2.1. Description of the robot structure

The structure of the proposed CR consists of the following modules, arranged from top to bottom: a replaceable tool module, an upper adsorption device, an upper rotating device, a kinematically redundant PM, a lower rotating device, and a lower adsorption device, as shown in Figure 1.

Figure 1.

Structure of the CR.

Figure 1 Long description

A diagram of a climbing robot structure showing its labeled components and their relationships. Panel A: A side view of the climbing robot structure. The diagram shows an upper suction cup connected to an upper rotating device, which is linked to a parallel mechanism. The parallel mechanism connects to a lower rotating device, which is attached to a lower suction cup. Panel B: A front view of the climbing robot structure. The upper suction cup is connected to the upper rotating device, which is part of the parallel mechanism. The parallel mechanism connects to the lower rotating device, which is attached to the lower suction cup. The tool module is mounted on the upper part of the structure.

As shown in Figure 2, the kinematically redundant PM proposed in this paper essentially functions as a planar mechanism when the rotating devices are not considered. It consists of a moving platform, an intermediate platform, a fixed platform, and four RPR branches. The revolute joint axes of the mechanism are projected onto a common plane for simplified kinematic analysis. The centers of the R-joints attached to the fixed platform are designated as A ₁, A ₂, A ₃, and A ₄, all of which lie in the same horizontal plane. On the moving platform side, the R-joint axes of Branch 1 and Branch 2 coincide spatially. The projection of their intersection onto the simplified planar mechanism is defined as point P. Similarly, the coincident axes of Branch 3 and Branch 4 project to point C, while the intersection of the moving platform and the intermediate platform projects to point B.

Figure 2.

Simplified diagram of the kinematically redundant PM.

In order to facilitate the subsequent analysis, a fixed coordinate frame O ₁-XY is established at Point A ₁, with the X-axis along the direction of A ₁ A ₂ and the Y-axis perpendicular to A ₁ A ₂, directed upwards. A moving coordinate frame o ₂ -xy is established at Point P, with the x-axis along the direction of PB and the y-axis perpendicular to PB, directed upwards. The distances from A ₂, A ₃, and A ₄ to the Y-axis are denoted as l ₂, l ₃, and l ₄, respectively, and their distances to the X-axis are zero. The length of PB is denoted as l ₅ , and the length of BC as l ₆ . The four prismatic joints serve as the driving joints of the mechanism, and the displacements of the prismatic joints are represented by q _i (i = 1,2,3,4).

2.2. Climbing process

The climbing process of the robot is illustrated in Figure 3. The fundamental gait cycle is as follows: In the initial state, both suction cups of the CR remain adhered to the wall surface. Subsequently, one suction cup releases its adhesion. Driven by the PM and the rotary device, the released end of the robot moves to a designated position, where the suction cup re-adheres. Then, the second suction cup releases its adhesion. Guided by the PM and the rotary device, this end subsequently moves to the second predefined position and re-adheres, completing one full climbing cycle.

Figure 3.

Schematic of the robot climbing process: (a) Plane climbing gait; (b) Wall transition gait.

This climbing process is categorized into two scenarios based on the working environment: (1) Plane climbing: As shown in Figure 3(a), the robot performs translational and rotational motions within a single wall surface. In this mode, the PM primarily adjusts the position, while the rotary devices adjust the orientation within the plane. (2) Wall transition: As shown in Figure 3(b), the robot traverses between two intersecting planes. This process requires the coordination of the PM and the rotary devices to rotate the robot’s body around the intersection line, allowing the suction cups to align with the normal vector of the new surface.

2.3. Kinematically redundant PM DOF analysis

In order to determine the configuration of the CR, it is necessary to analyze the working environment of the CR. The CR requires the coordination of multiple DOFs for climbing on walls and transitioning between walls. During the climbing process, the translational DOF along the normal direction of the wall is denoted as TX, as shown in Figure 4(b). The direction perpendicular to the wall normal and pointing upwards is denoted as TY, as shown in Figure 4(c). The rotation DOF along the wall normal direction is denoted as RX, as shown in Figure 4(d). Therefore, through the coordination of these three DOFs, motion within a single wall surface can be achieved. For transitioning between walls, an additional rotation DOF denoted as RZ is required, as shown in Figure 4(e).

Figure 4.

The minimum DOFs required for a CR: (a) Climbing environment 1; (b) Translational DOF TX; (c) Translational DOF TY; (d) Rotational DOF RX; (e) Climbing environment 2 and rotational DOF RZ.

Through the above analysis, it is determined that the CR requires a minimum of four DOFs to achieve the climbing task. Conventional 2T2R PMs, due to their structural limitations, cannot be directly applied to CR. For example, a 2-RPU/2-SPU PM [Reference Fan, Liu and Zhang30] possesses two translational DOFs: one along the normal direction of the moving platform and the other along the radial direction of the revolute joints on the fixed platform, as well as two rotational DOFs about the two axes of the universal joint on the moving platform. The mechanism is structurally complex, and the branch assembly requirements are high. The fixed platform size of the 2PRS/2PUS mechanism [Reference Pu and Jin31] is large, which is suitable for scenarios such as machine tools. 2UPU&R (2R (RR&RPR)R) [Reference Fang, Liu, Yang and Jiang32] is complex in structure and large in size, which is not suitable for CRs.

