Synthesis and mode transformation analysis of double loop 7R mechanism with motion and locking modes

Abstract

Analyzing the different characteristics of tangent lines of the coupler point curve of 4R mechanism, a 3R kinematic chain in a singular configuration is set up, planar and spherical 7R mechanisms with motion and locking modes are synthesized. Based on the spherical 4R mechanism with threefold and quadruple points on the coupler point curve, a spherical 7R mechanism with multiple motion and locking mode is ã synthesized. By analyzing the spherical 7R mechanism using screw theory, the motion mode transformation configuration of the mechanism when the coupler point curve of the spherical 4R loop is at a cusp are analyzed. The results show that this type of 7R mechanism can achieve the transformation of motion and locking modes. By changing the configuration of the singular 3R kinematic chain, the motion mode of the 4R mechanism loop under different motion bifurcation configurations can be transformed. The proposed method also has certain application value in designing, analyzing, and optimizing multi-mode mechanisms that meet the transformation requirements of motion and locking production operations.

Introduction

Multi-mode mechanisms ¹ can adapt to different work requirements through mode transformation without reassembly. During the transformation of motion modes, some multi-mode mechanisms have their links and kinematic pairs in a locking state. In some applications, by transforming the motion mode to a locking state, the rigidity of the mechanism can be improved. For example, in legged landers, the leg structure needs to have both folding and walking motion modes. During landing, the leg structure is in a locking state, thus providing the ability to bear high loads and resist strong impacts. ² When some robots with mobile bases are working under load, the mobile base has a locking mode that can maintain the stability of the robot’s body posture during operation. ³

When the motion mode of a multi-mode mechanism changes or the number of degrees of freedom is altered, the components and kinematic pairs of the mechanism may transition from a motion state to a locking state. Lin et al. ⁴ proposes a method based on the geometric characteristics of elliptical bifocal points, synthesis a spatial single-loop mechanism with the axis of the rotational pair passing through the geometric intersection of ellipses, to analyze the joint locking characteristics under different motion modes and apply it to the main frame of a wall-climbing robot. Li et al. ⁵ designs a material with negative Poisson’s ratio characteristics and high folding ratio characteristics. The material structure consists of a single-loop 6R mechanism module with multi-mode characteristics. When the multi-mode 6R mechanism undergoes mode transformation, some joints are in a locking state. Nie et al. ⁶ inserts two intersecting rotational pairs into a parallelogram mechanism to obtain a spatial 6R mechanism with six motion modes. Using screw theory analysis and Denavit–Hartenberg parameter kinematic modeling, it is analyzed to have one 2-joint locking mode and three 4-joint locking modes, which are applied to a variable structure wheeled frame. Zhang et al. ⁷ increases the working space of the mechanism through mode transformation and applies it to human rehabilitation equipment. Zhang et al. ⁸ uses the spatial 6R mechanism as a generalized variable motion joint, locks its motion pairs, and achieves the transformation of multiple motion modes, establishing a unified kinematic model for multiple motion modes. Kang et al. ⁹ adds four rotational pairs to the Bennett mechanism to obtain an 8R mechanism. Using screw theory analysis, different motion modes of the spatial 7R mechanism are obtained by locking different joints of the 8R mechanism. This method provides a new research perspective for the synthesis of new metamorphic mechanisms. Zhang et al. ¹⁰ uses an equivalent kinematic chain to simulate the crease motion of traditional folding mechanisms and designs a rigid-flexible coupled folding robot. By using flexible materials, it avoids the motion uncertainty issues of traditional rigid robots, and by using rigid kinematic chains, it solves the problems of low load capacity and low accuracy of traditional flexible folding robots. Yang et al. ¹¹ uses an 8R mechanism as the torso of a walking robot. Through the transformation of the motion mode of the 8R mechanism, some motion pairs are locked to enhance the passing performance of the walking robot in different environments. Zhao et al. ¹² uses a 3RRU-type parallel mechanism as a unit module of a grasping manipulator. When the manipulator transform between deployment and grasping modes, the parallel mechanism’s moving platform is in a locking mode. In the Rosyid et al., ¹³ a 3T parallel machining robot featuring both non-isotropy and isotropy modes was designed by utilizing a reconfigurable frame and lockable joints. This design expanded the robot’s machining workspace and the types of workpieces that can be processed. In the Tang and Dai, ¹⁴ a spatial 6R mechanism, consisting of two spherical 3R kinematic chains connected in series, was proposed. When its two joints are locked, the mechanism operates in either a spherical 4R or Bennett mechanism motion mode. Based on the analysis of this metamorphic mechanism’s motion modes, a novel method for designing multi-mode mechanisms was introduced.

The design and analysis of multi-mode mechanisms with locking modes remains a complex problem. The existing design methods include: (1) merging two overconstrained mechanisms to construct a single-loop multi-mode mechanism. (2) adding a kinematic pair to an overconstrained mechanism to design a multi-mode single-loop mechanism. (3) using high-order kinematic equations ¹⁵ to design a single-loop mechanism with concealed motion bifurcation characteristics. (4) by merging a multi-mode multi-loop mechanism with a serial kinematic chain, ¹⁶ a parallel mechanism with motion modes of different dimensions is designed. Based on the analysis of tangents of planar 4R and spherical 4R link trajectories, this paper proposes a method for synthesizing mechanisms with both motion and locking modes by setting up a folded 3R kinematic chain. When such mechanisms move to a specific working pose, they can further enhance their load capacity and stiffness by switching to a locking state through motion mode transformation, adapting to different operational requirements.

A planar 7R mechanism with motion and locking modes

As shown in Figure 1(a), the link 2 of the parallelogram mechanism can rotate around the axis R₁. When the mechanism moves to the configuration shown in Figure 1(b), it is in a constrained singular configuration, and the axes of the rotational pairs R₁, R₂, R₃, and R₄ are in a coplanar configuration. At this time, the axes of the rotational pairs R₂ and R₄ coincide, and the kinematic chain R₄R₃R₂ can rotate as a whole around the axis R₄ when the link 2 is in a locking mode, as shown in Figure 1(c). When the link 2 is in a rotational motion mode, the speed v₁ on the link 2 and v₂ on the coupler 3 of the coincident point A should have the same direction of movement, as shown in Figure 1(a) and (b). When the axes of joints R₂ and R₄ coincide and links 3 and 4 rotate about R₄ as a whole which the link 2 remains stationary, the velocity of point A is zero and does not have any direction. At this time, only the speed direction of point A as v₁ on the link 2 and v₂ on the coupler 3 are permitted. When the permitted directions of v₁ and v₂ are different, the movement speed of point A should be zero, and the link 2 is in a locking mode and cannot change its motion mode, as shown in Figure 1(c). As shown in Figure 1(b), when the coupler 3 rotates around the axis of the rotational pair R₄ and reaches the mechanism configuration shown in Figure 1(c), the directions of v₁ and v₂ are different, and the link 2 is in a locking mode. As shown in Figure 1(b), when the link 2 rotates around the axis of the rotational pair R₁ and reaches the mechanism configuration shown in Figure 1(a). The mechanism configuration shown in Figure 1(a) is not a constrained singular configuration. The directions of v₁ and v₂ are the same, and the link 2 is in a motion mode.

