Approach Parameters in Marine Navigation Graphical Interpretations

Transcription

1 the Intenational Jounal on Maine Navigation and Safe of Sea anspotation olume 11 Numbe 3 Septembe 017 OI: / ppoach Paametes in Maine Navigation Gaphical Intepetations.S. Lenat Gdynia Maitime Univesi, Gdynia, Poland SRC: In this pape appoach paametes widely used collision avoidance systems such as the distance at closest point of appoach and time to the closest point of appoach and less known and used as the distance on couse, the distance abeam and any distance and the times intevals to thei occuences ae deived, analyzed and gaphically intepeted in the combined coodinate system fo position and motion. hey can be used in collision avoidance systems and fo evesed puposes manoeuving to equied appoach paametes, intentional appoaches and naval tactical manoeuves. 1 INROUCION fte the intoduction of maine navigation adas fo collision avoidance puposes, appoach paametes of tacked objects wee detemined in a gaphical manne by manual ada plots. t the beginning, analytical fomulae fo detemination of motion and appoach paametes and collision avoidance manoeuves wee deived in a pola coodinate system, natual fo ada plots, with input values such as distances, beaings, velocities, couses and thei changes. he intoduction of compute contolled utomatic Rada Plotting ids (RPs has ceated the need fo algoithms fo detemination of motion and appoach paametes but calculations in such systems ae system specific because they use mainly a Catesian coodinate system. his is caused by: simple equations of motion in a system of Catesian coodinates, simple estimation algoithms fo motion paametes in digital tacking filtes (because fo objects tavelling with constant velocities and couses thei pola position changes adial and angula velocities ae not constant and in Catesian coodinates ae constant, eduction of numbe of tigonometic and cicula functions which, when used in numeical calculations, ae connected with longe and less accuate calculations. Publication of such algoithms is ve ae Jakševič (1967 and Lod (1968 ae two of the ve few that have been published. Only the second has some deivations and all of them ae limited to the pedicted object CP (Closest Point of ppoach distance and the time inteval to its occuence. hese paametes ae well established appoach paametes used in collision avoidance systems featuing RPs as well as in manual ada plots. In this pape othe appoach paametes such as: the pedicted object distance on couse and the time inteval to its occuence, the pedicted object distance abeam and the time inteval to its occuence, the pedicted object distance and the time inteval to its occuence 51

2 ae pesented in analytical and gaphical fom. his pape is mainly a combined and shotened vesion of Lenat 1999a, 1999b, 000a, 000b and 010 with emphasis to gaphical intepetations. SSUMPIONS N INPU PRMEERS Fo the puposes of this analysis, own vessel and extaneous objects of inteest ae egaded as if the mass of each object was concentated at a point. It will be assumed that all moving extenal objects ae tavelling at constant veloci and couse. In the movable plane tangential to the Eath s suface Catesian coodinates system Ox, Oy (Fig. 1, with Oy pointing Noth, O is the pesent position of own vessel. It is also be assumed that manual plots o the ada pocessing and tacking (RP o IS (utomatic Identification System has yielded: the pesent elative position of each object of inteest X, Y, the components of its tue veloci, and/o the components of its elative veloci,. x = sin (5 y = cos (6 y x (7 = own couse (the angle measued clockwise fom Oy to. Fom the above = + sin (8 = + cos (9 and = sin (10 = cos (11 Own and an object s motion paametes should be eithe gound o sea efeenced and a dift angle is assumed to be zeo. 3 COMINE COORINES SYSEM FOR POSIION N MOION conventional PPI displays the position of each object by plotting them in pola (, distance, beaing o Catesian (x, y coodinates. If we apply a scaling facto τ to the veloci coodinates (, o (x, y such that = τ (1 = (13 Figue 1. Input paametes he elationship of own and an object s velocities can be descibed by equations = + x (1 = + y ( t (3 x = x τ (14 y = y τ (15 then the position and veloci coodinates coupled by time τ can be plotted on a common display. On such a display, besides own veloci vecto (, and positions of objects (X, Y, vectos of theis tue (, o elative motion (, can be plotted in a coodinate system paallel shifted to the point (X, Y. his coesponds to equations x = X + τ (16 (4 y = Y + τ (17 x, y = own veloci components, o x = X + τ (18 5

