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外文文獻(xiàn)：
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(jī)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(jī)器人路徑規(guī)劃，這種規(guī)劃主要運(yùn)用的是避碰算法。對(duì)于未來(lái)的空間機(jī)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(biāo)，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(biāo)本身的碰撞，一旦地點(diǎn)、形狀和目標(biāo)的控制點(diǎn)得到確認(rèn)，這些點(diǎn)將在配置空間中表示出來(lái)。為了避免死鎖現(xiàn)象的發(fā)生，本文利用了一種勢(shì)場(chǎng)域算法，也就是將拉普拉斯勢(shì)函數(shù)的應(yīng)用在配置空間中獲取路徑。通過(guò)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(jìn)。這種方法主要是利用樣條函數(shù)插值，它減少了計(jì)算負(fù)荷和產(chǎn)生空間機(jī)器人的平滑路徑，這種方法的有效性可通過(guò)幾個(gè)數(shù)字模擬來(lái)展現(xiàn)。
關(guān)鍵字：空間機(jī)器人、路徑規(guī)劃、避碰、勢(shì)函數(shù)、樣條內(nèi)插
1 介紹
未來(lái)的空間發(fā)展中，空間機(jī)器人及其自主性能將成為航天科技的關(guān)鍵特征。這種空間機(jī)器人將在構(gòu)建空間站和執(zhí)行航天器的檢查和維護(hù)方面發(fā)揮重要的作用。這些機(jī)器人將以自主的形式取代航天員進(jìn)行艙外活動(dòng)。上述機(jī)器人運(yùn)行的一個(gè)基本和重要的任務(wù)就是由機(jī)械臂捕獲在軌道上自由飛行的目標(biāo)，為了這項(xiàng)捕獲操作的正常進(jìn)行，要求將機(jī)械臂從初始位置移動(dòng)到末位置而不與目標(biāo)發(fā)生碰撞。
這種空間配置和人工勢(shì)場(chǎng)的方法通常應(yīng)用于普通機(jī)器人的運(yùn)行規(guī)劃當(dāng)中，使機(jī)器人的機(jī)械臂能夠回避障礙物和朝目標(biāo)移動(dòng)。Khatib提出了一種運(yùn)動(dòng)規(guī)劃的方法，在這種方法中定義了障礙物與機(jī)器人的每個(gè)鏈接的排斥勢(shì)，還定義了機(jī)器人的末端執(zhí)行器與目標(biāo)的吸引勢(shì)，并通過(guò)計(jì)算勢(shì)場(chǎng)和勢(shì)場(chǎng)的梯度而生成了最優(yōu)路徑。根據(jù)這種實(shí)時(shí)操作的簡(jiǎn)單性和適應(yīng)性，我們得知該方法是有效的。
但是在吸引勢(shì)場(chǎng)和排斥勢(shì)場(chǎng)的共同作用下會(huì)產(chǎn)生局部極值點(diǎn)，這將導(dǎo)致所謂的死鎖現(xiàn)象。為了解決上述問(wèn)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(shì)場(chǎng)域不存在局部極值點(diǎn)，即無(wú)死鎖現(xiàn)象。勢(shì)場(chǎng)域分為很多小格，勢(shì)場(chǎng)域的每個(gè)節(jié)點(diǎn)的離散值將被唯一確定。
本文對(duì)上述缺陷的消除，提出了樣條插值技術(shù)。給定的節(jié)點(diǎn)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(biāo)，空間機(jī)器人的路徑規(guī)劃運(yùn)用了數(shù)字模擬技術(shù)，它是通過(guò)對(duì)勢(shì)場(chǎng)域求解拉普拉斯函數(shù)來(lái)實(shí)現(xiàn)的，并且從最初的位置到末尾位置的樣條插值來(lái)產(chǎn)生連續(xù)光滑的路徑。
2. 機(jī)器人模型
空間機(jī)器人的模型如圖1所示：機(jī)器人被安裝在航天器和兩個(gè)旋轉(zhuǎn)接頭上，這兩個(gè)旋轉(zhuǎn)接頭可以實(shí)現(xiàn)末端執(zhí)行器的平面運(yùn)動(dòng)。這種情況下，我們的航天器的姿態(tài)角有一個(gè)額外的自由度，我們將這個(gè)額外的自由度視為額外的旋轉(zhuǎn)接頭。這意味著空間機(jī)器人有三個(gè)自由度的鏈接，每個(gè)鏈路的長(zhǎng)度和每個(gè)旋轉(zhuǎn)關(guān)節(jié)角度，分別由 (i = 1,2,3)表示。為了簡(jiǎn)化這個(gè)討論，本文做了一些假設(shè)：
（1）空間機(jī)器人的運(yùn)動(dòng)是平面的，即二維；
（2）機(jī)器人機(jī)械臂的運(yùn)動(dòng)對(duì)航天器姿態(tài)的影響是可以忽略的；
（3）機(jī)器人運(yùn)動(dòng)給出了靜態(tài)幾何關(guān)系，并沒(méi)有明確的依賴時(shí)間；
（4）目標(biāo)衛(wèi)星在慣性的作用下是很穩(wěn)定的；
一般情況下，平面運(yùn)動(dòng)和空間運(yùn)動(dòng)將分別進(jìn)行，所以我們可以假設(shè)上面的第一個(gè)不失一般性，第二個(gè)假設(shè)來(lái)自機(jī)械臂和航天器質(zhì)量比的比較，對(duì)于第三個(gè)假設(shè)，我們專注于生成機(jī)器人的路徑規(guī)劃，這基本上是由幾何關(guān)系的靜態(tài)性質(zhì)決定，因此并不依賴明確的時(shí)間，最后一個(gè)就是合作衛(wèi)星。
圖1 雙鏈路空間機(jī)器人
3 路徑規(guī)劃算法
拉普拉斯勢(shì)場(chǎng)域?qū)б?
