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CN104483975B - A kind of UUV adaptive turning speed methods of adjustment based on way point - Google Patents

A kind of UUV adaptive turning speed methods of adjustment based on way point
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CN104483975B
CN104483975BCN201410617653.4ACN201410617653ACN104483975BCN 104483975 BCN104483975 BCN 104483975BCN 201410617653 ACN201410617653 ACN 201410617653ACN 104483975 BCN104483975 BCN 104483975B
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陈涛
张耕实
严浙平
徐达
张宏瀚
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Harbin Engineering University
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Abstract

Translated fromChinese

本发明公开了一种基于航路点的UUV转弯速度自适应调整方法。从使命文本读取UUV的前一航路点、当前航路点、下一航路点以及规划速度,将规划速度作为速度指令输出;计算出当前航路角;计算当前航路的转弯半径;惯性导航仪实时采集UUV的实际位置;判断UUV是否满足启用转弯速度的条件,如果满足条件,计算并输出转弯速度指令;光纤罗经实时采集UUV的实际航向;判断UUV是否到达当前航路点,如果到达当前航路点,判断使命文本中的航路点是否都已到达,如果都到达,任务结束。本发明可以减少UUV转弯时的航路超调,改善航路跟踪效果。

The invention discloses an adaptive adjustment method for UUV turning speed based on waypoints. Read the UUV's previous waypoint, current waypoint, next waypoint and planned speed from the mission text, and output the planned speed as a speed command; calculate the current waypoint; calculate the turning radius of the current waypoint; real-time acquisition by the inertial navigator The actual position of the UUV; judge whether the UUV meets the conditions for enabling the turning speed, and if the conditions are met, calculate and output the turning speed command; the fiber optic compass collects the actual heading of the UUV in real time; judge whether the UUV has reached the current waypoint, and if it reaches the current waypoint, judge Whether all the waypoints in the mission text have been reached, and if so, the mission ends. The invention can reduce the route overshoot when the UUV turns and improves the effect of route tracking.

Description

Translated fromChinese
一种基于航路点的UUV转弯速度自适应调整方法A Waypoint-Based UUV Turning Speed Adaptive Adjustment Method

技术领域technical field

本发明属于一种UUV速度调整方法,尤其涉及UUV航行转弯时的一种基于航路点的UUV转弯速度自适应调整方法。The invention belongs to a UUV speed adjustment method, in particular to a waypoint-based UUV turning speed adaptive adjustment method when the UUV is sailing and turning.

背景技术Background technique

目前,水下无人航行器(Unmanned Underwater Vehicle—UUV)执行使命时最常用的航行路径是以多个航路点联接而成的直线段路径。当UUV从一个航路转换到另一个航路时就必须要执行转弯动作。目前UUV进行水下航行任务时,接收的使命文本中仅包含航路点的位置信息和规划出的UUV速度信息,使命文本中并没有包含转弯速度信息,所以通常的做法是设定一个固定的转弯半径,当UUV与当前航路点的距离小于等于转弯半径时,UUV开始转弯,但并不对转弯时的速度进行调整,主要是不降速,整个航行过程均采用统一的规划速度。这样的做法存在以下两个问题:(1)固定的转弯半径并不适合所有的转弯角(两个航路之间的航路角)情况;(2)将会导致UUV以较大的速度进行转弯。以上两个问题综合起来将会造成UUV出现急转弯,使转弯效果变差,能耗增加并且产生航路超调。特别地,当转弯角度较大时(直角或钝角度时),更加明显。At present, the most commonly used navigation path for an Unmanned Underwater Vehicle (UUV) to perform a mission is a straight-line path formed by connecting multiple waypoints. Turning maneuvers are necessary when the UUV transitions from one route to another. At present, when the UUV conducts underwater navigation missions, the received mission text only includes the position information of the waypoint and the planned UUV speed information, and the mission text does not include the turning speed information, so the usual practice is to set a fixed turning Radius, when the distance between the UUV and the current waypoint is less than or equal to the turning radius, the UUV starts to turn, but does not adjust the speed when turning, mainly does not slow down, and adopts a unified planning speed throughout the navigation process. This approach has the following two problems: (1) The fixed turning radius is not suitable for all turning angles (the route angle between two routes); (2) it will cause the UUV to turn at a higher speed. The combination of the above two problems will cause the UUV to make a sharp turn, which will make the turning effect worse, increase energy consumption and cause route overshoot. Especially, when the turning angle is large (right angle or obtuse angle), it is more obvious.

