Coupling analysis of a 5-degree-of-freedom hybrid manipulator based on a global index

Abstract

Parallel kinematic machines have been applied in aerospace and automotive manufacturing due to their potentials in high speed and high accuracy. However, there exists coupling in parallel kinematic machines, which makes dynamic analysis, rigidity enhancement, and control very complicated. In this article, coupling characteristics of a 5-degree-of-freedom (5-dof) hybrid manipulator are analyzed based on a local index and a global index. First, velocity analysis as well as acceleration analysis of the robot is conducted to provide essential information for dynamic modeling. Then the dynamic model is built based on the principle of virtual work. Whereas the mass matrix is off-diagonal, a local coupling index as well as a global index is defined, based on which coupling characteristics of the robot are analyzed. Results show that distributions of coupling indices are symmetric due to its structural features. And dimensional parameters, structural parameters, as well as mass parameters have a large influence on the system’s coupling characteristics. Research conducted in the article is of great help in optimal design and control. Meanwhile, the method proposed in the article can be applied to other types of parallel kinematic machines or hybrid manipulators.

Introduction

Parallel kinematic machines (PKMs) have received significant attention both in terms of theoretical research and practical applications due to their merits of high loads, high accuracy, and high speed when compared with serial kinematic machines (SKMs). The typical success can be verified by the application of the Tricept and Sprint Z3 in assembly and milling in aerospace and automotive industry.^1–4 As is known, PKMs have large stiffness and large loading capability due to their closed-loop structural features. However, it is the closed-loop structural features that make limbs of a PKM coupled, which makes it very complicated for topological synthesis, dimensional design, performance enhancement, and control system design. Limbs of PKMs are coupled, which demonstrates that the driving force/torque of one certain limb is related to other limbs. Despite that there are many references focusing on kinematic analysis,^5–7 stiffness modeling,^8–10 dynamic analysis,^11–13 and calibration^14–16 of PKMs, references about coupling analysis are very scarce. As far as the authors are concerned, coupling analysis of PKMs involves two basic issues, one is how to establish a coupling model, and the other is how to quantify coupling strength.

In general, the dynamic model of a PKM is a highly nonlinear complex system with multiple inputs/outputs, which can be regarded as the basis of coupling analysis. There are mainly several methods of dynamic modeling, that is, Lagrange method,^17,18 Newton-Euler method,^19,20 Kane’s method,^21,22 the principle of virtual work.^23,24 The Lagrange method is based on the system’s energy. Bonnemains et al. ²⁵ built a dynamic model of Tripteor X7, based on which inertia parameters were indentified. The Newton-Euler method describes inertia parameters, force, and acceleration of a PKM by establishing one Newton equation and one Euler equation of each part. Based on the Newton-Euler method, Wang et al. ²⁶ proposed a simplified dynamic model which was used to the optimal design of the driving force. Relatively speaking, the Kane’s method is complicated, which calculates the generalized driving force and generalized inertia force of a system by defining partial velocity and partial angular velocity. The principle of virtual work is the most commonly used method due to its easy principle. Kalani et al. ²⁷ presented an improved dynamic model of a 6-UPS PKM based on the principle of virtual work. In addition, the Gibbs-Appell method is usually applied to establish dynamic equations of PKMs. ²⁸ As for the coupling strength, there are no too much references talking about how to quantify. Bergerman et al. ²⁹ developed a dynamic coupling index for an underactuated manipulator, which can express the coupling degree between passive joints and active joints. Inspired by the work conducted in Bergerman et al., ²⁹ this article attempts to analyze coupling in PKMs.

According to discussions above, this article deals with the coupling strength of a 5-dof hybrid manipulator. The remainder of the article is organized as follows. Kinematic analysis is described in Section 2, which can provide essential information for the following dynamic modeling. Based on the principle of virtual work, the rigid dynamic model is established in Section 3. Then a local coupling index and a global index are proposed, based on which coupling characteristics are analyzed in Section 4. Finally, main conclusions are drawn in Section 5.

Description of mechanical structures and inverse kinematic analysis

Figure 1 shows a 3D model of the Trimule robot proposed in Dong et al., ¹⁰ which contains a PKM module and a 2R module. In general, the PKM module is composed of one base, one platform, three active universal–prismatic–spherical (UPS) limbs and one passive universal–prismatic (UP) limb. The three active UPS limb assemblages have the same topological structures, which has an oscillating limb connected to the base by a universal joint and a stretchable limb connected to the platform by a spherical joint. And the passive UP limb assemblage mainly consists of a thin limb and a thick limb as shown in Figure 2. The thin limb is connected to the base by a universal joint while the thick limb is fixedly connected to the platform. The 2R module consists of two revolute joints which are defined as the A axis and the C axis, respectively, which can improve flexibility of the robot.

Figure 1.

A CAD model of the Trimule robot.

Figure 2.

Assemblage of active and passive limbs.

In order to facilitate derivation, the schematic diagram of the hybrid manipulator and several coordinate systems are depicted in Figure 3. Herein, C is the end point of the machine tool. A denotes the intersection point of A axis and C axis. A_i (i = 1 to 3) denotes the geometrical center of the ith spherical joint. A₄ is the connection point on the interface of the platform and the passive limb. B_k (k = 1 to 4) denotes the geometrical center of the kth universal joint. C_k (k = 1 to 4) is the end point of the kth limb. The machine tool body-fixed frame C-x_Cy_Cz_C is attached to the end point of the tool C, with the x_C axis parallel to the C axis’s rotation axis, the z_C axis parallel to the machine tool’s axis, and the y_C axis can be decided by the right-hand rule. The A axis body-fixed frame A-x_Ay_Az_A is built at the point A, where the x_A axis parallel to the C axis’s rotation axis, the z_A axis parallel to the axis of the A axis, and the y_A axis can be decided by the right-hand rule. B₄-xyz is the global frame attached to the point B₄, where the x axis points from B₄ to B₂ and the y axis is vertically up to B₃B₂. Then direction of the z axis can be determined by the right-hand rule. The platform body-fixed frame P-uvw is set at the midpoint of A₃A₂, with the u axis parallel to A₃A₂ and the w axis vertical to the platform. Accordingly, the v axis can be decided by the right-hand rule. The limb body-fixed frame B_k-x_ky_kz_k is set at the point B_k, where the z_k axis is along the kth limb, the y_k axis is along one rotation axis of the kth universal joint, and the x_k axis can be determined by the right-hand rule. To be specific, B₁ – x₁y₁z₁ in the 1st UPS limb is shown in Figure 3.

