A path planner enabling second-order continuity across multiple motion modes of the generalized bicycle model

Abstract

In mobile robotics, the generalized bicycle model is notable for its agility and diverse motion capabilities. However, current path planning methods often overlook its potential for multiple motion modes, leading to impractical paths and collision risks in complex environments. To address this, our study introduces a novel path planning algorithm that utilizes the generalized bicycle model and ensures second-order continuity, crucial for smooth and safe navigation. We establish rigorous second-order continuity conditions for seamless motion mode transitions and derive criteria for each mode. Our algorithm integrates these conditions with the rapidly exploring random tree concept to generate optimal paths that accommodate the model’s mode-changing capabilities. Through tests in challenging environments, our method demonstrates improved performance, especially in narrow passages, enhancing path planning safety and efficiency for mobile robots. This advancement significantly contributes to the field of robotic path planning by offering a more dynamic and reliable approach to autonomous navigation.

Introduction and review

Mobile robots have become an integral part of modern factories, significantly enhancing efficiency and flexibility. They autonomously handle and transport objects, navigate complex environments, and perform repetitive tasks. Mobile robots are also crucial in healthcare and in hazardous environments, assisting with a range of tasks from delivering medication to conduct search and rescue. The versatile functionality of mobile robots is enabled by their underlying design. In the following sections, we will provide an overview of the various forms of robot mobility, along with an in-depth discussion of the mechanisms that facilitate their movement, as detailed in existing literature.

Existing models of mobile robots

Numerous vehicle models have been proposed, each with distinct characteristics, and these models can be represented in a generalized form. For example, Figure 1 illustrates a generalized model featuring two wheels, providing a simplified representation for studying vehicle dynamics. This model abstracts the vehicle’s behavior using two wheels—one at the front (f) and one at the rear (r)—each with its own velocity vector (V) and steering angle (θ). Based on the controllability of wheel steering and driving, models in the literature are categorized as shown in Table 1.

Figure 1.

A generalized bicycle model.

Table 1.

Vehicle research with different controllabilities.

	Front drive	Rear drive	Double drive
Front steer	Chen et al. 1	Rajamani 2	Bonci et al. 3
Rear steer	Li et al. 4	Tamba et al. 5	Bonci et al. 3
Double steer	Yuan et al. 6		Kokot et al. 7

Table 1 classifies the studies in the literature according to the controllability of the models depicted in Figure 1. The “Bicycle Model,” recognized for its front wheel steering and rear wheel driving, stands out as one of the most studied variants. This particular model is frequently represented by car-like mobile robots, as described by Rajamani. ² Alternatively, some vehicles, especially those resembling forklifts, adopt a reverse configuration with the front wheel for driving and the rear wheel for steering, as investigated by Li et al. ⁴ Moreover, Chen et al. ¹ have highlighted potential issues such as steering deadlocks that can arise when both steering and driving functions are allocated to the front wheel, or both to the rear. ⁵

Yuan et al.’s work is particularly relevant to our exploration of dual-wheel steering systems, especially within the context of tractor-trailer mobile robots (TTMRs). According to Yuan et al., ⁶ they developed control laws for managing the driving speed and steering angles of the tractor, along with the steering angles of the trailer, closely paralleling the dual-wheel steering with front wheel driving design in Table 1. Their subsequent study ⁸ advanced this framework by integrating multiple steerable trailers, thus evolving the design into a more complex structure that simulates a system with several steering wheels all governed by a single front driving wheel. Although we have not come across a model that combines dual-wheel steering with rear wheel driving, this setup could theoretically be envisioned as a TTMR operating in reverse. While there are studies like those by Morales et al. ⁹ and Widyotriatmo et al. ¹⁰ that delve into the reverse movements of TTMR, they do not extend to cover dual-wheel steering functionality. This omission marks it as a relatively uncharted domain in the research.

In the context of double-wheel driving integrated with either front or rear wheel steering, this setup effectively represents an all-wheel-driving bicycle model. Bonci et al. ³ have not only introduced but also thoroughly examined the dynamics of this particular configuration. Despite their contributions, the authors acknowledge that this field of study still holds many opportunities for further development and exploration.

Kokot et al. ⁷ addressed the less-explored model of double steering and double driving in Table 1, suggesting its analysis through the generalized bicycle model (GBM). Equipping all wheels of GBM robots with both steering and driving capabilities is theorized to significantly enhance their agility and flexibility, making them comparable to omni-directional mobile robots equipped with Mecanum wheels. However, unlike their omni-directional counterparts, GBM robots avoid the maintenance complexities and increased costs associated with sophisticated wheel designs. Importantly, the control strategies and analytical approaches developed for GBM robots can be adapted to other models by accounting for specific wheel constraints. Consequently, our research focuses on exploring GBM-based mobile robots, analyzing their models, and effectively leveraging their features in practical implementations.

Additionally, to effectively enhance the agility of mobile robots, a well-designed path is crucial. Therefore, in the next section, we will outline the path requirements for a GBM robot and review the relevant literature.

Related works of path planning for GBM robots

Path planning, which determines a route or trajectory for a robot to move from a starting position to a desired target location while avoiding obstacles, involves various strategies such as graph search techniques (Dijkstra and A*) and sampling-based methods (rapidly exploring random tree (RRT) and probabilistic roadmaps (PRM)). ¹¹ Among these existing methods, RRT’s random nature and node expansion capabilities make it particularly adept at integrating the motion modes and constraints specific to the GBM. Consequently, this research adopts the RRT algorithm as the primary method for planning with GBM-based vehicles. Moreover, the continuity of the path is of paramount importance as it seeks to improve the controllability of local trajectories ¹² and lessen the demand on the vehicle’s control mechanisms.

A smoother path effectively diminishes control oscillations and reduces the margin of tracking errors. ⁷ For example, Huh and Chang ¹³ studied the various definitions of path continuity, highlighting that G₂ continuity signifies a path with continuous curvature, implying continuous acceleration and a reduction in the burden on the dynamic system. To address this concern, the authors parameterized each set of path points using a second-order polynomial. They employed a membership function from fuzzy control theory to manage the overlap of adjacent sets of points, enabling their combination into a third-order polynomial and thus ensuring G₂ continuity. In addition, Yang et al. ¹⁴ introduced a spline-based RRT (SRRT) planning method. The authors incorporated conditions such as the maximum curvature and the maximum angle for node expansion, thereby constraining the range of expanded RRT nodes. They further smoothed the path points using a third-order Bezier curve, ensuring G₂ continuity. This approach not only diminishes the control burden on the robot but also satisfies the constraints of non-holonomic motion.

Vehicle kinematics has also been incorporated into path planning to ensure the selected global path appropriately accounts for the vehicle’s motion behavior. For instance, in the study by Ghosh et al., ¹⁵ the authors integrated the kinematic constraints of non-holonomic mobile robots into the bi-directional RRT method, ensuring that the kinematic relations between adjacent nodes are maintained during node expansion. However, when connecting two RRTs, the authors noted potential issues with kinematic continuity. To address this, they employed the parametrized trajectory generator (PTG) ¹⁶ method to reconstruct a trajectory for the connection. Despite the consideration of kinematic constraints, this method only guarantees G₁ continuity, indicating continuous tangential orientation. Additionally, Wang et al. ¹⁷ also introduced a bi-directional RRT planning method that considers kinematic constraints. While the authors incorporated kinematic constraints into the node expansion process, there is a lack of careful consideration for continuity in the candidates of discrete control inputs. Consequently, this approach can only ensure continuous curvatures within a single node period and does not guarantee overall planning time continuity, ultimately achieving only G₁ continuity. Besides, Kang et al. ¹⁸ introduced a dynamic smoothing method that considers kinematic constraints. The authors, based on the kinematic limitations, modified the connection between points by combining arcs and lines as a rewiring technique. Despite incorporating kinematic constraints, the junctions between arc and line segments cannot ensure curvature continuity but only tangential direction continuity, resulting in the satisfaction of only G₁ continuity.

