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Abstract

The bogie 6-degree-of-freedom dynamic simulation test bench can simulate various operating conditions of the bogie through coordinated movement of the three parallel 6-degree-of-freedom motion platforms; however, the motion of the platforms will be interfered by some space obstacles. In order to obtain the collision-free workspace of the 6-degree-of-freedom motion platforms, a limit boundary–search method based on the collision-detection model is proposed. The structure of three parallel 6-degree-of-freedom motion platforms is analyzed, and its inverse kinematics model is established. Considering its internal constraints, the constraint models of the actuators’ stroke and the spherical hinges angle are established. Based on the collision-detection method between convex polygons, the minimum distance-detection models between platforms and obstacles as well as between actuators and obstacles are established. Then the limit boundary–search method is used to search for all boundary points that satisfy the constraint conditions. Finally, the correctness and feasibility of the proposed method and models are verified by some simulation examples. The proposed research avoids collisions effectively and provides a reference for the optimal design of the test bench.

Keywords

6-degree-of-freedom motion platform constraint collision detection limit boundary search workspace analysis

Introduction

Workspace analysis plays a significant role for parallel mechanisms. The shape, size, and volume of the workspace of the mechanism are limited by many factors.^1–2 For a long time, many scholars have performed extensive research on determining the workspace of parallel mechanisms.^3–4 In recent years, the research on the workspace characteristics of various parallel robots has become a hot topic.⁵

The workspace of a parallel robot is mainly solved by the graphical,⁶ analytical,⁷ and numerical methods.⁸ For examples, Abdel-Malek et al.⁹ obtained the workspace of the tandem-type manipulator by using the quadric surface. Liu et al.¹⁰ obtained the boundary point of the robot’s workspace using the Monte Carlo method. Castelli et al.¹¹ proposed a boundary-computation algorithm based on the binary representations of the manipulator workspaces. Bohigas et al.¹² introduced the method of determining the workspace boundary of general structural manipulator. Furthermore, the limit boundary–search method¹³ is a representative numerical method and is applied more in practice, for examples, Zhang et al.¹⁴ obtained the workspace of the seed implantation robot for prostate tumors by the limit boundary–search method. Zhang et al.¹⁵ obtained the workspace of a 6-prismatic-spherical-spherical (PSS) parallel machine tool by solving the limit value of the constraint condition. However, most studies only consider the influence of internal factors on their workspace and seldom consider the influence of external obstacles. Therefore, to avoid collisions and improve efficiency, some scholars have begun to study the collision-free workspace.^16–18 A notable example is an obstacle-free workspace of the parallel mechanisms with an interval-based approach obtained Farzanehkaloorazi et al.¹⁹

In this article, a bogie 6-degree-of-freedom (6-DOF) dynamic simulation test bench²⁰ was analyzed. It is a parallel mechanism composed of three 6-DOF motion platforms, there are 21 actuators to drive it. The test bench can simulate the motion pose of the bogie and complete the test of its performance parameters under various dynamic conditions.²¹ For convenience, some mating devices were installed on the test bench, like the beginning and the intermediate transition platforms and the braced frames. Thus, the workspace of the 6-DOF motion platform is bound by many factors, such as telescopic stroke, spherical hinge angle, and the surrounding obstacles. Therefore, it is essential to explore the collision-free workspace of 6-DOF motion platform and establish the internal constraint and collision-detection models between mechanisms and obstacles.

At present, two well-known collision-detection algorithms to calculate the minimum distance between convex bodies are Lin–Canny (LC) algorithm²² and Gilbert–Johnson–Keerthi (GJK) algorithm.²³ In order to optimize and improve the efficiency of detection, many scholars have carried out further research. Li et al.²⁴ proposed the projection separation surface method to achieve collision detection by constructing a quasi-projective separation surface. Zhang et al.²⁵ proposed a real-time collision-detection algorithm with a ridge-projection separation for convex polyhedrons. Jin et al.²⁶ proposed an algorithm to calculate the shortest distance between the static–static convex bodies, which could judge whether two objects collide or not. By means of the above algorithms, the problem of collision between static–static convex bodies can be solved. However, given that the actuators, platforms, and obstacles all belong to a convex body and the actuators also belong to a variable-size convex body, it is still necessary to solve the problem of collision between the moving and static objects.

This article proposed a limit boundary–search method to solve a collision-free workspace of the bogie 6-DOF dynamic simulation test bench. The inverse kinematics model of the 6-DOF motion platforms were established by the kinematics analysis. The interference models of the workspace of the test bench, such as the constraint models of the actuators’ stroke and the spherical hinges angle, the distance-detection model between vertical actuators of the upper platform and obstacles, as well as the distance-detection model between the vertical actuators of the lower platform and obstacles, were established. Based on the built models, the steps to solve the workspace by the limit boundary–search method were given. The results via the simulation runs reveal that it is feasible when used to solve the workspace of the mechanism.

Kinematics analysis

As shown in Figure 1, the bogie 6-DOF dynamic simulation test bench mainly includes three 6-DOF motion platforms, a mechanical frame, an electro-hydraulic vibration system, a bogie ferry assembly, and so on. The upper 6-DOF motion platform installed on the mechanical frame is used to load on the bogie, and the two lower 6-DOF motion platforms installed on the foundation are used to simulate the road condition of bogie operation. Every 6-DOF motion platform is driven by seven actuators, and each actuator is connected to the platform by a spherical joint.

Figure 1.

Overall structure of bogie dynamic simulation test bench.

