2.1. Introduction of screw theory

In screw theory, a unit screw is defined as

(1) \begin{align}\$ = ({\boldsymbol{s}};\,{{\boldsymbol{s}}_{{0}}}) = ({\boldsymbol{s}};\,{\boldsymbol{r}} \times {\boldsymbol{s}} + h{\boldsymbol{s}}) = \left( {l,m,n;\,p,q,r} \right)\end{align}

where s denotes the unit vector along the axis of screw, r denotes the position vector of arbitrary point on the axis of screw, and h denotes the pitch of screw. If the real unit is orthogonal to the dual unit, namely, pitch h =0, the screw will degenerate to be a line vector. Meanwhile, if real unit is null vector, the screw will degenerate to be a couple. The product of angular velocity ω and $\$_{i}$ represents the instantaneous twist of the rigid body, and the product of scalar f and $\$_{j}^{r}$ represents the constraint wrench acting on the rigid body. Furthermore, a line vector can be utilized to denote a rotation or a constraint force, and a couple can be utilized to denote a translation or a constraint couple. The reciprocal product of two screws is defined as

(2) \begin{align}{\$ _i} \circ \$ _j^r = {{\boldsymbol{s}_i} \cdot {{\boldsymbol{s}}_{0j}} + {{\boldsymbol{s}}_j} \cdot {{\boldsymbol{s}}}_{0i}}\end{align}

which represents the instantaneous power generated by them. The reciprocal product of 0 means the constraint wrench screw does not work on the twist screw; in other words, the twist screw is the permitted motion when the constraint wrench screw is acting. In 1T2R lower mobility PMs researched in this study, permitted twist screws ${\$}_{i} (i = 1, 2, 3)$ are reciprocal to constraint wrench screws ${\$}_{j}^{r}(j = 1, 2, 3)$ . For the sake of the graphical representation, some symbols are utilized to denote different screws, as shown in Table I.

Table I.

Notation of different screws.

Figure 1.

PU virtual chain.

2.2. Constraint analysis of PU virtual-chain and constraint characteristics of PU-equivalent PMs

Based on virtual-chain approach [Reference Kong and Gosselin8], the kinematic characteristics of PU-equivalent PMs can be expressed by a PU virtual-chain, as shown in Fig. 1, where the position of I ₁-axis changes with the translation of the prismatic joint and I ₂-axis rotates around I ₁-axis. These two rotational axes are coplanar and intersect at a common point M under any configuration. For convenience, the coordinate of intersection point M is defined as (0, 0, z _m ), the rotational angle around I ₁-axis is set as α, so the twist system of the PU virtual-chain can be written as

(3) \begin{align}\left\{ \begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l}{{\rm{\$ }}_1} = (0 & 0 & 0 & ; & 0 & 0 & 1)\\[4pt]{{\rm{\$ }}_2} = (1 & 0 & 0 & ; & 0 & {{z_m}} & 0)\\[4pt]{{\rm{\$ }}_3} = (0 & {{\rm c}\alpha}& {{\rm s}\alpha} & ; & - {z_m}{{\rm c}\alpha} & 0 & 0)\end{array} \right.\end{align}

where c and s denote the sine and cosine functions, respectively. Using reciprocal screw theory [Reference Huang, Zhao and Zhao33], the wrench system of PU virtual-chain under general configuration with respect to the coordinate system O-xyz can be figured out as

(4) \begin{align}\left\{\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l}{\rm{\$ }}_1^{{r}} = (1 & 0 & 0 & ; & 0 & {{z_m}} & 0)\\[4pt]{\rm{\$ }}_2^{{r}} = (0 & 1 & 0 & ; & { - {z_m}} & 0 & 0)\\[4pt]{\rm{\$ }}_3^{{r}} = (0 & 0 & 0 & ; & 0& {{{\rm s}} \alpha } & { - {{\rm c}} \alpha } )\end{array} \right.\end{align}

