Advances in the Kinematics of Hexapod Robots: An Innovative Approach to Inverse Kinematics and Omnidirectional Movement

Abstract

Hexapod robots have gained significant attention due to their potential applications in complex terrains and dynamic environments. However, traditional inverse kinematics approaches often face challenges in meeting the precision required for adaptive omnidirectional movement. This work introduces a novel approach to addressing these challenges through the Directed Angular Restitution (DAR) method. The DAR method offers significant innovation by simplifying the calculation of rotational transformations necessary for aligning vectors across different planes, thus enhancing control, stability, and accuracy in robotic applications. Unlike conventional methods, the DAR method extends the range of trigonometric functions and incorporates spin functions to ensure continuous and smooth trajectory tracking. This innovative approach has been rigorously tested on a hexapod robot model, demonstrating superior performance in movement precision and stability. The results confirm that the DAR method provides a robust and scalable solution for the inverse kinematics of hexapod robots, making it a critical advancement for applications in robotics and automation where precise control and adaptability are paramount.

1. Introduction

Hexapod robots have gained significant attention due to their potential applications in complex terrains and dynamic environments. Understanding and optimizing their locomotion patterns is crucial for enhancing their performance and adaptability. Despite various advancements in hexapod robot design and kinematics, traditional methods for solving inverse kinematics often encounter challenges in achieving the necessary precision and flexibility, particularly in scenarios requiring omnidirectional movement.

To address these challenges, this work introduces the Directed Angular Restitution (DAR) method, a novel approach designed to simplify the calculation of angular transformations necessary for aligning vectors across different coordinate planes. Unlike conventional methods that rely heavily on trigonometric functions or iterative numerical solutions, the DAR method employs a geometric framework that enhances the accuracy of angular adjustments. This innovation is particularly crucial for ensuring precise control of movement and orientation in hexapod robots, especially in complex environments.

The research presented in this paper focuses on the kinematic analysis of hexapod robots, specifically in scenarios where the terrain is uniform. However, the scalability of the DAR method suggests the potential for extension to dynamic analyses involving non-uniform terrains and varying conditions. This adaptability makes the DAR method a valuable tool not only for current kinematic applications but also for future exploration in more complex dynamic scenarios.

Numerous studies have explored various aspects of hexapod robot kinematics and gait design to improve efficiency and effectiveness. For instance, ref. [1] details the manufacturing process of a low-weight hexapod robot with 2 degrees of freedom per leg, featuring a kinematic design inspired by insect movement. Similarly, ref. [2] discusses the development of a neural network that emulates the locomotion kinematics of a hexapod robot, considering impairments that affect leg movement.

Further studies, such as [3], analyze specific movement patterns for hexapod robots, noting that while bio-inspired patterns are commonly used, they are not always the most effective. This research emphasizes the pursuit of the fastest movement pattern using evolutionary algorithms. In another study, ref. [4] proposes a dynamic model to improve hexapod robot locomotion under challenging conditions by using load sensors to detect the robot’s pose, again employing bio-inspired movement patterns.

Moreover, ref. [5] investigates posture patterns (tripod and bipod) for a hexapod robot, with a mechanical composition that includes two rotational degrees of freedom and a six-bar mechanism to extend the range of leg motion. Meanwhile, ref. [6] examines the locomotion of symmetrical hexapod robots, analyzing simple and mixed patterns based on insects and mammals, and studying a stability margin for maximum displacement distance under various conditions, such as inclinations and uneven terrain.

Recent work by [7] provides a comprehensive model for hexapod robot kinematics, focusing on the design of tripod gaits based on leg trajectories. This research offers a robust framework for understanding and optimizing hexapod locomotion. Similarly, ref. [8] concentrates on tripod gait planning and kinematic analysis, crucial for enhancing the performance and adaptability of hexapod robots in diverse environments.

Other significant contributions include [9], which explores dynamic modeling and control of hexapod robots using Matlab SimMechanics, and [10], which analyzes inverse kinematics for motion prediction in hexapod robots. Ref. [11] employs quadratic programming for inverse kinematics control with inequality constraints, providing a mathematical approach to enhance hexapod movement.

Additionally, ref. [12] implements the inverse kinematics method for self-moving hexapod robots, while [13] investigates leg coordination using a geometrical tripod-gait and inverse kinematics approach. Ref. [14] applies artificial intelligence for experimental modeling of hexapod robots, demonstrating the integration of advanced computational techniques in robotic control.

In [15], a path-tracking controller is designed for omnidirectional gait in hexapod robots, and [16] focuses on leg design, which is essential for ensuring efficient and adaptable movement. Motion control is explored in [17] using model-based design, emphasizing the importance of precise control algorithms.

Furthermore, ref. [18] introduces a model-based real-time motion tracking approach using dynamical inverse kinematics, relevant for improving the control and accuracy of hexapod robots. Additionally, the textbook [19] provides comprehensive insights into inverse kinematics, modeling, motion planning, and control of manipulators and mobile robots, offering valuable theoretical background and practical applications.

The DAR method introduced in this work stands out as an innovative solution to the key challenges identified in these studies. By offering a more accurate approach to inverse kinematics in scenarios with uniform terrain, the DAR method has the potential to significantly advance the field of robotics. Its scalability also makes it a promising candidate for future exploration in dynamic analyses, where precise control and adaptability are essential.

The structure of this paper is as follows: Section 2 begins with an introduction to the Directed Angular Restitution (DAR) method, laying the foundation for the subsequent analyses. Section 3 then provides a comparative analysis of inverse kinematics methods, demonstrating the performance of the DAR method in comparison to traditional approaches. Building on this, Section 4 presents the mathematical model of the closed-loop trajectory, which is essential for understanding the kinematic behavior of the system. Section 5 offers a detailed kinematic analysis of the hexapod robot, covering both direct and inverse kinematics. The discussion continues in Section 6 with the results and simulations, where the effectiveness of the DAR method is evaluated. In Section 7, the focus shifts to stability and support polygons, exploring their critical role in maintaining the robot’s balance. Finally, Section 8 concludes the paper, summarizing the key findings and proposing directions for future research.

