2.1. Unipedal workspace analysis of the hexapod robot

When analyzing the unipedal workspace of the hexapod bionic robot, the hardware structure was first converted into a 3D parametric model to determine the relationship between structural units. The unipedal spatial model is shown in Figure 1. In the hardware configuration, joint servos D ₀, D ₁, and D ₂ resemble the hip, knee, and ankle joints of organic limbs, respectively. The rotation angles of these joints and the link lengths between them directly define the spatial position of the foot end (Point O ₃), which provides a fundamental basis for leg trajectory planning and real-time posture estimation.

Figure 1.

Unipedal D−H coordinate system.

As the experimental robot adopts a cross-symmetrical leg configuration centered at the center of mass, each leg is identical in structure and function. Thus, a unipedal analysis is sufficient to represent the entire robot’s leg kinematics, enabling a modular approach to gait generation and stability analysis under hybrid disturbances. Before establishing the D–H parameter table for the unipedal kinematic chain, each parameter is defined as follows: θ _n denotes the rotation of the X_{n -1} axis about the Z_{n -1} axis to align with X_n , δn the rotation of Z_{n -1} about X_{n -1} to align with Z_n , the link length between Z_{n -1} and Z_n , and d _n the offset along the Z-axis between successive frames. The D–H coordinate system setup is illustrated in Figure 1.

The D–H coordinate axes are determined as follows: the Z ₀ axis is chosen first because all base servos lie on a common horizontal plane. Based on the unipedal spatial configuration, the direction of the D ₀ servo drive axis – aligned with the robot’s chassis plane – is designated as the Z ₀ axis, and the O ₀ coordinate system is defined so that the X ₀−Z ₀ plane is perpendicular to the robot’s main plate. The direction of the D ₁ servo axis defines the Z ₁ axis of the O ₁ coordinate system, and the linkage between D ₁ and D ₂ determines the X ₁ axis. The O ₂ coordinate system is established similarly, with the Z ₂ and X ₂ axes derived from the orientation of D ₂ and the next connecting link. Finally, the O ₃ system is anchored at the foot tip, with Z ₃ and X ₃ aligned in direction and level with the preceding O ₂ system, ensuring coordinate consistency across the entire unipedal chain.

This modeling process provides the structural foundation for mapping leg motion to body stability dynamics and is essential for implementing the posture feedback control strategy described later in this paper [Reference Saranli, Buehler and Koditschek30].

The D–H coordinate systems for each leg of the hexapod robot can be replicated across all limbs due to their standardized mechanical structures and identical servo configurations. Based on the coordinate system illustrated in Figure 1, the unipedal kinematic model is established by applying classical D–H methodology. The corresponding parameters are systematically organized into a unified unipedal D–H parameter table, in accordance with the previously defined variable specifications (Table I). This unified representation enables scalable control design and efficient inverse kinematics computation for real-time gait execution.

Table I.

Unipedal D–H parameter table.

¹ Variables measured in degrees (°).

² Variables measured in centimeters (cm).

To facilitate the subsequent computation of each servo’s rotation angle, the trigonometric functions are redefined for notational simplicity. Specifically, we define Cos(θ_n ) as C_n and Sin(θ_n ) as S_n , thereby enabling compact analytical expressions without altering the mathematical accuracy or final outcomes. By substituting the corresponding parameters into the unipedal D–H coordinate system illustrated in Figure 1, the rotation angles for the three joint servos are derived and expressed as θ ₁, θ ₂, and θ ₃, respectively, as shown in Eqs. (3)–(5).

