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Synthesis of a mechanism for the stabilisation of a radar antenna
Mechanism and Machine Thaoty, 1978, Vol. 13, pp. 369-389. Pergamon Press. Printed in Great Bdtain
Synthesis of a Mechanism for the Stabilisation of a Radar Antenna R. M. Peterst
Received 17 December 1976; received for publication 1 December 1977 Abstract A mechanism designed to stabilise radar antennae on sea-going vessels is analysed geometrically and equations derived which relate the mechanism parameters with the vessel attitude. The analysis is developed to give a method by which configurations of the mechanism may be selected which are compatible with the maximum pitch and roll of the vessel. Further development of the analysis yields a method of synthesis which enables desired kinematic conditions to be achieved. Examples of compatibility and of synthesis are given.
1. Introduction STABIL1SATIONof radar antennae carried aboard sea-going vessels may be obtained in a number of ways, some of which are shown by Arnold and Maunder[l]. Recently a novel method has been proposed by Delany [2], discussed by Mountain [3], and analysed by Morgan and Peters [4]. In this system, stabilisation of the antenna is achieved in conditions of pitching and rolling by means of a mechanism called TASP (tilted axes stable platform), which has two non-orthogonal operational axes. This is shown diagrammatically in Fig. I. One of the operational axes, M, is fixed at angle ~b to the deck perpendicular and in a plane which is at angle 8 to the vessel centre plane. The other axis, A, is set at angle/3 to the M axis and carries the stable platform upon which the radar antenna rotates, the antenna axis making an angle 7 with the A axis. In operation, a conventional stable element, which is mounted aboard the vessel, gives electrical signals which are proportional to the angles of pitch, A, and roll, p, of the vessel. These are suitably modified, amplified, and fed to servomotors on the TASP axes. The consequent rotation about the M and A axes (to angles /z and a, respectively, from data positions) maintains the antenna axis vertical. The advantages claimed for this arrangement are that the centre of gravity of the structure and its associated equipment will be significantly lower than for existing mechanisms, and the operational axes may be arranged to reduce torque requirements. The equations which relate the angles/z and a with the angles of pitch and roll, A and p, have been derived by a vectorial method[4]; but for a fuller understanding, a solid geometry analysis is shown in the following section. 2. Analysis by Solid Geometry Since the mechanism is fixed to the vessel, it is convenient to consider antenna axis movements as relative to the vessel, e.g. from a vessel deck perpendicular. In a general instantaneous vessel position of finite pitch and roll, in order to bring the antenna axis to a truly vertical position, it must be moved to a position as shown in Fig. 2(a). This figure shows, in various views of the vessel, the antenna axis moved through the pitch and roll angles from the perpendicular, to such a position. If the antenna axis is considered as of unit length, then its perpendicul~ height after movement through pitch angle A and roll angle p will be os' = cos A cos p
(1)
tDepartment of Mechanical Engineering and Engineering Production, University of Wales Institute of Science and Technology, King Edward VII Avenue, Cardiff, Wales.
369
370
~Antenna
,,,,,,.,,is
Axis
-AAx
"'~,~' "~ '--~imuth Arm Oo,urn
=o
~..1~_~-MastArm (art a.aes in ~y~ pi.an. ) " ~ x ~
| [
stable I;r IB p t a t f o r m " ~ . A Axis
TASP
|
A, , . . o ,
I //;/'
~
LA'
\k--_
z~
la about M ~ai,)
LAot,o., I dri,,
"b . . . . . . . . . . . . . . . . Xi
Figure 1. Schematic of TASP in datum and general positions. and the distance tu =
(sin e A + cos 2 A sin e p) '/2.
(2)
In the auxiliary view of Fig. 2(a), which is orthogonal with the frame F8 shown in Fig. 1, the antenna axis makes an angle ~ with the deck perpendicular, where O = tan -t E =
[tu
cos (8 + ¢)/(cos X cos p)]
tan -~ (cot ~ sin p)
(3) (4)
and o e = [tu 2 cos 2 (8
+ E) + cos 2 A cos 2 0] 1/2.
(5)
However, to move the antenna axis from its datum position, shown in Fig. 1, to the truly vertical position, shown as oe in Fig. 2(a), it must rotate about the A and M axes. Figure 2(b) shows three views of the three principal axes of the TASP mechanism (A, M, and antenna) in their data positions, the frame of reference again being the Fo frame shown in Fig. 1 and in the auxiliary view of Fig. 2(a); (i) Is the view normal to the axes, i.e. normal to the xsys plane. (ii) Is a view looking along the A axis. (iii) Is a view looking along the M axis. Since the rotations about the A and M axes are commutative, they may be considered
371
(iii)
(i0
fb)
Rgure 2. Geometry of antenna axis in general pitched and rolled position. sequentially. Further, if the sequence is assumed as A rotation followed by M rotation, the geometry is simplified, and such rotations will cause the point a to move perpendicularly to the A and M axes in view (i); from a to / due to a rotation and from / to e by # rotation. It should be noted that points a, f, e, etc. in view (i) bear the subscripts 1 and 2 when they appear in views (ii) and (iii). In triangle eoa, since oa is 1
ea = [I + oe 2- 2oe cos (~ + O + y - fl)]ii2
(6)
~"= sin-l[oe sin (#i + O + y - ~)Jea].
(7)
and
In triangle aef
af = sin y(l -cos a),
(8)
v = ~'+ y - 9 0 °,
(9)
~:=/3- ~'- y + g o °,
(10)
ealsin (180° - ~) = sin y(l - cos a)/sin ~:.
(II)
a = cos -I [I - ea cos ~:)/(sin ~ sin y)].
(12)
and since
372
View (iii) indicates the trajectory of a as an elliptical path moving from a2 to f2. This corresponds with the path of at rotating through a to f, in view (ii). Angle r defines the point/2 z = tan-'
(f2f~/f~o2)
(13)
= tan-' [sin a sin -//(sin/3 cos 3t - cos a cos/3 sin -/)].
(14)
Rotation about the M axis will now take the antenna axis from 02/2 to 02e2, where X = tan-'
(eze~/e~02)
= tan t [tu sin (~ +
(15)
8)/(oe sin (~b + @))1.
(16)
Hence since /z=r+X
(17)
/z = tan -t [sin a sin -//(sin/3 cos - / - cos a cos/3 sin 1')] +tan -t [tu sin (~ + 8)/(oe sin (@ + @))].
(18)
Computation of eqns (12) and (18) and comparison with similar equations derived by the vectorial method [4] has proved that the corresponding equations are identities.
3. a/it Envelopes The results of a computer program giving a and/~ values for a selection of/3, y, 8 and @ angles with A and p as running parameters in the typical ranges ;t-+6 °, p_+18 °, give a/iz envelopes typically as shown in Fig. 3. Such envelopes indicate that considerable flexibility in 180O,
X~= 20* y =
30*
B=60* @=30*
~--40* ~'-" 30* B=45" ~ : 4 5 "
90 °
18*
p=lS* ~
p= 18"
0
150*
0
Q
Figure 3. Typical all~ envelopes design is possible; the overall a/l~ ratio and symmetry of the a/tx curves can both be modified to a considerable extent by varying the configuration of the mechanism. It is evident also that the range of angular rotations about the A and M axes necessary to maintain the antenna vertical can be made significantly larger than the range of pitch and roll angles encountered by the vessel. This fact would influence the accuracy with which the antenna can be positioned since angular errors in a/l~ would then correspond to much smaller errors of antenna position than is the case in present mechanisms.
4. Limits of Compatibility of Combinations of Conflgurational Angles The range of a and ix angles required for rectification of the antenna during a given range of pitch and roll depend on the configurational angles fl, 7, 8 and ~. It is clear that some configurations will be unsuitable since for some parts of the range of pitch and roll there will be no values of a and/z which will set the antenna vertical. For example, it will be readily seen that with angles/3 = 0, 7 = 0, any 8 and ~b angles, movement about the A and M axes merely maintains the axis in its datum position and it is not possible to set the antenna vertical
373
whatever the pitch and roll. On the other hand, with angles 4~ = 90°,/3 = 90°, 3' = 90°, 8 = 0°, the configuration becomes identical with the orthogonal axes mechanism and this configuration is clearly suitable, and the antenna axis may be maintained vertical, for any values of pitch and roll. For the purpose of computation of an optimisation process it is useful to determine the boundary configurations, i.e. to be able to determine configurations which are compatible with a required range of pitch and roll duties and which are not. In the following sections the governing kinematic relationships are investigated to establish these boundaries. The area of employment Figure 4 corresponds with Fig. 2(b) and shows the antenna axis oa in the datum position, (a = 0,/~ = 0). This axis is required to move to any position within the range of pitch and roll angles likely to be encountered by the vessel. This range is shown in chain-line; it is the area generated by the end of the axis oa during pitching and rolling and is defined the "area of employment". When the vessel is in a level attitude (zero pitch, zero roll) the axis must be in the perpendicular position od; while the position indicated by the dotted line oe is the position required when the vessel is in a general attitude of pitch A and roll p. In order to be able to take up such positions the axis may be rotated sequentially from the datum position as described in the analysis above, namely, from oa to of by virtue of rotation about the A axis and then by rotation about the M axis to move from of to oe. Considered in this way, rotations of the antenna axis to any instantaneous position in fact generate curved surfaces of two imaginary cones. Rotation of oa about the A axis to a position such as of generates part of one cone (the "A cone") whilst its subsequent rotation, of to oe, about the M axis generates part of the other, (the " M cone".) It will be noted that the A cone is common to Area of
~
- ~...
