MEV Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 Journal of Mechatronics, Electrical Power, and Vehicular Technology e-ISSN: 2088-6985 p-ISSN: 2087-3379 mev.lipi.go.id doi: https://dx.doi.org/10.14203/j.mev.2022.v13.113-124 2088-6985 / 2087-3379 ©2022 National Research and Innovation Agency This is an open access article under the CC BY-NC-SA license (https://creativecommons.org/licenses/by-nc-sa/4.0/) MEV is Scopus indexed Journal and accredited as Sinta 1 Journal (https://sinta.kemdikbud.go.id/journals/detail?id=814) How to Cite: M. L. Ramadiansyah et al., “Numerical investigation of the effect of ocean depth variations on the dynamics of a ship mounted two- DoF manipulator system,” Journal of Mechatronics, Electrical Power, and Vehicular Technology, vol. 13, no. 2, pp. 113-124, Dec. 2022. Numerical investigation of the effect of ocean depth variations on the dynamics of a ship mounted two-DoF manipulator system Mohamad Luthfi Ramadiansyah a, *, Edwar Yazid a, Cheng Yee Ng b a Research Center for Smart Mechatronics, National Research and Innovation Agency (BRIN) Kawasan Bandung Cisitu, Jl. Sangkuriang, Dago, Coblong, Bandung, 40135, Indonesia b Department of Civil and Environmental Engineering, Universiti Teknologi PETRONAS 32610 Seri Iskandar, Perak, Malaysia Received 17 October 2022; Revised 16 November 2022; Accepted 17 November 2022; Published online 29 December 2022 Abstract The dynamics of a ship need to be considered in the development of a manipulator system that will be applied to the ocean-based operation. This paper aims to investigate the effect of ocean depth variations on the ship motion as disturbances to a ship-mounted two-DoF (Degrees of Freedom) manipulator joint torque using an inverse dynamics model. Realization is conducted by deriving the mathematical model of a two-DoF manipulator system subject to six-DoF ship motion, which is derived by using Lagrange-Euler method. It is then combined with numerical hydrodynamic simulation to obtain the ship motions under ocean depth variations, such as shallow (50 m), intermediate (750 m), and deep (3,000 m) waters. Finding results show that randomness of the ship motions appears on the manipulator joint torque. In the azimuth link, maximum joint torque is found in shallow water depth with an increment of 8.271 N.m (285.69 %) from the undisturbed manipulator. Meanwhile, the maximum joint torque of the elevation link is found in intermediate water depth with an increment of 53.321 N.m (6.63 %). However, the difference between depth variations is relatively small. This result can be used as a baseline for sizing the electrical motor and developing the robust control system for the manipulator that is mounted on the ship by considering all ocean depth conditions. ©2022 National Research and Innovation Agency. This is an open access article under the CC BY-NC-SA license (https://creativecommons.org/licenses/by-nc-sa/4.0/). Keywords: two-DoF manipulator; inverse dynamics; ship motion; ocean depth; hydrodynamic response. I. Introduction It is widely known that robotic systems may be easily found in many engineering applications. Design and analysis of such a system has been carried out in some areas, such as industrial application [1][2], underwater [3], vehicle [4], satellite antenna [5], and humanoid robot [6][7] with their specified objectives. The essential task in developing a robotic system is kinematics and dynamics modeling. Kinematics modeling is commonly carried out to determine the position and orientation of manipulator links. Tavassolian et al. [8] had employed the forward kinematics model of a parallel robot using a combined method based on the neural network. Dewandhana et al. [9] had similar work but with a different application, i.e., a full-arm robot. Inverse kinematics had been performed