1 Introduction

Variable geometry truss manipulator (VGTM) is one kind of serial-parallel hyper redundant robotic manipulators. This type of manipulator promises a significant increase in the stiffness-to-weight ratio,the deployment-to-compacting ratio,the dexterity,and so on over conventional anthropomorphic manipulators. These advantages make VGTM have potential to work in complex work environment. Since Miura et al.^[1] and Miura and Furuya^[2] first described the possibilities of this type of robotic manipulator for future space application in the 1980s,many scholars have paid attention to the VGTM's research^[3-17].

A VGTM usually consists of a periodic sequence of fundamental truss units and an end-effector,each of which has several adjustable-length actuators. There are several candidate topologies for the fundamental truss unit,such as the tetrahedron,octahedron, cube,decahedron,and dodecahedron. Hughes et al.^[3] pointed out that the ``standard" octahedron was the superior candidate in their paper. Comparing with the link unit (fundamental unit of the conventional anthropomorphic manipulators),the truss unit has higher stiffness-to-weight ratio and dexterity. However,these advantages have a price: a complex and unorthodox kinematic and dynamic description. Therefore,much attention was paid to the kinematics,dynamic analysis,and control of the VGTM. In 1988, Naccarato and Hughes^[4] proposed an inverse kinematic algorithm for a hyper-redundant VGTM based on the spline-like reference curve. Hughes et al.^[3] gave a brief but comprehensive description of the kinematics,dynamics,and control of the VGTM. Hertz and Hughes^[5] presented the forward kinematics of an octahedral type variable geometry manipulator. Furuya and Higashiyama^[6] used a Jacobian matrix and Kane's method to drive the dynamic equations of a VGTM. Huang et al.^[7-8] conducted a study on the forward kinematics and inverse kinematics of a free-floating octahedral variable geometry truss and presented an approach on controlling the motion of this variable geometry truss from the standpoint of kinematics. Tsou and Shen^[9] designed a fuzzy control system for the kinematic maneuvers of a variable geometry truss. Xu et al.^[10] presented the closed-form solutions for the inverse kinematic analysis of a new triple-octahedron variable geometry truss manipulator. From the above descriptions,we can know that there are a rich variety of research results on the kinematics and dynamics of the VGTM. However,these studies did not consider the elastic characteristics. Recently,some researchers have attempted to study dynamic characteristics and structural optimization of the VGTM. For example,MacAreno et al.^[11] adopted the conventional finite element method (FEM) to study dynamic characteristics of a variable geometry truss (VGT). Because VGTs may have highly different configurations throughout their operation,the elastic characteristics (natural frequencies and vibration modes) can be changed to a large extent. When using the FEM to calculate the elastic characteristics of VGTs,the computational cost will be much higher. To avoid this,Aguirrebeitia et al.^[12] used the metamodeling methodology to reduce the computational cost. Bilbao et al.^[13-15] presented a method for the estimation of natural frequencies and vibration modes of a VGT by the frequencies and vibration modes of several configurations of the VGT. Besides, they used the real Eigen-sensitivity and complex Eigen-sensitivity to search optimal damper distributions in the VGT. From the above research,we can see that the dynamics problem of VGTM has gained much attention,and some research results have been achieved. However,there are still some problems that need further study. For example,there is not a unified and convenient method that can establish the dynamic equations of various VGTM systems by independent variables. The dynamic equations are used to describe the motion of VGTM systems. From the perspective of multi-body dynamics,we can get that the Cartesian method^[16] and the recursive method^[17-19] can both be used to model the dynamic equations for variables VGTM systems. However,in the above two methods,the Cartesian coordinates of body and joint coordinates are chosen as the generalized variables. As a result,the dynamic equations of system cannot be used to design the controller. If we want to get high position control accuracy of the VGTM,the exact dynamics model of VGTM using the independent variables as the generalized variables is the basis for controller design. Therefore,the study of dynamic modeling for the VGTM is of high importance.

In this paper,dynamics modeling and control of a VGTM with two double-octahedral truss units and a 3-revolute-prismatic-spherical (3-RPS) parallel manipulator are investigated. The dynamic model of the system is established by Jourdain's velocity variation principle,and the active controller is designed by the computed torque control method. Numerical simulations demonstrate the validity of the proposed methods. This paper is organized as follows. Section 2 gives a geometrical description of the VGTM. Section 3 briefly presents the expressions of dynamics model of the system,including the kinematical recursive relations of bodies,the geometry constrains,and the dynamic equation of the system. The control design is given in Section 4. Section 5 provides simulations and comparison studies. Finally, concluding remarks are given in Section 6.

