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Evaluation of the effects of the communication cable on the dynamics of an underwater flight vehicle
Ocean Engineering 31 (2004) 1019–1035 www.elsevier.com/locate/oceaneng
Evaluation of the effects of the communication cable on the dynamics of an underwater flight vehicle Z. Feng , R. Allen Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK Received 26 June 2003; accepted 12 November 2003
Abstract This paper presents a numerical scheme to evaluate the effects of the communication cable attached to an underwater flight vehicle. Both simulation and model validation results show that the numerical scheme is effective and provides a means for developing a feed-forward controller to compensate for the cable effects when developing an autopilot for the tethered vehicle. Moreover, the numerical scheme can also be applied to predict the effects of the ROVs umbilical during its deployment. # 2004 Elsevier Ltd. All rights reserved. Keywords: Underwater flight vehicles; Towed cable; Cable deployment; Finite difference method; Least squares solution; Model validation
1. Introduction Underwater vehicles are playing an increasingly important role in exploration and exploitation of the underwater environment. Automatic control of underwater vehicles presents a challenge for control engineers due to the strong nonlinearity of the dynamics, the high degree of model uncertainty resulting from poor knowledge of hydrodynamic coefficients and the effect of external, un-measurable disturbances such as underwater currents. To provide a test-bed for control techniques of underwater vehicles, the Subzero II flight vehicle (see Fig. 1) was constructed (Lea, 1998a,b) and has recently been
Corresponding author. Tel.: +44-23-80594935; fax: +44-23-80593190. E-mail address: [email protected] (Z. Feng).
0029-8018/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2003.11.001
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Fig. 1. Subzero II.
developed further (Feng et al., 2003). It is a low cost, self-powered, torpedo-shaped flight vehicle with a length of 1 m and a maximum diameter of 10 cm. The power is from an on-board Ni–Cad battery package. The vehicle is propelled by a propeller and guided by four control fins, which include two rudders and two stern-planes. While the two rudders are linked together, the stern-planes are separated and thus provide a means for roll control. Vehicle control is achieved using a personal computer on the shore which communicates with the underwater vehicle over a thin and neutrally buoyant wire. Connected at one end to the serial port of the microcontroller on board the vehicle and at the other to the serial port of the personal computer, the communication cable transmits the manoeuvring commands and sensor data bi-directionally. The cable provides a means for development and assessment of autopilot controllers without modifying the software running in the microcontroller on board the underwater vehicle, but obviously affects the motion of the vehicle. To reduce the cable effects, the cable is deployed by a drum on the shore with negligible tension when it is pulled by the vehicle. The forward-moving vehicle experiences additional forces and moments which are induced by the cable. Intuitively, these forces and moments produce more significant effects on the motion of vehicle as the immersed cable lengthens. Due to the existence of the cable, the vehicle is really a tethered vehicle. Thus, to develop a model-based autopilot for the tethered vehicle, not only does the dynamics of the un-tethered vehicle need to be identified, but also the cable dynamics has to be modelled as well. Note that it is also true for remotely operated vehicles (ROVs) which are deployed by umbilical cables. Treated as a long, thin, flexible circular cylinder in arbitrary motions of gravity, inertial forces, driving forces and hydrodynamic loading which is taken to be the sum of independently operating normal and tangential drags, each given by a single coefficient, the dynamics of the cable can be described by the partial differential
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equations over time and arc length of the cable (Albow and Schechter, 1983). And for a typical towed cable system, where the length of the cable is fixed, numerical solutions of the equations can be obtained by finite difference methods (Albow and Schechter, 1983; Milinazzo et al., 1987). However, these numerical schemes cannot be applied to our situation directly due to the variable cable length. In this paper, the numerical scheme developed by Milinazzo et al. (1987) is extended and then applied to evaluate the effects of the communication cable on the Subzero II vehicle. The rest of the paper is organised as follows: cable dynamics, as well as boundary conditions, are described in Section 2. A numerical scheme which deals with cables of non-fixed length is proposed in Section 3. The cable effect, as well as the dynamic model of the tethered vehicle, is verified in Section 4 with conclusions in Section 5.
2. Cable dynamics 2.1. Three coordinate systems To analyse the motion of the cable as well as its effect on the vehicle, it is convenient to define three coordinate systems, i.e. the earth-fixed frame and the local frames along the cable and the vehicle-fixed frame. As shown in Fig. 2, the earth-fixed frame (i, j, k) is selected with k pointing vertically downwards. The vehicle-fixed frame (iv, jv, kv) is located at the centre of the hull, with iv coinciding with the longitudinal axis, and jv pointing to starboard. The relationship between the vehicle-fixed frame and the earth-fixed frame can be expressed in terms of Euler angles (Fossen, 1994), i.e. ½ iv
jv
kv ¼ ½ i
j k Rð/; h; wÞ
Fig. 2. Test configuration of Subzero II.
ð1Þ
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with 2
c/sw þ s/shcw c/cw þ s/shsw s/ch
chcw Rð/; h; wÞ ¼ 4 chsw sh
3 s/sw þ c/shcw s/cw þ c/shsw 5 c/ch
ð2Þ
where c ¼ cos , s ¼ sin and /, h, w are the roll, pitch and heading angle of the vehicle, respectively. The local frames (t, n, b) are located at points along the cable with t tangent to the cable in the direction of increasing arc length from the tow-point, and b in the plane of (i, j). They are obtained by three rotations of the earth-fixed frame in the following order: (1) a counter-clockwise rotation through angle a about the k axis to bring the i axis into the plane of t and n; (2) a counter-clockwise rotation about the new the i axis through p=2 to bring the k axis into coincidence with b; and (3) a clockwise rotation about b through b to bring i and j into coincidence with t and n. Thus, the relationship between the local frames and the earth-fixed frame can be expressed as follows (Albow and Schechter, 1983; Milinazzo et al., 1987): ½t
n b ¼ ½i j
k W ða; bÞ
ð3Þ
with 2
3 casb sa sasb ca 5 cb 0
cacb W ða; bÞ ¼ 4 sacb sb
ð4Þ
In terms of (1)–(4), the relationship between the local frames and the vehiclefixed frame can be written as ½t
n b ¼ ½ iv
jv
kv RT ð/; h; wÞW ða; bÞ
ð5Þ
where the orthogonal property of R has been used. 2.2. Dynamic equations Define the vector yðs; tÞ :¼ ½ T
Vt
Vn
Vb
a
b T
ð6Þ
where t and s denote the time and the arc length of the cable measured from the tow-point, respectively, T is the tension and Vc ¼ ½ Vt Vn Vb T denotes the velocity vector in the local frames along the cable. The cable dynamics can be expressed as the following partial differential equation (Albow and Schechter, 1983): M
@y @y ¼N þq @s @t
ð7Þ
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with 3 1 0 0 0 0 0 6 0 1 0 0 Vb cosb Vn 7 7 6 6 0 0 1 0 Vb sinb Vt 7 7 6 M¼6 7 7 6 0 0 0 1 Vn sinb Vt cosb 0 5 4 0 0 0 0 Tcosb 0 0 0 0 0 0 T 3 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi w qd 1 þ eT pC sinb þ U U j j c t t t 7 6 2 7 60 7 6 7 60 7 6 7 q¼6 7 6 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 61 7 6 qd 1 þ eT Cn Ub Un2 þ Ub2 qAJ_ b 7 62 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 4 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi wc cosb þ qd 1 þ eT Cn Un Un2 þ Ub2 qAJ_ n 2 2
2
Vt 6 me 1 þ eT 6 6e 6 60 6 0 N¼6 6 6 ðm1 Vb qAJb Þ 6 e 6 ð1 þ eTÞ 6 4 ðm1 Vn qAJn Þ e ð1 þ eTÞ
m 0
0
0 0 0 0 0 0
0 0 0
0 0
m1
0 m1 0
3 ðm1 Vn qAJn Þ 7 7 7 0 0 7 7 0 1 þ eT 7 7 ð1 þ eTÞcosb 0 7 7 7 ðm1 Vn qAJn Þsinb mVt cosb 0 7 7 5 ðm1 Vb qAJb Þsinb mVt ðm1 Vb qAJb Þcosb
where q is the fluid density (kg/m3), m is the mass per unit length of cable (kg/m), A is the cross-sectional area of the unstretched cable (m2), m1 is m þ qA (virtual mass per unit length) (kg/m), g is the gravitational acceleration (9.81 kg m/s2), wc is ðm qAÞg (immersed weight per unit length) (N/m), e is 1=EA where E is Young’s modulus, Ct, Cn are the tangential and normal drag coefficients, respectively, d is the diameter of the stretched cable (m), J ¼ ½ Jt Jn Jb T is the current velocity given in the local frame (t, n, b) (m/s) and U ¼ ½ Ut Un Ub T is Vc J (velocity relative to the current) (m/s). 2.3. Boundary conditions The initial configuration of the cable, as well as six boundary conditions, is required to specify the solution of (7). The immersed part of the cable has two ends: the vehicle-end and the drum-end. The vehicle-end, or the tow-point, is located at the tail of the vehicle, while the drum-end is located at the cable drum which deploys the cable into the water due to pull of the vehicle.