The aim of integrating kinematic constraints into path planning is to derive a path that aligns with the robot’s motion capabilities. Yet, current methodologies typically achieve only G₁ continuity. Research, such as that by Kokot et al., ⁷ has shown that navigating a robot along a G₁ continuous path can induce oscillatory behaviors, which are notably reduced when following a G₂ continuous path. This finding highlights the critical need for research focused on developing paths with G₂ continuity, particularly for enhancing stability and efficiency in robot navigation within the framework of the GBM.

The main contribution of this article is the definition of the motion modes and constraints of the GBM robot based on its kinematics, along with the proposal of an RRT-based path planner that ensures G₂ continuity to enhance the robot’s agility and flexibility. In summary, we present two key contributions:

An elaborate definition of the motion modes of the GBM is provided.

A RRT-based path planner, which considers these motion modes and ensures G₂ continuity by using second-order polynomials, is developed.

The remainder of this article is organized as follows: the “GBM and motion modes” section introduces the kinematics of the GBM and explores the GBM’s motion modes and the constraints of transitioning between these modes, crucial for understanding robot maneuverability. The “Requirements for path planners” section illustrates the requirements for paths and the focused problem of this paper. The “Methodology” section introduces our proposed path planning methodology along with a continuity analysis, ensuring that robots move smoothly and efficiently. The “Case studies and result discussions” section describes the specific case environments used to test our path planning approach, highlighting practical applications and challenges. Finally, the “Conclusion and future work” section summarizes the key contributions of this study, discusses ongoing considerations in the field of robotics, and outlines directions for future research, emphasizing the real-world impact of improved robotic path planning.

GBM and motion modes

Kinematics of GBM

The kinematic model of the GBM, equipped with two steerable and drivable wheels, as articulated by Kelly ¹⁹ and shown in Figure 2, is expressed inequations (1) and (2). Here, the position vectors and velocities of the wheels are represented as $(x_i,y_i{)}_{i=f,r}$ and $(V_{ix},V_{iy}{)}_{i=f,r}$ , while $(V_x,V_y,\omega )$ denote the linear and angular velocities of the vehicle.

Forward kinematics:

$$\begin{array}{rl}{v}_c^w& =\left[\begin{array}{c}{V}_{fx}\\ V_{fy}\\ V_{rx}\\ V_{ry}\end{array}\right]=\left[\begin{array}{ccc}1& 0& -y_f\\ 0& 1& x_f\\ 1& 0& -y_r\\ 0& 1& x_r\end{array}\right]\left[\begin{array}{c}{V}_x\\ V_y\\ \omega \end{array}\right]=H_c^v{\dot{x}}_v^w\end{array}$$

(1)

Inverse kinematics:

$$\begin{array}{rl}{\dot{x}}_v^w& ={[(H_c^v{)}^T(H_c^v)]}^{-1}(H_c^v{)}^T{v}_c^w\end{array}$$

(2)

In our study, the wheels are placed at the front and back of the vehicle, respectively, and it is assumed that no offset of the wheels occurs during movement, thereby avoiding singularity in equation (2). Additionally, based on the kinematics described above, multiple motion modes and relevant constraints can be defined, which are then applied to our path planner.

Figure 2.

Position vectors of the wheels of generalized bicycle model.

Definitions of motion modes

In Kokot’s research, ⁷ it was demonstrated that adjusting the orientation of the GBM robot along the same path can significantly influence its motion behavior, potentially causing it to move closer to obstacles such as walls. Consequently, the author developed motion modes for the GBM robot based on its orientation. However, the study did not explicitly mention the conditions governing the transitions between different motion modes during movement. For example, the condition under which the “Crab” mode, which enables parallel movement, transitions to the “Tangential” mode, aligning the robot’s orientation tangent to the path, was not discussed. Additionally, definitions based solely on orientation may overlook the continuous nature of wheel controls during mode transitions.

To address these issues, we utilize these modes as a foundation to develop a global planner. Our research defines motion modes based on wheel states, thoroughly examines motion behaviors, establishes switch conditions, and ultimately clarifies the intricate relationship between motion modes and path planning. Consequently, the motion modes are defined and discussed below.

Given the robot’s characteristics of low-speed motion, our analysis is based on kinematics. Figure 3 denotes the positions of the wheels and the vehicle’s wheelbase L. The standard analysis of the bicycle model, based on triangular relations between the turning radius and the wheelbase, is shown in the followingequation:

$$\begin{array}{rl}\frac{L}{\sin ({\theta }_f-{\theta }_r)}& =\frac{R_f}{\sin (\frac{\pi }{2}+{\theta }_r)}=\frac{R_r}{\sin (\frac{\pi }{2}-{\theta }_f)}\\ \frac{V_f}{R_f}& =\frac{V_r}{R_r}=\omega \end{array}$$

(3)

Figure 3.

The triangular relations for a standard bicycle model.

In equation (3), ${\theta }_f$ and ${\theta }_r$ represent the steering angles of the front and rear wheels, respectively; $V_f$ and $V_r$ are the translational velocities of the front and rear wheels. $\omega$ is the pure rolling angular velocity of the wheels; $R_f$ and $R_r$ are the distances from the front and rear wheels to the instantaneous center of rotation (ICR). Following these relationships, we consider the variables $\{V_f,V_r,{\theta }_f,{\theta }_r\}$ as control inputs and accordingly define the motion modes for GBM robots in the following sections.

Ackermann and tangential modes

Figure 4 illustrates an Ackermann steering geometry with its kinematic relationship expressed by equation (4), where $R_f\ne 0$ and $R_r\ne 0$ according to equation (3).

$$\begin{array}{rl}& \frac{\cos ({\theta }_f)}{\cos ({\theta }_r)}=\frac{R_r}{R_f}=\frac{V_r}{V_f}\\ & V_r=V_f\cdot \frac{\cos ({\theta }_f)}{\cos ({\theta }_r)}\end{array}$$

(4)

The Ackermann steering geometry mode adjusts the vehicle’s orientation based on the position of the ICR, as depicted in Figure 4(a). A far clearer illustration is when the line connecting the ICR and the center of the vehicle is perpendicular to the vehicle (refer to Figure 4(b)), resulting in the orientation being tangent to the path during motion. This specific scenario is referred to as the “Tangential” mode, with its conditions defined by the following equation:

$$\begin{array}{rl}{V}_f(t)& =V_r(t)\\ {\theta }_f(t)& =-{\theta }_r(t)\\ t& \ge 0\end{array}$$

(5)

The Ackermann mode may have three scenarios that disobey equations (3) and (4) depending on the steering geometry, namely:

• Two wheels with parallel steering angles: ${\theta }_f={\theta }_r$ .

• Both wheels with steering angles at ${90}^{\circ }$ : ${\theta }_f={\theta }_r=\pm \frac{\pi }{2}$ .

• One wheel with a steering angle at ${90}^{\circ }$ and the other without, indicating $R_f\ne 0$ and $R_r=0$ or $R_f=0$ and $R_r\ne 0$ .

In response to the first two scenarios, we introduce the “Crab” and “Differential Drive” modes. However, the third situation results in the GBM becoming immobile, and thus, it should be actively avoided during planning and control phases.

Figure 4.

Ackermann mode. (a) General Ackermann steering geometry visualization. (b) Specific condition of Ackermann steering geometry, defined as “Tangential” mode.

Crab mode

Figure 5 depicts the crab mode of GBM robots, characterized by parallel steering angles of the wheels that deviate from the traditional Ackermann steering geometry. This unique configuration enables a special motion control mechanism where the robot propels the wheels at a uniform velocity, enabling translational movement without changing its heading angle, reminiscent of a crab’s sideway locomotion. Consequently, the detailed definition of the “Crab” mode is articulated in the following equation:

$$\begin{array}{rl}{V}_f(t)& =V_r(t)\\ {\theta }_f(t)& ={\theta }_r(t)\\ t& \ge 0\end{array}$$

(6)

Figure 5.

