Optimal Design and Development of a Decoupled A/B-Axis Tool Head with Parallel Kinematics

Abstract

This paper is an attempt to design a decoupled A/B-axis tool head with parallel kinematics, due to the increasing demand for A/B-axis tool heads in industry, particularly in thin wall machining applications for structural aluminium aerospace components. In order to carry out further analysis, the method of orientation description based on the azimuth and tilt angles is introduced, which is a convenient method describing kinematics, parasitic motions, and orientation workspace. For the purpose of optimal design, three indices are defined to evaluate the force transmission performance of the tool head. They have obvious physical significance and are dependent of any coordinate system. Based on the indices and their performance atlases, the optimization process is presented in detail. The parasitic motions and orientation capability of the designed tool head are analyzed finally. The results show that the designed device is far from singularity, has good force transmissibility, and has very high tilting angle. The indices and analysis and design method used here should be said to be extended to other parallel robots.

1. Introduction

Generally, a tool head that can rotate about the x-axis and y-axis is defined as A/B-axis tool head, and that can rotate about the x-axis and z-axis is defined as A/C-axis tool head. Most of the multifunction milling machines have a tool head with two rotational degrees of freedom (DOFs), and the tool head usually adopts the serial architecture, such as the A/C-axis tool head proposed in [1]. In order to reduce the machining time and deformation of the work piece, A/B-axis tool heads are used instead of the A/C-axis tool head in the manufacture of structural aircraft parts with thin walls. Such kind of serial tool head usually has complicated transmission system, huge structure and high weight, it is difficult to install, and moreover, the manufacture cost and maintenance fee is relatively high. And, in practical engineering application, this serial architecture has many disadvantages such as inconvenience for locus planning and control difficulty since its two rotations are not coupled motions due to its serial kinematic structure. Although the A/B-axis tool head with parallel kinematics has relatively small rotational angle compared with the serial tool head, it has high manufacturing efficiency and much smaller deformation made to work piece. What is more, the parallel manipulator has the advantages of high stiffness, high velocity, compactness, time-saving of machining, high load/weight ratio, and low moving inertia. So, the A/B-axis tool head with parallel kinematics has the absolute advantage in the manufacture of structural aircraft parts with thin walls which needs mass cutting and high speed. Recently, parallel A/B-axis tool heads with high tilting angle have become the most important and fundamental part in the manufacturing field of key structural aircraft parts. DS Technology in Germany has developed a machining tool head [2], the Sprint Z3, and FATRONIK in Spain has developed the Space 5H tool head [3]. Both of them have made great success in engineering application. The Z3 tool head is based on a 3-DOF spatial 3-PRS (P, R, and S stand for prismatic, revolute, and spherical joints, resp.) parallel manipulator. It has a translational DOF along the z-axis and two rotational DOFs about the A and B axes, respectively. In this kind of tool head, the most important is the two rotational DOFs that provide the coupled tilting motion. And, generally, some machining centers just need the two rotational DOFs. So, the development of an A/B-axis tool head with parallel kinematics is becoming more and more popular.

For parallel manipulator, optimal design is one of the most important and challenging problems and is attracting more and more efforts [4–7]. There are two issues involved: performance evaluation and dimension synthesis. Having designed a mechanism, it is necessary to evaluate its performance. So, the first problem is of most importance and should be reconsidered due to the doubt of the mostly used index, that is, local conditioning index (LCI) [8]. The second problem is to determine the dimensions (link lengths) of the mechanism, which is suitable for the task at hand. It is one of the most difficult issues in the field.

Several well-defined performance indices which are popular in the field of serial mechanism, such as manipulability, workspace, singularity, dexterity, stiffness, and accuracy, have developed extensively and applied to the design of parallel manipulator. The LCI, which is the reciprocal of the condition number of Jacobian matrix, is usually used to evaluate the accuracy, dexterity, and distance to singularity of a parallel manipulator. For this reason, the LCI has drawn much more attention. However, a recent study [8] reviewed the LCI and global conditioning index (GCI) that is the computation over a kind of workspace of the manipulator. The study found serious inconsistencies when these indices are applied to parallel manipulators with combined translational and rotational degrees of freedom and conclude that these indices should not be used in parallel manipulators with mixed types of DOFs (translational and rotational). To eliminate the singularity and its near configurations, most researchers use the local conditioning index (LCI). Usually, a good-condition workspace [9] or effective workspace [10] was defined with respect to a specified minimum LCI. However, the minimum is still arbitrary or comparative since we cannot give it a defined value due to its frame-depended characteristic. Generally, it is impossible to define a mathematical distance to a singularity for a parallel manipulator. Thus, the authors think that the LCI cannot be used in parallel manipulator, but not only those with mixed type of DOFs.

