TW201228214A - Method for controlling a permanent synchronous motor with low resolution rotor position feedback sensors - Google Patents

Abstract

This invention is related to motor control method for a permanent synchronous motor with only three Hall-effect sensors as rotor position feedback sensors. First, the permanent synchronous motor is controlled with 180 DEG six-step current control at start-up and lower speed, and then synchronous frame current control method in middle to high speed range is proposed. Therefore, high torque in low speed and less vibration and noisy in high speed can be obtained. Furthermore, overall system efficiency is improved. Further enhance low speed regulation characteristics, a speed observer with load feedback is added to improve the resolution of the rotor position feedback sensors.

Description

201228214 VI. Description of the Invention: [Technical Field] The present disclosure relates to a permanent magnet synchronous motor driving material, and more particularly to a permanent magnet synchronous motor driving system for returning a Hall magnetic field as a rotor magnetic field position. [Prior Art] Generally, a permanent magnet synchronous motor needs to use a sensing element to detect a rotor magnetic field position during control to control the motor speed or output torque, which is especially important for electric vehicles that emphasize safety spot efficiency. However, many permanent magnet synchronous motors are used as power. Many low-resolution rotor magnetic field position feedback devices such as Hall-sensing 7L parts are used to replace the rotor magnetic % position feedback of the 咼j shell. This kind of permanent magnet synchronous motor uses the six-step wave current limiting driving method. Although this method is easy to control, it will cause the motor output torque to be chopped and the noise and vibration are too large and the efficiency is not good. Although many documents [2] 3] and patents [4, 5] are improving this shortcoming but have limited success. In addition, if this method is applied to an electric vehicle, it will cause a large chopper current output from the battery, which makes the processing of the battery management system more complicated. Figure 1 is a six-step wave current limiting control commonly used in electric vehicles. Method, it is apparent that there is a large battery output current chopping.

Yamaha Motor's patent [7] uses a low-resolution rotor field position feedback device to perform vector control of the motor in the synchronization frame, but provides information on how to generate sufficient rotor position resolution from the low-resolution rotor field position feedback. The description of the use of vector controls is limited. Furthermore, this method has its limitations if it is used to start a large torque or a low speed load torque change, such as when the electric vehicle is driven in a crowded metropolitan area. [1] R. Carlson, M. Lajoie-Mazenc, and JCS Fagundes, ''Analysis of torque ripple due to phase commutation in brushless DC machines, 5, IEEE Trans. Ind. Applicat., vol.28, no. 3, Pp 632-638, 1992.

[2] T.D. Batzel and K.Y. Lee, “Commutation torque ripple minimization for permanent magnet synchronous machines with Hall effect position feedback, 5, Trans. Energy Conversion, vol. 13, no. 3, pp 257-262, 1998.

[3] T. Hui ^ Controllability Analysis of torque ripple due to phase commutation in brushless DC motors, 5, Proc. of 2009 International Conference on 201228214

Electrical Machines and Systems, 2009, pp. 1317-1322.

[4] Republic of China Patent, 1229972, “Low-cost DC brushless motor digital position drive control system, [5] Republic of China patent, 1294717, “Motor drive, motor controller and method of controlling electric motor”. Patent of the Republic of China, 1315598, "Brushless motor drive circuit for reducing electromagnetic noise and its method". [7] "Wheel driving apparatus and electric vehicle including the same" EP2093098A1 ' ¥ SUMMARY OF THE INVENTION In summary of the above disadvantages, the present invention The low-resolution rotor magnetic field position sensor is used as the permanent magnet synchronous motor driving device of the feedback element. The sensing component used is the permanent magnet synchronous motor rotor magnetic field position, and generates a square wave signal with an electrical angle of 12 degrees. The invention uses the three signals as the input motor current control. Firstly, when the motor starts, that is, at low speed, the six-step wave output current with current loop control is applied to the motor 'refer to FIG. 2; and the motor is high. When the time is lost, the sinusoidal current is output to the motor by the synchronous frame current control. Referring to FIG. 3, the motor output torque is respectively performed. The control, in the figure, is the U-phase current of the permanent magnet synchronous motor, the U-phase current command and the output current of the power supply device. Comparing Fig. 1 to Fig. 3, the battery output current of the invention under the same conditions can be found. The wave is much lower than the traditional method. Figure 4 shows the waveform of the proposed invention switched from the six-step current control to the synchronous frame current control. It can be clearly found that the switching to the synchronous frame current control greatly improves the vibration of the motor. Noise suppression is also quite effective. / To improve the low-speed regulation of the motor, the speed observer is used as an input to estimate the position and degree of the current rotor of the motor using a low-resolution rotor field position sensor. Time as Time base, for three; square wave signal // „, 扎 and 圪 output, after the fine difference of the reading position, then the wheel and the second detector will be able to obtain a more stable speed and rotor position. Feedback. Fig. 5 and Fig. 6 show the speed at which the conventional method and the estimated speed after the implementation of the present invention are the same, and the stable speed can be obtained at 90 rpm. x 201228214 [Embodiment] Embodiments of the present invention refer to FIG. 7, and the apparatus includes: a control device 1GG. A micro-processing|| is constructed with a hardware circuit to implement a control strategy of the present invention as a read body. A software control mechanism 150: implements the proposed control method. Taking the microprocessor of the control device 100 as the core, the software is

