3.1.1. The kinematic pairs composition and connection sequence

(i) Limb configuration

In Fig. 3, the base twist screw system of the limb DOF space in the micro-motion stage can be denoted by $\boldsymbol{\$ }_{1}-\boldsymbol{\$ }_{6}$ , which is a six-dimensional twist screw system in the Cartesian coordinate system. $\boldsymbol{\$ }_{1}-\boldsymbol{\$ }_{3}$ denote rotational motion around the three coordinate axes, and $\boldsymbol{\$ }_{4}-\boldsymbol{\$ }_{6}$ denote translational motion along the three coordinate axes.

The twist screw system of different kinematic pairs can be obtained by performing linear combinations of $\boldsymbol{\$ }_{1}-\boldsymbol{\$ }_{6}$ . The limb configuration can be classified according to the number of prismatic joints (PN). This paper explores limb configurations with 6-DOF under conditions where limbs have 0, 1, 2, or 3 prismatic joints. Consequently, the limb configurations are shown in Table III.

Table III. Limb configurations.

In Table III, elements in the limb configuration are represented from left to right as follows: the joints connected to the fixed platform, intermediate joints, and the joints connected to the moving platform.

To simplify the limb configurations, there are three equivalence replacement rules as follows:

(1) When three revolute joints (R-joints) are mutually perpendicular, the three R-joints can be combined into a spherical joint (S-joint), as shown in Fig. 4(a).

Figure 4. Equivalent replacement rules.

(2) When two revolute joints are mutually perpendicular, the two R-joints can be combined into a universal joint (U-joint), as shown in Fig. 4(b).

(3) When a prismatic joint (P-joint) is parallel to a revolute joint, the P-joint and the R-joint can be combined into a cylindrical joint (C-joint), as shown in Fig. 4(c).

The first equivalence replacement rule has been applied in Table III to simplify the three mutually perpendicular revolute joints (R-joints) into a spherical joint (S-joint). In addition, rules two and three can further simplify the configurations of kinematic pairs that satisfy the conditions in Table III. The specific cases are as follows:

(1) Limb with zero prismatic joints:

Apart from the S-joint, the limb still has three R-joints. When two R-joints are mutually perpendicular, the limb configuration can be simplified into a combination of S, U, and R joints.

(2) Limb with one prismatic joint:

Apart from the S-joint, the limb has two R-joints and one P-joint. When two R-joints are mutually perpendicular, the limb can be simplified into a combination of S, U, and P joints. When the R-joint is parallel to one of the R-joints, the limb is simplified into a combination of S, C, and R joints.

(3) Limb with two prismatic joints:

Apart from the S-joint, the limb has one R-joint and two P-joints. When the R-joint is parallel to one of the P-joints, the limb can be simplified into a combination of S, C, and P joints.

The simplified configurations of limbs are shown in Table IV.

Table IV. Simplified configurations of limbs.

(ii) Configuration selection

As shown in Table IV, 30 types of 6-DOF limb configurations are generated in this paper. The characteristics of different configuration limbs are distinct [Reference Liu, Fan and Ding36–Reference Demjen, Lovasz, Ceccarelli, Sticlaru, Lupuţi, Oarcea and Silaghi-Perju38], and scholars have adopted various methods to select a reasonable limb configuration. In this paper, the selection is based on the functional requirements for PAMs. Since the PAM should weigh 10 tons, the limb configuration is selected based on the mechanism’s load-bearing capacity.

A mathematical model between the non-RPM’s topological structure and structural parameters is established in the paper, which can be denoted as:

(1) \begin{equation} \left\{\begin{array}{l} F=\sum\limits_{i=1}^{n}q_{i}\\[5pt] LN=m+n\\[5pt] n=F-\sum\limits_{i=1}^{n}\left(q_{i}-1\right)\\[5pt] q_{i}\leq F\left(i=1,2,\cdots,n\right)\\[5pt] n\leq F \end{array}\right. \end{equation}

where F is the DOF of PAM, LN is the number of limbs, m is the number of passive limbs, n is the number of active limbs, and q _i is the DN on the limb i.

In Fig. 2, the moving platform of the PAM is connected to limbs through joints. Geometrically, at least three non-coplanar points can determine a plane. Therefore, the moving platform needs at least three limbs for support. Maintaining structural symmetry can satisfy isotropy requirements according to the topological principles of limbs. Therefore, the topological structure of all limbs should be the same to ensure the structural symmetry of PAM. Consequently, using five limbs in a 6-DOF PAM is not advisable. Moreover, the greater the number of limbs (LN) for a determined mechanism size, the smaller the operational space of the PAM, and the more challenging the structural design. In summary, the LN should be considered as one of the selection criteria. This paper will compare the load-bearing capacity of PAMs with three, four, and six limbs.

