2.1. System overview

The heavy-duty hexapod robot utilized for validating the proposed tipping detection and recovery methods is depicted in Fig. 1. The robot possesses dimensions of 4.7 m in length, 2.5 m in width, and 2 m in height in its initial position. It has a total weight of 5 t, 4.4 t for the body, and 0.1 t for each leg. The robot is mounted with six legs, with the left three legs designated as leg 1, leg 2, and leg 3, and the right three legs as leg 4, leg 5, and leg 6. Each leg has a parallelogram pantograph mechanism structure, which offers superior rigidity and minimal deformation under high loads compared to traditional tandem leg designs. Each leg is facilitated by three electrically actuated joints, horizontal, vertical, and swing joints, employed for controlling the corresponding movements, as shown in Fig. 1. An inertial measurement unit (IMU) is mounted on the robot body to gather crucial information about the robot’s physical state. This IMU enables the acquisition of body attitude values such as angular velocities and linear accelerations. Joint position sensors are implemented to capture the joint positions, and 3D contact-force sensors are used to measure the foot-ground contact forces. These sensor inputs, combined with the robot’s physical parameters, provide comprehensive measurements of the robot’s operational status.

Figure 1. Robot structure and schematic of FASM. The figure depicts the main reference frames of the robot (world frame $\mathcal{\{W\}}$ and body frame $\mathcal{\{C\}}$ ). The stability polyhedron used in FASM is constructed by leg 1, leg 3, leg 5, leg 6, and the CoG collectively, and the ${}^{{B_i}}{\boldsymbol{P}_G}$ is the CoG of the robot. The leg structure is partially decoupled, as in the section view of leg 3.

As shown in Fig. 1, $\mathcal{\{W\}}$ is the inertia frame, $\mathcal{\{C\}}$ is the body frame. The Coxa frame of the $ i$ th leg is $\{\mathcal{B}_i\}$ , orientated in the same direction as the body frame. The position of the origin of $\{\mathcal{B}_i\}$ in $\mathcal{\{C\}}$ is $ \boldsymbol{l}_i={\left [{\begin{array}{*{20}{c}}{{l_{ix}}}&{{l_{iy}}}&{{l_{iz}}} \end{array}} \right ]^{\mathrm{T}}}$ , the $ i$ th leg joint position is $ \boldsymbol{J}_i ={\left [{\begin{array}{*{20}{l}}{{J_{ix}}}&{{J_{iz}}}&{{J_{i\theta }}} \end{array}} \right ]^{\mathrm{T}}}$ , the foot position in Coxa frame of $ i$ th leg is ${}^{{B_i}}{\boldsymbol{P}_i} ={\left [{\begin{array}{*{20}{l}}{{}^{B_i}{P_{ix}}}&{{}^{{B_i}}{P_{iy}}}&{{}^{{B_i}}{P_{iz}}} \end{array}} \right ]^{\mathrm{T}}}$ and in body frame is ${}^{C}{\boldsymbol{P}_i} ={\left [{\begin{array}{*{20}{l}}{{}^{C}{P_{ix}}}&{{}^{C}{P_{iy}}}&{{}^{C}{P_{iz}}} \end{array}} \right ]^{\mathrm{T}}}$ . Since the leg structure is partially decoupled, the forward and inverse kinematics of the leg can be obtained through the geometric analysis method. The forward and inverse kinematics in the Coxa frame are expressed as follows:

(1) \begin{equation} \left \{{\begin{array}{*{20}{l}}{^{{B_i}}{P_{ix}} ={K_x}{J_{ix}}}\\[4pt]{^{{B_i}}{P_{iy}} ={L_1}\cos \left ({{J_{i\theta }}} \right ) + \left ({{K_z}{J_{iz}} +{L_2}} \right )\sin \left ({{J_{i\theta }}} \right )}\\[4pt]{^{{B_i}}{P_{iz}} ={L_1}\sin \left ({{J_{i\theta }}} \right ) - \left ({{K_z}{J_{iz}} +{L_2}} \right )\cos \left ({{J_{i\theta }}} \right )} \end{array}} \right. \end{equation}

(2) \begin{equation} \left \{{\begin{array}{*{20}{c}}{{J_{ix}} = \dfrac{{{}^{{B_i}}{P_{ix}}}}{{{K_x}}}}\\[12pt]{{J_{iz}} = \dfrac{{\sqrt{{}^{{B_i}}{P_{iy}}^2{ + ^{{B_i}}}{P_{iz}}^2 - L_1^2 -{L_2}} }}{{{K_z}}}}\\[12pt]{{J_{i\theta }} = \arcsin \left ({\dfrac{{{}^{{B_i}}{P_{iz}}{L_1} +{}^{{B_i}}{P_{iy}}\sqrt{{}^{{B_i}}{P_{iy}}^2 +{}^{{B_i}}{P_{iz}}^2 -{L_1}^2} }}{{{}^{{{B_i}}}{P_{iy}}^2 +{}^{{B_i}}{P_{iz}}^2}}} \right )} \end{array}} \right. \end{equation}

where $ K_x =5$ and $ K_z = 4$ represent the scale factor of the horizontal joint and vertical joint, respectively. $ L_1$ represents the distance between the leg plane and the origin of frame $ \{ \mathcal{B}_i \}$ , with the left leg being positive and the right leg being negative. $ L_2$ represents the distance between the central axes of the horizontal joint and the origin of frame $ \{ \mathcal{B}_i \}$ .