2 Description of structure

A variable geometry truss manipulator considered in this paper consists of two double-octahedral truss units and a 3-RPS parallel manipulator truss unit,as shown in Fig. 1. Each truss unit has three degrees of freedom. The 3-RPS parallel manipulator truss unit has three actuators,and it has a stationary platform and a motion platform. The double-octahedral truss unit is composed of three actuators,six petal triangular trusses,and a pair of lateral triangular trusses. Each double-octahedral truss shares a lateral triangular truss unit with an adjacent truss unit. The structures of the lateral triangular truss and the petal triangular truss are equilateral triangle and isosceles triangle,respectively. Besides, each petal triangular truss can be regarded as one rigid body that can rotate about an adjacent side of the lateral triangular truss. Therefore,the structure of the VGTM can be equivalent to the structure shown in Fig. 1(b),where $B_0 $ is the base of the VGTM, which is the ground in this paper,and $B_i\,(i=1,2,\cdots,46)$ denotes the $i$th body. The bodies $B_1 $,$B_{16}$,and $B_{35} $ are the lateral triangular trusses. The bodies $B_{35} $ and $B_{40} $ are the stationary and motion platforms of the 3-RPS parallel manipulator truss unit,respectively. The topology structure of the system is shown in Fig. 2,where $H_i\,(i=1,2,\cdots ,46)$ denotes the inboard joint of the $i$th body,and $J_i\,(i=1,2,\cdots ,13)$ denotes the $i$th cut-off joint that will be elaborated in Section 3. The joint $H_1 $ is a fixed joint. The joints $H_4 $,$H_{11} $,$H_{13} $, $H_{14} $,$H_{17} $,$H_{19}$,$H_{23} $,$H_{30} $,$H_{32} $, $H_{33} $,$H_{36}$,and $H_{38} $ are all universal joints. The joints $H_5 $,$H_7 $,$H_9 $,$H_{24} $,$H_{26} $,and $H_{28} $ are all cylinder joints. The joints $H_{42} $,$H_{44} $,and $H_{46} $ are all prismatic joints. The joint $H_{40} $ is an imaginary virtual joint that has six degrees of freedom,which is introduced to connect with the bodies $B_{35} $ and $B_{40} $ for deducing the kinematics of the 3-RPS parallel manipulator truss unit conveniently. Except for the joints defined above,the other joints in Fig. 2 are all revolute joints.

3 Dynamic modeling

The variable geometry truss manipulator is a serial-parallel hyper redundant mechanical system. Up to now,there is not a unified and convenient method for dynamic modeling for the system. In this section,we present an approach to establish the dynamic model of a VGTM. First,we convert the serial-parallel system into a serial system by breaking the cut-off joints $J_i\,(i=1,2,\cdots ,13)$ (see Fig. 2) and deduce the kinematics of the serial system using the joint coordinates as the generalized variables. Then,we formulate the relations between the joint coordinates and the independent generalized variables with the help of the geometry constrains of the system to express the kinematics of the system by the independent generalized variables. Finally,the dynamic equations of a VGTM are established by Jourdain's velocity variation principle and the system kinematics.

3.1 Kinematics expressed by joint coordinates

As we can see from Fig. 3,the new topology structure of VGTM is one of the typical tree structures. Based on the kinematical recursive relations between a pair of adjacent bodies,we can formulate the kinematics of the system by joint coordinates. Here, we present the derivation process briefly.

For a system composed of $N$ rigid bodies,we assume that the number of bodies and joints are regularly labeled^[17],or precisely, the body $i$ is connected to its inboard body $ j$ by the joint $i$, as shown in Fig. 4,where $Q$ and $P$ are the joint definition points on the body $j$ and the body $i$,respectively. The body-fixed frames of the body $i$ and the body $j$ are represented by the ${{ e}}^h$ system and the ${{ e}}^{h_0 }$ system, respectively,and then the origins of the two frames are $P$ and $Q$,respectively. The attitude of the body $i$ with respect to the body $j$ can be described using the direction cosine matrix of the ${{ e}}^h$ system with respect to the ${{ e}}^{h_0 }$ system. The absolute reference frame is established on the body $B_0 $. It is assumed that ${ \omega} _i $ and ${ \omega} _j $ are the vectors of the absolute angular velocity of the body $i$ and the body $j$,respectively,and ${ \omega} _{\mathrm ri} $ is the relative angular velocity of the body $i$ with respect to the body $j$. The vector ${ h}_i =QP$ describes the relative motion of the joint $i$. ${ r}_i $ and ${ r}_j $ are the position vectors of the origins of the body-fixed frames attached to body $i$ and body $j$,respectively,as shown in Fig. 1,and $C_i $ and $C_j $ are the mass centers of the two bodies. In this paper,the inboard body number of body $i$ is expressed using ${L}$($i)$.