where $\$_{1}^{r}$ and $\$_{2}^{r}$ denote constraint forces passing point M and parallel to x- and y-axes, respectively, $\$_{3}^{r}$ denotes a constraint couple perpendicular to I ₁- and I ₂-axis and related to the rotational angle α of I ₂-axis. General wrench system shown in Fig. 2(a) demonstrates the basic constraint situation of PU-equivalent PMs without redundant constraints, namely, their constraint characteristics under general configuration. For convenience, this general wrench system is abbreviated as GWS 11. Herein, the constraint plane is defined, which is perpendicular to the constraint couple and contains the intersection point of the constraint forces, including basic information of constraints in GWS 11. Obviously, the constraint plane is coincident with the plane determined by I ₁- and I ₂-axis, and it will be further utilized to illustrate the cause of one parasitic motion. According to screw theory [Reference Huang, Zhao and Zhao33], two parallel constraint forces constrain a translation in the same direction of them and a rotation whose axis is perpendicular to the plane determined by them, as shown in Fig. 2(b), which equals $\$_{1}^{r}$ and $\$_{3}^{r}$ of GWS 11. Thus, an equivalent wrench system of GWS 11 can be obtained as shown in Fig. 2(c), which is denoted byGWS 12, and the constraint plane is now coincident with the plane determined by these two parallel constraint forces.

Figure 2.

General wrench system of PU-equivalent PMs. (a) General wrench system 11 (GWS 11),(b) an equivalent constraint case, (c) general wrench system 12 (GWS 12).

2.3. Cause of parasitic motion and constraint characteristics of P^*U^*-equivalent PMs with one parasitic motion

To illustrate the occurrence of the parasitic motion in a clearer manner, the difference between moving platform and constraint plane is further analyzed by taking PU-equivalent PMs with GWS 11 as example. As shown in Fig. 3(a) and (b), the constraint plane and moving platform are coincident with each other when PU-equivalent PMs are under initial configuration or rotating around I ₁-axis. However, they will become different planes after PU-equivalent PMs rotating around I ₂-axis or I ₁- and I ₂-axis, as shown in Fig. 3(c) and (d).

Figure 3.

Difference between the constraint plane and moving platform. (a) Under initial configuration, (b) rotating around I ₁-axis, (c) rotating around I ₂-axis, (d) rotating around I ₁- and I ₂-axis.

The only intersection line of the constraint plane and moving platform is the I ₂-axis, that is to say, if the constraint force $\$_{2}^{r}$ does not intersect with the moving platform by I ₂-axis, it will change its position with respect to the constraint plane when the PU-equivalent PMs rotate around I ₁- and I ₂-axis, as shown in Fig. 4(a). Under this situation, one parasitic motion occurs along with the rotation of the moving platform, and the process of proof is as following.

Figure 4.

Cause of one parasitic motion and corresponding general wrench systems. (a) Position change of $\$_2^{r}$ , (b) GWS 21, (c) twist system of GWS 21, (d) GWS 22.

Wrench system of Fig. 4(a) is denoted by general wrench system 21 (GWS 21) in Fig. 4(b), where point C is the intersection point of $\$_{2}^{r}$ with the O-xz plane, and its coordinate is set to be (x _c , 0, z _c ). Based on screw theory, GWS 21 can be written as

(5) \begin{align}\left\{\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}c@{\quad}c@{\quad}c}\$ _1^r = ( 1 & 0 & 0 & ; & 0 & {{z_m}} & 0)\\[4pt]\$ _2^r = ( 0 & 1 & 0 & ; & { - {z_c}} & 0 & {{x_c}})\\[4pt]\$ _3^r = ( 0 & 0 & 0 & ; & 0 & {\sin \alpha } & { - \cos \alpha })\end{array} \right.\end{align}

The twist system of GWS 21 can be figured out as

(6) \begin{align}\left\{\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}c@{\quad}c@{\quad}c}{\$ _1} = ( 0 & 0 & 0 & ; & 0 & 0 & 1 )\\[4pt]{\$ _2} = ( 1 & 0 & 0 & ; & 0 & {{z_c}} & 0 )\\[4pt]{\$ _3} = ( 0 & {{{\rm c}} \alpha } & {{{\rm s}} \alpha } & ; & { - {z_m}{{\rm c}} \alpha } & { - {x_c}{{\rm s}} \alpha } & 0 )\end{array} \right.\end{align}

where $\$_1$ represents a translation along the z-axis, $\$_2$ denotes a rotation that is parallel to the x-axis and goes through point C, $\$_3$ is a twist that can be divided into a translation $\$_{31}$ and a rotation $\$_{32}$ by rule of algebraic sum of two screws [Reference Huang, Zhao and Zhao33], where

