Research on position analysis of a new partly-DOF parallel stable platform 1

Abstract

Based on POC set theory and SOC unit, a 1T3R parallel stable platform is proposed. Topology structure design of low-DOF parallel stable platform is studied. Both inverse and direct kinematics and Jacobean matrix of this mechanism are analyzed. Based on the structure of the whole platform, a three-dimensional solid model is established with Pro/Engineer software, and kinematics simulation analysis is performed in ADAMS. It establishes base for the application of this kind of parallel stable platform and the follow-up research for analysis of kinematics and workspace. The results of this study on the stability of the institutional platform have an important significance.

Keywords

Position analysis inverse and direct kinematics Jacobean matrix parallel stable platform three-dimensional solid model

1 Introduction

During the last decade, structure synthesis of robot mechanism has become one of the focuses for mechanism research. Parallel Robot Mechanisms, in which one or more parallel organs are included, is a new type of manufacturing mechanisms thriving in recent years of the world. Comparing with the series configuration, the parallel configuration has special advantage like better rigidity, better precision, higher speed and acceleration which in all manufacturing fields it enjoy wider application prospect. Partly degrees of freedom parallel robot mechanisms has become a significant tendency in the field of parallel configuration equipment development.

Comparing with 6-degrees of freedom, the partly degrees of freedom parallel robot mechanisms is more complicated in the kinematical features and this has results in serious lack of the latter. So to research on the partly degrees of freedom parallel configuration is of more significance and values both in theory and in the application. Besides, the performance of the parallel robot mechanisms is directly influenced by the factors like constraint chains’ configuration types and joints and the freedom degree configuration, precision and rigidity. Under such circumstances, the research on all types of sub-chains and their kinematical features is imperative.

Parallel robot mechanisms with fewer DOFs, usually two to five, are especially prospective because of their simpler structures and lower production costs. Many researchers have carried out extensive studies in parallel mechanisms with pure three-translation output. On one hand, new types of mechanisms have been developed. Delta robot (1998) is the first proposal of a 3-DOF parallel manipulator that produces spatial translation [1]. In 1991 a comprehensive enumeration of the numerous structural types of limbs for Delta-like robots was published [2]. In his patent (1992), Appleberry described a truly new translational parallel manipulator of structural type 3-URU [3] (U-universal joint made of two revolute joints RR). Later (1996), Tsai presented a 3-UPU translational platform [4], which is a small modification of the Appleberry device. These translational parallel manipulators are not overconstrained. They have three limbs, each limb having 5-DOF. Frisoli et al. gave an enumeration of non-overconstrained translational manipulators [5].

On the other hand, design and analysis including kinematics, singularity, tolerance, workspace, dynamics of these new mechanisms or their modifications have been studied extensively [6, 7]. These works suggest that developing parallel robot mechanisms with pure three-translation output which are structurally simple and easy to manufacture is significant both academically and practically.

In the paper, a new 1T3R parallel stable platform is proposed based on POC set theory and SOC unit as shown in references [8, 9]. And both inverse and direct kinematics of this mechanism is made. Then the Jacobean matrix is studied. An example is given, and nonlinear equations are listed. By using of Homotopy Continuation, some real solutions of the mechanism are found. Finally based on the stable platform for the actual situation, the prototype simulation analysis method is proposed, which provides a practical method of analysis for kinematic analysis. The paper makes the guide sense for the follow-up research and the application of the 1T3R parallel stable platform.

2 Topology structure design of the 1T3R platform

Figure 1 shows a 3SPS+1PS parallel platform. Three identical limbs connect the moving platform to the fixed base by spherical joints at points B_i and A_i, i = 1, 2, 3, 4, j = 1, 2, 3, 4, respectively. A spherical joint B₄ is located at the centric of the moving platform. And the limb OB₄ is perpendicular to the fixed base at the point O, the centric of the fixed base. Each limb consists of an upper member and a lower member connected by a prismatic joint. There four prismatic joints are used as inputs to the parallel platform. Ball screw or hydraulic jacks can vary the lengths of the prismatic joints to control the location of the moving platform.

Fig.1

1T3R parallel stable platform.