2.2. Tipping analysis

As to hexapod robots, tipping can generally occur in two fundamental scenarios, namely corner tipping and sideline tipping, as illustrated in Fig. 2(a) and (b), respectively [Reference Tian, Fang, Zhou, Li and Kou33]. Corner tipping refers to the robot rotating around the foot tip of a support leg when tipping, while sideline tipping involves the robot revolving around an edge of the support polygon before overturned. Since corner tipping represents a specific case and is relatively less common than sideline tipping, this investigation focuses primarily on addressing the challenges posed by sideline tipping.

Figure 2. Tipping schematic. (a) is corner tipping, and the robot will tip around point $A_i$ . (b) is sideline tipping, and the robot will tip around line $A_iA_j$ .

The tipping phenomenon in the robot is characterized by the FASM. Tipping occurs when the minimum angle between the net force at CoG projected on the vertical plane of a side of the stability polyhedron and the vertical line of that side from CoG to support the polygon falls below the threshold set for stability margin. During destabilization, the robot rotates around the straight line formed by the feet of the two supporting legs – these two legs are the effective supporting legs. The two effective support legs can be determined by the foot force in $z$ direction and is verified in Section 5. The forces at the foot of the effective support legs partially counterbalancing the destabilizing force. The remaining unbalanced force induces robot rotation around the edge of the support polygon. Typically, the robot’s support polygon is a convex polygon. In the counterclockwise direction, considered as positive, the robot rotates along the edge line following the right-hand rule, resulting in a positive value of $\phi$ . Furthermore, it is assumed that during tipping, the foot tip forces remain within the friction cone, ensuring that no foot sliding occurs.

4.1. Adjustment leg selection

In the tripod gait, the robot operates with three legs in the stance phase while the other three legs are in the swinging phase. Referring to the leg numbering in Fig. 2(b), the points $ A_i$ , $ A_j$ , and $ A_k$ represent the foot tip of leg 6, leg 4, and leg 2. During tipping events, the effective support legs are leg 4 and leg 6, while leg 5 is the adjustment leg. To determine the adjustment leg when the robot tips over, consult Table II. In this table, $ \boldsymbol{a}_{ij}$ represents the tipping axis formed by the footholds of leg $ i$ and leg $ j$ , $\surd$ represents the corresponding leg to be chosen.

Table II. Adjustment leg selection when tipping.

4.2. Reachable space calculation and optimal foothold selection

The terrain is fitted using a linear least squares method to enhance computational efficiency, as described in ref. [Reference Xufan34]. Let ${}^{C}\boldsymbol{n}_s$ denote the unit normal vector of the fitted slope in the body frame, and ${}^{C}\boldsymbol{P}_i^\prime$ represent the foot position projection onto the fitted terrain in the body frame. The following calculations and derivations are conducted in the body frame, and “ground” refers to the fitted terrain.

Figure 3 illustrates the schematic diagram of the reachable space for the adjusting leg just before tipping. Points $A$ , $B$ , $C$ , $D$ , $L$ , and $R$ are the boundary points of the reachable space. The swing joint range of motion for the legs is denoted as $ J_\theta \in \left [ J_{\theta \min }\space, \space J_{\theta \max } \right ]$ . The origin of the leg frame, represented by $ B_i$ , is projected onto the ground along the $ z$ axis of the body frame to determine point $H$ . Additionally, the maximum length of the leg in the $ y_CCz_C$ plane is denoted as $ L_{\max }$ . If $ \left \|\vec{r}_{B_iH}\right \| \gt L_{\max }$ , no viable ground points are identified. If $ \left \|\vec{r}_{B_iH}\right \| = L_{\max }$ , point $H$ becomes the target location. If $ \left \|\vec{r}_{B_iH}\right \| \lt L_{\max }$ , the approximate reachable space of the foot on the ground needs to be solved first and then selects the optimal point. Next, the reachable workspace of the foot on the ground under condition $\left \|\vec{r}_{B_iH}\right \| \lt L_{\max }$ is derived.

Figure 3. Adjusting leg reachable space right before tipping. The robot will tip around line $P_3P_5$ , and the figure depicts the reachable space of leg 6. The projected point of point $B_6$ on the ground is $H$ . $L$ and $R$ are the left and right boundary points. $A_f$ and $B_f$ are the front and rear boundary points, and $A$ and $B$ are the projected point on the ground, respectively.

We define $\vec{r}_{HR} \bot{\boldsymbol{x}_{C}}$ , where ${\boldsymbol{x}_{C}}$ represents the unit vector of the body frame in the $ x$ direction. Based on geometric relationships, we know that $\vec{r}_{HR} \bot{}^{C}\boldsymbol{n}_s$ , and ${\boldsymbol{x}_{C}}$ is the normal vector of the plane $ B_iHR$ . The projection of $ \boldsymbol{y}_C$ – the unit vector of the body frame in $ y$ direction – onto the ground is opposite to $\vec{r}_{HR}$ . The angle $ \beta$ between $\vec{r}_{HR}$ and $\vec{r}_{HB_i}$ can be determined using the law of cosines. Let $\vec{r}_{B_iR}=L_{\max }$ , the length of $ \vec{r}_{HR}$ is

(10) \begin{equation} \left \| \vec{r}_{HR} \right \| = \left \| \vec{r}_{B_iH} \right \| \cos \beta + \left ({{\left \| \vec{r}_{B_iR} \right \|}^2} -{{\left ({\left \| \vec{r}_{B_iH} \right \|\sin \beta } \right )}^2}\right )^\frac{1}{2} \end{equation}

The angle of $ \angle HB_iR$ can be determined using the law of sines. If $\angle HB_iR \lt \theta _{\max }$ , $\left \| \vec{r}_{HR} \right \|$ remains unchanged. If $\angle HB_iR \gt \theta _{\max }$ , set $\angle HB_iR = \theta _{\max }$ and recalculate the length of $\left \| \vec{r}_{HR} \right \|$ .