The force rate unit 200 is composed of a power supply 2 i 〇, a three-phase power crystal and a current sensor 230, and is used to receive the power crystal driving signal transmitted by the control device to generate a voltage to drive the permanent magnet. Synchronous motor 3〇〇. A permanent magnet synchronous motor 300. - Hall sensor 310: Detects the position of the permanent magnet synchronous motor rotor magnetic field, and the return signal is sent to the sensor processing circuit 740. A sensor processing circuit 740: adjusts the signal input by the Hall sensor 31〇 via the level, and sends out three square wave signals, which are separated by an electrical angle of 120 degrees, and outputs, and the waveform is as shown in FIG. 8. Where 7 is the electrical cycle time, [~[ is the timing of the commutation in one electrical cycle, the six time points are located at the rising or falling edge of the 乂, 仏 and ~ signals, so an electrical cycle can be made Γ/〜Γ(5 time points are divided into six sections, the knife is zone 1 to zone 6, and each zone is 6 〇. Electrical angle. The present invention provides two motor drive modes, such as torque and speed control, The source of the torque command is selected by the drive mode switch 800 via switch S1, which is output by the speed controller 4 (7) or by the torque command τ > Γ, converted from the torque _ current gain 81 〇 ° to the current command /丨. Subsequently, the current mode selection 82〇 selects the six-step wave current control 5〇〇 via the switch 32 and the medium-low speed use mode, or the high-speed mode 2 the same step block current control 600. The current mode selection 820 is the basis for using speed as the basis for selecting the error to prevent the speed detection and avoiding the mode switching point. 0 0 ~ 丨 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Accelerate to high Under the condition, when |ω| 丨 mode one of the six-step current control 500, and when μ> _|| is the same as the mode two frame current control 600. 201228214 Tian motor by idle speed (1 "2| ) Deceleration to low speed synchronous frame current control 600, and 1/1 sub-type is 杈 type 2 with - Γ Γ 1/1 mode - the six-step wave current control track /,, and the conversion time of the formula 1 is divided according to Figure 9. The speed shown is later. The timing of the switching of the current command is φ. The extension is fixed. The extension is fixed. J1: It is fixed at any specific point of Γ/~Γ6, for example, the rising edge of the solid. 7) 'Let the money in the mode switching process be more stable. Mode - the six-step wave current control (10) is via the sensor processing circuit 7

__, number as a reference, respectively output waveform ^, slice: and <, as shown in Figure 10, the current command is M C = ^> K (1)

Lv~IpxHv (2) lw = Ip X Hw (3) Ten, these three commands are respectively sent to the u-phase f-flow controller 521 after being subtracted from the current feedback signals ^, : and /, respectively. v phase current controller 522 Xiao