发明内容Contents of the invention

本发明的目的是提供一种能够减少转弯航路超调、改善航路跟踪效果的基于航路点的UUV转弯速度自适应调整方法。The purpose of the present invention is to provide a waypoint-based UUV turning speed adaptive adjustment method that can reduce the overshoot of the turning route and improve the route tracking effect.

本发明是通过以下技术方案实现的:The present invention is achieved through the following technical solutions:

一种基于航路点的UUV转弯速度自适应调整方法,包括以下几个步骤:A method for adaptive adjustment of UUV turning speed based on waypoints, comprising the following steps:

步骤一:从使命文本读取UUV的前一航路点pk-1的位置(xk-1,yk-1)、当前航路点pk的位置(xk,yk)、下一航路点pk+1的位置(xk+1,yk+1)以及规划速度信息,将规划速度信息作为UUV速度指令输出,根据读取的信息计算出当前航路角αkStep 1: Read the position of UUV's previous waypoint pk-1 (xk-1 , yk-1 ), the position of current waypoint pk (xk , yk ), the next route from the mission text The position of point pk+1 (xk+1 , yk+1 ) and the planned speed information, the planned speed information is output as the UUV speed command, and the current route angle αk is calculated according to the read information;

步骤二:根据当前航路角αk,计算UUV当前航路的转弯半径RkStep 2: Calculate the turning radius Rk of the UUV's current route according to the current route angle α k;

步骤三:惯性导航仪实时采集UUV的实际位置(xuuv,yuuv);Step 3: The inertial navigator collects the actual position of the UUV (xuuv , yuuv ) in real time;

步骤四:根据UUV的实际位置(xuuv,yuuv)和当前航路点pk的位置(xk,yk),判断UUV是否满足启用转弯速度的条件,如果满足条件,进行步骤五,如果不满足条件,重复步骤三~步骤四;Step 4: According to the actual position of the UUV (xuuv , yuuv ) and the position of the current waypoint pk (xk , yk ), determine whether the UUV satisfies the conditions for enabling the turning speed. If the conditions are met, proceed to step 5. If If the conditions are not met, repeat steps 3 to 4;

步骤五:根据当前航路角αk,计算并输出UUV转弯速度指令;Step 5: Calculate and output the UUV turning speed command according to the current route angle αk ;

步骤六:光纤罗经实时采集UUV的实际航向ψuuvStep 6: The fiber optic compass collects the actual heading ψuuv of the UUV in real time;

步骤七:根据UUV的实际航向ψuuv、实际位置(xuuv,yuuv)、当前航路点pk的位置(xk,yk)、下一航路点pk+1的位置(xk+1,yk+1)以及当前航路的转弯半径Rk,判断UUV是否到达当前航路点,如果到达当前航路点,转步骤八,如果没有到达当前航路点,重复步骤六~步骤七;Step 7: According to the actual heading ψuuv of the UUV, the actual position (xuuv , yuuv ), the position of the current waypoint pk (xk , yk ), the position of the next waypoint pk+1 (xk+ 1 ,yk+1 ) and the turning radius Rk of the current route to determine whether the UUV has reached the current waypoint, if it has reached the current waypoint, go to step 8, if not, repeat steps 6 to 7;

步骤八:判断使命文本中的航路点是否都已到达,如果都到达,任务结束,如果没有,重复步骤一~步骤八。Step 8: Determine whether all the waypoints in the mission text have been reached, if they are all reached, the mission ends, if not, repeat steps 1 to 8.

本发明一种基于航路点的UUV转弯速度自适应调整方法,还可以包括:A method for adaptively adjusting UUV turning speed based on waypoints of the present invention may also include:

1、当前航路角αk为:1. The current route angle αk is:

其中,N为航路点的总个数,uk=[xk-xk-1,yk-yk-1]T,uk+1=[(xk+1-xk),(yk+1-yk)]T,并且有αk∈[-π,π]。Among them, N is the total number of waypoints, uk =[xk -xk-1 ,yk -yk-1 ]T , uk+1 =[(xk+1 -xk ),( yk+1 -yk )]T , and there is αk ∈[-π,π].