3 y = Y + τ (19 In a gaphical intepetation the above equations mean that vectos of veloci ae plotted in this coodinates system of position as τ minutes vectos of pedicted motion dawn fom the pesent positions of own vessel and objects. he full aea of (x, y o (, is the aea ou manoeuves which can be limited by ou maximum veloci max the cicle cented at (0, 0 and having adius max. 4 EQUIONS OF RELIE MOION he elative position of an object, at time t, is given by X(t = X + t (0 Y(t = Y + t (1 If (t is the distance to an object at time t, then (t R X (t Y (t t (X Y t ( CP X Y (7 6. eivation of Equation = f(, Fom Equations (6 and (4, by squaing both sides and eaanging tems, we obtain a quadatic equation in (X CP XY+(Y CP = 0 (8 whose solution is = (9 CP CP XY R (30 X substitution of Equations (10, 11 to Equation (9 and eaanging yields o afte squaing both sides and eaangements t (X Y t R (t 0 (3 R X Y (4 5 EQUION OF RUE MOION substitution of Equations (10 and (11 to equation of elative motion (3 and eaanging yields equation of tue motion ( +t [(X+sin+(Y+cos] +(X+Y+R = 0 (5 6 CP ISNCE N IME 6.1 Equations fo and CP In equation of elative motion (3 the distance eaches a minimum (Lenat 1983 CP X Y and time to achieve CP CP (6 Figue. Lines =const. and cicles CP=const. =0. h, X=Y=5 n.m., = 10 kt, =10 kt sin cos (31 = (3 and eal solutions exist if (Equation (30 53

4 R CP (33 Equation (31 gives the veloci which own vessel must adopt to achieve the equied CP distance (in espect to the selected object fo vaious assumed own couses, but we should seach fo solution 0 (34 and, fo which CP 0 ( Gaphical intepetation gaphical intepetation of and can be obtained on a plotting in Catesian coodinates of own veloci (x, y by substituting Equations (5 and (6 to Equation (31 y = x (36 In these coodinates all points coesponding to a given value of will lie on two staight lines having slopes + and (values with + o in the numeato of Equation (30 espectively, intesecting in the point (, and cutting the y axis at + and (the values of obtained on putting espective values of in Equation (3. In the combined coodinate system (Equations (1 15 Equations (3 and (36 tansfom espectively to and sin cos (37 y = x τ (38 Figue illustates a family of lines (36 o (37 and (38 fo vaious fo an exempla object. teminates inside the CP then thee is a collision theat. ny manoeuve, by change of couse and/o veloci, which deflects the end of this vecto out of the CP is a possible means of avoiding the given theat. 6.5 eivation of equation = f(, CP substitution of Equations (4, (10 and (11 to Equation (7 gives a quadatic equation in CP [(X+CPsin+(Y+CPcos] +(t CP+X+Y = 0 (40 whose solution is ( C CP CP CP CP sin sin CP CP CP cos cos C CP (41 X (4 Y (43 X CP Y CP t (44 CP Real solutions exist if ( CP sin CP cos CCP (45 Equation (41 can yield up to two velocities 0 which own vessel must adopt to achieve the equied time to CP CP (in espect to the selected object fo vaious assumed own couses. 6.4 Collision heat Paametes ea We can use Equations (36 o (38 and (30, (3 with a substitution of = S (39 S = assumed safe value of (thesholds set by the system s opeato, to daw lines (=S. Fo that pat of the aea bounded by these lines and within which CP>0, own vessel s motion paametes ae leading to a theat of collision. his egion is named the Collision heat Paametes ea CP (Lenat 1983 a new ada display and plot technique. he size and the position of CP ae independent of own vessel s motion paametes. If we also plot own vessel s veloci vecto (actual o simulated and this 6.6 Gaphical intepetation gaphical intepetation of solutions given by Equation (41 can be obtained in Catesian coodinates of own veloci (x, y substituting Equations (5 7 to Equation (41 ( R x CP (y CP CP (46 he above equation eveals that the locus of points, fo which CP is a constant, is a cicle cented at (CP, CP and having adius R /(CP. Figue illustates a family of cicles fo vaious values of CP 0 fo an exempla object. 54

5 ansfomation of Equation (46 to (x, y coodinates (Equations (14 15 yields cossing time. hese paametes ae given by equations (Lenat 1999b ( R x CP (y CP CP (47 c X Y sin cos ( eivation of equation = g(, If we seach fo own couse which will lead to the equied CP distance at an assumed own speed then we can get an invese function =g(, to the function =f(, by a substitution to Equation (31 the tigonometic identities tan sin (48 1 tan 1 tan cos (49 1 tan which will esult in equation c X cos Ysin sin cos (54 c>0 means that an object will coss the couse of own vessel ahead and c<0 that an object will coss the couse asten. Intepetation of the sign of c is simila to fo c<0 c has taken place in the past. 7. eivation of equation = f(, c substitution of Equations (10 and (11 to Equation (53 and eaanging yields c sin cos c (55 ( ( tan 0 and its solution tan (50 Y c cos c (56 X sin c tan ( 1 (51 Real solutions exist if and R CP 1 (5 and Equation (51 can give up to fou own couses, which will lead to the equied CP distance at an assumed own speed if they additionally fulfil Condition (35. Gaphically these solutions ae the intesection points of lines (=const. with a cicle =const. (a cicle cented at (0, 0 and having adius. 7 ISNCE N IME ON COURSE 7.1 Equations fo c and c he pedicted object distance on couse c (Figue 1 and the time inteval to its occuence c ae sometimes used as additional citeia fo collision theat. hey ae used in some RPs fo calculation of CR the bow cossing ange and C the bow Figue 3. Lines c=const. and c=const. =0. h, X=Y=5 n.m., = 10 kt, =10 kt 55