(7) \begin{align}\left\{ \begin{array}{l}{\$ _{31}} = \left( 0\quad0\quad0\quad;\quad0\quad{ - {x_c}{{\rm s}} \alpha }\quad0 \right)\\[5pt]{\$ _{32}} = \left( 0\quad{{\mathop{\rm c}} \alpha }\quad{{\mathop{\rm s}\nolimits} \alpha }\quad;\quad{ - {z_m}{\mathop{\rm c}\nolimits} \alpha }\quad0\quad0\right)\end{array} \right.\end{align}

As shown in Fig. 4(c), $\$_{31}$ represents a motion of one constrained DOF, namely, a translation parallel to the y-axis, while $\$_{32}$ represents a rotation whose axis goes through the center of the constraint plane, and the axial direction is related to the rotational angle of the constraint plane. Additionally, the rotational axis of $\$_2$ is instantaneous for the coordinate of intersection point C will change with the rotation of moving platform. Similarly, we could obtain an equivalent wrench system of GWS 21, which is denoted by GWS 22, as shown in Fig. 4(d).

According to screw theory [Reference Huang, Zhao and Zhao33], if the real unit of a constraint force contains the components on z-axis, the translation along z-axis will be prohibited. Thus, all constraint forces of P^*U^*-equivalent PMs with one parasitic motion should be parallel to the O-xy plane. However, if $\$_2^{r}$ of GWS 21 and 22 could change its position while remains to be parallel to the O-xy plane as shown in Fig. 5(a) and (b), one parasitic motion will occur as well, which can be proved as above and the generated parasitic motion will be different in kinematics. These two wrench systems are denoted by GWS 31 andGWS 32, respectively.

Figure 5.

Other general wrench systems of P^*U^*-equivalent PMs with one parasitic motion.(a) GWS 31, (b) GWS 32.

2.4. Constraint characteristics of P^*U^*-equivalent PMs with three parasitic motions

Constraint characteristics of P^*U^*-equivalent with three parasitic motions are more complicated for the complex spatial distributions of the two rotational axes after continuous rotation of moving platform. Thus, the research on general wrench systems of P^*U^*-equivalent PMs with three parasitic motions will be conducted based on the constraint analysis of existing PMs. According to the conclusion proposed by Li [Reference Li, Chen, Chen, Wu and Hu24], when three LPs (limb planes) of 3-PRS PM are arranged symmetrically and intersect at aline with noncollinear spherical centers, parasitic motions appear in three constrained DOFs. For more distinct expression of the properties of constraint forces, the 3-PRS PM is projected to the O-xy plane, as shown in Fig. 6. It is noticeable that three constraint forces are restricted to be parallel to J ₁-, J ₂- and J ₃-axis and go through the spherical centers B _i (i =1, 2, 3) for their geometric constraints.

Figure 6.

Projection view on O-xy plane of constraint forces for 3-PRS PM with three parasitic motions.

The connecting line of any two spherical centers of 3-PRS PM can be considered as a continuous rotational axis of the moving platform [Reference Huang, Tao and Fang21]. If the moving platform rotates around one of these three continuous rotational axes, the constraint wrench system and the twist system of 3-PRS PM are shown in Fig. 7(a), where point E is the intersection point of the constraint force $\$_1^{r}$ and $\$_2^{r}$ . Herein, $\$_1$ denotes a translation parallel to z-axis, $\$_2$ represents a rotation whose axis intersect the constraint force $\$_1^{r}$ and $\$_2^{r}$ and $\$_3$ stands for an instantaneous rotation whose axis intersect point E and the constraint force $\$_3^{r}$ . Obviously, there is no parasitic motion at this time.

Figure 7.

Constraint analysis and general wrench system of P^*U^*-equivalent PMs with three parasitic motions. (a) Rotating around one continuous axis, (b) rotating around two axes, (c) GWS 41.

However, when the moving platform rotates around two different axes, namely, the displacements of three actuated prismatic joints are different from each other, three constraint forces will not be on a common plane, as shown in Fig. 7(b), and three parasitic motions appear under this situation. Thus, we could draw the conclusion that the wrench system of P^*U^*-equivalent PMs with three parasitic motions consists of three linearly independent constraint forces. Furthermore, three of them are coplanar on the moving platform under initial configuration, two of them are coplanar when rotating around one continuous axis, and they are not on a common plane and do not intersect the same line under general configuration, as shown in Fig. 7(c). Such a wrench system is denoted by GWS 41.

