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Abstract

This article focuses on the forward kinematic analysis of a class of asymmetrical parallel manipulators by the proposed elimination approach. To solve the key forward kinematic constraint equations with transcendental parameters of the manipulator, an improved elimination algorithm is presented. First, by analyzing the geometry structure of the manipulator, we find the inherent triangular-topology relations of the manipulator. Further, by utilizing the parameter transformation of angular, the key transcendental equations of forward kinematic analysis are formulated into compact polynomial ones. In this context, comparing with the screw approach by Gallardo-Alvarado suggested that the computation efficiency of our proposed approach is superior. Finally, an example of the asymmetrical variable geometry truss manipulator illustrates the effectiveness of the proposed approach.

Keywords

Forward kinematics analysis geometry auxiliary line analysis improved elimination algorithm asymmetrical VGT manipulator

Introduction

Forward kinematics is a primary problem for the concept design and analysis of asymmetrical parallel manipulators (PMs). As a representative of asymmetrical PMs, asymmetrical variable geometry truss (VGT) manipulators have received a great deal of attention due to their inherent advantages over the conventional PMs, and these advantages involved simpler structure, higher stiffness, safer device, and had been gained in potential applications.¹

As far as most asymmetrical PMs, the key problem of forward kinematic analysis (FKA) is concluded as two aspects: establishing the main kinematic model in its structure topology and solving the main equations in its kinematics mechanism. In the last few decades, the FKA of asymmetrical PMs had stimulated the interest of many researchers.^{2

–15}

On the one hand, to set up the main kinematic model of PMs, several geometrical methods^4

–11 are introduced. As early as in 2002, Tsai and Joshi¹² investigated a hybrid kinematic machine by defining the notion of modular. In additional, the authors addressed that geometry centers of the modular can be reconfigured in series or parallel, but the machine did not provide the norm orientation of the platform so as to have trouble in analyzing complex or parasitic motions with more freedom. Further, Ceccarelli et al.³ presented a more suitable model through utilizing natural coordinates of suitable points and vectors. Because of the considered model without using any angular variables, forward kinematics are easier obtained via determining position and orientation of geometry centers of the model. Such a method is applied broadly, but it is inadequate for velocity and acceleration analysis. Extending this work, Correa and Crane⁴ put forward the forward and reverse analyses for a three-degree-of-freedom (DOF) compliant platform by using compliant joints referred to the velocity of the moving platform and the velocity of the actuators.

Recently, the theory of screws for PMs had been applied widely in displacement, workspace, velocity, and acceleration analyses. For instance, Gallardo-Alvarado et al.^5,6 presented forward displacement analysis (FDA) of a serial PM with two asymmetrical limited-DOF PMs^7,8 assembled in series connection. But the FDA approach is merely adapted in a limited DOF of PMs. Further, Zhao et al.⁹ developed a tetrahedron coordinate algorithm only by utilizing four noncoplanar points’ coordinates of the end effector to build up the linear kinematics model of a spatial PM. The advantage of this algorithm is explicit when establishing the kinematics models for complex spatial PMs with three to six DOFs.

However, the above-mentioned methods may be not applied directly to certain asymmetrical PMs with more DOFs, since the number of investigated points will be increased vastly when there is a slight increase in DOFs of the manipulators. As a result, too many variables lead to calculation of collapse.

On the other hand, in the solution of FKA problem for asymmetrical PMs, it is inescapable to generate nonlinear kinematic constraint equations which led to multiple solutions.^10,11 To obtain an interesting solution, several algebraic methods^12

–15 are introduced to solve the main nonlinear kinematic equations with its kinematic model. Griffis and Duffy¹² and Innocenti and Parenti-Catelli¹³ derived a 16th degree univariate polynomial on the general 3-6 Stewart platform. But the approach is considered as only closure configurations in the complex field. Further, Dhingra et al.¹⁴ used Gröbner–Sylvester hybrid method to obtain a 20th degree input–output polynomial directly from the 15 × 15 Sylvester’s matrix. Moreover, to yield a unique current solution, Kong et al.¹⁵ represented a new FDA approach for a linearly actuated quadratic spherical PM by formulating the kinematic equations.

But to the best of our knowledge, the mentioned approaches of solving nonlinear equations are very difficult to avoid the occurring of transcendental functions in the FKA. Here, taking consideration of the key nonlinear kinematic equations with some unknown transcendental parameters, the solution of FKA for asymmetrical PMs may be in a nonunique form or an interesting solution (singularity) loss of kinematic performance at its spatial configuration. Therefore, it is of significant to obtain a good solution by changing the original equation into a univariate polynomial equation. Furthermore, how to deal the class of equations with some transcendental parameters in maintaining the performance of the kinematics mechanism is still an open problem.

In this article, the FKA of the asymmetrical PM is investigated by means of the proposed elimination approach. First of all, by comparing with the topology structure of the manipulator in the studies of Williams¹⁷ and Gallardo-Alvarado et al.,¹⁹ we reveal the inherent geometrical triangular-topology relations. Secondly, the key transcendental equations of FKA are transformed into compact polynomial ones by selecting some suitable angular parameters from the transformed function proposed. Further, a modified elimination algorithm is utilized to solve the key constraint kinematic equations of the manipulator. In this context, the computation efficiency of the proposed approach is higher than the screw approach of Gallardo-Alvarado et al.¹⁹ Finally, a numerical example of the VGT manipulator is provided to illustrate the effectiveness of the proposed approach.

