2.1. Design of the proposed biomimetic underwater robot

Inspired by the common squid and cuttlefish, URSULA is comprised of 7 main body parts: (1) soft limbs, (2) head, (3) pen, (4) expandable mantle, (5) tail, (6) visible-light communication system, and (7) flexible pectoral fins (see Fig. 1). The overall system length (excluding the limbs) is less than 1200 mm while the limbs are 600 mm long. The largest diameter is 250 mm at the head. The total dry weight of the URSULA is around 30 kg, and the robot is designed to be slightly positively buoyant. The head, pen, and tail form the rigid, watertight components of the main body. Tendon-driven soft limbs are attached to the head. The head houses the tendon actuators as well as several navigation sensors. The pen bridges the head to the tail, functioning as an internal shell that hosts the electronics housing and batteries. The connector for the (optional) umbilical cable or tether and other waterproof connections are located at the tail. The port and starboard pectoral fins (each grouped into three sections) form the main actuation system of the robot and are attached to the two rails on the port and starboard sides that run along the length of the pen, outside of the mantle. These lateral rails are complemented with a pair of identical dorsal and ventral rails, which are used to attach several navigational sensors and the visible-light communication system. The visible-light communication system, in the form of a dorsal fin, provides a wireless communication interface required for the haptic teleoperation of the limbs. The jet propulsion system is designated as the secondary propulsion system and is located toward the front part of the pen, inside of the expandable mantle. Sea water is pumped into the double-walled soft mantle. This causes the mantle to expand inwards increasing water pressure between the mantle and pen. When sufficient water pressure is obtained, a solenoid valve is opened forcing the water out through a steerable nozzle. A 3-D render of the robot is presented in Fig. 1.

Although the initial design has only four limbs, future designs may incorporate more. A sonar is expected to be integrated, primarily to complement the existing navigation sensors in underwater SLAM-based navigation. The head will also host a 3-D stereo vision system for teleoperation and generating a virtual 3-D model of the surrounding environment. The information gathered from this camera will be used to regenerate obstacles in the master side of the teleoperation system. There is an additional camera in one of the limbs to capture views close to the tip of the limbs.

2.2. Soft-robotic limb design

The primary motivation behind URSULA was to develop a robot that can dexterously interact with the underwater environment. Therefore, the soft-robotic limbs are the most crucial subsystem of the robot. In the current design, the robot has four soft, tendon-actuated robotic limbs protruding from the head. Two of these limbs are designated as arms and are designed to provide functionality such as reaching, gripping, and pulling to perform manipulation tasks such as palpation, grasping, and retrieval. The remaining two limbs are designated as tentacles. These tentacles host an underwater camera and lighting system embedded at their distal ends. The camera system supplements the on-board 3-D vision system for visual feedback assisted teleoperation of the limbs as well as environment mapping. The motion of the light tentacle is synchronized with the camera tentacle to effectively illuminate the workspace of the arms. The tentacles are fully functional limbs and can also be used for other manipulation tasks such as anchoring the robot. The breakdown of the four limbs is grasper, palpation, camera, and light arms. The grasper arm is designated to be used in tasks such as holding or picking an intervened object. The palpation arm’s task is assigned as palping motion. A force sensor is mounted on the end of the limb. The manipulation of the palpation arm is similar to the manipulation of the grasper arm. The palping motion is achieved by the limited elongation of the soft arm. Both the camera and light arms are used to provide appropriate visual feedback to the operator.

Inspired by squid and cuttlefish tentacles, the hyper-redundant soft robot limbs are molded from Smooth-On Ecoflex 00-30 elastomer material. Ecoflex is almost neutrally buoyant in water with a mass density of 1070 kg/m $^3$ . It has Young’s modulus of 69 kPa and tensile strength of 1.38 MPa. Poisson’s ratio is assumed to be $\sim$ 0.5 (nearly incompressible) since the bulk modulus of rubber is much larger compared to the modulus of rigidity. Owing to the highly elastic, soft material used in their construction, the robotic limbs can safely interact with the fragile environment in tasks, such as palping and grasping, and handle delicate payloads and underwater objects.

