Type synthesis of uncoupled translational parallel manipulators based on actuation wrench screw theory

Abstract

In this article, a new methodology for type synthesis of uncoupled translational parallel manipulators with 3-degrees of freedom is proposed based on actuation wrench screw theory. Mapping matrix between outputs of the moving platform and the inputs of the actuators for uncoupled translational parallel manipulators is derived. The forms of both the actuated twist screws and the actuation wrench screws of the limbs are determined by means of the condition that the Jacobian is a diagonal matrix with full rank. The steps used to confirm the non-actuated screws of the limbs are also established. Then, procedures for structure synthesis of the limbs are set up and all possible basic structure limbs are enumerated. Some new uncoupled translational parallel manipulators are synthesized by selecting three limbs connecting the platform to the base and two examples are given. The approach proposed is applicable to the type synthesis of uncoupled parallel manipulators with rotational mobility as well.

Introduction

Parallel manipulators are typically closed-loop mechanisms composed of a moving platform and a fixed base which are linked by several serial limbs.^1–3 Sometimes, in order to obtain certain special characteristics, these serial limbs can be replaced by one or more hybrid chains,^4–6 in which there is at least one limb consisting of a closed-loop structure. Compared to traditional serial devices, parallel manipulators have some merits of higher stiffness, larger payload capacity, higher accuracy, and lower inertia. Therefore, parallel manipulators have been applied in many fields, such as virtual machine tools, medical equipments, industrial robots, rehabilitation robots, radar antennas, and sensors. Stewart ⁷ and Delta ⁸ are the most famous parallel manipulators applied successfully.

High kinematic coupling is one of the intrinsic characteristics for parallel mechanisms. One independent output motion of the platform is always controlled simultaneously by no less than two actuators or the input of an actuator influences more than two outputs of the platform. Although the coupling contributes high stiffness and large loading capability, it also leads to complicated problems in kinematics, controlling design and path planning, which enhances passive effects for the real applications of the parallel manipulators. Recently, many scholars have focused on research of the reduced parallel manipulators.^9–15 This kind of parallel manipulators can be classified into three types in terms of the different forms of Jacobian matrices. The first type is called decoupled parallel manipulator when its Jacobian is a triangular matrix. The second is named uncoupled parallel manipulator (UPM) when its Jacobian is a diagonal one. The third is defined as fully isotropic parallel manipulator as its Jacobian is an identical one. Especially, for the last two types, there exists a one-to-one mapping relationship between the outputs of the platform and the inputs of actuators, which means one output motion of the platform is only controlled by one actuator of the mechanism. So, these manipulators perform well with regard to motion and force transmission capabilities.

Based on the geometrical approach, Kong and Gosselin ¹⁶ performed type synthesis of linear translational parallel manipulators, where the Jacobian matrices were diagonal or identical, respectively. Carricato ¹⁷ addressed type synthesis of fully isotropic 3T1R parallel manipulators in terms of the screw theory. Gogu^18,19 put forward a method for type synthesis of fully isotropic parallel manipulators via linear transformations theory. Furthermore, many new decoupled or uncoupled parallel mechanisms were designed as well. Yu et al. ²⁰ enumerated a kind of orthogonal translational parallel manipulator by Lie group method. Zeng and Huang ²¹ synthesized some decoupled parallel mechanisms based on screw theory. Zhang and Ting ²² designed a family of uncoupled 2T2R parallel manipulators which were composed of a serial open-single kinematic chain and a hybrid chain.

In this article, a new methodology for type synthesis of the uncoupled translational parallel manipulators (UTPMs) with 3-degrees of freedom (DOFs) is proposed based on actuation wrench screw theory. Basic theory for the method of type synthesis is reviewed briefly in section “Theoretical background.” Mathematic model mapping relationship between the inputs and the outputs of UTPMs is discussed in section “Mapping matrix between the inputs and the outputs of the UTPMs.” Structural synthesis of the kinematic limbs is performed in section “Structural synthesis of the limbs for UTPM,” where the selection principles of the actuated joints driving the platform translation along the given direction are derived as well. In section “Type synthesis of the UTPMs,” type synthesis of UTPMs is dealt with. Finally, some conclusions are drawn in section “Conclusion.”

Theoretical background

Screw

A unit screw is defined by a straight line with an associated pitch, written as ²³

$$\mathit{\$}=\left[\begin{array}{c}\mathit{s}\hfill \\ \multicolumn{1}{c}{{\mathit{s}}_0}\end{array}\right]=\left[\begin{array}{c}\mathit{s}\hfill \\ \multicolumn{1}{c}{\mathit{r}\times \mathit{s}+\rho \mathit{s}}\end{array}\right]$$

(1)

where $\mathit{s}$ is a unit vector along the direction of the screw axis, ${\mathit{s}}_0=[\mathit{r}\times \mathit{s}+\rho \mathit{s}{]}^{\mathrm{T}}$ is the dual part of the direction vector $\mathit{s}$ , r denotes the position vector from the origin to an arbitrary point on the screw axis, and $\rho$ is the pitch of the screw.

At the same time, a screw also can be represented by a six-dimensional Plücker coordinate as follows

$$\mathit{\$}=[L,M,N;P,Q,R{]}^{\mathrm{T}}$$

(2)

where the first three components stand of the direction vector $\mathit{s}$ of the screw $\mathit{\$}$ , and last three denote the dual part of $\mathit{s}$ .

