Keywords

1 Introduction

Most of the 6-DOF PM consist of six limbs (Merlet 2000; Gogu 2008–2014 and others). These PM possess the advantages of high stiffness, low inertia, and large payload capacity. However, such six–limbed fully PM have a limited workspace and complex kinematic singularities, which are their major drawbacks. Therefore in robotics literature (Yang et al. 2004; Mianovski 2007; Jin et al. 2009; Glazunov 2010 and other) a great interest to the PM with few number of limbs and larger workspace is observed.

Following the above-mentioned trends in the development of PM, we proposed a novel structure of six-DOF three-limbed PM with cylindrical joints (PM 3CCC) (Baigunchekov et al. 2009), as shown in Fig. 1. This PM is formed by connection of a mobile platform 3 with a base 0 by three spatial dyads ABC, DEF and GHI of type CCC (C—cylindrical joint). Each of spatial dyads of type CCC do not impose restrictions on motion of the mobile platform, and six-DOF of the mobile platform are remained.

Fig. 1
figure 1

PM with cylindrical joints

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Each cylindrical joint has two-DOF: one rotation and one translation. In the considered PM the joints A, F and I are active joints, and the joints B, C, D, E, G and H are passive joints. Six variable parameters s 7, θ 7, s 8, θ 8, s 9, θ 9 of active joints A, F and I are the generalized coordinates. The results of singularity analysis of the PM 3CCC are presented (Baigunchekov et al. 2012). In this paper the geometry of this PM is described and its direct kinematics is solved.

2 Geometry of the PM 3CCC

To describe the geometry of the PM 3CCC two right-hand Cartesian coordinate systems UVW and XYZ are attached to each element of each joint. The W and Z axes of the coordinate systems UVW and XYZ are directed along the axes of rotation and translation of the cylindrical joints.

Transformation matrix \( {\mathbf{T}}_{jk} \) between the coordinate systems U j V j W j and X k Y k Z k , attached to the ends of the binary link with the j-th and k-th joints, has a form

$$ {\mathbf{T}}_{jk} = \left[ {\begin{array}{*{20}c} {t_{11} } & {t_{12} } & {t_{13} } & {t_{14} } \\ {t_{21} } & {t_{22} } & {t_{23} } & {t_{24} } \\ {t_{31} } & {t_{32} } & {t_{33} } & {t_{34} } \\ {t_{41} } & {t_{42} } & {t_{43} } & {t_{44} } \\ \end{array} } \right], $$
(1)

where \( t_{11} = 1 \), \( t_{12} = t_{13} = t_{14} = 0 \),

$$ \begin{aligned}t_{21} &= a_{jk} \cdot \cos \gamma_{jk} + b_{jk} \cdot \sin \gamma_{jk} \cdot \sin \alpha_{jk} ,\\ t_{22} &= \cos \gamma_{jk} \cdot \cos \beta_{jk} - \sin \gamma_{jk} \cdot \cos \alpha_{jk} \cdot \sin \beta_{jk} , \\ t_{23} &= - \cos \gamma_{jk} \cdot \sin \beta_{jk} - \sin \gamma_{jk} \cdot \cos \alpha_{jk} \cdot \cos \beta_{jk}, \\t_{24} &= \sin \gamma_{jk} \cdot \sin \alpha_{jk} , \\ t_{31} &= a_{jk} \cdot \sin \gamma_{jk} - b_{jk} \cdot \cos \gamma_{jk} \cdot \sin \alpha_{jk} , \\ t_{32} & = \sin \gamma_{jk} \cdot \cos \beta_{jk} + \cos \gamma_{jk} \cdot \cos \alpha_{jk} \cdot \sin \beta_{jk} , \\ t_{33} &= \cos \gamma_{jk} \cdot \cos \alpha_{jk} \cdot \cos \beta_{jk} - \sin \gamma_{jk} \cdot \sin \beta_{jk} , \\ t_{34} &= - \cos \gamma_{jk} \cdot \sin \alpha_{jk} ,\,t_{41} = c_{jk} + b_{jk} \cdot \cos \alpha_{jk}, \\ t_{42} &= \sin \alpha_{jk} \cdot \sin \beta_{jk} ,\,t_{43} = \sin \alpha_{jk} \cdot \cos \beta_{jk} ,\end{aligned}$$

\( t_{44} = \cos \alpha_{jk} \), \( a_{jk} \)—a distance from the W j axis to the Z k axis measured along the direction of the common perpendicular t jk between the W j and Z k axes; \( \alpha_{jk} \)—an angle between positive directions of the W j and Z k axes measured counterclockwise about positive direction of t jk ; b jk —a distance from direction of t jk to direction of the X k axis measured along the positive direction of the Z k axis; \( \beta_{jk} \)—an angle between positive directions of t jk and X k axis measured counterclockwise about the positive direction of the Z k axis; c jk —a distance from direction of U j axis to direction of t jk measured along the positive direction of the W j axis; \( \gamma_{jk} \)—an angle between positive directions of the U j axis and t jk measured counterclockwise about the positive direction of the W j axis.

