Design and analysis of multi-mode Rectangular-8R-like mechanism

Abstract

To address the configuration design problem of multi-mode mechanisms, this paper proposes a method for designing a multi-motion mode mechanism based on planar mechanisms and analyzes its multiple motion modes, aiming to provide effective design and analysis methods for applications in mobile robots and deployable mechanisms. The designed multi-mode Rectangular-8R-like mechanism is mainly composed of two different groups of anti-parallelogram units connected by revolute joints. Firstly, the design method and characteristics of the multi-mode Rectangular-8R-like mechanism are introduced. Secondly, according to the characteristics of the anti-parallelogram unit and the rectangle, the degrees of freedom (DOF) of the mechanism in three modes (symmetrical tumbling mode, quasi-spherical rolling mode, and bifurcation mode) are analyzed using Screw theory. Then, the motion modes and characteristics of the multi-mode Rectangular-8R-like mechanism are presented according to the characteristics of the multi-mode mechanism. Finally, a 3D-printed prototype model is fabricated to verify the correctness of the analysis results and the feasibility of the proposed motion modes. The experimental results demonstrate that the designed multi-mode mechanism can achieve multiple motion modes and can switch at the bifurcation position. This design method provides a new idea for the design of multi-mode mechanisms and has broad application prospects in mobile robots, large deployable mechanisms, and other aspects.

Introduction

The multi-mode robot has multiple motion modes and can switch between different motion modes under different working conditions to adapt to different terrains. At present, research on multi-mode and reconfigurable mechanisms mainly focuses on linkage mechanisms, such as deployable mechanisms, origami mechanisms, metamorphic mechanisms, parallel mechanisms. In the aspect of deployable mechanisms, Liu et al.^1,2 designed a reconfigurable deployable spatial Bricard-like mechanism and an 8R-like mechanism based on angulated elements, which can complete multiple motion modes after reconfiguration. Gu et al. ³ designed a class of deployable polyhedral mechanisms based on the Sarrus mechanism and analyzed their mobility and kinematics. Cheng et al. ⁴ proposed a class of deployable mechanisms with two-dimensional deploying-folding motion and analyzed their kinematics and dynamics. In the aspect of parallel mechanisms, Liu et al. ⁵ utilized the refined virtual chain approach to synthesize a series of multi-mode mobile parallel mechanisms with multiple forms of motion. Ping et al. ⁶ designed a multi-mode mobile handling mechanism that can carry data collectors, cameras, and other devices to better adapt mobile robots to patrol and monitor industrial substation areas. Tian et al. ⁷ proposed the design method for a class of multi-loop mechanisms, where substituting sub-mechanisms for the Bennett mechanism resulted in the design method for a multi-configurable morphing wing mechanism, which can meet different flight requirements. Gao et al. ⁸ proposed configuration synthesis method for multi-mode mechanisms and provided the generation principle of parallel mechanisms. Kong et al. ⁹ presented configuration synthesis method for one degree of freedom (DOF) multi-mode parallel mechanisms based on mechanism symmetry. In the aspect of multi-mode mobile robots, a series of mobile mechanisms with multiple motion modes have been designed through various polyhedra and single-loop mechanisms, and the designed mechanisms can switch between different motion modes under different conditions.^10–13 A series of origami mechanisms have been designed based on origami theory, and the designed mechanisms can achieve deformation through reconfiguration and metamorphosis.^14–17

Single-loop mechanisms and metamorphic mechanisms have significant advantages in mode switching, which has led to extensive research by scholars. Pfurner et al. ¹⁸ analyzed a one-DOF multi-mode seven-bar mechanism based on two overconstrained four-bar spatial mechanisms connected in series, using only revolute and prismatic joints, and employed algebraic methods to analyze its multiple modes. Müller et al. ¹⁹ analyzed the high-order mobility of metamorphic 8R mechanisms and obtained two types of 6R mechanisms by locking a revolute joint. Kong ²⁰ designed a class of variable-DOF single-loop 7R mechanisms with five motion modes and conducted a detailed analysis. Chen et al. ²¹ presented two design methods for a single-DOF single-loop 7R deployable polygon mechanism. Chai et al. ²² proposed six novel 6R metamorphic mechanisms induced from three series connected Bennett linkages based on the method of constructing Goldberg 6R mechanisms. Kong ²³ investigated the construction method and reconfiguration analysis problem of a spatial mechanism composed of four circular translation joints. Kang et al., ²⁴ based on the parallelogram mechanism, designed and analyzed a class of multi-loop metamorphic mechanisms. Liu et al., ²⁵ based on the dual quaternion, proposed a configuration synthesis method for variable-DOF single-loop mechanisms. Subsequently, multiple types of 6R, 8R and other metamorphic and multi-mode mechanisms were designed, and their configuration synthesis methods, analysis methods and bifurcation characteristics of multi-mode mechanisms were analyzed in detail.^26–30

