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Abstract

The path planning of free-floating space robot in space on-orbit service has been paid more and more attention. The problem is more complicated because of the interaction between the space robot and base. Therefore, it is necessary to minimize the base position and attitude disturbance to improve the path planning of free-floating space robot, reducing the fuel consumption for the position and attitude maintenance. In this paper, a reasonable path planning method to solve the problem is presented, which is feasible and relatively simple. First, the kinematic model of 6 degrees of freedom free-floating space robot is established. And then the joint angles are parameterized using the 7th order polynomial sine functions. The fitness function is defined according to the position and attitude of minimizing base disturbance and constraints of space robot. Furthermore, an improved chaotic particle swarm optimization (ICPSO) is presented. The proposed algorithm is compared with the standard PSO and CPSO algorithm in the literature by the experimental simulation. The simulation results demonstrate that the proposed algorithm is more effective than the two other approaches, such as easy to find the optimal solution, and this method could provide a satisfactory path for the free-floating space robot.

Keywords

ICPSO Path Planning Base Disturbance Space Robot On-orbit Service

1. Introduction

For space on-orbit service, the space robot plays an irreplaceable role [1]. Because of the unstable system of free-floating space robot, the movement of free-floating space robot has a disturbance on the base position and attitude when performing a task in the space. On the contrary, the changing of base position and attitude will have an impact on the space robot, which is likely to lead to the failure of task, raising concerns about an immeasurable loss [2]. To maintain the position and attitude of base remaining the same, to reduce the mutual influence between them, it is important for the space robot to have an optimized path planning, which reduces the disturbance of base during the movements.

It is a problem that the path planning for minimizing base disturbance has been one of the research hotspots and difficulties [3]. In 1991, a method of Disturbance Map (DM) about the virtual robot was developed by Dubowsky co-sponsored by Torres [4, 5]. The path planning of space robot can be optimized by this method, and thus the disturbance of base is reduced, but it is only efficient for the two degrees of freedom space robot; in other words, it is cannot be widely applied in the multidimensional space robot. In 2001, an idea of zero reaction manoeuvre (ZRM) was put forward by Yoshida et al. [6], which was based on the generalized Jacobi matrix. With the undisturbance of base, it can be used for the path planning of space robot. However, it is not for the redundant space robot. In 2006, Panfeng Huang et al. came up a method for path planning for minimizing disturbance of space robot based on the genetic algorithm (GA) was introduced by [7]. But the disadvantage is that the realization of genetic algorithm is more complex and convergence for a long time. In 2011, a path planning method for minimizing base reaction of space robot based on the chaotic particle swarm optimization was proposed by Ming Wang et al. [8]. But the position and attitude of base were not taken into account. In 2014, a path planning method for minimizing base disturbance of space robot based on the chaotic particle swarm optimization was developed by Hongwei Xia et al. [9]. However, the position of base is out of consideration.

In this paper, the problem on the path planning for minimizing base disturbance of free-floating space robot is discussed. It is considered that the position and attitude of base should be minimum disturbance. So the 6 degrees of freedom free-floating space robot is established by the kinematics equation [10]. Then the fitness function is defined, including four limit constraint conditions that are the base position, the base attitude, joint angular velocity and acceleration. And the joint angles should be parameterized in using of the 7th order polynomial sine functions [11]. The chaotic particle swarm optimization (CPSO) is a stochastic global optimization algorithm, which is proposed for applying to the path planning [12]. It is proposed that an improved chaotic particle swarm optimization (ICPSO) algorithm is applied to the path planning for minimizing base disturbance of free-floating space robot. According to the simulation results, compared with the standard particle swarm optimization (PSO) algorithm and chaos particle swarm optimization (CPSO) proposed in the literature, the proposed ICPSO is more efficient on the problem of path planning for minimizing base disturbance of free-floating space robot.

The rest of this paper is organized as follows. In Section 2, the kinematics of the free-floating space robot is overviewed. Then the standard PSO and the proposed ICPSO are stated in the Section 3. Furthermore, experiment simulations and comparisons are provided in the Section 4. Conclusions are drawn in the article finally.