The limbs are designed as thin and slender frustums with a length of 600 mm. Due to the size limitations of URSULA, the diameter at the base of the limbs (where the limbs are attached to the head) is 60 mm and reduces linearly to 20 mm at the tip. The limbs are actuated by a set of tendons channeled longitudinally through the body of the limbs (see Fig. 2b). Although 3-tendon configurations are investigated with promising initial results, a 4-tendon configuration is preferred for the preliminary design due to the simpler shape control strategies associated with the decoupling of the tendon actuation. In fact, more tendons (e.g., 6 tendons) can be used to increase the control accuracy. However, there is a tradeoff between the complexity of the actuation-tendon number and system size. In order to fit the actuation systems of the four limbs into the confined space of the head, the number of actuators and tendons per arm is limited to four, sacrificing some control accuracy. Thin channels made from soft silicone tubes are used to guide the tendons along the length of the limbs and provide a protective sheath to prevent the tearing of the soft limbs. The tendons are anchored to a rigid disk embedded into the limb at the tip and are connected to electro-mechanical actuators at the other end. Another rigid disk embedded at the base of the limb serves as the mechanical interface to 80 mm diameter flanges used to connect the limbs to the bow plate of the head. Aside from the tendon channels, another larger silicone tube is placed centrally along the spine of the arm and serves as the cable channel for the camera and lighting system. The solid model of the limbs is provided in Fig. 2a.

Figure 2. (a) The 3-D solid model of the limb system (showing only a single limb) and (b) the cross-sectional cut of a robotic limb labeled with the main parts.

3.1. Kinematic modeling

The robot arm, as presented in Fig. 3, can only have one or two segments mounted on top of each other. In Fig. 3a, the mounting location of two segments on top of each other is indicated by the green dashed line. Here, each segment is chosen to be discretized into 10 pieces, $N=10$ . Each segment is indicated by the thin dashed blue lines in Fig. 3a. Each section is composed of a spherical arm with RRP serial manipulator architecture. The kinematic sketches of RRP manipulators representing the motion of each segment are presented in Fig. 3b and c. Each segment architecture is created with zero offset for $a_{i,1}$ and $a_{i,4}$ parameters; hence, this architecture represents the spherical coordinates with variables { $\alpha _{i,1}, \alpha _{i,2}, a_{i,3}$ } for segment 1 and { $\alpha _{i,4}, \alpha _{i,5}, a_{i,6}$ } for segment 2. The offsets of robot arms are represented by $a$ parameters, and $\alpha$ parameters represent the R joint angles of the RRP manipulator representing a segment of a soft arm.

Figure 3. Kinematic sketches of the robot arms: (a) Representation of soft robot arm; (b) Soft robot arm first segment; (c) Soft robot arm second segment.

In our case, there are four soft robot arms with one and two segments: grasper and palpation arms have two segments, and camera and light arms have one segment. The $\bar{\alpha }$ and $\bar{a}$ column matrices are defined as $\bar{\alpha }$ = [ $\bar{\alpha }_g^T$ $\bar{\alpha }_p^T$ $\bar{\alpha }_c^T$ $\bar{\alpha }_l^T$ ] $^T$ , $\bar{a}$ = [ $\bar{a}_g^T$ $\bar{a}_p^T$ $\bar{a}_c^T$ $\bar{a}_l^T$ ] $^T$ , where $\bar{\alpha }_i$ = [ ${\alpha }_{i,1}$ ${\alpha }_{i,2}$ ${\alpha }_{i,3}$ ${\alpha }_{i,4}$ ${\alpha }_{i,5}$ ${\alpha }_{i,6}$ ] $^T$ , $\bar{a}_i$ = [ ${a}_{i,1}$ ${a}_{i,2}$ ${a}_{i,3}$ ${a}_{i,4}$ ${a}_{i,5}$ ${a}_{i,6}$ ] $^T$ and $i = g,p,c,l$ define the arm type as grasper, palpation, camera, and light arms of the slave robot. The ${\alpha }_{i,4},{\alpha }_{i,5},{\alpha }_{i,6}$ and ${a}_{i,4},{a}_{i,5},{a}_{i,6}$ parameters are simply not valid for soft robot arms with only one segment.