As is well known, the planar four-bar mechanisms have been studied for a very long time, longer than that of serial robots and much longer than that of parallel manipulators. The transmission angle is an important index for the design of such a mechanism as was pointed out by Alt [11], who defined the concept, using the forces tending to move the driven link and tending to apply pressure to the driven link bearings as a simple index, to judge the force-transmission characteristics of a mechanism. So, the transmission angle is an index evaluating the quality of motion/force transmission. By means of the index of transmission angle, the quality of motion/force transmission in a mechanism can be judged in the design stage. And it helps to decide the best from a family of possible mechanisms for the most effective force transmission [12]. Also, the transmission angle of a mechanism provides a very good indication of the quality of its motion, the accuracy of its performance, its expected noise output, and its cost in general [13]. Although a good transmission angle is not a cure-all for all design problems, as is pointed out in [14], for many mechanical applications it can guarantee the performance of a linkage at higher speed without unfavorable vibrations. The study [15] shows that when the transmission angle equals to 90°, the most effective force transmission takes place and the output motion becomes less sensitive to the manufacturing tolerances on the link lengths, clearance between joints, and change of dimensions due to thermal expansion. A large transmission angle usually leads to reasonable mechanical advantages and a high quality of motion transmission. The study of link mechanisms shows that transmission angle is significant not only as an indicator of good force and motion transmission but also as a prime factor in the linkage sensitivity to small design parameter errors. The smaller the transmission angle is, the more sensitive the linkage will be [16]. Mechanisms having a transmission angle too far from 90° exhibit poor operational characteristics such as noise and jerk at high speeds [17, 18], and many studies have reached the conclusion that if the transmission angle becomes too small, the mechanical advantage becomes small, even a very small amount of friction will lead the mechanism to jam, and if it is 0, self-locking takes place. For the purpose of high speed, high accuracy, and high quality of motion transmission, the most widely accepted design limits for the transmission angle are (45°,135°) [18] or (40°,140°) [11].

A planar four-bar mechanism is a single-closed-loop system. A parallel manipulator is a multiclosed-loop system. Usually, a fully parallel manipulator has more or less the characteristic of a planar four-bar mechanism. We suggest that the design concept of the four-bar mechanisms could be used in the design of a parallel manipulator. In this paper, the local and global transmission indices (defined in Section 5.2) based on the concept of transmission angle will be proposed as indices in the optimal design of an A/B-axis tool head with parallel kinematics, which can be kinematically considered as the combination of two slider-crank mechanisms at any moment.

Many methods have been proposed for the dimensional optimization of a specified mechanism. The most common method is the objective-function-based optimal design. According to this method, an objective function with specified constraints must be established, and then a search is conducted to find the result utilizing an optimum algorithm. Not only is this method time consuming, but it is difficult to reach the globally optimum target because of the infiniteness of the individual parameters, the antagonism of multiple criteria, and the assignment of initial values. The most serious drawback is that it provides only one solution for a design problem. This is actually unreasonable for a practical design, since it is impossible to predict any application in advance and to know whether a particular design is the only solution.

The ideal dimensional optimization method would be that using the performance charts (atlas), which is widely used in classical design. A performance atlas can show, visually and globally, the relationship between a performance index and the associated design parameters in a limited space [19]. Moreover, it can show how antagonistic the involved indices actually are. Compared with the result achieved by the objective-function-based method, the result of this optimal method is comparative and fuzzy. However, it is more flexible, because it provides not only a single solution, but also all possible solutions to a design problem. This means that the designer can adjust the optimum result appropriately according to the particular design conditions he is dealing with [20]. This method will be extended to the dimensional optimization of the A/B-axis tool head with parallel kinematics.

The remainder of this paper is organized as follows. The next section describes the structure of the proposed tool head with different configurations. Section 3 introduces the method of orientation description of the tool head and analyzes the relationship between orientational angles and output angles of the two legs, and then, the parasitic motions are given. Section 4 investigates the inverse kinematics of the tool head. Section 5 recalls the classical concept of transmission angle, defines the forward and inverse transmission angle, proposes some indices, and then, plots performance atlases for these new indices and presents the optimal design using the atlases. Section 6 analyzes the orientation capability and parasitic motions of the tool head given in Section 5. Development of the tool head is presented in Section 7 and conclusions are given in Section 8.

2. Structure Description

The A/B-axis tool head with parallel kinematic, shown in Figure 1, contains a moving platform which connects to the base through two legs. The first leg contains a fixed length link and a bracket, the bracket can rotate about the y-axis through a revolute joint, another revolute joint connects the link to the bracket, the other end of the link is connected to an active slider through a revolute joint, and the slider is attached to the base by a prismatic joint whose move direction is vertical (Figures 1(a), 1(c), and 1(d)) or horizontal (Figure 1(b)). The second leg contains a fixed length link (Figures 1(a), 1(b), and 1(c)) or an extensible link which is active (Figure 1(d)). For the models shown in Figures 1(a), 1(b), and 1(c), the link is connected to the active slider through a universal joint. For the model shown in Figure 1(d), the active link is connected to the base through a screw joint and a universal joint. The moving platform is connected with the two legs by two revolute joints, respectively. In order to make it move, the axes of the revolute joint fixed to the bracket and that of the universal joint in y-axis direction must be collinear. The rotational axes of the moving platform are orthogonal but not coplanar for the models shown in Figures 1(a) and 1(b); they are orthogonal and coplanar for the models shown in Figures 1(c) and 1(d). It is not difficult to find that the two rotations of any one of the structures are decoupled.

The CAD models of the A/B-axis tool head with parallel kinematics (P-prismatic joint, R-revolute joint, U-universal joint, H-helical joint, and the underlined joints are active.).