This article is outlined as follows. In the next section, the FKA of a class of asymmetrical PMs (VGT) mechanism is discussed. “The elimination approach for the FKA problem of the asymmetrical PM” section is devoted to using the improved elimination approach to solve for the FKA problem, and the simulation result is shown in “Simulation” section. The conclusion is drawn in the final section.

FKA of a class of asymmetrical PMs

Description of the asymmetrical manipulator module

Taking into coordinate transformation account for traditional approaches of kinematic mechanisms, the actual pose, position, and orientation of the moving platform are determined with respect to the fixed platform. At first, we give the geometry structural description of the manipulator module. As shown in Figure 1, the standard single module for a class of asymmetrical PMs is a relative independent one as a triple octahedron, and then, the whole manipulator is connected by combined plane (such as the mobile platform referred to center O₂).

Figure 1.

The single module of asymmetrical PM for the Stewart platform analogy. PM: parallel manipulator.

In our discussion, each typical single module includes two triangular platforms which are adjacent to six fixed links. Here, different from symmetrical Stewart platform¹⁸ or asymmetrical Stewart platform,¹⁹ the manipulator is assumed to be a statically determinate truss.²⁰ Specially, a three-DOF asymmetrical PM is designed in the following.

(1) As three points of a fixed triangle $Δ A_{1} B_{1} C_{1}$ , the joint centers $A_{1}, B_{1}, and C_{1}$ of the base platform lie on the triangle circumscribed circle centered at O₁ with the radius r₁. Also, each side length of the triangle satisfies $| \vec{B_{1} C_{1}} | = a$ , $| \vec{A_{1} C_{1}} | = b,$ $| \vec{A_{1} B_{1}} | = c$ , and $a, b, c \in [L_{min}, L_{max}]$ , respectively, as shown in Figure 1.

(2) As three points of an arbitrary triangle $Δ A_{2} B_{2} C_{2}$ , the joint centers $A_{2}, B_{2}, and C_{2}$ of the mobile platform lie on the triangle circumscribed circle centered at O₂ with the radius r₂. Each side length of the triangle satisfies the norm distance conditions given in the study of Williams.¹⁷

(3) In general, the triangle $Δ A_{1} B_{1} C_{1}$ is acute one that lies on the base platform. If the existing of an obtuse ϕ is situated at the fixed triangle $Δ A_{1} B_{1} C_{1}$ , the angular is replaced by a translation transformation π − ϕ (π is the radian of semicircle), since the geometry shape of the manipulator can be controlled by adjusting the orientation angular of the “triangular-topology structure” with a statically determinate character.

(4) For the convenient of analysis, the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ and the relative mobile frame system $O_{2} - X_{2} Y_{2} Z_{2}$ are fixed to the center point O₁ on the base platform and O₂ on the mobile platform, respectively. The Y₁-axis is chosen to pass through the joint center B₁ and to perpendicular to the vector $\vec{A_{1} C_{1}}$ . In a similar way, the Z₁-axis is set to intersect with the center point O₁ and to parallel in the vector $\vec{A_{1} C_{1}}$ , and thus, the X₁-axis is determined thoroughly by the right-handed coordinate rule.

Forward kinematic problem for single asymmetrical PM module

Given the lengths of fixed links for a class of asymmetrical PMs, corresponding to the conditions of constraint length of each virtual links L_i (implied two-norm distance conditions in the study of Williams¹⁷), the kinematic constraint equations of joints $A_{2}, B_{2}, and C_{2}$ are satisfied

{|| \vec{A_{2} B_{2}} ||}_{2} = \sqrt{{(B_{2} - A_{2})}^{T} \cdot (B_{2} - A_{2})} = L_{1}

{|| \vec{B_{2} C_{2}} ||}_{2} = \sqrt{{(C_{2} - B_{2})}^{T} \cdot (C_{2} - B_{2})} = L_{2}

|| {\vec{C_{2} A_{2}} ||}_{2} = \sqrt{{(A_{2} - C_{2})}^{T} \cdot (A_{2} - C_{2})} = L_{3}

where the dot ⋅ denotes the inner product of the usual three-dimensional vectorial algebra. It states that $L_{1}, L_{2}, and L_{3}$ are the information distances depended on separating points $A_{2}, B_{2}, and C_{2}$ with the angular parameters $θ_{1}, θ_{2}, and θ_{3}$ . Given the lengths of the virtual links $L_{1}, L_{2}, and L_{3}$ and the original point O₁, the distance P will be represented easily as

{|| \vec{O_{1} O_{2}} ||}_{2} = P

where P denotes the norm vector distance from original point O₁ to point O₂ for the single module.

From equations (1) to (3), the position center vector $\vec{X}$ of the mobile platform can be expressed by designing of joints center A₂, B₂, and C₂, when the base platform is fixed. Therefore, the position center vector $\vec{X}$ is satisfied by

\vec{X} = \frac{1}{τ} (A_{2} + B_{2} + C_{2})

where the value of $τ (= 3)$ is equivalent to the number of the vertexes at the mobile platform; the general expression of the position center vector $\bar{X}$ is presented by $\bar{X} = (1 / τ) \sum_{t = 1}^{τ} Q_{t}$ , if the tth vertex at the mobile platform is Q_t.