by Kusmenko and Schmidt [10] for developing a 5-DoF (Degrees of Freedom) robot arm and by Chen et al. [11] for an underwater propeller cleaning application. Amundsen et al. [12] had performed inverse kinematics for manipulator control that was implemented on a non-fixed based. Inverse dynamics model is commonly used to obtain the dynamic characteristics of a manipulator system. It was implemented by Polydoros et al. [13] for torque control manipulator, Awatef and Mouna [14] for motion control of the unicycle mobile robot, and Crenna and Rossi [15] for measurement of internal torques in the articulations of the human body during a gesture. The calculation of inverse dynamics using computational methods is currently well-known for its efficient purpose. Farah and Shaogang [16] had introduced an efficient approach * Corresponding Author. Tel: +62-878-2460-7227 E-mail address: moha057@brin.go.id https://dx.doi.org/10.14203/j.mev.2022.v13.113-124 https://dx.doi.org/10.14203/j.mev.2022.v13.113-124 http://u.lipi.go.id/1436264155 http://u.lipi.go.id/1434164106 https://mev.lipi.go.id/mev https://mev.lipi.go.id/mev https://dx.doi.org/10.14203/j.mev.2022.v13.113-124 https://dx.doi.org/10.14203/j.mev.2022.v13.113-124 https://creativecommons.org/licenses/by-nc-sa/4.0/ https://sinta.kemdikbud.go.id/journals/detail?id=814 https://crossmark.crossref.org/dialog/?doi=10.14203/j.mev.2022.v13.113-124&domain=pdf https://creativecommons.org/licenses/by-nc-sa/4.0/ M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 114 for modeling robot dynamics. They used Matlab and SimMechanics instead of an analytical approach, but the study was limited to ground-based applications. Meanwhile, Müller [17] had compared classical and computational methods. Besides to fulfill its function, the development of a robotic system must consider the behavior of its base, in which the system will be operated. Dynamics of the non-inertial base certainly affect the positioning of manipulator arms which is the main task of a robotic system. Wei et al. [18] had performed a dynamic analysis of a mobile manipulator that operated on the 3-DoF floating base. Similar work of non-inertial base manipulator had been done by [19], which introduced the modeling and control of a soft robotic arm on the aerial vehicle. To the best of the author’s knowledge, dynamic analysis of a manipulator excited by ship motions is relatively rare. Some literature that relates to this topic may be found in [20][21][22], although they mainly focused on the control system development. Research by Qian and Fang in [23] is regarded as the closest work where they had analyzed the regular ocean waves effect to the dynamic analysis of a ship- mounted crane system. Further, dynamics of manipulator systems subject to irregular ocean waves induced ship motions had been performed by [24] under variations of sea states. Important finding results show that the maximum joint torque of a manipulator is proportional to the increment of significant wave height and greatly affected by the direction of ocean wave propagation. To date, there has not yet been research working on the effect of ocean depths on manipulator dynamics. Ahmed et al. [25] presented that the water particle force on the oceanic structure depends on the ocean depth. Hence, this paper investigates the effect of ocean depth variations on the manipulator joint torque with contributions as follows: • To develop a mathematical model of a ship- mounted two-DoF manipulator considering the ship dynamics. • To characterize the ship motions as excitations to the base of a manipulator system subject to random ocean waves under variations of ocean depth using numerical hydrodynamic simulation and propose its methodology. • To perform a parametric study in terms of variations of ocean depth to the