Kinematic Chain Equivalent Method for Tube Model and Elastodynamic Optimization for Parallel Mechanism Based on Matrix Structural Analysis

Abstract

The platforms of parallel mechanisms usually suffer vibration loads. In these cases, structure elastodynamic analysis and elastodynamic optimization of parallel mechanisms are important. A tube structure is very common for parallel mechanisms. This work establishes the model of a tube structure based on matrix structural analysis. The kinematic pair equivalent method is used to simulate the surface contact between the inner and outer tubes. The corresponding mass and stiffness matrices are derived through the strain energy minimization method. The reconfigurable legged lunar lander has been used as an example to verify the effectiveness of this method. By adding the mechanism configuration to the optimization process, the equivalent static load method and the desirability approach are combined and modified. A procedure for the multi-objective elastodynamic optimization of parallel mechanisms is proposed. The optimization procedure is implemented on the lander and the results show a reduction in mass and an increase in natural frequency.

1. Introduction

A parallel mechanism (PM) is a closed-loop mechanism with a moving platform and a fixed base connected by at least two limbs. PMs have the advantage of higher stiffness, higher bearing capacity, and better dynamic response compared to serial mechanisms [1]. PMs have been used as machine tools [2], legged robots [3], pick-and-place applications [4], etc. In many applications, for example, machining processes, the end effector of the PM suffers high frequency vibration and a large load [5]. In other cases, the link of the PM is moving at a fast speed, so the elastic deformation of the mechanism needs to be considered to compensate for the error [6]. Therefore, the elastodynamic modeling of the PM is important to evaluate and optimize its performance.

The dynamics of PMs can be classified into three classes. The first class is rigid body dynamics. When the links and joints of the PM are rigid enough under external force, all parts of the mechanism can be considered rigid bodies. Screw theory [7,8] and the Newton–Euler method [9] have been widely used to establish rigid body dynamics. In the second class, the PM is moving at a low speed while subjected to external vibration force, for example, friction stir welding [10]. In this case, the rigid body inertia force is relatively low, whereas the structure vibration has a great impact on the performance. Usually, the mechanism in the workspace is considered a varying structure with vibration force on the end effector. Matrix structural analysis (MSA) [11] and the virtual joint method (VJM) [12,13] have been developed to construct the model. In the third class, the PM goes through both large rigid body motion and large deformation, and neither of their impacts can be ignored, for example, the Delta robot in the fast pick-and-place process. The kineto-elastodynamics (KED) [14,15] method has been proposed to construct the model in the last century. However, KED neglects the coupling terms of rigid body movement and flexible deformation. The floating frame of reference formulation (FFRF) [16] and the absolute nodal coordinates formulation (ANCF) [17] were later proposed to handle this problem. However, the process of modeling is extremely complicated and has a higher computational cost compared to the first two classes. In our work, we focused on the second class, the structural elastodynamics of the PM.

There are two main methods for the structural elastodynamics of PMs—the VJM and the MSA. With finite element analysis (FEA) of separate links, the VJM can obtain the numerical stiffness matrix of links with complex shapes [18]. Then, the overall stiffness matrix is derived using screw theory [19]. However, the force analysis in this method becomes very complicated when dealing with multiloop PMs [20]. The MSA is based on computational structural mechanics and can uniformly acquire the overall stiffness and mass matrix by summing all the energy together. Although traditional MSA simplifies all links into beam elements with the analytic element matrix, it can also combine the FEA result to obtain the numerical stiffness and mass matrix of complex shape links [21]. The MSA was first introduced by Deblaise [22]. Different kinds of kinematic joints between links are handled using Lagrange multipliers. Later, strain energy minimization was proposed by Cammarata [23,24] to eliminate the Lagrange multipliers, and only the differential vibration equation is presented in the final result. With this method, all kinds of kinematic joints, including prismatic joints, rotational joints, universal joints, etc., can be well assembled into the stiffness and mass matrices. However, the long prismatic joints composed of outer and inner tubes have not been modeled. Tube structure has been widely used for pneumatic cylinders or screwballs in PMs. Many researchers tie the outer and inner tubes together as a single beam [25] or only add a single prismatic constraint on the tube [26]. Therefore, the existing methods are still not accurate enough for the tube structure. Using FEA, the surface contact between the outer and inner tubes requires a lot of computational resources and is hard to converge sometimes [27]. We propose the kinematic pair equivalent method to solve this contact problem.

Elastodynamic optimization can improve the stiffness, dynamic performance, and lightness of PMs in the initial design stage [28]. The elastodynamic optimization process consists of adjusting the values of the design parameters to derive better elastodynamic performance defined by the designers [29]. One big challenge is the huge computational cost. The conventional structure optimization method under dynamic loading needs to repetitively solve the nonlinear differential equation. This process is time-consuming [30]. What is worse, the optimization process for a parallel mechanism needs to consider mechanism configurations. Usually, the mechanism under different configurations is considered a series of structures. Therefore, the configuration of the mechanism adds another iteration dimension for elastodynamic optimization. One solution to the computational problem is determining a particular trajectory in the workspace and evaluating the performance on the trajectory [26,31]. However, this cannot reflect the PM characteristics in the whole workspace [32]. We introduce and modify the equivalent static loads (ESLs) [33] in structural optimization to solve this problem. ESLs transfer the dynamic response to a series of equivalent static loads. Then, the dynamic optimization problem becomes the classical static optimization problem. As a result, only the algebraic equation needs to be solved in the sub-optimization procedure.

In application, more than one objective function needs to be considered simultaneously, such as the mass, fundamental frequency, energy, displacement, and so on. Therefore, linear static response optimization is a multi-objective optimization process. We introduce the desirability approach [34] to implement the multi-objective optimization process. The desirability approach transfers different objective functions into a single desirability objective function by assigning different weight factors. It is widely used in a multi-objective optimization design since it is easy to implement and has a low computational cost.

In this paper, the outer tube–inner tube model is added to the MSA with the kinematic pair equivalent method. Equivalent static loads and the desirability approach are introduced and modified in the elastodynamic optimization of PMs. In Section 2, the elastodynamics of a PM with tubes is constructed with the kinematic pair equivalent method based on matrix structural analysis. In Section 3, a reconfigurable lunar lander is used as an example. The theoretical elastodynamic model is compared to the FEA results to verify the effectiveness of the method. In Section 4, elastodynamic optimization with ESLs and the desirability approach are introduced, modified, and finally, applied to the lander.

2. Elastodynamic Model of Parallel Mechanism with Tube Structure

When PMs undergo a vibration process, such as machining and collision, the structural vibration force is comparatively much larger than the inertia force. In these situations, the rigid body movement can be neglected and the elastodynamics of the PM are structural elastodynamics. A PM in the workspace can then be regarded as a series of structures. We improved the MSA method by adding a tube structure to construct the elastodynamics of the PM. In MSA, the PM in a configuration is considered a structure. As shown in Figure 1, it contains both rigid bodies and flexible bodies, which are connected by different kinds of kinematic joints. The moving platform and the base of the PM are usually considered rigid bodies due to much higher stiffness. Links are treated as flexible bodies. Straight links are usually simulated by 3D Euler–Bernoulli beam elements.