Crab mode.

Differential drive mode

When both steering angles of a GBM reach $\pm \frac{\pi }{2}$ , as depicted in Figure 6, the robots can behave similar to differential drive mobile robots by driving the wheels with different velocities. As a result, this configuration is called “Differential drive” mode, and is defined as the following equation:

$$\begin{array}{rl}{\theta }_f(t)& =\pm \frac{\pi }{2}\\ {\theta }_r(t)& =\pm \frac{\pi }{2}\\ t& \ge 0\end{array}$$

(7)

Figure 6.

Differential drive mode.

Switch conditions of motion modes

In this section, we define the transition relationships between the modes, as depicted in Figure 7. Central to this figure is the Ackermann steering geometry, which serves as a pivot for transitioning between the modes. The “Crab” mode is indicated where both front ( ${\theta }_f$ ) and rear ( ${\theta }_r$ ) wheel angles are aligned, facilitating lateral movement without a change in orientation. The “Differential Drive” mode is specified when both wheels are steered to angles of $\pm \pi /2$ , allowing the vehicle to rotate around its center. Additionally, the “Tangential” mode is a subset of the Ackermann steering geometry where the velocity of the front wheel ( $V_f$ ) equals the velocity of the rear wheel ( $V_r$ ), and the wheel angles are equal but with opposite signs, enabling the vehicle to execute smooth, curvilinear paths. The switch between these modes can be achieved by ensuring the continuity of wheel control states. Taking “Crab” and Ackermann steering geometry modes as an example, the boundary conditions for both modes are $V_f=V_r$ and ${\theta }_f={\theta }_r$ . Therefore, the vehicle can seamlessly switch its behavior by adjusting its wheel states to meet these conditions.

Figure 7.

Relationship of motion modes.

Summary and visualization of GBM modes

Building upon the defined motion modes, we illustrate their respective behaviors in Figure 8, with the Ackermann steering geometry mode exemplified by the intuitive “Tangential” mode.

Figure 8.

Visualization of the generalized bicycle model (GBM) motions. Purple rectangles represent poses of GBM vehicle. (a) Tangential mode motions. During each movement, the robot steers its wheel angles from $0^{\circ }$ to a target value θ, where θ ranges from $0^{\circ }$ to ${50}^{\circ }$ . (b) Crab mode motions. During each movement, the robot steers its wheel angles from $0^{\circ }$ to a target value θ, where θ ranges from $0^{\circ }$ to ${50}^{\circ }$ . (c) Differential drive mode motions. During each movement, the robot adjusts its wheel velocities from $0^{\circ }$ to a target value V, where V ranges from $0$ to $0.3$  m/s.

Figure 8 provides graphical representations of the trajectories followed by a GBM robot employing the “Tangential,” “Crab,” and “Differential Drive” modes, respectively. By varying the steering angles of the wheels, as shown in Figure 8(a) and (b), and adjusting the velocities of the wheels, as depicted in Figure 8(c), a range of motion behaviors for each mode can be observed. In the “Tangential” and “Crab” modes, small variations in the steering angles can result in significantly different paths. For instance, a GBM robot with ${\theta }_f=-{\theta }_r=\pi /4$ in the “Tangential” mode can almost make a “U-turn,” whereas a GBM robot with ${\theta }_f={\theta }_r=\pi /4$ in the “Crab” mode can only move in a straight line. This illustrates the distinction between the two modes as mentioned. Furthermore, adjusting the velocities of the wheels allows a GBM robot in “Differential Drive” mode to trace curvilinear paths similar to those in the “Tangential” mode. Therefore, employing different motion modes can effectively enhance the agility and flexibility of a robot.

Requirements for path planners

With the defined motion modes, the next step is to integrate them into a path planner while satisfying the mentioned G₂ continuity requirement. Therefore, we will discuss path continuity and the relevant requirements for the path planner.

In the work of Huh et al., ¹³ a $G_n$ continuous path signifies the equality of the nth derivative at each point. For example, a $G_0$ continuous path represents continuously connected points throughout the entire section; a G₁ continuous path refers to a smooth curve where the direction (or the first derivative) is continuous at every point along the path; and a G₂ continuous path is a curve where both the first and second derivatives are continuous. This means that not only is the direction of the tangent vector continuous (G₁ continuity), but the curvature (the rate of change of the direction) is also continuous along the path. Additionally, G₁ continuity can be interpreted as continuous velocity, while G₂ continuity indicates continuous acceleration. According to the author, G₂ paths can reduce dynamic problems, a point also discussed in the work of Kokot et al. ⁷

Creating paths that capitalize on a robot’s agility is essential for enhancing operational efficiency and adaptability in varied environments. Not all paths are equally viable across all modes of transport; certain routes may be less suitable or inaccessible for specific methods of travel. While many path planning methods are tailored for non-holonomic vehicles, accommodating a single motion mode such as “Ackermann steering geometry” or “Differential Drive” modes, it becomes imperative to develop planners that consider diverse kinematics, especially when dealing with mobile robots like GBM robots and omni-directional robots. Traditional path planning methods often fall short when applied to these robots, primarily due to the undefined nature of their motion modes, posing significant challenges to their effectiveness.

To illustrate this challenge more clearly, Figure 9 depicts a comparison of covered areas, which we refer to as the “coverage area,” when two GBM robots with different sizes trace a path with different modes. It becomes apparent that the robots occupy different amounts of coverage areas according to the motion mode. These coverage area represent the space that must be free from obstacles. Therefore, a larger coverage area implies greater space requirements, making it less preferable during movement.

Figure 9.

Visualization of covered areas by different movements. (a) Covered areas with different movements and a shorter robot length. (b) Covered areas with different movements and a longer robot length.

The path depicted in Figure 9 consists of three linear segments and two turns. To further explore the coverage area issue, we specifically examine the path segments—straight lines and corners—as shown in Figure 10. The coverage areas resulting from the robot’s navigation through these segments are comprehensively detailed in Tables 2 and 3.

Figure 10.

Extracted path segments for robots navigation. (a) Straight lines navigation. Poses in the “Crab” mode overlap with poses in the “Differential Drive” mode. (b) Corners navigation.

Table 2.

Coverage area comparison across various modes and robot sizes during straight line navigation (refer to Figure 10(a)).

	Length × Width= (1.0 m × 0.6 m)	Length × Width= (1.5 m × 0.6 m)
Crab	3.7168 m2	5.5568 m2
Tangential	2.4480 m2	2.7480 m2
Differential drive	3.7168 m2	5.5568 m2

Table 3.

Coverage area comparison across various modes and robot sizes during corner navigation (refer to Figure 10(b)).

	Length × Width= (1.0 m × 0.6 m)	Length × Width= (1.5 m × 0.6 m)
Crab	0.8630 m2	1.2369 m2
Tangential	1.0433 m2	1.8116 m2
Differential drive	1.5549 m2	2.2904 m2

Observing the figures and tables, unlike moving in the “Crab” mode, the robot in the “Tangential” and “Differential Drive” modes requires a more significant amount of space to navigate corners because adjustments to the robot’s orientation are needed. However, these two modes have their own advantages. For example, the “Differential Drive” mode allows the robot to rotate around its center, a capability unique to this mode, while the “Tangential” mode leads to the least coverage area when the robot traces a straight line.

As conventional path planning methods are not designed to accommodate multiple motion modes, these issues remain unaddressed, impacting both collision detection and obstacle avoidance during planning. Therefore, our objective is to directly incorporate the consideration of motion modes into our path planning process. This approach not only resolves the issue of collision detection with obstacles but also enables the explicit definition of which mode to use in different path segments, thereby eliminating confusion in mode selection.

Therefore, this study employs the GBM as the basis for examining motion modes and developing a path planner that ensures smooth (G₂) continuity and enables mode-changing within GBM. To sum up, we develop a path planning that satisfies the following points:

Motion modes of the GBM robots are considered when planning paths.