2. Directed Angular Restitution (DAR)

In this section, the Directed Angular Restitution (DAR) method is developed. This proposed method allows for the alignment of links connected by rotational joints, facilitating the calculation of inverse kinematics in trajectory planning. It establishes continuity in the direction of rotation using spin functions, which are crucial for ensuring precise movement in robotic applications.

The basic concept in the DAR method starts with the calculation of the angle between vectors, as illustrated in Figure 1 (in the $xz$ plane). To determine $\mathsf{\Delta }q$ between the vectors $r_{a,b}$ and $r_{a,c}$, the following three trigonometric expressions are used, incorporating the elimination and selection vectors to ensure correctness in ${\mathbb{R}}^3$:

$$\mathsf{\Delta }q={cos}^{-1}\left({\displaystyle \frac{\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\right)\otimes \left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}∥}}\right)$$

(1)

$$\mathsf{\Delta }q={sin}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{y}}·\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}∥}}\right)$$

(2)

$$\mathsf{\Delta }q={tan}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{y}}·\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}{\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\right)·\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}}\right)$$

(3)

where ${\mathbf{E}}_{\mathbf{y}}={\left[\begin{array}{ccc}1& 0& 1\end{array}\right]}^T$ is the elimination vector, ${\mathbf{S}}_{\mathbf{y}}={\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T$ is the selection vector for the $xz$ plane and $(·)\otimes (·)$ is the Hadamar product. The elimination and selection vectors for the other planes are summarized in

Table 1. Elimination and selection vectors for different planes in ${\mathbb{R}}^3$.

Plane	Elimination Vector E	Selection Vector S
$yz$	${\mathbf{E}}_{\mathbf{x}}={\left[\begin{array}{ccc}0& 1& 1\end{array}\right]}^T$	${\mathbf{S}}_{\mathbf{x}}={\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T$
$xz$	${\mathbf{E}}_{\mathbf{y}}={\left[\begin{array}{ccc}1& 0& 1\end{array}\right]}^T$	${\mathbf{S}}_{\mathbf{y}}={\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T$
$xy$	${\mathbf{E}}_{\mathbf{z}}={\left[\begin{array}{ccc}1& 1& 0\end{array}\right]}^T$	${\mathbf{S}}_{\mathbf{z}}={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T$

.

To ensure the correct direction of rotation, spin functions are introduced. These functions use the components of the vectors $r_{a,b}$ and $r_{a,c}$ to determine the direction:

$$d_x=1-{\displaystyle \frac{2}{exp(\infty ·(ϵ+y_{a,b}·z_{a,c}-y_{a,c}·z_{a,b}))+1}}$$

(4)

$$d_y=1-{\displaystyle \frac{2}{exp(\infty ·(ϵ-x_{a,b}·z_{a,c}+x_{a,c}·z_{a,b}))+1}}$$

(5)

$$d_z=1-{\displaystyle \frac{2}{exp(\infty ·(ϵ+x_{a,b}·y_{a,c}-x_{a,c}·y_{a,b}))+1}}$$

(6)

where $r_{a,b}={\left[\begin{array}{ccc}{x}_{a,b}& y_{a,b}& z_{a,b}\end{array}\right]}^T$, $r_{a,c}={\left[\begin{array}{ccc}{x}_{a,c}& y_{a,c}& z_{a,c}\end{array}\right]}^T$ and $ϵ$ is the epsilon machine “eps” [20]. These spin functions are saturated sigmoid functions of the type:

$$sf\left(x\right)=1-{\displaystyle \frac{2}{1+e^{(x-ϵ)·\infty }}}=\left\{\begin{array}{cc}\hfill 1;\hfill & \hfill x>0,\hfill \\ \hfill -1;\hfill & \hfill x<0,\hfill \\ \hfill 1;\hfill & \hfill x=0.\hfill \end{array}\right.$$

(7)

The function $sf\left(x\right)$ is a saturated sigmoid function, which behaves similarly to a sign function but excludes zero from its results. This ensures that the direction of rotation is unambiguous. The function $sf\left(x\right)$ ensures the continuity of the rotational direction, providing a clear indication of whether the rotation should be clockwise or counterclockwise. By separating the calculation of the angular displacement from the determination of its direction, the method maintains the consistency of rotational sense, avoiding the potential ambiguities that could arise from the direct use of trigonometric functions.

3. Comparative Analysis of Inverse Kinematics Methods

In the field of robotics, the accuracy and robustness of inverse kinematics solutions are considered critical for ensuring precise motion and control. Traditional inverse kinematics methods are typically based on trigonometric functions such as inverse sine (${sin}^{-1}$), inverse cosine (${cos}^{-1}$), and inverse tangent (${tan}^{-1}$) to compute the angles between vectors. These angles are fundamental for determining the joint configurations required to achieve a desired end-effector position. However, inherent limitations in these standard trigonometric functions often lead to significant issues in practical applications, particularly when dealing with complex trajectories or when the system is sensitive to initial conditions.

In this section, a detailed comparative analysis is presented between traditional trigonometric approaches and the Directed Angular Restitution (DAR) method enhanced with spin functions. The objective of this comparison is to highlight the challenges associated with the standard methods and to demonstrate the superior performance of the DAR method in addressing these issues.

