Compliance analysis and lightweight design of a two-degree-of-freedom rotating head

Abstract

This article presents an approach for the compliance analysis and lightweight design of a two-degree-of-freedom rotating head by considering both gravity and joint/link compliances, which provides a comprehensive understanding on the posture adjustment mechanism of five-degree-of-freedom hybrid manipulator. A kinetostatic analysis is carried out to consider both externally applied wrench imposed upon the end-effector and gravity of all movable components. Then, a deflection analysis integrating both joint and link compliances and formulation of component compliance matrices are completed by using a semi-analytical approach. Finally, the lightweight design of two-degree-of-freedom rotating head is realized by considering the deflection constraints. This approach enables to effectively evaluate the deflections of end-effector caused by both payload and gravity under given operation conditions. Moreover, the established method provides reliable guidelines for the design of two-degree-of-freedom rotating head with superior static rigidities and dynamic behaviors.

Introduction

In modern industries, for example, aerospace, shipbuilding, and precision instruments, five-axis computer numerical control (CNC) machine tools play a critical role in manufacturing machinery components with superior quality.^1–3 Two-degree-of-freedom (2-DOF) rotating head is an essential part to fabricate these five-axis CNC machine tools because of its excellent performance, such as compact structure, flexible motion, high accuracy, and high rigidity. In conjunction with 1T2R (T denotes a translation DOF and R a rotational DOF) three-degree-of-freedom (3-DOF) parallel mechanism, 2-DOF rotating head is also used to construct five-degree-of-freedom (5-DOF) hybrid robots, Tricept robot, ⁴ Exechon robot, ⁵ and TriVariant robot, ⁶ for example. Due to their large workspace, high accuracy, and rigidity, they are preferable to developing plug and play module, and building customized manufacturing units or systems. With the aid of these robots, it is convenient to achieve automatic drilling and riveting for large components of an aircraft, field processing for fuselage and wing surface, interference assembling for cylinder pin holes in automobile engines, and wire-electrode cutting for large steel structures. Thus, they have been widely used to machine, weld, and assemble large structures in the fields of aerospace, automotive, and construction. ⁷ It should be pointed out that 2-DOF rotating head is necessary to realize the postural adjustments in five-axis CNC machine tools and 5-DOF hybrid robots when they are used in the applications mentioned above.

As a key functional component of five-axis CNC machine tools, 2-DOF rotating head, for example, biaxial rotary milling head, ⁸ has gained massive research interests. Extensive efforts have been made to explore the design methods of mechanical structure and transmission, the manufacturing and assembly techniques of the mechanism, and the performances, including stiffness, accuracy, dynamic characteristics, and reliability.^8–18 Gao et al.^14–16 overcame the difficulties of large volume, low rigidity, and low carrying capacity in higher pair transmission by investigating four types of biaxial rotary milling heads, for example, six-rod whole hinge double-row drive type, six-rod whole hinge singe-row drive type, single screw drive type, and double screw drive type. Cai et al.^17,18 established a technique system for heavy loading biaxial rotary milling head by integrating the analysis of kinematics, statics, dynamics, heat property, and thermomechanical coupling. They also carried out a systematic study on modal test, thermal analysis, and precision experiments to confirm the validity of the developed system. Hence, their works lay a theoretical and experimental foundation for optimizing the heavy loading biaxial rotary milling head. In addition, some 2-DOF parallel mechanisms are designed to a 2-DOF rotating head as special device, for example, solar tracker. ¹⁹

As the main body of 5-DOF hybrid robot, a great deal of attention has been paid to 3-DOF parallel mechanism of the hybrid robot.^20–30 However, as the postural adjustment mechanism of 5-DOF hybrid robot, 2-DOF rotating head is rarely studied. Yet, some works ³¹ have demonstrated that the gravity of 2-DOF rotating head has an important influence on the positioning accuracy of hybrid robots, especially when they have horizontal layout, leading to an issue in need of considerable further investigation for achieving a lightweight yet rigid design of the 2-DOF rotating head. In addition, dynamic characteristics of 2-DOF rotating head should be incorporated into the analysis when it is attached to the five-axis machine tools involving high frequencies of acceleration–deceleration cycles. To improve the dynamic characteristics of the mechanism and performance of machine tools, lightweight design has been proposed as it enables the mechanism to either keep the stiffness unchanged while reducing its weight or keep the weight unchanged while improving the stiffness. ³² Such lightweight design can be achieved by optimizing the topology, material, and structural parameter of the mechanism, and is widely applied to the optimal design of machine tool structure components.^33–37 For example, Zulaika et al. ³⁶ proposed a lightweight design method to minimize the mass of structural components of a milling machine by considering the maximum of the material removal rate. These results demonstrate that the lightweight design can avoid elastic dynamics modeling but implicitly improve the dynamic performance. Unfortunately, it has rarely been used in optimizing the structural parameters of 2-DOF rotating head.

This article presents an approach for the compliance analysis and lightweight design of a 2-DOF rotating head that can be used as the posture adjustment mechanism of 5-DOF hybrid manipulators. Having addressed the significance of lightweight design, the article is organized as follows. First, the system description and frame establishment are briefly introduced. Then, the procedures that realize the compliance analysis and lightweight design are developed systematically. Finally, a numerical example is given by using the established method to confirm its validity. It is expected that the approach presented in this article could provide a reliable guideline for the lightweight design of 2-DOF rotating head.