Figure 3.

Schematic diagram of the PKM module.

Assume that the transformation matrix of the platform body-fixed frame P-uvw with respect to the global frame B₄-xyz can be defined as

$${}^{B_4}{\mathit{R}}_P:\mathrm{Trans}(P-\mathrm{uvw}\to B_4-\mathrm{xyz})$$

(1)

Similarly, transformation matrices of frames B_k-x_ky_kz_k and C-x_Cy_Cz_C with respect to the global frame B₄-xyz can be expressed, respectively, as

$${}^{B_4}{\mathit{R}}_{\mathrm{Bk}}:\mathrm{Trans}(B_k-x_k{y}_k{z}_k\to B_4-\mathrm{xyz})$$

(2)

$${}^{B_4}{\mathit{R}}_C:\mathrm{Trans}(C-x_C{y}_C{z}_C\to B_4-\mathrm{xyz})$$

(3)

For content limitation, detailed inverse kinematics is not described in the article. Since there is no coupling in the 2R module, the following will focus on kinematic and dynamic modeling of the PKM module.

Velocity analysis

Position of the point P r _P can be expressed in the global frame B₄-xyz as

$${\mathit{r}}_P={\mathit{b}}_i+q_i{\mathit{w}}_i-{\mathit{a}}_i,(i=1~3)$$

(4)

$${\mathit{r}}_P=q_4{\mathit{w}}_4$$

(5)

where b _i is the vector pointing from B₄ to B_i; q_i and w _i denote the length and unit vector of the ith UPS limb, respectively; a _i is the vector pointing from P to A_i expressed in the global frame B₄-xyz; q₄ and w ₄ denote the length and unit vector of the UP limb, respectively.

Taking derivation of equations (4) and (5) yields

$${\mathit{v}}_P={\stackrel{·}{q}}_i{\mathit{w}}_i+q_i({\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{w}}_i)-{\mathit{\omega }}_P\times {\mathit{a}}_i,(i=1~3)$$

(6)

$${\mathit{v}}_P={\stackrel{·}{q}}_4{\mathit{w}}_4+q_4({\mathit{\omega }}_P\times {\mathit{w}}_4)$$

(7)

where v _P and w _P are the linear velocity of the point P and the angular velocity of the platform, respectively; ${\stackrel{·}{q}}_i$ and ω _Li denote the velocity of the ith prismatic joint and the angular velocity of the ith UPS limb, respectively.

Taking inner dot of equation (6) by w _i leads to

$${\stackrel{·}{q}}_i={\mathit{w}}_i^{\mathrm{T}}{\mathit{v}}_P+({\mathit{a}}_i\times {\mathit{w}}_i{)}^{\mathrm{T}}{\mathit{\omega }}_P,(i=1~3)$$

(8)

Taking inner dot of equation (7) by u and v ( u and v are unit vectors of u and v axes), respectively, yields

$${\mathit{u}}^{\mathrm{T}}{\mathit{v}}_P+q_4(\mathit{u}\times {\mathit{w}}_4{)}^{\mathrm{T}}{\mathit{\omega }}_P=0$$

(9)

$${\mathit{v}}^{\mathrm{T}}{\mathit{v}}_P+q_4(\mathit{v}\times {\mathit{w}}_4{)}^{\mathrm{T}}{\mathit{\omega }}_P=0$$

(10)

It is noted that the rotation matrix ${}^{B_4}{\mathit{R}}_P$ can be determined by rotating about the x axis with an angle ψ first and then rotating about the v axis with an angle θ. According to the principle of linear superposition, the angular velocity of the platform can be expressed as

$${\mathit{\omega }}_P=\stackrel{·}{\psi }\mathit{x}+\stackrel{·}{\theta }\mathit{v}$$

(11)

Taking inner dot of equation (11) by $\mathit{n}=\mathit{v}\times \mathit{x}$ yields

$${\mathit{n}}^{\mathrm{T}}{\mathit{\omega }}_P=0$$

(12)

Equations (8)–(10) and (12) can be written in a matrix form as follows

$$\mathit{J}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]=\stackrel{·}{\mathit{q}}$$

(13)

where $\stackrel{·}{\mathit{q}}={\left[\begin{array}{cccccc}{\stackrel{·}{q}}_1\hfill & {\stackrel{·}{q}}_2\hfill & {\stackrel{·}{q}}_3\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}^{\mathrm{T}}$ ; $\mathit{J}=\left[\begin{array}{c}{\mathit{J}}_a\hfill \\ \multicolumn{1}{c}{{\mathit{J}}_c}\end{array}\right]$ is the generalized Jacobian matrix of the PKM module, J _a and J _c are defined as the active Jacobian matrix and the constraint Jacobian matrix, respectively, which can be expressed as

$${\mathit{J}}_a={\left[\begin{array}{ccc}{\mathit{w}}_1\hfill & {\mathit{w}}_2\hfill & {\mathit{w}}_3\hfill \\ \multicolumn{1}{c}{{\mathit{a}}_1\times {\mathit{w}}_1}& {\mathit{a}}_2\times {\mathit{w}}_2\hfill & {\mathit{a}}_3\times {\mathit{w}}_3\hfill \end{array}\right]}^{\mathrm{T}}$$

(14)

$${\mathit{J}}_c={\left[\begin{array}{ccc}\mathit{u}\hfill & \mathit{v}\hfill & 0\hfill \\ \multicolumn{1}{c}{q_4(\mathit{u}\times {\mathit{w}}_4)}& q_4(\mathit{v}\times {\mathit{w}}_4)\hfill & \mathit{n}\hfill \end{array}\right]}^{\mathrm{T}}$$

(15)

Taking cross product of equation (6) by w _i , the angular velocity of the ith UPS limb ω _Li can be expressed as

$${\mathit{\omega }}_{\mathrm{Li}}={\mathit{J}}_{\omega i}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right],(i=1~3)$$

(16)

where ${\mathit{J}}_{\omega i}=1/q_i[{\mathit{w}}_i\times ](\begin{array}{cc}{\mathit{E}}_{3\times 3}\hfill & -[{\mathit{a}}_i\times ]\hfill \end{array})$ , $[{\mathit{w}}_i\times ]$ and $[{\mathit{a}}_i\times ]$ are skew matrices of vectors w _i and a _i .