With the energy method, by summing each element’s kinematic energy, elastic potential energy, and external force virtual work, the vibration equation of the PM can be obtained through the Lagrange formulation. The core of the MSA is to obtain the stiffness and mass matrices of the flexible elements, which are connected by complex kinematic joints. According to the connection type, the elements are classified into two main cases by Cammarata [11]: the rigid body–flexible beam and the flexible beam–flexible beam. Flexible elements in these two cases are two-node beam elements. They are connected at the top node and the end node. We added the outer tube–inner tube case to simulate the tube structure in a PM.

2.1. Tube Model in MSA

The first case is the outer tube–inner tube, as shown in Figure 2. The small sliding and small deflection hypotheses of the inner tube and outer tube were used, considering the structural vibration condition. The kinematic pair equivalent method was used to construct the tube model. According to the tube’s cross-section shape, we used either a prismatic pair or a cylinder pair to simulate the contact of the tube surface. If the tube cross-section is in a circle shape, the inner tube can both rotate and translate along the outer tube. Then, a series of cylinder pairs were used to simulate the contact behavior between them. As shown in Figure 2, the cylinder pairs are added at the red dotted lines. If the cross-section is rectangular, it can only translate a small distance and cannot rotate. Therefore, a series of prismatic pairs can be used.

In the following, cylinder pairs are used as an example. Both the outer and inner tubes are considered 3D Euler–Bernoulli beam elements. They are connected by a series of passive cylinder pairs. We cut the tube into ten pieces, as shown in Figure 2, in which the cylinder pairs are placed at the red dotted lines.

We call four adjacent elements near a cylinder pair a unit. For example, elements $1,2,3$ and $4$ comprise the first unit. The stiffness and mass matrices of these four elements are deduced as follows. The elements in the other units are the same. For the first unit, since the cylinder pair is passive, we choose nodes ${\mathit{u}}_1^1,{\mathit{u}}_2^2,{\mathit{u}}_3^1,{\mathit{u}}_4^2$ as independent nodes. Once this node displacement is determined, the elastic potential energy $V$ of the first unit is determined. ${\mathit{u}}_1^2,{\mathit{u}}_3^2$ are dependent nodes, which are irrelevant to $V$. Their displacement is determined by the independent node displacement.

We first write the elastic potential energy $V$ of the first unit with both the dependent and independent nodes in Equation (1). ${\mathit{K}}_i\left(i=1~4\right)$ is the stiffness matrix of the Euler–Bernoulli beam in the global coordinate. ${\overline{\mathit{K}}}_i\left(i=1~4\right)$ is the stiffness matrix in the local coordinate. $\mathit{R}$ is the rotation matrix of the local coordinate in the global coordinate. The expression of ${\overline{\mathit{K}}}_i$ is shown in Appendix A.

$$\begin{gathered} V=\frac{1}{2}{\left[\begin{array}{c}{\mathit{u}}_1^1\\ {\mathit{u}}_1^2\end{array}\right]}^T{\mathit{K}}_1\left[\begin{array}{c}{\mathit{u}}_1^1\\ {\mathit{u}}_1^2\end{array}\right]+\frac{1}{2}{\left[\begin{array}{c}{\mathit{u}}_1^2\\ {\mathit{u}}_2^2\end{array}\right]}^T{\mathit{K}}_2\left[\begin{array}{c}{\mathit{u}}_1^2\\ {\mathit{u}}_2^2\end{array}\right]+\frac{1}{2}{\left[\begin{array}{c}{\mathit{u}}_3^1\\ {\mathit{u}}_3^2\end{array}\right]}^T{\mathit{K}}_3\left[\begin{array}{c}{\mathit{u}}_3^1\\ {\mathit{u}}_3^2\end{array}\right]+\frac{1}{2}{\left[\begin{array}{c}{\mathit{u}}_3^2\\ {\mathit{u}}_4^2\end{array}\right]}^T{\mathit{K}}_4\left[\begin{array}{c}{\mathit{u}}_3^2\\ {\mathit{u}}_4^2\end{array}\right] \\ {\mathit{K}}_i=\left[\begin{array}{c}{\mathit{K}}_i^{11}\\ {\mathit{K}}_i^{21}\end{array}\begin{array}{c}{\mathit{K}}_i^{12}\\ {\mathit{K}}_i^{22}\end{array}\right]\left(i=1~4\right) \\ {\mathit{K}}_i=\left[\begin{array}{cccc}\mathit{R}& & & \\ & \mathit{R}& & \\ & & \mathit{R}& \\ & & & \mathit{R}\end{array}\right]{\overline{\mathit{K}}}_i{\left[\begin{array}{cccc}\mathit{R}& & & \\ & \mathit{R}& & \\ & & \mathit{R}& \\ & & & \mathit{R}\end{array}\right]}^T \end{gathered}$$

(1)

For the first unit, node ${\mathit{u}}_3^2$ and node ${\mathit{u}}_1^2$ are connected by a passive cylinder pair. Therefore, node ${\mathit{u}}_3^2$ can slide a small distance ${\theta }_1$ and rotate a small angle ${\theta }_2$ along the axis of the tube $\mathit{n}$. Their displacement has the following relationship:

$$\begin{gathered} {\mathit{u}}_1^2={\mathit{u}}_3^2+\mathit{H}\mathit{\theta } \\ \mathit{H}=\left[\begin{array}{c}\mathit{n}\\ \mathbf{0}\end{array} \begin{array}{c}\mathbf{0}\\ \mathit{n}\end{array}\right], \mathit{\theta }=\left[\begin{array}{c}{\theta }_1\\ {\theta }_2\end{array}\right] \end{gathered}$$

(2)

Adding Equation (2) to Equation (1), the elastic potential energy $V$ of the first unit becomes a function of six variables, as shown below. ${\mathit{u}}_1^1,{\mathit{u}}_2^2,{\mathit{u}}_3^1,{\mathit{u}}_4^2$ are the independent nodes and ${\mathit{u}}_3^2,\mathit{\theta }$ are the dependent nodes.

$$V=V\left({\mathit{u}}_1^1,{\mathit{u}}_2^2,{\mathit{u}}_3^1,{\mathit{u}}_3^2,{\mathit{u}}_4^2,\mathit{\theta }\right)$$

(3)

Since $V$ is irrelevant to $\mathit{\theta }$ and ${\mathit{u}}_3^2$, the partial derivative with respect to these two variables is zero.

$$\frac{\partial V}{\partial \mathit{\theta }}=\mathbf{0}$$

(4)

$$\frac{\partial V}{\partial {\mathit{u}}_3^2}=\mathbf{0}$$

(5)

After simplifying Equation (4), the irrelevant variable $\mathit{\theta }$ can be linearly represented by other variables.