In mechanical transmission systems, the more kinematic pairs that are involved, the lower the mechanical efficiency due to factors such as friction. Therefore, the number of kinematic pairs (KSN) should also be considered as one of the selection criteria.

In addition, prismatic and revolute joints are commonly used kinematic pairs for actuation. Compared to the revolute joint, the prismatic joint has a larger load-bearing surface, making it more suitable for bearing heavy loads. Therefore, the PN also needs to be considered as a selection criterion.

Figure 5. Configuration selection.

As a result, the LN and the sum of the KSN, as well as the PN, are adopted as the criteria for selecting configuration in this paper, as shown in Fig. 5(a)(b). In Fig. 5(a)(b), it can be seen that the more LN there is, the larger both PN and KSN are. The expression for each criterion is as follows:

(2) \begin{equation} \begin{array}{l} LN=\textrm{the number of the limbs}\\[5pt] KSN=i\times LN\\[5pt] PN=\sum\limits_{j=\mathrm{1}}^{LN}\mathrm{P}_{j} \end{array} \end{equation}

where

\begin{equation*} i=\left\{\begin{array}{ll} 3 & \textrm{I},\textrm{II},\textrm{III}.1\\[5pt] 4 & \textrm{III}.2,\textrm{IV} \end{array}\right.,\, j=3,4,6 \end{equation*}

However, LN, PN, and KSN are not as good as more for the load-bearing capability of PAM. The configuration of the PAM cannot be precisely selected based on the three criteria mentioned above alone. Therefore, the load-bearing capability is defined as:

(3) \begin{equation} LC=\frac{PN}{LN\times KSN} \end{equation}

The comprehensive load-bearing capability is defined as:

(4) \begin{equation} CLC=k\times \frac{PN}{LN\times KSN} \end{equation}

where

\begin{equation*} k=\left\{\begin{array}{l} 0.7\mathrm{3}-\text{limb PAM}\\[5pt] 1\mathrm{4}-\text{limb PAM}\\[5pt] 1.3\mathrm{6}-\text{limb PAM} \end{array}\right. \end{equation*}

The coefficient k is determined by factors such as load, motor power, motor volume, overall volume of the mechanism, and so on.

The load-bearing capability and comprehensive load-bearing capability of the limb are shown in Fig. 5(c)(d). In Fig. 5(c), it can be seen that among the limbs with the same LN, those with three prismatic joints, such as SPPP, PSPP, PPSP, and PPPS, have better load-bearing capacity. In Fig. 5(d), it can be seen that among PAMs with the same limb configuration, those with four limbs have stronger comprehensive load-bearing capacity. Therefore, for the functional requirements of this paper, suitable decentralized parallel limb configurations are 4-SPPP, 4-PSPP, 4-PPSP, and 4-PPPS.

When the 6-DOF PAM is subjected to a load, the limbs will carry a large additional moment in the posture adjustment process, which may impact structural safety and reliability. From the mechanical performance perspective, passive kinematic pairs like spherical joints (S) should be connected to the fixed platform or the moving platform of the PAM to free one end of the limb, preventing it from bearing a substantial bending moment. In conclusion, the 4-SPPP is chosen as the subject of this paper.

Figure 6. Schematic diagram of force increase in horizontal direction.

(iii) The connection sequence and layout of kinematic pairs

In the layout of horizontal prismatic joints, this paper sets the direction of the second prismatic joint P₂ and the third prismatic joint P₃ of each limb perpendicular to each other and parallel to the upper platform, as shown in Fig. 6(a). They also form a 45° angle with both the first principal motion direction (such as the x-axis) and the second principal motion direction (such as the y-axis), defined as $\varphi _{1},\varphi _{2}$ , as shown in Fig. 6(b). When P₂ and P₃ serve as active joints, according to the principle of force composition, the horizontal driving force can be denoted as:

(5) \begin{equation} F_{x}=F_{y}=\sqrt{F_{\mathrm{P}_{2}}^{2}+F_{\mathrm{P}3}^{2}} \end{equation}

When $F_{{\mathrm{P}_{2}}}=F_{{\mathrm{P}_{3}}}$ , there is $F_{x}=F_{y}=\sqrt{2}F_{\mathrm{P}2}=\sqrt{2}F_{\mathrm{P}3}$ .