Supposing that the mass center of the ith UPS limb is defined as D_i and the vector pointing from B_i to D_i is r _i in the global frame B₄-xyz, the velocity of the mass center D_i can be expressed in the frame B₄-xyz as

$${\mathit{v}}_{\mathrm{Li}}={\mathit{v}}_{\mathrm{Bi}}+{\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{r}}_i,(i=1~3)$$

(17)

where v _Bi denotes the velocity of point B_i and there exists ${\mathit{v}}_{\mathrm{Bi}}=0$ .

Substituting equation (16) into equation (17), the velocity of D_i can be calculated by

$${\mathit{v}}_{\mathrm{Li}}={\mathit{J}}_{\mathrm{vi}}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right],(i=1~3)$$

(18)

where ${\mathit{J}}_{\mathrm{vi}}=-1/q_i[{\mathit{r}}_i\times ][{\mathit{w}}_i\times ](\begin{array}{cc}{\mathit{E}}_{3\times 3}\hfill & -[{\mathit{a}}_i\times ]\hfill \end{array})$ and $[{\mathit{r}}_i\times ]$ are the skew matrix of r _i .

Combining equations (16) with (18), the velocity of the mass center D_i and the angular velocity of the ith UPS limb can be written in a matrix form as follows

$$\left[\begin{array}{c}{\mathit{v}}_{\mathrm{Li}}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{\mathrm{Li}}}\end{array}\right]={\mathit{J}}_{\mathrm{Li}}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right],(i=1~3)$$

(19)

where ${\mathit{J}}_{\mathrm{Li}}=\left[\begin{array}{c}{\mathit{J}}_{\mathrm{vi}}\hfill \\ \multicolumn{1}{c}{{\mathit{J}}_{\omega i}}\end{array}\right]$ can be defined as the transformation matrix of the platform’s velocity with respect to that of the ith UPS limb.

Similarly, define r ₄ as the position vector of the UP limb’ mass center D₄, whose velocity can thus be expressed as

$${\mathit{v}}_{L4}={\mathit{\omega }}_P\times {\mathit{r}}_4$$

(20)

Thus, the velocity of the mass center D₄ and angular velocity of the UP limb can be written as

$$\left[\begin{array}{c}{\mathit{v}}_{L4}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{L4}}\end{array}\right]={\mathit{J}}_{L4}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]$$

(21)

where ${\mathit{J}}_{L4}=\left[\begin{array}{cc}{0}_{3\times 3}\hfill & -[{\mathit{r}}_4\times ]\hfill \\ \multicolumn{1}{c}{0_{3\times 3}}& {\mathit{E}}_{3\times 3}\hfill \end{array}\right]$ , $[{\mathit{r}}_4\times ]$ denotes the skew matrix of vector r ₄.

Acceleration analysis

Taking inner dot of equation (7) by w ₄ yields

$${\stackrel{·}{q}}_4={\mathit{w}}_4^{\mathrm{T}}{\mathit{v}}_P$$

(22)

Taking derivation of equation (13) yields

$$\mathit{J}\left[\begin{array}{c}{\mathit{a}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_P}\end{array}\right]+\stackrel{·}{\mathit{J}}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]=\stackrel{··}{\mathit{q}}$$

(23)

where $\stackrel{·}{\mathit{J}}=\left[\begin{array}{c}{\stackrel{·}{\mathit{J}}}_a\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{J}}}_c}\end{array}\right]$ is the derivative matrix of J with respect to time. And there exist

$${\stackrel{·}{\mathit{J}}}_a={\left[\begin{array}{ccc}{\mathit{\omega }}_{L1}\times {\mathit{w}}_1\hfill & {\mathit{\omega }}_{L2}\times {\mathit{w}}_2\hfill & {\mathit{\omega }}_{L3}\times {\mathit{w}}_3\hfill \\ \multicolumn{1}{c}{({\mathit{\omega }}_P\times {\mathit{a}}_1)\times {\mathit{w}}_1+{\mathit{a}}_1\times ({\mathit{\omega }}_{L1}\times {\mathit{w}}_1)}& ({\mathit{\omega }}_P\times {\mathit{a}}_2)\times {\mathit{w}}_2+{\mathit{a}}_2\times ({\mathit{\omega }}_{L2}\times {\mathit{w}}_2)\hfill & ({\mathit{\omega }}_P\times {\mathit{a}}_3)\times {\mathit{w}}_3+{\mathit{a}}_3\times ({\mathit{\omega }}_{L3}\times {\mathit{w}}_3)\hfill \end{array}\right]}^{\mathrm{T}}$$

(24)

$${\stackrel{·}{\mathit{J}}}_c={\left[\begin{array}{ccc}{\mathit{\omega }}_P\times \mathit{u}\hfill & {\mathit{\omega }}_P\times \mathit{v}\hfill & 0\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_4(\mathit{u}\times {\mathit{w}}_4)+q_4({\mathit{\omega }}_P\times \mathit{u}\times {\mathit{w}}_4+\mathit{u}\times ({\mathit{\omega }}_P\times {\mathit{w}}_4))}& {\stackrel{·}{q}}_4(\mathit{v}\times {\mathit{w}}_4)+q_4({\mathit{\omega }}_P\times \mathit{v}\times {\mathit{w}}_4+\mathit{v}\times ({\mathit{\omega }}_P\times {\mathit{w}}_4))\hfill & {\mathit{\omega }}_P\times \mathit{v}\times \mathit{x}\hfill \end{array}\right]}^{\mathrm{T}}$$