Therefore, the required four DOFs are decomposed into two parts: the rotational DOF RX is provided by the serial rotary device, while the remaining DOFs can be achieved using a 3-RPR PM. However, the RZ rotational capability of this mechanism is highly limited and insufficient for wall-climbing applications that require the robot to traverse right-angle or acute-angle wall transitions. Such maneuvers typically demand at least 90° of rotation in RZ, which exceeds the range a conventional 3-RPR PM can provide [Reference Kong and Gosselin33]. To overcome this limitation and expand the rotational workspace, an intermediate platform and an additional RPR branch are introduced. This modification effectively upgrades the system into a kinematically redundant PM, as shown in Figure 2.

Based on the previous description, the aforementioned PM is a planar mechanism. According to the DOF formula [Reference Li, Wang, Liu and Huang34] for planar mechanisms, its mobility can be calculated as follows:

(1) \begin{align}F=3n-2P_{L}-P_{H}=4 \end{align}

where: n denotes the number of movable links in the mechanism, P _L represents the number of lower pairs, and P _H indicates the number of higher pairs.

In the mechanism under discussion, the specific identification of these components is illustrated in Figure 5. Specifically, Figure 5(a) labels the ten movable links as n _i (n = 10), while Figure 5(b) denotes the thirteen lower kinematic pairs as P _Lj (P _L = 13). There are no higher pairs in this configuration (P _H = 0).

Figure 5.

Kinematic diagram of the PM for DOF analysis:(a) Identification of the 10 movable links (n ₁–n ₁₀); (b) Distribution of the 13 lower kinematic pairs (P _L1–P _L13).

In summary, the mechanism possesses four DOFs. Due to its inherent planar nature, the moving platform contributes only three DOFs within the plane. The mechanism is actuated by four actuators, which match the total DOFs of the system but exceed the number of DOFs exhibited by the moving platform. Therefore, this configuration should be categorized as a kinematically redundant PM.

3.1. Position analysis

3.1.1. Inverse kinematics analysis

Inverse kinematics is the process of determining the required actuator variables to achieve a specific pose of the end-effector. In the context of the proposed kinematically redundant PM, this problem specifically involves calculating the linear displacements of the four prismatic driving joints, denoted as q _i (i = 1, 2, 3, 4). The inputs for this calculation are the position of the reference point p on the moving platform (represented by coordinates x, y), the orientation of the moving platform ( $\alpha$ ), and the angle of the intermediate platform ( $\beta$ ), as explicitly identified in Figure 2.

Based on the coordinate frame established in the previous section, the position vector of point p in the fixed coordinate frame O ₁-XY can be obtained as:

(2) \begin{align}{\bf{p}} = {\left[ {\begin{array}{cc}{x \quad y } \end{array}} \right]^{\rm{T}}} \end{align}

Similarly, the position vectors of points A _i (i = 1,2,3,4) in the fixed coordinate frame are:

(3) \begin{align} \begin{cases} \boldsymbol{a}_{1}=\left[\begin{array}{l@{\quad}l} 0 & 0 \end{array}\right]^{\mathrm{T}}\\[4pt] \boldsymbol{a}_{2}=\left[\begin{array}{l@{\quad}l} l_{2} & 0 \end{array}\right]^{\mathrm{T}}\\[4pt] \boldsymbol{a}_{3}=\left[\begin{array}{l@{\quad}l} l_{3} & 0 \end{array}\right]^{\mathrm{T}}\\[4pt] \boldsymbol{a}_{4}=\left[\begin{array}{l@{\quad}l} l_{4} & 0 \end{array}\right]^{\mathrm{T}} \end{cases} \end{align}

The position vectors of points B and C in the moving coordinate frame o ₂ -xy are denoted as:

(4) \begin{align}\boldsymbol{b}^{{o_{2}}}=\left[\begin{array}{l@{\quad}l} l_{5} & 0 \end{array}\right]^{\mathrm{T}} \end{align}

(5) \begin{align}\boldsymbol{c}^{{o_{2}}}=\boldsymbol{b}^{{o_{2}}}+\boldsymbol{R}_{f}\left[\begin{array}{l@{\quad}l} l_{6} & 0 \end{array}\right]^{\mathrm{T}} \end{align}

Where: $\boldsymbol{R}_{f}=\left[\begin{matrix} \begin{matrix} \cos \beta \\ \sin \beta \end{matrix} & \begin{matrix} -\sin \beta \\ \cos \beta \end{matrix} \end{matrix}\right], \beta$ represents the angle between the moving platform and the intermediate platform.

The position vectors of points B and C in the fixed coordinate frame can be represented as:

(6) \begin{align}\boldsymbol{b}=\boldsymbol{p}+\boldsymbol{R}\boldsymbol{b}^{{o_{2}}} \end{align}

(7) \begin{align}\boldsymbol{c}=\boldsymbol{b}+\boldsymbol{R}\boldsymbol{R}_{f}\left[\begin{array}{l@{\quad}l} l_{6} & 0 \end{array}\right]^{\mathrm{T}} \end{align}

Where: $\boldsymbol{R}=\left[\begin{array}{cc} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right], \alpha$ represents the angle of rotation of the moving platform around the Z-axis.