In comparision with the Denavit–Hartenberg transformation matrix, having four parameters, the transformation matrix (1) has six parameters fully characterizing the relative locations of the coordinate systems U j V j W j and X k Y k Z k , because a free rigid body in space has six generalized coordinates.

A binary link jk of type CC is shown in Fig. 2. Axes of the coordinate systems U j V j W j and X k Y k Z k , attached to the ends of this binary link, are chosen as follows: the W j and Z k axes are located along the axes of rotation and translation of the cylindrical joints j and k; the origins O j and O k of the coordinate systems U j V j W j and X k Y k Z k are located in points of intersection of the W j and Z k axes with the common perpendicular \( t_{jk} \) between these axes; the U j and X k axes are located along the common perpendicular \( t_{jk} \); the V j and Y k axes are completed the right-hand Cartesian coordinate systems U j V j W j and X k Y k Z k .

Fig. 2
figure 2

Binary link jk of type CC

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At such choice of the coordinate systems U j V j W j and X k Y k Z k nonzero parameters of the matrix \( {\mathbf{T}}_{jk} \) are \( a_{jk} \) and \( \alpha_{jk} \). Then from the matrix (1) we obtain a matrix of the binary link jk of type CC

$$ {\mathbf{G}}_{jk}^{{\text{CC}}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ {a_{jk} } & 1 & 0 & 0 \\ 0 & 0 & {\cos \alpha_{jk} } & { - \sin \alpha_{jk} } \\ 0 & 0 & {\sin \alpha_{jk} } & {\cos \alpha_{jk} } \\ \end{array} } \right], $$
(2)

where parameters \( a_{jk} \) and \( \alpha_{jk} \) are constant, and they characterize the geometry of the binary link jk of type CC.

Nonzero parameters of the cylindrical joint j shown in Fig. 3 are \( \theta_{j} \) and \( s_{j} \). Then from the matrix (1) we obtain a matrix of the cylindrical joint j

Fig. 3
figure 3

Cylindrical joint j

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$$ {\mathbf{P}}_{j}^{\text{C}} (\theta_{j} ,s_{j} ) = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & {\cos \theta_{jk} } & { - \sin \theta_{j} } & 0 \\ 0 & {\sin \theta_{j} } & {\cos \theta_{j} } & 0 \\ {s_{j} } & 0 & 0 & 1 \\ \end{array} } \right], $$
(3)

where \( s_{j} \)—a distance from the \( X_{j} \) axis to the \( U_{j} \) axis measured along the directions of the \( Z_{j} \) and \( W_{j} \) axes; \( \theta_{j} \)—an angle between the positive directions of the \( X_{j} \) and \( U_{j} \) axes measured counterclockwise about the positive directions of the \( Z_{j} \) and \( W_{j} \) axes. Parameters \( s_{j} \) and \( \theta_{j} \) are variable, and they characterize relative translation and rotation motions of the j-th cylindrical joint elements.

Choosing the coordinate systems UVW and XYZ, as shown in Figs. 2 and 3, the constant and variable parameters of the PM 3CCC have been obtained. Constant and variable parameters of the first limb ABC of the PM 3CCC are shown in Fig. 4, where θ 7 and s 7 are the generalized coordinates of the active joint A; θ 2, s 2 and θ 3, s 3 are the variable parameters of the passive joints B and C; all other parameters are the constant parameters characterizing the geometry of links. Constant and variable parameters of two other symmetrical legs are defined similarly.

Fig. 4
figure 4

The first limb

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3 Direct Kinematics

In direct kinematics of the PM 3CCC position of coordinate system X P Y P Z P attached to the mobile platform 3 are defined with respect to the base frame U o V o W o by known constant geometrical parameters of the links and the generalized coordinates \( s_{i} \), \( \theta_{i} \), (i = 7, 8, 9).