For convenience, the local coordinate system was introduced into the kinematic analysis²⁷ of the test bench. The intersection of the transverse and vertical symmetrical section of the upper 6-DOF motion platform projected at the horizontal plane was selected as the origin of the global coordinate system. The global coordinate system O-XYZ was established, and its x, y, and z axes were the longitudinal direction, the transverse motion direction, and the vertical motion direction of the train, respectively. Next, three local coordinate systems were established. The directions of the x, y, and z axes of the local coordinate systems are as same as the global coordinate system, and the origins of the three local coordinate systems are shown in Figures 2 and 3.

Figure 2.

Left view of 6-DOF motion simulation platforms system.

Figure 3.

Front view of 6-DOF motion simulation platforms system.

The three 6-DOF motion platforms of the test bench have the same structure, and their pose inverse solution models are similar. Therefore, kinematic analysis was performed by taking the upper platform as an example. According to its geometric structure parameters, the coordinate vector of the upper hinge points of actuators 1.1–1.7 in the local coordinate system can be expressed as follows

A_{j} = [\begin{matrix} l_{4} & l_{4} & - l_{4} & - l_{4} & 0 & 0 & 0 \\ - h_{5} & h_{5} & h_{5} & - h_{5} & - h_{6} & h_{6} & h_{7} \\ v_{3} & v_{3} & v_{3} & v_{3} & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]

(1)

The coordinate vector of the lower hinge points of actuators 1.1–1.7 in the local coordinate system can be expressed as follows

B_{j} = [\begin{matrix} l_{4} & l_{4} & - l_{4} & - l_{4} & l_{5} & l_{5} & 0 \\ - h_{5} & h_{5} & h_{5} & - h_{5} & - h_{6} & h_{6} & h_{8} \\ v_{4} & v_{4} & v_{4} & v_{4} & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]

(2)

The coordinate vector of the lower hinge points of actuators 1.1–1.7 in the global coordinate system was expressed as follows

B_{j}^{'} = B_{j} + [\begin{matrix} 0 \\ 0 \\ H \\ 1 \end{matrix}] * Λ (1, 7)

(3)

where $Λ (1, 7) = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]$ , and H is the initial height of the local coordinate system relative to the global coordinate system.

The pose of the local coordinate system relative to the global coordinate system can be described as $q = (θ_{x}, θ_{y}, θ_{z}, q_{x}, q_{y}, q_{z})$ , the parameters of q are expressed as the rotational and translational variable quantity around and along X, Y, Z. The following T₁ is a homogeneous transformation matrix in rotating space

T_{1} = [\begin{matrix} c θ_{y} c θ_{z} & - c θ_{x} s θ_{z} + s θ_{x} s θ_{y} c θ_{z} & s θ_{x} s θ_{z} + c θ_{x} s θ_{y} c θ_{z} & q_{x} \\ c θ_{y} s θ_{z} & c θ_{x} c θ_{z} + s θ_{x} s θ_{y} s θ_{z} & - s θ_{x} c θ_{z} + c θ_{x} s θ_{y} s θ_{z} & q_{y} \\ - s θ_{y} & s θ_{x} c θ_{y} & c θ_{x} c θ_{y} & q_{z} \\ 0 & 0 & 0 & 1 \end{matrix}]

(4)

where c and s denote cos and sin, respectively.

The expansion amount $Δ L_{1 j}$ of the actuator j can be defined by the distance between upper and lower hinge point and its initial length $L_{1 j}^{0}$ , that is to say, the pose inverse solution model of the upper 6-DOF platform can be expressed as follows

Δ L_{1 j} = ‖ T_{1} \cdot A_{j} - B'_{j} ‖ - L_{1 j}^{0} (for j = 1, 2, \dots, 7)

(5)

Constraint factors of the workspace

Actuators’ expansion amount

The test bench realized the movement of the platforms by controlling the expansion of the actuators. The design range of the actuators’ stroke was constant due to the restriction of its mechanical structure. Therefore, the stroke of the actuator needs to meet the following length constraints

Δ L_{ij min} \leq Δ L_{ij} \leq Δ L_{ij max}

(6)

where, $Δ L_{ij min}$ and $Δ L_{ij max}$ are the minimum and maximum expansions of the actuator j of the platform i, respectively.

In this article, the design range of all actuators’ expansion amount is $[- 300 mm, 300 mm]$ . If one or more actuators reach the limit position, the motion reference points of the platform reach the boundary of the workspace.

Spherical hinges’ angle

The working angle of spherical hinge at both ends of the actuator can only be within a certain range because of its structural characteristics. The geometric structure of the spherical hinge is shown as Figure 4. The workspace of the spherical hinge is a cone with the center of the spherical hinge as the apex. Its axial maximum swing angle along the shaft pin is $δ_{max}$ , and its radial maximum swing angle along the shaft pin is $ε_{max}$ . Since $δ_{max}$ is less than $ε_{max}$ . The real-time angle $δ_{ij}$ of the upper spherical hinge and the real-time angle $δ'_{ij}$ of the lower spherical hinge should meet the constraints, such that $δ_{ij} \leq δ_{max}$ and $δ'_{ij} \leq δ_{max}$ .

Figure 4.

Geometric model and structural sketch of the spherical hinge: (a) geometric model and (b) structural sketch.

With one of the actuators as an example, it is known that the distance between the center point of the lower spherical hinge and the borderline of the hinged support is L_s = 65 mm; the distance between the center line of the lower spherical hinge and the inner side of the hinge support is L_v = 72 mm; and the width of the spherical hinge connecting rod is D = 80 mm; then the maximum rotation angle of spherical hinge is $δ_{max} = 23 . 57^{°}$ .