The traditional methods of inverse kinematics, rooted in geometric analysis, rely heavily on the use of trigonometric functions. These methods are prevalent because they offer straightforward solutions based on the geometric relationships between joint vectors and desired positions. However, despite their simplicity, these approaches are prone to several significant drawbacks, including:

• Limited Range: Trigonometric functions like ${cos}^{-1}$, ${sin}^{-1}$, and ${tan}^{-1}$ are restricted by their mathematical domains, which can result in incorrect angle calculations when the actual angles fall outside these ranges.
• Sign Ambiguity: Determining the correct sign during the computation of these angles is critical for ensuring the correct direction of motion. A misinterpretation can lead to incorrect rotational directions and potential instability in the robot’s movement.
• Sensitivity to Initial Conditions: The performance of these methods is highly sensitive to initial conditions. Even slight variations in the starting configuration can lead to significant discrepancies in the resulting joint trajectories.

The analysis in this section focuses on two primary scenarios:

(i)

Initial Trajectory Calculations: In these simulations, the inverse kinematics is performed without updating the initial position of the vector $r_{a,b}$. Each point in the desired trajectory is analyzed independently, simulating a non-cumulative approach where the current state does not influence subsequent calculations.

(ii)

Cumulative Trajectory Calculations: In contrast, these simulations involve updating the position of the vector $r_{a,b}$ as the system evolves over time. This cumulative approach reflects a more realistic scenario where the position of the vector is continuously adjusted based on prior movements.

The comparison with standard trigonometric functions is crucial because these functions are widely used in basic geometric analyses for solving inverse kinematics problems. However, they are prone to the issues previously mentioned, which can undermine the accuracy and stability of robotic motion. By comparing these traditional methods with the DAR method, the analysis aims to underscore the importance of addressing these limitations to achieve more accurate and stable motion in robotic applications.

The following comparative analysis is based on the generalized system structure illustrated in Figure 1. This figure, presented in Section 2, outlines the arrangement of the four reference frames: $o_0$, $o_a$, $o_b$, and $o_c$. The system $o_c$ represents the points of three specific trajectories—circular, oval, and logarithmic—each of which is chosen to highlight particular characteristics that are essential for demonstrating the advantages and disadvantages of each function being analyzed. The link $r_{a,b}$ connects the reference frames $o_a$ and $o_b$, while the vector $r_{a,c}$ connects the system $o_a$ with the points along the given trajectory.

The following models for comparison, which assess the effectiveness of the Directed Angular Restitution (DAR) method, are all structured according to this generalized setup. The primary objective is to determine whether the vectors $r_{a,b}$ and $r_{a,c}$ are collinear after applying the rotational transformation as a function of $\mathsf{\Delta }q$. Collinearity is not about making $r_{a,b}$ and $r_{a,c}$ identical but ensuring they point in the same direction. This collinearity is visually represented by the orange dashed lines relative to the link $r_{a,b}$.

Two variants are analyzed for each trajectory type: cumulative displacement and non-cumulative displacement. The cumulative variant involves updating the vector $r_{a,b}$ with each transformation, changing its initial condition with each angular displacement $\mathsf{\Delta }q$. In the non-cumulative variant, $r_{a,b}$ does not accumulate transformations, retaining its initial condition throughout, thus requiring larger angular displacements to achieve collinearity.

For each trajectory and its respective variants, the error between the desired joint trajectory and the calculated joint trajectory is presented. This error analysis is crucial for evaluating the accuracy and effectiveness of the different methods in ensuring that the vectors remain collinear after the transformations are applied. By comparing the errors across the different methods, the significance of the DAR method in maintaining precise and stable motion is underscored.

Figure 2, Figure 3 and Figure 4 present the results of non-cumulative trajectory tracking for circular, oval, and logarithmic trajectories, respectively. In these simulations, it is observed that without the DAR functions, the trajectories exhibit inconsistencies and abrupt changes in the direction of rotation. In contrast, when the DAR functions are applied, the trajectories become smoother and demonstrate a continuous and consistent direction of rotation. This improvement is particularly notable with the directed inverse cosine function (DAR), which provides correct results due to its range of $[0,\pi ]$, compared to the inverse sine and tangent functions, which have ranges of $[-\pi /2,\pi /2]$. The discrepancy in directional changes is clearly visible in the orange dashed lines that represent the connection between the systems $o_b$ and $o_c$. These lines should be collinear with the link represented by the black lines, which remain of constant magnitude. The collinearity of the orange dashed lines indicates a correct calculation of the joint variable.

The importance of employing DAR functions in ensuring smooth and precise movement in robotic applications is underscored by these comparisons. The associated errors between the desired and calculated joint trajectories are shown in Figure 5, Figure 6 and Figure 7. These error analyses further highlight the effectiveness of the DAR method in maintaining accurate trajectory tracking, with significantly lower errors when the DAR functions are applied.

Figure 8, Figure 9 and Figure 10 present the results of cumulative trajectory tracking for circular, oval, and logarithmic trajectories, respectively. In practical trajectory planning, large displacements are uncommon as they tend to induce high acceleration values or uncertainties. Instead, small deltas are preferred to ensure smooth transitions and avoid over-accelerations. In this new simulation, the conditions from the previous experiment were maintained. However, in the cumulative variant, the vector $r_{a,b}$ undergoes continuous transformations, with its initial condition changing at each iteration. This cumulative adjustment reflects more realistic scenarios in robotic applications, where precise alignment and smooth movement are critical.

The associated errors between the desired and calculated joint trajectories are shown in Figure 11, Figure 12 and Figure 13. These error analyses further demonstrate the importance of the DAR functions in maintaining accurate and stable trajectory tracking under cumulative displacement conditions.

The comparative analysis presented across the various trajectory types—circular, oval, and logarithmic—highlights several critical aspects of inverse kinematics in robotic applications. The simulations demonstrate that the standard trigonometric functions, such as inverse sine, inverse cosine, and inverse tangent, are susceptible to significant errors due to their limited range and sensitivity to initial conditions. These limitations often result in inconsistencies and abrupt directional changes, as evidenced in the non-cumulative trajectory tracking scenarios (Figure 2, Figure 3 and Figure 8).