System description and frames establishment

An A/C rotating head generally consisted of an A-axis, a C-axis, and an end-effector. As illustrated in Figure 1, an electric spindle serves as the end-effector, and it rotates with respect to the A-axis which further rotates with respective to C-axis, forming a 2-DOF rotating head.

Figure 1.

Body fixed frames of the 2-DOF rotating head.

The coordinate systems of the body fixed frames of electric spindle {R_E}, A-axis {R_A}, and C-axis {R_C} are shown in Figure 1. Assuming that α is the rotating angle between A-axis and C-Axis, and β is the rotating angle between electric spindle and A-axis, the orientation matrices of {R_E} and {R_A} with respect to {R_C} can be expressed as

$${\mathit{R}}_A=\mathrm{Rot}(z_C,\alpha )=\left[\begin{array}{ccc}{\mathit{u}}_A\hfill & {\mathit{v}}_A\hfill & {\mathit{w}}_A\hfill \end{array}\right]$$

(1)

$${\mathit{R}}_E=\mathrm{Rot}(z_C,\alpha )\mathrm{Rot}(x_A,\beta )=\left[\begin{array}{ccc}{\mathit{u}}_E\hfill & {\mathit{v}}_E\hfill & {\mathit{w}}_E\hfill \end{array}\right]$$

(2)

where u _x , v _x , and w _x are the unit vectors of the three orthogonal axes of {R_X} evaluated in {R_C}; R _A and R _E are the orientation matrices of {R_E} and {R_A} with respect to {R_C}, respectively.

Compliance analysis

To integrate the component compliances and the gravity-induced deflections into analysis, a semi-analytic compliance model of the 2-DOF rotating head is presented in this section.

Force analysis

To establish a linear map between the externally applied wrench imposed on the end-effector and the joint reaction forces, the static analysis of the rotating head is essential. To solve the joint reaction forces of A-axis, it requires investigating the forces acting on the electric spindle as shown in Figure 2. Integrating these forces and moments with respect to point E, the static equilibrium equations can be derived as follows

$$\begin{array}{c}\left(\begin{array}{c}{\mathit{f}}_E\hfill \\ \multicolumn{1}{c}{{\mathbf{\tau }}_E}\end{array}\right)-m_{E}g\left(\begin{array}{c}\hat{\mathit{y}}\hfill \\ \multicolumn{1}{c}{\left({\mathit{R}}_E{\mathit{r}}_{G_E}^E\right)\times \hat{\mathit{y}}}\end{array}\right)\hfill \\ \multicolumn{1}{c}{=\left(\begin{array}{c}{\mathit{R}}_A{\mathit{f}}_P\hfill \\ \multicolumn{1}{c}{\left({\mathit{R}}_E{\mathit{r}}_P^E\right)\times \left({\mathit{R}}_A{\mathit{f}}_P\right)+{\mathit{R}}_A{\mathbf{\tau }}_P}\end{array}\right)}\\ {\mathit{f}}_P={\left(\begin{array}{ccc}{f}_{x_A}^P\hfill & f_{y_A}^P\hfill & f_{z_A}^P\hfill \end{array}\right)}^{\mathrm{T}},{\mathbf{\tau }}_P={\left(\begin{array}{ccc}{\tau }_{x_A}^P\hfill & {\tau }_{y_A}^P\hfill & {\tau }_{z_A}^P\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{\hat{\mathit{y}}={\left(\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\end{array}$$

(3)

where f _E and τ _E are the externally applied force and moment acting on the point E of the end-effector, respectively; m_E is the mass; ${\mathit{r}}_{G_E}^E$ is the position vector of mass center G_E of end-effector evaluated in $\{R_E\}$ ; ${\mathit{r}}_P^E$ is the position vector from point E to point P evaluated in $\{R_E\}$ ; $\hat{\mathit{y}}$ is the unit vector along the inverse direction of the gravitational field; $f_{x_A}^P$ , $f_{y_A}^P$ , and $f_{z_A}^P$ are the reaction forces acting on point P along the axis of x_A, y_A, and z_A, respectively; ${\tau }_{x_A}^P$ , ${\tau }_{y_A}^P$ , and ${\tau }_{z_A}^P$ are the corresponding moments of the reaction forces. Solving equation (3) leads to

$$\begin{array}{c}{\mathbf{\rho }}_{w,A}={\mathit{T}}_A^{\mathrm{T}}{\mathbf{\$}}_{w,A}\hfill \\ {\mathbf{\rho }}_{w,A}=\left(\begin{array}{c}{\mathit{f}}_P\hfill \\ \multicolumn{1}{c}{{\mathbf{\tau }}_P}\end{array}\right),{\mathbf{\$}}_{w,A}={\mathbf{\$}}_{w,E}+{\mathbf{\$}}_{w,G_E},{\mathbf{\$}}_{w,E}=\left(\begin{array}{c}{\mathit{f}}_E\hfill \\ \multicolumn{1}{c}{{\mathbf{\tau }}_E}\end{array}\right)\hfill \\ {\mathbf{\$}}_{w,G_E}=-m_{E}g\left(\begin{array}{c}\hat{\mathit{y}}\hfill \\ \multicolumn{1}{c}{\left({\mathit{R}}_E{\mathit{r}}_{G_E}^E\right)\times \hat{\mathit{y}}}\end{array}\right)\hfill \\ {\mathit{T}}_A=\left[\begin{array}{cc}{\mathit{R}}_A\hfill & [{\mathit{r}}_P\times ]{\mathit{R}}_A\hfill \\ \multicolumn{1}{c}{0}& {\mathit{R}}_A\hfill \end{array}\right],{\mathit{r}}_P={\mathit{R}}_E{\mathit{r}}_P^E\hfill \end{array}$$

(4)

where ${\mathrm{\$}}_{w,A}$ is the resultant externally applied wrench acting on $E$ , ${\mathbf{\$}}_{w,G_E}$ is the wrench resulted from the gravity of end-effector, and ${\mathit{T}}_A$ is the transformation between the wrench acting on the end-effector and the joint reaction forces of A-axis. Apparently, the gravity of end-effector can be equivalent to an externally applied wrench in the calculations of joint reaction forces, which significantly affects the joint reaction forces of A-axis, while the gravity of A-axis is not balanced by any joint reaction force of A-axis.

