## 1. Introduction

A typical parallel mechanism consists of a moving platform, a fixed base, and several kinematical chains (also called the legs or limbs) which connect the moving platform to its base. Only some kinematical pairs are actuated, whose number usually equals to the number of degrees of freedom (dofs) that the platform possesses with respect to the base. Frequently, the number of legs equals to that of dofs. This makes it possible to actuate only one pair per leg, allowing all motors to be mounted close to the base. Such mechanisms show desirable characteristics, such as large payload and weight ratio, large stiffness, low inertia, and high dynamic performance. However, compared with serial manipulators, the disadvantages include lower dexterity, smaller workspace, singularity, and more noticeable, coupled geometry, by which it is very difficult to determine the initial value of actuators while the end effector stands at its original position.

In an engineering point of view, it is always important to develop a simple and efficient original position calibration method to determine initial values of all actuators. This calibration method usually becomes one of the key techniques that a type of mechanism can be simply and successfully used to the precision applications. Accordingly, few have been reported that the parallel manipulators being applied to high precision situations except micro-movement ones.

The study of movement decoupling for parallel manipulators shows an opportunity to simply the original position calibration and to improve the precision of parallel manipulators in a handy way. One of the most important things in the study of movement decoupling of parallel manipulators is how to design a new type with decoupled geometry.

Decoupled parallel manipulators with less mobility (LM-DPMs) are parallel mechanisms with less than six dofs and with decoupled geometry. This type of manipulators has attracted more and more attention of academic researchers in recent years. Till now, it is difficult to design a decoupled parallel manipulator which has translational and rotational movement simultaneously (Zhang et al., 2006a, 2006b, 2006c). Nevertheless, under some rules, it is relatively easy to design a decoupled parallel manipulator which can produce pure translational (Baron & Bernier, 2001; Carricato, & Parenti-Castelli, 2001a; Gao et al., 2005; Hervé, & Sparacino, 1992; Kim & Tsai, 2003; Kong & Gosselin, 2002; Li et al., 2005a, 2005b, 2006a; Tsai, 1996; Tsai et al., 1996; Zhao & Huang, 2000) or rotational (Carricato & Parenti-Castelli, 2001b, 2004; Gogu, 2005; Li et al., 2006b, 2007a, 2007b) movements.

This chapter attempts to provide a unified frame for the type design of decoupled parallel manipulators with pure translational or rotational movements.

The chapter starts with the introduction of the LM-DPMs, and then, introduce a general idea for type design. Finally, divide the specific subjects into two independent aspects, pure translational and rotational. Each of them is discussed separately. Special attention is paid to the kinds of joins or pairs, the limb topology, the type design, and etc.

## 2. The general idea for decoupled parallel manipulators with less mobility

The general idea for the type design of decoupled parallel manipulators with less mobility can be expressed as the following theory.

Theory: A movement is independent with others if one of the following conditions is satisfied:To the pure translational mechanisms, the translational actuator is orthogonal with the plane composed of other translational actuators.

To the pure rotational mechanisms (spherical mechanisms), the translational actuator is parallel with the axis of rotational actuator.

Depend on part (1) of the theory, we can design some kinds of 3-dofs pure translational decoupled parallel manipulators. Also we can get some kinds of 2-dofs spherical mechanism based on part (2) of the theory.

For the convenience, first, let us define some letters to denote the joints (or pairs). They are the revolute joint (R), the spherical joint (S), the prismatic pair (P), and the planar pair or flat pair (F). They possess one revolute dof, three revolute dofs, one translational dof and three dofs (two translational and one revolute) respectively. Then the theory can be expressed by figure 1 and figure 2 separately.

Figure 1 illustrates the limb topology. The actuator should be installed with the prismatic pair. The flat pair can be composed in deferent way. Using this kind of limb, we can design some kinds of 3-dofs pure translational decoupled parallel manipulators.

Figure 2(a) illustrates the general one geometry of a decoupled 2-dofs spherical mechanism. The moving platform is anchored to the base by two legs. A leg consists of two revolute joints, *R*_{1} and *R*_{2}, whose axes, *z*_{1} and *z*_{2}, intersect at point *o* and connect to each other perpendicularly to form a universal joint; so the value of* * is /2. The other leg consists of a revolute joint, R_{3}, a flat pair, F, and a prismatic pair P, in which the moving direction of P is perpendicular to the working plane of F and the axis of R_{3}. The revolute joints R_{2} and R_{3} are mounted on the moving platform in parallel. The prismatic pair P and the revolute joint R_{1} are assembled to the base, in which the moving direction of P is parallel to the axis of R_{1}.

