Leg Mechanism Design and Motion Performance Analysis for an Amphibious Crab-like Robot

Abstract

Bionic-legged robots draw inspiration from animal locomotion methods and structures, demonstrating the potential to traverse irregular and unstructured environments. The ability of Portunus trituberculatus (Portunus) to run flexibly and quickly in amphibious environments inspires the design of systems and locomotion methods for amphibious robots. This research describes an amphibious crab-like robot based on Portunus and designs a parallel leg mechanism for the robot based on biological observations. The research creates the group and sequential gait commonly used in multiped robots combined with the form of the robot’s leg mechanism arrangement. This research designed the parallel leg mechanism and modeled its dynamics. Utilizing the outcomes of the dynamics modeling, we calculate the force and torque exerted on each joint of the leg mechanism during group gait and sequential gait when the robot is moving with a load. This analysis aims to assess the performance of the robot’s motion. Finally, a series of performance evaluation experiments are conducted on land and underwater, which show that the amphibious crab-like robot has good walking performance. The crab-like robot can perform forward, backward, left, and right walking well using group and sequential gaits. Simultaneously, the crab-like robot showcases faster movement in group gaits and a more substantial load capacity in sequential gaits.

1. Introduction

The interface between land and sea, commonly referred to as the amphibiotic zone, establishes a vital link between terrestrial and marine ecosystems. This zone is distinguished by optimal light conditions, promoting the proliferation of marine flora, and providing a habitat for a diverse array of fauna. Encompassing extensive areas, this environment exhibits intricate complexity and biodiversity, necessitating the execution of myriad operational tasks, including comprehensive environmental monitoring and the systematic observation of plant and animal life. In response to the multifaceted and diverse nature of this environment, the development of an amphibious robotic platform emerges as a strategic initiative. Such a platform should possess the capacity to adapt to the intricacies and diversities of the surroundings, complemented by robust locomotion capabilities. This technological advancement aims to optimize efficiency and supplant manual labor in the execution of tasks within the expansive and challenging amphibiotic zone.

Animals with legs are usually characterized by a high speed of movement, the ability to adapt to complex environments, and resistance to interference [1,2]. The flexible locomotion of humans and animals depends on close cooperation between legs [3]. Researchers consistently prefer bionic-legged robots because of their advantages in environmental adaptability and resistance to interference. After decades of research, various multi-legged bionic robots have been developed worldwide and applied to various fields. Han et al. constructed a control strategy based on the classical Raibert controller for legged locomotion under lateral impact disturbances [4]. Bian et al. proposed a novel type of wall-climbing robot to support the inspections of different contact walls (rough and smooth) [5]. Badri et al. demonstrated an avian-inspired robot leg design, BirdBot, that challenges the reliance on rapid feedback control for joint coordination and replaces active control with an intrinsic, mechanical coupling reminiscent of a self-engaging and disengaging clutch [6]. Wu et al. proposed a novel closed-chain elastic-bionic leg (CEL) with one actuated degree of freedom (DoF) based on a spring-loaded inverted pendulum (SLIP) [7]. Zhang et al. developed a two-loop framework incorporating a backstepping strategy for trajectory tracking in the presence of model errors and external disturbances [8].

Solving motion planning is one of the critical aspects for legged robots. Raiola et al. proposed a generic locomotion framework that can generate different gaits, ranging from very dynamic gaits like the trot to more static gaits like the crawl, without planning the Center of Mass (CoM) trajectory [9]. Chen et al. proposed a novel gait transition hierarchical control framework for the developed hexapod wheel-legged robot (BIT-6NAZA). The framework proposes a flexible gait planner (FGP) and gait feedback regulator (GFR) with behavioral rules [10]. Chen et al. proposed a bio-inspired reinforcement learning method that eliminates the need to model the motion of underwater robots, improves training efficiency, and implements a self-learning motion strategy that mimics the discrete swimming motion of beaver robots [11].

Gait planning is also a necessary aspect that needs to be addressed. Xu et al. proposed a new method based on metamorphosis and an equivalent mechanism to analyze the stability margin and stable workspace of multi-legged robots [12]. Chen et al. proposed a novel gait planning method for six-legged robots to optimize terrain adaptability and walking speed [13]. Xu et al. proposed a sliding Monte Carlo tree search (MCTS) method to effectively balance optimization and search operations by introducing a moving root node and controlling the sampling time [14]. Chakraborty et al. proposed a new approach for multi-legged robots to utilize leg–obstacle collisions to generate desired dynamics [15]. Bai et al. proposed a novel CPG-based gait generation for the curved-leg hexapod robot that can enable smooth gait transitions between multi-mode gaits [16]. Hao et al. addressed a stability-guaranteed and highly terrain adaptable static gait for quadruped robots to guarantee the stability of quadruped locomotion and improve their terrain adaptability [17]. Mao et al. proposed a method of gait switching so that six-legged robots can flexibly generate multiple gaits to adapt to complex terrain [18].

