An Indirect Foot-End Touchdown Detection Method for the Underwater Hexapod Robot

Abstract

The underwater hexapod robot has advantages such as lower energy consumption and reduced environmental interference compared to ROVs and AUVs. The foot-end contact detection with the seabed is the key technology for adapting to complex terrains. This paper focuses on the ‘Dragon Crab’ underwater hexapod robot developed by Shanghai Jiao Tong University and proposes an indirect detection method that does not require foot-end contact sensors. By establishing the kinematic and dynamic models of the robot’s legs, combined with multi-order polynomial trajectory planning to reduce non-contact force interference, the foot-contact determination condition is defined. Through simulation experiments and force analysis of the legs, the contact detection parameters are estimated. Then, single-leg contact tests are conducted to obtain joint motor torque variation curves and foot-end height variation curves through the kinematic model, verifying the proposed contact detection conditions and parameters. Finally, the method is applied to underwater obstacle-crossing experiments of the underwater hexapod robot using triangular and wave gait patterns. Experimental results show that the method can accurately identify the foot-end contact state and has high applicability in complex underwater terrains.

1. Introduction

With the human exploration of the marine environment, the development of underwater devices has become a hot topic in recent years. These mainly include ROVs (Remotely Operated Vehicles), AUVs (Autonomous Underwater Vehicles), and bio-inspired underwater vehicles [1,2,3,4]. ROVs are utilized for underwater target detection tasks [5]. AUVs are used for complex marine environment missions [6]. The manta ray-inspired underwater vehicles are utilized for an energy-saving strategy through bio-inspired intermittent locomotion [7]. The modular bio-inspired autonomous underwater vehicles are used for close subsea inspection [8]. The manta ray species is used for the bio-inspired underwater vehicles to achieve a flexible propulsor [9]. These devices use propellers or thruster-based motion structures to remain suspended in water and carry out large-scale patrol tasks. However, they disturb the environment during operation, and their stability is poor in high current conditions. When performing fixed-point operations, they need to maintain posture stability, which leads to high energy consumption. In contrast, crawling robots make contact with the seabed through their feet, providing better stability [10,11,12,13]. The underwater crawling robots include underwater crawler robot, soft underwater robot, underwater hexapod robot, etc. The underwater crawler robot uses tracks instead of traditional wheels or legs, which allows it to move steadily over rough underwater terrain [14,15,16]. The static stability of underwater hexapod robots is beneficial for stable locomotion, such as navigation across complex environments [17,18,19]. When performing fixed-point tasks, crawling robots move slowly and do not produce significant noise, which helps maintain environmental stability [20,21,22].

However, when encountering special seabed terrains, crawling robots need to assess the surrounding environment to ensure that their feet land accurately and stably, ensuring overall robot stability [23], since the foot-end contact detection with the seabed is the key technology for adapting to complex terrains for the underwater hexapod robots. Common methods for foot-end contact detection include using external sensors or indirect judgment through onboard sensors.

Using external sensors is a direct and reliable method, primarily relying on various force sensors. Land-based force sensors have rapidly developed in recent years, with sensors made from elastic materials that can be flexibly installed on various curved surfaces while maintaining high accuracy [24,25,26]. These sensors have a wide range of applications and can also be integrated with other body sensors for comprehensive ground contact detection. In recent years, there has been significant research on underwater force sensors [27]. The design of tactile sensors is proposed for underwater exploration tasks. The tactile sensor must withstand both pressure increase and water intrusion when operating underwater [28]. Subad designed and manufactured a simple, low-cost multi-directional force-sensing system based on flexible sensors, where single-point contact was identified on a 3 × 3 grid in two dimensions [29]. The system has an IP67 waterproof rating but can only measure forces below 10 N, making it suitable for small devices [30]. Zhang introduced a capsule-like structure using differential pressure technology to eliminate hydrostatic pressure for underwater force sensing, completing object grasping with a mechanical finger at a depth of 6 m. Wang presents a sensing method for the tire stress field of wheel-legged robots [31]. Cao uses a ground reaction force sensor for a hexapod robot on rough terrain [32]. Jun uses external sensors for the foot-contact detection of multi-legged underwater robots in a high tidal current environment [33]. However, external sensors for the foot-contact detection suffer from complex structures, high costs, and low reliability for underwater robots.

Compared to external sensors, body sensors indirectly estimate foot-end force by using data from body-mounted sensors such as motor encoders, torque sensors, and IMUs for kinematic and dynamic analysis. These sensors allow for the estimation of foot-end force and indirect acquisition of the robot’s ground contact information. Underwater hexapod robots face more challenging environmental factors (pressure, corrosion, noise, geology, etc.) compared to land-based robots in foot-contact detection [34,35]. The research on the indirect foot-end touchdown detection method for the underwater hexapod robot includes the control system design for the complex underwater terrains [36] and the conceptual overview of the underwater hexapod robot [37]. The indirect methods for the foot-contact detection without external sensors are few for underwater hexapod robots.