3.2. Modeling the robot arms with piecewise constant curvature approach

The mathematical model of a segment of the virtual robot arms is derived using the PCC approach, described in Fig. 4. Three parameters are required to model the soft arms: the arc length ( $\ell$ ), the angle ( $\theta$ ) through which the arc bends, and the angle ( $\phi$ ) defining the relative motion of the plane containing the arc about the z-axis.

Figure 4. PCC approach: (a) $\phi$ = 0, arc in the x-z plane; (b) $\phi \neq$ 0 (adapted from [Reference Webster and Jones18]).

The curve defined in the PCC approach is discretized into 10 pieces with equal lengths, which are represented as sections. A curve equation is selected to mimic the soft robot motion characteristics. In this work, a quarter of a circle is selected as the curve. The quarter of a circle-based curve equations, presented in Eqs. (1)–(6), are used to derive the PCC parameters for a two-segment soft robot arm. If the soft robot arm has one segment, only the Eqs. (1)–(3) are used. In Eqs. (1)–(6), PCC parameters are related to the kinematic model parameters presented in Section 3.1 as follows: $\phi =({\alpha }_{i,1},{\alpha }_{i,4})$ and $\theta = ({\alpha }_{i,2},{\alpha }_{i,5})$ and ${x}_{e,1}$ , ${z}_{e,1}$ , ${x}_{e,2}$ , and ${z}_{e,2}$ parameters define the tip-point position of the soft robot arm in the x, y, and z-axes. Here, ${r}_{1}$ and ${r}_{2}$ define the arc radii for the first and second segments based on $\theta$ and $\ell$ values for the respective segment where $\ell = ({a}_{i,3},{a}_{i,6})$ , ${r}_{1} ={a}_{i,3}/{\alpha }_{i,2}$ , and ${r}_{2} ={a}_{i,6}/{\alpha }_{i,5}$ for segment 1 and 2, respectively. The initial position of the segments is obtained when $\theta = 0^\circ$ and $\phi = 0^\circ$ . In this condition, the segments are in their initial lengths ${a}_{i,3}={a}_{i,6}$ = 300 mm for a two-segment arm and ${a}_{i,3}$ = 600 mm for a single-segment arm.

(1) \begin{equation} x_{e,1} = (\!-\!r_1 \cos (\alpha _{i,2})+r_1) \cdot \cos (\alpha _{i,1}) \end{equation}

(2) \begin{equation} y_{e,1} = (\!-\!r_1 \cos (\alpha _{i,2})+r_1) \cdot \sin (\alpha _{i,1}) \end{equation}

(3) \begin{equation} z_{e,1} = r_1 \sin (\alpha _{i,2}) \end{equation}

(4) \begin{equation} x_{e,2} = (\!-\!r_2 \cos (\alpha _{i,5})+r_2) \cdot \cos (\alpha _{i,4}) \end{equation}

(5) \begin{equation} y_{e,2} = (\!-\!r_2 \cos (\alpha _{i,5})+r_2) \cdot \sin (\alpha _{i,4}) \end{equation}

(6) \begin{equation} z_{e,2} = r_2 \sin (\alpha _{i,5}) \end{equation}

The initial position of the soft arms is defined in the +z-axis for this work with $\theta = 0^\circ$ and $\phi = 0^\circ$ . Given an arc that cuts through $\theta$ degrees extending from the origin tangent to the z-axis and bending about the y-axis, the angle between the z-axis and a line connecting the origin to the arc tip is $\theta/2$ , described in Fig. 5. To ease the implementation of the PCC approach, the bending angle of the arc, $\theta$ , is defined from the initial position of the arms as $\theta/2$ . Unit vectors along the x and z-axes on the base frame $F_0$ , shown in Fig. 5a, are used to construct a rigid-link representation of the soft robot with the same tip frame as the actual soft robot.

Figure 5. Geometrical representation of the PCC approach for a single segment. (a) Soft robot as represented with discrete sections. Here, $\vec{u}_{1}^{(0)}$ represents the unit vector along the x-axis, $\vec{u}_{3}^{(0)}$ represents the unit vector along the z-axis of the base frame $F_0$ , $\vec{u}_{1}^{(3N-1)}$ represents the unit vector along the x-axis, and $\vec{u}_{3}^{(3N-1)}$ represents the unit vector along the z-axis of the frame $F_{3N-1}$ . (b) The definition of the polar angle (the angle from the z-axis to the line connecting the tip point to the origin) (adapted from [Reference Webster and Jones18]).