To eliminate potential risks, a safety margin $\varrho ( 1\gt \varrho \gt 0$ ) is introduced, and $ \left \| \vec{r}_{HR} \right \|_{new}=\varrho ^{*}\left \| \vec{r}_{HR} \right \|$ . Then, we can get the position of $R$

(11) \begin{equation}{}^C\boldsymbol{R}={}^C\boldsymbol{H}+\left \| \vec{r}_{HR} \right \|_{new}*\hat{\vec{r}}_{HR} \end{equation}

Similarly, the length of $\vec{r}_{HL}$ and the angle of $\angle HB_iL$ can be calculated. If $\angle HB_iL \lt \theta _{\max }$ , its value remains unchanged. If $\angle HB_iL \gt \theta _{\max }$ , let $\angle HB_iL = \theta _{\max }$ and recalculate the length of $\left \| \vec{r}_{HL} \right \|$ . The position of $L$ is

(12) \begin{equation}{}^C\boldsymbol{L}={}^C\boldsymbol{H}-\varrho *\left \| \vec{r}_{HL} \right \|*\hat{\vec{r}}_{HR} \end{equation}

Considering that the foot’s range of motion is independent in the $ x$ direction, assuming that $\vec{r}_{LA_f}=\left ({{}^{Bi}P_{ix\max }} \enspace 0 \enspace 0 \right )$ and $\vec{r}_{LB_b}=\left ({{}^{Bi}P_{ix\min }} \enspace 0 \enspace 0 \right )$ , the projection vectors of them are

(13) \begin{equation} \left \{\begin{array}{l}{ \vec{r}_{LA} = \varrho \left ({\textbf{1} -{}^{C}{\boldsymbol{n}}_s{}^{C}{\boldsymbol{n}}_s^{\mathrm{T}}} \right ) \vec{r}_{LA_f} } \\[5pt]{ \vec{r}_{LB} = \varrho \left ({\textbf{1} -{}^{C}{\boldsymbol{n}}_s{}^{C}{\boldsymbol{n}}_s^{\mathrm{T}}} \right ) \vec{r}_{LB_b} } \end{array}\right. \end{equation}

Then, the position of $A$ , $B$ , $C$ , and $D$ can be got.

When the heavy-duty hexapod robot tips around the support polygon edge, the forces exerted by the adjusting leg must counteract the tipping wrench for the robot to regain stability. Considering the relationship between force and torque and aiming to minimize the required force on the foot, the adjusting foothold should be selected from within the reachable workspace on the ground, maximizing the lever arm. Simultaneously, it is crucial to balance reducing adjustment time and minimizing foot force when addressing tipping scenarios in a heavy hexapod robot. A tradeoff between adjustment time and foot force reduction must be carefully considered.

As shown in Fig. 3, segments $AB$ , $BC$ , $CD$ , and $DA$ are generally not parallel to the tipping axis. To prioritize reducing adjustment time and minimizing foot forces, the angle between the foot’s reachable space boundary and the tipping axis is calculated. If the angle falls below the threshold value, the $R$ or $L$ point – whichever offers a shorter adjustment time and a more considerable stability margin – as the target footing point. Conversely, when the angle exceeds the threshold, the distance from each boundary point of the reachable space to the tipping axis is calculated, and the point with the largest distance becomes the chosen footing point. Additionally, if point $H$ lies on the tipping axis, the distances from points $B$ and $D$ , $A$ and $C$ , and $L$ and $R$ to the tipping axis are equal. In such cases, we prioritize selecting the point on the right side of the tipping axis. For instance, considering point $A$ , we can use the following formula to determine whether the boundary point is located to the left or right of the tilting axis.

(14) \begin{equation} \Theta = \vec{r}_{{{}^{C}P_i^\prime }A} \times \vec{r}_{{{}^{C}P_i^\prime }{{}^{C}P_j^\prime }} \cdot{\boldsymbol{z}_{C}} \end{equation}

where ${}^{C}P_i^\prime$ represents the first support leg projected onto the fitted terrain along the counterclockwise tipping axis, and ${}^{C}P_j^\prime$ corresponds to the second support leg. $ \boldsymbol{z}_C$ denotes the unit vector of body frame in the $ z$ direction. If $ \Theta \gt 0$ , point $A$ lies to the right of the tipping axis. Conversely, if $ \Theta \lt 0$ , point $A$ is positioned to the left of the tipping axis. Based on this methodology, the adjustment leg target foothold can be determined, denoted as ${}^{C}\boldsymbol{P}_{tar}$ .

When the stability margin falls below the threshold $ \xi _f$ , the rate at which the stability margin decreases, denoted as $ v_s$ , and the corresponding acceleration, denoted as $ a_s$ , can be determined. This information allows us to estimate the time required for the stability margin to decrease from $ S_{\text{FASM}}\lt \xi _f$ to $ S_{\text{FASM}}=0$ .