2. Kinematics Equations of Free-floating Space Robot

2.1. Forward kinematics equations

Owing to free of outside interference, the whole system keeps the conservation of momentum. With regards to the 6 degrees of freedom of free-floating space robot, its kinematics equation can be formulated as follows [13]:

$[\begin{matrix} ν_{e} \\ ω_{e} \end{matrix}] = J_{b} [\begin{matrix} ν_{0} \\ ω_{0} \end{matrix}] + J_{m} \overset{\cdot}{θ}$ (1)

where, $ν_{e} \in R^{3}$ is called the velocity of end effector; $ω_{e} \in R^{3}$ , the angular velocity of end effector; $ν_{0} \in R^{3}$ , the velocity of base; $ω_{0} \in R^{3}$ , the angular velocity of base; $θ \in R^{6}$ , the matrix of joint angles, which consists of six joint angles of the space robot; $J_{b} \in R^{6 \times 6}$ , which represents the Jacobi matrix of base and $J_{m} \in R^{6 \times 6}$ , which represents the Jacobi matrix of space manipulator.

The position vector, which is of end effector on the free-floating space robot, can be represented as follows:

$p_{e} = p_{0} + b_{0} + \sum_{i = 1}^{6} (p_{i + 1} - p_{i})$ (2)

where, $b_{0} \in R^{3}$ , which represents from the base vector of centroid point to the first joint position; $p_{0} \in R^{3}$ , which represents the position vector of base relative to the inertial coordinate system;. $p_{e} \in R^{3}$ , which represents the position vector of end effector and $p_{i} \in R^{3} (i = 1, \dots, n)$ , which is the ith joint relative to the position vector of the inertial coordinate system.

The formula (2) is differential, and then the linear velocity, which is of the end effector on free-floating space robot, is as follows:

$ν_{e} = ν_{0} + ω_{0} \times (p_{e} - p_{0}) + \sum_{i = 1}^{6} [k_{i} \times (p_{e} - p_{i})] \cdot \overset{\cdot}{θ}$ (3)

where, $k_{i} \in R^{3}$ , which represents the unit direction vector of z axis.

The angular velocity, which is of the end effector on free-floating space robot, is as follows:

$ω_{e} = ω_{0} + \sum_{i = 1}^{6} k_{i} \overset{\cdot}{θ}$ (4)

According to the law of conservation of momentum, it is assumed that the initial momentum is zero. It is shown as follows:

$Ι_{b} [\begin{matrix} ν_{0} \\ ω_{0} \end{matrix}] + Ι_{m} \overset{\cdot}{θ} = 0$ (5)

where, $Ι_{b} \in R^{6 \times 6}$ is called the inertia matrix of base and $Ι_{m} \in R^{6 \times 6}$ , the inertia matrix of space manipulator.

2.2. Constraint conditions and parameters of the joint

In the process of movement, the velocity and acceleration, which is the joint angular of space robot, should be limited. In addition, it is in favour of the path planning of space robot smooth. The constraints of joint angle are as follows:

$θ_{i} (t_{0}) = θ_{i 0}, θ_{i} (t_{f}) = θ_{i f}$ (6)

${\overset{\cdot}{θ}}_{i} (t_{0}) = {\overset{\cdot}{θ}}_{i} (t_{f}) = 0, {\overset{\cdot\cdot}{θ}}_{i} (t_{0}) = {\overset{\cdot\cdot}{θ}}_{i} (t_{f}) = 0$ (7)

$θ_{i_m i n} \leq θ_{i} (t) \leq θ_{i_m a x}$ (8)

where, $i = 1, \dots, 6$ , $θ_{i 0}$ is called the initial joint angle; $θ_{i f}$ , the expecting joint angle; $θ_{i_m i n}$ , the minimum value of the ith joint angle and $θ_{i_m a x}$ , the maximum value of the ith joint angle.

In addition, the limiting conditions, such as the joint angular velocity and acceleration of space robot, are as follows:

$| {\overset{\cdot}{θ}}_{i} (t) | \leq {\overset{\cdot}{θ}}_{i_l i m i t}, | {\overset{\cdot\cdot}{θ}}_{i} (t) | \leq {\overset{\cdot\cdot}{θ}}_{i_l i m i t}$ (9)

where, ${\overset{\cdot}{θ}}_{i_l i m i t}$ is called the limited velocity of joint angular and ${\overset{\cdot\cdot}{θ}}_{i_l i m i t}$ ,the limited acceleration of joint angular.

The constraint conditions of the formula (8) need to be met by the joint angles. Because the sine function has the feature of boundedness, the joint angle can be directly constrained. In this paper, the joint angles are parameterized by the 7th order sine polynomial functions. As shown in the following formula:

$\begin{array}{l} θ_{i} (t) = Δ_{i 1} sin (α_{i 7} t^{7} + α_{i 6} t^{6} + α_{i 5} t^{5} + α_{i 4} t^{4} \\ + α_{i 3} t^{3} + α_{i 2} t^{2} + α_{i 1} t + α_{i 0}) + Δ_{i 2} \end{array}$ (10)

where, $i = 1, \dots, 7$ , $α_{i 0} ~ α_{i 7}$ is called the polynomial coefficient, combining with the limited conditions of formula (8): $Δ_{i 1} = \frac{θ_{i_m a x} - θ_{i_m i n}}{2}, Δ_{i 2} = \frac{θ_{i_m a x} + θ_{i_m i n}}{2}$ .