If the pitch is equal to zero, that is, $\rho =0$ , the screw reduces to a linear vector, and

$$\mathit{\$}={[\mathit{s};\mathit{r}\times \mathit{s}]}^{\mathrm{T}}$$

(3)

Otherwise, if the pitch tends to infinite, that is, $\rho \to \infty$ , the screw becomes a pure couple vector, and

$$\mathit{\$}={[\mathit{0};\mathit{s}]}^{\mathrm{T}}$$

(4)

A screw multiplied by a scalar $\lambda$ is written as $\overline{\mathit{\$}}=\lambda \mathit{\$}$ , which is regarded as a twist when it denotes the instantaneous motion of a rigid body or a wrench if it denotes a system of forces and couples exerting on a rigid body. For example, a zero-pitch screw can be used to express the motion screw of a revolute joint as a twist, or a pure linear force vector as a wrench. Similarly, an infinite-pitch screw can also be used to represent the motion screw of a prismatic pair as a twist or a pure couple vector as a wrench.

Reciprocal screws

If two screws, $\mathit{\$}$ and ${\mathit{\$}}^r$ , satisfy the following condition

$$\mathit{\$}\circ {\mathit{\$}}^r=0$$

(5)

They are said to be reciprocal screws. Here “∘” denotes the reciprocal product.

Physically speaking, if a screw represents an instantaneous twist of a rigid body, its reciprocal screw represents a wrench imposed on the rigid body. When the virtual power developed by the wrench while it moves along the direction of the twist is equal to zero, both of them are reciprocal. Based on equation (5), some conclusions will be obtained as follows: (1) arbitrary two infinite-pitch screws are reciprocal to each other, (2) two zero-pitch screws are reciprocal if their axes are parallel or intersect at a point, and (3) a zero-pitch screw is reciprocal to an infinite-pitch screw if and only if their axes are normal to each other.

Actuation wrench screw

Actuation wrench screw, or called actuation screw, is defined as a screw which is reciprocal to all twist screws within a limb except for the actuated one. Physically speaking, an actuation screw is a wrench exerted on the platform by the actuated twist screw of the limb. In other words, it is a driving wrench imposed on the platform by the actuated joint.

The procedures to solve the actuation wrench screw of the open-single kinematic chain are as follows:

• Step 1: to list the twist-system of the kinematic chain;

• Step 2: to derive the wrench system, ${\mathit{\$}}_{\omega }^r$ , of the twist system based on the reciprocal principle;

• Step 3: to work out the wrench system, ${\mathit{\$}}_{\varsigma }^r$ , of the twist system after locking the actuated twist;

• Step 4: to find the different item between ${\mathit{\$}}_{\omega }^r$ and ${\mathit{\$}}_{\varsigma }^r$ , which is the actuation screw, ${\mathit{\$}}_a$ , of the limb.

Thus, it is obvious that the relations of ${\mathit{\$}}_{\omega }^r$ , ${\mathit{\$}}_{\varsigma }^r$ , and ${\mathit{\$}}_a$ can be expressed as

$${\mathit{\$}}_{\varsigma }^r={\mathit{\$}}_{\omega }^r\cup {\mathit{\$}}_a$$

(6)

Mapping matrix between the inputs and the outputs of the UTPMs

The instantaneous output motions of the platform of the parallel manipulator can be described using the twist system of its limbs as follows ²⁴

$$\mathit{V}=\sum _{j=1}^{F_{\mathrm{ci}}}{\stackrel{·}{q}}_{\mathrm{ji}}{\mathit{\$}}_{\mathrm{ji}}(i=1,2,\dots ,n)$$

(7)

where V is the generalized velocity of the platform, including three angular and three linear velocities, $F_{\mathrm{ci}}$ is the connectivity of the ith limb, ${\mathit{\$}}_{\mathrm{ji}}$ is the twist associated with the jth single-DOF joint of the ith limb, ${\stackrel{·}{q}}_{\mathrm{ji}}$ denotes the instantaneous velocity of the corresponding joint, and n denotes the number of limbs.

Taking the reciprocal product of both sides of equation (7) with the actuation wrench screw, ${\mathit{\$}}_{\mathrm{ai}}$ , of the ith limb, we have

$${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{\mathrm{ai}}^{\mathrm{T}}(\Pi \mathit{V})={\stackrel{·}{q}}_{1i}{\begin{array}{c}\mathit{\$}\hfill \end{array}}_{\mathrm{ai}}^{\mathrm{T}}(\Pi {\begin{array}{c}\mathit{\$}\hfill \end{array}}_{1i})(i=1,2,\dots ,n)$$

(8)

and

$$\Pi =\left[\begin{array}{cc}\mathit{0}\hfill & {\mathit{I}}_3\hfill \\ \multicolumn{1}{c}{{\mathit{I}}_3}& \mathit{0}\hfill \end{array}\right]$$

(9)

where ${\mathit{\$}}_{1i}$ denotes the twist screw of the actuated joint in the ith limb which is assembled on the base, ${\stackrel{·}{q}}_{1i}$ represents the angular or the linear velocity associated with the actuated joint, “ $\Pi$ ” is the dual operator, $\mathit{0}$ is the $3\times 3$ zero matrix, and ${\mathit{I}}_3$ is the $3\times 3$ identity matrix.

Rewriting equation (8) into matrix form yields

$${\mathit{J}}_{\mathrm{dir}}\mathit{V}={\mathit{J}}_{\mathrm{inv}}\stackrel{·}{\mathit{q}}$$