2、UUV当前航路的转弯半径Rk为:2. The turning radius Rk of the UUV's current route is:

其中,Rmax为UUV转弯半径的最大限值,Rmin为UUV转弯半径的最小限值,σR为半径因子。Among them, Rmax is the maximum limit of the UUV turning radius, Rmin is the minimum limit of the UUV turning radius, and σR is the radius factor.

3、判断UUV是否满足启用转弯速度的条件为:3. The conditions for judging whether the UUV meets the enabling turning speed are:

4、转弯速度指令vcmd_k为:4. The turning speed command vcmd_k is:

其中,vmax为UUV转弯速度指令的最大限值,vmin为UUV转弯速度指令的最小限值,σv为速度因子。Among them, vmax is the maximum limit value of the UUV turning speed command, vmin is the minimum limit value of the UUV turning speed command, and σv is the speed factor.

5、判断UUV是否到达当前航路点的条件为:5. The conditions for judging whether the UUV has reached the current waypoint are:

其中,Ψ为航向偏差阈值。Among them, Ψ is the heading deviation threshold.

本发明的有益效果:Beneficial effects of the present invention:

可以使UUV在航行转弯时自适应的采取合理的转弯半径,并进行速度调整以降低转弯速度,避免了UUV急转弯的出现。It can make the UUV adaptively adopt a reasonable turning radius when sailing and turning, and adjust the speed to reduce the turning speed, avoiding the occurrence of UUV sharp turns.

通过对UUV转弯速度的自适应调整,可以消除UUV转弯时的航路超调,减少了能耗并且可以获得更好的航路跟踪效果。Through the adaptive adjustment of the UUV turning speed, the route overshoot when the UUV turns can be eliminated, energy consumption can be reduced and better route tracking effect can be obtained.

计算UUV的转弯半径和转弯速度时,只用到了航路点的位置信息,所需信息量少,计算简单,易于工程实现。When calculating the turning radius and turning speed of UUV, only the position information of the waypoint is used, which requires less information, simple calculation, and easy engineering implementation.

附图说明Description of drawings

图1为UUV航路点及航路角示意图;Figure 1 is a schematic diagram of UUV waypoints and route angles;

图2为UUV转弯速度自适应调整方法流程图;Fig. 2 is a flow chart of the UUV turning speed adaptive adjustment method;

图3为UUV转弯半径生成示意图;Figure 3 is a schematic diagram of UUV turning radius generation;

图4为UUV转弯速度指令生成示意图;Fig. 4 is a schematic diagram of UUV turning speed command generation;

图5为UUV转弯速度自适应调整系统示意图;Figure 5 is a schematic diagram of the UUV turning speed adaptive adjustment system;

图6为未利用本发明的UUV航行转弯效果图;Fig. 6 is an effect diagram of UUV sailing and turning without utilizing the present invention;

图7为利用本发明的UUV航行转弯效果图。Fig. 7 is an effect diagram of UUV sailing and turning utilizing the present invention.

具体实施方式detailed description

下面将结合附图对本发明做进一步详细说明。The present invention will be described in further detail below in conjunction with the accompanying drawings.

结合图1,UUV的航路点和航路角描述如下:Combined with Figure 1, the UUV waypoints and waypoints are described as follows:

定义UUV当前将要到达的航路点称为当前航路点pk,其位置信息为(xk,yk);定义前一航路点为pk-1,其位置信息为(xk-1,yk-1);定义下一航路点为pk+1,其位置信息为(xk+1,yk+1)。航路称为当前航路、航路称为下一航路,之间的方向角αk称为当前航路角。Define the waypoint that the UUV will reach at present is called the current waypoint pk , and its position information is (xk , yk ); define the previous waypoint as pk-1 , and its position information is (xk-1 , yk-1 ); define the next waypoint as pk+1 , and its position information as (xk+1 ,yk+1 ). route current route is called the next route, with The direction angle αk between is called the current route angle.