manipulator dynamics. This paper is organized as follows: system description and the underlying method, as well as the governing equations, are described in Section II. Results and discussions of derived governing equations, numerical simulations, and manipulator dynamic characteristics are presented in Section III. Conclusion and recommendations are put in Section IV. II. Materials and Methods The underlying manipulator construction is illustrated in Figure 1, where it has two degrees of freedom, namely azimuth and elevation links. The former is designed to be able to fully rotate in the horizontal plane with a maximum angle of 360° C(C)W, while the elevation angle can rotate in the vertical plane with a range angle at -20° ~ 60° C(C)W. The end-effector is designed to aim and lock on the target on the ocean water surface. A control system must be applied to move the arms at certain angles precisely when the base of the manipulator is excited by random ocean waves induced ship motions. This is in order to enable the end-effector to stick to the target. Moreover, ship motions are treated as a six-DoF rigid body, as visualized in Figure 2 under non-propelled conditions. Respectively in the X, Y, and Z axis, translational motions are called Surge, Sway, and Heave, and rotational motions are Roll, Pitch, and Yaw. To calculate manipulator joint torque, the inverse dynamics model is applied, and the process flow is shown in Figure 3. The ship motions can be measured with a motion sensor unit, which has three accelerometers for detecting surge, sway, and heave and three rotation rate sensors for measuring roll, pitch, and yaw [26]. In this paper, a combination Figure 1. Schematic of a manipulator system M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 115 of analytical and numerical methods is proposed to simulate ship motions and manipulator joint trajectory from the sensor system. The former is analytical simulations of joint trajectory and manipulator dynamics, and the latter is a numerical simulation of the ship motions using ANSYS Aqwa in the variations of ocean depth. Here, equations of motion of a ship-mounted manipulator system are derived by using Lagrange-Euler method. Thus, discussion with regard to control system design and analysis, including sensor system, is out of this paper’s range. To begin with, the main parameters of the manipulator system and ship geometry are given with certain conditions applied. A. Forward kinematics Kinematics of a ship-mounted manipulator system as an early step in the dynamic analysis is realized in the form of forward and inverse kinematics. The former is defined from the base to the end-effector using manipulator joint parameters and coordinates, as noted in Figure 1. A homogeneous transformation matrix of the system is then built based on the widely adopted Denavit- Hartenberg (DH) method [27] by multiplying each homogeneous transformation matrix of the joint from the base into the end-effector. Homogeneous transformation matrix (𝑇) consists of a rotational matrix (𝑅) and position matrix (𝑃), which is defined as equation (1), 𝑇 = �𝑅𝑛,𝑠,𝑎 𝑃𝑋,𝑌,𝑍 0 1 � = � 𝑛𝑋 𝑠𝑋 𝑎𝑋 𝑃𝑋 𝑛𝑌 𝑠𝑌 𝑎𝑌 𝑃𝑌 𝑛𝑍 𝑠𝑍 𝑎𝑍 𝑃𝑍 0 0 0 1 � (1) The terms, 𝑠 and 𝑎 denote normal, shear, and approach vectors in the XYZ-axes. Using manipulator frame coordinates in Figure 1, the total homogeneous transformation matrix can be written as equation (2), 𝐻 = 𝑇𝑆. 