Generate an obstacle-free path from a starting position to a target location.

Generated path satisfies the requirements of G₂ continuity.

Methodology

Based on the discussion in the “Generalized bicycle model and motion modes” section, this section proposes a path planner that incorporates the motion modes of GBM kinematics and satisfies the G₂ continuity requirement. Unlike the classic RRT method, which randomly samples points in flexible regions and connects them to form a path without ensuring continuity or exploiting the characteristics of robot kinematics, our method considers motion modes and G₂ continuity to enable GBM robots to smoothly navigate a calculated route.

The motion mode constraints are contingent upon the wheel states, leading us to formulate the path planner utilizing the RRT framework. Our approach is inspired by the work of Wang et al., ¹⁷ incorporating node expansion with control inputs while addressing these constraints. The control inputs encompass the steering angles and velocities of the wheels.

The subsequent section will begin by elucidating the continuity analysis, followed by an in-depth discussion on the design of polynomial functions for G₂ paths, and ultimately, a comprehensive introduction of our RRT-based path planner.

Continuity analysis

According to equation (1), the wheel velocity can be alternatively expressed using the following equation:

$$\begin{array}{rl}{V}_{fx}& =V_f\cdot \cos ({\theta }_f)\\ V_{fy}& =V_f\cdot \sin ({\theta }_f)\\ V_{rx}& =V_r\cdot \cos ({\theta }_r)\\ V_{ry}& =V_r\cdot \sin ({\theta }_r)\end{array}$$

(8)

Combining equations (1), (2) and equation (8), a substitution for the inverse kinematics can be derived as follows:

$$\begin{array}{r}\left[\begin{array}{c}{V}_x\\ V_y\\ \omega \end{array}\right]=\left[\begin{array}{cccc}\frac{1}{2}& 0& \frac{1}{2}& 0\\ 0& \frac{1}{2}& 0& \frac{1}{2}\\ 0& \frac{1}{L}& 0& \frac{-1}{L}\end{array}\right]\left[\begin{array}{c}{V}_f\cos {\theta }_f\\ V_f\sin {\theta }_f\\ V_r\cos {\theta }_r\\ V_r\sin {\theta }_r\end{array}\right]\end{array}$$

(9)

This shows how one can obtain the velocities of the GBM robot given the control of the velocity and steering angles of the front and rear wheels.

G₂ continuity refers to the uninterrupted progression of curvature, ensuring a smooth transition at the juncture of differentiable segments. ¹³ To investigate the continuity of a path by wheel command, we first define the curvature of a path as the following equation:

$$\kappa =\frac{|\ddot{x}\dot{y}-\dot{x}\ddot{y}|}{({\dot{x}}^2+{\dot{y}}^2{)}^{\frac{3}{2}}}$$

(10)

In equation (10), the curvature depends on the first and second-order derivatives of the robot poses in the global coordinate system, which can be calculated using the fundamental kinematics of vehicles and their derivatives as presented in the following equations:

$$\begin{array}{rl}\dot{x}& =V_x\cos \theta -V_y\sin \theta \\ \dot{y}& =V_x\sin \theta +V_y\cos \theta \\ \dot{\theta }& =\omega \end{array}$$

(11)

$$\begin{array}{rl}\ddot{x}& =\dot{V_x}\cos \theta -V_x\omega \sin \theta -\dot{V_y}\sin \theta -V_y\omega \cos \theta \\ \ddot{y}& =\dot{V_x}\sin \theta +V_x\omega \cos \theta +\dot{V_y}\cos \theta -V_y\omega \sin \theta \\ \ddot{\theta }& =\dot{\omega }\end{array}$$

(12)

Additionally, obtaining the first-order derivative of $(V_x,V_y,\omega )$ is necessary for curvature calculation. Hence, we differentiate equation (9) and present the result in the following equation:

$$\begin{array}{rl}\dot{V_x}& =\frac{1}{2}(\dot{V_f}\cos {\theta }_f-V_f\dot{{\theta }_f}\sin {\theta }_f+\dot{V_r}\cos {\theta }_r-V_r\dot{{\theta }_r}\sin {\theta }_r)\\ \dot{V_y}& =\frac{1}{2}(\dot{V_f}\sin {\theta }_f+V_f\dot{{\theta }_f}\cos {\theta }_f+\dot{V_r}\sin {\theta }_r+V_r\dot{{\theta }_r}\cos {\theta }_r)\\ \dot{\omega }& =\frac{1}{L}(\dot{V_f}\sin {\theta }_f+V_f\dot{{\theta }_f}\cos {\theta }_f-\dot{V_r}\sin {\theta }_r-V_r\dot{{\theta }_r}\cos {\theta }_r)\end{array}$$

(13)

Integrating equations (9) to (13), a new expression for curvature can be computed as shown in equation (14). In this equation, continuous curvature is achievable as long as the wheel states are designed to be continuous, and the robots avoid zero velocity, thereby preventing the occurrence of a zero denominator in the equation.

$$\begin{array}{rl}\kappa & =\frac{|\ddot{x}\dot{y}-\dot{x}\ddot{y}|}{({\dot{x}}^2+{\dot{y}}^2{)}^{\frac{3}{2}}}=\frac{1}{\frac{1}{8}(V_f^2+V_r^2+2V_f{V}_r\cos ({\theta }_f-{\theta }_r){)}^{\frac{3}{2}}}\\ & \times |\frac{-1}{4L}(V_f^3\sin {\theta }_f+V_f{V}_r^2\sin {\theta }_r-V_f^2{V}_r\sin {\theta }_r-V_r^3\sin {\theta }_r)\\ & +\frac{-1}{4L}(2V_f^2{V}_r\sin {\theta }_f\cos ({\theta }_f-{\theta }_r)-2V_f{V}_r^2\sin {\theta }_r\cos ({\theta }_f-{\theta }_r))\\ & +\frac{1}{4}(-V_f^2\dot{{\theta }_f}-V_r^2\dot{{\theta }_r}+(V_f\dot{V_r}-V_r\dot{V_f})\sin ({\theta }_f-{\theta }_r))\\ & +\frac{1}{4}(-V_f{V}_r(\dot{{\theta }_f}+\dot{{\theta }_r})\cos ({\theta }_f-{\theta }_r))|\end{array}$$

(14)

Based on the aforementioned derivations, G₂ continuous paths can be achieved through path planning that considers appropriate continuous wheel states. Consequently, in our proposed path-finding algorithm, second-order polynomial functions are formulated for the wheel states to meet the G₂ continuity requirement, facilitating a seamless control of GBM robots. Further elaboration on the specifics of these polynomial functions will be provided in the subsequent section.

Design of polynomial functions for G₂ continuous paths

The analysis of continuity underscores the necessity for wheel states to be governed by polynomial functions that uphold continuous first-order derivatives. Moreover, within each polynomial function, a fixed time period T is taken into consideration to regulate the wheel states. Consequently, we employ a second-order polynomial function, as outlined in equation (15), to describe the function $f_N(t)$ and its derivatives $f_N^{\mathrm{\prime }}(t)$ , with N denoting the number of nodes in a path during planning. This variable also signifies the enumeration of polynomials originating from the initial state.

$$\begin{array}{rl}{f}_N(t)& =a_N{t}^2+b_{N}t+c_{N}t\in [0,T)\\ f_N^{\mathrm{\prime }}(t)& =2a_{N}t+b_{N}t\in [0,T)\end{array}$$

(15)

Moreover, the connection between polynomial functions across adjacent time intervals is expressed as follows:

$$\begin{array}{rl}{f}_N(t=T)& =f_{N+1}(t=0)\\ f_N^{\mathrm{\prime }}(t=T)& =f_{N+1}^{\mathrm{\prime }}(t=0)\end{array}$$

(16)