The velocity is obtained from the formula (10). As shown in the following formula:

$\begin{array}{l} {\overset{\cdot}{θ}}_{i} (t) = Δ_{i 1} cos (α_{i 7} t^{7} + α_{i 6} t^{6} + α_{i 5} t^{5} + α_{i 4} t^{4} + \\ α_{i 3} t^{3} + α_{i 2} t^{2} + α_{i 1} t + α_{i 0}) (7 α_{i 7} t^{6} + 6 α_{i 6} t^{5} + \\ 5 α_{i 5} t^{4} + 4 α_{i 4} t^{3} + 3 α_{i 3} t^{2} + 2 α_{i 2} t + α_{i 1}) \end{array}$ (11)

The acceleration is obtained from the formula (11). As shown in the following formula:

$\begin{array}{l} {\overset{\cdot\cdot}{θ}}_{i} (t) = - Δ_{i 1} sin (α_{i 7} t^{7} + α_{i 6} t^{6} + α_{i 5} t^{5} + α_{i 4} t^{4} + \\ α_{i 3} t^{3} + α_{i 2} t^{2} + α_{i 1} t + α_{i 0}) (7 α_{i 7} t^{6} + 6 α_{i 6} t^{5} + \\ 5 α_{i 5} t^{4} + 4 α_{i 4} t^{3} + 3 α_{i 3} t^{2} + 2 α_{i 2} t + α_{i 1}) + \\ Δ_{i 1} cos (α_{i 7} t^{7} + α_{i 6} t^{6} + α_{i 5} t^{5} + α_{i 4} t^{4} + \\ α_{i 3} t^{3} + α_{i 2} t^{2} + α_{i 1} t + α_{i 0}) (42 α_{i 7} t^{5} + \\ 30 α_{i 6} t^{4} + 20 α_{i 5} t^{3} + 12 α_{i 4} t^{2} + 6 α_{i 3} t + 2 α_{i 2}) \end{array}$ (12)

The formulas (6) and (7) are plugged into the formulas (10)–(12). As shown in the following formula:

$α_{i 0} = arcsin (\frac{θ_{i 0} - Δ_{i 2}}{Δ_{i 1}})$ (13)

$α_{i 1} = 0, α_{i 2} = 0$ (14)

$α_{i 3} = 10 (\begin{matrix} arcsin \frac{θ_{i f} - Δ_{i 2}}{Δ_{i 1}} - \\ arcsin \frac{θ_{i 0} - Δ_{i 2}}{Δ_{i 1}} \end{matrix}) - (3 α_{i 7} t^{7} - α_{i 6} t^{6}) / t^{3}$ (15)

$α_{i 4} = - 15 (\begin{matrix} arcsin \frac{θ_{i f} - Δ_{i 2}}{Δ_{i 1}} - \\ arcsin \frac{θ_{i 0} - Δ_{i 2}}{Δ_{i 1}} \end{matrix}) + (8 α_{i 7} t^{7} + 3 α_{i 6} t^{6}) / t^{4}$ (16)

$α_{i 5} = 6 (\begin{matrix} arcsin \frac{θ_{i f} - Δ_{i 2}}{Δ_{i 1}} - \\ arcsin \frac{θ_{i 0} - Δ_{i 2}}{Δ_{i 1}} \end{matrix}) - (6 α_{i 7} t^{7} - 3 α_{i 6} t^{6}) / t^{5}$ (17)

It is derived from the above formulas. Only the variables $α_{i 6}$ and $α_{i 7}$ cannot be certain, which are in the functions of joint angles, the velocity and acceleration of joint angles. As shown in the following formula:

$α = (α_{16}, α_{17}, \dots, α_{i 6}, α_{i 7})$ (18)

When $α$ is confirmed, the path of free-floating space robot should be planned according to the formulas (10)–(12).