This paper proposes an indirect foot-end touchdown detection method using the output torque at each joint of the mechanical leg for underwater hexapod robots. The parameters of the underwater hexapod robots ‘Dragon Crab’ are shown in Section 2. A kinematic and dynamic model of the underwater hexapod robot is established in Section 3. The gait planning to reduce interference from non-contact force interference is proposed in Section 4. The foot-end touchdown determination conditions and detection parameters are obtained by simulation experiments and force analysis of the legs in Section 5. The experiments of the foot-end contact detection of the ‘Dragon Crab’ are shown in Section 6.

2. The Parameters of the Underwater Hexapod Robots ‘Dragon Crab’

The underwater hexapod robot ‘Dragon Crab’ developed by the Underwater Engineering Research Institute of Shanghai Jiao Tong University is used as the research object. The top view of the ‘Dragon Crab’ is shown in Figure 1a; the front view of the ‘Dragon Crab’ is shown in Figure 1b. The design parameters are summarized in

Table 1. The design parameters of the ‘Dragon Crab’.

Name	Value
Standard size	1.7 × 2.3 × 0.7 m3
Retracted size	1.4 × 1.7 × 0.7 m3
Total weight in air	335 kg
Total weight in water	25 kg
Depth level	3000 m
Host power	4.5 kw
Single-leg body weight	30.7 kg
Single-leg support force	10 kgf
Single-leg rated power	1.5 kw
Maximum pitch angle	±15°
Maximum speed	0.5 m/s
Maximum climbing angle	37.5°
Maximum obstacle clearance	0.5 m

. The ‘Dragon Crab’ features a sealed body design, with the electronic control and communication systems installed inside the cabin. The chassis of the ‘Dragon Crab’ has a symmetric, irregular octagonal shape, with each mechanical leg’s hip joint connected to the chassis. Buoyancy materials are installed in the thigh segments, and the six legs are symmetrically distributed on both sides. Each mechanical leg has three degrees of freedom, and the tibia adopts a curved design. The foot-end is wrapped with rubber material. The joint movement units use harmonic drive motors paired with high-speed reduction gears.

The equipment parameters of the ‘Dragon Crab’ are shown in Figure 2. The length of the link of Joint 1 is 0.2 m; the length of the link of Joint 2 is 0.5 m; and the length of the link of Joint 3 is 0.6 m, which is shown in Figure 2a. The installation angles of the robotic arms are 30°, 90°, 150°, −30°, −90°, −150°, which is shown in Figure 2b.

3. The Establishment of the Mechanism Model of the Underwater Hexapod Robot

The mechanism model of the underwater hexapod robot includes the kinematic modeling of the underwater hexapod robots and the dynamic modeling of the underwater hexapod robots. The reference coordinate systems for the underwater hexapod robot are established by the Modified Denavit–Hartenberg coordinate system, which is shown below. The Modified Denavit–Hartenberg method has the advantages of better describing the kinematic characteristics of robotic arms and avoiding the occurrence of ambiguous points [38].

The reference coordinate systems for the underwater hexapod robot’s body and its joints are defined as shown in Figure 3. The body coordinate system of the ground is denoted as ${\Sigma }_G(X_G{Y}_G{Z}_G)$. The body coordinate system of the underwater hexapod robot is denoted as ${\Sigma }_B(X_B{Y}_B{Z}_B)$. The leg (LEG1 to LEG6) is numbered by $j=1,2,3,4,5,6$. The length of the link in each leg is represented by $l_i$, $i=1,2,3$. The joint angles are represented by ${\theta }_i$, $i=1,2,3$. The fixed coordinate system of each leg is ${\Sigma }_{0j}(X_{0j}{Y}_{0j}{Z}_{0j})$. The coordinate system at the root joint is ${\Sigma }_{1j}(X_{1j}{Y}_{1j}{Z}_{1j})$, the coordinate system at the hip joint is ${\Sigma }_{2j}(X_{2j}{Y}_{2j}{Z}_{2j})$, the coordinate system at the knee joint is ${\Sigma }_{3j}(X_{3j}{Y}_{3j}{Z}_{3j})$, and the coordinate system at the foot end is ${\Sigma }_{4j}(X_{4j}{Y}_{4j}{Z}_{4j})$. The origin points of different coordinate systems are expressed by $O_G$, $O_B$ and $O_{ij}$.

3.1. The Establishment of the Kinematic Modeling of the Underwater Hexapod Robot

The kinematic modeling of the underwater hexapod robots includes forward kinematics analysis and the inverse kinematics analysis.

3.1.1. The Forward Kinematics Analysis of the Underwater Hexapod Robots

The forward kinematics analysis of a single leg solves for the position of the foot-end in the coordinate system ${\Sigma }_0$, given the joint angles of each joint on the leg.

The transformation relationship between the joint angles $\theta$ and the position $\mathit{P}$ of the foot-end is established to facilitate the foot-end position control. Based on the modified Denavit–Hartenberg method, which is used to describe the relationship between the joints and links of the underwater hexapod robot, the parameters of the method include $a_i,{\alpha }_i,d_i,{\theta }_i$. $a_i$ is the length of the link $i$($l_i$); ${\alpha }_i$ is the link twist angle; $d_i$ is the link offset; and ${\theta }_i$ is the joint angle (${\theta }_i$). $a_i,{\alpha }_i,d_i$ are the constants; ${\theta }_i$ is the variable.