3.1. Classification of branch chains and its rules of synthesis

Based on screw theory [Reference Huang, Zhao and Zhao33], the twist system of PM is an intersection set of twist system of each branch, while the wrench system of PM is a union set of wrench system of each branch. Thus, each branch chain of PU- or P^*U^*-equivalent PMs should have at least two rotational and one translational three DOFs and could provide at most two constraint forces and one constraint couple. Therefore, we could classify branches into six categories: 1T2R, 1T3R, 2T2R, 2T3R, 3T2R and 3T3R, which could generate two constraint forces and one constraint couple, two constraint forces, one constraint force and one constraint couple, one constraint force, one constraint couple and no constraint, respectively.

Analysis above has demonstrated all feasible basic wrench systems of PU- and P^*U^*-equivalent PMs, revealing their constraint characteristics, namely, distributions and properties of constraints. The reciprocal product of constraint wrench screw and twist screw can be further written as

(8) \begin{align}{{{\$}}_i} \circ {\$}_j^r = {l_i}{p_j} + {m_i}{q_j} + {n_i}{r_j} + {p_i}{l_j} + {q_i}{m_j} + {r_i}{n_j} {\rm{ }} = {\omega _i}{f_j}\left[ {({h_i} + {h_j})\cos {\alpha _{ij}} - {a_{ij}}\sin {\alpha _{ij}}} \right] \end{align}

where α _ij denotes the torsion angle of axes of two screws, and a _ij denotes the mutual moment of axes of two screws. According to Eq. (8), two line vectors (h = 0) are reciprocal only when they are parallel (sinα _ij = 0) or intersecting (a _ij = 0); two couples are certainly reciprocal; and a line vector and a couple are reciprocal only when they are perpendicular (cosα _ij = π/2). Based on this and constraint characteristics, rules of synthesis of structures of different kinds of branch chains are determined as following:

1. (a) If a branch chain generates a constraint force, all rotational axes of revolute joints in this branch chain should be parallel to or intersect with the constraint force, and axes of prismatic joints should be perpendicular to the constraint force. Additionally, the intersection point of rotational axes and constraint force should locate at the end of the branch chain;

2. (b) If a branch chain generates a constraint couple, all rotational axes of revolute joints in this branch chain should be perpendicular to the constraint couple, and there is no limit for the axes of prismatic joints;

3. (c) If a branch chain generates constraint force and couple at the same time, it should meet the above two conditions at the same time.

3.2. Feasible structures of different branch chains

3.2.1. Structures of 1T2R and 1T3R branch chain

Both 1T2R and 1T3R branch chain could generate two linearly independent constraint forces, and they should locate on the moving platform under initial configuration. According to rule (a), structure of 1T2R branch chain could only be $\mathrm{P}_z^x \mathrm{U}^{y}$ , and structure of 1T3R branch chain could only be P _z S, as shown in Fig. 8, where f _i denotes corresponding constraint forces and c _i denotes corresponding constraint couple. These two structures also restrict the rotational center of 1T2R PMs contained them, namely, center of universal or spherical joint. Thus, P^*U^*-equivalent PMs without determined rotational center could not contain 1T2R or 1T3R branch chain.

Figure 8.

Structures of 1T2R and 1T3R branch chain. (a) ${P}_z^x {U}^{y}$ , (b) P _z S.

3.2.2. Structures of 2T2R branch chain

2T2R branch chain could generate a constraint force and a constraint couple and contain two linearly independent revolute joints. According to rule (a), the plane determined by two translational DOFs must be perpendicular to the constraint force, and revolute joint with axis intersecting at the constraint force must locate at the end of the branch chain. According to rule (b), the axial direction of the last revolute joints could and only could change once. Hence, the limb plane (LP) of 2T2R branch chain is fixed and perpendicular to O-xy plane, and there only exists one revolute joint which is not parallel to the constraint force. As shown in Fig. 9, feasible structures of 2T2R branch chain for PU- and P^*U^*-equivalent PMs are P _z R _x ^x U ^y , R _x P ^x U ^y , R _x R _x ^x U ^y , P _z R _x C _y , R _x PC _y and R _x R _x C _y , where cylindrical joint can be considered to be the combination of a prismatic joint and a revolute joint with coincident axis. Note that structures with other prismatic joints are out of consideration herein.

Figure 9.

Feasible structures of 2T2R branch chain. (a) P _z $R_x^x$ U ^y , (b) R _x P ^x U ^y , (c) R _x $R_x^x$ U ^y , (d) P _z R _x C _y , (e) R _x PC _y , (f) R _x R _x C _y .