Rgure 4. Geometry of ,, and/z rotations.
374 all final positions of the antenna axis (its generator is of unit length and its angle ,/), while the M cone, although its generator must be of unit length, has an angle which varies according to the final position, e.g. oa to of to oe, or, oa to oc to od. The range of the base diameters for the M cone is given by diameters jn and gm in Fig. 4. It may be seen that to be able to move the end of the axis to any position within the area of employment, there must always be a general position f on the line ab (the A cone base) in Fig. 4. The existence of such a position indicates the existence of finite and therefore suitable angles a and/~. It follows, therefore, that in order to be able to move the antenna axis to any position within the area of employment, all points within this area must be served by lines perpendicular to the M axis such as ef. Conversely, if a line, perpendicular to the M axis, is produced from a point on the curved surface of the area of employment does not intersect the line ab on or between the points a and b, then that particular set of values of/3, 3', 8 and ~ do not constitute a suitable configuration for the pitch and roll duty required of that mechanism.
Band of deployment All suitable configurations are therefore identified by the existence of the two extreme cases of such lines ef in Fig. 4, namely ph and qk. These lines extended as gm and in, constitute a band, which is defined as the band of deployment, whose slope is 4) - 90° and whose width and position are functions of the required area of. employment. The definition of a suitable configuration, therefore, without reference to its kinematic characteristics, is that the base of the A cone must completely straddle the band of deployment. Such straddling falls into the types shown in Fig. 5(a)-(c). From this diagram it can be seen that, for each type, the angles/3
Ceh/o~~00, x.
(a)
b
a
(d)
Figure 5. Extreme and general types of configuration.
375 and 3' are defined by the general line ab as follows 3' = sin-I (ab/2)
(19)
= 3' + 4 , - o,,
(20)
and
where ~o = sin -1Xa.
(21)
Since the band of deployment is automatically defined by the area of employment and the angles 8 and ~, the extreme values of ~ and ~ are related by /3 = y + a constant angle.
(22)
For a given band of deployment, therefore, mechanisms having suitable and unsuitable and 3' angles can be indicated by a diagram of the pattern shown in Fig. 6. In this diagram the
Unsu.ob" / l e / / / .
/ yo0./
/ Figure 6. Typical graph for the configuration compatibility of given angles 8 and ~. three extreme types identified above are indicated by the three graphs rs, st and tu. The lines sr and tu have a slope of 1, the line st has a slope of - 1, and the intersections s and t indicate the extreme conditions corresponding to the chain lines in Fig. 5(b). It, therefore, follows that values of fl and 3' which fall within the area enclosed by the lines rs, st and tu indicate suitable mechanisms of the type shown in Fig. 5(d) and those which are outside this area are not. Hence in Fig. 4, for suitable configurations oh' < og'
(23)
cos (~ + 3') < og'
(24)
oa' >- oj'
(25)
cos (~ - 3') > oj'.
(26)
and
Both (24) and (26) must, therefore, apply, for the mechanism having the particular values of 8 and ~b corresponding to that band of deployment to be suitable.
376
Patterns of bands ,9/deployment The band of deployment is defined in Fig. 4 by the lines gm and jn which are projected perpendicular to the M axis from extreme points in the area of employment. The points from which these lines are projected originate in different parts of the area of employment. Bands of deployment, as a consequence, fall into the three patterns shown in Fig. 7. The equations governing these patterns are derived in Appendix 1.
~
I
(o)
|
PotternB ( ~
~
|
(c)
PolternC
Figure 7. The three patterns of the band of deployment. 5. identification of Suitable Combinations of Configurational Angles A computer program based on the equations derived above and which evaluates/3 and 3' according to the limits of (24) and (26) for different values of 8 and ~ has been prepared. The results of this program for a particular duty of pitch and roll are given in Fig. 8. This diagram indicates the loci of the two corners s and t of the graph shown in Fig. 6 for changes in 8 and Oh, and thus combines the variation of the four angles/3, 31, 8 and ~b. This diagram may be used as illustrated by the following examples.
Example 1. If 8 and ~b angles are selected for a TASP mechanism as 20° and 45 °, respectively, what values of/3 and 3' may be selected to ensure suitability? Solution. Draw lines from the values 8 = 20° and ~ = 45° as indicated by the dotted lines in Fig. 8. All simultaneous values of/3 and 3' within the area bounded by these dotted lines then imply suitable configurations, e.g., respectively, 20°, 45°; or 45°, 20°; or 35°, 35° etc. Example 2. If angle/3 is selected as 15° and 3' as 30°, what angles 8 and ~ are suitable? Solution. The lines rs, st and tu must enclose the point/3 = 15°, T = 30° (indicated in Fig. 8 as X), hence simultaneous values of 8 and ~ which are within the quadrant lying to the left of the point give suitable configurations, since such values of 8 and 4' imply the s points of the graphs given in Fig. 6, e.g., respectively, 0°, 30*; or 20°, 25° etc.
37"/
B=o=,
7O°
.3/1' 11lit "ll III1| 60 =
G )"
50°
4 ~ o ~
," /
/
4°.
f / 3 D°
\
f
f
/ /
\
~o-
7
0°
I
~
I0°
20°
i
\
i
i
40=
30"
II11 /
J
50=
I
I
60 =
7~'
B Figure 8. Composite graph for compatible configurations.
6. Kinematic Synthesis of a TASP Mechanism Any point may be selected at random within the rstu lines implied by the loci of Fig. 8 to give simultaneous values of/3, 1', 8 and @ of a TASP configuration which is compatible within the pitch and roll duty for which the diagram Fig. 8 is constructed. Such a random selection, of course, would take no account of the kinematic characteristics of the mechanism so obtained. Design considerations however demand that desirable kinematic characteristics should be obtainable more directly. This requires a more ordered method of selection or of synthesis in which the required shape and size of the alp envelope, typified in Fig. 3, may be obtained by the identification of suitable configurational angles• In this section such a method of synthesis is derived which is developed from the geometrical nature of the analyses given above. The a range
From Section 4 above it is clear that the range of a angles required for a given duty of pitch and roll may be varied by adjustment of the angles/3 and 1'. For example, in the configuration of given B and @shown in Fig. 4, rotation about the A axis is required to operate from point k, the intersection of the ab line (the base of the A cone), with the upper limit of the band of deployment (the smallest base diameter of the M cone) and point h, a similar intersection with the lower limit (the largest base diameter of the M cone). The auxiliary view gives the specific A angle range (o~k,-oth~). It follows that, for maximum range of angle a, the points h and k should be as far apart as possible, i.e. coincident with points b and a respectively; coincidence of h and b implies an angle a of 180° and coincidence of k with a means a = 0°. There are two configurations possible for maximum range of angle a; these are shown by the chain lines in Fig. 5(b) which correspond with the points s and t in Fig. 6. It is obviously useful to relate the a angle range with the graphs separating suitable and unsuitable configurations (Fig. 6). Configurations corresponding to the line rs in Fig. 6 correspond with ab lines in the positions shown in Fig. 5(a). It will be seen that the range of the a angles, being 180" at point s, decreases as values of 8 (and 1') increase. The line tu in Fig. 6 corresponds with configurations such as shown in Fig. 5(c); again the range of the angle a decreases from 180" at point t for configurations further along the tu line• The line st
378 corresponds with configurations further along the tu line. The line st corresponds with configurations shown in Fig. 5(b), while all practical configurations represented by points within the boundary rstu of Fig. 6 correspond with Fig. 5(d), i.e. the base of the A cone overlapping the extreme bases of the M cones (the band of deployment). The equations which related to the actual ranges are derived in Appendix 2. For the smallest range of ~, the M cone base should cut the band of deployment perpendicularly. This most nearly occurs whea large/~ and 3' angles are employed. Figure 9
G
91~
0¢ 0"
i 20 ~
i 40"
I 60"
I i 80"
180" -
(:~ O"
i
I
I
I
180" -
(a) i
0"
B
B
(b)
Figure g. Variation of range -,with angles/~ and 3'. shows typical plots of the equations derived in Appendix 2 for particular values of ~ when 8 = 0 and for a particular set of maximum pitch and roll conditions. The vertical lines in this diagram indicate the s and t cases. From Figs. 4 and 5 it can be seen that the extreme A cones are indicated by the values 3'. = sin-' (gj/2),
(27)
3'. = sin-' (gn/2).
