Analysis of a Five-Degree-of-Freedom Hybrid Robot RPR/RP + RR + P

Abstract

This work proposes a hybrid machining robot RPR/RP + RR + P based on a planar parallel mechanism. Based on the screw theory, the characteristics of the degree of freedom of the hybrid robot are analyzed, and then the inverse and forward kinematics are solved by geometric methods. And size optimization is carried out by taking the workspace and driving forces as the objects. Then the working space of the hybrid robot is calculated by the Monte Carlo method. Finally, the three-dimensional model of the mechanism is established, and through ADAMS simulation, the accuracy of the inverse kinematics solution was verified.

1 Introduction

With the development of aerospace, aviation, automotive industries, and so on, the demands for high-efficiency machining of large-scale structural parts with high material removal ratio and complex parts with curved surfaces increase rapidly. At the same time, traditional serial machine tools have insufficient flexibility, large inertia, and parallel machine tools have small working space, which fails to meet the requirements. The new types of machining equipment are urgently needed. Compared with serial robots and parallel robots, hybrid robots combine the advantages of these two robots and have attracted the attention of many researchers and companies. Hybrid robots can be divided into planar hybrid robots (formed by connecting a swing head with a certain degree of freedom (DOF) in serial on a planar parallel mechanism) and spatial hybrid robots (formed by connecting a swing head with a certain DOF in serial on a spatial parallel mechanism). Compared with the spatial hybrid mechanism, the planar hybrid mechanism has the advantages of simple structure, simple kinematics and dynamics models and low manufacturing cost.

At present, there are many spatial hybrid robots, such as Tricept hybrid robots [1, 2], Exechon hybrid machining center [3, 4], TriMule [5, 6] and TriVariant [7, 8] hybrid robots as well as other robots [9,10,11,12,13,14] etc. And there are relatively few planar hybrid mechanisms. Yang and Wang [15] proposed to develop the XNZD755 planar hybrid machine tool and analyzed the dynamics of the machine tool. A redundantly actuated 3-DOF planar PM was developed in [16] for a 4-DOF hybrid machine tool and again in [17] for 5-DOF one.

This work will propose a new five-DOF planar hybrid robot RPR/RP + RR + P, which is used to machine complex aluminum alloy parts for new energy vehicles.

The remainder of this article is organized as follows: Sect. 2 describes the composition of the hybrid robot, and introduces the DOF of the mechanism. Section 3 analyzes the robot’s inverse and forward kinematic. Section 4 carries out the size optimization by taking the workspace and driving forces as the objects, and analyzes the workspace of the hybrid robot under the given mechanism parameters. Section 5 gives the three-dimensional model of the mechanism, and the verifies the correction of inverse kinematics analysis. Finally, conclusions are drawn in Sect. 6.

2 Description of the Five-Axis Hybrid Robot

2.1 Mechanism Composition

The RPR/RP + RR hybrid robot proposed in this article is shown in Fig. 1. The robot consists of three parts: a planar parallel mechanism RPR/RP, a serial BC swing head, and a serial P joint. The moving platform of the parallel mechanism RPR/RP is supported by two limbs A₁a₁ and A₂a₂. The limb A₁a₁ is connected to the fixed platform and moving platform both through the R joint. The limb A₂a₂ is connected to the fixed platform through an R joint, which is connected to the moving platform through a prismatic joint directly. The axes of three R joints are parallel to each other, and the axes of the prismatic are perpendicular to the axes of the R joints.

Fig. 1

Schematic diagram of the hybrid robot

As shown in Fig. 1, the fixed platform and the moving platform of the parallel mechanism are equivalent to two straight lines. The length of the fixed platform is 2a, and the length of the moving platform is 2b. Point G is the intersection point of the BC swing head. The straight-line Go₃ is perpendicular to the moving platform, the vertical foot is the o₂ point, the length of Go₃ is k, and the length between the intersection point of the BC swing head and the tip of the tool is f.

2.2 Coordinate System Establishment

In order to establish the position model of the mechanism, the global frame O-XYZ is established at point O on the fixed platform, where the X-axis is parallel to the line A₂A₁, the Z-axis is parallel to the axis of the R joint, and the right-hand system rule determines the Y-axis. The local frame o₁-x₁y₁z₁ is established at o₁ on the fixed platform, where the x₁-axis points from point A₂ to A₁, the z₁-axis is collinear with the Z-axis, and the right-hand system rule determines the y₁-axis.

