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外文翻译 空间机器人防碰路径规划


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Space Robot Path Planning for Collision Avoidance

r />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 expressed 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. Some 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 manner in place of extravehicular activities by astronauts. 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. 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 issue, a few methods are proposed where the solution of Laplace equation is utilized. This method assures 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 by li and ? i (i = 1,2,3) , respectively. In order to simplify the discussions a few assumptions 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. -robot motion is given by the relation of static geometry and not explicitly depending on time. -the target satellite is inertially stabilized. In general in-plane motion and out-of-plane motion will be separately performed. So we are able to assume the above first one without loss of generality. The second assumption derives from the comparison of the ratio of mass between the robot arm and the spacecraft body. With respect to the third assumption we focus on generating the path planning of the robot and this is basically given by the static nature of geometry relationship and is therefore not depending on the time explicitly. The last one means the satellite is cooperative.

Fig.1 Model of Two-link Space Robot

III. PATH PLANNING GALGORITHM A. Laplace Potential Guidance The solution of the Laplace equation (1) is called a Harmonic potential function, and its and minimum values take place only on the boundary. In the robot path generation the boundary means obstacle and goal. Therefore inside the region where the potential is defined, no local minimum takes place except the goal. This eliminates the deadlock phenomenon for path generation.

????
2 i ?1

n

? 2? ?0 ?xi2

(1)

The Laplace equation can be solved numerically. We define two dimensional Laplace equation as below:

? 2? ? 2? ? ?0 ?x 2 ?y 2

(2)

And this will be converted into the difference equation and then solved by Gauss -Seidel method. In equation (2) if we take the central difference formula for second derivatives, the following equation will be obtained:

? 2? ? 2? ? ?0 ?x 2 ?y 2 ?( x ? ?x, y ) ? 2?( x, y ) ? ?( x ? ?x, y ) ? ?x2 ?( x, y ? ?y ) ? 2?( x, y ) ? ?( x, y ? ?y ) ? ? y2

(3)

where ?x , ?y are the step (cell) sizes between adjacent nodes for each x, y direction. If the step size is assumed equal and the following notation is used:

?( x ? ?x, y ) ? ?i ?1, j
Then equation (3) is expressed 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 ? 4

(5)

In 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

(6)

In order to solve the above equations we apply Gauss-Seidel method and have equations as follows:
1 ?in,? j ?

1 n 1 n n ?1 ??i ?1, j ? ?in??1, j ? ?i , j ?1 ? ?i , j ?1 ? 4

(7)

1 where ?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 also given for all of the free nodes. By this approach during iterative computations the value of the boundary nodes will not change and the values only for free nodes will be varying. Applying the same potential values as the obstacle and in accordance with the iterative computational process, the small potential around the goal will be gradually propagating like surrounding the obstacle. The potential field will be built based on the above procedure. Using the above potential field from 4 nodal points adjacent to the node on which the space robot exists, the smallest node is selected for the point to move to. This procedure finally leads the space robot to the goal without collision. B. Spline Interpolation The path given by the above approach does not assure the smoothly connected one. And if the goal is not given on the nodal point, we have to partition the cells into much more smaller cells. This will increase the computational load and time. In order to eliminate the above drawbacks we propose the utilization of spline interpolation technique. By assigning the nodal points given by the solution to via points on the path, we try to obtain the smoothly connected path with accurate initial and final points. In this paper the cubic spline was applied by using MATLAB command. C. Configuration Space When we apply the Laplace potential, the path search is assured only in the case where the robot is expressed to be a point in the searching space. The configuration space(C-Space), where the robot is expressed as a point, is used for the path search. To convert the real space into the C-Space the calculation to judge the condition of collision is performed and if the collision exists, the corresponding point in the C-space is regarded as the obstacle. In this paper when the potential field was generated, the conditions of all the points in the real space, corresponding to all the nodes, were

calculated. The judgment of intersection between a segment constituting the robot arm and a segment constituting the obstacle at each node was made and if the intersection takes place, this node is treated as the obstacle in the C-Space. IV.NUMERICAL SIMULATIONS Based on the above approach the path planning for capturing a target satellite was examined using a space robot model. In this paper we assume the space robot with two dimensional and 2 DOF robotic arm as shown in Fig.1. 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 assumed 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 assume 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 assumed 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 assumed to coincide with the desirable angle from the beginning. The coordinate system was

assumed as shown in Fig.2.

?1 was taken into consideration for the calculation of the initial condition of the Link 2
and its goal angles: Innitial condition: ?2 ? ?64.3? ,?3 ? 90o Goal condition:

?2 ? ?166.5? ,?3 ? 76.5o

In this case the potential field was computed for the C-Space with 180 segments. Fig.3 shows the C-Space and the hatched large portion in the center is given by the obstacle mapped by the spacecraft body. The left side portion is a mapping of the target satellite. Fig.4 shows a generated path and this was spline-interpolated curve by using alternate points of discrete data for smoothing.