6 Equation (57 is simila to Equation (31 with (3 but c is dependent on. Equation (57 gives the speed, which own vessel must adopt to achieve the equied distance on couse c (in espect to the selected object fo vaious assumed own couses, but we should seach fo solution 0 (57 and fo which c 0 (58 8 ISNCE N IME EM 8.1 Equations fo ab and ab he pedicted object distance abeam ab and the time inteval to its occuence ab ae sometimes used additional citeia fo collision theat. hese paametes ae given by equations (Lenat 000a ab X Y sin cos (6 Condition (58 means that the appoach on couse is at pesent o will be in the futue, not in the past. ab Xsin Y cos sin cos ( eivation of Equation = g(c Substituting Equations (10 and (11 to Equation (54 esults in Equation c Ysin X cos sin cos (59 his equation eveals that the time to distance on couse c is independent of own veloci. heefoe fom the above tan X c (60 Y c and Equation (60 gives own couse, which will lead to the equied time to distance on couse c. 7.4 Gaphical intepetation gaphical intepetation of solutions given by Equation (60 can be obtained in Catesian coodinates of own veloci (x, y X c y x (61 Y c and the locus of points, fo which c is a constant, is a staight line cossing the oigin of coodinates. Figue 3 illustates a family of lines (53 fo vaious equied c and a family of staight lines (61 fo vaious values of c 0 fo an exempla object. 7.5 Sign of c It can poved (Lenat 010 that if fo a given own couse exists own veloci (=0>0 with (=0>0 then fo >(=0 an object will pass asten (c<0, and fo <(=0 an object will pass ahead (c>0. his sign of c is illustated in Figue. ab>0 means that an object will be abeam on the staboad side of own vessel, and ab<0 that an object will be abeam on the pot side. Intepetation of the sign ab is simila to fo ab<0 ab has taken place in the past. 8. eivation of equation = f(, ab Substituting Equations (10 and (11 to Equation (6 and eaanging yields ab ab sin cos (64 Y ab sin (65 ab X cos ab Equation (64 is simila to Equation (31 with (3 but ab is dependent on. Equation (64 gives the veloci, which own vessel must adopt to achieve the equied distance abeam ab (in espect to the selected object fo vaious assumed own couses, but we should seach fo solution 0 (66 and, fo which ab 0 (67 Condition (67 means that the appoach abeam is at pesent o will be in the futue, not in the past. 8.3 eivation of equation = f(, ab Substituting Equations (10 and (11 to Equation (63 esults in Equation ab Xsin Y cos (68 sin cos hence 56

7 ab ab sin cos ab ab (69 X ab Y ab (70 (71 Equation (69 can yield the veloci 0, which own vessel must adopt to achieve the equied time to the distance abeam ab (in espect to the selected object fo vaious assumed own couses. 8.4 Gaphical intepetation gaphical intepetation of solutions given by Equation (69 can be obtained in Catesian coodinates of own veloci (x, y substituting Equations (5 though (7 to Equation ( Sign of ab It can be poved (Lenat 010 that the sign of fomula unde the modulus in Equation (6 is the sign opposite to the sign of the distance abeam ab, if own couse is equal to beaing to an object (when ab>0 and CP>0. 9 ISNCE N IME 9.1 eivation of equation = f(, Solving a quadatic equation of elative motion in (3 we obtain (fo t= (X Y ( (X Y (73 o (Equations (6 and (7 CP CP (74 ( x ab ab ( y ab 1 ab (7 and eal solutions exist if CP (75 he above equation eveals that the locus of points, fo which ab is a constant, is a cicle cented 1 1 at (, ab ab and cossing the oigin of a coodinates system. Figue 4 illustates a family of lines (64 fo vaious equied ab and a family of cicles (7 fo vaious values of ab 0 fo an exempla object. Equation (73 o (74 gives time to achieve the distance to the selected object. 9. ime to safe distance Since in Equation (73 o (74 can be any distance, we can substitute =S (as in Section 6.5 and this time can be named the time to safe distance and have been poposed analyzed and applied to detection of dangeous objects and to display the possible evasive manoeuves (accuate Pedicted eas of ange instead of thei geometical appoximations in Lenat ( eivation of equation = f(,, Solving a quadatic equation in (5 we obtain ( d d sin sin d d cos cos C d (76 d X (77 d Y (78 Figue 4. Lines ab=const. and cicles ab=const. =0. h, X=Y=5 n.m., = 10 kt, =10 kt 57