2.3. Definition of objective function

In this paper, it is the path planning for minimizing base disturbance of space robot, including the position and attitude of base. As shown in the following formula:

$F (α) = \frac{{‖ δ p_{b} ‖}_{2}}{ω_{p}} + \frac{‖ δ q_{b} ‖}{ω_{q}} + \frac{F_{\overset{\cdot}{θ}}}{ω_{\overset{\cdot}{θ}}} + \frac{F_{\overset{\cdot\cdot}{θ}}}{ω_{\overset{\cdot\cdot}{θ}}}$ (19)

where, ${‖ δ p_{b} ‖}_{2} = \sqrt{δ p_{b}^{T} δ p_{b}}$ ; $δ p_{b}$ , which represents the position error between the initial and final state; $δ q_{b}$ , which represents the attitude error between the initial and final state; $ω_{p}$ , called the weight coefficient of position error; $ω_{q}$ , called the weight coefficient of attitude error; $ω_{\overset{\cdot}{θ}}$ , called the weight coefficient of joint angular velocity constraints; $ω_{\overset{\cdot\cdot}{θ}}$ , called the weight coefficient of joint angular acceleration constraints and $F_{\overset{\cdot}{θ}}$ and $F_{\overset{\cdot\cdot}{θ}}$ can be respectively, defined as follows:

$F_{\overset{\cdot}{θ}} = max_{1 \leq i \leq 6} (J_{{\overset{\cdot}{θ}}_{i}}), J_{{\overset{\cdot}{θ}}_{i}} = {\begin{cases} \begin{matrix} 0 & \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} {\overset{\cdot}{θ}}_{i_max} \leq {\overset{\cdot}{θ}}_{i_lim i t} \end{matrix} \\ \begin{matrix} ({\overset{\cdot}{θ}}_{i_max} - {\overset{\cdot}{θ}}_{i_lim i t}) / {\overset{\cdot}{θ}}_{i_lim i t} & e l s e \end{matrix} \end{cases}$ (20)

$F_{\overset{\cdot\cdot}{θ}} = max_{1 \leq i \leq 6} (J_{{\overset{\cdot\cdot}{θ}}_{i}}), J_{{\overset{\cdot\cdot}{θ}}_{i}} = {\begin{cases} \begin{matrix} 0 & \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} {\overset{\cdot\cdot}{θ}}_{i_max} \leq {\overset{\cdot\cdot}{θ}}_{i_lim i t} \end{matrix} \\ \begin{matrix} ({\overset{\cdot\cdot}{θ}}_{i_max} - {\overset{\cdot\cdot}{θ}}_{i_lim i t}) / {\overset{\cdot\cdot}{θ}}_{i_lim i t} & e l s e \end{matrix} \end{cases}$ (21)

where, ${\overset{\cdot}{θ}}_{i_m a x}$ , which represents the maximum velocity of the ith joint and ${\overset{\cdot\cdot}{θ}}_{i_m a x}$ , which represents the maximum acceleration of the ith joint.

As a consequence, the issues of this paper convert to that solving the two unknown parameters of the formula (18), with a kind of proper method. And the fitness function of formula (19) can be achieved using the minimum value by the optimal parameters.

3. Improved Chaotic Particle Swarm Optimization

3.1. Standard particle swarm optimization

In 1995, the standard particle swarm optimization (PSO) was developed by Kennedy and Eberhat [14], inspired by the behaviour of birds when they were preying. There are many advantages of the standard particle swarm optimization, such as simple structure, less parameters, fast convergence rate and easy to operate. In the particle swarm optimization (PSO), each particle is described by the two variables of velocity and position. When the particle finds an optimal position, the particle will move towards the position, and other particles also move towards to the same position. The position of the ith particle should be expressed as in the formula $x_{i} = [x_{i 1}, \dots, x_{i d}]$ . The velocity of the ith particle should be expressed as in the formula $v_{i} = [v_{i 1}, \dots, v_{i d}]$ . It can be written as follows [15]:

$ν_{i j} (t + 1) = ω v_{i j} (t) + c_{1} r_{1} [p_{i j} - x_{i j} (t)] + c_{2} r_{2} [p_{g j} - x_{i j} (t)]$ (22)

$x_{i j} (t + 1) = x_{i j} (t) + v_{i j} (t + 1)$ (23)

where, $j = 1, \dots, d$ ; d is called the dimensions of the search space; ω, the inertia weight; c₁ and c₂, called acceleration coefficients; r₁ and r₂, which represent the two independently uniformly distributed random variables with range $[0, 1]$ ; $p_{i j}$ , called the optimal value of particle itself and $p_{g j}$ , called the optimal value of global.