(10)

where $\mathit{V}=[\begin{array}{cccc}{v}_1\hfill & v_2\hfill & \cdots \hfill & v_6\hfill \end{array}{]}_{1\times 6}^{\mathrm{T}}$ denotes the generalized velocity of the moving platform, the first three components represent the linear velocity and the last three are the angular velocity, $\stackrel{·}{\mathit{q}}=[\begin{array}{cccc}{\stackrel{·}{q}}_{11}\hfill & {\stackrel{·}{q}}_{12}\hfill & \cdots \hfill & {\stackrel{·}{q}}_{1n}\hfill \end{array}{]}_{1\times n}^{\mathrm{T}}$ is the input velocity vector of the actuated joints, and ${\mathit{J}}_{\mathrm{dir}}$ and ${\mathit{J}}_{\mathrm{inv}}$ represent the direct and the inverse Jacobian matrices of parallel mechanism, respectively, and they can be expressed as

$$\begin{array}{c}{\mathit{J}}_{\mathrm{dir}}=\left[\begin{array}{c}{\mathit{\$}}_{a1}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\mathit{\$}}_{a2}^{\mathrm{T}}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\mathit{\$}}_{\mathrm{an}}^{\mathrm{T}}}\end{array}\right]={\left[\begin{array}{cccccc}{L}_{a1}\hfill & M_{a1}\hfill & N_{a1}\hfill & P_{a1}\hfill & Q_{a1}\hfill & R_{a1}\hfill \\ \multicolumn{1}{c}{L_{a2}}& M_{a2}\hfill & N_{a2}\hfill & P_{a2}\hfill & Q_{a2}\hfill & R_{a2}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{L_{\mathrm{an}}}& M_{\mathrm{an}}\hfill & N_{\mathrm{an}}\hfill & P_{\mathrm{an}}\hfill & Q_{\mathrm{an}}\hfill & R_{\mathrm{an}}\hfill \end{array}\right]}_{n\times 6}\hfill \\ {\mathit{J}}_{\mathrm{inv}}={\left[\begin{array}{cccc}{\mathit{\$}}_{a1}^{\mathrm{T}}(\Pi {\mathit{\$}}_{11})\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{\$}}_{a2}^{\mathrm{T}}(\Pi {\mathit{\$}}_{12})\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \cdots \hfill & {\mathit{\$}}_{\mathrm{an}}^{\mathrm{T}}(\Pi {\mathit{\$}}_{1n})\hfill \end{array}\right]}_{n\times n}\hfill \end{array}$$

Evidently, the inverse matrix ${\mathit{J}}_{\mathrm{inv}}$ is a diagonal one. When it is full rank, equation (10) can be arranged as

$${\left[\begin{array}{cccc}{L}_{a1}/g_1\hfill & M_{a1}/g_1\hfill & \cdots \hfill & R_{a1}/g_1\hfill \\ \multicolumn{1}{c}{L_{a2}/g_2}& M_{a2}/g_2\hfill & \cdots \hfill & R_{a2}/g_2\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \hfill & ⋮\hfill \\ \multicolumn{1}{c}{L_{\mathrm{an}}/g_n}& M_{\mathrm{an}}/g_n\hfill & \cdots \hfill & R_{\mathrm{an}}/g_n\hfill \end{array}\right]}_{n\times 6}{\left[\begin{array}{c}{v}_1\hfill \\ \multicolumn{1}{c}{v_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{v_6}\end{array}\right]}_{6\times 1}={\left[\begin{array}{c}{\stackrel{·}{q}}_{11}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{12}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{1n}}\end{array}\right]}_{n\times 1}$$

(11)

and ${\begin{array}{c}{g}_i=\mathit{\$}\hfill \end{array}}_{\mathrm{ai}}(\Pi {\begin{array}{c}\mathit{\$}\hfill \end{array}}_{1i}{)}^{\mathrm{T}}$ .

As for n-DOF parallel manipulator, if its output motions of the point on the platform are independent, there are $6-n$ zero components in the vector V . Removing the zero components in V and the corresponding column in Jacobian matrix, equation (11) is reduced as

$${\left[\begin{array}{cccc}{L}_{a1}/g_1\hfill & M_{a1}/g_1\hfill & \cdots \hfill & R_{a1}/g_1\hfill \\ \multicolumn{1}{c}{L_{a2}/g_2}& M_{a2}/g_2\hfill & \cdots \hfill & R_{a2}/g_2\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \hfill & ⋮\hfill \\ \multicolumn{1}{c}{L_{\mathrm{an}}/g_n}& M_{\mathrm{an}}/g_n\hfill & \cdots \hfill & R_{\mathrm{an}}/g_n\hfill \end{array}\right]}_{n\times n}{\left[\begin{array}{c}{v}_1\hfill \\ \multicolumn{1}{c}{v_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{v_n}\end{array}\right]}_{n\times 1}={\left[\begin{array}{c}{\stackrel{·}{q}}_{11}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{12}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{1n}}\end{array}\right]}_{n\times 1}$$

(12)

If the UPM is considered, one of the outputs of the platform is only relative to one input of the corresponding actuated joint and irrelevant to other inputs. Then, the inputs and outputs of the mechanism satisfy the function, $v_i=f({\stackrel{·}{q}}_{1i})$ . Therefore, the form of Jacobian of the UPM must be a diagonal matrix, and the mathematic model between the inputs and the outputs can be obtained in terms of equation (12), and we have

$${\left[\begin{array}{cccc}{L}_{a1}/g_1\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& M_{a2}/g_2\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \cdots \hfill & R_{\mathrm{an}}/g_n\hfill \end{array}\right]}_{n\times n}{\left[\begin{array}{c}{v}_1\hfill \\ \multicolumn{1}{c}{v_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{v_n}\end{array}\right]}_{n\times 1}={\left[\begin{array}{c}{\stackrel{·}{q}}_{11}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{12}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{1n}}\end{array}\right]}_{n\times 1}$$

(13)

According to the definition of the actuation wrench screw mentioned above, we know that it is a driving wrench exerted on the platform by the actuated joint of the limb. When the actuation wrench is a zero-pitch screw (linear force vector), it will push the platform to translate along its direction vector. Similarly, if the actuation wrench is an infinite-pitch screw (pure couple vector), it will force the platform to rotate around the direction of the couple vector. Since any point on the platform of UTPM can only translate linearly in the Cartesian space, the actuation wrench screws imposed on the platform by the actuated joints must be the zero-pitch screws. Then referring to equation (13), kinematics equation of the UTPM is derived as

$$\left[\begin{array}{ccc}{L}_{a1}/g_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& M_{a2}/g_3\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & N_{a3}/g_3\hfill \end{array}\right]\left[\begin{array}{c}{v}_1\hfill \\ \multicolumn{1}{c}{v_2}\\ \multicolumn{1}{c}{v_3}\end{array}\right]=\left[\begin{array}{c}{\stackrel{·}{q}}_{11}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{12}}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{13}}\end{array}\right]$$

(14)

where $v_1$ , $v_2$ , and $v_3$ denote the linear velocities of one point on the platform along three axes of the Cartesian coordinate, respectively.