Fig.3 2 DOF C-Space

Fig.4 Path in C-Space(2 DOF)

The conversion of the generated path into the real space is given by Fig.5. The path has no collision with target satellite and is expressed as smoothed curve.

Fig.5 Path of Robotic Arm in Real Space(2 DOF) B. 3 DOF Robot Fig.6 shows a path planning case in which the spacecraft attitude motion is incorporated: Initial Conditions: ?1 = -90°,

Fig.6 Path Planning Problem (3 DOF)

Fig.7 C-Space for 3 DOF Case

In this example the potential field was computed by generating the C-Space with 36 segments of joint angles. Fig.7 illustrates the C-Space and the surrounding of this space derives from the mapping of the spacecraft body. The central portion is given by the mapping of the target satellite. The white colored volume is free space for joint travel. When we consider the rotation of spacecraft body, -180 degrees are equal to +180 degrees and, then, the state over -180 degrees will be started from +180 degrees and again back to the C-Space. For this reason the periodic boundary condition was applied in order to assure the continuity of the rotation. For the simplicity to look at the path, the mapped volume by the spacecraft body was omitted. Also for the simplicity of the path expression the chart which has the connection of -180 degrees in the ?1 direction 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.

Fig.8 Path in C-Space (3 DOF)

Fig.9 Path (Enlarged) in C-Space (3 DOF)

The same spline interpolation was carried out as in 2 DOF case for the generation of the smooth path. Fig. 9 shows the enlarged path from another view angle.

The chart shows no collision with the obstacle and the pathreached the goal. And when we convert the path into the reaspace, smooth path is given by Fig. 10 without collision.

Fig.10 Robotic Arm Path in Real Space

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.

REFERENCES [1] Khatib, O, “Real-Time Obstacle Avoidance for Manipulators and Mobile Robots”, International Journal of Robotics Research, Vol.5, No.1,1986 [2] C. I. Connoly, J. B. Burns and R. Weiss, “ Path Planning Using Laplace’s Equation”, Proceedings of the IEEE International Conference on Robotics and Automation, pp.2102-2106, 1990 [3] Sato, K.,”Deadlock-Free Motion Planning Using the Laplace Potential Field”, Advanced Robotics, Vol.7, No.5,pp.449-461, 1993

空间机器人防撞路径规划
Yuya Yanoshita and Shinichi Tsuda

摘要:本文主要论述的是空间机器人路径规划,其主要运用的是一种防撞算 法。对于未来的太空机器人操作,独立自主的路径规划方法是自主的支配去捕获 目标,而不用一直受地面站的控制。尤其是防撞与目标本身必须始终考虑。一旦 位置、形状和掌握目标识别的点,这些点将在配置空间中表示出来。为了避免死 锁现象的发生,本文利用了一种势场域算法,也就是将拉普拉斯势函数的应用在 配置空间中获取路径。通过利用平滑路径的方法,我们已经在路径生成方面做了 一定的改进。这种方法主要是利用样条函数插值,它减少了计算负荷和产生空间 机器人的平滑路径,这种方法的有效性可通过几个数字模拟来展现。 关键字:空间机器人、路径规划、防撞、势函数、样条插值

一、绪论
未来的空间发展中,空间机器人及其自主性能将成为航天科技的主要特点。 这种空间机器人将扮演构建空间结构和航天器执行检查和维护的角色。这些操作 将被自主的执行代替宇航员在舱外的活动。在上面的空间机器人操作,一个基本的 和重要的任务就是机械手臂捕获自由飞行目标轨道的。为了这项捕获操作的正常 进行,要求将机械臂从初始位置移动到末位置而不与目标发生碰撞。 这种配置空间和人工势场的方法通常应用于机器人的操作计划。这使机器人 手臂躲避障碍和向目标移动。Khatib 提出了一种运动规划的方法,每个链接的机 器人和障碍物排斥势的定义,还定义机器人末端执行器与目标的有吸引力,利用 势场和势场梯度的生成路径。并通过计算势场和势场梯度而生成了最优路径。根 据这种实时操作的简单性和适应性,我们得知该方法是有效的。但是可能有排斥 力和吸引力指向是相等的,这将导致所谓的僵局。 为了解决上述问题,科研人员提出了一些方法,例如拉普拉斯算法的使用。 这种方法保证了势场域不存在局部极值点,即无死锁现象。势场域分为很多小格, 势场域的每个节点的离散值将被唯一确定。

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。

二、 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

三、路径规划算法
1.拉普拉斯势场域导引:
? 的拉普拉斯方程求解称为谐波的势场域功能,并且最大值和最小值仅发生

在边界处,在生成的机器人路径中,边界处代表障碍物和目标,因此在此范围中 定义势场域,除了目标处其他位置不会发生局部极值点的问题,这为路径的生 成消除了死锁现象。

????
2 i ?1

n

? 2? ?0 ?xi2

(1)

拉普拉斯方程可以数值求解,我们定义了二维拉普拉斯方程,如下公式所示:

? 2? ? 2? ? ?0 ?x 2 ?y 2

(2)