Figure 2.

Free body diagram of the electric spindle.

The joint reaction forces of C-axis can be obtained in a similar approach as illustrated in Figure 3. Considering the forces and moments acting on point E, the static equilibrium equations can be derived as follows

$$\begin{array}{l}\left(\begin{array}{c}{\mathit{f}}_E\hfill \\ {\mathit{\tau }}_E\hfill \end{array}\right)-m_{E}g\left(\begin{array}{c}\widehat{\mathit{y}}\hfill \\ \left({\mathit{R}}_E{\mathit{r}}_{G_E}^E\right)\times \widehat{y}\hfill \end{array}\right)-m_{A}g\left(\begin{array}{c}\widehat{\mathit{y}}\hfill \\ \left({\mathit{R}}_E{\mathit{r}}_P^E+{\mathit{R}}_A{\mathit{r}}_{G_A}^A\right)\times \widehat{\mathit{y}}\hfill \end{array}\right)\hfill \\ =\left(\begin{array}{c}{\mathit{f}}_D\hfill \\ \left({\mathit{R}}_E{\mathit{r}}_P^E+{\mathit{R}}_A{\mathit{r}}_{PD}^A\right)\times \left({\mathit{f}}_D\right)+{\mathit{\tau }}_D\hfill \end{array}\right)\hfill \\ {\mathit{f}}_D={\left(\begin{array}{ccc}{f}_{x_A}^D\hfill & f_{y_A}^D\hfill & f_{z_A}^D\hfill \end{array}\right)}^{\mathrm{T}},{\mathit{\tau }}_D={\left(\begin{array}{ccc}{\tau }_{x_A}^D\hfill & {\tau }_{y_A}^D\hfill & {\tau }_{z_A}^D\hfill \end{array}\right)}^T\hfill \end{array}$$

(5)

where m_A is the mass and ${\mathit{r}}_{G_A}^A$ is the position vector of the mass center $G_A$ of A-axis evaluated in {R_A}; ${\mathit{r}}_{\mathrm{PD}}^A$ is the position vector from point P to point D evaluated in {R_A}; $f_{x_C}^D$ , $f_{y_C}^D$ , and $f_{z_C}^D$ are the reaction forces acting on point D along the directions of x_C, y_C, and z_C axes, respectively; and ${\tau }_{x_C}^D$ , ${\tau }_{y_C}^D$ , and ${\tau }_{z_C}^D$ are the corresponding moments of the reaction forces. Solving equation (5) results in

$$\begin{array}{c}{\mathbf{\rho }}_{w,C}={\mathit{T}}_C^{\mathrm{T}}{\mathbf{\$}}_{w,C}\hfill \\ {\mathbf{\rho }}_{w,C}=\left(\begin{array}{c}{\mathit{f}}_D\hfill \\ \multicolumn{1}{c}{{\mathbf{\tau }}_D}\end{array}\right),{\mathbf{\$}}_{w,C}={\mathbf{\$}}_{w,E}+{\mathbf{\$}}_{w,G_E}+{\mathbf{\$}}_{w,G_A}\hfill \\ {\mathbf{\$}}_{w,G_A}=-m_{A}g\left(\begin{array}{c}\hat{\mathit{y}}\hfill \\ \multicolumn{1}{c}{({\mathit{R}}_E{\mathit{r}}_P^E+{\mathit{R}}_A{\mathit{r}}_{G_A}^A)\times \hat{\mathit{y}}}\end{array}\right)\hfill \\ {\mathit{T}}_C=\left[\begin{array}{cc}{\mathit{E}}_3\hfill & [{\mathit{r}}_D\times ]\hfill \\ \multicolumn{1}{c}{0}& {\mathit{E}}_3\hfill \end{array}\right],{\mathit{r}}_D={\mathit{R}}_E{\mathit{r}}_P^E+{\mathit{R}}_A{\mathit{r}}_{\mathrm{PD}}^A\hfill \end{array}$$

(6)

where ${\mathbf{\$}}_{w,C}$ is the resultant externally applied wrench acting on point E; ${\mathbf{\$}}_{w,G_E}$ and ${\mathbf{\$}}_{w,G_A}$ are the components caused by the gravity of end-effector and A-axis, respectively; and ${\mathit{T}}_C$ is the transformation between the wrench imposed on the end-effector and the joint reaction forces of C-axis. Clearly, both the gravity of end-effector and A-axis can be treated as an externally applied wrench in the calculations of joint reaction forces of C-axis. However, the gravity of C-axis is not balanced by any joint reaction force of C-axis.

Figure 3.

Free body diagram of the electric spindle and A-axis.