(15) \begin{equation} t_s=\varkappa \frac{-v_s+\sqrt{v_s^2-4a_s{\xi _f}}}{2a_s} \quad (0\lt \varkappa \lt 1) \end{equation}

When the heavy-duty hexapod robot is judged to be tipping, the start position, velocity, and acceleration of the adjusting leg can be determined. After setting the target velocity and acceleration of the adjusting leg, the foot trajectory can be planned using a quintic polynomial. Then, we can get the adjusting leg foot trajectory.

Thus far, we have completed the foothold reachable space calculation and optimal foothold selection.

4.3. Foot force distribution considering stability

To effectively restrain the ongoing instability of the heavy-duty hexapod robot and regain stability upon adjusting leg touchdown while simultaneously preventing damage to the robot caused by internal forces, it is necessary to distribute the foot forces. As the system is overdetermined, it is necessary to introduce additional constraints or define optimization objectives to get an optimal solution. In this context, QP is used to achieve foot force distribution. Notably, the stability margin of the robot, as defined by FASM, is closely linked to foot forces. By integrating QP and FASM, we can consider the robot’s stability while obtaining the optimal foot force solution at the moment the adjusting leg makes contact with the terrain.

Considering the high inertia of the heavy-duty hexapod robot, the impact of inertial forces is not negligible, and cannot simplify the calculation of expected force at the CoG as in ref. [Reference Gehring, Coros, Hutter, Bloesch, Hoepflinger and Siegwart35] or [Reference Focchi, Prete, Havoutis, Featherstone, Caldwell and Semini36]. The force and torque in CoG can be formulated as

(16) \begin{equation} \left \{ \begin{aligned} \boldsymbol{F}_b = -{m_b}{\boldsymbol{a}_b} -{m_b}{\textbf{R}_W^C} \boldsymbol{g} \\ \boldsymbol{T}_b = -\boldsymbol{I}{\omega _b} -{\omega }_b \times \boldsymbol{I}{\omega _b} \end{aligned} \right. \end{equation}

where ${\textbf{R}_W^C} \in \mathbb{R}^{3 \times 3}$ represents the transformation from inertial to body frame. $ \boldsymbol{I}$ represents the inertia of the body.

To achieve accurate tracking of the desired force and torque at the CoG when mapping the foot forces, the problem can be formulated as

(17) \begin{equation} \begin{aligned} \text{min} \quad \quad &\left ({\boldsymbol{A}}\boldsymbol{x}-\boldsymbol{b}\right )^{\mathrm{T}}{\textbf{S}}\left ({\boldsymbol{A}} \boldsymbol{x}-\boldsymbol{b}\right )+\kappa \boldsymbol{x}^{\mathrm{T}} \textbf{W} \boldsymbol{x} \\ \text{s.t.}\quad \quad &Constraints \end{aligned} \end{equation}

where $ \boldsymbol{A} \in \mathbb{R}^{6k}$ is the transform matrix, and $ \boldsymbol{A} ={\left [{\begin{array}{*{20}{c}} \boldsymbol{E}&\boldsymbol{E}& \cdots &\boldsymbol{E}\\{{}^C\boldsymbol{P}_1 }&{{}^C\boldsymbol{P}_2 }& \cdots &{{}^C\boldsymbol{P}_k} \end{array}} \right ]}$ . $ \boldsymbol{x} \in \mathbb{R}^{3k}$ is the stance foot force vector. $ \boldsymbol{b} \in \mathbb{R}^6$ is the desired force and moment of the robot body. $ \textbf{S} \in \mathbb{R}^{6 \times 6}$ and $ \textbf{W} \in \mathbb{R}^{3k \times 3k}$ are the positive definite weight matrix. $ \kappa$ is the weight factor for the second optimization item.

By transforming equation (17) to the standard form of QP, we can obtain

(18) \begin{equation} \begin{aligned} \text{min} \quad \quad &\frac{1}{2} \boldsymbol{x}^{\mathrm{T}}\textbf{U}\boldsymbol{x}+\boldsymbol{f}^{\mathrm{T}}\boldsymbol{x} +\boldsymbol{b}^{\mathrm{T}}{\textbf{S}}\boldsymbol{b} \\ \text{s.t.}\quad \quad &Constraints \end{aligned} \end{equation}

where $ \textbf{U} \in \mathbb{R}^{3k \times 3k}$ and $ \textbf{U}=2{\boldsymbol{A}}^{\mathrm{T}}{\textbf{S}}{\boldsymbol{A}}+2\kappa W$ . $ \boldsymbol{f} \in \mathbb{R}^{3k}$ and $ \boldsymbol{f}^{\mathrm{T}} = -2\boldsymbol{b}^{\textbf{T}}{\textbf{S}}{\boldsymbol{A}}$ . Since $ \boldsymbol{b}^{\mathrm{T}}{\textbf{S}}\boldsymbol{b}$ is a constant term and has no effect on the optimization result, it can be ignored.

The constraints in equation (18) are force and stability margin constraints. To ensure the robot’s normal operation on the ground, the ground reaction forces in the $z$ direction in the body frame are always positive. It is necessary to prevent sliding between the support feet and the ground. This requires that the tangential force at the robot’s feet not exceed the maximum friction force. We adopt the method described in ref. [Reference Guanyu, Liang, Haibo, Yiqun, Yufei, Zhen and Zhongquan37] to establish the force constraint for stance feet.