Crabs, common organisms in amphibious environments, are mostly used as observation subjects for amphibious robots. Crab-like robots improve the robot’s movement speed or environmental adaptability by imitating crabs’ appearance and behavioral characteristics. Using a hexapod robot, Chen et al. simulated insect-like forward walking and crab-like lateral walking morphology. Crab-like lateral walking performed better on hard surfaces and dry sand [19]. Graf et al. improved the robot’s grip on the beach by designing the crab leg joint with a tapered bent leg at the tip of the foot, based on their study of the behavioral characteristic that crabs bend their toes when resisting wave forces in the surf zone [20]. A gripping quadrilateral gait was subsequently developed and applied to a 12-degree-of-freedom crab-like robot, and the results showed that the gait could save between 15% and 34% of energy during walking [21]. The research mentioned above demonstrates that by imitating the behavior or shape of a crab and applying it to a robot, researchers can effectively improve the robot’s performance.

Most of these studies on bionic multi-legged robots focus on environmental adaptability, and most of the designed robot leg structures are series structures. There is a lack of sufficient introduction and research on the load capacity of robots. The load capacity of the robot determines the number of sensors and operating tools carried by the robot, which plays a decisive role in the robot’s environment perception and autonomous operation.

Our previous research has demonstrated the ability of robotic solutions to move well in amphibious environments. Robots can walk straight, walk laterally, turn in place on land, swim, and jump underwater. Previous studies investigated the design principles and gait experimental results of the undersea propulsion mode of a crab-like robot called “crab bounding gait” [22], introducing the design of the swimming paddle of the imitation crab-like robot with the swimming prize swimming simulation process [23] as well as swimming paddle structural parameters on hydrodynamics [24]. This research initiates by presenting the design and dynamics modeling process of the parallel leg structure for a crab-like robot, wherein the parallel leg structure outperformed the series leg structure in terms of load capacity. Subsequently, we designed some motion gait for the robot and proposed a performance evaluation method based on torque changes during the motion. Finally, we designed experiments to verify the motion and load capacity of the crab-like robot, thus establishing a theoretical foundation for the subsequent application and popularization.

The research aims to apply the physiological properties and motions of Portunus’s legs as a bionic prototype in a crab-like robot, providing a new underwater platform for exploring amphibious environments. The main work and contributions of this research are as follows:

• Simplify the design of the leg mechanism based on observing the motion state of the walking leg of the Portunus. Simplify the single walking leg into a parallel mechanism with three degrees of freedom while obtaining a similar walking ability and performing detailed dynamic modeling based on the designed parallel leg mechanism.
• Design the group and sequential gait commonly used in crab-like robots based on the parallel leg structure and robot form. According to the dynamics model of the parallel leg mechanism, evaluate the robot’s kinematic performance by analyzing the forces on each leg and the torque required at each joint on the leg when moving using the two gaits.
• Evaluate and simulate the robot’s kinematic performance for robot group gait and sequential gait. Design the experiments to validate the results of the kinematic performance evaluation.

When combined with the kinematic model, the proposed leg mechanism design scheme allows for the replication of Portunus’s walking gait. By adjusting a few control parameters, the robot can achieve stride length and gait adjustments, enabling it to adapt to various terrain environments on land and underwater. The authors designed the control strategy for an amphibious bionic multi-legged robot. However, with a few modifications, the proposed method can be applied to other multi-legged robots with leg designs.

Using Portunus as a bionic prototype, the paper applies physiological properties and locomotion methods to the field of amphibious crab-like robots. In Section 2, we construct a motion observation, image recognition, and data processing platform to observe the walking process of the crab and determine the critical parameters of the walking legs of the crab-like robot. In Section 3, we design the parallel leg mechanism based on biological data and model the statics analysis of a single parallel leg. In Section 4 we design the group gait and sequential gait for the crab-like robot structure, calculate the force on each leg, and the torque required by each joint during the motion of the two gaits according to the hydrostatic modeling, and evaluate the motion performance of the two gaits based on this. In Section 5, we show the performance evaluation experiments performed on the various gaits of the robot and present the capabilities realized by the crab-like robot as well as the performance metrics. Finally, in Section 6, the conclusions of this research and our future work are presented.

2. Motion and Structure Characteristics of Natural Portunus

2.1. The Portunus and Walking Leg

The Portunus (as shown in Figure 1a) is a shoal crab with superior underwater locomotion capability. They crawl on the sea floor with the first three pairs of walking legs and swim with the last two swimming legs. The creature possesses three propulsion methods: land walking, sea crawling, and underwater swimming, resulting in excellent adaptability to ocean waves and currents. Therefore, Portunus widely exists in shoal, surge, and other environments [22,25].

The crab walking leg (as shown in Figure 1b) comprises six sections: coxa, basis, merus, carpus, propus, and dactyl. The arthroses between sections are TC, CB, BM, MC, CP, and PD. The arthrosis, TC, links the chest of the crab and can rotate in 2 DOF and the others in a single DOF.

To design a crab-like robot walking leg, the length of the crab leg arthrosis and the real-time angle changes between these arthroses need to be measured and analyzed to judge the degree of participation of each arthrosis in walking. Because of the complexity of the leg structure and motions, it is still tricky or unnecessary for the crab-like robot walking leg to mimic all the native crab leg characteristics and movements, so only a few critical features of crabs have been studied to inspire the robotic design.

