Abstract
The kinematics analysis method of a novel 3-DOF wind tunnel mechanism based on cable-driven parallel mechanism is provided. Rodrigues' parameters are applied to express the transformation matrix of the wire-driven mechanism in the paper. The analytical forward kinematics model is described as three quadratic equations using three Rodridgues' parameters based on the fundamental theory of parallel mechanism. Elimination method is used to remove two of the variables, so that an eighth-order polynomial with one variable is derived. From the equation, the eight sets of Rodridgues' parameters and corresponding Euler angles for the forward kinematical problem can be obtained. In the end, numerical example of both forward and inverse kinematics is included to demonstrate the presented forward-kinematics solution method. The numerical results show that the method for the position analysis of this mechanism is effective.
1. Introduction
Parallel manipulators have separate serial kinematic chains that are linked to the ground and the moving platform at the same time. They have some potential advantages over serial robot manipulators such as accuracy, greater load capacity, higher velocities, and accelerations. Parallel manipulators have been developed for applications in many fields [1–4].
In the past few decades, parallel manipulators using cable transmission have been enthusiastically studied in a number of areas. In a cable suspended parallel robot, the moving platform is suspended and manipulated by the attached cables that are connected to the base; for example cable-suspended robots are Robocrane [5, 6], ultra-high-speed cable robot [7], dexterous hands [8, 9], parallel cable-suspended manipulators [10, 11], teleoperating robots [12], and robots for biological use [13]. The major advantage of using tendon transmission lies in that actuators can be installed on the remote base such that a lightweight and compact design can be realized.
In recent years, many researchers paid great attention to the use of cable-driven mechanism in wind tunnel test due to its fewer interference on the streamline flow. Some successful achievements have been made in the Suspension ACtive pour Soufflerie (SACSO) project about the cable-driven parallel suspension system in low-speed wind tunnels with 8 cables [14], another application is WDPSS-8 developed by Huaqiao university for use in wind tunnel test [15].
However, recent works about cable-driven wind tunnel mechanism (CDWTM) have focused on the design of 6 degree-of-freedom (DOF); further research is still under way to meet practical application. This article puts forward a new 3-DOF wind tunnel equipment based on cable-driven parallel mechanism, which can provide 3-DOF pure rotation for the scale model in wind tunnel. Because position analysis is one of the key complicated and important problems for the cable-driven parallel mechanism, the forward and inverse kinematics of the aforementioned mechanism is the main concern of the paper. In general, a numerical iterative scheme, such as Newton-Raphson method, can be applied to this problem. But such method not only demands an initial estimate that should be fairly close to the solution of the current configuration but also cannot guarantee the convergence to the actual solution. As is well known there are many methods that can be used to express transformation matrix in the closed-form solution for position analysis; in [16] the closed-form solution of a 3-DOF parallel manipulator is investigated by the Euler angles, resulting in a 16th degree polynomial expression in one single variable; screw theory [17, 18] is employed for the forward kinematics of the parallel mechanism; Husty [19] developed an algorithm for solving the direct kinematics of general Stewart-Gough platforms by using Euler parameters; an univariate polynomial of 40th degree is obtained; Lee and Shim et al. [20]. presented the closed-form forward kinematics of the 6-6 Stewart platform with Rodridgues’ parameters; 40 sets of solutions to describe the posture of the moving platform have been determined. Our goal is to develop an algebraic algorithm to provide all the solutions of forward kinematics for the 3-DOF mechanism by means of Rodridgues’ parameters, and a concise forward kinematics equation can be achieved.
The reminder of the paper is the following. Section 2 briefly outlines the system. Section 3 presents a method to obtain the analytical solution of forward kinematics. Sections 4 and 5 cover a numerical example, and conclusions are given in the last section.