The configuration velocity recursive relation between the body $i$ and the body $j$ can be expressed as^[17-19]

(1)

where ${{ q}}_i \in {\mathfrak{R}} ^{\delta _i \times 1}$ is the coordinate of the $i$th joint,$\delta _i $ is the number of degrees of freedom of the $i$th joint,${{ T}}_{ij} $ and ${{ U}}_i $ can be expressed as^[17-19]

(2)

in which and throughout the text the symbol $-$ denotes a skew symmetric matrix such as

(3)

In Eq.(2),${{ H}}_i^{h{\rm T}} $ is the matrix whose non-zero columns are composed of the joint translational axis vectors,and ${ { H}}_i^{\Omega {\rm T}} $ is the matrix whose non-zero columns are composed of the joint rotational axis vectors^[17-19].

Similarly,the configuration acceleration recursive relation between the body $i$ and the body $j$ can be expressed as

(4)

where ${{ \beta }}_i \in {\mathfrak{R}} ^{6\times 1}$ is the convected configuration acceleration^[17-19]. For the multibody system with $N$ rigid bodies,the configuration velocity and acceleration can be expressed as^[19]

(5)

where

(6)

In Eq.(6),$B_k $ represents the $k$th body in the system,$B_k <B_i $ means that the body $k$ is on the way from the root body to the body $i$,$B_k =B_i $ means that the body $k$ is the body $i$,and $B_k \ne <B_i $ the body $k$ is not on the way from the root body to the body $i$.

Writing Eq.(5) in a matrix form,we have

(7)

where ${{ v}}=[{{ v}}_1^{\rm T} ,u^{\mathrm T}_2,\cdots ,{{ v}}_N^{\rm T}]^{\rm T}$ is the total vector of configuration velocity of the system,${{ y}}$ is the total coordinate vector of all the joints,and ${{\widehat { I}}}_N \in {\mathfrak{R}} ^{N\times 1}$ is an $N$-dimensional vector with all elements being 1. The parameters ${{ G}}$ and ${{ g}}$ in Eq.(7) are

3.2 Kinematics expressed by independent generalized variables

As we can see from Fig. 1,the VGTM consists of two double-octahedral truss units and a 3-RPS parallel manipulator truss unit,and each truss unit has three actuators. The confi-guration of the VGTM can be changed by adjusting the length of the actuators. In other words,the configuration of the VGTM is determined by the length of the actuators. Therefore,we can use the actuator lengths as the independent variables to express the kinematics of the VGTM. In this section,we first deduce the relation between the actuator lengths and the joint coordinates in the double-octahedral truss unit,then present the relation between the actuator lengths and the joint coordinates in the 3-RPS parallel manipulator truss unit,and finally give the kinematics of the VGTM expressed by the actuator lengths.

3.2.1 Kinematics of double-octahedral truss unit

Because of the symmetry of the double-octahedral truss unit (see Fig. 1),the kinematical relation of the truss unit between the joint coordinates and the independent generalized variables can be obtained by the analysis of coordinate relation of bottom half of the truss unit. Below,we give the derivation process of the kinematical relation between the joint coordinates and the independent generalized variables. A simplified scheme of a half of the double-octahedral truss unit is shown in Fig. 5. The angles $\theta _1 $,$\theta _2 $,and $\theta _3 $ are the joint coordinates of the revolute joints $H_2 $,$H_{10} $,and $H_{12} $, respectively. The angles $\alpha _{1,2} $,$\alpha _{4,5} $,and $\alpha _{7,8} $ are the joint coordinates of the universal joints $H_3 $,$H_{11} $,and $H_{13} $,respectively. The angles $\alpha _3 $,$\alpha _6 $,and $\alpha _9 $ are the joint coordinates of the revolute joints $H_6 $,$H_8 $,and $J_1 $,respectively. The parameters $y_1 $,$y_2 $,and $y_3 $ are the lengths of $KL $,$LM $,and $MK$,respectively,which can be regarded as the translational coordinates of the cylinder joints $H_5 $,$H_7$,and $H_9 $,and these three parameters are the independent variables of the VGTM. The frame ${{e}}$ is the absolute reference frame. The frame ${{{e}}}_1$ is the body-fixed one of the body $B_1 $,and $O_{1}$ is the origin. The frame ${{{e}}}_{KLM}$ is the body-fixed frame of the plane of the triangle $KIM$. The frame ${{ {e}}}_3 $ is the body-fixed frame of the body $B_{3}$. When $y_1 =y_2 =y_3 $,the vectors ${ e}_3^y $ and ${ e}_3^z $ are parallel to the vectors ${ e}_{KLM}^y $ and ${ e}_{KLM}^z $,respectively. The unit vectors ${ e}_{KLM}^y $ and ${ e}_{KLM}^z $ are perpendicular to the triangle $KIM$ and the rectangle $ILMG$, respectively. The frame ${{{e}}}_4 $ is the body-fixed frame of the body $B_{4}$,in which the vector ${ e}_4^z $ is perpendicular to the rectangle $FILK$. The bodies $B_3 $,$B_{11} $,and $B_{13} $ are,respectively,connected with the bodies $B_4 $,$B_6$,and $B_8 $ by a revolute joint in turn,and the vectors ${ {FK}} $,$ { {IL}} $,and ${ {GM}} $ are all perpendicular to the plane of the triangle $KLM$. Thus,the triangles $KLM$ and $FIG$ are congruent. Therefore,we have