$$\begin{gathered} \begin{array}{c}\mathit{\theta }={\mathit{F}}^1{\mathit{u}}_1^1+{\mathit{F}}^2{\mathit{u}}_2^2+{\mathit{F}}^3{\mathit{u}}_3^2\end{array} \\ {\mathit{F}}^1=-{\mathit{D}}_1^{-1}{A}_1,{\mathit{F}}^2=-{\mathit{D}}_1^{-1}{\mathit{B}}_1,{\mathit{F}}^3=-{\mathit{D}}_1^{-1}{\mathit{C}}_1 \\ {A}_1={\mathit{H}}^T{\mathit{K}}_1^{21},{\mathit{B}}_1={\mathit{H}}^T{\mathit{K}}_2^{12} \\ {\mathit{C}}_1={\mathit{H}}^T\left({\mathit{K}}_1^{22}+{\mathit{K}}_2^{11}\right),{\mathit{D}}_1={\mathit{H}}^T\left({\mathit{K}}_1^{22}+{\mathit{K}}_2^{11}\right)\mathit{H} \end{gathered}$$

(6)

Adding Equation (6) to Equation (3), $V$ becomes a function of five variables $V\left({\mathit{u}}_1^1,{\mathit{u}}_2^2,{\mathit{u}}_3^1,{\mathit{u}}_3^2,{\mathit{u}}_4^2\right)$. Adding $V\left({\mathit{u}}_1^1,{\mathit{u}}_2^2,{\mathit{u}}_3^1,{\mathit{u}}_3^2,{\mathit{u}}_4^2\right)$ to Equation (5), ${\mathit{u}}_3^2$ can be expressed by the independent variables, as shown below. ${\mathit{E}}_6$ is a six-order unit matrix.

$$\begin{gathered} {\mathit{u}}_3^2={\mathit{G}}^1{\mathit{u}}_1^1+{\mathit{G}}^2{\mathit{u}}_2^2+{\mathit{G}}^3{\mathit{u}}_3^1+{\mathit{G}}^4{\mathit{u}}_4^2 \\ {\mathit{G}}^1=-{\mathit{C}}_2^{-1}{A}_2,{\mathit{G}}^2=-{\mathit{C}}_2^{-1}{\mathit{B}}_2 \\ {\mathit{G}}^3=-{\mathit{C}}_2^{-1}{\mathit{K}}_3^{21},{\mathit{G}}^4=-{\mathit{C}}_2^{-1}{\mathit{K}}_4^{12} \\ {A}_2={\mathit{F}}^T{\mathit{K}}_1^{21}+{\mathit{F}}^T\left({\mathit{K}}_1^{22}+{\mathit{K}}_2^{11}\right)\mathit{H}{\mathit{F}}^1 \\ {\mathit{B}}_2={\mathit{F}}^T\left({\mathit{K}}_1^{22}+{\mathit{K}}_2^{11}\right)\mathit{H}{\mathit{F}}^2+{\mathit{F}}^T{\mathit{K}}_2^{12} \\ {\mathit{C}}_2={\mathit{F}}^T\left({\mathit{K}}_1^{22}+{\mathit{K}}_2^{11}\right)\mathit{F}+{\mathit{K}}_3^{22}+{\mathit{K}}_4^{11} \\ \mathit{F}={\mathit{E}}_6+\mathit{H}{\mathit{F}}^3 \end{gathered}$$

(7)

Adding Equation (7) to Equation (6), $\mathit{\theta }$ can be expressed by the independent nodes.

$$\begin{gathered} \mathit{\theta }={\mathit{Y}}^1{\mathit{u}}_1^1+{\mathit{Y}}^2{\mathit{u}}_2^2+{\mathit{Y}}^3{\mathit{u}}_3^1+{\mathit{Y}}^4{\mathit{u}}_4^2 \\ {\mathit{Y}}^1={\mathit{F}}^1+{\mathit{F}}^3{\mathit{G}}^1,{\mathit{Y}}^2={\mathit{F}}^2+{\mathit{F}}^3{\mathit{G}}^2 \\ {\mathit{Y}}^3={\mathit{F}}^3{\mathit{G}}^3,{\mathit{Y}}^4={\mathit{F}}^3{\mathit{G}}^4 \end{gathered}$$

(8)

Taking Equations (7) and (8) into Equation (2), ${\mathit{u}}_1^2$ can be expressed by the independent nodes.

$$\begin{gathered} \begin{array}{c}{\mathit{u}}_1^2={\mathit{X}}^1{\mathit{u}}_1^1+{\mathit{X}}^2{\mathit{u}}_2^2+{\mathit{X}}^3{\mathit{u}}_3^1+{\mathit{X}}^4{\mathit{u}}_4^2\end{array} \\ {\mathit{X}}^i={\mathit{G}}^i+\mathit{H}{\mathit{Y}}^i\left(i=1~4\right) \end{gathered}$$

(9)

Adding Equations (7) and (9) to Equation (1), the elastic potential energy $V$ of the first unit is expressed by the independent node displacement ${\mathit{u}}_1^1,{\mathit{u}}_2^2,{\mathit{u}}_3^1,{\mathit{u}}_4^2$.

$$\begin{gathered} \begin{array}{c}V={\displaystyle \sum }_{i=1}^4\frac{1}{2}{\left[\begin{array}{c}{\mathit{u}}_1^1\\ {\mathit{u}}_2^2\\ {\mathit{u}}_3^1\\ {\mathit{u}}_4^2\end{array}\right]}^T{\mathit{J}}_i^T{\mathit{K}}_i{\mathit{J}}_i\left[\begin{array}{c}{\mathit{u}}_1^1\\ {\mathit{u}}_2^2\\ {\mathit{u}}_3^1\\ {\mathit{u}}_4^2\end{array}\right]\end{array} \\ {\mathit{J}}_1=\left[\begin{array}{c}{\mathit{E}}_6\\ {\mathit{X}}^1\end{array}\begin{array}{c}{\mathbf{0}}_6\\ {\mathit{X}}^2\end{array}\begin{array}{c}{\mathbf{0}}_6\\ {\mathit{X}}^3\end{array}\begin{array}{c}{\mathbf{0}}_6\\ {\mathit{X}}^4\end{array}\right],{\mathit{J}}_2=\left[\begin{array}{c}{\mathit{X}}^1\\ {\mathbf{0}}_6\end{array}\begin{array}{c}{\mathit{X}}^2\\ {\mathit{E}}_6\end{array}\begin{array}{c}{\mathit{X}}^3\\ {\mathbf{0}}_6\end{array}\begin{array}{c}{\mathit{X}}^4\\ {\mathbf{0}}_6\end{array}\right] \\ {\mathit{J}}_3=\left[\begin{array}{c}{\mathbf{0}}_6\\ {\mathit{G}}^1\end{array}\begin{array}{c}{\mathbf{0}}_6\\ {\mathit{G}}^2\end{array}\begin{array}{c}{\mathit{E}}_6\\ {\mathit{G}}^3\end{array}\begin{array}{c}{\mathbf{0}}_6\\ {\mathit{G}}^4\end{array}\right],{\mathit{J}}_4=\left[\begin{array}{c}{\mathit{G}}^1\\ {\mathbf{0}}_6\end{array}\begin{array}{c}{\mathit{G}}^2\\ {\mathbf{0}}_6\end{array}\begin{array}{c}{\mathit{G}}^3\\ {\mathbf{0}}_6\end{array}\begin{array}{c}{\mathit{G}}^4\\ {\mathit{E}}_6\end{array}\right] \end{gathered}$$