3.2.3. Structures of 2T3R branch chain

2T3R branch chain could only generate a constraint force, and its feasible structures can be obtained by adding another linearly independent revolute joint to 2R2T branch chains shown in Fig. 9. Based on Section 2, they can be divided into two categories: one generating direction-fixed constraint force, the other generating direction-varying constraint force, and both of them should be parallel to the O-xy plane. For the former category, the linearly independent revolute joint should be added at the end of feasible 2T2R branch chains, and thus we can obtain feasible structures of 2T3R branch chain: P _z $\mathrm{R}_x^x\mathrm{U}^y\mathrm{R}_z$ , R _x P ^x U ^y R _z , R _x $\mathrm{R}_x^x\,\mathrm{U}^y\,\mathrm{R}_z$ , P _z R _x C _y R _z , R _x PC _y R _z , R _x R _x C _y R _z . Figure 10 shows P _z $\mathrm{R}_x^x\mathrm{U}^y\mathrm{R}_z$ and P _z R _x C _y R _z branch chain as examples, and the generating constraint force must go through the intersection point of last two rotational axes. Furthermore, P _z $\mathrm{R}_x^x\mathrm{U}^y\mathrm{R}_z$ , R _x P ^x U ^y R _z and R _x $\mathrm{R}_x^x\mathrm{U}^y\mathrm{R}_z$ branch chain will translate into P _z R _x S, R _z PS and R _x R _x S branch chain when d (distance of center of universal joint and revolute joint, as shown in Fig.10(a)) equals 0.

Figure 10.

Two feasible structures of 2T3R branch chain. (a) P _z ${R}_x^x{U}^y{R}_z$ , (b) P _z R _x C _y R _z .

Obviously, if the above six 2T3R branch chains are utilized to provide $\$_2^{r}$ of GWS 11 and 12, centers of last revolute joints of branch chains generating two basic constraint forces must be collinear, and three branch chains should satisfy “T” shape layout, which may lower performance of corresponding PMs [Reference Sun and Huo25]. Hence, it is necessary to develop 2T3R branch chain whose constraint force are with distal acting point. Meanwhile, the acting point is determined by the intersection point of other two linearly independent rotational axes, and thus expected branch chain: P _z R _x ( ^x UR _y )_N, R _x P( ^x UR _y )_N, R _x R _x ( ^x UR _y )_N can be obtained by keeping these two revolute joints are angled in structure, as shown in Fig. 11(a). Additionally, the last revolute joint shown in Fig. 10(a) can be replaced by an arc slider whose center line is collinear with corresponding rotational axis, as shown in Fig. 11(b), and we can obtain P _z R _x ^x U ^y Pc _z , R _x P ^x U ^y Pc _z and R _x R _x ^x U ^y Pc _z branch chain.

Figure 11.

Two feasible structures of 2T3R branch chain with distal constraint force acting point.(a) P _z R _x ( ^x UR _y )_N, (b) P _z $R_x^x$ U ^y Pc _z .

For 2T3R branch chain generating direction-varying constraint force, the limb plane (LP) could rotate around a fixed axis and should be perpendicular to O-xy plane. Thus, axis of the linearly independent revolute joint must be parallel to z-axis and added at the beginning of 2R2T branch chain shown in Fig. 9, and we could obtain expected branch chain: P _z ^z U ^xx U ^y , ^z U ^x P ^x U ^y , ^z U ^x R _x ^x U ^y , P _z ^z U ^x C _y , ^z U ^x PC _y and ^z U ^x R _x C _y . Figure 12 shows ^z U ^x R _x ^x U ^y and P _z ^z U ^x C _y branch chain as examples. According to rule (a), the generating constraint force is perpendicular to LP that could rotate around the first rotational axis and goes through the intersection point of the additional rotational axis and the last rotational axis, namely, point G in Fig. 12.

Figure 12.

Two feasible structures of 2T3R branch chain with direction-varying constraint force.(a) ^z U ^x $R_x^x$ U ^y , (b) $P_z^z$ U ^x C _y .