Deflection analysis

To develop the linear map between the deflection twist of end-effector and the joint deflections of the A-axis and C-axis, the deflection analysis of the rotating head is carried out in this section. With the aid of angular velocity superposition theorem and small deformation synthesis principle, the following relationship can be obtained

$$\begin{array}{c}\delta \mathit{r}={\mathit{R}}_A\delta {\mathit{r}}_A+\left({\mathit{R}}_A\delta {\mathit{\alpha }}_A\right)\times \left(-{\mathit{r}}_P\right)+{\mathit{R}}_C\delta {\mathit{r}}_C+\left({\mathit{R}}_C\delta {\mathit{\alpha }}_C\right)\times (-{\mathit{r}}_D)\hfill \\ \delta \mathit{\alpha }={\mathit{R}}_A\delta {\mathit{\alpha }}_A+{\mathit{R}}_C\delta {\mathit{\alpha }}_C\hfill \\ \delta {\mathit{r}}_A={\left(\begin{array}{ccc}{\delta }_{x_A}^P\hfill & {\delta }_{y_A}^P\hfill & {\delta }_{z_A}^P\hfill \end{array}\right)}^T,\delta {\mathit{\alpha }}_A={\left(\begin{array}{ccc}\delta {\alpha }_{x_A}\hfill & \delta {\alpha }_{y_A}\hfill & \delta {\alpha }_{z_A}\hfill \end{array}\right)}^T\hfill \\ \delta {\mathit{r}}_C={\left(\begin{array}{ccc}{\delta }_{x_C}^D\hfill & {\delta }_{y_C}^D\hfill & {\delta }_{z_C}^D\hfill \end{array}\right)}^T,\delta {\mathit{\alpha }}_C={\left(\begin{array}{ccc}\delta {\alpha }_{x_C}\hfill & \delta {\alpha }_{y_C}\hfill & \delta {\alpha }_{z_C}\hfill \end{array}\right)}^{\mathrm{T}}\hfill \end{array}$$

(7)

where δa and δr are linear deflection at point $E$ and angular deflection of the end-effector, respectively; ${\delta }_{x_A}^P$ , ${\delta }_{y_A}^P$ , and ${\delta }_{z_A}^P$ are the linear deflections caused by the elasticity of A-axis at point P along the directions of x_A, y_A, and z_A, respectively; $\delta {\alpha }_{x_A}$ , $\delta {\alpha }_{y_A}$ , and $\delta {\alpha }_{z_A}$ are the corresponding angular deflections of A-axis; ${\delta }_{x_C}^D$ , ${\delta }_{y_C}^D$ , and ${\delta }_{z_C}^D$ are the linear deflections caused by the elasticity of C-axis at point D along the directions of x_C, y_C, and z_C, respectively; and $\delta {\alpha }_{x_C}$ , $\delta {\alpha }_{y_C}$ , and $\delta {\alpha }_{z_C}$ are the corresponding angular deflections of C-axis. Rewriting equation (7) results in the following equation

$$\begin{array}{c}{\mathbf{\$}}_t={\mathit{T}}_A{\mathbf{\rho }}_{t,A}+{\mathit{T}}_C{\mathbf{\rho }}_{t,C}\hfill \\ {\mathbf{\$}}_t=\left(\begin{array}{c}\delta \mathit{r}\hfill \\ \multicolumn{1}{c}{\delta \mathbf{\alpha }}\end{array}\right),{\mathbf{\rho }}_{t,A}=\left(\begin{array}{c}\delta {\mathit{r}}_A\hfill \\ \multicolumn{1}{c}{\delta {\mathbf{\alpha }}_A}\end{array}\right),{\mathbf{\rho }}_{t,C}=\left(\begin{array}{c}\delta {\mathit{r}}_C\hfill \\ \multicolumn{1}{c}{\delta {\mathbf{\alpha }}_C}\end{array}\right)\hfill \end{array}$$

(8)

where ${\mathrm{\$}}_t$ denotes the deflection twist of the end-effector at point E; ${\mathbf{\rho }}_{t,A}$ $({\mathbf{\rho }}_{t,C})$ is the joint deflection twist of A-axis (C-axis) at point P (D); and ${\mathbf{\rho }}_{t,A}$ and ${\mathbf{\rho }}_{t,C}$ can be further decomposed into two components, that is

$$\begin{array}{c}{\mathbf{\rho }}_{t,A}={\mathit{\rho }\prime }_{t,A}+{\mathit{\rho }″}_{t,A},{\mathbf{\rho }}_{t,C}={\mathit{\rho }\prime }_{t,C}+{\mathit{\rho }″}_{t,C}\hfill \\ {\mathit{\rho }}_{t,A}^{\prime }=\left(\begin{array}{ccc}{\delta }_{x_A}^{\prime P}\hfill & {\delta }_{y_A}^{\prime P}\hfill & \begin{array}{cccc}{\delta }_{z_A}^{\prime P}\hfill & \delta \alpha {\prime }_{x_A}\hfill & \delta \alpha {\prime }_{y_A}\hfill & \delta \alpha {\prime }_{z_A}\hfill \end{array}\hfill \end{array}\right)\hfill \\ {\mathit{\rho }″}_{t,A}=\left(\begin{array}{ccc}{\delta }_{x_A}^{″P}\hfill & {\delta }_{y_A}^{″P}\hfill & \begin{array}{cccc}{\delta }_{z_A}^{″P}\hfill & \delta {\alpha }_{x_A}\hfill & \delta {\alpha ″}_{y_A}\hfill & \delta {\alpha ″}_{z_A}\hfill \end{array}\hfill \end{array}\right)\hfill \\ {\mathit{\rho }\prime }_{t,C}=\left(\begin{array}{ccc}{\delta }_{x_C}^{\prime D}\hfill & {\delta }_{y_C}^{\prime D}\hfill & \begin{array}{cccc}{\delta }_{z_C}^{\prime D}\hfill & \delta \alpha {\prime }_{x_C}\hfill & \delta \alpha {\prime }_{y_C}\hfill & \delta \alpha {\prime }_{z_C}\hfill \end{array}\hfill \end{array}\right)\hfill \\ {\mathit{\rho }″}_{t,C}=\left(\begin{array}{ccc}{\delta }_{x_C}^{″D}\hfill & {\delta }_{y_C}^{″D}\hfill & \begin{array}{cccc}{\delta }_{z_C}^{″D}\hfill & \delta {\alpha ″}_{x_C}\hfill & \delta {\alpha ″}_{y_C}\hfill & \delta {\alpha ″}_{z_C}\hfill \end{array}\hfill \end{array}\right)\hfill \end{array}$$

