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機(jī)械專業(yè)外語文獻(xiàn)翻譯
2.2.5 Vector fields
A vector-field is essentially a 2-Dimentional field with vectors. A vector consists of magnitude and angle, which represent importance or speed and demanded heading angle respectively.
The magnitude interpretation as a an importance is useful when vector fields are combined by addition of each vector. The more important vector is longer, and therefore the resulting heading angle will be more into the direction of the more important vector.
Some vector field generators, such as basic potential field methods, are not concerned about the magnitude. In this case the magnitude is often normalized.
2.2.5.1 Potential fields
The theory of potential fields as trajectories is derived from an electrical field of a
sphere in physics.
The ormulae for an attractive field is as follows:
where
is a vector from the origin to the target position
is a vector from the origin to the robot
is the resultant vector with normalized length, indicating the direction of
the field at the robots current position.
The resultant is pointing towards the target. The attractive potential field is therefore related to line of sight guidance.
A repulsive field is generating vectors pointing away from T.
The formulae is
Fig 6: attractive potential field
Figure 6 shows a attractive potential field with a target point at T(0,0) . In an application usually only the vector at the current position of the robot is calculated, for demonstration graphs such as Figure 6 the robot is assumed to be in every possible position in the field, and therefore generating the vectors at each point.
2.2.5.2 Limit cycle based vector fields
Limit cycles are part of nonlinear control theory. However the properties of a graph representing a limit cycle, Figure 7, can be adopted for path generation. For further reading see D-H Kim (2000). The limit-cycle characteristics of the 2nd order nonlinear function can be represented as a vector field containing a unit circle. Vectors outside the circle will be directed tangentially onto the circle. It can be seen as an arc/circle trajectory generator that lines up the robot coming in from any direction automatically. The resulting vector-field can be used like a arc trajectory generator or for obstacle avoidance.
The disadvantage of the limit cycle method is that once the robot crosses the unit circle, the vector pointing towards a singularity in the centre. Therefore, a practical implementation is not easy, since is likely to overshoot the circle border slightly when arriving at the circle. A modification of the field within the circle is a proposed solution to the problem.
Fig 7: limit cycle
2.2.5.3 Vector field fusion
All discussed vector field methods can be applied at the same time. The author
developed a way of combining (fusing) vector fields, which is published in
Robinson P. (2004).
Constraints and requirements:
-Two or more vector fields are given
-These vector fields contain normalized vectors
The method is best described in an example. A typical combination of vector-field shall be analyzed where a Robot R avoids and obstacle Robot O on the way to a target point T. See figure 8 below.
A weighting function is required to fuse the vector fields together. Experiments have shown that the Gaussian normal distribution function is an acceptable method of combining these fields. (A cylinder or cone would create a sudden change in heading angle and excites instability.)
The angle Δ is the difference between the instant heading angle of the robot and the vector ro which points from robot to obstacle.
Fig 8: avoidance scenario
ThusΔ is an indication of how much the robot is on collision course with the obstacle. The smaller the angle, the more it is on collision course and the importance to avoid the obstacle is high.
The mission of the robot is to go to T. In order to take into account the obstacle on
its way towards the target it must consider how close the obstacle is. The distance to the obstacle is defined as ro
. A smaller distance to an obstacle means that is more important to avoid it.
An avoidance vector field VOshall be defined which is normal tot he mission
vector field rt .The normalized target vector is VT.
Suppose two vectors VTand VOare added together – ‘fused’ -with a Gaussian
weighting function m*G(d).
Where:
is the resultant modified target vector
Mis a additional constant weighting factor
G() is the Gaussian distribution function.
μ is the offset of the Gaussian hat
is the distribution of the Gaussian hat
We just learned that there are essentially two factors that define how important it is to avoid the obstacle. Δ and ro .
The author will base the principle of vector field fusion by relating the length of
each vector to importance towards the mission at a particular point in the field. Thus Δ and ro can be modelled as follows to influence the length of VO.
-r1 is the maximum offset that Δ can cause. θ is steepness of the slope ( relationship of μ and Δ ) A larger θ will result in higher angles already to be considered as important.
And the distance of the robot to the obstacle ro is modelled as the position parameter in the Gaussian function.