(28)
(Subscripts m and n refer to maximum and minimum values, respectively). Figure 9(b) shows a 3-dimensional graph of the variation of the a range with/3 and 7 for given angles 3' and 4P, while the effect on the a range of varying 8 may be observed from the examination of Fig. 10. The ~t range
Displacement of the antenna axis about the A a~s implies a displacement relative to the M axis. In Fig. 4, displacement of the antenna through a to position o=ft in the lower auxiliary
379 a Range
I
~-o G
i Q
~3 2
P
Q I
i
0
8"90 ° 4
I Q
Rgure 10. Effect of 8 variation on a range.
view corresponds with displacement from oa to of in the main view, and in the upper auxiliary view with displacement through/Zo to ozf, (with/~0 given by a~ozf2). Such equivalent displacements about the M axis may be considered as the true operating data positions subsequent to rotation/~ to the required antenna axis position. The overall lz range is given by the difference between the largest and smallest/~ angles (/zm and/~., respectively) required to reach all parts of the area of employment. Since these angles are, in general, measured from different data positions, it is not possible to get a simple and clear picture of the variations of the/~ range with the different geometrical parameters, as it is with the a range. However, it is possible to take a line of approach to the problem which is based on the two-term form of eqn (18) and to consider a/~ rotation as having two components X and ~-; both measured from a datum of o2d2. One component is measured from this datum to the required position in the area of employment, and the other componentt from the datum o2d2 to the equivalent displacement about the M axis which is due to the a rotation, i.e. through angles X and ~- (Fig. 2) to the positions o2e2 and ozf,. From the auxiliary view of Fig. A5 it can be seen that the/~ rotation to reach p2 of the area of employment consists of X2 (component 1 of p2) + ~, (component 2 of P2), similarly/~ rotation for p3 consists of - X3+ ~'3. It will be observed that the first component X being angle to a point in the area of employment from point o5 is a function of 8 and 0 and is independent of the angles/~ and y. The second component, 1"being a measure of a point on the base of the A cone is a function of /3 and y and is independent of 8 and 4), It will thus be seen that by selecting components in this manner, the proble m of synthesis is simplified, and some control of t h e / t values of the a/~ envelopes may be obtained. Derivation of the equations governing these components is given in Appendix 3. Computation of the equations giving the X components in respect of the four corners of the area of employment when Am= 6° and p., = 18° for various values of ~b and 8 produces the
380 Ioo °
so"
I O0 °
N i\
(8 =oo) Cornersof area
(8 =20o) 4O"
×
20°
2O o
. ~
x o" 2o =
. ..~.."
4o = 60"
60 •
•
i '/
8(7)
¢a)
140" 120" I00 °
120" ( ~ - 4 0 =)
i00o 8O"
80 = 60 •
×4o.
4O"
(hi
~
(8- 6o')
2O" I
O"
1_4 _~__.-b" r-'3L-'-.~"--
I
20 ° 4O"
4O" (el
i...~'~ (d)
Figure 11. Variation of # range component % with 8 and ~. graphs shown in Fig. 11, while Fig. 12 shows graphs of the ~- values against angle 8(= 1' + the constant for given 8 and ~b values) for various 8 and ~ values. The graphs in Fig. 12 should be read in the same way as those of Fig. 9. The graphs of Figs. 11 and 12 may thus be used to obtain the/~ range for any given fl, 1', 8 and 4) values by adding the two pertinent graph values from each figure, e.g. for a system of /3 = 40°, 8 = 40°, ~ = 20°: Fig. 1l(c), (8 = 40°) the chain line (4' = 20°) gives the values of % for the various corners of the area of employment. Figure 12(c) shows the four families of graphs of • related to 8 = 40°.
Summary of the synthesis a angle ranges can be selected from Figs. 8 and 9 (see Example 1 below) and the all~ envelope corner position arranged by adjustment of the 8 angle shown in Fig. 10. /~ angle ranges can be deduced from Figs. 11 and 12 which give the complementary terms X and ~-, % being f(& ~) and r being f(/3, 1'). When 8 = 0, the # range is given by the range of the g components only since the/~ components for the principal points (p~ and p4), i.e. those which define the range, are equal. (see Example 2 below). When 84=0, however, synthesis of /~ ranges is less easy since it is not clear, without computing values, which corners of the area of employment are principal points.
Example 1. What configurations will give large and small values of a range? Solution. F r o m Fig. 9 it can be seen that a range of 180~ can be obtained with any 4) angle over 7° and any 8 angle if the/3 and 1' angles given by the points s and t are selected. These points are given in Fig. 8. For example, if ~ = 45° and 8 = 20° then/] = 12°, 1' = 47½° will give an ot range of 180°. Examination of Fig. 7 shows that small a ranges (<40 °) result when configurations in which/3 = 1' > 40° are selected. Example2. What configurations will give a value of/~ range of 120°?
381 (~ " 5 & 140" 12~Y'
V
I(~ °
Comes l~,q, of
t
(~ = 0 °)
eo*
T
s 2~3
I •
60 •
~
2O*
40 °
(o)
20"
I
I
I
/ (7
0•
20*
40"
6&
8(7
160 ° ~--20
~
°
1213"
Corner I
(~-20") T ~°
40*
I
I
I
40* 60*
~:~°
O*
~o
(b)
~o
I~ •
-- -
, ,o.
40"
o.i-
--
.,o. 6o.
-
~
(c)
.4o. do-
B Figure 12. Variation of t~ range component r with/3 and 3'. Solution. When 6 = 0 the r components of the principal points are equal, hence from Fig. 11 since/~ components of the principal points are equal in magnitude, a X component of 60° is, therefore, required; this is given by ~ = 18", # = 0,/3 = any, 3' = any. 7. Conclusions A geometrical analysis of a tilted axes stable platform mechanism has been presented; the results of the analysis confirm a vectorial analysis derived previously. The geometrical nature of the analysis has lead to the formulation of a method whereby the suitability of any configuration of the mechanism may be quickly determined. The method also allows numerous alternative suitable configurations to be selected. A method of synthesis, which is also based on the geometrical character of the analysis, has been evolved; use of the method enables required kinematic characteristics to be obtained. The paper illustrates the possibilities of synthesis which are generally inherent in a geometrical analysis. Other methods, such as vector analysis may have advantages in manipulation of the computations aspects but frequently fail in this respect. Acknowledgements--Theauthor acknowledges the help and inspiration of his colleagues in UWIST, in particular Mr. Colin Morgan, and of his associates in the Admiralty Surface Weapons Establishment, Cosham. This work has been carried out
382
with the support of the Procurement Executive, Ministry of Defence, ASWE, Cosham, Portsmouth, Gt. Britain. The help of Mrs. W. Scott is gratefully acknowledged in the presentation of this paper.
References l. R. N. Arnold and L. Maunder, Gyrodynamics and its Engineering Applications. pp. 484. Academic Press. New York (1961). 2. D.Delany, M. J. Durbin, C. Morgan, D. S. Mountain, R. M. Peters and R. Wilson, The Feasibility of a Non-Orthogonal Axes Stable Platform for Ship-Borne Use. UWIST Technical Note MMI6 pp. 43 (1975). 3. D. S. Mountain, A novel mechanism for antenna stabilisation. Naval Electrical Review, 29 22 (1975). 4. C. Morgan and R. M. Peters, The tilted axes stable platform mechanism: kinematic and kinetic aspects. Proc. Inst. Mech. Engrs 191 277 (1977).
Appendix 1 In each case the lower limit of a band of deployment is defined by the corner of the area of employment corresponding to maximum roll and maximum pitch, (point p in Fig. 4). Hence eqn (24) may always be written as (see Figs. 4 and 7) cos (~ + Y) <-op cos (4 + ~b,,),
(1.1)
(note, subscript m refers to maximum value). For the upper limit of the band of deployment however three different patterns are possible. Pattern A. When the angle ~ is small the upper limit of the area of employment is tangential to the unit axis sphere, i.e. it is defined by a point within the area of employment. Pattern B. Here, when the angle ¢, is larger, the upper limit originates within the segment of the ellipse which corresponds to the condition of rolling during maximum pitch (point q in Fig. 4). Pattern C. When the angle 4) is large, the upper limit originates at the corner of the area of employment icondition of maximum roll and maximum pitch). When the angle ~b is such that Pattern A pertains, since on' cannot be greater than o]', eqn 26 may be written as cos (/3 - 3') = I,
(I.2)
//= 3,.
(I.3)
i.e.
When the M axis is arranged such that Pattern B pertains, the eqn (26) may he written as (see Appendix IA) cos (d - v) = cos # cos A,(tan z 4) sinz/i + I) m + sin 6 sin A,~ cos 8.
(I.4)
But if d~> tan
-t f xcosec2 8 8)lt2"l,j, ~.(cos2 it,. - x 2 cosec2
(see Appendix 1B) where x = tu,, cos (~,, - 8) - sin A cos a, then Pattern C pertains and eqn (26) is written as cos (13 - y) = op cos (#~ - ~").
0.5)
Appendix 1A Condition [or upper limit for Pattern B When Pattern B pertains, ~ is sufRciently large that the line jn in Fig. 4 becomes tangential to the line indicating condition of rolling during maximum pitch, a boundary of the area of employment. This boundary line is circular when 8 is 0 and when viewed along the ships axis, and is a vertical straight line when 8 is 0 and viewed athwart ship. The latter condition is shown in Fig. AI. When O< 8 <90L the line is elliptical, as shown by the chain line in Fig. AI and its major axis swings in a circular path of radius sin ,~, about a vertical axis through 0. The ellipse, origin 0' is y:/a 2 + x2/b 2 = I,
(AI)
where a = c o s Am,
b = cos A,, sin 8. A general tangent to an ellipse is given by the equation y =mx 4- (m2b 2 + a2) ~1"-.