3.2.4. Structures of 3T2R and 3T3R branch chain

3T2R branch chain could only generate a constraint couple, and the number of linearly independent revolute joints is two. To satisfy rule (b), we could add another parallel revolute joint at the end of 2T2R branch chain shown in Fig. 9 and obtain expected 3T2R branch chain: P _z $\mathrm{R}_x^x$ U ^y R _y , R _x P ^x U ^y R _y , R _x $\mathrm{R}_x^x$ U ^y R _y , P _z R _x C _y R _y , R _x PC _y R _y and R _x R _x C _y R _y , as shown in Fig. 13(a) and (b). Furthermore, by meeting the condition that the direction of the last rotational axis could only change once, other feasible 3T2R branch chain can be obtained: $\mathrm{R}_x^x$ U ^y PR _y , $\mathrm{R}_x^x$ U ^y R _y R _y , R _x R _x PR _y R _y , R _x C _y R _y R _y , C _x $\mathrm{R}_x^x$ U ^y and C _x ^x U ^y R _y , as shown in Fig. 13(c) and (d).

Figure 13.

Four feasible structures of 3T2R branch chain. (a) P _z $R_x^x$ U ^y R _y , (b) P _z R _x C _y R _y , (c) $R_x^x$ U ^y PR _y , (d) R _x C _y R _y R _y .

3T3R branch chain generates no constraints, we will not go any further herein since its feasible structures such as UPS, SPS, PUS and PUS are well known to us.

3.2.5. Summary

According to aforementioned analysis, detailed structures of branch chains have been demonstrated under their local coordinate systems. As shown in Table II, since some 2T3R branch chain and 3T2R branch chain are composed of the same kinematic joints and the difference is the layout of axes, so 2T3R branch chains are denoted by superscript “f” and 3T2R branch chains are denoted by superscript “c”. 2T3R branch chain can be further divided into three categories, and they are denoted by 2T3R ⁿ , 2T3R ^d and 2T3R ^v branch chain, respectively. Note that the underlined kinematic joints are the actuated joints of each branch chain.

Table II.

Summary of feasible branch chains.

4.1. Type synthesis of PU-equivalent PMs

The key of type synthesis is to obtain all kinds of constraints distributions according to different general wrench systems, and then determine assembly conditions based on chosen branch chains. Since PU and PS branch chain generates two basic linearly independent constraint forces and restrict the rotational center of the PMs, we will discuss constraint distributions of PU-equivalent PMs with or without PU/PS branch chain separately. Based on numbers and properties of redundant constraints, all constraints distributions (abbreviated as CD) of PU-equivalent PMs with PU/PS branch chain are obtained, as shown in Fig. 14. For simplification, redundant constraint force is denoted by RCF, and redundant constraint couple is denoted by RCC.

Figure 14.

All constraint distributions of PU-equivalent PMs with PU/PS branch chain. (a) CD 11 (0), (b) CD 12 (1 RCF), (c) CD 13 (1 RCC), (d) CD 14 (2 RCFs), (e) CD 15 (2 RCCs), (f) CD 16 (1 RCF and 1 RCC), (g) CD 17 (2 RCFs and 1 RCC), (h) CD 18 (1 RCF and 2 RCCs).

Similarly, all constraints distributions of PU-equivalent PMs without PU or PS branch chain are obtained in the same way, as shown in Fig. 15. Although some constraint distributions are the same as above, they are grouped into different categories for their different compositions of branch chains.

Figure 15.

All constraint distributions of PU-equivalent PMs without PU/PS branch chain. (a) CD 21 (0), (b) CD 22 (1 RCC), (c) CD 23 (2 RCCs).

Table III.

Type synthesis of PU-equivalent PMs with PU/PS branch chain.

4.1.1. Type synthesis of PU-equivalent PMs with PU/PS branch chain

Based on the constraint distributions shown in Fig. 14, all cases of different combinations of branch chains and corresponding assembly conditions are listed in Table III, and type synthesis of PU-equivalent PMs with PU/PS branch chain is finished by mechanism configuration. Obviously, existing 1T2R PMs can be systematically categorized and obtained by this method, e.g., PU-RPUR-UPS [Reference Song, Han and Wang27] belongs to CD 13-1, PU-RPU-UPS [Reference Xu, Zhang, Yao and Zhao28] and PU-PRC-UPS [Reference Fan, Liu and Zhang26] belong to CD 16-3, PS-RPUR-UPS [Reference Song, Han and Wang27] belongs to CD 11-2, PS-RPS-UPS [Reference Song, Han and Wang27] belongs to CD 11-3 and PU-RPS-UPS [Reference Song, Han and Wang27] belongs to CD 12-1. Many novel 1T2R PMs can also be obtained using this type synthesis method, e.g., PS-PRUR ^f -PRUR ^c ,PS-PRU-PR(UR)_N ^f and PU-PRU-PRUR ^c , as shown in Fig. 16, where the red arrows indicate the two rotational axes of novel PMs. Other novel PMs be obtained in the same way by replacing branch chains of a sort, and there is no more description for space saving.