To obtain ten sections in equal lengths, the curve is divided into 11 uniformly spaced points and denoted with ${x}_{s,k}$ , ${y}_{s,k}$ , and ${z}_{s,k}$ parameters. These parameters are the tip-point position of the $s\text{th}$ segment and $k\text{th}$ section of the robot arm on the x, y, and z-axis. By using the tangent of these points, the bending angle of the arc $\theta _{i,s,k}$ and angle of the plane $\phi _{i,s,k}$ on the $s\text{th}$ segment and $k\text{th}$ section, and in total in each section for each segment, is acquired, presented in Eqs. (7)–(9), which results in a constant curvature arc representation of each segment. The arc length $\ell _{i,s,k}$ is obtained from the base frame to the tip-point position of each section, as presented in Eq. (9).

(7) \begin{equation} \theta _{i,s,k} = \mathrm{atan2}\left (\left [(x_{s,k+1}-x_{s,k})^2+(y_{s,k+1}-y_{s,k})^2\right ]^{1/2},(z_{s,k+1}-z_{s,k})\right ) \end{equation}

(8) \begin{equation} \phi _{i,s,k} = \mathrm{atan2}\left ((y_{s,k+1}-y_{s,k}),(x_{s,k+1}-x_{s,k})\right ) \end{equation}

(9) \begin{equation} \ell _{i,s,k} = \left [(x_{s,k+1}-x_{s,k})^2 + (y_{s,k+1}-y_{s,k})^2 + (z_{s,k+1}-z_{s,k})^2\right ]^{1/2} \end{equation}

3.3. Kinematic modeling with Denavit-Hartenberg parameters

The discretized geometrical representation of a soft robot arm is illustrated in Fig. 6. The soft robot arms are modeled with segments having 10 sections with 3-DoF that are assembled on top of each other. As a consequence, each segment of the soft robot arm is represented with 30-DoF hyper-redundant rigid-link robot arm.

Figure 6. The architecture of the robot arm.

$\zeta _{i,s,k}$ , $s_{i,s,k}$ , $\gamma _{i,s,k}$ , and $a_{i,s,k}$ parameters denote relative rotation of links, effective link length, twist angle, and relative offset of the link, respectively. In each section, the relative rotation between the $1\text{st}$ and $2\text{nd}$ links { $\zeta _{i,s,2}, \zeta _{i,s,5},\ldots, \zeta _{i,s,3N-1}$ } is equal to the corresponding bending angle of the arc parameter { $\theta _{i,s,1}, \theta _{i,s,2},\ldots, \theta _{i,s,N}$ }. The relative rotation between the last joint of the previous section and the first joint of the following section { $\zeta _{i,s,1}, \zeta _{i,s,4},\ldots, \zeta _{i,s,3N-2}$ } is equal to the corresponding angle of the plane parameters { $\phi _{i,s,1}, \phi _{i,s,2},\ldots, \phi _{i,s,N}$ }. The PCC approach represents the soft arm as an arc on a plane that is rotated by an angle of $\phi$ . Therefore, only the first section of a segment is used to represent the rotation of the plane $\zeta _{i,s,1} = \phi _{i,s}$ and the rest maintains to be zero { $\zeta _{i,s,4}, \zeta _{i,s,7},\ldots, \zeta _{i,s,3N-2}$ } = 0. The effective link length parameter in each section { $s_{i,s,3}, s_{i,s,6},\ldots, s_{i,s,3N}$ }is equal to the corresponding arc length parameters { $\ell _{i,s,1}, \ell _{i,s,2},\ldots, \ell _{i,s,N}$ }. These parameters are presented in Table I with numerical values and their operating ranges, where $N$ is equal to the total number of sections of the corresponding arm’s one segment (i.e., $N=10$ in our study). The limitations of each parameter are denoted in a way that each soft arm’s PCC parameter $\theta$ does not exceed $\pi/2$ radians and $\phi$ does not exceed $2\pi$ radians. Since the material selected is almost incompressible, the range of the arc length is limited to $\ell _{i,s,\min}$ = 580 mm and $\ell _{i,s,\max}$ = 600 mm for a single-segment arm and $\ell _{i,s,\min}$ = 290 mm and $\ell _{i,s,\max}$ = 300 mm for a two-segment arm. The ranges of motion for all joints of the arms, given in Table I, can be updated for different materials that are used for producing the soft limbs by considering the material characteristics.