After the controller 523 is operated, the reverse 屮啻 人 人 人 人 L ^ ^ ^ ^ ^ ^ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , The crystal 22 〇 sends a varying voltage to achieve the purpose of controlling the input permanent magnet synchronous motor 3 〇〇 current. The mode II synchronous frame current control 600 is generated by the DQ current command generation mechanism 440 ’. The current command is I*px COS φ (4) lds = I*px sin φ (5) where ', 4 is the phase lead angle. The two commands are respectively subtracted from the current feedback signals V and L, and then sent to the Q-axis current controller 621 and the D-axis current controller 622, respectively, to 'send the voltage command ~ and ~, and then through the 2-to-3 conversion. The device 63 sends the voltage command ν "2,, and ~. Then, the pulse width modulation mechanism 720, the power crystal driving circuit 710 and the three-phase power crystal 220 send the varying voltage to control the input permanent magnet synchronous motor 300 current. The current feedback signal b and the heart are calculated by the current feedback signal, /v and via the 3-to-2 converter 610. The pulse signal source selection 730 is an instruction to accept the current mode selection 820, and the input voltage command is selected. 201228214 The pulse width modulation mechanism 720 is a signal that converts the voltage to a wide voltage. The command is converted to a fixed frequency with different pulse power. The crystal drive circuit 710 is a fixed bottle that can push the power crystal to be reduced. W-shirt with the signal of « The conversion to the speed controller 410 is to use the speed command (7) ♦ and the speed feedback, the error as a change to change the output permanent magnet, the motor __ size of the square wire reduction speed = angle information fine difference is %, ^The signal is finely differentiated, as shown in Figure η. 'The implementation method is based on the interruption of the current loop ^ as the time base, assuming = the synchronous speed of the magnetic synchronous motor 稳定 or stable (four), so the required interruption time °; As the current electrical angle estimate, suppose the number of interruptions in f intervals is =, the average electrical (four) of the county breaks is yeah-...dead (6), so the current estimated electrical angle is eca) G(J) 9c{k)^e{J) + ^ec{J-\) (7) The electrical angle of the k-th current loop interruption in the current interval. The electrical angle of the current interval, J=1~6, and, • β(2 ) = 60., Μ), ., θ (4 b gift, β (5) = 240., (10) household pass.; that is, the heart of interval 1 = state continued. Similarly, interval 6 te (J) = e(6) = 3QQR. When the transition from the -interval to the lower-interval, set k to Q and load the electrical angle _ of the current interval, and perform the operation of (7). In addition, when ^() is added Take the average electrical angle change of the interval 6 of the previous electrical cycle. The speed observer 4 3 〇, referring to Figure 12, is the output of the angular information fine differential 4 2 为 as the command and the Q-axis current "or current 々乍 is the input of the load estimation, and the current position of the rotor of the permanent magnet synchronous motor 300 is estimated to provide the synchronous frame current control of the f-type 2 _ 2 f" converter 63 〇 and 3-to-2 conversion The utility model uses the 610. The observer reduces the error between the expected rotor position 201228214 W and the observer rotor output position by the closed loop control mechanism, and increases the motor load variation to improve the accuracy of the measurement. Instantly estimate the motor torque. In the speed observer 43G, '& is the estimated motor torque constant, the heart is the motor spot whose combined inertia is the control parameter, 2_, indicating _ current: sampling time 7; When < and Λ are closer to the actual state of the permanent magnet synchronous motor 3 〇 , the estimated value of the velocity observer 430 will be closer to the actual value. In addition, it is also estimated that the rotor speed controller 410 of the present permanent magnet synchronous motor is used. < 乂 ί工 φ [Simple diagram of the diagram] Figure 1 shows the current flow of the six-step current limiting control method commonly used in electric vehicles. Figure 2 shows the 5(9) output current with a six-step current control at the start of the permanent magnet synchronous motor and at low speed. Figure 3 shows the output current of the synchronous frame current control 600 at high speed in a permanent magnet synchronous motor. FIG. 4 is an