Stanisaw Radkowski, Przemysaw Szulim
Warsaw University of Technology, Institute of VehicleUSE OF DIAGNOSTIC INFORMATION
IN THE ALGORITHM USED FOR PLANNING
THE MOVEMENT OF AN AUTONOMOUS VEHICLE
The manuscript delivered: June 2013
Summary: The goal of the paper is to present a concept of an algorithm which plans the motion of an
autonomous vehicle travelling in a known environment and the preliminary results of the work on such an algorithm. Thanks to this, through relevant inclusion of the information on a vehicle’s technical condition, it is possible to enhance the security of mission completion and expend energy in a more efficient manner.
Keywords: diagnosis, brushless motor, motion planning
1. INTRODUCTION
The concept of construction of autonomous vehicles has been known for a very long time. Research teams in many scientific centers continue efforts aimed at more in-depth exploration of the topic. Autonomization is common in human life and manifests itself in diverse ways. There is no indication of the process coming to a stop. On the contrary, contests and scientific programs contribute to people becoming even more familiar with the issue. Automotive market is a good example where top manufacturers introduce increasingly more advanced systems which assist the driver, from power steering, through park assist solutions to experiments involving vehicles covering routes automatically and autonomously.
Though the area of application of autonomous vehicles is still a very specialized, while the notion “autonomous” means operator-supporting operation in many cases, still the area continues to develop very dynamically, both in scientific and application terms. The paper presents a discussion on the possibility of using algorithms to optimize the trajectory of an autonomous electric vehicle where the optimization criterion is minimization of energy. Related research topics are discussed in many research centers. In paper [1], [2] [5] authors present the solution of the problem of minimizing energy consumption for a small three-wheeled mobile robot. The task was divided into two stages: first determined a global path based on graph algorithm, while in the second stage the path was smoothed.
Authors assumed that the sources of energy losses are rolling resistance. In the paper [3] the authors focus on the problem of local optimization of robot trajectory. They used a simple mathematical model to focus mainly on the problem of presence of nonholonomic constraints. In the paper [4] authors investigated problem of optimal trajectory planning of robot which was powered by batteries and solar cells. They built solar map that was used by algorithm to do trajectory for the robot. The solution of the problem of minimizing energy consumption and time was presented as well. In the present paper, authors attempts to solve the task of finding the optimal trajectory for a mobile robot. It was assumed that the goal is to find a global, optimum trajectory that minimize energy. Important aspect related to the kinematic constraints of the vehicle was skipped. This topic will be more detailed examine in the later stage of the project. In the proposed solution, the source of energy loss is the electric motor drive of the robot. As a result of the occurrence of certain defects, characteristics of the engine and the ability to energy transformation is changing. According to the Cempel model, energy associated with damage is dispersed partly in the form of heat and partly is used for evolution of the damage. The next stage of work should allows answer the question, does through appropriate control of an autonomous vehicle, we are able to reduce the energy dissipation and in that way reduce the growth of damage. Phenomena occurring in BLDC electric motor as a result of the occurrence of certain errors are the subject of research by our team. The paper presents examples of modeling the impact of selected defects on the efficiency of energy conversion. The remainder of this article will be presented proposal of algorithm that solves optimization of simplified problem and operation results.
2. ENERGY DISSIPATION RELATED TO DEFECTS
One of several types of defects occurring in brushless motors is misalignment. The phenomenon occurs when the rotor axis is not aligned with the axis of a motor’s stator. The situation is presented in Fig. 1. In addition two types of misalignment are identified: static and dynamic. The differentiation is associated with the behavior of the characteristic space between the rotor and the stator.