(9)

where ${\mathit{\rho }\prime }_{t,A}$ $({\mathit{\rho }\prime }_{t,C})$ is the joint deflection caused by the compliance of A-axis (C-axis) under the action of the joint force ${\mathbf{\rho }}_{w,A}$ $({\mathbf{\rho }}_{w,C})$ , and ${\mathit{\rho }″}_{t,A}$ $({\mathit{\rho }″}_{t,C})$ is the joint deflection caused by the compliance of A-axis (C-axis) under the action of the distributed gravity of the A-axis (C-axis) itself. Since the structure of A-axis and C-axis are relatively chunky, ${\mathit{\rho }″}_{t,A}$ $({\mathit{\rho }″}_{t,C})$ is much smaller than ${\mathit{\rho }\prime }_{t,A}$ $({\mathit{\rho }\prime }_{t,C})$ , which can be ignored in the evaluation.

Compliance modeling

With the aforementioned linear maps to hand, the deflection model of the rotating head can be formulated. By assuming a linear elasticity of component and using Hooke’s law, the following equation can be obtained

$${\mathit{\rho }\prime }_{t,A}={\overline{\mathit{C}}}_A{\mathbf{\rho }}_{w,A},{\mathit{\rho }\prime }_{t,C}={\overline{\mathit{C}}}_C{\mathbf{\rho }}_{w,C}$$

(10)

where ${\overline{\mathit{C}}}_A$ and ${\overline{\mathit{C}}}_C$ represent the joint compliance matrices of A-axis and C-axis, respectively, which can be evaluated based on the compliance matrix of the cantilever beam structure, that is

$${\overline{\mathit{C}}}_X=\left[\begin{array}{cccccc}{c}_{11,X}\hfill & \hfill & \hfill & \hfill & c_{15,X}\hfill & \hfill \\ c_{22,X}\hfill & \hfill & c_{24,X}\hfill & \hfill & \hfill \\ \multicolumn{1}{c}{}& \hfill & c_{33,X}\hfill & \hfill & \hfill & \hfill \\ \multicolumn{1}{c}{}& c_{42,X}\hfill & \hfill & c_{44,X}\hfill & \hfill & \hfill \\ \multicolumn{1}{c}{c_{51,X}}& \hfill & \hfill & \hfill & c_{55,X}\hfill & \hfill \\ \multicolumn{1}{c}{}& \hfill & \hfill & \hfill & \hfill & c_{66,X}\hfill \end{array}\right]$$

(11)

$$c_{15,X}=c_{51,X},c_{24,X}=c_{42,X}$$

The elements in ${\overline{\mathit{C}}}_X$ can be obtained by finite element analysis. Then, by considering the gravity of the movable components, the compliance model of the rotating head can be formulated using equations (4), (6), and (8) to (10)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{\$}}_t={\mathit{T}}_A{\overline{\mathit{C}}}_A{\mathit{T}}_A^{\mathrm{T}}{\mathbf{\$}}_{w,A}+{\mathit{T}}_C{\overline{\mathit{C}}}_C{\mathit{T}}_C^{\mathrm{T}}{\mathbf{\$}}_{w,C}+{\mathit{T}}_A{\mathit{\rho }″}_{t,A}}\\ \multicolumn{1}{c}{+{\mathit{T}}_C{\mathit{\rho }″}_{t,C}}\end{array}$$

(12)

It can be found that the established model contains two components, that is, the deflection caused by the resultant externally applied wrench and that caused by the gravity-induced deflections ${\mathit{\rho }″}_{t,A}$ and ${\mathit{\rho }″}_{t,C}$ . Obviously, ${\mathbf{\$}}_t$ in equation (12) can be decomposed into two components as well, that is

$${\mathbf{\$}}_t={\mathbf{\$}}_{t,E}+{\mathbf{\$}}_{t,G}$$

(13)

$${\mathbf{\$}}_{t,E}=\mathit{C}{\mathbf{\$}}_{w,E},\mathit{C}={\mathit{C}}_A+{\mathit{C}}_C$$

$${\mathit{C}}_A={\mathit{T}}_A{\overline{\mathit{C}}}_A{\mathit{T}}_A^{\mathrm{T}},{\mathit{C}}_C={\mathit{T}}_C{\overline{\mathit{C}}}_C{\mathit{T}}_C^{\mathrm{T}}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{\$}}_{t,G}={\mathit{T}}_A{\overline{\mathit{C}}}_A{\mathit{T}}_A^{\mathrm{T}}{\mathbf{\$}}_{w,G_E}+{\mathit{T}}_C{\overline{\mathit{C}}}_C{\mathit{T}}_C^{\mathrm{T}}({\mathbf{\$}}_{w,G_E}+{\mathbf{\$}}_{w,G_A})}\\ \multicolumn{1}{c}{+{\mathit{T}}_A{\mathit{\rho }″}_{t,A}+{\mathit{T}}_C{\mathit{\rho }″}_{t,C}}\end{array}$$