(25)

Taking derivation of equation (19) yields

$$\left[\begin{array}{c}{\mathit{a}}_{\mathrm{Li}}\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_{\mathrm{Li}}}\end{array}\right]={\mathit{J}}_{\mathrm{Li}}\left[\begin{array}{c}{\mathit{a}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_P}\end{array}\right]+{\stackrel{·}{\mathit{J}}}_{\mathrm{Li}}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right],(i=1~3)$$

(26)

where ${\stackrel{·}{\mathit{J}}}_{\mathrm{Li}}$ is the derivative matrix of J _Li with respect to time, which can be expressed as ${\stackrel{·}{\mathit{J}}}_{\mathrm{Li}}=\left[\begin{array}{c}{\stackrel{·}{\mathit{J}}}_{\mathrm{vi}}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{J}}}_{\mathrm{wi}}}\end{array}\right]$ . And there exist

$${\stackrel{·}{\mathit{J}}}_{\omega i}=\frac{1}{q_i}\{([({\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{w}}_i)\times ]-\frac{{\stackrel{·}{q}}_i}{q_i}[{\mathit{w}}_i\times ])\left(\begin{array}{cc}{\mathit{E}}_{3\times 3}\hfill & -[{\mathit{a}}_i\times ]\hfill \end{array}\right)+[{\mathit{w}}_i\times ]\left(\begin{array}{cc}{0}_{3\times 3}\hfill & [({\mathit{a}}_i\times {\mathit{\omega }}_P)\times ]\hfill \end{array}\right)\}$$

(27)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathit{J}}}_{\mathrm{vi}}=\frac{1}{q_i}\{(\frac{{\stackrel{·}{q}}_i}{q_i}[{\mathit{r}}_i\times ]-[({\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{r}}_i)\times ])[{\mathit{w}}_i\times ]\left(\begin{array}{c}{\mathit{E}}_{3\times 3}-[{\mathit{a}}_i\times ]\hfill \end{array}\right)}\\ -[{\mathit{r}}_i\times ]([({\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{w}}_i)\times ]\left(\begin{array}{c}{\mathit{E}}_{3\times 3}-[{\mathit{a}}_i\times ]\hfill \end{array}\right)+[{\mathit{w}}_i\times ]\left(\begin{array}{cc}{0}_{3\times 3}\hfill & [({\mathit{a}}_i\times {\mathit{\omega }}_P)\times ]\hfill \end{array}\right))\}\hfill \end{array}$$

(28)

Similarly, taking derivation of equation (21) yields

$$\left[\begin{array}{c}{\mathit{a}}_{L4}\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_{L4}}\end{array}\right]={\mathit{J}}_{L4}\left[\begin{array}{c}{\mathit{a}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_P}\end{array}\right]+{\stackrel{·}{\mathit{J}}}_{L4}\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]$$

(29)

where ${\stackrel{·}{\mathit{J}}}_{L4}=\left[\begin{array}{cc}0\hfill & [{\mathrm{\Delta }}_4\times ]\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$ , $[{\mathrm{\Delta }}_4\times ]$ denotes the skew matrix of vector ${\mathrm{\Delta }}_4={\mathit{r}}_4\times {\mathit{\omega }}_P$ .

Formulation of rigid dynamics

In this section, the principle of virtual work is applied to formulate the rigid dynamic model of the PKM module. And the 2R module will be treated as the platform’s load whose inertia matrix is position-dependent. In order to facilitate derivation, the force diagram of the PKM module is depicted in Figure 4.

Figure 4.

Force diagram of the PKM module.

Virtual work conducted by the platform

Assume that the external load $[\begin{array}{cc}\mathit{F}\hfill & \mathit{M}\hfill \end{array}{]}^{\mathrm{T}}$ is applied at the point P in the platform, of which the virtual work can be expressed as

$$W_P^F={\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}\mathit{F}\hfill \\ \multicolumn{1}{c}{\mathit{M}}\end{array}\right]$$

(30)

The inertial force and moment of the platform can be expressed as

$$\left[\begin{array}{c}{\mathit{F}}_P^a\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_P^a}\end{array}\right]=\left[\begin{array}{c}-m_{\mathrm{P}}{\mathit{a}}_P\hfill \\ \multicolumn{1}{c}{-{\mathit{I}}_{\mathrm{P}}{\mathit{\xi }}_P-{\mathit{\omega }}_P\times {\mathit{I}}_{\mathrm{P}}{\mathit{\omega }}_P}\end{array}\right]$$

(31)

where m_P and I _P are the equivalent mass and inertia matrix of the platform measured in the global frame B₄-xyz, respectively. It is noted that mass of the 2R module can be regarded as load applied at the platform. Thus there exists

$$m_{\mathrm{P}}=m_{\mathrm{P}0}+m_{\mathrm{A}}+m_{\mathrm{C}},{\mathit{I}}_{\mathrm{P}}={\mathit{I}}_{\mathrm{P}0}+{\mathit{I}}_{\mathrm{A}}+{\mathit{I}}_{\mathrm{C}}$$

(32)

where m_P0, m_A, and m_C are the mass of the platform itself, the A axis, and the C axis, respectively; I _P0, I _A, and I _C are the inertia matrices of the platform itself, the A axis and the C axis expressed in the global frame B₄-xyz. It is noted that I _P0, I _A, and I _C are calculated about the platform’s mass center P.

Then the virtual work conducted by the platform’s inertial force and moment can be expressed as

$$W_P^a={\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{\mathit{F}}_P^a\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_P^a}\end{array}\right]$$

(33)

Substituting equation (31) into equation (33) yields

$$W_P^a=-{\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]}^{\mathrm{T}}(\left[\begin{array}{cc}{m}_{\mathrm{P}}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{\mathrm{P}}\hfill \end{array}\right]\left[\begin{array}{c}{\mathit{a}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_P}\end{array}\right]+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P\times {\mathit{I}}_{\mathrm{P}}{\mathit{\omega }}_P}\end{array}\right])$$

(34)

where e is an identity matrix in 3×3.