It is evident that the design of multi-mode mechanisms has long been a hot topic and a challenging issue in the field of mechanism science. In the research on multi-mode mechanisms, significant progress has been made in the study of multi-mode and reconfigurable mechanisms, particularly in the areas of deployable mechanisms, origami mechanisms, metamorphic mechanisms, parallel mechanisms. However, these mechanisms still have certain limitations. For example, deployable mechanisms exhibit excellent deployable performance but lack sufficient mobility; single-loop mechanisms possess multiple modes, but their structural rigidity needs to be further improved. Therefore, in this paper, a multi-mode mechanism is designed by integrating a planar mechanism with a spatial single-loop mechanism. In this design, the single bar of the single-loop mechanism is replaced by a planar mechanism, thereby combining the multiple modes of the single-loop mechanism with the rigidity advantages of the multi-loop mechanism. Moreover, the proposed multi-mode mechanism is capable of folding and unfolding, enabling it to undergo deformation and achieve multiple motion modes. This provides new ideas and methods for the study of multi-mode mechanisms.

This paper mainly focuses on the design of a multi-mode mechanism and the analysis of its motion modes and DOF. The specific chapters are arranged as follows: Section “The design of multi-mode mechanism” mainly presents the design methodology for the multi-mode mechanisms; Section “DOF analysis of multi-mode mechanism” analyzes the DOF of the multi-mode mechanism; Section “Motion modes analysis of multi-mode mechanism” discusses its motion modes and motion characteristics; Section “Prototype model verification” fabricates a principle prototype model to verify the correctness of the motion mode analysis; Section “Conclusions” summarizes the entire paper.

The design of multi-mode mechanism

The four-bar linkage is a kind of typical mechanism in the mechanical systems. A rectangle is a general form of four-bar linkage. As shown in Figure 1, a Rectangle-8R-like mechanism composed of two groups of anti-parallelogram units is constructed based on the shape of a rectangle. In this design, the corresponding sides of the rectangle are replaced by identical set of anti-parallelogram units, and the vertices of each rectangle are connected by revolute joints. Each anti-parallelogram unit can be regarded as two open-loop linkage mechanisms with variable link lengths, hinged together by revolute joints. Therefore, the mechanism composed of two groups of anti-parallelogram units can be called Rectangular-8R-like mechanism, in which the link lengths vary periodically during motion.

Figure 1.

Construction method of Rectangular-8R-like mechanism.

Combined with Figure 1, the specific design method can be obtained as follows: Firstly, determine the length of the long side (L₁) and the short side (L₂) of the rectangle, as well as the length (l₀) of the short side (the connecting link) of the anti-parallelogram unit. Then, calculate the lengths of the two sets of diagonal links (l₁ and l₂) of the anti-parallelogram using the Pythagorean theorem, resulting in two sets of anti-parallelograms. Finally, connect the midpoints of the short sides through four revolute joints to form a rectangular-like mechanism with a long side of L₁, a short side of L₂, and the length of the short side (the connecting link) of the anti-parallelograms being l₀.

