Introduction

Traditional mobile robots mainly include: wheeled, tracked, legged mobile robots, and hybrid mobile robots. These robots, due to their unique structural design and mobility mechanisms, demonstrate their respective adaptability and characteristics when facing different terrain environments1. Wheeled robots are suitable for high-speed movement on flat ground, tracked robots are adept at handling complex terrain, while legged robots exhibit exceptional flexibility and adaptability in extreme environments. Hybrid robots can combine the advantages of wheeled, tracked, and legged robots. The body of wheeled, tracked, legged, and their hybrid mobile mechanisms remains in a relatively fixed state during movement, with the mechanism achieving mobility through the motion of its mobile units or articulated joints. To enhance mobility, the design of some wheels, tracks, or legged mechanisms has already incorporated the concept of deformability. To fully utilize the deformable design concept and further enhance the robot’s mobility, the concept of deformability is extended from a single mobile unit to the entire mobile mechanism, thereby exploring and developing an overall deformable mobile mechanism. This type of mechanism achieves movement through regular deformation of the entire structure by eliminating the fixed frame, enhancing the mechanism’s deformation and mobility capabilities. To improve the mobility of robots, new types of mobile robots distinct from traditional mobile robots, such as biomimetic mobile robots2, reconfigurable mobile robots3, and origami robots4, are continuously being researched and developed.

The tensegrity system is a spatial stable structure composed of the interaction between discontinuous compressed components and continuous tensioned components. Shibata et al.5 proposed a rolling robot based on the tensegrity structure, which verified that the six-bar tensegrity structure has a rolling movement mode. NASA proposed the SuperBall planetary exploration robot based on the tensegrity structure, which achieves movement by deforming the entire robot6. Böhm et al.7 designed a spherical tensegrity rolling robot that achieves rolling by changing the internal mass distribution through a driving module. Subsequently, scholars have conducted extensive research on the structural design, control algorithms, gait and path planning of tensegrity robots8,9,10,11. In the field of polyhedral linkage robots, Hamlin et al.12 combined concentric multi-link spherical joints with parallel mechanisms, and proposed the concept of a variable geometry truss with walking function, through different combinations, its appearance can exhibit the geometric characteristics of tetrahedrons and octahedrons. NASA proposed the TET-walker space exploration robot with tetrahedrons as the basic unit13. Liu et al.14 proposed a modular reconfigurable variable topology truss robot, which can be applied to rescue work in disaster scenes and other scenarios. Tian et al.15 proposed a modular polyhedral multi-loop rover, which forms a polyhedral shape through a multi-loop linkage mechanism. Li et al.16 proposed a class of deformable tetrahedral rolling mechanisms with URU branched chains, and proposed an equivalent planar mechanism analysis method. Liu et al.17 designed a class of 8-DOF deformable tetrahedral mobile robots based on tetrahedrons and Sarrus linkages, which can obtain multiple movement modes through deformation. Wang et al.18 designed and analyzed a fully rotatable folding tetrahedral rolling mechanism, the proposed tetrahedron has four vertices and three RRR branched chains. Cheng et al.19 designed a class of mobile mechanisms based on the double parallelogram, which has two movement modes: tracked rolling mode and obstacle-climbing mode, and can switch between different terrains. Tang et al.20 designed a class of 4-DOF mobile mechanisms with Oloid-like paddlewheels based on the Schatz linkage mechanism, which can complete omni-directional movement on land and in water. Zhang et al.21,22 designed a centrally-driven single DOF rolling mechanism based on the hexagonal mechanism and crank linkage mechanism, and analyzed its terrain adaptability, including its climbing and passive rolling modes. Zhao et al.23 designed a class of highly terrain-adaptive polyhedral robots based on the tetrahedral mechanism, and proposed a concave polyhedral structure for vertical obstacle enveloping and climbing motion planning. Bian et al.24 proposed a new type of wheeled rolling robot composed of a planar 3-RRR parallel mechanism and a spoke-type variable-diameter wheel, and established kinematic and dynamic models. It can be seen that a large number of polyhedral linkage mobile mechanisms have been designed based on tetrahedral, parallelogram, hexagonal, and other mechanisms.

The mobile performance is reflected by its mobile mechanism, so the research and development of the mobile mechanism are conducive to enhancing the robot’s mobility. From the perspective of mechanism theory, the mobile mechanism can be regarded as a complete set of linkage mechanisms, and the movement forms such as rolling, walking, crawling, sliding, and squirming can be regarded as the combined movement of the linkages. Therefore, the linkage mechanism possesses flexible and rich deformation capabilities and motion characteristics. In-depth research on the combination methods and techniques of the linkage mechanism can be conducted to enhance the mobility of the mechanism. Previously, the design of Bricard-like mechanism with single DOF and 8R-like mechanism with multi-mode based on anti-parallelogram mechanism was proposed, and the construction method and analysis method were proposed25,26,27. The application of the constructed linkage mechanism to the mobile mechanism has not yet been studied. Therefore, this paper is mainly based on the idea and method of loop-construction, that is, the planar single DOF mechanisms (two parallelograms and two anti-parallelograms) are connected through revolute joints to construct a new type of spatial mechanism with multiple motion modes. This paper analyzes the construction, DOF, motion characteristics, etc. of the multi-mode mobile mechanism from the perspective of mechanism science. The paper is organized as follows: “The construction of the multi-mode mobile mechanism” introduces the morphological change characteristics of parallelogram mechanism unit and anti-parallelogram mechanism unit, and obtain the construction method of the multi-mode mobile mechanism. In Section “DOF analysis of multi-mode mobile mechanism”, the motion screw and the screw constraint topological diagram are obtained through screw theory and graph theory, then the DOF is obtained. The motion feasibility and motion characteristics of the three motion modes are discussed in Section “Analysis of motion feasibility and characteristics”. A simulation model and prototype model are made to verify the correctness of the analysis results and the feasibility of motion in Section “Verification of the principle prototype model”. Section “Conclusion and discussion” concludes the paper and discusses subsequent research work and applications.