From the above description, one can see that, at any moment, this kind of A/B-axis tool head can be considered as the combination of two slider-crank mechanisms. The slider-crank mechanism, shown in Figure 2, is a classical planar four-bar mechanism. And such a combination can make sure that this A/B-axis tool head is decoupled. As is well known, a decoupled mechanism is easier to control and achieve higher accuracy.

Figure 2

A slider-crank mechanism.

We may see that the models shown in Figures 1(a), 1(b), and 1(c) are similar in kinematics but are different from that shown in Figure 1(d). In this paper, we will mainly focus on the models shown in Figures 1(a) and 1(d). For the former model, its kinematic scheme is shown in Figure 3, where the two rotational axes of the moving platform are orthogonal but not coplanar. Thus, such a mechanism has parasitic motion which will be analyzed in the next section. We call such a case the case I in this paper. For the later model in Figure 1(d), its kinematic scheme is shown in Figure 4, where the two rotational axes of the moving platform are orthogonal and coplanar. There is no parasitic motion for this mechanism. This is a great advantage for application. We call such a case the case II.

Figure 3

kinematical scheme of the model with parasitic motions (Figure 1(a)).

Figure 4

kinematical scheme of the model with no parasitic motion (Figure 1(d)).

3. Orientation Description of the Parallel A/B-Axis Tool Head

Most researchers have used Euler angles to describe the orientation of a parallel manipulator. For the manipulator discussed in this paper, this method is inconvenient to describe the orientation and to analyze the kinematic problems. We will use another method introduced in [21], which is suitable for the task at hand. For any feasible orientation of the moving platform of a mechanism with two rotational DOFs, its two rotations can be obtained from the reference (home) orientation by a single rotation about a line passing through the center of the moving platform. Here, two angular parameters are involved. The azimuth angle, denoted by ϕ, defines an a-axis passing through the platform center o′ (Figure 3) or o (Figure 4) and lying in the o′- $x^{'} y^{'}$ or o-xy plane, and the tilt angle, denoted by θ, describes the swing angle about the a-axis. And, the corresponding rotation matrix can be written as

\begin{matrix} R_{a} (θ) = [\begin{bmatrix} \cos^{2} ϕ 프 + \cos θ & \sin ϕ \cos ϕ 프 & \sin ϕ \sin θ \\ \sin ϕ \cos ϕ 프 & \sin^{2} ϕ 프 + \cos θ & - \cos ϕ \sin θ \\ - \sin ϕ \sin θ & \cos ϕ \sin θ & \cos θ \end{bmatrix}], \end{matrix}

(1)

where 프 denotes $(1 - \cos θ)$ ,ϕ is measured between the x′-axis and the a-axis, and $ϕ ∊ (0,36 0^{\circ}), θ ∊ (0,9 0^{\circ})$ .

As analyzed above, this mechanism can be considered as the combination of two slider-crank mechanisms. Then, there are two independent rotational angles about the y-axis and x-axis or x′-axis, respectively, and they are referred to as α and β, correspondingly.

In order to carry on further analysis, we need to find out the relationship between $(ϕ, θ)$ and $(α, β)$ and that between the parasitic motions and $(ϕ, θ)$ . The projective relationship is shown in Figure 5.

Figure 5

The projective relationship between (ϕ,θ) and (α,β).

Point P denotes the tip of the tool, and o′ is the center of the moving platform. Angular parameter α is defined as the rotational angle from home orientation of $o^{'} P$ about the y′-axis in the π1 plane, and β is referred to the rotational angle from current orientation of $o^{'} P$ to the orientation of $o^{'} P^{'}$ in the π3 plane. The planes π1 and π3 are orthogonal to each other. θ is the angle mentioned above, which is the rotational angle from home orientation of $o^{'} P^{'}$ about a-axis in the π2 plane, and the a-axis is orthogonal to the π2 plane. π0 denotes the o′- $x^{'} y^{'}$ plane. ϕ is the angle between the x′-axis and a-axis in π0 plane.

Letting $o^{'} P = o^{'} P^{'} = l$ , we can get

\begin{gathered} l \cos (9 0^{\circ} - β) = l \cos (9 0^{\circ} - θ) \cos (ϕ), \\ l \cos (9 0^{\circ} - θ) \sin (ϕ) \tan (α) = l \sin (9 0^{\circ} - θ) . \end{gathered}

(2)

That is

\begin{matrix} \begin{matrix} β = 9 0^{\circ} - arccos [\sin (θ) \cos (ϕ)], β ∊ (- 9 0^{\circ}, 9 0^{\circ}), \\ α = ta n^{- 1} [\tan (θ) \sin (ϕ)], α ∊ (- 9 0^{\circ}, 9 0^{\circ}) . \end{matrix} \end{matrix}

(3)

For the case I (shown in Figure 3), ${o o}^{'} = R$ , and $(x, y, z)$ is the position of the point o′.

Please note that, for the second leg shown in Figure 7(a), ${o o}^{'} = R = D_{3}$ is required to allow the first leg shown in Figure 6 to rotate about y-axis.

Figure 6

The first leg of the tool head.

The second leg of the tool head: (a) for the model in Figure 1(a) and 1(b) for the model in Figure 1(d).