Obviously, the axes of the revolute joints in the Rectangular-8R-like mechanism are not perfectly parallel, and the simplified virtual revolute joints of the anti-parallelogram unit are always perpendicular to the plane of the unit and perpendicular to the axes of the original revolute joints of the rectangle. On the other hand, during motion, the axes of the original revolute joints of the rectangle exhibit intersections, parallelism, and coincidence. As shown in Figure 2, during the whole motion, both the position and direction of the revolute joints will change, except at specific positions. Consequently, the mechanism is only an approximate rectangle, not a strict one. The link length of the simplified 8R mechanism is related to the link length of the anti-parallelogram unit. As shown in Figure 3, when the diagonal link AB = CD = l₁, and the connecting link AD = BC = l₀, the angle between BC and AD is θ, the virtual link length can be expressed as follows:

$$\begin{array}{c}{l}_{\mathrm{EG}}=l_{\mathrm{FG}}\hfill \\ =\frac{\sqrt{{l_0}^4+2{l_1}^4-4l_0{l_1}^3\cos \theta -{l_0}^4\cos 2\theta +2{l_0}^2{l_1}^2\cos 2\theta }}{2\sqrt{2}(l_1-l_0\cos \theta )}\hfill \end{array}$$

(1)

Figure 2.

Change of Rectangular-8R-like mechanism.

Figure 3.

The relationship between the virtual link length and the angle during motion.

Therefore, the constructed Rectangular-8R-like mechanism can be simplified as an 8R-like mechanism with real-time changes of virtual link length during motion. Moreover, the mechanism has two symmetric surfaces, enabling it to perform tumbling movement by using the symmetric characteristics to adapt to different terrain and working conditions.

DOF analysis of multi-mode mechanism

According to the characteristics of the traditional 8R mechanism and the Rectangular-8R-like mechanism constructed in this paper, the mechanism mainly consists of three distinct motion modes: the symmetrical tumbling mode, the quasi-spherical rolling mode, and the bifurcation mode of both. The DOF of the three modes are analyzed below.

DOF of symmetrical tumbling mode

As shown in Figure 4, the eight revolute joints are denoted as S _i (i = 1, …, 8). During the motion process, the intersection of axes S₁ and S₇ is denoted as J₁, the intersection of axes S₃ and S₅ is denoted as J₂, the intersection of axes S₄ and S₈ is denoted as J₃, the intersection of axes S₂ and S₆ is denoted as J₄. When J₁ and J₂ do not coincide, J₃ and J₄ also do not coincide, the planes I and II of this Rectangular-8R-like mechanism are symmetrical planes, respectively. This configuration can be simplified to an 8R mechanism with two symmetrical planes. A Cartesian coordinate system O-XYZ is established as follows: the origin O is located at the midpoint of the line between intersection points J₃ and J₄, the X-axis is parallel to the line between intersection points J₁ and J₂ through the origin O, the Y-axis is aligned along the line connecting J₃ and J₄, and the Z-axis is determined by the right-hand rule. According to the characteristics of Rectangular-8R-like mechanism and anti-parallelogram unit, it is evident that any two adjacent revolute joints among the eight are perpendicular to each other.

Figure 4.

Coordinate establishment of symmetrical tumbling mode.

Let the length of the line segment connecting the intersections points J₁ and J₂ be 2f, and the length of the line segment connecting the intersections points J₃ and J₄ be 2g. Let h denote the distance between the line on which the X-axis lies and the line segment connecting J₁ and J₂. Let 2ν₁ indicate the angle between the axes of the revolute joints S₄ and S₈, and let 2ν₂ indicate the angle between the axes of the revolute joints S₂ and S₆. According to the Screw theory, ³¹ the screw coordinates of the eight revolute joints can be obtained as follows:

$${\mathbf{S}}_{1i}=\{\begin{array}{c}{S}_1=(m_2{n}_1,-m_1{n}_1,-m_1{n}_2,h{m}_1{n}_2,n{m}_1{n}_2,h{m}_2{n}_1-f{m}_1{n}_1{)}^{\mathrm{T}}\hfill \\ S_2=(0,-n_2,n_1,g{n}_1,0,0{)}^{\mathrm{T}}\hfill \\ S_3=(-m_2{n}_1,-m_1{n}_1,-m_1{n}_2,h{m}_1{n}_2,-f{m}_1{n}_2,-h{m}_2{n}_1+f{m}_1{n}_1{)}^{\mathrm{T}}\hfill \\ S_4=(-m_1,m_2,0,0,0,-g{m}_1{)}^{\mathrm{T}}\hfill \end{array}$$