结合图2,介绍UUV转弯速度自适应调整方法:Combined with Figure 2, the UUV turning speed adaptive adjustment method is introduced:

步骤一:初始化设定Rmax、Rmin、σR、vmax、vmin、σv和Ψ;Step 1: Initialize and set Rmax , Rmin , σR , vmax , vmin , σv and Ψ;

步骤二:从使命文本读取UUV的前一航路点pk-1的位置(xk-1,yk-1)、当前航路点pk的位置(xk,yk)、下一航路点pk+1的位置(xk+1,yk+1)以及规划速度信息,将规划速度信息作为UUV速度指令输出,根据读取的信息计算出当前航路角αkStep 2: Read the position of UUV's previous waypoint pk-1 (xk-1 , yk-1 ), the position of current waypoint pk (xk , yk ), and the next route from the mission text The position of point pk+1 (xk+1 , yk+1 ) and the planning speed information, the planning speed information is output as the UUV speed command, and the current route angle αk is calculated according to the read information:

其中,N为航路点的总个数,uk=[xk-xk-1,yk-yk-1]T,uk+1=[(xk+1-xk),(yk+1-yk)]T,并且有αk∈[-π,π]。Among them, N is the total number of waypoints, uk =[xk -xk-1 ,yk -yk-1 ]T , uk+1 =[(xk+1 -xk ),( yk+1 -yk )]T , and there is αk ∈[-π,π].

步骤三:根据当前航路角αk,计算UUV当前航路的转弯半径RkStep 3: Calculate the turning radius Rk of the current UUV route according to the current route angle α k:

其中,Rk为当前航路的转弯半径,Rmax为UUV转弯半径的最大限值,Rmin为UUV转弯半径的最小限值,σR为半径因子。Rmax、Rmin和σR在初始化时设置。Among them, Rk is the turning radius of the current route, Rmax is the maximum limit of UUV turning radius, Rmin is the minimum limit of UUV turning radius, and σR is the radius factor. Rmax , Rmin and σR are set at initialization.

步骤四:惯性导航仪实时采集UUV的实际位置(xuuv,yuuv);Step 4: The inertial navigator collects the actual position of the UUV (xuuv , yuuv ) in real time;

步骤五:根据UUV的实际位置(xuuv,yuuv)和当前航路点pk的位置(xk,yk),判断UUV是否满足启用转弯速度的条件,如果满足条件,进行步骤六,如果不满足条件,转步骤四;Step 5: According to the actual position of the UUV (xuuv , yuuv ) and the position of the current waypoint pk (xk , yk ), determine whether the UUV meets the conditions for enabling the turning speed. If the conditions are met, proceed to step 6. If If the conditions are not met, go to step 4;

判断方法如式(3)所示。The judgment method is shown in formula (3).

如果式(3)满足,则UUV启用转弯速度;If formula (3) is satisfied, the UUV enables the turning speed;

步骤六:根据当前航路角αk,计算并输出转弯速度指令;Step 6: Calculate and output the turning speed command according to the current route angle αk ;

其中,vcmd_k为当前航路的转弯速度指令,vmax为UUV转弯速度指令的最大限值,vmin为UUV转弯速度指令的最小限值,σv为速度因子。vmax、vmin和σv在初始化时设置。Among them, vcmd_k is the turning speed command of the current route, vmax is the maximum limit of the UUV turning speed command, vmin is the minimum limit of the UUV turning speed command, and σv is the speed factor. vmax , vmin and σv are set at initialization.

步骤七:光纤罗经实时采集UUV的实际航向ψuuvStep 7: The fiber optic compass collects the actual heading ψuuv of the UUV in real time;

步骤八:根据UUV的实际航向ψuuv、实际位置(xuuv,yuuv)、当前航路点pk的位置(xk,yk)、下一航路点pk+1的位置(xk+1,yk+1)以及当前航路的转弯半径Rk,判断UUV到达当前航路点,判断方法如式(5)所示。Step 8: According to the UUV's actual heading ψuuv , actual position (xuuv , yuuv ), the position of the current waypoint pk (xk , yk ), the position of the next waypoint pk+1 (xk+ 1 ,yk+1 ) and the turning radius Rk of the current route to judge that the UUV has reached the current waypoint, the judgment method is shown in formula (5).