where ${\mathbf{\$}}_{t,E}=\mathit{C}{\mathbf{\$}}_{w,E}$ is the deflection twist of the end-effector caused by ${\mathbf{\$}}_{w,E}$ , leading to a conventional compliance model without considering the gravity. Here, $C$ denotes the compliance matrix in the Cartesian space. ${\mathit{C}}_A$ $({\mathit{C}}_C)$ is the compliance matrix induced by only considering elasticity of A-axis (C-axis). ${\mathbf{\$}}_{t,G}$ is the deflection twist caused by the gravity of components, which can be written as

$${\mathbf{\$}}_{t,G}={\mathbf{\$}}_{t,G_E}+{\mathbf{\$}}_{t,G_A}+{\mathbf{\$}}_{t,G_C}$$

(14)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{\$}}_{t,G_E}=({\mathit{C}}_A+{\mathit{C}}_C){\mathbf{\$}}_{w,G_E}=\mathit{C}{\mathbf{\$}}_{w,G_E},{\mathbf{\$}}_{t,G_A}={\mathit{C}}_C{\mathbf{\$}}_{w,G_A}}\\ \multicolumn{1}{c}{+{\mathit{T}}_A{\mathit{\rho }″}_{t,A},{\mathbf{\$}}_{t,G_C}={\mathit{T}}_C{\mathit{\rho }″}_{t,C}}\end{array}$$

where ${\mathbf{\$}}_{t,G_E}$ , ${\mathbf{\$}}_{t,G_A}$ , and ${\mathbf{\$}}_{t,G_C}$ are the deflection twists of the end-effector, A-axis, and C-axis caused by their gravity and elasticity, respectively. It should be noted that ${\mathit{\rho }″}_{t,A}$ and ${\mathit{\rho }″}_{t,C}$ are negligible because of the chunky structure of A-axis and C-axis. Equation (14) demonstrates that the influence of gravity on the deflection of end-effector is dependent on the center of mass. For instance, the gravity of end-effector can be treated as an external wrench since the deflection twist caused by it has the same format as the deflection twist ${\mathbf{\$}}_{t,E}=\mathit{C}{\mathbf{\$}}_{w,E}$ caused by external wrench ${\mathbf{\$}}_{w,E}$ , which causes the elastic deflection of A-axis and C-axis; gravity of A-axis can cause the elastic deflection of C-axis, but gravity of C-axis cannot cause the elastic deflection of A-axis. It can be concluded that the transmission of component’s gravity is from the connected joint to the fixed frame.

Lightweight design

As discussed above, the lightweight design of the 2-DOF rotating head is of great importance to improve the performance of a posture adjusting mechanism of 5-DOF hybrid robot. To achieve such an aim, the motor and reducer of C-axis are connected directly to reduce the axial size of its supporting component as shown in Figure 4. Then, a motor-synchronous belt-reducer is employed as the drive system of A-axis to reduce the width of rotating head. In addition, the exterior geometry of the rotating head’s outer cover is designed to be approximately convex, which can avoid interference with the synchronous belt and reduce both the weight and supporting width of such rotating head. Furthermore, aluminum alloy is used to manufacture the supporting components of the A-axis and C-axis whose strength is reinforced by using some strengthening ribs in their inner space.

Figure 4.

Structure diagram of the rotating head.

To meet both excellent static rigidities and dynamic behaviors of the system, this article presents an approach to implement the lightweight design by considering the deflection constraints of the end-effector. For convenience, let ${\mathbf{\$}}_{t,G}(i)$ $(i=1,2,\dots ,6)$ be the ith element of ${\mathbf{\$}}_{t,G}$ , which represents the linear deflection along the three orthogonal axes of {R_C} or angular deflection with respect to these axes. For example, ${\mathbf{\$}}_{t,G}(1)$ and ${\mathbf{\$}}_{t,G}(4)$ represent the translational deflection along $x_C$ axis and angular deflection about this axis, respectively. Then, the following index can be defined

$${\delta }_t=\sqrt{{\mathbf{\$}}_t^2\left(1\right)+{\mathbf{\$}}_t^2\left(2\right)+{\mathbf{\$}}_t^2\left(3\right)}$$

(15)

where ${\delta }_t$ is the total linear deflection of the end-effector caused by the applied wrench. Hence, an optimization design can be performed by

$$m\to min$$

(16)

$$\mathrm{s}.\mathrm{t}.{{\delta }_t}_{max}⩽{\delta }_T$$

(16)

where ${{\delta }_t}_{max}=max({\delta }_t)$ . δ_tmax is the largest deflection within the workspace whose allowable deflection is δ_T, and m is the total mass of the rotating head.

Since the rigidities of supporting structures and shafts are much higher than those of supporting bearings and reducers, the overall rigidity of rotating head is dominated by the rigidities of bearings and reducers. Therefore, the optimal design of rotating head requires selecting appropriate supporting bearings and reducers. It can be implemented by exploring the performance of different combination of bearings, reducers, and corresponding structure. The solution with lightest weight will be the final design. In this process, it is necessary to ensure the maximal deflection caused by the elasticity of rotating head to meet the predefined constraints. For simplicity, a set of feasible designs of the A-axis and C-axis are predetermined, and their mass and compliance coefficients are evaluated in advance. It is worth pointing out that the maximal specifications of bearings and reducers might not meet the deflection constraints under the given scope of structure size. Whenever this occurs, it requires relaxing constraints of deflection or increasing structure size.