Suppose that the input parameters, q_{1} and q_{2}, represent the positions of the revolute joint R_{1} and the prismatic pair P, which are driven by a rotary actuator and a linear actuator separately. The pose of the moving platform is defined by the Euler angles _{1} and _{2} of the platform. When the value of q_{1} changes and q_{2} holds the line, only _{1} alters. On the other hand, when the value of q_{2} changes, only _{2} changes. So, _{1} and _{2} are independently determined by q_{1} and q_{2} respectively, i.e., one output parameter only relates to one input parameter. In other words, the platform rotations around two axes are decoupled.

Figure 2(b) is an improved idea of figure 2(a). Using this idea, we can get a decoupled 2-dofs spherical mechanism with a hemi-sphere work space.

## 3. Design of 3-dofs translational manipulators with decoupled geometry

### 3.1. Type design

The Type design of 3-dofs translational manipulators is based on the analysis of limb topology shown in figure 1.

Firstly, we construct deferent structures to replace the flat pair and the prismatic pair. Some substitutes for the flat pair and the prismatic pair are shown in figure 3. Then, using the pairs to form variational kinds of limbs. Figure 4 shows three examples. Finally, we can constitute the 3-dofs translational manipulators by installing the specified limbs in orthogonal as shown in figure 5, 6 and 7.

### 3.2. Kinematics

The forward and inverse kinematic analyses for the 3-PPP manipulator shown in figure 5 are trivial since there exists a one-to-one correspondence between the moving platform position and the input pair displacements. So the velocity jacobia matrix is a 33 identity matrix.

The kinematics of 3-7R manipulator can be analysed as follows. Referring to figure 6(b), each limb constrains point P to lie on a plane which passes through points M_{j2}, M_{j3}, and B_{j}, and is perpendicular to the axis of x, y, and z, respectively. The position of j^{th} plane is determined only by _{j} whenever the length l_{j1} is given. Consequently, the position of P is determined by the intersection of three planes, i.e., the intersection of _{j} for j=1,2,3. If the distance from M_{j1} to M_{j2} is m_{0j}, then a simple kinematic relation can be written as

where p=[p_{x} p_{y} p_{z}]^{T} denotes the position vector of the end-effector. Taking the time derivative of equation (1) yields

where J is a diagonal matrix that holds

The kinematics of the modified 3-7R manipulator are the same.

### 3.3. Original position calibration

The calibration of 3-PPP manipulator is the same as a pure translational 3-dofs serial manipulator. So we just consider the manipulator of 3-7R and modified 3-7R, they can be expressed in the same way as shown in figure 8(a).

For convenience, we suppose,

The input

_{j}(j=1,2,3) is within [-_{jm},_{jm}], where_{jm}>0, and_{j}=_{jm}denotes the initial position of the j^{th}limb;In the initial position (see figure 8), the angle between the link l

_{j1}(j=1,2,3) and the axis u(u=x,y,z) is_{0j}(j=1,2,3).

Then the initial value _{jm} of _{j} can be determined as

So we can determine _{0j} first, then_{jm}, the steps of the calibration can be as follows. From the initial position _{0j} of the arm in figure 8, rotate the driving arm twice in a specified angle _{s}, which satisfies

During the process, record the two moving distances l_{0} and l of the platform in the direction of axis u(u=x,y,z), they satisfy

expand

The geometric signification of the equation (8) is shown in figure 8(b), which is very sententious and convenient to industrial applications.

### 3.4. Singularity

The 3-PPP manipulator has no singularity, so we just discuss the manipulator of 3-7R and modified 3-7R, they can be expressed in the same.

From equation (2) we can find out that the rotational actuator speed is nonlinear to the velocity of the end-effector. Moreover, if

Suppose the desired velocity of the end-effector is

Then

Whenever the mechanism design satisfies equation (11), no singularity will exist.

## 4. Design of 2-dofs spherical manipulators with decoupled geometry

### 4.1. Type Design

The Type design of 2-dofs spherical manipulators is based on the general idea shown in figure 2. Using the 3R and 4R pairs in figure 3 to replace the F and P pairs separately, a new structure (2R&8R manipulator) for figure 2(a) is constructed as shown in figure 9. Similarly, figure 10 shows the improved configuration of figure 2(b), a 2R&PRR manipulator, but distinguishingly, additional modification is that a through hole is added to the center of the revolute joint R_{1}, so the prismatic pair P can be set in the center of the hole and rotates with R_{1}. As a result, the workspace of _{1} can reach 2.