When the DAR functions are applied, the simulations show a marked improvement in the smoothness and consistency of the trajectories. This is particularly evident in the cumulative tracking scenarios (Figure 3, Figure 4 and Figure 10), where the DAR functions help maintain correct alignment even as the initial conditions of the vectors evolve through continuous transformations. The error analyses further underscore the effectiveness of the DAR method in reducing discrepancies between the desired and calculated joint trajectories, ensuring precise and stable motion.

The importance of considering the range of the trigonometric functions is highlighted by the fact that only the directed inverse cosine function (DAR) provides correct results across the full range of $[0,\pi ]$, avoiding the pitfalls of the limited ranges of $[-\pi /2,\pi /2]$ in the inverse sine and tangent functions. Additionally, the sensitivity to initial conditions is mitigated through the use of the DAR method, which allows for more reliable and consistent trajectory tracking across different scenarios.

The DAR method, with its various implementations, is generalized as shown in

Table 2. Variants of the Directed Angular Restitution (DAR) method in different planes.

Type	Angular Delta	Range
DAR in the $yz$ plane	$d_x\left|{cos}^{-1}\left({\displaystyle \frac{\left({\mathbf{E}}_{\mathbf{x}}\otimes r_{a,b}\right)\otimes \left({\mathbf{E}}_{\mathbf{x}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{x}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{x}}\otimes r_{a,c}∥}}\right)\right|$	$[0,\pi ]$
	$d_x\left|{sin}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{x}}·\left({\mathbf{E}}_{\mathbf{x}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{x}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{x}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{x}}\otimes r_{a,c}∥}}\right)\right|$	$[-\pi /2,\pi /2]$
	$d_x\left|{tan}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{x}}·\left({\mathbf{E}}_{\mathbf{x}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{x}}\otimes r_{a,c}\right)}{\left({\mathbf{E}}_{\mathbf{x}}\otimes r_{a,b}\right)·\left({\mathbf{E}}_{\mathbf{x}}\otimes r_{a,c}\right)}}\right)\right|$	$[-\pi /2,\pi /2]$
DAR in the $xz$ plane	$d_y\left|{cos}^{-1}\left({\displaystyle \frac{\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\right)\otimes \left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}∥}}\right)\right|$	$[0,\pi ]$
	$d_y\left|{sin}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{y}}·\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}∥}}\right)\right|$	$[-\pi /2,\pi /2]$
	$d_y\left|{tan}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{y}}·\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}{\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,b}\right)·\left({\mathbf{E}}_{\mathbf{y}}\otimes r_{a,c}\right)}}\right)\right|$	$[-\pi /2,\pi /2]$
DAR in the $xy$ plane	$d_z\left|{cos}^{-1}\left({\displaystyle \frac{\left({\mathbf{E}}_{\mathbf{z}}\otimes r_{a,b}\right)\otimes \left({\mathbf{E}}_{\mathbf{z}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{z}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{z}}\otimes r_{a,c}∥}}\right)\right|$	$[0,\pi ]$
	$d_z\left|{sin}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{z}}·\left({\mathbf{E}}_{\mathbf{z}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{z}}\otimes r_{a,c}\right)}{∥{\mathbf{E}}_{\mathbf{z}}\otimes r_{a,b}∥∥{\mathbf{E}}_{\mathbf{z}}\otimes r_{a,c}∥}}\right)\right|$	$[-\pi /2,\pi /2]$
	$d_z\left|{tan}^{-1}\left({\displaystyle \frac{{\mathbf{S}}_{\mathbf{z}}·\left({\mathbf{E}}_{\mathbf{z}}\otimes r_{a,b}\times {\mathbf{E}}_{\mathbf{z}}\otimes r_{a,c}\right)}{\left({\mathbf{E}}_{\mathbf{z}}\otimes r_{a,b}\right)·\left({\mathbf{E}}_{\mathbf{z}}\otimes r_{a,c}\right)}}\right)\right|$	$[-\pi /2,\pi /2]$

, where all the variants are described. This generalization provides a robust framework for addressing the challenges posed by traditional trigonometric functions, making it a critical tool for ensuring accuracy and stability in robotic applications.

4. Mathematical Model of the Closed-Loop Trajectory

In this section, the mathematical model of a closed-loop trajectory composed of a rotated semi-oval and a straight-line segment is presented. This model is designed to control the movement of a hexapod robot, taking into account the phase shift between different groups of legs. The closed-loop trajectory shown in Figure 14 is a mathematical proposal to represent the displacements discussed in [1,2,3]. This trajectory consists of two types of segments: $\left\{x_1,z_1\right\}$ and $\left\{x_2,z_2\right\}$. The function $x_1\left(t\right)$ represents the x-coordinate of the straight-line segment, while $z_1\left(t\right)$ is its corresponding z-coordinate, which remains zero. The functions $x_2\left(t\right)$ and $z_2\left(t\right)$ represent the x and z coordinates of the rotated semi-oval, respectively. These segments are defined as follows:

$$x_1\left(t\right)=x_2\left({\tau }_2\left(t\right)\right)+\left(x_2\left({\tau }_3\left(t\right)\right)-x_2\left({\tau }_2\left(t\right)\right)\right){\displaystyle \frac{t-{\tau }_2\left(t\right)}{{\tau }_3\left(t\right)-{\tau }_2\left(t\right)}}$$

(8)

$$z_1\left(t\right)=0$$

(9)

$$x_2\left(t\right)=cos\left(\theta \right)acos(\omega t+\varphi )+sin\left(\theta \right)bsin(\omega t+\varphi )$$

(10)

$$z_2\left(t\right)=-sin\left(\theta \right)acos(\omega t+\varphi )+cos\left(\theta \right)bsin(\omega t+\varphi )$$