The virtual work conducted by the platform’s gravity can be expressed as

$$W_P^g={\left[\begin{array}{c}{\mathit{v}}_P\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{m}_{\mathrm{P}}\mathit{g}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$$

(35)

where g denotes the gravity’s acceleration.

Virtual work conducted by the ith UPS limb

Assume that the external force f_i (actually caused by the driving torque) is applied at the ith UPS limb, whose virtual work can be expressed as

$$W_{\mathrm{Li}}^{\mathrm{fi}}={\stackrel{·}{q}}_i{f}_i$$

(36)

The inertial force and moment of the ith UPS limb can be expressed as

$$\left[\begin{array}{c}{\mathit{F}}_{\mathrm{Li}}^a\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{\mathrm{Li}}^a}\end{array}\right]=\left[\begin{array}{c}-m_{\mathrm{Li}}{\mathit{a}}_{\mathrm{Li}}\hfill \\ \multicolumn{1}{c}{-{\mathit{I}}_{\mathrm{Li}}{\mathit{\xi }}_{\mathrm{Li}}-{\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{I}}_{\mathrm{Li}}{\mathit{\omega }}_{\mathrm{Li}}}\end{array}\right]$$

(37)

where m_Li and I _Li are the equivalent mass and inertia matrix of the ith UPS limb measured in the global frame B₄-xyz. It is noted that m_Li and I _Li are contributed by the stretchable limb and oscillating limb. And the inertia matrix I _Li is position-dependent.

Then the virtual work conducted by the ith UPS limb’s inertial force and moment can be expressed as

$$W_{\mathrm{Li}}^a={\left[\begin{array}{c}{\mathit{v}}_{\mathrm{Li}}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{\mathrm{Li}}}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{\mathit{F}}_{\mathrm{Li}}^a\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{\mathrm{Li}}^a}\end{array}\right]$$

(38)

Substituting equation (37) into equation (38) yields

$$W_{\mathrm{Li}}^a=-{\left[\begin{array}{c}{\mathit{v}}_{\mathrm{Li}}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{\mathrm{Li}}}\end{array}\right]}^{\mathrm{T}}(\left[\begin{array}{cc}{m}_{\mathrm{Li}}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{\mathrm{Li}}\hfill \end{array}\right]\left[\begin{array}{c}{\mathit{a}}_{\mathrm{Li}}\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_{\mathrm{Li}}}\end{array}\right]+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{I}}_{\mathrm{Li}}{\mathit{\omega }}_{\mathrm{Li}}}\end{array}\right])$$

(39)

The virtual work conducted by the ith UPS’s gravity can be expressed as

$$W_{\mathrm{Li}}^g={\left[\begin{array}{c}{\mathit{v}}_{\mathrm{Li}}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{\mathrm{Li}}}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{m}_{\mathrm{Li}}\mathit{g}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$$

(40)

Virtual work conducted by the UP limb

The inertial force and moment of the UP limb can be expressed as

$$\left[\begin{array}{c}{\mathit{F}}_{L4}^a\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{L4}^a}\end{array}\right]=\left[\begin{array}{c}-m_{L4}{\mathit{a}}_{L4}\hfill \\ \multicolumn{1}{c}{-{\mathit{I}}_{L4}{\mathit{\xi }}_{L4}-{\mathit{\omega }}_{L4}\times {\mathit{I}}_{L4}{\mathit{\omega }}_{L4}}\end{array}\right]$$

(41)

where m_L4 and I _L4 are the equivalent mass and inertia matrix of the UP limb measured in the global frame B₄-xyz.

Then the virtual work conducted by the UP limb’s inertial force and moment can be expressed as

$$W_{L4}^a={\left[\begin{array}{c}{\mathit{v}}_{L4}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{L4}}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{\mathit{F}}_{L4}^a\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{L4}^a}\end{array}\right]$$

(42)

Substituting equation (41) into equation (42) yields

$$W_{L4}^a=-{\left[\begin{array}{c}{\mathit{v}}_{L4}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{L4}}\end{array}\right]}^{\mathrm{T}}(\left[\begin{array}{cc}{m}_{L4}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{L4}\hfill \end{array}\right]\left[\begin{array}{c}{\mathit{a}}_{L4}\hfill \\ \multicolumn{1}{c}{{\mathit{\xi }}_{L4}}\end{array}\right]+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{L4}\times {\mathit{I}}_{L4}{\mathit{\omega }}_{L4}}\end{array}\right])$$

(43)

The virtual work conducted by the UP limb’s gravity can be expressed as

$$W_{L4}^g={\left[\begin{array}{c}{\mathit{v}}_{L4}\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{L4}}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{m}_{L4}\mathit{g}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$$

(44)

Dynamic equation of the PKM module

Based on the principle of virtual work, there exists

$$W_P^a+W_P^g+W_P^F+\sum _{i=1}^3(W_{\mathrm{Li}}^{\mathrm{Fi}}+W_{\mathrm{Li}}^a+W_{\mathrm{Li}}^g)+W_{L4}^a+W_{L4}^g=0$$

(45)

Substituting equations (13), (19), (21), (23), (26), (29), (30), (34)–(36), (39), (40), (43) and (44) into equation (45) yields

$$\mathbf{M}\stackrel{··}{U}+\mathbf{C}\stackrel{·}{U}+\mathbf{G}+\mathbf{Q}-\left[\begin{array}{c}\mathbf{F}\\ \mathbf{M}\end{array}\right]={\mathbf{J}}^{\mathrm{T}}\mathbf{f}$$

(46)

where U is the displacement of the platform, C is the coefficient matrix of $\stackrel{·}{\mathit{U}}$ , G is the system’s gravity, Q is the Coriolis force which can be ignored if the PKM works in a low speed, $\mathit{f}=[\begin{array}{cccccc}{f}_1\hfill & f_2\hfill & f_3\hfill & f_{1p}\hfill & f_{2p}\hfill & f_{3p}\hfill \end{array}{]}^{\mathrm{T}}$ , M is the mass matrix of the PKM module, which can be expressed as

$$\mathit{M}={\mathit{M}}_{\mathrm{P}}+\sum _{i=1}^3{\mathit{M}}_{\mathrm{Li}}+{\mathit{M}}_{L4},\mathit{C}=\sum _{i=1}^3{\mathit{C}}_{\mathrm{Li}}+{\mathit{C}}_{L4}$$