The transformation matrix ${}_i{}^{i-1}\mathit{T}$ between the coordinate system ${\Sigma }_i$ and the coordinate system ${\Sigma }_{i-1}$ is expressed in Equation (1); ${}_i{}^{i-1}\mathit{R}$ is the rotation matrix of the transformation matrix. ${}_i{}^{i-1}\mathit{T}$ is expressed in Equation (2); ${}_i{}^{i-1}\mathit{P}$ is the translation vector of the transformation matrix. ${}_i{}^{i-1}\mathit{T}$ is expressed in Equation (3).

$${}_i{}^{i-1}\mathit{T}=\left[\begin{array}{cc}{}_i{}^{i-1}\mathit{R}& {}_i{}^{i-1}\mathit{P}\\ 0& 1\end{array}\right]$$

(1)

$${}_i{}^{i-1}\mathit{R}=\left[\begin{array}{ccc}c{\theta }_i& -s{\theta }_i& 0\\ s{\theta }_{i}c{\alpha }_{(i-1)}& c{\theta }_{i}c{\alpha }_{(i-1)}& -s{\alpha }_{(i-1)}\\ s{\theta }_{i}s{\alpha }_{(i-1)}& c{\theta }_{i}s{\alpha }_{(i-1)}& c{\alpha }_{(i-1)}\end{array}\right]$$

(2)

$${}_i{}^{i-1}\mathit{P}={\left[\begin{array}{ccc}{a}_{(i-1)}& -s{\alpha }_{(i-1)}{d}_i& c{\alpha }_{(i-1)}{d}_i\end{array}\right]}^{\mathrm{T}}$$

(3)

$s$ is represented as $\sin$; $c$ is represented as $\cos$; the number of the subscript $i$, $i=1,2,3,4$.

In the underwater hexapod robots, the transformation matrix ${}_0{}^4\mathit{T}$ between the fixed coordinate system ${\Sigma }_0$ and the foot-end coordinate system ${\Sigma }_4$ is expressed in Equation (4).

$${}_4{}^0\mathit{T}={}_1{}^0\mathit{T}{}_2{}^1\mathit{T}{}_3{}^2\mathit{T}{}_4{}^3\mathit{T}$$

(4)

The position ${}_4{}^0\mathit{P}$ of the foot-end in the fixed coordinate system ${\Sigma }_0$ is expressed in Equation (5).

$${}_4{}^0\mathit{P}=\left[\begin{array}{c}{}_4{}^{0}P_x\\ {}_4{}^{0}P_y\\ {}_4{}^{0}P_z\end{array}\right]=\left[\begin{array}{c}c{\theta }_1(l_1+l_{2}c{\theta }_2+l_{3}c({\theta }_2+{\theta }_3))\\ s{\theta }_1(l_1+l_{2}c{\theta }_2+l_{3}c({\theta }_2+{\theta }_3))\\ l_{2}s{\theta }_2+l_{3}s({\theta }_2+{\theta }_3)\end{array}\right]$$

(5)

3.1.2. The Inverse Kinematics Analysis of the Underwater Hexapod Robots

The inverse kinematics solution is to obtain the joint angles ${\theta }_i$ through the position ${}_4{}^0\mathit{P}$ of the foot-end.

The inverse kinematics solution is related to the joint range of motion. Thus, the joint range of motion of the underwater hexapod robots is shown below:

${\theta }_3\le 0$, which means that the link $l_3$ is facing only the seabed.

${\theta }_2>0$, which means the low stance of the underwater hexapod robots.

${\theta }_2\le 0$, which means the high stance of the underwater hexapod robots.

The low stance and high stance of the underwater hexapod robots are shown in Figure 4.

According to Equation (5), ${}_4{}^{0}P_x,{}_4{}^{0}P_y,{}_4{}^{0}P_{\mathrm{z}}$ is represented as $P_x,P_y,P_{\mathrm{z}}$, and the inverse kinematics of the underwater hexapod robots is determined in Equations (6)–(8).

$${\theta }_1=\arctan 2\left(P_y/P_x\right)$$

(6)

$${\theta }_2=\arctan 2{\displaystyle \frac{P_z\left(l_3\mathrm{c}{\theta }_3+l_2\right)-b_1\left(-l_3\mathrm{s}{\theta }_3\right)}{P_z(-l_3\mathrm{s}{\theta }_3)+b_1\left(l_3\mathrm{c}{\theta }_3+l_2\right)}}$$

(7)

$${\theta }_3=\pi -\arccos \left(\left(l_2^2+l_3^2-{\left|O_2{O}_4\right|}^2-P_z^2\right)/2l_2{l}_3\right)$$

(8)

In Equation (7), $b_1=P_x\mathrm{c}{\theta }_1+P_y\mathrm{s}{\theta }_1-l_1$; In Equation (8), ${\left|O_2{O}_4\right|}^2={\left(\sqrt{P_x^2+P_y^2}-l_1\right)}^2+P_z^2$.

3.2. The Establishment of the Dynamic Modeling of the Underwater Hexapod Robot

The underwater hexapod robot increases the support force of the foot-end compared with the underwater legs. The dynamic model of the underwater hexapod robot can be analyzed in two parts: the underwater legs and the joint torque generated by the support force of the foot-end [39].

$$\mathit{\tau }={\mathit{\tau }}_N+{\mathit{\tau }}_E$$

(9)

In Equation (9), $\mathit{\tau }$ is the dynamic model of the underwater hexapod robot; ${\mathit{\tau }}_N$ is the dynamic model of the underwater legs; and ${\mathit{\tau }}_E$ is the joint torque generated by the support force of the foot-end.