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Since the cable shares the velocity of the vehicle at the tow-point, three boundary conditions can be obtained. Denote the position of the tow-point in the vehicle-fixed frame with rc ¼ ½ xc yc zc T , and the linear and angular velocity of the vehicle with V ¼ ½ u v w T and X ¼ ½ p q r T , where (u, v, w) denote surge, sway and heave speeds , while (p, q, r) denote roll, pitch and yaw rates of the vehicle. Then the velocity of the tow-point in the vehicle-fixed frame is ðV þ X rc Þ, where denotes the cross product of two vectors. In terms of (5), the velocity of the towpoint can be repressed in the local frame of the cable at the vehicle end, i.e. Vc ð0; tÞ ¼ W T ða; bÞRð/; h; wÞðV þ X rc Þ
ð8Þ
which involves three boundary conditions. At the drum-end, since the drum always deploys the cable in the tangential direction, the normal components of the cable velocity must be zero, i.e. Vn ðSt ; tÞ ¼ 0
ð9Þ
Vb ðSt ; tÞ ¼ 0
ð10Þ
where St is the total arc length of the immersed cable at time t. To determine the sixth boundary condition, consider the rolling motion of the cable drum. In terms of Newton’s second law, one has d ðId Xd Þ ¼ C Cf dt where Id is the moment of inertia of the drum, Xd is the angular speed of the drum, C is the driving torque produced by the cable tension at the drum-end, and Cf is the resisting torque caused by the sliding friction between the drum and the pivot. Since Xd ¼ Vt ðSt ; tÞ=Rd and C ¼ TðSt ; tÞRd where Rd is the radius of the cable drum, the dynamic equation of the drum can be rewritten as Id V_ t ðSt ; tÞ þ TðSt ; tÞR2d ¼ Cf Rd
ð11Þ
So far, the six boundary conditions have been determined by (8)–(11). 2.4. Cable effect The tension of the cable at the tow-point, i.e. Tð0; tÞtð0; tÞ, results in the additional forces and moments that affect the motion of the vehicle. The additional forces can be obtained by expressing the cable tension in the vehicle-fixed frame in
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terms of (5), i.e. 2
3 2 3 FcX Tð0; tÞ 5 Fc ðtÞ :¼ 4 FcY 5 ¼ RT ð/; h; wÞW ðað0; tÞ; bð0; tÞÞ4 0 0 FcZ
ð12Þ
where FcX ¼ Tð0; tÞðcwchcacb swchsacb þ shsbÞ FcY ¼ Tð0; tÞðchs/sb þ s/cwshcacb s/swshsacb cwc/sacb swc/cacbÞ FcZ ¼ Tð0; tÞðchc/sb þ s/swcacb þ s/cwsacb þ cwshc/cacb swshc/sacbÞ In terms of the position of the tow-point in the vehicle-fixed frame, the cableinduced moments are 2 3 2 3 McX yc FcZ zc FcY ð13Þ Mc ðtÞ :¼ 4 McY 5 ¼ rc Fc ðtÞ ¼ 4 zc FcX xc FcZ 5 McZ xc FcY yc FcX
3. Numerical scheme The numerical solutions of the partial differential equations (7) under the boundary conditions (8)–(11) will be obtained by the finite difference method which discretises the equations over time and space. However, unlike a typical towed cable system where the length of the cable is fixed, the length of the immersed communication cable is not fixed as it is gradually pulled into the water by the vehicle. Moreover, the total length of the wet cable at time t is determined by the deployment speed Vt ðSs ; sÞ with 0 s < t, i.e. ðt St ¼ Vt ðSs ; sÞds ð14Þ 0
where the initial length of the wet cable has been assumed to be zero. Therefore, it is impossible to divide the wet cable into fixed number segments each of which has a fixed length. Since the length of each segment (or the position of each node) needs to be fixed for finite difference methods, the number of segments (nodes) of the wet cable must increase with increasing time as shown in Fig. 3. This is achieved by discretising (14) over the time step Dt, i.e. k1 ð ðiþ1ÞDt X Sk ¼ Vt ðSs ; sÞds ð15Þ i¼0
iDt
where Sk (k ¼ 1; 2; . . .) denotes the total length of the immersed cable at time kDt.
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Fig. 3. Evolution of cable configuration from tk to tk þ1 .