2.2. The Motion Observation Platform

Understanding the accurate motion of crabs requires information on real-time precise angle-changing data. We used the high-speed camera (Chronos 1.4, Software v0.3.0) to record the real-time position of joints. We recorded motions at 500 frames with shutter speeds of 1/1000 s. We conducted the experiments in an enclosed observation tank (1500 mm × 600 mm × 600 mm). The length of the observation tank is more than 20 times the average body width of the crabs we used for observation to ensure that the crabs can complete a complete motion cycle in the observation tank. We obtained the body size of sea crabs by measuring the body shape of 16 sea crabs (mean value of 71.32 mm). Figure 2 shows the motion observation platform.

2.3. The Observation Result and DOF Simplified for Walking Leg

The researchers observed the right front walking leg of the crab and counted the changes in the angles of its joints (as shown in Figure 3) to summarize some characteristics of its motion when it walked freely along a straight line. Video S1 of the Supplementary Material shows the biological walking observations.

• The angles of the CB, MC, and PD joints changed significantly during the walking process. The CB joint had the smaller overall angular variation range, with a maximum variation range of about 35° in the three motion cycles. The MC and PD joints had a more extensive variation range, with a maximum variation range of about 84° for the MC joint and 88° for the PD joint.
• A complete cycle of walking motion includes a support phase and oscillation phase, each accounting for about 1/2 of the total cycle; the angle of each joint changes slowly during the support phase and abruptly during the oscillation phase. In Figure 3a, the angle of the PD joint is the largest when the crab is first braced and then decreases at a slow angular velocity, and the angular velocity is negative during this process; as in Figure 3b, when the crab’s foot endpoint leaves the ground, the angular velocity suddenly accelerates and the bracing process ends; until it reaches the highest point, as in Figure 3c, when the angle reaches the minimum value and the angular velocity reaches the maximum negative value. After that, the angular velocity becomes positive and reaches the maximum positive value first, then the angular velocity decreases until it reaches the maximum angle and the minimum positive value when the crab braces again, as in Figure 3d. The overall angle-decreasing process accounts for 2/3 of the total period. The increasing process accounts for 1/3 of the total period from the graph.
• The trends of the CB and MC joint angles are similar, and the trends of the root joint and PD joint are almost opposite. The root joint reaches a maximum when the PD joint reaches a minimum, and the root joint reaches a minimum when the PD joint reaches a maximum. When the foot reaches its highest point, as shown in Figure 3b, the upward displacement at the MC joint point is the largest, and correspondingly, the root joint angle reaches its maximum. Conversely, at the moment when the foot endpoint touches the ground, as shown in Figure 3c, the downward displacement at the MC joint point is the largest, leading to the smallest root joint angle, which is also reasonable. The MC joint angle reaches the minimum value about 1/12 cycle before the PD joint angle reaches the minimum value, and the MC joint angle reaches the maximum value about 1/12 cycle before the PD joint angle reaches the minimum value.

3. Design, Analysis, and Modeling for the Walking Leg

3.1. Design of the Crab-like Robot Walking Leg

Based on our observations, we have simplified the degrees of freedom for the crab’s walking leg, retaining three specific degrees denoted as $M_1$, $M_2$, and $M_3$. As shown in Figure 3, $M_1$ rotates around the z-axis, $M_2$ rotates around the x-axis to execute the function of the CB joint, and $M_3$ rotates around the x-axis to perform the role of the MC joint. $M_2$ and $M_3$ dominate the two movements of raising and stretching the leg, respectively. For the angle change in the PD joint, the primary function is to adjust the contact angle between the leg and the ground when touching the ground. Therefore, we adopted the curvature of the PD joint during the crab’s stepping on the ground in Figure 3c when designing the leg shape.

The walking legs are designed as a parallel leg mechanism tailored to the robot’s working environment and load capacity requirements, drawing inspiration from biological observations of pike crabs, as shown in Figure 4. The second and third joints of the walking foot constitute a planar five-link mechanism connected in series with the first joint. The rotation axis of the first joint is securely connected to the body, offering advantages in terms of robust load-bearing capacity and stability compared to the walking foot in a series connection. Three servo-steering gears were integrated into a steering gear unit. One steering gear, $M_1$, was applied on joint one rotating around z in the (−90°, +90°) range. The other two steering gears, $M_2$ and $M_3$, are applied to joints 2 and 3, rotating around x to drive the five-link mechanism to move the foot tip back and forth, and up and down, in the ranges of (−30°, +30°) and (−60°, +60°), respectively, in the leg coordinates system of $O-xyz$.

The diagram of the five-link mechanism is shown in Figure 4b. The crab leg structure determines the initial length ratio of each rod of the planar five-link. The length of each rod of the planar five-link is optimized by taking the flexibility, light mobility, and the minimum joint driving torque as the optimization indexes, to obtain the link length parameters of the planar five-link mechanism as shown in

Table 1. Link lengths of walking leg.

Link	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$	${\mathit{l}}_5$	${\mathit{l}}_6$
Length,m	0.029	0.070	0.040	0.075	0.060	0.120

.

The servo-steering gear’s specification is the SAVOX SW-1210SG with a profile of 40.6 mm × 20.7 mm × 42 mm.

Table 2. Parameters of crab-like robot and SAVOX SW-1210SG.