As shown in Figure 5, it is assumed that $A_{ij}$ and $A_{ij}^{*}$ are the coordinates of the upper hinge point at the initial position and the position after exercise, respectively. Then, the real-time vector of the actuator’s center line can be expressed as

\vec{{B'}_{ij} A_{ij}^{*}} = A_{ij}^{*} - B'_{ij} = T_{i} \cdot A_{ij} - B'_{ij}

(7)

where i expresses numbers of the three moving platforms; j expresses the numbers of the seven actuators; $T_{i}$ expresses the rotation matrix of the platform i; and $B'_{ij}$ expresses the coordinate of the lower hinge points of actuator j of platform i.

Figure 5.

Angle diagram of the spherical hinge.

The initial vector between the upper and the lower hinge point can be expressed as

\vec{{B'}_{ij} A_{ij}} = A_{ij} - B'_{ij}

(8)

According to the angle theorem of space vector, the included angle $δ_{ij}$ between the real-time center line of the actuator j and the center line of the lower spherical hinge at initial position can be obtained by formula (9)

δ_{ij} = ar c \cos (\frac{\vec{{B'}_{ij} A_{ij}^{*}} • \vec{{B'}_{ij} A_{ij}}}{| {B'}_{ij} A_{ij}^{*} | | {B'}_{ij} A_{ij} |})

(9)

Then the lower spherical hinge should satisfy the constraint condition as follows

0 \leq δ_{ij} \leq δ_{max} (for i = 1, 2, 3 and j = 1, 2, \dots, 7)

(10)

During the movement of the platform, the upper spherical hinge support is always vertical to the upper platform. As shown in Figure 5, it is assumed that $C_{ij}$ and $C'_{ij}$ are the intersection points between the center line of the upper spherical hinge support and the top surface of the platform at the initial position and the position after exercise, respectively. Hence, according to the space vector angle theorem and the inverse cosine method, the included angle $δ'_{ij}$ between the real-time center line of the actuator j, and the center line of the upper spherical hinge can be calculated as follows

δ'_{ij} = arc \cos (\frac{\vec{A_{ij}^{*} {C'}_{ij}} • \vec{{B'}_{ij} A_{ij}^{*}}}{| A_{ij}^{*} {C'}_{ij} | | {B'}_{ij} A_{ij}^{*} |})

(11)

Then the upper spherical hinge should satisfy the constraint condition as follows

0 \leq δ'_{ij} \leq δ_{max} (for i = 1, 2, 3 and j = 1, 2, \dots, 7)

(12)

Interference between platforms and obstacles

In order to ensure that the test bogie or test vehicle can be pushed into the test station, the transition device has been designed. It includes the beginning and the intermediate transition platforms that can move along the longitudinal direction and the intermediate support platform with the guide rails, as shown in Figure 6. If the amplitude of the side roll and translation of the moving platform exceed the space limit, the motion platforms and the transition device may have interference problems.

Figure 6.

Structural model of the lower platforms and the transition device.

Therefore, in order to avoid collisions in the process of movement, the minimum distance between platforms and obstacles should be greater than zero at any time. This can be obtained by using the algorithm for calculating the shortest distance between convex polyhedrons in article.²⁶

From the perspective of space mathematics, if both points $x_{1}$ and $x_{2}$ are two different points in the spatial polyhedron W, any point $λ_{1} x_{1} + λ_{2} x_{2} (λ_{i} \geq 0, λ_{1} + λ_{2} = 1)$ on the line segment belongs to W, then the polyhedron W is a convex polyhedron, otherwise it is a concave body. If $χ_{i} (i = 1, \dots, n)$ are $n$ different points in a given space $R^{3}$ , then the point set $χ = \sum_{i = 1}^{n} λ_{i} χ_{i}$ is a convex combination of $χ_{i}, \dots, χ_{n}$ , where the coefficient $λ_{i}$ satisfies the following constraint

\begin{matrix} s . t λ_{1} + λ_{2} + λ_{3} + λ_{4} + \dots + λ_{n} = 1 \\ λ_{i} \geq 0 \end{matrix}

(13)

In general, a spatial convex polyhedron can be represented by a convex combination of convex polyhedral vertices. If the convex polyhedral vertex set is $χ = {χ_{1}, χ_{2}, χ_{3}, \dots, χ_{n}}$ , then any point on the convex polyhedron can be represented by a convex combination of vertex sets.

In this article, the intermediate transition platform is simplified as the convex polyhedron U with eight vertices. Supposing that $η_{1}, η_{2}, \dots, η_{8} (η_{1} + η_{2} + \dots + η_{8} = 1, η_{1}, η_{2}, \dots, η_{8} \geq 0)$ are its function parameters and ${u_{1}, u_{2}, u_{3}, \dots, u_{8}}$ is its vertex set, then the coordinate representation model of any point on it can be expressed as follows

U = η_{1} u_{1} + η_{2} u_{2} + \dots + η_{8} u_{8}

(14)

For the beginning transition platform which is simplified as the convex polyhedron S, supposing that $μ_{1}, μ_{2}, \dots, μ_{8} (μ_{1} + μ_{2} + \dots + μ_{8} = 1, μ_{1}, μ_{2}, \dots, μ_{8} \geq 0)$ are its function parameters and ${s_{1}, s_{2}, s_{3}, \dots, s_{8}}$ is its vertex set, then the coordinate representation model of any point on it can be expressed as follows