In order to manipulate and navigate the virtual soft arm model, necessary equations for forward and inverse kinematics of the robotic arms that represent soft arms are derived by using the Denavit-Hartenberg parameters. The tip-point position $\bar{p}_{i,s}$ of a single segment and transformation matrix $\hat{C}_{i,s}$ between coordinate frames $F_0$ and $F_{3N-1}$ are given in Eqs. (10) and (11).

(10) \begin{equation} \begin{split} \bar{p}_{i,s} = d_{i,s,1}\bar{u}_3+s_{i,s,3}\hat{C}_{i}^{(0,2)}\bar{u}_3+ d_{i,s,4}\hat{C}_{i}^{(0,3)}\bar{u}_3+s_{i,s,6}\hat{C}_{i}^{(0,5)}\bar{u}_3+\ldots +\\[5pt] d_{i,s,3N-2}\hat{C}_{i}^{(0,3N-3)}\bar{u}_3+ s_{i,s,3N}\hat{C}_{i}^{(0,3N-1)}\bar{u}_3 \end{split} \end{equation}

(11) \begin{equation} \begin{split} \hat{C}_{i,s} = e^{\tilde{u}_3\zeta _{i,s,1}} e^{-\tilde{u}_1\pi/2} e^{\tilde{u}_3\zeta _{i,s,2}} e^{\tilde{u}_1\pi/2} e^{\tilde{u}_3\zeta _{i,s,4}} e^{-\tilde{u}_1\pi/2} e^{\tilde{u}_3\zeta _{i,s,5}} e^{\tilde{u}_1\pi/2} \\[5pt] \ldots e^{\tilde{u}_3\zeta _{i,s,3N-1}} e^{\tilde{u}_1\pi/2} \end{split} \end{equation}

$\bar{p}_{i,s}$ and $\hat{C}_{i,s}$ equations can be further simplified as given in Eqs. (12) and (13). Here, ${\vec{u}_i}^{(k)}$ is defined as the $i^{th}$ unit vector of frame $F_k$ and ${\bar{u}_i}^{(k/j)}$ defines the resolution of ${\vec{u}_i}^{(k)}$ vector in the $j^{th}$ frame $F_j$ .

(12) \begin{equation} \begin{split} \bar{p}_{i,s} = d_{i,s,1}\bar{u}_3+s_{i,s,3}\bar{u}_3^{(2/0)}+ d_{i,s,4}\bar{u}_3^{(3/0)}+s_{i,s,6}\bar{u}_3^{(5/0)}+\ldots +\\[5pt] d_{i,s,3N-2}\bar{u}_3^{(3N-3/0)}+ s_{i,s,3N}\bar{u}_3^{(3N-1/0)} \end{split} \end{equation}

(13) \begin{equation} \hat{C}_{i,s} = e^{\tilde{u}_3\zeta _{i,s,1}} e^{\tilde{u}_2\zeta _{i,s,2}} e^{\tilde{u}_3\zeta _{i,s,4}} e^{\tilde{u}_2\zeta _{i,s,5}} \ldots e^{\tilde{u}_3\zeta _{i,s,3N-2}} e^{\tilde{u}_2\zeta _{i,s,3N-1}} \end{equation}

Table I. Denavit-Hartenberg parameters of a single segment.

Figure 7. The section architecture of a segment of the soft robotic arms: (a) the first section of the camera arm with one segment, (b) the first section of the grasper arm with two segments (all dimensions are in mm).