Finally, the resultant vector field VMTindicates the new instant heading angle for
the robot.
Test results at different speeds with a robot football robot. The maximum speed is 100% corresponding to 3.0 m/sec. The coordinate system is in inches.
Fig 9: avoidance path at 0.36 m/sec Fig 10: avoidance path at 0.51 m/sec Fig 11: avoidance path at 0.84 m/sec
2.2.6 Matching the trajectories to the dynamic model of mobile robots
A current attempt of the author is to compare a path through a potential field with
the robots dynamics model in order to determine if the robot can follow it. This can be done in frequency domain, by comparing the bandwidth of the robot plus controller model to the bandwidth of the input signal when trying to follow the path. This approach can be taken further. This could provide a basis of matching a vector-field by design to the robot’s bandwidth.
2.3 Modelling mobile robots
This chapter is concerned with developing and understanding models of mobile
robot kinematics and the control of each individual motor actuating the links within
the kinematic model. Further reading is available in McKerrow P J (1991) chapter
8.1 which references to Muir P F and Neuman C P (1986). Muir and Neuman
introduced a way of model ling wheeled mobile robots. It is related to model ling the
kinematics of robot arms (manipulator kinematics).
Differential driven Robot
Differential driving is one of the simplest methods of model ling a mobile robot. This is probably why it is so common. The robot consists of 2 diagonally opposing
wheels, see Fig. 12. If both wheels have the same velocity, the robot will go straight. If one wheel goes faster than the other the robot will follow a circular trajectory. If one wheel turns in the opposite direction of the other but with the same magnitude in speed, the robot will turn around its cent re, “on the spot”.
The wheel Jacobian matrix is given and can be used as follows:
Where v is the velocity forward of the centre of the robot and . is the angular
velocity around the centre of the robot, see Fig 12. p& wheel Jacobian. p is the posture of the robot. The posture gives information about how the robot moves with respect to the floor.
indicates the instant heading angle of the robot.
Assuming no slip, the direction the vehicle is facing towards, is the same as the direction of the velocity vector (at and instant in time). An advantage of this fact, it simplifies calculations. A disadvantage however is that it can not move side wards.
Fig 12: Differential driven Robot
3 DESIGN AND IMPLEMENTATION
3.1 Specification
for fast autonomous mobile platform:
faster than 1m/sec
large enough for real world application, such as picking up goods
space for a onboard laptop
enough sensors for autonomous movements
battery life for several hours
inexpensive ( < ￡1000)
3.2 Mechanical Design
Every part of the mechanical design is build from basic materials, only the caster
wheels are a ready made construction. One focus of the project was to build the mechanical construction rather than buy a ready made gearbox and frame. As a benefit the authors machining skills has improved.
3.2.1 Frame
The robot body consists of a steel frame that is welded together forming a box. Initially the frame was screwed together until the design was fully developed. Then the screws and brackets have been replaced by welded joints. The top rectangle can be taken of in order to do repair work. A large orange plastic sheet is mounted on top as a base for the circuit boards and the notebook. The battery is placed on top of the bottom frame. The key point is here that the bottom frame is lower than the wheel axis. It is placed just 2 cm above ground to prevent the robot from toppling at high speed.
3.2.2 Steering
The steering consists of 2 links, i.e. 2 wheels.
Fig 13: Explosion picture of one steering link
One steering link consists of a medium duty caster wheel that has been welded to a plate. The plate and the underlying caster-wheel have a 12 mm shaft welded on in order to enable steering of the wheel. The wheel is not offset its centre, unlike on a shopping trolley for example. Therefore it must be controlled by active steering to line it up with the direction of movement. Both steering shafts are driven by a motor-gearbox combination (gear-ratio 1:50) over a belt system (ratio 1:2). The motor is a 12 Volt DC Motor. A potentiometer on the top of one shaft is read by a micro con troller to determine the current steering angle. The overall system is a servo system, since it has positional feedback, see section 3.3.5 for a description of the control.
The above design, is the finally implemented one, the initial design had a stepper
motor with controller circuit. However, the stepper motor was not powerful enough to turn the steering on rough surfaces. The implemented system responds quick and accurate within a fraction of a second to any angle.