S = μ_{1} s_{1} + μ_{2} s_{2} + \dots + μ_{8} s_{8}

(15)

For the intermediate support platform which is simplified as the convex polyhedron E, supposing that $σ_{1}, σ_{2}, \dots, σ_{8} (σ_{1} + σ_{2} + \dots + σ_{8} = 1, σ_{1}, σ_{2}, \dots, σ_{8} \geq 0)$ are its function parameters and ${e_{1}, e_{2}, e_{3}, \dots, e_{8}}$ is its vertex set, then the coordinate representation model of any point on it can be expressed as follows

E = σ_{1} e_{1} + σ_{2} e_{2} + \dots + σ_{8} e_{8}

(16)

For computational convenience, the platform i is simplified to the union of three convex hexahedrons, as shown in Figure 7. The three convex hexahedrons of platform i are represented as $W^{i 1}$ , $W^{i 2}$ , and $W^{i 3}$ , and their vertex sets are represented as ${w_{11}^{i}, w_{12}^{i}, \dots, w_{18}^{i}}$ , ${w_{21}^{i}, w_{22}^{i}, \dots, w_{28}^{i}}$ , and ${w_{31}^{i}, w_{32}^{i}, \dots, w_{38}^{i}}$ . When the platform i is moving, the spatial mathematical models of the convex polyhedrons in the global coordinate system can be expressed as follows

{\begin{matrix} W^{i 1} = T_{i} [ξ_{1}^{i} w_{11}^{i} + ξ_{1}^{i} w_{12}^{i} + \dots + ξ_{8}^{i} w_{18}^{i}] + o_{i} \\ W^{i 2} = T_{i} [ξ_{9}^{i} w_{21}^{i} + ξ_{10}^{i} w_{22}^{i} + \dots + ξ_{16}^{i} w_{28}^{i}] + o_{i} \\ W^{i 3} = T_{i} [ξ_{17}^{i} w_{31}^{i} + ξ_{18}^{i} w_{32}^{i} + \dots + ξ_{24}^{i} w_{38}^{i}] + o_{i} \end{matrix} (for i = 2, 3)

(17)

where T_i is the homogeneous transformation matrix of the platform i in rotation space, $o_{i}$ is the coordinate representation of the origin i of the local coordinate system relative to the global coordinate system, $ξ_{1}^{i}, ξ_{2}^{i}, \dots, ξ_{24}^{i} (ξ_{1}^{i} + ξ_{2}^{i} + \dots + ξ_{8}^{i} = 1, ξ_{9}^{i} + ξ_{10}^{i} + \dots + ξ_{16}^{i} = 1, ξ_{17}^{i} + ξ_{18}^{i} + \dots + ξ_{24}^{i} = 1, ξ_{1}^{i}, ξ_{2}^{i}, \dots, ξ_{24}^{i} \geq 0)$ are the function parameters of the convex hexahedrons of the platform i.

Figure 7.

Split model of the platform i.

Therefore, the distance-detection models between the platform 2 and the intermediate transition platform with the intermediate support platform, respectively, can be expressed as follows

\begin{array}{l} {\begin{matrix} d_{W^{2 m}, U} = M i n (\sqrt{{(x_{W^{2 m}} - x_{U})}^{2} + {(y_{W^{2 m}} - y_{U})}^{2} + {(z_{W^{2 m}} - z_{U})}^{2}}) \\ d_{W^{2 m}, E} = M i n (\sqrt{{(x_{W^{2 m}} - x_{E})}^{2} + {(y_{W^{2 m}} - y_{E})}^{2} + {(z_{W^{2 m}} - z_{E})}^{2}}) \end{matrix} \\ (for m = 1, \dots, 9) \end{array}

(18)

The distance-detection model between the platform 3 and the beginning transition platform can be expressed as follows

d_{W^{3 m}, S} = M i n (\sqrt{{(x_{W^{3 m}} - x_{S})}^{2} + {(y_{W^{3 m}} - y_{S})}^{2} + {(z_{W^{3 m}} - z_{S})}^{2}})

(19)

When the platforms are moving, it is necessary to make sure the distances ( $d_{W^{2 m}, U}$ , $d_{W^{2 m}, E}$ , and $d_{W^{3 m}, S}$ ) are greater than zero.

Interference between actuators and obstacles

As shown in Figure 6, there may be movement interference between the vertical actuator j of the lower motion simulation platform i and the intermediate support platform during exercise. In addition, the main crossbeam is large in size and complex in structure, so it is easy to cause movement interference between the main crossbeam and the vertical actuator j of the upper platform, as shown in Figure 8.

Figure 8.

Position between the vertical actuators and main crossbeam.

Therefore, in order to avoid collisions in the movement, the minimum distance between actuators and obstacles should be greater than zero at any time. Based on the algorithm for calculating the minimum distance between convex polyhedrons, the distance-detection model between the vertical actuators of the upper platform and obstacles, as well as the distance-detection model between the lower platforms and obstacles, are established.

The distance-detection model between the vertical actuators of the upper platform and obstacles

In this article, we ignored the geometrical characteristics which have no influence on the collision and simplified the actuator as a simple polyhedron with nine vertices. As shown in Figure 9, this polyhedron’s upper vertex stands for the origin of the upper spherical hinge, which can be represented by the point $p_{1 j}$ ; the intersection of the cylinder’s upper surface and central axis is represented as the point $D_{1 j}$ ; the intersection of the cylinder’s lower surface and central axis is represented as the point $C_{1 j}$ ; and the vertex set of this convex polyhedron is represented as ${p_{1 j}, p_{2 j}, \dots, p_{9 j}} (j = 1, \dots, 4)$ .