Figure 1.

Parallelogram mechanism with both locking and motion modes: (a) unlocking mode, (b) motion mode transform configurations, and (c) locking mode.

Assembly conditions with motion and locking modes

The curve of point B₁ on link 2 of kinematic chain R₁ shown in Figure 2(a) is a circle centered at R₁. In kinematic chain R₄R₃R₂, the lengths of link 4 and coupler 3 are the same, that is, l₄ = l₃, so the workspace of point B₂ on coupler 3 is a circle centered at the rotation center of R₄ with a radius of 2l₃. When there is an intersection between the workspace of point B₁ on kinematic chain R₁ and the workspace of point B₂ on kinematic chain R₄R₃R₂ as shown in Figure 2(b), the planar 4R mechanism can be assembled, as shown in Figure 2(b). At this time, the planar 4R mechanism is a non-instantaneous mechanism.

Figure 2.

Assembly condition of planar 4R mechanism with motion and locking modes: (a) planar 4R mechanism can not be assembled, (b) planar 4R mechanism can be assembled normally, And (c) planar 4R mechanism with locking mode can be assembled.

Specifically, when the curve of point B₁ on link 2 in Figure 2(b) passes through the axis of the revolute pair R₄, the assembly of point B₁ on link 2 and point B₂ on coupler 3 can coincide with the rotational center of the revolute pair R₄, as shown in Figure 2(c). In this configuration, point B₁ coincides with the rotational center of the revolute pair R₄, and the entire kinematic chain R₄R₃R₂ can rotate around the revolute pair R₄.

Based on the four-bar mechanism with locking and motion modes shown in Figure 1, it can be seen that the end point B₂ of the kinematics chain R₄R₃R₂ passes through the rotation center of the rotating pair R₄, and the end point B₁ of the kinematics chain R₁ can also pass through the rotation center of the rotating pair R₄, thus assembling a planar four-bar mechanism that can be in a folded state, that is, the position where the axis of the rotating pair of R₂ and R₄ coincide, thus achieving the transformation of locking and motion modes.

Difference between locking mode and dead center mode

The parallelogram mechanism shown in Figure 1(c) has its link 2 in a locking mode configuration, which is fundamentally different from the dead-center configuration of a typical planar four-bar mechanism. The simplified diagram of the mechanism in Figure 1(c) is illustrated in Figure 3(a).

Figure 3.

3R kinematic chain in constraint singular configuration and dead point configuration of planar 4R mechanism: (a) locking mode and (b) “dead point” configuration.

Figure 3(a) shows the coupler 3 in a locking mode. The screw system of the mechanism in Figure 3(a) is

$$\{\begin{array}{c}{\mathit{S}}_{\mathrm{R}4}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\mathit{S}}_{\mathrm{R}3}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & {\mathrm{a}}_3\hfill & {\mathrm{b}}_3\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{\mathit{S}}_{\mathrm{R}2}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{\mathit{S}}_{\mathrm{R}1}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & 0\hfill & {\mathrm{b}}_1\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\end{array}$$

(1)

In equation (1), a _i , b _i , and c _i represent the dual part of the screw of the i-th rotational pair, respectively. The velocity of the end of kinematic chain R₄R₃R₂ can be expressed as J ω, as shown in equation (2)

$$\mathit{J}\omega =\left[\begin{array}{ccc}{\mathit{S}}_{\mathrm{R}4}\hfill & {\mathit{S}}_{\mathrm{R}3}\hfill & {\mathit{S}}_{\mathrm{R}2}\hfill \end{array}\right]\left[\begin{array}{c}{\omega }_{\mathrm{R}4}\hfill \\ \multicolumn{1}{c}{{\omega }_{\mathrm{R}3}}\\ \multicolumn{1}{c}{{\omega }_{\mathrm{R}2}}\end{array}\right]$$

(2)

The velocity of the end of kinematic chain R₁ can be expressed as J ω, as shown in equation (3)

$$\mathit{J}\omega ={\omega }_{\mathrm{R}1}{\mathit{S}}_{\mathrm{R}1}$$

(3)

The velocity constraint are shown in equations (4)–(6)

$${\omega }_{\mathrm{R}4}+{\omega }_{\mathrm{R}3}+{\omega }_{\mathrm{R}2}={\omega }_{\mathrm{R}1}$$

(4)

$${\omega }_{\mathrm{R}3}{\mathrm{a}}_3=0$$

(5)

$${\omega }_{\mathrm{R}3}{\mathrm{b}}_3={\omega }_{\mathrm{R}1}{\mathrm{b}}_1$$

(6)

The velocity constraint equations (5) and (6) indicates that the velocity of the end of kinematic chain R₄R₃R₂ v ₂ should be the same as that of the kinematic chain R₁ v ₁.

Equation (5) is factored which is leads to equations (7) and (8)

$${\omega }_{\mathrm{R}3}=0$$

(7)

or

$${\mathrm{a}}_3=0$$

(8)

In the configuration shown in Figure 3(a), the direction of v ₂ at the end of the kinematic chain R₄R₃R₂ is different from that of the end of the kinematic chain R₁ v ₁, and equation (8) does not hold. Therefore, only equation (7) holds.

However, the kinematic chain R₄R₃R₂ shown in Figure 3(a) requires the rotational pair R₃ to produce rotation, that is, ${\omega }_{\mathrm{R}3}$ is not zero, in order to leave the singular configuration. Therefore, in the configuration shown in Figure 3(a), the kinematic chain R₁ is in a locking mode. When the velocity of the end of kinematic chain R₄R₃R₂ is the same as the velocity v ₁ of the end of kinematic chain R₁, the link 2 can leave the locking mode configuration.

Figure 3(b) shows the coupler 3 in the dead center configuration. The twist system of the mechanism in Figure 3(a) is

$$\{\begin{array}{c}{\mathit{S}}_{\mathrm{R}4}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\mathit{S}}_{\mathrm{R}3}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & {\mathrm{a}}_3\hfill & {\mathrm{b}}_3\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{\mathit{S}}_{\mathrm{R}2}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & {\mathrm{a}}_2\hfill & {\mathrm{b}}_2\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{\mathit{S}}_{\mathrm{R}1}={\left(\begin{array}{cccccc}0\hfill & 0\hfill & 1\hfill & 0\hfill & {\mathrm{b}}_1\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\end{array}$$

(9)

Similarly, the velocity constraint are shown in equations (10)–(12)

$${\omega }_{\mathrm{R}4}+{\omega }_{\mathrm{R}3}+{\omega }_{\mathrm{R}2}={\omega }_{\mathrm{R}1}$$

(10)

$${\omega }_{\mathrm{R}2}{\mathrm{a}}_2+{\omega }_{\mathrm{R}3}{\mathrm{a}}_3=0$$

(11)

$${\omega }_{\mathrm{R}2}{\mathrm{b}}_2+{\omega }_{\mathrm{R}3}{\mathrm{b}}_3={\omega }_{\mathrm{R}1}{\mathrm{b}}_1$$