3.2.1. The Related Motion Parameters of the Dynamic Modeling

The linear velocity and angular velocity of the link are calculated before establishing the dynamic model. The motion parameters of the coordinate system ${\Sigma }_0$ can be calculated from the trajectory of the underwater hexapod robot. The motion parameters of the coordinate system ${\Sigma }_0$ include the position ${}_0{}^G\mathit{P}\in {ℝ}^{3\times 1}$ and rotation matrix ${}_0{}^G\mathit{R}\in {ℝ}^{3\times 3}$ of the coordinate system relative to the reference ground coordinate system ${\Sigma }_G$, and the linear velocity ${}_0{}^G\mathit{v}\in {ℝ}^{3\times 1}$ and angular velocity ${}_0{}^G\mathit{w}\in {ℝ}^{3\times 1}$ of the link.

The position of the coordinate system ${\Sigma }_i$ in the ground coordinate system ${\Sigma }_G$ is

$${}_i{}^G\mathit{P}={}_0{}^G\mathit{P}+{}_0{}^G\mathit{R}{}_i{}^0\mathit{P}$$

(10)

The position of the center of mass of the link $i$ in the ground coordinate system ${\Sigma }_G$ is

$${}_i{}^G\mathit{P}_c={}_{i-1}{}^G\mathit{P}+{}_i{}^G\mathit{R}{}_i{}^i\mathit{P}_c$$

(11)

In Equation (11), ${}_i{}^i\mathit{P}_c$ is the position of the center of mass of the link $i$ in the coordinate system ${\Sigma }_i$.

The angular velocity of the link $i$ in the ground coordinate system ${\Sigma }_G$ is

$${}_i{}^G\mathit{w}={}_{i-1}{}^G\mathit{w}+{}_i{}^G\mathit{R}{}_i{}^i\mathit{w}$$

(12)

In Equation (12), ${}_i{}^i\mathit{w}={\dot{\theta }}_i{\dot{\mathit{z}}}_i$, ${\dot{\mathit{z}}}_i={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T$.

The linear velocity of the link $i$ in the ground coordinate system ${\Sigma }_G$ is

$${}_i{}^G\mathit{v}={}_{i-1}{}^G\mathit{v}+{}_i{}^G\mathit{w}\times {}_i{}^G\mathit{P}_c$$

(13)

3.2.2. The Establishment of the Dynamic Model of the Underwater Legs

Based on the Lagrange equation, the energy of the mechanical legs completely submerged in water is denoted as $E=E_k+E_p$, where $E_k$ represents the kinetic energy of the mechanical leg, $E_p$ represents the potential energy of the mechanical leg, and the buoyancy is included in the potential energy term. The Lagrangian function is

$$L\left({\theta }_i,{\dot{\theta }}_i\right)=E_k\left({\theta }_i,{\dot{\theta }}_i\right)-E_p\left({\theta }_i\right)$$

(14)

In Equation (14), ${\theta }_i$ is the motion variable, and ${\dot{\theta }}_i$ is the generalized velocity of the joint $i$.

The dynamic equation of the underwater legs is

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\mathsf{\partial }L}{\mathsf{\partial }{\dot{\theta }}_i}}\right)-{\displaystyle \frac{\mathsf{\partial }L}{\mathsf{\partial }{\theta }_i}}+{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }{\dot{\theta }}_i}}={\tau }_{Ni};i=1,2,3$$

(15)

In Equation (15), $f$ is the Rayleigh dissipation function.

The kinetic energy formula for the link $i$ is

$$\begin{gathered} E_{ki}=0.5m_i{}_i{}^G\mathit{v}^T{}_i{}^G\mathit{v}+0.5{}_i{}^G\mathit{\omega }^T{}_i{}^G\mathit{I}{}_i{}^G\mathit{\omega }; \\ {}_i{}^G\mathit{I}={}_i{}^G\mathit{R}{}^i\mathit{I}_i{}_i{}^G\mathit{R}^T \end{gathered}$$

(16)

In Equation (16), $m_i$ is the mass of the link $i$; ${}_i{}^G\mathit{I}\in {ℝ}^{3\times 3}$ is the inertia matrix of the link $i$ in the ground coordinate system; ${}^i\mathit{I}_i$ is the inertia matrix of the link $i$.

The alternative expression for the kinetic energy formula of the link $i$ is

$$\begin{gathered} E_{ki}=tr\left(0.5\left|{\mathit{U}}_i\right|{\mathit{U}}_i^T\left[\begin{array}{cc}{m}_i{\mathit{E}}_{3\times 3}& 0\\ 0& {}_i{}^G\mathit{I}\end{array}\right]\right) \\ {\mathit{U}}_i=\left[\begin{array}{c}{}_i{}^G\mathit{v}\\ {}_i{}^G\mathit{\omega }\end{array}\right] \end{gathered}$$

(17)

In Equation (17), ${\mathit{E}}_{3\times 3}\in {ℝ}^{3\times 3}$ is the identity matrix; $tr(\ast )$ is the trace of the matrix.