When $F_{{P_{2}}}\neq F_{{P_{3}}}$ , there is $F_{x}\geq F_{\mathrm{P}2},F_{x}\geq F_{\mathrm{P}3},F_{y}\geq F_{\mathrm{P}2},F_{y}\geq F_{\mathrm{P}3}$ .

From Fig. 6(b) and Eq. (6), it can be seen that this layout of horizontal prismatic joints can increase the horizontal driving force without increasing the individual driving force in the horizontal direction.

Let the angle between the first prismatic joint P₁ of each limb in the initial posture and the footpad be $\beta$ ( $\beta$ < 90°), as shown in Fig. 6(c), where θ denotes the friction angle. In this paper, if $\beta =85^{\circ}$ , then $\gamma =5^{\circ}$ . According to reference [Reference Han, He and Zheng39], it is known that $\theta \approx 8-10^{\circ}$ when the footpad is made of rubber. Therefore, $\gamma \lt \theta$ , satisfying the self-locking condition where no matter how heavy the load is, there will be no sliding on the floor.

According to Fig. 6(c), when the first prismatic joint P₁ serves as an active joint, it not only provides a vertical driving force but also a horizontal driving force, which can be denoted as:

(6) \begin{equation} F_{xoy}=F_{{P_{1}}}\cos \beta \end{equation}

In conclusion, in order to provide the greater driving force in the horizontal principal motion direction under limited actuation, the DOF space and kinematic pairs layout of the decentralized limb of PAM as shown in Fig. 7, where $\alpha =95\mathbf{^{\circ}}$ . The structure of the DLPM is shown in Fig. 8.

Figure 7. Decentralized limb DOF space and structure.

Figure 8. The structure of DLPM.

3.1.2. Number and position of driving pairs

If there are passive kinematic pairs of the limbs, the ends of decentralized limbs will exhibit random motion in the macro-motion stage. Consequently, the footpad’s position would not be the same during each landing, leading to the non-uniqueness of the upper platform posture during mode transitions. For the DLPM, it is necessary to ensure that the translational DOF is zero in the macro-motion stage. Then, the DN should satisfy the following condition:

(7) \begin{equation} DN=PN\times LN \end{equation}

The configuration of the DLPM in this paper is 4-SPPP, with three prismatic joints per limb and four limbs. Therefore, the required DN is 12. However, the more driving pairs there are, the greater the difficulty in control [Reference Kakehashi, Okada and Inaba40], so the driving approach is unsuitable.

3.2.1. Number and position of driving pairs

The 4-SPPP mechanism has 6-DOF, so theoretically, only six inputs are needed. In this paper, the selection rules for the DN are as follows to meet the functional requirements:

(1) Redundant drives can be used to avoid singular configurations.

(2) To achieve structural symmetry in design, the DN for four-limb PAM should be a multiple of four, that is $DN=4i(i=1,2,\cdots )$ and $DN\leq PN\times LN$ .

Therefore, this paper selects eight driving pairs for the next design. In order to ensure load-bearing capacity in the vertical direction, driving motors are assigned to the first prismatic joints of all four limbs. Simultaneously, to avoid generating additional torques, the arrangement of the four horizontal drives should be in the same plane. From Fig. 6, it can be seen that the second prismatic joint and the third prismatic joint of each limb have the same effect as the active drive. Therefore, the second prismatic joints of the four limbs are selected as the driving pair in this paper.

3.2.2. Reconfigurable mechanism design

The type synthesis of a reconfigurable mechanism is the reverse process of state change. In the transformation process from the initial state (macro-motion) to the final state (micro-motion), the number of effective links in a reconfigurable mechanism generally decreases, leading to changes in the DOF. The type synthesis of a reconfigurable mechanism is based on the topological structure of the final state (micro-motion), that is, the DLPM designed above. By adding series-coupled kinematic pairs, the translational DOFs of the moving platform are further constrained to finally obtain the initial state (macro-motion) configuration.

In this paper, a coupled mechanism model with multi-loop consists of serial coupling sub-chains is formed by adding series coupling sub-chains between decentralized limbs as shown in Fig. 9. Figure 9(a) illustrates the topology of the PM with coupling sub-chains between limbs, and Fig. 9(b) shows the planar topological structure with coupling sub-chains. In macro-motion mode, any footpad, such as link i, link j, or link k, can serve as an independent moving platform.