3.2. Improved chaotic particle swarm optimization

Like the most optimization algorithms, the standard particle swarm optimization (PSO) is born with some shortcomings, including premature convergence and easily plunged into local optimum or others. To make up for the defect, in 2005, the chaotic particle swarm optimization was developed by Bo Liu et al. [16]. The chaotic particle swarm optimization is a combination of chaotic theory and particle swarm optimization. It refers to the chaotic local search algorithm. For the fitness function, the search process is transformed into the orbital ergodic process of chaos. The advantage of the universality and ergodicity of chaos makes the particles avoid falling into local minimum in the search process, but the calculation accuracy of chaos particle swarm optimization (CPSO) is not enough. The main reason is that the uncertainty of random factors, especially the random generation of 80% particles. And on this basis, an improved chaotic particle swarm optimization (ICPSO) is brought up: (1) Adopting the non-linear dynamic inertia weight, in other words, the inertia weight changes automatically with the value of objective function; (2) The idea of golden ratio is added up to the CPSO, that is to say, 38.2% of the best particles are retained and (3) Logistic function is replaced by sinusoidal function as the chaotic sequence generator.

The value of the particles is influenced by its current speed, which is determined by the inertia weight. The particles will have balanced exploration ability and development ability by the appropriate selection of inertia weight. The larger inertia weight can enhance the global search ability; on the contrary, the smaller inertia weight can improve the local search ability. It includes a constant method, a linear gradient method and adaptive weighting method. In this paper, an improved adaptive weighting method is adopted, as shown in the following formula [17]:

$ω = {\begin{cases} \begin{matrix} ω_{max} - i \frac{ω_{max} - ω_{min}}{max_g e n} & \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} f < f_{a v g} / 2 \end{matrix} \\ \begin{matrix} ω_{min} + \frac{(f - f_{min}) (ω_{max} - ω_{min})}{f_{a v g} / 2 - f_{min}} & f_{a v g} / 2 \leq f \leq f_{a v g} \end{matrix} \\ \begin{matrix} ω_{max} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} f \geq f_{a v g} / 2 \end{matrix} \end{cases}$ (24)

where, $ω_{m i n}$ is the minimum value of ω; $ω_{m a x}$ , the maximum value of ω; f, the value of fitness function, which is the current particle; $f_{m i n}$ , the minimum value of fitness function, which is all the particles; $f_{a v g}$ , the average value of fitness function, which is all the particle; i, the current number of iterations and $m a x_g e n$ , the maximum iterations. As shown in the formula (24), the inertia weight is along with the value of fitness function. When the target value of each particle is close to the local optimum, the inertia weight will increase to avoid trapping in a local optimum; on the other hand, the inertia weight will reduce.

Different chaotic sequence generators in the chaos particle swarm optimization (CPSO) are adopted. Many scholars have studied on the choice of chaotic sequence generator. To enhance CPSO for optimization and to consider the operability, one-dimensional, non-invertible maps were adopted. In 2013, the different chaotic mapping functions were tested through the test functions in the article by Amir Hossein Gandomi et al. [18]. This paper discusses the performing of different chaotic maps. From the analysis and results of this paper, it can be seen that the sinusoidal and singer functions are more effective than others. They perform well in dealing with most optimization problems, whereas sometimes the sinusoidal function slightly outperforms the sine function. In this paper, the sinusoidal function is adopted. As shown the in following formula [19]:

$x_{k + 1} = a x_{k}^{2} sin (π x_{k})$ (25)

where, $a = 2.3, x_{0} = 0.7$ . As shown in the following simplified formula:

$x_{k + 1} = sin (π x_{k})$ (26)

The chaotic particle swarm optimization consists of chaos optimization and particle swarm optimization. Furthermore, the chaotic local search is involved in the process of algorithm. In the process of proposed algorithm, firstly the adaptive weighting is adopted to the global searching of particle swarm, using the formula as shown in the equation (24). Then the idea of golden ratio is brought in, where the 38.2% particles are reserved for the swarm. And then the sinusoidal function is adopted as a chaotic sequence generator, where the 38.2% particles are searched by the chaotic local. Finally, to keep the diversity of particles, the remaining 61.8% are randomly produced in the dynamic contraction area. The dynamic contraction area is shown as follows:

$\begin{array}{l} x_{min, j} = max {x_{min, j}, x_{g, j} - r * (x_{max, j} - x_{min, j})}, 0 < r < 1 \\ x_{max, j} = min {x_{min, j}, x_{g, j} + r * (x_{max, j} - x_{min, j})}, 0 < r < 1 \end{array}$ (27)

where, $x_{g, j}$ represents the jth dimensional variable value of the current pbest.