Since the position vector $\vec{X}$ referred to the vectors of vertexes A₂, B₂, and C₂, the orientation vector $\vec{n}$ of the mobile platform is described by

\vec{n} = \frac{\bar{n}}{∥ \bar{n} ∥}

where

\bar{n} = | \begin{matrix} i & j & k \\ B_{2} (1) - C_{2} (1) & B_{2} (2) - C_{2} (2) & B_{2} (3) - C_{2} (3) \\ A_{2} (1) - C_{2} (1) & A_{2} (2) - C_{2} (2) & A_{2} (3) - C_{2} (3) \end{matrix} |

= (\begin{matrix} (B_{2} (2) - C_{2} (2)) (A_{2} (3) - C_{2} (3)) - (A_{2} (2) - C_{2} (2)) (B_{2} (3) - C_{2} (3)) \\ (B_{2} (3) - C_{2} (3)) (A_{2} (1) - C_{2} (1)) - (A_{2} (3) - C_{2} (3)) (B_{2} (1) - C_{2} (1)) \\ (B_{2} (1) - C_{2} (1)) (A_{2} (2) - C_{2} (2)) - (A_{2} (1) - C_{2} (1)) (B_{2} (2) - C_{2} (2)) \end{matrix}),

and

A_{2} (m)

m = 1, 2, 3

, denotes the x-axis, y-axis, and z-axis coordinates of joints center A₂ in the absolute frame system

O_{1} - X_{1} Y_{1} Z_{1}

. In a similar way, the x-axis, y-axis, and z-axis coordinates of joint centers B₂ and C₂ are also defined.

In practice, the essence of the FKA problem of the asymmetrical manipulator is to find the geometry center of position vector and orientation vector at the mobile platform via designing several controllable parameters related to angular $θ_{1}, θ_{2}, and θ_{3}$ , since these vertexes $A_{2}, B_{2}, and C_{2}$ referred merely to the cross longeron face angles $θ_{1}, θ_{2}, and θ_{3}$ and the vertexes $A_{1}, B_{1}, and C_{1}$ referenced in the study of Williams.¹⁷ Once we obtain the geometry relations of these parameters, the vectors $\vec{X} and \vec{n}$ can be figured out by using equations (5) and (6). Therefore, in this way, the FKA of the single module is suggested in the rest part of “FKA of a class of asymmetrical PMs” section.

Space position of joint center A₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$

Here, joint center A₂ will be first given in a set of generalized coordinates by introducing certain auxiliary lines as shown in Figure 2. To simplify the complexity of constituting auxiliary lines, we recreate a reference frame system $O_{1} - X_{1} Y_{1} Z_{1}$ as follows.

Figure 2.

Space position of joint center A₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ by constituting auxiliary lines.

First, if the original center O₁ is located at the orthocenter of the based platform, Y₁-axis is defined by keeping the same orientation as the vector $\vec{O_{1} B_{1}}$ ; Z₁-axis is fixed by passing through the center O₁ and perpendicular to the Y₁-axis; the system implies that X₁-axis is gained by the right-handed coordinate rule.

Second, in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ , connecting joint C₁ to center point O₁, the newborn vector $\vec{C_{1} O_{1}}$ interconnects with the line A₁B₁ at point H₁.

Third, the point O_a is taken to be the midpoint of line A₁B₁, and then, an auxiliary line A₂I₁ (paralleled with X₁-axis) is produced by passing through the vertex A₂ and being orthogonal to the bottom triangle ΔA₁B₁C₁ at point I₁ which falls into the extension line E₁O_a (paralleled to Z₁-axis).

The angle expression of joint center A₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$

From Figure 2, once given the coordinates of points $A_{1}, B_{1}, C_{1}, and O_{1}$ , the vector of joint A₂ is able to be determined in a unique closed form, since the coordinates of points E₁, O_a, and I₁ are only referred to cross longeron face angle θ₁, so a suitable strategy for geometry auxiliary line analysis is summarized in the following steps.

Step 1. Determining the coordinates of points H₁, F₁ in the new absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ by using the similarity theorem of triangle.

From the coordinates of points E₁, O_a displayed in Figure 3, there are the pedals H₁, H₂, and H₃ located on sidelines A₁B₁, B₁C₁, and A₁C₁, respectively. On one hand, since the bottom platform is regarded as a fixed triangle, thus all angles $A_{1}, B_{1}, and C_{1}$ at $Δ A_{1} B_{1} C_{1}$ can be calculated from triangle cosine theorem. On the other hand, since the line E₁O_a is paralleled to line H₁F₁ and Z₁-axis, the auxiliary line H₁F₁ is yielded by passing through pedal F₁ along Y₁-axis and being orthogonal to perpendicular line B₁H₃.

Figure 3.

The coordinates of points E₁, O_a in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ .

Therefore, two right-angled triangles $R_{t} Δ A_{1} B_{1} H_{3}$ and $R_{t} Δ H_{1} B_{1} F_{1}$ meet the similarity of triangles; then in $R_{t} Δ A_{1} C_{1} H_{1}$ , one writes

| \vec{H_{1} B_{1}} | = | \vec{A_{1} B_{1}} | - | \vec{A_{1} H_{1}} | = | c - b \cos A_{1} |

In $R_{t} Δ A_{1} B_{1} H_{3}$ , one can write

| \vec{H_{1} F_{1}} | = | \vec{A_{1} H_{3}} | \cdot \frac{| \vec{B_{1} H_{1}} |}{| \vec{B_{1} A_{1}} |} = | c \cos A_{1} - b \cos^{2} A_{1} |

and further, the vector $\vec{B_{1} F_{1}}$ is duplicated

| \vec{B_{1} F_{1}} | = | \vec{B_{1} H_{3}} | \cdot \frac{| \vec{B_{1} H_{1}} |}{| \vec{B_{1} A_{1}} |} = sin A_{1} | c - b \cos A_{1} |

Concerning the collinearity of three joints A₁, H₃, and C₁, one has

| \vec{H_{3} C_{1}} | = | \vec{A_{1} C_{1}} | - | \vec{A_{1} H_{3}} | = | b - c \cos A_{1} |

In $R_{t} Δ A_{1} B_{1} H_{3}$ and $R_{t} Δ A_{1} C_{1} H_{1}$ , the vector $\vec{O_{1} H_{3}}$ can be replaced as

| \vec{O_{1} H_{3}} | = | \vec{H_{3} C_{1}} | \cdot \frac{| \vec{A_{1} H_{1}} |}{| \vec{H_{1} C_{1}} |} = \csc A_{1} | b \cos A_{1} - c \cos^{2} A_{1} |

For simplicity, let $O_{A} = \csc A_{1} | b \cos A_{1} - c \cos^{2} A_{1} |$ . Moreover, the vector $\vec{B_{1} O_{1}}$ is computed as

| \vec{B_{1} O_{1}} | = | \vec{B_{1} H_{3}} | - | \vec{O_{1} H_{3}} | = | c sin A_{1} - O_{A} |

From equations (9) to (12), we obtain

| \vec{O_{1} F_{1}} | = | \vec{O_{1} B_{1}} | - | \vec{F_{1} B_{1}} | = | | c sin A_{1} - O_{A} | + O_{B} |

where $O_{B} = - (c - b) sin A_{1} \cdot \cos A_{1}$ .

Since $O_{1} = {(0, 0, 0)}^{T}$ is defined, substituting equation (13) into equation (8), the coordinate of point H₁ is expressed as

H_{1} = (\begin{matrix} 0 \\ | | c sin A_{1} - O_{A} | + O_{B} | \\ | c \cos A_{1} - b \cos^{2} A_{1} | \end{matrix})

Step 2. Figuring out the coordinates of points E₁, O_a in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ .

From step 1, the auxiliary line O_aE₁ passed through midpoint O_a at Y₁-axis is orthogonal to line B₁H₃, and we obtain $R_{t} Δ A_{1} B_{1} H_{3} \sim R_{t} Δ H_{1} B_{1} F_{1}$ and $R_{t} Δ H_{1} B_{1} F_{1} \sim R_{t} Δ O_{a} B_{1} E_{1}$ .

Substituting equations (7) and (8) into equation (14), one gets

| \vec{O_{a} E_{1}} | = | \vec{H_{1} F_{1}} | \cdot \frac{| \vec{B_{1} O_{a}} |}{| \vec{B_{1} H_{1}} |} = \frac{c}{2} | \cos A_{1} |

Substituting equations (7) to (9) into equation (14), one writes

| \vec{B_{1} E_{1}} | = | \vec{B_{1} F_{1}} | \cdot \frac{| \vec{B_{1} O_{a}} |}{| \vec{B_{1} H_{1}} |} = \frac{c}{2} sin A_{1}

Subtracting equation (16) from equation (12), the vector $\vec{O_{1} E_{1}}$ can be written

| \vec{O_{1} E_{1}} | = | \vec{O_{1} B_{1}} | - | \vec{E_{1} B_{1}} | = | \frac{c}{2} sin A_{1} - O_{A} |

Further, by observing from equations (15) to (17), the coordinate of point O_a is represented as

O_{a} = (\begin{matrix} 0 \\ | \frac{c}{2} sin A_{1} - O_{A} | \\ \frac{c}{2} | \cos A_{1} | \end{matrix})

Step 3. Calculating the coordinate of point I₁ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ .

To build up an angle formula of point I₁, geometry relationships of the lateral triangle $Δ A_{1} A_{2} B_{1}$ for the asymmetrical manipulator are plotted in Figure 4.

Figure 4.

Geometry relationships of the lateral triangle $Δ A_{1} A_{2} B_{1}$ .

In $Δ A_{1} A_{2} B_{1}$ , we have

| \vec{A_{2} O_{a}} | = N_{1} = \sqrt{M^{2} - \frac{c^{2}}{4}}

for $| \vec{A_{1} A_{2}} | = | \vec{B_{1} A_{2}} | = M$ , where M is a known fixed length.