Figure 9. Coupled mechanism model with multi-loop consists of serial coupling sub-chains.

To ensure that all four footpads can have the same DOF when functioning as the moving platform, the coupled mechanism should possess the following characteristics:

(1) $\mathrm{Limb}\;2+\mathrm{Coupling\;sub}-\mathrm{chain\;5}=\mathrm{Limb\;}3+\mathrm{Coupling\;sub}-\mathrm{chain\;}7$ , from $\mathrm{Limb\;}2=\mathrm{Limb\;3}$ , it follows that $\mathrm{Coupling\;sub}-\mathrm{chain\;5}=\mathrm{Coupling\;sub}-\mathrm{chain\;}7$ .

(2) Coupling sub-chain is equal to shared sub-chain, that is, $\mathrm{Coupling\;sub}-\mathrm{chain\;6}=\mathrm{Coupling\;sub}-\mathrm{chain\;}8$ .

From this, it can be inferred that the dual-mode PAM with coupling sub-chains between limbs is a symmetrical coupled mechanism. The flowchart of the configuration synthesis process is shown in Fig. 10.

Figure 10. The flowchart of configuration synthesis process of symmetrical coupled mechanism.

The detailed steps for configuration synthesis are as follows:

Step 1: Select a PM with the desired final state DOF (a-DOF). Analyze the motion and constraint properties of the PM based on screw theory.

Step 2: Analyze the twist screw system $\boldsymbol{\$ }_{\!\mathrm{PM}i}$ of the mechanism with the desired initial state DOF (b-DOF). Using reciprocal screw theory, the constraint wrench screw system $\boldsymbol{\$ }_{\!\mathrm{PM}}^{r}$ or the basic constraint wrench screw system $\boldsymbol{\$ }_{\!\mathrm{PM}i}^{r}$ of the initial state mechanism can be obtained.

Step 3: Construct a coupled mechanism model with multi-loop consists of serial coupling sub-chains. Analyze the motion properties $\boldsymbol{\$ }_{\!\mathrm{L}i}$ of each limb and then add the appropriate serial coupling sub-chain (coupling sub-chain i) between Limb m and Limb n.

Step 4: Decouple the coupled mechanism model and establish a motion-splitting model using the motion partitioning method. Divide the basic model into 2LN-1 individual limbs, which are denoted by Limb Ci.

Step 5: Under the initial configuration, take Limb Ci as the research object. Based on screw theory, analyze the motion and constraint properties of Limb Ci. Establish appropriate coordinate systems, and the twist screw system $\boldsymbol{\$ }_{\!\mathrm{CL}i}$ of Limb Ci can be determined. Using reciprocal screw theory, the constraint wrench screw system $\boldsymbol{\$ }_{\!\mathrm{CL}i}^{r}$ of Limb Ci can be obtained.

Step 6: Confirming the correctness of the PM structure with coupled limbs can be done through two methods:

a. Check if the union of constraint wrench screw systems $\boldsymbol{\$ }_{\!\mathrm{CL}}^{r}$ of coupled limbs is equal to the expected constraint wrench screw system $\boldsymbol{\$ }_{\!\mathrm{PM}}^{r}$ , that is, $\boldsymbol{\$ }_{\!\mathrm{CL}}^{r}=\boldsymbol{\$ }_{\!\mathrm{PM}}^{r}$ .

b. Check if the union of constraint wrench screw systems $\boldsymbol{\$ }_{\!\mathrm{CL}}^{r}$ of coupled limbs can be linearly combined from the expected basic constraint wrench screw system $\boldsymbol{\$ }_{\!\mathrm{PM}i}^{r}$ , that is, $\boldsymbol{\$ }_{\!\mathrm{CL}}^{r}=g_{1}\boldsymbol{\$ }_{\!\mathrm{PM1}}^{r}+g_{2}\boldsymbol{\$ }_{\!\mathrm{PM2}}^{r}+\cdots +g_{k}\boldsymbol{\$ }_{\!\mathrm{PM}k}^{r}=\sum\limits_{k=1}^{6}g_{k}\boldsymbol{\$ }_{\!\mathrm{PM}k}^{r}$ , where $g_{k}$ is the linear combination coefficient, when $\boldsymbol{\$ }_{\!\mathrm{PM}k}^{r}=0$ , it follows that $g_{k}=0$ .

The analysis process of this article is as follows:

Step 1: The 4-SPPP 3T3R symmetrical PAM with 6-DOF based on the decentralized limb configuration mentioned above is selected to study in this paper. The motion and constraint properties of the PAM are analyzed based on screw theory.

