- Original Article
- Open Access
Design and Analysis for a Three-Rotational-DOF Flight Simulator of Fighter-Aircraft
- Chang-Chun Zhou^{1} and
- Yue-Fa Fang^{1}Email author
https://doi.org/10.1186/s10033-018-0256-z
© The Author(s) 2018
- Received: 3 March 2016
- Accepted: 13 June 2018
- Published: 2 July 2018
Abstract
Most of researchers focused on traditional six degrees of freedom (DOF) Stewart flight simulator, which can not be adaptive in fighter-aircraft flight simulator. A three rotational DOF flight simulator of fighter-aircraft based on double parallel manipulator and hybrid structure is presented. The flight simulator is composed of two identical 3-RRS (revolute-revolute-spherical) spherical parallel manipulators and one cabin, called Twins. The cabin has an additional independent DOF for 360° continuous rotation, so it can be applied as a flight simulator for a fighter-aircraft to achieve spin maneuvering. Because of the introduction of the hybrid structure and double parallel manipulator of the mechanism, the redundancy exists with respect to both kinematics and actuation. Kinematics is carried out and Jacobian matrix is established by means of screw theory. The inverse kinematics is given out by the analytical method. 64 groups inverse solutions are showed in a table by permutation. Forward kinematics is solved by an effectively numerical method. The forward numerical method is realized based on the analytically inverse kinematics and Jacobian matrix. The numerical examples show that the forward numerical method can be used in real-time control. The rolling motion is considered in forward kinematics and a numerical example is given out. The proposed flight simulator can spin and there are three rotational DOF with a hybrid structure so that the novel flight simulator can be used in the field of the fighter-aircraft for pilots to train.
Keywords
- Parallel manipulator
- Hybrid mechanisms
- Actuation redundancy
- Flight simulator
- Kinematics
- Screw theory
- Jacobian matrix
1 Introduction
Flight simulators are devices in which air crews and pilots can train without the use of an actual aircraft [1]. In flight training, they are used mainly to reduce costs and increase safety. In their most sophisticated form, they simulate an aircraft’s vehicular motion, instrumentation and sounds, gravitational forces, radar and electro-optical sensor displays, and out-the-window views. According to the USA’s Federal Aviation Administration (FAA) regulations, any device called a Flight Simulator must have at least one motion platform, otherwise it can only be termed a Flight Training Device [2]. Therefore, the vehicular motion platform in flight simulator is one of the most important parts.
Since Stewart’s initial use of a 6-DOF parallel manipulator as a flight simulator in 1965 [3], this approach has become standard. Over the past five decades, the Stewart parallel manipulator has been used to make significant contributions to aeronautical research [4–7]. Even so, there is still the disadvantage of the Stewart-type flight simulator in that its posture rotation range is less than 30°. The limitation of the posture range of a Stewart parallel manipulator hinders its ability to serve as a flight simulator of a fighter-aircraft. The motion of a fighter-aircraft involves continuous 360° rolling frequently. In order to achieve continuous 360° rolling, Kim et al. [8] presented an innovative motion base as a flight simulator, based on a 6-DOF parallel mechanism, called Eclipse-II. The Eclipse-II allows continuous 360° rotation in A, B, and C-axes as well as translational motions in X, Y, and Z-axes. However, the rotations of the Eclipse-II parallel manipulator are achieved by two circular guide rails, which severely affect the dynamical performance. As a matter of fact, in both the Stewart and Eclipse-II simulators, translational movements are nearly inoperative. Low-cost flight simulator with a reduced-DOF (less than six) platform was proposed by Pouliot and Gosselin [9], which revealed result comparable to those of a 6-DOF Stewart platform. Subsequently, Shui et al. [10] proposed a more advanced and innovative reconfigurable spherical motion generator to enable continuous spherical motion of the flight simulator. The mechanism enables unlimited workspace with respect to 3-DOF spherical motion with rapid, continuous, and precise motion capabilities. However, it is actuated by an electromagnetic motor with a highly complicated configuration that causes the volume of manipulator to be huge. So far, no parallel manipulator can be used as a flight simulator to accomplish a 360° continuous rotation in practice.