(2)

$${\mathbf{S}}_{2i}=\{\begin{array}{c}{S}_5=(-m_2{n}_1,-m_1{n}_1,m_1{n}_2,-h{m}_1{n}_2,f{m}_1{n}_2,-h{m}_2{n}_1+f{m}_1{n}_1{)}^{\mathrm{T}}\hfill \\ S_6=(0,-n_2,-n_1,-g{n}_1,0,0{)}^{\mathrm{T}}\hfill \\ S_7=(m_2{n}_1,-m_1{n}_1,m_1{n}_2,-h{m}_1{n}_2,-f{m}_1{n}_2,h{m}_2{n}_1-f{m}_1{n}_1{)}^{\mathrm{T}}\hfill \\ S_8=(m_1,m_2,0,0,0,g{m}_1{)}^{\mathrm{T}}\hfill \end{array}$$

(3)

where m₁ denotes sinν₁, m₂ denotes cosν₁, n₁ denotes sinν₂, n₂ denotes cosν₂.

The constraint screw of eight revolute joints can be obtained from equations (2) and (3).

$${\mathbf{S}}_{1i}^r=\{\begin{array}{c}{S}_{11}^r=(-1,-\frac{m_2{n}_2^2(g+h)(m_{1}f-m_{2}h)}{m_1^2\mathrm{gf}},-\frac{m_2{n}_1{n}_2(g+h)}{m_{1}g},0,\hfill \\ -n_1{n}_2(g+h),n_1^{2}g-n_2^{2}h{)}^{\mathrm{T}}\hfill \\ S_{12}^r={(0,-\frac{n_1(m_{2}g+m_{1}f-m_{2}h)}{m_1{n}_2\mathrm{gf}},-\frac{1}{g},1,0,0)}^{\mathrm{T}}\hfill \end{array}$$

(4)

$${\mathbf{S}}_{2i}^r=\{\begin{array}{c}{S}_{21}^r=(-1,\frac{m_2{n}_2^2(g+h)(m_{1}f-m_{2}h)}{m_1^2\mathrm{gf}},-\frac{m_2{n}_1{n}_2(g+h)}{m_{1}g},\hfill \\ 0,n_1{n}_2(g+h),n_1^{2}g-n_2^{2}h{)}^{\mathrm{T}}\hfill \\ S_{22}^r={(0,\frac{n_1(m_{2}g+m_{1}f-m_{2}h)}{m_1{n}_2\mathrm{gf}},-\frac{1}{g},1,0,0)}^{\mathrm{T}}\hfill \end{array}$$

(5)

Therefore, according to equations (4) and (5), it is obtained that dim( $S_i^r$ ) = 4 and card( $S_i^r$ ) = 4 (where card( $S_i^r$ ) represents the cardinal number of the constraint screw system and dim( $S_i^r$ ) denotes the dimension of the constraint screw system). Let m be the DOF of the mechanism, g be the number of joints, f _i be the DOF of the ith joint, and l be the number of independent loops. Based on screw theory, ³¹ the DOF of the symmetric tumbling mode is as m =  $\sum _{i=1}^g{f}_i-6l+\mathrm{card}〈S_i^r〉-\dim 〈S_i^r〉$  = 8 − 6 + 4 − 4 = 2. This indicates that, in this mode, only two drives are required to control the movement of the entire mechanism.

DOF of quasi-spherical rolling mode

As shown in Figure 5, during the motion process, the axes of revolute joints S₁, S₃, S₅, and S₇ intersect at point P₃, the axes of S₄ and S₆ intersect at point P₂, and the axes of S₂ and S₈ intersect at point P₁. When the mechanism has no symmetry plane and is symmetrical about its center, the Rectangular-8R-like mechanism transitions into a quasi-spherical mode. In this mode, it can be simplified as a quasi-spherical 8R mechanism. A Cartesian coordinate system O-XYZ is established, in which the origin O is located at the intersection point P₃, the X-axis passes through the origin O and is perpendicular to the line connecting points P₁ and P₂, the Y-axis is parallel to the line connecting points P₁ and P₂, and the Z-axis is determined by the right-hand rule. According to the characteristics of the quasi-spherical Rectangular-8R-like mechanism and the anti-parallelogram unit, the two adjacent revolute joints among the eight are perpendicular to each other.