(3) \begin{align}\theta_1&=arctan(p_y/p_x)\end{align}

(4) \begin{align}\theta_2&=arccos\frac{(C_3a_2+a_2)(p_z-d_1)-(C_1p_x+S_1p_y-a_1)S_3a_3}{(a_3C_3+a_2)^2+a_3^2S_3^2}\end{align}

(5) \begin{align}\theta_3&=arccos\frac{p_x^2+p_y^2-2\times a_1\times\sqrt{p_x^2+p_y^2}+a_1^2+p_z^2-2\times p_z\times d_1+d_1^2-a_3^2-a_2^2}{2\times a_2a_3}\\[6pt] \nonumber \end{align}

The expressions of p _x , p _y , and p _z are shown in Eqs. (6)–(8):

(6) \begin{align} p_{x}&=a_{3}C_{1}C_{2}C_{3}-a_{3}C_{1}S_{2}S_{3}+a_{2}C_{1}C_{2}+a_{1}C_{1} \end{align}

(7) \begin{align} p_{y}&=a_{3}S_{1}C_{2}C_{3}-a_{3}S_{1}S_{2}S_{3}+a_{2}S_{1}C_{2}+a_{1}S_{1} \end{align}

(8) \begin{align} p_{z}&=a_{3}S_{2}C_{3}-a_{3}C_{2}S_{3}+a_{2}S_{2}\\[6pt] \nonumber \end{align}

With the rotation angles of all joints determined, the foot-end position of the unipedal model can be calculated via the forward kinematics solution based on classical formulations [Reference Soliman, Ribeiro, Gan and Rastgaar31–Reference Zhang, Liu and Tan33]. By applying the D–H parameterization method, the unipedal spatial model not only enables analytical mapping of joint angles to Cartesian space but also provides quantitative support for the development of the hexapod robot’s gait library. This modeling process ensures the accuracy and repeatability of generated gaits, thereby forming the foundational architecture for the gait control system, which is an integral component of the proposed control strategy.

2.2. Design and optimization of gait library

The constructed hardware experimental platform for the hexapod robot integrates both the mechanical system architecture and the basic gait library, which collectively form a crucial component of the proposed attitude stability control framework. This integrated platform not only enables algorithm verification but also serves as the physical foundation for the control system implementation. Grounded in the mathematical modeling of the hexapod robot from a bionic perspective, a set of action group-based gait patterns is designed to satisfy motion stability requirements under various disturbance conditions.

A heuristic–analytic–refinement scheme is employed to optimize gait parameters. Heuristic rules initialize the cost-function weights based on stability sensitivity; analytical constraints derived from the zero-moment point (ZMP) criterion define the feasible parameter space; and a deterministic refinement process is subsequently applied to identify the optimal gait configuration.

Each gait is parameterized analytically from the unipedal kinematics to ensure that the ZMP remains within the support polygon throughout the support phase. Within this analytically constrained parameter space, candidate gait configurations are systematically evaluated according to the defined cost function until convergence is achieved or no further improvement is observed:

(9) \begin{equation} J=\omega _{1}| \Delta \phi| +\omega _{2}| \Delta \theta | +\omega _{3}T_{cmd} \end{equation}

The weighting coefficients in the cost function are initialized according to heuristic stability-prioritization principles. Terms that directly influence static stability, such as support polygon margin and body inclination, are assigned higher relative weights, whereas secondary objectives related to smoothness or efficiency are weighted lower. This heuristic initialization ensures that stability-related criteria dominate the optimization objective while preserving numerical robustness.

Where ${\unicode[Arial]{x0394}} \phi ,{\unicode[Arial]{x0394}} \theta$ denote attitude error bounds and $T_{cmd}$ is command duration. This formulation allows theoretical convergence toward minimal attitude deviation without requiring stochastic search or physical iteration.

While the mathematical model provides the theoretical foundation for control algorithm design, the hardware experimental platform offers the necessary empirical validation environment for implementation and performance testing. The gait library invoked by the attitude stability control system consists of multiple predefined action group modules, each associated with specific disturbance scenarios and recovery strategies. The structural configuration of the gait library, along with the servo ID coding schematic, is illustrated in Figure 2.

Figure 2.

Gait analysis and design of the hexapod robot. (a) Structure of the gait library. (b) Schematic diagram of the ID coding of the servos.