### 4.2. Kinematics

Firstly, the 2R&8R manipulator in figure 9 will be discussed. Let e be the distance between the axes of R_{2} and R_{3}, m be the distance between the axes of R_{8} and R_{10} (or R_{7} and R_{9}). Also, suppose that, when the moving platform is on the initial position, the axis of R_{1} is perpendicular to the plane consisting of the axes of R_{2} and R_{3}. Then the displacement relationships between input and output for the 2R&8R manipulator are:

In the structure design, it is easy to set the length m of _{2} and R_{3} so as to get the one-to-one input-output mapping. Let m = e, it follows that:

This implies that the direct linear one-to-one input-output correlation, so the velocity jacobia matrix becomes an identity one.

Now we discuss the the 2R&PRR manipulator shown in figure 10. Suppose that the input parameters, q_{1} and q_{2}, represent the angular displacement of the revolute joint R_{1} and the distance between the axes of R_{2} and R_{4} separately. They are driven by a rotary actuator and a linear actuator. The pose of the moving platform is defined by the Euler angles _{1} and _{2} of the platform. Let e be the distance between the axes of R_{2} and R_{3}, m be the distance between the axes of R_{3} and R_{4}. Also suppose that, axis z_{3} is through the point o and always perpendicular to the plane of z_{1}-z_{2} and moreover, define the value of _{2} is zero whenever the axis of R_{3} is on the plane of z_{1}-z_{2}. Then the coordinates of R_{4} and R_{3} for the axes z_{1} and z_{3} are

The displacement relationship between input and output is:

Taking the derivative of equation (15), it follows that

Where,

### 4.3. Singularity and workspace

The 2R&8R manipulator shown in figure 9 has two legs. The first leg (R_{1} to R_{2}) produces the Euler angle _{1} of the platform by the input of q_{1}; while the second one (R_{10} to R_{3}) produces _{2} by q_{2}. To illustrate the motional relationship, let us introduce a transition parameter z to equation (12), it follows that:

where, z is the displacement of F-pair (R_{4} to R_{6}) in the direction of z_{1}.

From equation (18), it is seen that the Euler angle _{1} is produced from the input of q_{1} directly by the first leg; while _{2} is produced from q_{2} by the second leg through two transformations, which include (1) rotary to linear motion _{2}, which is determined by the product of the radius r of the revolute joint’s axis and the equivalent friction coefficient as follows.

Let denote the angle between z_{1} and the link_{r} and the tangent component F_{t} (see figure 11). Then the force F acts on R_{2} is equivalent to a force Q and a torque M, which can be calculated from the following equations.

As a basic law in mechanics, the effect of a force Q and a torque M acting on a rigid body is equivalent to a force Q_{h} with an offset h, which is shown in figure 12 and can be calculated as follows

where, h is the distance between the action lines of force Q_{h} and Q.

There exist three instances for the different relationship between h and , which are (1) h < , the revolute joint R_{2} will never rotate regardless the value of Q_{h}; (2) h > , revolute joint R_{2} can rotate; and (3) h = , the critical condition. In the critical condition of h = , using equation (21), it follows that:

Then the workspace of _{2} satisfies:

On the other hand, the angle _{1} produced by the first leg is limited only by the structure design of the F-pair and the base, so the workspace of _{1} can reach a designated area through proper design. Assume that the workspace of _{1} is from – /2 to /2, then the workspace of the spherical mechanism can be depicted by the reachable range of the point P as shown in figure 13. The workspace is smaller than a hemisphere, so it would be limitted in some applications.

When the mechanism is running, the direction of axis z_{1} keeps unchanged, while the direction of axis z_{2} alters according to _{1}. So the workspace represented by spherical surface in figure 13 can be interpreted as follows: point P draws latitude lines when only _{1} changes and draws longitude lines while only _{2} alters.

Now we examine the 2R&PRR manipulator in figure 10. The only limitation of this mechanism is caused by the friction circle of R_{2}. This limitation can be described by figure 14, from which we can see that the work space of

Where

It means that the workspace of the mechanism can not reach a hemisphere. Clearly, this is not desirable.

In fact, because the workspace of

Figure 15 shows the critical instances for both of them; each one uses the similar technique to offset the axis of R_{4} from the axis z_{1}. Let n denotes the axis offset of R_{4} (or the length of AR_{4}), and n_{c} is the special value of n for the critical configurations as shown in figure 15, then n should be chosen equation (27). Using this technique, a hemisphere work space can be obtained.

The improved architectures are shown in figure 16 and figure 17, in which the workspace of

A prototype model of the mechanism for the condition of _{5} is added to connect the prismatic pair with the nut of the ball screw.

## 5. Conclusions

A general idea for type design of decoupled parallel manipulators with less mobility is introduced. A unified frame for the type design is provided and divided into two independent aspects. Some kinds of decoupled parallel manipulators with 3-dofs pure translational and 2-dofs pure rotational movements are obtained.