It can be static (invariable) when the rotor’s axis does not change its position with respect to the stator’s axis. In such a case we talk about the static misalignment. When the axis of the rotor revolves around the axis of the stator, it is also the air-gap that rotates. In such a case we talk about the dynamic misalignment. As a result of shift of the axis, the space between the rotor and the stator is deformed. If the nominal width of the air-gap is d0, then
following the dynamic misalignment the resultant width of the gap is expressed by the following formula:
݀ௗ؆ ݀൫ͳ െ ߜௗܿݏሺߠ െ ߱ݐሻ൯ (1)
While in the case of static misalignment, the resultant air-gap can be modeled with the use of the following formula:
݀௦؆ ݀൫ͳ െ ߜ௦ܿݏሺߠሻ൯ (2)
Ër is the angular velocity of the rotor, Ìd,Ìs – shift of the rotor and stator axis, – the angle
for which the width of the gap is measured. Changes in the width of the gap lead to changes of reluctance of the magnetic track which is directly proportionate to the length of the air gap. Permeance, which the opposite of reluctance, can be expressed while using the following formula: ߉ௗൌ ߉ σஶୀଵ߉ܿݏሺ݅ߠ െ ݅߱ݐሻ (3) where: ߉ൌ ଶఓబ൫ଵିξଵିఋమ൯ ௗబఋξଵିఋమ (4) ߉ൌఓௗబబ (5)
The paper [6] demonstrates that for Ìd<40% the influence of the components from the series for ݅>2 is negligibly small and that is why it will be disregarded for simplification. A non-homogenous gap results in change of the magneto-motoric force, and further in changes of magnetic induction, which can be approximated by means of the following formulas for the case with a static gap (7) and for the one with a revolving gap (6):
ܤௗൌ ܤଵሾͳ ߜௗܿݏሺߠ െ ߱ݐሻሿܿݏሺߠ െ ߱௦ݐ െ ߮௧ሻ (6)
ܤ௦ൌ ܤଵሾͳ ߜ௦ܿݏሺߠሻሿܿݏሺߠ െ ߱௦ݐ െ ߮௧ሻ (7)
where p – the number of pairs of poles, Ës – angular velocity of the stator field, Ër –
angular velocity of the rotor.
At this place it should be noted that the derived formulas concern a case with an induction asynchronous motor. In the case of a BLDC synchronous motor, the angular velocities of
the shaft and of the rotating field are permanently linked. It is also worth noting that formula (6) presents a description of amplitude modulation where ߜௗ is the parameter which talks about the depth of modulation. Such modulation never occurs in the case of static misalignment. The electrical current signal for the BLDC motor with a defect caused by dynamic misalignment can be expressed with the use of the below formula [7].
ܫௗൌ ܫଵሾͳ ߙܿݏሺ߱ݐሻሿܿݏሺ߱௦ݐ െ ߮ሻ (8)
is the parameter associated with ߜௗ which tells us about the shift of the rotor’s and stator’s axis. It is also known that the mean value of the current is constant and it does not depend on the depth of modulation. This means that the mean power used by the motor remains unchanged. It is the power which is dissipated by the motor, which is associated with loss in the area of electrical system’s resistance, that changes. If we present the power by means of the following formula:
ܧௌൌ ܴ כ ܫௗଶ݀ݐ (9)
Then the mean value of the power dissipated by the motor will be equal to:
ܧௌൌ ሺͲǤͷ ͲǤʹͷߙଶሻܴܫௗଶ (10)
Thus it can be seen that the presence of a defect of the dynamic misalignment type results in increase of loss in the motor’s electrical circuitry.
Evaluation of the loss is important not only due to the possibility of taking actions which increase efficiency of energy expending but also due to the threat of causing temperature-related demagnetization. It is an issue which is critical for electrical motors with permanent magnets. Depending on the elements of which magnets are made, the permitted operating temperature can be very important in many applications. For some magnets, made of rare earths elements, the permitted (maximum) operating temperature can be around 80OC. Exceeding the temperature could result in permanent loss of magnetic properties of permanent magnets, thus leading to deterioration of operating parameters. Hence, it could be essential to control the drive/motor of an electrical vehicle in such a way so as not to allow the permitted operating conditions to be exceeded. Since there exist many causes of demagnetization, hence it is important to monitor such types of defects so as to estimate their impact on efficiency of energy.