The following step is to obtain the expression for the stability margin constraint. As depicted in Fig. 4, the main idea of the stability margin constraint is to construct a stability constraint cone and then to constrain the net forces acting at the CoG during the FASM calculation to be within the stability constraint cone.

Figure 4. Stability margin constraint. Based on the geometry relationship, a stability constraint cone with vertex ${}^{C}\boldsymbol{P}$ and base $ \odot \mathrm{G}_s$ is constructed. $\boldsymbol{f}_i^*$ is the net force acting on the CoG. Keeping the $\boldsymbol{f}_i^*$ within the stability constraint cone will be a hard constraint.

Based on the formulation of (7), the tipping net force in CoG is composed by $ \boldsymbol{f}_{{\mathrm{r}}i}$ and ${\boldsymbol{f}}_{ni}$ . Transforming the $ \boldsymbol{f}_{{\mathrm{r}}i}$ and ${\boldsymbol{f}}_{ni}$ to include the foot force vector $ \boldsymbol{x}$

(19) \begin{equation} \left \{ \begin{array}{l} \boldsymbol{f}'_{{\mathrm{r}}i}=-\left (\boldsymbol{E}-\hat{\boldsymbol{a}}_{i}\hat{\boldsymbol{a}}_{i}^{\mathrm{T}}\right )\left [{\begin{array}{*{20}{c}} \boldsymbol{E}&\boldsymbol{E}& \cdots &\boldsymbol{E} \end{array}} \right ]\cdot \boldsymbol{x} \\[9pt]{\boldsymbol{f}}'_{ni}=-\dfrac{1}{\left |\boldsymbol{I}_i\right |} \left (\hat{\boldsymbol{a}}_{i}\hat{\boldsymbol{a}}_{i}^{\mathrm{T}}\right )\left [{\begin{array}{*{20}{c}}{\boldsymbol{T}_1}&{\boldsymbol{T}_2}& \cdots &{\boldsymbol{T}_k} \end{array}} \right ]\cdot \boldsymbol{x} \end{array}\right. \end{equation}

where $ \boldsymbol{T}_i$ is the equivalent couples of $ i$ th leg, and $ \boldsymbol{T}_i = \left [{\begin{array}{c@{\quad}c@{\quad}c} 0&{ - \left ({{{}^CP^\prime _{iz}} -{x_G}} \right )}&{{{}^CP^\prime _{iy}} -{y_G}}\\[5pt]{{{}^CP^\prime _{iz}} -{z_G}}&0&{ - \left ({{}^C{P^\prime _{ix}} -{x_G}} \right )}\\[5pt]{ - \left ({{{}^CP^\prime _{iy}} -{y_G}} \right )}&{{{}^CP^\prime _{ix}} -{x_G}}&0 \end{array}} \right ]$ .

Based on the idea of FASM, the new formula of net force in CoG is expressed as

(20) \begin{equation} \boldsymbol{f}_{i}^{*\prime }=\boldsymbol{f}^{*\prime }_{{\mathrm{r}}i}+{\boldsymbol{f}}^{*\prime }_{ni}=-\boldsymbol{V}_i \boldsymbol{x} \end{equation}

where $ \boldsymbol{V}_i \in \mathbb{R}^{3\times 3k}$ is the transform matrix of CoG net force, and $ \boldsymbol{V}_i=\left (\textbf{1}-\hat{\boldsymbol{a}}_{i}\hat{\boldsymbol{a}}_{i}^{\mathrm{T}}\right )\left [{\begin{array}{*{20}{c}} \boldsymbol{E}&\boldsymbol{E}& \cdots &\boldsymbol{E} \end{array}} \right ]+\frac{1}{\left |\boldsymbol{I}_i\right |}\cdot \left (\hat{\boldsymbol{a}}_{i}\hat{\boldsymbol{a}}_{i}^{\mathrm{T}}\right )\left [{\begin{array}{*{20}{c}}{\boldsymbol{T}_1}&{\boldsymbol{T}_2}& \cdots &{\boldsymbol{T}_k} \end{array}} \right ]$ .

As the support polygon is convex, and the CoG of a triangle is easy to calculate, the centroid $ \boldsymbol{G}_{s}$ of the support polygon can be determined by dividing it into several triangles. Then the minimum value from the $ \boldsymbol{G}_{s}$ to the edges of the support polygon is

(21) \begin{equation}{d_{\min }} = \min \left ( \left \| \textbf{1}-{ \left \|{}^{C}\boldsymbol{P}_j^\prime -{}^{C}\boldsymbol{P}_i^\prime \right \| }^{-1} \left ({}^{C}\boldsymbol{P}_j^\prime -{}^{C}\boldsymbol{P}_i^\prime \right ) ^{\mathrm{T}} \left ({}^{C}\boldsymbol{P}_j^\prime -{}^{C}\boldsymbol{P}_i^\prime \right ) \left ({}^{C}\boldsymbol{P}_i^\prime - \boldsymbol{G}_{s} \right ) \right \| \right ) \end{equation}