𝑇𝑀. (2) The term 𝐻 is the total, 𝑇𝑆 is the ship, and 𝑇𝑀 is the manipulator homogeneous transformation matrices, respectively. The homogeneous transformation matrix of a ship can be expressed as equation (3), 𝑇𝑠 = � 𝑐𝑐𝑐𝑐 𝑠𝑐𝑐𝑐 𝑐𝑐𝑠𝑐𝑠𝑐 − 𝑠𝑐𝑐𝑐 𝑠𝑐𝑠𝑐𝑠𝑐 + 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑠𝑐 + 𝑠𝑐𝑠𝑐 𝑋𝑠 𝑠𝑐𝑠𝑐𝑐𝑐 − 𝑐𝑐𝑠𝑐 𝑌𝑠 −𝑠𝑐 𝑐𝑐𝑠𝑐 𝑐𝑐𝑐𝑐 𝑍𝑠 0 0 0 1 � (3) The term 𝑐 and 𝑠 represent cos and sin; 𝑋𝑠, 𝑌𝑠, and 𝑍𝑠 are the translational ship motions, such as surge, Figure 2. Visualization of ship motions Figure 3. Steps for simulating the inverse dynamics of a manipulator system M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 116 sway, and heave, respectively; and 𝑐, 𝑐, and 𝑐 are the rotational ship motions, namely roll, pitch, and yaw, respectively. The homogeneous transformation matrix of manipulator system can be written as equation (4) and equation (5), 𝑇𝑀 = 𝑇𝑀1 0 . 𝑇𝑀 212 0 (4) 𝑇𝑖 𝑖−1 𝑀 = ⎣ ⎢ ⎢ ⎡ 𝑐𝜃𝑙𝑖 𝑐𝛼𝑖−1𝑠𝜃𝑙𝑖 −𝑠𝜃𝑙𝑖 𝑐𝛼𝑖−1𝑐𝜃𝑙𝑖 0 𝑎𝑖−1 −𝑠𝛼𝑖−1 −𝑠𝛼𝑖−1𝑑𝑖 𝑠𝛼𝑖−1𝑠𝜃𝑙𝑖 𝑐𝜃𝑙𝑖𝑠𝛼𝑖−1 𝑐𝛼𝑖−1 −𝑐𝛼𝑖−1𝑑𝑖 0 0 0 1 ⎦ ⎥ ⎥ ⎤ (5) The term 𝑇𝑖 𝑖−1 𝑀 is the transformation matrix from the 𝑖 − 1 frame to 𝑖 frame, 𝜃𝑙 is joint angle, 𝛼𝑖−1 is the rotational link angles in X-axis, 𝑎 and 𝑑 are respective link distances in the X- and Z-axes. Using manipulator kinematic parameters of the link in Table 1, equation (5) can be rewritten for each joint as equation (6) and equation (7), 𝑇𝑀 = � 𝑐𝜃𝑙𝑖 −𝑠𝜃𝑙𝑖 0 0 𝑠𝜃𝑙𝑖 𝑐𝜃𝑙𝑖 0 0 0 0 0 0 1 0 0 1 �1 0 (6) 𝑇𝑀 = � 𝑐𝜃𝑙2 −𝑠𝜃𝑙2 0 0 0 0 −1 0 𝑠𝜃𝑙2 0 𝑐𝜃𝑙2 0 0 0 0 1 �21 (7) B. Inverse dynamics An inverse dynamics model is used to define the manipulator joint torque with predefined joint trajectories. The torque can be expressed in several terms, such as inertia, centrifugal, Coriolis, and gravity as equation (8), 𝜏 = 𝑀(𝛩)�̈� + 𝑉�𝛩, �̇�� + 𝐺(𝛩) (8) The term 𝜏 is manipulator joint torque, 𝑀 is the mass matrix that contributes to the torque due to inertia, 𝑉 is the matrix of centrifugal and Coriolis terms, and 𝐺 is the matrix of gravity term [28]. Recall the Euler’s equation, the torque value is defined as equation (9), 𝑑 𝑑𝑑 𝜕𝜕 𝜕�̇� − 𝜕𝜕 𝜕𝛩 = 𝜏 (9) 𝐿�𝛩, �̇�� = 𝐾�𝛩, �̇�� − 𝑃(𝛩) (10) The term 𝐿 is the Lagrange operator, 𝛩 is position, �̇� is velocity, and �̈� is acceleration of the joint. Lagrange formulation is defined as the difference between kinetic energy (𝐾) and potential energy (𝑃) following equation (10). Substituting equation (10) into equation (9), it can be written as equation (11), 𝜏𝑖 = 𝑑 𝑑𝑑 𝜕𝐾𝑖 𝜕�̇� − 𝜕𝐾𝑖 𝜕𝛩 + 𝜕𝑃𝑖 𝜕𝛩 (11) Kinetic energy is obtained from translational and rotational motions, while potential energy is due to gravity effect as equation (12) and equation (13), 1 1 2 2 i i i cT i T i i i c c i i iK m v v Iω ω= + (12) 0 i T i i cP m g P= (13) The term 𝑚𝑖 is mass, 𝑣𝑐 is linear velocity at the centre of gravity, 𝜔 is angular velocity, 𝐼 is the moment of inertia of the link, 𝑔 is gravity, and 𝑃𝑐 is position matrix from the homogeneous transformation matrix. All variables in equations (12)(13) are transformed into matrix form where the mass and moment of inertia of each link follow the manipulator design parameters. Linear and angular velocities can be obtained from velocity propagation as equation (14) and equation (15), ( )1 1 11 1 1 1ˆi i i i i ii i i i i i iv R v P d Zω+ + ++ + + += + × +  (14) 1 1 1 1 1 1 ˆi i i i i i i i iR Zω ω θ + + + + + += +  (15) The term 𝑅 is rotational matrix from homogeneous transformation matrix, �̇� is linear velocity, �̇� is angular velocity, and �̂� is direction vector of the joint. It should be noted that the manipulator joint motion is predefined using a 5th- order spline function. C. Ship Motions Ship motions are carried out by numerical simulation using commercial software. In this paper, ANSYS Aqwa is employed to simplify the process from the ship modelling until the ship motions analysis. This type of software provides a toolset for investigating the effects of environmental loads on floating and fixed offshore as well as marine structures. This software can also be used to analyze the hydrodynamic diffraction and responses of a body subject to ocean waves. Overall, hull modelling, meshing process, and ocean waves generation subject to the ocean depth are evaluated by ANSYS Aqwa. 1) Ship hull modelling In practice, a ship