The construction of the multi-mode mobile mechanism

Both anti-parallelogram mechanism and parallelogram mechanism have one DOF, but their motion characteristics are different. As shown in Fig. 1a, the anti-parallelogram mechanism has two sets of identical links that can undergo morphological changes with the variation of angle δ1. According to the deformation characteristics, the anti-parallelogram mechanism can be used to design a tumbling mechanism, which completes the tumbling motion by changing the center of mass through angular variations. As shown in Fig. 1b, the parallelogram mechanism has four identical links that can undergo morphological changes with the variation of angle δ2. According to the deformation characteristics, two types of mobile mechanisms can be designed: when all four links touch the ground, the ground can be creeped through the variation of angle δ2. When only one link touches the ground, the center of mass can be changed through the variation of angle δ2 to complete rolling. Therefore, combining the motion characteristics of anti-parallelogram mechanism unit and parallelogram mechanism unit, a multi-mode mobile mechanism that can complete rolling, creeping, and tumbling can be designed.

Fig. 1
figure 1

Motion characteristics of anti-parallelogram unit and parallelogram unit. (a) Anti-parallelogram. (b) Parallelogram.

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According to the characteristics of parallelogram mechanism, a quasi-regular hexahedron mechanism composed of two symmetrically arranged anti-parallelogram units and two symmetrically arranged parallelogram units is designed. In this article, let the short links of the anti-parallelogram be the same length as the links of the parallelogram, denoted as l, and the long links of the anti-parallelogram be denoted as \(\sqrt 2\) l. As shown in Fig. 2, all the mechanism units are connected by revolute joints. The multi-mode mechanism has two types of units (anti-parallelogram units and parallelogram units). Figure 2 shows its initial position, which is a quasi-regular hexahedron. The multi-mode mechanism has a total of 16 links and 20 revolute joints.

Fig. 2
figure 2

Configuration composition of multi-mode mobile mechanism.

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DOF analysis of multi-mode mobile mechanism

The DOF refers to the number of independent motion parameters that must be given when determining the motion of the mechanism, that is, the minimum number of inputs required. The multi-mode mobile mechanism has multiple loops. As shown in Fig. 3, a Cartesian coordinate system for the initial position of the multi-mode mechanism is established, let the origin be the revolute joint point A1, the line connecting the revolute joint point A1 and the revolute joint point B1 be the x-axis, and the line perpendicular to the x-axis in the plane where the parallelogram unit A1B1C1D1 is located be the z-axis (at the initial position, the z-axis is the line where A1D1 is located), the direction of the y-axis can be obtained by using the right-hand rule, and the four revolute joints connecting the two parallelogram and two anti-parallelogram units be denoted as J1, J2, J3, and J4 respectively.

Fig. 3
figure 3

Establishment of coordinate systems for multi-mode mobile mechanisms.

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According to the established coordinate system, the motion screws of each revolute joint are respectively denoted as SAi, SBi, SCi, SDi, and SJi (i = 1, 2, 3, 4). Ignoring the size of the connecting joints, the motion screws of the 20 revolute joints can be obtained as follows:

$$\begin{gathered} \left\{ \begin{gathered} {\mathbf{S}}_{A1} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{A2} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{A3} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{A4} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right\} \hfill \\ \end{gathered} \right. \, \left\{ \begin{gathered} {\mathbf{S}}_{B1} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 & l \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{B2} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & l & { - l} \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{B3} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 & l \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{B4} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & l & { - l} \\ \end{array} } \right\} \hfill \\ \end{gathered} \right. \, \left\{ \begin{gathered} {\mathbf{S}}_{C1} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & { - l} & 0 & l \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{C2} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & { - l} \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{C3} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & { - l} & 0 & l \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{C4} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & { - l} \\ \end{array} } \right\} \hfill \\ \end{gathered} \right. \, \\ \left\{ \begin{gathered} {\mathbf{S}}_{D1} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & { - l} & 0 & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{D2} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & l & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{D3} = \left\{ {\begin{array}{*{20}c} 0 & 1 & 0 & { - l} & 0 & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{D4} = \left\{ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & l & 0 \\ \end{array} } \right\} \hfill \\ \end{gathered} \right. \, \left\{ \begin{gathered} {\mathbf{S}}_{J1} = \left\{ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 & 0 & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{J2} = \left\{ {\begin{array}{*{20}c} 0 & 0 & 1 & l & 0 & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{J3} = \left\{ {\begin{array}{*{20}c} 0 & 0 & 1 & l & { - l} & 0 \\ \end{array} } \right\} \hfill \\ {\mathbf{S}}_{J4} = \left\{ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 & { - l} & 0 \\ \end{array} } \right\} \hfill \\ \end{gathered} \right. \\ \end{gathered}$$
(1)