Each action group library corresponds to a discrete movement primitive, allowing the hexapod robot to execute one complete motion cycle. These action groups are classified into two categories: closed-loop action groups – such as High-Posture Forward, Low-Posture Backward, and Medium-Posture Turn Left – and open-loop action groups, including No-Displacement Down and Reset Recovery. Closed-loop action groups are designed to ensure continuity between sequential frames and are constructed by decomposing each robot movement into multiple single-frame actions, where each frame contains synchronized servo instructions. In contrast, open-loop groups define only the final pose without imposing continuity constraints between frames.

The motion state of the hexapod robot is determined by the rotation angles and speeds of the 18 servos distributed across its 6 legs. As such, each single-frame action must include a servo-level control database, aggregating three core parameters for each servo: pluse width modulation (PWM), Bias, and Time. The PWM value directly encodes the servo’s rotation angle, with operational values ranging from 500 to 2,500, corresponding to physical rotation angles of 0° to 270°. However, due to mechanical limitations, the effective PWM range is narrower. The Bias term refers to a constant calibration offset, which ensures that all servos maintain a consistent default angle in the reset state and serves as the baseline for control alignment.

The Time parameter, expressed in milliseconds, defines the actuation period between consecutive frames, determining servo speed. A larger Time value reflects slower rotational movement, allowing for adjustable action tempo (e.g., fast vs. slow responses). Each joint’s Time value can be independently configured, enabling motion diversity and gait complexity. To maintain inter-frame synchronization, the maximum Time value within a single-frame instruction set is used as the timing reference. For instance, if the Time values for servos 000, 015, and 023 are 100, 200, and 1000 ms, respectively, the next action frame is not issued until 1000 ms have elapsed, even if the other servos complete earlier.

Considering the robot’s multiple leg configuration, a core design principle of the experimental platform is inter-leg coordination, ensuring gait consistency and bio-inspired parallel motion. This necessitates the introduction of a gait framework, which governs the order and timing of leg movements along predefined trajectories [Reference Chen, San, Wu and Xiong34]. Accordingly, the construction of a gait library is essential. Fundamental descriptors such as gait period, duty factor, gait mode, and stride length are used to characterize and implement the hexapod robot’s locomotion patterns [Reference Kapsalyamov, Hussain, Goecke, Brown and Jamwal35]. Recent studies also emphasize integrating gait transitions and time synchronization in real-time embedded control, highlighting the necessity of servo-level timing alignment and priority-based scheduling [Reference Kalra, Singla and Saini36, Reference Martinez and Tsai37].

During continuous locomotion, the gait periodicity (Figure 2a) enforces the ZMP criterion:

(10) \begin{equation} x_{\mathrm{zmp}}=\frac{\sum _{i-1}^{6}m_{i}(\ddot{z}_{i}+g)x_{i}-\sum _{i-1}^{6}m_{i}\ddot{x}_{i}z_{i}}{\sum _{i-1}^{6}m_{i}(\ddot{z}_{i}+g)} \end{equation}

The ZMP criterion is incorporated as an analytical constraint to ensure static stability during gait execution. Specifically, the projected ZMP is required to remain within the support polygon throughout each support phase, thereby restricting the admissible range of gait parameters and eliminating unstable configurations from the optimization space. The “Time” parameter in Table I controls the step duration to maintain this condition. For example, the “Mid-Pose Fast Forward” gait (Table III) exemplifies this principle by synchronizing leg movements with center-of-mass adjustment during attitude correction.

While this study primarily addresses static attitude stability, the gait library design inherently supports dynamic balancing during continuous movement. Each gait action is planned to ensure the robot’s center of mass remains within the support polygon formed by the legs in contact with the ground. This approach, based on the static stability margin principle, provides a foundation for dynamic stability that will be explored in future research.