For branches 1 and 2, the constraint equations for the limbs are expressed as:

(8) \begin{align}{q}_{i}^{2}=(\,\boldsymbol{p}-\boldsymbol{a}_{i})^{\mathrm{T}}(\,\boldsymbol{p}-\boldsymbol{a}_{i}),(i=1,2) \end{align}

Similarly, for branches 3 and 4, the constraint equations are established as:

(9) \begin{align}{q}_{i}^{2}=(\boldsymbol{c}-\boldsymbol{a}_{i})^{\mathrm{T}}(\boldsymbol{c}-\boldsymbol{a}_{i}),(i=3,4) \end{align}

Based on Eqs. (8) and (9), the inverse position solution of the mechanism is obtained as follows:

(10) \begin{align} \begin{cases} q_{1}=\sqrt{x^{2}+y^{2}}\\[2pt] q_{2}=\sqrt{(x-l_{2})^{2}+y^{2}}\\[2pt] q_{3}=\sqrt{(y+e_{1})^{2}+(e_{2}-x)^{2}}\\[2pt] q_{4}=\sqrt{(y+e_{1})^{2}+(e_{3}-x)^{2}} \end{cases} \end{align}

Where: $e_{1}=\mathrm{s}_{\alpha }(l_{5}+l_{6}c_{\beta })+l_{6}c_{\alpha }s_{\beta },e_{2}=l_{3}-c_{\alpha }(l_{5}+l_{6}c_{\beta })+l_{6}s_{\alpha }s_{\beta },e_{3}=l_{4}-c_{\alpha }(l_{5}+l_{6}c_{\beta })+l_{6}s_{\alpha }s_{\beta }$ .

Since this mechanism is a kinematically redundant PM, if only the position and orientation of the moving platform are given, the inverse positions of branches 1 and 2 are uniquely determined, but the inverse positions of branches 3 and 4 are not unique. This is due to the existence of multiple possible values for the $\beta$ angle between the moving platform and the intermediate platform, leading to a multitude of inverse positions of branches 3 and 4. Consequently, to uniquely determine the inverse kinematics solutions for branches 3 and 4, the $\beta$ angle must be specified.

3.1.2. Forward kinematics analysis

The forward kinematics problem involves solving for the position and orientation of the moving platform and the angle between the moving platform and the intermediate platform using the driving values q _i of the branches. Based on the first two equations in Eq. (10), the position of the moving platform can be derived as:

(11) \begin{align}{\bf{p}} = \left[ \begin{array}{c}{p_x} \\{p_y} \end{array} \right] = \left[ \begin{array}{c}{\dfrac{{q_2}^2 - {q_1}^2 - {l_2}^2} { - 2{l_2}}} \\ {\sqrt {{q_1}^2 - {{\left( {\dfrac{{{q_2}^2 - {q_1}^2 - {l_2}^2}}{{ - 2{l_2}}}} \right)}^2}} } \end{array} \right] \end{align}

Combined with Eq. (7) and the latter two equations in Eq. (10), the position vector of point C in the O ₁-XY frame can be expressed as:

(12) \begin{align}{\bf{c}} = \left[ \begin{array}{c}{{c_x}} \\{{c_y}} \end{array} \right] = \left[ \begin{array}{c}\dfrac{q_4^2 - q_3^2 + l_3^2 - l_4^2} {2({l_3} - {l_4})} \\ \sqrt {q_3^2 - {{\left( {{l_3} - \dfrac{{q_4^2 - q_3^2 + l_3^2 - l_4^2} }{ {2({l_3} - {l_4})}}} \right)}^2}} \end{array} \right] \end{align}

Let k denote the distance between points p and C:

(13) \begin{align}k=\overline{pc}=\sqrt{(p_{x}-c_{x})^{2}+(p_{y}-c_{y})^{2}} \end{align}

Using triangle PBC, angle $\beta$ can be expressed as:

(14) \begin{align}\beta = - \left( {\pi - \arccos \left( {{{{l_5}^2 + {l_6}^2 - {k^2}} \over {2{l_5}{l_6}}}} \right)} \right) \end{align}

Let θ ₁ denote the angle between PB and PC, and θ ₂ denote the angle between PC and the X-axis. Both angles can be expressed as:

(15) \begin{align} \begin{cases}{\theta _1} = \arccos \left( {{\dfrac{l_5^2 + {k^2} - l_6^2} {2{l_5}k}}} \right) \\[8pt] {\theta _2} = \arctan \left( \dfrac{{c_y} - {p_y}} {{c_x} - {p_x}} \right) \end{cases} \end{align}

Angle $\alpha$ can be expressed as:

(16) \begin{align}\alpha =\theta _{1}+\theta _{2} \end{align}

The forward kinematics of the PM can be solved using Eqs. (11), (14), and (16).

3.2. Velocity and singularity analysis

Velocity analysis is the foundation for subsequent singularity and workspace analysis. In the subsequent analysis, the velocity v _p at the point P, the angular velocity of the moving platform with respect to the Z-axis $\dot{\alpha }$ and the relative angular velocity of the moving platform to the intermediate platform are $\dot{\beta }$ known.