The local frame o₂-x₂y₂z₂ is established at point A₂, where the x₂-axis is collinear with A₁A₂, the z₂-axis is collinear with the axis of the R₂₁, and the right-hand system rule determines the y₂-axis. The local frame o₃-x₃y₃z₃ is established at the midpoint of a₁a₂, where the x₃-axis is collinear with a₁a₂, the y₃-axis is collinear with the C axis of the BC swing head, and the right-hand system rule determines the z₃-axis. Moreover, the local frame o₄-x₄y₄z₄ is established at the point G, where the y₄-axis is collinear with the axis of the end handle, and the z₄-axis is collinear with the B axis. The right-hand system rule determines the x₄-axis, as shown in Fig. 1.

2.3 Degree of Freedom Analysis

The hybrid robot is composed of a parallel part and two serial parts. The parallel part mechanism is RPR/RP, composed of limbs A₁a₁ and A₂a₂. The schematic diagram of the parallel mechanism RPR/RP is shown in Fig. 2.

Fig. 2

Schematic diagram of the parallel mechanism

The DOF of the parallel mechanism can be calculated from the revised G-K formula:

$$K = d\left( {n - g - 1} \right) + \sum\limits_{i = 1}^{g} {f_{i} } + v$$

(1)

d:

The order of the parallel mechanism, $d = 6 - \lambda$;

n:

Number of parallel mechanism components;

g:

Number of joints of parallel mechanism;

$f_{i}$:

The number of DOFs of the ith joint;

v:

Number of redundant constraints of parallel mechanism.

The expression of redundant constraint v is:

$$v = l - k$$

(2)

where l is the number of remaining constraint wrenches after removing the public constraints of the mechanism, and k is the rank of the remaining constraint wrenches. It can be obtained that l is 3, k is 3, the DOF of the RPR/RP mechanism can be obtained as:

$$\begin{aligned} K & = d\left( {n - g - 1} \right) + \sum\limits_{i = 1}^{g} {f_{i} } + v \\ & = 3 \times \left( {5 - 5 - 1} \right) + 5 \\ & = 2 \\ \end{aligned}$$

(3)

The DOF of the parallel mechanism is 2, including the translation along the rod A₂a₂ and the rotation around the axis of the R joint at A₂.

The mechanism is a parallel mechanism in serial with a serial two-DOF BC swing head and a serial one-DOF P joint, so the hybrid robot is a five-DOF hybrid robot.

3 Inverse and Forward Kinematics Analysis

3.1 Inverse Kinematics of Parallel Mechanism

In the inverse solution, it is considered that the coordinates of the tool tip point ${\varvec{F}} = [F_{x} ,F_{y} ,F_{z} ]^{T}$ and the unit direction vector S of the end tool rod are known, and the purpose is to calculate the input of each actuator, i.e., h, l₁, l₂, α, β, which are the inputs of P₃, P₁, P₂, C, B respectively.

Because the tool tip point coordinate F and the unit direction vector S are known, the coordinates can be easily obtained from the established coordinate system:

$$\begin{gathered} G_{x} = F_{x} - f \cdot [1,0,0] \cdot \varvec{S} \hfill \\ G_{y} = F_{y} - f \cdot [0,1,0] \cdot \varvec{S} \hfill \\ G_{z} = F_{z} - f \cdot [0,0,1] \cdot \varvec{S} \hfill \\ \end{gathered}$$

(4)

The expression of point G in the global frame O-XYZ and local frame o₂-x₂y₂z₂ can be obtained:

$${\varvec{G}} = \left( {G_{x} ,G_{y} ,G_{z} } \right)^{{\text{T}}} \quad {\varvec{G}}_{2} = \left( {G_{x} + a,G_{y} ,0} \right)^{{\text{T}}}$$

(5)

Then the input of P₃ is G_z, that is: h = G_z.

As shown in Fig. 1, the angle $\theta_{1}$ can be obtained by the coordinate relationship of point G in local frame o₁-x₁y₁z₁:

$$\begin{gathered} \sin \theta_{1} = \frac{{G_{x} + a}}{{\left| {G_{2} } \right|}} \hfill \\ \cos \theta_{1} = \frac{{G_{z} }}{{\left| {G_{2} } \right|}} \hfill \\ \end{gathered}$$

(6)

where $\theta_{1}$ is the angle of ∠A₂Go₁, l_G is the length of GA₂.

As shown in Fig. 1, in the trapezoid $GKa_{2} o_{2}$, ∠1 is known, o₃a₂ = $b$, Go₃ = $k$, then through geometric relations, the length of $GK$ and $Ka_{2}$ can be calculated. In the triangle $GKA_{2}$, according to the law of sine

$$\sin \theta_{2} = \frac{GK\sin (\angle 1)}{{\left| {G_{2} } \right|}}$$

(7)

where $\theta_{2}$ is angle of ∠KA₂G.