Equation (16) describes the continuity between two adjacent polynomials, which can be utilized to calculate the parameters, $b_N$ and $c_N$ . For example, for the two adjacent polynomials $f_0$ and $f_1$ , we have:

$$\begin{array}{rl}{f}_0(t)& =a_0{t}^2+b_{0}t+c_0\\ f_0^{\mathrm{\prime }}(t)& =2a_{0}t+b_0\\ f_1(t)& =a_1{t}^2+b_{1}t+c_1\\ f_1^{\mathrm{\prime }}(t)& =2a_{1}t+b_1\end{array}$$

(17)

Because $f_0$ represents the initial polynomial, hence $f_0(t=0)=0$ and $f_0^{\mathrm{\prime }}(t=0)=0$ is assumed, resulting in $b_0=0$ and $c_0=0$ . Then equation (16) shows that

$$\begin{array}{rl}{f}_0(t=T)=f_1(t=0)& ⟹a_0{T}^2=c_1\\ f_0^{\mathrm{\prime }}(t=T)=f_1^{\mathrm{\prime }}(t=0)& ⟹2a_{0}T=b_1\end{array}$$

(18)

Thus, by iteratively applying the continuity conditions to the N polynomials, the parameters $b_N$ and $c_N$ can be derived through the recursive relation outlined in the following equation:

$$\begin{array}{rl}{b}_0& =0,c_0=0\\ b_N& =2a_{N-1}T+b_{N-1}N>0\\ c_N& =a_{N-1}{T}^2+b_{N-1}T+c_{N-1}N>0\end{array}$$

(19)

However, it is noteworthy that the parameter $a_N$ remains independent of the preceding polynomials; its determination is governed by the boundary conditions. These conditions are derived from both the kinematic constraints of GBM and the capabilities of the wheel actuators. To determine the value of $a_N$ , we start by defining the boundary conditions for the polynomial, as indicated in the following equation:

$$\begin{array}{r}{f}_{\text{min},N}\le f_N(t)\le f_{\text{max},N}\\ f_{\text{min},N}^{\mathrm{\prime }}\le f_N^{\mathrm{\prime }}(t)\le f_{\text{max},N}^{\mathrm{\prime }}\end{array}$$

(20)

Subsequently, the first-order derivative of the polynomial is designed as a symmetric piece-wise linear function, as described by equation (21) and illustrated in Figure 11.

$$f_N^{\mathrm{\prime }}(t)=\{\begin{array}{cc}2a_{N}t,& 0\le t\le \frac{T}{2}\\ -2a_N(t-T),& \frac{T}{2}<t\le T\end{array}$$

(21)

Figure 11.

The calculation of polynomial’s extreme values.

The integration of the first-order derivative, representing the function’s change, facilitates a straightforward control of the variation in $f(t)$ . Consequently, it becomes more manageable to meet the boundary conditions outlined in equation (20). Ultimately, by leveraging these boundary conditions, the range of $a_N$ can be determined, as expressed in the following equation:

$$\begin{array}{rl}& f_{\text{min},N}^{\mathrm{\prime }}\le a_{N}T\le f_{\text{max},N}^{\mathrm{\prime }}\\ & f_{\text{min},N}\le f_{N-1}(T)+\frac{a_N{T}^2}{2}\le f_{\text{max},N}\end{array}$$

(22)

On the other hand, as the symmetric function is defined by two sub-equations, leading $f_N$ to involve two sub-equations as well. Therefore, we reformulate $f_N$ as expressed in the following equation:

$$\begin{array}{rl}& f_N(t)\\ & =\{\begin{array}{ccc}& a_N{t}^2+b_N^{\text{1st}}t+c_N^{\text{1st}}& 0\le t\le \frac{T}{2}\\ & -a_N(t-\frac{T}{2}{)}^2+b_N^{\text{2nd}}(t-\frac{T}{2})+c_N^{\text{2nd}}& \frac{T}{2}\le t\le T\end{array}\end{array}$$

(23)

For the parameters $b_N^{\text{1st}},c_N^{\text{1st}},b_N^{\text{2nd}},c_N^{\text{2nd}}$ , equation (19) can still be used, and each parameter is computed as shown in the following equation:

$$\begin{array}{rl}{b}_0^{\text{1st}}& =0,c_0^{\text{1st}}=0\\ b_0^{\text{2nd}}& =a_{0}T,c_0^{\text{2nd}}=\frac{T^2}{4}{a}_0\\ b_N^{\text{1st}}& =-a_{N-1}T+b_{N-1}^{\text{2nd}}N>0\\ c_N^{\text{1st}}& =-\frac{T^2}{4}{a}_{N-1}+\frac{T}{2}{b}_{N-1}^{\text{2nd}}+c_{N-1}^{\text{2nd}}N>0\\ b_N^{\text{2nd}}& =a_{N}T+b_N^{\text{1st}}N>0\\ c_N^{\text{2nd}}& =\frac{T^2}{4}{a}_N+\frac{T}{2}{b}_N^{\text{1st}}+c_N^{\text{1st}}N>0\end{array}$$

(24)

By discretizing $a_N$ , the corresponding parameters of $b_N^{\text{1st}},c_N^{\text{1st}},b_N^{\text{2nd}},c_N^{\text{2nd}}$ can be computed, and feasible polynomial candidates for the wheel states can be established. The resulting polynomial functions, denoted as $F=\{f_0,f_1,\dots ,f_N\}$ , allow the derivation of paths that are guaranteed to satisfy the G₂ continuity requirement. Implementing this design in the path planner facilitates the generation of G₂ continuous paths. Consequently, we integrate this design for the control inputs, specifically wheel states, into our proposed RRT-based planner to ensure continuity during path generation.

Path planning

By adopting a continuous wheel input commands, from previous design approach, we proposed an RRT path planning method, which incorporates the diverse motion modes of GBM kinematics and meets the G₂ continuity requirement. This algorithm has three key components: random state sampling, node selection, and node expansion, as detailed in Algorithm 1. Subsequent sections will provide a step-by-step elucidation of the planner.

Random state sampling

Our algorithm begins with a randomly sampled state $X_{\text{sample}}$ within the obstacle-free areas of the known environment. Additionally, to encourage the path to extend toward the destination, a probability $P_{\text{sample}}$ is introduced to decide whether to consider the goal position as the sampled state. The specific sampling method is outlined in Algorithm 2.

Node selection

During the node selection phase, we establish a sorted node set $X_{\text{sorted}}$ by organizing each node in the RRT tree based on its distance to the sampled state $X_{\text{sample}}$ . Subsequently, k nodes are chosen iteratively in sequence from $X_{\text{sorted}}$ . If a node $x_{\text{sorted},i}$ meets the condition where the angle between the velocity direction in the world coordinate system and the position angle relative to the node is below a predefined threshold ${\theta }_{\text{thres}}$ , as described in equation (25), it is included in the neighboring node set $X_{\text{near}}$ . The pseudocode for this node selection is presented in Algorithm 3.

$$|\arctan \frac{V_{\text{sorted},iy}}{V_{\text{sorted},ix}}+{\theta }_{\text{sorted},i}-\arctan \frac{y_{\text{sample}}-y_{\text{sorted},i}}{x_{\text{sample}}-x_{\text{sorted},i}}|\le {\theta }_{\text{thres}}$$

(25)

Node expansion

Utilizing information from the nearest nodes set $X_{\text{near}}$ , we iteratively compute the constraints of motion modes for each node, which can be used to calculate possible polynomials for steering and velocity. By considering these polynomials, we can obtain several predictive states. Subsequently, based on their distances to the sample state, the optimal node can be identified and become the result of the node expansion process. Through this node expansion, these connected nodes not only satisfy the constraints of motion modes and kinematics but also form a G₂ continuous path. The comprehensive concept is detailed in Algorithm 4.

Case studies and result discussions

In this section, we will establish test scenarios to emphasize the impact of motion modes on path planning. Initially, we employ the SRRT method ¹⁴ to acquire global paths. Subsequently, we pinpoint the issue arising from the oversight of motion modes. Following that, we test our proposed method in the same scenarios and conduct a thorough analysis of the results.