(47)

$$\mathit{G}=-{\mathit{G}}_{\mathrm{P}}-\sum _{i=1}^3{\mathit{G}}_{\mathrm{Li}}-{\mathit{G}}_{L4},\mathit{Q}={\mathit{Q}}_{\mathrm{P}}+\sum _{i=1}^3{\mathit{Q}}_{\mathrm{Li}}+{\mathit{Q}}_{L4}$$

(48)

where ${\mathit{M}}_{\mathrm{P}}=\left[\begin{array}{cc}{m}_{\mathrm{P}}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{\mathrm{P}}\hfill \end{array}\right]$ , ${\mathit{M}}_{\mathrm{Li}}={\mathit{J}}_{\mathrm{Li}}^{\mathrm{T}}\left[\begin{array}{cc}{m}_{\mathrm{Li}}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{\mathrm{Li}}\hfill \end{array}\right]{\mathit{J}}_{\mathrm{Li}}$ , ${\mathit{M}}_{L4}={\mathit{J}}_{L4}^{\mathrm{T}}\left[\begin{array}{cc}{m}_{L4}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{L4}\hfill \end{array}\right]{\mathit{J}}_{L4}$ , ${\mathit{C}}_{\mathrm{Li}}={\mathit{J}}_{\mathrm{Li}}^{\mathrm{T}}\left[\begin{array}{cc}{m}_{\mathrm{Li}}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{\mathrm{Li}}\hfill \end{array}\right]{\stackrel{·}{\mathit{J}}}_{\mathrm{Li}}$ , ${\mathit{C}}_{L4}={\mathit{J}}_{L4}^{\mathrm{T}}\left[\begin{array}{cc}{m}_{L4}\mathit{e}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\mathit{I}}_{L4}\hfill \end{array}\right]{\stackrel{·}{\mathit{J}}}_{L4}$ , ${\mathit{G}}_{\mathrm{P}}=\left[\begin{array}{c}{m}_{\mathrm{P}}\mathit{g}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$ , ${\mathit{G}}_{\mathrm{Li}}={\mathit{J}}_{\mathrm{Li}}^{\mathrm{T}}\left[\begin{array}{c}{m}_{\mathrm{Li}}\mathit{g}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$ , ${\mathit{G}}_{L4}={\mathit{J}}_{L4}^{\mathrm{T}}\left[\begin{array}{c}{m}_{L4}\mathit{g}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$ , ${\mathit{Q}}_{\mathrm{P}}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_P\times {\mathit{I}}_{\mathrm{P}}{\mathit{\omega }}_P}\end{array}\right]$ , ${\mathit{Q}}_{\mathrm{Li}}={\mathit{J}}_{\mathrm{Li}}^{\mathrm{T}}\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{\mathrm{Li}}\times {\mathit{I}}_{\mathrm{Li}}{\mathit{\omega }}_{\mathrm{Li}}}\end{array}\right]$ , and ${\mathit{Q}}_{L4}={\mathit{J}}_{L4}^{\mathrm{T}}\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_{L4}\times {\mathit{I}}_{L4}{\mathit{\omega }}_{L4}}\end{array}\right]$ .

If the generalized Jacobian matrix J is invertible, the dynamic model can be expressed in the joint space as

$${\mathit{M}}^{*}\stackrel{··}{\mathit{q}}+{\mathit{C}}^{*}\stackrel{·}{\mathit{q}}+{\mathit{G}}^{*}+{\mathit{Q}}^{*}-{\mathit{J}}^{-\mathrm{T}}\left[\begin{array}{c}\mathit{F}\hfill \\ \multicolumn{1}{c}{\mathit{M}}\end{array}\right]=\mathit{f}$$

(49)

where M ^* denotes the mass matrix of the PKM module in the joint space, which can be expressed as

$${\mathit{M}}^{*}={\mathit{J}}^{-\mathrm{T}}\mathit{M}{\mathit{J}}^{-1},{\mathit{C}}^{*}={\mathit{J}}^{-\mathrm{T}}\mathit{C}{\mathit{J}}^{-1}-{\mathit{M}}^{*}\stackrel{·}{\mathit{J}}{\mathit{J}}^{-1}$$

(50)

$${\mathit{G}}^{*}={\mathit{J}}^{-\mathrm{T}}\mathit{G},{\mathit{Q}}^{*}={\mathit{J}}^{-\mathrm{T}}\mathit{Q}$$

(51)

Coupling analysis

In general, the driving force caused by ${\mathit{C}}^{*}\stackrel{·}{\mathit{q}}$ and Q ^* is very small and the term G ^* can be compensated by feedforward control. Therefore, if the external force is zero, the driving force can be mostly determined by the inertial force as follows

$$\mathit{f}\approx {\mathit{M}}^{*}\stackrel{··}{\mathit{q}}$$

(52)

In order to extract the three driving force of the three UPS limbs, the following is given

$$\tilde{\mathit{f}}=\tilde{\mathit{M}}\stackrel{··}{\tilde{\mathit{q}}}$$

(53)

where $\tilde{\mathit{f}}=[\begin{array}{ccc}{f}_1\hfill & f_2\hfill & f_3\hfill \end{array}{]}^{\mathrm{T}}$ , $\tilde{\mathit{M}}=\mathit{E}{\mathit{M}}^{*}{\mathit{E}}^{\mathrm{T}}$ , $\mathit{E}=\left[\begin{array}{cc}\mathit{e}\hfill & \hfill \\ \hfill & 0\hfill \end{array}\right]$ , $\tilde{\mathit{q}}=[\begin{array}{ccc}{q}_1\hfill & q_2\hfill & q_3\hfill \end{array}{]}^{\mathrm{T}}$ .