(8)

As we can see from Fig. 5,the positions of the points $F$,$I$, and $G$ can be expressed as

(9)

where $\|{CF}\|=\|{DI}\| =\|{EG} \|=l_1 $. The matrix forms of Eq.(9) in the frame ${{{e}}}_1 $ are

(10)

where ${{ {CF}'}}$,${{ {DI}'}}$,and ${{ {EG}'}}$ can be expressed by $y_1$,$y_2$,$y_3$,$\theta_1$,$\theta_2$,and $\theta_3$. The coordinate vectors of ${FI} $,${IG} $,and ${GF} $ in the frame ${e}_1$ can be expressed as

(11)

Considering Eqs.(8) and (11),we can obtain the geometry constrains about the joint coordinates $\theta _i\,(i=1,2,3)$ and the independent variables $y_i\,(i=1,2,3)$ in the absolute frame ${{{ e}}}$ given by

(12)

where $\| \ast \|$ denotes the Euclidean norm. Because Eq.(12) is nonlinear,the iterative algorithm is needed to calculate $\theta_i\,(i=1,2,3)$ when $y_i\,(i=1,2,3)$ are known.

Taking the derivative of Eq.(12),we have

(13)

where

and the matrix ${{ I}}_3 \in \mathfrak{R} ^{3\times 3}$ is an unit matrix.

From Eq.(13),we can obtain

(14)

Taking the derivative of Eq.(14),we have

(15)

Equations(14) and (15) are the kinematical relations between the joint coordinates ${{ \theta }}$ and the independent variables ${{ y}}_{123} $.

As shown in Fig. 5,the vectors ${ e}_{KLM}^y $ and ${ e}_{KLM}^z $ can be expressed as

(16)

Considering that the unit vectors ${ e}^y_3$ and ${ e}^z_3$ are,respectively,parallel to the unit vectors ${ e}_{KLM}^y $ and ${ e}_{KLM}^z $ when $y_1 =y_2 =y_3 $,the vector coordinates of ${ e}^y_3$ and ${ e}^z_3$ in the frame ${{{e}}}_1 $ are

(1)

where ${{ A}}_3^{\mathrm r} $ is the direction cosine matrix relating to the frames ${{ {e}}}_3 $ and ${{e}}_1 $,which can be expressed by the joint coordinates $\theta _1 $ and $\alpha_{1,2}$, given by

(18)

where

Substituting Eq.(18) into Eq.(16),we can obtain

(19)

From Eqs.(11) and (19),we can get

(20)

where $F_{s_{\alpha _1 } } $, $F_{c_{\alpha _1 } } $,$F_{s_{\alpha _2 } } $,and $F_{c_{\alpha _2 } } $ are the expressions of $\sin \alpha _1 $, $\cos \alpha _1 $,$\sin \alpha _2 $,and $\cos \alpha _2$, respectively.

Solving the above equation,we can obtain

(21)

Taking the derivative of Eq.(21),we have

(22)

where

From Eqs.(14) and (22),we have

(23)

Taking the derivative of Eq.(23),we have

(24)

where ${{ A}}_4^{\mathrm r} $ is the direction cosine matrix relating to the frames ${{{e}}}_4 $ and ${{{e}}}_1 $,which can be expressed as

(26)

where

is the direction cosine matrix relating to the frames ${{{e}}}_4 $ and ${{{e}}}_3 $,in which $c_{\alpha _3 } =\cos \alpha _3 $ and $s_{\alpha _3 } =\sin \alpha _3 $. Substituting Eq.(26) into Eq.(25),we can obtain

(27)

From Eq.(27),we have

(28)

where $F_{s_{\alpha _{_3 } } } $and $F_{s_{\alpha _{_3 } } } $are the expressions of $\sin \alpha _3 $ and $\cos \alpha _3 $, respectively.