Figure 9.

Kinematic simplified model of vertical actuator j of the upper platform.

Supposing the initial vector between the upper and lower hinge points of actuator j is $b_{j} = (b_{j 1}, b_{j 2}, b_{j 3}, 0)$ , which can be obtained by formula (8); and when the upper platform moved to a certain position, supposing the real-time vector of the center line of actuator j is $a_{j} = (a_{j 1}, a_{j 2}, a_{j 3}, 0)$ , which can be obtained by formula (7); therefore, the angle $α_{1 j}$ between the center axis of the vertical actuator j before and after the movement can be obtained by formula (20)

α_{1 j} = arc \cos (\frac{b_{j} • a_{j}}{| b_{j} | | a_{j} |}) (for j = 1, \dots, 4)

(20)

Thus, the following relationship can exist

z_{p_{tj}} - z_{C_{1 j}} = \pm \frac{1}{2} d_{1} \sin α_{1 j} (t = 2, \dots, 9)

(21)

where $d_{1}$ represented the section width of the cylinder.

The first component $a_{j 1}$ in vector $a_{j} = (a_{j 1}, a_{j 2}, a_{j 3}, 0)$ has two cases: one is $a_{j 1} = 0$ and the other is $a_{j 1} \neq 0$ . Next, for the two different cases, the coordinate of the vertices of actuator polyhedron j in the global coordinate system was calculated.

1. When $a_{j 1} = 0$ .

For the first case, the axis after motion is in the YOZ plane; then, according to the trigonometric function and the circular motion characteristic, the coordinate of intersection point $C_{1 j}$ in the global coordinate system can be expressed by formula (22)

{\begin{matrix} x_{C_{1 j}} = x_{p_{1 j}} \\ y_{C_{1 j}} = - d_{C_{1 j}} \sin α_{1 j} + y_{p_{1 j}} \\ z_{C_{1 j}} = - d_{C_{1 j}} \cos α_{1 j} + z_{p_{1 j}} \end{matrix} (for j = 1, \dots, 4)

(22)

where $d_{C_{1 j}}$ represented the distance between the vertex $p_{1 j}$ and intersection $C_{1 j}$ .

Therefore, the coordinate of the vertices $(p_{6 j}, p_{7 j}, p_{8 j}, p_{9 j})$ in the global coordinate system can be expressed as follows

P'_{j} = [\begin{matrix} x_{C_{1 j}} \\ y_{C_{1 j}} \\ z_{C_{1 j}} \\ 1 \end{matrix}] \times Λ (1, 4) + G_{1}

(23)

where $Λ (1, 4) = [\begin{matrix} 1 & 1 & 1 & 1 \end{matrix}]$ ; $G_{1}$ represents the position matrix of the vertices $(p_{6 j}, p_{7 j}, p_{8 j}, p_{9 j})$ when $a_{j 1} = 0$ , which is relative to the intersection point $C_{1 j}$ and can be obtained by formula (24)

G_{1} = \frac{d_{1}}{2} [\begin{matrix} 1 & - 1 & - 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ - \sin α_{1 j} & - \sin α_{1 j} & \sin α_{1 j} & \sin α_{1 j} \\ 0 & 0 & 0 & 0 \end{matrix}]

(24)

In the same way, the coordinates of $(p_{2 j}, p_{3 j}, p_{4 j}, p_{5 j})$ , in the global coordinate system when $a_{j 1} = 0$ , can be obtained.

2. When $a_{j 1} \neq 0$ .

The vector between point $C_{1 j}$ and the origin of the spherical hinge is parallel to the axis vector, according to the necessary and sufficient condition of vector parallelism, the existence of unique real number $λ$ satisfies the following condition

(x_{C_{1 j}} - x_{p_{1 j}}, y_{C_{1 j}} - y_{p_{1 j}}, z_{C_{1 j}} - z_{p_{1 j}}) = λ (a_{j 1}, a_{j 2}, a_{j 3})

(25)

The distance between point $C_{1 j}$ and the origin of the spherical hinge can be expressed as

(x_{C_{1 j}} - x_{p_{1 j}})^{2} + (y_{C_{1 j}} - y_{p_{1 j}})^{2} + (z_{C_{1 j}} - z_{p_{1 j}})^{2} = {d_{C_{1 j}}}^{2}

(26)

The coordinate of intersection point $C_{1 j}$ in the global coordinate system can be obtained by combining formulas (25) and (26), as shown below

{\begin{matrix} x_{C_{1 j}} = x_{p_{1 j}} + ϕ_{1} a_{j 1} \\ y_{C_{1 j}} = y_{p_{1 j}} + ϕ_{1} a_{j 2} \\ z_{C_{1 j}} = z_{p_{1 j}} + ϕ_{1} a_{j 3} \end{matrix} (for j = 1, \dots, 4)

(27)

where $ϕ_{1} = \sqrt{({d_{C_{1 j}}}^{2}) / ({(a_{j 1})}^{2} + {(a_{j 2})}^{2} + {(a_{j 3})}^{2})}$ .