4.1. Verification of the kinematic representation

The accuracy of the representation is investigated through the comparison of tip-point positions and orientation calculated by using both forward kinematics formulations represented in Eqs. (10) and (11) and the quarter circle equation that is used to derive the PCC parameters. To validate that the presented modeling approach is suitable for this simulation, the following equations must satisfy each other: (1) The tip-point position calculation method given in Eq. (12) and the results of the curve equations presented in Eqs. (1)–(6), and (2) The orientation of the limb end calculation presented in Eqs. (13) and (20).

(14) \begin{equation} \bar{p}_{i,s} ={p}_{i,s,1}\bar{u}_1+{p}_{i,s,2}\bar{u}_2+{p}_{i,s,3}\bar{u}_3 \end{equation}

(15) \begin{equation}{p}_{i,s,1}^{*} = -r_1 \cos (\theta )+r_1 \end{equation}

(16) \begin{equation}{p}_{i,s,1} ={p}_{i,s,1}^{*} \cos (\phi ) \end{equation}

(17) \begin{equation}{p}_{i,s,2} ={p}_{i,s,1}^{*} \sin (\phi ) \end{equation}

(18) \begin{equation}{p}_{i,s,3} = r_1 \sin (\theta ) \end{equation}

(19) \begin{equation} \hat{C}_{i,s}^{(0,N-1)} = [\bar{u}_1^{{(N-1}/0)} \bar{u}_2^{{(N-1}/0)} \bar{u}_3^{{(N-1}/0)}] \end{equation}

(20) \begin{equation} \hat{C}_{i,s}^{(0,N-1)} = \left[\begin{array}{c@{\quad}c@{\quad}c} \cos (\theta ) \cos (\phi ) & -\sin (\phi ) & \sin (\theta ) \cos (\phi )\\[5pt] \cos (\theta ) \sin (\phi ) & \cos (\phi ) & \sin (\theta ) \sin (\phi )\\[5pt] -\sin (\theta ) & 0 & \cos (\theta ) \end{array}\right] \end{equation}

The validation process is carried out for the camera arm with one segment. Figure 9 represents the comparison of the positions of each discrete section for a single-segment arm with $\theta = 45^\circ$ and $\phi =30^\circ$ . The corresponding DH values of each section’s parameters are as follows: $\bar \zeta _{i}$ = [ $30$ $4.5$ $0$ $0$ $9$ $0$ $0$ $9$ $0$ … $0$ $9$ $0$ ] $^T$ , $\bar{s}_{i}$ = [ $0$ $0$ $60$ $0$ $0$ $60$ … $0$ $0$ $0$ ] $^T$ , where the unit of $\bar \zeta _{i}$ values are given in degrees, and $\bar{s}_{i}$ are given in millimeters. The location of blue circles is obtained from the curve equation by dividing it into 10 equal lengths. The red points represent the section positions acquired by the DH parameters.

Figure 9. Discretization of a single-segment arm.

In addition to the visual representation of the validation through Fig. 9, the tip-point position and orientation of the camera arm obtained by the forward kinematics and the corresponding PCC parameters are related to each other via Eqs. (14)–(20). By comparing the results of these equations for different $\theta$ , $\phi$ , and $\ell$ parameters, a perfect match of the tip-point position and limb end orientation acquired by the Denavit-Hartenberg parameters and the PCC approach is observed.

The illustration with different values of $\theta$ , $\phi$ , and $\ell$ is given in Fig. 10. The camera arm is manipulated with $\theta$ = 90 $^\circ$ , $\phi$ = $-$ 5 $^\circ$ , and $\ell$ = 0 mm, while the grasper arm’s segments are manipulated with $\theta$ = 30 $^\circ$ , $\phi$ = 10 $^\circ$ , and $\ell$ = 10 mm for segment 1, $\theta$ = 60 $^\circ$ , $\phi$ = 30 $^\circ$ , and $\ell$ = 10 mm for segment 2, and palpation arm is manipulated with $\theta$ = 60 $^\circ$ , $\phi$ = 20 $^\circ$ , and $\ell$ = 0 mm for segment 1, $\theta$ = $-$ 30 $^\circ$ , $\phi$ = 15 $^\circ$ , and $\ell$ = 0 mm for segment 2.

Figure 10. Manipulation of the soft arms with different parameters.