model can be simplified into a hull model, which can be seen in Figure 4. The hull interacts with ocean waves so that it becomes the main geometry that must be modeled properly. In Aqwa, modelling is based on surface geometry that may be designed from a geometry editor in ANSYS or other modelling softwares in the form of *.stp or *.igs files. In this paper, Solidworks in *.stp format is used and then imported into a geometry editor in Aqwa. 2) Meshing process The meshing process is performed in Aqwa. Boundary conditions and parameters of the ship are tabulated in Table 2. However, the current meshing process is different from the common CFD mesh, where the working fluid is set as the object. Here, the ship hull is the mesh object, as seen in Figure 5. It shows the surface mesh of the ship hull and the Table 1. Manipulator kinematic parameters i θl α a d 1 𝜃𝑙1 0 0 0 2 𝜃𝑙2 90 o 0 0 3 0 0 0 r M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 117 grid independence test result that corresponds to the hydrostatic heave as the parameter for determining the effective total elements. At the 22,210 elements, the hydrostatic value has approached the correct value and it becomes the meshing parameter hereafter. Using the higher elements can result in long-time iteration in the simulation process. 3) Ship and ocean random waves interaction modelling Random ocean waves are applied rather than regular ocean waves since they represent the actual ocean waves. Adopted from Linear airy wave theory [29], random ocean wave height is a summation of regular waves with different frequencies. JONSWAP type spectrum is used, which is the standard ocean wave model and more versatile than other spectrums. Its spectral ordinate at a frequency (𝜔) is expressed as equation (16), 𝑆(𝜔) = 𝛼𝐻𝑠2 � 𝜔𝑝4 𝜔5 � exp �−5 4 � 𝜔𝑝 𝜔 � 4 �𝛾 exp�− �𝜔−𝜔𝑝� 2 2𝜎2𝜔𝑝 2 � (16) The term 𝛼 is a Phillip’s constant, 𝐻𝑠 is significant wave height, 𝜔𝑝 is peak frequency, 𝛾 is peakedness parameter, 𝜎 is shape parameter [24]. By taking the values of spectral ordinate ( )ωS , the amplitude of the i-th ocean wave component can be calculated by using equation (17), 𝑎𝑖 = �2𝑆(𝜔𝑖)𝛥𝜔 (17) From equation (17) and the values of 𝑎𝑖, the time series of wave height can be generated as equation (18), 𝜉(𝑥, 𝑦, 𝑡) = ∑  𝑎𝑖 𝑠𝑖𝑛(𝜔𝑖𝑡 + 𝜃𝑖 − 𝑘𝑖𝑥𝑐𝑐𝑠𝜒 − 𝑘𝑖𝑦𝑠𝑖𝑛𝜒)𝑁𝑖=0 (18) The term 𝜉 is wave elevation, 𝑁 is number of wave component, 𝑘 is wave number, and 𝜒 is wave propagating direction. From linear wave theory, wave particle kinematics can be expressed as equations (19)–(23), 𝑣𝑥 = 𝜔𝜁𝑎 𝑐𝑐𝑠ℎ[𝑘(𝑧+ℎ)] 𝑠𝑖𝑛ℎ(𝑘ℎ) 𝑐𝑐𝑠(𝑘𝑥 − 𝜔𝑡) (19) 𝑣𝑧 = 𝜔𝜁𝑎 𝑠𝑖𝑛ℎ[𝑘(𝑧+ℎ)] 𝑠𝑖𝑛ℎ(𝑘ℎ) 𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡) (20) 𝑎𝑥 = 𝜔2𝜁𝑎 𝑐𝑐𝑠ℎ[𝑘(𝑧+ℎ)] 𝑠𝑖𝑛ℎ(𝑘ℎ) 𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡) (21) 𝑎𝑧 = 𝜔2𝜁𝑎 𝑠𝑖𝑛ℎ[𝑘(𝑧+ℎ)] 𝑠𝑖𝑛ℎ(𝑘ℎ) 𝑐𝑐𝑠(𝑘𝑥 − 𝜔𝑡) (22) 𝑝𝐷 = 𝜌𝑔𝜁𝑎 𝑐𝑐𝑠ℎ[𝑘(𝑧+ℎ)] 𝑐𝑐𝑠ℎ(𝑘ℎ) 𝑐𝑐𝑠(𝑘𝑥 − 𝜔𝑡) (23) The term 𝑣𝑥 and 𝑣𝑧 are horizontal and vertical water particle velocity, 𝑎𝑥 and 𝑎𝑧 are horizontal and Figure 4. Geometry of ship hull surface Table 2. Ship parameters in meshing process Parameter Value Defeaturing tolerance (m) 0.15 Maximum element size (m) 0.35 Total nodes 22,484 Total elements 22,210 Density of water (kg/m3) 1025 Water length and width (m) [300; 200] Gravitational acceleration (m/s2) 9.81 Draught (m) 3.13 Breadth (m) 9.5 Length between perpendiculars (m) 53.25 Mass (kg) 600858 Radius of gyration (m) [3.179; 13.313; 13.845] Center of gravity (m) [0; 0; 0] Center of buoyancy (m) [0; 0; 0.83] M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 118 vertical water particle acceleration, 𝑝𝐷 is dynamic pressure, 𝜌 is water density, 𝜁𝑎 is wave height, 𝑡 is time, and ℎ is water depth. Impulse of the wave particles will cause motion of the ship hull. The equation of motion is expressed in a convolution integral form as equation (24), (m + A∞)X ¨(𝑡) + c X ˙(𝑡) + KX(𝑡) + ∫ R(𝑡 − 𝑑 0 𝜏)X ˙(𝜏)𝑑𝜏 = F(𝑡) (24) The term 𝑚 is structural mass matrix, 𝐴∞ is fluid added mass matrix at infinite frequency, 𝑐 is damping matrix including the linear radiation damping effects, 𝐾 is total stiffness matrix, 𝑅 is velocity impulse function matrix, and 𝑋, �̇�, and �̈� are respectively matrices of position, velocity, and acceleration of the ship. An integration of equation (24) is held numerically by Aqwa using parameters in Table 2. Ship motion analysis is carried out in three classifications of ocean depth, such as shallow, intermediate, and deep waters, as shown in Table 3, which have an effect on the speed of ocean waves Table 3. Random ocean waves parameters Ocean classification Depth (m) Significant wave height (m) Wave frequency (Hz) Sea state Shallow 50 2 0.50 Moderate Intermediate 750 0.48 Deep 3000 0.46 (a) (b) Figure 5. Result of meshing process, (a) hull surface mesh; (b) grid independence test M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 119 [30]. Equations (19)-(23) fortify that the kinematics of the wave particle is the function of ocean depth. Once the simulation is completed, six-DoF ship motion can be obtained and applied to the calculation in equation (11). III. Results and Discussions The results of the analytical derivation of manipulator joint torque using the Lagrange-Euler method are tabulated in Table 4 for the azimuth angle and Table 5 for the elevation angle. The equations are classified into dynamic terms for clarity. They have been validated by excluding the terms of ship motions, and the results are similar to those without ship motions. Those terms are then utilized by substituting manipulator parameters in Table 6 and joint trajectories in Figure 6, which consist of both azimuth and elevation joints position, velocity, and acceleration over 30 s. The joint angles are obtained from the inverse kinematics process [31] and their trajectories are generated using the 5th-order spline function. As can be seen, smoothness of joint position, velocity, and acceleration can be obtained. As mentioned in the previous section, ship motions are obtained using numerical simulation through hydrodynamic time response analysis in Table 4. Dynamic terms for azimuth angle Term Torque Inertia �𝐼𝑧𝑧1 + 𝐼𝑥𝑥2 𝑠𝑖𝑛 2 𝜃𝑙2 + �𝐼𝑦𝑦2 + 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠2 𝜃𝑙2��̈�𝑙1 +�−𝑚2𝑟𝑥2 𝑠𝑖𝑛𝜃𝑙1 𝑐𝑐𝑠 𝜃𝑙2��̈�𝑆 +�𝑚2𝑟𝑥2 𝑐𝑐𝑠 𝜃𝑙1 𝑐𝑐𝑠 𝜃𝑙2��̈�𝑆 + � 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 𝜃𝑙1 𝑠𝑖𝑛 2 𝜃𝑙2��̈� + � 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 𝜃𝑙1 𝑠𝑖𝑛 2 𝜃𝑙2��̈� +�𝐼𝑧𝑧1 + 𝐼𝑥𝑥2 𝑠𝑖𝑛 2 𝜃𝑙2 + �𝐼𝑦𝑦2 + 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠2 𝜃𝑙2��̈� Coriolis ��𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 2 𝜃𝑙2��̇�𝑙1�̇�𝑙2 +���𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 2 𝜃𝑙2 − 𝐼𝑧𝑧2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 𝜃𝑙1��̇�𝑙2�̇� +���𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 2 𝜃𝑙2 − 𝐼𝑧𝑧2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 𝜃𝑙1��̇�𝑙2�̇� +��𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 2 𝜃𝑙2��̇�𝑙2�̇� +�𝑚2𝑟𝑥2 𝑐𝑐𝑠 𝜃𝑙1 𝑐𝑐𝑠 𝜃𝑙2��̇�𝑆�̇� +�𝑚2𝑟𝑥2 𝑠𝑖𝑛 𝜃𝑙1 𝑐𝑐𝑠 𝜃𝑙2��̇�𝑆�̇� +�−𝑚2𝑟𝑥2 𝑐𝑐𝑠𝜃𝑙1 𝑐𝑐𝑠 𝜃𝑙2��̇�𝑆�̇� +�−𝑚2𝑟𝑥2 𝑠𝑖𝑛𝜃𝑙1 𝑐𝑐𝑠 𝜃𝑙2��̇�𝑆�̇� +�−�𝐼𝑥𝑥1 − 𝐼𝑦𝑦1 − 𝐼𝑧𝑧2 − 𝑚2𝑟𝑥2 2 + �𝐼𝑦𝑦2 + 𝑚2𝑟 2�𝑠𝑖𝑛2 𝜃𝑙2 + 𝐼𝑥𝑥2 𝑐𝑐𝑠 2 𝜃𝑙2�𝑐𝑐𝑠 2 𝜃𝑙1��̇��̇� + � 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 𝜃𝑙1 𝑠𝑖𝑛 2 𝜃𝑙2��̇��̇� + �− 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 𝜃𝑙1 𝑠𝑖𝑛 2 𝜃𝑙2��̇��̇� Centripetal � 1 2 �𝐼𝑥𝑥1 − 𝐼𝑦𝑦1 − 𝐼𝑧𝑧2 − 𝑚2𝑟𝑥2 2 + �𝐼𝑦𝑦2 + 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛2 𝜃𝑙2 + 𝐼𝑥𝑥2 𝑐𝑐𝑠 2 𝜃𝑙2�𝑠𝑖𝑛 2 𝜃𝑙1��̇� 2 + �− 1 2 �𝐼𝑥𝑥1 − 𝐼𝑦𝑦1 − 𝐼𝑧𝑧2 − 𝑚2𝑟𝑥2 2 + �𝐼𝑦𝑦2 + 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛2 𝜃𝑙2 + 𝐼𝑥𝑥2 𝑐𝑐𝑠 2 𝜃𝑙2�𝑠𝑖𝑛 2 𝜃𝑙1��̇� 2 Gravity 𝑚2𝑔𝑟𝑥2 𝑐𝑐𝑠 𝜃𝑙2 �𝑠𝑖𝑛𝜃𝑙1 