There are 3 ball bearings per link: one in the axis of the wheel and two in line with the 12mm steering shaft. This two ball-bearings shift the weight of the robot onto
the wheel. One steering link is designed to carry a weight of 120Kg. One could argue that axial-ball bearings would have been better, but the axial load of the radial ball-bearings chosen is much higher than the maximum weight that the robot will ever experience. The two ball-bearings are placed in a machined al u minium housing. All the machining for the slot and the place to fit the bearing was done with a lathe and a milling machine.
3.2.3 Gearbox Fig 14: Gearbox in AutoCAD
The two gearboxes are constructed out of 4 solid al u minium bars each, which are bolted together. On the bottom bar two slots are milled out, increasing the accuracy of their alignment with the other bars. During construction the bars where clamped together, in order to align the shaft holes of both bars precisely. The surfaces of the bars have been milled straight at the beginning, to have accurate reference during construction. The gearbox has 2 ball bearings on the shaft that is connected to the wheel. The other two shafts are for transmission gears. Each shaft has sleeves to adapt to the different diameters of the gears.
The gear ratio is:
n.b.
Wheel diameter = 125mm
Wheel circumference 392.7mm
A further ball bearing with housing is mounted onto the frame. Thus the frame is
connected to the housing and the housing to the gearbox.
The holes marked with stripes in figure 15 are for fixing frame an housing together.
Fig 15: Housing with 3 holes for gearbox-mount
3.2.4 Accuracy
For the construction of the gearbox, only machine tools such as a lathe and a
milling machine can achieve the accuracy. A stand drill is already problematic. The
machines should be calibrated with a dial indicator. A dial indicator is a dial gauge
that can measure distance in fractions of millime tres. It is mounted onto the lathe or milling machine to align the tool with the work piece.
3.3 Electronic Hardware Design
Every circuit in the robot has been designed from basic principles. The design
consists of two modular Micro controllers, the power electronics and the ultrasonic
sensors.
3.3.1 Power Supply circuit
The robot runs of a 12Volt battery. In the cent re of the frame is place to strap on a car battery or motor-cycle battery. With a car battery, the robot runs approximately
3-4 hours in constant action. The power is split up into signal power and motor power from the battery on wards to minimize noise distribution. The motor power goes through an emergency stop button before being fed to the electronics board. All circuits can be switched of through a lever switch added next to the emergency stop. A bipolar capacitor with 4700uF is placed on the power electronics board. Each power regulator is surrounded by capacitors as well. The larger electrolytic capacitors are always accompanied by a bipolar 10nF or 100nF ceramic capacitor. The tracks on the power electronics board have a diameter of 6mm. The motor power cables have a diameter of 4.4mm. The cable is originally designed for speakers. The noise amplitude on the 12Volt rail is less than 100mV.
3.3.2 Micro controller Module
The modular micro controllers was designed to be an improvement from the popular robot football circuit, which is used by many students at the university. Unfortunately the chip used in the old circuit (90S8515) is discontinued and the new generation, the Atmel Mega series usually comes as surface mount device). At a development stage, surface mount is a problem. Firstly, it is not easy to unsolder asurface mount chip and secondly, a surface mount chip can not be stuck into a breadboard to do a quick design check.
The module was designed with the following specification in mind:
-similar amount of ports as the 90S8515
-only a bare minimum on components on board
-serial and programming connector (Robot football compatible)
-Avoid extra features such as test LEDs, I2C connector etc. since they are application dependant
-Power LED for quick confirmation
-Crystal with build-in capacitors
-Plug-in design with a Pin distance usable for bread-boards
The specification is appreciated by the technicians and other students of the University.
Several other students already applied this design to their final year project, which proves the flexibility of the design. The author is currently writing a guide on how to develop with an At mel Mega and the new g cc 3.X compiler. A draft version of the guide can be found in the Appendix.
Fig 16: At mel Mega16 Micro controller board used for designing the motor
controllers
Technical Details of the Microcontroller Module
-Atmel Mega16-AI in TQFP package (Atmel Package Code 44A)
-16 MHz Crystal
-Atmel ISP Programming connector (IDC10, right angled)
-Robot Football 4-Pin Molex Serial Port connector
-3x 10Pin Single-in-Line connectors for IO-Ports
3.3.3 Ultra Sonic Sensors design
The final design of the sensor is more simple than the original. The flexibility has
increased since modulation and signal decoding is part of the software. Faster
sensing is made possible through the changes. However, it demands more
computing time.