Numerical example

A lightweight design of the 2-DOF rotating head shown in Figure 1 is taken as an example to verify the effectiveness of the proposed method. Table 1 lists the selected five schemes of the lightweight design for the rotating head and the corresponding specifications of standard products. Tables 2–4 give the dimensions, component compliance coefficients, masses, and mass center of the selected components. The data are obtained from the three-dimensional (3D) model, or evaluated by using finite element analysis, and from product’s specifications.

Table 1.

Five design schemes for the rotating head and the relevant standard products.

Designscheme	Reducer of C-axis	Bearing ofC-axis	Reducer ofA-axis	Bearing ofA-axis	Mass ofC-axis (kg)	Mass ofA-axis (kg)
1	F2C-A25	RB15025	F2C-A25	7015AC	35.5	33.1
2	F2C-A35	RB19025	F2C-A25	7015AC	50.3	33.1
3	F2C-A35	RB19025	F2C-A35	7018AC	50.3	47.2
4	F2C-A45	RB22025	F2C-A35	7018AC	61.3	47.2
5	F2C-A45	RB24025	F2C-A45	7021AC	61.3	56.7

The specification of electric spindle is IBAG HF140, and its mass is 46.8 kg (including the mass of the mounting flange of the electric spindle).

Table 2.

The dimensions, masses, and center of mass locations of the relevant components.

Design scheme	${\mathit{r}}_{G_E}^E(\mathrm{mm})$	${\mathit{r}}_{G_A}^A(\mathrm{mm})$	${\mathit{r}}_P^E(\mathrm{mm})$	${\mathit{r}}_{\mathrm{PD}}^A(\mathrm{mm})$
1	$(0.3-15.9-213.2{)}^{\mathrm{T}}$	$(20.2-9.3-126.4{)}^{\mathrm{T}}$	$(0-165-235{)}^{\mathrm{T}}$	$(0-0-378.1{)}^{\mathrm{T}}$
2	$(0.3-15.9-213.2{)}^{\mathrm{T}}$	$(21.1-9.8-131.3{)}^{\mathrm{T}}$	$(0-165-235{)}^{\mathrm{T}}$	$(0-0-378.1{)}^{\mathrm{T}}$
3	$(0.3-15.9-213.2{)}^{\mathrm{T}}$	$(21.1-9.8-131.3{)}^{\mathrm{T}}$	$(0-165-235{)}^{\mathrm{T}}$	$(0-0-391.4{)}^{\mathrm{T}}$
4	$(0.3-15.9-213.2{)}^{\mathrm{T}}$	$(21.6-10.1-143.3{)}^{\mathrm{T}}$	$(0-165-235{)}^{\mathrm{T}}$	$(0-0-391.4{)}^{\mathrm{T}}$
5	$(0.3-15.9-213.2{)}^{\mathrm{T}}$	$(21.6-10.1-143.3{)}^{\mathrm{T}}$	$(0-165-235{)}^{\mathrm{T}}$	$(0-0-411.5{)}^{\mathrm{T}}$

Table 3.

The component compliance coefficients of A-axis in the body fixed frames (unit: $\times {10}^{-3}$ ).

Designscheme	$c_{11,A}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{N})$	$c_{22,A}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{N})$	$c_{33,A}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{N})$	$c_{44,A}$ $(\mathrm{rad}/\mathrm{Nm})$	$c_{55,A}$ $(\mathrm{rad}/\mathrm{Nm})$	$c_{66,A}$ $(\mathrm{rad}/\mathrm{Nm})$	$c_{51,A}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{Nm})$	$c_{42,A}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{Nm})$
1	8.27	7.25	2.53	9.2 × 10−4	8.89 × 10−4	7.25 × 10−4	31.73	27.81
2	8.27	7.25	2.53	9.2 × 10−4	8.89 × 10−4	7.25 × 10−4	31.73	27.81
3	5.92	4.18	2.04	7.61 × 10−4	5.53 × 10−4	3.68 × 10−4	22.71	16.03
4	5.92	4.18	2.04	7.61 × 10−4	5.53 × 10−4	3.68 × 10−4	22.71	16.03
5	4.85	3.04	1.28	5.79 × 10−4	3.77 × 10−4	2.93 × 10−4	18.61	11.66

Table 4.

The component compliance coefficients of C-axis in the body fixed frames (unit: $\times {10}^{-3}$ ).

Designscheme	$c_{11,C}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{N})$	$c_{22,C}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{N})$	$c_{33,C}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{N})$	$c_{44,C}$ $(\mathrm{rad}/\mathrm{Nm})$	$c_{55,C}$ $(\mathrm{rad}/\mathrm{Nm})$	$c_{66,C}$ $(\mathrm{rad}/\mathrm{Nm})$	$c_{51,C}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{Nm})$	$c_{42,C}$ $(\mathrm{\mu }\mathrm{m}/\mathrm{Nm})$
1	2.38	2.77	3.61	5.49 × 10−4	5.25 × 10−4	6.22 × 10−4	9.23	11.08
2	1.02	1.14	1.52	2.41 × 10−4	2.11 × 10−4	2.91 × 10−4	4.08	4.56
3	1.02	1.14	1.52	2.41 × 10−4	2.11 × 10−4	2.91 × 10−4	4.08	4.56
4	0.81	0.91	1.20	1.98 × 10−4	1.65 × 10−4	2.03 × 10−4	3.24	3.64
5	0.81	0.91	1.20	1.98 × 10−4	1.65 × 10−4	2.03 × 10−4	3.24	3.64