(12)

Similarly, for the kinematic chain R₄R₃R₂ shown in Figure 3(b), to leave the dead center configuration of the mechanism, rotation is required to be generated by the rotational pair R₃, that is, ${\omega }_{\mathrm{R}3}$ is not zero. Since the mechanism is in the dead center configuration, substituting ω_R1 = 0 into equations (11) and (12) which leads to equation (13)

$$\left[\begin{array}{cc}{\mathrm{a}}_2\hfill & {\mathrm{a}}_3\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_2}& {\mathrm{b}}_3\hfill \end{array}\right]\left[\begin{array}{c}{\omega }_{\mathrm{R}2}\hfill \\ \multicolumn{1}{c}{{\omega }_{\mathrm{R}3}}\end{array}\right]=0$$

(13)

Based on the mechanism configuration shown in Figure 3(b) and in conjunction with equation (9), it can be inferred that [a₂, b₂]^T in equation (13) is linearly related to [a₃, b₃]^T. Therefore, the coefficient matrix in equation (13) is linearly dependent and ${\omega }_{\mathrm{R}3}$ , ${\omega }_{\mathrm{R}2}$ has a non-zero solution.

The velocity of the end of the kinematic chain R₄R₃R₂ v ₂ is in a different direction from that of the velocity v ₁ of the kinematic chain R₁ depicted in Figure 3(b), which is similar with the situation of Figure 3(a). However, since ${\omega }_{\mathrm{R}3}$ has a non-zero solution, indicating that the kinematic chain R₄R₃R₂ shown in Figure 3(b) can move away from the dead center configuration of the mechanism, link 2 is not in a locking mode in the mechanism configuration depicted in Figure 3(b).

In summary, under both the locking mode configuration and the dead center configuration, the velocity at the end of the kinematic chain R₄R₃R₂ is different in direction from the velocity at the end of the kinematic chain R₁, and both velocities should be zero in these two cases. The locking mode configuration analyzed in this article differs from the dead center mode in that it cannot achieve the motion of a specified component by changing the drive settings. The locking mode configuration analyzed in this article can leave the locking mode configuration, only when the velocities of the two ends of the kinematic chain are the same.

The point on the coupler point curve of the 4R mechanism is a double point

The velocity direction of the coincide point at the ends of the two kinematic chains of the planar 4R mechanism determines the configuration of the mechanism during the transition between motion and locking modes. Specifically, when constructing a mechanism with locking and motion mode by using coupler of a planar 4R mechanism as a kinematic chain. The velocity direction of the coupler, that is the tangent direction of the coupler point curve of the 4R mechanism, determines the configuration when the mode of the mechanism transforms from locking mode to motion mode.

When the point on the coupler point curve of the 4R mechanism is a double point, the tangent direction of the coupler point curve may not be unique.

As shown in Figure 4, the planar 4R mechanism consists of four links with lengths l₁ = 200 mm, l₂ = 50 mm, l₃ = 220 mm, and l₄ = 30 mm, respectively. That is, l₁ + l₂ = l₃ + l₄, and link 1 serves as the frame. As shown in Figure 4(b) and (d), the points C₂ and C₄ on the curve of coupler point C have two tangents τ ₂ and τ ₄, respectively. As shown in Figure 4(c), due to the mechanism being in a constrained singular configuration, motion bifurcation occurs, and the coupler 3 has two instantaneous centers, which are collinear with point C₃. The curve of the coupler point C has two coincident tangents τ ₃ and τ ₃′.

Figure 4.

The coupler point curve of planar 4R mechanism has two double points: (a) coupler point has one tangen, (b) coupler point has two tangents I, (c) coupler point has two coincide tangents, and (d) coupler point has two tangents II.

The curve of coupler point D of the planar 4R mechanism is shown in Figure 5. In the mechanism configuration shown in Figure 5(a), the curve of coupler point D coincides with the axode of the coupler 3. At this time, the moving speed of point D₁ on coupler 3 is 0. The literature ¹⁷ refers to this point on the curve of coupler point as a cusp, where there are two coincident tangents τ ₅ and τ ₅′ of the point D₁. According to mechanism configuration as shown in Figure 5(a), (c), and (d), it can be seen that point D₂ on the coupler point curve in Figure 5(b) corresponds to only one mechanism configuration.

Figure 5.

The coupler point curve of planar 4R mechanism has a double point and a cusp: (a) coupler point has two coincide tangents, (b) coupler point has two tangents, (c) configuration I is close to double point of coupler point curve, and (d) configuration II is close to double point of coupler point curve.

In the mechanism configuration shown in Figure 5(b), point D₂ on the coupler point curve has two tangents τ₆ and τ₇, indicating that the mechanism is in a constrained singular configuration. Comparing with Figure 5(b) and (d), it can be found that points D₂ and D₄ in the mechanism configurations of Figure 5(b) and (d) correspond to two mechanism configurations, while points D₂ has only one tangent, indicating that the mechanism is in a general configuration. The point D₂ in the configuration of the mechanism shown in Figure 5(b) has two tangents, which correspond to one type of mechanism configuration, namely the constrained singular configuration of the mechanism.

The point on the coupler point curve is a threefold points

Specifically, as shown in Figure 6(d), the curve of coupler point E has a threefold Point E₄. The dimensions of this mechanism are l₁ = 100 mm, l₂ = 150 mm, l₃ = 100 mm, and l₄ = 150 mm. The distance from point E to the rotational pair R₂ is 25 mm. As shown in Figure 6(d), Point E₄ coincides with the instantaneous center of coupler 3, and the point E₄ on the coupler point curve has a cusp point. The mechanism shown in Figure 6(c) could transform to the motion mode shown in Figure 6(b) passing the common mechanism configuration as shown in Figure 6(d). Point E₄ is a threefold point, and its trajectory has two tangents, τ ₈ and τ ₉ in Figure 6(d). Point E₅ is a double point, and its trajectory has two coincident tangents τ ₁₀ and τ ₁₀′.

Figure 6.

The coupler point curve of planar 4R mechanism is a threefold point which is constructed by a cusp and a common point: (a) coupler point has one tangent I, (b) coupler point has two tangents, (c) coupler point has one tangent II, and (d) coupler point has 3 tangents.

A method for constructing a four-bar mechanism with a threefold points on the coupler point curve is presented in Hernandez et al., ¹⁸ as shown in Figure 7(a). Point F₁ represents the threefold point on the coupler point curve. By combining the mechanism configurations shown in Figures 7(b) to (d), it can be observed that the configuration depicted in Figure 7(a) is in a motion bifurcation state and can be transformed into the configurations shown in Figure 7(c) and (d). Meanwhile, the mechanism configuration shown in Figure 7(b) is not near the motion bifurcation configuration. When the mechanism moves from the configuration shown in Figure 7(b) to a position where the point F₂ of the link coincides with F₁ in Figure 7(a), the motion of the mechanism does not exhibit bifurcation. When the mechanism moves from the configurations shown in Figure 7(c) and (d) to a position where the points F₃ and F₄ coincide with F₁ in Figure 7(a), the motion of the mechanism exhibits bifurcation.