(A2)
In this application y =mx + (m z cos 2 A,, sinz 8 + cos2 A.) ~/:,
(A3)
383
to ellipse
Figure A1. Upper limit of band of deployment as tangent to ellipse. where
m = -tan 4',
The intercept on the O'Y axis is cos Am(m 2 sin s 8 + I) '12,
(A4)
hence the intercept in the oy axis is cos A.(m z sin s 8 + I) jlz- m sin A, cos &
(A5)
Thus the top limit in this case is cos 4) cos ,~.,(m 2 sin s 8 + I) 't2 - m sin A., cos 8 cos 4,
(A6)
and the required condition is cos (/3 - 4') > cos d' cos ~,,(tan z 4, sin s 8 + I) 'lz + sin 4' sin ,~,. cos 8.
(A7)
Appendix 1B Condition/or upper limit/or Pattern C When PatlBrn C pertains, 6 is so large that the top limit is projected from the corner of the area of employment. From Appendix IA, the ellipse corresponding to rolling in a condition of maximum pitch is given by yZ/cos2 A,, + x21(cos2 Am sin 2 B)= I.
(BI)
The radius of the extreme end of the elliptical segment shown in Fig. A2 has been derived as Radius = tu. cos ( , . - 8), where
tu. = (sin 2 A,. + cos 2 A= sin 2 p.),l~
(B2)
384
o
\
Figure A2. Upper limit of band of deployment as tengent to corner of ~trea of
employment. and
~m = tan-' (cos ~,. sin p,./sin ~,.). A tangent at this extreme corner is given by d(y)/dx at (turn cos (e,. - 8) - sin A,,, cos 8, cos pm cos A,. ). From equation B1 y2 = (1 - x2/cos A,~ sin z 81 cos 2 Am y = (cos 2 A,,, - x: cosec 2 8) ~/z dy = -x
dx
c o s e c z 8 ( c o s ~ ;t,. - x 2 c o s e c 2 8 ) -'12.
(B3) (B4) (851
Thus if ~b> tan-L (x cosec 2 6 (cos z a,. - x 2 cosec: 8)- L/z).
(B61
then the top limit is dictated by the corner of the area of employment. The top limit is then given by
op cos (~b - ip.. ),
(B7)
where op = (turn cos z (8 + e,,,)+ cos z ,~,,, cos2 p.,.)~l~ and the required condition is cos (# - y) > op cos (,/, - Om1.
(88)
Appendix lC Derivation of equations giving band of deployment The top and bottom limits are given by the two lines jn and gm as shown in Fig. A3. From Appendix 1A. the equation for the top limit line for medium values of ~ (Pattern B) is given as y = mx + cos A,.(m 2 sin 2 8 + 1t ./2 - m sin ,~,. cos 8,
(A31
and the equation for the circle y = (1 - x2) '/2,
(C1)
385 .Y
\i/. 0
Im
x
Figure A3. Identification of points defining the boundaries of deployment. hence points xj and x~ are given by (1 - x) tl2 = m x + cos A.(m 2 sin 2 8 + I) t/2 - m sin A,. cos 8, = mx + K, where
m = - tan
(C2) (C3)
1 - x 2 = m2x 2 + 2 K m x + K 2
(C4)
(m 2 + l)x 2 + 2 K m x + ( K 2 - I) = 0
(C5)
[2Km \
K 2-
X2 + L m - - " ~ ) X "1-~
I
_
= U,
(C6)
the roots of which are xi and x.. The equation for the bottom limit line is H-L y = m x +COS Am(m 2 sin 2 8 + 1) j/z- m sin A,. cos 8 - cos----~'
=rex+J,
(C7) (CS)
points x~ and x,. are given by
+J
(C9)
I - x 2 = m2x 2 + 2mJx + j2
(CIO)
(I - x2) I/z=mx
(m 2 + 1)x 2 + 2mJx + (j2 _ I) = 0
/ 2mJ \ +j2+l
_
x+ Lm--~+l}x ---~-~-- u,
(Cll) (CI2)
the roots of which are xg and x,.. Equations for xj and x. for the Pattern C may be arrived at by substituting K = tu,~ cos (4) - ~,. ) sec
(CI3)
instead of eqn C6.
Appendix 2 The manner in which the a range varies as ~, (and/3) changes depends on the manner in which the base of the A cone intersects the band of deployment: Fig. 5(a)-(c) show the three extreme cases, corresponding to the rs, st and tu types and F!g. 5(d) shows the general case. The four types of case are shown also in Fig. A4. For a rs configuration, Fig. A4(a), it is deduced that maximum a angle = cos-t((sin ~/- W/sin/3)/sin ~,),
(2.1)
hence the a range = 0 to cos-tO - W/(sin 0 sin y),
(2.2)
where W is the width of the band of deployment, measured as a fraction of the radius of the circle. For a st configuration, Fig. A4(b), the range = 180° to cos -t (W/(sin ~8 sin 3,)- I).
(2.3)
For a tu type, Fig. A4(c) the a range is 0 to cos-tO - W/(sin ~ sin y)), i.e. eqn (2.2) and for the general configuration, Fig. A4(d)
386
~"
Figure A4.
a
Y
?
tu Type
range in relation to types of suitable configurations.
the a range = a' to a", where a ' = cos -I (KWl(sin 8 sin y)),
(2.4)
K is a fraction of W and a "= cos-'(I - W(I - K)/(sin # sin 3')).
(2.5)
Appendix 3 The X components of the four corners of the area of employment are deduced from Fig. A5 as angles X~, X2, X3 and g4. From this diagram it will be seen that the four angles are given by x~ = tan-~(Ql/( sin 4) - Q2 cos d) - Q5)),
(3.1)
x2 = tan-I(Q3/( sin 4) - Q4 cos 4) - Q5)),
(3.2)
£3 = tan-~(QIl( sin 4) + Q2 cos 4) - QS)),
(3.3)
x4 = tan-~(Q3/( sin 4) + Q4cos 4) - Q5)),
(3.4)
QI = tu~ sin (~,. - 8),
(3.5)
Q2 = tu~ cos (~m -- 8),
(3.6)
Q3 = tu,, sin (~,. + 8),
(3.7)
Q4 = tu,, cos (e,. +8),
(3.8)
Q5 = (1 - op cos Ore) sin 4).
(3.9)
where
The • component for each point is given by the intersection of a circle running through the point centered at 02 and the ellipse which is the auxiliary view of the base of the A cone. In Fig. A6(a) the ellipse
y2/a2 + (x -
Z ) 2 / b 2 = 1,
(3.10)
(x is assumed positive leftwards from 02), where a = sin 3' and b = sin y cos 8 and Z the distance of the ellipse centre
from 02.
b2y2 + a2xr- 2a2xZ + a 2 Z
= a2b 2.
(3.11)
387
J
I
Figure AS. Geometry of/~ range related to r - s cases of A cone. For the circle
y2 = R2_ x 2.
(3,12)
At intersectionof ellipseand circle R 2 b 2 _ b2x 2 + a2x 2 - 2 a 2 x Z + a 2 Z 2 = a 2 b 2
(3.13)
(a 2 - b2)x 2 - ( 2 a 2 Z ) x + ( a 2 Z + R 2 b 2 - a2b 2) = 0
(3.14)
x = a2Z/(a 2 - b 2) +- ( a 4 Z 2 - (a 2 - b2)(a2Z 2 + R 2 b 2 - a2b2))l121(a2 - b2).
(3.15)
hence
For
r-
S cases, shown in Fig. A6(a),
(3.16)
Z=R5+b,
R5, r, s and the general case are as shown in Fig. AS. s - t cases are shown in Fig. A7. For these cases Z -- R 6 - b, as shown in Fig. A6(b).
(3.17)
t - u cases are shown in Fig. A8. For these cases
Z = b -
R5, as shown in Fig. A6(c).
In all cases the only root ~pplicable is the smaller of the two.
(3.18)
388
- i
I
(a)
i
{b) Oz
Figure A6. Patterns of A cone position.
x.G
Figure A7. s - t cases of A cone.
389 t-u .
-
Case
-
.
Figure A8. t - u cases of A c o n e . In each of these patterns the ¢ components are given by
COS-' (X,/O2p,),
(3.19)
"r2= COS-I (x2/o,p2).
(3.20)
¢, =
¢3 = cos-I
(x31o2pD,
(3.21)
¢4 = COS-I (X4/o2P4),
(3.22)
o2pl = (QI 2 + (sin 4' - Q 2 - Q5)2) '12,
(3.23)
o2p2 = (Q3 2 + (sin 4' + Q4 - Q5)2//2,
(3.24)
o2p3 = (Q12 + (sin 4' + Q2 - Q5)2) '12,
(3.25)
o2p4 = (Q3 2 + (sin 4' - Q4 - Q5)2) '/2.
(3.26)
where
SYNTHESE D'UN MECANISME POUR LA STABILISATION D'UNE ANTENNE DE RADAR. R. M. Peters Un m4canisme destind ~ stabiliser les antennee de radar sur les navires en m e r e s t analys4 g4ometriquement et les 4quations qui en d4coulent mettent en relation lee param~tres du m4canisme avec l'attitude du navire. On developpe une m4thode par laquelle on peut choisir les configurations du m4canisme compatibles avec le tangage et le roulis maximum du navir~. Le d4veloppement ult4rieur de l'analyse produit une m~thode de synth~ss qui permet d'arriver aux conditions cin4matiques d4sir4es. Des exemples de compatibilit4 et de eynth~se R4sum4
sont dorh~ s.