Noting that in Fig. 1, there are 9 links connected by 4 prismatic joints and 7 spherical joints. Using a Kutzbach Grűbler equation, the degrees of freedom of the 3SPS+1PS mechanism is calculated as $\begin{matrix} F & = & λ (n - j - 1) + \sum_{i} f_{i} \\ = & 6 (9 - 11 - 1) + (7 \times 3 + 4) = 7 \end{matrix}$

However, there are 3 passive degrees of freedom associated with the three SPS limbs. Therefore, the moving platform possesses 4 degrees of freedom.

Without losing generality, As shown in Fig. 1, we assume that a reference coordinate system P (u, v, w) is attached to the Center P (B₄) of the moving platform, and its u and v axes lie on the moving plane, the w-axis points up vertically to the moving platform at B₄. v-axis is in line with $\vec{B_{4} B_{1}}$ . The coordinate system O (x, y, z) is attached to the center of fixed base, its x and y axes lie on the fixed base, the z-axis points up vertically. y-axis is in line with $\vec{{OA}_{1}}$ . △B₁B₂B₃ and △A₁A₂A₃ are equilateral triangles.

3 Position analysis of the 1T3R platform

Defining three rotation angles α, β and γ as roll, pitch and yaw about the u, v and w axes. The rotation transformation matrix from moving platform to fixed base is $T = [\begin{matrix} C γ C β & C γ S β S α - S γ C α & C γ S β C α + S γ S α \\ S γ C β & S γ S β S α + C γ C α & S γ S β C α - C γ S α \\ - S β & C β S α & C β C α \end{matrix}]$ (1)

Where S is a shorthand notation for sin and C for cos. As a matter of convenience for the equation of position analysis, let $T = [\begin{matrix} u_{x} & v_{x} & w_{x} \\ u_{y} & v_{y} & w_{y} \\ u_{z} & v_{z} & w_{z} \end{matrix}]$ (2)

The nine orientation parameters have following constrained equations:

$\begin{matrix} u_{x}^{2} + u_{y}^{2} + u_{z}^{2} = 1, u_{x} v_{x} + u_{y} v_{y} + u_{z} v_{z} = 0, \\ w_{x}^{2} + w_{y}^{2} + w_{z}^{2} = 1, u_{x} w_{x} + u_{y} w_{y} + u_{z} w_{z} = 0, \\ v_{x}^{2} + v_{y}^{2} + v_{z}^{2} = 1, v_{x} w_{x} + v_{y} w_{y} + v_{z} w_{z} = 0 \end{matrix}$ (3)

The coordinate of a_i in fixed base and b_i in moving platform B are expressed as $a_{i} = [\begin{matrix} a_{ix} \\ a_{iy} \\ a_{iz} \end{matrix}], b_{i}^{B} = [\begin{matrix} b_{iu} \\ b_{iv} \\ b_{iw} \end{matrix}]$ (4)

They are derived as

$\begin{matrix} a_{1} & = & [\begin{matrix} 0 \\ r_{f} \\ 0 \end{matrix}], a_{2} = \frac{r_{f}}{2} [\begin{matrix} \sqrt{3} \\ - 1 \\ 0 \end{matrix}], a_{3} = \frac{r_{f}}{2} [\begin{matrix} - \sqrt{3} \\ - 1 \\ 0 \end{matrix}] \\ b_{1}^{B} & = & [\begin{matrix} 0 \\ r_{m} \\ 0 \end{matrix}], b_{2}^{B} = \frac{r_{m}}{2} [\begin{matrix} \sqrt{3} \\ - 1 \\ 0 \end{matrix}], b_{3}^{B} = \frac{r_{m}}{2} [\begin{matrix} - \sqrt{3} \\ - 1 \\ 0 \end{matrix}] \end{matrix}$ (5)

Where r_m and r_f are the radius of the moving platform and the fixed base respectively.