output waveform of the present invention which is controlled by a six-step current control and is switched to a synchronous pivot current of 60 。. Figure 5 is the speed information estimated by the traditional method. Figure 6 is the speed information estimated after the implementation of the present invention. Figure 7 is a block diagram of the system of the present invention. Figure 8 is a voltage output waveform of the sensor processing circuit 74. Figure 9 is the basis for the conversion timing of mode one and mode two. FIG. 1 is a perspective information fine differential 420 output signal waveform of the present invention. Figure 11 is an angle angle of the information fine differential 420 output of the present invention. Figure 12 is a velocity observer 430 of the present invention. 201228214 [Main component symbol description] Add device name 210: power supply device 220: three-phase power crystal 230: current sensor 310: Hall sensor 410: speed controller 420: angle information fine differential 430: speed observer 440: DQ current command generation mechanism 510: Six-step current command generator 521: U-phase current controller 522: V-phase current controller 523: W-phase current controller 610: 3-to-2 converter 621: Q-axis current control 622: D-axis current controller 630: 2-to-3 converter 710: power crystal drive circuit 720: pulse width modulation mechanism 730: pulse signal source selection 740 · sensor processing circuit 800: drive mode switching 810 : Torque-current gain 820: current mode selection / „ : u phase current of permanent magnet synchronous motor 300 iv : v phase current of permanent magnet synchronous motor 300 : w phase current of permanent magnet synchronous motor 300 : six step current Control u-phase current command of 500 /;: v-phase current command of six-step wave current control 500 201228214 C: w-phase current command of six-step wave current control 500: power supply device 210 output current: sensor processing circuit 740 Output u / / v: sensor phase processing circuit 740 output v phase signal / / w: sensor processing circuit 740 output w phase signal β: angle information fine differential 420 output u phase signal: angle information fine differential 420 output ν phase signal < : angle information fine differential 420 output w phase signal Te : a 360 degree electrical cycle time of the permanent magnet synchronous motor 300 T wide T6 : a commutation of a permanent magnet synchronous motor 300 in a 360 degree electrical cycle Time: Torque command output by speed controller 410 ': Torque command 4: Torque-current gain 810 Output current command 51: Switch 52 of drive mode switch 800: Switch of current mode selection 820 ω: Speed observer 430 Output motor speed φ: DQ current command generation mechanism 440 current phase lead voltage phase angle νΜ / · Six-step wave current control 500U phase voltage command vw: six-step wave current control 500V phase voltage command vw / : six-step wave Current Control 500W Phase Voltage Command 4: Synchronous Frame Current Control 600Q Axis Current Command 4: Synchronous Frame Current Control 600D Axis Current Command b: Q-Axis Current L Generated by 3-to-2 Converter 610: Generated by 3-to-2 Converter 610 D-axis v : Synchronous frame current control 600Q axis voltage command: Synchronous frame current control 600D axis voltage command vu2 : Synchronous frame current control 600U phase voltage command vv:: Synchronous frame current control 600V phase voltage command 10 201228214 : Synchronous frame current control 600W phase voltage Command J: Input speed command Tc of speed controller 410: Interruption time of current loop N: Number of interruptions per section of current loop k: Current interval interrupt number 6»c^): Current k-th phase current loop interruption occurs Electrical angle θ ε : (= pJ3 to 2 converter 610 and 2 to 3 converter 630 conversion required electrical angle A: The motor rotor position estimated by the speed observer 430 is: the current angle / / electrical angle k, : motor torque constant Λ : combined inertia of the motor and its load A: control parameter of the speed observer 430 2: control parameter of the speed observer 430 3: control parameter Z-1 of the speed observer 430: one current loop Sampling time 7; delay ω, hysteresis function first speed point %: hysteresis function second speed point h| : absolute value of motor speed output by speed observer 430, * = 1 or 2 T : permanent magnet synchronous motor 300 Output turn B: coefficient of friction difference TL: load disturbance P: the number of poles of the rotor of the permanent magnet synchronous motor 300