Synthesis of a Mechanism for the Stabilisation of a Radar Antenna R. M. Peterst
Received 17 December 1976; received for publication 1 December 1977 Abstract A mechanism designed to stabilise radar antennae on sea-going vessels is analysed geometrically and equations derived which relate the mechanism parameters with the vessel attitude. The analysis is developed to give a method by which configurations of the mechanism may be selected which are compatible with the maximum pitch and roll of the vessel. Further development of the analysis yields a method of synthesis which enables desired kinematic conditions to be achieved. Examples of compatibility and of synthesis are given.
1. Introduction STABIL1SATIONof radar antennae carried aboard sea-going vessels may be obtained in a number of ways, some of which are shown by Arnold and Maunder[l]. Recently a novel method has been proposed by Delany [2], discussed by Mountain [3], and analysed by Morgan and Peters [4]. In this system, stabilisation of the antenna is achieved in conditions of pitching and rolling by means of a mechanism called TASP (tilted axes stable platform), which has two non-orthogonal operational axes. This is shown diagrammatically in Fig. I. One of the operational axes, M, is fixed at angle ~b to the deck perpendicular and in a plane which is at angle 8 to the vessel centre plane. The other axis, A, is set at angle/3 to the M axis and carries the stable platform upon which the radar antenna rotates, the antenna axis making an angle 7 with the A axis. In operation, a conventional stable element, which is mounted aboard the vessel, gives electrical signals which are proportional to the angles of pitch, A, and roll, p, of the vessel. These are suitably modified, amplified, and fed to servomotors on the TASP axes. The consequent rotation about the M and A axes (to angles /z and a, respectively, from data positions) maintains the antenna axis vertical. The advantages claimed for this arrangement are that the centre of gravity of the structure and its associated equipment will be significantly lower than for existing mechanisms, and the operational axes may be arranged to reduce torque requirements. The equations which relate the angles/z and a with the angles of pitch and roll, A and p, have been derived by a vectorial method[4]; but for a fuller understanding, a solid geometry analysis is shown in the following section. 2. Analysis by Solid Geometry Since the mechanism is fixed to the vessel, it is convenient to consider antenna axis movements as relative to the vessel, e.g. from a vessel deck perpendicular. In a general instantaneous vessel position of finite pitch and roll, in order to bring the antenna axis to a truly vertical position, it must be moved to a position as shown in Fig. 2(a). This figure shows, in various views of the vessel, the antenna axis moved through the pitch and roll angles from the perpendicular, to such a position. If the antenna axis is considered as of unit length, then its perpendicul~ height after movement through pitch angle A and roll angle p will be os' = cos A cos p
(1)
tDepartment of Mechanical Engineering and Engineering Production, University of Wales Institute of Science and Technology, King Edward VII Avenue, Cardiff, Wales.
369
370
~Antenna
,,,,,,.,,is
Axis
-AAx
"'~,~' "~ '--~imuth Arm Oo,urn
=o
~..1~_~-MastArm (art a.aes in ~y~ pi.an. ) " ~ x ~
| [
stable I;r IB p t a t f o r m " ~ . A Axis
TASP
|
A, , . . o ,
I //;/'
~
LA'
\k--_
z~
la about M ~ai,)
LAot,o., I dri,,
"b . . . . . . . . . . . . . . . . Xi
Figure 1. Schematic of TASP in datum and general positions. and the distance tu =
(sin e A + cos 2 A sin e p) '/2.
(2)
In the auxiliary view of Fig. 2(a), which is orthogonal with the frame F8 shown in Fig. 1, the antenna axis makes an angle ~ with the deck perpendicular, where O = tan -t E =
[tu
cos (8 + ¢)/(cos X cos p)]
tan -~ (cot ~ sin p)
(3) (4)
and o e = [tu 2 cos 2 (8
+ E) + cos 2 A cos 2 0] 1/2.
(5)
However, to move the antenna axis from its datum position, shown in Fig. 1, to the truly vertical position, shown as oe in Fig. 2(a), it must rotate about the A and M axes. Figure 2(b) shows three views of the three principal axes of the TASP mechanism (A, M, and antenna) in their data positions, the frame of reference again being the Fo frame shown in Fig. 1 and in the auxiliary view of Fig. 2(a); (i) Is the view normal to the axes, i.e. normal to the xsys plane. (ii) Is a view looking along the A axis. (iii) Is a view looking along the M axis. Since the rotations about the A and M axes are commutative, they may be considered
371
(iii)
(i0
fb)
Rgure 2. Geometry of antenna axis in general pitched and rolled position. sequentially. Further, if the sequence is assumed as A rotation followed by M rotation, the geometry is simplified, and such rotations will cause the point a to move perpendicularly to the A and M axes in view (i); from a to / due to a rotation and from / to e by # rotation. It should be noted that points a, f, e, etc. in view (i) bear the subscripts 1 and 2 when they appear in views (ii) and (iii). In triangle eoa, since oa is 1
ea = [I + oe 2- 2oe cos (~ + O + y - fl)]ii2
(6)
~"= sin-l[oe sin (#i + O + y - ~)Jea].
(7)
and
In triangle aef
af = sin y(l -cos a),
(8)
v = ~'+ y - 9 0 °,
(9)
~:=/3- ~'- y + g o °,
(10)
ealsin (180° - ~) = sin y(l - cos a)/sin ~:.
(II)
a = cos -I [I - ea cos ~:)/(sin ~ sin y)].
(12)
and since
372
View (iii) indicates the trajectory of a as an elliptical path moving from a2 to f2. This corresponds with the path of at rotating through a to f, in view (ii). Angle r defines the point/2 z = tan-'
(f2f~/f~o2)
(13)
= tan-' [sin a sin -//(sin/3 cos 3t - cos a cos/3 sin -/)].
(14)
Rotation about the M axis will now take the antenna axis from 02/2 to 02e2, where X = tan-'
(eze~/e~02)
= tan t [tu sin (~ +
(15)
8)/(oe sin (~b + @))1.
(16)
Hence since /z=r+X
(17)
/z = tan -t [sin a sin -//(sin/3 cos - / - cos a cos/3 sin 1')] +tan -t [tu sin (~ + 8)/(oe sin (@ + @))].
(18)
Computation of eqns (12) and (18) and comparison with similar equations derived by the vectorial method [4] has proved that the corresponding equations are identities.
3. a/it Envelopes The results of a computer program giving a and/~ values for a selection of/3, y, 8 and @ angles with A and p as running parameters in the typical ranges ;t-+6 °, p_+18 °, give a/iz envelopes typically as shown in Fig. 3. Such envelopes indicate that considerable flexibility in 180O,
X~= 20* y =
30*
B=60* @=30*
~--40* ~'-" 30* B=45" ~ : 4 5 "
90 °
18*
p=lS* ~
p= 18"
0
150*
0
Q
Figure 3. Typical all~ envelopes design is possible; the overall a/l~ ratio and symmetry of the a/tx curves can both be modified to a considerable extent by varying the configuration of the mechanism. It is evident also that the range of angular rotations about the A and M axes necessary to maintain the antenna vertical can be made significantly larger than the range of pitch and roll angles encountered by the vessel. This fact would influence the accuracy with which the antenna can be positioned since angular errors in a/l~ would then correspond to much smaller errors of antenna position than is the case in present mechanisms.
4. Limits of Compatibility of Combinations of Conflgurational Angles The range of a and ix angles required for rectification of the antenna during a given range of pitch and roll depend on the configurational angles fl, 7, 8 and ~. It is clear that some configurations will be unsuitable since for some parts of the range of pitch and roll there will be no values of a and/z which will set the antenna vertical. For example, it will be readily seen that with angles/3 = 0, 7 = 0, any 8 and ~b angles, movement about the A and M axes merely maintains the axis in its datum position and it is not possible to set the antenna vertical
373
whatever the pitch and roll. On the other hand, with angles 4~ = 90°,/3 = 90°, 3' = 90°, 8 = 0°, the configuration becomes identical with the orthogonal axes mechanism and this configuration is clearly suitable, and the antenna axis may be maintained vertical, for any values of pitch and roll. For the purpose of computation of an optimisation process it is useful to determine the boundary configurations, i.e. to be able to determine configurations which are compatible with a required range of pitch and roll duties and which are not. In the following sections the governing kinematic relationships are investigated to establish these boundaries. The area of employment Figure 4 corresponds with Fig. 2(b) and shows the antenna axis oa in the datum position, (a = 0,/~ = 0). This axis is required to move to any position within the range of pitch and roll angles likely to be encountered by the vessel. This range is shown in chain-line; it is the area generated by the end of the axis oa during pitching and rolling and is defined the "area of employment". When the vessel is in a level attitude (zero pitch, zero roll) the axis must be in the perpendicular position od; while the position indicated by the dotted line oe is the position required when the vessel is in a general attitude of pitch A and roll p. In order to be able to take up such positions the axis may be rotated sequentially from the datum position as described in the analysis above, namely, from oa to of by virtue of rotation about the A axis and then by rotation about the M axis to move from of to oe. Considered in this way, rotations of the antenna axis to any instantaneous position in fact generate curved surfaces of two imaginary cones. Rotation of oa about the A axis to a position such as of generates part of one cone (the "A cone") whilst its subsequent rotation, of to oe, about the M axis generates part of the other, (the " M cone".) It will be noted that the A cone is common to Area of
~
- ~...