Solving the above equation,we can obtain

(29)

Taking the derivative of Eq. (29), we have

(30)

where

From Eqs.(14) and (30),we can obtain

(31)

Taking the derivative of Eq. (31), we have

(32)

Equations (31) and (32) are the kinematical relations between the joint coordinate $\alpha _3 $ and the independent variables ${ { y}}_{123} $.

From Fig. 5,we can get

(33)

Taking the derivative of Eq.(33),we have

(34)

where

From Eq.(34),we obtain

(35)

Taking the derivative of Eq.(35),we have

(36)

Equations (35) and (36) are the kinematical relations between the joint coordinates $\gamma _{2,3} $ and the independent variables ${{ y}}_{123} $.

Because of the symmetry of the truss unit shown in Fig. 5,if the truss unit rotates 60$^{\circ}$ about the vector ${ e}_1^y $,the bodies $B_{12}$,$B_{13}$,and $B_{8}$ can be,respectively, regarded as the bodies $B_{2}$,$B_{3}$,and $B_{4}$,the joint coordinates $\theta _3 $,$\theta _1 $,$\theta _2 $,$\alpha _7 $, $\alpha _8 $,and $\alpha _9 $ can be,respectively,regarded as $\theta _1 $,$\theta _2 $,$\theta _3 $,$\alpha _1 $,$\alpha _2 $,and $\alpha _3 $,and the independent variables $y_3 $,$y_1 $, and $y_2 $ can be regarded as $y_1 $,$y_2 $,and $y_3 $, respectively. According to the above derivation process,we can obtain

(37)

Similarly,if the truss unit rotate $-$60$^{\circ}$ about the vector ${ e}^y_1$,we can obtain

(38)

Define the joint coordinates of $H_{i}$ ($i=1,2,\cdots$, 20) by ${{ q}}^1=[{{ q}}_1^{{\rm 1T}} ,{ q}^{\rm 2T}_2,\cdots,{{ q}}_{20}^{{\rm 1T}}]^{\rm T}$. According to the symmetry of a double-octahedral truss unit,we can obtain the kinematical relation between the joint coordinates ${ { q}}_{14}^{1} ,{ { q}}_{15}^{1},\cdots ,{{ q}}_{20}^{1} $ of $H_{i}$ ($i$=14,15,$\cdots$,20) and the independent variables ${{ y}}_{123} $. Considering Eqs.(14),(15),(23),(24),(31), (32),and (35)--(38),the kinematical relation between the joint coordinates ${{ q}}^1$ and the independent variables ${{ y}}_{123} $ is

(39)

where ${ { q}}^2 =[{{ q}}_{21}^{{\rm 2T}} ,{ q}_{22}^{2{\rm T}}\cdots ,{{ q}}_{39}^{{\rm 2T}}]^{\rm T}$,${ { y}}_{456} =[y_4 ,y_5 ,y_6]^{\rm T}$,and ${ { J}}_{q^2} $ are the joint coordinates of $H_{i}$ ($i$=21,22,$\cdots$,39),the actuator lengths,and the Jacobian matrix of the second double-octahedral truss unit,respectively. Similar to ${{ J}}_{q^1} $,the expression of ${ { J}}_{q^2} \in{\mathfrak{R}} ^{10\times 3}$ can be obtained by the above derivation process.

3.2.2 Kinematics of 3-RPS parallel manipulator

From Fig. 1,we can see that the actuators of the 3-RPS parallel manipulator are connected with the stationary platform $B_{35} $ by the cut-off joints $J_{11}$-$J_{13}$. Joint types of $J_{11}$-$J_{13}$ are all spherical joints. In order to deduce the kinematics of the 3-RPS parallel manipulator,we introduce a virtual joint connecting the stationary platform $B_{35}$ and the motion platform $B_{40}$,which has three translational degrees of freedom and three rotational degrees of freedom. Below,the derivation process of the kinematical relation of the 3-RPS parallel manipulator between the joint coordinates and the independent generalized variables will be given in detail.

