Transmission Index Research of Parallel Manipulators Based on Matrix Orthogonal Degree

Parallel manipulator usually consists of the base and the end effector, which are connected together with at least two identical kinematic chains (branches) [1, 2]. With the structure feature of closed-loop kinematic chains, the parallel mechanism is recognized as a significant complementary to the traditional serial mechanism [3]. Correspondingly, parallel manipulators possess stable, compact and simple structures with low moving inertia and mass, and show advantages in terms of rigidity, precision, load, and speed, according to Mrthy’s theory [4]. Above statement can be verified by the applications as machine tools [5], positioning devices [6], motion simulators [7] and high-speed pick-and-place robots [8, 9]. Although application number and field of parallel manipulators increase rapidly, the large-scale commercial application is scarce. As Merlet [10] once pointed out, the performance advantages of parallel manipulators seem to be just potential for now. The industrial performance of the parallel manipulator is one of the most concerned issues for both researchers and engineers. Thus, further study on performance analysis and optimal design should be carried out.

Performance index is the standard to evaluate the manipulator capability and the basis for performance analysis and optimal design. A good index is expected to be finite, independent of the coordinate system, dimensionally homogeneous and with clear physical significance. Besides, ease of calculation and intuitive concept will make the index more convenient and likely to be implemented. However, even without considering the mechanism dynamics and elasticity of the parallel manipulator, it is still a challenging issue to seek the suitable kinematic indices [11]. Nature of the mechanism is transmitting force and motion [12], and the parallel mechanism is not an exception. Force/motion transmissibility is the key to the kinematic performance for the parallel mechanism. Intensive efforts have been made to study related indices, which can be classified into three categories.

Index researches with the Jacobian matrix belong to the first category. In kinematics of non-redundant parallel manipulators, force Jacobian matrix is the transpose of kinematic Jacobian matrix, which reflects the truth that force and motion are two equivalent aspects of transmissibility. Based on the condition number of the matrix, Local Conditioning Index (LCI) [13, 14] and Global Conditioning Index (GCI) [15] of the parallel manipulator are developed, which are originally proposed for the serial mechanism [16]. When these indices are applied to the parallel manipulator with combined translational and rotational Degrees of Freedoms (DOFs), dimensional inconsistency appears, which leads to evaluation failure on performance [17, 18]. Then, characteristic length-based method [19] and characteristic point-based method [20] are investigated to formulate the dimensionally homogeneous Jacobian matrix. Recently, MA et al. [21] and Liu et al. [22] found that LCI values could converge when the index is applied to the translational parallel manipulator, which results in poor performance resolution. In addition, the LCI value is frame-related [23, 24].

The second category is referred as angle-based indices, such as pressure angle and transmission angle. Pressure angle is the angle between driving force vector and the velocity vector of force application point. Transmission angle is the complementary angle of the pressure angle, which is defined by the angle between input and output forces of a joint [25]. These indicators are powerful for the planar mechanism [26] with clear significance. Huang et al. [27] evaluated the performance of a two DOFs translational parallel manipulator with the transmission angle. The sine value of the transmission angle is adopted to develop a global force transmission index, and has been applied to performance analysis of the planar 5R parallel manipulator [28]. Philipp et al. carried out the transmission angle analysis of the planar seven-bar mechanism [29]. Based on the fictitious limb composed of two physical limbs, transmission angles among limbs were proposed, attempting to extend the application of the transmission angle to the spatial parallel mechanism [30]. However, with the increase in limb number as well as the transformation from planar to spatial mechanism, definitions of angle-based indices become difficult and complex with vague physical meanings.

The last category is the transmission indices based on the screw theory [31, 32]. Considering the virtual coefficient is dimensionless and independent of the coordinate system, the virtual coefficient between input and output wrench screws is defined as the transmission factor, and used to evaluate kinematic performance of the spatial mechanism [33]. Then, the transmission factor is unitized [34], and the generalized transmission wrench screw is proposed [35, 36]. Chen et al. introduced the generalized transmission index and analyzed spatial linkages [37]. Further, input and output transmission indices as well as the total transmission index are proposed [38]. Above indices focus on the analysis of the input power efficiency of the driving joint and the output power efficiency of the branch. The power is calculated through the reciprocal product of twist and wrench screws. Screw theory is a powerful and systematic mathematical tool. However, definition and calculation of these indices are quite complicated, and not intuitive.