The crab-like robot	Parameters	Value
	Mass (kg)	6.5
	Length (m)	0.66
	Width (m)	0.49
	Height (m)	0.38
The SAVOX SW-1210SG	Parameters	Value
	Item no.	SW-1210SG
	Dimension (mm)	40.6 × 20.7 × 42
	Weight (g)	71.0
	Torque (kg·cm)	32
	Speed (s/60°)	0.13
	Gear	Unique steel
	Waterproof	IP67

lists the parameters. The servo steering employs Pulse Width Modulation (PWM) control with a single control cycle lasting 20 ms and an input voltage of 7.4 V. The pulse width from 0.5 ms to 2.5 ms corresponded to 0° to 180° output angles.

3.2. Statics Modeling and Analysis

Figure 5a shows the force diagram of the walking leg. If no extra load is present, the the crab-like robot stands on the ground, and the legs share the weight $Mg$, of the crab-like robot. The force on the foot tip, F, is:

$$F=Mg/n$$

(1)

where M represents the mass of the crab-like robot (6.50 kg), g denotes the gravitational acceleration (9.8 m/s${}^2$), and n signifies the number of legs in contact with the ground. When n = 3 or 5, the load is not equally shared.

The forces applied to link BD are depicted in Figure 5c, where $F_1$ and $F_2$ represent the forces acting on joints B and C, respectively, exerted by links $l_1$ and $l_3$.

The equation of static equilibrium for link BD is:

$$\left\{\begin{array}{c}{F}_{x_1}+F_{x_2}=0\hfill \\ F_{y_1}+F_{y_2}+F=0\hfill \\ \frac{F_{y_2}}{F_{x_2}}=\tan {\theta }_1\hfill \\ F_{y_2}{l}_5\sin \left({\theta }_2-\frac{\pi }{2}\right)+F_{x_2}{l}_{5}cos\left({\theta }_2-\frac{\pi }{2}\right)+F\left(l_5+l_6\right)sin\left({\theta }_2-\frac{\pi }{2}\right)=0\hfill \end{array}\right.$$

(2)

where $F_{x_1}$ and $F_{x_2}$ are the force components of $F_1$ and $F_2$ in the x direction, $F_{y_1}$ and $F_{y_2}$ are the force components of $F_1$ and $F_2$ in the y direction. ${\theta }_1$ and ${\theta }_2$ are the phase angles of the link BD and AC.

The force components are obtained from Equation (2) as:

$$\left\{\begin{array}{c}\begin{array}{c}{F}_{x_1}=\frac{F\left(l_5+l_6\right)sin\left({\theta }_2-\frac{\pi }{2}\right)}{tan{\theta }_1{l}_{5}sin\left({\theta }_2-\frac{\pi }{2}\right)+l_{5}cos\left({\theta }_2-\frac{\pi }{2}\right)}\hfill \\ F_{y_1}=-F+\frac{F\left(l_5+l_6\right)sin\left({\theta }_2-\frac{\pi }{2}\right)tan{\theta }_1}{tan{\theta }_1{l}_{5}sin\left({\theta }_2-\frac{\pi }{2}\right)+l_{5}cos\left({\theta }_2-\frac{\pi }{2}\right)}\end{array}\hfill \\ \begin{array}{c}{F}_{x_2}=-\frac{F\left(l_5+l_6\right)sin\left({\theta }_2-\frac{\pi }{2}\right)}{tan{\theta }_1{l}_{5}sin\left({\theta }_2-\frac{\pi }{2}\right)+l_{5}cos\left({\theta }_2-\frac{\pi }{2}\right)}\\ F_{y_2}=-\frac{F\left(l_5+l_6\right)sin\left({\theta }_2-\frac{\pi }{2}\right)tan{\theta }_1}{tan{\theta }_1{l}_{5}sin\left({\theta }_2-\frac{\pi }{2}\right)+l_{5}cos\left({\theta }_2-\frac{\pi }{2}\right)}\end{array}\hfill \end{array}\right.$$

(3)

The moments, the torque $T_1$ and $T_2$ on the joints of $O_1$ and $O_2$, are expressed as:

$$\left\{\begin{array}{c}{T}_1=F_{x_1}{l}_{2}sin{\theta }_{01}-F_{y_1}{l}_{2}cos{\theta }_{01}\hfill \\ T_2={-F}_{x_2}{l}_{3}sin{\theta }_{02}-F_{y_2}{l}_{3}cos{\theta }_{02}\hfill \end{array}\right.$$

(4)

$T_1$ and $T_2$ torque are positive in a clockwise direction and negative in an anticlockwise direction. The dynamic model of the leg mechanism in the robot can be employed to compute the joint torques at each node during the motion gait. Analyzing the magnitudes and variations of the joint torques at each node enables the assessment of the robot’s motion performance in different gaits.

4. Sports Performance Evaluation

4.1. Group Gait and Sequential Gait

The structure and coordinate system of the crab robot are displayed in Figure 6. The coordinate system in the middle of the figure represents the robot’s coordinate system, defining the robot’s forward motion as X-positive, the leftward motion as Y-positive, and adhering to the right-hand rule for the Z-direction. To facilitate the robot’s control, the leg coordinate system is utilized for all walking foot gait planning. It is specified that the Z-axis direction of the foot endpoint aligns with the Z-direction of the robot’s coordinate system, the Y-positive direction of the foot endpoint faces away from the body in the robot’s Y-coordinate system, and the X-positive direction of the foot endpoint follows the right-hand rule. As per this specification, the coordinate system of the robot’s right and left leg mechanisms differs, labeled accordingly on the left and right sides of the figure.