3. ALGORITHM
Rational management of energy requires many information items, which includes two basic groups: information on the sources of energy dissipation and information on limitations. The first type of information could be used for updating an object’s model, the autonomous vehicle in this case. The second type provides information regarding the existing limitations of a model. Having such information at one’s disposal, it is possible to
create a model which will combine all the interesting parameters with the dissipated energy. Analytical solution of this issue would be very difficult or even impossible, mainly due to numerous instances of non-linearity or the nature of the input data. It is known that the surface on which a vehicle travels, where optimum solutions should be sought, is presented in the form of a discrete, two-dimensional table. However, similar problems have already been solved in a different way, which could be adopted to solve the presented issue.
The task of searching for the shortest paths has been described well in many publications and the problem has been successfully resolved in many applications [8]. There are many methods of finding the optimum paths, among which the analytical, field and the so-called grid-based methods should be distinguished. The relatively good properties of the latter methods suggest their use for solving the issue of finding the optimum path and the optimum parameters of a vehicle’s motion, namely the ones which will minimize electrical power consumption.
The grid-based methods owe their name to the method of searching for the shortest path. To put it simply, the map of the terrain on which a vehicle will move is sampled by overlaying a grid of points on it. The algorithm which determines the path analyzes only the points of the grid while examining the cost of covering the distance from one point to another. The general method has many modifications. Numerous tools from the area of mathematics which deals with graphs have been used to solve the problem. That is why, many notions in the further part of the article will come from this field of science. Among several algorithms which have the task of searching for the shortest path, and which have been created for the purpose of searching the graphs, there the algorithms under the names of Djikstra and A* [9]. Both of these algorithms find the shortest path, however algorithm A*, thanks to some additional information, is capable of determining the path faster. Further considerations regarding the algorithm will use the elements of the graph theory. In the case of this class of methods, the solution of each problem associated with search for the optimum path can thus be reduced to finding a relevant path on the graph. A graph is an abstract structure, strongly linked to the terrain map (for this class of a problem). It consists of vertexes and edges. Let us consider a simple case of finding the path from two random points in a town. While using the grid approach, one should overlay the grid on the map of a town and then create a graph involving only the points which are in the area where the streets are. Specific points on the map of the town will correspond to the points of the grid which have been selected in the above way, while the vortexes of the graph will correspond to the points on the map. The edges will connect the neighboring vortexes. In addition weights will be assigned to all vortexes, which will be a parameter corresponding to the cost of moving from one vortex of the graph to its neighbor. In this case, a weight can be the distance between two points on the map. Based on such a representation, the shortest path between any two points can be determined while using the two above mentioned algorithms. An example of a solution of this problem for a labyrinth is presented in Fig. 2.
Fig. 2. Shortest path in maze
A feature of a graph-type representation is the possibility of representing any complex problem. Fig. 3 presents the solution of the problem involving finding the shortest path for a 3D map.
Fig. 3. Search for the shortest path on a 3D map
Representation of many interesting problems with the use of graphs offers the possibility of solving them, including the problem of finding the optimum path and the parameters of movement of a vehicle. To simplify the problem we assumed that the vehicle will be treated as a material point, so as not to include the properties of many vehicles which are associated with non-holonomic constraints. A model of a vehicle was reduced to a certain point with a finite mass of m, aerodynamic drag of C, and rolling friction of f. The vehicle is powered by an electrical motor which is permanently coupled with the wheels by means of a gear. The motion equation describing the vehicle is presented by means of the following formula:
ݒሶ ߦଵݒ ൌ ߦଶ (11) where: ߦଵൌ మഘమఎ మோ (12) ߦଶൌఎோഘݑ െ ݃൫݂ܿݏሺߙሻ ݏ݅݊ሺߙሻ൯ (13)
C – air-drag coefficient m – vehicle mass i – reduced gear ratio
kË – the electric motor parameter
Í – mechanical system efficiency r – radius of the vehicle’s wheels R – engine resistance
f – rolling friction coefficient g – acceleration of gravity
– slope (angle) of the hill along which the vehicle travels
U – is the controlling input, which in the case of this model corresponds to the voltage applied to the motor. Voltage will be taken as the factor controlling the vehicle’s motion in the first stage. In such a case it is expected that the algorithm will find the path which will ensure minimization of energy consumption on the Îs section of the road, from a certain starting point s to a certain end point t:
ܧ ൌ ݑሺݐሻ݅ሺݐሻ݀ݏ௦௧ (14)
The algorithm should search the three-dimensional space, whose two dimensions correspond to the location while the third one corresponds to the control voltage. The first problem emerges once the precise relation (amount) of the energy drawn from the power source is determined. For a selected section Îs (a section of the road connecting two neighboring points on the map), with a slope of and initial vehicle speed of v0, the energy drawn from the source is determined by the following formula:
ܧ ൌ ቀ௨ோమെ௨కమ కభ ቁ ݐ ௨ కభቀ కమ కభെ ݒቁ ൫ͳ െ ݁ ିకభ௧൯ (15)
x The first problem is that the energy drawn from the source can have a negative value. This can happen, e.g. when a vehicle was braking. Negative energy value means that the cost of driving between two points could be negative. None of the proposed algorithms is able to solve in a correct way the task of finding a path with the smallest sum of weights in the case of negative weights. Thus change of the algorithm is required.