Figure 7.

The coupler point curve of planar 4R mechanism is a threefold point which has three tangent lines: (a) coupler point has 3 tangents I, (b) configurations close to threefold coupler point, (c) coupler point has 3 tangents II, and (d) coupler point has 3 tangents III.

Synthesis a Planar 7R mechanism with locking and motion mode

According to the curve of coupler point F of 4R mechanism in Figure 7, a planar 7R mechanism can be constructed as shown in Figure 8. A point G₁ on the base in Figure 8(a) is taken as the rotation center of the rotational pair R₅. The motion chain R₅R₆R₇ is connected to point G₁ on coupler 3 in Figure 7(a), and the rotation center of the rotating pair R₇ coincides with point G on coupler 3. As shown in Figure 8(c), under this mechanism configuration, the velocity direction of the end point G₃ of the kinematic chain R₅R₆R₇ is the same as the direction of the trajectory tangent τ₁₂ of point G₃ on coupler 3, allowing point G₃ to undergo slight displacement, thereby leaving the singular configuration and achieving the transformation between locking and motion modes of coupler 3. As shown in Figure 8(e), under this mechanism configuration, the velocity direction of the end point G₅ of the kinematic chain R₅R₆R₇ is the same as the direction of the trajectory tangent τ₁₃ of point G₅ on coupler 3, allowing point G₅ to undergo slight displacement, thereby leaving the singular configuration and achieving the transformation between locking and motion modes due to the kinematic chain R₅R₆R₇ could rotate freely. This results in a mechanism with six configurations that transition from locking mode to planar 7R motion mode.

Figure 8.

Planar 7R mechanism with two locking modes: (a) motion mode transform configuration I, (b) unlocking configuration I, (c) motion mode transform configuration II, (d) unlocking configuration II, and (e) motion mode transform configuration III, (f) unlocking configuration III.

The coupler 3 of the mechanism shown in Figure 8 has two locking modes, as illustrated in Figure 8(a) and (c), where the coupler 3 is in two different configurations. When the velocity direction of the end point G of the kinematic chain R₅R₆R₇ does not coincide with the tangent of the coupler point curve of the coupler 3, the coupler 3 is in a locking mode.

If the axes of the joints in the R₅R₆R₇ chain become coplanar, this chain imposes one constraint on the coupler 3 and could locks its motion, and the mechanism cannot escape the locking mode. When the whole of the mechanism is in a singular state, the mechanism can escape the locking mode and continue its motion.

Spherical 7R mechanism with motion and locking modes

Transition between motion and locking modes

The method for synthesizing a planar 7R mechanism with locking and motion modes can be applied to synthesize a spherical 7R mechanism with locking and motion modes. As shown in Figure 9, point H₁ on the coupler point curve of R₁R₂R₃R₄ Spherical 4R loop is a non-singular point, and there is only one tangent line at this point on the link curve. In the mechanism configuration shown in Figure 9(a), the axes of the rotational pairs R₇ and R₅ coincide, and the kinematic chain R₅R₆R₇ can rotate freely. At this time, the spherical 4R loop R₁R₂R₃R₄ is in a locking state. The direction of the velocity v ₄ of point H₁ at the end of the kinematic chain R₅R₆R₇ does not coincide with the direction of the velocity v ₃ of point H₁ on the coupler of the spherical 4R loop R₁R₂R₃R₄, and point H₁ is in a locking mode at this time. When the kinematic chain R₅R₆R₇ rotates around the axis of the rotational pair R₅ as a whole, forming the mechanism configuration shown in Figure 9(b), the velocity v ₄ direction of the end point H₂ of the kinematic chain R₅R₆R₇ is the same as the velocity direction v ₃ of the point H₂ on the link rod of the spherical 4R loop R₁R₂R₃R₄. At this time, the spherical 4R loop R₁R₂R₃R₄ can be transformed from the locking mode to the motion mode, and can move to the configuration shown in Figure 9(d). When the mechanism shown in Figure 9(b) is in the motion mode transformation configuration, after the kinematic chain R₅R₆R₇ rotates 180°, the velocity v ₄ direction of the end point H₃ of the kinematic chain R₅R₆R₇ is still the same as the velocity direction v ₃ of the point H₃ on the link rod of the spherical 4R mechanism R₁R₂R₃R₄. In addition to the motion mode transformation configuration shown in Figure 9(b), the mechanism shown in Figure 9 also has one more motion mode transformation configuration, as shown in Figure 9(c).

Figure 9.

Spherical 7R mechanism with one motion mode and one locking mode: (a) locking mode, (b) motion mode transform configuration I, (c) motion mode transform configuration II, and (d) unlocking mode.

Assembly conditions with motion and locking modes

Based on the assembly conditions of the planar 7R mechanism with motion and locking modes, the assembly conditions of the spherical 7R mechanism with motion and locking modes can be similarly derived, as shown in Figure 10. The spherical workspace of the kinematic chain R₅R₆R₇ allows the point I₁ on the link 6 to rotate around the axis R₆ as a fixed axis, resulting in a circle. This circle rotates around the axis R₅, forming a spherical surface, which represents the spherical workspace that the point I₁ at the end of the kinematic chain R₅R₆R₇ can reach. In Figure 10, the angles between the rotational pairs R₅ and R₆, and R₆ and R₇ are both 90°, so the point I₁ at the end of the kinematic chain R₅R₆R₇ can reach the entire spherical surface. The curve of the coupler point I₂ on the coupler 3 of the spherical 4R mechanism is shown as the black curve in Figure 10(b). The workspace reachable by point I₃ in the mechanism shown in Figure 10(c) is the intersection of the workspace reachable by the point I₁ at the end of the kinematic chain R₅R₆R₇ and the workspace reachable by the point I₂ on the coupler of the spherical 4R mechanism R₁R₂R₃R₄, which is represented by the black curve which is the same as the coupler point curve of the spherical 4R in Figure 10(b).

Figure 10.

Schematic diagram of assembly conditions for spherical 7R mechanism: (a) spherical 3R kinematic chain, (b) coupler curve has 3 double points, (c) spherical 7R mechanism at unlocking mode.

Coupler point curve has a double point during mode transformation

The curve of the coupler point I₂ on the coupler 3 of the spherical 4R mechanism R₁R₂R₃R₄ in Figure 10(b) is shown in Figure 11. In Figure 11(a), J₁ is a double point, and there are two tangents at the point J₁ on the coupler point curve, corresponding to two different mechanism configurations. Specifically, when the velocity direction of point J₁ is along the tangent τ ₁₄ of the coupler point curve, the mechanism is in the configuration shown in Figure 11(a). When the velocity direction of point J₂ is along the tangent τ ₁₅ of the coupler point curve, the mechanism is in the configuration shown in Figure 11(b). In Figure 11(c) and (d), J₃ and J₄ are double points, and there are two coincident tangents at points J₃ and J₄ on the curve of the coupler point, both corresponding to the folded configuration of the spherical 4R mechanism.

Figure 11.