It is noted that the mass matrix $\tilde{\mathit{M}}$ is off-diagonal, which can be regarded as the root of the PKM’s coupling. In general, the mass matrix $\tilde{\mathit{M}}$ can be expressed as

$$\tilde{\mathit{M}}=\left[\begin{array}{ccc}{m}_{11}\hfill & m_{12}\hfill & m_{13}\hfill \\ \multicolumn{1}{c}{m_{21}}& m_{22}\hfill & m_{23}\hfill \\ \multicolumn{1}{c}{m_{31}}& m_{32}\hfill & m_{33}\hfill \end{array}\right]$$

(54)

If the PKM is not coupling, off-diagonal elements should be zero. Therefore, κ_i can be regarded as the local coupling index, which can be expressed as

$${\kappa }_i=\frac{\sum _{j=1,j\ne i}^3\left|m_{\mathrm{ij}}\right|}{m_{\mathrm{ii}}}$$

(55)

It can be seen from equation (55) that the larger the local coupling index κ_i is, the more coupled the system is.

Parameters of the hybrid manipulator

Main geometrical parameters and mass parameters are listed in Tables 1 and 2, respectively. It is noted that inertia of all parts listed in Table 2 is measured about their mass centers in their body-fixed frames.

Table 1.

Geometrical parameters of the Trimule robot.

Length of the triangle A1A2A3a (mm)	135
Length of the triangle B1B2B3b (mm)	320
Stroke of the platform s (mm)	200
Outer diameter of the stretchable limb Da1 (mm)	60
Inner diameter of the stretchable limb da1 (mm)	40
Outer diameter of the oscillating limb Da2 (mm)	89
Inner diameter of the oscillating limb da2 (mm)	73
Outer diameter of the thick limb Dp1 (mm)	205
Inner diameter of the thick limb dp1 (mm)	189
Outer diameter of the thin limb Dp2 (mm)	130
Inner diameter of the thin limb dp2 (mm)	106

Table 2.

Mass parameters of the Trimule.

Mass of the oscillating limb mo (kg)	34.7504
Inertia of the oscillating limb in the xi axis Iox0 (kg·m2)	4.8210
Inertia of the oscillating limb in the yi axis Ioy0 (kg·m2)	4.8186
Inertia of the oscillating limb in the zi axis Ioz0 (kg·m2)	0.0496
Mass of the stretchable limb ms (kg)	13.1696
Inertia of the stretchable limb in the xi axis Isx0 (kg·m2)	1.2023
Inertia of the stretchable limb in the yi axis Isy0 (kg·m2)	1.2023
Inertia of the stretchable limb in the zi axis Isz0 (kg·m2)	0.0081
Mass of the thick limb mtk (kg)	18.2863
Inertia of the thick limb in the x4 axis Itkx0 (kg·m2)	0.3203
Inertia of the thick limb in the y4 axis Itky0 (kg·m2)	0.3203
Inertia of the thick limb in the z4 axis Itkz0 (kg·m2)	0.1604
Mass of the thin limb mtn (kg)	33.4241
Inertia of the thin limb in the x4 axis Itnx0 (kg·m2)	2.3216
Inertia of the thin limb in the y4 axis Itny0 (kg·m2)	2.3216
Inertia of the thin limb in the z4 axis Itnz0 (kg·m2)	2.3380
Mass of the platform mP (kg)	27.0335
Inertia of the platform in the u axis IPx0 (kg·m2)	0.2725
Inertia of the platform in the v axis IPy0 (kg·m2)	0.2345
Inertia of the platform in the w axis IPz0 (kg·m2)	0.3244
Mass of the A axis mA (kg)	28.9460
Inertia of the A axis in the xA axis IAx0 (kg·m2)	1.4970
Inertia of the A axis in the yA axis IAy0 (kg·m2)	1.6407
Inertia of the A axis in the zA axis IAz0 (kg·m2)	0.3441
Mass of the C axis mC (kg)	40.6882
Inertia of the C axis in the xC axis ICx0 (kg·m2)	1.0051
Inertia of the C axis in the yC axis ICy0 (kg·m2)	0.5346
Inertia of the C axis in the zC axis ICz0 (kg·m2)	0.6243

Coupling analysis at a typical configuration

Taking a typical configuration as an example, where x = y = 0 m, z = 1.35 m, α = 90°, β = 0°, the calculated mass matrix is shown in equation (56). Herein, α and β are Euler angles of the machine tool fame C-x_Cy_Cz_C with respect to the global frame B₄-xyz.

$$\tilde{\mathit{M}}=\left[\begin{array}{ccc}{m}_{11}\hfill & m_{12}\hfill & m_{13}\hfill \\ \multicolumn{1}{c}{m_{21}}& m_{22}\hfill & m_{23}\hfill \\ \multicolumn{1}{c}{m_{31}}& m_{32}\hfill & m_{33}\hfill \end{array}\right]=\left[\begin{array}{ccc}559.1358\hfill & -267.7605\hfill & -267.7605\hfill \\ \multicolumn{1}{c}{-267.7605}& 470.5595\hfill & -162.5702\hfill \\ \multicolumn{1}{c}{-267.7605}& -162.5702\hfill & 470.5595\hfill \end{array}\right]$$

(56)

As can be seen obviously, the mass matrix is not diagonal. Based on equation (55), three local coupling indices are calculated, that is, κ₁ = 0.9578, κ₂ = κ₃ = 0.9145. It can be found that κ₁ is larger than κ₂ and κ₃, demonstrating that the 1st UPS limb has a larger coupling strength with the other two UPS limbs. And κ₂ is equal to κ₃, which is caused by that the 2nd UPS limb and the 3rd UPS limb are distributed symmetrically at this configuration.

Distributions of local coupling indices over a working plane

Distributions of three local coupling indices over the working plane of z = 1.35 m are demonstrated in Figure 5. Herein, both of x and y vary from −0.2 m to 0.2 m.

Figure 5.

Distributions of coupling indices over the working plane z = 1.35 m, α = 90°, β = 0°.

