LMF Algorithm Based on Hyper-Chaos for the Solving of Forward Displacement in a Parallel Robot Mechanism

Abstract

The forward displacement problem of the parallel robot mechanism can be converted to nonlinear equations in order to find solutions, but it is very difficult to find all solutions because of the strong coupling of the nonlinear equations. Given the problems of having only one solution and sometimes no convergence when solving the nonlinear equations with the Newton method and quasi-Newton method, a LMF algorithm based on hyper-chaos is proposed to solve the general 6-6 platform parallel mechanism, based on the combination of the hyper-chaos system and the Levenberg-Marquardt-Fletcher (abbreviated as LMF) algorithm. This method uses the hyper-chaotic system to produce the initial point of the LMF algorithm, and takes advantage of the characteristics of the chaotic sequence and the LMF algorithm to find all the real solutions. The numerical example shows that the new method has some characteristics such as that it runs in the initial value range, it has fast convergence, it finds all the real solutions that can be found, and it proves the correctness and validity. It provides a new approach to mechanism design.

Keywords

Hyper-Chaos System Parallel Mechanism LMF Algorithm Nonlinear Equations

1. Introduction

The Stewart platform is something new in the field of robotics and its design theory presents an extremely difficult problem. The forward displacement of the Stewart platform poses questions for mechanism, requiring work in the fields of robotics, algebraic geometry, differential geometry, symbolic computation and numerical computation in the field of mathematics [1 –3]. In the mathematical model of the Stewart platform, there is a class of strongly nonlinear algebraic equations with many variables so it is extremely difficult to solve, since nonlinear science is at the forefront of technological development and still relatively little is understood about it. The Stewart mechanism is a general 6-6 type parallel mechanism. Its upper and lower platforms are flat arbitrary hexagons connected by 6 sliding pairs with spherical pairs at both ends. The forward displacement ultimately boils down to solving a set of nonlinear equations. Solving these equations is extremely difficult and we are faced with which is another difficult problem of mechanism even after completing the displacement analysis in the space 6R series mechanical arm [4]. The methods for solving forward displacement in the Stewart platform generally have an analytic and a numerical method. Using an analytic method to find the closed form of the forward displacement can give the exact solution, but often generates the middle polynomial which is too large and so is hard to calculate [5]. For the kinematics of the Stewart platform, steps were taken to obtain an estimated number of solutions, some of the real solutions and the closed form of some special configuration, but this is still a long away from completely solving all the real solutions of the equations [6 –13]. The numerical methods used for the Stewart platform are mainly the Newton-Rapbson iterative method (referred to as the NR method) [14], the homotopy method [15 –17], the neurons algorithm [18], the additional sensor method [19], etc.. The NR method does not need to solve the complex nonlinear equations with n-order. But in the NR method the calculation speed is unstable, and its convergence of the calculation results and the speed of convergence all are dependent on the initial value. The homotopy and neurons methods can find all the solutions, but involve large amounts of calculations. The additional sensor method uses the necessary number of additional sensors and a certain arrangement to simplify the solution process of the forward displacement position, but the simple additional sensor method has a very high demand with regards to the level of error in the processing and assembly in platform parts which makes it difficult to calibrate the platform structure.

With the numerical iteration method or the constraint optimization method, if the initial value selected is improper, the result is not easy to converge and all solutions are more difficult to obtain. This problem has not been fully resolved until now. The Newton iterative method as the traditional numerical iterative method has second-order convergence and high performance, but this method is extremely sensitive to initial values. Sometimes this phenomenon is considered to be the arithmetic singularity or inevitable singularity, in fact, the reason for the numerical instability is that the NR method is a nonlinear discrete dynamic system where the chaos and fractal phenomena will be generated in the sensitive area. In mechanism and optimal design, the chaos phenomenon is considered to be incomprehensible and unusable, or it is regarded as a random figment of the imagination to be ignored. The rapid development of chaos theory is one of the major achievements of the last century [20]. With the continuous development of society, many nonlinear phenomena and models continue to emerge and so research in this field is on the rise. The international academic community generally sees non-linear mathematics, nonlinear natural science and social science and their technological development as part of the mainstream in the 21st century, while chaos is the basic pattern of almost all sports phenomenon in the natural world. Chaos is considered to be a phenomenon that is seemingly irregular and similar to a ‘random’ in a deterministic system. The most essential characteristic of chaotic behaviour is extreme sensitivity to the initial conditions of the nonlinear system. It also has many basic characteristics such as boundedness, ergodicity, intrinsic randomness, scaling, universality, fractal dimension, the positive Lyapunov exponent, unlimited broadband power spectrum and sub-dimensional power spectrum. The Lyapunov exponent is one of a number of effective methods for depicting the chaos specific property of a nonlinear system. If one of the Lyapunov exponents is positive, the system is chaotic, and if a system has two or more positive Lyapunov exponents, the system is hyper-chaotic. The greater the number of positive Lyapunov exponents, the higher the degree of instability in the system [21–22]. It is of important theoretical and practical significance that the chaotic and hyper-chaotic systems are used to calculate kinematics.