(10)

Based on the small sliding and small deformation hypothesis, the coefficient matrices in Equations (7) and (9) are constant with respect to the time variable. Therefore, the time derivative is only implemented on the node displacement, as shown in Equations (11) and (12). Then, we have the kinematic energy of the first unit, as shown in Equation (13). ${\mathit{M}}_i$ and ${\overline{\mathit{M}}}_i$ are the mass matrices in the global and local coordinates, respectively. The expression of ${\overline{\mathit{M}}}_i$ is shown in Appendix A. The elastic potential energy and the kinematic energy of other units can be obtained in a similar manner.

$${\dot{\mathit{u}}}_3^2={\mathit{G}}^1{\dot{\mathit{u}}}_1^1+{\mathit{G}}^2{\dot{\mathit{u}}}_2^2+{\mathit{G}}^3{\dot{\mathit{u}}}_3^1+{\mathit{G}}^4{\dot{\mathit{u}}}_4^2$$

(11)

$$\begin{array}{c}{\dot{\mathit{u}}}_1^2={\mathit{X}}^1{\dot{\mathit{u}}}_1^1+{\mathit{X}}^2{\dot{\mathit{u}}}_2^2+{\mathit{X}}^3{\dot{\mathit{u}}}_3^1+{\mathit{X}}^4{\dot{\mathit{u}}}_4^2\end{array}$$

(12)

$$\begin{gathered} T=\frac{1}{2}{\left[\begin{array}{c}{\dot{\mathit{u}}}_1^1\\ {\dot{\mathit{u}}}_1^2\end{array}\right]}^T{\mathit{M}}_1\left[\begin{array}{c}{\dot{\mathit{u}}}_1^1\\ {\dot{\mathit{u}}}_1^2\end{array}\right]+\frac{1}{2}{\left[\begin{array}{c}{\dot{\mathit{u}}}_1^2\\ {\dot{\mathit{u}}}_2^2\end{array}\right]}^T{\mathit{M}}_2\left[\begin{array}{c}{\dot{\mathit{u}}}_1^2\\ {\dot{\mathit{u}}}_2^2\end{array}\right]+\frac{1}{2}{\left[\begin{array}{c}{\dot{\mathit{u}}}_3^1\\ {\dot{\mathit{u}}}_3^2\end{array}\right]}^T{\mathit{M}}_3\left[\begin{array}{c}{\dot{\mathit{u}}}_3^1\\ {\dot{\mathit{u}}}_3^2\end{array}\right]+\frac{1}{2}{\left[\begin{array}{c}{\dot{\mathit{u}}}_3^2\\ {\dot{\mathit{u}}}_4^2\end{array}\right]}^T{\mathit{M}}_4\left[\begin{array}{c}{\dot{\mathit{u}}}_3^2\\ {\dot{\mathit{u}}}_4^2\end{array}\right] \\ ={\displaystyle \sum }_{i=1}^4\frac{1}{2}{\left[\begin{array}{c}{\dot{\mathit{u}}}_1^1\\ {\dot{\mathit{u}}}_2^2\\ {\dot{\mathit{u}}}_3^1\\ {\dot{\mathit{u}}}_4^2\end{array}\right]}^T{\mathit{J}}_i^T{\mathit{M}}_i{\mathit{J}}_i\left[\begin{array}{c}{\dot{\mathit{u}}}_1^1\\ {\dot{\mathit{u}}}_2^2\\ {\dot{\mathit{u}}}_3^1\\ {\dot{\mathit{u}}}_4^2\end{array}\right] \\ {\mathit{M}}_i=\left[\begin{array}{cccc}\mathit{R}& & & \\ & \mathit{R}& & \\ & & \mathit{R}& \\ & & & \mathit{R}\end{array}\right]{\overline{\mathit{M}}}_i{\left[\begin{array}{cccc}\mathit{R}& & & \\ & \mathit{R}& & \\ & & \mathit{R}& \\ & & & \mathit{R}\end{array}\right]}^T \end{gathered}$$

(13)

2.2. Other Two Cases

The other two cases are the rigid body–flexible beam case and the flexible beam–flexible beam case, as shown in Figure 3. The kinematic and elastic energy was explained by Cammarata and can be found in reference [5]. We corrected the small calculation errors in [5] and simply list the results in Appendix B.

2.3. Elastodynamic Equation of the PM

As shown in Equations (14)–(16), the elastic energy $V$, kinematic energy $T$, and external force virtual work $\mathsf{\delta }W$ of all elements can be expressed by all independent node displacements $\mathit{u}=\left[{\mathit{u}}_1;{\mathit{u}}_2;\cdots ;{\mathit{u}}_n\right]$ and node velocities $\dot{\mathit{u}}=\left[{\dot{\mathit{u}}}_1;{\dot{\mathit{u}}}_2;\cdots ;{\dot{\mathit{u}}}_n\right]$.

$$\begin{array}{c}V={\displaystyle {\displaystyle \sum }_{i=1}^m}\frac{1}{2}{\mathit{u}}^T{\mathit{K}}_i\mathit{u}\end{array}$$

(14)

$$T={\displaystyle \sum }_{i=1}^m\frac{1}{2}{\dot{\mathit{u}}}^T{\mathit{M}}_i\dot{\mathit{u}}$$

(15)

$$\begin{array}{c}\mathsf{\delta }W={\mathit{F}}^T\mathsf{\delta }\mathit{u}\end{array}$$

(16)

Then, according to the Lagrange formulation in Equation (17), the elastodynamic equation of the PM is expressed in Equation (18). Then, we add displacement boundary conditions to solve the equation. We divide the independent node displacement $\mathit{u}$ into two parts $\mathit{u}=\left[{\mathit{u}}_1;{\mathit{u}}_2\right]$${\mathit{u}}_2$ contains the known boundary condition ${\mathit{u}}_2={\mathit{u}}_2^0$. Then, the mass matrix $\mathit{M}$, stiffness matrix $\mathit{K},$ and external force matrix $\mathit{F}$ are also divided into corresponding block matrices, as shown below.

$$\frac{d}{dt}\frac{\partial T}{\partial \dot{\mathit{u}}}+\frac{\partial V}{\partial \mathit{u}}=\mathit{F}$$

(17)

$$\begin{gathered} \begin{array}{c}\mathit{M}\ddot{\mathit{u}}+\mathit{K}\mathit{u}=\mathit{F}\end{array} \\ \mathit{M}={\displaystyle \sum }_{i=1}^m{\mathit{M}}_i=\left[\begin{array}{c}{\mathit{M}}_{11}\\ {\mathit{M}}_{21}\end{array}\begin{array}{c}{\mathit{M}}_{12}\\ {\mathit{M}}_{22}\end{array}\right],\mathit{K}={\displaystyle \sum }_{i=1}^m{\mathit{K}}_i=\left[\begin{array}{c}{\mathit{K}}_{11}\\ {\mathit{K}}_{21}\end{array}\begin{array}{c}{\mathit{K}}_{12}\\ {\mathit{K}}_{22}\end{array}\right] \\ \mathit{u}=\left[\begin{array}{c}{\mathit{u}}_1\\ {\mathit{u}}_2^0\end{array}\right],\mathit{F}=\left[\begin{array}{c}{\mathit{F}}_1\\ {\mathit{F}}_2\end{array}\right] \end{gathered}$$