The coupler point curve of spherical 4R mechanism has three double points: (a) coupler point has two tangens I, (b) coupler point has two tangens II, (c) coupler point has two coincidetangens I, and (d) coupler point has two coincide tangens II.

When the spherical 4R mechanism is in the folded configuration, the point on the coupler is generally the double point on the coupler point curve. As shown in Figure 12(a), point K₁ on the coupler point curve has double point when the spherical 4R mechanism is in the folded configuration. Point K₁ on the coupler point curve of the spherical 4R mechanism has two coincident tangents, τ₁₈ and τ₁₈′.

Figure 12.

The coupler curve of spherical 4R mechanism has 2 kinds of double point: (a) coupler point has two coincide tangens, (b) coupler point has two tangens I, (c) coupler point has two tangens II.

When the mechanism moves to the configuration shown in Figure 12(b), coupler point K reaches point K₂, which is the double point on the coupler point curve. There are two non-coincident tangents to its trajectory. In the configuration of the mechanism shown in Figure 12(b), one tangent to the curve of the coupler point K₂ is τ₁₉. The other mechanism configuration corresponding to point K₂ on the link trajectory is shown in Figure 12(c). In this configuration, another tangent to the curve of the coupler point K₃ is τ₁₉′.

Based on the two double points on the coupler point curve shown in Figure 12(a), the kinematic chain R₅R₆R₇ is set up, ensuring that the rotational pairs R₅ and R₇ are collinear, and their axes pass through point K₁ on coupler 3 in the mechanism configuration of Figure 13(a). Figures 13(a) and (d) depict the mechanism configurations of the spherical 4R loop R₁R₂R₃R₄ transitioning from the locking mode to the motion mode. As shown in Figure 13(a) and (d), when the curve of the coupler point L₁ on the coupler 3 of the spherical 4R loop R₁R₂R₃R₄ is tangent to τ₂₀, the tangent line τ₂₀ is perpendicular to the folding plane of the spherical 4R loop R₁R₂R₃R₄. In this configuration, when the tangent line of the curve of the point L₁ on the kinematic chain R₅R₆R₇ coincides with the tangent line τ₂₀, the kinematic chain R₅R₆R₇ can leave the folded configuration, and the coupler 3 can achieve the transition to the motion mode.

Figure 13.

Spherical 7R mechanism has two motion modes and one locking mode: (a) motion mode tranform configuration I, (b) unlocking mode configuration I, (c) unlocking mode configuration II, (d) motion mode tranform configuration II, (e) unlocking mode configuration III, (f) unlocking mode configuration IV, and (g) coupler point on the coupler curve which are corresponding to mechanism configurations.

Under the mechanism configurations of Figure 13(c) and (e), points L₃ and L₅ are double point on the coupler point curve. At this time, there are two different mechanism configurations, neither of which can be switched between motion mode and locking mode.

Figure 13(a) shows the mechanism configuration during the counterclockwise rotation of the rotational pair R₁, moving from the mechanism configuration shown in Figure 13(b) to (d), the mechanism undergoes motion bifurcation in the configuration of Figure 13(d), allowing it to move to the configurations of Figure 13(e) and (f). The mechanism configuration from Figure 13(a) to (f) are corresponds to the point L₁ to L₆ on the coupler point curve in Figure 13(g), respectively.

The point on the coupler point curve of spherical 4R mechanism is a threefold point

Under the folded configuration of the spherical 4R mechanism, the coupler possesses two non-coincident instantaneous axes of rotation. If a point on the coupler lies on its instantaneous axis of rotation, and this point is a cusp, it is a special type of double point. Given that the mechanism is in a folded configuration, this point inherently serves as a double points, thereby rendering it a threefold point on the coupler point curve at this moment.

As shown in Figure 14(a), there is only one tangent line to point M₁ on the coupler 3 of the spherical 4R mechanism. When the mechanism moves to the folded configuration shown in Figure 14(b), one instantaneous rotation axis l ₁ of the coupler 3 passes through point M₂. At this time, point M₂ is the threefold point on the coupler point curve, and it has three tangent lines τ₂₂, τ₂₃, and τ₂₃′. Among them, two tangent lines τ₂₃ and τ₂₃′ coincide, which are the tangent lines corresponding to the cusp point M₂ in Figure 14(b). Based on the threefold point on the curve of the coupler point M shown in Figure 14, as depicted in Figure 15(a), a kinematic chain R₅R₆R₇ is established, ensuring that the rotational pairs R₅ and R₇ are collinear, and their axes pass through point M₂ on coupler 3 in the mechanism configuration of Figure 14(b). Figure 15(a) to (d) respectively represent the mechanism configurations when the mode of spherical 4R loop R₁R₂R₃R₄ transforms from the locking mode to the motion mode.

Figure 14.

The coupler point curve of spherical 4R mechanism has a threefold point which is constructed by a cusp and a common point: (a) coupler point has one tangent and (b) coupler point has 3 tangents, 2 of them are coincide.

Figure 15.

Spherical 7R mechanism with one locking mode and one motion modes: (a) motion mode transform configuration I, (b) motion mode transform configuration II, (c) motion mode transform configuration II, (d) motion mode transform configuration IV.

As shown in Figure 15(a) and (b), when the trajectory tangent of point N on the coupler 3 of the spherical 4R mechanism R₁R₂R₃R₄ is along τ₂₄, the tangent τ₂₄ is perpendicular to the folding plane of the spherical 4R loop R₁R₂R₃R₄. Under this configuration, when the trajectory tangent of the end point N of the kinematic chain R₅R₆R₇ coincides with the tangent τ₂₄, the kinematic chain R₅R₆R₇ can leave the folding configuration, and the coupler 3 can achieve a change in motion mode.

As shown in Figure 15(c) and (d), when the trajectory tangent of point N on the coupler 3 of the spherical 4R mechanism R₁R₂R₃R₄ is along τ₂₅, in this configuration, when the trajectory tangent of the end point N of the kinematic chain R₅R₆R₇ coincides with the tangent τ₂₅, the kinematic chain R₅R₆R₇ can leave the folded configuration, and the coupler 3 can achieve a change in motion mode.

The specific process of the mechanism transitioning from the locking mode to the motion mode is shown in Figure 16(h). The mechanism transforms from the configuration shown in Figure 16(a), corresponding to point N₁ on the link trajectory in Figure 16(h), to the configuration shown in Figure 16(b), corresponding to point N₂ on the link trajectory in Figure 16(h).

Figure 16.

Mode transformation of spherical 7R mechanism from locking mode to motion modes: (a) motion mode transform configuration I, (b) unlocking mode configuration I, (c) unlocking mode configuration II, (d) motion mode transform configuration II, (e) motion mode transform configuration III, and (f) coupler point on the coupler curve which are corresponding to mechanism configurations.

Figure 16(a) depicts the mechanism configuration, where during the counterclockwise rotation of the revolute pair R₁, the mechanism sequentially transforms from the configurations shown in Figure 16(b) to (g), corresponding to the link trajectories N₁ to N₇ in Figure 16(h).