The chaos method can calculate all the real solutions within the scope of the real number, and it has high computational efficiency because it does not seek plural results. The method described in Y.X. Luo, D.Z. LI (2003) [23] posits that the points of Julia centralization in the Newton iteration method will appear when the Jacobian matrix of the equations is equal to zero. But this guess has not been proven. For the multivariable Jacobian matrix, first, its symbolic expression is found; second, all the variable values are determined except one variable; finally, the chaos zone is searched for the variable to be determined. So the matrix is quite complex to solve. The chaotic sequence method is a new method, in which the initial point of the Newton iteration is generated using the chaotic and hyper-chaotic system and all the real solutions of the mechanism synthesis can be effectively solved [22 –27]. But the Hénon hyper-chaotic Newton iteration method cannot solve the mechanism synthesis problem of 6-SPS. When the solutions do not converge using the Newton or quasi-Newton method, the mathematical programming method can be adopted [28]. The mathematical programming method with the hyper-chaotic system was put forward to solve the synthesis problem of 6-SPS by transforming it into nonlinear equations with six variables [29]. The Hénon super-chaotic sequence, combined with the Newton descent method, created the super-chaotic Newton descent method to solve this problem by transforming it into nonlinear equations with nine variables [30]. The quaternion method was used for the synthesis problem of 6-SPS: it established nonlinear equations with eight variables and it was solved by using the hyper-chaotic neural networks damp least-square method where the hyper-chaotic neural network produces a hyper-chaotic sequence as the initial iteration value in the damped least-square method [31]. The mathematical programming method and the damped least square method based on chaos anti-control were proposed respectively as ways to solve the forward displacement of the general 6-6 type parallel mechanism [32–33].

In this paper, based on the combination of the hyper-chaos system and the LMF algorithm, the LMF algorithm based on hyper-chaos was proposed as a means to solve the general 6-6 platform parallel mechanism. This method uses the hyper-chaotic system to produce the initial point of the LMF algorithm, and takes advantage of the characteristics of the chaotic sequence and the LMF algorithm to find all the real solutions. The numerical example shows that the new method has some characteristics such as that it runs in the initial value range, it has fast convergence, it can find all the potential real solutions and it proves the correctness and validity. It provides a new approach to mechanism design.

2. The Hénon hyper-chaotic system

The Lyapunov exponent is one of a number of effective methods for depicting the chaos specific property of a nonlinear system, and the number of Lyapunov exponents is the same as the dimension n of the state space of the system. If one of the Lyapunov exponents is positive, the system is chaotic. Furthermore, if a system has two or more positive Lyapunov exponents, the system is hyper-chaotic. The greater the number of positive Lyapunov exponents, the higher the degree of instability in the system [19–20]. In general, if the systematic state variable number is higher (for high dimension system, e.g. the discrete system, n>2), the level of unsteadiness will probably be higher.

A general Hénon mapping was designed as follows in Y.X. Luo et al. [32,34]:

{\begin{matrix} x_{1, k + 1} \\ x_{i, k + 1} \end{matrix}} = {\begin{matrix} a - x_{n_{1} - 1, k}^{2} - b x_{n_{1}, k} \\ x_{i - 1, k} \end{matrix}}

(1)

where, i = 2,3,…,n₁ expresses the dimension of the system, k is the discrete time, and a and b are adjustable parameters. When i=2, the above mapping is known as the famous Hénon mapping. When the fixed parameters are a=1.76, b=0.1 and the dimensions vary from 2 to 10, after computing it was found that by increasing n ₁, the simple relation of the number of positive Lyapunov exponents n to the system dimension n₁ is n = n₁−1 [32,34], in other words, when a system the dimension of system is larger than two, the system is hyper-chaotic. We also did a simulating study for n>10 and obtained the same result.