3.2.1. Jacobian matrix analysis

The Jacobian matrix of PM plays an important role in singular position analysis, so it is necessary to analyze the Jacobian matrix of this mechanism [Reference Gosselin, Laliberté and Veillette16].

In branches 1 and 2, differentiating Eq. (8) with respect to time gives:

(17) \begin{align}q_{i}\dot{q}_{i}=\left(\,\boldsymbol{p}-\boldsymbol{a}_{i}\right)^{\mathrm{T}}\dot{\boldsymbol{p}},\left(i=1,2\right) \end{align}

Similarly, in branches 3 and 4, differentiating Eq. (9) with respect to time gives:

(18) \begin{align}q_{i}\dot{q}_{i}=\left(\boldsymbol{c}-\boldsymbol{a}_{i}\right)^{\mathrm{T}}\dot{\boldsymbol{c}},\left(i=3,4\right) \end{align}

The constraint equation for the intermediate platform BC can be expressed as follows:

(19) \begin{align}{l}_{6}^{2}=(\boldsymbol{c}-\boldsymbol{b})^{\mathrm{T}}(\boldsymbol{c}-\boldsymbol{b}) \end{align}

Taking the derivative of Eq. (19) with respect to time, we obtain:

(20) \begin{align}\left(\boldsymbol{c}-\boldsymbol{b}\right)^{\mathrm{T}}\dot{\boldsymbol{b}}=\left(\boldsymbol{c}-\boldsymbol{b}\right)^{\mathrm{T}}\dot{\boldsymbol{c}} \end{align}

By rearranging Eq. (18), we can obtain:

(21) \begin{align}\boldsymbol{G}\dot{\boldsymbol{c}}=\boldsymbol{H} \end{align}

Where: $\boldsymbol{G}=\left[\begin{matrix} (\boldsymbol{c}-\boldsymbol{a}_{3})^{\mathrm{T}}\\ (\boldsymbol{c}-\boldsymbol{a}_{4})^{\mathrm{T}} \end{matrix}\right],\boldsymbol{H}=\left[\begin{matrix} q_{3}\dot{q}_{3}\\ q_{4}\dot{q}_{4} \end{matrix}\right]$ .

For the matrix G , as long as branches 3 and 4 do not coincide (i.e., l ₃≠l ₄), the matrix remains full-rank. Therefore, Eq. (21) can be rearranged to obtain:

(22) \begin{align}\dot{\boldsymbol{c}}=\boldsymbol{KH} \end{align}

Where: $\boldsymbol{K}=\frac{1}{(\boldsymbol{c}-\boldsymbol{a}_{3})^{\mathrm{T}}\boldsymbol{L}(\boldsymbol{c}-\boldsymbol{a}_{4})}\left[\begin{matrix} \boldsymbol{L}(\boldsymbol{c}-\boldsymbol{a}_{4}) & -\boldsymbol{L}(\boldsymbol{c}-\boldsymbol{a}_{3}) \end{matrix}\right], \boldsymbol{L}=\left[\begin{matrix} \begin{matrix} 0\\ 1 \end{matrix} & \begin{matrix} -1\\ 0 \end{matrix} \end{matrix}\right]$ .

By taking the derivative of Eq. (6) with respect to time and substituting it along with Eq. (22) into Eq. (20), we obtain:

(23) \begin{align}\left(\boldsymbol{c}-\boldsymbol{b}\right)^{\mathrm{T}}\left(\dot{\boldsymbol{p}}+\dot{\alpha }\boldsymbol{LR}\boldsymbol{b}^{{o_{2}}}\right)=\left(\boldsymbol{c}-\boldsymbol{b}\right)^{\mathrm{T}}\boldsymbol{KH} \end{align}

Based on the analysis in the previous section, the moving platform has three DOFs (2T1R). The velocity of the moving platform is defined as $\boldsymbol{t}_{p}=[\begin{matrix} \dot{x} && \dot{y} && \dot{\alpha } \end{matrix}]^{\mathrm{T}}$ , and the velocity of the driving joint is defined as $\dot{\boldsymbol{q}}=[\begin{matrix} \dot{q}_{1} && \dot{q}_{2} && \dot{q}_{3} && \dot{q}_{4} \end{matrix}]^{\mathrm{T}}$ .