Claims (1)

201228214, the scope of patent application: VII. Rotor motor drive device' is characterized by simple three Hall elements as the mold modulo f. Both control modes provide output permanent magnet synchronous horse current control. When the permanent magnet synchronous motor is in the =2 = part of the feedback signal, the rotor magnetic field position is finely differentiated and the load is compensated - to change the speed of the 1 2 3 4 money (four) secret. The proposed system includes, apparatus, apparatus for controlling the control strategy of the present invention constructed by a microprocessor and a hardware circuit. (4) It is expected that the software control mechanism of the #blade body is realized: the control method and the speed estimation are taken as the core of the control step 罟 ' ' ' ' ' ' 以 以 以 以 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源 电源The power crystal and current sensing circuit μ receives the power crystal driving signal from the control device to generate a signal. Dynamic water magnetic synchronous motor. Among them, the current sensor can output a three-phase current of a permanent magnet synchronous motor. , sensor, detecting permanent magnet synchronous motor rotor magnetic sensor processing circuit. Ϊ «送入 - =1: Processing circuit: The signal input from the Hall sensor is output via two square wave signals 乂, ^ and ^ separated by 120 degrees electrical angle. The hydromagnetic synchronous motor drive unit k 1 has a soft body control mechanism that provides speed j or output torque control. The software control mechanism includes - speed controller, one drive system, one torque current gain, one current mode selection, one six step wave current control 2 synchronous frame current control, one pulse signal source selection, one pulse width adjustment Change the software function module such as 3, , angle information fine difference and a speed observer. 4 = Item 2 software control mechanism, the software function of the speed controller includes a speed 栌 = benefit, - angle information fine difference, a speed observer, a drive mode switch, two moment current gain, a current mode selection, one six Step wave current control, - synchronization box 10 201228214 system, - pulse signal source selection and - pulse width modulation mechanism and other software power month t» chess group. 4.=Solution 2 software control mechanism, the software function of output torque control includes a torque = increase ^, - angle information fine differential, - current mode selection, - six-step wave current i mechanism Rong It box current control, one Pulse signal source selection and a pulse width modulation mechanism 荨 software function module. • V 1 permanent magnet synchronous motor drive, whose speed or output torque controls the preset operating mode of the modular device, or by external device via the signal interface & ° command for speed or output torque control mode Switching. Hf, 1 permanent magnet synchronous motor drive device, current control dragon type switching is to use _', = as the basis for selection and add delay Wei set the first - speed point ^ and the first = "2" as hysteresis The judgment speed of the function, wherein the first speed ^ is less than or the first speed point %, that is, the | called |_吟 When the motor is started in the low speed zone and the second speed speed is called, the current control mode is six steps. Wave current control. When the speed exceeds the second speed point %, the current control mode is switched to the sync box current (4). And _b 2| respectively, the table "with the absolute value of %. ^ Please ^ item 6 of the current control mode switching, when the motor is in the high-speed zone and the speed is higher than the degree point' current control mode for the synchronous frame current control. Then when the motor When the demon is less than the first degree, the current control mode is switched to the six-step current path. 8'=The switching of the current control mode of the solution 6 is when the motor speed is positive. It means that the motor speed ω is reversed and the speed is represented by a negative value. The hydromagnetic synchronous motor drive device of the month 1 provides six current control modes such as six-step current control and the same two-flow control. The detector processing circuit extracts three square wave signals and I, the three square wave signals 乂, 仏, and fire 1 〇 information fine differential n module 'generates three square wave signals to open: good: good: output. π ° π 'the sensor processing circuit of the item 9 outputs three read-only signals A, sense and chopping> can be represented by an electrical angle of 0 to 360 degrees, which is divided into six intervals. 13 201228214 "1", Hv = ''0,,, Η,, ='' 1 ,, . "1", Η, 0 ", =" 0 = "1 ,, > collapse = 'T ,, =" "//." 0 ,,. //„=“0”, "ν = “1,,, Κ: =“ο”. //„=“0”, “Γ,, Κ: =“Γ, ο //„=“0,,, “0”, Κ: ='' 1 ,, 〇(8) Interval 1 is 0~60 degrees: (b) Interval 2 is 60 to 120 degrees: (c) Interval 3 is 12〇~18〇 (4) Interval 4 is 180~240 degrees (4) Interval 5 is 240~300 degrees (1) Interval 6 is 300~360 degrees _ and " 1" indicates that the logic level is high, "〇" indicates the logic level (low). The m is the low 11. As in the request item 9, the angle information fine differential module outputs three signals " 0.5, ":=-1·0, κ= 0.5. "„* 1.0, ugly:=-0.5, κ= :-0.5. 