Then the loop-closure equation for the ith limb is: $A_{i} B_{i} = d_{i} = p + b_{i} - a_{i} = p + {Tb}_{i}^{B} - a_{i}$ (6)

By taking the dot product of vector A_iB_i with itself, the length of the ith limb is:

$\begin{matrix} d_{i}^{2} & = & {[p + {Tb}_{i}^{B} - a_{i}]}^{T} [p + {Tb}_{i}^{B} - a_{i}] \\ = & p^{T} p + {[b_{i}^{B}]}^{T} [b_{i}^{B}] + a_{i}^{T} a_{i} + 2 p^{T} [{Tb}_{i}^{B}] \\ - 2 p^{T} a_{i} - 2 {[{Tb}_{i}^{B}]}^{T} a_{i}, i = 1, 2, 3 \end{matrix}$ (7)

For the inverse kinematics, the vector p and rotation matrix T are given. And the limb lengths d_i, i = 1, 2, 3, are easy to be found.

For the direct kinematics, the limb lengths d_i are given, and the vector p [x, y, z, α, β, γ] ^T is to be found. It is easy to have the vector p = [0, 0, z] ^T = [0, 0, d₄] ^T. Based on the Equations (4–7) above, we obtain ${\begin{matrix} f (1) = - 2 r_{m} r_{f} v_{y} + 2 d_{4} r_{m} v_{z} + d_{4}^{2} + r_{m}^{2} + r_{f}^{2} - d_{1}^{2} \\ f (2) = - \frac{1}{2} r_{m} r_{f} (3 u_{x} - \sqrt{3} v_{x} + 3 u_{y} - \sqrt{3} v_{y}) \\ + d_{4} r_{m} (\sqrt{3} u_{z} - v_{z}) + d_{4}^{2} + r_{m}^{2} + r_{f}^{2} - d_{2}^{2} \\ f (3) = - \frac{1}{2} r_{m} r_{f} (3 u_{x} + \sqrt{3} v_{x} + \sqrt{3} u_{y} + v_{y}) \\ - d_{4} r_{m} (\sqrt{3} u_{z} + v_{z}) + d_{4}^{2} + r_{m}^{2} + r_{f}^{2} - d_{3}^{2} \end{matrix}$ (8)

So there are only nine scalar unknowns in T that are related to (α, β, γ). To find the (α, β, γ), only u_v, u_z and v_z need to be found.

In general, the solution for these nonlinear equations is not unique, see references [10 –14] for examples. Professor Liu Anxin presents an effective real continuation method which can find all real solutions to arbitrary nonlinear equations in real field [15].

4 Jacobean matrix

The vector of actuated joint positions is defined as $t = {[\begin{matrix} d_{1} & d_{2} & d_{3} & d_{4} \end{matrix}]}^{T}$ (9)

And the moving platform angular rate is defined as $ω_{m} = [\dot{α} \dot{β} \dot{γ}]^{T}$ (10)

Differentiating Equation (6) with respect to time, as follows: $\frac{d d_{i}}{dt} = \frac{d (p + b_{i} - a_{i})}{dt}$ (11)

With d a _i/dt = 0, it results in $d_{i} ω_{i} \times e_{i} + {\dot{d}}_{i} e_{i} = \dot{p} + ω_{m} \times b_{i}$ (12) where ω_i is the angular velocity of the ith limb and e_i is the unit vector pointing along the direction of A_iB_i. And they are expressed in the coordinate system O (x, y, z). e_i can be expressed: $e_{i} = \frac{d_{i}}{| d_{i} |}$ (13)

To eliminate ω_i, dot-multiplying both sides of Equation (12) by e_i and obtain $(b_{i} \times e_{i}) ω_{m} + e_{i} \dot{p} = {\dot{d}}_{i}$ (14)

Where i = 1, 2, 3, 4. Equation (14) can be expressed in matrix form: $JV = {\dot{t}}_{i}$ (15) where J and V have the form $J = [\begin{matrix} {(b_{1} \times e_{1})}^{T} & e_{1}^{T} \\ {(b_{2} \times e_{2})}^{T} & e_{2}^{T} \\ {(b_{3} \times e_{3})}^{T} & e_{3}^{T} \\ {(b_{4} \times e_{4})}^{T} & e_{4}^{T} \end{matrix}]$ (16) $V = [\begin{matrix} ω_{m} \\ \dot{p} \end{matrix}]$ (17)

Equation (16) is the Jacobean matrix of the parallel mechanism, and $\dot{t} = [{\dot{d}}_{1} {\dot{d}}_{2} {\dot{d}}_{3}]^{T}$ is the vector of the joint velocities.