The mass matrix of the link $i$ with the supplementary mass term is

$${\mathit{M}}_i=\left[\begin{array}{cc}\left(m_i+C_a\rho V\right){\mathit{E}}_{3\times 3}& 0\\ 0& \left(1+C_A\right){}_i{}^G\mathit{I}\end{array}\right]$$

(18)

In Equation (18), $C_a$ is the additional mass coefficient; $C_A$ is the coefficient of the additional inertia matrix; $\rho$ is the density of seawater $(\mathrm{kg}/{\mathrm{m}}^3)$; $V$ is the volume of the link $({\mathrm{m}}^3)$.

The total kinetic energy of the underwater legs is

$$E_k={\displaystyle \sum _{i=1}^3{E}_{ki}=}{\displaystyle \sum _{i=1}^{3}tr(0.5\left|{\mathit{U}}_i\right|{\mathit{U}}_i^T{\mathit{M}}_i)}$$

(19)

The gravitational direction is along the Z-axis of the ground coordinate system, and the direction of buoyancy is opposite to gravity. The action points of both forces are at the center of mass. Therefore, the potential energy of the link is

$$E_{pi}={\overline{E}}_p-\left(m_i-\rho V_i\right)g{}_i{}^{G}P_{cz}$$

(20)

In Equation (20), the constant ${\overline{E}}_p$ represents the potential energy in the reference ground coordinate system.

The total potential energy of the underwater legs is

$$E_p={\displaystyle \sum _{i=1}^3{E}_{pi}}$$

(21)

Besides the action of conservative and non-conservative forces, the object is also subjected to viscous damping. Viscous damping is a linear resistance acting on the object, and because this resistance dissipates mechanical energy, it is also called a dissipative force. During the rotation of the link, the forces dissipating mechanical energy include viscous damping and the drag force acting on an object moving in water, both of which are non-conservative forces. Therefore, the non-conservative forces acting on the underwater link are

$${\mathit{F}}_{Di}=-\left(C_{d1i}+C_{d2i}\right)\left|{}_i{}^G\mathit{v}\right|{}_i{}^G\mathit{v}$$

(22)

In Equation (22), $C_{d1i}$ is the viscous damping coefficient; $C_{d2i}$ is the drag force coefficient.

The sum of the virtual work done by the linear resistance acting on the underwater legs in any virtual displacement in the particle system is

$$\begin{gathered} \delta W_D={\displaystyle \sum _{i=1}^3\delta W_{Di}}=-{\displaystyle \sum _{i=1}^3{\mathit{F}}_{Di}·\delta q_i} \\ \delta q_i={\displaystyle \sum _{e=1}^3\frac{\delta q_i}{\delta {\theta }_e}}\delta {\theta }_e={\displaystyle \sum _{e=1}^3\frac{\delta {\dot{q}}_i}{\delta {\dot{\theta }}_e}}\delta {\theta }_e \\ \delta W_D=-{\displaystyle \sum _{e=1}^3\frac{\delta {\mathit{F}}_{Di}}{\delta {\dot{\theta }}_e}}\delta {\theta }_e \\ {\mathit{F}}_D={\displaystyle \frac{1}{3}}{\displaystyle \sum _{i=1}^3\left(C_{d1i}+C_{d2i}\right){}_i{}^G\mathit{v}^3} \end{gathered}$$

(23)

Based on the above equation, the dynamic equation of the underwater legs is derived, and the final dynamic equation is

$$\begin{gathered} {\tau }_N={\displaystyle \sum _{k=i}^{3}tr\left(\left[\frac{\mathsf{\partial }{U}_k}{\mathsf{\partial }{\dot{\theta }}_k}\frac{d(U_k^T)}{dt}{M}_k\right]\right)}+{\tau }_P+{\displaystyle \frac{\mathsf{\partial }{F}_D}{\mathsf{\partial }{\dot{\theta }}_i}} \\ {\tau }_P={\displaystyle \sum _{k=i}^3(m_k-\rho V_k)g\frac{\mathsf{\partial }{}_k{}^{G}P_{cz}}{\mathsf{\partial }{\theta }_i}} \end{gathered}$$

(24)

3.2.3. The Calculation of the Torque Generated by the Foot-End Force

Based on the force Jacobian matrix of the underwater legs, the torque generated at each joint by the foot-end force is

$${\tau }_E={}_3{}^{0}J^T{F}_E={\left[\begin{array}{ccc}{\displaystyle \frac{\mathsf{\partial }{P}_x}{\mathsf{\partial }{\theta }_1}}& {\displaystyle \frac{\mathsf{\partial }{P}_x}{\mathsf{\partial }{\theta }_2}}& {\displaystyle \frac{\mathsf{\partial }{P}_x}{\mathsf{\partial }{\theta }_3}}\\ {\displaystyle \frac{\mathsf{\partial }{P}_y}{\mathsf{\partial }{\theta }_1}}& {\displaystyle \frac{\mathsf{\partial }{P}_y}{\mathsf{\partial }{\theta }_2}}& {\displaystyle \frac{\mathsf{\partial }{P}_y}{\mathsf{\partial }{\theta }_3}}\\ {\displaystyle \frac{\mathsf{\partial }{P}_z}{\mathsf{\partial }{\theta }_1}}& {\displaystyle \frac{\mathsf{\partial }{P}_z}{\mathsf{\partial }{\theta }_2}}& {\displaystyle \frac{\mathsf{\partial }{P}_z}{\mathsf{\partial }{\theta }_3}}\end{array}\right]}^T{F}_E$$

(25)

In Equation (25), $F_E$ is the support force from the foot-end.