Although it is not necessary to be able to continuously rotate in all orientations, a flight simulator of a fighter-aircraft needs to be able to roll in an additional 360° to perform most motions of a fighter-aircraft. Therefore, the purpose of this study is to achieve continuous 360° rotation of a fighter-aircraft flight simulator.
Many researchers have focused on hybrid platforms such as the famous Tricept by Neumann [11] and Trivariant by Huang et al. [12] that provide a good workspace volume. They are typically a combination of a parallel manipulator and a serial manipulator. As a hybrid alternative, a double parallel manipulator was proposed for enlarging the workspace by Lee et al. [13, 14]. Tsai et al. [15] used a double parallel manipulator to enlarge the workspace of the manipulator and analyzed its kinematic properties. Furthermore, many researchers have noted that the redundancy of a double parallel manipulator can improve its abilities and performance, for example, enlarging the volume of the workspace. There are two main types of redundancy for the parallel manipulator: (a) kinematic redundancy and (b) actuation redundancy [16, 17]. Zanganeh and Wang et al. [18–20] studied the kinematic redundancy of a parallel manipulator by analyzing its kinematics, and pointed out that the extra DOF not only allow for execution of the original output task, but also additional tasks such as increasing the workspace. It is notable that the Eclipse-II is a redundant actuation parallel manipulator. Redundant actuation has been proven to be a good method to enhance performance of parallel manipulators. Nokleby et al. [21] proved that redundant actuation can improve force capabilities. Kim et al. [22] investigated the redundantly actuated parallel manipulator and proved that redundant actuation of a parallel manipulator not only can improve force capabilities, but can also enhance the stiffness of a manipulator. Li et al. [23] derived the conclusion that redundant actuation has little effect on the stiffness when the actuators are on prismatic joints and enhances stiffness value greatly when the actuators are on revolute joints.
Although the novel mechanisms are proposed one by one, they don’t adapt for the flight simulator. Regarding existing parallel manipulators, the 3-DOF spherical parallel manipulator is a compact configuration with large rotational posture [24]. Therefore, given the above conclusions, a flight simulator of a fighter-aircraft based on a double and hybrid 3-RRS [25–28] spherical parallel manipulator was chosen for this study.
The remainder of this paper is organized as follows. In Section 2, the structure of the Twins flight simulator is described. In Section 3, the kinematic properties of the Twins are discussed, including the development of the direction-cosine matrixes in Section 3.1 with analytical spherical theory, the analysis of DOF in Section 3.2, and the development of the Jacobian matrix via screw theory in Section 3.3. Inverse kinematics and forward kinematics are conducted in Sections 4 and 5. Numerical examples are provided in Section 6. Lastly, the conclusions are discussed.
2 Architectural Description
Looking at Figure 2, based on the symmetrical rule [29], γ_{1} = 120°, γ_{2} = 120°, γ_{3} = 120°, γ_{1}′ = 120°, γ_{2}′ = 120°, γ_{3}′ = 120°. α_{1} is the angle between the B_{i}-axis and C_{i}-axis, and similarly α _{1} ^{′} is the angle between the B_{i}′ axis and C_{i}′ axis. α_{2} is the angle between the B_{i}-axis and A_{i}-axis, α_{2}′ is the angle between the B_{i}′ axis and A_{i}′ axis. β_{1} is the angle between line MN and line OC_{i}. β_{2} is the angle between line QV and OA_{i}. 0° < α_{1 }= α_{1}′ < 90°, 0° < α_{2 }= α_{2}′ < 90°, 0° < β_{1} < 90°, 0° < β_{2} < 90°.
3 Degrees of Freedom and Jacobian Matrix
3.1 Development of Direction-cosine Matrices
3.2 Degrees of Freedom Analysis
Therefore, the mechanism has three degrees of freedom, and the cabin’s spin becomes a redundant degree of freedom with regard to the w-axis.
3.3 Jacobian Matrix
- (1)
\(^{r} \varvec{\$ }_{1}\) is through the center of the spherical joint of the A_{1}-axis and B_{1}-axis;
- (2)
\(^{r} \varvec{\$ }_{1}\) does not intersect with the C_{1}-axis.