(18)

After simplification with the block matrix, Equation (18) becomes the equivalent Equations (19) and (20). The unknown variables ${\mathit{u}}_1$ and ${\mathit{F}}_2$ can be obtained by solving these ordinary differential equations.

$${\mathit{M}}_{11}{\ddot{\mathit{u}}}_1+{\mathit{K}}_{11}{\mathit{u}}_1={\mathit{F}}_1-{\mathit{K}}_{12}{\mathit{u}}_2^0-{\mathit{M}}_{12}{\ddot{\mathit{u}}}_2^0$$

(19)

$${\mathit{F}}_2={\mathit{M}}_{21}{\ddot{\mathit{u}}}_1+{\mathit{K}}_{21}{\mathit{u}}_1+{\mathit{K}}_{22}{\mathit{u}}_2^0+{\mathit{M}}_{22}{\ddot{\mathit{u}}}_2^0$$

(20)

Then, the independent node displacement $\mathit{u}=\left[{\mathit{u}}_1;{\mathit{u}}_2^0\right]$ is derived. The node displacement $\left[{\mathit{u}}_1^1;{\mathit{u}}_1^2\right]$ at two sides of any beam element can also be obtained in the global coordinate. Coordinate transformation can be used to derive the node displacement $\left[{\overline{\mathit{u}}}_1^1;{\overline{\mathit{u}}}_1^2\right]$ in the beam local coordinate, as shown in Equation (21). $\mathit{R}$ is the rotation matrix of the local coordinate expressed in the global coordinate.

$$\begin{gathered} \left[\begin{array}{c}{\overline{\mathit{u}}}_1^1\\ {\overline{\mathit{u}}}_1^2\end{array}\right]={\tilde{\mathit{R}}}^T\left[\begin{array}{c}{\mathit{u}}_1^1\\ {\mathit{u}}_1^2\end{array}\right] \\ \tilde{\mathit{R}}=\mathrm{diag}\left(\left[\mathit{R},\mathit{R},\mathit{R},\mathit{R}\right]\right) \end{gathered}$$

(21)

According to elastic theory [35], for any interpolation point in a beam element, the strain field $\mathit{\epsilon }$, the stress field $\mathit{\sigma }$, and the stress tensor $\mathit{Q}$ can be obtained as follows. The details of Equations (22) and (23) are shown in Appendix C.

$$\begin{array}{c}\mathit{\epsilon }=\mathit{D}\left(\mathit{S}\right)\left[\begin{array}{c}{\overline{\mathit{u}}}_1^1\\ {\overline{\mathit{u}}}_1^2\end{array}\right]\end{array}$$

(22)

$$\mathit{\sigma }=\mathit{f}\left(\mathit{\epsilon }\right)$$

(23)

$$\begin{gathered} \mathit{Q}=\left[\begin{array}{c}{\sigma }_x\\ {\mathsf{\tau }}_{yx}\\ {\mathsf{\tau }}_{\mathit{z}x}\end{array}\begin{array}{c}{\mathsf{\tau }}_{xy}\\ {\sigma }_y\\ {\mathsf{\tau }}_{\mathit{z}y}\end{array}\begin{array}{c}{\mathsf{\tau }}_{x\mathit{z}}\\ {\mathsf{\tau }}_{y\mathit{z}}\\ {\sigma }_{\mathit{z}}\end{array}\right] \\ \mathit{\epsilon }={\left[{\mathsf{\epsilon }}_x,{\mathsf{\epsilon }}_y,{\mathsf{\epsilon }}_{\mathit{z}},{\mathsf{\gamma }}_{xy},{\mathsf{\gamma }}_{x\mathit{z}},{\mathsf{\gamma }}_{y\mathit{z}}\right]}^T \\ \mathit{\sigma }={\left[{\sigma }_x,{\sigma }_y,{\sigma }_{\mathit{z}},{\mathsf{\tau }}_{y\mathit{z}},{\mathsf{\tau }}_{\mathit{z}x},{\mathsf{\tau }}_{xy}\right]}^T \end{gathered}$$

(24)

The principal stress ${\sigma }_1,{\sigma }_2,{\sigma }_3$ at an interpolation point is three eigenvalues of matrix $\mathit{Q}$. Thus, Von Mises stress ${\sigma }_s$ at an interpolation point is expressed as follows:

$${\sigma }_s=\sqrt{\frac{1}{2}\left({\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right)}$$

(25)

3. Simulation and Verification

A reconfigurable legged lander was used as an example to verify the tube model. The lander, in its symmetric configuration, is shown in Figure 4a. The lander has three modes—the adjusting, landing, and walking mode. In the adjusting mode, the lander will hover several hundred meters above the landing site. It will select suitable terrain for landing and avoid hazardous rocks or large craters. At the same time, adjusting links will rotate around the axis $R_i\left(i=1~3\right)$ to a suitable position according to the selected landing site. In the landing mode, the lander will descend to several meters above the lunar surface and axis $R_i\left(i=1~3\right)$ will be fixed. Then, the engine will be cut off and the lander will fall down freely. When hitting the lunar surface, the impact force will be cushioned by the aluminum honeycomb in the primary and two secondary struts of the lander. After landing, other joint axes will be released. The lander can walk on the lunar surface. For more detail on this reconfigurable mechanism, readers can look at reference [36].

Since the landing vibration is crucial to the lander, the elastodynamics of the landing mode are analyzed below. After adjusting the axis $R_i$, the lander goes into the landing mode, and the axis $R_i$ is fixed. Figure 4b shows the symmetric configuration of the lander. From the main body to the footpad are the primary and two secondary struts composed of $UPS$ limbs, where $U$ denotes the universal joint, $P$ denotes the prismatic joint, and $S$ denotes the spherical joint. Three prismatic joints are formed by the outer and inner tubes. Among the joints, all links are regarded as 3D Euler–Bernoulli beams. The link numbers are shown in Figure 4b. A connection graph of all the links is shown in Figure 4c. The outer tube of the main strut $L4$ is fixed on the platform $L13$. The platform $L13$, the foot pad $L14,$ and the main body $L15$ are regarded as rigid bodies. All other parts are considered flexible beams.