In this paper, the flowchart of improved chaotic particle swarm optimization (ICPSO) is shown in Figure 1.

Figure 1.

The flowchart of ICPSO

3.3. Path planning based on ICPSO

In this paper, the proposed algorithm is used to plan the path of space robot under the constrain conditions of minimizing base disturbance. The process, which solves the optimal parameters $α = (α_{16}, α_{17}, \dots, α_{i 6}, α_{i 7})$ , is as shown the following steps:

Step 1: The parameters of each particle are set up: N is the size of particles; $ω_{m a x}$ denotes the maximum value of inertia weight; $ω_{m i n}$ , the minimum value of inertia weight; c₁ and c₂, acceleration coefficients; $x_{m a x}$ , the maximum of independent variable search domain; $x_{m i n}$ ; the minimum of independent variable search domain; M, the maximum number of iterations; $M a x C$ , the maximum step number of chaos search; D, the number of independent variable; $x m$ , the independent variable when the minimum value is got by the fitness function and f_v, the minimum value of fitness function.

Step 2: The position and velocity of each particle in the population are initialized randomly.

Step 3: The value of fitness function, of each particle, is evaluated. The current position and values of each particle are stored in the pbest. The best value and its individual position of the fitness function are stored in the gbest.

Step 4: The velocity and position of particles are updated using the formulas (22) and (23). And the weight of particles is updated using the formula (24). For each particle, the current value of the fitness function is compared with the best location. If the current value of the fitness function is better than the best one, it will be stored as the best position. All the values of the pbest are compared with the gbest, and the gbest is updated.

Step 5: The fitness function value of each particle is calculated. The 38.2% better particles are saving selected.

Step 6: The optimal particles of population are searched by chaotic local. And at the same time, the pbest of particles and the gbest of population are updated.

Step 7: If the condition of stopping (usually the maximum number of iterations defaulted or computation accuracy) is met, the search is broken, the optimal result $α = (α_{16}, α_{17}, \dots, α_{i 6}, α_{i 7})$ is the output, therefore determining the optimal path planning; if not satisfied, then go back to the step (8);

Step 8: The contraction of dynamic regions is as in the formula (27).

Step 9: The remaining 61.8% particles are produced in the contraction of dynamic regions, and then go back to the step (3).

4. Experiment Simulation and Results

To solve the problem of minimizing base disturbance of space robot, we have adopted a proposed method. First, as shown in the formula (1), the forward kinematics equation is established. Then, the joint angles are parameterized by the 7th order sine polynomial functions. Through derived by the formulas, only the variables $α_{i 6}$ and $α_{i 7}$ cannot be certain. At last, the fitness function is defined according to the different constraints. An actual problem is transformed into a math problem. Using the ICPSO (CPSO and PSO) algorithm, we find the minimum value of the fitness function, and the formula (18) is confirmed. Then the path of free-floating space robot should be planned according to the formulas (10)–(12). The flowchart of simulation setup is shown in Figure 2.

Figure 2.

The flowchart of simulation setup

To validate whether the proposed method is effective, which is on the path planning for minimizing base disturbance of space robot, a kinematics model of the system has been established. The system consists of the free floating base and 6 degrees of freedom space robot. The proposed algorithm is simulated and verified on the platform of MATLAB R2013a. And the result of the proposed algorithm is compared with the standard PSO and the CPSO algorithm [16] proposed by Bo Liu et al. in 2005. It is assumed that the time of path planning is $t = 10 s$ , the quality parameters of the space robot are as shown in Table 1, the D-H parameters of space robot are as shown in Table 2 [20].

Table 1.

The quality parameters of space robot

	Link 1	Link 2	Link 3
Mass/kg	4.239	6.3585	21.195
Link/m	0.2	0.3	1
$I_{x x}$ /kg.m²	0.168	0.0517	1.7804
$I_{y y}$ /kg.m²	0.168	0.0517	1.7804
$I_{z z}$ /kg.m²	0.0009	0.0013	0.0044
	Link 4	Link 5	Link 6
Mass/kg	21.195	6.3585	4.239
Link/m	1	0.3	0.2
$I_{x x}$ /kg.m²	1.7804	0.0517	0.0168
$I_{y y}$ /kg.m²	1.7804	0.0517	0.0168
$I_{z z}$ /kg.m²	0.0044	0.0013	0.0009

Table 2.