Based on the definition of the cross longeron face angle, the angle θ₁ can be stated as the angle E₂O_aA₂, as shown in Figure 2. In $R_{t} Δ A_{2} O_{a} I_{1}$ , the perpendicular line from vertex A₂ to a base platform can be defined as

h_{1} = | \vec{A_{2} I_{1}} | = N_{1} sin θ_{1}

and

| \vec{A_{2} O_{a}} | = | - N_{1} \cos θ_{1} |

Arranging from equations (15) to (21), the length of the vector $\vec{E_{1} I_{1}}$ satisfies

| \vec{E_{1} I_{1}} | = | \vec{E_{1} O_{a}} | + | \vec{O_{a} I_{1}} | = | \frac{c}{2} | \cos A_{1} | - N_{1} \cos θ_{1} |

Subtracting equation (17) from equation (22), the coordinate of point I₁ is unfold as

I_{1} = (\begin{matrix} 0 \\ | \frac{c}{2} sin A_{1} - O_{A} | \\ | \frac{c}{2} | \cos A_{1} | - N_{1} \cos θ_{1} | \end{matrix})

Step 4. Providing the coordinate of points A₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ .

Once equations (20) and (23) are held, then the coordinate of vertex A₂ will be yielded as

A_{2} = (\begin{matrix} N_{1} sin θ_{1} \\ | \frac{c}{2} sin A_{1} - O_{A} | \\ | \frac{c}{2} | \cos A_{1} | - N_{1} \cos θ_{1} | \end{matrix})

where $O_{A} = \csc A_{1} | b \cos A_{1} - c \cos^{2} A_{1} |$ .

Therefore, the angle expression of the joint vector A₂ can be stated as

A_{2} = A_{2} (θ_{1})

For the purpose of understanding well the context of this article, the derivation process of the coordinates of two vertexes $B_{2} and C_{2}$ is given in Appendices 1 and 2. Here, their angle expressions are marked as $B_{2} = B_{2} (θ_{2})$ and $C_{2} = C_{2} (θ_{3})$ . By utilizing these angle expressions of joint vectors $A_{2}, B_{2}, and C_{2}$ , a desired or an interesting solution may be derived from the key kinematic equations (1) to (3), and further, several main results and methodologies to solve FKA problem of the asymmetrical PM will be presented.

The elimination approach for the FKA problem of the asymmetrical PM

Description of the elimination process

For most PMs, there are always multiple solutions of the forward kinematics. An algebraic elimination method¹⁸ is adopted to search the correct and feasible solution from multiple solutions of the FKA problem.

When the angle expressions $A_{2} (θ_{1}), B_{2} (θ_{2}), and C_{2} (θ_{3})$ of the joint vectors $A_{2}, B_{2}, and C_{2}$ are implanted into the closed formulae (1) to (3), the key kinematic constraint equations are rewritten as

f_{1} = \sqrt{{(B_{2} (θ_{2}) - A_{2} (θ_{1}))}^{T} \cdot (B_{2} (θ_{2}) - A_{2} (θ_{1}))} - L_{1}

f_{2} = \sqrt{{(C_{2} (θ_{3}) - B_{2} (θ_{2}))}^{T} \cdot (C_{2} (θ_{3}) - B_{2} (θ_{2}))} - L_{2}

f_{3} = \sqrt{{(A_{2} (θ_{1}) - C_{2} (θ_{3}))}^{T} \cdot (A_{2} (θ_{1}) - C_{2} (θ_{3}))} - L_{3}

Naturally, we will acquire several transcendental equations with angular parameters θ_i, if these equations (26) to (28) are expanded thoroughly. But the derivation of solution of transcendental equations is complex and difficult as usual. Here, by utilizing the algebra elimination process, we give some of angular parameters transformation formulae, and thus, these transcendental equations (26) to (28) are transformed into such compact polynomial equations with 12th degree as

f_{1} = A_{11} x_{1}^{2} x_{2}^{2} + A_{12} x_{1}^{2} + A_{13} x_{2}^{2} + A_{15} x_{1} x_{2} + A_{14}

f_{2} = A_{21} x_{2}^{2} x_{3}^{2} + A_{22} x_{2}^{2} + A_{23} x_{3}^{2} + A_{25} x_{2} x_{3} + A_{24}

f_{3} = A_{31} x_{1}^{2} x_{3}^{2} + A_{32} x_{1}^{2} + A_{33} x_{3}^{2} + A_{35} x_{1} x_{3} + A_{34}

where some angular parameters s_i and c_i are substituted by the scalar $x_{i} \in [0, 1]$ ; A_ij, i,j = 1, 2, 3 referred to known parameters such as $L, S, and N$ .

Proof

See Appendix 3. □

The improved elimination algorithm for the FKA problem

Throughout the strategy for the geometry auxiliary line analysis, all the feasible locations of the mobile platform, with respect to the based platform, are able to be computed by giving a set of generalized coordinates, if the kinematic constraint equations (29) to (31) are solved. In the studies of Wu and Huang¹⁶ and Huang et al.,¹⁸ an improved elimination process is presented.

The improved elimination algorithm is as follows:

Step 1. Initialize $L, L_{i}, O_{1}, X, and n$ .

Step 2. Establish geometry relationships of parameters $L_{i} and θ_{i}$ by using equations (1) to (3).

Step 3. Extract angle expressions of joint points $A_{2}, B_{2}, and C_{2}$ from the strategy for the proposed auxiliary line analysis.