0_5, ":=0.5, κ=. -1.0 . Κ= -Q.5, Κ= lo , κ= :-0.5 〇 //:= -1.0, ":=0.5, κ= =0.5. Η[= -0.5 > Ην= -0.5, κ = 1.0 〇.β, //: and //: the value, -1.0 1 -0.5 > : The electrical angle of ~360 degrees is divided into 6 intervals. It is (0) & 1 is 0~60 degrees: r (8) Interval 2 is 6〇~12〇: (C) Interval 3 is 120 ~180 degrees: (d) Interval 4 is 180~240 degrees: (e) Interval 5 is 240~3 degrees: (7) Interval 6 is 300~360 degrees: 12 discs are as requested by the three signals of the U : The value, heart 2, 5., w (4) In the fixed-point operation using software, its representative meaning is (bX5 means one of the largest negative values (4) 〇.5 means the largest positive value - half means the maximum positive value of 13 as ^ mountain Seeking a 9-member permanent magnet synchronous motor drive device, the six-step wave current command generates a write message: the difference is the torque current gain output current size; and the angle / /.. / _, and output two signals <, 圮 and 圮 multiply, respectively, sequentially generate /:, life = three-phase current command. That is, 'u phase Flow command «V phase current 14. If request 2: and w phase current command 3 turn 榼 Wu's permanent magnet synchronous motor drive device, synchronous box current control 2 bis and ^ 3 to 2 converter required electrical angle ", Can be angle information fineness ^ angle see multiplier with the permanent magnet synchronous motor rotor pole number and then divide 2 or speed 14 201228214 degree view: the wheel angle of the device θ 〆 interrupt ^ software control mechanism, angle information fine difference is Take the number of the current circuit as: :; Electric base:: Sense: 处理 处理 处理 ... _ _ _ _ _ _ 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以 以For the second angle of the gas angle, except for the angle of the current circuit required for the previous rotation of the interval: the interval of the day is increased by the interruption time of a current loop ^ u 4 degrees; the frequency of the differential differential output is here. Interval current;: the number of interruptions is multiplied by the angular increment and then the number of interruptions of the electrical circuit from the starting angle of the interval - as the current angular increment of - interval increment - ί When the motor is running forward, this 1 7 ^Bb value, the S motor is The angle increment is negative when turning. In the angle/item 15 angle information fine differential software module, taking the signal of the sensor processing circuit as input, the degree electrical angle is divided into interval i = interval 6 Each interval is 6 〇. Electrical angle. When the motor is forward rotation, the starting angle of the 2nd 2 is 0 degrees, the starting angle of the interval 2 is 60 degrees, and the mouth of the interval 3 is 120 degrees. The starting angle is 18 degrees, the starting angle of interval 5 is 240 degrees, and the starting angle of interval 6 is 3 degrees. = explicit item 15 t angle information fine differential software module, with the sensor processing circuit \ 乂, 乂 and heart signal as input, the 36 电气 electrical angle is divided into interval 丄 = interval 6 'each interval is 6Q . Electrical angle. & When the motor is reversed, the starting angle of interval 1 is 60 degrees, the starting angle of interval 2 is 12 degrees, the starting angle of interval 3 is 180 degrees, and the starting angle of interval 4 is 24 inches. Degree, the starting angle of the interval 5曰 is 300 degrees, and the starting angle of the interval 6 is 36 degrees. A, for example, the velocity observer of claim 2, wherein the input command outputs the electrical angle at the angle of the request item 15. The output of the speed observer is the estimated permanent magnet synchronous electrical angle 6 and speed α, which respectively provide the angle information of the current control and the speed controller. In the six-step wave current mode, the torque current is increased by two currents/) as input, or the synchronous frame current is controlled by the Q-axis current command γ^201228214. The drive mode switching can be performed by the microprocessor preset. When operating in the torque mode, the permanent magnet synchronous motor drive program of the item 1 is input or the command mode is externally input via the communication interface and the selected mode. Han Yutuo Yuehe 1 item 1 water magnetic 151 step motor drive device, | in the torque mode, the life program can be preset by the microprocessor to execute the program or input via the communication interface or by analog signal. ° 22. As requested in item 2, the torque command is converted to a current command via the torque current gain. Torque current benefits can be pre-stored in the memory of the microprocessor or via external inputs. 23. As requested in the permanent magnet synchronous motor drive, the current control mode maintains a six-step current control mode at all speeds and uses a speed observer to provide feedback information for the turbulence and rotor position. ’, 16