For n₁ = 5, we put together a program using Matlab with a time series method in order to solve the Lyapunov exponents, and we obtained four positive Lyapunov exponents, shown in Fig.1. When n₁ =13, the simulating result is that the system has 12 positive Lyapunov exponents as shown in Fig.2, from Y.X. Luo (2009) [30].

Figure 1.

Lyapunov exponent of Hénon maps where n₁=5

Figure 2.

Lyapunov exponent of Hénon maps where n₁=13

3. Principle of LMF algorithm

Let us say we have a general over-determined system of nonlinear algebraic equations

f (x) \approx y \Rightarrow f (x) - y = r

(2)

The solution, optimal in the least squares sense, is sought by minimizing‖r‖²₂ =2r^Tr. Necessary conditions for the optimum solution are zero values of partial derivatives of ‖r‖²₂ due to unknown variables x, i.e.

\frac{\partial {| | r | |}_{2}^{2}}{\partial x} = 2 \frac{\partial r^{T}}{\partial x} r = 2 J^{T} r = 2 v .

(3)

Elements of J, which is called a Jacobian matrix, are $J_{i j} = \frac{\partial r_{i}}{x_{j}}$ . Vector v should be equal to the zero vector in the point of optimal solution x*. It is sought after k^th iteration in the form

x_{​^{k + 1}} = x_{​^{k}} + Δ x_{​^{k}}

(4)

Let residuals r(x) be smooth functions, then it holds that:

r_{​^{k + 1}} = r_{​^{k}} + \frac{\partial r_{k}}{\partial x} Δ x_{​^{k}} + \dots

(5)

After some manipulations, the equation for the solution increment takes the form

A_{​^{k}} Δ x_{​^{k}} - {[J_{​^{k}}]}^{T} r_{​^{k + 1}} = - v_{​^{k}}

(6)

in which A = J^TJ. The solution would be found quite easily if r_k+1were known. Unfortunately, this is not the case. It was the reason why Levenberg substituted the second term in equation (6), -[J_k]^Tr_k+1, by λΔAx_k. The scalar λ serves for scaling purposes. For λ = 0, the method transforms into the fast Newton method, which may diverge, while for λ → ∞, the method approaches the stable steepest descent method. Later, Marquardt changed λ into λ_k, so that the equation for Δx_k became

(A_{​^{k}} + λ_{k} I) Δ x_{​^{k}} = - v_{​^{k}}

(7)

Values of λ_k+1 vary in dependence depending on the behaviour of the iteration process. For a slow stable convergence of iterations, the new value λ_k−1=λ_k/v is set, which accelerates the process. If an indication of divergence is observed, the value is changed to λ_k+1=λ_kv. The usual value of v is between 2 and 10.

Fletcher improved the Marquardt strategy of adaptation significantly. He substituted the unity matrix I with a diagonal matrix D of scales in the formula (7). Furthermore, he introduced a new quotient R which expresses how the forecasted sum of squares agrees with the real sum in the current iteration step. If R falls between preset limits (Rlo, Rhi), parameters of iteration do not change; otherwise changes of λ and v follow. The value of λ is halved if R>Rhi. Provided λ reaches a value which is lower than the critical value λ_c, it is cleared which causes the next iteration to proceed like in the Newton method. If the R < Rlo, parameter v is set so that it holds2≤v≤;10, and if λ were zero, a modification of λ_c and λ would follow. A complete reconstruction of the function named LMF was recently undertaken [35].

4. LMF algorithm based on hyper-chaos for solving nonlinear equations

Using the LMF algorithm based on hyper-chaos, all solutions of nonlinear equations can be obtained. The calculation steps are as follows:

Construction of the chaos set x₀(i,j) according to Eq. (1), i = 1,2,-,n where n is the number of variables and is also the number of the positive Lyapunov exponents of the Hyper-chaotic Hénon system, and j = 1,2,…;N where N is the length of the chaos sets.