Establish the coordinate system O _xyz ; the coordinate origin O is the connection point of the four S-joint centers. The x-axis points from S-joint 4 to S-joint 1, the y-axis points from S-joint 4 to S-joint 3, and the z-axis is determined by the right-hand rule, as shown in Fig. 11. In the final state, the moving platform is the upper platform. Figure 11(a) shows the constraint wrench analysis of the PM with decentralized limbs. Based on the number and positions of the driving pairs determined in the previous section, lock the active driving pairs, and the constraint wrench analysis is shown in Fig. 11(b).

Figure 11. The constraint wrench analysis of decentralized limbs PAM.

Step 2: This paper focuses on ground adaptability in micro-motion mode, with an expected DOF for the footpad being 2R or 1R. Taking 1R as an example, analyze the twist screw system $\boldsymbol{\$ }_{\!\mathrm{PM}i}$ of the mechanism with 1R DOF. Using reciprocal screw theory, the basic constraint wrench screw system $\boldsymbol{\$ }_{\!\mathrm{PM}i}^{r}$ of the initial state mechanism can be obtained.

To ensure that the limb landing points remain at the same position during mode transition, the target basic constraint wrench screw system at the center S _i of the spherical joint should be as follows:

(8) \begin{equation} \begin{array}{l} \boldsymbol{\$ }_{\!\mathrm{s}i1}^{r}=\left(1,0,0;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{s}i2}^{r}=\left(0,1,0;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{s}i3}^{r}=\left(0,0,1;\;0,0,0\right) \end{array} \end{equation}

According to the reciprocity of the screw, the basic twist screw system at point S _i can be obtained as:

(9) \begin{equation} \begin{array}{l} \boldsymbol{\$ }_{\!\mathrm{s}i1}=\left(1,0,0;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{s}i2}=\left(0,1,0;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{s}i3}=\left(0,0,1;\;0,0,0\right) \end{array} \end{equation}

where $\boldsymbol{\$ }_{\!\mathrm{s}i3}$ does not affect the ground adaptation requirements of the mechanism and can be considered a local DOF. Additionally, $\boldsymbol{\$ }_{\!\mathrm{s}i3}$ would lead to the generation of movement within the xy plane for the DLPM, which means that it cannot constrain the motion of the passive pairs in macro-motion stage. Therefore, it should be discarded.

In the initial state, the moving platform contacts the ground through various footpads. Therefore, the footpads possess a basic twist screw system with 1R DOF, which is given by:

(10) \begin{equation} \boldsymbol{\$ }_{\!\mathrm{PM}1}=\left(1,0,0;\;0,0,0\right)\text{or }\boldsymbol{\$ }_{\!\mathrm{PM2}}=\left(0,1,0;\;0,0,0\right) \end{equation}

Then, the basic constraint wrench screw system of the footpads is:

(11) \begin{equation} \begin{array}{ll} \boldsymbol{\$ }_{\!\mathrm{PM}1}^{r}=\left(1,0,0;\;0,0,0\right) & \boldsymbol{\$ }_{\!\mathrm{PM}1}^{r}=\left(1,0,0;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{PM}2}^{r}=\left(0,1,0;\;0,0,0\right) & \boldsymbol{\$ }_{\!\mathrm{PM}2}^{r}=\left(0,1,0;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{PM}3}^{r}=\left(0,0,1;\;0,0,0\right)\mathrm{or} & \boldsymbol{\$ }_{\!\mathrm{PM}3}^{r}=\left(0,0,1;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{PM}4}^{r}=\left(0,0,0;\;0,1,0\right) & \boldsymbol{\$ }_{\!\mathrm{PM}4}^{r}=\left(0,0,0;\;1,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{PM}5}^{r}=\left(0,0,0;\;0,0,1\right) & \boldsymbol{\$ }_{\!\mathrm{PM}5}^{r}=\left(0,0,0;\;0,0,1\right) \end{array} \end{equation}

Step 3: Construct a coupled mechanism model with primary nodes, analyze the twist screw system $\boldsymbol{\$ }_{\!\mathrm{L}i}$ of each limb, and then add serial coupling sub-chain (Coupling sub-chain 5) between the decentralized limbs Limb 1 and Limb 2, similarly, add Coupling sub-chain 6 between Limb 2 and Limb 3, Coupling sub-chain 7 between Limb 3 and Limb 4, and Coupling sub-chain 8 between Limb 4 and Limb 1, as shown in Fig. 9.