Figure 5 shows the variation of ${\mathbf{\$}}_{t,G}(i)$ versus α and β. It can be seen that ${\mathbf{\$}}_{t,G}(2)$ , ${\mathbf{\$}}_{t,G}(3)$ , and ${\mathbf{\$}}_{t,G}(4)$ symmetrically distribute with respect to the plane with an angle α of 180°. In contrast, ${\mathbf{\$}}_{t,G}(1)$ , ${\mathbf{\$}}_{t,G}(5)$ , and ${\mathbf{\$}}_{t,G}(6)$ distribute asymmetrically in the same plane. ${\mathbf{\$}}_{t,G}(2)$ , ${\mathbf{\$}}_{t,G}(3)$ , and ${\mathbf{\$}}_{t,G}(4)$ are the dominant factors determining the linear and angular deflections. Figure 5 also indicates that the gravity-induced deflection reduces gradually from schemes 1, 2, 3, 5 to 4. These results demonstrate that the gravity-induced deflection initially decreases before inclining with the increase in the specifications of the bearings and gear reducers. Hence, the stiffness of the rotating head can be improved with a slight increase in the specifications of the standard products which is beneficial to reduce the deflection of the end-effector. Moreover, a large increase in the specifications of the standard products can lead to a large mass, which in turn brings about a large applied wrench and deflection.

Figure 5.

The variation of ${\mathbf{\$}}_{t,G}$ versus $\alpha$ and $\beta$ : (a) ${\mathbf{\$}}_{t,G}$ (1), (b) ${\mathbf{\$}}_{t,G}$ (2), (c) ${\mathbf{\$}}_{t,G}$ (3), (d) ${\mathbf{\$}}_{t,G}$ (4), (e) ${\mathbf{\$}}_{t,G}$ (5), (f) ${\mathbf{\$}}_{t,G}$ (6).

To compare the gravity-induced deflection ${\mathbf{\$}}_{t,G}(i)$ with the deflection ${\mathbf{\$}}_{t,E}(i)$ caused by external force, a set of operation parameters for a typical groove milling process is listed as presented in Klimchik et al. ³⁸

$$F_x=215\mathrm{N},F_y=-10\mathrm{N},F_z=-25\mathrm{N},M_x=1\mathrm{Nm}$$

(17)

By taking scheme 4 as an example, Table 5 shows the deflection at a given configuration ( $\alpha =0^{°}$ , $\beta =\mathrm{\pi }/4$ ) of the rotating head. It can be found that the translational deflection along y_c axis and rotational deflection with respect to $x_C$ axis cannot be neglected since it is even larger than the cutting-force-induced deflection.

Table 5.

The deflection, respectively, caused by the cutting forces and gravity at a given configuration.

$i$	1 (mm)	2 (mm)	3 (mm)	4 $(10^{-4}\mathrm{rad})$	5 $(10^{-4}\mathrm{rad})$	6 $(10^{-4}\mathrm{rad})$
${\mathbf{\$}}_{t,E}(i)$	0.0374	−0.0022	−3.501 × 10−4	0.0544	0.786	0.0608
${\mathbf{\$}}_{t,G}(i)$	1.265 × 10−4	−0.0592	−0.0053	1.070	1.01 × 10−5	0.0256

F Furthermore, the groove milling is used to demonstrate the lightweight design of rotating head by assuming the allowable linear deflection δ_T to be 0.08 mm. Table 6 lists the maximal deflection within the workspace when considering both groove-milling-induced forces and gravity of components. It shows that the performance of scheme 4 outweighs the other schemes in terms of deflection. Figure 6 shows the distributions of the first four-order nature frequencies of the scheme 4 within the workspace, which indicates that the rotating head has high dynamic behaviors.

Table 6.

The maximum value of the deflection, respectively, caused by the cutting forces and gravity within the workspace.

Scheme	1	2	3	4	5
${\delta }_{tmax}(\mathrm{mm})$	0.1593	0.0925	0.0838	0.0747	0.0732

Figure 6.

The distributions of the first four-order natural frequencies of the scheme 4 within workspace $W_t$ : (a) the first-order natural frequency, (b) the second-order natural frequency, (c) the third-order natural frequency, and (d) the fourth-order natural frequency.

Conclusion

By considering the gravity, this article presents an approach for the compliance analysis and lightweight design of a 2-DOF rotating head. The following conclusions can be drawn:

1. A linear map is established between the joint reaction forces and the externally applied wrench, including the gravity of components. The gravity of end-effector can be treated as external loads which can cause the elastic deformation of A-axis and C-axis. The gravity of A-axis can cause the elastic deformation of C-axis; however, the gravity of C-axis cannot cause the elastic deformation of A-axis. Hence, it can be concluded that the transmission of gravity is in a straightforward manner, that is, from connected joints to fixed frame.

2. The method enables to effectively evaluate the end-effector’s deflection caused by the payload and gravity within the given task workspace. The example demonstrates that the gravity-induced deflection initially decreases before inclining with the increase in the specifications of the bearings and gear reducers.

3. An approach for the lightweight design of a 2-DOF rotating head with excellent static rigidities and dynamic behaviors is proposed by considering the deflection constraints of the end-effector.