A simplified scheme of the 3-RPS parallel manipulator is shown in Fig. 6. The angles $\beta _1 $,$\beta _2 $,and $\beta _3 $ are the joint coordinates of the revolute joints $H_{41} $,$H_{43} $, and $H_{45} $,respectively. The parameters $y_7 $,$y_8 $,and $y_9 $ are the lengths of the vectors ${P_1 Q_1 } $,${P_2 Q_2 } $,and ${P_3 Q_3 } $,respectively,and they are the joint coordinates of the prismatic joints $H_{42} $,$H_{44}$,and $H_{46} $. The triangles $Q_1 Q_2 Q_3 $ and $P_1 P_2 P_3 $ are both equilateral triangles,and their radii of circumcircle are represented by $R$ and $r$,respectively. The frame $\overline {O}$-$\overline {X}\overline {Y}\overline {Z}$ is a body-fixed one of the body $B_{35} $,and the origin $\overline {O}$ is the center of the triangle $Q_1Q_2Q_3$. The frame $\widehat{O}$-$\widehat{X}\widehat{Y}\widehat{Z}$ is a body-fixed one of the body $B_{40} $,and the origin $\widehat {O} $ is the center of the triangle $P_1P_2P_3$. The unit vectors ${\overline {O}\overline{Y}} $ and ${\widehat{O}\widehat{Y}}$ are the normal vectors of the triangles $Q_1Q_2Q_3$ and $P_1P_2P_3$,respectively. From Fig. 6,we can obtain

(41)

Since the triangle $Q_1Q_2Q_3$ is an equilateral triangle,we can obtain

(42)

Substituting Eq.(41) into Eq.(42),we can obtain the geometrical constrain equations about the joint coordinates $\beta _i\, (i=1,2,3)$ and the independent generalized variables $y_i\, (i=7,8,9)$,given by

(43)

Since Eq.(43) is nonlinear,the iterative algorithm is needed to calculate $\beta_i\,(i=1,2,3)$ when $y_i\,(i=7,8,9)$ are known.

Taking the derivative of Eq.(43),we have

(44)

where

From Eq.(44),we can obtain

(45)

Taking the derivative of Eq. (45), we have

(46)

Equations (45) and (46) describe the kinematical relations between the joint coordinates ${{ \beta }}$ and the independent variables ${ { y}}_{789} $.

Next,we deduce the kinematical relation between the coordinates of joint $H_{40} $ and the independent variable ${ y}_{789}$. As shown in Fig. 6,we can get

(47)

In the frame $\widehat{O}$-$\widehat{X}\widehat{Y}\widehat{Z}$,the matrix forms of Eq.(47) are

(48)

where ${{ A}}_{40}^{35} $ is the direction cosine matrix relating to the frames $\overline{O}$-$\overline{X}\overline {Y}\overline {Z}$ and $\widehat{O}$-$\widehat{X}\widehat{Y}\widehat{Z}$. When ${{ \beta }}$ and ${{ y}}_{789} $ are known,we can obtain the direction cosine matrix ${{ A}}_{40}^{35}$ by solving Eq.(48). The direction cosine matrix ${{ A}}_{35}^{40} $ relating to the frames $\widehat{O}$-$\widehat{X}\widehat{Y}\widehat{Z}$ and $\overline{O}$-$\overline{X}\overline{Y}\overline{Z}$ can be obtained by ${{ A}}_{35}^{40} =({{ A}}_{40}^{35} )^{\rm T}$. Define that ${{ q}}_{40}^3 =[\eta _1 ,\eta _2 ,\eta _{3},\varphi _1 ,\varphi _2,\varphi _3]^{\rm T}$ is the joint coordinate of the joint $H_{40} $. We can get

(49)

where $S_1 =\sin \varphi _1 $,$C_1 =\cos \varphi _1$,$S_2 =\sin \varphi _2 $,$C_2 =\cos \varphi _2 $,$S_3 =\sin \varphi _3 $,and $C_1 =\cos\varphi _3 $. From Eq.(49),we can get

(50)

where $\varphi =[\varphi _1 ,\varphi _2 ,\varphi _3 ]^{\rm T}$.

By taking the derivative of Eq.(50), we have

(51)

where

From Eq.(51),we get

(52)

By taking the derivative of Eq.(52),we can obtain

(53)

Equations(42) and (53) describe the kinematical relations between the rotational coordinates $\varphi $ and the independent variables ${ y}_{789}$.

Next,we deduce the relation between the translational coordinates ${{ \eta }}=[\eta _1 ,\eta _2 ,\eta _3]^{\rm T}$ and the independent variables ${ y}_{789}$. As shown in Fig. 6,we can get

(54)

The matrix form of Eq.(54) in the frame $\overline{O}$-$\overline{X}\overline{Y}\overline{Z}$ can be written as

(55)

From Eq.(55),we can get

(56)

By taking the derivative of Eq.(56),we have

(57)

where

(57)

From Eq.(57),we get

(58)

By taking the derivative of Eq.(52),we can get

(59)

Considering Eqs. (52), (53), (58), and (59), we can get

(60)

where ${{ q}}^3 =[{ { q}}_{40}^{\rm T} ,{ q}^{\rm T}_{41},\cdots ,{{ q}}_{46}^{\rm T}]^{\rm T}$ and ${{\bf J}}_{q^3} =[{ { J}}_\eta ^{\rm T} ,{{ J}}_\varphi ^{\rm T} ,{ { J}}_\beta ^{\rm T} ,{{ I}}_3]^{\rm T}\in {\mathfrak{R}} ^{12\times 3}$ are the joint coordinates of $H_{i}$ ($i$ = 40,41, $\cdots$,46) and the Jacobian matrix of the 3-RPS parallel manipulator,respectively.

