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外文文獻：
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 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. 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.
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 byand(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.
（1）
The Laplace equation can be solved numerically. We define two dimensional Laplace equation as below:
（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:
（3）
where ， 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:
Then equation (3) is expressed in the following manner:
（4）
And as a result, two dimensional Laplace equation will be converted into the equation (5) as below:
（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:
（6）
In order to solve the above equations we apply Gauss-Seidel method and have equations as follows:
（7）
where 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 is defined for the obstacle and 0 for the goal. And as the initial conditions the same number 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. 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:
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.
was taken into consideration for the calculation of the initial condition of the Link 2 and its goal angles:
Innitial condition:
Goal condition:
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)
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 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.
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.
17
中文譯文：
空間機器人避碰路徑規(guī)劃
Yuya Yanoshita and Shinichi Tsuda
摘要：本文論述的是空間機器人路徑規(guī)劃，這種規(guī)劃主要運用的是避碰算法。對于未來的空間機器人操作，自主控制的路徑規(guī)劃方法可以受到固定指令的支配去捕獲目標，不用一直受地面站的控制。尤其是從始至終要考慮到避免與目標本身的碰撞，一旦地點、形狀和目標的控制點得到確認，這些點將在配置空間中表示出來。為了避免死鎖現(xiàn)象的發(fā)生，本文利用了一種勢場域算法，也就是將拉普拉斯勢函數(shù)的應用在配置空間中獲取路徑。通過利用平滑路徑的方法，我們已經(jīng)在路徑生成方面做了一定的改進。這種方法主要是利用樣條函數(shù)插值，它減少了計算負荷和產(chǎn)生空間機器人的平滑路徑，這種方法的有效性可通過幾個數(shù)字模擬來展現(xiàn)。
關鍵字：空間機器人、路徑規(guī)劃、避碰、勢函數(shù)、樣條內插
1 介紹
未來的空間發(fā)展中，空間機器人及其自主性能將成為航天科技的關鍵特征。這種空間機器人將在構建空間站和執(zhí)行航天器的檢查和維護方面發(fā)揮重要的作用。這些機器人將以自主的形式取代航天員進行艙外活動。上述機器人運行的一個基本和重要的任務就是由機械臂捕獲在軌道上自由飛行的目標，為了這項捕獲操作的正常進行，要求將機械臂從初始位置移動到末位置而不與目標發(fā)生碰撞。
這種空間配置和人工勢場的方法通常應用于普通機器人的運行規(guī)劃當中，使機器人的機械臂能夠回避障礙物和朝目標移動。Khatib提出了一種運動規(guī)劃的方法，在這種方法中定義了障礙物與機器人的每個鏈接的排斥勢，還定義了機器人的末端執(zhí)行器與目標的吸引勢，并通過計算勢場和勢場的梯度而生成了最優(yōu)路徑。根據(jù)這種實時操作的簡單性和適應性，我們得知該方法是有效的。
但是在吸引勢場和排斥勢場的共同作用下會產(chǎn)生局部極值點，這將導致所謂的死鎖現(xiàn)象。為了解決上述問題，科研人員提出了一些方法，例如拉普拉斯算法的使用。這種方法保證了勢場域不存在局部極值點，即無死鎖現(xiàn)象。勢場域分為很多小格，勢場域的每個節(jié)點的離散值將被唯一確定。
本文對上述缺陷的消除，提出了樣條插值技術。給定的節(jié)點作為路徑的一部分將被定義為平滑樣條函數(shù)的一部分。為了捕獲到目標，空間機器人的路徑規(guī)劃運用了數(shù)字模擬技術，它是通過對勢場域求解拉普拉斯函數(shù)來實現(xiàn)的，并且從最初的位置到末尾位置的樣條插值來產(chǎn)生連續(xù)光滑的路徑。
2. 機器人模型