Step 4. Derive compact polynomial equations (29) to (31) by transforming from transcendental equations (26) to (28).

Step 5. Solve equations (29) to (31) by using algebraic elimination process to make angular parameters be iterative, and further find a feasible or an interesting solution from multiple solutions with constraint conditions of lengths of the actuators ( $L_{min} \leq L_{i} \leq L_{max}$ ). Otherwise, transfer to step 2 and change norm distance conditions of the triangle of the mobile platform.

Step 6. Compute the coordinates of joint vectors $A_{2}, B_{2}, and C_{2}$ .

Step 7. Give the position vector P from equation (4).

Step 8. Obtain the position vector X and the normal orientation n at the mobile platform from equations (5) and (6).

Step 9. Update $k : = k + 1$ , transfer to step 2.

In a nutshell, the forward kinematic problem for single module of the asymmetrical manipulator is solved by using the proposed elimination approach. The algorithm is of global convergence if a desired initial value is demanded in such an iterative process, and further, a feasible or an interesting solution should be close to the solution of the current expected configuration.

Simulation

The purpose of this simulation experiment is to synthetically verify the improvement in computational efficiency using polynomial equations by the proposed elimination algorithm instead of transcendental ones used by Gallardo-Alvarado et al.¹⁹ In our discussion, the forward kinematics of the asymmetrical PM (chosen VGT manipulator) is to present the center position vector $\vec{X}$ and norm orientation $\vec{n}$ of the mobile platform by taking advantage of the obtained desired solution of the improved elimination algorithm with the limited length of the actuators.

In the framework of the improved elimination algorithm, the experiment setup is presented as follows. When the initial parameters are given in Table 1, a forward kinematic of VGT manipulator is utilized to calculate the vectors of the vertexes of platform. Further, by utilizing the parameter transformation of angular, the polynomial equations of forward kinematic are used to solve for the vertexes. Without loss of generality, the model dimension parameters of the manipulator are consistent with physical prototypes. The specific simulation parameters are recorded.

In this experiment, by analyzing the suggested elimination algorithm, steps 2 to 4 will be calculated on off-line, and the results are saved in the prescribed path so as to analyze conveniently in the next step. In step 5, substituting equations (29) and (31) into equation (30), when we choose x₁ defined as a fixed variable (x = x₁) as shown in Appendix 3, the key compact polynomial equations (29) to (31) are converted as a fourth univariate polynomial with a unknown variable x

F (x) = [((a_{5} x + a_{4}) x + a_{3}) x + a_{2}] x + a_{1}

where a_p, $p = 1, \dots, 5$ , are constants. On the basis of the above analysis of polynomial equations, the complexity of the proposed approach in this article is $O (8 \times n^{4})$ better than $O (11 \times n^{5})$ considered an unknown transcendental parameter in the study of Gallardo-Alvarado et al.¹⁹

For the purpose of proving the following two conclusions:

The improved elimination algorithm is reasonable.

There is indeed an improvement in computational efficiency by employing polynomial equations instead of transcendental ones.

Selecting the same initial parameters as in Table 1, we compare two approaches in variables $\vec{X}, \vec{n}$ , and t with respect to four solutions of the forward kinematics for the asymmetrical VGT manipulator as follows.

Table 1.

Parameters of VGT physical prototype.

Parameters in the simulation model	Value
L	0.475 m
S	0.082 m
N	0.390 m
L _min	0.45 m
L _max	0.75 m
a	0.49 m
b	0.51 m
c	0.53 m
Moving frequency of the modular	1 Hz

VGT: variable geometry truss.

Here, t is called the consumption time of the algorithm, which is also regarded as elapsed times of the computer, when the program for one of two approaches runs once. It is calculated as $t = t_{a} + t_{b}$ , where t_a and t_b are run times with online and off-line of the algorithm program, respectively. In general, the average consumption time t_ave is measured as

t_{ave} = \frac{1}{K_{N}} (t_{1} + \dots + t_{K_{N}})

where the number K_N is taken to be the number of solutions of the forward kinematics.

As shown in Table 2, by computing formula (33), the average consumption time of the screw approach of Gallardo-Alvarado et al.¹⁹ and the improved elimination algorithm is 4.03437 s and 3.4699 s, respectively. In other words, the computation efficiency of the latter approach is increased by 16.27% than the former one.

Table 2.

The comparison of several main parameters between the proposed approach and the screw approach of Gallardo-Alvarado et al.¹⁹

Solution	Variables	Screw approach	Proposed approach
1	$\vec{X_{1}}$ (m)	(0.2143; 0.9813; 0.2695)	(0.2101; 0.9781; 0.2711)
	$\vec{n_{1}}$ (m)	(0.0281; 0.9983; 0.0516)	(0.0282; 1.001; 0.0517)
	t₁ (s)	3.9768	3.3376
2	$\vec{X_{2}}$ (m)	(0.67; −0.7487; 0.2695)	(0.6471; −0.7454; 0.2637)
	$\vec{n_{2}}$ (m)	(0.794; −0.6057; 0.0514)	(0.8083; −0.6166; 0.0523)
	t₂ (s)	3.9825	3.3378
3	$\vec{X_{3}}$ (m)	(−0.1067; −0.7463; 0.7163)	(−0.0956; −0.7428; 0.7103)
	$\vec{n_{3}}$ (m)	(−0.3519; −0.5967; 0.7212)	(−0.3478; −0.5897; 0.7128)
	t₃ (s)	4.1726	3.6525
4	$\vec{X_{4}}$ (m)	(0.3447; 0.67; 0.7163)	(0.3425; 0.654; 0.7143)
	$\vec{n_{4}}$ (m)	(0.5273; 0.4486; 0.7215)	(0.5337; 0.4540; 0.7302)
	t ₄ (s)	4.0056	3.5517