Using Eqs.(39),(40),and (60),the kinematical relation between the joint coordinates and the independent variables can be expressed as

(61)

where ${{ q}}=[{ { q}}^{1{\rm T}},{ { q}}^{2{\rm T}},{ { q}}^{3{\rm T}}]^{\rm T}$ is the joint coordinates,${{ y}}=[{ { y}}_{123}^{\rm T} ,{ { y}}_{456}^{\rm T} ,{ { y}}_{789}^{\rm T}]^{\rm T}$ is the independent variables,and ${ { J}}_q ={\rm diag}[{ { J}}_{q^1} ,{{ J}}_{q^2} ,{ { J}}_{q^3}]$ is the Jacobian matrix of the VGTM system.

3.3 Dynamic equation

Here,we establish the dynamic equation of the VGTM system. Based on Jourdain's velocity variation principle,the dynamic equation of the system can be expressed as^[19]

(62)

where ${{ v}}_i $ and ${{ \dot { v}}}_i $ are the velocity and the acceleration of mass center of the $i$th rigid body, respectively,$-{ { M}}_i {{ \dot { v}}}_i $ denotes the acceleration-dependent inertia force acting on the mass centre of the $i$th rigid body,${ { f}}_i^{\rm w} $ is the velocity-dependent inertia force acting on mass centre of the $i$th rigid body,${ { f}}_i^{\rm o} $ is the external force of system acting on mass centre of the $i$th rigid body,and $\Delta P$ is the sum of virtual power of the inner forces of system.

In Eq.(62),${{ M}}_i $ can be expressed as

(63)

where $m_i $ is the mass of the $i$th rigid body,and ${{ \xi }}_i \in {\mathfrak{R}} ^{3\times 3}$ the matrix of moment of inertia of $i$th rigid body with respect to the mass centre.

The parameters ${{ f}}_i^{\mathrm w} $ and ${ { f}}_i^{\mathrm o} $ in Eq.(62) can be expressed as

(64)

where ${{ F}}_i^{\mathrm o} $ and ${ { M}}_i^{\mathrm o} $ are the vectors of resultant external force and torque of system on the mass center of the $i$th rigid body,respectively.

In order to express Eq.(62) conveniently,we set

(65)

Substituting Eq. (65) into Eq. (62) yields

(66)

In the matrix form, Eq. (66) can be written as

(67)

where ${ { v}}=[{{ v}}_1^{\rm T} ,{ v}^2_{\rm T},\cdots ,{ { v}}_N^{\rm T}]^{\rm T}$ is the total configuration velocities of the system,and ${{ M}}={\rm diag}({{ M}}_1 ,{ M}_2,\cdots ,{ { M}}_N )$ and ${ { f}}=[{ { f}}_1^{\rm T} ,{ f}_2^{\rm T},\cdots ,{{ f}}_N^{\rm T}]^{\rm T}$ are the matrices of generalized mass and generalized force of the system, respectively.

Using Eq.(23),Eq.(67) can be written as

(68)

where ${{ Z}}={ { G}}^{\rm T}{{ {MG}}}$,and ${ { z}}={{ G}}^{\rm T}({{ f}}-{ {{ Mg}\widehat { I}}}_N )$.

Substituting Eq.(61) into Eq.(68),one can obtain

(69)

where ${ { U}}={{ J}}_{ { q}}^{\rm T} { { {ZJ}}}_{ { q}} $, ${{ u}}={ { J}}_{ { q}}^{\rm T} (-{ { Z\dot { J}}}_q { {\dot { y}}}+{{ z}})$, and ${{ f}}^{ey}$ is the generalized force of the system.

Since the joint coordinates ${{ y}}$ are independent, Eq.(69) becomes

(70)

This formula is the dynamic equation of the system.