This paper considers the serial coupling sub-chain without drivers and forced to undergo determined motion as passive constraint limbs. Their primary function is to constrain one or several DOFs of the moving platform to achieve the expected motion. In addition, constraint limbs can also improve the stiffness of the mechanism.

The constraint wrench screw systems of limbs are the same as the constraint wrench screw systems of mechanism. Therefore, the standard basis for the initial state limb twist screw system can be obtained as follows:

(12) \begin{equation} \begin{array}{l} \boldsymbol{\$ }_{\!\mathrm{L}1}=\boldsymbol{\$ }_{\!\mathrm{PM}1}=\left(1,0,0;\;0,0,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{L}2}=\boldsymbol{\$ }_{\!\mathrm{PM2}}=\left(0,1,0;\;0,0,0\right) \end{array} \end{equation}

where $\boldsymbol{\$ }_{\!i1}$ and $\boldsymbol{\$ }_{\!i2}$ are the mechanism constraint wrench line vectors along the x and y directions, respectively, intersecting at the origin of the reference coordinate system, can be generated in various forms of limb motion limbs through linear combinations and different arrangements of two twist screw systems.

The general expression of a kinematic pair is:

(13) \begin{equation} \boldsymbol{\$ }_{\!\mathrm{L}i}=\boldsymbol{a}\boldsymbol{\$ }_{\!\mathrm{L}1}+b\boldsymbol{\$ }_{\!\mathrm{L}2}=\left(a,b,0;\;0,0,0\right) \end{equation}

where a and b are not both 0, they can form a 2R limb that rotates in the x and y directions.

When the kinematic pair refers to a rotating joint, it is evident that $a^{2}+b^{2}=1$ .

According to the limb comprehensive method, the direction of the rotational axis of the revolute joint can be arbitrary. The rotational axis of the revolute joint is perpendicular to the constraint force. By analyzing the standard basic twist screw system, it can be inferred that the rotation axis for the revolute joint can be in any direction within the xy plane. Therefore, it can be inferred that there will be R _x , R _y , and U _xy in the limb. In this paper, R _x and R _y denote a revolute joint with the x-axis and y-axis as the rotational axis, respectively, and U _xy denotes joints with rotational axes in the x and y directions. As a result, the arrangement and combination of kinematic pairs in serial constraint limbs are presented in Table V.

Table V. Hinge arrangement and combination of kinematic pairs in serial constraint limbs.

Note^a: RR^* is equivalent to R. When the four coupling sub-chains are not completely identical, in order to satisfy the symmetry of the mechanism, both R and RR^* are represented as RR^*.

Note^b: RR^ is equivalent to U. When the four coupling sub-chains are not completely identical, for simplicity of expression, both RR^ and U are represented as U.

Due to the functional requirements of this paper, it is necessary for the four coupling sub-chains to have opposite sides pairwise identical. Therefore, the available combinations for serial coupling sub-chain in this paper include 4R, 4RR*, 4RR^, 4U, 4RU, 4UU, 2(RR*-RU), 2(RR*-U), 2(RR*-UU), 2(U-RU), 2(U-UU), and 2(RU-UU).

Step 4: According to Fig. 12(a), the coupled mechanism model is decoupled into seven individual limbs, as shown in Fig. 12(b).

Figure 12. Motion distribution model of coupled mechanism.

Among them, Limb 1 is one limb of the original DLPM, Limb 2→Limb 5, Limb 3→Limb 6→Limb 5, Limb 4→Limb 7→Limb 6→Limb 5, Limb 2→Limb 6→Limb 7→Limb 8, Limb 3→Limb 7→Limb 8, and Limb 4→Limb 8 are six coupled limbs containing serial coupling sub-chains. The decoupled limbs are denoted as Limb Ci (i = 1–7), from left to right, labeled as Limb C1 to Limb C7.

Step 5: In the initial configuration, taking the individual limb (Limb Ci) containing serial coupling sub-chain as the research object, based on screw theory, analyze the motion and constraint of the decoupled coupled limbs in the coupled mechanism. Utilizing the coordinate system construction method shown in Fig. 11(b), the twist screw system $\boldsymbol{\$ }_{\!\mathrm{CL}i}$ of Limb Ci is determined. Using reciprocal screw theory, the constraint wrench screw system $\boldsymbol{\$ }_{\!\mathrm{CL}i}^{r}$ of Limb Ci can be obtained.