Since the point $p_{tj} (t = 6, \dots, 9; j = 1, \dots, 4)$ is in the same plane as the point $C_{1 j}$ , the following relationship can be obtained according to the point normal form equation of a plane

a_{j 1} (x_{p_{tj}} - x_{C_{1 j}}) + a_{j 2} (y_{p_{tj}} - y_{C_{1 j}}) + a_{j 3} (z_{p_{tj}} - z_{C_{1 j}}) = 0

(28)

When $a_{j 1} \neq 0$ , formula (28) is further organized as formula (29)

x_{p_{tj}} - x_{C_{1 j}} = \frac{- a_{j 2} (y_{p_{tj}} - y_{C_{1 j}}) - a_{j 3} (z_{p_{tj}} - z_{C_{1 j}})}{a_{j 1}}

(29)

The distance formula between point $p_{tj} (t = 6, \dots, 9; j = 1, \dots, 4)$ and point $C_{1 j}$ can be obtained as follows

(x_{p_{tj}} - x_{C_{1 j}})^{2} + (y_{p_{tj}} - y_{C_{1 j}})^{2} + (z_{p_{tj}} - z_{C_{1 j}})^{2} = {(\frac{\sqrt{2}}{2} d_{1})}^{2}

(30)

Therefore, the coordinate of the vertices $(p_{6 j}, p_{7 j}, p_{8 j}, p_{9 j})$ in the global coordinate can be expressed as follows

P'_{j} = [\begin{matrix} x_{C_{1 j}} \\ y_{C_{1 j}} \\ z_{C_{1 j}} \\ 1 \end{matrix}] \times Λ (1, 4) + G'_{1} (for j = 1, \dots, 4)

(31)

where $G'_{1}$ represents the position matrix of the vertices $(p_{6 j}, p_{7 j}, p_{8 j}, p_{9 j})$ when $a_{j 1} \neq 0$ , which is relative to the intersection point $C_{1 j}$ and can be obtained by formula (32)

G_{1}^{'} = [\begin{matrix} \frac{a_{j 2}}{a_{j 1}} M_{1} + \frac{a_{j 2}}{a_{j 1}} N_{1} - \frac{a_{j 3}}{a_{j 1}} U_{1} & - \frac{a_{j 2}}{a_{j 1}} M_{1} - \frac{a_{j 2}}{a_{j 1}} N_{1} + \frac{a_{j 3}}{a_{j 1}} U_{1} & \dots \\ M_{1} + N_{1} & M_{1} + N_{1} & \dots \\ - U_{1} & - U_{1} & \dots \\ 0 & 0 & \dots \end{matrix} \begin{matrix} \frac{a_{j 2}}{a_{j 1}} M_{1} + \frac{a_{j 2}}{a_{j 1}} N_{1} - \frac{a_{j 3}}{a_{j 1}} P_{1} \\ - M_{1} - N_{1} \\ U_{1} \\ 0 \end{matrix}]

(32)

where $M_{1} = (a_{j 1} d_{1} \sqrt{2 ({a_{j 1}}^{2} + {a_{j 2}}^{2}) - ({a_{j 1}}^{2} + {a_{j 2}}^{2} + {a_{j 3}}^{2}) \sin^{2} α_{1 j}}) / (2 ({a_{j 1}}^{2} + {a_{j 2}}^{2}))$ ; $N_{1} = (a_{j 2} a_{j 3} d_{1} \sin α_{1 j}) / (2 ({a_{j 1}}^{2} + {a_{j 2}}^{2}))$ ; and $U_{1} = (1 / 2) d_{1} \sin α_{1 j}$ .

In the same way, the coordinates of $(p_{2 j}, p_{3 j}, p_{4 j}, p_{5 j})$ in the global coordinate system when $a_{j 1} \neq 0$ can be obtained.

In summary, the spatial mathematical representation of the vertices in the global coordinate system can be obtained by formulas (22)–(32). Supposing that $τ_{1 j}, τ_{2 j}, \dots, τ_{9 j} (τ_{1 j} + τ_{2 j} + \dots + τ_{9 j} = 1, τ_{1 j}, τ_{2 j}, \dots, τ_{9 j} \geq 0)$ are the function parameters of the vertical actuator j simplified model of the upper platform, then the coordinate representation model of any point on it can be expressed as follows

P_{j} = τ_{1 j} p_{1 j} + τ_{2 j} p_{2 j} + \dots + τ_{9 j} p_{9 j}

(33)

Supposing that ${f_{1}, f_{2}, \dots, f_{8}}$ is the vertex set of the crossbeam simplified convex polyhedron and $ζ_{1}, ζ_{2}, \dots, ζ_{8} (ζ_{1} + ζ_{2} + \dots + ζ_{8} = 1, ζ_{1}, ζ_{2}, \dots, ζ_{8} \geq 0)$ are its function parameters vertex set, then the coordinate representation model of any point on it can be expressed as follows

F = ζ_{1} f_{1} + ζ_{2} f_{2} + \dots + ζ_{8} f_{8}

(34)

Therefore, the distance-detection model between the vertical actuator j of the upper platform and the crossbeam can be expressed as follows

{d_{P_{j}}}_{, F} = Min (\sqrt{{(x_{P_{j}} - x_{F})}^{2} + {(y_{P_{j}} - y_{F})}^{2} + {(z_{P_{j}} - z_{F})}^{2}})

(35)

The distance-detection model between the vertical actuators of the lower platform and obstacles

As shown in Figure 10, we simplified the vertical actuator of the lower platform as a convex polyhedron with nine vertices, the polyhedron’s lower vertex stands for the origin of the lower spherical hinge, which can be represented by $p_{1 j}^{i}$ , and its vertex set is represented as ${p_{1 j}^{i}, p_{2 j}^{i}, \dots, p_{9 j}^{i}} (i = 2, 3, j = 1, \dots, 4)$ . With the actuator moved to a certain position, the intersection of cylinder’s lower surface and central axis moved to point $D'_{ij}$ from point $D_{ij}$ , and the intersection of the force sensor’s upper surface and the central axis moved to point $C'_{ij}$ from point $C_{ij}$ , so the distance between the origin of the spherical hinge and the cylinder’s lower surface had no change; however, the distance between the origin of the spherical hinge and the force sensor’s upper surface increased about $Δ L_{ij}$

d_{{C'}_{ij}} = d_{C_{ij}} + Δ L_{ij} (for i = 2, 3 and j = 1, \dots, 4)

(36)

Figure 10.