Then, the parasitic motions can be described as

\begin{gathered} x = R \cos (9 0^{\circ} + α), \\ y = 0, \\ z = R \sin (9 0^{\circ} + α) . \end{gathered}

(4)

For the case II (shown in Figure 4), ${o o}^{'} = R = 0$ ; so, there is no parasitic motion.

4. Inverse Kinematic Analysis of the Tool Head

4.1. Inverse Kinematics of the Tool Head with Parasitic Motions

For the case I, a kinematical scheme of the tool head is developed as shown in Figure 3, and the first and second legs are shown in Figures 6 and 7(a), respectively. A fixed global reference frame o- $x y z$ is located at the center of the bracket with the y-axis being collinear with the rotational axis of the first leg and the plane o-xy being horizontal.

A mobile reference frame o′- $x^{'} y^{'} z^{'}$ is located at the center of the rotational axis of the second leg with the x-axis being collinear with this rotational axis and the z-axis directing along ${o o}^{'}$ , which is also the direction of the tool.

As shown in Figures 6 and 7(a), the planar schemes of the first and second legs are similar in configuration. The geometric parameters will be $s_{1} c_{1} = R_{1}$ , $c_{1} o = R_{2}$ and the normal distance from point o to the straight-line path of the joint point s₁ is R₃. And the adjusting angles ε₁ and ε₂ are constants which will be given in the optimal process.

Vectors $s_{1}$ , $c_{1}$ , $s_{2}$ , and $c_{2}$ are defined as the position vectors of points s₁, c₁, s₂, and c₂ in the reference frame o- $x y z$ and can be written as

\begin{matrix} s_{1} = (R_{3}, 0, z), \\ c_{1} = (R_{2} \cos (\frac{π}{2} - ε_{1} + α), 0, R_{2} \sin (\frac{π}{2} - ε_{1} + α)), \\ s_{2} = (0, y, - D_{3}), \\ c_{2} = (0, D_{2} \cos (\frac{π}{2} - ε_{2} - β), D_{2} \sin (\frac{π}{2} - ε_{2} - β) + R) . \end{matrix}

(5)

The kinematic problem of the tool head can be solved by writing the constraint equations:

\begin{matrix} | s_{1} c_{1} | & = R_{1}, \\ | s_{2} c_{2} | & = D_{1} . \end{matrix}

(6)

Then, there are

\begin{matrix} {[R_{2} \cos (\frac{π}{2} - ε_{1} + α) - R_{3}]}^{2} + {[R_{2} \sin (\frac{π}{2} - ε_{1} + α) - z]}^{2} \\ = R_{1}^{2}, \\ {[D_{2} \cos (\frac{π}{2} - ε_{2} - β) - y]}^{2} + {[D_{2} \sin (\frac{π}{2} - ε_{2} - β) + R + D_{3}]}^{2} \\ = D_{1}^{2} . \end{matrix}

(7)

The inputs y and z can be obtained as

\begin{gathered} y \end{gathered} = \frac{- b \pm \sqrt{b^{2} - 4 c}}{2},

(8)

\begin{gathered} z = \frac{- e \pm \sqrt{e^{2} - 4 g}}{2}, \end{gathered}

(9)

where

\begin{matrix} b & = - 2 D_{2} \cos (\frac{π}{2} - ε_{2} - β), \\ c & = D_{2}^{2} - D_{1}^{2} + {(R + D_{3}^{})}^{2} + 2 D_{2} (R + D_{3}^{}) \sin (\frac{π}{2} - ε_{2} - β), \\ e & = - 2 R_{2} \sin (\frac{π}{2} - ε_{1} + α), \\ g & = R_{2}^{2} + R_{3}^{2} - R_{1}^{2} - 2 R_{3} R_{2} \cos (\frac{π}{2} - ε_{1} + α) . \end{matrix}

(10)

For a given pose $(ϕ, θ)$ , substituting (3) into (10), we can get the inputs y and z easily. The parasitic motions can be got from (4).

4.2. Inverse Kinematics of the Tool Head without Parasitic Motions

For the case II, a kinematical scheme of the tool head is developed as shown in Figure 4, and the first and second legs are shown in Figures 6 and 7(b), respectively.

The fixed global reference frame o- $x y z$ is the same as that in Section 4.1. A mobile reference frame o′- $x^{'} y^{'} z^{'}$ is located at the intersecting point of the two rotational axes with the x-axis being collinear with the rotational angle of the second leg and the y-axis being collinear with the rotational angle of the first leg. It is clear that the fixed and mobile reference frames are identical at this moment.

Compared with case I, only the second leg is different. So, we will focus on the mechanism shown in Figure 7(b).

The point c₁ locates at the intersecting line of the o-yz plane and the horizontal plane passing through the point o. The geometric parameters will be $o c_{2} = l_{1}$ and $o c_{1} = l_{2}$ . Then, the extensible link length l can be expressed as follows:

l = \sqrt{l_{1}^{2} + l_{2}^{2} - 2 l_{1} l_{2} \cos (\frac{π}{2} + β) .}

(11)

Therefore, for case II the inputs z and l can be obtained through (9) and (11).