This suggests that the immersed cable at kDt be divided into k segments with the segment lengths ð ðiþ1ÞDt DSiþ1 ¼ Vt ðSs ; sÞds; i ¼ 0; 1; . . . ; k 1 ð16Þ iDt
Provided the time step Dt is sufficiently small, the segment lengths can then be approximated by DSiþ1 ¼ Vt ðSi ; ti ÞDt;
i ¼ 0; 1; . . . ; k 1
ð17Þ
where ti ¼ iDt. Thus, the lengths of all the k segments are fixed as they are determined by the past deployment speeds. From (15)–(17), it can also be obtained DSkþ1 ¼ Skþ1 Sk ¼ Vt ðSk ; tk ÞDt
ð18Þ
This implies a new segment DSkþ1 emerges once a time step Dt has elapsed since tk, and the total cable length at tkþ1 is Skþ1 ¼ Sk Vt ðSk ; tk ÞDt
ð19Þ
After the segments (nodes) along the immersed cable have been specified, the finite difference method of second order approximation (Milinazzo et al., 1987) can be applied. Suppose that the configuration of the cable at time tk, i.e. yðSi ; tk Þ with i ¼ 0; 1; . . . ; k, is known, the problem is then to predict the configuration of the cable at time tkþ1 , i.e. yðSi ; tkþ1 Þ with i ¼ 0; 1; . . . ; k þ 1, where Skþ1 is determined by (19). Define DSj :¼ Sj Sj1 1 Sj1=2 :¼ ðSj þ Sj1 Þ ; 2 tkþ1=2 :¼ ðk þ 1=2ÞDt
j ¼ 1; 2; . . . ; k þ 1
ð20Þ
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and 1 yðSj ; tk Þ þ yðSj1 ; tk Þ 2 1 yðSj1=2 ; tkþ1 Þ :¼ yðSj ; tkþ1 Þ þ yðSj1 ; tkþ1 Þ 2 ; 1 yðSj1 ; tkþ1=2 Þ :¼ yðSj1 ; tk Þ þ yðSj1 ; tkþ1 Þ 2 1 yðSj ; tkþ1=2 Þ :¼ yðSj ; tk Þ þ yðSj ; tkþ1 Þ 2
yðSj1=2 ; tk Þ :¼
j ¼ 1; 2; . . . ; k þ 1
ð21Þ
where S 0 ¼ 0. Note that since the new segment DSkþ1 is not in the water at tk, yðSkþ1 ; tk Þ in (21) is prescribed according to the rolling motion of the drum which is assumed to be uniform, i.e. yðSkþ1 ; tk Þ ¼ ½ TðSk ; tk Þ Vt ðSk ; tk Þ 0 0 aðSk ; tk Þ bðSk ; tk Þ DSkþ1 =Rd T . Moreover, the governing Eq. (7) will still be applied to approximate the dynamics of the new segment although it is only partly true for the new segment which is gradually pulled into the water. Obviously this may result in modelling error. However, as will be verified, the modelling error can be negligible when the new segment is not too long. Applying the governing Eq. (7) at the points ðSj 1=2 ; tkþ1=2 Þ, j ¼ 1; 2; . . . ; k þ 1 yields the 6ðk þ 1Þ difference equations as follows (Milinazzo et al., 1987):
yðSj ; tkþ1 Þ yðSj1 ; tkþ1 Þ M yðSj1=2 ; tkþ1 Þ DSj
yðSj ; tk Þ yðSj1 ; tk Þ þ M yðSj1=2 ; tk Þ DSj
yðSj ; tkþ1 Þ yðSj ; tk Þ ¼ N yðSj ; tkþ1=2 Þ Dt
yðSj1 ; tkþ1 Þ yðSj1 ; tk Þ
þ N yðSj1 ; tkþ1=2 Þ þ q yðSj1=2 ; tkþ1 Þ Dt
þ q yðSj1=2 ; tk Þ for j ¼ 1; 2; . . . ; k þ 1
ð22Þ
Note that the six equations under j ¼ k þ 1 approximate the dynamics of the new segment. The remaining six equations are determined by the boundary conditions (8)–(11), i.e. Vc ðS0 ; tkþ1 Þ ¼ W T ðaðS0 ; tkþ1 Þ; bðS0 ; tkþ1 ÞÞRð/; h; wÞðV þ X rc Þ
ð23Þ
Vn ðSkþ1 ; tkþ1 Þ ¼ 0
ð24Þ
Vb ðSkþ1 ; tkþ1 Þ ¼ 0
ð25Þ
Id
Vt ðSkþ1 ; tkþ1 Þ Vt ðSk ; tk Þ þ TðSkþ1 ; tkþ1 ÞR2d ¼ Cf Rd Dt
ð26Þ
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Combining (22)–(26), 6ðk þ 2Þ nonlinear equations with 6ðk þ 2Þ unknowns can be obtained. And the cable effects can be determined by solving these equations. It should be noted that the dimension of (22)–(26) expands with time until the deployment of the cable finishes. Therefore, an increasing computation effort has to be undertaken with time before the deployment of cable stops. One might conclude that this scheme is not suitable for on-line analysis of the cable effect. However, this is not true since the computational effort is limited due to the limited length of a practical cable. Once the solutions to difference equations (22)–(26) have been obtained, the cable effects at time tkþ1 can be determined from (12) and (13) where t is replaced with tkþ1 . To summarise, the numerical scheme to evaluate the effects of the communication cable comprises the following steps: (1) Initialisation. Set the time step Dt and stop time TIME of the simulation. Set the time index to k ¼ 0. Since no cable is in water at tk ¼ t0 ¼ 0, the vehicle-end coincides with the drum-end. Therefore, the initial configuration y(Sk, tk) can be obtained by solving (8)–(11) with t ¼ 0. Note that the initial states of the vehicle (u, v, w, p, q, r, /, h, w) need to be given to specify a solution for (8)–(11). (2) Given the vehicle states (u, v, w, p, q, r, /, h, w) at time tkþ1 ¼ ðk þ 1ÞDt The states can be measured by the on-board sensors or predicted by the dynamical model of the vehicle. (3) Solve the nonlinear equations (22)–(26). The equations can be solved efficiently using Matlab Optimization Toolbox (The Mathworks, 2000). For example, the least squares solution to the nonlinear equations P fi ðx1 ; x2 ; :::; xn Þ ¼ 0, (i ¼ 1; 2; . . . ; n) which minimises ni¼1 fi2 ðx1 ; x2 ; . . . ; xn Þ can be obtained by invoking the command fsolve. (4) Determine the cable effects. This is achieved by substituting yð0; tkþ1 Þ into (12) and (13) where t is replaced with tkþ1 . (5) Update time index k k þ 1. (6) If k TIME=Dt, then go to step (2), otherwise stop the simulation.
4. Assessment of the cable effects for Subzero II Subzero II’s communication cable is a thin, lightweight and neutrally buoyant wire which attached to the vehicle at the tail with coordinates rc ¼ ð0:5; 0; 0Þ relative to the vehicle-fixed frame. It is deployed by a cable drum of 0.2 m diameter with negligible tension, i.e. Cf ¼ 0. The cable specifications and fluid parameters are listed in Table 1.
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Table 1 Cable and fluid parameters (units as given in Section 2.2) q
m
A
m1
wc
e
Ct
Cn
d
1000
0.0049
4.9e6
0.0098
0
0
0.01
1.0
0.0025
4.1. Evaluation of the cable effects As stated in Section 3, the vehicle states (u, v, w, p, q, r, /, h, w) are required to specify the boundary conditions (8) at the tow-point and to determine the cable effects (12) and (13). These states can be measured by the sensors on board Subzero II or estimated via the dynamic model of the vehicle. For Subzero II, they are obtained by the latter method since the sway and heave speeds are not currently measured. Therefore, the reliability of the cable effect predicted by the numerical scheme not only depends on the numerical scheme itself, but also on the dynamic model of the vehicle due to the coupling between the cable and the vehicle as described by (8), (12) and (13). Since model uncertainty involved in dynamic models of underwater vehicles can never be avoided due to inaccurate model parameters, e.g. hydrodynamic coefficients. In order to assess the cable effect independently of the vehicle dynamics, an ‘idealised’ vehicle is assumed where the dynamics are ignored. Since the flight control of Subzero II comprises three subsystems: forward speed, heading and depth, three situations need to be considered: (i) The vehicle is moving straight ahead with prescribed states: u ¼ 1, v ¼ w ¼ 0, p ¼ q ¼ r ¼ 0 and / ¼ h ¼ w ¼ 0 (ii) The vehicle is diving while moving ahead with prescribed states: u ¼ 1, v ¼ w ¼ 0, p ¼ r ¼ 0, / ¼ w ¼ 0 and q ¼ h_ ðtÞ. The pitch angle h changes with the time (see Fig. 6(a)), i.e. 8 v 0 t 10 s <0 ; v hðtÞ ¼ 6 ðt 10Þ; 10 < t 20 s : v 60 ; t > 20 s (iii) The vehicle is turning while moving ahead with prescribed states: u ¼ 1, v ¼ w ¼ 0, / ¼ h ¼ 0 and r ¼ w_ ðtÞ. The heading angle w changes with the time. Since the cable effects of (iii) can be deduced from that of (ii) due to the neutral buoyancy of the cable, only the cable effect under case (i) and case (ii) will be evaluated. Setting the time step Dt ¼ 1 s and TIME ¼ 30 s, the cable effects are calculated by applying the evaluation procedure in Section 3. The results are shown in Figs. 4 and 5. Fig. 4 illustrates the results of straight ahead motion: the tension at the towpoint, i.e. the cable drag experienced by the vehicle is plotted by the dotted line.
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Fig. 4. Simulation results of cable drag under straight ahead motion.
Note that the remaining cable-induced forces and moments are all zero. It can be seen that the cable drag is increasing with the time. This indicates that the vehicle is slowing down if the thrust remains invariant. In other words, the thrust needs to be increasing for the vehicle to hold a constant forward speed. Fig. 5 shows the cable effects on the vehicle’s diving. The configurations of the cable at 10, 15, 20, 25 and 30 s are shown in Fig. 5(b). It can be seen that the immersed part of the cable is lengthening and descending. The cable tension at the tow-point is plotted in Fig. 5(c), the velocity components of the tow-point with respect to the cable local frame are shown in Fig. 5(d). The cable-induced forces and moments are shown in Fig. 5(e) and (f). As can be seen from Fig. 5(e), the cable tends to slow down the vehicle due to the increasing drag in surge motion. Moreover, the cable tends to push the vehicle in heave direction as a positive heave force is produced. From Fig. 5(f), it can be seen that the cable tends to make the vehicle head up due to the positive pitch moment. However, the cable effects on the heave and pitch motions are much smaller than that on the surge motion. Normally, it is difficult to verify the numerical solutions of partial derivative equations since they do not always admit analytical solutions. Fortunately, as will be derived, the partial derivative equation (7) subject to boundary conditions (8)–(11) admits an analytical solution under case (i). Being pulled into the still water by the vehicle which retains a pure surge motion of constant speed, the neutrally buoyant cable must maintain a configuration of a straight line which extends in the direction of vehicle motion. Thus, orientation angles between the local frames along the cable and the inertial frame are invariant over space and time, i.e. @a @b ¼ ¼ 0; @s @s
@a @b ¼ ¼0 @t @t
ð27Þ
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Fig. 5. Cable effects on vehicle diving. (a) Pitch angle; (b) cable configurations at different times; (c) cable tension at the tow-point; (d) local velocity at the tow-point; (e) cable-induced forces; (f) cableinduced moments.
According to the definition of the local frame (t, n, b), the orientation angles of level motion are a ¼ 0;
b¼p
ð28Þ
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Fig. 6. Model validation of the tethered vehicle. (a) Propeller command; (b) forward speed; (c) rudder command; (d) heading angle.
Moreover, since the cable is supposed to be inelastic, the local velocities along the straight cable must be constant, i.e. @Vt @Vn @Vb ¼ ¼ ¼0 @s @s @s
ð29Þ
Inserting (28) and the vehicle parameters into the boundary conditions (8) yields Vt ¼ 1;
Vn ¼ Vb ¼ 0
ð30Þ
which implies that the time derivative of local velocities along the straight cable are zero, i.e. @Vt @Vn @Vb ¼ ¼ ¼0 @t @t @t
ð31Þ
Substituting (27)–(31) into (7) yields @T 1 ¼ qdpCt @s 2
ð32Þ
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with boundary condition TðSt ; tÞ ¼ 0
ð33Þ
which is obtained from the boundary condition (11) where the resisting torque Cf is assumed to be zero. Thus, the analytical solution to (32) and (33) is 1 Tðs; tÞ ¼ qdpCt ðs St Þ 2
ð34Þ
Therefore, the tension at the tow-point is Tð0; tÞ ¼
1 qdpCt St 2
ð35Þ
which states that the tension at the tow-point is proportional to the total length of the immersed cable. Furthermore, from (14), it can be shown that St ¼ t
ð36Þ
Thus, the tension at the tow-point is Tð0; tÞ ¼
1 qdpCt t ¼ 0:0393tðNÞ 2
ð37Þ
which is illustrated by the solid line in Fig. 5. From Fig. 4, it can be seen that the numerical solution reproduces the exact solution with negligible error which may be caused by the approximation of the dynamics of the new segment by the governing equation (7). Thus, the effectiveness of the numerical scheme in Section 3 has been verified. 4.2. Model validation of the tethered vehicle By coupling the cable effects with the dynamic model of the un-tethered vehicle (Lea, 1998b; Feng and Allen, 2001), the dynamic model of the tethered vehicle can be built. Here, the quality of the model is assessed by checking the reproduction error between the test data (measured by the sensors onboard the vehicle) and simulated data (predicted by the model). The results of a water trial of the vehicle accelerating and turning are shown in Fig. 6. The vehicle was at rest before the manoeuvring commands were applied. While the stern-planes remained the central position, the propeller command in revolutions per second and the rudder command in degrees are shown in Fig. 6(a) and (c), respectively. The forward speed and the heading of the vehicle are shown in Fig. 6(b) and (d) (dotted lines), respectively. From Fig. 6(b), it can be seen that the vehicle accelerated in the early stage (0–10 s) and decelerated in the final stages (20–30 s). The acceleration at early stage is due to the constant propeller thrust and negligible effects of the short cable. The deceleration at final stage is due to continuously increasing drag of the extending cable. v From Fig. 6(d), it can be seen that for a small turn (27–35 ), the cable produces slight influence on the heading at final stage. Therefore, the cable effects on the
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heading of a flight vehicle can be neglected due to its small range of turn. However, this is not true for the umbilical attached to a ROV which can turn in a large range. The simulation results of the models of the tethered vehicle (solid lines) and the un-tethered vehicle (dashed lines) under the test commands are also shown in Fig. 6 (b) and (d). From Fig. 6(b), it can be seen that the model of the tethered vehicle can reproduce the decreasing speed while the model of the un-tethered vehicle cannot. From Fig. 6(d), it can be seen that the model of the tethered model can predict the cable effects on vehicle turning while the model of the un-tethered vehicle cannot. However, while the steady state error of the model is acceptable, the dynamic mismatch cannot be ignored. This is believed to be caused by the model uncertainty of the control surfaces and will be improved by feeding back the actual positions of the control fins to their servo’s.
5. Conclusions A numerical scheme which deals with cables of non-fixed lengths is presented in this paper. It has been verified by comparing the numerical solution with an exact solution in a special case. By applying this scheme to assess the effect of the communication cable attached to the Subzero II vehicle, it can be seen that the most significant effect to translational motions of the vehicle is to reduce the vehicle’s forward speed since the cable-induced drag in surge motion dominates all other induced forces. Moreover, the rotational motions of the vehicle are also affected by the cable since the towpoint does not coincide with the mass centre of the vehicle. To take the cable effect into account while designing an autopilot for the tethered vehicle, it is preferable to adopt a composite structure which contains a feedback path and a feed-forward path. While the feedback controller compensates the dynamics of the un-tethered vehicle, the feed-forward controller compensates the cable effect in terms of the cable dynamics. This enables the feedback controller to be transplanted from the computer on the shore to the microcomputer on board the vehicle when the communication cable is removed, i.e. when the vehicle becomes an autonomous underwater vehicle. Future work will focus on development of such an autopilot and verification via water tests.
Acknowledgements The authors are very grateful for the support of this study by the Engineering and Physical Sciences Research Council (UK) under the IMPROVES project (Improving the Performance of Remotely Operated Vehicles).
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References Albow, C.M., Schechter, S., 1983. Numerical simulation of undersea cable dynamics. Ocean Engineering 10 (6), 443–457. Feng, Z., Allen, R., 2001. Modelling of Subzero II. Tech. Memo. 880, ISVR, University of Southampton. Feng, Z., Stansbridge, R., White, D., Wood, A., Allen, R., 2003. Subzero III—a low cost underwater flight vehicle. In: Proceedings of the First Workshop on Guidance and Control of Underwater Vehicles, IFAC, Newport, Wales, United Kingdom, pp. 215–219. Fossen, T.I., 1994. Guidance and Control of Ocean Vehicles. John Wiley & Sons, Chichester. Lea, R., 1998a. A comparative study by simulation and experimentation of control techniques for autonomous underwater flight vehicles. PhD Thesis, ISVR, University of Southampton. Lea, R., 1998b. AutoRov—an underwater flight vehicle simulation program. Tech. Memo. 828, ISVR, University of Southampton. Milinazzo, F., Wilkie, M., Latchman, S.A., 1987. An efficient algorithm for the dynamics of towed cable systems. Ocean Engineering 14 (6), 513–526. The Mathworks, 2000. Optimization toolbox for use with Matlab.
Evaluation of the effects of the communication cable on the dynamics of an underwater flight vehicle Z. Feng , R. Allen Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK Received 26 June 2003; accepted 12 November 2003
Abstract This paper presents a numerical scheme to evaluate the effects of the communication cable attached to an underwater flight vehicle. Both simulation and model validation results show that the numerical scheme is effective and provides a means for developing a feed-forward controller to compensate for the cable effects when developing an autopilot for the tethered vehicle. Moreover, the numerical scheme can also be applied to predict the effects of the ROVs umbilical during its deployment. # 2004 Elsevier Ltd. All rights reserved. Keywords: Underwater flight vehicles; Towed cable; Cable deployment; Finite difference method; Least squares solution; Model validation
1. Introduction Underwater vehicles are playing an increasingly important role in exploration and exploitation of the underwater environment. Automatic control of underwater vehicles presents a challenge for control engineers due to the strong nonlinearity of the dynamics, the high degree of model uncertainty resulting from poor knowledge of hydrodynamic coefficients and the effect of external, un-measurable disturbances such as underwater currents. To provide a test-bed for control techniques of underwater vehicles, the Subzero II flight vehicle (see Fig. 1) was constructed (Lea, 1998a,b) and has recently been
Corresponding author. Tel.: +44-23-80594935; fax: +44-23-80593190. E-mail address: [email protected] (Z. Feng).
0029-8018/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2003.11.001
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Fig. 1. Subzero II.
developed further (Feng et al., 2003). It is a low cost, self-powered, torpedo-shaped flight vehicle with a length of 1 m and a maximum diameter of 10 cm. The power is from an on-board Ni–Cad battery package. The vehicle is propelled by a propeller and guided by four control fins, which include two rudders and two stern-planes. While the two rudders are linked together, the stern-planes are separated and thus provide a means for roll control. Vehicle control is achieved using a personal computer on the shore which communicates with the underwater vehicle over a thin and neutrally buoyant wire. Connected at one end to the serial port of the microcontroller on board the vehicle and at the other to the serial port of the personal computer, the communication cable transmits the manoeuvring commands and sensor data bi-directionally. The cable provides a means for development and assessment of autopilot controllers without modifying the software running in the microcontroller on board the underwater vehicle, but obviously affects the motion of the vehicle. To reduce the cable effects, the cable is deployed by a drum on the shore with negligible tension when it is pulled by the vehicle. The forward-moving vehicle experiences additional forces and moments which are induced by the cable. Intuitively, these forces and moments produce more significant effects on the motion of vehicle as the immersed cable lengthens. Due to the existence of the cable, the vehicle is really a tethered vehicle. Thus, to develop a model-based autopilot for the tethered vehicle, not only does the dynamics of the un-tethered vehicle need to be identified, but also the cable dynamics has to be modelled as well. Note that it is also true for remotely operated vehicles (ROVs) which are deployed by umbilical cables. Treated as a long, thin, flexible circular cylinder in arbitrary motions of gravity, inertial forces, driving forces and hydrodynamic loading which is taken to be the sum of independently operating normal and tangential drags, each given by a single coefficient, the dynamics of the cable can be described by the partial differential
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equations over time and arc length of the cable (Albow and Schechter, 1983). And for a typical towed cable system, where the length of the cable is fixed, numerical solutions of the equations can be obtained by finite difference methods (Albow and Schechter, 1983; Milinazzo et al., 1987). However, these numerical schemes cannot be applied to our situation directly due to the variable cable length. In this paper, the numerical scheme developed by Milinazzo et al. (1987) is extended and then applied to evaluate the effects of the communication cable on the Subzero II vehicle. The rest of the paper is organised as follows: cable dynamics, as well as boundary conditions, are described in Section 2. A numerical scheme which deals with cables of non-fixed length is proposed in Section 3. The cable effect, as well as the dynamic model of the tethered vehicle, is verified in Section 4 with conclusions in Section 5.
2. Cable dynamics 2.1. Three coordinate systems To analyse the motion of the cable as well as its effect on the vehicle, it is convenient to define three coordinate systems, i.e. the earth-fixed frame and the local frames along the cable and the vehicle-fixed frame. As shown in Fig. 2, the earth-fixed frame (i, j, k) is selected with k pointing vertically downwards. The vehicle-fixed frame (iv, jv, kv) is located at the centre of the hull, with iv coinciding with the longitudinal axis, and jv pointing to starboard. The relationship between the vehicle-fixed frame and the earth-fixed frame can be expressed in terms of Euler angles (Fossen, 1994), i.e. ½ iv
jv
kv ¼ ½ i
j k Rð/; h; wÞ
Fig. 2. Test configuration of Subzero II.
ð1Þ
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with 2
c/sw þ s/shcw c/cw þ s/shsw s/ch
chcw Rð/; h; wÞ ¼ 4 chsw sh
3 s/sw þ c/shcw s/cw þ c/shsw 5 c/ch
ð2Þ
where c ¼ cos , s ¼ sin and /, h, w are the roll, pitch and heading angle of the vehicle, respectively. The local frames (t, n, b) are located at points along the cable with t tangent to the cable in the direction of increasing arc length from the tow-point, and b in the plane of (i, j). They are obtained by three rotations of the earth-fixed frame in the following order: (1) a counter-clockwise rotation through angle a about the k axis to bring the i axis into the plane of t and n; (2) a counter-clockwise rotation about the new the i axis through p=2 to bring the k axis into coincidence with b; and (3) a clockwise rotation about b through b to bring i and j into coincidence with t and n. Thus, the relationship between the local frames and the earth-fixed frame can be expressed as follows (Albow and Schechter, 1983; Milinazzo et al., 1987): ½t
n b ¼ ½i j
k W ða; bÞ
ð3Þ
with 2
3 casb sa sasb ca 5 cb 0
cacb W ða; bÞ ¼ 4 sacb sb
ð4Þ
In terms of (1)–(4), the relationship between the local frames and the vehiclefixed frame can be written as ½t
n b ¼ ½ iv
jv
kv RT ð/; h; wÞW ða; bÞ
ð5Þ
where the orthogonal property of R has been used. 2.2. Dynamic equations Define the vector yðs; tÞ :¼ ½ T
Vt
Vn
Vb
a
b T
ð6Þ
where t and s denote the time and the arc length of the cable measured from the tow-point, respectively, T is the tension and Vc ¼ ½ Vt Vn Vb T denotes the velocity vector in the local frames along the cable. The cable dynamics can be expressed as the following partial differential equation (Albow and Schechter, 1983): M
@y @y ¼N þq @s @t
ð7Þ
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with 3 1 0 0 0 0 0 6 0 1 0 0 Vb cosb Vn 7 7 6 6 0 0 1 0 Vb sinb Vt 7 7 6 M¼6 7 7 6 0 0 0 1 Vn sinb Vt cosb 0 5 4 0 0 0 0 Tcosb 0 0 0 0 0 0 T 3 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi w qd 1 þ eT pC sinb þ U U j j c t t t 7 6 2 7 60 7 6 7 60 7 6 7 q¼6 7 6 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 61 7 6 qd 1 þ eT Cn Ub Un2 þ Ub2 qAJ_ b 7 62 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 4 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi wc cosb þ qd 1 þ eT Cn Un Un2 þ Ub2 qAJ_ n 2 2
2
Vt 6 me 1 þ eT 6 6e 6 60 6 0 N¼6 6 6 ðm1 Vb qAJb Þ 6 e 6 ð1 þ eTÞ 6 4 ðm1 Vn qAJn Þ e ð1 þ eTÞ
m 0
0
0 0 0 0 0 0
0 0 0
0 0
m1
0 m1 0
3 ðm1 Vn qAJn Þ 7 7 7 0 0 7 7 0 1 þ eT 7 7 ð1 þ eTÞcosb 0 7 7 7 ðm1 Vn qAJn Þsinb mVt cosb 0 7 7 5 ðm1 Vb qAJb Þsinb mVt ðm1 Vb qAJb Þcosb
where q is the fluid density (kg/m3), m is the mass per unit length of cable (kg/m), A is the cross-sectional area of the unstretched cable (m2), m1 is m þ qA (virtual mass per unit length) (kg/m), g is the gravitational acceleration (9.81 kg m/s2), wc is ðm qAÞg (immersed weight per unit length) (N/m), e is 1=EA where E is Young’s modulus, Ct, Cn are the tangential and normal drag coefficients, respectively, d is the diameter of the stretched cable (m), J ¼ ½ Jt Jn Jb T is the current velocity given in the local frame (t, n, b) (m/s) and U ¼ ½ Ut Un Ub T is Vc J (velocity relative to the current) (m/s). 2.3. Boundary conditions The initial configuration of the cable, as well as six boundary conditions, is required to specify the solution of (7). The immersed part of the cable has two ends: the vehicle-end and the drum-end. The vehicle-end, or the tow-point, is located at the tail of the vehicle, while the drum-end is located at the cable drum which deploys the cable into the water due to pull of the vehicle.
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Since the cable shares the velocity of the vehicle at the tow-point, three boundary conditions can be obtained. Denote the position of the tow-point in the vehicle-fixed frame with rc ¼ ½ xc yc zc T , and the linear and angular velocity of the vehicle with V ¼ ½ u v w T and X ¼ ½ p q r T , where (u, v, w) denote surge, sway and heave speeds , while (p, q, r) denote roll, pitch and yaw rates of the vehicle. Then the velocity of the tow-point in the vehicle-fixed frame is ðV þ X rc Þ, where denotes the cross product of two vectors. In terms of (5), the velocity of the towpoint can be repressed in the local frame of the cable at the vehicle end, i.e. Vc ð0; tÞ ¼ W T ða; bÞRð/; h; wÞðV þ X rc Þ
ð8Þ
which involves three boundary conditions. At the drum-end, since the drum always deploys the cable in the tangential direction, the normal components of the cable velocity must be zero, i.e. Vn ðSt ; tÞ ¼ 0
ð9Þ
Vb ðSt ; tÞ ¼ 0
ð10Þ
where St is the total arc length of the immersed cable at time t. To determine the sixth boundary condition, consider the rolling motion of the cable drum. In terms of Newton’s second law, one has d ðId Xd Þ ¼ C Cf dt where Id is the moment of inertia of the drum, Xd is the angular speed of the drum, C is the driving torque produced by the cable tension at the drum-end, and Cf is the resisting torque caused by the sliding friction between the drum and the pivot. Since Xd ¼ Vt ðSt ; tÞ=Rd and C ¼ TðSt ; tÞRd where Rd is the radius of the cable drum, the dynamic equation of the drum can be rewritten as Id V_ t ðSt ; tÞ þ TðSt ; tÞR2d ¼ Cf Rd
ð11Þ
So far, the six boundary conditions have been determined by (8)–(11). 2.4. Cable effect The tension of the cable at the tow-point, i.e. Tð0; tÞtð0; tÞ, results in the additional forces and moments that affect the motion of the vehicle. The additional forces can be obtained by expressing the cable tension in the vehicle-fixed frame in
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terms of (5), i.e. 2
3 2 3 FcX Tð0; tÞ 5 Fc ðtÞ :¼ 4 FcY 5 ¼ RT ð/; h; wÞW ðað0; tÞ; bð0; tÞÞ4 0 0 FcZ
ð12Þ
where FcX ¼ Tð0; tÞðcwchcacb swchsacb þ shsbÞ FcY ¼ Tð0; tÞðchs/sb þ s/cwshcacb s/swshsacb cwc/sacb swc/cacbÞ FcZ ¼ Tð0; tÞðchc/sb þ s/swcacb þ s/cwsacb þ cwshc/cacb swshc/sacbÞ In terms of the position of the tow-point in the vehicle-fixed frame, the cableinduced moments are 2 3 2 3 McX yc FcZ zc FcY ð13Þ Mc ðtÞ :¼ 4 McY 5 ¼ rc Fc ðtÞ ¼ 4 zc FcX xc FcZ 5 McZ xc FcY yc FcX
3. Numerical scheme The numerical solutions of the partial differential equations (7) under the boundary conditions (8)–(11) will be obtained by the finite difference method which discretises the equations over time and space. However, unlike a typical towed cable system where the length of the cable is fixed, the length of the immersed communication cable is not fixed as it is gradually pulled into the water by the vehicle. Moreover, the total length of the wet cable at time t is determined by the deployment speed Vt ðSs ; sÞ with 0 s < t, i.e. ðt St ¼ Vt ðSs ; sÞds ð14Þ 0
where the initial length of the wet cable has been assumed to be zero. Therefore, it is impossible to divide the wet cable into fixed number segments each of which has a fixed length. Since the length of each segment (or the position of each node) needs to be fixed for finite difference methods, the number of segments (nodes) of the wet cable must increase with increasing time as shown in Fig. 3. This is achieved by discretising (14) over the time step Dt, i.e. k1 ð ðiþ1ÞDt X Sk ¼ Vt ðSs ; sÞds ð15Þ i¼0
iDt
where Sk (k ¼ 1; 2; . . .) denotes the total length of the immersed cable at time kDt.
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Fig. 3. Evolution of cable configuration from tk to tk þ1 .
This suggests that the immersed cable at kDt be divided into k segments with the segment lengths ð ðiþ1ÞDt DSiþ1 ¼ Vt ðSs ; sÞds; i ¼ 0; 1; . . . ; k 1 ð16Þ iDt
Provided the time step Dt is sufficiently small, the segment lengths can then be approximated by DSiþ1 ¼ Vt ðSi ; ti ÞDt;
i ¼ 0; 1; . . . ; k 1
ð17Þ
where ti ¼ iDt. Thus, the lengths of all the k segments are fixed as they are determined by the past deployment speeds. From (15)–(17), it can also be obtained DSkþ1 ¼ Skþ1 Sk ¼ Vt ðSk ; tk ÞDt
ð18Þ
This implies a new segment DSkþ1 emerges once a time step Dt has elapsed since tk, and the total cable length at tkþ1 is Skþ1 ¼ Sk Vt ðSk ; tk ÞDt
ð19Þ
After the segments (nodes) along the immersed cable have been specified, the finite difference method of second order approximation (Milinazzo et al., 1987) can be applied. Suppose that the configuration of the cable at time tk, i.e. yðSi ; tk Þ with i ¼ 0; 1; . . . ; k, is known, the problem is then to predict the configuration of the cable at time tkþ1 , i.e. yðSi ; tkþ1 Þ with i ¼ 0; 1; . . . ; k þ 1, where Skþ1 is determined by (19). Define DSj :¼ Sj Sj1 1 Sj1=2 :¼ ðSj þ Sj1 Þ ; 2 tkþ1=2 :¼ ðk þ 1=2ÞDt
j ¼ 1; 2; . . . ; k þ 1
ð20Þ
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and 1 yðSj ; tk Þ þ yðSj1 ; tk Þ 2 1 yðSj1=2 ; tkþ1 Þ :¼ yðSj ; tkþ1 Þ þ yðSj1 ; tkþ1 Þ 2 ; 1 yðSj1 ; tkþ1=2 Þ :¼ yðSj1 ; tk Þ þ yðSj1 ; tkþ1 Þ 2 1 yðSj ; tkþ1=2 Þ :¼ yðSj ; tk Þ þ yðSj ; tkþ1 Þ 2
yðSj1=2 ; tk Þ :¼
j ¼ 1; 2; . . . ; k þ 1
ð21Þ
where S 0 ¼ 0. Note that since the new segment DSkþ1 is not in the water at tk, yðSkþ1 ; tk Þ in (21) is prescribed according to the rolling motion of the drum which is assumed to be uniform, i.e. yðSkþ1 ; tk Þ ¼ ½ TðSk ; tk Þ Vt ðSk ; tk Þ 0 0 aðSk ; tk Þ bðSk ; tk Þ DSkþ1 =Rd T . Moreover, the governing Eq. (7) will still be applied to approximate the dynamics of the new segment although it is only partly true for the new segment which is gradually pulled into the water. Obviously this may result in modelling error. However, as will be verified, the modelling error can be negligible when the new segment is not too long. Applying the governing Eq. (7) at the points ðSj 1=2 ; tkþ1=2 Þ, j ¼ 1; 2; . . . ; k þ 1 yields the 6ðk þ 1Þ difference equations as follows (Milinazzo et al., 1987):
yðSj ; tkþ1 Þ yðSj1 ; tkþ1 Þ M yðSj1=2 ; tkþ1 Þ DSj
yðSj ; tk Þ yðSj1 ; tk Þ þ M yðSj1=2 ; tk Þ DSj
yðSj ; tkþ1 Þ yðSj ; tk Þ ¼ N yðSj ; tkþ1=2 Þ Dt
yðSj1 ; tkþ1 Þ yðSj1 ; tk Þ
þ N yðSj1 ; tkþ1=2 Þ þ q yðSj1=2 ; tkþ1 Þ Dt
þ q yðSj1=2 ; tk Þ for j ¼ 1; 2; . . . ; k þ 1
ð22Þ
Note that the six equations under j ¼ k þ 1 approximate the dynamics of the new segment. The remaining six equations are determined by the boundary conditions (8)–(11), i.e. Vc ðS0 ; tkþ1 Þ ¼ W T ðaðS0 ; tkþ1 Þ; bðS0 ; tkþ1 ÞÞRð/; h; wÞðV þ X rc Þ
ð23Þ
Vn ðSkþ1 ; tkþ1 Þ ¼ 0
ð24Þ
Vb ðSkþ1 ; tkþ1 Þ ¼ 0
ð25Þ
Id
Vt ðSkþ1 ; tkþ1 Þ Vt ðSk ; tk Þ þ TðSkþ1 ; tkþ1 ÞR2d ¼ Cf Rd Dt
ð26Þ
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Combining (22)–(26), 6ðk þ 2Þ nonlinear equations with 6ðk þ 2Þ unknowns can be obtained. And the cable effects can be determined by solving these equations. It should be noted that the dimension of (22)–(26) expands with time until the deployment of the cable finishes. Therefore, an increasing computation effort has to be undertaken with time before the deployment of cable stops. One might conclude that this scheme is not suitable for on-line analysis of the cable effect. However, this is not true since the computational effort is limited due to the limited length of a practical cable. Once the solutions to difference equations (22)–(26) have been obtained, the cable effects at time tkþ1 can be determined from (12) and (13) where t is replaced with tkþ1 . To summarise, the numerical scheme to evaluate the effects of the communication cable comprises the following steps: (1) Initialisation. Set the time step Dt and stop time TIME of the simulation. Set the time index to k ¼ 0. Since no cable is in water at tk ¼ t0 ¼ 0, the vehicle-end coincides with the drum-end. Therefore, the initial configuration y(Sk, tk) can be obtained by solving (8)–(11) with t ¼ 0. Note that the initial states of the vehicle (u, v, w, p, q, r, /, h, w) need to be given to specify a solution for (8)–(11). (2) Given the vehicle states (u, v, w, p, q, r, /, h, w) at time tkþ1 ¼ ðk þ 1ÞDt The states can be measured by the on-board sensors or predicted by the dynamical model of the vehicle. (3) Solve the nonlinear equations (22)–(26). The equations can be solved efficiently using Matlab Optimization Toolbox (The Mathworks, 2000). For example, the least squares solution to the nonlinear equations P fi ðx1 ; x2 ; :::; xn Þ ¼ 0, (i ¼ 1; 2; . . . ; n) which minimises ni¼1 fi2 ðx1 ; x2 ; . . . ; xn Þ can be obtained by invoking the command fsolve. (4) Determine the cable effects. This is achieved by substituting yð0; tkþ1 Þ into (12) and (13) where t is replaced with tkþ1 . (5) Update time index k k þ 1. (6) If k TIME=Dt, then go to step (2), otherwise stop the simulation.
4. Assessment of the cable effects for Subzero II Subzero II’s communication cable is a thin, lightweight and neutrally buoyant wire which attached to the vehicle at the tail with coordinates rc ¼ ð0:5; 0; 0Þ relative to the vehicle-fixed frame. It is deployed by a cable drum of 0.2 m diameter with negligible tension, i.e. Cf ¼ 0. The cable specifications and fluid parameters are listed in Table 1.
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Table 1 Cable and fluid parameters (units as given in Section 2.2) q
m
A
m1
wc
e
Ct
Cn
d
1000
0.0049
4.9e6
0.0098
0
0
0.01
1.0
0.0025
4.1. Evaluation of the cable effects As stated in Section 3, the vehicle states (u, v, w, p, q, r, /, h, w) are required to specify the boundary conditions (8) at the tow-point and to determine the cable effects (12) and (13). These states can be measured by the sensors on board Subzero II or estimated via the dynamic model of the vehicle. For Subzero II, they are obtained by the latter method since the sway and heave speeds are not currently measured. Therefore, the reliability of the cable effect predicted by the numerical scheme not only depends on the numerical scheme itself, but also on the dynamic model of the vehicle due to the coupling between the cable and the vehicle as described by (8), (12) and (13). Since model uncertainty involved in dynamic models of underwater vehicles can never be avoided due to inaccurate model parameters, e.g. hydrodynamic coefficients. In order to assess the cable effect independently of the vehicle dynamics, an ‘idealised’ vehicle is assumed where the dynamics are ignored. Since the flight control of Subzero II comprises three subsystems: forward speed, heading and depth, three situations need to be considered: (i) The vehicle is moving straight ahead with prescribed states: u ¼ 1, v ¼ w ¼ 0, p ¼ q ¼ r ¼ 0 and / ¼ h ¼ w ¼ 0 (ii) The vehicle is diving while moving ahead with prescribed states: u ¼ 1, v ¼ w ¼ 0, p ¼ r ¼ 0, / ¼ w ¼ 0 and q ¼ h_ ðtÞ. The pitch angle h changes with the time (see Fig. 6(a)), i.e. 8 v 0 t 10 s <0 ; v hðtÞ ¼ 6 ðt 10Þ; 10 < t 20 s : v 60 ; t > 20 s (iii) The vehicle is turning while moving ahead with prescribed states: u ¼ 1, v ¼ w ¼ 0, / ¼ h ¼ 0 and r ¼ w_ ðtÞ. The heading angle w changes with the time. Since the cable effects of (iii) can be deduced from that of (ii) due to the neutral buoyancy of the cable, only the cable effect under case (i) and case (ii) will be evaluated. Setting the time step Dt ¼ 1 s and TIME ¼ 30 s, the cable effects are calculated by applying the evaluation procedure in Section 3. The results are shown in Figs. 4 and 5. Fig. 4 illustrates the results of straight ahead motion: the tension at the towpoint, i.e. the cable drag experienced by the vehicle is plotted by the dotted line.