Since the outer tube of the main strut is partly tied to the platform, we defined the tied nodes of the main strut. The tied outer tube nodes $13,14,15$ are shown in Figure 4b. Their displacement is determined by the platform node ${\mathit{u}}_p$. $r_{ip}$ is the vector from the tied node ${\mathit{u}}_i$ to the platform node ${\mathit{u}}_p$.

$$\begin{gathered} {\mathit{u}}_i=\mathit{G}{\mathit{u}}_p \\ \mathit{G}=\left[\begin{array}{c}{\mathit{E}}_3\\ \mathbf{0}\end{array}\begin{array}{c}\widetilde{{\mathit{r}}_{ip}}\\ {\mathit{E}}_3\end{array}\right]\left(i=13~15\right) \end{gathered}$$

(26)

The virtual work $\mathsf{\delta }{W}_i$ of the buffer force on the $i\mathrm{th}$ strut is shown as follows. ${\mathit{n}}_i$ is the direction vector of the $i\mathrm{th}$ buffer limb. ${\mathit{u}}_{Ii}$ is the node that buffer force acts on for the $i$th inner tube. ${\mathit{u}}_{Oi}$ is the node that buffer force acts on for the $i$th outer tube. ${\mathit{f}}_{im}$ is the magnitude of the $i$th buffer force.

$$\begin{gathered} \begin{array}{c}\mathsf{\delta }{W}_i={\mathit{F}}_i^T\mathsf{\delta }{\mathit{u}}_{Ii}+\left(-{\mathit{F}}_i^T\right)\mathsf{\delta }{\mathit{u}}_{Oi}\end{array} \\ {\mathit{F}}_i=\left[\begin{array}{c}{f}_{im}{\mathit{n}}_i\\ \mathbf{0}\end{array}\right]\left(i=1~3\right) \end{gathered}$$

(27)

We first verified the statics of the lander. The displacement boundary condition and the magnitude of the buffer force are set below. ${\mathit{u}}_b$ is the node of the main body, and ${\mathit{u}}_f$ is the node of the footpad.

$$\begin{gathered} {\mathit{u}}_b={\left[0,0,0,0,0,0\right]}^T\mathrm{m} \\ {\mathit{u}}_f={\left[-5,5,10,0,0,0\right]}^T\times {10}^{-3}\mathrm{m} \\ {f}_{1m}=2000\mathrm{N},f_{2m}=10000\mathrm{N},f_{3m}=5000\mathrm{N} \end{gathered}$$

By setting the acceleration terms ${\ddot{\mathit{u}}}_1$ and ${\ddot{\mathit{u}}}_2^0$ in Equations (19) and (20) to $\mathbf{0}$, the theoretical boundary force ${\mathit{F}}_f$ of node ${\mathit{u}}_f$ were solved. These results were compared with the FEA results in Abaqus. In Abaqus, all links use the C3D8R element. The surface contact was set between the outer and inner tube surfaces as hard and frictionless contact. The $U$ and $S$ pairs were simulated by the U JOINT connector and JOIN + CARDAN connecter, respectively. The boundary force ${\mathit{F}}_f$ of the theoretical results and simulation results are compared in

Table 1. The boundary force ${\mathit{F}}_f$ of the theoretical statics results and FEA simulation results.

	Theoretical Results	Simulation Results	Relative Error (%)
$F_x\left(\mathrm{N}\right)$	−3525.19	−3541.30	−0.46
$F_y\left(\mathrm{N}\right)$	−2060.13	−2073.82	−0.66
$F_z\left(\mathrm{N}\right)$	435.27	425.617	2.27
$M_x\left(\mathrm{N}\cdot \mathrm{M}\right)$	48.07	48.53	−0.95
$M_y\left(N\cdot M\right)$	−82.26	−82.63	−0.46
$M_z\left(\mathrm{N}\cdot \mathrm{M}\right)$	3.04 × 10−7	−0.28	------

. The relative error is less than $3\%$. The Von Mises stress distributions of the two results are compared in Figure 5, which shows a good match.

With stiffness matrix ${\mathit{K}}_{11}$ and mass matrix ${\mathit{M}}_{11}$, the $i\mathrm{th}$ theoretical natural frequency ${\mathsf{\omega }}_i$ and the corresponding modal ${\mathit{\varphi }}_i=\left[{\mathit{\varphi }}_1;{\mathit{u}}_2^0\right]$ can be obtained as follows.

$$\left|{\mathit{K}}_{11}-{\mathsf{\omega }}_i^2{\mathit{M}}_{11}\right|=\mathbf{0}$$

(28)

$$\left({\mathit{K}}_{11}-{\mathsf{\omega }}_i^2{\mathit{M}}_{11}\right){\mathit{\varphi }}_1=\mathbf{0}$$

(29)

The first $10$ natural frequency and modalities between the FEA results and theoretical results are compared in Figure 6 and Figure 7, which show a good match.

4. Multi-Objective Optimization with Equivalent Static Loads for Parallel Mechanism

4.1. Equivalent Static Loads

In MSA, the PM in the workspace is considered a series of structures. Therefore, we added the configuration of the PM as a new dimension to the structural optimization procedure. The equivalent static loads were modified for the elastodynamic optimization of the PM. In ESLs, the dynamic response is transferred into a series of equivalent static loads. Therefore, in the sub-optimization procedure, only static equilibrium algebraic equations need to be solved. The modified ESL procedure is shown in Figure 8.

As shown in Figure 8, we first chose a set of initial design variables ${\mathit{b}}^k$. The design variables are usually the dimension parameters of the PM, including the structural parameters, such as the radii of the links, and mechanism parameters, such as the length of the base and platform.

In the first step, the configuration sample points ${\mathit{d}}^i\left(k=1,2,\cdots ,{\mathrm{num}}_{\mathrm{c}}\right)$ under the current design variables ${\mathit{b}}^k$ were sorted from the mechanism workspace. For each configuration point ${\mathit{d}}^i$, the vibration equation was solved and the dynamic response $\mathit{z}\left({\mathit{b}}^k,{\mathit{d}}^i,t^j\right)\left(j=1,2,\cdots ,{\mathrm{num}}_{\mathrm{t}}\right)$ was obtained. The dynamic response $\mathit{z}$ was sampled at time sampling points $t^j$. The vibration equation was obtained from Equations (19) and (20). Since the dimension of the vibration equation is relatively large, usually several hundred, the process is very time-consuming, even within a small time span. However, the vibration equation only needs to be solved one time in each cycle. Therefore, the ESLs greatly reduce the optimization time.

In the next step, a series of equilibrium static loads ${\mathit{f}}_{eq}\left({\mathit{b}}^k,{\mathit{d}}^i,s^j\right)$ is calculated by multiplying the dynamic response $\mathit{z}\left({\mathit{b}}^k,{\mathit{d}}^i,t^j\right)$ with the stiffness matrix $\mathit{K}\left({\mathit{b}}^k,{\mathit{d}}^i\right)$. The sampling points are equal to the time sampling points $s^j=t^j$. When this series of loads ${\mathit{f}}_{eq}$ is statically added to the structure, it can create the same dynamic response in the dynamic vibration process.