式中,Ψ为航向偏差阈值,在初始化时设置,一般可取Ψ<10°。如果式(5)满足,表明UUV的航向已经基本调整到下一航路的航向,即UUV转弯完成,已到达当前航路点,准备向下一航路点航行,转步骤九,如果不满足转步骤七;In the formula, Ψ is the heading deviation threshold, which is set during initialization, and generally Ψ < 10°. If formula (5) is satisfied, it indicates that the course of the UUV has basically been adjusted to the next route heading, that is, the UUV has completed the turn, has reached the current waypoint, and is ready to sail to the next waypoint, go to step 9, if not satisfied, go to step 7;

步骤九:判断使命文本中的航路点是否都已到达,如果都到达,任务结束,如果没有,UUV向下一航路点航行,重复步骤一~步骤九。Step 9: Determine whether all the waypoints in the mission text have been reached, if all have been reached, the mission ends, if not, the UUV sails to the next waypoint, and repeat steps 1 to 9.

结合图3介绍UUV的转弯半径生成示意图:Combined with Figure 3, a schematic diagram of the UUV turning radius generation is introduced:

根据式(2)生成UUV当前航路的转弯半径。由式(2)可以知道,UUV的当前航路的转弯半径Rk和Rmax、Rmin、σR以及当前航路角αk有关。其中,Rmax和Rmin是固定值,Rmin与UUV的固有转弯特性有关,可根据实验确定;Rmax一般可取Rmin的1.5~2倍;半径因子σR一般取0.5~2,可以根据实际需要进行设定。一旦Rmax、Rmin、σR设定,那么UUV的转弯半径将由当前航路角αk决定。图3给出了Rmax=35m、Rmin=20m,σR分别设定为0.5、1和2三种情况下,转弯半径与航路角αk的关系。从图中可以看出,无论哪种半径因子,当前航路角的绝对值越大,UUV的转弯半径越大,即UUV越应该提前转弯并进行速度调整。The turning radius of the UUV's current route is generated according to formula (2). It can be known from formula (2) that the turning radius Rk of the UUV's current route is related to Rmax , Rmin , σR and the current route angle αk . Among them, Rmax and Rmin are fixed values, and Rmin is related to the inherent turning characteristics of UUV, which can be determined according to experiments; Rmax is generally 1.5 to 2 times of Rmin ; the radius factor σR is generally 0.5 to 2, which can be determined according to actually need to be set. Once Rmax , Rmin , and σR are set, the turning radius of the UUV will be determined by the current route angle αk . Figure 3 shows the relationship between the turning radius and the route angle αk in the three cases where Rmax =35m, Rmin =20m, and σR are set to 0.5, 1 and 2 respectively. It can be seen from the figure that, regardless of the radius factor, the greater the absolute value of the current route angle, the greater the turning radius of the UUV, that is, the earlier the UUV should turn and adjust its speed.

结合图4,介绍UUV的转弯速度指令生成示意图:Combined with Figure 4, a schematic diagram of UUV turning speed command generation is introduced:

根据式(4)生成UUV当前航路的转弯速度指令。由式(4)可以知道,UUV的当前航路的转弯速度指令vcmd_k和vmax、vmin、σv以及当前航路角αk有关。其中,vmax和vmin与UUV的固有航速特性有关,可根据实验确定;速度因子σv一般也可取0.5~2,可以根据实际需要进行选择设定。一旦vmax、vmin、σv设定,那么UUV的转弯速度指令将由当前航路角αk决定。图4给出了vmax=5m/s、vmin=0.5m/s,σv分别设定为0.5、1和2三种情况下,转弯速度指令与航路角αk的关系。从图中可以看出,无论哪种速度因子,当前航路角的绝对值越大,UUV的转弯速度指令越小,即UUV越应该降低速度进行转弯。According to formula (4), the turning speed instruction of UUV's current route is generated. It can be known from formula (4) that the UUV's current route turning speed command vcmd_k is related to vmax , vmin , σv and the current route angle αk . Among them, vmax and vmin are related to the inherent speed characteristics of UUV, which can be determined according to experiments; the speed factor σv can generally be 0.5-2, and can be selected and set according to actual needs. Once vmax , vmin , and σv are set, the UUV's turning speed command will be determined by the current route angle αk . Figure 4 shows the relationship between the turning speed command and the route angle αk when vmax =5m/s, vmin =0.5m/s, and σv are set to 0.5, 1 and 2 respectively. It can be seen from the figure that, regardless of the speed factor, the greater the absolute value of the current route angle, the smaller the UUV's turning speed command, that is, the UUV should reduce its speed to make a turn.