4 Active control 4.1 Trajectory planning

In order to complete an assigned mission,such as point to point assignment and continuous path assignment,the motion trajectory planning of the VGTM needs to be planned in the actuator-space that is expressed by the actuator lengths. In this paper,the third-order polynomial trajectory planning method is used to design the motion of actuators. Set the actuator trajectory as

(71)

where $a_{i0} $, $a_{i1} $, $a_{i2} $, and $a_{i3} $ are the multinomial coefficients, and $j_n $ is the amount of joints. The initial and final conditions of $\theta _i (t)$ and $\dot {\theta }_i (t)$ are

(72)

Taking the first derivative of Eq. (72), we get

(73)

Substituting the initial and final conditions into Eq.(72) yields

(74)

4.2 Controller design

As stated in the above, ${{ f}}_{\rm e}^{ey} $ in Eq.(70) is the force element of the system which acts on the actuators of VGTM. In this paper, it is the active control forces. Below, we design this controller using the computed torque control method. Equation (68) may be rewritten as

(75)

where ${{ \Gamma }}( {{{ \dot { y}}},{{ y}}})={ { U}}^{-1}{{ u}}$,${{ B}}={{ U}}^{-1}$,and ${ { \tau }}={ { F}}_{\rm e}^{ey} $.

Define the desired trajectory of ${{ y}}$ by ${ { y}}_{\mathrm d} $ and the error vector ${{ e}}={ { y}}_{\mathrm d} -{ { y}}$. Let the controller be

(76)

where

(77)

in which ${{ K}}_{\rm D} $ and ${{ K}}_{\rm P} $ are the differential and proportional gain matrices,respectively,which are both positive-definite matrices. Substituting Eq.(77) into Eq.(76),we have

(78)

From Eq.(78),we know that ${{ e}}$ will tend to zero when ${{ K}}_{\mathrm D} >0$ and ${{ K}}_{\mathrm P} >0$. Thus,${ { y}}$ will approach to ${ { y}}_{\mathrm d} $.

5 Numerical simulations

In this section,numerical simulations are carried out to demonstrate the effectiveness of the methods in this paper. The considered VGTM is shown in Fig. 3,which has nine degrees of freedom. The physical parameters of the VGTM robot are given in Table 1. The initial lengths of the VGTM actuators are chosen as [0.279 7 m,0.279 7 m,0.279 7 m,0.279 7 m,0.279 7 m, 0.279 7 m,0.314 1 m,0.314 1 m,0.314 1 m]. The validity of the dynamics model proposed in this paper is verified firstly through the comparison with the ADAMS software. It is assumed that the forces exerted on the actuators are $\tau = {[0{\mkern 1mu} {\text{N}}, - 0.1{\mkern 1mu} {\text{N}}, - 0.1{\mkern 1mu} {\text{N}},0{\text{N}}, - 0.05{\text{N}}, - 0.1{\text{N}},0{\text{N}},0{\text{N}},0{\text{N}}]^{\text{T}}}$. Under these forces,the system moves. Figures 7 and 8 are the time histories of lengths and velocities of the actuators. It is clearly observed from Figs. 7 and 8 that the proposed model can obtain the same results with the ADAMS software,which proves the validity of the proposed model.

The results above show that the proposed model is effective in describing the dynamic behavior of the VGTM system. Next,we verify the controller given in this paper. In this simulation,the VGTM will be controlled to track the planned trajectories in the actuator space by the computed torque method and the classical PD control method,respectively. The desired actuator displacements of the VGTM are [0.379 7 m,0.179 7 m,0.329 7 m,0.379 7 m, 0.179 7 m,0.329 7 m,0.414 1 m,0.214 1 m,0.264 1 m], and the desired actuator velocities are all zero. The desired trajectories of actuators can be designed with the method in Section 4.1. In the controller design of computed torque method,${ K}_{\mathrm P}$ and ${ K}_{\mathrm D}$ in Eq.(78) are chosen as ${ K}_{\mathrm P}$ = diag (50 50 50 50 50 50 50 50 50) and ${ K}_{\mathrm D}$ = 0.1${ K}_{\mathrm P}$. When using the PD control,the same ${ K}_{\mathrm P}$ and ${ K}_{\mathrm D}$ are used. The control simulation results are given in Figs. 9 and 10,which show the time histories of lengths and velocities of the actuators in the process of trajectory tracking. It is observed from Figs. 9 and 10 that the control design by the computed torque method can achieve good control effectiveness,but the PD cannot.

6 Concluding remarks

In this paper,the dynamics and control of a VGTM are studied. A dynamic model using the actuator lengths as the generalized variables is established by Jourdain's velocity variation principle. The modeling approach given in this paper can also be applicable to the other kinds of variable geometry truss mechanisms or parallel mechanisms. The validity of the proposed dynamic model is verified by the comparison with the ADAMS software. Simulation results prove that the proposed model is effective to describe the dynamic behavior of the VGTM. Besides,a controller for trajectory tracking of the system is designed by the computed torque control method in this paper. Control simulation results show the validity of this controller.