Rgure 4. Geometry of ,, and/z rotations.
374 all final positions of the antenna axis (its generator is of unit length and its angle ,/), while the M cone, although its generator must be of unit length, has an angle which varies according to the final position, e.g. oa to of to oe, or, oa to oc to od. The range of the base diameters for the M cone is given by diameters jn and gm in Fig. 4. It may be seen that to be able to move the end of the axis to any position within the area of employment, there must always be a general position f on the line ab (the A cone base) in Fig. 4. The existence of such a position indicates the existence of finite and therefore suitable angles a and/~. It follows, therefore, that in order to be able to move the antenna axis to any position within the area of employment, all points within this area must be served by lines perpendicular to the M axis such as ef. Conversely, if a line, perpendicular to the M axis, is produced from a point on the curved surface of the area of employment does not intersect the line ab on or between the points a and b, then that particular set of values of/3, 3', 8 and ~ do not constitute a suitable configuration for the pitch and roll duty required of that mechanism.
Band of deployment All suitable configurations are therefore identified by the existence of the two extreme cases of such lines ef in Fig. 4, namely ph and qk. These lines extended as gm and in, constitute a band, which is defined as the band of deployment, whose slope is 4) - 90° and whose width and position are functions of the required area of. employment. The definition of a suitable configuration, therefore, without reference to its kinematic characteristics, is that the base of the A cone must completely straddle the band of deployment. Such straddling falls into the types shown in Fig. 5(a)-(c). From this diagram it can be seen that, for each type, the angles/3
Ceh/o~~00, x.
(a)
b
a
(d)
Figure 5. Extreme and general types of configuration.
375 and 3' are defined by the general line ab as follows 3' = sin-I (ab/2)
(19)
= 3' + 4 , - o,,
(20)
and
where ~o = sin -1Xa.
(21)
Since the band of deployment is automatically defined by the area of employment and the angles 8 and ~, the extreme values of ~ and ~ are related by /3 = y + a constant angle.
(22)
For a given band of deployment, therefore, mechanisms having suitable and unsuitable and 3' angles can be indicated by a diagram of the pattern shown in Fig. 6. In this diagram the
Unsu.ob" / l e / / / .
/ yo0./
/ Figure 6. Typical graph for the configuration compatibility of given angles 8 and ~. three extreme types identified above are indicated by the three graphs rs, st and tu. The lines sr and tu have a slope of 1, the line st has a slope of - 1, and the intersections s and t indicate the extreme conditions corresponding to the chain lines in Fig. 5(b). It, therefore, follows that values of fl and 3' which fall within the area enclosed by the lines rs, st and tu indicate suitable mechanisms of the type shown in Fig. 5(d) and those which are outside this area are not. Hence in Fig. 4, for suitable configurations oh' < og'
(23)
cos (~ + 3') < og'
(24)
oa' >- oj'
(25)
cos (~ - 3') > oj'.
(26)
and
Both (24) and (26) must, therefore, apply, for the mechanism having the particular values of 8 and ~b corresponding to that band of deployment to be suitable.
376
Patterns of bands ,9/deployment The band of deployment is defined in Fig. 4 by the lines gm and jn which are projected perpendicular to the M axis from extreme points in the area of employment. The points from which these lines are projected originate in different parts of the area of employment. Bands of deployment, as a consequence, fall into the three patterns shown in Fig. 7. The equations governing these patterns are derived in Appendix 1.
~
I
(o)
|
PotternB ( ~
~
|
(c)
PolternC
Figure 7. The three patterns of the band of deployment. 5. identification of Suitable Combinations of Configurational Angles A computer program based on the equations derived above and which evaluates/3 and 3' according to the limits of (24) and (26) for different values of 8 and ~ has been prepared. The results of this program for a particular duty of pitch and roll are given in Fig. 8. This diagram indicates the loci of the two corners s and t of the graph shown in Fig. 6 for changes in 8 and Oh, and thus combines the variation of the four angles/3, 31, 8 and ~b. This diagram may be used as illustrated by the following examples.
Example 1. If 8 and ~b angles are selected for a TASP mechanism as 20° and 45 °, respectively, what values of/3 and 3' may be selected to ensure suitability? Solution. Draw lines from the values 8 = 20° and ~ = 45° as indicated by the dotted lines in Fig. 8. All simultaneous values of/3 and 3' within the area bounded by these dotted lines then imply suitable configurations, e.g., respectively, 20°, 45°; or 45°, 20°; or 35°, 35° etc. Example 2. If angle/3 is selected as 15° and 3' as 30°, what angles 8 and ~ are suitable? Solution. The lines rs, st and tu must enclose the point/3 = 15°, T = 30° (indicated in Fig. 8 as X), hence simultaneous values of 8 and ~ which are within the quadrant lying to the left of the point give suitable configurations, since such values of 8 and 4' imply the s points of the graphs given in Fig. 6, e.g., respectively, 0°, 30*; or 20°, 25° etc.
37"/
B=o=,
7O°
.3/1' 11lit "ll III1| 60 =
G )"
50°
4 ~ o ~
," /
/
4°.
f / 3 D°
\
f
f
/ /
\
~o-
7
0°
I
~
I0°
20°
i
\
i
i
40=
30"
II11 /
J
50=
I
I
60 =
7~'
B Figure 8. Composite graph for compatible configurations.
6. Kinematic Synthesis of a TASP Mechanism Any point may be selected at random within the rstu lines implied by the loci of Fig. 8 to give simultaneous values of/3, 1', 8 and @ of a TASP configuration which is compatible within the pitch and roll duty for which the diagram Fig. 8 is constructed. Such a random selection, of course, would take no account of the kinematic characteristics of the mechanism so obtained. Design considerations however demand that desirable kinematic characteristics should be obtainable more directly. This requires a more ordered method of selection or of synthesis in which the required shape and size of the alp envelope, typified in Fig. 3, may be obtained by the identification of suitable configurational angles• In this section such a method of synthesis is derived which is developed from the geometrical nature of the analyses given above. The a range
From Section 4 above it is clear that the range of a angles required for a given duty of pitch and roll may be varied by adjustment of the angles/3 and 1'. For example, in the configuration of given B and @shown in Fig. 4, rotation about the A axis is required to operate from point k, the intersection of the ab line (the base of the A cone), with the upper limit of the band of deployment (the smallest base diameter of the M cone) and point h, a similar intersection with the lower limit (the largest base diameter of the M cone). The auxiliary view gives the specific A angle range (o~k,-oth~). It follows that, for maximum range of angle a, the points h and k should be as far apart as possible, i.e. coincident with points b and a respectively; coincidence of h and b implies an angle a of 180° and coincidence of k with a means a = 0°. There are two configurations possible for maximum range of angle a; these are shown by the chain lines in Fig. 5(b) which correspond with the points s and t in Fig. 6. It is obviously useful to relate the a angle range with the graphs separating suitable and unsuitable configurations (Fig. 6). Configurations corresponding to the line rs in Fig. 6 correspond with ab lines in the positions shown in Fig. 5(a). It will be seen that the range of the a angles, being 180" at point s, decreases as values of 8 (and 1') increase. The line tu in Fig. 6 corresponds with configurations such as shown in Fig. 5(c); again the range of the angle a decreases from 180" at point t for configurations further along the tu line• The line st
378 corresponds with configurations further along the tu line. The line st corresponds with configurations shown in Fig. 5(b), while all practical configurations represented by points within the boundary rstu of Fig. 6 correspond with Fig. 5(d), i.e. the base of the A cone overlapping the extreme bases of the M cones (the band of deployment). The equations which related to the actual ranges are derived in Appendix 2. For the smallest range of ~, the M cone base should cut the band of deployment perpendicularly. This most nearly occurs whea large/~ and 3' angles are employed. Figure 9
G
91~
0¢ 0"
i 20 ~
i 40"
I 60"
I i 80"
180" -
(:~ O"
i
I
I
I
180" -
(a) i
0"
B
B
(b)
Figure g. Variation of range -,with angles/~ and 3'. shows typical plots of the equations derived in Appendix 2 for particular values of ~ when 8 = 0 and for a particular set of maximum pitch and roll conditions. The vertical lines in this diagram indicate the s and t cases. From Figs. 4 and 5 it can be seen that the extreme A cones are indicated by the values 3'. = sin-' (gj/2),
(27)
3'. = sin-' (gn/2).
