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外文文献：

Space Robot Path Planning for Collision Avoidance

Yuya Yanoshita and Shinichi Tsuda

Abstract — This paper deals with a path planning of space robot which includes a collision avoidance algorithm.For the future space robot operation, autonomous and self-contained path planning is mandatory to capture a target without the aid of ground station.Especially the collision avoidance with target itself must be always considered.Once the location, shape and grasp point of the target are identified, those will be expreed in the configuration space.And in this paper a potential method.Laplace potential function is applied to obtain the path in the configuration space in order to avoid so-called deadlock phenomenon.Improvement on the generation of the path has been observed by applying path smoothing method, which utilizes the spline function interpolation.This reduces the computational load and generates the smooth path of the space robot.The validity of this approach is shown by a few numerical simulations.Key Words —Space Robot, Path Planning, Collision Avoidance, Potential Function, Spline Interpolation

I.INTRODUCTION

In the future space development, the space robot and its autonomy will be key features of the space technology.The space robot will play roles to construct space structures and perform inspections and maintenance of spacecrafts.These operations are expected to be performed in an autonomous.In the above space robot operations, a basic and important task is to capture free flying targets on orbit by the robotic arm.For the safe capturing operation, it will be required to move the arm from initial posture to final posture without collisions with the target.山东建筑大学毕业论文外文文献及译文

The configuration space and artificial potential methods are often applied to the operation planning of the usual robot.This enables the robot arm to evade the obstacle and to move toward the target.Khatib proposed a motion planning method, in which between each link of the robot and the obstacle the repulsive potential is defined and between the end-effecter of the robot and the goal the attractive potential is defined and by summing both of the potentials and using the gradient of this potential field the path is generated.This method is advantageous by its simplicity and applicability for real-time operation.However there might be points at which the repulsive force and the attractive force are equal and this will lead to the so-called deadlock.In order to resolve the above iue, a few methods are proposed where the solution of Laplace equation is utilized.This method aures the potential fields without the local minimum, i.e., no deadlock.In this method by numerical computation Laplace equation will be solved and generates potential field.The potential field is divided into small cells and on each node the discrete value of the potential will be specified.In this paper for the elimination of the above defects, spline interpolation technique is proposed.The nodal point which is given as a point of path will be defined to be a part of smoothed spline function.And numerical simulations are conducted for the path planning of the space robot to capture the target, in which the potential by solving the Laplace equation is applied and generates the smooth and continuous path by the spline interpolation from the initial to the final posture.II.ROBOT MODEL The model of space robot is illustrated in Fig.1.The robot is mounted on a spacecraft and has two rotary joints which allow the in-plane motion of the end-effecter.In this case we have an additional freedom of the spacecraft attitude angle and this will be considered the additional rotary joint.This means that the space robot is three linked with 3 DOF(Degree Of Freedom).The length of each link and the angle of each rotary joint are given byliandi(i = 1,2,3), respectively.In order to simplify the discuions a few aumptions are made in this paper:-the motion of the space robot is in-plane,i.e., two dimensional one.-effect of robot arm motion to the spacecraft attitude is negligible.山东建筑大学毕业论文外文文献及译文

2220

（2）2xyAnd this will be converted into the difference equation and then solved by Gau-Seidel method.In equation(2)if we take the central difference formula for second derivatives, the following equation will be obtained: 220x2y2(xx,y)2(x,y)(xx,y)

（3）x2(x,yy)2(x,y)(x,yy)y2where x，y are the step(cell)sizes between adjacent nodes for each x, y direction.If the step size is aumed equal and the following notation is used:

(xx,y)i1,j

Then equation(3)is expreed in the following manner: i1,ji1,ji,j1i,j1i,j0

（4）

And as a result, two dimensional Laplace equation will be converted into the equation(5)as below: i,j1i1,ji1,ji,j1i,j1

（5）4In the same manner as in the three dimensional case, the difference equation for the three dimensional Laplace equation will be easily obtained by the following:

i,j,k1i1,j,ki1,j,ki,j1,ki,j1,ki,j,k1i,j,k1

（6）6In order to solve the above equations we apply Gau-Seidel method and have equations as follows: n1i,j1n1nn1i1,jin1,ji,j1i,j1

（7）41where in,j is the computational result from the(n +1)-th iterative calculations of the potential.In the above computations, as the boundary conditions, a certain positive number 0 is defined for the obstacle and 0 for the goal.And as the initial conditions the same number 0 is

山东建筑大学毕业论文外文文献及译文

The length of each link is given as follows:

l1 =1.4[m], l2 = 2.0[m], l3 = 2.0[m] , and the target satellite was aumed 1m square.The grasp handle, 0.1 m square, was located at a center of one side of the target.So this handle is a goal of the path.Let us explain the geometrical relation between the space robot and the target satellite.When we consider the operation after capturing the target, it is desirable for the space robot to have the large manipulability.Therefore in this paper the end-effecter will reach the target when the manipulability is maximized.In the 3DOF case, not depending on the spacecraft body attitude, the manipulability is measured by2,3.And if we aume the end-effector of the space robot should be vertical to the target, then all of the joints angles are predetermined as follows:

1160.7o,232.8o,376.5o

As all the joints angles are determined, the relative position between the spacecraft and the target is also decided uniquely.If the spacecraft is aumed to locate at the origin of the inertial frame(0, 0), the goal is given by(-3.27,-2.00)in the above case.Based on these preparations, we can search the path to the goal by moving the arm in the configuration space.Two simulations for path planning were carried out and the results are shown below.A.2 DOF Robot In order to simplify the situation, the attitude angle(Link 1 joint angle)is aumed to coincide with the desirable angle from the beginning.The coordinate system was aumed as shown in Fig.2.1 was taken into consideration for the calculation of the initial condition of the Link 2 and its

山东建筑大学毕业论文外文文献及译文

the connection of-180 degrees in the 1direction was illustrated.From this figure it is easily seen that over-180 degrees the path is going toward the goal C.B and C are the same goal point.V.CONCLUSION In this paper a path generation method for capturing a target satellite was proposed.And its applicability was demonstrated by numerical simulations.By using interpolation technique the computational load will be decreased and smoothed path will be available.Further research will be recommended to incorporate the attitude motion of the spacecraft body affected by arm motion.山东建筑大学毕业论文外文文献及译文

本文对上述缺陷的消除，提出了样条插值技术。给定的节点作为路径的一部分将被定义为平滑样条函数的一部分。为了捕获到目标，空间机器人的路径规划运用了数字模拟技术，它是通过对势场域求解拉普拉斯函数来实现的，并且从最初的位置到末尾位置的样条插值来产生连续光滑的路径。

2.机器人模型

空间机器人的模型如图1所示：机器人被安装在航天器和两个旋转接头上，这两个旋转接头可以实现末端执行器的平面运动。这种情况下，我们的航天器的姿态角有一个额外的自由度，我们将这个额外的自由度视为额外的旋转接头。这意味着空间机器人有三个自由度的链接，每个链路的长度和每个旋转关节角度，分别由li和i(i = 1,2,3)表示。为了简化这个讨论，本文做了一些假设：（1）空间机器人的运动是平面的，即二维；

（2）机器人机械臂的运动对航天器姿态的影响是可以忽略的；（3）机器人运动给出了静态几何关系，并没有明确的依赖时间；（4）目标卫星在惯性的作用下是很稳定的；

一般情况下，平面运动和空间运动将分别进行，所以我们可以假设上面的第一个不失一般性，第二个假设来自机械臂和航天器质量比的比较，对于第三个假设，我们专注于生成机器人的路径规划，这基本上是由几何关系的静态性质决定，因此并不依赖明确的时间，最后一个就是合作卫星。

图1 双链路空间机器人

0

山东建筑大学毕业论文外文文献及译文

为了解决上述方程，我们应用了高斯赛德尔算法和求解方程，如下：

n1i,j1n1nn1i1,jin1,ji,j1i,j1

(8)4in,j1表示势场域的迭代计算结果。

在上述的计算中，作为边界条件，定义特定的正数0来表示障碍物和目标。为保证初始条件相同，给所有的自由节点赋同样的数值0。通过这种方法，在迭代计算的边界节点获得的的值将不会改变，而且自由节点的值是不同。我们应用相同的域值作为障碍物，并且按照迭代计算方法，则目标周围较小的势场域会像障碍物一样缓慢的向周围传播，势场域就是根据上述方法建立的。采用4节点相邻的空间机器人存在的节点上的势场，最小的节点选择移动到另一点，这个过程最终引导机器人无碰撞的到达目标的位置。

样条内插法：

通过上述方法给出的路径不能保证能够与另一个目标顺利连接，如果节点上没有给定目标，我们会将栅格划分成的更小，但这将增加计算量和所用时间。为了消除这些弊端，我们提出利用样条插值技术。通过在将节点解给出的通过点的道路上，我们试图获得顺利连接路径与准确获取最初的和最后的点。本文主要是通过MATLAB命令应用样条函数。

配置空间：

当我们在应用拉普拉斯势域的时候，路径搜索只能在当机器人在搜索空间过程中表示成一个点的情况下才能保证实现。配置空间(C空间)中机器人仅表示为一个点，主要是用于路径搜索。将真正的空间转换到C空间，必须执行判断碰撞条件的计算，如果碰撞存在,相应的点在c空间被认为是障碍。本文中，在生成势场域时，所有现实空间的点的生成条件对应于所有的节点都是经过计算的。在构成的机械臂和生成的节点的障碍物出现判断选择时，该节点可以看作是在c空间的障碍点。

数值仿真：

基于上述方法对于捕获目标卫星路径规划的检查是使用空间机器人模型进行的。在本文中，我们假设空间机器人二维和2自由度机械手臂见图1。每个链接的长度给出如下:

山东建筑大学毕业论文外文文献及译文

初始角度：264.3,390o 目标角度：2166.5,376.5o

在这种情况下，势场域分成180段计算成C空间。图3显示的C空间和计划中的很大一部分的中心是由航天器本体映射的障碍了，左边部分是目标卫星的映射。图4显示的是生成的路径，这是通过利用离散数据点平滑交替生成的样条插值曲线。当我们考虑航天器本体的旋转时，-180度相当于+180度状态，然后，状态超过-180度时，它将从180度再次转到C-空间当中。正是由于这个原因，为了保证旋转的连续性，我们需要充分利用周期性的边界条件。为方便观察路径，航天器机体的映射体积忽略不计。同时为了路径表述的更加简单，附有在1方向上-180度范围的连接的插图，并做了说明。从图中可以很容易看出在-180度的范围内，沿着路径走向目标C，B和C是走向相同的目标点。

图3 两个自由度的C空间

图4 C空间的路径（2个自由度）