Vertical actuator’s kinematic simplified model of the lower platform.

According to formulas (5) and (36), we can know the distance $d_{{C'}_{ij}}$ between the origin of the spherical hinge j and the force sensor’s upper surface during the movement of the platform i. Then, referring to formulas (20)–(32), the coordinates of the vertices $(p_{6 j}^{i}, p_{7 j}^{i}, p_{8 j}^{i}, p_{9 j}^{i})$ in the global coordinate can be expressed as follows

P^{i^{'}} = {\begin{matrix} [\begin{matrix} x_{p_{j 1}^{i}} \\ - (d_{{C^{'}}_{i j}} + Δ L_{i j}) \sin α_{i j} + y_{p_{j 1}^{i}} \\ - (d_{{C^{'}}_{i j}} + Δ L_{i j}) \cos α_{i j} + z_{p_{j 1}^{i}} \\ 1 \end{matrix}] \times Λ (1, 4) + G_{2} & (a_{j 1}^{i} = 0) \\ [\begin{matrix} x_{p_{j 1}^{i}} + ϕ_{2} (a_{j 1}^{i}) \\ y_{p_{j 1}^{i}} + ϕ_{2} (a_{j 2}^{i} \\ y_{p_{j 1}^{i}} + ϕ_{2} (a_{j 3}^{i}) \\ 1 \end{matrix}] \times Λ (1, 4) + {G^{'}}_{2} & (a_{j 1}^{i} \neq 0) \end{matrix}

(37)

where $ϕ_{2} = \sqrt{({(d_{{C'}_{ij}} + Δ L_{ij})}^{2}) / ({{(a}_{j 1}^{i})}^{2} + {(a_{j 2}^{i})}^{2} + {(a_{j 3}^{i})}^{2})}$ .

The coordinates of the vertices $(p_{2 j}^{i}, p_{3 j}^{i}, p_{4 j}^{i}, p_{5 j}^{i})$ in the global coordinate can be obtained by the same way. Supposing that $τ_{1 j}^{i}, τ_{2 j}^{i}, \dots, τ_{9 j}^{i} (τ_{1 j}^{i} + τ_{2 j}^{i} + \dots + τ_{9 j}^{i} = 1, τ_{2 j}^{i}, \dots, τ_{9 j}^{i} \geq 0)$ are the vertical actuator j simplified model’s function parameters of the lower platform i, then the coordinate representation model of any point on it can be expressed as follows

P_{j}^{i} = τ_{1 j}^{i} p_{9 j}^{i} + τ_{9 j}^{i} p_{9 j}^{i} + \dots + τ_{9 j}^{i} p_{9 j}^{i}

(38)

Supposing that ${e_{1}, e_{2}, \dots, e_{8}}$ is the vertex set of the simplified convex polyhedron of intermediate support and $σ_{1}, σ_{2}, \dots, σ_{8} (σ_{1} + σ_{2} + \dots + σ_{8} = 1, σ_{1}, σ_{2}, \dots, σ_{8} \geq 0)$ are its function parameters, then the coordinate representation model of any point on it can be expressed as follows

E = σ_{1} e_{1} + σ_{2} e_{2} + \dots + σ_{8} e_{8}

(39)

Therefore, the distance-detection model between the vertical actuator j of the lower platform i and the intermediate support platform can be expressed as follows

{d_{P_{j}^{i}}}_{, E} = Min (\sqrt{{(x_{P_{j}^{i}} - x_{E})}^{2} + {(y_{P_{j}^{i}} - y_{E})}^{2} + {(z_{P_{j}^{i}} - z_{E})}^{2}})

(40)

In summary, in order to ensure that the actuators avoid the obstacles effectively, ${d_{P_{j}}}_{, F}$ and ${d_{P_{j}^{i}}}_{, E}$ should be greater than zero at the movement of the motion platforms.

Workspace analysis

Workspace search algorithm

According to the inverse kinematics model and structural parameters of the parallel motion platforms, all the constraint models can be calculated when the motion platform maintains a certain pose. If the results of the calculations satisfied all the constraint conditions, it can be judged that the position is within the workspace. Thus, if the boundary points on all sections can be searched, the workspace of the 6-DOF motion platform is exactly the space surrounded by these points.

The limit boundary–search method to solve the platform’s fixed pose workspace consists of the six steps as follows:

Step 1: Determine the initial position of the 6-DOF motion platform and give it a fixed pose $(θ_{x}, θ_{y}, θ_{z})$ .

Step 2: According to the bar length constraint of the vertical actuators of the test bench, the lowest point $Z_{min}$ and the highest point $Z_{max}$ of the search space can be determined.