5. Kinematic Optimum Design

5.1. Transmission Angle

The transmission angle is something we are very familiar with without realizing it; it is a classical concept in the field of four-bar mechanism design. In fact, in our daily life, we frequently try to move something which is constrained in some way and cannot be moved freely, such as the handle of a crank, a curtain on a rail, and a sliding door. In all of these cases, the object may not be able to be moved even when we exert pressure against it. Let us take the case of the handle of a crank as an example. As shown in Figure 8, the crank is attached to the base with a constant counterclockwise moment M; to move it, we must apply a right-hand force F at the end of the crank; when the direction of the force is constant, depending on the position of the end point it may be more or less easy to start rotating. This is actually a matter of the transmission angle. Since the direction of motion of a crank is always perpendicular to the crank, the smaller angle between the force and crank is defined as the transmission angle, denoted as μ. When the force is normal to the crank, that is, identical to the direction of motion, force transmission is most effective; when the force is perpendicular to the direction of motion, force transmission is very inefficient.

Figure 8

The handle of a crank.

For the planar four-bar mechanism shown in Figure 9, $O_{1} A$ is the input link, and the force applied to the output link $B O_{2}$ is transmitted through the coupler link AB. For sufficiently slow motions (negligible inertia forces), the force in the coupler link is pure tension or compression (negligible bending action) and is directed along AB. For a given force in the coupler link, the torque transmitted to the output bar (about point O₂) is at a maximum when the angle μ between the coupler bar AB and the output bar $B O_{2}$ is $9 0^{\circ}$ . Therefore, angle $A B O_{2}$ is called the transmission angle. In [17], the transmission angle is defined as the smaller angle between the direction of the velocity difference vector of the driving link and the direction of the absolute velocity vector of the output link, both taken at the point of connection. Although there are other definitions (see [11, 22], e.g.), all these definitions are somehow related to a joint variable(s) of the mechanism.

Figure 9

Transmission angle.

When the transmission angle deviates significantly from $9 0^{\circ}$ , the torque on the output bar decreases and may not be sufficient to overcome the friction in the system. For this reason, the deviation angle $α = | 90^{°} - μ |$ should not be too great.

Meanwhile, at the moment that the angle γ between the input link $O_{1} A$ and the coupler link AB is 0 or 180°, as shown in Figure 9, the output point B will not move whatever the input is. Then, the motion of the input cannot be transmitted to the output effectively. This means that the output point will lose a degree of freedom. Therefore, the deviation angle $β = | 90^{°} - γ |$ should not be too great either. In this paper, the angle μ is defined as the forward transmission angle, and the angle γ is referred to as the inverse transmission angle. If $B O_{2}$ is an active link, the angle $O_{1} A B$ is the forward transmission angle and angle $A B O_{2}$ theinverse transmission angle.

Take the slider-crank mechanism shown in Figure 2 as an example, where angle $O^{'} P_{3} B_{3}$ is the forward transmission angle and the angle between the coupler link and the normal to the straight-line path of the slider is the inverse transmission angle if the slider is actuated.

5.2. Definition of Some Indices

5.2.1. Local Transmission Index (LTI)

The condition number of the Jacobian matrix is an index that has been used successfully in the design of serial robots. Although the condition number is depended on the coordinate frame, it has the main advantage of describing the kinematic behavior of a robot by means of a number. The index has also been applied in the analysis and design of parallel robots. It was used as an index to evaluate the accuracy/dexterity [23, 24], to describe the closeness of a pose to a singularity of a parallel robot [25, 26]. In an optimal design, the condition number (or its reciprocal) was used to define a useful, good-condition, or effective workspace with a specified minimum LCI [10, 19, 27]. However, the minimum is still arbitrary or comparative since we cannot give it a definite value. Generally, it is not possible to define a mathematical distance to a singularity for a parallel robot [8]. Instead of LCI, another index will be defined here.

Following the definition of transmission index proposed in [28], an index based on the transmission angle is defined as

χ = \sin (T A),

(12)

where $T A = μ$ or $T A = γ$ . Then, there is

0 \leq χ \leq 1 .

(13)

A larger χ indicates better motion/force transmission. Since at a different pose the transmission angles will be different, the index χ is referred to as the local transmission index (LTI) in this paper. The angle is defined as the figure formed by two lines diverging from a common point, or as that formed by two planes diverging from a common line. Thus, the angle is usually measured by the ratio of two linear parameters. Therefore, the LTI is definitely independent of any coordinate frame. This is one of its advantages and is the most important for the optimal design of mechanisms.

For the purpose of high speed and high quality of motion/force transmission, the most widely accepted range for the transmission angle is $(45^{°}, 135^{°})$ [18] or $(40^{°}, 140^{°})$ [11]. Therefore, the LTI limit will correspondingly be

χ > s in (\frac{π}{4}) or χ > s in (\frac{2 π}{9}) .

(14)

Then, unlike the LCI, the LTI has a significative limitation to its application.

It is worth mentioning that, in [26, 29], the authors defined a transmission index, that is, cosα, by using the concept of the pressure angle α. Since the pressure angle is the complement of the transmission angle, the two definitions have the same significance. However, the authors did not define indices to evaluate a workspace with good transmission and to judge the effectiveness of motion/force transmission of a robot over a whole workspace. The two kinds of indices will be very important in the design of a parallel robot.