Structural synthesis of the limbs for UTPM

Structural synthesis procedure of the kinematic limbs

Structural synthesis of the kinematic limbs is to determine the type, number, and order of the joints within a chain by means of the output-motion characteristics of the mechanism. As far as UTPM is concerned, it is necessary that its one limb only provides independently direct actuation to the platform translation along a specific direction. So, how to set up the synthesis method of the limbs becomes the essentially theoretical basis for type synthesis of the manipulators.

Based on the results in section “Mapping matrix between the inputs and the outputs of the UTPMs,” the general procedures are proposed for structural synthesis of the kinematic limbs of UTPMs as follows:

• Step 1: to distribute the controlling target of each limb, such as translation along the given direction. According to the output-motion characteristics of the platform, to determine the actuation wrench screw of each limb based on equation (14);

• Step 2: to identify the actuated twist screw of each limb based on the principle that the reciprocal product between the actuation wrench screw and the actuated twist screw is non-zero, that is, $g_i\ne 0$ ;

• Step 3: to figure out all types of the non-actuated screws of the limbs using the principle that the actuation wrench screw is reciprocal to the all twist screw except for the actuated one. Then discuss the possible number of the different types of screws based on the maximum linear independent groups of the twist screw within a limb;

• Step 4: to determine the type, number, and assemble orientation of the actuated and the non-actuated joints in a limb by means of step 3;

• Step 5: to enumerate all possible kinematic limbs according to the different connectivity and complete the structural synthesis of the limbs.

The first limb

Without loss of generality, provided that the first limb only supports the driving wrench to the platform translation along the X-axis, referring to equation (14), we get $v_x=v_1$ . To simplify the problem analysis, here we only choose two kinds of basic joints, revolute (R) and prismatic (P) joints to discuss. Other complex joints can be obtained by combining R and P joints. For example, cylindrical joint (C) is regarded as the combination of one R and one P joint whose axes are parallel to each other; universal joint (U) as the combination of two R joints whose axes are perpendicular and intersect at one point; and spherical joint (S) as the combination of three R joints whose axes intersect at one point but not coplanar.

Referring to Figure 1, the coordinate system O-XYZ is attached on the fixed base and o-xyz is mounted on the moving platform. Three axes of the o-xyz are parallel to the corresponding axes of the O-XYZ, respectively.

Figure 1.

Actuation wrench screws and their corresponding actuated twist screws within limbs.

Since the first element on the diagonal line of the Jacobian matrix must be non-zero, see equation (14), the actuation wrench screw, ${\mathit{\$}}_{a1}$ , is imposed on the platform by the first limb and must have the form

$${\mathit{\$}}_{a1}={[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & Q_{a1}\hfill & R_{a1}\hfill \end{array}]}^{\mathrm{T}}$$

(15)

Evidently, ${\mathit{\$}}_{a1}$ is a pure wrench screw with zero-pitch and parallel to X-axis. Provided that it passes through the point $A_1(0,y_{a1},z_{a1})$ on the platform, we get $Q_{a1}=z_{a1}$ , $R_{a1}=-y_{a1}$ .

Furthermore, the first element on the diagonal line should also satisfy the following condition

$${\begin{array}{c}{g}_1=\mathit{\$}\hfill \end{array}}_{a1}^{\mathrm{T}}(\Pi {\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11})\ne 0$$

(16)

where ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}$ is the actuated twist associated with the actuated joint of the first limb. It may have two types, that is, indefinite-pitch or zero-pitch.

Type 1: indefinite-pitch screw

In this case, the actuated twist screw of the first limb can be defined as $\begin{array}{c}{\mathit{\$}}_{11}^1\hfill \end{array}={[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}{L}_{11}^1\hfill & M_{11}^1\hfill & N_{11}^1\hfill \end{array}]}^{\mathrm{T}}$ . Substituting ${\mathit{\$}}_{a1}$ and ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^1$ into $g_1$ yields

$${\begin{array}{c}{g}_1=\mathit{\$}\hfill \end{array}}_{a1}^{\mathrm{T}}(\Pi \begin{array}{c}{\mathit{\$}}_{11}^1\hfill \end{array})=L_{11}$$

(17)

Obviously, the value of $g_1$ is only related to the component ( $L_{11}^1$ ) of ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^1$ on the X-axis and without any relation to the other components along the directions Y and Z ( $M_{11}^1$ and $N_{11}^1$ ). In terms of equation (16), we get $L_{11}^1\ne 0$ and $M_{11}^1=N_{11}^1=0$ . So, the form of the actuated twist screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^1$ can be expressed as

$${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^1={[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}]}^{\mathrm{T}}$$

(18)

Substituting equation (18) into equation (17), we have $g_1=1$ , which holds equation (16). Thus, the actuated twist screw of the first limb may be an indefinite-pitch screw parallel to the X-axis (see Figure 1). That is to say, the actuated joint can be a prismatic pair and its axis is along the X-axis. This can be defined as a selection principle of the actuated joints for UTPMs.