As can be seen clearly from Figure 5, the three local coupling indices are strongly position-dependent. To be specific, κ₁ varies from 0.9403 to 0.9736; κ₂ and κ₃ change from 0.9104 to 0.9304. Further observation can be found that distributions of κ₁ are symmetric about the plane of x = 0 while distributions of κ₂ are symmetric to that of κ₃ about the plane of x = 0. To make it clear, distributions of κ₂ and κ₃ over the x-y plane are shown in Figure 6. The reason for this phenomenon is that the system is distributed symmetrically. To be specific, the 2nd UPS limb and the 3rd UPS limb are distributed symmetrically about the plane of x = 0. It can be found from Figure 4 that κ₁ increases monotonously with the increment of y and the three limbs demonstrate the most coupling strength when the system works at the configuration where x = ±0.2 m, y = 0.2 m.

Figure 6.

Distributions of local coupling indices over the x-y plane.

Parametric analysis

As can be seen from Figure 5, three local coupling indices are strongly position-dependent. For the convenience of analysis, a global coupling index which demonstrates the average maximum local coupling index throughout the whole workspace is defined as

$$\kappa =\frac{\int {\kappa }_{max,\rho }\mathrm{dV}}{V}$$

(57)

where κ_max,ρ is the largest value among κ₁, κ₂, and κ₃ at a typical configuration ρ; V denotes the volume of the entire workspace. Workspace of the manipulator is the combination of a cylinder portion and a spherical portion as depicted in Dong et al. ¹⁰ For the convenience of analysis, the followings only analyze coupling strength in a prescribed task workspace where z = 1.15 to 1.35 m, x = –0.2 to 0.2 m, y = –0.2 to 0.2 m.

Based on the global coupling index, the followings are analyzed.

Dimensional parameters effects

Variations of κ with respect to the size of the platform and the base are shown in Figure 7. Herein, a denotes the size of the platform, which varies from 0.08 m to 0.16 m. And b denotes the size of the base, varying from 0.28 m to 0.36 m.

Figure 7.

Variations of κ with respect to the size of the platform and base.

As shown in Figure 7, the global coupling index κ increases monotonously with the increment of a while decreases with that of b. Further observation shows that the size of the base has a larger effect on the coupling strength than that of the platform. Therefore, increasing the size of the base is suggested preferentially to have a low coupled system so that the real control performance can be improved.

Structural parameters effects

Variations of κ with respect to the length of active UPS and passive UP limbs are depicted in Figure 8. Herein, l_a1 and l_a2 are length of the stretchable limb and the oscillating limb while l_p1 and l_p2 are that of the thick limb and the thin limb as shown in Figure 2, respectively.

Figure 8.

Variations of κ with respect to the length of active and passive limbs.

It can be seen from Figure 8 that the global coupling index κ increases monotonously with the increment of the oscillating limb’s length l_a2 while decreases with that of the stretchable limb’s length l_a1. And it seems that the length of the oscillating limb l_a2 has a larger effect on κ than that of the stretchable limb l_a1. Further observation demonstrates that the global coupling index κ decreases monotonously with the increment of the thick limb’s length l_p1. And the global coupling index κ seems to keep unchanged with the increment of the length of the thin limb l_p2. By comparing, it can be found that length of the active limbs has a larger effect on κ than that of the passive limb. Therefore, it is suggested to decrease length of the oscillating limb preferentially to decrease the coupling strength of the system.

Figure 9 shows variations of κ with respect to the depth of active UPS and passive UP limbs. Herein, δ_a1 and δ_a2 are depth of the stretchable limb and oscillating limb while δ_p1 and δ_p2 are that of the thick limb and thin limb, respectively. Depth of the active limbs and passive limb varies from 2 mm to 14 mm. For the convenience of analysis, external diameters of the active limbs and passive limb keep unchanged.

Figure 9.

Variations of κ with respect to the depth of active and passive limbs.

It can be seen from Figure 9 that the global coupling index κ increases monotonously with the increment of the UPS limbs’ depth. By comparing, the global coupling index κ is more sensitive to the depth of the stretchable limb δ_a1. And the effect of the oscillating limb’s depth δ_a2 on κ is very small. On the contrary, the global coupling index κ decreases monotonously with the increment of the UP limb’s depth. And depth of the thick limb and thin limb seems to have the same effect on the system’s coupling strength.

In general, effects of the structural parameters on the system’s coupling strength are very weak, which can be found from Figures 8 and 9.

Mass parameters effects

Variations of κ with respect to mass of the 2R module are depicted in Figure 10. Herein, m_A and m_C are mass of the A axis and C axis. As can be seen, the global coupling index κ decreases monotonously with the increment of mass of the 2R module. Therefore, increasing mass of the 2R module properly is advised to reduce coupling strength of the system.

Figure 10.

Variations of κ with respect to mass of the 2R module.

Combining Figures 7–10, it can be concluded that sizes of the platform and the base have an obvious effect on the global coupling strength. And effects of structural parameters of the active and passive limbs as well as mass of the 2R module are relatively low. Despite that, coupling may affect the real control performance or other indices in a large degree. Therefore, optimal design of the parallel/hybrid manipulator should be synthesized with kinematic, stiffness, and dynamic performance together.

Conclusion

Based on the rigid dynamic model, this article analyzes coupling characteristics of the Trimule robot. The main contributions of the article are drawn as follows.

1. A local coupling index is defined based on the off-diagonal mass matrix. Distributions of three local coupling indices are strongly position-dependent and are symmetric to the plane of x = 0 due to the mechanical features of the Trimule robot. And the three UPS limbs are coupled most when the robot works at the workspace boundaries.

2. A global coupling index is proposed which indicates the average maximum coupling index throughout the whole workspace. Parametric analysis demonstrates that dimensional parameters and structural parameters as well as mass parameters can affect the coupling strength to an extent. And the dimensional parameters of the PKM seem to have the largest influence.

3. It is worth noting that optimal design of the Trimule should be conducted by considering kinematic performance, stiffness, dynamic characteristics, and coupling together, which will be published in the near future.