Based on leg kinematics, we can derive the three joint rotation angles for each leg of a robot at every moment within a complete motion cycle. Storing these angle values in a gait repository, when the robot receives specific gait commands, it accesses this repository to retrieve the joint rotation angles for each moment and communicates these angles via serial port to the joint drivers. Consequently, the robot can initiate control over each leg’s movement, cyclically executing a specified gait motion cycle, thereby accomplishing the intended motion. Hence, the study of motion gaits emerges as particularly crucial. Continuously enriching the robot’s repertoire of motion gaits allows us to enhance the robot’s selectivity during movement, enabling its greater adaptability to diverse task contexts and environments. This serves as a foundational research aspect for the future autonomous operations of robots.

The most prevalent motion gaits utilized by hexapod robots are group and sequential gaits. As shown in Figure 7, the group gait involves categorizing legs 1, 3, and 5 as Group 1 and legs 2, 4, and 6 as Group 2. The robot alternates movement between Group 1 and Group 2, maintaining consistent movement patterns among the three legs within each group throughout the motion cycle. In contrast, as shown in Figure 8, in a sequential gait, legs 1, 3, 5, 2, 4, and 6 individually complete their respective movements in sequence, with no instance during the motion cycle where the movement processes of any two legs are identical.

4.2. Walking Forward/Backward

When employing a group gait for forward motion, as shown in Figure 9, the first row delineates the trajectory of the robot’s foot endpoints.The trajectory forms a semi-circle, with points 0, 1, 2, and 3 representing four positions in one gait cycle, where point 0 corresponds to the location of the robot’s foot endpoint when it is in a static state. Due to the distinct coordinate systems of the parallel legs on the left and right sides of the robot, during the advancement with a group gait, the foot endpoints of the left-walking legs 1, 2, and 3 exhibit a cyclic motion pattern of 0-3-2-1, whereas the foot endpoints of the right-walking legs 4, 5, and 6 exhibit a cyclic motion pattern of 0-1-2-3.

The second row in Figure 9 shows the states of the six legs at four instances (F1–F4) during one forward movement cycle using a group gait. The numbers adjacent to each leg indicate the corresponding foot endpoint positions at the respective instances. The four instances (F1–F4) cyclically repeat, facilitating the robot’s forward motion. Throughout this cyclic process, at each instance, the robot’s body moves forward by $l/2$ distance. In a complete motion cycle, the robot moves a total distance of $2l$ forward. Within the four instances of a motion cycle, the robot has all six legs on the ground in two instances and only three legs on the ground in the remaining two instances.

When employing a sequential gait for forward motion, as shown in Figure 10, the first row illustrates the trajectory of the robot’s foot endpoints. The trajectory forms a semi-circle, with points 0, 1, 2, 3, 4, and 5 representing six positions in one gait cycle, where point 0 corresponds to the location of the robot’s foot endpoint when it is in a static state. During forward motion with a sequential gait, the foot endpoints of the left-walking legs 1, 2, and 3 exhibit a cyclic motion pattern of 0-5-4-3-2-1, whereas the foot endpoints of the right-walking legs 4, 5, and 6 exhibit a cyclic motion pattern of 0-1-2-3-4-5.

The second row in Figure 10 illustrates the states of the six legs at six instances (F1–F6) during one forward movement cycle using a sequential gait. The numbers adjacent to each leg indicate the corresponding toe point positions at the respective instances. The six instances (F1–F6) cyclically repeat, facilitating the robot’s forward motion. Throughout this cyclic process, at each instance, the robot’s body moves forward by $l/2$ distance. In a complete motion cycle, the robot moves a total distance of $1.5l$ forward. Within the six instances of a motion cycle, at each instance, five walking legs of the robot make contact with the ground.

Similarly, the group and sequential gait of the robot moving backward can be deduced.

4.3. Side Walking

Crabs are well known to be good at side-walking due to leg arrangement. Sideways walking allows more room for movement of the robot’s joints than forward/backward walking, which can increase speed/stride length.

Similar to forward walking, one robot cycle is divided into R1–R4, four instances when using group gait for rightward walking. The robot sequentially switches between the four instances through each foot mechanism to complete rightward walking. After each state switching, the robot body moves to the right $s/2$ distance. A complete cycle of the robot’s movement is the forward movement of $2s$. In the four instances of a motion cycle, the robot landed on six feet in two instances, and in two instances, it landed on three feet. Figure 11 illustrates the foot endpoint and gait.

When employing a sequential gait to walk to the right, the robot divides a cycle into six instances: R1–R6. The robot sequentially switches between the six instances through each foot mechanism to complete rightward walking. After each state switching, the robot body moves to the right $s/4$ distance. The complete cycle of the robot’s forward instance is $1.5s$. In the six instances of a motion cycle, the robot always maintains the five-walking foot landing situation. Figure 12 illustrates the foot endpoint and gait.

Similarly, the group and sequential gait of the robot moving to the left can be deduced.

4.4. Initial Posture Analysis

Table 3. Phase angles with initial posture.