In the next step, namely linear static optimization, ${\mathit{f}}_{eq}\left({\mathit{b}}^k,{\mathit{d}}^i,s^j\right)$ is considered a constant in this sub-optimization procedure. In this procedure, the classical static optimization process is implemented. The objective function is $D$. The constraint equation is $\mathit{K}\left({\mathit{b}}^k,{\mathit{d}}^i\right)\mathit{z}={\mathit{f}}_{eq}\left({\mathit{b}}^k,{\mathit{d}}^i,s^j\right)$ and $\mathit{g}\left(\mathit{b},{\mathit{d}}^i,\mathit{z}\right)\le \mathbf{0},$ and the boundary of the design parameter is ${\mathit{b}}_{min}\le \mathit{b}\le {\mathit{b}}_{max}$. In this sub-optimization procedure, optimized design parameters $\mathit{b}$ will be found. Then, they are used in the next iteration process, ${\mathit{b}}_{k+1}=\mathit{b}$. The iteration converges and stops when the discrepancy of the adjacent two sets of parameters is within an acceptable range, $\left|{\mathit{b}}_k-{\mathit{b}}_{k+1}\right|\le \mathit{\epsilon }$.

4.2. Desirability Approach

In application, designers usually optimize more than one aspect of the PM. For the machining process, designers pursue higher stiffness and lower vibration amplitude. For a fast pick-and-place process, a lower mass, lower energy cost, and higher accuracy are considered. Therefore, more than one objective function needs to be considered simultaneously. The linear static response optimization step in ESLs should be a multi-objective optimization process. The desirability approach is added to the ESLs to realize it. Since it is easy to implement and has a low computational cost, the desirability approach is widely used in multi-objective optimization design. The desirability approach transfers multiple objective functions into a single function by assigning different weight factors. In the first step, each objective function is transformed into a dimensionless variable. The objective function to be minimized is denoted as $f_i,$ and the objective function to be maximized is denoted as $f_j$. Each one has the lower bounds $L_i,L_j$ and upper bounds $H_i,H_j$. Then, according to Equations (30) and (31), $f_i$ and $f_j$ are transformed into dimensionless individual desirability functions $D_i$ and $D_j$, ranging from $0$ to $1$. $w_i\mathrm{and}{w}_j$ are the weight coefficients on each objective function [34].

$$\begin{array}{c}{D}_i=\{\begin{array}{ll}1& f_i<L_i\\ {\left(\frac{H_i-f_i}{H_i-L_i}\right)}^{w_i}& L_i<f_i<H_i\\ 0& f_i>H_i\end{array}\end{array}$$

(30)

$$\begin{array}{c}{D}_j=\{\begin{array}{ll}0& f_j<L_j\\ {\left(\frac{f_j-L_j}{H_j-L_j}\right)}^{w_j}& L_j<f_j<H_j\\ 1& f_j>H_j\end{array}\end{array}$$

(31)

Then, the overall desirability function $D$ is obtained as the geometric mean of all individual desirability functions $D_k\left(k=1,2,\cdots ,n\right)$.

$$\begin{array}{c}D={\left({\displaystyle {\displaystyle \prod }_{k=1}^n}{D}_k\right)}^{\frac{1}{n}}\end{array}$$

(32)

4.3. Application in the Lander

In this section, the lander introduced in Section 3 is optimized with modified ESLs and the desirability approach. Before landing on the ground, the legged lander will adjust the axis $R_i\left(i=1~3\right)$ to fit the landing surface environment. Therefore, the lander has a different configuration to land on the ground. We added the landing configuration as a new dimension in the structural optimization. When hitting on the ground, the footpad suffers severe vibration force. To reduce the mass and have better vibration performance, the leg of the lander should be optimized. The design parameter includes the tube radius in the main strut $r_{mO},r_{mI}$ and the tube radius in the secondary struts $r_{sO},r_{sI}$. It also includes mechanism parameters of the main body $\theta ,a,b$ and mechanism parameters of the platform $a_p,b_p,$ as shown in Figure 4b. The main body ${\mathit{u}}_b$ is fixed. The rotational DOF (degree of freedom) of the footpad is fixed. The landing force exerted on the footpad ${\mathit{F}}_f$ is chosen as the force boundary condition. The parameter value of ${\mathit{F}}_f$ is listed in

Table 2. Parameter values related to the landing force ${\mathit{F}}_f$ , the buffer force magnitude $f_{im}$, the desirability approach, and the ESLs.

${\mathit{A}}_{\mathit{x}}\left(kN\right)$	${\mathit{A}}_{\mathit{y}}\left(kN\right)$	${\mathit{A}}_{\mathit{z}}\left(kN\right)$	${\mathsf{\xi }}_{\mathit{x}}$	${\mathsf{\xi }}_{\mathit{y}}$	${\mathsf{\xi }}_{\mathit{z}}$	${\mathsf{\omega }}_{\mathit{x}}$	${\mathsf{\omega }}_{\mathit{y}}$	${\mathsf{\omega }}_{\mathit{z}}$	${\mathit{x}}_{\mathit{c}1}^1\left(mm\right)$	${\mathit{x}}_{\mathit{c}1}^2\left(mm\right)$	${\mathit{x}}_{\mathit{c}2}^2\left(mm\right)$
$5$	$5$	$15$	$0.05$	$0.05$	$0.05$	$600\mathsf{\pi }$	$600\mathsf{\pi }$	$600\mathsf{\pi }$	$0.01$	$0.01$	$0.01$
${\mathit{f}}_{\mathbf{1}\mathit{m}\mathit{a}\mathit{x}}\left(kN\right)$	${\mathit{f}}_{\mathbf{2}\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{1}}\left(kN\right)$	${\mathit{f}}_{\mathbf{2}\mathit{m}\mathit{a}\mathit{x}}^{\mathbf{2}}\left(kN\right)$	${\mathit{L}}_{\mathit{m}}\left(kg\right)$	${\mathit{H}}_{\mathit{m}}\left(kg\right)$	${\mathit{w}}_{\mathit{m}}$	${\mathit{L}}_{\mathsf{\omega }}\left(Hz\right)$	${\mathit{H}}_{\mathsf{\omega }}\left(Hz\right)$	${\mathit{w}}_{\mathsf{\omega }}$	$nu{m}_c$	$nu{m}_t$	$\left[{\mathit{\sigma }}_{\mathit{s}} \right]\left(MPa\right)$
$25$	$10$	$15$	$30$	$90$	$2$	$30$	$300$	$1$	$20$	$30$	$200$

. The lunar gravity is also considered, which is along the opposite direction of the $z$ axis. $\mathit{g}$ is the gravitational acceleration.

$$\begin{gathered} {\mathit{u}}_b={\left[0,0,0,0,0,0\right]}^T \\ {\mathit{u}}_f={\left[u_{fx},u_{fy},u_{fz},0,0,0\right]}^T \\ {\mathit{F}}_f={\left[F_{fx},F_{fy},F_{fz},0,0,0\right]}^T \\ {F}_{fx}=A_x{e}^{-{\mathsf{\xi }}_x{\mathsf{\omega }}_{x}t}\sin \left({\mathsf{\omega }}_{x}t\right) \\ {F}_{fy}=A_y{e}^{-{\mathsf{\xi }}_y{\mathsf{\omega }}_{y}t}\sin \left({\mathsf{\omega }}_{y}t\right) \\ {F}_{fz}=A_z\left(1-e^{-{\mathsf{\xi }}_z{\mathsf{\omega }}_{z}t}\sin \left({\mathsf{\omega }}_{z}t\right)\right) \\ \mathit{g}={\left[0,0,-1.63\right]}^T \end{gathered}$$