(11)

where a is the semi-major axis of the semi-oval, b is the semi-minor axis of the semi-oval, $\omega$ is the angular frequency, $\theta$ is the rotation angle of the semi-oval, and t is the time. The compensation angle $\varphi$ is given by:

$$\varphi ={tan}^{-1}\left({\displaystyle \frac{a}{b}}tan\left(\theta \right)\right).$$

(12)

The interception times ${\tau }_1\left(t\right)$, ${\tau }_2\left(t\right)$, and ${\tau }_3\left(t\right)$, which mark the transitions between different segments of the trajectory, are defined as follows:

$${\tau }_1\left(t\right)={\displaystyle \frac{2k\left(t\right)\pi }{\omega }}$$

(13)

$${\tau }_2\left(t\right)={\displaystyle \frac{2k\left(t\right)\pi }{\omega }}+{\displaystyle \frac{\pi }{\omega }}$$

(14)

$${\tau }_3\left(t\right)={\displaystyle \frac{2\left(k\right(t)+1)\pi }{\omega }}.$$

(15)

In these expressions, to manage the periodic nature of the trajectory, a cycle function $k\left(t\right)$ is defined as:

$$k\left(t\right)=⌊{\displaystyle \frac{\omega t}{2\pi }}⌋.$$

(16)

To control the phase shift between the two groups of legs of the hexapod robot, we define the upper time for group 1 and the lower time for group 2 as:

$$t_{g1}={\displaystyle \frac{3\pi }{2\omega }}$$

(17)

$$t_{g2}={\displaystyle \frac{1}{\omega }}\left({tan}^{-1}\left(-{\displaystyle \frac{cos\left(\theta \right)a}{sin\left(\theta \right)b}}\right)-{tan}^{-1}\left({\displaystyle \frac{a}{b}}tan\left(\theta \right)\right)+\pi \right).$$

(18)

Time vector t is then adjusted to include the phase shift $\eta$:

$$t^{\prime }=t+t_{g2}+\eta (t_{g1}-t_{g2}).$$

(19)

Finally, the trajectory is plotted by combining the semi-oval and the straight-line segments. For each time step t, the position vector $p\left(t\right)$ is calculated as:

$$p\left(t\right)=\left\{\begin{array}{cc}\hfill {\left[\begin{array}{ccc}{x}_2\left(t\right)& 0& z_2\left(t\right)\end{array}\right]}^T,\hfill & \hfill \mathrm{if}{\tau }_1\left(t\right)\le t\le {\tau }_2\left(t\right)\hfill \\ \hfill {\left[\begin{array}{ccc}{x}_1\left(t\right)& 0& z_1\left(t\right)\end{array}\right]}^T,\hfill & \hfill \mathrm{if}{\tau }_2\left(t\right)<t\le {\tau }_3\left(t\right).\hfill \end{array}\right.$$

(20)

This model provides a comprehensive framework for generating closed-loop trajectories with precise control over the phase shifts between different groups of legs, ensuring smooth and coordinated movement for hexapod robots.

Previous studies have explored similar trajectories for hexapod robots to evaluate their locomotion capabilities and control algorithms. For instance, in [1], a biologically inspired locomotion pattern was proposed for a lightweight hexapod robot, emphasizing the use of curved and linear paths to mimic natural insect movements. In another study, ref. [2], a neural network-based approach was used to replicate the kinematics of hexapod locomotion, utilizing both curved and straight trajectories to test the system’s adaptability and efficiency. Additionally, ref. [3] discussed the application of evolutionary algorithms to find the most effective movement patterns for hexapod robots, including the use of semi-circular paths to optimize speed and stability.

These previous works justify the choice of a rotated oval and linear trajectory in our study, as they provide a comprehensive framework to evaluate the performance of the Directed Angular Restitution (DAR) method in maintaining precise vector alignment through both curved and linear segments. The chosen trajectory ensures that the DAR method is rigorously tested under various movement scenarios, validating its application in complex robotic movements.

5. Kinematic Analysis of the Hexapod Robot

In this section, a detailed kinematic analysis of the hexapod robot is presented. The analysis is conducted on the HexV3-MythBot model, as depicted in Figure 15. This model provides a comprehensive platform for examining the kinematic behavior of hexapod robots, allowing for the evaluation of both direct and inverse kinematics.

The geometrical structure of a single leg of the hexapod robot is shown in Figure 16. The analysis begins with the direct kinematics, where the position and orientation of the leg end-effector are derived based on the joint parameters. This involves calculating the forward transformation matrices that relate the base frame to the end-effector frame.

Following this, the joint equations are defined using inverse kinematics, incorporating the Directed Angular Restitution (DAR) method to ensure accurate and smooth movements of the robot. The inverse kinematics process involves determining the necessary joint angles that achieve a desired end-effector position and orientation, starting from the specified target and working backward to the joint parameters. This section provides a step-by-step breakdown of the kinematic equations and their application to the HexV3-MythBot, highlighting the effectiveness of the DAR method in solving complex kinematic problems.

5.1. Direct Kinematics

In this subsection, the direct kinematics of the hexapod robot are analyzed. Figure 17 shows the free-body diagram of the leg joints and the closed-loop trajectory whose origin corresponds to the system $o_5$. The equations for the direct kinematics are as follows.

The vector describing the kinematic chain $o_0\prec o_1\prec o_2\prec o_3\prec o_4\prec o_5$ is ${\mathbf{r}}_{0,5}$. Therefore, the formula is:

$${\mathbf{r}}_{i,j}={\mathbf{r}}_{i,j-1}+{\mathbf{R}}_{i,j-1}{\mathbf{r}}_{j-1,j}.$$

(21)

The position vectors and rotation matrices for different points in the hexapod leg and the corresponding rotation matrices are summarized in

Table 3. Direct kinematics equations, recursive notation.