2.3. Optimization of the pose determination strategy

The inertial measurement unit (IMU) coordinate system of the designed attitude sensor is precisely aligned with the body-fixed coordinate system of the hexapod robot, allowing direct mapping between sensor readings and robot orientation. This alignment facilitates the use of the right-hand rule to determine the positive rotational direction for each axis: pointing the right thumb along the positive direction of an axis, the curl of the remaining fingers indicates the positive direction of angular rotation. Accordingly, a positive angular offset along the X-axis corresponds to a forward tilt, while a positive offset along the Y-axis indicates a rightward tilt.

The drawback of using a gyroscope for attitude estimation lies in its insufficient integration accuracy, while accelerometer-based estimation is prone to interference from high-frequency noise. Therefore, data fusion via Kalman filtering is required during attitude estimation to achieve optimal results. To improve robustness under hybrid disturbances, a quaternion-based extended Kalman filter (EKF) is used for real-time attitude estimation:

(11) \begin{equation} x=\left[\frac{q}{b_{g}}\right] \end{equation}

With quaternion q and gyro-bias b _g . Prediction:

(12) \begin{equation} \hat{q}_{k+1}^{-}=quat\_ \textit{propagate}(\hat{q}_{k}^{+},\varpi _{k}-\hat{b}_{g,k}) \end{equation}

(13) \begin{equation}\hat{b}_{g,k+1}^{-}=\hat{b}_{g,k}^{+} \end{equation}

Update:

(14) \begin{align} K_{k+1}&=P_{k+1}^{-}H_{k+1}^{T}(H_{k+1}P_{k+1}^{-}H_{k+1}^{T}+R_{k+1})^{-1} \end{align}

(15) \begin{align} \hat{x}_{k+1}^{+}&=\hat{x}_{k+1}^{-}+K_{k+1}(y_{k+1}-h(\hat{x}_{k+1}^{-}))\\[10pt] \nonumber\end{align}

Tuning: $Q_{\varpi }\in [10^{-6},10^{-3}] , R_{acc}\in [10^{-3},10^{-1}]$ .

The quaternion-based EKF is configured with process covariance $Q=\text{diag}(q_{att},q_{bias})$ and measurement covariance $R=\text{diag}(r_{acc})$ . The tuning principle follows $q_{att}\ll r_{acc}$ to prioritize short-term inertial consistency while rejecting accelerometer noise. The filter converges when attitude residuals satisfy $| \phi _{k}-\phi _{k-1}| \lt 0.5^{\circ }$ and $| \theta _{k}-\theta _{k-1}| \lt 0.5^{\circ }$ for at least three cycles. This design ensures a balance between responsiveness and noise suppression without over-smoothing and has been validated to yield consistent attitude trajectories under laboratory hybrid disturbances. The tuned EKF reduced attitude-angle variance from 0.21 to 0.095 and improved classifier robustness under vibration.

In an ideal disturbance-free scenario, pose identification can be achieved using single-threshold values A and B for the X-axes and Y-axes, respectively. However, in the presence of hybrid environmental noise, this single-threshold method lacks robustness and may lead to misclassification or unintended gait responses. To improve noise resilience, a double-threshold mechanism is introduced, enabling more reliable pose classification across disturbance boundaries (Table II). In this approach, threshold intervals [A1, A2] and [B1, B2] are defined such that A1 $\lt$ 0 $\lt$ A2, A1 $\lt$ 0 $\lt$ A2, A1 $\lt$ 0 $\lt$ A2 and B1 $\lt$ 0 $\lt$ B2, B1 $\lt$ 0 $\lt$ B2, B1 $\lt$ 0 $\lt$ B2, where minor fluctuations within the tolerance band do not trigger gait switching, thus reducing false positives in control execution. The double-threshold classification requires only four floating-point comparisons per control cycle (200 Hz), consuming $\lt$ 1% of the STM32H743’s computational budget. PWM generation and servo updates are hardware-accelerated, ensuring real-time operation even with 18 servos. It should be emphasized that this event-triggered mechanism is designed for quasi-static stabilization scenarios, in which disturbances occur during support phases or between gait transitions, rather than during continuous dynamic locomotion.