Step 3: The search space is divided into k differential spaces by using a planar cluster that is parallel to the XOY plane with a thickness about $Δ z = (Z_{max} - Z_{min}) / k$ , and then we start to search step by step along the z axis. When searching for step i along the z axis, the value of Z can be expressed as follows

Z = Z_{min} + (i - 1) * Δ z

(41)

Step 4: Each tiny subspace is treated as a section of the workspace, the polar coordinate search method is used to determine the workspace boundary of each section, throughout the search process, the polar angle $θ$ increments from 0 to $2 π$ and the polar radius $ρ$ is incremented from 0 to $ρ_{max}$ , and $ρ_{max}$ should be large enough to prevent workspace boundary search omission, which is due to inaccurate boundary estimation. According to the transformation relationship between polar coordinate and rectangular coordinate, the rectangular coordinates $x_{m, j}$ and $y_{m, j}$ of the searching points can be obtained by formula (42) when step m of the polar radius search and step j of the polar angle search are being carried out

{\begin{matrix} x_{m, j} = m Δ ρ \cos (j Δ θ) \\ y_{m, j} = m Δ ρ \sin (j Δ θ) \end{matrix}

(42)

Step 5: The pose point searched from one section plane is substituted into the pose inverse solution model and the spatial mathematical model of the moving platforms; all the constraint models can be calculated. If the calculations satisfied all the constraint conditions, the point is included within the workspace.

Step 6: Record all the points satisfying step 5, and the space composed by these points is the collision-free workspace of the 6-DOF motion platform.

Simulation and analysis

The fixed-pose workspace is the space formed by the translation quantity of the reference points along the x, y, and z axes, when the motion platform maintains a certain pose. Based on the inverse kinematics model and the various constraint conditions, the position workspace of platform i can be obtained with the limit boundary–search method. Factors affecting the workspace of platform 1 are the actuators’ expansion amount, the upper and lower hinge angles, and the minimum distance between the vertical actuator j and the crossbeam, then the positions in the workspace should satisfy the constraints of formulas (6), (10), (12), and (35), as shown in the following

{\begin{matrix} Δ L_{1 j min} \leq Δ L_{1 j} \leq Δ L_{1 j max} (j = 1, 2, \dots, 7) \\ Min (\sqrt{{(x_{p_{1 j}} - x_{F})}^{2} + {(y_{p_{1 j}} - y_{F})}^{2} + {(z_{p_{1 j}} - z_{F})}^{2}}) \geq 0 \\ 0 \leq δ_{1 j} \leq δ_{max} \\ 0 \leq δ_{1 j}^{'} \leq δ_{max} \end{matrix}

(43)

It is known that the structural parameters of the upper platform and the crossbeam at the initial position. Then, the workspaces of platform 1 at the fixed pose $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 0)$ and $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 3^{°})$ can be searched and plotted by MATLAB, and the results were shown in Figures 11 and 12.

Figure 11.

Position workspace of platform 1 at pose $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 0)$ .

Figure 12.

Position workspace platform 1 at pose $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 3^{°})$ .

In Figure 11, the searched result show that the workspace of the upper platform has the following features: (1) The fixed-pose workspace is approximated as a hexahedron with a trapezoidal cross-section, which conforms to the theory that the longer the radius, and the longer the trajectory. (2) The four vertical actuators are symmetrically distributed on both sides of the main crossbeam, so the entire fixed-pose workspace is symmetrical about the XOZ section and the YOZ section, respectively.

By comparing Figures 11 and 12, we can see that the platform 1 has maximum position workspace at pose $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 0)$ . The workspace of the 6-DOF platform will change with the change of its pose angle, and the greater the absolute value of the pose angle, the smaller the workspace.

The factors affecting the workspace of the lower motion platforms are relatively complex. They include the actuators’ expansion amount, the upper and lower hinge angles, the beginning and the intermediate transition platforms, as well as the intermediate support platform. According to the constraint equations (6), (10), (12), (18), (19), and (40) described above, the workspace of platforms 2 and 3 at pose $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 0)$ can be obtained, and the results were shown in Figures 13 and 14.

Figure 13.

Position workspace of platform 2 at pose $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 0)$ .

Figure 14.

Position workspace of platform 3 at pose $(θ_{x}, θ_{y}, θ_{z}) = (0, 0, 0)$ .

Conclusion

This article presents a method for solving the collision-free workspace of the bogie 6-DOF dynamic simulation test bench. Through the kinematics analysis, the pose inverse solution model of the 6-DOF motion platform is established, which provides the basis for the further establishment of the motion constraint model and the minimum distance-detection model. Based on the collision-detection method between convex polygons, the minimum distance-detection models between platform and obstacles, as well as between actuators and obstacles are established, which can effectively prevent collisions. Based on all the models that have been built, the limit boundary–search method is introduced to solve the workspace of the 6-DOF platform, which lays the foundation for the optimal design of the motion platforms and the smooth progress of the bogie dynamic simulation test. The modeling method for variable-size object constrained by spherical hinge and the collision-detection model between dynamic and static polyhedrons has good reference for the spatial motion analysis and collision detection of other multi-degree-of-freedom mechanisms.

Footnotes

Handling Editor: Jose Ramon Serrano

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 work was financial supported by the National Science Foundation of China (NSFC) (grant no.: 51575232) and the National Science Foundation of Jilin Province, China (grant no.: R20160204018GX).

ORCID iD

Jiaojiao Zhuang

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Workspace analysis of the bogie 6-degree-of-freedom dynamic simulation test bench

Abstract

Keywords

Introduction

Kinematics analysis

Constraint factors of the workspace

Actuators’ expansion amount

Spherical hinges’ angle

Interference between platforms and obstacles

Interference between actuators and obstacles

The distance-detection model between the vertical actuators of the upper platform and obstacles

The distance-detection model between the vertical actuators of the lower platform and obstacles

Workspace analysis

Workspace search algorithm

Simulation and analysis

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References