Therefore, rearranging Eqs. (17) and (23) we obtain:

(24) \begin{align}\boldsymbol{J}_{A}\boldsymbol{t}_{p}=\boldsymbol{J}_{q}\dot{\boldsymbol{q}}_{i} \end{align}

Where: $\boldsymbol{J}_{A}=\left[\begin{matrix} \begin{matrix} (\,\boldsymbol{p}-\boldsymbol{a}_{1})^{\mathrm{T}}\\ (\,\boldsymbol{p}-\boldsymbol{a}_{2})^{\mathrm{T}}\\ (\boldsymbol{b}-\boldsymbol{c})^{\mathrm{T}} \end{matrix} & \begin{matrix} 0\\ 0\\ (\boldsymbol{b}-\boldsymbol{c})^{\mathrm{T}}\boldsymbol{LR}\boldsymbol{b}^{{o_{2}}} \end{matrix} \end{matrix}\right]_{3\times 3}\!,\,\boldsymbol{J}_{q}=\left[\begin{matrix} \begin{matrix} q_{1}\\ 0\\ 0 \end{matrix} & \begin{matrix} 0\\ q_{2}\\ 0 \end{matrix} & \begin{matrix} 0\\ 0\\ \boldsymbol{n}_{1} \end{matrix} & \begin{matrix} 0\\ 0\\ \boldsymbol{n}_{2} \!\end{matrix} \end{matrix}\right]_{3\times 4}\!, \boldsymbol{n}_{1}=\dfrac{(\boldsymbol{c}-\boldsymbol{b})^{\mathrm{T}}\boldsymbol{L}(\boldsymbol{c}-\boldsymbol{a}_{4})q_{3}}{(\boldsymbol{c}-\boldsymbol{a}_{3})^{\mathrm{T}}\boldsymbol{L}(\boldsymbol{c}-\boldsymbol{a}_{4})},\\ \boldsymbol{n}_{2}=\dfrac{-(\boldsymbol{c}-\boldsymbol{b})^{\mathrm{T}}\boldsymbol{L}(\boldsymbol{c}-\boldsymbol{a}_{3})q_{4}}{(\boldsymbol{c}-\boldsymbol{a}_{3})^{\mathrm{T}}\boldsymbol{L}(\boldsymbol{c}-\boldsymbol{a}_{4})}$ .

3.2.2 Velocity analysis

Since the angular velocity is perpendicular to the XOY plane, subsequent analysis and calculations need to be conducted in a three-dimensional space. For the coordinate frame established in the previous section, all points have zero linear velocity components in the Z-axis direction. Therefore, the position vector of point p can be represented as $\boldsymbol{p}=[\begin{matrix} x && \begin{matrix} y && 0 \end{matrix} \end{matrix}]^{\mathrm{T}}$ , and the position vectors of other points can be transformed into three-dimensional vectors using the same method as for point P.

Therefore, the velocity of point p is represented as:

(25) \begin{align}\boldsymbol{V}_{p}=\left[\begin{matrix} \boldsymbol{v}_{p}\\ \boldsymbol{\omega }_{p} \end{matrix}\right]=\left[\begin{matrix} \left[\begin{matrix} \dot{x} &\quad \dot{y} &\quad 0 \end{matrix}\right]^{\mathrm{T}}\\[2pt] \left[\begin{matrix} 0 &\quad 0 &\quad \dot{\alpha } \end{matrix}\right]^{\mathrm{T}} \end{matrix}\right] \end{align}

Similarly, the velocity of point B is:

(26) \begin{align}\boldsymbol{V}_{b}=\left[\begin{matrix} \boldsymbol{v}_{b}\\ \boldsymbol{\omega }_{b} \end{matrix}\right]=\left[\begin{matrix} \boldsymbol{v}_{p}+\boldsymbol{\omega }_{p}\times \boldsymbol{s}_{b}\\ \boldsymbol{\omega }_{p} \end{matrix}\right] \end{align}

Where: s _b represents the vector from the point p to the point B in a fixed coordinate system.

The linear velocity of the point P can be expressed in another way, as Eq. (27):

(27) \begin{align}\boldsymbol{v}_{p}=\dot{q}_{i}\boldsymbol{l}_{i}+\boldsymbol{\omega }_{li}\times \left(q_{i}\cdot \boldsymbol{l}_{i}\right)\!,\left(i=1,2\right) \end{align}

Where: l _i represents the unit vector from A _i to P, $\boldsymbol{\omega}$ _li denotes the angular velocities of the driving joints.

By multiplying both sides of Eq. (27) by l _i , we can derive the velocity of the driving joints 1 and 2 as:

(28) \begin{align}\dot{q}_{i}={\boldsymbol{l}}_{i}^{T}\boldsymbol{v}_{p},\left(i=1,2\right) \end{align}

In order to determine the angular velocities of parts in branches 1 and 2, we perform a cross product with l _i on both sides of the Eq.(27), and the following angular velocity expression is obtained:

(29) \begin{align}\boldsymbol{\omega }_{li}=\frac{\boldsymbol{l}_{i}\times \boldsymbol{v}_{p}}{q_{i}},\left(i=1,2\right) \end{align}

For branches 3 and 4, the linear velocity of the point C can be expressed as shown in Eq. (30). Alternatively, it can be formulated in a different way as demonstrated in Eq. (31):

(30) \begin{align}\boldsymbol{v}_{c}=\boldsymbol{v}_{b}+\boldsymbol{\omega }_{c}\times \boldsymbol{s}_{c} \end{align}

(31) \begin{align}\boldsymbol{v}_{c}=\dot{q}_{i}\boldsymbol{l}_{i}+\boldsymbol{\omega }_{li}\times \left(q_{i}\cdot \boldsymbol{l}_{i}\right),\left(i=3,4\right) \end{align}

Where: s _c represents the vector from the point B to the point C in a fixed coordinate system, l _i represents the unit vector from A _i to C.