And the length of l₂ can be calculated based on sine low:

$$l_{2} = \frac{{GK\sin \left( {\pi - \angle 1 - \theta_{2} } \right)}}{{\sin (\theta_{2} )}} - Ka_{2}$$

(8)

After obtaining $\theta_{1}$ and $\theta_{2}$, the rotation angle $\theta_{3}$ of the mechanism around the axis of R₂₁ can be obtained. Then the point of ${}^{3}a_{i}$ (i = 1, 2) can be expressed in the local frame o₁-x₁y₁z₁.

$${}^{1}a_{i} = R(\theta_{3} ){}^{3}a_{i} + {}_{3}^{1} P$$

(9)

where $R(\theta_{3} )$ is the transformation matrix of local frame o₃-x₃y₃z₃ respect to local frame o₁-x₁y₁z₁, and ${}_{3}^{1} P$ is origin of local frame o₃-x₃y₃z₃ respect to local frame o₁-x₁y₁z₁, which can be gotten through the closed-loop vector.

The length of two limbs of the parallel part can be obtained:

$$l_{i} = \left\| {{\varvec{a}}_{i1} - {\varvec{A}}_{i} } \right\|\quad \left( {i = 1,2} \right)$$

(10)

Substituting the coordinates of points $a_{11}$, $a_{21}$, $A_{1}$, and $A_{2}$, the length of two limbs can be calculated.

3.2 Solving the Rotational Angles of the Serial Part

The unit direction vector of the hilt is S, S = (m, n, q)^T, and set two rotating angles of the BC swing head in the local frame O-XYZ: α′ is the rotating angle around the x₃ axis, and β′ is the rotating angle around the z₄ axis. Then the angle α′ and angle β′ are:

$$\begin{gathered} \alpha^{\prime} = \arctan \left( \frac{n}{m} \right) \hfill \\ \beta^{\prime} = \arccos \left( q \right) \hfill \\ \end{gathered}$$

(11)

Then through transformation matrix ${\varvec{R}}\left( {\theta_{3} } \right)$, the input angle α of head C and the input angle β of head B can be obtained.

3.3 Forward Kinematics Analysis

In the forward solution, the inputs of h, l₁, l₂, α and β are known. The purpose is to solve the coordinate $F = \left[ {F_{x} ,F_{y} ,F_{z} } \right]^{T}$ of the tool tip point and the unit direction vector S of the hilt.

Because the length l₂ is known, the length of l_G can be obtained based on cosine law:

$$l_{G} = \sqrt {KG^{2} + KA_{2}^{2} - 2\cos (\angle 1) \cdot KG \cdot KA_{2} }$$

(12)

As shown in Fig. 2, in the quadrilateral A₁A₂a₁a₂, the length of four sides and $\angle 2$ are known. According to the law of cosines, the length a₁A₁ and the angle of $\angle a_{1} A_{2} a_{2}$ and $\angle a_{1} A_{2} A_{1}$ can be calculated.

According to formula (7), the angle $\theta_{2}$ can be obtained. Then the angle of $\angle A_{1} A_{2} G$ and $\theta_{3}$ can be obtained by Eq. (13).

$$\begin{gathered} \angle A_{1} A_{2} G = \angle a_{1} A_{2} a_{2} + \angle a_{1} A_{2} A_{1} - \theta_{2} \hfill \\ \theta_{3} = \angle a_{1} A_{2} a_{2} + \angle a_{1} A_{2} A_{1} + \angle 1 - \pi \hfill \\ \end{gathered}$$

(13)

The coordinate of G in the frame O-XYZ can be expressed:

$$G = \left[ {l_{G} \cos \left( {\angle A_{1} A_{2} G} \right) - a,l_{G} \sin \left( {\angle A_{1} A_{2} G} \right),h} \right]$$

(14)

And the input angles α and β are known, the unit direction vector S can be obtained by the rotation transformation matrix $R\left( {\theta_{3} } \right)$.