x The formula contains relation to time. The time of driving between the two points selected on the map, with a relevant control of the drive, can be determined while using a differential equation, however it cannot be reduced to an analytical form. Thus an approximate solution has to be determined while using one of several known methods, which complicates the solution.
x In accordance with the formula, the energy drawn from the source is also dependent on the initial speed v0. It is the speed which the vehicle reached while moving from a preceding point to the point currently contemplated. Thus the energy, and so the cost of moving between the two nodes is not constant in the sense that it only depends on the selected section of the route as well as on the adopted control, but it also depends on the way the motion was managed at preceding sections of the
route. Reducing this to the level of graphs, we obtain a graph in which the weights of the edges are the functions of the weights of preceding edges of the path, which in a simplified manner is presented by Fig. 4. No algorithm which can determine the optimum solution is known for such a problem.
Fig. 4. The issue of weights of the edges
To solve the problem, one should make several simplifying assumptions. Let the task of finding the optimum path and control be reduced to a case of searching only for the optimum control for the defined path. This way the three-dimensional problem is reduced down to a two-dimensional problem. The next assumption is the possibility of moving only in one direction, from the starting point to the end point. Thus a directed graph is created. Fig. 5 presents an example of such a graph. What is characteristic in this graph is that it is not possible to move between the vertexes which are located above and below the contemplated vertex.
Fig. 5. A two-dimensional graph
This fact has its physical justification, since change of the level of control is not possible when one simultaneously does not start to move. The graph has a matrix structure.
Fig. 6. A modified two-dimensional graph
A modified graph (Fig. 6) can be used to solve the problem of lack of constant weights in the graph. One should note that moving from the first column of the graph to the second one is clear, since the initial conditions are constant and do not change. Thus the indicated problem does not exist in this case. Moving from the second column to the third one is no longer so clear, since while contemplating every pair of points of which one belongs to column 2 and the other belongs to column 3, we need to take into account the speed with which the vehicle reached the contemplated point while determining the cost. There are as many such possibilities as the number of lines R in the graph. Thus while contemplating each and every pair from columns 2, 3, whose total number is R2, one should additionally analyze each such a pair for R cases. The observation can be presented in the form of a graph, by adding (R-1) columns to column 3. When contemplating column 3 now, together with the columns added to it, one can note that all of them correspond to one and the same point on the map. Similarly when analyzing column 4, the R columns for each column from the 3rd row should be added. The number of edges between the 1st and the 2nd column is R2, between the 2nd and the 3rd it is R3, while between the 3rd and the 4th it is R4. If we wanted to determine the total number of edges in such a graph, then while assuming that the number of points contemplated on the map is N (that is it corresponds to the original number of columns in the graph), the relevant formula could be presented in the following way: ܮ ൌ ܴଶ ܴଷ ܴସ ܴହ ڮ ൌ σே ܴൌ ୀଶ σ ܴଶܴିଵൌ ܴଶ ଵିோ ಿషభ ଵିோ ൎ ܴே ேିଵ ୀଵ (16)