8 C (X Y d t (79 Real solutions exist if ( d sin d cos Cd (80 Equation (76 can yield up to two own velocities 0, which own vessel must adopt to achieve the equied distance at the equied time (in espect to the selected object fo vaious assumed own couses. R It should be noted fom Equations (73 and (74 that thee can exist two times of appoach at distance : shote appoach at the point (Figue. 1 and longe appoach at the point. If only the ealie (the fist appoach is inteesting fo us, then, fo this time condition CP> fo selected own motion paametes, should be fulfilled. his citeion fulfil points of cicle (,, which lie inside a cicle (CP=const. fo the same time (maked in Figue 5 by the thicke line. Gaphically solutions of Equation (76 ae the intesection points of the cicle =f(, with a line of an assumed own couse. 9.4 Gaphical intepetation gaphical intepetation of solutions given by Equation (76 can be obtained in Catesian coodinates of own veloci (x, y substituting Equations (5 7, 4 to Equation (5 ( x d (y d (81 he above equation eveals that the locus of points fo which and ae constants is a cicle cented at (d, d and having adius /. Figue 5 illustates a family of cicles (81 fo vaious equied and fo an exempla object as well as, fo compaison, cicles (CP=const. (Equation (47 and the line (=0 (Equation (36. Figue 5. Cicles, =const. =0. h, X=Y=5 n.m., = 10 kt, =10 kt 10 IME O MNOEURE It has to be emphasized that the manoeuves calculated in the pevious Sections ae kinematic and should be undetaken immediately. If we equie to have the time lapse t fo calculations, fo the decision to initiate a manoeuve and fo the execution of the calculated manoeuve then (X, Y in the pevious equations should be eplaced by (Xt, Yt espectively, given by equations Xt = X + t (8 Yt = Y + t (83 11 CONCLUSIONS Fomulae fo such appoach paametes as the pedicted object CP distance, the distance on couse, the distance abeam, any distance and the times intevals to thei occuences in a Catesian coodinates system have been deived, analyzed and gaphically intepeted in the combined coodinate system fo position and motion. Moe than 80 diectly applicable fomulae fo collision avoidance and quite evesed puposes manoeuving to equied appoach paametes, intentional appoaches and naval tactical manoeuves have been povided almost all of them ae deived fom one basic equation of elative motion. he intoduction such auxilia paametes as the distance on couse and the distance abeam, apat fom the main appoach paamete the distance to CP, makes possible: a esignation fom assumption (Section that the mass of each object was concentated at a point which can have significance when distances ae compaable to objects dimensions, moe complete analysis of the main paametes (e.g. conclusions in Sections 7.5, 8.5 and 9.. Intepetation and plotting of deived fomulae in the combined coodinate system of position and motion enable thei applications as well in compute contolled ada systems as in manual ada plots some of them ae ve simple in manual plots as it has been shown in Lenat (

9 It must be emphasized that owing to the fact that in the deived fomulae tigonometic and invese tigonometic functions of extaneous objects paametes ae not used, compute calculations can be faste and moe accuate. REFERENCES Jakševič, E lgoitm posledovatel noj obabotki adiolokacionnych izmeenij. udy CNIMF Sudovoždenie i svjaz, 83: Lenat,.S Collision heat Paametes fo a New Rada isplay and Plot echnique. Jounal of Navigation, 36: Lenat,.S. 1999a. Manoeuving to Requied ppoach Paametes CP istance and ime. nnual of Navigation, 1: Lenat,.S. 1999b. Manoeuving to Requied ppoach Paametes istance and ime on Couse. nnual of Navigation. 1: Lenat,.S. 000a. Manoeuving to Requied ppoach Paametes istance and ime beam. nnual of Navigation. : Lenat,.S. 000b. Manoeuving to Requied ppoach Paametes istance, ime and eaings. nnual of Navigation. : Lenat,.S ppoach Paametes in Maine Navigation heo and pplications. Gdynia: Gdynia Maitime Univesi Publishing. Lenat,.S nalysis of Collision heat Paametes and Citeia. Jounal of Navigation, 68: Lod, R.N Contolled zimuth voidance Manoeuve. Jounal of the Institute of Navigation, 1: