Dynamics modeling and experiments of wave driven robot

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Dynamics modeling and experiments of wave driven robot Ye Li, Kaiwen Pan, Yulei Liao∗, Weixin Zhang, Leifeng Wang Science and Technology on Underwater Vehicle Laboratory, Harbin Engineering University, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 24 November 2018 Revised 10 October 2019 Accepted 21 October 2019 Available online xxx Keywords: Wave driven robot Dynamic model Motion simulation Tense umbilical Tank test

a b s t r a c t Wave driven robots (WDRs) take ocean energies as the power sources and are often used for long-term monitoring of the marine environment. The unique multi-body joint structure and special operation mechanism of a WDR make the dynamics modeling problem unusual. The dynamic model of a WDR was put forward by taking the interconnection of forces and motions between the float body (float) and the submerged glider (glider) into account. Numerical simulation of longitudinal motion and the comparison between simulation and tank test of reciprocating steering motion of the "Ocean Rambler" WDR were carried out. The dynamic model proposed in this paper was consistent with the motion characteristics of "Ocean Rambler" WDR. Simulations of PID heading control demonstrated the unique control characteristics of the WDR, which proved the significance of the established dynamic model of the WDR in control algorithm design. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Traditional ocean environmental monitoring platforms take fuel oil or battery as power sources and the endurances are limited by energy capacities. In recent years, more and more researchers pay attention to take ocean energies as the power sources of marine vehicles, mainly including solar powered underwater vehicles [1], thermal powered underwater vehicles [2], wind powered or solar-powered unmanned surface vehicles [3,4], and wave powered vehicles. Wave driven robots (WDRs) use their special multi-body structure to convert wave energy into navigational power. WDRs are particularly suited for large-scale marine environmental monitoring tasks because of their indefinite endurance. The Liquid Robotics Inc. has successfully developed the SV2 and SV3 wave gliders, which provide a novel means of observation for various marine research [5–12]. The WDR operating principle of is shown in Fig. 1. A WDR is usually comprised of three parts: a surface float body (float), a submerged glider (glider) and a flexible umbilical. The glider is Equipped with multiple groups of hydrofoils. The thrust comes from the wave potential energy with different depths. Surface waves cause the heave motion of the float and glider. The relative motion direction between the hydrofoil and the water changes repeatedly, causing the hydrofoil to rotate repeatedly around the fixed axis, which makes the lift always forward. The rudder on the back of the glider is the steering actuator. The autonomous control is the main topic for an unmanned marine vehicle. At present, commonly adopted control methods of WDRs are the improvements of control methods of single-body autonomous marine vehicles based on the intuition of the weak maneuvering, large time-lag and large disturbance characteristics of WDRs [13,14]. It is not really rigorous since the influences of the multi-body structure of the WDR are not considered. Simulation results in Section 4.3 will show that the multi-body structure makes the heading control problem of the WDR special. A reasonable and explicit dynamic ∗

Corresponding author. E-mail address: [email protected] (Y. Liao).

https://doi.org/10.1016/j.apm.2019.10.046 0307-904X/© 2019 Elsevier Inc. All rights reserved.

Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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Fig. 1. The operating principle of a WDR.

model of the WDR is asked to provide researchers the theoretical basis and the simulation platform for dynamic analysis and control algorithm design. WDRs are a special kind of multi-body joint system which is very different from the structures of conventional marine vehicles. Some researchers have discussed the dynamic characteristics of this special structure in recent years:

(1) Kraus and Bingham [15] built mathematical model of WDRs in both the along-track and heave directions. Qi et al. [16] applied Kane equation to dynamic analysis of WDRs in vertical plane and a multi-rigid model was proposed [17]. On this basis, Zhou et al. [18] considered the vertical motion of the float in the dynamic model, so that the thrust and resistance were coupled with wave motions. On the base of the Newton–Euler equation and the Lagrange equation, Tian et al. [19,20] established the dynamic model of WDRs in longitudinal profile. The researches mentioned above mainly focused on the longitudinal velocities of WDRs and are beneficial for the speed analysis but are helpless for control algorithm design. (2) Wang et al. [21] proposed the 4-DOF (degree of freedom) mathematical model of a WDR by using the Newton–Euler approach, which taking the wave drift force on the horizontal plane and the wave force in the vertical direction into account. Wang et al. [22] established the planar motion model of a WDR and identified the model parameters. In these investigations, the yawing motions of the float and the glider were treated consistent. In practice, however, the movements of the two are not identical, which influence the control performance of a WDR deeply. (3) Kraus [23] improved the mathematical model of space motion of a ship to the model of a WDR. Tian and Yu [24] applied the Denavit–Hartenberg (DH) method to describe the relative motion of each part of a WDR. A dynamic model on the basis of the Lagrange equation was proposed. In these researches, the yawing motions of the float and the glider are separated, which is beneficial for the dynamic analysis and control algorithm design. However, the physical meanings of some elements, like the inertial mass of the WDR, are only with abstract meaning but are not available in the actual calculation when the heading of the float and the glider are inconsistent. Besides, the vertical motions of the float and the glider are not taken into account. (4) Wang et al. [29] proposed a mathematical model of a WDR considering the characteristics of flexible umbilical. This paper mainly focused on analyzing the switching process of tension and relaxation of the umbilical during the motion of WDR, and a judgment principle of state switching was proposed. This model was suitable for the dynamic analysis of WDR under high sea conditions. Under low sea conditions, the flexible umbilical of WDR is almost always tensioned. However, the motion parameters of the dynamic model under the assumption that the umbilical always being in tension were not deduced and analyzed in detail. Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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Fig. 2. Coordinate frames of a WDR.

In this paper, a WDR dynamic model under the assumption that the umbilical always being in tension was established through the conversion of forces and motions between the parts and the whole system of the WDR. The kinematic analysis and the dynamic model of a WDR were presented. Numerical simulation and tank test were implemented to prove the validity of the proposed WDR dynamic model. 2. Kinematic analysis of a WDR 2.1. Coordinate frames Four coordinate frames are presented in Fig. 2 [23]: (1) System frame (x0 − y0 − z0 ): The origin is located on the center of gravity of the WDR (on the umbilical). The axes defined as x0 is locally normal to the umbilical and is positive in the direction of forward velocity; the axes defined as z0 lies along with umbilical and points to the glider; and y0 satisfies the right-hand rule. (2) Float frame (xF − yF − zF ): The origin is located on the hinged joint between float and umbilical. xF points to the bow of the float; yF points to the starboard of the float; zF lies vertically downward. (3) Glider frame (xG − yG − zG ): The origin is located on the hinged joint between glider and umbilical. xG points to the bow of the glider; yG points to the starboard of the glider; zG lies vertically downward. (4) Earth-fixed frame (ξ − η − ζ ): ξ points to the positive north, η points to the positive east, and ζ points to vertically downward. Six independent DOFs were mainly considered: the surge and sway of the center of gravity of WDR, the roll and pitch of the umbilical, and the yawing motions of the float and the glider. The heave motion of the float was viewed as the known input, assuming that it is the same as that of surface waves. The heave motions of the centers of gravity of the WDR and the glider can be calculated according to the above mentioned DOFs. Remark: To simplify the dynamic model, the rolling and pitching motions of the float and the glider were ignored for following reasons: (1) In heading control, course control and way point tracking tasks, rolling and pitching motions of the float and the glider were not the main factors; (2) The influences of the rolling and pitching motions of the float and the glider could be treated as additional resistances; (3) The rolling and pitching motions of the glider were reduced as far as possible in architecture design by most closely coinciding the connecting point with umbilical, the center of gravity of the glider and the center of hydrodynamic forces of the glider. The location of the center of gravity of the WDR in the earth-fixed frame was described as (ξ , η, ζ ). The relative positions of the float and the glider were represented by ϕ and θ (respectively corresponding to the rolling and pitching angle of the umbilical). The heading of the float and the glider were not same for most of time, and were described by ψ F and ψ G respectively. Furthermore, the heading of the system frame was set as ψ 0 = ψ G . ψ = ψ G − ψ F represented the difference between the heading of the float and the glider. The acceleration, velocity and position vector of WDR are shown in Eq. (1):







a = u˙ v= u

 η= ξ

v˙

p˙

q˙

r˙ F

r˙ G

v

p

q

rF

rG

η

φ

θ

ψF

T

T ψG

T

(1)

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Remark: The first and second elements of the vector a (or v or η) describe the surge and the sway motions of the center





of gravity of WDR in the system frame respectively. The third and fourth elements of the vector a (or v or η) describe the rolling and the pitching motions of the umbilical respectively. ϕ and θ are the spatial Euler angles between the system





frame and the earth-fixed frame. The fifth and sixth elements of the vector a (or v or η) describe the yawing motions of the float and the glider in the float frame and the glider frame respectively. For the sake of simplicity, the sine, cosine and tangent function is represented by s(•), c(•), t(•) in this paper. 2.2. Coordinate transformation

The transformation matrix from (x0 − y0 − z0 ) frame to (xF − yF − zF ) frame, from (x0 − y0 − z0 ) frame to (xG − yG − zG ) frame, and from (x0 − y0 − z0 ) frame to (ξ − η − ζ ) frame were shown in Eqs. (2)–(5), respectively.:

⎡c(θ )c(ψ )

⎢ F ⎢ O R = ⎣c (θ )s (ψ ) −s(θ )

G OR

=

c (θ ) 0 −s(θ )

s(θ )s(φ ) c (φ ) c(θ )s(φ )

⎡ c ( θ )c ( ψ G )

NED O R

s(θ )c(ψ )s(φ ) −s(ψ )c(φ ) s(θ )s(ψ )s(φ ) +c(ψ )c(φ ) c(θ )s(φ )

⎢ ⎣

= ⎢ c ( θ )s ( ψ G ) −s(θ )



s(θ )c(φ ) −s(φ ) = c(θ )c(φ )

c ( ψ G )s ( θ )s ( φ ) −s(ψ G )c(φ ) s ( ψ G )s ( θ )s ( φ ) +c(ψ G )c(φ ) c ( θ )s ( φ )

⎤

s(θ )c(ψ )c(φ )

+s(ψ )s(φ ) ⎥ α s(θ )s(ψ )c(φ )⎥ = γ ⎦ −c(ψ )s(φ ) −s(θ ) c(θ )c(φ )

β χ κ

∂ μ



   c(φ ) −s(φ ) κ μ ⎤ G c ( ψ )s ( θ )c ( φ ) +s(ψ G )s(φ ) ⎥ s ( ψ G )s ( θ )c ( φ ) ⎥ ⎦ −c(ψ G )s(φ ) c ( θ )c ( φ )

(2)



c (θ ) 0 −s(θ )

(3)

(4)

Now, O FR

=(FO R )T

O GR

=(GO R )T

O NED R

T =(NED O R)

(5)

Some parameters in Eqs. (2)–(5) (like α , β etc.) are substitutions to simplify the expressions of subsequent equations. 2.3. Motion states transformation Motion parameters of the float and the glider in (x0 − y0 − z0 ) frame and the body-fixed frames were derived as Eq. (6):

η¯ F = η¯ + r¯ F O G η¯ = η¯ + r¯ G O F ¯ × r¯ F v¯ = v¯ + ω O G ¯ × r¯ G v¯ = v¯ + ω O





¯ × ω ¯ × r¯ F a¯ = a¯ + α¯ × r¯ F + ω

O F

¯ × ω ¯ × r¯ G a¯ = a¯ + α¯ × r¯ G + ω

O G

v¯ F =FO R · O v¯ F v¯ G =GO R · O v¯ G a¯ F =FO R · O a¯ F a¯ G =GO R · O a¯ G where



(6)

r¯ F = 0

0

−dF

r¯ G = 0

0

dG

q

r

q˙

r˙



 ω¯ = p  α¯ = p˙

T

T

T

T

(7)

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5

where, the meaning of right superscripts is a particular subassembly of WDR and the meaning of left superscripts is the reference coordinate frame that the motion parameter is “expressed in”. For the motion parameters without left superscripts, it means that they are expressed in the body-fixed frame of itself. ω ¯ and α¯ are, respectively, the angular velocity and the angular acceleration vectors of the system frame, d is the distance between the origin of the system frame and the hinged joint on particular subassembly of WDR represented by the right superscript. The yawing velocity and the yawing acceleration of (x0 − y0 − z0 ) frame (r and r˙ ) in Eq. (7) could be derived in Eqs. (8)–(11): According to the law of Euler angle transformation:

ψ˙ =

s (φ ) c (φ ) q+ r c (θ ) c (θ )

(8)

It was defined that:

ψ = ψG

(9)

Therefore,

ψ˙ = ψ˙ G = rG

(10)

Then,

r=

c ( θ )r G − s ( φ )q c (φ )

(11)

Further, r˙ was the derivation of r. Eqs. (12) and (13) could be deducted by the expansions of the 7th line and the 8th line of Eq. (6):

w= w˙ =

wF + s(θ )(u − qdF ) − κ (v + pdF )

(12)

μ

w˙ F + s(θ )(u˙ − dF (q˙ + r p)) − κ (v˙ + dF ( p˙ − rq ))

μ

− dF ( p × p + q × q )

(13)

From Eqs. (12) to (13), the heave velocity and acceleration of the center of gravity of the WDR were calculated according to the assumed heave motion of the float. Further, the heave velocity and acceleration of the center of gravity of the glider were obtained. The attitude angles were derived by Eq. (14):

φ˙ = p + t(θ )s(φ )q + t(θ )c(φ )r θ˙ = c(φ )q − s(φ )r ψ˙ F = rF ψ˙ G = rG

(14)

The locations of the centers of gravity of WDR, the float and the glider in earth-fixed frame were derived by Eq. (15):



ξ˙

η˙

ζ˙

 η¯ F = ξ  NED G η¯ = ξ NED

T



= NED O R· u

η

ζ

η

ζ

T

T

v

T

w

¯F + NED O R·r ¯G + NED O R·r

(15)

3. Dynamic mathematical model of a WDR The dynamic mathematical model of WDR was established under these assumptions: (1) The earth-fixed frame was treated as an inertial frame. (2) The umbilical was always in tension since the glider was heavy enough under water. (3) The orientation stability caused by yawing motions of WDR’s subassemblies was ignored owing to the low yawing velocities during actual sailing. (4) When calculating forces relevant to holistic motions of the WDR, the shapes of the float and the glider were ignored. (5) The accelerations and angular accelerations of all DOFs changed slowly in the motion process, and the forces were irrelevant with historical motions. The dynamic equation of WDR was presented as Eq. (16): ˙ MRB v + CRB (ν )ν + FMA (ν r ) + FCA (ν r ) + D(ν r ) + g(η ) = τ

(16)

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where MRB ∈ R6 × 6 is the rigid mass matrix; CRB (ν ) ∈ R6×6 is the Coriolis-centripetal coefficient matrix; F MA (ν r ) ∈ R6×6 is





the inertial hydrodynamic force vector; FCA (ν r ) ∈ R6×6 is the hydrodynamic Coriolis-centripetal force vector; D(ν r ) ∈ R6×1





is the damping force vector; g (η ) ∈ R6×1 is the restoring force vector; τ ∈ R6×1 is the active control force vector. D(ν r )

contains viscous hydrodynamic force, and τ contains thrust and rudder force of the glider. When considering ocean current, the inertial hydrodynamic force, hydrodynamic Coriolis-centripetal and viscous hydrodynamic forces should be calculated relative to ocean current, and the inertial force and Coriolis-centripetal force should be calculated in the earth-fixed frame. The structure of the WDR is unique and there exists complex coupling among different DOFs. The elements of the matrix of inertial, Coriolis-centripetal matrixes etc. reflect the coupling effect among different DOFs, which are discussed in the following sections. 3.1. The rigid mass matrix Fossen [25] discussed the 6-DOFs dynamic model of single-body marine vehicle. The velocity vector and position vector and the rigid mass matrix are as Eqs. (17) and (18): ship

ν

ship

η



= u

v

w

= x

y

z



⎡

MRB

m ⎢ 0 ⎢ 0 =⎢ ⎢ 0 ⎣ mzG −myG

p

φ 0 m 0 −mzG 0 mxG

T

q

r

θ

ψ

T

0 0 m myG −mxG 0

(17) 0 −mzG myG Ix −Ixy −Ixz

mzG 0 −mxG −Ixy Iy −Iyz

⎤

−myG mxG ⎥ 0 ⎥ ⎥ −Ixz ⎥ ⎦ −Iyz Iz

(18)

In Eq. (18), the elements of MRB reflect the effects on inertial forces of rigid body of motions of each DOFs. For instance, MRB (1, 6) reflects the effect on inertial force in surge direction of the yawing motion. According to the assumption (4), when discussing forces relevant to holistic motions of the WDR, holistic movements of the WDR were the major factors and gestures of the float and the glider were secondary factors. Therefore, the float and the glider were treated as particles connected by the umbilical. The 4 × 4 elements in the top left corner of the rigid mass matrix of WDR were conducted through intersecting the 1st, 2nd, 4th and 5th lines and rows in Eq. (18), as shown in Eq. (19):

⎡

m ⎢ 0 MRB (1 : 4, 1 : 4 ) =⎣ 0 mzG

0 m −mzG 0

0 −mzG Ix −Ixy

⎤

mzG 0 ⎥ −Ixy ⎦ Iy

(19)

where m represents the total mass of the WDR (m = mF + mG ,mF is the mass of the float, mG is the mass of the glider); xG ,yG ,zG are coordinates of the center of gravity of WDR in (x0 − y0 − z0 ) frame; Ix and Iy are the moments of inertia of the WDR to x0 and y0 axes; Ixy is the product of inertia of the WDR to x0 and y0 axes. The elements of the rigid mass matrix in the 5th and 6th rows and 1st and 2nd lines reflect the effects on the longitudinal and transverse forces of the center of gravity of WDR from the yawing motions of the float and the glider. Referring to Eq. (18), the longitudinal, transverse and vertical forces related to the yawing motion of the float in (xF − yF − zF ) frame were derived as:

X F = −mF yFG r˙ F Y F = mF xFG r˙ F ZF = 0

(20)

The longitudinal, transverse and vertical forces related to yawing motion of the glider in (xG − yG − zG ) frame were derived as:

X G = −mG yGG r˙ G Y G = mG xGG r˙ G ZG = 0

(21)

In Eqs. (20) and (21), the right subscript ‘G’ represents the coordinates of center of gravity. For example, yGG is the transverse coordinate of the center of gravity of the glider in (xG − yG − zG ) frame. Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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The longitudinal, transverse and vertical forces of the center of gravity of WDR related to the yawing motions of WDR’s subassemblies in (x0 − y0 − z0 ) frame were shown in Eq. (22):



XO YO ZO



F

=

As a result,





O FR

X · YF ZF



+

O GR

XG · YG ZG



(22)



mF (−yFG α + xFG γ ) XO = YO mF (−yFG β + xFG χ )

Therefore,



MRB (1 : 2, 5 : 6 ) =

−mG yGG c(θ )

(

mG −yGG

+

mF (−yFG α + xFG γ )

xGG c

 

(φ ) )

r˙ F r˙ G

−mG yGG c(θ )

mF (−yFG β + xFG χ )

(23)

 (24)

mG (−yGG  + xGG c(φ ) )

The pitching, rolling and yawing moments of the umbilical (connected with the float and the glider “particles”) related to the yawing motions of WDR’s subassemblies in (x0 − y0 − z0 ) frame were shown in Eq. (25):



MO NO KO





= r¯ F ×

F 



X O F FR· Y ZF

Therefore,



MRB (3 : 4, 5 : 6 ) =

+ r¯ G ×



XG O R · YG G ZG



(25)



mF dF (−yFG β + xFG χ )

−mG dG (−yGG  + xGG c(φ ) )

−mF dF (−yFG α + xFG γ )

−mG dG yGG c(θ )

(26)

Referring to Eq. (18) and ignoring the rolling and pitching motions of the float, the yawing moment of the float in the float frame was derived as:

MF = −mF yFG u˙ F + mF xFG v˙ F + IzF r˙ F

(27)

Combining Eq. (6),

⎡

⎤T ⎡ ⎤ u˙ ⎢ −mF yF β + mF xF χ ⎥ ⎢ v˙ ⎥ G G ⎢ ⎥ ⎥ ⎢−m yF β dF + m xF χ dF ⎥ ⎢ p˙ ⎥ F G ⎢ F G ⎥⎢ F ⎢ ⎥ + w˙ × (−mF yFG + mF xFG ∂ ) + ( p × p + q × q )dF mF (−yFG + xFG ∂ ) M =⎢ ⎥ q˙ ⎥ ⎢ mF yFG α dF − mF xFG γ dF ⎥ ⎢ ⎢ ⎢ ⎥ ⎣ F⎥ ⎣ ⎦ r˙ ⎦ IF −mF yFG α + mF xFG γ

z

r˙ G

0 + ( r × p )d Therefore,

⎡

F

( α−

mF yFG

xFG

γ ) + (r × q )dF mF (yFG β − xFG χ )

−mF yFG α + mF xFG γ

(28)

⎤T

⎢ −mF yF β + mF xF χ ⎥ G G ⎢ ⎥ ⎢−m yF β dF + m xF χ dF ⎥ F G ⎢ F G ⎥ MRB (5, : ) = ⎢ ⎥ ⎢ mF yFG α dF − mF xFG γ dF ⎥ ⎢ ⎥ ⎣ ⎦ IzF

(29)

0 and there was also an additional force:

M¯ F = w˙ × (−mF yFG + mF xFG ∂ ) + ( p × p + q × q )dF mF (−yFG + xFG ∂ ) + (r × p)dF mF (yFG α − xFG γ ) + (r × q )dF mF (yFG β − xFG χ ) Similarly,

⎡

−mG yGG c(θ )

(30)

⎤T

⎢ −mG yG  + mG xG c(φ ) ⎥ G G ⎢ ⎥ ⎢m yG  d − m xG c(φ )d ⎥ G G G⎥ ⎢ G G G MRB (6, : ) = ⎢ ⎥ −mG yGG c(θ )dG ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0

(31)

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and the additional force was:

M¯ G = w˙ × (−mG yGG  − mG xGG s(φ ) ) + ( p × p + q × q )dG mG (G yGG  + G xGG s(φ ) ) − r pdG mG yGG c(θ ) + rqdG mG (−yGG  + xGG c(φ ) )

(32)

The vertical acceleration of the center of gravity of WDR w˙ in system frame in the above equations could be calculated according to Eq. (13). 3.2. The Coriolis-centripetal coefficient matrix Similarly with Section 3.1, the Coriolis-centripetal coefficient matrix CRB was presented in Eqs. (33)–(36):

⎡

0

0

⎤

−m(xG q − w )

myG q

⎢ ⎥ 0 0 −m(yG p + w ) mxG p ⎥ ⎣ −myG q m ( yG p + w ) 0 −Iyz q − Ixz p ⎦ m ( xG q − w ) −mxG p Iyz q + Ixz p 0 ⎡ ⎤ F F F G G G −mF (xG r + v )α −mG (xG r + v )c(θ ) ⎢−mF (yFG rF − uF )γ ⎥ ⎥ CRB (1 : 2, 5 : 6 ) = ⎢ G G G ⎣ −mF (xF rF + vF )β −mG (xG r + v ) ⎦ G −mF (yFG r F − uF )χ −mG (yGG r G − uG )c(φ ) CRB (1 : 4, 1 : 4 ) = ⎢

⎡

dF mF (−(xFG r F + vF )β

dG mG ((xGG r G + vG )

dF mF ((xFG r F + vF )α

−dG mG (xGG r G + vG )c(θ ) + (yFG r F − uF )γ )

CRB (3 : 4, 5 : 6 ) = ⎣

 CRB (5 : 6, : ) =

−(yFG r F − uF )χ )

+(yGG r G − uG )c(φ ))

(33)

(34)

⎤ ⎦

(35)



0

0

0

0

mF (xFG uF + yFG vF )

0

0

0

0

0

0

mG (xGG uG + yGG vG )

(36)

The effects of the yawing motions of the float and the glider were considered in the 5th and 6th lines of CRB , while the elements in the top left 4 × 4 parts related to the yawing motions in CRB were valued as 0. The vertical velocity of the center of gravity of the WDR w could be calculated according to Eq. (12). uF ,vF ,uG ,vG should be replaced according to Eq. (6). 3.3. Vectors of other forces The solving processes of the vectors of inertial hydrodynamic force, hydrodynamic Coriolis-centripetal force, damping force, restoring force and active control force were presented as following steps: (1) The longitudinal, transverse and vertical forces and yawing moments of WDR’s subassemblies were calculated their body-fixed frame, respectively, while the effects caused by the yawing and pitching motions of WDR’s subassemblies were neglected. (2) The longitudinal, transverse and vertical forces of WDR’s subassemblies were transformed into the system frame multiplied by coordinate transformation matrixes. (3) The resultant force in the system frame was the summation of the component of forces in step (2). The resultant moment was the summation of the vector products of component forces and radius vectors. (4) The surging and swaying forces of the center of gravity of WDR in the system frame were respectively obtained as the longitudinal and transverse resultant forces in step (3). The rolling and pitching moments of the umbilical were respectively obtained as the rolling and pitching components of the resultant moments in step (3). The 5th and 6th elements of the force vector were directly the yawing moments of the float and the glider in step (1). The detailed solving process was shown as follows: ¯ were derived as: Then the resultant force F¯ and the resultant moment M



F¯ = OF R · X F

YF

ZF



¯ = r¯ F × (O R · X F M F

T

YF



+ OG R · X G ZF

T

YG

ZG

T

 ) + r¯ G × (OG R · X G

YG

ZG

T

)

(37)



The vector of force of the WDR X ∈ R6×1 was derived as:

X(1 : 2 ) = F¯ (1 : 2 )

¯ (1 : 2 ) X (3 : 4 ) = M



X (5 : 6 ) = N F

N

 G T

(38)

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Further, the expansion of X was as:

⎡

⎤ α X F + γ Y F + c ( θ )X G ⎢ ⎥ β X F + χ Y F +  X G + c(φ )Y G ⎢ ⎥ ⎢ ⎥ F F F G G G ⎢ dF (β X + χ Y + κ Z ) − dG ( X + c(φ )Y + κ Z )⎥ X=⎢ ⎥ ⎢−dF (α X F + γ Y F − s(θ )Z F ) + dG (c(θ )X G − s(θ )Z G )⎥ ⎢ ⎥ ⎣ ⎦ NF

(39)

NG The corresponding longitudinal, transverse, vertical forces and yawing moment of WDR’s subassemblies in their bodyfixed frames (XF etc.) could be calculated according to the motion states in the body-fixed frames. The required motion states including velocities and accelerations of the float and the glider were obtained by Eq. (6). Then the forces and moments in their body-fixed frames were essentially same with the one of the traditional single-body marine vehicles. As to the “Ocean Rambler” WDR, which is the experimental objects in Section 4, the calculation of longitudinal, transverse, vertical forces and yawing moment of WDR’s subassemblies in their body-fixed frames are listed in Section 3.4 of Ref. [26], including the computational formula and hydrodynamic coefficients of longitudinal, transverse, vertical forces and yawing moment of WDR’s subassemblies. There are two special cases: (1) As to the restoring force vector, the gravity and buoyance of WDR generate only rolling and pitching moments of the umbilical. The force of the float in the float frame was:

F¯ Fg = [0 0 −W ]T

(40)

where W refers to the gravity of the glider in the air minus the buoyancy in the water, that is, the actual gravity in the water. The force of the glider in its body-fixed frame was:



F¯ Gg = 0

0

W

T

(41)

The moment about the center of gravity of WDR was obtained by transforming F¯ Fg and F¯ Gg into system frame:

¯ g = r¯ F × (OF R · F¯ Fg ) + r¯ G × (OG R · F¯ Gg ) = −(dF + dG ) M

 κW

s(θ )W

0

T

(42)



As a result, g (η ) was expressed as:





g (η ) = 0

0

− ( dF + dG )κ W

−(dF + dG )s(θ )W

0

0

T

(43)

(1) As to the active control force vector, only the glider makes contributions. The active forces contain the thrust provided by hydrofoils hinged on the glider and the force and moment provided by the rotating rudder mounted on the glider. As a result, the active control force vector was derived as:

⎡

c(θ )XτG

⎤

⎢  X G + c(φ )Y G ⎥ τ τ ⎢ ⎥ ⎢−dG ( X G + c(φ )Y G )⎥ ⎢ τ τ ⎥ τ =⎢ ⎥ dG c(θ )XτG ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0

(44)

NτG where

XτG = T +

1 ρ S(vδr )2CD (δ ) 2

1 ρ S(vδr )2CL (δ ) 2 NτG = YτG Lδ

YτG =

(45)

where T is the thrust provided by the glider, ρ is the density of sea water; S is rudder’s area; Lδ is the steering arm generated by the rudder; vδr is the flow velocity of rudder; CL (δ ) and CD (δ ) are coefficients of lift and drag of the rudder respectively. Both CL (δ ) and CD (δ ) depend on the rudder angle δ . The origin of the system frame coincided with the center of gravity of the WDR. Therefore,

xG = yG = zG = 0

(46)

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Y. Li, K. Pan and Y. Liao et al. / Applied Mathematical Modelling xxx (xxxx) xxx Table 1 Part of the main parameters of the “Ocean Rambler” WDR. Parameters

Parameter values

Mass of float mF Mass of glider mG X coordinate of the center of gravity of the float xFG Moment of inertia of float IzF Moment of inertia of glider IzG Underwater gravity of glider W Length of umbilical dumbilical Rudder’s area S

55 kg 40 kg 25 cm 90 kg m2 10 kg m2 239 N 4m 350 cm2

Table 2 Part of CFD results of thrust with 0.2 m wave height of the “Ocean Rambler” WDR. Wave length

Length of umbilical 2m

Length of umbilical 4m

6m 8m 10 m

16.12 N 16.75 N 15.08 N

16.86 N 17.51 N 15.75 N

Length of umbilical 7m 16.83 N 17.44 N 16.17 N

Similarly, the shapes of WDR’s subassemblies were neglected when discussing the holistic motions of the WDR. Therefore,

Ixy = 0

(47)

The pitching motion of the glider would decrease the efficiency of wave energy. So the centers of gravity and of hydrodynamic force of the glider, and the hinged joint on the glider should be on the same vertical line in architecture design. Besides, the connection points from umbilical to the float and the glider are located in the longitudinal section in center planes of the float and the glider respectively. Therefore,

xGG = yGG = zGG = yFG = zGF = 0 Ix = Iy = mF (dF )2 + mG (dG )2 The hinged point on the float is located near the bow to provide the force arm for the steering of the float; thus The additional forces ware simplified as:

(48) xFG

> 0.