𝑠𝑖𝑛 𝑐 + 𝑐𝑐𝑠 𝜃𝑙1 𝑠𝑖𝑛 𝑐 𝑐𝑐𝑠𝑐� Figure 6. Manipulator joint trajectories 0 10 20 30 -100 0 100 200 300 400 l ( o ) azimuth elevation 0 10 20 30 -10 0 10 20 30 l( o /s ) 0 10 20 30 Time (s) -3 -1.5 0 1.5 3 l ( o /s 2 ) M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 120 ANSYS Aqwa subject to the three ocean depths. As part of the hydrodynamic response study, an exhaustive time domain response analysis examines the various effects of irregular wave loads on the dynamic responses of the ship [32]. Figure 7 shows three time series of ocean wave height and its corresponding six-DoF ship motion. It is clearly seen that the wave height is inversely proportional to the ocean depth following equation (20) and equation (22). Shallow water produces higher ocean wave height so that the amplitudes of the surge, sway, and yaw motions become higher than the other motions. On the contrary, the dynamic pressure of the ocean wave is directly proportional to the ocean depth following equation (23), implying that lifting motions such as heave, roll, and pitch have higher amplitudes in deep water. Those results are then fed into dynamic terms in Table 4 and Table 5, along with predefined joint trajectories. Torque comparisons between an undisturbed manipulator (without ship motions) and a disturbed manipulator (with ship motions) in shallow water are then investigated and displayed in Figure 8 and Figure 9, respectively. Those figures present the distributions of joint torque for each dynamic term of azimuth and elevation links. As can be observed, the inertia term is the most dominant torque to the manipulator for the azimuth link, while the gravity term is found to be dominant in the elevation link. This is to be expected since gravity works on the axis of rotation of the elevation link. Further, a comparison between undisturbed and disturbed manipulators under variations of ocean depth is revealed in Figure 10. It is apparent that the ship motions greatly affect the values of manipulator joint torque. The values fluctuate around the value of the undisturbed manipulator for all ocean depths, become unstable and increase to certain maximum values in order to maintain the position of the end- effector. It is found that shallow water produces the highest torque value in azimuth angle, where the increment is around 8.271 N.m or 285.69 % from the undisturbed manipulator. Intermediate water produces the highest torque in elevation angle, where the increment is around 53.321 N.m or 6.63 %. The performance of manipulator joints in terms of angular speed and torque is then compiled in Table 7 to support the results in Figure 10. Table 5. Dynamic terms for elevation angle Term Torque Inertia �𝐼𝑧𝑧2 + 𝑚2𝑟𝑥2 2��̈�𝑙2 +�−𝑚2𝑟𝑥2 𝑐𝑐𝑠𝜃𝑙1 𝑠𝑖𝑛 𝜃𝑙2��̈�𝑆 +�−𝑚2𝑟𝑥2 𝑠𝑖𝑛𝜃𝑙1 𝑠𝑖𝑛 𝜃𝑙2��̈�𝑆 +�𝑚2𝑟𝑥2 𝑐𝑐𝑠 𝜃𝑙2��̈�𝑆 +��𝐼𝑧𝑧2 + 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 𝜃𝑙1��̈� +�−�𝐼𝑧𝑧2 + 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠𝜃𝑙1��̈� Coriolis ��−�𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 2 𝜃𝑙2 + 𝐼𝑧𝑧2 + 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 𝜃𝑙1��̇�𝑙1�̇� +��−�𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 2 𝜃𝑙2 + 𝐼𝑧𝑧2 + 𝑚2𝑟 2�𝑠𝑖𝑛𝜃𝑙1��̇�𝑙1�̇� +�−�𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 2 𝜃𝑙2��̇�𝑙1�̇� +�−𝑚2𝑟𝑥2 𝑐𝑐𝑠𝜃𝑙2��̇�𝑆�̇� +�−𝑚2𝑟𝑥2 𝑠𝑖𝑛𝜃𝑙1 𝑠𝑖𝑛 𝜃𝑙2��̇�𝑆�̇� +�𝑚2𝑟𝑥2 𝑐𝑐𝑠 𝜃𝑙2��̇�𝑆�̇� +�𝑚2𝑟𝑥2 𝑐𝑐𝑠 𝜃𝑙1 𝑠𝑖𝑛 𝜃𝑙2��̇�𝑆�̇� +�𝑚2𝑟𝑥2 𝑠𝑖𝑛 𝜃𝑙1 𝑠𝑖𝑛𝜃𝑙2��̇�𝑆�̇� +�−𝑚2𝑟𝑥2 𝑐𝑐𝑠𝜃𝑙1 𝑠𝑖𝑛 𝜃𝑙2��̇�𝑆�̇� + � 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 2 𝜃𝑙1 𝑠𝑖𝑛 2 𝜃𝑙2��̇��̇� +�−�𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠 𝜃𝑙1 𝑐𝑐𝑠 2 𝜃𝑙2��̇��̇� +�−�𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 𝜃𝑙1 𝑐𝑐𝑠 2 𝜃𝑙2��̇��̇� Centripetal �− 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 2 𝜃𝑙2��̇�𝑙1 2 + � 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑐𝑐𝑠2 