2. Description of the 3-DOF CDWTM
Figure 1 shows the kinematic model of the CDWTM with the scale model, spherical joint, five cables, and the base (the wind tunnel). For convenience sake, the scale model is substituted for moving platform . The spherical joint is fixed in the center of moving platform, one end of the cables , is connected to the spherical joint, and the other ends and are attached to the upper and lower walls of the wind tunnel, respectively. The CDWTM can achieve 3-DOF rotational motion about point when the other three cables are actuated cooperatively via the motors and pulleys fixed in wind tunnel, which is very suitable for changing the three angles of the scale model in wind tunnel test. The base coordinate frame is located in spherical joint, with origin placed in the center of the spherical joint. The -axis is against the streamline flow, the -axis is along lift force, and the -axis is coincident with side force. The frame is attached to the moving platform, origin is the same with the base frame. Here, vector connects the couple vertices and (). The forward kinematics of the CDWTM is to determine the orientation of the moving platform while the lengths of the three cables, are known. are orientation of the scale model, which is formed by rotating the axis , the axis , and the axis , respectively.
3. Analysis of Forward Kinematics
3.1. Rodridgues’ Parameter Description of the Rotational Matrix
Choose Rodridgues vector , and it’s opposite corresponding symmetry matrix is here are called Rodridgues’ parameters. According to Cayley’s formula [21], the transformational matrix can be written as follows:
Here is a unit 3 × 3 matrix.
Furthermore, the inverse of can be represented as a function of vector , that is, Taking (2) into (1), then we have
3.2. Solution Procedure
From the geometric relationship in Figure 1, we can have where and denote the position vectors and , respectively.
So the following equation can be obtained: where, , , and are the norms for the corresponding vectors , , and , respectively.
For the convenience, let and be the unit vector for the vertices vectors and , respectively, that is,
And also set
Here, the physical meaning of is the initial angle of . So (5) can be rewritten in a concise form as
Substitute (3) into (8), and multiply both sides by the nonzero factor (). (8) becomes where,
There are three unknowns in (9), which are , and . By eliminating two of the variables, the eighth polynomial in one variable can be algebraically achieved.
In order to obtain the polynomial, we let and
; taking the initial conditions into (10), we have Here, and are known from the initial conditions and (10).
According to (11), (9) can be rewritten in the following scalar equations: with .
To derive a univariate equation in , simplify (12) as if is a known constant; we have where are all the function of ; the details can be found in the appendix.
From (15), we have
Amplify (14) by ; we can get
We also have (18) and (19) from (16) and (17) where Similarly, amplify (15) by ; we can get that is, with In matrix form, Equations (18)–(22) can be arranged as follows:
To get the nontrivial solution of (24), the determinant of the coefficient must be zero, that is,
From the appendix and (18)–(22), is a polynomial of , while are all polynomials of , and are only polynomials of , so (25) is an eighth polynomial in one variable.
4. Example
As an example, we consider a 3-DOF CDWTM with the following initial structural parameters, the vertices vectors of the base are , , and , the vertices vectors of the moving platform are , , and . The lengths of the three cables are , and . The final eighth equation about in the example is
Solving the equation, the eight sets of Rodridgues’ parameters can be achieved in Table 1.
According to (3), their eight corresponding orientations are showed in Table 2.
Next we will demonstrate the validity of the analytic forward kinematics by the numerical example of inverse kinematics. Let the initial orientations of the scale model be zero, the corresponding vertices vectors of the moving platform are , , and . When the moving platform has a posture of , , and , the transformation matrix expressed by means of Euler angles can be showed as following where , , , , ,
With the help of , we can calculate the corresponding vertices vectors of the moving platform in the base frame
From the above equation, we have , , and .
So the lengths of the three cables are the norms of vectors , , , that is, , , and .
The inverse kinematic results are the same with the initial structural parameters of the forward kinematics, which verifies that the analysis of forward kinematics is correct.
5. Conclusion
This paper presents the kinematics analysis method of a novel 3-DOF cable-driven parallel mechanism used in wind tunnel test, and then we employ elimination method to solve the analytic forward kinematics of the mechanism by using of Rodridgues’ parameters, so an eighth polynomial in one variable is derived finally. A numerical example is included to verify the effectiveness and accuracy of the developed algorithm for real-time computation and control.
Appendix
We have the following:
Acknowledgment
This work was Supported by the National Defense Pre-Research Foundation of China.