Type 2: zero-pitch screw

In this case, the form of the actuated twist screw of the first limb can be written as ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^2={[\begin{array}{ccc}{L}_{11}^2\hfill & M_{11}^2\hfill & N_{11}^2\hfill \end{array};\begin{array}{ccc}{P}_{11}^2\hfill & Q_{11}^2\hfill & R_{11}^2\hfill \end{array}]}^{\mathrm{T}}$ . Then, we have

$${\begin{array}{c}{g}_1=\mathit{\$}\hfill \end{array}}_{a1}^{\mathrm{T}}(\Pi \begin{array}{c}{\mathit{\$}}_{11}^2\hfill \end{array})=P_{11}^2+z_{a1}{M}_{11}^2-y_{a1}{N}_{11}^2$$

(19)

Equation (19) shows that the value of $g_1$ is relative to the components $M_{11}^2$ and $N_{11}^2$ of screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^2$ , and irrelevant to the component $L_{11}^2$ , so we obtain $L_{11}^2=0$ . Since ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^2$ is attached directly on the fixed base, its axis can be arranged through the original point O of the fixed system, and we have $P_{11}^2=Q_{11}^2=R_{11}^2=0$ . Therefore, equation (19) is reduced as

$$\begin{array}{c}{g}_1=\hfill \end{array}{z}_{a1}{M}_{11}^2-y_{a1}{N}_{11}^2$$

(20)

where $y_{a1}$ and $z_{a1}$ are the coordinates of point $A_1$ on the platform with respect to the base. In the general configurations, both $y_{a1}$ and $z_{a1}$ are not equal to zero and their values are variable with the motion of the mechanism. If both $M_{11}^2$ and $N_{11}^2$ are not equal to zero as well, it may produce the result, $g_1=0$ . This is contradictory to the condition $g_1\ne 0$ in equation (16). In order to ensure $g_1$ being non-zero, we have to select $M_{11}^2=1$ or $N_{11}^2=1$ . In fact, regardless of which one is selected as 1, it only represents the different directions of the actuated twist screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^2$ , but the property of the ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^2$ has no change. In this article, we choose the case that $M_{11}^2=1$ and $N_{11}^2=0$ . So, the form of the actuated twist screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^2$ can be determined as follows

$${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}^2={[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}]}^{\mathrm{T}}$$

(21)

Substituting equation (21) into equation (20), we get $g_1=z_{a1}$ . If actuation wrench screw ${\mathit{\$}}_{a1}$ does not locate on the O-XY plane, equation (16) will be satisfied, that is, $g_1\ne 0$ . Therefore, the actuated twist screw of the first limb can also be selected as a zero-pitch screw parallel to the Y-axis (see Figure 1). In other words, the actuated joint may be a revolute joint with the axis along the direction of Y-axis. This is another principle for selecting the actuated joints for UTPMs.

In terms of processes above, the possible forms of the actuated twist screws of the first limb are obtained: one is an infinite-pitch screw parallel to X-axis and other is a zero-pitch screw parallel to Y-axis. Then, based on the rule that the actuation wrench screw is reciprocal to all the twists except for the actuated screw, the non-actuated twist screws of the first limb will be determined.

• Case I—when the actuated twist screw is an infinite-pitch screw parallel to X-axis, the possible forms of the non-actuated twist screws can be

• I-1—zero-pitch screw parallel to X-axis. This kind of screw forms a set connected directly to the platform. Its number is at least two and no more than three.

• I-2—infinite-pitch screws perpendicular to X-axis. This kind of screws can be placed at any positions in the limb and its number is no more than two.

• I-3—zero-pitch screw intersected with the axis of ${\mathit{\$}}_{a1}$ and not parallel to X-axis. Its number is no more than two. It is remarkable that this class of screws is called idle twist because they contribute nothing to the instantaneous kinematics of the mechanism.

• Case II—when the actuated twist screw is a zero-pitch screw parallel to Y-axis, the possible forms of the non-actuated twist screws can be

• II-1—zero-pitch screw parallel to X-axis. This kind of screw forms a set connected directly to the platform. The number is at least two and no more than three. However, any zero-pitch screw cannot be inserted in this set.

• II-2—zero-pitch screw parallel to Y-axis. Its axis is intersected with the axis of ${\mathit{\$}}_{a1}$ . There is one and only one in the limb.

• II-3—zero-pitch screw (or called idle screw) perpendicular to Y-axis. This kind of screw intersects with the axis of ${\mathit{\$}}_{a1}$ and its number is no more than two.

• II-4—infinite-pitch screw perpendicular to X-axis. This kind of screw cannot be placed between two zero-pitch screws parallel to Y-axis. The number is no more than two.

After determining the actuated twist screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{11}$ and all possible non-actuated twist screws, the types, number and assembly configurations of the actuated joint, and the non-actuated joints in the first kinematic limb will be confirmed. According to the different connectivity, $F_c$ , the basic structure forms without idle joints can be determined. There are five types, such as 3P-, 2P2R-, 1P3R-, 1P4R-, and 5R-type. When the idle revolute joint is inserted in the appropriate position of the basic limbs, some new structure forms can be designed. Although the idle joints have no influence on the kinematics, they increase the connectivity of the limb and lower the dimension of the overconstraints of the mechanisms.

In order to simplify the assembly forms, provided that the axes of the two adjacent joints are parallel or perpendicular to each other in the limbs. The basic kinematic limbs are enumerated by means of the rules proposed (see the fourth column of Table 1). The limbs including the complex joints are listed in the fifth column. The subscripts u, v, and w represent the translational direction of the prismatic pairs or the rotational axis direction of the revolute joints, respectively. The symbols with the same subscript denote their axes are parallel, otherwise, perpendicular to each other. The subscript m denotes the axis of the corresponding prismatic pair which is normal to the axes of both adjacent joints. The superscripts t and r represent to select the translational or rotational displacement of the actuated cylindrical joint as the input of that limb. The symbol underlined indicates that there is a rotational idle DOF associated with the joint.