For automation of calculation of the direct kinematics we use following designations: \( n \)—number of mobile links, \( m \)—number of input links, \( L \)—number of the closed loops, \( n_{l} \)—number of links in the l-th loop (l = 1, 2, …, \( L \)), \( (l,a) \)—index of the a-th link in the \( l \)-th loop. Dependent variable parameters of the passive joints \( h_{(l,a)}^{(\omega )} \) are numerated 1, 2,…, n − m, independent variable parameters of the active joints \( q_{(l,a)}^{(\omega )} \) are numerated n − m + 1, n − m + 2, …, n, where \( \omega \)—number of DOF of the joint \( (l,a) \). For the considered PM 3CCC n = 7, m = 3, \( L \) = 2, \( n_{1} = n_{2} = 6 \), \( \omega \) = 2. PM 3CCC has two closed loops ABCDEFA and ABCGHIA.

Let made a Table 1 of conformity between the numbers of links and joints of the PM 3CCC.

Table 1 Conformity between the numbers of links and joints
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Using the Table 1 of conformity between the numbers of links and joints we can write the loop-closure equations of the loops ABCDEFA and ABCGHIA

$$ \left. \begin{aligned} {\mathbf{M}}_{71} \cdot {\mathbf{M}}_{12} \cdot {\mathbf{M}}_{23} \cdot {\mathbf{M}}_{34} \cdot {\mathbf{M}}_{48} \cdot {\mathbf{M}}_{87} = {\mathbf{E}} \hfill \\ {\mathbf{M}}_{71} \cdot {\mathbf{M}}_{12} \cdot {\mathbf{M}}_{25} \cdot {\mathbf{M}}_{56} \cdot {\mathbf{M}}_{69} \cdot {\mathbf{M}}_{97} = {\mathbf{E}} \hfill \\ \end{aligned} \right\} $$
(4)

or

$$ {\mathbf{M}}_{l,1} \cdot {\mathbf{M}}_{l,2} \cdot \ldots \cdot {\mathbf{M}}_{l,a} \cdot \ldots \cdot {\mathbf{M}}_{{l,n_{l} }} = {\mathbf{E}};\,\,l = 1,\, 2, $$
(5)

where E is a unit matrix, \( {\mathbf{M}}_{jk} = {\mathbf{P}}_{jk}^{\text{C}} \cdot {\mathbf{G}}_{jk}^{{\text{CC}}} \), j, k =1, 2,…, 9.

For the direct kinematics of the PM 3CCC the iterative method (Uicker et al. 1964) is used. According to this method unknown dependent variable parameters \( h_{(l,a)}^{(\omega )} \) are written through their initial approaches \( h_{(l,a)}^{(\omega )*} \) and deviations \( dh_{(l,a)}^{(\omega )} \) by expression

$$ h_{(l,a)}^{(\omega )} = h_{(l,a)}^{(\omega )*} + dh_{(l,a)}^{(\omega )} , $$
(6)

where \( h_{(l,a)}^{(\omega )} \in \theta_{l,a} ,\,s_{l,a} \).

The system of matrix Eq. (5) are transformed to the system of linear equations, from which the deviations \( dh_{(l,a)}^{(\omega )} \) are determined. Adding the determined deviations to their previous values, more exact values of the passive joints variable parameters have been obtained.

Position of the coordinate system X P Y P Z P attached to the mobile platform 3 with respect to the base coordinate system U o V o W o can be defined by matrix \( {\mathbf{S}}_{P} \)

$$ {\mathbf{S}}_{P} = {\mathbf{G}}_{07} \cdot {\mathbf{M}}_{71} \cdot {\mathbf{M}}_{12} \cdot {\mathbf{M}}_{23} \cdot {\mathbf{G}}_{3P} , $$
(7)

where G 07 is a transformation matrix between the coordinate system X 7 Y 7 Z 7 and the absolute coordinate system U 0 V 0 W 0, G 3P is a transformation matrix between the coordinate systems X P Y P Z P and X 3 Y 3 Z 3 of the mobile platform 3.

4 Conclusions

A novel six-DOF three-limbed PM 3CCC is formed by connection of the mobile platform with the base by three spatial dyads with cylindrical joints. Constant and variable parameters of the PM 3CCC are defined on the basis of the transformation matrix of two systems of coordinates attached to each element of each joints. Constant parameters characterize geometry of links, and variable parameters characterize relative motions of the joint elements. The loop-closure matrix equations of the PM 3CCC are made up. The direct kinematics of the PM 3CCC is solved by iterative method of solution of the loop-closure matrix equations.