Drawing a circle $ \odot \mathrm{G}_s$ on the ground with $ \boldsymbol{G}_{s}$ as the center and a radius of $ \lambda d_{\min } \space (1\gt \lambda \gt 0)$ . Constructing a stability constraint cone with vertex ${}^C\boldsymbol{P}_G$ and base $ \odot \mathrm{G}_s$ , as depicted in Fig. 4. The intersection of $ \odot \mathrm{G}_s$ with the plane perpendicular to the $ y_CCz_C$ plane and passing through ${}^C\boldsymbol{P}_G \boldsymbol{G}_{s}$ occurs at points $ \boldsymbol{T}_1$ and $ \boldsymbol{T}_3$ . Similarly, the intersection of $ \odot \mathrm{G}_s$ with the plane perpendicular to the $ x_CCz_C$ plane and passing through ${}^C\boldsymbol{P}_G \boldsymbol{G}_{s}$ occurs at points $ \boldsymbol{T}_2$ and $ \boldsymbol{T}_4$ . As shown in Fig. 3, the projected unit vector of the $ y$ axis of the body frame onto the fitted ground is denoted as $\hat{\vec{r}}_{HR}$ . From equation (13), the projected unit vector of the $ x$ axis onto the fitted ground is $\hat{ \vec{r} }_{LA} ={ \vec{r}_{LA} } \mathord{\left/{\vphantom{{ \vec{r}_{LA} }{\left \|{ \vec{r}_{LA} } \right \|}}} \right. }{\left \|{ \vec{r}_{LA} } \right \|}$ . Finally, the vectors from the CoG to $ \boldsymbol{T}_1$ , $ \boldsymbol{T}_2$ , $ \boldsymbol{T}_3$ and $ \boldsymbol{T}_4$ can be obtained.

Let $\angle{G_s}{{}^CP_G}{T_1} ={\alpha _1}$ , $\angle{G_s}{{}^CP_G}{T_2} ={\alpha _2}$ , $\angle{G_s}{{}^CP_G}{T_3} ={\alpha _3}$ , $\angle{G_s}{{}^CP_G}{T_4} ={\alpha _4}$ , and $\alpha _1,\alpha _2,\alpha _3,\alpha _4\in \left (0,90^\circ \right )$ , then we have

(22) \begin{equation} \tan{\alpha _i} ={{\left \|{{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{G}}_{\mathrm{s}}}}} \times{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{T}}_{\mathrm{i}}}}}} \right \|} \mathord{\left/{\vphantom{{\left \|{{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{G}}_{\mathrm{s}}}}} \times{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{T}}_{\mathrm{i}}}}}} \right \|}{\left |{{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{G}}_{\mathrm{s}}}}} \cdot{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{T}}_{\mathrm{i}}}}}} \right |}}} \right. }{\left |{{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{G}}_{\mathrm{s}}}}} \cdot{{\vec r}_{{}^{\mathrm{C}}{{\mathrm{P}}_{\mathrm{G}}}{{\mathrm{T}}_{\mathrm{i}}}}}} \right |}} \quad \left ({i = 1, \cdots,4} \right ) \end{equation}

For $ \boldsymbol{f}_{i}^{*\prime }$ to be within the stability margin constraint cone, it is necessary to satisfy

(23) \begin{equation} \sqrt{f_{xi}^{*\prime 2} + f_{yi}^{*\prime 2} } \le -\tan{\alpha _i}f_{zi}^{*\prime } \quad \left ({i = 1, \cdots,4} \right ) \end{equation}

as $ \boldsymbol{f}_{i}^{*\prime } \le 0$ , the negative sign is added to equation (23).

Equation (23) is a nonlinear inequality. In optimization problems, nonlinear constraints can significantly increase the complexity of the problem. To simplify the nonlinear constraints to linear constraints, we use an inscribed pyramid instead of the stability constraint cone expressed in equation (23). The stability constraint formula of the robot based on the FASM can be expressed as

(24) \begin{equation} \boldsymbol{S}_M^{\mathrm{T}} \boldsymbol{V}_i \boldsymbol{x} \le 0 \end{equation}

where $ \boldsymbol{S}_M=\frac{1}{\sqrt{2}}\left [{\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \sqrt{2}&0&-\sqrt{2}&0\\[4pt] 0&\sqrt{2}&0&-\sqrt{2}\\[4pt] \tan{\alpha _1}&\tan{\alpha _2}&\tan{\alpha _3}&\tan{\alpha _4} \end{array}} \right ]$ is stability margin constraint transformation matrix.

Putting the constraints into (18), the optimal foot forces vector $ \boldsymbol{x}$ is solved after the adjusting leg makes contact with the ground.

5.1. Stability margin measurement and tipping judgment

To demonstrate the correctness of stability margin measurements and the feasibility of the tipping instability detection method, the hexapod robot is controlled to walk on unilateral cliff terrain with a tripod gait. Initially, the robot moves on flat terrain, after which it moves towards a unilateral cliff terrain where it is expected to tip over. This allows for observing and validating the stability margin and robot state during both stable and unstable conditions. Set the robot to stand for 3 s after starting the simulation to ensure a stable initial state before movement. The duty factor is set to $ \beta = 14/24$ , the step length is 500 mm for all gaits, the gait cycle is set as T = 6 s, and the sample time is 0.02 s.