Based on screw theory and graph theory to calculate the DOF, let the links are represented by circles, the revolute joints are represented by lines, and the links are respectively denoted as:

$$\left\{ \begin{gathered} A_{i} B_{i} = 1{\mathbf{ + }}4(i - 1) \hfill \\ B_{i} C_{i} = 2{\mathbf{ + }}4(i - 1) \hfill \\ C_{i} D_{i} = 3{\mathbf{ + }}4(i - 1) \hfill \\ D_{i} A_{i} = 4{\mathbf{ + }}4(i - 1) \hfill \\ \end{gathered} \right.$$
(2)

As shown in Fig. 4, the screw constraint topological diagram of the multi-mode mobile mechanism can be obtained, which shows that the mechanism has five loops (I, II, III, IV, V).

Fig. 4
figure 4

Screw constraint topological diagram.

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Let ω represents the angular velocity of the corresponding revolute joint, and establish screw constraint equations for the five closed loops I to V shown in Fig. 4. Then, the screw constraint equation set corresponding to the multi-mode mobile mechanism is:

$$\left\{ \begin{gathered} \omega_{11} {\mathbf{S}}_{A1} + \omega_{12} {\mathbf{S}}_{B1} - \omega_{13} {\mathbf{S}}_{C1} - \omega_{14} {\mathbf{S}}_{D1} = {\mathbf{0}} \hfill \\ \omega_{21} {\mathbf{S}}_{A2} + \omega_{22} {\mathbf{S}}_{B2} - \omega_{23} {\mathbf{S}}_{C2} - \omega_{24} {\mathbf{S}}_{D2} = {\mathbf{0}} \hfill \\ \omega_{31} {\mathbf{S}}_{A3} + \omega_{32} {\mathbf{S}}_{B3} - \omega_{33} {\mathbf{S}}_{C3} - \omega_{34} {\mathbf{S}}_{D3} = {\mathbf{0}} \hfill \\ \omega_{41} {\mathbf{S}}_{A4} + \omega_{42} {\mathbf{S}}_{B4} - \omega_{43} {\mathbf{S}}_{C4} - \omega_{44} {\mathbf{S}}_{D4} = {\mathbf{0}} \hfill \\ \omega_{11} {\mathbf{S}}_{A1} + \omega_{51} {\mathbf{S}}_{J1} + \omega_{24} {\mathbf{S}}_{D2} + \omega_{23} {\mathbf{S}}_{C2} + \omega_{52} {\mathbf{S}}_{J2} + \omega_{34} {\mathbf{S}}_{D3} \hfill \\ + \omega_{33} {\mathbf{S}}_{C3} - \omega_{53} {\mathbf{S}}_{J3} - \omega_{42} {\mathbf{S}}_{B4} - \omega_{41} {\mathbf{S}}_{A4} - \omega_{54} {\mathbf{S}}_{J4} - \omega_{12} {\mathbf{S}}_{B1} = {\mathbf{0}} \hfill \\ \end{gathered} \right.$$
(3)

ωji represents the angular velocity of the revolute joint i in the loop j of the mechanism, and 0 is a 6-dimensional zero vector.

Write Eq. (3) in matrix form as:

$${\varvec{MN}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {\mathbf{0}} \\ {\mathbf{0}} \\ {\mathbf{0}} \\ {\mathbf{0}} \\ \end{array} } \right]$$
(4)