Joint Angle	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$	${\mathit{\theta }}_3$	${\mathit{\theta }}_{01}$	${\mathit{\theta }}_{02}$
Value (deg)	17	105	135	14	74

presents the phase angles of the parallel leg with the initial posture. File S1 of the Supplementary Materials shows the changes in leg phase angle during group gait and sequential gait rightward travel.

When the crab stands with three and five legs, the locations of the foot tips in the initial state are as shown in Figure 13. The distance of two front foot tips or rear foot tips is $L_1=0.4$ m, the distance of the two middle legs is $L_2=0.49$ m, the distance between the front leg and rear leg is $L_3=0.37$ m, and O is the gravity center. There are two ways to stand with three legs as well as five legs. For standing with three legs, one side has two legs and another side has one leg, as shown in Figure 13a. When standing with five legs, it is divided into two cases: one side has three legs and another side has two legs, when the lifted leg is the center leg, as shown in Figure 13b, and when the lifted leg is not the center leg, as shown in Figure 13c.

Assuming the same force on each leg of the robot. When standing with three legs, the force on the leg of a single leg side is:

$$F_{31}=\frac{Mg{L}_1}{L_1+L_2}$$

(5)

The force on the legs of two legs side is:

$$F_{32}=\frac{Mg{L}_2}{2(L_1+L_2)}$$

(6)

When standing with five legs and the raised leg is the middle leg, the force on the legs of three legs side is:

$$F_{53}=\frac{Mg{L}_2}{5L_1+L_2}$$

(7)

The force on the legs of two legs side is:

$$F_{52}=\frac{Mg(2L_1+L_2)}{2(5L_1+L_2)}$$

(8)

When standing with five legs and the raised leg is not the middle leg, the force on the legs of three legs side is:

$$F_{53}=\frac{Mg(L_1+L_2)}{7L_1+5L_2}$$

(9)

The force on the legs of two legs side is:

$$F_{52}=\frac{Mg(2L_1+L_2)}{7L_1+5L_2}$$

(10)

Table 4. Instances $T_1$ and $T_2$ with three, five, and six legs standing on ground.

Standing with n Legs	Three	Five (Mid)	Five (Not-Mid)	Six
Side	two legs/one leg	three legs/two legs	three legs/two legs	-
$T_1$ (N·m)	1.1423/1.8649	0.6667/1.0749	0.7034/1.0196	0.6916
$T_2$ (N·m)	0.5454/0.8905	0.3183/0.5132	0.3359/0.4868	0.3302

displays the instances $T_1$ and $T_2$ when three to six legs are standing.

The maximum load-carrying capacity of the crab-like robot is defined as the maximum weight/mass to be carried when any of the motors (steering gear) reaches the maximum instance. In the walking state, the worst situation is the three legs for standing. The load-carrying capacity with three legs is a good reference for dynamic motion load. Furthermore, we recommend reducing the proportion of torque used by each joint to less than 80% of the maximum torque of the joint during the three-legged exercise to enhance safety.

4.5. Motion Performance Evaluation

The robot can use group and sequential gait to realize forward/backward and right/left walking. In sequential and group gait, the robot moves its body similarly, and the main effect on the speed is observed in the time it takes to complete a motion cycle. As far as the motion is concerned, a motion cycle of the leg mechanism in group gait consists of four instances, translating a total of $2l$ or $2s$ distance. A motion cycle of the leg mechanism in the sequential gait consists of six instances, translating a distance of $1.5l$ or $1.5s$. In the case of the same frequency of robot movement, continuous gait takes longer to complete a motion cycle, and the group gait speed should be twice as fast as the continuous gait theoretically. Based on the static force analysis in the previous section, we calculate the leg forces and joint instances for each instance in the four gait banks of the robot: group gait forward, sequential gait forward, group gait rightward, and sequential gait rightward. Table S1 of the Supplementary Material shows the torque of each leg at each instance of the four gaits. File S2 of the Supplementary Material shows the procedure for calculating the torque of each joint of the leg at each instance of group gait and sequential gait.

As shown in Figure 14, the two-joint torques of the leg mechanism are calculated according to the dynamics at various moments in a complete motion cycle when the robot walks to the right using the group gait and walks forward using the forward gait, respectively. It can be seen that the total torques of the two joints of the robot are essentially the same at all instances when moving forward and to the right using the group gait. During walking to the right, there was a large disparity in the T2 joint torque of each leg. Through analysis, this situation is related to the change in the foot endpoint. When walking to the right, the foot endpoint mainly moves along the y-axis; in this instance, the phase angle of the leg model changes more. When walking forward, the foot endpoint mainly moves along the x-axis, and the phase angle of the leg model changes less.

As shown in Figure 15, the two-joint torques of the leg mechanism are calculated according to the dynamics at various moments in a complete motion cycle when the robot walks to the right using the sequential gait and walks forward using the sequential gait. Same as group gait, the total torques of the two joints of the robot are essentially the same at all instances when moving forward and to the right using the sequential gait. During walking to the right, there was a large disparity in the T2 joint torque of each leg. The main reason for the differences in joint motion stability between the two movement patterns is the variation in torque. The time required to complete a movement cycle also differs, with the most noticeable difference being movement speed. A comparison of Figure 14 and Figure 15 shows that the legs are subjected to more torque during group gait movement, which means the robot has a weaker load capacity. Sequential gait motion has less torque variation from instance to instance in the joints, which means that the robot moves more smoothly during this gait.

