輪被鎖止，司機將失去控制車輛前進方向，將無法有效控制車輛。然而，這并不意味著不穩(wěn)定是完全發(fā)生。因此，前軸自控力從而防止側(cè)向不穩(wěn)定[14]。
圖5 理想—實際制動曲線圖
根據(jù)這些討論，似乎后輪的鎖止更危險，尤其是在小道路。因為在這樣的道路上，制動力逐漸減少，動能逐漸降低。所以，狀況不穩(wěn)定、車輛本身變得更遠。因此，汽車設計必須首先保證后輪不鎖止。由于限制，產(chǎn)生的轉(zhuǎn)矩和牽引電機的供應用于電動汽車和很多不同ja0。圖6所示，電氣和機械轉(zhuǎn)矩比根據(jù)路滑條件可以調(diào)整。如圖6所示，例如在冷凍條件下，扭矩為200 N.m和有關電氣制動電機功率值的定義。Ho越?電氣部分總比，其值會越高[14]。
圖6 u在不同條件下的分布圖
四、直接轉(zhuǎn)矩和磁通控制(DTFC)
直接轉(zhuǎn)矩控制來源于重構磁通量的直接控制，而且比磁通量矢量控制其實現(xiàn)容易。磁通和轉(zhuǎn)矩通常由磁滯控制器控制。由PWM調(diào)制器造成的延誤在這個方法可以削減并且PWM調(diào)制器可以進行替換為一個最優(yōu)切換技術。傳統(tǒng)轉(zhuǎn)矩控制方法的總結在表中提出了。
本文提出的方法是一個商業(yè)名稱叫做DTFC技術使牽引電機自我控制。該方法首先提出了感應電動機由PWM電壓源控制，開發(fā)成一個控制扭矩的矢量，應用于交流電機由電壓或電流源逆變器提供的。事實上，基于定子磁通矢量的大小、轉(zhuǎn)矩誤差及其數(shù)量和矢量的定子磁通在每6個活躍狀態(tài) (或12個狀態(tài))，某一電壓向量 (或電壓向量的組合) 直接或有特定的時間表適用于逆變器。為了計算磁通、定子轉(zhuǎn)矩的大小和相位誤差對應的值應該估計。因此，一個合適的轉(zhuǎn)矩和磁通估計量或者速度傳感器直接控制的磁通量向量和DTFC（磁通控制）是必要的。DTFC法和直接矢量控制的基本框圖圖7所示。
圖7 a)直接磁通量控制圖 b)直接轉(zhuǎn)矩控制圖
圖7所示，DTFC是一種直接電流矢量控制。兩種控制策略，圖7所示需要磁通和轉(zhuǎn)矩的觀察器。然而，直接轉(zhuǎn)矩控制和定子磁通控制具有良好的精度。因此用直流電流設計PI控制器不是必需的。以及開環(huán)PWM用于該方法被一個最佳的轉(zhuǎn)換表代替了。這些簡化意味著DTFC只使用磁通、扭矩和速度觀測器控制電機。此外，定子磁通作用的DTFC和不需要轉(zhuǎn)子磁通從而導致控制系統(tǒng)更簡單。雖然動態(tài)特性圖7所示的方法都是一樣的,直接矢量控制一般緩慢是由于轉(zhuǎn)子磁通分析的必要。
id=(L+sr)*λr/Lm (12)
等式 (12)顯示，對于轉(zhuǎn)子通過其適應機制工作，有一個時間常數(shù)(rr)。所以很明顯，DTFC間接充當直接矢量對噪聲控制更簡單和更強大。此外，它具有更好的動態(tài)轉(zhuǎn)矩響應速度范圍。
五、模擬結果
模擬是由兩部分組成：第一部分包括動態(tài)響應的模擬車輛制動模式和基于p和道路情況的滑移變化。在第二部分，獲得值與預期值、速度參考和計算電機轉(zhuǎn)矩值比較，這些值應用于DTFC。電機轉(zhuǎn)矩模擬控制系統(tǒng)的電氣防抱死制動系(EABS)統(tǒng)圖8所示。模擬中使用的電機是一個三相感應電動機，其參數(shù)表II所示
圖8 EABS電機轉(zhuǎn)矩控制系統(tǒng)圖
表Ⅱ 模擬電機參數(shù)
在實踐中，考慮到應用制動力和最大可持續(xù)車輪力之間的差異，滑移變化。涉及這些變化相關的方程可以計算。在提到的控制系統(tǒng)中，滑動選擇參考0.2。重要的是要注意，滑移曲線分為兩個部分：滑移率大于0.2的曲線和小于0.2的曲線。輸入這些方程的制動轉(zhuǎn)矩比率和最大可持續(xù)車輪扭矩比率并且輸出就是滑移值。計算滑移后，它的值與0.2相比。如果是大于0.2，它將減去0.2。如果是小于0.2，它將被增加到0.2。這些值的變化可以依賴或獨立于滑移值。這些計算的步驟框圖如圖9所示。
為了使模擬更真實，最大可持續(xù)車輪轉(zhuǎn)矩和機械轉(zhuǎn)矩是在230 Nm和400 Nm分別在1和3的規(guī)定，隨意波動?；?轉(zhuǎn)矩-速度子系統(tǒng)在車輛制動模式下的相關結果，如圖10—17所示。
圖9 滑移—轉(zhuǎn)矩—速度控制及計算子系統(tǒng)圖
圖10 電子轉(zhuǎn)矩圖
圖11 最大可持續(xù)道路—車輪轉(zhuǎn)矩圖
圖12 可應用轉(zhuǎn)矩減圖及可持續(xù)道路—車輪轉(zhuǎn)矩圖
圖13 滑移率圖
圖14 沒有控制器的滑移圖
圖15 每個車輪上的轉(zhuǎn)矩圖