Figure 17(a) illustrates the spherical 4R mechanism with angles α₁₄ = 90°, α₁₂ = 30°, α₂₃ = 60°, α₃₄ = 60°, α₂₇ = 45°, and α₃₇ = 45°. The curve of the coupler point P of this spherical mechanism has a special threefold point. The mechanism shown in Figure 17(a) is in a constrained singular configuration, leading to motion bifurcation. Under normal constrained singular configuration, the point on the coupler point curve has a double point. When the mechanism transforms to the configuration shown in Figure 17(b), the point P₂ on the coupler point curve coincides with the double point on the curve of coupler point P₁ in the configuration shown in Figure 17(a). Based on the threefold point on the coupler point curve, a spherical 7R mechanism with two locking modes can be designed.

Figure 17.

The coupler point curve of spherical 4R mechanism has a threefold point which has three tangent lines: (a) configuration I is corresponding to coupler point has 3 tangents, and (b) configuration II is corresponding to coupler point has 3 tangents.

As shown in Figure 18(a), the motion R₅R₆R₇ is set up so that the axes of the rotational pairs R₅ and R₇ coincide and pass through the threefold point P₁ of the coupler point curve in Figure 17(a). In the mechanism configuration shown in Figure 18(a), the velocity direction of point P₃ v ₅ on coupler 3 is the same as that of point P₃ v ₆ on link 6. At this time, the mechanism is in a transformation configuration between locking and motion modes, and can move to the mechanism configuration shown in Figure 18(b).

Figure 18.

Spherical 7R mechanism has two locking modes and one motion modes: (a) motion mode transform configuration I, (b) unlocking mode configuration I, (c) motion mode transform configuration II, (d) unlocking mode configuration II, (e) motion mode transform configuration III, (f) unlocking mode configuration III, (g) locking mode configuration I, and (h) locking mode configuration II.

Similarly, for the mechanism configurations shown in Figure 18(c) and (e), the velocity directions of points P v ₅ on coupler 3 are the same as the velocity directions of P on link 5, allowing for movement to the mechanism configurations shown in Figure 18(d) and (f).

The mechanism configurations shown in Figure 18(g) and (h), where the velocity directions v ₅₁, v ₅₂, and v ₅₃ of the three motion bifurcations at points P₉ and P₁₀ on coupler 3 are different from those on link 5, indicate that the spherical 4R loop R₁R₂R₃R₄ is in a locking mode. The kinematic chain R₅R₆R₇ possesses one local rotational degree of freedom, allowing it to freely rotate around the axis of the rotational pair R₅.

Coupler point curve has a quadruple point during mode transformation

As shown in Figure 19(a), the structural parameters of the spherical 4R mechanism are α₁₄ = 60°, α₁₂ = 30°, α₂₃ = 60°, α₃₄ = 30°, α₂₇ = 90°, and α₃₇ = 90°. The spherical 4R mechanism is in a folded configuration, and it possesses two variable-axis motion modes. ¹⁶ The point Q₁ on the coupler 3 is located perpendicular to the plane of the mechanism’s folded configuration and passes through the normal line of the mechanism’s rotation center as shown in Figure 19. When the mechanism has two folded configurations as shown in Figure 19(a) and (b), the points Q₁ and Q₂ on the coupler are at the same point in both folded configurations.

Figure 19.

The coupler point curve of planar 4R mechanism has a quadruple point which has four tangent lines: (a) configuration I is corresponding to coupler point has 4 Tangents, and (b) configuration II is corresponding to coupler point has 4 tangents.

In the folded configuration of the mechanism, as Figures 17 and 18, the point on the couple is either a double point or threefold point on the coupler point curve. When the point Q on the coupler passes through the instantaneous axis of rotation of the coupler in the folded configuration, it becomes a threefold point on the coupler point curve. In Figure 19, the point Q on coupler 3 is not located on the folding plane of the mechanism, thus it is a double point on the coupler point curve. It can be observed that the mechanism in Figure 19 has two folded configurations, and the point Q on coupler 3 is a quadruple point on the coupler point curve the mechanism.

As shown in Figure 20, the motion R₅R₆R₇ is set to ensure that the axes of the rotational pairs R₅ and R₇ coincide and pass through the point Q₁ on the coupler point curve in Figure 19(a). In the mechanism configuration shown in Figure 20(a), the velocity direction of point Q₃ on coupler 3 is the same as that of point Q₃ on link 5. At this time, the mechanism is in a transformation configuration between locking and motion modes, and can move to the mechanism configuration shown in Figure 20(b). Similarly, for the mechanism configurations shown in Figure 20(c), (e), and (g), the velocity direction of points Q₅, Q₇, and Q₉ on coupler 3 is the same as that of point Q on link 5, allowing it to move to the mechanism configurations shown in Figure 20(d), (f), and (h).

Figure 20.

Spherical 7R mechanism featuring two locking modes and two motion modes: (a) motion mode transform configuration I, (b) unlocking mode configuration I, (c) motion mode transform configuration II (d) unlocking mode configuration II, (e) motion mode transform configuration III, (f) unlocking mode configuration III, (g) motion mode transform configuration IV, (h) unlocking mode configuration IV, (i) locking mode configuration I, and (j) locking mode configuration II.

The configurations of the mechanisms shown in Figure 20(g), (h), and (j), where the velocity directions of the four motion bifurcations at point Q on coupler 3 are all different from the velocity direction of point Q on link 5, indicating that the mechanism is in a locking mode. The kinematic chain R₅R₆R₇ possesses one local rotational degree of freedom, allowing it to freely rotate around the axis of the rotational pair R₅.

Analysis of the transformation of motion mode of spherical 7R mechanism

As shown in Figure 21(b), for the spherical 7R mechanism, the point T on the coupler point curve coincides with the instantaneous rotation center Saxode of the coupler 3 as Figure 21(a). ¹⁹ It can be observed that the velocity v _T of point T at this time is 0. The velocity direction of this point can be determined based on the tangent of its trajectory, thereby determining the configuration of the kinematic chain of the rotational pairs R₅, R₆, and R₇, ensuring that the mechanism is in a motion mode transition configuration. Generally, the mechanism’s instantaneous degrees of freedom increases at the motion mode transformation configuration. The motion mode transformation configuration of the mechanism shown in Figure 21 can be determined using screw theory to calculate the number of instantaneous degrees of freedom of the mechanism. By way of contras, obtaining the tangent of the curve of the coupler point T generally requires complex calculations.

Figure 21.

Spherical 7R mechanism with one locking mode and one motion mode: (a) coupler point has two coincide tangents and (b) locking mode configuration.