Table 1.

Type and structure of the kinematic limbs for UTPMs.

Fc	Type	Order	Basic limb structures	Limb structures including the complex joints
3	3P	1	PuPvPw	–
4	2P2R	2–10	PuPvRwRw; PuRuPmRu PuRuRuPv; RvRvPvPw RvPvRvPw	CtuPmRu; CtuRuPv RvCvPw; CrvRvPw
	1P3R	11–12	PuRuRuRu	CtuRuRu
5	1P4R	13–29	PuRuRuRuRv; PuRuRuRvRuPuRvRuRuRu; RvRvRuRuPwRvRvRuPmRu; RvRvPvRuRuRvPvRvRuRu	PuRuRuUuv;CtuRuRuRv;CtuRuUuvPuUvuRuRu; PuUuvRuRu; CrvUvuRuRvUvuRuPw; RvUvuPmRu; RvCvRuRuCrvRvRuRu;
	2P3R	30–61	PuPvRuRuRv; PuPvRuRvRuPuPvRvRuRu; PuRuPmRuRvPuRuPmRvRu; PuRuRvPmRuPuRuRuPvRv; PuRuRuRvPvPuRuRvRuPv; RvRvPwPvRuRvRvRuPwPv; RvPvRvPwRuRvPvRvRuPw	PuPvRuUuv; PuPvUvuRu; PuCvRuRuPuRuPmUuv; CtuPmUuv; CtuPmRuRvPuRuCvRu; CtuRvPmRu; CtuCmRu;PuUuvPmRu; PuRuRuCv; CtuRuPvRvCtuRuCv; PuRuUuvPu; CtuUuvPv;PuUuvRuPv; RvUvuPwPv; CrvRvPwRu;CrvUvuPw
	5R	62–63	RvRvRuRuRu	RvUvuRuRu
6	1P5R	64–78	RvRvRwPwRuRu; RvRvRwRuRuPwRvRvRwRuPmRu; RvPvRyRwRuRu	RvUvwPwRuRu; RvRvCwRuRuRvRvUwuRuPm; RvUvwRuRuPz; RvSRuPzRvUvwRuPmRu; RvSPmRu; RvRvUwuPmRuCrvRvRwRxRx; CrvUvwRuRu; CrvSRu
	6R	79–82	RvRvRwRuRuRu	RvUvwRuRuRu; RvRvUuwRuRu; RvSRuRu

If let $u=x$ , $v=y$ , and $w=z$ , Table 1 shows the first limb possible structures. Every one of them can provide the driving force for the platform translating along X-axis.

The second limb

This limb supports the driving wrench to the platform translating along Y-axis. According to the condition that the second element on the diagonal line of the Jacobian matrix is non-zero, see equation (14), the actuation wrench screw ${\mathit{\$}}_{a2}$ of the second limb will be determined and yields

$${\mathit{\$}}_{a2}={[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array};\begin{array}{ccc}{P}_{a2}\hfill & 0\hfill & R_{a2}\hfill \end{array}]}^{\mathrm{T}}$$

(22)

From equation (22), we know that ${\mathit{\$}}_{a2}$ is a pure wrench screw with zero-pitch parallel to Y-axis. Assumed that this screw passes through the point $A_2(x_{a2},0,z_{a2})$ , we get $P_{a2}=-z_{a2}$ , $R_{a2}=x_{a2}$ .

Similarly, the second element on the diagonal line also satisfies the following condition

$${\begin{array}{c}{g}_2=\mathit{\$}\hfill \end{array}}_{a2}^{\mathrm{T}}(\Pi {\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12})\ne 0$$

(23)

where ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12}$ is the actuated twist screw associated with the actuated joint. It has two possible forms as well, indefinite-pitch or zero-pitch screw.

By using the similar procedures in section “The first limb,” the possible forms of the actuated twist screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12}$ can be determined, and we have

$${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12}^1={[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}]}^{\mathrm{T}}$$

(24)

or

$${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12}^2={[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}]}^{\mathrm{T}}$$

(25)

which implies that the actuated twist screw of the second limb may be an infinite-pitch screw parallel to Y-axis or a zero-pitch screw parallel to Z-axis. That is to say, the actuated joint can be selected as a prismatic pair with axis parallel to Y-direction or a revolute joint with axis parallel to Z-direction.

As the forms of both the actuation wrench screw ${\mathit{\$}}_{a2}$ and the actuated twist screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12}$ are determined, all possible forms of the non-actuated screws will be confirmed in terms of the reciprocal principle, too.

• Case I—when ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12}$ is an infinite-pitch screw parallel to Y-axis, the possible forms of the non-actuated twist screws can be

• I-1—zero-pitch screw parallel to Y-axis. This kind of screw forms a set connected directly to the platform. Its number is at least two and no more than three.

• I-2—infinite-pitch screw normal to Y-axis. This kind of screw can be placed at any positions in the limb and its number is no more than two.

• I-3—zero-pitch screw intersected with the axis of ${\mathit{\$}}_{a2}$ . Its axis is not parallel to Y-axis and its number is no more than two. It is worth noting that this class of screws is idle twist.

• Case II—when ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{12}$ is the zero-pitch screw parallel to Z-axis, the possible forms of the non-actuated twist screws can be

• II-1—zero-pitch screw parallel to Y-axis. This kind of screw forms a set connected directly to the platform. The number is at least two and no more than three. Any zero-pitch screw cannot be inserted in this set.

• II-2—zero-pitch screw parallel to Z-axis. Its axis is intersected with the axis of ${\mathit{\$}}_{a2}$ . There is one and only one in the limb.

• II-3—zero-pitch screw (or called idle screw) perpendicular to Y-axis and intersecting with the axis of ${\mathit{\$}}_{a2}$ . Its number is no more than two.