𝜃𝑙1 𝑠𝑖𝑛 2 𝜃𝑙2��̇� 2 + � 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛2 𝜃𝑙1 𝑠𝑖𝑛 2 𝜃𝑙2��̇� 2 + �− 1 2 �𝐼𝑥𝑥2 − 𝐼𝑦𝑦2 − 𝑚2𝑟𝑥2 2�𝑠𝑖𝑛 2 𝜃𝑙2��̇� 2 Gravity 𝑚2𝑔𝑟𝑥2�𝑠𝑖𝑛𝜃𝑙2 �𝑐𝑐𝑠𝜃𝑙1 𝑠𝑖𝑛 𝑐 − 𝑠𝑖𝑛 𝜃𝑙1 𝑠𝑖𝑛 𝑐 𝑐𝑐𝑠𝑐� + 𝑐𝑐𝑠 𝜃𝑙2 𝑐𝑐𝑠𝑐 𝑐𝑐𝑠 𝑐� Table 6. Parameters of manipulator Parameter Value Mass (kg) m1 = 150; m2 = 128 Coordinates of CoG (m) rx1 = 0.00; ry1 = 0.19; rz1 = 0.00; rx2 = 0.64; ry2 = 0.45 rz2 = 0.40 Inertia moment at CoG (kg.m2) Ixx1 = 4.25; Iyy1 = 5.45; Izz1 = 5.98; Ixy1 = 0.043; Iyz1 = 0.553; Ixz1 = 0.012; Ixx2 = 0.108; Iyy2 = 14.745; Izz2 = 14.74; Ixy2 = 0.083; Iyz2 = 0.002; Ixz2 = 0.019 Gravity (m/s2) gx = 0; gy = 0; gz = 9.81 M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 121 This is to be expected since shallow water produces the highest amplitude in yaw motion, which is the variable of the inertia term in azimuth angle. The maximum torque of elevation angle is produced in intermediate water depth because the highest amplitude in roll motion is achieved in Figure 7. Ship motions subject to ocean depth variation (a) (b) Figure 8. Azimuth torque distribution: (a) undisturbed; (b) in shallow water 0 10 20 30 Time (s) -3 -2 -1 0 1 2 3 4 l 1 ( N .m ) inertia coriolis centripetal gravity total 0 10 20 30 Time (s) -15 -10 -5 0 5 10 l 1 ( N .m ) M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 122 intermediate water depth in the first 7 s of response since gravity term, as the main contributor of the elevation angle, contains roll motion. The results show that maximum torque between applied ocean depth variations has small differences. (a) (b) Figure 9. Elevation torque distribution: (a) undisturbed; (b) in shallow water (a) (b) Figure 10. Dynamic joint torques: (a) azimuth; (b) elevation 0 10 20 30 Time (s) -200 0 200 400 600 800 1000 l2 ( N .m ) inertia coriolis centripetal gravity total 0 10 20 30 Time (s) -200 0 200 400 600 800 1000 l2 ( N .m ) 0 5 10 15 20 25 30 Time (s) -15 -10 -5 0 5 10 l1 ( N .m ) shallow middle deep Undisturbed 0 5 10 15 20 25 30 Time (s) 300 400 500 600 700 800 900 l2 ( N .m ) M.L. Ramadiansyah et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 113-124 123 IV. Conclusion A ship-mounted two-DoF manipulator dynamics under the variations of ocean depth have been investigated in this paper. The results are obtained by combining the mathematical model of the manipulator system with the numerical simulation of ship motions. Finding results show that randomness of ship motions appears in joint torque in terms of oscillations, resulting in higher maximum torque values than the manipulator without ship motions. Shallow water produces maximum joint torque to the azimuth angle with an increment of 8.271 N.m (285.69 %) from the undisturbed manipulator. Meanwhile, intermediate water produces a maximum joint torque value to the elevation angle with an increment of 53.321 N.m (6.63 %). However, the difference between water depth variations is relatively small. Current results can be taken as a baseline for sizing the electrical motor of the manipulator system and the development of a robust control system. Experimental work is recommended as future work to validate simulation results. Acknowledgements The authors are grateful to the National Research and Innovation Agency (BRIN), especially Research Center for Smart Mechatronics for providing the research facility and also to the Ministry of Finance of the Republic of Indonesia for financial support through the LPDP scheme with the project no. PRJ- 92/LPDP/2020. Declarations Author contribution M.L. 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Materials and Methods Forward kinematics B. Inverse dynamics C. Ship Motions III. Results and Discussions Conclusion Acknowledgements Declarations Author contribution Funding statement Competing interest Additional information References