To begin, we introduce the environment settings and the robot’s specifications. The robot has dimensions of $1\text{m}\times 0.6\text{m}$ , with two steerable wheels located at the front and back of its center points, and a wheelbase of $0.8\text{m}$ . The wheel capabilities include a maximum speed of $0.3\text{m/s}$ , acceleration of $0.15{\text{m/s}}^2$ , and angular speed of $45\circ /\text{s}$ . These specifications play a crucial role in the path planning process.

Two distinct environments as shown in Figure 12 are considered. The first scenario involves a cluttered environment with various irregular obstacles. The second scenario features a narrow passage with a width of $0.8\text{m}$ that imposes constraints on the robot’s pose during traversal. In both cases, the overall environment size is $15.18\text{m}\times 8.18\text{m}$ . The paths are required to start from $(1,1)$ and end near $(15,8)$ , with a tolerance of $1\text{m}$ .

Figure 12.

Environments for the test scenarios. The gray point on the bottom left represents the start point, while the orange range on the top right signifies the acceptable range of the destination. (a) Environment 1: cluttered environment. (b) Environment 2: narrow passages.

Planned paths using SRRT method

As discussed in the “Requirements for path planners” section, neglecting to consider motion modes can impact obstacle detection, as the planned path is generated without accounting for orientation, making it challenging to predict the robot’s pose on the path. Therefore, without integrating motion modes or kinematic models, a common approach for obstacle avoidance involves using an inflation layer around obstacles. However, relying solely on the inflation layer may not fully resolve the issue. Typically, the inflation layer width is set to at least the radius of the circle that encloses the robot, which occupies considerable space, as depicted in Figure 13, and can pose challenges for path planners in computing feasible paths. To mitigate this, a reduced width, as shown in Figure 14, is sometimes used, but this approach still introduces challenges, as the robot may collide with obstacles when operating in different motion modes. We demonstrate this issue using SRRT and various motion modes.

Figure 13.

Obstacle inflation with a large width. Black areas represent the original obstacles, while the gray areas signifies the inflation layers. (a) Large inflation. The inflated areas occupy a significant amount of free space. (b) Large inflation. The inflated areas occupy a significant amount of free space.

Figure 14.

Obstacle inflation with a large width. Black areas represent the original obstacles, while the gray areas signify the inflation layers. (a) Small inflation that occupies less free space. (b) Small inflation that occupies less free space.

Scenario 1: Cluttered environment

In the cluttered environment, Figure 15(a) illustrates the planned path generated using SRRT. Meanwhile, Figure 15(b) to (d) depicts the robot poses with different motion modes considered. It is evident that by incorporating inflation layers, SRRT can calculate an obstacle-free global path. This path allows the robot in the tangential mode to navigate without colliding with obstacles. However, when the robot operates in crab and differential drive mode along the same path, collisions with obstacles occur, as indicated by the orange boxes in Figure 15(b). This issue stems from the lack of consideration for motion modes, kinematic models, or more specifically, the robot’s orientation. Inflation layers prove ineffective in preventing collisions if a proper width fully encircling robots is not applied.

Figure 15.

Graphical results by the spline-based rapidly exploring random tree (SRRT) ¹⁴ in the cluttered environment. (a) The planned paths by using SRRT in the cluttered environment. (b) The robot poses executing the crab movements along the planned path. The collision between the robot and the obstacle is highlighted by the orange box. (c) The robot poses executing the tangential movements along the planned path. (d) The robot poses executing the differential drive movements along the planned path. The collision between the robot and the obstacle is highlighted by the orange box.

Scenario 2: Narrow passage

In the case of the narrow passage, Figure 16 displays the SRRT-planned path and the robot poses with different motion modes along the path. However, a similar issue to the one mentioned earlier arises. As depicted in Figure 16(b) and (c), the robot successfully navigates the passage with crab and tangential modes. On the contrary, the robot employing differential drive mode collides with obstacles, as highlighted by the orange box in Figure 16(d), because the robot’s length exceeds the width of the passage, preventing it from passing through.

Figure 16.

Graphical results by the spline-based rapidly exploring random tree (SRRT) ¹⁴ in the narrow passage. (a) The planned paths by using SRRT in the narrow passage environment. (b) The robot poses executing the crab movements along the planned path. (c) The robot poses executing the tangential movements along the planned path. (d) The robot poses executing differential drive movements along the planned path. The collision between the robot and the obstacle is highlighted by the orange box.

As discussed in the “Requirements for path planners” section, various motion modes result in different area coverage for robots, thereby influencing obstacle avoidance. To address this, consideration of motion modes and kinematic models becomes imperative. Consequently, we integrate motion modes constraints into our proposed path planning method, enhancing obstacle detection and leveraging the agility and flexibility of GBM robots effectively.

Planned paths using proposed method

In comparison to SRRT, our proposed method, which integrates the constraints of motion modes and the kinematic model, enables the prediction of robot poses along the planned path. This prediction facilitates direct collision detection with obstacles, eliminating the need for inflation layer design. Additionally, as illustrated in Figure 7, our approach in these tests considers the following motion modes: crab, tangential, and differential drive. We employ the Tangential mode to represent Ackermann steering geometry, chosen for its ease in achieving the mode switch condition between crab and tangential modes.

Scenario 1: Cluttered environment

The paths generated by our method in the cluttered environment and the percentages of each mode are illustrated in Figure 17, while the corresponding continuity analysis and planned control inputs can be found in Figures 18 and 19, respectively.

Figure 17.

Graphical results in cluttered environment scenario. (a) The planned paths in the cluttered environment. The red line represents the differential drive mode, the green one represents the crab mode, and the blue one represents the Tangential mode. (b) The robot poses on the planned path. The purple rectangles indicate the robot’s poses. (c) Percentage of the used modes of the planned path. The slash (/) denotes that the robot node is in a transitional state. For example, “Crab/Tangential” mode indicates that the current node is transitioning between the states of “Crab” and “Tangential.”

Figure 18.

G₂ continuity. Due to scaling constraints, we utilize double axes to depict both the original and a specified range of curvature. It is evident that the curvature remains continuous at all times, thereby achieving G₂ continuity.

Figure 19.

The control inputs corresponding to the planned path. At $t=4$ s, the robot switched from the Tangential to the crab motion mode, and a similar behavior can be observed again after $t=44$ s.

In Figure 17, the crab and tangential modes are used to navigate the robot toward the target. Examining the planned control inputs illustrated in Figure 19, we can observe that from $t=0$ to $4$ s, the robot operates in tangential mode, characterized by opposite steering angles and identical wheel speeds. Afterward, the steering angles converge to $0$ , and subsequently, both steering angles remain consistent, signifying a transition to crab mode. This mode switch occurs at $t=4$ s. Furthermore, the continuity analysis in Figure 18 confirms that the curvature remains continuous throughout, satisfying the G₂ continuity requirement. The overall polynomial functions of the control inputs planned for the global path are listed in Table 4.

Table 4.

The polynomials of the control inputs referred to Figure 19.