Additionally, by inspection of the coordinate values referred to as the variable $\vec{X}$ for the mentioned two approaches, we find that center position vectors $\vec{X_{m}}$ ( $m = 1, 2, 3, 4$ ) of the mobile platform for the obtained four solutions by the proposed approach are almost consistent with the numerical results of the corresponding vectors in the study of Gallardo-Alvarado et al.¹⁹ In a similar comparison, the norm orientation $\vec{n}$ of the mobile platform has the same conclusion as the position vector $\vec{X}$ . It is stated that the proposed improved elimination approach is effective and reasonable.

Therefore, the simulation experiment has verified that the improved elimination algorithm in this article can effectively make up for the shortcomings of the large amount of computation in the study of Gallardo-Alvarado et al.^19,21 The scheme may be a feasible one to implement mechanics or dynamics analysis of the manipulator in the future.

Conclusions

In this article, we emphasize on an improved elimination algorithm with improving computational efficiency, after analyzing the geometry relationship of triangular-topology structure for a class of asymmetrical PMs and applying parameters transformation to the forward kinematics. Further, compared with the screw approach by Gallardo-Alvarado,¹⁹ the computation efficiency of the proposed approach is improved by 16.27%. Finally, an example of the VGT manipulator is given in order to prove the efficacy of the proposed method.

Footnotes

Acknowledgements

The authors are grateful to the anonymous reviewers for their helpful comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article is jointly supported by the Natural Science Foundation of China (61175028 and 61374161) and the Key Project of Natural Science Foundation of Shanghai (16JC1401100).

Appendix 1

The derivation process of the angle expression of the vertex B₂ is shown as follows.

Main derivation process: Here, joint center B₂ will be given in a set of generalized coordinates by constituting several proper auxiliary lines in the reference frame system $O_{1} - X_{1} Y_{1} Z_{1}$ . Since the points A₁ and O₁ are connected, this newborn line A₁O₁ will interconnect with sideline C₁B₁ at point H₂. O_b is taken to be the midpoint of sideline C₁B₁. Through the vertex B₂, an auxiliary line (X₁-axis) is created to be perpendicular to triangle $Δ A_{1} B_{1} C_{1}$ at point I₂, which falls through the point E₂ in the extension line E₂O_b, as shown in Figure 5.

From Figure 5, several auxiliary lines positioned the coordinate of points E₂, O_b, and I₂ are given with the cross longeron face angles θ₂. The detail derivative of auxiliary line analysis will be represented by the following fourth steps:

Step 1. Providing the coordinate of points H₂, F₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ by using the similarity of right-angled triangles.

On the one hand, the coordinate of points E₂, O_b displayed in Figure 6 are consistent with respect to the same geometry positions as drawn in Figure 5, since the auxiliary lines of construction are not altered at the plane of $Δ A_{1} B_{1} C_{1}$ .

From Figure 6, if the original point O₁ located at the orthocenter of $Δ A_{1} B_{1} C_{1}$ , then pedals H₂ and H₃ would lie on lines B₁C₁ and A₁C₁, respectively. Without loss of generality, the auxiliary line H₂F₂ can be created by being orthogonal to perpendicular line B₁H₃ and passing through pedal F₂ along Y₁-axis.

Because of the vector $\vec{E_{2} O_{b}}$ paralleled to the vector $\vec{H_{2} F_{2}}$ and Z₁-axis, two triangles $R_{t} Δ A_{1} O_{1} H_{3}$ ∼ $R_{t} Δ A_{1} C_{1} H_{2}$ , we have

Taking consideration of collinearity of three joints A₁, O₁, and H₂, one writes

Let $H_{A} = | c sin B_{1} - b \csc B_{1} | \cos A_{1} | |$ , since three triangles among $R_{t} Δ H_{2} O_{1} F_{2}$ , $R_{t} Δ A_{1} O_{1} H_{3}$ , and $R_{t} Δ A_{1} C_{1} H_{2}$ included the equivalent angle as $H_{2} O_{1} F_{2}, A_{1} O_{1} H_{3}$ , and C ₁, respectively, are similar. In $R_{t} Δ H_{2} O_{1} F_{2}$ , one can get

and

By combining equations (36) and (37), the coordinate of point H₂ is expressed as

Step 2. Figuring out the coordinate of points E₂, O_b in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ , if the coordinate of points H₂, F₂ are given.