的拉普拉斯方程求解稱為諧波的勢(shì)場(chǎng)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(jī)器人路徑中，邊界處代表障礙物和目標(biāo)，因此在此范圍中定義勢(shì)場(chǎng)域，除了目標(biāo)處其他位置不會(huì)發(fā)生局部極值點(diǎn)的問(wèn)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(zhuǎn)化成差分方程，并通過(guò)高斯-賽德?tīng)柗椒ㄇ蠼?，在方?3)中，如果采用的二階導(dǎo)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(gè)相鄰節(jié)點(diǎn)的X、Y的方向，假設(shè)長(zhǎng)度等同于使用以下符號(hào)：
然后，方程3用以下方程表達(dá)：
(5)
結(jié)果二維拉普拉斯方程轉(zhuǎn)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(yīng)用了高斯賽德?tīng)査惴ê颓蠼夥匠?，如下?
(8)
表示勢(shì)場(chǎng)域的迭代計(jì)算結(jié)果。
在上述的計(jì)算中，作為邊界條件，定義特定的正數(shù)來(lái)表示障礙物和目標(biāo)。為保證初始條件相同，給所有的自由節(jié)點(diǎn)賦同樣的數(shù)值。通過(guò)這種方法，在迭代計(jì)算的邊界節(jié)點(diǎn)獲得的的值將不會(huì)改變，而且自由節(jié)點(diǎn)的值是不同。我們應(yīng)用相同的域值作為障礙物，并且按照迭代計(jì)算方法，則目標(biāo)周圍較小的勢(shì)場(chǎng)域會(huì)像障礙物一樣緩慢的向周圍傳播，勢(shì)場(chǎng)域就是根據(jù)上述方法建立的。采用4節(jié)點(diǎn)相鄰的空間機(jī)器人存在的節(jié)點(diǎn)上的勢(shì)場(chǎng)，最小的節(jié)點(diǎn)選擇移動(dòng)到另一點(diǎn)，這個(gè)過(guò)程最終引導(dǎo)機(jī)器人無(wú)碰撞的到達(dá)目標(biāo)的位置。
樣條內(nèi)插法：
通過(guò)上述方法給出的路徑不能保證能夠與另一個(gè)目標(biāo)順利連接，如果節(jié)點(diǎn)上沒(méi)有給定目標(biāo)，我們會(huì)將柵格劃分成的更小，但這將增加計(jì)算量和所用時(shí)間。為了消除這些弊端，我們提出利用樣條插值技術(shù)。通過(guò)在將節(jié)點(diǎn)解給出的通過(guò)點(diǎn)的道路上，我們?cè)噲D獲得順利連接路徑與準(zhǔn)確獲取最初的和最后的點(diǎn)。本文主要是通過(guò)MATLAB命令應(yīng)用樣條函數(shù)。
配置空間：
當(dāng)我們?cè)趹?yīng)用拉普拉斯勢(shì)域的時(shí)候，路徑搜索只能在當(dāng)機(jī)器人在搜索空間過(guò)程中表示成一個(gè)點(diǎn)的情況下才能保證實(shí)現(xiàn)。配置空間(C空間)中機(jī)器人僅表示為一個(gè)點(diǎn)，主要是用于路徑搜索。將真正的空間轉(zhuǎn)換到C空間，必須執(zhí)行判斷碰撞條件的計(jì)算，如果碰撞存在,相應(yīng)的點(diǎn)在c空間被認(rèn)為是障礙。本文中，在生成勢(shì)場(chǎng)域時(shí)，所有現(xiàn)實(shí)空間的點(diǎn)的生成條件對(duì)應(yīng)于所有的節(jié)點(diǎn)都是經(jīng)過(guò)計(jì)算的。在構(gòu)成的機(jī)械臂和生成的節(jié)點(diǎn)的障礙物出現(xiàn)判斷選擇時(shí)，該節(jié)點(diǎn)可以看作是在c空間的障礙點(diǎn)。