Supposing that the variable interval of x(i) is [a(i),b(i)], the chaos set is mapped to the variable interval to generate the jth initial value of x(i), that is, x(i,j).

x(i,j) is regarded as the initial value of the LMF algorithm; once Eq. (2) has been iterated j times, then all the solutions x* are obtained.

5. Mathematical modelling of forward displacement

The structure diagram of the general 6-6 type 3-D parallel mechanism is shown in Fig.3. In this mechanism, the upper plane is connected to the lower by six branched chains with ball joints and sliding pairs. The fixed coordinate system O₁x₁y₁z₁ and the moving system O₂x₂y₂z₂ are established respectively in Fig.3, where the coordinates of A and B_i are Q_ia (a_xi,a_yi,a_zi) and Q_ib(b_xi,b_yi,b_zi) respectively, and the length of A_iB_i is L_i(i = 1,2 b_yi…,6). So as long as the rotation transformation matrix R and the translation vector P(x, y, z) from O₂x₂y₂z₂ to O₁x₁y₁z₁ are obtained, the spatial position of the upper plane relative to the lower can be determined.

Figure 3.

Structure diagram of the general 6-6 type 3-D parallel mechanism

According to the length of the rod, the following equations can be created:

\begin{array}{l} L_{i}^{2} = {[R Q_{i b} + P - Q_{i a}]}^{T} [R Q_{i b} + P - Q_{i a}] \\ \begin{matrix} ​ & ​ & ​ & ​ \end{matrix} (i = 2, 3, \dots, 6) \end{array}

(8)

L_{1}^{2} = P^{T} P

(9)

where

R = [\begin{matrix} l_{x} & m_{x} & n_{x} \\ l_{y} & m_{y} & n_{y} \\ l_{z} & m_{z} & n_{z} \end{matrix}]

After substituting Eq.(3) for Eq.(2) and expanding it, we can write the equation as:

\begin{array}{l} L_{1}^{2} - L_{i}^{2} + Q_{i b}^{T} Q_{i b} + Q_{i a}^{T} Q_{i a} + 2 P^{T} R Q_{i b} - 2 Q_{i a}^{T} P - 2 Q_{i b}^{T} R Q_{i a} = 0 \\ \begin{matrix} ​ & ​ & ​ \end{matrix} (i = 2, 3, \dots, 6) \end{array}

(10)

R is the unit orthogonal matrix, so the following equations can be obtained:

l_{x}^{2} + l_{y}^{2} + l_{z}^{2} = 1

(11)

m_{x}^{2} + m_{y}^{2} + m_{z}^{2} = 1

(12)

l_{x} m_{x} + l_{y} m_{y} + l_{z} m_{z} = 0

(13)

n_{x} = l_{y} m_{z} - l_{z} m_{y}

(14)

n_{y} = l_{z} m_{x} - l_{x} m_{z}

(15)

n_{z} = l_{x} m_{y} - l_{y} m_{x}

(16)

According to Eqs. (14 –16), n_x, n_y and n_z can be obtained. Therefore, Eqs.(6 –11) are the key and also the difficulty of the forward displacement in the general 6-6 type 3-D parallel mechanism [30, 32–30]. They are also the mathematical models in this paper which are denoted by the following equation:

F (x), F (x) = {[f_{1} (x), f_{2} (x), \dots, f_{n} (x)]}^{T} .

where, there are nine unknown variables as l_x, l_y, l_z, m_x, m_y, m_z, x, y and z, which are written by

x = {[l_{x}, l_{y}, l_{z}, m_{x}, m_{y}, m_{z}, x, y, z]}^{T} .

6. Numerical example

In Fig.3, the coordinate of point A_i in a static coordinate system x₁y₁z₁ is A₁(0,0,0), A₂(2,0,0), A₃(4,1,0), A₄(5,3,1), A₅(3,5,-1),A₆(−1,2,2) and the coordinate of point B_i in dynamic coordinate system x₂y₂z₂ is B₁(0,0,0), B₂(1,0,0), B₃(2,1,0),B₄(4,3,2), B₅(3,4,-1), B₆(−2,-1,3), l₁=11, l₂=12, l₃=13,l₄=15,l₅=14, l₆=10, find the mechanism's total position forward solutions.