圖16 汽車加速度變化圖
圖17 汽車速度變化圖
在顯示的數(shù)據(jù)，注意到轉(zhuǎn)矩跟隨最大轉(zhuǎn)矩和車輪滑移被控制在可接受的波動0.2。第二部分是與DTC和感應電動機有關圖18所示。
圖18 DTC子系統(tǒng)圖
感應電動機由PWM電壓源逆變器所控制。在速度控制回路，PI控制器是用于生成參考磁通和轉(zhuǎn)矩值。在DTC塊中，計算電機轉(zhuǎn)矩和估計磁通，然后與參考值比較?？紤]到特殊的情況，輸出脈沖比較器應用于切換到逆變器開關。這個子系統(tǒng)產(chǎn)生和遵循所需的扭矩高速度和準確度。
六、結論
電力和混合動力汽車可以使用電氣制動。然而，考慮到這種想法的限制，它用于機械制動。在嚴重和突然剎車時，再生制動是不可能的。因為在這種制動車輛停止是最快的。另外，電機本身不能產(chǎn)生制動轉(zhuǎn)矩。甚至如果電機有這種能力，電源和效率問題阻止這種情況。所以，最好的方式似乎是使用帶有ABS特色的并行制動。然而，應該探索更好的方法，因為ABS方法不減少制動距離。該方法是結合ABS和電氣制動，而ABS系統(tǒng)在汽車輪子上負責控制油壓，涉及電氣車輛特征。在實際情況下，赤潮的控制系統(tǒng)比ABS系統(tǒng)更簡單、更便宜，由于控制4泵減少到2泵。關于制動的性能質(zhì)量的適當?shù)姆椒ǎ瑥哪M結果來看，滑移控制及其根據(jù)期望的值是顯而易見的。這種方法的唯一缺陷可能是，在電氣制動使用能量，降低車輛的效率。然而，用于制動、增加制動和車輛安全的能量是微不足道的小值。
參考文獻：
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附件：外文資料原文
A NEW HYBRID ANTI-LOCK BRAKING SYSTEM FOR HEVS BASED ON OPTIMAL SLIP CONTROL USING DTFC
Sharifian, Mohammad Bagher Bannae;?Yousefi, Babak;?Ebadpour, Mohsen.?Science International26.1?(Jan-Mar 2014): 197-203.
Headnote
ABSTRACT: The?braking?performance of a?vehicle?is an important factor in?vehicle?safety. A successfully designed?braking system?for a?vehicle?must always meet the distinct demand of quickly reducing?vehicle speed and maintaining?vehicle?direction controllable by the?steering?wheel. It has been proved that for a certain ratio of wheel rotation and?vehicle?speed,?braking?force is maximum. The main function of nowadays braking systems?which is known as anti-lock?braking system?(ABS) first is to prevent wheels from locking, second bringing the?vehicles?slip to the ideal slip. The proposed method in this paper is a combination of the principles which are used in ABS and electrical?braking systems?and consists of two mechanical and electrical parts. Simulation of torque?control?and slip?control system?design has been performed with MATLAB/Simulink software. To prove the high performance of proposed?braking system?a road with variable and rather small adhesive coefficient, which conventional brakes face with problem on it, has been considered. The reliability of proposed method has been proved.