4. The Body Motion Planning and Control Process of the Underwater Hexapod Robot

4.1. The Design of the Foot-End Trajectory of the Swinging Leg

Common foot-end trajectories include cycloid trajectories and multi-order curves. While these can effectively enhance the robot’s running speed, they exhibit poor terrain adaptability. In this paper, a rectangular trajectory is adopted, which possesses superior obstacle-crossing capability. As shown in Figure 5, the foot-end trajectory of the swinging leg is divided into three parts: vertical leg lifting, horizontal movement, and vertical landing. At the corners of the rectangular trajectory, the acceleration undergoes abrupt changes. In this paper, a fifth-order polynomial is used to fit the trajectory, reducing the acceleration discontinuities at the corners. The constraints of the fifth-order polynomial include the starting position, ending position, starting velocity, ending velocity, starting acceleration, and ending acceleration.

4.2. The Design of the Body Motion Trajectory and the Gait Planning

The motion trajectory uses a movement pattern that alternates between moving and stationary states, and the motion trajectory is represented by a fifth-order polynomial trajectory function $Q(t,t_f,x(0),x(t_f),\dot{x}(0),\dot{x}(t_f),\ddot{x}(0),\ddot{x}(t_f))$ for fitting. $t$ is the independent variable; Constants include trajectory duration time $t_f$, starting point $x(0)$, ending point $x(t_f)$, starting velocity $\dot{x}(0)$, ending velocity $\dot{x}(t_f)$, starting acceleration $\ddot{x}(0)$, and ending acceleration $\ddot{x}(t_f)$. The fifth-order polynomial curve has smoothness and second-order differentiability, which can meet the smoothness requirements of the motion trajectory. The mathematical expression of the motion trajectory $k(t)$ is shown in Equation (26). When the body is stationary, the swinging leg swings, which reduces the impact of the body’s movement on the swinging leg.

$$k(t)=\left\{\begin{array}{c}0,t\le {\displaystyle \frac{5}{6}}{t}_f\\ Q\left(t,t_f,x\left(0\right),x\left(t_f\right),0,0,0,0\right),t>{\displaystyle \frac{5}{6}}{t}_f\end{array}\right.$$

(26)

In the motion state, the foot-ends touch the ground simultaneously, supporting the body’s movement. After the body moves a certain distance, the body stops, and the swinging leg swings. After touching the ground, one cycle ends. The gait uses common triangular and wave gaits. The schematic diagram of the body movement and gait planning is shown in Figure 6 and Figure 7. The body status and wave gait are shown in Figure 6, and the body status and triangular gait are shown in Figure 7. Wave gait employs five legs to support the body, with six legs moving in sequence. Triangular gait utilizes three legs to support the body, with three legs moving in alternating sequence.

4.3. The Design of the Motion Control Process of the Body

The overall motion control process of the body is shown in Figure 8.

The underwater hexapod robot cannot move continuously like wheeled and propulsive robots. The mechanical legs need to swing periodically to complete the movement. The path of the body is decomposed into small segments according to the motion mode and mechanical structure. The position of the foot-end is obtained through forward kinematics as the starting position for the foot-end trajectory planning. The foot-end trajectory planning method is described in Section 4.2. The trajectory needs to be mapped to the ${\Sigma }_0$ system of each leg, and inverse kinematics is then solved to convert it into joint angle information. This information is transmitted to the joint motor driver, which calculates the output torque using a dual-loop PID controller. The motor rotation is controlled by the current loop, and position control is achieved. During vertical landing of the swinging leg, the foot-end touchdown is detected using the underwater hexapod robot’s dynamics equation and the joint motor output torque. After all the foot-ends of the swinging leg touch the ground, the next small segment of the path decomposition begins.

5. Foot-End Contact Detection Method and Parameter Confirmation

5.1. Foot-End Contact Detection Determination Condition Defining

From Equation (9), it can be seen that before the foot-end touchdown, the dynamic model of the underwater hexapod robot is $\mathit{\tau }={\mathit{\tau }}_N$. ${\mathit{\tau }}_N$ can be calculated from Equation (24). The unknown parameters in Equation (24) include various hydrodynamic coefficients and viscous damping coefficients, which cannot be accurately determined. In the gait planning in Section 4, the potential energy term is a computable value, while the kinetic energy term and dissipative force term are velocity-dependent. By reducing the velocity, the fluctuation of ${\mathit{\tau }}_N$ can be minimized, maintaining a low-speed uniform motion during the foot-end landing. After the foot-end touchdown, the dynamic model of the underwater hexapod robot includes ${\mathit{\tau }}_E$, the value of which can be estimated from Equation (25).

From the above, it can be concluded that by stabilizing the value of ${\mathit{\tau }}_N$ by the gait planning in Section 4, and utilizing the sudden change characteristic of the value of ${\mathit{\tau }}_E$, the foot-end touchdown can be determined.