In summary, both gaits have advantages and disadvantages when the robot moves using group and sequential gaits. The robot moves faster when using group gait and has more load capacity and smoother motion when using sequential gait. The power is basically the same when both gaits are in motion, and we can choose the appropriate motion gait to complete the homework task according to the task requirements.

5. Experiment

5.1. Experimental Platform Construction

We build an experiment platform for robot walking experiments. Figure 16 shows a schematic diagram of the experimental platform for the crab-like robot, including the motion capture system (NOKOV MARS 2H) and the crab-like robot prototype in two parts. Motion commands are sent to the robot by radio, and the motion capture system captures the robot’s movement in real instances. Videos S2 and S3 of the Supplementary Material shows the robot’s land walking process, including the robot moving forward, walking left, walking right and turning in place.

Install Tornado 2.2 software on the host PC. The Tornado integrated development environment can debug the program, display the running results, and then communicate through the target server on the PC and the target agent on the target PC104 based on the TCP/IP protocol. The PC104 can download the compiled program, and the prototype of the crab-like robot has an installed servo drive connected to the serial port of the PC104 motherboard to receive control commands from the PC104 and generate the corresponding drive to operate the robot.

Figure 17 shows the control system and circuitry. The crab robot contains two power supplies, one for the control system and one for the walking legs, and the control system includes the attitude sensor, PC104, camera, and servo driver. The attitude sensor provides 3D attitude angle information and communicates with the PC104 main control board through RS232. The camera captures image information around the robot and communicates with the PC104 main control board through UAB. The servo driver communicates with the PC104 main control board through TTL protocol to control the servo driver to output PWM signals to realize the control of the walking foot. The robot’s autonomous motion control algorithm achieves the localization function by utilizing camera fusion and attitude sensor data information. The camera information can also actively accomplish local obstacle avoidance during robot motion. According to the specific tasks, additional sensors like GPS and LIDAR can also be incorporated into the robot’s system.

The motion data of the robot are acquired by the motion capture system in real instancesand saved by the software (Seeker 1.6.2.1). The camera can recognize reflective markers affixed to the robot body, as shown in Figure 18a. Before conducting the robot motion experiment, the motion capture system needs to be calibrated with an L-shaped calibration bar and a T-shaped calibration bar, as shown in Figure 18b.

5.2. The Walking Experiment and Result without Load

5.2.1. Walking Right/Left

Set the robot’s gait frequency and measure the robot’s average motion speed over some time. The motion capture system utilizes the real-time position it obtains to calculate the average speed of the robot throughout the entire motion cycle. We display the results of the right/left walking experiment in

Table 5. The experiment result of the right/left walking.

Gait	Parameters	Value
Group gait	Frequency (steps/s)	0.22	0.27	0.32	0.4	0.48
	Walking speed (m/s)	0.0968	0.121	0.152	0.17	0.224
Sequential gait	Frequency (steps/s)	0.22	0.27	0.32	0.4	0.48
	Walking speed (m/s)	0.0475	0.0558	0.0592	0.0763	0.0935

and Figure 19a.

5.2.2. Walking Forward/Backward

The result of the forward/backward walking experiment is shown in

Table 6. The experiment result of the forward/backward walking.

Gait	Parameters	Value
Group gait	Frequency (steps/s)	0.22	0.27	0.32	0.4	0.48
	Walking speed (m/s)	0.0439	0.0580	0.0783	0.0852	0.0935
Sequential gait	Frequency (steps/s)	0.22	0.27	0.32	0.4	0.48
	Walking speed (m/s)	0.0158	0.0200	0.0258	0.0283	0.0372

and Figure 19b.

It can be seen from the experimental results that as the skew frequency increases, the robot movement rate increases approximately uniformly. Considering the actual situation of the steering gear, the skew frequency cannot be increased indefinitely. Otherwise, the robot will be unstable, whether it is right/left walking or forward/backward walking, the average speed of the robot in group gait is higher than the sequential gait under the same skew frequency. For example, forward/backward walking skew frequency was 0.48 steps/s, the average speed of sequential gait was 0.0372 m/s, the average speed of group gait was 0.0935 m/s, and the group gait was 60.2% faster.

5.3. The Walking Experiment and Result with Load

5.3.1. Walking Right/Left with Load

Add load (1 kg∼5 kg) to the upper surface center point of the robot’s body, test the walking speed, and the result of the right/left walking experiment is shown in

Table 7. The experiment result of the right/left walking with load.

Gait	Parameters	Value
Group gait	Load (kg)	1	2	3	4	4.5	5
	Walking speed (m/s)	0.207	0.193	0.179	0.171	0.163	0.149
Sequential gait	Load (kg)	1	2	3	4	4.5	5
	Walking speed (m/s)	0.107	0.101	0.0967	0.0880	0.0839	0.0801

and Figure 20a.

5.3.2. Walking Forward/Backward with Load

The result of the forward/backward walking experiment is shown in

Table 8. The experiment result of the forward/backward walking with load.

Gait	Parameters	Value
Group gait	Load (kg)	1	2	3	4	4.5	5
	Walking speed (m/s)	0.103	0.0828	0.0817	0.0644	0.0602	0.0558
Sequential gait	Load (kg)	1	2	3	4	4.5	5
	Walking speed (m/s)	0.0401	0.0361	0.0358	0.0337	0.0322	0.0279

and Figure 20b.