4 Inverse Kinematics
In the equations of inverse kinematics, the posture parameters \((\lambda ,\varepsilon ,\upsilon )\) of the moving platform are given. The objective is to calculate the input angle \(\theta_{i}\) and \(\theta^{\prime}_{1}\), i = 1, 2, 3. In the inverse kinematic analysis, the kinematic equations of the 3-RRS spherical parallel manipulator are the same as those of the 3-RRR spherical parallel manipulator, but the 3-RRR spherical parallel manipulator is an over-constrained mechanism that is difficult to practically assemble. That is why we chose a flight simulator model based on 3-RRS spherical parallel manipulators.
When considering the independently redundant DOF of the cabin, because there are three outputs and eight inputs, it is innumerable about the inverse solutions. If we ignore the independently redundant DOF of the cabin, and fix the independent DOF of cabin, the analysis of the inverse kinematics is as follows.
All forms of the combination according to Eq. (46) and Eq. (47) are shown in the Table 5 of Appendix 1.
5 Forward Kinematics
For the forward kinematics, without considering the independently redundant DOF of the cabin, the angles of the input \(\theta_{i}\) and \(\theta^{\prime}_{i} ,i = 1,{ 2},{ 3}\) are given, and the posture \((\lambda ,\varepsilon ,\upsilon )\) of the moving platform is determined. However, considering the redundant actuation, the DOF of the moving platform of the flight simulator is three; thus, it can only choose three angles of the input at one time. Herein, we choose \(\theta_{i}\)(\(0 < \theta_{i} <\uppi,i = 1,{ 2},{ 3}.\)) as the inputs, and the remaining three angles of the input \(\theta^{\prime}_{i}\)(\(0 < \theta_{i}^{\prime } < \pi ,i = 1,{ 2},{ 3}\)) are obtained via inverse kinematics. In general, the forward kinematics of a parallel manipulator is fairly complicated when using the analytical method, and many solutions are derived. Moreover, in practical application, the configuration of a manipulator is just one of the forward solutions. The motion of the flight simulator is continuous and the control system is based on a dynamical model of MIMO (multiple input multiple out) that needs to obtain the posture of the moving platform in real-time using the forward equations. Thus, an effective numerical method is employed to solve the forward equations. The numerical method was first used in Stewart’s parallel manipulator [35].
The iteration is stopped by the constraint condition \(max\left| {\Delta \theta_{i}^{k} } \right| < \delta ,i = 1,2,3\). \(\Delta \theta_{i}^{k}\) is an element of \(\Delta \varvec{p}^{(k)}\), and \(\delta\) is the permissible error. \(\varvec{J}_{\text{s}}\) must be nonsingular when the manipulator performs a continuous motion and the iteration must converge to a unique solution. Eq. (53) is similar to the Newton-Raphson iteration, but the physical and geometrical meanings are more significant, and the whole process of the algorithm is different. Appendix 2 shows a flow chart of the algorithm.