where

$${\text{{\rm N} = [}}\omega_{{{11}}} \, \omega_{{{12}}} \, \omega_{{{13}}} \, \omega_{{{14}}} \, \omega_{{{21}}} \, \omega_{{{22}}} \, \omega_{{{23}}} \, \omega_{{{24}}} \, \omega_{{{31}}} \, \omega_{{{32}}} \, \omega_{{{33}}} \, \omega_{{{34}}} \,\omega_{{{41}}} \, \omega_{{{42}}} \, \omega_{{{43}}} \, \omega_{{{44}}} \, \omega_{{{51}}} \, \omega_{{{52}}} \, \omega_{{{53}}} \, \omega_{{{54}}} { ]}^{{\text{T}}}$$
(5)
$${\text{M = [M}}_{{1}} {\text{ M}}_{{2}} {\text{ M}}_{{3}} {\text{ M}}_{{4}} {\text{ M}}_{{5}} {]}$$
(6)
$$\begin{gathered} {\text{M}}_{{1}} { = }\left[ {\begin{array}{*{20}c} {{\text{S}}_{{{\text{A1}}}} } & {{\text{S}}_{{{\text{B1}}}} } & { - {\text{S}}_{{{\text{C1}}}} } & { - {\text{S}}_{{{\text{D1}}}} } \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {{\text{S}}_{{{\text{A1}}}} } & { - {\text{S}}_{{{\text{B1}}}} } & {0} & {0} \\ \end{array} } \right]{\text{, M}}_{{2}} { = }\left[ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} \\ {{\text{S}}_{{{\text{A2}}}} } & {{\text{S}}_{{{\text{B2}}}} } & { - {\text{S}}_{{{\text{C2}}}} } & { - {\text{S}}_{{{\text{D2}}}} } \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {{\text{S}}_{{{\text{C2}}}} } & {{\text{S}}_{{{\text{D2}}}} } \\ \end{array} } \right]{\text{,M}}_{{3}} { = }\left[ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {{\text{S}}_{{{\text{A3}}}} } & {{\text{S}}_{{{\text{B3}}}} } & { - {\text{S}}_{{{\text{C3}}}} } & { - {\text{S}}_{{{\text{D3}}}} } \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {{\text{S}}_{{{\text{C3}}}} } & {{\text{S}}_{{{\text{D3}}}} } \\ \end{array} } \right]{, } \\ {\text{M}}_{{4}} { = }\left[ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {{\text{S}}_{{{\text{A4}}}} } & {{\text{S}}_{{{\text{B4}}}} } & { - {\text{S}}_{{{\text{C4}}}} } & { - {\text{S}}_{{{\text{D4}}}} } \\ { - {\text{S}}_{{{\text{A4}}}} } & { - {\text{S}}_{{{\text{B4}}}} } & {0} & {0} \\ \end{array} } \right]{\text{, M}}_{{5}} { = }\left[ {\begin{array}{*{20}c} {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {{\text{S}}_{{{\text{J1}}}} } & {{\text{S}}_{{{\text{J2}}}} } & { - {\text{S}}_{{{\text{J3}}}} } & { - {\text{S}}_{{{\text{J4}}}} } \\ \end{array} } \right] \\ \end{gathered}$$
(7)

The screw constraint matrix M is a 30 × 20 matrix, and the DOF of the mechanism correspond to the dimension of the constraint matrix null space. The calculation result is:

$${\text{rank }}{\mathbf{M}} = 17$$
(8)

The dimension of the matrix null space is the number of columns minus the rank. Therefore, it can be obtained that the DOF of this mechanism is 3, that is, this mechanism requires 3 drives. According to the characteristics and movements of the parallelogram mechanism and the anti-parallelogram mechanism, it can be known that the three drives respectively control the angle θ1 between A1B1 and A1D1 in the parallelogram unit, the angle θ2 between A2D2 and A2B2 in the anti-parallelogram unit, and the angle θ3 between the parallelogram unit A1B1C1D1 and the anti-parallelogram unit A2B2C2D2. By changing the three angles, the mode switching and movement of the multi-mode mechanism can be completed, that is, the mechanism can complete rolling, creeping, and tumbling modes. As shown in Fig. 5, the three revolute joints A1, A2 and J1 are set as the driving positions of drive-1, drive-2, and drive-3, respectively. The three modes can be achieved by controlling the three drives: when drive-1 and drive-2 are fixed, it is parallel-like rolling mode; when drive-1 and drive-3 are fixed, it is anti-parallel-like tumbling mode; when drive-2 and drive-3 are fixed, it is parallel-like creeping mode.

Fig. 5
figure 5

Layout of driving methods.

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Analysis of motion feasibility and characteristics

Based on the analysis of DOF and the characteristics of the multi-mode mobile mechanism, the motion feasibility and characteristics of the mechanism with parallel-like rolling mode, anti-parallel-like tumbling mode, and parallel-like creeping mode are mainly analyzed.

Parallel-like rolling mode

When the angle between the parallelogram unit A1B1C1D1 and the anti-parallelogram unit A2B2C2D2 is θ3, the multi-mode mechanism undergoes deformation with the change of θ3. When the center of mass exceeds the parallelogram where the support area A1B1C1D1 is located, the mechanism will roll. As shown in Fig. 6, a coordinate system for the general state of the parallel-like rolling mode is established, let the origin be the revolute joint point A1, the line connecting the revolute joint point A1 and the revolute joint point B1 be the x-axis, and the line perpendicular to the x-axis in the plane where the parallelogram unit A1B1C1D1 is located be the z-axis (the z-axis is the line where A1D1 is located), and the direction of the y-axis can be obtained by using the right-hand rule. By evenly distributing the mass of the links, the center of mass is the midpoint of the link. By solving the coordinates of each vertex, the center of mass coordinates in the general state can be obtained.

Fig. 6
figure 6

Establishment of coordinate system for parallel-like rolling mode.

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Since the short links length of the anti-parallelogram and the links length of the parallelogram are both l, the long links length of the anti-parallelogram is \(\sqrt 2\) l, and the angle between the parallelogram unit A1B1C1D1 and the anti-parallelogram unit A2B2C2D2 is θ3, then it becomes a generalized parallelogram (θ1 = 90°, θ2 = 45°), which can be simplified to parallelogram J1J2J3J4. The center of mass coordinates can be obtained as:

$${\mathbf{r}}_{N3} = \left\{ {\begin{array}{*{20}c} {\frac{{l + l\cos \theta_{3} }}{2}} & {\frac{{l\sin \theta_{3} }}{2}} & \frac{l}{2} \\ \end{array} } \right\}$$
(9)

When l = 100 mm, the variation range of θ3 is (0, π). Obviously, the support area of this multi-mode mechanism is a square with a side length of 100 mm, and the relationship between the support area and the center of mass is independent of both the y-axis and z-axis coordinates, where the z-axis coordinates remain unchanged, while the y-axis coordinates represent the vertical change of center of mass and are independent of the support area. Then the centroid change curve (x-axis coordinates) and the centroid change curve in the vertical direction (y-axis coordinates) can be obtained as shown in Fig. 7.