The variation curves of the CoG position and stability margin, as well as the robot’s motion state snapshots, are shown in 5. During the flat terrain movement stage, the position of the CoG in the $ y$ axis is 0 due to the robot’s symmetry with respect to the $ x_CCz_C$ plane of the body frame. Since the robot’s movement is periodic, the position of the CoG in the $ x$ axis and $ z$ axis, and the stability margin of the robot vary periodically. With a duty cycle of $ \beta = 14/24$ greater than 0.5, there are moments when all six legs are in a stance state. At t = 3 s and t = 9.4 s, both represent six legs support states. It can be observed that the robot has the maximum stability margin during the six legs support, with ${S}_{\mathrm{FASM}}$ approximately equal to 48. As the supporting legs move backward, the CoG of the robot moves forward, causing the stability margin to decrease gradually. At t = 9.7 s, the robot transitions from a six legs support state to a three legs support state, and the stability margin rapidly decreases with ${S}_{\mathrm{FASM}}$ approximately equal to 31.41. The corresponding robot state is shown in Fig. 5 (c). When the stance legs reach the rear limits, the stability margin is minimal, and the corresponding robot state is shown in Fig. 5 (c) at t = 11.91 s, where the leg is about to transition from swing state to stance state. During the tripod gait, the stability margin of the robot varies periodically in the range of approximately 21–51.

Figure 5. Robot moves on flat and unilateral cliff terrain. (a) illustrates the CoG position and stability margin curve when robot moving with tripod gait. (b) shows the corresponding $z$ direction feet force curves. (c) depicts the corresponding motion capture snapshots.

At 21.06 s, leg 4 steps into the void, but the stability margin curve does not show significant fluctuations since the other five legs are still in a supporting state. As the supporting legs continue to move backward, the stability margin remains above 20 for a long time until 22.08 s. At this point, the robot’s posture has changed significantly, and the $z$ direction force at the foot of the effective supporting legs (legs 1 and 5) notably exceeds that of the remaining legs, as shown in Fig. 5(b) and (c), indicating that the robot has become unstable. At t = 22.36 s, ${S}_{\mathrm{FASM}} =0$ , however, due to the severe change in the robot’s posture, as depicted in Fig. 5(c), the stability margin of the robot sharply decreases, and this posture cannot be recovered using an instability recovery strategy. Subsequently, the controller reports an error and stops. To rescue the robot from instability in advance, the criteria given by Papadopoulos cannot be strictly followed to determine stable or unstable states. Instead, the threshold should be set based on the actual robot to judge instability. In this work, the stability margin threshold is set as $ \xi _f = 20$ .

5.2. Tipping recovery

Various types of sideline tipping can occur on the hexapod robot, characterized by the change in robot posture and rotation around one edge of the support polygon. To facilitate the validation of the validity of tipping recovery under various conditions and reduce complexity, a 6-UPS Stewart parallel mechanism is established. The heavy-duty hexapod robot is placed on the moving platform of the Stewart platform to establish a combined Stewart-hexapod robot simulation platform, as shown in Fig. 6(a). The radius of the moving platform is 3.5 m, while the fixed platform is 5.5 m. The moving frame $ \{ \mathcal{A}_s \}$ is attached to the moving platform with the $z$ axis normal to it, and the inertia frame $ \{ \mathcal{B}_s \}$ is attached to the fixed base with $z$ axis pointing vertically upward. The origins of the two frames, $ A _s$ and $ B_s$ , are located at the geometric centers of the moving and fixed platforms, respectively. Point $ b_{si}\left (i=1,2,\cdots,6\right )$ is the hinge joint connecting the fixed platform and actuators, while point $ a_{si}\left (i=1,2,\cdots,6\right )$ is the hinge joint connecting the actuators and moving platform. The six actuators are numbered corresponding to their hinge joints, such as $ a_{si}b_{si}$ corresponding to actuator $ i$ . The minimal length of these actuators is 9 m, and the maximum length of them is 14 m. By controlling the pose of the moving platform, tipping of the robot in different directions can be achieved to verify the effectiveness of the tipping recovery method. The Stewart platform regulates the robot’s pose, and the desired functionalities can be achieved through position control. The kinematic analysis of the platform is conducted based on the work in [Reference Yanhao, Dong, Ying, Meng, Zhenpeng and Defang38].

Figure 6. Stewart-hexapod platform and tipping recovery simulation. (a) represents the combined Stewart-hexapod robot simulation platform, consisting of a 6-UPS Stewart parallel mechanism and the heavy-duty hexapod robot. (b)–(d) illustrate the robot’s backward, forward, and sideline tipping recovery with legs 1, 3, and 5 in stance and the corresponding foot force in $z$ direction. The top chart of (e) depicts the stability margin, while the bottom chart depicts the tipping rotation axis. Similarly, (f)–(h) demonstrate the robot’s backward, forward, and sideline tipping recovery with legs 2, 4, and 6 stance and the corresponding foot force in $z$ direction. The top chart (i) showcases the stability margin variation curve, and the bottom illustrates the corresponding tipping rotation axis.

Control the Stewart platform to rotate counterclockwise about the $ y$ axis, rotate clockwise about the $ y$ axis, and rotate counterclockwise about the $ x$ axis, respectively. These three simulation scenarios simulate the robot’s tipping and recovery when climbing a steep slope, descending a steep slope, and moving on a steeply inclined slope at a large angle.