There are demands for simple and intuitive force/motion transmission indices for parallel manipulators, considering both planar and spatial mechanism. In this paper, based on the concept of the matrix (or vector group) orthogonal degree, a series of transmission indices are proposed, in terms of the mathematical description, definition and calculation for the fully parallel manipulator, which is considered as the non-redundant parallel manipulator with one actuator in each limb. Then, transmission performance analysis of typical parallel mechanisms is carried out to illustrate proposed indices and the analysis method. The remainder of this paper is organized as follows. In the next section, branch and end-effector transmission indices are introduced, and the local transmission index is defined to evaluate the overall transmissibility of parallel manipulators. Transmission performance analysis of typical branches and end effectors for fully parallel manipulators are illustrated in sections 3 and 4. In section 5, transmissibility of two typical parallel manipulators is discussed. Conclusions of this paper are given in section 6.

In this paper, transmission indices are deduced on the basis of orthogonal degree of matrix, which is defined with matrix volume [39]. First of all, matrix volume can be described as follows: assuming \(\varvec{X}_{m \times n} = \left[ {\begin{array}{*{20}l} {\varvec{x}_{1} } \hfill & {\varvec{x}_{2} } \hfill & \ldots \hfill & {\varvec{x}_{n} } \hfill \\ \end{array} } \right]\) is a real matrix (column vector group), composed of n real column vectors \({\varvec{x}_{i}}\). Then, the volume of the real matrix can be deduced as

Further, the matrix orthogonal degree can be described as following: if \(\hbox{min} \left\| {\varvec{x}_{i} } \right\| = 0\;(i = 1,2, \ldots, n)\), the matrix orthogonal degree \({\text{ort(}}\varvec{X} )\) equals 0. Otherwise,

Above definition is based on the column vector, and can be considered as the volume of matrix composed of unit column vectors. On the basis of above definition, it is easy to get the property that \(0 \le {\text{ort(}}\varvec{X} )\le 1\). When column vectors are orthogonal to each other, the maximum value 1 can be obtained. On the contrary, when multicollinearity of column vectors appears, the minimum value 0 can be deduced. Extreme values of the matrix orthogonal degree indicate two important geometric properties, multicollinearity and orthogonality of multiple vectors. Obviously, matrix orthogonal degree is an efficient mathematical tool to study force/motion vectors of spatial parallel mechanisms.

The force (or motion) transmission in mechanism could be understood as from the base (fixed on the ground) to the terminal. According to structural features of the parallel mechanism, identical branches connect the end effector with the base. Transmissibility of the parallel manipulator can be subdivided into two aspects, such as branch transmissibility and end-effector transmissibility. The former describes the manipulability of the limb, and the later indicates the complementarity of limbs’ contributions on the end effector.

Firstly, let us focus on the branch transmissibility. Each branch (or limb) of the parallel manipulator is a typical serial mechanism, and force is transmitted from the base to the end effector through the limb. Limb transmissibility depends on transmission performances of the mounted joints. Index of the joint transmissibility can be defined by the relation between the afferent force applying on the joint by the previous component and the efferent force that exerted on the next component through the joint. Specifically, Joint Transmission Index (JTI) of the ith joint located in the jth branch of the parallel manipulator can be written aswhere \(\varvec{Y}_{ji} = \left[ {\begin{array}{*{20}l} {F_{{ji{\text{A}}}} } \hfill & {F_{{ji{\text{E}}}} } \hfill \\ \end{array} } \right]\). \({\varvec{f}}_{{ji{\text{A}}}}\) and \({\varvec{f}}_{{ji{\text{E}}}}\) are the afferent and efferent force vectors, respectively. From the above definition, it can be found that when these force vectors are collinear, the force transmissibility is the best, and \(\varGamma_{\text{JTI}} { = }1\). On the contrary, when force vectors are mutually perpendicular, the force transmissibility is the worst, and \(\varGamma_{\text{JTI}} { = 0}\). Numerically, the JTI value equals the cosine value of the pressure angle, and indicates the ratio of the efferent force to the afferent force. For different limbs, JTI analysis is diverse. The JTI descriptions of typical branches of parallel manipulators will be introduced in Section 3.

The branch of the parallel manipulator is formed by a series of joints. On the basis of the JTI, the Branch Transmission Index (BTI) can be defined as the product of JTI values of all joints in the branch, and can be written as

Eq. (4) reflects the overall transmission performance of the limb from the base to the end-effector joints. When the BTI value of a branch equals 0, the end effector of the parallel manipulator will lose the support of the branch. In this case, the manipulator is at its singular configuration, and this kind of singularity is called the reverse singularity.