Keywords: hybrid electric?vehicles?(HEVs), anti-lock?braking system?(ABS), direct torque and flux?control(DTFC), slip?control
(ProQuest: ... denotes formulae omitted.)
INTRODUCTON
In recent decades, the research and development activities related to transportation have emphasized the development of high-efficiency, clean, and safe transportation. Electric?vehicles?(EVs), hybrid electric?vehicles(HEVs), and fuel cell?vehicles?have been typically proposed to replace conventional?vehicles?in the near future. Growing interest in HEVs is due to this fact that, in low speeds electrical motors and in high speeds diesel engines have high performance. In addition to high-efficiency of electrical motors due to generator behavior of these machines, they are capable of saving part of their electrical powers [1-3]In EVs or HEVs, it is possible to use electrical motors for?braking. Wheels' locking during?braking?process is causing a negative impact on the stability of the?vehicle. On the other hand, it also endangers the lives of passengers. That's why today is very common to use?vehicles?with anti-locking?system?[4].
The purpose of installing of anti-lock?braking system?(ABS) on the?vehicle?is to keep the wheels slip in a certain range to ensure maximum?braking?force and reduce?braking?distance. Furthermore, it increases?braking?force in direct and lateral friction force that helps to maintain?vehicle?stability [5]. By applying sensors on each wheel and using four regulator oil taps to adjust the?braking?force the best?braking?state will be obtained on ABS system. In [6], a method is presented to optimize the opening and closing oil valves to calculate the appropriate oil pressure on each wheel in least time. There is another type of electric brake that is suitable only for commonplace accidents, called, regenerative?braking. In the process of regenerative?braking?a part of kinetic energy is converted into electrical energy. This brake is used with conventional ones due to low reliability of it [7-9]. In ABS?braking systems, due to the variety of pressure in?braking?pipes, variable adhesive coefficient of roads and its dependence to speed, different wheel slips on road bends, and other parameters, the calculation method of slip has great importance. In [10], by using a feedback, a?control?loop has been formed which measures slip rates dynamically. By means of Lyapunov theory a strategy for opening and closing valves of oil pressure?control?has been presented to?control?slip. The algorithms, based on?intelligent techniques can be used.
In [11], an ABS?system?is provided to detect the road conditions and slip?control. To estimate the friction and road profile, LuGre model has been used. Then, an estimator based on fuzzy-neural network is applied. In some papers, the slip rate is estimated based on Gray model and the accuracy of them is confirmed by sliding-mode controller (SMC). In these methods slip estimation is combined with optimal slip regulator [12]. There are some methods in order to increase the?vehicle?stability which has principles analogous to the principles of ABS. In [13], a method based on fuzzy logic to?control?yaw and lateral slip has been presented. In this method by installing controllers on rear motors lateral stability of?vehicles?at high speed has been controlled.
In this paper, obtaining maximum brake force by using mechanical brake with constant force and electrical brake with the probability of adjusting slip at the same time is aimed. Considering the?braking?force is depend to the road profiles, increasing?braking?torque on the wheel causes wheel lock and consequently leads to approximately 30% reduction in?braking?force and 100% lateral stability. In proposed method, by getting feedback from wheels slip, the quantity of it is fixed at optimal amount (normally 0.2) to obtain maximum braking?force. In addition, some other problems which ABS?systems?on icy or dusty roads have, has been obviated to somewhat.
PRINCIPLES OF?BRAKING?FORCE
When the?vehicle?is moving with constant velocity, its velocity is proportional to its wheels speed, and wheels have not slip. But when the driver presses on the brake pedal to reduce speed, wheels speed gradually decreases and theirs proportion with?vehicle's body is destroyed. It should be noted that, in this case due to the force of inertia, the?vehicle's body is tendency to move. Thus, a small slip between the wheels and the road surface is created. The difference between wheel speed and?vehicle?speed indicates the amount of slip. Slip rate is calculated as follows:
Λ=(Vv-Vw)/Vv (1)
In Eq. (1), parameters Vv and Vw are indicates?vehicle?velocity and wheel speed, respectively. Zero percent of the slip rate shows the wheel moves freely and is not facing with any obstacles. One hundred percent of the slip rate shows the mode in which the wheel completely locks and wheel slipping extremely on the road. By increasing the difference between the wheel speed and?vehicle?velocity, the slip between the wheels and road surface is too high. This is lead to cause friction and?braking?force. Then, the?vehicle?velocity is reduced. Relation between the?braking?force and the slip rate is shown in Fig. 1. Brake force is not always associated with slip rate. However, when the slip rate is between 10 to 30 percent, the maximum?braking?force is obtained [14].