• II-4—infinite-pitch screw perpendicular to Y-direction. This kind of screw cannot be placed between two zero-pitch screws parallel to Y-axis. Its number is no more than two.

After the actuated twist screw and the non-actuated screws are determined, all possible structures of the second limb can be enumerated based on the differences of the connectivity.

If let $u=y$ , $v=z$ , and $w=x$ , Table 1 shows the possible structures of the second limb. Every one of them can provide the driving force to the platform translating along Y-axis.

The third limb

This limb provides the direct actuation to the platform along Z-axis. Similar to the process in section “The first limb” and section “The second limb,” the forms of both the actuation wrench screw ${\mathit{\$}}_{a3}$ and the actuated twist screw ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{13}$ can be derived as follows

$${\mathit{\$}}_{a3}={[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array};\begin{array}{ccc}{P}_{a3}\hfill & Q_{a3}\hfill & 0\hfill \end{array}]}^{\mathrm{T}}$$

(26)

and

$${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{13}^1={[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}]}^{\mathrm{T}}$$

(27)

$${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{13}^2={[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}]}^{\mathrm{T}}$$

(28)

Obviously, ${\mathit{\$}}_{a3}$ is a zero-pitch screw parallel to Z-axis, and ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{13}$ may be an infinite-pitch screw parallel to Z-axis or a zero-pitch screw parallel to X-axis.

Then, the forms of the non-actuated screws of the third limb are obtained as well.

• Case I—when ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{13}$ is the infinite-pitch screw parallel to Z-axis, the possible forms of the non-actuated twist screws can be

• I-1–zero-pitch screw parallel to Z-axis. This kind of screw forms a set connected directly to the platform. The number is at least two and no more than three.

• I-2–infinite-pitch screw normal to Z-axis. This kind of screw can be placed any positions in the limb and its number is no more than two.

• I-3–zero-pitch screw (idle screw) intersected with the axis of ${\mathit{\$}}_{a3}$ and not parallel to Z-axis. Its number is no more than two.

• Case II–when ${\begin{array}{c}\mathit{\$}\hfill \end{array}}_{13}$ is the zero-pitch screw parallel to X-axis, the possible forms of the non-actuated twist screws can be

• II-1–zero-pitch screw parallel to Z-axis. This kind of screw forms a set connected directly to the platform. The number is at least two and no more than three. Any zero-pitch screw cannot be inserted in these twist screws.

• II-2–zero-pitch screw parallel to X-axis and intersected with the axis of ${\mathit{\$}}_{a3}$ . There is one and only one in the limb.

• II-3–screws of zero-pitch (idle screw) intersected with the axis of ${\mathit{\$}}_{a3}$ and perpendicular to X-axis. Its number in the limb is no more than two.

• II-4–infinite-pitch screw perpendicular to Z-axis. This kind of screw cannot be intersected between two zero-pitch screws parallel to X-axis. The number is no more than two.

When the actuated twist screw and the non-actuated screws are determined, all possible structures of the third limb can be enumerated based on the different connectivity.

If let $u=z$ , $v=x$ , and $w=y$ , Table 1 illustrates the possible structures of the second limb. Every one of them can provide the driving force to the platform translating along the Z-axis.

Type synthesis of the UTPMs

Type synthesis

An UTPM can be generated by connecting the platform to the base by use of three limbs shown in Table 1. Each one of the limbs gives direct actuation to one DOF of the platform. However, the total connectivity of three limbs should be considered carefully when to construct the mechanisms. For non-overconstrained parallel mechanism, the total connectivity of its limbs must hold the following condition ²⁵

$$F=\sum _{i=1}^n{F}_{\mathrm{ci}}=M_p+\mu l$$

(29)

where $F$ is the total connectivity of the mechanism, $F_{\mathrm{ci}}$ is the connectivity of the ith limb, $n$ is the number of the limbs, $M_p$ is the DOF of the mechanism, $\mu$ is the order of the mechanism, and $l$ is the independent closed loops of the mechanism.

For the non-overconstrained translational parallel manipulator, $n=3$ , $M_p=3$ , $\mu =6$ , and $l=2$ , then we get $F=15$ . Thus, the total connectivity of three limbs of UTPM must be no more than 15. Moreover, when total connectivity is less than 15, the mechanism is called overconstrained parallel manipulator. Then, if the connectivity of a limb selected from Table 1 is equal to 6, it must be existed at least another one whose connectivity is less than 5. All kinematic limbs in Table 1, whose connectivity is no more than 5, can be chosen directly to design the UTPMs. It is worth remarking that if there are two limbs with idle joints in a mechanism, the axes of both idle joints must be not parallel to each other. As long as the conditions mentioned above are satisfied, an expected UTPM will be constructed by selecting three limbs in Table 1 connected to the moving platform and the fixed base.

For example, a novel non-overconstrained UTPM is obtained by assembling three R _v U _vu P _m R _u -type (revolute-, universal-, prismatic- and revolute-joint) kinematic chains, shown in Figure 2. The axes of three actuated revolute joints amounted on the base are perpendicular to each other. Similarly, thee revolute joints attached on the platform should also meet the same assembly condition. Then, the first limb, R _y U _yx P _m R _x , controls the motion along X-direction of the platform. The second and the third limbs control Y- and Z-direction motions, respectively. The velocity equation of the manipulator can be derived as

$$\left[\begin{array}{c}{v}_x\hfill \\ \multicolumn{1}{c}{v_y}\\ \multicolumn{1}{c}{v_z}\end{array}\right]=\left[\begin{array}{ccc}f\cos q_{11}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& f\cos q_{12}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & f\cos q_{13}\hfill \end{array}\right]\left[\begin{array}{c}{\stackrel{·}{q}}_{11}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{12}}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{13}}\end{array}\right]$$

(30)

where f is the length of the lower arm of every limb, measuring the distance between the first two revolute joints, and $q_{1i}$ is the angular displacement of the actuated joint of the ith limb.