5.2.2. Good-Transmission Workspace (GTW)

With the minimum of LTI, that is, $\sin (π / 4)$ or $\sin (2 π / 9)$ , we can identify a workspace for a mechanism. The corresponding workspace is referred to as a good-transmission workspace (GTW), which is defined as the set of pose where the transmission index for every transmission angle is greater than $\sin (π / 4)$ or $\sin (2 π / 9)$ . In other words, within the GTW, at every pose the sine of every transmission angle is subject to the condition given by (14).

5.2.3. Global Transmission Index (GTI)

The LTI χ can only judge the effectiveness of motion/force transmission of a robot at a single pose. Usually, a robot performs a task in a specified workspace but not at a pose. In a practical design, we should make a decision whether a robot is good or not by taking into account the behavior within a considering workspace. In order to measure the global behavior of the motion/force transmission over the whole GTW, following the definition of GCI suggested in [30], a global transmission index (GTI) is defined as

Λ = \frac{\int_{W} \sum_{i}^{n} χ_{i} / n d W}{\int_{W} d W}

(15)

in which W denotes the good-transmission workspace, n the number of transmission angles, and $Λ_{\min} < Λ < 1$ (Λ_min equals $\sin (π / 4)$ or $\sin (2 π / 9)$ ). It is obvious that the GTI is also independent of any coordinate frame.

Undoubtedly, for manipulators with different link lengths, their GTWs will be the same. If such a case occurs, we cannot judge which robot is better with respect to the GTW index itself. However, with the same GTWs, their GTIs may be different. The two indices, GTW and GTI, together will help us to design a robot optimally. Additionally, for a specified design problem, other performance indices such as stiffness and accuracy may be involved in identifying a better solution. This is not the content of the paper. To demonstrate the use of the proposed indices, that is, LTI, GTI, and GTW, the subsequent section will discuss the optimal design of the A/B-axis tool head with parallel kinematics proposed in Section 2.

5.3. Performance Atlas

As analyzed in Section 2, to optimize the proposed A/B-axis tool head, we should first analyze the slider-crank mechanism.

For the case I, the first and second legs are similar. We need just to analyze one of them; here, we consider the mechanism shown in Figure 6. By using the normalization method introduced in [20], we here have three normalization parameters r₁, r₂, and r₃ for R₁, R₂, and R₃. The parameter design space of the slider-crank mechanism is constrained by

r_{1} + r_{2} + r_{3} = 3, r_{1} + r_{2} > r_{3}, r_{2} \leq r_{3},

(16)

where $r_{i} = R_{i} / D (i = 1,2, 3)$ and $D = (R_{1} + R_{2} + R_{3}) / 3$ is the normalization factor. The parameter design space is shown in Figure 10.

Figure 10

Parameter design space of the slide-crank mechanism.

We can easily get the relationship between $s, t,$ and $r_{1}, r_{2}, r_{3}$ , that is,

\begin{gathered} s = r_{1}, \\ t = \sqrt{3} - \frac{\sqrt{3}}{3} r_{1} - \frac{2 \sqrt{3}}{3} r_{2}, \\ r_{1} = s, \\ r_{2} = \frac{3 - s - \sqrt{3} t}{2}, \\ r_{3} = 3 - r_{1} - r_{2} . \end{gathered}

(17)

We should first find out the usable workspace, which is defined as the maximum continuous workspace that contains no singular loci inside but bounded by singular loci outside, in the parameter design space. For the slider-crank mechanism, the usable workspace can be written as $W_{ω_UW} = | ω_{O 1} - ω_{O 2} |$ , where ω_O1 and ω_O2 are the output angles when the mechanism is singular, respectively. And the atlas of the usable workspace of the slider-crank mechanism is shown in Figure 11. It is clear that, when s is near 1.5, the mechanism has a relatively larger usable workspace.

Figure 11

The usable workspace of the slider-crank mechanism.

As shown in Figure 6, according to (16), the forward and inverse transmission angles of the normalized mechanism can be obtained as

\begin{matrix} μ & = \cos^{- 1} (\frac{r_{1}^{2} + r_{2}^{2} - l^{2}}{2 r_{1} r_{2}}), \\ γ & = \cos^{- 1} (\frac{r_{3} - r_{2} \cos (ω)}{r_{1}}), \end{matrix}

(18)

where ω is the output angle, and $l^{2} = {(r_{1} \sin γ + r_{2} \sin ω)}^{2} + r_{3}^{2}$ .

Suppose that when $ω ∊ (ω_{\min}, ω_{\max}),$ the two transmission angles μ and γ in (18) are all subject to the LTI constraint given by (14). Here, ω_min and ω_max are the output angles where one of sinμ and sinγ is equal to $\sin (π / 4)$ or $\sin (2 π / 9)$ and another one is subject to (14). Then, the GTW for the output angle is defined as the relative angle between the two orientations ω_min and ω_max. If the workspace is denoted as $W_{ω_GTW}$ , we get

W_{ω_GTW} = ω_{\max} - ω_{\min} .

(19)

Figure 12 gives the relationship between $W_{ω_GTW}$ and $s, t$ when $χ \geq \sin (π / 4)$ . It shows that, when s is near 1.5 and t is less than 0.5, that is, r₁ is near 1.5 and r₃ is more than 0.5 and less than 1, the mechanism has a larger GTW.

Figure 12

Atlas of the good transmission workspace (GTI) of the slider-crank mechanism.