Point	Position Vector	Matrix	Orientation
${\mathbf{r}}_{0,1}$	${\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$	$R_{0,1}$	$R_z\left(q_4\right)$ *
${\mathbf{r}}_{1,2}$	${\left[\begin{array}{ccc}{x}_{1,2}& 0& 0\end{array}\right]}^T$	$R_{1,2}$	$R_z\left(q_1\right)$
${\mathbf{r}}_{2,3}$	${\left[\begin{array}{ccc}{x}_{2,3}& 0& 0\end{array}\right]}^T$	$R_{2,3}$	$R_y\left(q_2\right)$
${\mathbf{r}}_{3,4}$	${\left[\begin{array}{ccc}{x}_{3,4}& 0& z_{3,4}\end{array}\right]}^T$	$R_{3,4}$	$R_y\left(q_3\right)$
${\mathbf{r}}_{4,5}$	${\left[\begin{array}{ccc}{x}_{4,5}& 0& z_{4,5}\end{array}\right]}^T$	$R_{4,5}$	I
${\mathbf{r}}_{1,6}$	${\left[\begin{array}{ccc}{x}_{1,6}& 0& z_{1,6}\end{array}\right]}^T$	$R_{1,6}$	$R_z(q_5-q_4+{\displaystyle \frac{\pi }{2}})$
${\mathbf{r}}_{6,p}$	${\left[\begin{array}{ccc}{p}_x& p_y& p_z\end{array}\right]}^T$	$R_{6,p}$	I
${\mathbf{r}}_{0,2}$	${\mathbf{r}}_{0,1}+R_{0,1}{\mathbf{r}}_{1,2}$	$R_{0,2}$	$R_{0,1}{R}_{1,2}$
${\mathbf{r}}_{0,3}$	${\mathbf{r}}_{0,2}+R_{0,2}{\mathbf{r}}_{2,3}$	$R_{0,3}$	$R_{0,2}{R}_{2,3}$
${\mathbf{r}}_{0,4}$	${\mathbf{r}}_{0,3}+R_{0,3}{\mathbf{r}}_{3,4}$	$R_{0,4}$	$R_{0,3}{R}_{3,4}$
${\mathbf{r}}_{0,5}$	${\mathbf{r}}_{0,4}+R_{0,4}{\mathbf{r}}_{4,5}$	$R_{0,5}$	$R_{0,4}{R}_{4,5}$
${\mathbf{r}}_{0,6}$	${\mathbf{r}}_{0,1}+R_{0,1}{\mathbf{r}}_{1,6}$	$R_{0,6}$	$R_{0,1}{R}_{1,6}$
${\mathbf{r}}_{0,p}$	${\mathbf{r}}_{0,1}+R_{0,6}{\mathbf{r}}_{6,p}$	$R_{0,p}$	$R_{0,6}{R}_{6,p}$

* This is a unique transformation for each leg, it is not part of the inverse kinematics analysis, where it is assumed as $R_z\left(0\right)$.

.

In this case, the variable $q_1$ represents a rotation around the z-axis in the system $o_1$, while $q_2$ and $q_3$ correspond to rotations around the y-axis in the systems $o_3$ and $o_4$, respectively. The variable $q_4$, however, is not a joint variable but it is used to represent the different legs of the robot. The values of $q_4$ are multiples of ${360}^{\circ }/6$, starting from $0^{\circ }$, with each value denoting a specific leg. This approach enables the analysis of a single leg while accounting for the variations represented by the different values of $q_4$. Additionally, $q_5$ is used to adjust the orientation of the trajectory by rotating it around the z-axis in the system $o_6$.

In addition to the primary kinematic chain, there is a secondary kinematic chain formed by the systems $o_0$, $o_6$, and the point p on the trajectory. Here, ${\mathbf{r}}_{6,p}$ represents the position vector of the point p in the system $o_6$, and ${\mathbf{R}}_{1,p}$ is the rotation matrix that defines the orientation of the point p relative to the system $o_1$. This rotation is determined by the angle $(q_5-q_4+{\displaystyle \frac{\pi }{2}})$, which accounts for the orientation difference between $q_5$ and $q_4$. The position vector ${\mathbf{r}}_{0,6}$ describes the location of the system $o_6$ relative to the base system $o_0$. This vector is defined with respect to the geometric constraints of the robot, as it represents the portion of the trajectory that each leg must follow. Similar to the parameters a and b of the trajectory, ${\mathbf{r}}_{0,6}$ must ensure that the trajectory exists within a safe operating zone for all rotations, avoiding collisions during the robot’s movement. This is crucial for maintaining a valid trajectory throughout the robot’s operation.

5.2. Inverse Kinematics

In this subsection, the inverse kinematics problem for the hexapod robot is addressed. The goal of inverse kinematics is to determine the joint angles $q_1$, $q_2$, and $q_3$ such that the position of the end effector ${\mathbf{r}}_{0,5}$ matches the desired position ${\mathbf{r}}_{0,p}$ on the trajectory. By applying the Directed Angular Restitution (DAR) method, the joint angles can be calculated to ensure the correct alignment and movement of the hexapod leg. The process involves solving for $q_1$, $q_3$, and $q_2$ sequentially, ensuring that the resulting configuration achieves the target position while maintaining smooth and continuous motion.