Figure 5.

Coordinate establishment of quasi-spherical mode.

Let the length of the line segment connecting the intersection points P₁ and P₂ be 2p, and the distance between the Y-axis and the line segment connecting the intersection points P₁ and P₂ be q. Let 2δ₁ indicate the angle between the axes of the revolute joints S₁ and S₅, and 2δ₂ indicate the angle between the axes of the revolute joints S₃ and S₇. According to the Screw theory, the screw coordinates of the eight revolute joints can be obtained as follows:

$${\mathbf{S}}_{1j}=\{\begin{array}{c}{S}_1=(a_2,-a_1,0,0,0,0{)}^{\mathrm{T}}\hfill \\ S_2=(a_1{b}_1,a_2{b}_1,a_1{b}_2,p{a}_1{b}_2,q{a}_1{b}_2,-p{a}_1{b}_1-q{a}_2{b}_1{)}^{\mathrm{T}}\hfill \\ S_3=(b_2,0,-b_1,0,0,0{)}^{\mathrm{T}}\hfill \\ S_4=(a_1{b}_1,-a_2{b}_1,a_1{b}_2,-p{a}_1{b}_2,q{a}_1{b}_2,p{a}_1{b}_1+q{a}_2{b}_1{)}^{\mathrm{T}}\hfill \end{array}$$

(6)

$${\mathbf{S}}_{2j}=\{\begin{array}{c}{S}_5=(a_2,a_1,0,0,0,0{)}^{\mathrm{T}}\hfill \\ S_6=(a_1{b}_1,-a_2{b}_1,-a_1{b}_2,p{a}_1{b}_2,-q{a}_1{b}_2,p{a}_1{b}_1+q{a}_2{b}_1{)}^{\mathrm{T}}\hfill \\ S_7=(b_2,0,b_1,0,0,0{)}^{\mathrm{T}}\hfill \\ S_8=(a_1{b}_1,a_2{b}_1,-a_1{b}_2,-p{a}_1{b}_2,-q{a}_1{b}_2,-p{a}_1{b}_1-q{a}_2{b}_1{)}^{\mathrm{T}}\hfill \end{array}$$

(7)

Where a₁ denotes sinδ₁, a₂ denotes cosδ₁, b₁ denotes sinδ₂, b₂ denotes cosδ₂.

The constraint screw of eight revolute joints can be obtained from equations (6) and (7).

$${\mathbf{S}}_{1j}^r=\{\begin{array}{c}{S}_{11}^r={(-\frac{a_2^2{b}_1^2}{{\mathrm{pa}}_1^2{b}_2^2},-\frac{b_1^2+b_2^2}{{\mathrm{qb}}_2^2},0,\frac{b_1}{b_2},\frac{a_2{b}_1}{a_1{b}_2},1)}^{\mathrm{T}}\hfill \\ S_{12}^r={(\frac{b_1(p{a}_1+q{a}_2)}{p{a}_1{b}_2},0,1,0,0,0)}^{\mathrm{T}}\hfill \end{array}$$

(8)

$${\mathbf{S}}_{2j}^r=\{\begin{array}{c}{S}_{21}^r={(\frac{a_2^2{b}_1^2}{{\mathrm{pa}}_1^2{b}_2^2},-\frac{b_1^2+b_2^2}{{\mathrm{qb}}_2^2},0,-\frac{b_1}{b_2},\frac{a_2{b}_1}{a_1{b}_2},1)}^{\mathrm{T}}\hfill \\ S_{22}^r={(-\frac{b_1(p{a}_1+q{a}_2)}{p{a}_1{b}_2},0,1,0,0,0)}^{\mathrm{T}}\hfill \end{array}$$

(9)

Therefore, according to equations (8) and (9), it is obtained that dim( $S_j^r$ ) = 4 and card( $S_j^r$ ) = 4, the DOF of the quasi-spherical rolling mode is 8 − 6 + 4 − 4 = 2. This indicates that, in this mode, only two drivers are needed to control the movement of the whole mechanism.