Nodes	Polynomials of ${\theta }_f$	Polynomials of ${\theta }_r$	Polynomials of $V_f$	Polynomials of $V_r$
0 (tangential)	(0.098175, 0.000000, 0.000000)(−0.098175, 0.196350, 0.098175)	(−0.098175, −0.000000, −0.000000)(0.098175, −0.196350, −0.098175)	(0.075000, 0.000000, 0.000000)(−0.075000, 0.150000, 0.075000)	(0.075000, 0.000000, 0.000000)(−0.075000, 0.150000, 0.075000)
1 (crab/tangential)	(−0.098175, 0.000000, 0.196350)(0.098175, −0.196350, 0.098175)	(0.098175, −0.000000, −0.196350)(−0.098175, 0.196350, −0.098175)	(0.075000, 0.000000, 0.150000)(−0.075000, 0.150000, 0.225000)	(0.075000, 0.000000, 0.150000)(−0.075000, 0.150000, 0.225000)
2 (crab)	(0.392699, 0.000000, 0.000000)(−0.392699, 0.785398, 0.392699)	(0.392699, 0.000000, 0.000000)(−0.392699, 0.785398, 0.392699)	(0.000000, 0.000000, 0.300000)(−0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(−0.000000, 0.000000, 0.300000)
3 (crab)	(−0.098175, 0.000000, 0.785398)(0.098175, −0.196350, 0.687223)	(−0.098175, 0.000000, 0.785398)(0.098175, −0.196350, 0.687223)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
4 (crab)	(−0.098175, 0.000000, 0.589049)(0.098175, −0.196350, 0.490874)	(−0.098175, 0.000000, 0.589049)(0.098175, −0.196350, 0.490874)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
5 (crab)	(−0.098175, 0.000000, 0.392699)(0.098175, −0.196350, 0.294524)	(−0.098175, 0.000000, 0.392699)(0.098175, −0.196350, 0.294524)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
6 (crab)	(0.000000, 0.000000, 0.196350)(0.000000, 0.000000, 0.196350)	(0.000000, 0.000000, 0.196350)(0.000000, 0.000000, 0.196350)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
7 (crab)	(0.098175, 0.000000, 0.196350)(−0.098175, 0.196350, 0.294524)	(0.098175, 0.000000, 0.196350)(−0.098175, 0.196350, 0.294524)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
8 (crab)	(0.392699, 0.000000, 0.392699)(−0.392699, 0.785398, 0.785398)	(0.392699, 0.000000, 0.392699)(−0.392699, 0.785398, 0.785398)	(−0.075000, 0.000000, 0.300000)(0.075000, −0.150000, 0.225000)	(−0.075000, 0.000000, 0.300000)(0.075000, −0.150000, 0.225000)
9 (crab)	(0.049087, 0.000000, 1.178097)(−0.049087, 0.098175, 1.227185)	(0.049087, 0.000000, 1.178097)(−0.049087, 0.098175, 1.227185)	(0.000000, 0.000000, 0.150000)(0.000000, 0.000000, 0.150000)	(0.000000, 0.000000, 0.150000)(0.000000, 0.000000, 0.150000)
10 (crab)	(−0.257709, 0.000000, 1.276272)(0.257709, −0.515418, 1.018563)	(−0.257709, 0.000000, 1.276272)(0.257709, −0.515418, 1.018563)	(−0.075000, 0.000000, 0.150000)(0.075000, −0.150000, 0.075000)	(−0.075000, 0.000000, 0.150000)(0.075000, −0.150000, 0.075000)
11 (crab)	(−0.294524, 0.000000, 0.760854)(0.294524, −0.589049, 0.466330)	(−0.294524, 0.000000, 0.760854)(0.294524, −0.589049, 0.466330)	(0.037500, 0.000000, 0.000000)(−0.037500, 0.075000, 0.037500)	(0.037500, 0.000000, 0.000000)(−0.037500, 0.075000, 0.037500)
12 (crab)	(−0.392699, 0.000000, 0.171806)(0.392699, −0.785398, −0.220893)	(−0.392699, 0.000000, 0.171806)(0.392699, −0.785398, −0.220893)	(−0.037500, 0.000000, 0.075000)(0.037500, −0.075000, 0.037500)	(−0.037500, 0.000000, 0.075000)(0.037500, −0.075000, 0.037500)
13 (crab)	(0.392699, 0.000000, −0.613592)(−0.392699, 0.785398, −0.220893)	(0.392699, 0.000000, −0.613592)(−0.392699, 0.785398, −0.220893)	(0.075000, 0.000000, 0.000000)(−0.075000, 0.150000, 0.075000)	(0.075000, 0.000000, 0.000000)(−0.075000, 0.150000, 0.075000)
14 (crab)	(0.392699, 0.000000, 0.171806)(−0.392699, 0.785398, 0.564505)	(0.392699, 0.000000, 0.171806)(−0.392699, 0.785398, 0.564505)	(0.075000, 0.000000, 0.150000)(−0.075000, 0.150000, 0.225000)	(0.075000, 0.000000, 0.150000)(−0.075000, 0.150000, 0.225000)
15 (crab)	(−0.392699, 0.000000, 0.957204)(0.392699, −0.785398, 0.564505)	(−0.392699, 0.000000, 0.957204)(0.392699, −0.785398, 0.564505)	(−0.037500, 0.000000, 0.300000)(0.037500, −0.075000, 0.262500)	(−0.037500, 0.000000, 0.300000)(0.037500, −0.075000, 0.262500)
16 (crab/tangential)	(−0.085903, 0.000000, 0.171806)(0.085903, −0.171806, 0.085903)	(−0.085903, 0.000000, 0.171806)(0.085903, −0.171806, 0.085903)	(0.037500, 0.000000, 0.225000)(−0.037500, 0.075000, 0.262500)	(0.037500, 0.000000, 0.225000)(−0.037500, 0.075000, 0.262500)
17 (crab)	(−0.098175, 0.000000, 0.000000)(0.098175, −0.196350, −0.098175)	(−0.098175, 0.000000, 0.000000)(0.098175, −0.196350, −0.098175)	(0.000000, 0.000000, 0.300000)(−0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(−0.000000, 0.000000, 0.300000)
18 (crab)	(0.392699, 0.000000, −0.196350)(−0.392699, 0.785398, 0.196350)	(0.392699, 0.000000, −0.196350)(−0.392699, 0.785398, 0.196350)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
19 (crab)	(−0.196350, 0.000000, 0.589049)(0.196350, −0.392699, 0.392699)	(−0.196350, 0.000000, 0.589049)(0.196350, −0.392699, 0.392699)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
20 (crab/tangential)	(−0.098175, 0.000000, 0.196350)(0.098175, −0.196350, 0.098175)	(−0.098175, 0.000000, 0.196350)(0.098175, −0.196350, 0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
21 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
22 (tangential)	(0.098175, 0.000000, 0.000000)(−0.098175, 0.196350, 0.098175)	(−0.098175, −0.000000, −0.000000)(0.098175, −0.196350, −0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
23 (crab/tangential)	(−0.098175, 0.000000, 0.196350)(0.098175, −0.196350, 0.098175)	(0.098175, −0.000000, −0.196350)(−0.098175, 0.196350, −0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
24 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
25 (tangential)	(−0.098175, 0.000000, 0.000000)(0.098175, −0.196350, −0.098175)	(0.098175, −0.000000, −0.000000)(−0.098175, 0.196350, 0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
26 (tangential)	(0.294524, 0.000000, −0.196350)(−0.294524, 0.589049, 0.098175)	(−0.294524, −0.000000, 0.196350)(0.294524, −0.589049, −0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
27 (tangential)	(−0.294524, 0.000000, 0.392699)(0.294524, −0.589049, 0.098175)	(0.294524, −0.000000, −0.392699)(−0.294524, 0.589049, −0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
28 (crab/tangential)	(0.098175, 0.000000, −0.196350)(−0.098175, 0.196350, −0.098175)	(−0.098175, −0.000000, 0.196350)(0.098175, −0.196350, 0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
29 (crab)	(0.098175, 0.000000, 0.000000)(−0.098175, 0.196350, 0.098175)	(0.098175, 0.000000, 0.000000)(−0.098175, 0.196350, 0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
30 (crab)	(−0.294524, 0.000000, 0.196350)(0.294524, −0.589049, −0.098175)	(−0.294524, 0.000000, 0.196350)(0.294524, −0.589049, −0.098175)	(−0.075000, 0.000000, 0.300000)(0.075000, −0.150000, 0.225000)	(−0.075000, 0.000000, 0.300000)(0.075000, −0.150000, 0.225000)

The computation of rapidly exploring random tree (RRT) nodes assumes a period of 2 s. Additionally, the modes listed in the first column represent the mode in which the robot moves at the end of each node.