这将转化成差分方程,并通过高斯-赛德尔方法求解,在方程(3)中,如果采用的二 阶导数的差分公式,可以得到以下的差分公式:

? 2? ? 2? ? ?0 ?x 2 ?y 2 ?( x ? ?x, y ) ? 2?( x, y ) ? ?( x ? ?x, y ) ? ?x2 ?( x, y ? ?y ) ? 2?( x, y ) ? ?( x, y ? ?y ) ? ? y2

(4)

?x , ?y 的代数值代表每个相邻节点的X、Y的方向,假设长度等同于使用以下符

号:

?( x ? ?x, y ) ? ?i ?1, j
然后,方程3用以下方程表达:

?i ?1, j ? ?i ?1, j ? ?i, j ?1 ? ?i, j ?1 ? ??i, j ? 0
结果二维拉普拉斯方程转变为方程5,如下:
?i , j ? 1 ??i ?1, j ? ?i ?1, j ? ?i , j ?1 ? ?i , j ?1 ? 4

(5)

(6)

同样的方式,在三维的情况下,三维的拉普拉斯方程的差分方程由下式易得:

?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
1 n 1 n n ?1 ??i ?1, j ? ?in??1, j ? ?i , j ?1 ? ?i , j ?1 ? 4

(7)

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:
1 ?in,? j ?

(8)

?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。 2.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。 3.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。

四、数值模拟
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: l1 =1.4[m], l2 = 2.0[m], l3 = 2.0[m] , 并假设目标卫星有1平方米。掌握处理1平方米的范围,是以目标中心的一侧为中 心的,所以这种处理方法就是最优路径的一个选择。 我们来解释一下空间机器人和目标卫星的几何关系,在捕捉到目标后,我们 再回想一下整个操作过程,让空间机器人有更大的可操作性是完全可行的。因此 在本文中,可操作性最大化的情况下,末端执行器将到达指定目标位置。在3个自 由度的情况下,并不是根据航天器机体的角度,可操纵性由 ?2,?3 来衡量。如果我 们假设空间机器人的末端应垂直于目标,然后所有的关节角度是预先确定的,数值 如下:

?1 ? 160.7o ,?2 ? 32.8o ,?3 ? 76.5o
因为所有的关节角度是确定的,航天器之间的相对位置和目标也唯一确定,如果 飞船被认为定位在原点的惯性坐标系(0,0),目标坐标在上面的情况下是给出的 (-3.27,-2.00)。基于这些准备,我们可以通过在配置空间中机械臂的移动搜索来到 达目标位置。 进行了两个路径规划模拟结果如下所示: A.两个自由度机器人 为了简化境况,一开始就假设姿态角(链接 1 关节角)符合理想情况。假定的坐 标系统图 2 所示

图 2 2 个自由度的路径规划问题

为计算初始条件的链接 2 和它的目标角度,应考虑 ?1 的大小: 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发出,左边部分是目标卫星的 映射。生成的路径如图 4 所示,这是通过利用离散数据点平滑交替生成的样条插 值曲线。

图 3 两个自由度的 C 空间

图 4 C 空间的路径(2 个自由度)

生成的路径转换成真正的空间如图 5 所示。路径与目标卫星无碰撞并表示为 平滑的曲线。

图 5 机械臂在现实空间的路径

B.三个自由度的机器人 图 6 显示了路径规划案例中的航天器姿态运动的结合:

图 6 路径规划问题(3 个自由度)

图 7 在 C-空间 3 个自由度的情况下

在本例中,计算了势场生成 C 空间 36 段的关节角度。在 C-空间及该空间的周 围的飞行器本体的映射推导如图 7 所示。中心部分是由目标卫星的映射给出的。 白颜色的卷是公共的的自由空间。 当我们考虑航天器主体的旋转时,-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。路径生成的结果如图 8 所 示。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 8 在 C-空间路径(3 个自由度)

相同的样条插值进行了 2 自由度平滑路径的生成。图 9 所示的是从另一个视 角扩大的路径。

图 9 在 C-空间路径(扩大) (3 自由度)

图表显示没有碰到障碍和到达目标的路径,当我们将它转换成真实的空间, 也将会如图 10 所示一样畅通无阻。

图 10 在现实空间机械臂的路径

五、结论
本文提出了捕捉目标卫星的路径生成方法。并用数字模拟的方法证明了其实用性。通过 使用插值技术使计算量减少和平滑路径可用。进一步的研究将会建议机械臂的运动来影响飞 行器的运动姿态。

参考文献 [1] Khatib, O, “Real-Time Obstacle Avoidance for Manipulators and Mobile Robots”, International Journal of Robotics Research, Vol.5, No.1,1986 [2] C. I. Connoly, J. B. Burns and R. Weiss, “ Path Planning Using Laplace’s Equation”, Proceedings of the IEEE International Conference on Robotics and Automation, pp.2102-2106, 1990 [3] Sato, K.,”Deadlock-Free Motion Planning Using the Laplace Potential Field”, Advanced Robotics, Vol.7, No.5,pp.449-461, 1993


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