空間機器人的模型如圖1所示：機器人被安裝在航天器和兩個旋轉接頭上，這兩個旋轉接頭可以實現(xiàn)末端執(zhí)行器的平面運動。這種情況下，我們的航天器的姿態(tài)角有一個額外的自由度，我們將這個額外的自由度視為額外的旋轉接頭。這意味著空間機器人有三個自由度的鏈接，每個鏈路的長度和每個旋轉關節(jié)角度，分別由 (i = 1,2,3)表示。為了簡化這個討論，本文做了一些假設：
（1）空間機器人的運動是平面的，即二維；
（2）機器人機械臂的運動對航天器姿態(tài)的影響是可以忽略的；
（3）機器人運動給出了靜態(tài)幾何關系，并沒有明確的依賴時間；
（4）目標衛(wèi)星在慣性的作用下是很穩(wěn)定的；
一般情況下，平面運動和空間運動將分別進行，所以我們可以假設上面的第一個不失一般性，第二個假設來自機械臂和航天器質量比的比較，對于第三個假設，我們專注于生成機器人的路徑規(guī)劃，這基本上是由幾何關系的靜態(tài)性質決定，因此并不依賴明確的時間，最后一個就是合作衛(wèi)星。
圖1 雙鏈路空間機器人
3 路徑規(guī)劃算法
拉普拉斯勢場域導引：
的拉普拉斯方程求解稱為諧波的勢場域功能，并且最大值和最小值僅發(fā)生在邊界處，在生成的機器人路徑中，邊界處代表障礙物和目標，因此在此范圍中定義勢場域，除了目標處其他位置不會發(fā)生局部極值點的問題，這為路徑的生
成消除了死鎖現(xiàn)象。
(1)
拉普拉斯方程可以數(shù)值求解，我們定義了二維拉普拉斯方程，如下公式所示：
(2)
這將轉化成差分方程，并通過高斯-賽德爾方法求解，在方程(3)中，如果采用的二階導數(shù)的差分公式，可以得到以下的差分公式：
(4)
，的代數(shù)值代表每個相鄰節(jié)點的X、Y的方向，假設長度等同于使用以下符號：
然后，方程3用以下方程表達：
(5)
結果二維拉普拉斯方程轉變?yōu)榉匠?，如下：
(6)
同樣的方式，在三維的情況下，三維的拉普拉斯方程的差分方程由下式易得：
(7)
為了解決上述方程，我們應用了高斯賽德爾算法和求解方程，如下：
(8)
表示勢場域的迭代計算結果。
在上述的計算中，作為邊界條件，定義特定的正數(shù)來表示障礙物和目標。為保證初始條件相同，給所有的自由節(jié)點賦同樣的數(shù)值。通過這種方法，在迭代計算的邊界節(jié)點獲得的的值將不會改變，而且自由節(jié)點的值是不同。我們應用相同的域值作為障礙物，并且按照迭代計算方法，則目標周圍較小的勢場域會像障礙物一樣緩慢的向周圍傳播，勢場域就是根據(jù)上述方法建立的。采用4節(jié)點相鄰的空間機器人存在的節(jié)點上的勢場，最小的節(jié)點選擇移動到另一點，這個過程最終引導機器人無碰撞的到達目標的位置。
樣條內插法：
通過上述方法給出的路徑不能保證能夠與另一個目標順利連接，如果節(jié)點上沒有給定目標，我們會將柵格劃分成的更小，但這將增加計算量和所用時間。為了消除這些弊端，我們提出利用樣條插值技術。通過在將節(jié)點解給出的通過點的道路上，我們試圖獲得順利連接路徑與準確獲取最初的和最后的點。本文主要是通過MATLAB命令應用樣條函數(shù)。
配置空間：
當我們在應用拉普拉斯勢域的時候，路徑搜索只能在當機器人在搜索空間過程中表示成一個點的情況下才能保證實現(xiàn)。配置空間(C空間)中機器人僅表示為一個點，主要是用于路徑搜索。將真正的空間轉換到C空間，必須執(zhí)行判斷碰撞條件的計算，如果碰撞存在,相應的點在c空間被認為是障礙。本文中，在生成勢場域時，所有現(xiàn)實空間的點的生成條件對應于所有的節(jié)點都是經(jīng)過計算的。在構成的機械臂和生成的節(jié)點的障礙物出現(xiàn)判斷選擇時，該節(jié)點可以看作是在c空間的障礙點。
數(shù)值仿真：
基于上述方法對于捕獲目標衛(wèi)星路徑規(guī)劃的檢查是使用空間機器人模型進行的。在本文中，我們假設空間機器人二維和2自由度機械手臂見圖1。每個鏈接的長度給出如下:
l1 =1.4[m], l2 = 2.0[m], l3 = 2.0[m]
并假設目標衛(wèi)星有1平方米。掌握處理1平方米的范圍，是以目標中心的一側為中心的，所以這種處理方法就是最優(yōu)路徑的一個選擇。
我們來解釋一下空間機器人和目標衛(wèi)星的幾何關系，在捕捉到目標后，我們再回想一下整個操作過程，讓空間機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況下，末端執(zhí)行器將到達指定目標位置。在3個自由度的情況下，并不是根據(jù)航天器機體的角度，可操縱性由來衡量。如果我們假設空間機器人的末端應垂直于目標,然后所有的關節(jié)角度是預先確定的，數(shù)值如下:
因為所有的關節(jié)角度是確定的，航天器之間的相對位置和目標也唯一確定，如果飛船被認為定位在原點的慣性坐標系(0,0),目標坐標在上面的情況下是給出的(-3.27,-2.00)?；谶@些準備，我們可以通過在配置空間中機械臂的移動搜索來到達目標位置。
為了簡化境況，一開始就假設姿態(tài)角(鏈接1關節(jié)角)符合理想情況。假定的坐標系統(tǒng)圖2所示
圖2 2個自由度的路徑規(guī)劃問題
為計算初始條件的鏈接2和它的目標角度，應考慮的大小：
初始角度：
目標角度：
在這種情況下，勢場域分成180段計算成C空間。圖3顯示的C空間和計劃中的很大一部分的中心是由航天器本體映射的障礙了，左邊部分是目標衛(wèi)星的映射。圖4顯示的是生成的路徑，這是通過利用離散數(shù)據(jù)點平滑交替生成的樣條插值曲線。當我們考慮航天器本體的旋轉時，-180度相當于+180度狀態(tài)，然后，狀態(tài)超過-180度時，它將從180度再次轉到C-空間當中。正是由于這個原因，為了保證旋轉的連續(xù)性，我們需要充分利用周期性的邊界條件。為方便觀察路徑，航天器機體的映射體積忽略不計。同時為了路徑表述的更加簡單，附有在方向上-180度范圍的連接的插圖，并做了說明。從圖中可以很容易看出在-180度的范圍內，沿著路徑走向目標C，B和C是走向相同的目標點。
圖3 兩個自由度的C空間
圖4 C空間的路徑（2個自由度）
5 結論
本文提出了捕獲目標衛(wèi)星的路徑生成方法，并用數(shù)字模擬的方法證明了它的實用性。使用差值技術，計算量將減小，平滑路徑完全可以是實現(xiàn)。進一步的研究將證明機械臂的運動將影響飛行器機身的角度。