The other aspect of transmission analysis for the parallel manipulator is the end-effector transmissibility, which reflects the integrated efficacy of all branches. The End-effector Transmission Index (ETI) reflects the ability of branch force vectors to span the whole required output force space, and is defined with the limb force vectors exerted on the end effector and the end-effector geometric characteristics. The ETI can be calculated aswhere \({\varvec{F}} = \left[ {\begin{array}{*{20}c} {{\varvec{\uptau}}_{1} } & {{\varvec{\uptau}}_{2} } & \ldots & {{\varvec{\uptau}}_{n} } \\ \end{array} } \right]\) is the branch force matrix. The limb force vector of the jth branch is \({\varvec{\uptau}}_{j} = \left[ {{\varvec{f}}_{j} ; \, {\varvec{c}}_{j} \times {\varvec{f}}_{j} } \right]\), composed of the unit force vector \({\varvec{f}}_{j}\) and unit torque vector \(\varvec{c}_{j} \times {\varvec{f}}_{j}\). \({\varvec{f}}_{j}\) is the unit force vector exerted on the end effector by the jth limb, while \(\varvec{c}_{j}\) is the unit radial vector pointing from the geometric center of the end effector to the rotational center of the end-effector joint of the jth limb. Rigid body in space possesses six DOFs, but some DOFs of the end effector are constrained to obtain the required DOFs. The matrix E is a diagonal matrix, used to describe the required DOFs of the parallel manipulator. When a degree of freedom of the end effector is required and movable, the corresponding diagonal element of matrix E is 1. On the contrary, if some DOFs are constrained, the corresponding diagonal elements of matrix E are 0. Take the 3-RRR parallel manipulator for example. The matrix E = diag (1, 1, 0, 0, 0, 1), which indicates the end effector of this manipulator possesses two translational DOFs along X and Y axes and a rotational degree of freedom along C axis (represents the rotational movement around the Z axis). The dimension of the matrix EF always equals the number of required end-effector DOFs, and the ETI is applicable to all non-redundant parallel manipulator. If the end effector of the parallel manipulator is a point-type, for example 5R parallel manipulator, \(\varvec{c}_{j}\) becomes zero. It is worth noting that when the value of ETI is 0, the parallel manipulator will lose the force output in a certain direction. In this case, the manipulator is at its forward singularity.

As mentioned above, force transmission in the parallel manipulator can be considered as from the base to the end effector. Thus, the total transmissibility of the parallel manipulator equals the product of BTI and ETI values. However, BTI values and branch transmission performances are different among various branches under the arbitrary pose. Considering that the main work in the design and analysis stage is to avoid configuration and pose with poor transmission performance, the minimal BTI value is adopted, and the Orthogonal degree based Local Transmission Index (OLTI) of the parallel manipulator can be defined as

When the value of OLTI is 0, the parallel mechanism is in a singularity locus. Conversely, when the value of OLTI is 1, the transmissibility of the mechanism is the best. As illustrated in the above definitions, the proposed transmission indices are not related to the established coordinate system, and are frame-free. The value range is finite and normalized [0, 1]. Besides, index significances are clear, while calculation is simple and concise.

In this section, based on the proposed BTI, transmission performances of typical branches for fully parallel manipulators are analyzed. Gravity, friction and inertial forces are not taken into account in this paper.

For the parallel manipulator, the base and the end effector are connected with multiple identical branches simultaneously. The output force of the end effector is the common effort of all branches. The typical planar and spatial parallel mechanisms, such as 5R and Stewart parallel manipulators, are analyzed in this section to reveal the end-effector transmissibility.

The structure of 5R mechanism is shown in Figure 5. Point P is the end effector, with two translational DOFs. Both branches are of the RRR configuration. According to the definition of ETI proposed in Section 2, the ETI expression of the 5R mechanism can be written aswhere f ₁ and f ₂ are unit force vectors exerted on the end effector by two branches, along l ₁₂ and l ₂₂ vectors respectively, which are unit structural vectors of passive links. According to the geometric relation, the transmission index can also be written as \(\varGamma_{\text{ETI}} = {\text{ort}}\left( {\varvec{L}} \right)\) and \(\varvec{L} = \left[ {\begin{array}{*{20}l} {\varvec{l}_{12} } \hfill & {\varvec{l}_{22} } \hfill \\ \end{array} } \right]\). For the 5R parallel manipulator, the ETI value equals the sine value of the angle between f ₁ and f ₂ vectors (l ₁₂ and l ₂₂ vectors). Obviously, when the angle becomes 0° or 180°, limb force vectors are collinear, and the end effector loses the force output capacity along the direction perpendicular to limb force vectors. Accordingly, a singularity occurs, and the ETI value is 0. If the angle is 90°, the transmissibility of the end effector is the best, and the ETI value equals 1.