Taking 2(RR*-UU) as an example for illustration in this paper. Let $\mathrm{Limb\;}5=\mathrm{Limb\;}7=\mathrm{UU}$ and $\mathrm{Limb\;}6=\mathrm{Limb\;}8=\mathrm{RR}*$ . In the coupling sub-chain, the two rotation axes of RR* are coaxial, which can be integrated into a single R. Taking Limb C3 as an example, composed of Limb 3, Limb 6, and Limb 5, where the twist screw system of Limb 3 is:

(14) \begin{equation} \begin{array}{l} \boldsymbol{\$ }_{\!\mathrm{L3}1}=\left(0,0,0;\;-d,e,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{L3}2}=\left(1,0,0;\;0,0,-b\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{L3}3}=\left(0,1,0;\;0,0,-a\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{L3}4}=\left(0,0,1;\;b,a,0\right) \end{array} \end{equation}

The twist screw system of Limb 6 is:

(15) \begin{equation} \boldsymbol{\$ }_{\!\mathrm{L6}}=\left(1,0,0;\;0,l_{1},-n_{1}\right) \end{equation}

The twist screw system of Limb 5 is:

(16) \begin{equation} \begin{array}{l} \boldsymbol{\$ }_{\!\mathrm{L51}}=\left(1,0,0;\;0,l_{2},-n_{2}\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{L52}}=\left(1,0,0;\;0,l_{2},n_{2}\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{L53}}=\left(0,1,0;\;-l_{2},0,m_{2}\right) \end{array} \end{equation}

Then, the twist screw system of Limb C3 is:

(17) \begin{equation} \begin{array}{l} \boldsymbol{\$ }_{\!\mathrm{CL3}1}=\left(0,0,0;\;-d,e,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL3}2}=\left(1,0,0;\;0,0,-b\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL3}3}=\left(0,1,0;\;0,0,-a\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL3}4}=\left(0,0,1;\;b,a,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL3}5}=\left(1,0,0;\;0,l_{1},-n_{1}\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL3}6}=\left(1,0,0;\;0,l_{2},-n_{2}\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL3}7}=\left(1,0,0;\;0,l_{2},n_{2}\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL3}8}=\left(0,1,0;\;-l_{2},0,m_{2}\right) \end{array} \end{equation}

Taking the inverse screw, the constraint wrench screw system of Limb C3 can be derived as:

(18) \begin{equation} \begin{array}{l@{\quad}l} \boldsymbol{\$ }_{\!\mathrm{CL3}1}^{r}=\left(0,0,1;\;b,a,0\right) & b=n,a=m\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL32}}^{r}=\left(e,d,0;\;-dl_{1},el_{2},be+ad\right) & l_{1}=l_{2} \end{array} \end{equation}

The detailed analysis of the twist/constraint wrench screw systems of the decoupled limbs of the PM with serial coupling sub-chains can be found in Table VII. The union of the constraint wrench screw system of each decoupled limb is:

(19) \begin{equation} \begin{array}{l} \boldsymbol{\$ }_{\!\mathrm{CL}1}^{r}=\left(0,0,1;\;-b,-a,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL}2}^{r}=\left(e,d,0;\;0,0,-\left(be+ad\right)\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL}3}^{r}=\left(0,0,1;\;b,-a,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL4}}^{r}=\left(e,-d,0;\;dl_{1},el_{2},-\left(be+ad\right)\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL5}}^{r}=\left(0,0,1;\;b,a,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL6}}^{r}=\left(e,d,0;\;-dl_{1},el_{2},-\left(be+ad\right)\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL7}}^{r}=\left(0,0,1;\;-b,a,0\right)\\[5pt] \boldsymbol{\$ }_{\!\mathrm{CL8}}^{r}=\left(-e,d,0;\;-dl_{1},-el_{2},-\left(be+ad\right)\right) \end{array} \end{equation}

Step 6: Through analysis, it can be determined that the union of the constraint wrench screw systems of the coupled limbs can be linearly combined to form the desired basic constraint wrench screw system. Therefore, the correctness of the PM structure with coupled limbs can be determined.