Figure 16.

Three novel PMs of PU-equivalent PMs with PU/PS branch chain. (a) PS-PRUR ^f -PRUR ^c (CD 12–2) (b) PS-PRU-PR(UR)_N ^f (CD 14–2) (c) PU-PRU-PRUR ^c (CD 18–1).

4.1.2. Type synthesis of PU-equivalent PMs without PU/PS branch chain

Table IV shows all combinations of branch chains and corresponding assembly conditions of PU-equivalent PMs without PU/PS branch chain. For existing 1T2R PMs, RPUR-RPU-UPS [Reference Song, Han and Wang27] belongs to CD 21-1, 2RPU-RPUR [Reference Song, Han and Wang27] belongs to CD 23-1. Figure 17 gives three novel PMs as examples, and they are 2PRCR ^f -PR(UR)_N ^f , RPU-RPUR ^c -RPUPc ^f and 2PRU-PRUPc ^f . Note that the distal acting point of 2T3R ^d branch chain is the rotational center of PMs.

Table IV.

Type synthesis of PU-equivalent PMs without PU/PS branch chain.

Figure 17.

Three novel PMs of PU-equivalent PMs without PU/PS branch chain. (a) 2PRCR ^f -PR(UR)_N ^f (CD 21–2), (b) RPU-RPUR ^c -RPUPc ^f (CD 22–1), (c) 2PRU-PRUPc ^f (CD 23–1).

4.2. Type synthesis of $P^{\ast}U^{\ast}$ -equivalent PMs with one parasitic motion

The difference between GWS 21, 22 and GWS 31, 32 is the properties of the time-varying constraint force, and thus both of them should contain at least one 2T3R branch chain to provide required constraint force, which means that their constraint distributions could have two redundant constraints at most. As shown in Figs. 18 and 19, their constraint distributions are quite similar.

Figure 18.

All constraint distributions of P^*U^*-equivalent PMs of GWS 21 and 22. (a) CD 31 (0),(b) CD 32 (1 RCC), (c) CD 33 (2 RCCs).

Figure 19.

All constraint distributions of P^*U^*-equivalent PMs of GWS 31 and 32. (a) CD 41 (0),(b) CD 42 (1 RCC), (c) CD 43 (2 RCCs).

4.2.1. Type synthesis of $P^{\ast}U^{\ast}$ -equivalent PMs of GWS 21 and 22

Table V shows all combinations of branch chains and corresponding assembly conditions of P^*U^*-equivalent PMs of GWS 21 and 22, and the existing 2PRU-PRS [Reference Xie, Liu and Li30, Reference Herrero, Pinto, Altuzarra and Mikel31] PM belongs to CD 33–1. Two novel synthesized P^*U^*-equivalent PMs, namely, 3-PRUR ^f and 2RRC-RRCR ^f are given as examples, as shown in Fig. 20.

Table V.

Type synthesis of P^*U^*-equivalent PMs of GWS 21 and 22.

Figure 20.

Two novel PMs of P^*U^*-equivalent PMs of GWS 21 and GWS 22. (a) 3-PRUR ^f (CD 31–1), (b) 2RRC-RRCR ^f (CD 33–1).

4.2.2. Type synthesis of $P^{\ast}U^{\ast}$ -equivalent PMs of GWS 31 and 32

Table VI shows all combinations of branch chains and corresponding assembly conditions of P^*U^*-equivalent PMs of GWS 31 and 32, and this type of P^*U^*-equivalent PMs have not been researched before. Two novel synthesized P^*U^*-equivalent PMs, namely, 2RRU-URC ^f and 2PRU-PUU ^f , are shown in Fig. 21 as examples.

Table VI.

Type synthesis of P^*U^*-equivalent PMs of GWS 31 and 32.

Figure 21.

Two novel PMs of P^*U^*-equivalent PMs of GWS 31 and GWS 32. (a) 2RRU-URC ^f (CD 43–1), (b) 2PRU-PUU ^f (CD 43–1).