Table II.

Optimized attitude determination for the hexapod robot.

¹ The reading of the X-axis deflection is M.

² The reading of the Y-axis deflection is N.

This method allows for smoother transitions between attitude states and improves the accuracy of selecting appropriate action groups from the gait library under varying field conditions. The refined rule-based structure mitigates unstable responses caused by transient measurement errors, ensuring the robot performs consistent and deliberate body motions. Similar dual-threshold strategies have been validated in real-time disturbance classification and robot pose estimation literature, confirming the effectiveness of bounded confidence zones for motion control [Reference Wu, Ma and Chen38].

Algorithm 1

$\mathrm{Real\text{-}time\, Attitude\, Stability\, Control\, Loop\, (executed\, at} \ f_{s} = 200\, \mathrm{Hz)}$

Subsequently, the control system executes targeted gait actions by invoking the appropriate action group from the predefined gait library, based on the final output of the pose classification module. This ensures that the robot body reacts in real time to instantaneous disturbances and posture changes, maintaining overall balance and motion integrity. The flow chart outlining the complete stability control process is provided in Figure 3.

Figure 3.

Flow chart of stability control strategy implementation.

The design of the action group invocation logic requires the robot to dynamically adjust to transient attitude deviations using the proposed pose-aware control algorithm. Specifically, the hexapod robot deploys body adjustments through gait primitives that offset transient instability caused by external hybrid disturbances. The selected action group must also adhere to the robot’s mechanical and motion logic and ensure time-efficient execution, so that recovery actions are both fast and mechanically feasible.

These considerations aim to minimize the load on the robot’s power subsystem, thereby preserving energy for high-level functional modules such as environment mapping or obstacle negotiation. The integration of gait library invocation into the real-time attitude stability control loop plays a pivotal role in enabling robust robot performance in unstructured environments. The full implementation scheme of this stability control strategy is summarized in Table III, which offers a detailed mapping between attitude classification results and corresponding corrective actions.

Table III.

Table of attitude stability control strategies implemented.

This layered control approach aligns with embedded system strategies for disturbance mitigation in legged robots, particularly where timing, energy, and computation are tightly constrained [Reference Gao, Liu and Zhang39].

The gait actions of the hexapod robot are categorized along three dimensions to support pose-matched control responses. First, based on execution speed and actuation timing, actions are divided into slow, medium, and fast categories. Second, with respect to the robot body height relative to the contact surface, the actions are further grouped into low-, mid-, and high-pose actions. Finally, actions are also classified by spatial displacement behavior, including advancing, retreating, turning left, and turning right.

During operation, the control system dynamically selects the appropriate action group based on the posture inferred from IMU data. For instance, when the X-axis deflection exceeds the upper threshold A2 while the Y-axis remains within [B1, B2], the system classifies the pose as “Tilt to the front” and invokes the “Mid-Pose Fast Backward” action group from the gait library to counteract the disturbance-induced forward lean. Conversely, if the X-axis deflection falls below the lower threshold A1 and the Y-axis remains within the valid range, the system classifies the posture as “Tilt to the back” and calls the “Mid-Pose Fast Forward” gait pattern to restore balance.

The control system operates in a high-frequency cyclic loop. At each iteration, the embedded processor acquires IMU data, evaluates the attitude deviation through threshold-based classification, and activates the corresponding compensatory gait from a predefined lookup table (LUT). This feedback-switching mechanism enables the robot to maintain or recover balance when subjected to compound or hybrid disturbances.