DOF of bifurcation mode

As shown in Figure 6, during the motion process, the axes of revolute joints S₁, S₃, S₅, and S₇ intersect at point Q₁, and the axes of S₂, S₄, S₆, and S₈ intersect at point Q₂. When the mechanism has two symmetrical planes and is symmetrical about its center, the Rectangular-8R-like mechanism transitions into the bifurcation position between the symmetric tumbling mode and the quasi-spherical mode. In this mode, it can be simplified as a symmetric-quasi-spherical 8R-like mechanism. A Cartesian coordinate system O-XYZ is established, where the origin O is located at the midpoint of the line segment between the intersection points Q₁ and Q₂, the X-axis is the line that perpendicularly bisects the line segment connecting Q₁ and Q₂ and is located on the plane of the revolute joints S₄ and S₈, the Y-axis is the line that passes through the origin O and connects the intersection points Q₁ and Q₂, and the Z-axis is determined by the right-hand rule. According to the characteristics of the quasi-spherical Rectangular-8R-like mechanism and the anti-parallelogram unit, the two adjacent revolute joints among the eight are perpendicular to each other.

Figure 6.

Coordinate establishment of bifurcation mode.

Let the length of the line segment connecting the intersection points Q₁ and Q₂ be 2c, let 2η₁ represent the angle between the axes of the revolute joints S₁ and S₅, let 2η₂ represent the angle between the axes of the revolute joints S₃ and S₇. The screw coordinates of the eight revolute joints can be obtained according to the Screw theory. Combining the characteristics of Rectangular-8R-like mechanism and anti-parallelogram unit, along with the symmetric tumbling mode and the quasi-spherical mode, the screw coordinates of eight revolute joints can be obtained as follows:

$${\mathbf{S}}_{1k}=\{\begin{array}{c}{S}_1=(0,d_2,-d_1,c{d}_1,0,0{)}^{\mathrm{T}}\hfill \\ S_2=(d_1{e}_2,-d_1{e}_1,-e_1{d}_2,-c{e}_1{d}_2,0,-c{d}_1{e}_2{)}^{\mathrm{T}}\hfill \\ S_3=(e_1,e_2,0,0,0,c{e}_1{)}^{\mathrm{T}}\hfill \\ S_4=(d_1{e}_2,-d_1{e}_1,e_1{d}_2,c{e}_1{d}_2,0,-c{d}_1{e}_2{)}^{\mathrm{T}}\hfill \end{array}$$

(10)

$${\mathbf{S}}_{2k}=\{\begin{array}{c}{S}_5=(0,d_2,d_1,-c{d}_1,0,0{)}^{\mathrm{T}}\hfill \\ S_6=(-d_1{e}_2,-d_1{e}_1,e_1{d}_2,c{e}_1{d}_2,0,c{d}_1{e}_2{)}^{\mathrm{T}}\hfill \\ S_7=(-e_1,e_2,0,0,0,-c{e}_1{)}^{\mathrm{T}}\hfill \\ S_8=(-d_1{e}_2,-d_1{e}_1,-e_1{d}_2,-c{e}_1{d}_2,0,c{d}_1{e}_2{)}^{\mathrm{T}}\hfill \end{array}$$

(11)

Where d₁ denotes sinη₁, d₂ denotes cosη₁, e₁ denotes sinη₂, e₂ denotes cosη₂.

The constraint screw of eight revolute joints can be obtained from equations (10) and (11).

$${\mathbf{S}}_{1k}^r=\{\begin{array}{c}{S}_{11}^r={(\frac{c(-e_1^2+e_2^2)}{e_1^2+e_2^2},-\frac{2c{e}_1{e}_2}{e_1^2+e_2^2},-\frac{c{d}_2{e}_1{e}_2}{d_1(e_1^2+e_2^2)},\frac{d_2{e}_1{e}_2}{d_1(e_1^2+e_2^2)},0,1)}^{\mathrm{T}}\hfill \\ S_{12}^r=(0,0,0,0,1,0{)}^{\mathrm{T}}\hfill \end{array}$$