4.3. Type synthesis of $P^{\ast}U^{\ast}$ -equivalent PMs with three parasitic motions

According to Subsection 2.4, GWS 41 consists of three linearly independent constraint forces, which should be coplanar on the moving platform under initial configuration. Thus, there exists no redundant constraints, and if three 3T2R ⁿ or 3T2R ^v branch chains are layout like the typical 3-PRS PM, the change of constraint forces will conform to the constraint characteristics of P^*U^*-equivalent PMs with three parasitic motions. Two novel PMs, namely, 3-PRCR ^f and 3-URC ^f , are shown in Fig. 22 as examples.

Figure 22.

Two novel PMs of P^*U^*-equivalent PMs of GWS 41. (a) 3-PRCR ^f , (b) 3-URC ^f .

Figure 23.

Mobility of 2PRU-PRUPc ^f PM. (a) Rotation around v ₁-axis, (b) rotation around u ₁-axis,(c) rotation around two continuous axes.

5.1. Mobility analysis of 2PRU-PRUPc^f PM

Figure 23(a) and (b) show the moving platform configurations of the 2PRU-PRUPc ^f PM after undergoing finite rotations around the v ₁-axis and the u ₁-axis, respectively. Figure 23(c) shows the moving platform configuration and constraint situation after undergoing rotation around two continuous axes. The distal acting point of PRUPc branch chain, namely, rotational center of the 2PRU-PRUPc ^f PM, could only move along the z ₁-axis, demonstrating the synthesized PM is exactly a PU-equivalent PM. In addition, the constraints distributions are truly the same as CD 23, which means that the mobility can be proved by reciprocal screw theory as well.

5.2. Mobility and parasitic motion analysis of 2PRU-PUU^f PM

Similarly, Fig. 24(a) and (b) show the moving platform configurations of the 2PRU-PUU ^f PM after undergoing finite rotations around the v ₂-axis and the u ₂-axis, respectively. Obviously, the rotational center locates on the z ₂-axis, and there is no parasitic motion under these two situations. However, one parasitic motion occurs when the moving platform rotates around two axes, and it is a parasitic translation d _p along the x ₂-axis, as shown in Fig. 24(c). Likewise, the constraints distributions of 2PRU-PUU ^f PM are the same as CD 33, and the mobility can be proved by reciprocal screw theory too.

Figure 24.

Mobility of 2PRU-PUU ^f PM. (a) Rotation around v ₁-axis, (b) rotation around u ₁-axis,(c) rotation around two axes.

As shown in Fig. 25(a), limb plane of two PRU branch chain (LP ₁) is fixed by the restriction of two revolute joints, while limb plane of the PUU ^f branch chain (LP ₂) could rotate around the first rotational axis of the first universal joint that leads to the generation of parasitic motion along with the change of constraints. To solve the analytical solution of the parasitic motion, the 2PRU-PUU ^f PM is projected to the P-x ₂ y ₂ plane, as shown in Fig. 25(b), where R _i (i =1, 2, 3) denotes centers of the first revolute or universal joints of three branch chains and D _i (i =1, 2, 3) denotes centers of the second universal joints of three branch chains. Obviously, the projection line of LP ₂, namely, R₃D₃ and center of the moving platform are collinear in the projection view, so we have

(9) \begin{align}{{\boldsymbol{k}}_{{R_3}{D_3}}} = {{\boldsymbol{k}}_{{D_3}p}}\end{align}

where k denote the slopes of line R₃D₃ and D₃p, respectively. By substituting into the coordinates of the corresponding points analytical solution of the parasitic motion can be obtained, which have been drawn in Fig. 26.

Figure 25.

Parasitic motion analysis of 2PRU-PUU ^f PM. (a) Change of LP ₂, (b) projection view of 2PRU-PUU ^f PM.

Figure 26.

Comparison of parasitic motion.

The existing 2PRU-PRS PM is a typical P^*U^*-equivalent PM with one parasitic motion, which has been further researched and applied [Reference Xie, Liu and Li30, Reference Herrero, Pinto, Altuzarra and Mikel31], including its parasitic motion. According to former analysis, the novel 2PRU-PUU ^f PM differs from the 2PRU-PRS PM in constraint characteristics, which reflects in their parasitic motion. In the case of the same parameters of these two PMs, it can be concluded that variation laws of their parasitic motion are similar, but the magnitude of parasitic motion of 2PRU-PUU ^f PM is larger than 2PRU-PRS PM, as shown in Fig. 26. However, manufacturing and assembly of 2PRU-PUU ^f PM are more convenient for the avoidance of the spherical joint.