M¯ F = w˙ mF xFG ∂ + ( p2 + q2 )dF mF xFG ∂ + r pdF mF xFG γ + rqdF mF xFG χ M¯ G = 0

(49)

4. Numerical simulation and tank test Aiming at the “Ocean Rambler” WDR designed by Harbin Engineering University, the motion simulation and tank test were carried out. Parts of the main parameters are listed in Table 1. The computational formula and hydrodynamic coefficients of the hydrodynamic forces of the float and the glider, and the coefficients related to the rudder are listed in Section 3.4 of Ref. [26]. The added masses of WDR in the horizontal and vertical directions were respectively set as 0.1 times and 2.1 times of the inherent mass [20,24,27]. Zhang calculated the thrust generated by the glider under several different regular sine waves by computational fluid dynamics (CFD) technique [28]. Parts of the results are listed in Table 2. The research results were applied in the motion simulation. In 2017, the tank test of the “Ocean Rambler” WDR was operated in the general deep water tank in Harbin Engineering University, as shown in Fig. 3. The weather station was mounted on the float and the magnetic compass was mounted on the glider. The sensors measured the headings and the yawing velocities of WDR’s subassemblies in tank test. 4.1. Simulation on longitudinal motion The wave height was set as 0.2 m; the wave length was set as 8 m; the length of umbilical was 4 m. The results are shown in Figs. 4–7. Fig. 4 shows a brief description of the longitudinal motion including information about sailing distance, pitching angle of the umbilical, trajectories of the float and the glider, etc. The detailed information is shown in Figs. 5–7. Fig. 5 shows the longitudinal velocities of the float and the glider. In steady state of motion, the velocity of the float oscillated in sine form with steady amplitude and with the same period of wave. The velocity of the glider also oscillated with the same period of wave. Their average velocities were the same, roughly 0.3 m/s. From Fig. 6, the pitching angle oscillated stably in steady state of motion. The average pitching angle was about 6° and oscillation amplitude was about 1°. Fig. 7 Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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Fig. 3. Tank test of “Ocean Rambler” WDR.

Fig. 4. The X–Z view of the WDR in longitudinal motion.

shows the vertical displacement of the float and the glider. The zero points of the float and the glider were set as the initial positions of themselves, so in the 1st wave period the vertical displacement of the float was exactly same with the glider.

4.2. Simulation and tank test on reciprocating steering motion In simulation, the rudder angle was set as same with the actual rudder angle in tank test. The results are shown in Figs. 8–10. Figs. 8 and 9 present the comparison of yawing velocities between in simulation and in tank test respectively of the float and the glider. Whether for float and glider, the trends of the heading response were similar in simulation and tank test when rudder angle changed. With rudder angle of 20°, the stable average angular velocities of the float and the glider were about 3°/s in simulation and tank test. For the float’s heading response, both in simulation and tank test, there was a lag of about 5 s relative to the glider’s heading response when the rudder angle changed, the heading of the float lagged behind that of the glider about 5 s. This was due to the driving characteristics of the float. Only when the float’s heading is Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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Fig. 5. Longitudinal velocities of WDR’s subassemblies.

Fig. 6. Pitching angle of the umbilical.

different from the glider’s heading, there would be a yawing moment drive the float’s heading to the glider’s heading when umbilical is tense. Fig. 10 shows the heading error between WDR’s subassemblies (ψ = ψ G − ψ F ) in simulation and tank test. The variation trends of the heading error were similar in simulation and tank test. If the rudder angle was stable, for the float’s heading response, there was a lag of about 15° relative to the glider’s heading response. In the preliminary stage, there was a deviation of the heading error between in simulation and in tank test. It happened because the WDR was close to Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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Fig. 7. Vertical displacements of the float and the glider.

Fig. 8. Yawing velocities of the float in simulation and tank test.

the wave-absorbing beach in the beginning of the movement and the metal in the wave-absorbing beach deteriorated the accuracy of the magnetic compasses. In Figs. 8–10, there were oscillations of the sampling points in tank test. There are two main reasons: (1) The wave attacked the float which strengthened the heading oscillation of the float, and conducted to the glider partly; (2) There were sensor noises in tank test. Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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Fig. 9. Yawing velocities of the glider in simulation and tank test.

Fig. 10. Heading error between WDR’s subassemblies in simulation and test.

4.3. Simulation of an example for PID heading control In this section, we provide an example of PID heading control to show the significance of the established dynamic model of WDR in control algorithm design. The wave input was set as 0.2 m wave height and 8 m wave length. The surface current was set as 0.1 m/s current velocity and 0° current orient which brought a yawing moment disturbance on the float. The desired heading was set as Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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Fig. 11. Heading control response of the float.

Fig. 12. Heading control response of the glider.

30°. The PID controller parameters were kp = 3, ki = 0.0 0 01, kd = 0.5. The simulation results of heading control of the float and heading control of the glider are shown in Figs. 11 and 12. From Fig. 11, it is hard to realize the stable heading control of the float using the basic PID controller, and the control performance cannot be improved only by adjusting the parameters of the PID controller. In essence, the instability of the control system is due to the lag response of the heading of the float. Further, the environmental disturbances deteriorate the control performance. In Fig. 12, the glider’s heading converged to the desired value. However, there was a steady error between the heading of WDR’s subassemblies caused by the environmental disturbances. The simulation example of PID Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046

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heading control of WDR reminds us that the control problem of WDR is unusual because of the special structure of WDR, which should be considered with great attention in control algorithm design. 5. Conclusion (1) The kinematic analysis and dynamic model of WDR under the assumption that the umbilical always being in tension were presented. The established dynamic model of WDR is proved reasonable and operable in practice by numerical simulation and test, which provides researchers a theoretical basis for further study. (2) The results in simulation and tank test show that the distinctive multi-body joint structure brings the WDR some particular dynamic characteristics. The velocities of the float and the glider oscillate with the undulation of waves. The yawing response of the float lags behind the glider. (3) The control characteristic of WDR is unusual because of the special structure of WDR, which should be considered with great attention in control algorithm design. (4) This paper only discussed the dynamic analysis and modeling of WDRs under regular waves. The dynamic analysis and modeling of the motion response of WDRs under complex waves and currents in the marine environment is the focus of follow-up research. Acknowledgments This work was supported by the National Natural Science Foundation of China [grant numbers 51779052, 51879057, U1806228]; the Research Fund from Science and Technology on Underwater Vehicle Laboratory (grant numbers 614221503091701 and 6142215180102); the Heilongjiang Postdoctoral Funds for Scientific Research Initiation [grant number LBH-Q17046]; the National Key R&D Program of China [grant number 2017YFC0305700]. Appendix Table A1. Table A1 Nomenclature of all state variables. Name