Fig. 7
figure 7

Centroid change curve of parallel-like rolling mode.

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As shown in Fig. 7, the projection of the center of mass of the mechanism on the ground is always within its support range, so movement can only be achieved through dynamic performance. The ZMP (Zero Moment Point) criterion is used to determine whether the mechanism can move the ZMP coordinates outside the stable support area under the action of deformation and drive to ensure rolling. Just consider using the x-axis of ZMP coordinates, and the calculation formula is:

$$x_{{{\text{ZMP}}}} = \frac{{\sum\limits_{i = 0}^{n} {\left[ {m_{i} x_{i} (\ddot{y}_{i} + g) - m_{i} y_{i} \ddot{x}_{i} } \right]} }}{{\sum\limits_{i = 0}^{n} {m_{i} (\ddot{y}_{i} + g)} }}$$
(10)

where mi is the mass of link i,\((x_{i} ,y_{i} ,z_{i} )\) and \((\ddot{x}_{i} ,\ddot{y}_{i} ,\ddot{z}_{i} )\) are the centroid coordinates of link i and the centroid acceleration of link i respectively, and g is the acceleration of gravity.

let the rotational angular velocity be ω, calculate the coordinates of each point and its centroid coordinates of the parallel-like rolling mode, and substitute them into the Eq. (10) to obtain:

$$x_{{{\text{ZMP}}}} = \frac{{l(2{\text{g}} + 2{\text{g}}\cos \theta_{3} - l\omega^{2} \sin^{2} \theta_{3} )}}{{4{\text{g}} - 2l\omega^{2} \sin \theta_{3} }}$$
(11)

When the projection of the mechanism on the plane xoz is a parallelogram, the support area of the mechanism is [0, l] in the x-axis direction. When the mechanism exceeds the support area, the mechanism will overturn. Therefore, the rolling condition of the mechanism is:

$$x_{ZMP} < 0,{\text{ or }}x_{ZMP} > l$$
(12)

As shown in Fig. 8, the xZMP change curve of parallel-like rolling mode is presented. The initial position is θ3 = 90°, at which point the parallelogram unit A1B1C1D1, also known as J1J4, is in contact with the ground. xZMP will change with the change of θ3. When the angular velocity ω = 0°/s, the centroid change curve is the same as Fig. 7, and the curve is always within the range of the support area (0 ≤ xZMP ≤ l), so rolling cannot occur at this time. When the angular velocity ω = 10°/s, the critical point is reached at xZMP = 17.7° or xZMP = 162.3°, that is, when xZMP < 17.7° or xZMP > 162.3°, the curve exceeds the support area, thus rolling occurs at this time. When the angular velocity ω = 20°/s, the critical point is reached at xZMP = 63.7° or xZMP = 116.3°, that is, when xZMP < 63.7° or xZMP > 116.3°, the curve exceeds the support area, and rolling occurs at this time. According to the symmetry of the parallelogram mechanism units, the multi-mode mechanism can complete continuous rolling motion.

Fig. 8
figure 8

xZMP change curve of parallel-like rolling mode.

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A 3D model is established to verify the parallel-like rolling gait of the multi-mode mechanism. Figure 9 shows the rolling gait from the initial position, during the completion of the periodic continuous rolling process, the contact plane change between the mechanism and the ground is I-IV-III-II-1. The initial position is θ3 = 90° (plane I is in contact with the ground), θ3 gradually decreases from 90°, the mechanism deforms to the direction of plane IV, and when xZMP exceeds the support area, the mechanism rolls to plane IV (plane IV is in contact with the ground), and the plane in contact with the ground changes from I to IV. As θ3 gradually becomes larger, when θ3 = 90°, it is similar to the initial position (plane IV is still in contact with the ground), with the increase of θ3, the mechanism deforms to the direction of plane III, and when xZMP exceeds the support area, the mechanism rolls to plane III (plane III is in contact with the ground), and the plane in contact with the ground changes from IV to III. As θ3 gradually decreases, when θ3 = 90°, it is similar to the initial position (plane III is still in contact with the ground), with θ3 decreasing, and when xZMP exceeds the support area, the mechanism rolls to plane II (plane II is in contact with the ground), and the plane in contact with the ground changes from III to II. As θ3 gradually increases, when θ3 = 90°, it is similar to the initial position (plane II is still in contact with the ground), as θ3 increases, the mechanism deforms to the direction of plane I, and when xZMP exceeds the support area, the mechanism rolls to plane I (plane I is in contact with the ground), and the plane in contact with the ground changes from II to I. Thus, the periodic continuous rolling process is completed. Conversely, the rolling gait from the initial position to another direction (the contact plane change between the mechanism and the ground is I-II-III-IV-1) can be obtained.

Fig. 9
figure 9

Rolling gait of parallel-like rolling mode (I-IV-III-II-1).