Figure 2.

3-RUPR non-overconstrained UTPM.

Since the Jacobian of the manipulator is a diagonal matrix, there exists a one-to-one mapping relationship between the output velocity vector of the platform and the input velocity vector of the actuated joints.

Another new manipulator is shown in Figure 3, which consists of three C ^t _u C _m R _u -type chains. The symbol underlined denotes that it is an idle joint. That is to say, there is an idle rotational DOF in the cylindrical joint of each limb. When the linear displacements of the actuated C-pairs attached to the base is selected as the inputs, the kinematics formula of the mechanism can be written as

$$\left[\begin{array}{c}{v}_x\hfill \\ \multicolumn{1}{c}{v_y}\\ \multicolumn{1}{c}{v_z}\end{array}\right]=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\left[\begin{array}{c}{\stackrel{·}{q}}_{11}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{12}}\\ \multicolumn{1}{c}{{\stackrel{·}{q}}_{13}}\end{array}\right]$$

(31)

Referring to equation (31), it is more interesting that all components on the diagonal line of the Jacobian are constantly equal to 1. Therefore, the condition number of the Jacobian is equal to 1 throughout its whole workspace as well. In other words, it is a fully isotropic translational parallel mechanism, which performs well with regard to force and motion transmission capabilities.

Figure 3.

3-CCR UTPM with idle joints.

Mobility analysis of the 3-RUPR parallel manipulator

The screw theory is a valid mathematic tool to analyze the mobility of parallel manipulator and it is used to deal with the following contents. Referring to Figure 2, the kinematic screw (or called twist) of each joint in the first limb in the fixed frame, O-xyz, can be described as

$$\begin{array}{c}{\mathit{\$}}_{11}=[\begin{array}{cccccc}0\hfill & 1\hfill & 0;\hfill & a_{11}\hfill & 0\hfill & c_{11}\hfill \end{array}{]}^{\mathrm{T}}\hfill \\ {\mathit{\$}}_{21}=[\begin{array}{cccccc}0\hfill & 1\hfill & 0;\hfill & a_{21}\hfill & 0\hfill & c_{21}\hfill \end{array}{]}^{\mathrm{T}}\hfill \\ {\mathit{\$}}_{31}=[{\begin{array}{cccccc}1\hfill & 0\hfill & 0;\hfill & 0\hfill & b_{31}\hfill & c_{31}]\hfill \end{array}}^{\mathrm{T}}\hfill \\ {\mathit{\$}}_{41}=[\begin{array}{cccccc}0\hfill & 0\hfill & 0;\hfill & 0\hfill & m_{41}\hfill & n_{41}\hfill \end{array}{]}^{\mathrm{T}}\hfill \\ {\mathit{\$}}_{51}=[\begin{array}{cccccc}1\hfill & 0\hfill & 0;\hfill & 0\hfill & b_{51}\hfill & c_{51}\hfill \end{array}{]}^{\mathrm{T}}\hfill \end{array}$$

(32)

where $m_{41}$ and $n_{41}$ are components of the direction vector of screw ${\mathit{\$}}_{41}$ associated with the prismatic pair along y- and z-axis, respectively. $p_{j1}$ , $q_{j1}$ , and $r_{j1}$ denote the position parameters corresponding to the jth single-DOF joint.

Then, using the reciprocal product principle, the constraint screw (or called wrench) of the first limb can be derived as follows

$${\mathit{\$}}_{11}^r=[\begin{array}{cccccc}0\hfill & 0\hfill & 0;\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}{]}^{\mathrm{T}}$$

(33)

Obviously, the wrench ${\mathit{\$}}_{11}^r$ of the first limb is an infinite-pitch screw along z-axis. Speaking in physical meaning, it is a constraint pure couple and restricts the rotational DOF of the platform around z-axis.

Similarly, the constraint screw system of the second and the third limbs can also be obtained, and we have

$$\begin{array}{c}{\mathit{\$}}_{12}^r=[\begin{array}{cccccc}0\hfill & 0\hfill & 0;\hfill & 1\hfill & 0\hfill & 0\hfill \end{array}{]}^{\mathrm{T}}\hfill \\ {\mathit{\$}}_{13}^r=[\begin{array}{cccccc}0\hfill & 0\hfill & 0;\hfill & 0\hfill & 1\hfill & 0\hfill \end{array}{]}^{\mathrm{T}}\hfill \end{array}$$

(34)

Both of them, ${\mathit{\$}}_{12}^r$ and ${\mathit{\$}}_{13}^r$ , are infinite-pitch screws and restrict two-rotational DOFs of the platform around x- and y-axis, respectively.

In terms of equations (33) and (34), we know that three constraint screws of limbs limit together three-rotational DOFs of the mechanism. Therefore, the platform can only translate along three axes of the Cartesian coordinate system. In addition, it is obvious that three constraint screws are independent linearly, so there is no redundant screw among them, which implies that this mechanism is non-overconstrained.

Conclusion

In this article, a new methodology for type synthesis of the UTPMs is proposed based on the actuation wrench screw theory. Selection principles of the actuated joints driving the platform to translate along the given direction are explored by means of the different types of the actuation wrench screws, zero-pitch, or infinite-pitch screws. The process for structure synthesis of limbs is established and many new limbs with idle joints are also enumerated. Two 3-DOF UTPMs are taken as examples to verify the correctness and feasibility of the type synthesis method presented. It is worth noting that the mechanisms have fully isotropic performances when the actuated joints are prismatic or cylindrical pairs. In addition, the method proposed here is also applicable to the type synthesis of lower mobility UPMs with the exception of 3-DOF spherical type.