The buffer force magnitude $f_1$ in the main strut is the function of buffer stroke $x$. Their relationship is shown in Figure 9a. The $x$ axis is the main strut buffer stroke. It is the relative translation distance between nodes ${\overline{\mathit{u}}}_{O1}$ and ${\overline{\mathit{u}}}_{I1}$, which are expressed in local coordinates.

$$x={\overline{\mathit{u}}}_{O1}\left(1\right)-{\overline{\mathit{u}}}_{I1}\left(1\right)$$

(33)

The buffer force magnitude in two secondary struts has the same function, $f_{2m}=f_{3m}$, which is shown in Figure 9b. The $x$ axis is the secondary strut buffer stroke computed by the relative translation distance between the ${\overline{\mathit{u}}}_{O2}$ and ${\overline{\mathit{u}}}_{I2}$, which are also expressed in local coordinates. The parameter values in Figure 9 are listed in Table 2.

$$\begin{array}{c}x={\overline{\mathit{u}}}_{O2}\left(1\right)-{\overline{\mathit{u}}}_{I2}\left(1\right)\end{array}$$

(34)

The vibration equation can be derived from Equations (19) and (20). The dimension of ${\mathit{M}}_{11}$ is $270$. The time span was set from $0$ to $0.02s$. The mass of the leg $m$ and the average fundamental frequency in the workspace $\mathsf{\omega }$ were chosen as the individual objective functions. The desirability approach was used to combine $m$ and ω into a single desirability function $D$. The overall objective function $f_{obj}$ is the negative of $D$. The parameter values in the desirability approach are listed in Table 2.

$$f_{obj}=-D$$

(35)

The maximum Von Mises stress $\left[{\sigma }_s\right]$ and the radius of the tubes are used as the constraint conditions. The boundaries of the design parameter are ${\mathit{b}}_{min},{\mathit{b}}_{max}$.

$$\begin{array}{c}{\sigma }_s\left(\mathit{b},{\mathit{d}}^i,\mathit{z}\right)-\left[{\sigma }_s\right]\le 0\\ r_{mI}+0.005-r_{mO}\le 0\\ r_{sI}+0.005-r_{sO}\le 0\end{array}$$

(36)

$$\begin{array}{c}{\mathit{b}}_{min}=0.8\times {\mathit{b}}_k\\ {\mathit{b}}_{max}=1.2\times {\mathit{b}}_k\end{array}$$

(37)

The optimization process and results are shown in

Table 3. Design parameters and objective functions in the optimization process.

	θ(∘)	$a\left(m\right)$	$\mathit{b}\left(m\right)$	$a_p\left(m\right)$	${\mathit{b}}_p\left(m\right)$	$r_{mI}\left(m\right)$	$r_{mO}\left(m\right)$	$r_{sI}\left(m\right)$	$r_{sO}\left(m\right)$	$\mathsf{\omega }\left(Hz\right)$	$m\left(kg\right)$	${\mathit{f}}_{o\mathit{b}j}$
Initial value	$30.00$	$0.3826$	$0.1439$	$0.0477$	$0.0318$	$0.0350$	$0.0600$	$0.0250$	$0.0500$	$83.83$	$61.61$	$-0.2016$
Iteration 1	$29.58$	$0.3759$	$0.1488$	$0.0485$	$0.0314$	$0.0368$	$0.0580$	$0.0253$	$0.0473$	$89.56$	$56.48$	$-0.2508$
Iteration 2	$28.79$	$0.3832$	$0.1417$	$0.0483$	$0.0305$	$0.0389$	$0.0538$	$0.0242$	$0.0438$	$93.95$	$48.53$	$-0.3219$
Iteration 3	$29.69$	$0.3823$	$0.1318$	$0.0470$	$0.0307$	$0.0391$	$0.0519$	$0.0256$	$0.0399$	$92.50$	$42.93$	$-0.3610$
Iteration 4	$29.62$	$0.3919$	$0.1302$	$0.0474$	$0.0298$	$0.0428$	$0.0550$	$0.0239$	$0.0367$	$103.1$	$43.36$	$-0.3876$
Iteration 5	$29.52$	$0.4064$	$0.1485$	$0.0468$	$0.0284$	$0.0482$	$0.0593$	$0.0224$	$0.0348$	$116.0$	$46.45$	$-0.3933$
Iteration 6	$27.97$	$0.3887$	$0.1398$	$0.0481$	$0.0291$	$0.0460$	$0.0613$	$0.0196$	$0.0302$	$117.1$	$45.45$	$-0.4052$
Iteration 7	$29.11$	$0.3859$	$0.1351$	$0.0478$	$0.0306$	$0.0477$	$0.0600$	$0.0163$	$0.0295$	$113.8$	$43.50$	$-0.4146$
Iteration 8	28.15	$0.3622$	$0.1510$	$0.0459$	$0.0321$	$0.0470$	$0.0576$	$0.0158$	$0.0274$	$111.1$	$39.53$	$-0.4426$
Iteration 9	$27.97$	$0.3644$	$0.1639$	$0.0478$	$0.0321$	$0.0433$	$0.0553$	$0.0172$	$0.0273$	$107.3$	$37.04$	$-0.4531$
Iteration 10	$28.06$	$0.3516$	$0.1626$	$0.0473$	$0.0332$	$0.0388$	$0.0503$	$0.0169$	$0.0272$	$96.11$	$32.05$	$-0.4575$
Iteration 11	$28.94$	$0.3566$	$0.1708$	$0.0486$	$0.0344$	$0.0424$	$0.0527$	$0.0168$	$0.0277$	$102.8$	$34.69$	$-0.4588$
Optimized value	$28.94$	$0.3566$	$0.1708$	$0.0486$	$0.0344$	$0.0424$	$0.0527$	$0.0168$	$0.0277$	$102.8$	$34.69$	$-0.4588$
Convergence $\mathit{\epsilon }$	1	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	-----	-----	-----

. The optimization process went through $11$ iterations and finally converged. The total optimization time was $5$ h (performed on Intel Xeon Dual CPU 2.10 GHz, 64.0GB RAM). After optimization, the mass of the lander was reduced by $43.7\%$, and the average fundamental frequency was increased by $22.6\%$.

5. Conclusions

This work used the kinematic pair equivalent method to simulate the surface contact between the outer and inner tubes. It expanded the connection type in MSA by adding the tube model. The strain energy minimization method was used to obtain the corresponding mass matrix and stiffness matrix. The reconfigurable lunar legged lander was used as an example to verify the effectiveness of the method. The static reaction force, natural frequency, and modal of the theoretical results show a good match with the FEA results. Equivalent static loads and the desirability approach were introduced and modified in the elastodynamic optimization of the PMs. A mechanism configuration was added as a new dimension in the structural optimization. The optimization procedure was implemented on the legged lander. The mass of the lander was reduced by $43.7\%$ and the average fundamental natural frequency was increased by $22.6\%$.