5 Simulation analysis

Based on the relationship between the structure of the whole platform assembly, a three-dimensional solid model is established with Pro/Engineer software; and check components interference and collision; and then put 3-SPS (PS) parallel platform virtual prototype model into ADAMS; finally kinematics simulation analysis is performed inADAMS.

5.1 Building the virtual prototype model

Depending on the design of parallel platform, its structure parameters: the following platforms radius is 3240 mm; upper platform radius is 2228 mm; electric cylinder bottom bit length is 3345 mm; bottom level height is 2078 mm; adjacent the upper and lower hinge point distance are: 300 mm; max stroke: 1630 mm.

3-SPS (PS) parallel platform in Pro/Engineer software is built by the three-dimensional model assembly drawings, which is shown in Fig. 2.

Fig.2

A parallel model in Pro/E software platform.

3-SPS (PS) parallel stable platform is completed in Pro/Engineer software, which is imported into the Adams. Then, boundary conditions and constraints conditions are set up based on the position analysis of the platform, including material, quality, moving platform centroid location, inertia moment and the inertial reference point location. So that the virtual prototype model of kinematics simulation analysis is shown in Fig. 3.

Fig.3

The parallel stable platform model in Adams.

5.2 Inverse kinematics simulation

Inverse kinematics simulation is to seek stable platform inverse kinematics solution, which is variation regular of the length of each branch.

The stable platform is fixed into the earth, the centroid point of stable platform is choosed as the critical point, which is added with a three-dimensional excitation -Point Motion. Point excitation function equation is:

X Axis: RotX = pi/12* sin(time) ;

Y Axis: RotY = pi/12* sin(time) ;

Z Axis: RotZ = pi/12* sin(time) ;

Z Axis: TraZ = 20* cos(time) ;

Simulation parameters is that t = (2 * pi) s, step size = 0.01. Then we can get displacement, velocity and acceleration variation with time of the three hydraulic rod, the driver and the hydraulic cylinder, which are shown in Figs. 4 to 12.

Fig.4

The displacement – time curve of three hydraulic rod.

Fig.5

The velocity – time curve of three hydraulic rod.

Fig.6

The acceleration – time curve of three hydraulic rod.

Fig.7

The displacement – time curve of three drive joints.

Fig.8

The velocity – time curve of three drive joints.

Fig.9

The acceleration – time curve of three drive joints.

Fig.10

The displacement – time curve of three hydraulic cylinders.

Fig.11

The velocity – time curve of three hydraulic cylinders.

Fig.12

The acceleration – time curve of three hydraulic cylinders.

5.3 Direct kinematics simulation

The direct kinematics simulation of stable platform is to get the solution of the parallel kinematics, which is the displacement, velocity and acceleration variation regular of each branchplatform.

Adams simulation data curve is determined by a number of discrete data points, which are provided by the AKISPL command in Adams. And simulation parameters is that t = (2 * pi) s, step size = 0.01, the angular velocity-tme and the angular acceleration-time curves of centroid point moving X axis, Y axis and Z axis separately are shown in Figs. 13–15.

Fig.13

The curve of centroid point moving X axis.

Fig.14

The curve of centroid point moving Y axis.

Fig.15

The curve of centroid point moving Z axis.

6 Conclusion

In this paper, a new 1T3R parallel stable platform is proposed based on POC set theory and SOC unit and the position analysis and the Jacobean matrix of a kind of 1T3R parallel platform are studied. Then the virtual prototype simulation analysis is proposed based on the application of Pro/Engineer and Adams. The proposed method has many advantages: (1) It is to achieve the stable platform mechanism interference and collision checking, which is intuitive and clear analysis of structure design. (2) The inverse and direct kinematics solution can get based on the simulation curves, which laid a certain foundation for realistic simulation 3SPS (PS) in parallel stable platform for dynamic simulation. (3) It is to avoid tedious programming and mathematical calculations, improving work efficiency, reducing development costs. And it establishes base for the application of this kind of parallel stable platform and the follow-up research for analysis of kinematics and workspace.

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