Similarly, the velocities of the driving joints 3 and 4 can be represented as:

(32) \begin{align}\dot{q}_{i}={\boldsymbol{l}}_{i}^{T}\boldsymbol{v}_{c},\left(i=3,4\right) \end{align}

The angular velocities of parts in branches 3 and 4 are represented as:

(33) \begin{align}\boldsymbol{\omega }_{li}=\frac{\boldsymbol{l}_{i}\times \boldsymbol{v}_{c}}{q_{i}},\left(i=3,4\right) \end{align}

3.3. Singularity analysis

When the mechanism is in a singular position, there is a significant alteration in its motion accuracy, stability, and DOFs. For example, instantaneous changes in the number of DOFs can cause the mechanism to jam, result in mechanical structure damage, or lead to loss of control. Therefore, studying the singularity of the mechanism can help us identify its singular configuration and ensure the safe operation of the mechanism.

The singular configurations of a mechanism can be obtained by analyzing the rank of the Jacobian matrix of Eq. (24). When the rank of the matrix J _q is reduced, inverse kinematic singularity occurs in the mechanism; When the rank of the matrix J _A is reduced, forward kinematic singularity occurs; When the ranks of both matrices are reduced, mixed singularity occurs in the mechanism [Reference Gosselin and Angeles35, Reference Yiğit, Breton and Gosselin36].

According to the Jacobian matrix, the rows within matrix J _A correspond to the position vectors of PA ₁, PA ₂, and BC; these position vectors are non-zero. In the design process, as long as the positions of points A ₁ and A ₂ do not coincide, PA ₁ and PA ₂ will not be collinear. Since PA ₁ and PA ₂ intersect at point P, it is necessary to ensure that BC and PB are not collinear to prevent forward kinematic singularity. In matrix J _q , the displacement of the driving joints q _i (i = 1, 2, 3, 4) will not be zero. Therefore, it is only necessary to analyze the position vectors of CA ₃ and CA ₄ when the positions of A ₃ and A ₄ do not coincide; inverse kinematic singularity will not occur, a condition that can be avoided during the design process. Through the above analysis, some singularities of the mechanism can be avoided through structural design. Therefore, during the operation of the mechanism, forward kinematic singularity will occur when BC and PB are collinear; the mechanism will have no forward kinematic singularity and mixed singularity. The robot must avoid entering these singular configurations during movement, as shown in Figure 6.

Figure 6.

Singular configurations: (a) Singular configuration 1; (b) Singular configuration 2.

3.4. Workspace analysis

The reachable workspace [Reference Ebrahimi, Carretero and Boudreau37, Reference Huang and Thebert38] of a PM refers to all the positions and orientations that the moving platform can reach under corresponding constraints. The solution of the workspace of the PM is based on the inverse kinematics of the PM, which is solved by the numerical search method, as shown in Figure 7. The parameters of the mechanism are as follows: l ₂ = 68 mm, l ₃ = 0, l ₄ = 68 mm, l ₅ = 43 mm, l ₆ = 25 mm, the motion range for driving q _i (i = 1, 2, 3, 4) is 120 ∼ 240 mm, the range of $\beta$ angle is 180°∼360°, the angular range of the joints is −50°∼50°.

Figure 7.

Workspace of the PM: (a) $\alpha$ = 0°; (b) $\alpha$ = 45°; (c) $\alpha$ = 90°.

The figure presents the workspace for $\alpha$ angles of 0°, 45°, and 90°, and it can be seen from the figure that as the $\alpha$ angle increases, the workspace of the mechanism gradually decreases in the y-axis direction. When the CR is in motion, the poses of the upper and lower adsorption devices relative to the upper and lower platforms remain constant; only simple coordinate transformations are needed, which will not be further elaborated here.

3.5. Simulation

After conducting the aforementioned analysis, the kinematics of the PM were simulated, encompassing both position and velocity simulations. The trajectory of a moving platform is defined as:

(34) \begin{align}\left\{\begin{matrix} \!\!\!\!\!\!\!\alpha =-\dfrac{\pi }{180}\left(45\dfrac{t^{5}}{10000}-25\dfrac{t^{4}}{400}+9\dfrac{t^{3}}{80}\right)\\[12pt] \begin{matrix} \beta =\dfrac{\pi }{180}\left(9\dfrac{t^{5}}{10000}-9\dfrac{t^{4}}{400}+3\dfrac{t^{3}}{20}\right)+11\dfrac{\pi }{9}\\[12pt]\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!x=\left(8\dfrac{t^{5}}{2500}-9\dfrac{t^{4}}{100}+3\dfrac{t^{3}}{5}\right)+25 \end{matrix}\\[25pt] \!\!\!\!\!\!\!\!\!\!\!y=\left(21\dfrac{t^{5}}{2500}-22\dfrac{t^{4}}{100}+7\dfrac{t^{3}}{5}\right)+125 \end{matrix}\left(t=0\sim 10s\right)\right. \end{align}

The simulation results for inverse position and velocity are shown in Figures 8 and 9, respectively. The simulated values are represented by circles, while the theoretical values are depicted by solid lines. Upon comparison, the data from the theoretical and simulated values align well, verifying the correctness of the kinematic modeling.