结合图5,介绍UUV转弯速度自适应调整系统:Combined with Figure 5, the UUV turning speed adaptive adjustment system is introduced:

航路角生成模块从使命文本读取航路点信息,并根据UUV前一航路点、当前航路点和下一航路点的位置信息,计算出当前航路角,输出到转弯半径生成模块和速度指令生成模块。The route angle generation module reads the waypoint information from the mission text, and calculates the current route angle according to the position information of the UUV’s previous waypoint, current waypoint and next waypoint, and outputs it to the turning radius generation module and speed command generation module .

转弯半径生成模块根据当前航路角计算出UUV沿当前航路进行速度调整时的转弯半径,并输出到启用转弯速度模块The turning radius generation module calculates the turning radius when the UUV adjusts the speed along the current route according to the current route angle, and outputs it to the enabling turning speed module

惯性导航仪采集UUV当前的位置信息,输出到启用转弯速度模块。The inertial navigator collects the current position information of the UUV and outputs it to the turning speed module.

启用转弯速度模块根据UUV当前位置信息和转弯半径判断UUV是否启用转弯速度。如果启用,向速度指令生成模块发送启用转弯速度信号。The module of enabling the turning speed judges whether the UUV enables the turning speed according to the current position information of the UUV and the turning radius. If enabled, an enable turn speed signal is sent to the speed command generation module.

光纤罗经采集UUV当前航向信息,输出到停用转弯速度模块。The fiber optic compass collects the current heading information of the UUV and outputs it to the deactivated turning speed module.

停用转弯速度模块从使命文本读取航路点信息,并根据UUV的当前航向信息、UUV当前位置信息、当前航路点信息、下一航路点信息和当前航路转弯半径判断UUV是否到达航路点并停用转弯速度,如果停用,向速度指令生成模块发送停用转弯速度信号。Disable the turning speed module to read the waypoint information from the mission text, and judge whether the UUV has reached the waypoint and stop according to the UUV’s current heading information, UUV’s current position information, current waypoint information, next waypoint information and current route turning radius. Use the turn speed, if disabled, send a deactivated turn speed signal to the speed command generation module.

速度指令生成模块如果收到启用转弯速度信号,则根据当前航路角信息计算并输出UUV的转弯速度指令;如果收到停用转弯速度信号,则从使命文本读取规划速度信息并输出规划速度指令。If the speed command generation module receives the turning speed signal enabled, it will calculate and output the UUV's turning speed command according to the current route angle information; if it receives the turning speed signal disabled, it will read the planning speed information from the mission text and output the planning speed command .

图6和图7呈现了未利用和利用本发明方法的两种情况下UUV航行转弯效果的对比结果。Fig. 6 and Fig. 7 present the comparative results of the UUV navigation and turning effects under the two situations of not utilizing and utilizing the method of the present invention.

其中,图6中呈现的是未利用本发明的UUV航行转弯效果,图7呈现的是利用本发明的UUV航行转弯效果。从对比结果可以明显的看出,采用本发明后,UUV转弯时的航路超调被消除了,UUV的航路跟踪效果更好。另外,UUV的转弯过程更加平滑、转弯效率提高,减少了UUV的能耗。Among them, Fig. 6 presents the effect of UUV sailing and turning without using the present invention, and Fig. 7 presents the effect of UUV sailing and turning using the present invention. It can be clearly seen from the comparison results that after the present invention is adopted, the route overshoot when the UUV turns is eliminated, and the route tracking effect of the UUV is better. In addition, the turning process of UUV is smoother, the turning efficiency is improved, and the energy consumption of UUV is reduced.

Claims (5)

<mrow> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mi>tan</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>|</mo> <mo>|</mo> </mrow> <mrow> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>i</mi> <mi>f</mi> </mrow> </mtd> <mtd> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>k</mi> <mo>&lt;</mo> <mi>N</mi> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>e</mi> <mi>l</mi> <mi>s</mi> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
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