Since O_b is the midpoint of sideline C₁B₁, and five points B₁, H₂, F₂, O_b, and C₁ are collinear, then the vector $\vec{H_{2} O_{b}}$ is satisfied as

Since $R_{t} Δ B_{1} H_{2} F_{2} \sim R_{t} Δ B_{1} O_{b} E_{2}$ , the expression of the vector $\vec{E_{2} O_{b}}$ can be rewritten as

where $H_{B} = [a / (2 c)] sin C_{1} \csc B_{1} | sec B_{1} |$ .

In $R_{t} Δ B_{1} H_{2} F_{2}$

Moreover, the vector $\vec{B_{1} E_{2}}$ is duplicated

where $H_{C} = (a / 2) sin C_{1} | \cos C_{1} | | sec B_{1} |$ .

Since three points B₁, E₂, and F₂ are colinear, and once equations (40) to (42) are held, one obtains

where $H_{D} = (1 / 2) sin C_{1} | \cos C_{1} | | sec B_{1} | | a - 2 c | \cos B_{1} | |$ .

On abstracting equations (37) and (43) into the vector $\vec{O_{1} E_{2}}$ , one can obtain

Let $O_{1} = {(0, 0, 0)}^{T}$ ; arranging equations (41) and (44), the coordinate of point O_b is defined as

where H_A, H_B, and H_D can be seen in equations (36), (40), and (43), respectively.

Step 3. Calculating the coordinate of point I₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ , if the coordinate of points E₂, O_b are provided.

In Figure 6, several auxiliary lines are established. For instance, a line O_bU₁ paralleled to Y₁-axis is plotted by intersecting with the line A₁C₁ at point U₁. Since $\vec{O_{b} U_{1}} / / \vec{B_{1} H_{3}}$ in $R_{t} Δ B_{1} C_{1} H_{3},$ the vectors $\vec{B_{1} E_{2}}$ and $\vec{B_{1} H_{3}}$ are orthogonal to the vector $\vec{O_{b} E_{2}}$ and the vector $\vec{C_{1} H_{3}}$ , respectively, one writes

and thus, the quadrilateral $O_{b} E_{2} H_{3} U_{1}$ is parallelogram and $\vec{O_{b} U_{1}} / / \vec{B_{1} H_{3}}$ ; the vector $\vec{O_{b} U_{1}}$ is suggested

In $Δ B_{1} B_{2} C_{1}$ , two lines B₁B₂ and B₂C₁ are the equivalent lengths, known as M mm. From the definition of cross longeron face angles, the angle θ₂ is equivalent to the angle E₂O_bB₂, as shown in Figure 5.

As shown in Figure 7, we have 48

| \vec{B_{2} O_{b}} | = N_{2} = \sqrt{M^{2} - \frac{a^{2}}{4}}

According to the angle bisector theorem, $R_{t} Δ C_{1} G_{1} O_{b} \sim$ $R_{t} Δ H_{3} K G_{1}$ ∼ $R_{t} Δ E_{2} K O_{b}$ ∼ $R_{t} Δ I_{2} O_{b} G_{2}$ , and then, a cosine equation is given

and

From the Pythagorean theorem, the height line from vertex A₂ to a base platform in $R_{t} Δ I_{2} O_{b} B_{2}$ can be written using equation (49)

What’s more, since three joints E₂, O_b, and I₂ are collinear, and once equations (40) and (50) are held, the vector $\vec{E_{2} I_{2}}$ along Z₁-axis is satisfied as

where $I_{A} = H_{B} \cdot H_{A}$ and $I_{B} = N_{2} \cdot sin C_{1} \cdot | \cos (π - θ_{2}) |$ .

Considering equations (44) and (52), the coordinate of point I₂ can be expressed as

Step 4. Determining the coordinate of points B₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$ , if the coordinate of vertex I₂ is known.

Substituting equation (51) into equation (53), the coordinate of vertex B₂ will be measured as

where I_A, I_B, H_A, and H_D can be seen in equations (44) and (53), respectively.

Further, the angle expression of the joint vector B₂ can be stated as

Therefore, the derivation process of the angle expression of the vertex B₂ is completed. □

Appendix 2

Here, the derivation process of the coordinates of the vertex C₂ will be ignored for the sake of brevity. For the purpose of understanding well the context of this article, the specific angle coordinates of joint vectors $C_{2} = C_{2} (θ_{3})$ are given as follows

where $N_{3} = \sqrt{M^{2} - (b^{2} / 4)}$ . □

Appendix 3

The derivation of polynomial equations (29) to (31) is shown as follows. To make transcendental equations (26) to (28) be formulated by using the parameter transformation of angular, some angular parameters are selected as

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Forward kinematics analysis for a class of asymmetrical parallel manipulators

Abstract

Keywords

Introduction

FKA of a class of asymmetrical PMs

Description of the asymmetrical manipulator module

Forward kinematic problem for single asymmetrical PM module

Space position of joint center A2 in the absolute frame system O 1 − X 1 Y 1 Z 1

The angle expression of joint center A2 in the absolute frame system O 1 − X 1 Y 1 Z 1

The elimination approach for the FKA problem of the asymmetrical PM

Description of the elimination process

Proof

The improved elimination algorithm for the FKA problem

Simulation

Conclusions

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

Appendix 1

Appendix 2

Appendix 3

References

Space position of joint center A₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$

The angle expression of joint center A₂ in the absolute frame system $O_{1} - X_{1} Y_{1} Z_{1}$