5.2. Foot-End Contact Detection Parameter Confirmation

The leg 5 of the ‘Dragon-Crab’ is taken as an example, the design of the foot-end swinging trajectory $P(t)$ includes $P_x(t),P_y(t),P_{\mathrm{z}}(t)$. Vertical leg lifting and horizontal movement use a fifth-order polynomial for fitting. During obstacle crossing, the body adopts a high-position motion, with the starting point as ${}_4{}^{B}P_5={[\begin{array}{ccc}-0.1& -0.6844& -0.7692\end{array}]}^T$ and the lifting height as $h=0.3$. The height from the ground to the root joint is 0.3 m. When the ‘Dragon-Crab’ descends, the foot-end speed is set to 0.02~0.03 m/s, with 0.02 m/s as an example. The foot-end swinging trajectory is shown in Figure 9. 0 to 5 s is the vertical leg lifting phase; 5 to 10 s is the horizontal movement phase; 10 to 40 s is the vertical landing phase.

The foot-end trajectory can be obtained through inverse kinematics, which provides the joint angles and joint angular velocities, as shown in Figure 10 and Figure 11.

It can be seen that during vertical landing, the angular velocity values of Joint 2 and Joint 3 reach a maximum of $0.15 \mathrm{rad}/\mathrm{s}$ and $-0.12 \mathrm{rad}/\mathrm{s}$, respectively, to reduce the fluctuation of ${\mathit{\tau }}_N$.

In the body’s contact with the ground in a stationary state, the forces on the single leg are shown in Figure 12:

The hip joint is connected to the main body of the robot, influenced by the potential energy from the main body as $F_{PB}$, and the potential energy from the mechanical leg itself as $F_{PL}$. The supporting force from the support potential energy is shown in Equation (27):

$$F_{PE}=F_{PB}+F_{PL}$$

(27)

Based on the ‘Dragon-Crab’ parameters $g=9.8 \mathrm{N}/\mathrm{kg}$, it can be concluded that the maximum value of $F_{PB}$ is the total potential energy of the main body, whose mass is $30 \mathrm{kg}$ and weight is $294.0 \mathrm{N}$. This is evenly distributed across the 6 legs, and $F_{PB}=49.0 \mathrm{N}$ can be calculated.

The potential energy of the mechanical leg (whose mass is $10 \mathrm{kg}$) is $F_{PL}=98.0 \mathrm{N}$. The joint angle at landing is ${\theta }_e={\left[\begin{array}{ccc}0& -0.3514& -1.1206\end{array}\right]}^T(\mathrm{rad})$.

When the joint angle at landing is ${\theta }_e={\left[\begin{array}{ccc}0& -0.3514& -1.1206\end{array}\right]}^T(\mathrm{rad})$, the torque generated to overcome the support force is calculated according to Equation (25) as ${\tau }_e={\left[\begin{array}{ccc}0& -63.07& -37.77\end{array}\right]}^T(\mathrm{Nm})$, which is the instantaneous change in the joint output torque at the moment of foot-contact with the ground. The Joint 2 has the maximum torques of the instantaneous change in the joint output torque. The instantaneous change in Joint 2 $\left|{\tau }_e\right|=63.07 \mathrm{Nm}$ is the determination criterion for foot-contact with the ground.

6. The Experiment of the Foot-End Contact Detection of the ‘Dragon Crab’

6.1. The Test Experimental Platform and Experimental Plan

The experimental location is the experimental pool at the Underwater Engineering Research Institute of Shanghai Jiao Tong University. Through foot-contact detection experiments, the obstacle-crossing capability of the ‘Dragon-Crab’ is observed, and real-time data from the torque sensors and encoders carried by the joint motors are analyzed to verify the correctness of the algorithm presented in this paper. The torque fluctuation characteristics at the moment of foot-contact are validated through single-leg landing detection in water, and an obstacle-crossing application experiment is designed by placing a rectangular object in the middle of the fake ground to simulate an obstacle, as shown in Figure 13. Triangular gait and wave gait are used for the passage ability test.

6.2. Single-Leg Landing Experiment

The ‘Dragon-Crab’ is suspended in the air, using a high-position movement, with the height from the foot tip to the root joint being 0.7658 m. The foot trajectory is the one after removing the horizontal displacement as described in Section 4.1. The mechanical leg is lifted 0.3 m vertically and then slowly drops at a speed of 0.02 m/s, repeating this process, and continuously increasing the ground height in an irregular manner for testing. The threshold for determining foot-end contact is when the output torque of Joint 2 exceeds 20 $\mathrm{Nm}$.

The testing process of the foot-end contact detection is shown in Figure 14. Figure 14a represents the single leg of the ‘Dragon-Crab’ in the air station. Figure 14b represents the foot-end of the single leg of the ‘Dragon-Crab’ that contacts two spacer blocks. Figure 14c represents the foot-end of the single leg of the ‘Dragon-Crab’ that contacts three spacer blocks. Figure 14d represents the foot-end of the single leg of the ‘Dragon-Crab’ that contacts four spacer blocks. The schematic diagram of the foot-end contact detection test is shown in Figure 15. The total foot-end contact detection test includes five different heights.

The foot tip trajectory in the vertical direction during testing is shown in Figure 16, and the output torque of each joint is shown in Figure 17.