The aforementioned union of constraint wrench screw systems for the coupled limbs is indeed the constraint wrench screw system for the moving platform. Thus, it follows that the twist screw system of the footpad is:

(20) \begin{equation} \boldsymbol{\$ }_{1}=\left(1,0,0;\;0,0,b\right) \end{equation}

In this paper, after introducing the serial coupling sub-chains 2(RR*-UU) between the decentralized limbs, the union of constraint wrench screw systems for the coupled limbs equals the desired constraint wrench screw system. The results show that the moving platform (footpad) has only 1R DOF, and the rotation axis direction for the footpad is determined by the R rotation axis direction. At this point, the translational freedom of the passive pairs can be locked, satisfying the requirement for consistency in the position of the limb ends during mode transition. Similarly, it can be determined that RxUxy-RyUxy-UxyRx-UxyRy, RxRx-RyUxy-RxRx-UxyRy/RyRy-RxUxy-RyRy-UxyRx and RxRx-Uxy-RxRx-Uxy/RyRy-Uxy-RyRy-Uxy all satisfy the requirement above.

In summary, to ensure consistency in the position of the limb ends during mode transitions and meet ground adaptation, it is necessary to ensure that the moving platform has 1R or 2R DOF in the macro-motion stage. To achieve this, coupling sub-chains are designed between the limb ends based on DLPM, utilizing multiple coupling kinematic pairs to constrain the translational motion of DLPM. Thus, to satisfy the symmetry requirements in the structure, the four decentralized limbs of DLPM are connected with two sets of RR*-UU in this paper. The DOFs of the mechanism in the macro-motion stage can be satisfied when the common axis of UU and the common axis of RR are perpendicular to each other. Ultimately, the configuration of the PAM is determined as 4-SPPP-2(RR*-UU), and the limb DOF space and structural layout are shown in Fig. 13.

Figure 13. Coupled limb DOF space and structure.

4.2.1. Trajectory planning

In the layout of horizontal prismatic joints, this paper sets the direction of the second prismatic joint P₂. It is assumed that the PAM needs to reach the maximum range of the posture adjustment within 5s, that is, moving 50 mm along the coordinate axis and rotating 5° around the coordinate axis. In order to reduce the impact and inertia force during the movement of the PAM, it is hoped that the movement of the posture process is smooth under the ideal state. Let the displacement, velocity, and acceleration of the moving and rotating of the RrPAP satisfy the following equations:

(21) \begin{equation} \begin{array}{l} \left\{\begin{array}{l} s_{t=0}=0\\[5pt] s_{t=5}=50 \end{array}\right.\left\{\begin{array}{l} \dot{s}_{t=0}=0\\[5pt] \dot{s}_{t=5}=0 \end{array}\right.\,\left\{\begin{array}{l} \ddot{s}_{t=0}=0\\[5pt] \ddot{s}_{t=5}=0 \end{array}\right.\\ \\ \left\{\begin{array}{l} \theta _{t=0}=0^{\circ}\\[5pt] \theta _{t=5}=5^{\circ} \end{array}\right.\left\{\begin{array}{l} \dot{\theta }_{t=0}=0\\[5pt] \dot{\theta }_{t=5}=0 \end{array}\right. \;\left\{\begin{array}{l} \ddot{\theta }_{t=0}=0\\[5pt] \ddot{\theta }_{t=5}=0 \end{array}\right. \end{array} \end{equation}

Hence, the movement driver function can be denoted as:

(22) \begin{equation} \mathrm{Y}=50\!\left(\frac{t}{5}-\frac{1}{2\pi }\sin \frac{2\pi t}{5}\right)\!\left(0\leq t\leq 5\right) \end{equation}

And the rotation driver function can be denoted as:

(23) \begin{equation} \mathrm{Y}=5\!\left(\frac{t}{5}-\frac{1}{2\pi }\sin \frac{2\pi t}{5}\right)\!\left(0\leq t\leq 5\right) \end{equation}

The trajectory is plotted according to the Eqs. (22) and (23), as shown in Fig. 16(a).

Figure 16. Motion simulation.

4.2.2. Motion simulation

The DOF of PAM is 6, and three typical motion modes, 3T, 3R, and 3T3R, were selected for motion simulation in ADAMMS in this paper. The trajectory equation of the center reference point of the moving platform is determined according to the Eqs. (22) and (23). The main structural parameters of the PAM are shown in Fig. 9, and their values are shown in Table VI.

Figure 16(b)–(d) shows the simulation curves of the relative displacement of the active driving pairs of each limb obtained by ADAMS simulation of the mechanism model in three typical motion modes, respectively. Figure 16(e)–(f) shows the simulation curves of the velocity and acceleration of the active driving pairs of each limb obtained by ADAMS simulation of the mechanism model with 3R motion modes.

Table VI. The main structural parameters of the PAM mechanism.

Figure 17. The workspace of PAM.