The D-H parameters of space robot

i	$θ_{i} (deg)$	$α_{i - 1} (deg)$	$a_{i - 1} (m m)$	$d_{i} (m m)$
1	θ ₁	0	0	$l_{0} / 2 + l_{1}$
2	θ ₂	$- π / 2$	0	0
3	θ ₃	$π / 2$	0	$l_{2} + l_{3}$
4	θ ₄	$- π / 2$	0	0
5	θ ₅	0	l ₄	0
6	θ ₆	$- π / 2$	0	0

4.1. Setting of the parameters

By experience, the initial joint angle of the space robot is set up as $θ_{i 0} = [0^{o}, 0^{o}, 0^{o}, 0^{o}, 0^{o}, 0^{o}]$ , the expected joint angle is set up as $θ_{i f} = [1 2^{o}, 3 1^{o}, - 1 7^{o}, - 5 0^{o}, 2 6^{o}, - 8 3^{o}]$ . The scope of the joint angle is as shown in the following formula:

$\begin{array}{l} - 160^{o} \leq θ_{1} \leq + 160^{o}, - 160^{o} \leq θ_{2} \leq + 160^{o} \\ - 160^{o} \leq θ_{3} \leq + 160^{o}, - 179^{o} \leq θ_{4} \leq + 70^{o} \\ - 179^{o} \leq θ_{5} \leq + 70^{o}, - 160^{\circ} \leq θ_{6} \leq + 160^{o} \end{array}$ (28)

The scope of the joint angular velocity is as shown in the following formula:

$| {\overset{\cdot}{θ}}_{i} (t) | \leq 60^{o} / s, i = 1, \dots, 6$ (29)

The scope of the joint angular acceleration is as shown in the following formula:

$| {\overset{\cdot\cdot}{θ}}_{i} (t) | \leq 60^{o} / s^{2} = 1.0472 r a d / s^{2}, i = 1, \dots, 6$ (30)

The coefficients of the fitness function are determined by the precision requirement. In this paper, we assume that the deviation of end position is less than $2 \cdot 10^{- 3}$ meter, and the rotation angle of base attitude is not more than one degree. So the coefficients of the fitness function should be defined, respectively, as shown in the following formula:

$ω_{p} = 2 \times 10^{- 3}, ω_{q} = sin (π / 360), ω_{\overset{\cdot}{θ}} = ω_{\overset{\cdot\cdot}{θ}} = 0.01$ (31)

The parameters of optimization algorithms are decided according to the usual value. For example, the size of particles N usually obtains 20–40 for the most optimization problems; the acceleration coefficients c₁ and c₂ often equal to 2; the value scope of inertia weight $ω_{}$ is 0–1. Considering the various factors in this paper, the parameters of ICPSO and CPSO are set up as in the formula (32); the parameters of PSO are set up as shown in the formula (33).

$\begin{array}{l} N = 40, c_{1} = c_{2} = 2, ω_{max} = 0.9, ω_{min} = 0.2, x_{max} = 0.1, \\ x_{min} = - 0.1, M = 100, M a x C = 0.1, D = 12 \end{array}$ (32)

$N = 40, c_{1} = c_{2} = 2, ω = 0.5, M = 100, D = 12$ (33)

4.2. Results and comparison

The time complexity of ICPSO is $Ο (0.0312 N^{2} \cdot M)$ , the standard PSO is $Ο (N \cdot M)$ and CPSO is $Ο (0.002 N^{2} \cdot M)$ . From the results of calculation, the time complexity of ICPSO is an order of magnitude higher than PSO. Because the value of N is not large, it can be ignored.

The proposed ICPSO algorithm, standard PSO algorithm, and CPSO algorithm are, respectively, used in the path planning for minimizing base disturbance of space robot. The optimal parameters and fitness function values are rounded as shown in the following results:

${\begin{cases} α_{_ICPSO} = [\begin{array}{l} - 0.2422, 0.0086, - 0.1115, 0.1953 \\ - 0.2527, - 0.2295, - 0.0466, - 0.0227 \\ - 0.0130, - 0.0816, 0.1270, 0.0089 \end{array}] * 1 0^{- 5} \\ f (α)_{_I C P S O} = 14.3820 \end{cases}$ (34)

${\begin{cases} α_{_PSO} = [\begin{array}{l} 0.2139, - 0.0026, 0.3550, 0.2086 \\ 0.3158, 0.2422, 0.1258, 0.0667 \\ - 0.0028, 0.0458, 0.2291, 0.2356 \end{array}] * 1 0^{- 5} \\ f (α)_{_P S O} = 15.0957 \end{cases}$ (35)

${\begin{cases} α_{_CPSO} = [\begin{array}{l} 0.1376, - 0.0143, 0.2325, 0.2691 \\ 0.0586, 0.2494, - 0.1014, 0.0775 \\ - 0.6430, 0.2348, - 0.3314, - 0.2002 \end{array}] * 1 0^{- 5} \\ f (α)_{_C P S O} = 15.0212 \end{cases}$ (36)

The black (dotted line) represents the results with ICPSO, cyan (diamond) with standard PSO, and red (pentagram) with CPSO.