The angle between the axis of the R _i axis of the rotational pair and the axis of the R _j axis in Figure 22(b) is α _ij . Figure 22(b) The spherical 4R loop R₁R₂R₃R₄ with α₁₂ = π/4, α₂₃ = π/3, α₃₄ = π/4, α₄₁ = π/2 is shown in Figure 22(a). The kinematic chain R₅R₆R₇ is shown in Figure 21(b), where α₂₇ = π/4, α₃₇ = π/4, α₇₆ = π/2, and α₆₅ = π/2. In Figure 22(b), when the four rotational pairs R₁, R₂, R₃, and R₄ of the spherical 4R mechanism correspond to the twist elements S ₁, S ₂, S ₃, and S ₄ in the fixed coordinate system, respectively, the first-order kinematic constraint of the spherical 4R loop equation is written as:

$${\omega }_1{\mathit{S}}_1+{\omega }_2{\mathit{S}}_2+{\omega }_3{\mathit{S}}_3+{\omega }_4{\mathit{S}}_4=0$$

(14)

Figure 22.

Twist of spherical 7R mechanism has one locking mode and one motion mode: (a) kinematic screw of spherical 4R and (b) kinematic screw of spherical 7R.

The first-order kinematic constraint equation for the spherical 5R loop of R₁R₂R₇R₆R₅ in Figure 22(b) is

$${\omega }_1{\mathit{S}}_1+{\omega }_2{\mathit{S}}_2+{\omega }_7{\mathit{S}}_7+{\omega }_6{\mathit{S}}_6+{\omega }_5{\mathit{S}}_5=0$$

(15)

Equation (16) is obtained by combining (14) and (15)

$$\begin{array}{c}\left[\begin{array}{ccccccc}{\mathit{S}}_1\hfill & {\mathit{S}}_2\hfill & {\mathit{S}}_3\hfill & {\mathit{S}}_4\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{{\mathit{S}}_1}& {\mathit{S}}_2\hfill & 0\hfill & 0\hfill & {\mathit{S}}_7\hfill & {\mathit{S}}_6\hfill & {\mathit{S}}_5\hfill \end{array}\right]·\hfill \\ {\left[\begin{array}{ccccccc}{\omega }_1\hfill & {\omega }_2\hfill & {\omega }_3\hfill & {\omega }_4\hfill & {\omega }_7\hfill & {\omega }_6\hfill & {\omega }_5\hfill \end{array}\right]}^T=0\hfill \end{array}$$

(16)

In Figure 22(b), because each moment part of the screws equals zero which could be ignored in order to simplify the synthesis. And then, the screw of the rotation axes in the coordinate system O-XYZ are

$$\begin{array}{c}{\mathit{S}}_1={\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]}^T\hfill \\ {\mathit{S}}_2={\left[\begin{array}{ccc}\cos \left({\alpha }_{12}\right)\hfill & 0\hfill & \sin \left({\alpha }_{12}\right)\hfill \end{array}\right]}^T\hfill \\ {\mathit{S}}_3={\left[\begin{array}{ccc}0\hfill & \cos \left({\alpha }_{34}\right)\hfill & \sin \left({\alpha }_{34}\right)\hfill \end{array}\right]}^T\hfill \\ {\mathit{S}}_4={\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]}^T\hfill \\ {\mathit{S}}_7={\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]}^T\hfill \\ {\mathit{S}}_6={\left[\begin{array}{ccc}\cos \left({\alpha }_{16}\right)\hfill & \sin \left({\alpha }_{16}\right)\hfill & 0\hfill \end{array}\right]}^T\hfill \\ {\mathit{S}}_5={\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]}^T\hfill \end{array}$$

(17)

Substituting the twist of each joint in equation (17) into equation (16), equation (18) is obtained.

$$\left[\begin{array}{ccccccc}1\hfill & \cos \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \cos \left({\alpha }_{34}\right)\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \sin \left({\alpha }_{12}\right)\hfill & \sin \left({\alpha }_{34}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& \cos \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & \cos \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & \sin \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \sin \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \end{array}\right]\left[\begin{array}{c}{\omega }_1\hfill \\ \multicolumn{1}{c}{{\omega }_2}\\ \multicolumn{1}{c}{{\omega }_3}\\ \multicolumn{1}{c}{{\omega }_4}\\ \multicolumn{1}{c}{{\omega }_7}\\ \multicolumn{1}{c}{{\omega }_6}\\ \multicolumn{1}{c}{{\omega }_5}\end{array}\right]=0$$

(18)

After simplifying, equation (19) is obtained.

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{ccccccc}1\hfill & \cos \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \cos \left({\alpha }_{34}\right)\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \sin \left({\alpha }_{12}\right)\hfill & \sin \left({\alpha }_{34}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & \cos \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & \sin \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill \end{array}\right]\left[\begin{array}{c}{\omega }_1\hfill \\ \multicolumn{1}{c}{{\omega }_2}\\ \multicolumn{1}{c}{{\omega }_3}\\ \multicolumn{1}{c}{{\omega }_4}\\ \multicolumn{1}{c}{{\omega }_7}\\ \multicolumn{1}{c}{{\omega }_6}\\ \multicolumn{1}{c}{{\omega }_5}\end{array}\right]=0}\end{array}$$

(19)

The rank of the coefficient matrix of equation (19) is 5. According to the analysis of equation (19), the mechanism have 2 degrees of freedom, and the mechanism undergoes motion bifurcation at any configurations when the axis of rotating pair R₅ coincides with the axis of rotating pair R₇. However, it is not true. By analyzing the velocity constraint of the mechanism in equation (19), it is not possible to determine the bifurcation configuration of the mechanism’s motion, therefore it is necessary to analyze the acceleration constraint of the mechanism.

Take the derivative of equation (14) with respect to time t yields

$$\begin{array}{c}{\omega \prime }_1{\mathit{S}}_1+{\omega \prime }_2{\mathit{S}}_2+{\omega \prime }_3{\mathit{S}}_3+{\omega \prime }_4{\mathit{S}}_4\hfill \\ +{\omega }_1{\mathit{S}}_1\times ({\omega }_2{\mathit{S}}_2+{\omega }_3{\mathit{S}}_3+{\omega }_4{\mathit{S}}_4)\hfill \\ +{\omega }_2{\mathit{S}}_2\times ({\omega }_3{\mathit{S}}_3+{\omega }_4{\mathit{S}}_4)\hfill \\ +{\omega }_3{\mathit{S}}_3\times \left({\omega }_4{\mathit{S}}_4\right)=0\hfill \end{array}$$

(20)

Substituting equation (14) into equation (20) yields

$$\begin{array}{c}{\omega \prime }_1{\mathit{S}}_1+{\omega \prime }_2{\mathit{S}}_2+{\omega \prime }_3{\mathit{S}}_3+{\omega \prime }_4{\mathit{S}}_4\hfill \\ -{\omega }_2{\omega }_1{\mathit{S}}_2\times {\mathit{S}}_1+{\omega }_3{\omega }_4{\mathit{S}}_3\times {\mathit{S}}_4=0\hfill \end{array}$$

(21)