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Anti-parallel-like tumbling mode

When the angle between links A2D2 and A2B2 in the anti-parallelogram unit A2B2C2D2 is θ2, the multi-mode mechanism undergoes deformation with the change of θ2. When the center of mass exceeds the parallelogram where the support area A1B1C1D1 is located, the mechanism will tumble. As shown in Fig. 10, a coordinate system for the general state of the anti-parallel-like tumbling mode is established, let the origin be the revolute joint point A1, the line connecting the revolute joint point A1 and the revolute joint point B1 be the x-axis, and the line perpendicular to the x-axis in the plane where the parallelogram unit A1B1C1D1 is located be the z-axis (the z-axis is the line where A1D1 is located), and the direction of the y-axis can be obtained by using the right-hand rule. By evenly distributing the mass of the links, the center of mass is the midpoint of the link. By solving the coordinates of each vertex, the center of mass coordinates in the general state can be obtained.

Fig. 10
figure 10

Establishment of coordinate system for anti-parallel-like tumbling mode.

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Since the short links length of the anti-parallelogram and the links length of the parallelogram are both l, the long links length of the anti-parallelogram is \(\sqrt 2\) l, and the angle between links A2D2 and A2B2 in the anti-parallelogram unit A2B2C2D2 is θ2, then it becomes a generalized anti-parallelogram (θ1 = 90°, θ3 = 90°), which can be simplified to anti-parallelogram Q1Q2Q3Q4. The center of mass coordinates can be obtained as:

$${\mathbf{r}}_{N2} = \left\{ {\begin{array}{*{20}c} \frac{l}{2} & { - \frac{{l(\sqrt 2 - \cos \theta_{2} )\sin \theta_{2} }}{{ - 3 + 2\sqrt 2 \cos \theta_{2} }}} & {\frac{{l( - \sqrt 2 \cos \theta_{2} + \cos 2\theta_{2} )}}{{ - 6 + 4\sqrt 2 \cos \theta_{2} }}} \\ \end{array} } \right\}$$
(13)

When l = 100 mm, the variation range of θ2 is (0, π). Obviously, the support area of this multi-mode mechanism is a square with a side length of 100 mm, and the relationship between the support area and the center of mass is independent of both the x-axis and y-axis coordinates, where the x-axis coordinates remain unchanged, while the y-axis coordinates represent the vertical change of center of mass and are independent of the support area. Then the centroid change curve (z-axis coordinates) and the centroid change curve in the vertical direction (y-axis coordinates) can be obtained as shown in Fig. 11.

Fig. 11
figure 11

Centroid change curve of anti-parallel-like tumbling mode.

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As shown in Fig. 11, the projection of the center of mass of the mechanism on the ground can exceed its support range, so the mechanism can achieve tumbling. The initial position is θ2 = 45°, at which point the parallelogram mechanism unit A1B1C1D1, also known as Q1Q4, is in contact with the ground. The center of mass will change with the change of θ2, when θ2 = 12.3° or θ2 = 115.9°, the critical point is reached. That is, when θ2 < 12.3° or θ2 > 115.9°, this curve exceeds the support area resulting in tumbling motion. Due to the symmetry of the anti-parallelogram mechanism units, this multi-mode mechanism can perform continuous tumbling motion.

A 3D model is established to verify the anti-parallel-like tumbling gait of the multi-mode mechanism. Figure 12 shows the tumbling gait from the initial position, during the completion of the periodic continuous tumbling process, the contact plane change between the mechanism and the ground is I-III-1. The initial position is θ2 = 45° (plane I is in contact with the ground), θ2 gradually decreases from 45°, the mechanism deforms to the direction of plane III, and when the center of mass exceeds the support area, the mechanism tumbles to plane III (plane III is in contact with the ground), and the plane in contact with the ground changes from I to III. As θ2 gradually becomes larger, when θ2 = 45°, it is similar to the initial position (plane III is still in contact with the ground), with the increase of θ2, the mechanism deforms to the direction of plane I, and when the center of mass exceeds the support area, the mechanism tumbles to plane I (plane I is in contact with the ground), and the plane in contact with the ground changes from III to I. Thus, the periodic continuous tumbling process is completed. Conversely, the tumbling gait from the initial position to another direction can be obtained.

Fig. 12
figure 12

Tumbling gait of anti-parallel-like tumbling mode (I-III-1).

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Parallel-like creeping mode

When the angle between links A1D1 and A1B1 in the parallelogram unit A1B1C1D1 is θ1, the multi-mode mechanism undergoes deformation with the change of θ1. As shown in Fig. 13, a coordinate system for the general state of the parallel-like creeping mode is established, let the origin be the revolute joint point A1, the line connecting the revolute joint point A1 and the revolute joint point D1 be the z-axis, and the line perpendicular to the z-axis in the plane where the parallelogram unit A1B1C1D1 is located be the x-axis, and the direction of the y-axis can be obtained by using the right-hand rule. By evenly distributing the mass of the links, the center of mass is the midpoint of the link. By solving the coordinates of each vertex, the center of mass coordinates in the general state can be obtained.

Fig. 13
figure 13

Establishment of coordinate system for parallel-like creeping mode.