Figure 3 compares the position of the base in this period of time with the ICPSO, standard PSO, and CPSO.

Figure 3.

The position of base

From Figure 3, the position of base when it completes the movement can be seen. The result obtained by ICPSO is [−0.0005, −0.3163, −3.0872], and the error is [−0.0005, 0.0170, −0.0039]. From the standard PSO, the result is [−0.0054, −0.3140, −3.0867], and the error is [−0.0054, 0.0193, −0.0034]. The position of base when it completes the movement with CPSO is observed. The result is [−0.0036, −0.3169, −3.0870], and the error is [−0.0036, 0.0164, −0.0037].

The comparison of the altered attitude of base in this period of time with the ICPSO, standard PSO, and CPSO is shown in Figure 4:

Figure 4.

The attitude of base

From Figure 4, the attitude of base when it completes the movement is seen. The result is [0.0168, −0.0100, 0.0497, 0.9993], and the error is [0.0168, −0.0100, 0.0497, −0.0007]. With standard PSO, the result is [0.0193, −0.0061, 0.0474, 0.9999], and the error is [0.0193, −0.0061, 0.0474, −0.0001]. The attitude of the base when it completes the movement with CPSO is [0.0380, −0.0058, 0.0522, 0.9991], and the error is [0.0380, −0.0058, 0.0522, −0.0009]. To reflect the changing of base attitude intuitively, the quaternion expressions are converted to the “z-y-x” Euler angle. The error is [−2.9661, −0.6193, −0.4534] (rad) with ICPSO. With standard PSO, the error is [−3.0141, −0.7591, −0.3069] (rad). Obtained by CPSO, the error is [−2.7550, −1.2266, −0.4951] (rad).

The solved optimal parameters are substituted into the formulas (10)–(12), then the Figures (5)–(7) are obtained. The comparison of altered joint path, velocity and acceleration of space robot in this period of time is shown in Figures 5–7:

Figure 5.

Joint path of the space robot

Figure 6.

Velocity of the space robot

Figure 7.

Acceleration of the space robot

Comparing equations (34)–(36), we can see that the fitness function value obtained by the proposed ICPSO algorithm is the smallest, Figures 3 and 4 are the detailed explanations. Figure 3 shows that the position of base is the most closest to the desired position. Figure 4 displays that the attitude of base is the most closest to the anticipated attitude. On the whole, the proposed ICPSO algorithm can find the optimal solution. Through Figure 5, we can see that the joint path is smooth. Furthermore, the limitation of velocity and acceleration are considered. The smaller scope of joint angle, velocity and acceleration implies less energy consumption to drive the joints. And the manipulator is also easy to control. It is proved that the method proposed in this paper has some practical significance.

5. Conclusion

This paper provides a reasonable path planning method as a solution for minimizing base disturbance of space robot. First, the joint angle should be parameterized in using of the 7th order polynomial sine functions. The fitness function is defined according to the position and attitude of minimizing base disturbance. And the optimization problem of path planning is converted into a non-linear problem. Because the standard PSO algorithm is difficult to find the optimal solution, easy to fall into local optimum and slow convergence speed, an improved chaotic particle swarm optimization (ICPSO) is put forward to optimize the unknown coefficients. The proposed algorithm is compared with the standard PSO and CPSO algorithms in the literature by the experimental simulation. It is confirmed that the proposed algorithm is more effective, such as fast convergence speed and easy to find the optimal solution.

The problem that the path planning for minimizing base disturbance of space robot is effectively solved by this proposed method. Furthermore, the joint angle, velocity and acceleration are limited in the process of planning. Then the method not only has a higher application, but also for the path planning problem of other space robots has a certain guiding significance, for example, it is can be extended to the seven degrees of freedom free-floating space robot.

Footnotes

6. Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 61425002,61300015),Program for Changjiang Scholars and Innovative Research Team in University (No.IRT_15R07),the Program for Liaoning Innovative Research Team in University (No. LT2015002),the Program for Dalian High-level Talent's Innovation,the Program for Technology Research in New Jinzhou District (No. KJCX-ZTPY-2014-0012),and the Program for Liaoning Key Lab of Intelligent Information Processing and Network Technology in University.

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