6 Numerical Examples
Parameters of the manipulator
α_{1} (rad) | α_{2} (rad) | β_{1} (rad) | β_{2} (rad) |
---|---|---|---|
π/5 | π/5 | π/20 | 29π/36 |
Forward kinematics example 1
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
\('\theta_{1} \;(\text{rad})\) | 1.89417 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\('\theta_{2} \;(\text{rad})\) | 1.89417 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\('\theta_{3} \;(\text{rad)}\) | 1.89417 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\lambda \;(\text{rad)}\) | 0.00000 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\varepsilon \;(\text{rad})\) | 0.00000 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\upsilon \;(\text{rad})\) | 0.00000 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\theta^{\prime}_{1} \;(\text{rad)}\) | 1.89417 | 1.89417 | 1.89417 | − 1.89417 | − 1.89417 | − 1.89417 | − 1.89417 | 1.89417 |
\(\theta^{\prime}_{2}\; (\text{rad})\) | 1.89417 | 1.89417 | − 1.89417 | − 1.89417 | − 1.89417 | 1.89417 | 1.89417 | − 1.89417 |
\(\theta^{\prime}_{3} \;(\text{rad})\) | 1.89417 | − 1.89417 | − 1.89417 | − 1.89417 | 1.89417 | 1.89417 | − 1.89417 | 1.89417 |
Forward kinematics example 2
Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
\('\theta_{1} \;(\text{rad})\) | 1.74548 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\('\theta_{2} \;(\text{rad})\) | 2.29808 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\('\theta_{3} \;(\text{rad)}\) | 2.05784 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\lambda \;(\text{rad)}\) | 0.10000 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\varepsilon \;(\text{rad})\) | 0.10000 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\upsilon \;(\text{rad})\) | 0.10000 | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ | ‒ |
\(\theta^{\prime}_{1}\; (\text{rad)}\) | 2.24689 | 2.24689 | 2.24689 | − 1.72881 | − 1.72881 | − 1.72881 | − 1.72881 | 2.24689 |
\(\theta^{\prime}_{2} \;(\text{rad})\) | 1.81504 | 1.81504 | − 2.02941 | − 2.02941 | − 2.02941 | 1.81504 | 1.81504 | − 2.02941 |
\(\theta^{\prime}_{3}\; (\text{rad})\) | 2.08712 | − 1.51476 | − 1.51476 | − 1.51476 | 2.08712 | 2.08712 | − 1.51476 | 2.08712 |
As shown in Tables 2 and 3, the results are obtained numerically by the forward algorithm and analytically by the inverse equations. The time costs of the numerical forward algorithm are less than 3 ms using MATLAB software by Intel Core i7-3.2G CPU, thus proving that the algorithm of the iteration meets the typical real-time control requirement of less than 6 ms.
To consider the independent DOF of cabin, the forward solution is obtained in two steps. The first step is to obtain the posture \((\lambda ,\varepsilon ,\upsilon )^{\text{T}}\) by the numerical method. Then the posture \((\lambda ,\varepsilon ,\upsilon )^{\text{T}}\) is changed by the cabin’s spin with reference to the w-axis of the moving coordinate.
Forward kinematics example with cabin’ spin
\(\theta^{\prime}_{1} (\text{rad)}\) | \(\theta^{\prime}_{2} (rad)\) | \(\theta^{\prime}_{3} (rad)\) | \(\lambda \;(\text{rad)}\) | \(\varepsilon \;(rad)\) | \(\upsilon \;(rad)\) | \(\varphi \;\left( {rad} \right)\) | \(\lambda^{\prime}\;(rad)\) | \(\varepsilon^{\prime}\;(rad)\) | \(\upsilon^{\prime}\;(rad)\) |
---|---|---|---|---|---|---|---|---|---|
1.74548 | 2.29808 | 2.05784 | 0.10000 | 0.10000 | 0.10000 | 3π/5 | − 0.12601 | 0.06420 | 0.10000 |
7 Conclusions
- (1)
The flight simulator for a fighter-aircraft with a hybrid configuration of symmetrically double parallel manipulators named Twins is presented.
- (2)
Twins is a multi-functional flight simulator. It can be used as a normal flight simulator like Stewart mechanism. And when the cabin spins that will be a flight simulator of fighter-aircraft.
- (3)
Screw theory is used to establish the Jacobian matrix that simplifies the process of establishing the Jacobian by the means of closed equations.
- (4)
A numerically forward kinematics is adopted by the inverse kinematics and Jacobian matrix and the method is more simple.
Declarations
Authors’ Contributions
Y-FF was in charge of the whole trial; C-CZ wrote the manuscript; C-CZ assisted with sampling and laboratory analyses. All authors have read and approved the final manuscript.
Authors’ Information
Chang-Chun Zhou, born in 1981, is currently a PhD candidate at Robot Research Center, Beijing Jiaotong University, China. His research interests include robotics and dynamics.
Yue-Fa Fang, born in 1958, is currently a professor at Beijing Jiaotong University, China. He received his PhD degree from Staffordshire University, UK, in 1994. His research interest is parallel manipulator.
Competing Interests
The authors declare no competing financial interests.
Funding
Supported by National Natural Science Foundation of China (Grant No. 51675037).
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