Features of the new design include:
-frequency can be set by software
-signal can be coded
-reliable range 1.3 m
The transmitter consists of a software running in a timer at 76 to 84 kHz and
toggling the transistor Q1. The toggling divides the frequency by two.
Unfortunately none of the timer frequency settings match the resonance frequency of the transducers. Therefore, the timer frequency must be programmed to sweep from a few kilohertz under the resonance frequency to a few over the resonance frequency.
Fig 17: Ultra sonic distance measurement electronics
The receiver end consists of a operational amplifier for signal boosting, a transistor Q2 for level shifting (12V to 5V) and a low pass filter R7,C7.
The Op Amp is configured with a only positive rail at 12V. The positive input is clamped to 6V. Feedback resistor RV1 is a 47KOhm potentiometer in the final version, thus creating a variable gain from 1 to 48.
Practically gain values over about 30 amplify noise created by the transmitter over the power rail. Even the extensive use of capacitors could not remove this problem. The sensor can detect flat objects, such as walls and boxes up to 3 meters away. Reliable detection of humans can only be achieved within 1.3 meters.
Fig 18: design of a ultra sonic distance sensor with 8-bit bus connector (original
design)
Low pass Filter
The micro controller recognizes a logical high at 3.5V and above, Atmel (2003), on an digital IO pin. The filter must be matched to give this voltage at the maximum acceptable frequency. Experiments show that, the a design with the 3dB point at 42KHz (Transducer frequency) has not enough safety margin and the micro controller does not always recognise the signal as high when it should be. Therefore the 3dB point is set to 49KHz.
The question is which R and C values to choose in order to have 3.5 Volt at the
output at 49 kHz.
Fig 19: low pass filter (used in ultra sonic circuit)
Initial formulae
(15)
rearranged for R.
(16)
n.b. the output impedance of the transistor circuit has been neglected, since it is lower than the low-pass circuit. The input impedance of the micro controller is much higher than the one of the low-pass circuit, and the impedance can be neglected in the calculation again.
Fig 20: Ultrasonic sensor electronics (final design)
快速自動(dòng)機(jī)器人人平臺(tái)-2
2.2.5向量場(chǎng)
一個(gè)向量場(chǎng)實(shí)質(zhì)是由一個(gè)2-維向量組成的區(qū)域。一個(gè)向量由大小和方向組成，向量對(duì)于速度和航向角而言相當(dāng)重要。