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Since the short links length of the anti-parallelogram and the links length of the parallelogram are both l, the long links length of the anti-parallelogram is \(\sqrt 2\) l, and the angle between links A1D1 and A1B1 in the anti-parallelogram unit A1B1C1D1 is θ1, then it becomes a generalized parallelogram (θ2 = 45°, θ3 = 90°), which can be simplified to parallelogram A1B1C1D1. The center of mass coordinates can be obtained as:

$${\mathbf{r}}_{N1} = \left\{ {\begin{array}{*{20}c} {\frac{{l\sin \theta_{1} }}{2}} & \frac{l}{2} & {\frac{{l + l\cos \theta_{1} }}{2}} \\ \end{array} } \right\}$$
(14)

According to Eq. (14), the center of mass coordinate is completely consistent with the x coordinate and z coordinate of the center of mass coordinate of the parallelogram unit A1B1C1D1, and the support area always changes with the change of θ1. However, the center of mass coordinate is always at the center of the parallelogram unit A1B1C1D1, that is, it will not exceed its support area, that is, it will not roll. The creeping of the multi-mode mechanism occurs based on the friction between the links and the ground.

A 3D model is established to verify the parallel-like creeping gait of the multi-mode mechanism. Figure 14 shows the creeping gait from the initial position, the mechanism undergoes deformation with the change of θ1. During the movement, the link A1D1 is regarded as the reference frame. The initial position is θ1 = 90°. As θ1 gradually increases from 90°, the link B1C1 creeps along the direction of vector D1A1, and the mechanism deforms. When θ1 reaches to a certain angle (less than 180°), the mechanism reaches the extreme position. Then θ1 gradually decreases, and when θ1 = 90°, the mechanism returns to the initial position. Subsequently, as θ1 decreases further, link B1C1 creeps along the direction of vector A1D1, and the mechanism deforms again. When θ1 is reduced to a certain angle (greater than 0°), the mechanism reaches another extreme position. Then θ1 gradually increases, when θ1 = 90°, the mechanism returns to the initial position again. Thus, the periodic creeping process is completed. Obviously, creeping relies on the friction between the links and the ground, and its movement efficiency is relatively low.

Fig. 14
figure 14

Creeping gait of parallel-like creeping mode.

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Verification of the principle prototype model

According to the motion feasibility and characteristics in the previous section, a simulation model is established for the simulation verification of this multi-mode mobile mechanism. Let the short links length of the anti-parallelogram and the links length of the parallelogram be both 100 mm, the long links length of the anti-parallelogram be 141.42 mm. As shown in Fig. 15, when θ1 = 90° and θ2 = 45°, the multi-mode mechanism transforms into parallel-like mechanism, which can complete rolling by adjusting the angle θ3. The initial position is θ3 = 90° (plane I is in contact with the ground), θ3 gradually decreases from 90°, when xZMP exceeds the support area and the mechanism rolls to plane IV, and the ground changes from I to IV. Therefore, the mechanism can complete the continuous parallel-like rolling. As shown in Fig. 16, when θ1 = 90° and θ3 = 90°, the multi-mode mechanism transforms into anti-parallel-like mechanism, which can complete tumbling by adjusting the angle θ2. The initial position is θ2 = 45° (plane I is in contact with the ground), θ2 gradually decreases from 45°, and when the center of mass exceeds the support area, the mechanism tumbles to plane III, and the plane in contact with the ground changes from I to III. Therefore, the mechanism can complete the continuous anti-parallel-like tumbling. As shown in Fig. 17, When θ2 = 45° and θ3 = 90°, the multi-mode mechanism transforms into parallel-like mechanism, which can complete creeping by adjusting the angle θ1. The initial position is θ1 = 90°, and θ1 gradually decreases or decreases from 90°, the mechanism deforms, and creeping can occur through the friction between the links and the ground. Therefore, the mechanism can complete the parallel-like creeping. According to the simulation results, it is known that the mechanism can complete parallel-like rolling, anti-parallel-like tumbling and parallel-like creeping.

Fig. 15
figure 15

Parallel-like rolling simulation of the multi-mode mechanisms.

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Fig. 16
figure 16

Anti-parallel-like tumbling simulation of the multi-mode mechanisms.

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Fig. 17
figure 17

Parallel-like creeping simulation of the multi-mode mechanisms.

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The terrain adaptability simulation verification is carried out for the parallel-like rolling mode and anti-parallel-like tumbling mode. Based on the respective motion characteristics, different terrains are designed. As shown in Fig. 18, to imitate the shape of the gullies, a sunken ground structure is designed. Through simulation, it can be seen that the mechanism can complete rolling on this terrain. As shown in Fig. 19, to mimic the shape of stones, a terrain with raised obstacles is designed. Through simulation, it can be seen that the mechanism can complete tumbling on this terrain. Obviously, the adaptability varies depending on the modes. For example, for gullies of a certain size, the anti-parallelogram mode cannot overcome the gullies, while the parallelogram mode can.

Fig. 18
figure 18

Parallel-like rolling simulation with sunken ground structure.

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Fig. 19
figure 19

Anti-parallel-like tumbling simulation with raised obstacles.