The relationship between good transmission index (GTI) and $s, t$ when $χ \geq \sin (π / 4)$ is shown in Figure 13, from which we can see that the GTI is inversely proportional to the parameter r₃ and is proportional to the parameter r₂.

Figure 13

Atlas of the good transmission index (GTI) of the slider-crank mechanism.

For the case II, the first leg is identified with that of the case I; what we need do is to analyze the extensible link mechanism shown in Figure 7(b).

Through similar process of normalization, we can get the normalization parameters l₁ and l₂ for L₁ and L₂. The parameter design space of the extensible link mechanism is constrained by

l_{1} + l_{2} = 2, l_{1} < l_{2},

(20)

where $l_{i} = L_{i} / D, (i = 1,2)$ and $D = (L_{1} + L_{2}) / 2$ is the normalization factor. From (20), we know that $l_{1} ∊ (0, 1)$ .

For the extensible link mechanism, only one transmission angle, that is, $μ = ∠ O C_{2} C_{1}$ should be considered. When $l_{1} < l_{2}$ and $4 5^{\circ} \leq μ \leq 13 5^{\circ}$ , we can get the two limit points $C_{2}^{'}$ and $C_{2}^{''}$ of point C₂, as shown in Figures 14 and 15.

Figure 14

The relationship between λ₁ and λ₂ when $μ = 45^{°}$ or 135°.

Figure 15

The rotational capability of the extensible link mechanism when $μ ∊ (45^{°}, 135^{°})$ .

From Figure 14, we can get

\begin{matrix} \frac{l_{1}}{\sin (λ_{1})} & = \frac{l_{2}}{\sin (4 5^{\circ})}, \\ \frac{l_{1}}{\sin (λ_{2})} & = \frac{l_{2}}{\sin (13 5^{\circ})} . \end{matrix}

(21)

Then

λ_{1} = λ_{2} .

(22)

That is to say that points C₁, $C_{2}^{'}$ , and $C_{2}^{''}$ are collinear as shown in Figure 15.

So, the angle between ${O C}_{2}^{'}$ and ${O C}_{2}^{''}$ is $9 0^{\circ}$ .

Figure 16 shows the relationship between GTW and l₁ ( $l_{1} ∊ (0, 1)$ ), from which we can see that $W_{gtw} = 9 0^{\circ}$ when $μ ∊ (4 5^{\circ}, 13 5^{\circ})$ . And Figure 17 gives the relationship between GTI and l₁ ( $l_{1} ∊ (0, 1)$ ).

Figure 16

The relationship between the good transmission workspace (GTW) and l₁.

Figure 17

The relationship between the good transmission index (GTI) and l₁.

5.4. Dimension Optimization Using the Performance Atlases

For the case I shown in Figure 3, we consider the design problem that the desired rotational angle is as much as $\pm 5 0^{\circ}$ at every point of the workspace. The optimization process of the first leg shown in Figure 6 based on the atlases of Figures 12 and 13 can be summarized as follows.

Step 1.

Identification of an optimum region in the parameter design space.

Please note that since all design conditions cannot be the same, the identification of an optimum region is up to the designer. Here, we just give an example.

For the slider-crank mechanism, an optimum region Ω will be determined with respect to the constraints of $W_{ω_GTW} \geq 100^{°}$ and better GTI, for example, $Λ > 0.92$ . This region is shown as the hatched parts in the parameter design space in Figure 18, where the region is divided two parts.

The identified optimum region contains all possible nondimensional solutions for the design problem. Since a nondimensional robot in the parameter design space and all of its corresponding dimensional mechanisms are similar in performance [20], the final design result based on the mechanism in the optimum region will be optimal. Therefore, the next step is to find an acceptable solution candidate in the optimum region.

Figure 18

An optimum region for the slider-crank mechanism when $W_{ω_GTW} \geq 100^{°}$ and $Λ > 0.92$ .

Step 2.

Selection of a solution candidate from the optimum region.

The obtained optimum region Ω contains all possible solutions for the design. Since there is no best but only a comparatively better solution for a design problem, one may pick up any nondimensional robot from the region.

We here pick up the nondimensional robot with parameters $r_{1} = 1.5$ , $r_{2} = 0.65,$ and $r_{3} = 0.85$ from the optimum region Ω. For the selected robot, there are $W_{ω_GTW} = {102.7}^{°}$ and $Λ = 0.9347$ . The orientation range for the robot is defined by $ω_{\min} = {- 69.5}^{°}$ and $ω_{\max} = {33.2}^{°}$ .

Step 3.

Determination of the dimensional parameters R₁, R₂, and R₃ with respect to the desired workspace.

According to the normalization method introduced in [20, 31], we should first determine the normalization factor, that is, D. This factor can be obtained by comparing the desired workspace of a design problem and the workspace of the nondimensional mechanism selected from the optimum region.

For the slider-crank mechanism, the output angle is only dependent on the ratio of related linear parameters but not on any one of the parameters itself, we cannot determine the normalization factor D with respect to the orientational workspace $\pm 50^{°}$ . In this case, we should first determine the parameter R₁ of the mechanism according to the practical application and let the parameter be as small as possible for reducing the occupation space of the manipulator. Here, we take $R_{1} = 150 mm$ , for example. There are $D = R_{1} / r_{1} = 100, R_{2} = D r_{2} = 65 mm,$ and $R_{3} = D r_{3} = 85 mm$ .