OPTIMUM DISTRIBUTION OF?BRAKING?FORCE
When the brake pedal is pressed, a?braking?torque is created on the brake disc. This?braking?torque makes a force in the contact area of wheel and ground. If this force exceeds the maximum?braking?torque, it will stop the?vehicle.?Braking?force and its maximum values can be obtained as follows:
Fb=Tb/rd (2)
Fbmax=Ub*w (3)
In equation (2) and (3), Tb, rd , and co refer to torque, effective radius of wheels and wheels speed, respectively. Hb is the adhesion coefficient of road and wheel and it varies by slip rate. Its maximum value can be obtained in the (1520) percentage of slip.?Braking?force increases with the increase in brake torque. As shown in Fig. 2, when?braking?force reaches to maximum tolerable road wheels amount, its value remains almost constant [14]. Maximum achievable?braking?force between the road and wheels is depended to /ib and vehicle's weight.
BRAKING?FORCE BETWEEN FRONT AN REAR AXLES
The forces into the?vehicle?on a flat road shown in Fig. 3.?Steering?wheel and aerodynamic resistances are neglected due to their small value in comparison with?braking?force. Negative acceleration of?vehicle?in braking?mode is defined as j which can be easily obtained as follow:
j=(Fbf+Fbr)/Mv (4)
Where, Fbr and Fbf are the?braking?forces which act on front and rear axles of?vehicle. The maximum?braking force is limited by the adhesion coefficient of wheel-road and mechanical load of each wheel. Therefore, the braking?force obtained from?braking?torque should be proportional to the loads of front and rear axles. In a result, front and rear axles achieve their maximum?braking?force at the same time. Neglecting the mass movement from rear axle to front axle during the?braking, wheels' weight in front and rear axles on the contact points A and B of Fig. 3 can be calculated as (5) and (6)[4].
Wf=Mv*g*(Lb+hg*j/g)/L (5) Wr= Mv*g*(La-hg*j/g)/L (6)
Moreover, the value of?braking?force in front and rear axles should be equal to nominal value of weight. So:
Fbf/Fbr= Wf/ Wr=(Lb+hg*j/g)/ (La-hg*j/g) (7) Comparing (7) to (4), in ideal?braking,?braking?forces in front and rear axles are shown in Fig. 4, where, j is the maximum negative acceleration of?vehicle?on road.
According to Fig. 4, ideal?braking?force distribution curve which is named I-curve is a non-linear hyperbolic curve. For locking the front and rear wheels simultaneously,?braking?forces in front and rear axles should follow the I-curve. In a real design, these forces and their ratio are considered linear and defined the ratio of?braking force of front axle to the total?braking?force, according to (8).
β=Fbf/Fbr (8)
Fig. 5 shows the ideal and actual?braking?curves[14]. In Fig. 5 clear that these curves meet together only in one point and in this point, front and rear wheels are locked in the same time. This point appears for a certain value of adhesion coefficient Ho and can be computed from the following equations:
Fbf/Fbr=β/(1-β) (9) β/(1-β)= (Lb+hg*j/g)/ (La-hg*j/g) (10)
u0=(L*β-Lb)/hg (11)
With?braking?in a road which it's ia is less than Ho (A region that the ? -curve is under I-curve), front wheels are locked before rear wheels and vice versa. When the rear wheels lock first, the?vehicle?loses its directional stability and lateral stability of rear wheels decreases to zero. In this condition, a small lateral force such as wind or centrifugal forces in road turns can cause lateral instability and?vehicle?will tum 90 and then 180 degrees around itself and will be taken out of the road. On the other hand, when the front wheels are locked due to?braking, the driver will lose the?vehicle control?on forward direction and will not be able to effective control?of?vehicle. However, it does not mean that the instability is completely happened. Since, there is a self correcting force in the front axle which prevents the lateral instability [14].
According to these discussions, it seems that the locking of rear wheels are more perilous, especially in the roads with small /a , because in such kind of roads?braking?force is less and kinetic energy of?vehicle decreases gradually and so, in instability condition,?vehicle?turns more distances around itself. Therefore, in vehicle?design, it must be ensured that the rear wheels do not lock first. Due to restrictions in the amount of generated torque and the supply of traction motors which used in electric?vehicles?and large differences of ja0 that shown in Fig. 6, electrical and mechanical torque ratios can be adjust by applying the slippery road condition. As shown in