For the robot climbing a steep slope scenario, adjust the robot to the state where legs 1, 3, and 5 are in the stance state. From 0 to 7 s is the robot status adjustment stage when the robot tips with legs 1, 3, and 5 in the stance state, after which the Stewart platform begins rotational motion. The change of the stability margin during the counterclockwise rotation about the $ y$ axis of the Stewart platform is shown by the red line in Fig. 6(e) of the top figure, and the red line of the bottom figure shows the corresponding supporting polygon edge. From the top figure of Fig. 6(e), we observe that the stability margin initially increases and then decreases due to the positioning of the robot’s center of gravity in the front of the body frame. The backward tipping of the robot causes the net force at the CoG to gradually shift towards the rear, resulting in an increase in the angle between the plane of the support polygon’s third edge (formed by the lines connecting leg 1 and leg 5). Simultaneously, the angle between the plane of the support polygon’s second edge (formed by the lines connecting leg 3 and leg 5) decreases. However, the minimum angle remains within the plane of the support polygon’s third edge. Once the critical value is exceeded, the minimum angle becomes the plane of the second edge of the support polygon. At this point, the Stewart platform continues counterclockwise rotation, and the net force further shifts rearward. The robot stability margin gradually decreases. At 14.48 s, the robot is determined to be about to lose stability when ${S}_{\mathrm{FASM}}\lt 20$ . At this point, legs 3 and 5 have the maximum leg forces. According to Table II, leg 6 should be selected as the adjustment leg. Based on equation (15), the adjustment time is 0.74 s, with the target foothold point at (−2390, −1908, −1741). After leg 6 lands as a stance leg, the robot stability margin rises from below 20 to around 40, significantly improving stability. The robot state during leg 6 landing as the adjustment leg and the feet force variation curves are shown in Fig. 6(b).

For the robot descending a steep slope, control the Stewart platform to rotate clockwise about the $ y$ axis to make the robot tip forward. The stability margin and corresponding support polygon edge variation curves are shown by the green line in the top figure and the black line in the bottom figure of Fig. 6(e). At 11.85 s, the robot is about to lose stability with ${S}_{\mathrm{FASM}}\lt 20$ . At this point, legs 1 and 5 have the maximum leg forces. Leg 4 should be selected as the adjustment leg, and the adjustment time is 1.0 s with the target foothold point at (−2490, −1908, −1741). After leg 4 lands as a stance leg, the robot stability margin rises from below 20 to around 45, significantly improving stability. The robot state during leg 4 landing as the adjustment leg and the stance feet force curves are shown in Fig. 6(c).

For the scenario of tipping recovery when the heavy-duty hexapod robot moves on a slope with a large inclination angle, control the Stewart platform to rotate counterclockwise around the $ x$ axis. The stability margin and corresponding support polygon edge variation curves are shown by the yellow line in the top figure and the blue line in the bottom figure of Fig. 6 (e). At 15.83 s, the robot is about to lose stability with ${S}_{\mathrm{FASM}}\lt 20$ . Currently, legs 1 and 3 have the maximum force at the foot, and leg 2 should be selected as the adjustment. Based on equation (15), the adjustment time is 2.1 s, with the target foothold point at (0, 1894, −1783). After leg 2 lands as a stance leg, the stability margin of the robot increases from below 20 to about 21.5. The robot’s state during leg 2 landing as the adjusting leg and the stance feet force curves are illustrated in Fig. 6 (d).

The recovery of the robot from tipping over when legs 2, 4, and 6 are in the stance state is similar to that when legs 1, 3, and 5 are in the stance state. From 0 to 5 s is the robot status adjustment stage when the robot tips with legs 2, 4, and 6 in the stance state, after which the Stewart platform begins rotational motion.

In the scenario of the robot tipping recovery when climbing a steep slope, the stability margin and corresponding changes in the support polygon’s edges during the recovery process are shown by the red line in the top figure of Fig. 6 (i) and by the red line in the bottom figure of it. Leg 3 is selected as the adjustment leg, the adjustment time is 2.31 s with the target foothold point at (−2390, 1908, −1741). The robot state and feet force curves after leg 3 lands on the ground are shown in Fig. 6 (f). In the scenario of the robot tipping recovery when descending a steep slope, the stability margin and corresponding changes in the support polygon’s edges during the recovery process are shown by the green line in the top figure and by the black line in the bottom figure of Fig. 6 (i). Leg 1 is selected as the adjustment leg, and the adjustment time is 1.3 s with the target foothold point at (2490, 1913, −1728). The state and the feet force in the $z$ direction of the robot after leg 1 lands on the ground are shown in Fig. 6 (g). In the scenario of the robot tipping recovery when moving on a slope with a large angle of inclination, the stability margin and corresponding changes in the support polygon’s edges during the recovery process are shown by the yellow line in the top figure and by the blue line in the bottom figure of Fig. 6 (i). Leg 5 is selected as the adjustment leg, and the adjustment time is 1.48 s with the target foothold point at (0, −1894, −1782). the robot state and stance feet force in $z$ direction after leg 5 lands on the ground are shown in Fig. 6 (h).

Based on the above simulations, it can be observed that the robot is able to achieve tipping recovery according to the pre-designed recovery methods. Particularly in cases of forward and backward tipping instability, the improvement in the robot’s stability margin becomes quite evident after the adjustment leg lands. This is primarily because the new stability cone formed after the adjusting leg lands significantly expands the original cone. As to the simulation scenery of moving on a steeply inclined slope at a large angle, although the robot has regained stability, the stability margin is not high. The primary reason is that the range of motion of the leg’s swing joint and the maximum length of the leg limit the lateral distance that the foot can reach, preventing the construction of a new, larger stability cone. Therefore, when the robot is moving on a large lateral slope, more attention should be paid to the changes in the robot’s stability margin.