It is a very important problem since already for a relatively small size the graph, e.g. 10x10, when the above method is used, the graph grows by so much that the number of edges in the graph will be 1010. Hence it is a problem which is virtually unsolvable in this form. Another solution is to select the parameter controlling the vehicle’s movement which will ensure that the weights of the edges of the graph will be constant. Adoption of controls in the form of electrical current or torque does not solve the problem. It is only adoption of the speed as the method of control that guarantees that the planned parameters will be reached. To implement such a form of control, one must make an assumption as regards the way in which transition from the peak of one column to the peak of another column will be interpreted. The paper proposes a variant where the velocity decreases in a straight line from vC do vC+1 where C is the number of the analyzed column. Such an assumption,
though it need not necessarily mean a solution which is optimum from energy point of view, basically simplifies the problem of controlling the current applied to the motor. In this case the basic motion equation (11) remains valid. It is necessary to add a controller which will ensure that the required velocity is achieved. The adopted solution about the velocity increasing in a straight line manner while covering a section of the road between to neighboring points, implies that a constant value of acceleration be adopted for Îs section of the road based on the following formula:
ܽ ൌ௩శభି௩
௧శభି௧ൌ
ο௩
ο௧ (17)
οݐ ൌ௩ଶο௦
ା௩శభ (18)
While introducing ݒሶ ൌ ܽ to formula (11) and while transforming (11) in such a way so as to determine u(t), we obtain a formula for the voltage which should be applied to the motor’s terminals so that for the defined model, for the specified parameters of a road section (length, slope), we can obtain the required increase of the velocity from vC to vC+1,
which are the parameters optimized by the algorithm. ݑሺݐሻ ൌఎோ ഘቂߦଵ ο௩ ο௧ݐ ߦଵݒ ο௩ ο௧ ݃൫݂ܿݏሺߙሻ ݏ݅݊ሺߙሻ൯ቃ (19)
It should be added that for the sake of simplicity, the restrictions associated with the permitted values of the controlling voltage have been omitted. The current fed from the source of power for the purpose of control (19) can be presented by means of the following formula: ݅ሺݐሻ ൌఎ ഘቂߦଵ ο௩ ο௧ݐ ߦଵݒ ο௩ ο௧ ݃൫݂ܿݏሺߙሻ ݏ݅݊ሺߙሻ൯ቃ െ ഘ ோ ቀݒ ο௩ ο௧ݐቁ (20)
The energy taken from the source for the purpose of control (19) will thus be equal to: ܧ ൌ ݑሺݐሻ݅ሺݐሻ݀ݐ ൌο௧ ଵଷఎோഘο௩ο௧మమߦଵቀ ఎഘߦଵെ ഘ ோቁ οݐଷ ଵ ଶ ோ ఎഘ ο௩ ο௧ቀ ఎഘߦଵ൬ߦଵݒ ο௩ ο௧ ݃൫݂ܿݏሺߙሻ ݏ݅݊ሺߙሻ൯൰ െ ߦଵோഘݒെ ൬ߦଵݒο௩ο௧ ݃൫݂ܿݏሺߙሻ ݏ݅݊ሺߙሻ൯൰ோഘቁ οݐଶ ோ ఎഘ൬ߦଵݒ ο௩ ο௧ ݃൫݂ܿݏሺߙሻ ݏ݅݊ሺߙሻ൯൰ ቀ ఎഘ൬ߦଵݒ ο௩ ο௧ ݃൫݂ܿݏሺߙሻ ݏ݅݊ሺߙሻ൯൰ െ ഘ ோ ݒቁ οݐ (21)
The feature which is characteristic for the energy determined on the basis of formula (21), which defines the cost of moving between two peaks of the graph, is that it depends only on the parameters which are associated with every peak of the graph, and more precisely with the pair of peaks connected by the contemplated edge. Since the graph is a directed graph, where the weights can adopt negative values, thus the Bellman-Ford algorithm has been applied to find the path with the smallest weight, and hence to find the optimum speeds at respective sections of the route which will ensure minimal energy consumption. It is worth noting that in the equation (21) the only parameter subject to optimization is the speed on a particular segment. In the paper 1.2 * robots moved in the flat area and time optimalization was required, otherwise the problem become trivial (minimum energy cost is obtained at the minimum speed). If the ground is not leveled, optimization problem of time may, but need not be taken into account. This effect can be seen in the next section.