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Fig. 4. Simulation results of cable drag under straight ahead motion.
Note that the remaining cable-induced forces and moments are all zero. It can be seen that the cable drag is increasing with the time. This indicates that the vehicle is slowing down if the thrust remains invariant. In other words, the thrust needs to be increasing for the vehicle to hold a constant forward speed. Fig. 5 shows the cable effects on the vehicle’s diving. The configurations of the cable at 10, 15, 20, 25 and 30 s are shown in Fig. 5(b). It can be seen that the immersed part of the cable is lengthening and descending. The cable tension at the tow-point is plotted in Fig. 5(c), the velocity components of the tow-point with respect to the cable local frame are shown in Fig. 5(d). The cable-induced forces and moments are shown in Fig. 5(e) and (f). As can be seen from Fig. 5(e), the cable tends to slow down the vehicle due to the increasing drag in surge motion. Moreover, the cable tends to push the vehicle in heave direction as a positive heave force is produced. From Fig. 5(f), it can be seen that the cable tends to make the vehicle head up due to the positive pitch moment. However, the cable effects on the heave and pitch motions are much smaller than that on the surge motion. Normally, it is difficult to verify the numerical solutions of partial derivative equations since they do not always admit analytical solutions. Fortunately, as will be derived, the partial derivative equation (7) subject to boundary conditions (8)–(11) admits an analytical solution under case (i). Being pulled into the still water by the vehicle which retains a pure surge motion of constant speed, the neutrally buoyant cable must maintain a configuration of a straight line which extends in the direction of vehicle motion. Thus, orientation angles between the local frames along the cable and the inertial frame are invariant over space and time, i.e. @a @b ¼ ¼ 0; @s @s
@a @b ¼ ¼0 @t @t
ð27Þ
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Fig. 5. Cable effects on vehicle diving. (a) Pitch angle; (b) cable configurations at different times; (c) cable tension at the tow-point; (d) local velocity at the tow-point; (e) cable-induced forces; (f) cableinduced moments.
According to the definition of the local frame (t, n, b), the orientation angles of level motion are a ¼ 0;
b¼p
ð28Þ
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Fig. 6. Model validation of the tethered vehicle. (a) Propeller command; (b) forward speed; (c) rudder command; (d) heading angle.
Moreover, since the cable is supposed to be inelastic, the local velocities along the straight cable must be constant, i.e. @Vt @Vn @Vb ¼ ¼ ¼0 @s @s @s
ð29Þ
Inserting (28) and the vehicle parameters into the boundary conditions (8) yields Vt ¼ 1;
Vn ¼ Vb ¼ 0
ð30Þ
which implies that the time derivative of local velocities along the straight cable are zero, i.e. @Vt @Vn @Vb ¼ ¼ ¼0 @t @t @t
ð31Þ
Substituting (27)–(31) into (7) yields @T 1 ¼ qdpCt @s 2
ð32Þ
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with boundary condition TðSt ; tÞ ¼ 0
ð33Þ
which is obtained from the boundary condition (11) where the resisting torque Cf is assumed to be zero. Thus, the analytical solution to (32) and (33) is 1 Tðs; tÞ ¼ qdpCt ðs St Þ 2
ð34Þ
Therefore, the tension at the tow-point is Tð0; tÞ ¼
1 qdpCt St 2
ð35Þ
which states that the tension at the tow-point is proportional to the total length of the immersed cable. Furthermore, from (14), it can be shown that St ¼ t
ð36Þ
Thus, the tension at the tow-point is Tð0; tÞ ¼
1 qdpCt t ¼ 0:0393tðNÞ 2
ð37Þ
which is illustrated by the solid line in Fig. 5. From Fig. 4, it can be seen that the numerical solution reproduces the exact solution with negligible error which may be caused by the approximation of the dynamics of the new segment by the governing equation (7). Thus, the effectiveness of the numerical scheme in Section 3 has been verified. 4.2. Model validation of the tethered vehicle By coupling the cable effects with the dynamic model of the un-tethered vehicle (Lea, 1998b; Feng and Allen, 2001), the dynamic model of the tethered vehicle can be built. Here, the quality of the model is assessed by checking the reproduction error between the test data (measured by the sensors onboard the vehicle) and simulated data (predicted by the model). The results of a water trial of the vehicle accelerating and turning are shown in Fig. 6. The vehicle was at rest before the manoeuvring commands were applied. While the stern-planes remained the central position, the propeller command in revolutions per second and the rudder command in degrees are shown in Fig. 6(a) and (c), respectively. The forward speed and the heading of the vehicle are shown in Fig. 6(b) and (d) (dotted lines), respectively. From Fig. 6(b), it can be seen that the vehicle accelerated in the early stage (0–10 s) and decelerated in the final stages (20–30 s). The acceleration at early stage is due to the constant propeller thrust and negligible effects of the short cable. The deceleration at final stage is due to continuously increasing drag of the extending cable. v From Fig. 6(d), it can be seen that for a small turn (27–35 ), the cable produces slight influence on the heading at final stage. Therefore, the cable effects on the
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heading of a flight vehicle can be neglected due to its small range of turn. However, this is not true for the umbilical attached to a ROV which can turn in a large range. The simulation results of the models of the tethered vehicle (solid lines) and the un-tethered vehicle (dashed lines) under the test commands are also shown in Fig. 6 (b) and (d). From Fig. 6(b), it can be seen that the model of the tethered vehicle can reproduce the decreasing speed while the model of the un-tethered vehicle cannot. From Fig. 6(d), it can be seen that the model of the tethered model can predict the cable effects on vehicle turning while the model of the un-tethered vehicle cannot. However, while the steady state error of the model is acceptable, the dynamic mismatch cannot be ignored. This is believed to be caused by the model uncertainty of the control surfaces and will be improved by feeding back the actual positions of the control fins to their servo’s.
5. Conclusions A numerical scheme which deals with cables of non-fixed lengths is presented in this paper. It has been verified by comparing the numerical solution with an exact solution in a special case. By applying this scheme to assess the effect of the communication cable attached to the Subzero II vehicle, it can be seen that the most significant effect to translational motions of the vehicle is to reduce the vehicle’s forward speed since the cable-induced drag in surge motion dominates all other induced forces. Moreover, the rotational motions of the vehicle are also affected by the cable since the towpoint does not coincide with the mass centre of the vehicle. To take the cable effect into account while designing an autopilot for the tethered vehicle, it is preferable to adopt a composite structure which contains a feedback path and a feed-forward path. While the feedback controller compensates the dynamics of the un-tethered vehicle, the feed-forward controller compensates the cable effect in terms of the cable dynamics. This enables the feedback controller to be transplanted from the computer on the shore to the microcomputer on board the vehicle when the communication cable is removed, i.e. when the vehicle becomes an autonomous underwater vehicle. Future work will focus on development of such an autopilot and verification via water tests.
Acknowledgements The authors are very grateful for the support of this study by the Engineering and Physical Sciences Research Council (UK) under the IMPROVES project (Improving the Performance of Remotely Operated Vehicles).
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