Description

mF , mG , m IzF , IzG

The masses of the float, the glider and the whole WDR. The moment of inertia about the z-axis of the body-fixed frame of the particular subassembly of WDR represented by the right superscript. The moments of inertial and the inertial product of WDR about the x-axis and y-axis of the system frame. The distance between the origin of the system frame and the hinged joint on particular subassembly of WDR represented by the right superscript. The length of umbilical. The x, y, z coordinates of the center of gravity of the particular subassembly of WDR represented by the right superscript in the corresponding body-fixed frames. The x, y, z coordinates of the center of gravity of WDR in system frame. The longitudinal, transverse, and vertical coordinates of the origin of system frame in the earth-fixed frame. The headings of the float, the glider, the system frame, and the heading error between the float and the glider. The rolling angle and the pitching angle of umbilical. The surge, sway, heave, yawing velocities of the float in the float frame. The surge, sway, heave, yawing velocities of the glider in the glider frame. The surge, sway, heave velocities of the center of gravity of WDR in the system frame. The rolling, pitching, yawing velocities of umbilical (system frame) in the system frame. The angular velocity vector and angular acceleration vector of the system frame. The vector of velocity relative to current of the WDR.

3×1 6×1

a, v, η

The acceleration vector, velocity vector, position vector of the WDR.

6×1

ν

The velocity vector and position vector of a ship in reference [25] The acceleration vector, velocity vector, position vector] of the center of gravity of WDR in the system frame. The vectors of position, velocity and acceleration. The meaning of right superscripts is a particular subassembly of WDR and the meaning of left superscripts is the reference coordinate frame that the motion parameter is “expressed in”. For the motion parameters without left superscripts, it means that they are expressed in the body-fixed frame of itself by default.

6×1 3×1

Ix , Iy , Ixy dF , dG umbilical

d xFG , yFG , zGF , xGG , yGG , zGG xG , yG , zG

ξ , η, ζ

ψ F , ψ G , ψ 0 , ψ ϕ, θ

uF , vF , wF , rF uG , vG , wG , rG u, v, w p, q, r

ω¯ , α¯ νr



ship



ship

η a¯ , v¯ , η¯

O

,

η¯ F , O η¯ G , v¯ F , v¯ G

NED F NED G O F O G η¯ , η¯ , v¯ , v¯ , a¯ , O a¯ G , a¯ F , a¯ G

O F

Dimension 1×1 1×1 1×1 1×1 1×1 1×1 1×1 1×1 1×1 1 1 1 1

× × × ×

1 1 1 1

1×1

3×1

(continued on next page)

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Table A1 (continued) Name

Description

r¯ F , r¯ G

The position vector of the part of the particular subassembly of WDR represented by the right superscript in the system frame. The transformation matrix from the frame represented by the left subscript to the frame represented by the left superscript. Substitutions (related to ϕ , θ , ψ ) to simplify the expression of other equations. Rigid body mass matrix. The matrix of Coriolis-centripetal coefficient.

F G NED O O O O R, O R, O R, F R, G R, NED R

α , β , ϖ, γ , χ , ∂ , κ , μ,  ,  MRB CRB









FCA , FMA , D, g, τ

M¯ F , M¯ G XF , YF , ZF , NF , XG , YG , ZG , NG

XτG , YτG , NτG ¯ F¯ , M

X F¯ Fg , F¯ Gg ¯g M T

ρ S

vδr δ

CL (δ ), CD (δ ) Lδ W

The vector of hydrodynamic Coriolis-centripetal force, the vector of inertial hydrodynamic force, the vector of damping force, the vector of restoring force, the vector of active control force. The additional force in the derivation process of rigid mass matrix. The longitudinal, transverse, vertical forces and yawing moment (of the kind of force being discussed) of the part of the particular subassembly of WDR represented by the right superscript in corresponding body-fixed frames. The longitudinal, transverse active control force and yawing active control moment of the glider in the glider frame. The resultant force and the resultant moment (of the kind of force being discussed). The vector of force of the WDR (of the kind of force being discussed). The force of the part of the particular subassembly of WDR represented by the right superscript in corresponding body-fixed frames in the restoring force vector. The vector contains the rolling and pitching moments of umbilical related to the glider’s actual gravity in the water. The thrust provided by the glider The density of sea water. The rudder’s area. The inflow velocity of rudder. Rudder angle. Coefficients of lift and drag of rudder. The steering arm generated by the rudder. The glider’s actual gravity in the water.

Dimension 3×1 6×6 1×1 6×6 6×6 6×1

1×1 1×1

1×1 3×1 6×1 3×1 3×1 1 1 1 1 1 1 1 1

× × × × × × × ×

1 1 1 1 1 1 1 1

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Please cite this article as: Y. Li, K. Pan and Y. Liao et al., Dynamics modeling and experiments of wave driven robot, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.046