First, to obtain $\mathsf{\Delta }{q}_1$, the vectors ${\mathbf{r}}_{2,5}$ and ${\mathbf{r}}_{2,p}$ are calculated as:

$$\begin{array}{cc}\hfill {\mathbf{r}}_{2,5}& \hfill ={\mathbf{r}}_{0,5}-{\mathbf{r}}_{0,2},\hfill \\ \hfill {\mathbf{r}}_{2,p}& \hfill ={\mathbf{r}}_{0,p}-{\mathbf{r}}_{0,2},\hfill \end{array}$$

(22)

$$\mathsf{\Delta }{q}_1=d_z\left|{cos}^{-1}\left({\displaystyle \frac{x_{2,5}{x}_{2,p}+y_{2,5}{y}_{2,p}}{∥x_{2,5},y_{2,5}∥∥x_{2,p},y_{2,p}∥}}\right)\right|.$$

(23)

where these vectors are the focal point of the DAR method. Next, the following equation is applied (23), which represents the directed angular compensation in the $xy$ plane. For a better understanding of these expressions, refer to Figure 18.

Once $\mathsf{\Delta }{q}_1$ is defined, the posture is updated so that the calculation of $\mathsf{\Delta }{q}_3$ takes this modification into account. The vectors ${\mathbf{r}}_{3,5}$ and ${\mathbf{r}}_{3,p}$ are calculated next, (24). The analysis of $\mathsf{\Delta }{q}_3$ involves adjusting the magnitude of the vector ${\mathbf{r}}_{3,5}$ to match the magnitude of ${\mathbf{r}}_{3,p}$. This adjustment is performed using Equation (25), which represents the difference between the current angle $\beta$ and the angle $\beta$ that ensures the equality of both magnitudes. This approach is illustrated in Figure 19.

$$\begin{array}{cc}\hfill {\mathbf{r}}_{3,5}& \hfill ={\mathbf{r}}_{0,5}-{\mathbf{r}}_{0,3},\hfill \\ \hfill {\mathbf{r}}_{3,p}& \hfill ={\mathbf{r}}_{0,p}-{\mathbf{r}}_{0,3},\hfill \end{array}$$

(24)

$$\mathsf{\Delta }{q}_3={cos}^{-1}\left({\displaystyle \frac{{∥r_{4,5}∥}^2+{∥r_{3,4}∥}^2-{∥r_{3,p}∥}^2}{2∥r_{4,5}∥∥r_{3,4}∥}}\right)-{cos}^{-1}\left({\displaystyle \frac{{∥r_{4,5}∥}^2+{∥r_{3,4}∥}^2-{∥r_{3,5}∥}^2}{2∥r_{4,5}∥∥r_{3,4}∥}}\right).$$

(25)

The posture is updated with the value of $\mathsf{\Delta }{q}_3$. Finally, $\mathsf{\Delta }{q}_2$ is determined as (26). This expression involves a directed angular compensation in the $xz$ plane (see Figure 20).

$$\mathsf{\Delta }{q}_2=d_y\left|{cos}^{-1}\left({\displaystyle \frac{x_{3,5}{x}_{3,p}+z_{3,5}{z}_{3,p}}{∥x_{3,5},z_{3,5}∥∥x_{3,p},z_{3,p}∥}}\right)\right|.$$

(26)

With $\mathsf{\Delta }{q}_2$ calculated, the inverse kinematics problem is fully resolved, providing the necessary joint angles $q_1$, $q_2$, and $q_3$ to position the hexapod leg correctly. By applying the Directed Angular Restitution (DAR) method, the angles are obtained in a manner that ensures smooth and continuous motion, while also maintaining precise alignment with the desired trajectory. The sequential approach, starting from $\mathsf{\Delta }{q}_1$ and progressing through $\mathsf{\Delta }{q}_3$ to $\mathsf{\Delta }{q}_2$, allows for a systematic adjustment of the leg’s posture, accounting for each rotational transformation required to achieve the target position ${\mathbf{r}}_{0,p}$. This methodology not only enhances the accuracy of the hexapod’s movements but also optimizes its performance in dynamic environments.

6. Results and Simulations

In this section, the complete model of the HexV3-Mythbots robot (Mythbots, Guadalajara, Jalisco, Mexico) is utilized, as shown in Figure 15. The geometry of each leg is detailed in Figure 16. Each leg is differentiated by the orientation defined by $q_4$ with respect to the z axis in the $o_0$ system.

Figure 21 illustrates the omnidirectional kinematics of the hexapod robot for different values of $q_5$. The end-effector trajectories of each leg are depicted on the surface. As discussed, these are closed-loop paths composed of a rotated oval and a straight-line segment (refer to Section 4). The overall direction of movement is the result of all these trajectories, and this direction is indicated by an arrow placed above the robot.

The parameter $q_5$ is the one that should be adjusted to change the orientation of the robot. For example, to make the robot rotate on its own axis in a counterclockwise direction, set $q_5=0^{\circ }$, so that the trajectories are perpendicular to the vector ${\mathbf{r}}_{0,6}$. To rotate the robot in a clockwise direction, set $q_5=\pi$. To control the robot’s speed, the parameter $\omega$ presented in Equations (8)–(11) should be adjusted.

Figure 22 presents the locomotion of the Hex-Kurumy robot, a hexapod with a morphology similar to the HexV3-Mythbots. Hex-Kurumy employs the DAR method and is controlled remotely by a joystick, which adjusts the variable $q_5$ and the angular velocity parameter $\omega$.

Figure 23, Figure 24, Figure 25 and Figure 26 show the behavior of the joint variables for different functions of $q_5$. Note that the peak values presented in Figure 23 and Figure 24 are attenuated with smooth changes of the variable $q_5$, as shown in Figure 25 and Figure 26.

The presented simulations and experimental results confirm the efficacy of the Directed Angular Restitution (DAR) method in controlling the locomotion of hexapod robots. By adjusting the parameters $q_5$ and $\omega$, the HexV3-Mythbots and Hex-Kurumy robots demonstrated smooth and coordinated movement across various trajectories. The ability to manipulate $q_5$ for directional control and $\omega$ for speed regulation was crucial in achieving the desired motion patterns. These findings highlight the potential of the DAR method for practical applications in robotic locomotion, paving the way for further developments and optimizations in future research.