The trajectory of the foot tip in the vertical direction during the test is shown in Figure 16, and the output torque of each joint is shown in Figure 17. Comparing Figure 16 and Figure 17, the joint output torque of the ‘Dragon-Crab’ leg remains relatively stable during the swing phase. Joint 2 experiences the greatest force, stabilizing between 100 and 130 $\mathrm{Nm}$. At the moment of foot-contact with the ground, there is a noticeable torque fluctuation, with a change greater than 120 $\mathrm{Nm}$. Since the single-leg ground contact test was conducted in the air, the torque variation is larger compared to the theoretical ground contact judgment value $\left|{\tau }_e\right|=63.07 \mathrm{Nm}$.

6.3. Triangular Gait Obstacle-Crossing Experiment

Based on the motion planning in Section 4.2 and the triangular gait shown in the figure, the experimental process is illustrated in Figure 18. The test process of the triangular gait of the ‘Dragon-Crab’ is shown from Figure 18a–f.

The trajectories of Joint 2 of each leg and the height from the foot tip to the root joint during testing are shown in Figure 19.

From the results, it can be seen that the ‘Dragon-Crab’ successfully crossed the obstacle in the triangular gait from Figure 18 and Figure 19. The total weight in water of the ‘Dragon-Crab’ is 25 kg in Table 1. Three legs are used to support the weight of the ‘Dragon-Crab’ in the triangular gait. The maximum output torque of Joint 2 was in a range of $(70~100) \mathrm{Nm}$, with noticeable fluctuations during the braking during the vertical leg lift and horizontal movement in the triangular gait. The motion trajectory and foot tip swing trajectory met the stability requirement of ${\mathit{\tau }}_N$. At the moment of foot-contact with the ground, Joint 2 experienced a torque fluctuation of approximately $50 \mathrm{Nm}$, allowing for accurate detection of the foot’s contact with the ground, halting the foot’s descent, and stabilizing the robot’s posture. The average $52.16 \mathrm{Nm}$ torque fluctuation in Joint 2 is used for the indirect foot-end touchdown detection of the underwater hexapod robot.

6.4. Wave Gait Obstacle-Crossing Experiment

Based on the motion planning in Section 4.2 and the wave gait shown in the figure, the experimental process is illustrated in Figure 20. The test process of the wave gait of the ‘Dragon-Crab’ is shown from Figure 20a–f.

From Figure 20, it can be seen that the ‘Dragon-Crab’ successfully crossed the obstacle. The output torque of Joint 2 of each leg and the trajectory of the height from the foot tip to the root joint during testing are shown in Figure 21.

The maximum output torque of Joint 2 ranged within $(70~120) \mathrm{Nm}$, with noticeable fluctuations for the braking during the vertical leg lift and horizontal movement in the wave gait. The average $53.24 \mathrm{Nm}$ torque fluctuation in Joint 2 is used for the indirect foot-end touchdown detection of the underwater hexapod robot. Figure 21 shows the output torque and the distance from the foot tip to the root joint. From Figure 21, it can be observed that the ‘Dragon-Crab’ used the wave gait, with the legs swinging sequentially. The foot tip detected the obstacle and stopped descending, completing the test.

7. Conclusions

Accurate determination of foot-contact with the seabed is a core technology for enabling terrain adaptation in underwater hexapod robots. However, conventional underwater bottom sensors suffer from complex structures, high costs, and low reliability. To address this, this paper proposes an indirect detection method that determines foot-contact by detecting sudden changes in joint motor torque, without relying on foot-end contact sensors. The specific research and verification process is as follows:

(1)

Considering underwater environmental characteristics, establish kinematic and dynamic models for the underwater hexapod robot. Systematically analyze the impact of non-end-effector forces—such as water flow interference and component inertial forces—on joint torque, and propose mitigation strategies. Further introduce multi-order polynomial optimization for end-effector motion trajectories to ensure stable joint torque during the swinging phase. This effectively suppresses interference from non-ground contact forces caused by motion abruptness, eliminating irrelevant disturbance factors for subsequent ground contact detection.

(2)

Single-leg ground contact experiments were conducted to obtain joint motor torque variation curves. The drive motor torque of the hip joint exhibited significant fluctuations (far exceeding the stable values during the swing phase) at the instant of foot-contact. This torque surge characteristic serves as the core basis for ground contact determination, demonstrating consistent and stable detection response, thereby preliminarily validating the method’s feasibility.

(3)

This contact detection method was applied to the self-developed ‘Dragon Crab’ underwater hexapod robot and validated through pool obstacle-crossing experiments to assess practical performance. Experimental results demonstrate that this method accurately identifies foot-contact status and adapts to various locomotion patterns, including triangular and wave-like gaits. During obstacle crossing, it recognizes contact states between the foot and obstacles or the seabed, effectively avoiding collisions or suspension risks. This ensures that the underwater robot can stably complete locomotion tasks in complex terrains.

Because the proposed method is affected by the disturbance of the water flow, the limitation of this method is that it is suitable for subsea environments with minimal changes in water flow.

In future work, the proposed method will be used in seabeds disturbed by water flow to conduct more dynamic analysis for reducing the influence of water flow on torque variations in mechanical legs.