(28)
(Subscripts m and n refer to maximum and minimum values, respectively). Figure 9(b) shows a 3-dimensional graph of the variation of the a range with/3 and 7 for given angles 3' and 4P, while the effect on the a range of varying 8 may be observed from the examination of Fig. 10. The ~t range
Displacement of the antenna axis about the A a~s implies a displacement relative to the M axis. In Fig. 4, displacement of the antenna through a to position o=ft in the lower auxiliary
379 a Range
I
~-o G
i Q
~3 2
P
Q I
i
0
8"90 ° 4
I Q
Rgure 10. Effect of 8 variation on a range.
view corresponds with displacement from oa to of in the main view, and in the upper auxiliary view with displacement through/Zo to ozf, (with/~0 given by a~ozf2). Such equivalent displacements about the M axis may be considered as the true operating data positions subsequent to rotation/~ to the required antenna axis position. The overall lz range is given by the difference between the largest and smallest/~ angles (/zm and/~., respectively) required to reach all parts of the area of employment. Since these angles are, in general, measured from different data positions, it is not possible to get a simple and clear picture of the variations of the/~ range with the different geometrical parameters, as it is with the a range. However, it is possible to take a line of approach to the problem which is based on the two-term form of eqn (18) and to consider a/~ rotation as having two components X and ~-; both measured from a datum of o2d2. One component is measured from this datum to the required position in the area of employment, and the other componentt from the datum o2d2 to the equivalent displacement about the M axis which is due to the a rotation, i.e. through angles X and ~- (Fig. 2) to the positions o2e2 and ozf,. From the auxiliary view of Fig. A5 it can be seen that the/~ rotation to reach p2 of the area of employment consists of X2 (component 1 of p2) + ~, (component 2 of P2), similarly/~ rotation for p3 consists of - X3+ ~'3. It will be observed that the first component X being angle to a point in the area of employment from point o5 is a function of 8 and 0 and is independent of the angles/~ and y. The second component, 1"being a measure of a point on the base of the A cone is a function of /3 and y and is independent of 8 and 4), It will thus be seen that by selecting components in this manner, the proble m of synthesis is simplified, and some control of t h e / t values of the a/~ envelopes may be obtained. Derivation of the equations governing these components is given in Appendix 3. Computation of the equations giving the X components in respect of the four corners of the area of employment when Am= 6° and p., = 18° for various values of ~b and 8 produces the
380 Ioo °
so"
I O0 °
N i\
(8 =oo) Cornersof area
(8 =20o) 4O"
×
20°
2O o
. ~
x o" 2o =
. ..~.."
4o = 60"
60 •
•
i '/
8(7)
¢a)
140" 120" I00 °
120" ( ~ - 4 0 =)
i00o 8O"
80 = 60 •
×4o.
4O"
(hi
~
(8- 6o')
2O" I
O"
1_4 _~__.-b" r-'3L-'-.~"--
I
20 ° 4O"
4O" (el
i...~'~ (d)
Figure 11. Variation of # range component % with 8 and ~. graphs shown in Fig. 11, while Fig. 12 shows graphs of the ~- values against angle 8(= 1' + the constant for given 8 and ~b values) for various 8 and ~ values. The graphs in Fig. 12 should be read in the same way as those of Fig. 9. The graphs of Figs. 11 and 12 may thus be used to obtain the/~ range for any given fl, 1', 8 and 4) values by adding the two pertinent graph values from each figure, e.g. for a system of /3 = 40°, 8 = 40°, ~ = 20°: Fig. 1l(c), (8 = 40°) the chain line (4' = 20°) gives the values of % for the various corners of the area of employment. Figure 12(c) shows the four families of graphs of • related to 8 = 40°.
Summary of the synthesis a angle ranges can be selected from Figs. 8 and 9 (see Example 1 below) and the all~ envelope corner position arranged by adjustment of the 8 angle shown in Fig. 10. /~ angle ranges can be deduced from Figs. 11 and 12 which give the complementary terms X and ~-, % being f(& ~) and r being f(/3, 1'). When 8 = 0, the # range is given by the range of the g components only since the/~ components for the principal points (p~ and p4), i.e. those which define the range, are equal. (see Example 2 below). When 84=0, however, synthesis of /~ ranges is less easy since it is not clear, without computing values, which corners of the area of employment are principal points.
Example 1. What configurations will give large and small values of a range? Solution. F r o m Fig. 9 it can be seen that a range of 180~ can be obtained with any 4) angle over 7° and any 8 angle if the/3 and 1' angles given by the points s and t are selected. These points are given in Fig. 8. For example, if ~ = 45° and 8 = 20° then/] = 12°, 1' = 47½° will give an ot range of 180°. Examination of Fig. 7 shows that small a ranges (<40 °) result when configurations in which/3 = 1' > 40° are selected. Example2. What configurations will give a value of/~ range of 120°?
381 (~ " 5 & 140" 12~Y'
V
I(~ °
Comes l~,q, of
t
(~ = 0 °)
eo*
T
s 2~3
I •
60 •
~
2O*
40 °
(o)
20"
I
I
I
/ (7
0•
20*
40"
6&
8(7
160 ° ~--20
~
°
1213"
Corner I
(~-20") T ~°
40*
I
I
I
40* 60*
~:~°
O*
~o
(b)
~o
I~ •
-- -
, ,o.
40"
o.i-
--
.,o. 6o.
-
~
(c)
.4o. do-
B Figure 12. Variation of t~ range component r with/3 and 3'. Solution. When 6 = 0 the r components of the principal points are equal, hence from Fig. 11 since/~ components of the principal points are equal in magnitude, a X component of 60° is, therefore, required; this is given by ~ = 18", # = 0,/3 = any, 3' = any. 7. Conclusions A geometrical analysis of a tilted axes stable platform mechanism has been presented; the results of the analysis confirm a vectorial analysis derived previously. The geometrical nature of the analysis has lead to the formulation of a method whereby the suitability of any configuration of the mechanism may be quickly determined. The method also allows numerous alternative suitable configurations to be selected. A method of synthesis, which is also based on the geometrical character of the analysis, has been evolved; use of the method enables required kinematic characteristics to be obtained. The paper illustrates the possibilities of synthesis which are generally inherent in a geometrical analysis. Other methods, such as vector analysis may have advantages in manipulation of the computations aspects but frequently fail in this respect. Acknowledgements--Theauthor acknowledges the help and inspiration of his colleagues in UWIST, in particular Mr. Colin Morgan, and of his associates in the Admiralty Surface Weapons Establishment, Cosham. This work has been carried out
382
with the support of the Procurement Executive, Ministry of Defence, ASWE, Cosham, Portsmouth, Gt. Britain. The help of Mrs. W. Scott is gratefully acknowledged in the presentation of this paper.
References l. R. N. Arnold and L. Maunder, Gyrodynamics and its Engineering Applications. pp. 484. Academic Press. New York (1961). 2. D.Delany, M. J. Durbin, C. Morgan, D. S. Mountain, R. M. Peters and R. Wilson, The Feasibility of a Non-Orthogonal Axes Stable Platform for Ship-Borne Use. UWIST Technical Note MMI6 pp. 43 (1975). 3. D. S. Mountain, A novel mechanism for antenna stabilisation. Naval Electrical Review, 29 22 (1975). 4. C. Morgan and R. M. Peters, The tilted axes stable platform mechanism: kinematic and kinetic aspects. Proc. Inst. Mech. Engrs 191 277 (1977).
Appendix 1 In each case the lower limit of a band of deployment is defined by the corner of the area of employment corresponding to maximum roll and maximum pitch, (point p in Fig. 4). Hence eqn (24) may always be written as (see Figs. 4 and 7) cos (~ + Y) <-op cos (4 + ~b,,),
(1.1)
(note, subscript m refers to maximum value). For the upper limit of the band of deployment however three different patterns are possible. Pattern A. When the angle ~ is small the upper limit of the area of employment is tangential to the unit axis sphere, i.e. it is defined by a point within the area of employment. Pattern B. Here, when the angle ¢, is larger, the upper limit originates within the segment of the ellipse which corresponds to the condition of rolling during maximum pitch (point q in Fig. 4). Pattern C. When the angle 4) is large, the upper limit originates at the corner of the area of employment icondition of maximum roll and maximum pitch). When the angle ~b is such that Pattern A pertains, since on' cannot be greater than o]', eqn 26 may be written as cos (/3 - 3') = I,
(I.2)
//= 3,.
(I.3)
i.e.
When the M axis is arranged such that Pattern B pertains, the eqn (26) may he written as (see Appendix IA) cos (d - v) = cos # cos A,(tan z 4) sinz/i + I) m + sin 6 sin A,~ cos 8.
(I.4)
But if d~> tan
-t f xcosec2 8 8)lt2"l,j, ~.(cos2 it,. - x 2 cosec2
(see Appendix 1B) where x = tu,, cos (~,, - 8) - sin A cos a, then Pattern C pertains and eqn (26) is written as cos (13 - y) = op cos (#~ - ~").
0.5)
Appendix 1A Condition [or upper limit for Pattern B When Pattern B pertains, ~ is sufRciently large that the line jn in Fig. 4 becomes tangential to the line indicating condition of rolling during maximum pitch, a boundary of the area of employment. This boundary line is circular when 8 is 0 and when viewed along the ships axis, and is a vertical straight line when 8 is 0 and viewed athwart ship. The latter condition is shown in Fig. AI. When O< 8 <90L the line is elliptical, as shown by the chain line in Fig. AI and its major axis swings in a circular path of radius sin ,~, about a vertical axis through 0. The ellipse, origin 0' is y:/a 2 + x2/b 2 = I,
(AI)
where a = c o s Am,
b = cos A,, sin 8. A general tangent to an ellipse is given by the equation y =mx 4- (m2b 2 + a2) ~1"-.