For clarity, the control law can be formally described as a discrete mapping between attitude error and gait selection:

(16) \begin{equation} case(k)=f_{cls}(\varphi (k),\theta (k))=\left\{\begin{array}{l@{\quad}c} 1, & A_{1}^{\varphi }\leq | \varphi (k)| \lt B_{1}^{\varphi }\wedge A_{1}^{\varphi }\leq | \theta (k)| \lt B_{1}^{\varphi }\\[3pt] 2, & A_{2}^{\varphi }\leq | \varphi (k)| \lt B_{2}^{\varphi }\wedge A_{2}^{\varphi }\leq | \theta (k)| \lt B_{2}^{\varphi }\\[0pt] \vdots & \vdots \\[3pt] N, & A_{N}^{\varphi }\leq | \varphi (k)| \lt B_{N}^{\varphi }\wedge A_{N}^{\varphi }\leq | \theta (k)| \lt B_{N}^{\varphi } \end{array}\right. \end{equation}

Equation (16) defines a double-threshold, piecewise classifier, where each discrete gait case corresponds to a predefined roll–pitch error region bounded by $[A_{i}^{\varphi },B_{i}^{\varphi }]$ and $[A_{i}^{\theta },B_{i}^{\theta }]$ in Table II. Where $\varphi$ and $\theta$ denote the roll and pitch angles, respectively, and $[A_{i},B_{i}]$ represent the predefined threshold bounds listed in Table II that define the piecewise classification regions. The gait command is then $u(k)=g(\text{case}(k))$ , where $g(\!\cdot\! )$ maps each discrete case to a predefined servo action group. This symbolic representation formalizes the closed-loop mapping between attitude error and control action, ensuring reproducibility even without real-time experiments.

The algorithm summarizes the complete closed-loop cycle executed on the embedded controller at 200 Hz. The function $f_{cls}$ maps real-time roll–pitch attitude angles ( $\varphi$ , θ) into discrete gait cases according to the piecewise rule defined in (16), where each case corresponds to a stored compensatory pattern in the gait LUT.

Let the instantaneous roll–pitch error be

(17) \begin{equation} e(t)=\left[\begin{array}{l} \varphi (t)\\[3pt] \theta (t) \end{array}\right] \end{equation}

The controller performs discrete classification and compensation according to

\begin{equation*} case=f_{cls}(e(t))\in \{1,\ldots ,9\} ,\quad u=g(case) ,\quad e^{+}={\Phi} (e,u) \end{equation*}

where $g(\!\cdot\! )$ denotes the gait-selection mapping and Φ(e, μ) represents the closed-loop attitude-transition function. The rule-based form ensures deterministic switching and computational efficiency for real-time embedded execution.

Let the closed-loop attitude error vector be $e(t)=[\phi ,\theta ]^{T}$ . Define a candidate Lyapunov function $V(e)=\frac{1}{2}e^{T}e$ . During each active control period, the change in $V$ satisfies:

(18) \begin{equation} \dot{V}=e^{T}\dot{e}=e^{T}(f(e,\varpi)-Ke) \end{equation}

where $f(e,w)$ denotes disturbance influence and $K$ is the effective feedback gain derived from gait response. Assuming $\parallel f(e,w)\parallel \leq \delta$ , we obtain

(19) \begin{equation} \dot{V}\leq -\lambda _{\min }(K)\| e\| ^{2}+\| e\| \delta \end{equation}

Thus, when $\parallel e\parallel \gt \delta /\lambda _{\min }(K)$ , $\dot{V}\lt 0$ , implying UUB. This establishes that the system converges to a bounded region proportional to the disturbance magnitude, ensuring global stability under hybrid perturbations.

This structured mapping between attitude states and corrective action groups enables the robot to maintain locomotion stability under hybrid disturbances in unstructured terrains. Such hierarchical action–response mechanisms have demonstrated effectiveness in disturbance-adaptive legged robot systems. During continuous movement, the gait library inherently maintains dynamic stability by enforcing periodic footfall patterns (Figure 2). Each action group’s time parameter (Table I) ensures the center of mass remains within the support polygon, satisfying the ZMP criterion [Reference Inoue, Yamada and Tokuda40]. While full dynamic modeling is beyond this study’s scope, the ‘Mid-Pose Fast Forward’ gait (Table III) demonstrates this principle by synchronizing leg movements with attitude corrections.