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Additionally, for more complex terrain where obstacles cannot be overcome, the modes can be combined for turning and advancing. As shown in Fig. 20, when encountering obstacles that cannot be overcome, and the start and goal points are set as A and B respectively, the mechanism can pass through points C, D, E, F and finally reach B by switching modes. The simulation results show that this type of multi-mode mobile mechanism exhibits a certain terrain adaptability.

Fig. 20
figure 20

Overcoming obstacles through mode switching.

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A 3D printing principle prototype model of the multi-mode mechanism is fabricated as shown in Fig. 21, let the short links length of the anti-parallelogram and the links length of the parallelogram be both 153.44 mm, the long links length of the anti-parallelogram be 217 mm. Three drives, installed at the revolute joints A1, A2, and J1, control the following angles respectively: the angle θ1 between links A1D1 and A1B1 in the parallelogram unit A1B1C1D1, the angle θ2 between links A2D2 and A2B2 in the anti-parallelogram unit A2B2C2D2, and the angle θ3 between the parallelogram unit A1B1C1D1 and the anti-parallelogram unit A2B2C2D2. Verification is achieved by controlling three angles to complete three modes: that is, when the angles θ1 and θ2 are fixed, it is parallel-like rolling mode; when the angles θ1 and θ3 are fixed, it is anti-parallel-like tumbling mode; when the angles θ2 and θ3 are fixed, it is parallel-like creeping mode.

Fig. 21
figure 21

Prototype model of a multi-mode mechanism.

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When θ1 = 90° and θ2 = 45°, the multi-mode mechanism transforms into parallel-like mechanism, which can complete rolling by adjusting the angle θ3. Figure 22 shows the parallel-like rolling gait of the multi-mode mechanism rolling from the initial position, indicating that the contact plane change between the mechanism and the ground is I-IV-III-II-1, and then completes the periodic continuous rolling process. Through the prototype model, it can be obtained that the multi-mode mechanism can complete the continuous parallel-like rolling.

Fig. 22
figure 22

Gait verification of parallel-like rolling mode.

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When θ1 = 90° and θ3 = 90°, the multi-mode mechanism transforms into anti-parallel-like mechanism, which can complete tumbling by adjusting the angle θ2. Figure 23 shows the anti-parallel-like tumbling gait of the multi-mode mechanism tumbling from the initial position, indicating that the contact plane change between the mechanism and the ground is I-III-1, and then completes the periodic continuous tumbling process. Through the prototype model, it can be obtained that the multi-mode mechanism can complete the continuous anti-parallel-like tumbling.

Fig. 23
figure 23

Gait verification of anti-parallel-like tumbling mode.

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When θ2 = 45° and θ3 = 90°, the multi-mode mechanism transforms into parallel-like mechanism, which can complete creeping by adjusting the angle θ1. Figure 24 shows the parallel-like creeping gait of the multi-mode mechanism creeping from the initial position, indicating that the mechanism can complete creeping process. Through the prototype model, it can be obtained that the multi-mode mechanism can complete the parallel-like creeping. However, due to its creeping relying on the friction between the links and the ground, the movement efficiency is relatively low.

Fig. 24
figure 24

Gait verification of parallel-like creeping mode.

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Through verification of the prototype model, it is known that the multi-mode mechanism prototype model can complete parallel-like rolling mode, anti-parallel-like tumbling mode and parallel-like creeping mode. Since the creeping mode of the parallel-like creeping mode mainly relies on the friction between the links and the ground, its creeping efficiency is relatively low. In future research, materials can be further selected to address this problem and achieve efficient creeping.

Conclusion and discussion

In this paper, a type of multi-mode mobile mechanism with multi-link is designed based on two parallelogram mechanism units and two anti-parallelogram mechanism units connected symmetrically through four revolute joints. The multi-mode mobile mechanism has three motion modes: parallel-like rolling, anti-parallel-like tumbling, and parallel-like creeping. Firstly, the configuration design method of the multi-mode mechanism is given. Secondly, the coordinate system of the initial position is established, and the motion screws of each revolute joint are obtained. Based on graph theory, the screw constraint topological diagram of the multi-loop mechanism is obtained, then the DOF (the number of drives) of the initial position is obtained, and the drives are laid out. Then, the motion feasibility and characteristics of the three motion modes are analyzed, and the motion gaits of the three modes are obtained. Finally, the motion feasibility and motion gaits of the multi-mode mechanism are verified through simulation model and principle prototype model.

The multi-mode mobile mechanism designed in this paper transforms into a spatial multi-mode mechanism through the loop-connection of planar mechanisms. It can integrate the motion characteristics of both the anti-parallelogram and the parallelogram, thereby enabling the transition from planar motion to spatial multi-mode motion. This mechanism possesses multiple motion modes, providing new ideas and methods for the design of mobile robots. The simulation results show that this type of multi-mode mobile mechanism exhibits a certain terrain adaptability. In the future, this multi-mode mobile mechanism can be further developed into a mobile robot for application in unstructured environments. For example, when performing exploration, rescue, or patrol tasks in complex terrains, the robot can switch between different motion modes to adapt to varying terrain and environmental requirements. Moreover, the design of the control system for this mobile robot is also an important research direction. How to achieve efficient, stable, and intelligent control strategies will directly affect the robot’s motion performance and its ability to complete tasks.