(A2)
In this application y =mx + (m z cos 2 A,, sinz 8 + cos2 A.) ~/:,
(A3)
383
to ellipse
Figure A1. Upper limit of band of deployment as tangent to ellipse. where
m = -tan 4',
The intercept on the O'Y axis is cos Am(m 2 sin s 8 + I) '12,
(A4)
hence the intercept in the oy axis is cos A.(m z sin s 8 + I) jlz- m sin A, cos &
(A5)
Thus the top limit in this case is cos 4) cos ,~.,(m 2 sin s 8 + I) 't2 - m sin A., cos 8 cos 4,
(A6)
and the required condition is cos (/3 - 4') > cos d' cos ~,,(tan z 4, sin s 8 + I) 'lz + sin 4' sin ,~,. cos 8.
(A7)
Appendix 1B Condition/or upper limit/or Pattern C When PatlBrn C pertains, 6 is so large that the top limit is projected from the corner of the area of employment. From Appendix IA, the ellipse corresponding to rolling in a condition of maximum pitch is given by yZ/cos2 A,, + x21(cos2 Am sin 2 B)= I.
(BI)
The radius of the extreme end of the elliptical segment shown in Fig. A2 has been derived as Radius = tu. cos ( , . - 8), where
tu. = (sin 2 A,. + cos 2 A= sin 2 p.),l~
(B2)
384
o
\
Figure A2. Upper limit of band of deployment as tengent to corner of ~trea of
employment. and
~m = tan-' (cos ~,. sin p,./sin ~,.). A tangent at this extreme corner is given by d(y)/dx at (turn cos (e,. - 8) - sin A,,, cos 8, cos pm cos A,. ). From equation B1 y2 = (1 - x2/cos A,~ sin z 81 cos 2 Am y = (cos 2 A,,, - x: cosec 2 8) ~/z dy = -x
dx
c o s e c z 8 ( c o s ~ ;t,. - x 2 c o s e c 2 8 ) -'12.
(B3) (B4) (851
Thus if ~b> tan-L (x cosec 2 6 (cos z a,. - x 2 cosec: 8)- L/z).
(B61
then the top limit is dictated by the corner of the area of employment. The top limit is then given by
op cos (~b - ip.. ),
(B7)
where op = (turn cos z (8 + e,,,)+ cos z ,~,,, cos2 p.,.)~l~ and the required condition is cos (# - y) > op cos (,/, - Om1.
(88)
Appendix lC Derivation of equations giving band of deployment The top and bottom limits are given by the two lines jn and gm as shown in Fig. A3. From Appendix 1A. the equation for the top limit line for medium values of ~ (Pattern B) is given as y = mx + cos A,.(m 2 sin 2 8 + 1t ./2 - m sin ,~,. cos 8,
(A31
and the equation for the circle y = (1 - x2) '/2,
(C1)
385 .Y
\i/. 0
Im
x
Figure A3. Identification of points defining the boundaries of deployment. hence points xj and x~ are given by (1 - x) tl2 = m x + cos A.(m 2 sin 2 8 + I) t/2 - m sin A,. cos 8, = mx + K, where
m = - tan
(C2) (C3)
1 - x 2 = m2x 2 + 2 K m x + K 2
(C4)
(m 2 + l)x 2 + 2 K m x + ( K 2 - I) = 0
(C5)
[2Km \
K 2-
X2 + L m - - " ~ ) X "1-~
I
_
= U,
(C6)
the roots of which are xi and x.. The equation for the bottom limit line is H-L y = m x +COS Am(m 2 sin 2 8 + 1) j/z- m sin A,. cos 8 - cos----~'
=rex+J,
(C7) (CS)
points x~ and x,. are given by
+J
(C9)
I - x 2 = m2x 2 + 2mJx + j2
(CIO)
(I - x2) I/z=mx
(m 2 + 1)x 2 + 2mJx + (j2 _ I) = 0
/ 2mJ \ +j2+l
_
x+ Lm--~+l}x ---~-~-- u,
(Cll) (CI2)
the roots of which are xg and x,.. Equations for xj and x. for the Pattern C may be arrived at by substituting K = tu,~ cos (4) - ~,. ) sec
(CI3)
instead of eqn C6.
Appendix 2 The manner in which the a range varies as ~, (and/3) changes depends on the manner in which the base of the A cone intersects the band of deployment: Fig. 5(a)-(c) show the three extreme cases, corresponding to the rs, st and tu types and F!g. 5(d) shows the general case. The four types of case are shown also in Fig. A4. For a rs configuration, Fig. A4(a), it is deduced that maximum a angle = cos-t((sin ~/- W/sin/3)/sin ~,),
(2.1)
hence the a range = 0 to cos-tO - W/(sin 0 sin y),
(2.2)
where W is the width of the band of deployment, measured as a fraction of the radius of the circle. For a st configuration, Fig. A4(b), the range = 180° to cos -t (W/(sin ~8 sin 3,)- I).
(2.3)
For a tu type, Fig. A4(c) the a range is 0 to cos-tO - W/(sin ~ sin y)), i.e. eqn (2.2) and for the general configuration, Fig. A4(d)
386
~"
Figure A4.
a
Y
?
tu Type
range in relation to types of suitable configurations.
the a range = a' to a", where a ' = cos -I (KWl(sin 8 sin y)),
(2.4)
K is a fraction of W and a "= cos-'(I - W(I - K)/(sin # sin 3')).
(2.5)
Appendix 3 The X components of the four corners of the area of employment are deduced from Fig. A5 as angles X~, X2, X3 and g4. From this diagram it will be seen that the four angles are given by x~ = tan-~(Ql/( sin 4) - Q2 cos d) - Q5)),
(3.1)
x2 = tan-I(Q3/( sin 4) - Q4 cos 4) - Q5)),
(3.2)
£3 = tan-~(QIl( sin 4) + Q2 cos 4) - QS)),
(3.3)
x4 = tan-~(Q3/( sin 4) + Q4cos 4) - Q5)),
(3.4)
QI = tu~ sin (~,. - 8),
(3.5)
Q2 = tu~ cos (~m -- 8),
(3.6)
Q3 = tu,, sin (~,. + 8),
(3.7)
Q4 = tu,, cos (e,. +8),
(3.8)
Q5 = (1 - op cos Ore) sin 4).
(3.9)
where
The • component for each point is given by the intersection of a circle running through the point centered at 02 and the ellipse which is the auxiliary view of the base of the A cone. In Fig. A6(a) the ellipse
y2/a2 + (x -
Z ) 2 / b 2 = 1,
(3.10)
(x is assumed positive leftwards from 02), where a = sin 3' and b = sin y cos 8 and Z the distance of the ellipse centre
from 02.
b2y2 + a2xr- 2a2xZ + a 2 Z
= a2b 2.
(3.11)
387
J
I
Figure AS. Geometry of/~ range related to r - s cases of A cone. For the circle
y2 = R2_ x 2.
(3,12)
At intersectionof ellipseand circle R 2 b 2 _ b2x 2 + a2x 2 - 2 a 2 x Z + a 2 Z 2 = a 2 b 2
(3.13)
(a 2 - b2)x 2 - ( 2 a 2 Z ) x + ( a 2 Z + R 2 b 2 - a2b 2) = 0
(3.14)
x = a2Z/(a 2 - b 2) +- ( a 4 Z 2 - (a 2 - b2)(a2Z 2 + R 2 b 2 - a2b2))l121(a2 - b2).
(3.15)
hence
For
r-
S cases, shown in Fig. A6(a),
(3.16)
Z=R5+b,
R5, r, s and the general case are as shown in Fig. AS. s - t cases are shown in Fig. A7. For these cases Z -- R 6 - b, as shown in Fig. A6(b).
(3.17)
t - u cases are shown in Fig. A8. For these cases
Z = b -
R5, as shown in Fig. A6(c).
In all cases the only root ~pplicable is the smaller of the two.
(3.18)
388
- i
I
(a)
i
{b) Oz
Figure A6. Patterns of A cone position.
x.G
Figure A7. s - t cases of A cone.
389 t-u .
-
Case
-
.
Figure A8. t - u cases of A c o n e . In each of these patterns the ¢ components are given by
COS-' (X,/O2p,),
(3.19)
"r2= COS-I (x2/o,p2).
(3.20)
¢, =
¢3 = cos-I
(x31o2pD,
(3.21)
¢4 = COS-I (X4/o2P4),
(3.22)
o2pl = (QI 2 + (sin 4' - Q 2 - Q5)2) '12,
(3.23)
o2p2 = (Q3 2 + (sin 4' + Q4 - Q5)2//2,
(3.24)
o2p3 = (Q12 + (sin 4' + Q2 - Q5)2) '12,
(3.25)
o2p4 = (Q3 2 + (sin 4' - Q4 - Q5)2) '/2.
(3.26)
where
SYNTHESE D'UN MECANISME POUR LA STABILISATION D'UNE ANTENNE DE RADAR. R. M. Peters Un m4canisme destind ~ stabiliser les antennee de radar sur les navires en m e r e s t analys4 g4ometriquement et les 4quations qui en d4coulent mettent en relation lee param~tres du m4canisme avec l'attitude du navire. On developpe une m4thode par laquelle on peut choisir les configurations du m4canisme compatibles avec le tangage et le roulis maximum du navir~. Le d4veloppement ult4rieur de l'analyse produit une m~thode de synth~ss qui permet d'arriver aux conditions cin4matiques d4sir4es. Des exemples de compatibilit4 et de eynth~se R4sum4
sont dorh~ s.