數(shù)值仿真：
基于上述方法對(duì)于捕獲目標(biāo)衛(wèi)星路徑規(guī)劃的檢查是使用空間機(jī)器人模型進(jìn)行的。在本文中，我們假設(shè)空間機(jī)器人二維和2自由度機(jī)械手臂見(jiàn)圖1。每個(gè)鏈接的長(zhǎng)度給出如下:
l1 =1.4[m], l2 = 2.0[m], l3 = 2.0[m]
并假設(shè)目標(biāo)衛(wèi)星有1平方米。掌握處理1平方米的范圍，是以目標(biāo)中心的一側(cè)為中心的，所以這種處理方法就是最優(yōu)路徑的一個(gè)選擇。
我們來(lái)解釋一下空間機(jī)器人和目標(biāo)衛(wèi)星的幾何關(guān)系，在捕捉到目標(biāo)后，我們?cè)倩叵胍幌抡麄€(gè)操作過(guò)程，讓空間機(jī)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(dá)指定目標(biāo)位置。在3個(gè)自由度的情況下，并不是根據(jù)航天器機(jī)體的角度，可操縱性由來(lái)衡量。如果我們假設(shè)空間機(jī)器人的末端應(yīng)垂直于目標(biāo),然后所有的關(guān)節(jié)角度是預(yù)先確定的，數(shù)值如下:
因?yàn)樗械年P(guān)節(jié)角度是確定的，航天器之間的相對(duì)位置和目標(biāo)也唯一確定，如果飛船被認(rèn)為定位在原點(diǎn)的慣性坐標(biāo)系(0,0),目標(biāo)坐標(biāo)在上面的情況下是給出的(-3.27,-2.00)。基于這些準(zhǔn)備，我們可以通過(guò)在配置空間中機(jī)械臂的移動(dòng)搜索來(lái)到達(dá)目標(biāo)位置。
為了簡(jiǎn)化境況，一開(kāi)始就假設(shè)姿態(tài)角(鏈接1關(guān)節(jié)角)符合理想情況。假定的坐標(biāo)系統(tǒng)圖2所示
圖2 2個(gè)自由度的路徑規(guī)劃問(wèn)題
為計(jì)算初始條件的鏈接2和它的目標(biāo)角度，應(yīng)考慮的大小：
初始角度：
目標(biāo)角度：
在這種情況下，勢(shì)場(chǎng)域分成180段計(jì)算成C空間。圖3顯示的C空間和計(jì)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(biāo)衛(wèi)星的映射。圖4顯示的是生成的路徑，這是通過(guò)利用離散數(shù)據(jù)點(diǎn)平滑交替生成的樣條插值曲線。當(dāng)我們考慮航天器本體的旋轉(zhuǎn)時(shí)，-180度相當(dāng)于+180度狀態(tài)，然后，狀態(tài)超過(guò)-180度時(shí)，它將從180度再次轉(zhuǎn)到C-空間當(dāng)中。正是由于這個(gè)原因，為了保證旋轉(zhuǎn)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(jī)體的映射體積忽略不計(jì)。同時(shí)為了路徑表述的更加簡(jiǎn)單，附有在方向上-180度范圍的連接的插圖，并做了說(shuō)明。從圖中可以很容易看出在-180度的范圍內(nèi)，沿著路徑走向目標(biāo)C，B和C是走向相同的目標(biāo)點(diǎn)。
圖3 兩個(gè)自由度的C空間
圖4 C空間的路徑（2個(gè)自由度）
5 結(jié)論
本文提出了捕獲目標(biāo)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(shí)用性。使用差值技術(shù)，計(jì)算量將減小，平滑路徑完全可以是實(shí)現(xiàn)。進(jìn)一步的研究將證明機(jī)械臂的運(yùn)動(dòng)將影響飛行器機(jī)身的角度。