Scenario 2: Narrow passage

For the scenario of narrow passage, Figure 20 showcases the planned paths and the percentage of each mode. Additionally, Figure 21 provides insights into the continuity analysis, while Figure 22 displays the control inputs planned for the scenario.

Figure 20.

Graphical results in narrow passage scenario. (a) The planned paths in the narrow passage. The red line represents the differential drive mode, the green one represents the crab mode, and the blue one represents the tangential mode. (b) The robot poses on the planned path. The purple rectangles indicate the robot’s poses. (c) Percentage of the used modes of the planned path. The slash (/) denotes that the robot node is in a transitional state. For example, “Crab/Tangential” mode indicates that the current node is transitioning between the states of “Crab” and “Tangential.”

Figure 21.

G₂ continuity. Due to scaling constraints, we utilize double axes to depict both the original and a specified range of curvature. It is evident that the curvature remains continuous at all times, thereby achieving G₂ continuity.

Figure 22.

The control inputs corresponding to the planned path. The robot operates within the boundary of crab and tangent modes from $t=26$ to $40$ s.

From the planned paths in Figure 20(a), the robot traversed the narrow passage using the tangential mode, represented by the blue lines. The corresponding control inputs in Figure 22 were recorded from $t=16$ to $40$ s. During this period, the robot maintained the steering angles of its two wheels at $0$ , effectively operating at the boundary between crab and tangential modes. Additionally, similar to Scenario 1, the planned path also satisfies G₂ continuity, as demonstrated in Figure 21. Furthermore, the overall polynomial function of the control inputs is also documented and listed in Table 5.

Table 5.

The polynomials of the control inputs referred to Figure 22.

Nodes	Polynomials of ${\theta }_f$	Polynomials of ${\theta }_r$	Polynomials of $V_f$	Polynomials of $V_r$
0 (crab)	(0.294524, 0.000000, 0.000000)(−0.294524, 0.589049, 0.294524)	(0.294524, 0.000000, 0.000000)(−0.294524, 0.589049, 0.294524)	(0.075000, 0.000000, 0.000000)(−0.075000, 0.150000, 0.075000)	(0.075000, 0.000000, 0.000000)(−0.075000, 0.150000, 0.075000)
1 (crab)	(−0.098175, 0.000000, 0.589049)(0.098175, −0.196350, 0.490874)	(−0.098175, 0.000000, 0.589049)(0.098175, −0.196350, 0.490874)	(0.075000, 0.000000, 0.150000)(−0.075000, 0.150000, 0.225000)	(0.075000, 0.000000, 0.150000)(−0.075000, 0.150000, 0.225000)
2 (crab)	(0.098175, 0.000000, 0.392699)(−0.098175, 0.196350, 0.490874)	(0.098175, 0.000000, 0.392699)(−0.098175, 0.196350, 0.490874)	(0.000000, 0.000000, 0.300000)(−0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(−0.000000, 0.000000, 0.300000)
3 (crab)	(0.000000, 0.000000, 0.589049)(0.000000, 0.000000, 0.589049)	(0.000000, 0.000000, 0.589049)(0.000000, 0.000000, 0.589049)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
4 (crab)	(0.392699, 0.000000, 0.589049)(−0.392699, 0.785398, 0.981748)	(0.392699, 0.000000, 0.589049)(−0.392699, 0.785398, 0.981748)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
5 (crab)	(−0.208621, 0.000000, 1.374447)(0.208621, −0.417243, 1.165825)	(−0.208621, 0.000000, 1.374447)(0.208621, −0.417243, 1.165825)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
6 (crab)	(−0.305262, 0.000000, 0.957204)(0.305262, −0.610524, 0.651942)	(−0.305262, 0.000000, 0.957204)(0.305262, −0.610524, 0.651942)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
7 (crab/tangential)	(−0.173340, 0.000000, 0.346680)(0.173340, −0.346680, 0.173340)	(−0.173340, 0.000000, 0.346680)(0.173340, −0.346680, 0.173340)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
8 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
9 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
10 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
11 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
12 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
13 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
14 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
15 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
16 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
17 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
18 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
19 (crab/tangential)	(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)	(−0.000000, −0.000000, −0.000000)(−0.000000, −0.000000, −0.000000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
20 (crab)	(0.098175, 0.000000, 0.000000)(−0.098175, 0.196350, 0.098175)	(0.098175, 0.000000, 0.000000)(−0.098175, 0.196350, 0.098175)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
21 (crab)	(0.392699, 0.000000, 0.196350)(−0.392699, 0.785398, 0.589049)	(0.392699, 0.000000, 0.196350)(−0.392699, 0.785398, 0.589049)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
22 (crab)	(0.036816, 0.000000, 0.981748)(−0.036816, 0.073631, 1.018563)	(0.036816, 0.000000, 0.981748)(−0.036816, 0.073631, 1.018563)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
23 (crab)	(0.176408, 0.000000, 1.055379)(−0.176408, 0.352816, 1.231787)	(0.176408, 0.000000, 1.055379)(−0.176408, 0.352816, 1.231787)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
24 (crab/differential)	(0.081301, 0.000000, 1.408194)(−0.081301, 0.162602, 1.489495)	(0.081301, 0.000000, 1.408194)(−0.081301, 0.162602, 1.489495)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
25 (crab)	(−0.392699, 0.000000, 1.570796)(0.392699, −0.785398, 1.178097)	(−0.392699, 0.000000, 1.570796)(0.392699, −0.785398, 1.178097)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
26 (crab)	(0.098175, 0.000000, 0.785398)(−0.098175, 0.196350, 0.883573)	(0.098175, 0.000000, 0.785398)(−0.098175, 0.196350, 0.883573)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
27 (crab)	(−0.049087, 0.000000, 0.981748)(0.049087, −0.098175, 0.932660)	(−0.049087, 0.000000, 0.981748)(0.049087, −0.098175, 0.932660)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)
28 (crab/differential)	(0.343612, 0.000000, 0.883573)(−0.343612, 0.687223, 1.227185)	(0.343612, 0.000000, 0.883573)(−0.343612, 0.687223, 1.227185)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)	(0.000000, 0.000000, 0.300000)(0.000000, 0.000000, 0.300000)

The computation of rapidly exploring random tree (RRT) nodes assumes a period of 2 s. Additionally, the modes listed in the first column represent the mode in which the robot moves at the end of each node.

In both test cases, our proposed method showcases its ability to compute a global path by integrating the diverse motion modes of GBM robots. Throughout the planning phase, the RRT expands its nodes, considering the current state, obstacles, and available modes. This highlights the flexibility and agility of GBM robots, effectively addressing the previously mentioned obstacle detection issue. Additionally, our planning method utilizes smooth second-order polynomials for control inputs, ensuring controllability. This is further substantiated by our analysis of G₂ continuity.

Conclusion and future work

This work initially reviewed existing research on vehicle types and highlighted the potential for further development of GBM robots. Subsequently, we defined detailed motion modes for GBM robots, underscoring their significance in path planning. However, for general path planners, neglecting motion modes can impact both area coverage and obstacle avoidance. Therefore, we developed an RRT-based path planner that integrates the motion modes of GBM robots and ensures G₂ continuous paths by designing control inputs with second-order polynomials. We also established a continuity analysis to validate the planned path. In two distinct scenarios, our proposed method successfully computes global paths that meet the G₂ continuity requirement.

However, several issues still exist, warranting further study. For example, incorporating asymptotic optimality considerations into path planners based on RRTs will be a future focus, aiming to compute the best path while accounting for motion modes. Additionally, while bi-directional expansion is commonly used in RRT methods to enhance efficiency, its application in our method poses challenges due to the need for continuity in both path and control inputs. Moreover, path tracking methods are essential for GBM robots, given the likelihood of encountering control errors or the need to avoid dynamic obstacles, which can limit reliance on information from the planned path. Despite these challenges, we believe that this work contributes significantly and will propose solutions to address these issues in the future.