Then the coordinates of the tool tip point F can be obtained.

$$\begin{gathered} F_{x} = G_{x} + f \cdot [1,0,0] \cdot \mathop{S}\limits^{\rightharpoonup} \hfill \\ F_{y} = G_{y} + f \cdot [0,1,0] \cdot \mathop{S}\limits^{\rightharpoonup} \hfill \\ F_{z} = G_{z} + f \cdot [0,0,1] \cdot \mathop{S}\limits^{\rightharpoonup} \hfill \\ \end{gathered}$$

(15)

4 Dimensional Optimization and Workspace Analysis

Taking the working space and driving force as optimization objective, the robot’s key dimensions are optimized. The parallel mechanism part is optimized as the main optimization object of the hybrid robot. Firstly, the parallel mechanism is determined based on the size of the BC swing head, and the size of its moving platform is 420 mm. Secondly, based on the limit of the drive joint, the change range of the PM’s limb is determined to be [390 mm, 760 mm], and the optimization range of the fixed platform’s size is [640 mm, 840 mm]. Under the action of external force F = [10 N, 10 N, 10 N, 10 N mm, 10 N mm, 10 N mm]^T, finding the optimal size of the fixed platform through optimization. The flow chart of the optimization procedure is shown in Fig. 3. Table 1 lists the force and workspace indexes with different sizes of the fixed platform.

Fig. 3

Flow chart of platform parameter scaling optimization

Table 1 The force and workspace indexes with different sizes of the fixed platform

Finally, through comparison, it is determined that the fixed platform size of the parallel mechanism is 740 mm. Therefore, the key dimensions of the hybrid robot are shown in Table 2.

Table 2 The key dimensions of hybrid robot RPR/RP + RR

According to the dimensions given in Table 2, the Monte Carlo method is used to analyze the new workspace of the hybrid robot, and its dexterity space is shown in Fig. 4.

Fig. 4

The workspace of the five-axis hybrid robot RPR/RP + RR + P

Through observation, the robot can translate in a large range in the Z direction, which is convenient for machining narrow and long parts. At the same time, due to the existence of the BC swing head, the flexibility of the robot end tool in the workspace can be ensured, which is convenient for machining various complex curved surfaces, holes, etc. Therefore, it can meet the machining workspace requirements of aluminum alloy parts with complex curved surfaces for new energy vehicles.

5 Simulation Verification of Inverse Kinematic Analysis

Based on the obtained dimensions of the hybrid robot RPR/RP + RR + P shown in Table 2. As shown in Fig. 5, the three-dimensional model of the hybrid robot RPR/RP + RR + P is designed.

Fig. 5

Three-dimensional model of the hybrid robot

In ADAMS, a motion trajectory is given for the tool tip of the hybrid robot, as shown in Eq. (16).

$$\left\{ {\begin{array}{*{20}l} {X = 20t\,({\text{mm}})} & {t \in [1,5]} \\ {Y = 10(t - 5)\,({\text{mm}})} & {t \in [5,10]} \\ {Z = 40t\,({\text{mm}})} & {t \in [1,5]} \\ {\alpha^{\prime} = \pi t/45\,(^\circ )} & {t \in [1,5]} \\ {\beta^{\prime} = \pi t/90\,(^\circ )} & {t \in [1,5]} \\ \end{array} } \right.$$

(16)

After simulation, the lengths l₁, l₂, h, and angles α, β of the BC swing head can be measured. Then substituting the inputs into forward kinematics model, the coordinate curve of the tool tip can also be obtained again, as shown in Fig. 6. It can be found that the results are consistent with the values given by Eq. (16), indicating the kinematics analysis is correct.

Fig. 6

The coordinates variation curve of end tip measured in the simulation model

6 Conclusions and Future Works

A new type of five-degree-of-freedom hybrid robot RPR/RP + RR + P is proposed for machining of aluminum alloy plates for new energy vehicles. The modified hybrid robot includes a two-degree-of-freedom parallel mechanism and two series parts, which are the translation along the limb A₂a₂ and the rotation around the axis of the R joint.

Through the screw theory, the degree of freedom of the robot is analyzed, and the result shows that the robot has five DOFs, and can achieve five-axis hybrid processing.

Through the geometric relationship, the forward and inverse position solutions of the hybrid robot are analyzed. Compared with the general hybrid robot, the forward and inverse position model are simple, thus the hybrid robot is easy to control.

Taking the limb driving forces and working space as the optimization objective, the dimensions of the mechanism is optimized, and the fixed platform size is determined to be 370 mm. After that, the working space of the robot is obtained by the Monte Carlo method, which is a cylindrical body with an approximate triangular section.

Finally, the three-dimensional model of the mechanism is also constructed through Solidworks software, and the forward and inverse kinematic models of the hybrid robot are verified by using the software ADAMS and MATLAB.

The force and stiffness analysis of the five-axis hybrid machining robot will be carried out in future work. And the prototype will be developed, the prototype experiments will be carried out.