(12)

$${\mathbf{S}}_{2k}^r=\{\begin{array}{c}{S}_{21}^r={(\frac{c(-e_1^2+e_2^2)}{e_1^2+e_2^2},\frac{2c{e}_1{e}_2}{e_1^2+e_2^2},-\frac{c{d}_2{e}_1{e}_2}{d_1(e_1^2+e_2^2)},\frac{d_2{e}_1{e}_2}{d_1(e_1^2+e_2^2)},0,1)}^{\mathrm{T}}\hfill \\ S_{22}^r=(0,0,0,0,1,0{)}^{\mathrm{T}}\hfill \end{array}$$

(13)

Therefore, according to equations (12) and (13), it is obtained that dim( $S_k^r$ ) = 3 and card( $S_k^r$ ) = 4, the DOF of the bifurcation mode is 8 − 6 + 4 − 3 = 3, that is, the mode is the intermediate state between the symmetric tumbling mode and the quasi-spherical mode.

Motion modes analysis of multi-mode mechanism

According to the DOF analysis of the multi-mode mechanism presented in section “DOF analysis of multi-mode mechanism,” the three main motion modes of the constructed Rectangular-8R-like mechanism (symmetrical tumbling mode, quasi-spherical rolling mode, and bifurcation mode) and their characteristics can be obtained.

Motion mode of symmetrical tumbling mode

According to the symmetry characteristics of Rectangular-8R-like mechanism and the motion characteristics of the anti-parallelogram unit mechanism, as shown in Figure 7, the multi-mode mechanism has four anti-parallelogram units. In this mechanism, unit-1 and unit-3 are the identical (group 1), unit 2 and unit 4 are the identical (group 2). Due to its symmetry and the motion characteristics of the anti-parallelogram unit, the two groups of units behave identically in real time. In the symmetrical tumbling mode, the mechanism can exhibit two distinct motion modes: transverse motion and longitudinal motion. The transverse motion is achieved through the movement of group 1, while the longitudinal motion is achieved through the movement of group 2. As shown in Figure 8, a schematic diagram of lateral motion shows that when the unit of group 2 remains unchanged, the transverse movement can be realized with the left and right movement of group 1. Similarly, as shown in Figure 9, when the unit of group 1 remains unchanged, the longitudinal movement can be realized with the up and down movement of group 2. Furthermore, even the combination of transverse and longitudinal motion can generate more complex surging motion.

Figure 7.

The three-dimensional model of Rectangular-8R-like mechanism.

Figure 8.

Transverse motion 3D model of symmetrical tumbling mode.

Figure 9.

Longitudinal motion 3D model of symmetrical tumbling mode.

Motion mode of quasi-spherical rolling mode

According to the analysis of the DOF of the quasi-spherical rolling mode, the characteristics of the quasi-spherical Rectangular-8R-like mechanism, and the motion characteristics of the anti-parallelogram unit, the mechanism becomes a quasi-spherical four-bar mechanism in this quasi-spherical rolling mode. At each position, the anti-parallelogram can be regarded as a simple link, however, due to the characteristics of the anti-parallelogram unit mechanism, the link length changes in real time. That is, each link length corresponds to a specific spherical four-bar mechanism. As shown in Figures 10 and 11, two distinct motion modes can be observed under different link lengths in the quasi-spherical rolling mode.

Figure 10.

3D model for the position-I of quasi-spherical rolling mode.

Figure 11.

3D model for the position-II of quasi-spherical rolling mode.

Motion mode of bifurcation mode

The bifurcation mode of the Rectangular-8R-like mechanism is the intermediate mode between the symmetrical tumbling mode and the quasi-spherical mode. Therefore, in this symmetric-quasi-spherical 8R-like configuration, the mechanism can perform the longitudinal motion mode, the transverse motion mode, and the quasi-spherical rolling mode at this position, as shown in Figure 12. In this bifurcation mode, the longitudinal motion mode, the transverse motion mode, and the quasi-spherical rolling mode can be selected and switched by adjusting the angle of a single anti-parallelogram unit and the inter-unit angles.

Figure 12.

3D model of bifurcation mode.