Take the derivative of equation (15) with respect to time t yields

$$\begin{array}{c}{\omega \prime }_1{\mathit{S}}_1+{\omega \prime }_2{\mathit{S}}_2+{\omega \prime }_7{\mathit{S}}_7+{\omega \prime }_6{\mathit{S}}_6+{\omega \prime }_5{\mathit{S}}_5\hfill \\ +{\omega }_1{\mathit{S}}_1\times ({\omega }_2{\mathit{S}}_2+{\omega }_7{\mathit{S}}_7+{\omega }_6{\mathit{S}}_6+{\omega }_5{\mathit{S}}_5)\hfill \\ +{\omega }_2{\mathit{S}}_2\times ({\omega }_7{\mathit{S}}_7+{\omega }_6{\mathit{S}}_6+{\omega }_5{\mathit{S}}_5)\hfill \\ +{\omega }_7{\mathit{S}}_7\times ({\omega }_6{\mathit{S}}_6+{\omega }_5{\mathit{S}}_5)\hfill \\ +{\omega }_6{\mathit{S}}_6\times {\omega }_5{\mathit{S}}_5=0\hfill \end{array}$$

(22)

Substituting equation (17) into equation (15) yields ω₆ = 0, substituting it into equation (22) yields

$$\begin{array}{c}{\omega \prime }_1{\mathit{S}}_1+{\omega \prime }_2{\mathit{S}}_2+{\omega \prime }_7{\mathit{S}}_7+{\omega \prime }_6{\mathit{S}}_6+{\omega \prime }_5{\mathit{S}}_5\hfill \\ -{\omega }_2{\mathit{S}}_2\times {\omega }_1{\mathit{S}}_1=0\hfill \end{array}$$

(23)

Substitute equation (17) into equations (21) and (23), equation (24) is obtained by combining equations (21)–(23)

$$\begin{array}{c}\left[\begin{array}{ccccccc}1\hfill & \cos \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \cos \left({\alpha }_{34}\right)\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \sin \left({\alpha }_{12}\right)\hfill & \sin \left({\alpha }_{34}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& \cos \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & \cos \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & \sin \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \sin \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}{\omega \prime }_1\hfill \\ \multicolumn{1}{c}{{\omega \prime }_2}\\ \multicolumn{1}{c}{{\omega \prime }_3}\\ \multicolumn{1}{c}{{\omega \prime }_4}\\ \multicolumn{1}{c}{{\omega \prime }_5}\\ \multicolumn{1}{c}{{\omega \prime }_6}\\ \multicolumn{1}{c}{{\omega \prime }_7}\end{array}\right]=\left[\begin{array}{c}{\omega }_3{\omega }_4\frac{\sqrt{2}}{2}\hfill \\ \multicolumn{1}{c}{{\omega }_1{\omega }_2\frac{\sqrt{2}}{2}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{{\omega }_1{\omega }_2\frac{\sqrt{2}}{2}}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \end{array}$$

(24)

After simplification

$$\begin{array}{c}\left[\begin{array}{ccccccc}1\hfill & \cos \left({\alpha }_{12}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \cos \left({\alpha }_{34}\right)\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \sin \left({\alpha }_{12}\right)\hfill & \sin \left({\alpha }_{34}\right)\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & \cos \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & \sin \left({\alpha }_{16}\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}{\omega \prime }_1\hfill \\ \multicolumn{1}{c}{{\omega \prime }_2}\\ \multicolumn{1}{c}{{\omega \prime }_3}\\ \multicolumn{1}{c}{{\omega \prime }_4}\\ \multicolumn{1}{c}{{\omega \prime }_5}\\ \multicolumn{1}{c}{{\omega \prime }_6}\\ \multicolumn{1}{c}{{\omega \prime }_7}\end{array}\right]=\left[\begin{array}{c}{\omega }_3{\omega }_4\frac{\sqrt{2}}{2}\hfill \\ \multicolumn{1}{c}{{\omega }_1{\omega }_2\frac{\sqrt{2}}{2}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{{\omega }_3{\omega }_4\frac{\sqrt{2}}{-2}}\\ \multicolumn{1}{c}{{\omega }_1{\omega }_2\frac{\sqrt{2}}{2}}\\ \multicolumn{1}{c}{{\omega }_1{\omega }_2\frac{\sqrt{2}}{2}}\end{array}\right]\hfill \end{array}$$

(25)

When equation (25) has a solution, it is necessary to ensure that the rank of the coefficient matrix is the same as that of the augmented matrix. When the mechanism shown in Figure 22(b) undergoes motion bifurcation, the mechanism must have at least 2 degrees of freedom. According to observation, only the rank of matrix in equation (26) affects the rank of the coefficient matrix and the augmented matrix in equation (25)

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{cc}\cos \left({\alpha }_{16}\right)\hfill & -{\omega }_3{\omega }_4\frac{\sqrt{2}}{2}\hfill \\ \multicolumn{1}{c}{\sin \left({\alpha }_{16}\right)}& {\omega }_1{\omega }_2\frac{\sqrt{2}}{2}\hfill \end{array}\right]}\end{array}$$

(26)

When the determinant of the matrix in equation (26) is zero, the rank of both the coefficient matrix and the augmented matrix of the linear equation system (25) is 5. At this point, the mechanism has two instantaneous degrees of freedom and is in a motion bifurcation configuration. The determinant of the matrix in equation (26) is

$$\cos \left({\alpha }_{16}\right){\omega }_1{\omega }_2+\sin \left({\alpha }_{16}\right){\omega }_3{\omega }_4=0$$

(27)

According to equation (19), we can obtain equation (28)

$$\begin{array}{c}{\omega }_1+\frac{\sqrt{2}}{2}{\omega }_2=0\hfill \\ \frac{\sqrt{2}}{2}{\omega }_3+{\omega }_4=0\hfill \\ {\omega }_2+{\omega }_3=0\hfill \\ {\omega }_6=0\hfill \\ {\omega }_4+{\omega }_7+{\omega }_5=0\hfill \end{array}$$

(28)

By combining equation (28) with algebraic expression (27) yields

$${{\omega }_2}^2\sin ({\alpha }_{16}+{45}^{°})=0$$

(29)

According to equation (29), when α₁₆ is either −45° or 135°, the rank of both the coefficient matrix and the augmented matrix of the linear equation system (25) is 5. At this point, the mechanism possesses two instantaneous degrees of freedom and is in a motion bifurcation configuration.

Conclusion

A planar 4R mechanism and a spherical 4R mechanism are equipped with a folding 3R kinematic chain on their links, thereby integrating a planar 7R and spherical 7R mechanism with both motion and locking modes. Based on the tangent of the coupler point curve, the configuration characteristics for switching between the mechanism’s motion and locking modes are determined. Based on the coupler point curve of the planar 4R mechanism with double and threefold point, and the coupler point curve of the spherical 4R mechanism with threefold and quadruple point, a spherical 7R mechanism with different mode-switching configurations is designed.

When the point on the curve of coupler point of the spherical 4R loop is a cusp, it is difficult to determine the tangent of the trajectory. The screw theory is used to analyze the first-order and second-order motion constraint equation of the spherical 7R mechanism, and determine the transformation configuration of the mechanism with motion and locking modes. When it is necessary to analyze the tangent of the multiple points of on the coupler point curve of the 4R mechanism, the 7R double loop mechanism can be constructed using the method proposed in this paper, and the screw theory is used to determine the transformation configuration of the mechanism’s motion mode, thereby determining the direction of tangents of the multiple points on the curve of coupler point.