7. Stability and Support Polygons

The stability of a hexapod robot’s motion is primarily ensured by maintaining the center of mass (CoM) within the support polygon formed by the legs in contact with the ground. During omnidirectional walking, tripod gaits are employed, where three legs are in contact with the ground at any time, forming a triangular support polygon. The variation in $q_5$, which defines the robot’s orientation and direction, contributes to generating a smooth trajectory, as illustrated in Figure 27. However, the key factor in ensuring that the CoM remains within the support polygons, thus maintaining stability and avoiding collisions, lies in the geometric design of the closed-loop trajectory.

The trajectory design in this study utilizes the following parameters:

• $a=40$ mm—Semi−major axis of the elliptical trajectory.
• $b=20$ mm—Semi−minor axis of the elliptical trajectory.
• $\omega =2\pi$ rad/s—Angular frequency of the trajectory.
• $\theta =\pi /4$ rad—Rotation angle defining the trajectory’s orientation.
• ${\eta }_1=0$—Phase shift constant for the first group of legs, ranging from 0 to 1.
• ${\eta }_2=1$—Phase shift constant for the second group of legs, ranging from 0 to 1.

These parameters define the shape and orientation of the closed-loop trajectory, ensuring that it fits within the robot’s workspace and supports stable locomotion.

Figure 27 illustrates the perimeters of the support polygons during an omnidirectional walk of the hexapod robot. The polygons shown represent the space covered by the legs in contact with the ground as the trajectory parameter $q_5$ varies linearly. These results correspond to kinematic walks performed on a uniform surface, which is essential for understanding the behavior of the support polygons under controlled conditions.

In this article, the frequency of this tripod gait pattern is adjusted by modifying the parameter $\eta$. This adjustment is reflected in the time vector t, which includes a phase shift $\eta$ as described in (19). This modification allows for precise control over the timing of the leg movements, ensuring that the robot’s trajectory remains smooth and consistent with the desired gait pattern (see

Table 4. Sequence diagram of tripod gait during a walking cycle.

Phase	Leg 1	Leg 2	Leg 3	Leg 4	Leg 5	Leg 6
1	Swing	Support	Swing	Support	Swing	Support
2	Support	Swing	Support	Swing	Support	Swing
3	Swing	Support	Swing	Support	Swing	Support
4	Support	Swing	Support	Swing	Support	Swing

).

Let ${\mathrm{Leg}}_i$ represent the position of leg i in a local reference frame of the robot. For a set of three legs in contact with the ground, the support polygon is defined by the vertices $({\mathrm{Leg}}_1,{\mathrm{Leg}}_3,{\mathrm{Leg}}_5)$ for one set of legs and $({\mathrm{Leg}}_2,{\mathrm{Leg}}_4,{\mathrm{Leg}}_6)$ for the complementary set. The condition for stability is established when the projection of the CoM onto the ground plane, denoted as $\mathbf{C}$, is contained within the area of these polygons, such that:

$$\mathrm{Stability}\equiv \mathbf{C}\in \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{Support}\mathrm{Polygon}.$$

(27)

The design of the closed-loop trajectory is critical for maintaining stability throughout the robot’s movement. While the variation in the parameter $q_5$ contributes to a smooth trajectory, it is the comprehensive design of the closed-loop trajectory that ensures that the CoM remains within the support polygons, preventing collisions and guaranteeing stability. This design must also consider the feasible and safe workspace of the robot, avoiding self-collisions and maintaining the stability condition defined in Equation (27). The importance of these considerations is emphasized in studies such as [5,7], where the geometric design of the trajectory directly influences the robot’s stability.

While the Directed Angular Restitution (DAR) method provides precise control over the angular orientation of the legs, it is the closed-loop trajectory design that plays a fundamental role in ensuring stability. Traditional approaches, such as those described in [13], focus on predefined gait patterns, which may not dynamically adjust to changes in the robot’s environment. In contrast, the trajectory design approach, as discussed in [6,15], allows for continuous adaptation to varying terrains, ensuring that the CoM remains within the support polygons even under challenging conditions.

In summary, the stability of the hexapod robot is ensured by the proper application of support polygon principles in combination with the careful design of the closed-loop trajectory. This design ensures that the CoM remains within the support polygons, avoiding collisions and ensuring robust locomotion. This approach, as supported by [8,13], and others, is essential for achieving stable and adaptable walking in dynamic environments.

8. Conclusions

This study presented an in-depth analysis of the Directed Angular Restitution (DAR) method and its application to the locomotion of hexapod robots. The DAR method was developed to address the challenges of inverse kinematics and ensure smooth and continuous movement, allowing hexapod robots to achieve precise control over their trajectories.

The HexV3-Mythbots model served as an exemplary platform to validate the DAR method. Detailed simulations demonstrated the ability of the robot to perform omnidirectional locomotion, with the capability to adjust its orientation and speed through the parameters $q_5$ and $\omega$. This flexibility allows the robot to maneuver effectively in any direction, highlighting the robustness of the method in controlling multi-legged robots.

Experimental results with the Hex-Kurumy robot further validated the practical application of the DAR method. The remote control via a joystick, allowing for real-time adjustments of $q_5$ and $\omega$, showcased the ease with which the movement of the robot can be fine-tuned to meet various operational needs. The ability to switch between different motion patterns seamlessly underscored the adaptability and effectiveness of the method in real-world scenarios.

In conclusion, the directed angular restitution method provides a significant advancement in the control of hexapod robots, enabling precise, smooth, and omnidirectional movement. The contributions of this work include the development and validation of the DAR method for inverse kinematics, the demonstration of enhanced omnidirectional locomotion capabilities, and practical implementation in physical robots, confirming its applicability and robustness. This work paves the way for further research into optimizing the DAR method for more complex terrains and integrating additional sensory inputs to enhance environmental interaction and adaptability.