Prototype model verification

According to the analysis of the first two sections, a principle prototype model is fabricated, the 3D-printed model is shown in Figure 13. Based on a rectangular with width L₂ = 66 mm and length L₁ = 88 mm, and two sets of anti-parallelogram units with connecting link l₀ = 66 mm, the Rectangular-8R-like mechanism is designed. Using the Pythagorean theorem, the lengths of the diagonal links are calculated as l₁ = 110 mm for unit-3 and l₂ = 93.34 mm for unit-4. It can be observed that the constructed Rectangular-8R-like mechanism is rectangular in the illustrated state. The following section verifies the motion modes of the mechanism, including symmetrical tumbling mode, quasi-spherical rolling mode, and bifurcation mode.

Figure 13.

3D-printed model.

As shown in Figures 14 and 15, the transverse and longitudinal motion modes of the Rectangular-8R-like mechanism in the symmetrical tumbling mode are presented. It can be observed that, when the units of group 2 (units 2 and 4 in Figure 13) remain unchanged, the transverse movement can be realized with the left and right movements of group 1 (units 1 and 3 in Figure 13) (Figure 14). Similarly, when the unit of group 1 is unchanged, the longitudinal motion can be realized with the up and down movement of group 2 (Figure 15).

Figure 14.

Verification of transverse motion mode of symmetrical tumbling mode.

Figure 15.

Verification of longitudinal motion mode of symmetrical tumbling mode.

As shown in Figure 16, the motion mode of Rectangular-8R-like mechanism in the quasi-spherical rolling mode shows that the mechanism becomes a quasi-spherical four-bar mechanism. In this mode, the anti-parallelogram can be regarded as a simple link, but because of the characteristics of the anti-parallelogram unit mechanism, the link length changes in real time, that is, each specific length corresponds to a distinct spherical four-bar mechanism.

Figure 16.

Verification of motion mode of the quasi-spherical rolling mode.

Obviously, in the bifurcation mode, the Rectangular-8R-like mechanism can exhibit the motion characteristics of both the symmetrical tumbling mode and the quasi-spherical mode, as shown in Figure 17. Moreover, the bifurcation mode can be converted into the longitudinal motion mode, the transverse motion mode, and the quasi-spherical rolling mode. By adjusting the angle of a single anti-parallelogram unit and the angles between units, selecting and witching among three modes can be achieved. It can be indicated that the mechanism can be used as a multi-mode mobile mechanism for further research in the future. In addition, due to its characteristics such as overturning and folding, it can also be used as a structural subunit of large deployable mechanisms.

Figure 17.

Verification of motion mode of bifurcation mode.

Conclusions

In this paper, a multi-mode Rectangular-8R-like mechanism is designed. The mechanism is composed of two different groups of anti-parallelogram units and possesses multiple motion modes, including the symmetrical tumbling mode, the quasi-spherical rolling mode, and the bifurcation mode. Among these, the symmetrical tumbling mode has two sub-modes: the longitudinal motion mode and the transverse motion mode. The bifurcation mode is an intermediate state between the symmetrical tumbling mode and the quasi-spherical rolling mode. The DOF, motion modes, and motion characteristics of the three modes are obtained. A 3D model is designed to verify its motion modes based on the analysis results. Finally, a prototype model is fabricated to verify the correctness of the analysis results and the feasibility of movement. It is concluded that this kind of multi-mode Rectangular-8R-like mechanism can subsequently utilize the quasi-spherical rolling mode for rapid rolling, the symmetrical tumbling mode to surmount the obstacle, and the bifurcation mode to switch. Therefore, the multi-mode mechanism designed in this paper has certain application prospects in mobile mechanisms and deployable mechanisms.

The multi-mode mechanism designed in this paper has been verified its multiple motion modes through prototypes and experiments. However, only the design method and feasibility study have been completed at the mechanism science level. If it is to be applied in the mobile mechanisms or the deployable mechanisms, further exploration is still needed. In the future, it could be further developed into a mobile robot, and the layout and design of the driving system should be studied first. In addition, the control strategy for mode switching, especially the collaborative control method, is also a key issue that urgently needs to be addressed. In terms of network design for deployable mechanisms, this multi-mode mechanism can be used as the basic sub-unit to construct a large deployable mechanism. However, its network methods and corresponding analysis approaches still need in-depth research.