Based on the LMF algorithm of the Hénon hyper-chaotic mapping, Eqs. (8 –13) were converted into the mathematical form of Eq.(2) (when y = 0). After a random number generated an initial point of the hyper-chaotic set, the number of hyper-chaotic variables n ₁ = 10 from the generalized hyper-chaos produced a hyper-chaotic sequence variable with the positive Lyapunov exponents n = 9, and then 1180 hyper-chaotic variables were obtained. Taking 20 latter pieces of data as the initial values of the LMF algorithm as shown in Table 1, the algorithm runs for 1.0984s to find the corresponding Euler angles as shown in Table 3.

If we adopt the Euler method in order to solve the similar problem, it will cost 327.5 s, which is shown in Table 3, which is the same as Table 1 [29], and, based on the quaternion and hyper-chaotic LMF algorithm, it will cost 3.8s, which is shown in Table 4 and can be changed into Table 3.

Table 1.

Four examples of real solutions

No	x (1)	x (2)	x (3)	x (4)	x (5)
1	−0.6539	0.2181	0.7245	−0.6314	−0.6849
2	−0.5958	0.4756	0.6472	−0.6076	−0.7939
3	0.0291	−0.8060	−0.5912	0.9820	−0.0872
4	−0.5410	−0.7491	−0.3823	0.6784	−0.6574

No	x (6)	x (7)	x (8)	x (9)

1	−0.3637	0.2301	3.8573	10.2989
2	0.0240	0.3682	5.3866	9.5838
3	0.1673	−1.1889	−0.3137	−10.9311
4	0.3280	−1.8184	1.1055	−10.7922

Table 2.

Corresponding initial values to four real solutions

No	x ₀ (1)	x ₀ (2)	x ₀ (3)	x ₀ (4)	x ₀ (5)
1	−0.9313	−0.0624	−0.2810	0.4387	0.4037
2	1.4331	1.5889	0.7098	1.4863	−0.7063
3	1.7589	−1.2483	1.1747	−0.0452	−0.7305
4	−1.6941	1.5795	1.1037	1.5728	−0.7410

No	x₀(6)	x₀(7)	x₀(8)	x₀(9)

1	−0.6850	0.2135	−0.2603	1.7201
2	0.9940	1.7497	−0.3151	1.5887
3	−0.4166	−1.1095	1.3506	−0.4995
4	1.6044	1.1655	0.9266	0.1389

Table 3.

Corresponding Euler angles to four real solutions

No.	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆
1	0.2768	0.5113	−0.6170	0.2301	3.8573	10.2989
2	0.5116	0.3675	0.9636	0.3682	5.3866	9.5838
3	−0.1554	0.3434	−0.7563	−1.1889	−0.3137	−10.9311
4	−0.8487	0.2703	−0.4596	−1.8184	1.1055	−10.7922

Table 4.

Examples of four real results based on the quaternion hyper-chaotic LMF algorithm

No.	q ₀	q ₁	q ₂	q ₃
1	0.2484	0.3337	−0.3096	0.8550
2	0.3051	0.3302	−0.1000	0.8876
3	−0.4079	−0.2512	0.0703	0.8750
4	−0.6578	−0.2861	−0.1538	0.6796

No.	g ₀	g ₁	g ₂	g ₃

1	−7.6883	6.5440	4.1984	1.1998
2	−8.0901	5.8517	4.4810	1.1085
3	8.9082	2.4680	3.8515	4.5518
4	7.0399	−1.1127	4.1411	7.2837

7. Conclusions

For the problem of the forward displacement of parallel robots, we need to find solutions for a class of strongly nonlinear algebraic equations with many variables which are consequently extremely difficult to solve. Based on a combination of the hyper-chaos system and the LMF algorithm, a LMF algorithm based on hyper-chaos was proposed in order to solve the general 6-6 platform parallel mechanism. This method uses the hyper-chaotic system to produce the initial point of the LMF algorithm, and takes advantage of the characteristics of the chaotic sequence and the LMF algorithm to find all the real solutions. The steps in this calculation were shown. The numerical example shows that the new method has some characteristics such as that it runs in the initial value range, it has fast convergence, it can find all the potential real solutions and it proves the correctness and validity of this method when compared with other methods. It provides a new approach to mechanism design.

Footnotes

8. Acknowledgments

This research is supported by the National Natural Science Foundation of P.R. China (No:51075144) and the grant of the 12th Five-Year Plan for the construct program of the key discipline (Mechanical Design and Theory) in Hunan province(XJF2011[76]).

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