4. THE RESULTS
This part of the article will present the results that demonstrate the operation of the algorithm. The only sources of energy dissipation will be the sources which have been included in the model, namely the loss in the area of resistance of the motor, the loss associated with air drag and the loss associated with rolling friction. Fig. 7 presents the result of the algorithm’s operation for a theoretical section of the route whose profile is presented by the first graph. Such a profile has been adopted on purpose due to its regular nature. One could expect that the shape of such a profile should be at least partly reflected in the control profile. The lower graph presents the profile of the velocity which was determined for the predefined profile of the route and for certain parameters of the model. The road section was divided into 400 equal parts. The algorithm determined the speeds for respective sections. The speed could change in the range from 0.1 to 30 [m/s]. The area of changes of the speed was divided into 100 equal sections, which resulted in speed resolution of around 0.3 m/s, with the route sections equal to 20 m. What is characteristic is that in accordance with the movement plan defined by the algorithm the vehicle should start accelerating at a uniform rate after around 1000 meters of distance. This agrees quite well with the intuitions which tell us that due to the presence of obstacles the vehicle should react early enough. The algorithm proposed that the movement be realized in a manner which is optimum from energy consumption point of view. The final phase of the movement is also characteristic. It is true that the terrain profile is sinusoid starting from the 2000 meter mark, still due to the fact that the vehicle is to come to a stop at the 8000-meter mark, the algorithm manages the vehicles movement in a different way over the last around 1000 meters, while accounting for the necessity of the vehicle finally coming to a stop.
The further part of the paper will present the influence that respective parameters, which are associated with energy dissipation, have on the optimum speed profile.
Fig. 8. Influence of resistance on the velocity profile
Figure 8 presents the influence of resistance change on the resultant velocity profile. The blue color shows the basic profile, the same as in Fig. 7.
Fig. 9 presents the influence of increase of the rolling friction factor on the resultant optimum velocity profile. It can be seen that for the road sections with the lowest height, the algorithm selected the highest possible speeds. In such cases it can be seen that increase of the permitted speed could increase the vehicles energy efficiency.
Fig. 10. Impact of air-drag on the velocity profile
Fig. 10 presents the influence that increase of the aerodynamic factor of a vehicle has on the optimum velocity profile. As the intuition tells us, the vehicle will be moving with lower speeds.
5. CONCLUSIONS
The article presents the general concept of creating an algorithm which enables determination of the optimum parameters of a vehicle’s movement. The defects of the motor and of the power unit have been assumed to be the sources of energy dissipation. The second part of the paper presents the essence of the algorithm’s operation while pointing to some problems which have emerged and while proposing an algorithm which determines the optimum movement parameters for a simplified case in which the vehicle moved along a predefined route. The results of the algorithm’s operation for a simple case are also presented, along with the influence that individual parameters of the model, which are associated with energy dissipation, have on the optimum velocity profile of the vehicle. In the further stage of the work we expect to estimate the impact that other defects, such as bearing defects and demagnetization have on energy dissipation, and we plan to continue the work on the algorithm which enables simultaneous determination of the optimum route and velocity profile.
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WYKORZYSTANIE INFORMACJI DIAGNOSTYCZNEJ W ALGORYTMIE PLANOWANIA RUCHU POJAZDU AUTONOMICZNEGO
Streszczenie: Celem pracy jest przedstawienie koncepcji algorytmu planujcego ruch pojazdu
autonomicznego poruszajcego si w znanym rodowisku oraz wstpnych rezultatów prac nad nim. Dziki niemu, poprzez odpowiednie uwzgldnienie informacji o stanie technicznym pojazdu, mo na bdzie zwikszy bezpieczestwo wykonania misji pojazdu oraz efektywniej wydatkowa energi.