Figure 8.

Comparison between theoretical and simulated values for position analysis: (a) drive joint 1; (b) drive joint 2; (c) drive joint 3; (d) drive joint 4.

Figure 9.

Comparison between theoretical and simulated values for velocity analysis: (a) drive joint 1; (b) drive joint 2; (c) drive joint 3; (d) drive joint 4.

4.1. Global path planning

Before conducting global path planning, the transitional capability between two walls for the CR must be analyzed. If the suction cup at one end of the robot is anchored within a plane, and the robot’s workspace intersects with another plane, then the inter-wall transition is deemed feasible. In order to quickly analyze the transition between wall surfaces, the model is simplified as shown in Figure 10. For ease of representation and subsequent analysis, we define a method for representing walls in a three-dimensional environment. The coordinates of wall vertices are expressed in the base coordinate system as follows:

(35) \begin{align}D_{i}=\left\{P| P\in \Gamma _{i}\right\} \end{align}

Where: D _i denotes the vertices of the wall surface, Γ_i denotes the wall surface. The normal vector of the wall surface is denoted as m , which can be calculated using the coordinates of the wall vertices.

Figure 10.

Simplified model of walls and CR.

To ensure that the entire adsorption area of the suction cup lies within the wall surface, the wall surface is contracted inwards by a distance of R _l (the radius of the suction cup). The position of the suction cup on the wall surface can be simplified to a point (the geometric center of the suction contact area). When the suction cup is securely adhered to the wall surface, the normal of the suction surface, the axis of the rotating device, and the normal of the wall surface are parallel to each other, so the inwardly retracted walls are moved a distance of h along the normal direction of the walls. The wall surfaces in contact with the adsorption devices on the fixed and moving platform sides are denoted as $\Gamma_1$ and $\Gamma_2$ , respectively; their corresponding point-simplified versions are designated as $\Gamma_1'$ and $\Gamma_2'$ ; and the surfaces mapped onto the fixed and moving platforms of PM are referred to as $\Gamma_1''$ and $\Gamma_2''$ , respectively.

If a transition is possible between the two walls, after translation, the coordinate origins of the upper and lower platforms of the PM will be respectively positioned within these two walls. The rotation range of the moving platform in the PM is from −20° to 110°, and it can achieve climbing on walls with large angles. After simplified modeling, taking a point on one of the simplified wall surfaces as the coordinate origin, if the workspace of the PM intersects with the other wall plane, it indicates the feasibility of inter-wall traversal between the two surfaces.

During the search process, the wall surfaces in the environment are first assigned numerical labels. Subsequently, the transitions between each pair of wall surfaces are analyzed. Specifically, transition analysis is performed on point pairs near the boundary regions of two wall surfaces. The results of this analysis are stored in the adjacency matrix. The adjacency matrix is a square matrix used to represent the connection relationships between vertices in the graph.

Following the transitional feasibility analysis, the wall surface at the starting point is designated as the root node. The tree is then traversed in a depth-first manner, visiting all wall surfaces until a leaf node is reached. The search then backtracks to the previous wall surface and continues searching along other branches, until all possible wall paths from the starting wall surface to the target wall surface are identified.

For each accessible wall path, the transition point pairs are computed in a pairwise manner according to the path sequence. During the computation of transition point pairs, the robotic constraint equations (e.g., Eq. (36)) are incorporated. If a point pair satisfies the constraints, it is stored as a candidate transition point pair. After solving for the candidate transition point pairs, we optimize the selection of transition points to minimize the total global path length. The global path solution equation is presented as Eq. (37).

(36) \begin{align} \begin{cases} \boldsymbol{q}=IK(x,y,\alpha, \beta )\\ \boldsymbol{J}_{A}\neq 0\\ {\boldsymbol{P}}_{i}^{S}\in \Gamma _{i+1}\\ {\boldsymbol{P}}_{i}^{E}\in \Gamma _{i} \end{cases} \end{align}

(37) \begin{align}\boldsymbol{P}_{\min}\!\left(x,y,z\right)=\sum_{i=1}^{N}\left| {\boldsymbol{P}}_{i}^{E}-{\boldsymbol{P}}_{i}^{S}\right| +\sum_{i=1}^{N-1}\left| {\boldsymbol{P}}_{i+1}^{B}-{\boldsymbol{P}}_{i}^{B}\right| \end{align}

where: N represents the number of walls, ${\boldsymbol{P}}_{i}^{S}$ and ${\boldsymbol{P}}_{i}^{E}$ denote the coordinate positions of the transition points within the wall surfaces, IK represents the inverse kinematic of the PM, q represents the displacements of the driving joints in the PM, q = [q ₁, q ₂, q ₃, q ₄], and Γ_i denotes the i-th wall. J _A denotes the Jacobian matrix.

4.2. Local path planning and Single-step motion planning

In the previous section, the globally shortest path was identified, but the path within the wall surfaces has not yet been processed. On the wall surfaces, there exist obstacles of different sizes, and different strategies are adopted for these obstacles. Therefore, a cost function is incorporated into the original A * algorithm [Reference Hart, Nilsson and Raphael39] to handle various obstacles. The obstacles on the walls are classified according to the robot’s movement capabilities, with step over obstacles and step on obstacles illustrated in Figure 11(a) and (b).