大小被認(rèn)為是向量場(chǎng)中很重要的問題，大小對(duì)于通過每個(gè)向量組合成為向量場(chǎng)是很有用的。越重要的向量越長(zhǎng)，航向角貼近的是更加重要的向量。
一些向量場(chǎng)產(chǎn)生器，像是基礎(chǔ)的勢(shì)場(chǎng)產(chǎn)生法，是不考慮大小的。這種情況下大小經(jīng)常被忽略。
2.2.5.1勢(shì)場(chǎng)
勢(shì)場(chǎng)的一些理論像是軌跡的概念是從物理領(lǐng)域中的電學(xué)部分中分化而出的。
引力場(chǎng)公式如下：
這里 是一個(gè)沖起始到目標(biāo)位置的向量
是一個(gè)從起始指向機(jī)器人的向量
是一個(gè)表征機(jī)器人當(dāng)前位置的單位化的長(zhǎng)度和預(yù)計(jì)的角度。結(jié)果是指向目標(biāo)的。引力場(chǎng)是關(guān)聯(lián)其中的可視的指引。斥力場(chǎng)產(chǎn)生背向目標(biāo)的向量。等式是
表格6展示了一個(gè)引力場(chǎng)指向目標(biāo)點(diǎn)（0.0）。在當(dāng)前應(yīng)用的機(jī)器人僅有當(dāng)前位置的向量才加入計(jì)算 ，對(duì)于多為圖表像表格6這樣，機(jī)器人可以在場(chǎng)中任何可能的地方，同時(shí)也可以在任何點(diǎn)長(zhǎng)生向量。
表格6：引力場(chǎng)
2.2.5.2基于極限環(huán)的向量場(chǎng)
極限環(huán)是非線性控制理論的一部分。但是一個(gè)表格能夠表現(xiàn)極限環(huán)的屬性，像是表格7，那么這個(gè)表格便可以適應(yīng)路徑生成。此問題更深入的解讀請(qǐng)閱讀D-H Kim（2000）。極限環(huán)的非線性功能的第二位表現(xiàn)為一個(gè)向量場(chǎng)包含一個(gè)單位環(huán)。單位環(huán)外的向量將產(chǎn)生于單位環(huán)相切的方向。這可以看成是一個(gè)圓弧/圓軌跡生成率可以引導(dǎo)機(jī)器人自動(dòng)從任何方向進(jìn)入該圓。最終生成的向量場(chǎng)可以用來產(chǎn)生圓弧軌跡或者是用于避障。
極限環(huán)的缺點(diǎn)在于一旦機(jī)器恩跨過了單位元，向量場(chǎng)將指向中心。所以，具體實(shí)現(xiàn)極限環(huán)控制并不容易，因?yàn)闄C(jī)器人在接近單位圓時(shí)可能會(huì)稍稍的越過邊界。
這種場(chǎng)在單位環(huán)內(nèi)進(jìn)行修改時(shí)一個(gè)解決此種問題的可行的措施。
對(duì)圖表7：極限環(huán)
2.2.5.3矢量場(chǎng)的融合
所有的可提供向量場(chǎng)都可以在同一時(shí)間進(jìn)行討論。作者開發(fā)了一種可供合并向量場(chǎng)合并的方法，該方法在Robinson P(2004)中論述。
約束和要求：
-兩個(gè)或者更多的向量場(chǎng)。
-這些向量場(chǎng)包含標(biāo)準(zhǔn)化的向量。
這種方法最好用一個(gè)例子來描述。一個(gè)典型的需要向量場(chǎng)合并的地方在于當(dāng)一個(gè)機(jī)器人R需要避免和機(jī)器人O在路上相遇去目標(biāo)T時(shí)?？聪旅鎴D表8。
在融合向量場(chǎng)過程中，需要一個(gè)加權(quán)函數(shù)。經(jīng)驗(yàn)已經(jīng)證明，高斯正態(tài)分布函數(shù)在合并兩個(gè)場(chǎng)域是很合適的方式。（一個(gè)圓柱體或是一個(gè)椎體都可能產(chǎn)生一個(gè)突然的沖擊以使航向角發(fā)生變化并產(chǎn)生激發(fā)不穩(wěn)定現(xiàn)象。）
圖表8：回避方案
角和當(dāng)前的機(jī)器人航向角q是不同的，ro向量是指向機(jī)器人回避方向。
那個(gè)角是表征機(jī)器人和障礙物碰撞程度的量。這個(gè)角度越小，碰撞事件發(fā)生的情況就越小，同時(shí)避開障礙物的可能性越高。
這個(gè)機(jī)器人的任務(wù)是走到T點(diǎn)。為了將攔在它和目標(biāo)點(diǎn)之間的障礙物也納入考慮，那個(gè)機(jī)器人就必須計(jì)算障礙物和自己的距離。這個(gè)距離在式子中是以矢量ro定義的。和障礙物的距離越短就意味著避開障礙物的重要性越大。
一個(gè)回避向量場(chǎng)應(yīng)該被定義的和任務(wù)向量場(chǎng)rt場(chǎng)一樣。標(biāo)準(zhǔn)化后的目標(biāo)向量是。
提供的兩個(gè)向量和是用高斯加權(quán)方程加在一起的-融合。
這里：是結(jié)果典型的目標(biāo)向量
M是一個(gè)固定的加權(quán)因素
G()是高斯方程。
m , 是對(duì)高斯方程的安全系數(shù)
s 是高斯方程的分布
我們可以知道要回避一個(gè)障礙，本質(zhì)上有兩個(gè)因素。,和ro。
作者按對(duì)于在場(chǎng)域內(nèi)特定點(diǎn)完成這個(gè)任
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