As shown in Figure 6, the Stewart parallel manipulator consists of six extensible UPS branches, which is a typical 6 DOFs parallel manipulator with combined translational and rotational DOFs. According to the branch analysis, the unit force f _j acting on the end effector by the jth limb is along the limb, through rotational center of the spherical joint. As illustrated in Figure 6, the geometric center of the end effector is point o. Thus, direction of the unit radial vector c _j is from point o to the rotational center of the jth spherical joint. Considering the geometric structure, the unit branch vector l _j2 can be used instead of the limb force vector f _j . The matrix E becomes the unit diagonal matrix, and all diagonal elements are 1. Then, we can deduce the \(\varGamma_{\text{ETI}}\) expression of the Stewart manipulator as

And, singularity of the Stewart parallel manipulator will be discussed in the next section.

In order to illustrate the analysis method and the proposed indices in detail, the transmission performances of two typical parallel manipulators are analyzed with proposed indices in this section, and comparative discussion with the traditional LCI is developed.

Firstly, the 5R parallel manipulator is analyzed, and adopted kinematic parameters are listed in Table 1. The global coordinate system is established at the midpoint between two actuated joints, as shown in Figure 5. With Eqs. (6), (7) and (10), value distributions of the proposed indices, such as BTI, ETI and OLTI, can be determined, as shown in Figures 7, 8, and 9 respectively. The value distribution of traditional LCI is exhibited in Figure 10.

As shown in Figure 8, in the reachable workspace (illustrated with red curve), ETI values are relatively large, which indicates that the transmissibility of the end effector for the 5R mechanism is quite good. With comparative analysis of Figures 7, 8, and 9, we can find that the overall transmission performance of the 5R mechanism is mainly determined by the branch transmissibility. On the boundary of the reachable workspace, reverse singularity occurrence, introduced by the branch transmission performance, as shown in Figure 7. With the proposed indices, it is convenient to determine the singularity type and analyze the cause.

As shown in Figures 9 and 10, overall trends of LCI and OLTI atlases for the 5R parallel manipulator are similar. Large index values which mean good transmission performance appear in the center of the reachable workspace. With the end effector moves away from the center region, the transmission performance becomes worse. However, there are some significant differences between LCI and OLTI atlases. In the LCI atlas, contour curves with different LCI values converge, which reduces the performance resolution capacity and fails to reflect the distance off singularity. In contrast, contour curves with different OLTI values never intersect, and the offset of each curve is quite clear. The proposed index is relative to the singularity.

In addition, the transmission performance of the Stewart parallel manipulator is evaluated, and adopted kinematic parameters are shown in Table 2. Joint distribution angles α and β of the Stewart manipulator are used to describe locations of spherical and universal joints in the end effector and the base respectively, as shown in Figure 11. The global coordinate system is established at the geometrical center of the base, as illustrated in Figures 6 and 11.

The distributions of LCI and OLTI for the Stewart parallel manipulator are given in Figures 12 and 13, when the end effector is located in the horizontal plane of Z = 0.45 m without any rotation. Since the BTI value of the Stewart manipulator is the constant 1, its OLTI atlas is the same with its ETI atlas. In the calculation of LCI value, considering the dimensional homogeneity, the end-effector radius r is adopted as the characteristic length to deduce the normalized Jacobian matrix.

Generally, overall trends of LCI and OLTI atlases for the Stewart parallel manipulator are similar, as illustrated in Figures 12 and 13. Both LCI and OLTI values decrease while the end effector moves away from the center of the workspace, which indicates that the transmission performance becomes worse. The minimum and maximum values of OLTI are 0.146 and 0.636 respectively. The minimum and maximum values of LCI are 0.178 and 0.456 respectively. The value variation of OLTI (0.49) is greater than that of the LCI (0.278), and the OLTI possesses better performance resolution consequently.

Figure 14 illustrates LCI and OLTI value changes when the end effector of the Stewart manipulator moves along Z axis from 0 m to 0.7 m with X and Y coordinate values being 0. The manipulator obtains the best transmission performance when the end effector is located around Z = 0.31 m. The maximum value of OLTI is 0.77, while the maximum value of LCI is 0.59. Minimum values of OLTI and LCI are both zero, which indicate the singularity. In the left region, with the end effector approaching the base, both index values decrease, and the value declining speed is faster than that in the right region when the end effector moves away from the base.

Figure 15 indicates LCI and OTLI value changes when the end effector rotates along C axis with angle γ from 0° to 90° and is located at the position of (0, 0, 0.45) m. Both LCI and OTLI values gradually decrease. When the rotation angle γ equals 90°, values of OTLI and LCI are both 0, which means the Stewart parallel manipulator is at its singular pose.