Figure 11.

Obstacle analysis: (a) step over obstacles (b) step on obstacles.

Based on the three-dimensional dimensions of the obstacles, for those that can be stepped over, their length can be disregarded, while their width and height dimensions should meet the following conditions:

(38) \begin{align} \begin{cases} w_{o}\leq L-2R_{l}\\ h_{o}\leq l \end{cases} \end{align}

Where: w _o and h _o indicate the length and height of the obstacle, respectively. R _l is the radius of the suction cup, L represents the robot’s maximum movement range, and l denotes the distance between the lowest point of the connecting rod in a PM and the adsorption surface of the suction cup.

For the case that can step on an obstacle, the dimensions of the obstacle should meet the following constraints:

(39) \begin{align} \begin{cases} w_{o}\geq 2R_{l}\\ l_{o}\geq 2R_{l}\\ h_{o}\leq L\cos \phi \end{cases} \end{align}

where: $\phi$ denotes the rotation angle at the joint, and l _o denotes the length of the obstacle.

For obstacles of other sizes, the CR can only choose to bypass them. The improved A * algorithm is shown in Eq. (40):

(40) \begin{align}f\!\left(u\right)=g(u)+h(u)+o(u) \end{align}

where: g(u) represents the actual distance to the goal point, h(u) represents the Manhattan distance to the goal point, o(u) represents the cost function, and is used to deal with different obstacles. For obstacles that require detouring, their cost function is set to infinity. For traversable or step on obstacles, their cost function is determined by multiplying the obstacle’s height by a coefficient(k * h _o ).

The path searched by the improved A* algorithm is continuous, whereas the movement of the CR is in a discrete point-to-point form; therefore, the path needs to be discretized. During the discretization process, since the rotating device of the CR can rotate 240°, an annular sector region is formed, centered at the starting point with a radius equivalent to CR’s motion range, as illustrated in Figure 12. The extent of this region can be mathematically described by Eq. (41).

Figure 12.

Schematic diagram for solving discrete points.

The path points within the annular region can be considered as potential discrete points. If a potential discrete point intersects with the path, it is designated as a discrete point. If there are multiple intersects within the annular region, the point nearest to the target position should be selected as the discrete point while avoiding obstacles. Then, this discrete point is taken as the new starting point, and the above operations are repeated. If there are no path points within the annular area, the range of the annular area is adjusted, and the search is redone. Finally, to ensure safety, collision detection is performed on the above discrete points.

(41) \begin{align} \begin{cases} x_{a}=x_{0}+(L_{\max }-L_{\min })\cdot \sin \varphi \\ y_{a}=y_{0}+(L_{\max }-L_{\min })\cdot \cos \varphi \\ z_{a}=z_{0} \end{cases} \end{align}

Where: [x _a y _a z _a ] represents the coordinates of potential discrete points within the region, [x ₀ y ₀ z ₀] denotes the coordinates of the current point, and $\varphi$ indicates the rotation angle.

Through the search process outlined above, we obtain all the discrete points from the starting point to the target point.

4.3. Simulation

After a theoretical analysis, each layer of the three-layer search framework was simulated, including global path planning, local path planning, and collision-free, singularity-free single-step motion.

4.3.1 Global path planning simulation

In Global Path Planning, a given set of 50 sets of walls (number of walls is 2–6) is analyzed using the Global Path Planner. During global path planning experiments, 50 sets of planes (each containing 2–6 planes) were analyzed. To optimize computational efficiency, the mechanism workspace was precomputed offline. The number of transition point pairs sampled within each wall was set to 100. The total search time for all transition point pairs remained approximately 6 s, with search time increasing proportionally as the number of walls (planes) grew. Table I lists the search time and path length.

Table 1.

Global path search time and length

4.3.2 Local path planning simulation

During the local path-finding process, a local path search within the plane was conducted for 50 sets of walls from the global path planning. Taking two of these groups as examples, the results are shown in Figure 13. In the figure, obstacles are differentiated by color: green denotes traversable obstacles, blue indicates obstacles that can be stepped on, and cyan represents obstacles requiring bypass. During path planning, the robot’s end-effector suction cup is treated as a point. To ensure collision avoidance, obstacle boundaries undergo expansion processing: (1) Bypass-required obstacles have their boundaries expanded outward by half the robot’s upper/lower platform width; (2) For traversable/step-on obstacles, boundaries are adjusted by expanding/contracting outward/inward by the suction cup radius. Solid lines depict actual obstacle boundaries, while dashed lines indicate processed boundaries after expansion/contraction adjustments.

Figure 13.

Comparison of path search between the traditional A* algorithm and the improved A* algorithm: (a) traditional A* algorithm (b) improved A* algorithm.

As shown in the figure, the traditional A* algorithm can only bypass obstacles, while the improved A* algorithm fully leverages the CR’s obstacle-surmounting capabilities. When encountering traversable or step-on obstacles, the algorithm calculates the cost of crossing/stepping versus bypassing, prioritizing paths with lower costs.