Step 4.

Calculation of the input range and adjusting angle.

The input range to reach the desired workspace can be calculated by means of the inverse kinematic equations (9) and (10). In Step 2, we have got $ω_{\min} = - {69.5}^{°}$ and $ω_{\max} = {33.2}^{°}$ , please note that ω_min and ω_max are not symmetric about zero, and to make the output angle is α (shown in Figure 6) bilateral symmetry about O-yz plane, we choose the adjusting angle $ε_{1} = 9 0^{\circ} + (ω_{\min} + ω_{\max}) / 2 \approx 7 2^{\circ}$ . Then, when the output angle $α = \pm 5 0^{\circ}$ , $ω ∊ (- 6 8^{\circ}, 3 2^{\circ})$ and the input range for the actuator is $z ∊ [- 76.92 mm, - 181.44 mm]$ .

Step 5.

Checking the design result and adjusting, if necessary, the design solution.

In this step, what should be checked depends on the application. For example, the designer may check whether the input range is suitable for his preferred commercial actuator. Whatever is checked, if the solution is not satisfactory, the designer can return to Step 2 and pick up another nondimensional mechanism, or even return to Step 1 to identify another optimum region, adjusting the design solution until the solution is satisfactory. This is actually the advantage of this design method.

Similarly, we still pick up the nondimensional robot with parameters $r_{1} = 1.5$ , $r_{2} = 0.65$ , and $r_{3} = 0.85$ from the optimum region Ω. According to the practical problem, the geometric parameters of the second leg can be got as $D_{1} = 120 mm$ , $D_{2} = 52 mm$ , and $D_{3} = R = 68 mm$ .

Then, the adjusting angle is $ε_{2} = 16 2^{\circ}$ . When the output angle is $β = \pm 5 0^{\circ}$ , the input range for the actuator is $y ∊ [- 104.72 mm, 19.53 mm]$ .

For the case II shown in Figure 4, the optimization process and the parameters of the first leg are identified with those in the case I. For the second leg, we pick up the nondimensional robot with parameters $l_{1} = 0.8$ and $l_{2} = 1.2$ as shown in Figures 16 and 17. From the two atlases, we can see that $W_{ω − g t w} = 9 0^{\circ}$ and $Λ = 0.8958$ . To compare it with the second leg in the case I, we take $L_{1} = 52 mm$ and $L_{2} = 78 mm$ , for example. Then, the adjusting angle is $ε_{3} = 15 2^{\circ}$ and the output angle is $β = \pm 4 5^{\circ}$ . The specified information is shown in Figure 19.

Figure 19

The second leg in the case II.

6. Orientation Capability and Parasitic Motions Analysis

In Section 3, we have introduced the method of orientation description of the tool head. In this section, we will investigate the relationship between theazimuth and tilt angles and thegood transmission workspace of the designed tool head in Section 5.

We can get the output angles (α and β) of the first and second legs from (3).

For the case I, as analyzed in Section 5.4, the good transmission workspace is $α = \pm 5 0^{\circ}$ and $β = \pm 5 0^{\circ}$ . The orientation workspace of the tool head is shown in Figure 20.

Figure 20

The orientation workspace of the tool head in the case I.

The parasitic motions can be got from (4). In order to know the relationship between the orientation angles and the parasitic motions, we need first to find out the relationship between the orientation angles and the output angle α of the first leg, as shown in Figure 21. Figure 22 shows the relationship between the orientation angles and the output angle α. Then, from Figures 21 and 22, we can clearly know how the parasitic motions are.

Figure 21

The relationship between the orientation angles and the output angle α of the first leg in the case I.

Figure 22

The relationship between parasitic motions and the output angle α of the first leg in the case I.

For the case II, as analyzed in Section 5.4, the good transmission workspace can be described as $α = \pm 5 0^{\circ}$ and $β = \pm 4 5^{\circ}$ . Similarly, we can get the orientation workspace of the tool head as shown in Figure 23.

Figure 23

The orientation workspace of the tool head in the case II.

7. Development of the Tool Head

We have built the real model of this kind of mechanism, as shown in Figures 24 and 25 and may have them manufactured and carry out some experiments further.

Figure 24

The real model of the mechanism in the case I.

Figure 25

The real model of the mechanism in the case II.

8. Conclusion

In this paper, we have introduced and analyzed an A/B-axis tool head with parallel kinematics, which is kinematically decoupled. Firstly, we have introduced a method of orientation description of the tool head, that is, the azimuth and tilt angles. Then, the transmission angle was recalled and the forward and inverse transmission angles were defined accordingly. Based on the concept of transmission angle, we proposed and defined some new indices, that is, local transmission index (LTI), good transmission workspace (GTW), and global transmission index (GTI). Then, the performance atlases and optimal design of the tool head based on these indices are presented. Finally, we analyzed the orientation capability and the parasitic motions of the designed tool head. The results show that the azimuth and tilt angles are convenient to describe the orientation of the tool head discussed here, and the indices proposed in this paper have obvious physical significance and could be extended to the analysis and design of other parallel robots.

Footnotes

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 50775118, the National Basic Research Program (973 Program) of China (no. 2007CB714000), and by Program for New Century Excellent Talents in the University of China.

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