The computational complexity of the proposed algorithm was analyzed to confirm real-time feasibility.

The double-threshold classification requires only four simple comparisons per control cycle, and gait selection is implemented via the LUT (Table III), resulting in negligible computational overhead. On the embedded STM32 microcontroller, the entire control loop executes within 2 ms, well below the 10 ms threshold required for real-time operation at 100 Hz.

3.1. Verification experiments of the attitude stability control system

To evaluate the reliability and effectiveness of the proposed attitude stability control algorithm, we conduct experimental validation under both single-axis and multi-axis hybrid disturbance scenarios. Key performance indicators, including response time, recovery duration, and spatial displacement coordinate trajectories, are adopted as quantitative metrics for performance assessment.

When the contact platform tilts along the positive Y-axis direction, the control system activates the “Case 4” stability control algorithm. As shown in Figure 5(b), the robot initially transitions from a static state to executing a “Fast Move Left” maneuver (Figures 5bi–biii), aimed at rebalancing the lateral displacement caused by the tilt. This directional motion is derived from the algorithm’s classification logic based on detected attitude deviation and disturbance direction.

Figure 5.

Validation experiments on the attitude stability control algorithm for robots in the “Tilt to the right” condition. (a) Robot spatial displacement coordinate trajectories. (b) Robot attitude and motion at different durations.

As the robot moves, the platform’s center of gravity gradually re-centers, prompting the control system to switch to the “Case 9” neutral stability algorithm, thereby guiding the robot back to a “No tilt” condition (Figures 5biii–biv). Red trajectory markers in Figure 5(b) indicate the robot’s physical motion path, while the blue trajectory markers denote the corresponding stability control actions determined by the disturbance recognition and posture feedback loop.

When the contact platform tilts in the direction formed by the positive X-axis and negative Y-axis, the control strategy corresponding to “Case 7” in Table III is activated. As shown in Figure 6(a), the robot transitions from a static state to executing a “High-Pose Fast Turn Right” maneuver (Figures 6ai–aiii). This maneuver is selected due to the controller’s design, which prioritizes the shortest angular rotation for posture correction. Subsequently, as the robot aligns with the platform’s tilt direction, the “Case 2” algorithm is triggered, resulting in a “Mid-Pose Fast Forward” motion (Figures 6aiii–av). Once the platform’s center of gravity re-centers due to the robot’s movement, the “Case 9” neutral stability algorithm is engaged, and the robot returns to a “No tilt” condition (Figures 6av–avi). The corresponding averaged displacement trajectory is plotted in Figure 6(b), which confirms the robot’s gradual convergence back to stability.

Figure 6.

Attitude stability control algorithm validation experiments for robots under hybrid disturbances, including “Tilt to the left-backward” and “Tilt to the right-backward” conditions. (a) Robot attitude and motion at different durations in the “Tilt to the left-backward” condition. (b) Robot spatial displacement coordinate trajectories in the “Tilt to the left-backward” condition. (c) Robot attitude and motion at various durations in the “Tilt to the right-backward” condition. (d) Robot spatial displacement coordinate trajectories in the “Tilt to the right-backward” condition.

In contrast, when the platform tilts in the direction formed by the negative X-axis and negative Y-axis – a scenario geometrically opposite to the previous – “Case 8” is activated. Figure 6(c) shows that the robot again transitions from “STOP” to “High-Pose Fast Turn Left” (Figures 6ci–ciii), mirroring the previous scenario in symmetry. Upon alignment with the tilt vector, the “Case 2” algorithm governs a “Mid-Pose Fast Forward” response (Figures 6ciii–cv). As the center of gravity shifts back toward equilibrium due to this motion, the “Case 9” stability policy is re-engaged, restoring the system to “No tilt” (Figures 6cv–cvi). The displacement trajectory in Figure 6(d) similarly illustrates successful dynamic stabilization.