It can be seen from the results of the load experiment that with the increase in the load, the limit walking speed of the robot decreases, but the degree of decrease is different. Under the same load conditions, the change in the limit walking speed of sequential gait is smaller than that of group gait. The experimental results of loaded movement show that with the increase in load from 1kg to 5kg, the right/left walking speed in group gait decreased from 0.207 m/s to 0.149 m/s, it decreased by 28%. The forward/backward walking speed decreased from 0.103 m/s to 0.0558 m/s. It decreased by 45.8%. In sequential gait, the right/left walking speed decreased from 0.107 m/s to 0.0801 m/s with increasing load. It decreased by 25.1%. The forward/backward walking speed decreased from 0.0401 m/s to 0.0279 m/s. It decreased by 30.4%. It is consistent with our gait performance assessment trend based on kinematic and kinetic modeling. However, due to the influence of the load distribution on the robot’s movement ability during the experiment, determining the magnitude of this part needs further experimental verification.

5.4. The Walking Experiment Underwater

The underwater walking ability of the crab-like robot is verified in a comprehensive experimental pool. Figure 21a shows the experiment of the underwater walking speed at a depth of 3 m. The motion capture system cannot work underwater, so we calculate the speed of the robot based on the time it takes to pass a distance of 3 m. The floor of the test pool is a hard surface. Figure 21b displays the experiment result. It can be seen that the speed underwater is significantly lower than on land due to the slippage between the robot’s walking legs, and the ground in the water is more pronounced than on land. Video S4 of the Supplementary Material shows the robot underwater walking experiment.

The walking experiments on land and underwater demonstrated that the designed crab-like robot can easily walk in an amphibious environment. The two gaits are compared and analyzed through experimental research, which provides a basis for selecting the robot’s walking mode under different working conditions. It was shown that by using sequential gait for walking, the robot moves slower, but the robot has more load capacity, less variation in joint torque during the gait cycle, and a more stable motion process. Using this gait, the robot can have more load capacity, which means that the robot can carry more sensors, and the better stability of the motion can reduce the sensor measurement error, so the sequential gait is suitable for the robot to complete more complex tasks. When walking with a group gait, the robot moves faster, but the load capacity of the robot is worse, the joints are more likely to reach peak torque, the joint torque varies more during the gait cycle, and the stability of the motion is worse, so this gait is suitable for robots to accomplish simpler and less demanding tasks. When walking underwater, because of buoyancy, although the load capacity of the robot is no longer required, the water resistance will amplify the instability of the robot’s movement process, so it is recommended that the robot use the sequential gait to move underwater.

6. Conclusions and Future Work

Firstly, this research designs a crab-like robot and a walking leg mechanism scheme according to Portunus’s motion and structural characteristics and carries out static analysis modeling for a single leg. Then, we propose group and sequential gaits based on the robot’s motion requirements. These gaits are utilized to achieve the robot’s forward/backward and left/right walking. Additionally, we analyze the support leg force and calculate the joint torque of the crab-like robot during these two motion gaits. Based on the above analysis, we evaluate the motion performance of the two gaits. Finally, we measure the amphibious bionic multipedal robot’s right/left and forward/backward walking speeds in the two gaits under loaded or unloaded conditions through experiments and obtain the experimental data. The conclusions are as follows:

• The leg mechanism scheme designed based on biological observations can well reproduce the walking gait of Portunus, and the designed crab-like robot has good mobility and load capacity.
• The robot’s mobility requirements are met by designing the configuration, group, and sequential gait. During the design process, S, L, and H parameters can be adjusted to fine-tune the robot’s gait. Different motion environments can have new gait libraries added to the robot.
• Based on the calculation of the leg force at each moment of group and sequential gait and the required torque of each joint, we evaluate the motion performance of the two gaits. The walking performance of these two gaits is compared and analyzed through experimental studies, which proved that the group gait is more advantageous in terms of robot motion speed compared to the sequential gait, and the sequential gait is more suitable for motion in scenarios where the load or the required joint torque variations are small. It is consistent with the previous motion performance analysis.

This research provides motion methods and an amphibious crab-like robot platform with practical application capabilities in environment detection, concealed reconnaissance, and aquaculture. In this research, work has been conducted on the performance evaluation of the leg design and gait planning of a crab-like robot, which theoretically requires the need to replace the kinematics and dynamics model depending on the specific scheme of the robot, a process that can be applied to other crab-like robots. The basic idea of this process is generalized. Through previous studies, we have investigated the crab-like robot’s swimming abilities, and in this research, we have examined the crab-like robot’s walking capabilities, but how to fuse the two movements has yet to be studied. During the experimental process, we also identified a limitation in our research. Specifically, although the parallel leg mechanism exhibits superior performance in terms of load capacity compared to the series leg mechanism, it falls short in the range of motion at the foot tip. This implies that, in terms of the robot’s maximum speed and obstacle-surmounting capabilities, the parallel leg mechanism is not as effective as the series leg mechanism.

In future work, we will conduct in-depth research on the motion methods of the crab-like robot, including path planning, the force laws of walking legs and sandy ground, intelligent motion control, etc.