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fuzzy logic applications full report
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Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise. Fuzzy set theory, originally introduced by Lotfi Zadeh in the 1960's, resembles human reasoning in its use of approximate information and uncertainty to generate decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. By contrast, traditional computing demands precision down to each bit. Since knowledge can be expressed in a more natural by using fuzzy sets, many engineering and decision problems can be greatly simplified.
Fuzzy set theory implements classes or groupings of data with boundaries that are not sharply defined (i.e., fuzzy). Any methodology or theory implementing "crisp" definitions such as classical set theory, arithmetic, and programming, may be "fuzzified" by generalizing the concept of a crisp set to a fuzzy set with blurred boundaries. The benefit of extending crisp theory and analysis methods to fuzzy techniques is the strength in solving real-world problems, which inevitably entail some degree of imprecision and noise in the variables and parameters measured and processed for the application. Accordingly, linguistic variables are a critical aspect of some fuzzy logic applications, where general terms such a "large," "medium," and "small" are each used to capture a range of numerical values. While similar to conventional quantization, fuzzy logic allows these stratified sets to overlap (e.g., a 85 kilogram man may be classified in both the "large" and "medium" categories, with varying degrees of belonging or membership to each group). Fuzzy set theory encompasses fuzzy logic, fuzzy arithmetic, fuzzy mathematical programming, fuzzy topology, fuzzy graph theory, and fuzzy data analysis, though the term fuzzy logic is often used to describe all of these.
Fuzzy logic emerged into the mainstream of information technology in the late 1980's and early 1990's. Fuzzy logic is a departure from classical Boolean logic in that it implements soft linguistic variables on a continuous range of truth values which allows intermediate values to be defined between conventional binary. It can often be considered a superset of Boolean or "crisp logic" in the way fuzzy set theory is a superset of conventional set theory. Since fuzzy logic can handle approximate information in a systematic way, it is ideal for controlling nonlinear systems and for modeling complex systems where an inexact model exists or systems where ambiguity or vagueness is common. A typical fuzzy system consists of a rule base, membership functions, and an inference procedure. Today, fuzzy logic is found in a variety of control applications including chemical process control, manufacturing, and in such consumer products as washing machines, video cameras, and automobiles.
Boolean Vs Fuzzy
300 years B.C., the Greek philosopher, Aristotle came up with binary logic(0,1), which is now the principle foundation of Mathematics. It came down to one law: A or not-A, either this or not this. For example, a typical rose is either red or not red. It cannot be red and not red. Every statement or sentence is true or false or has the truth value 1 or 0. This is Aristotle's law of bivalence and was philosophically correct for over two thousand years.
Two centuries before Aristotle, Buddha, had the belief which contradicted the black-and-white world of worlds, which went beyond the bivalent cocoon and see the world as it is, filled with contradictions, with things and not things. He stated that a rose, could be to a certain degree completely red, but at the same time could also be at a certain degree not red. Meaning that it can be red and not red at the same time. Conventional(Boolean) logic states that a glass can be full or not full of water. However, suppose one were to fill the glass only halfway. Then the glass can be half-full and half-not-full. Clearly, this disprove's Aristotle's law of bivalence. This concept of certain degree or multivalence is the fundamental concept which propelled Zader Lofti of University Berkely in the 1960's to introduce fuzzy logic. The essential characteristics of fuzzy logic founded by him are as follows.
¢ In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning.
¢ In fuzzy logic everything is a matter of degree.
¢ Any logical system can be fuzzified
¢ In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables
¢ Inference is viewed as a process of propagation of elastic constraints.
The third statement hence, define Boolean logic as a subset of Fuzzy logic.
Fuzzy Sets
Fuzzy Set Theory was formalised by Professor Lofti Zadeh at the University of California in 1965. What Zadeh proposed is very much a paradigm shift that first gained acceptance in the Far East and its successful application has ensured its adoption around the world.
A paradigm is a set of rules and regulations which defines boundaries and tells us what to do to be successful in solving problems within these boundaries. For example the use of transistors instead of vacuum tubes is a paradigm shift - likewise the development of Fuzzy Set Theory from conventional bivalent set theory is a paradigm shift.
Bivalent Set Theory can be somewhat limiting if we wish to describe a 'humanistic' problem mathematically. For example, Fig 1 below illustrates bivalent sets to characterise the temperature of a room.
The most obvious limiting feature of bivalent sets that can be seen clearly from the diagram is that they are mutually exclusive - it is not possible to have membership of more than one set ( opinion would widely vary as to whether 50 degrees Fahrenheit is 'cold' or 'cool' hence the expert knowledge we need to define our system is mathematically at odds with the humanistic world). Clearly, it is not accurate to define a transiton from a quantity such as 'warm' to 'hot' by the application of one degree Fahrenheit of heat. In the real world a smooth (unnoticeable) drift from warm to hot would occur.
This natural phenomenon can be described more accurately by Fuzzy Set Theory. Fig.2 below shows how fuzzy sets quantifying the same information can describe this natural drift.
Fuzzy Set Operations.
The membership function of the Union of two fuzzy sets A and B with membership functions and respectively is defined as the maximum of the two individual membership functions

The Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra.
The membership function of the Complement of a Fuzzy set A with membership function is defined as
The following rules which are common in classical set theory also apply to Fuzzy set theory.
De Morgans law

Time dependent fuzzy logic
¢ Crisp logics:
Traditional combinatorial logic is a static logic. It provides a logic output 1 or 0 based on the binary values at the inputs. Sequential logic takes combinatorial logic a step further; it considers events or states in sequential order. It is a process/state-based logic that, based on current states and input/output parameters, determines the next state. This may be encapsulated in an if¦.then, else statement: if {this happens}, then {do this}, else {do that}. The next stat is merely the next ˜step™of a process that takes place at a later, unspecified,time. The state diagram in figure (a) substantiates exactly this point: the next state {A ,B or C} in the state diagram depends on the current state{ A,B,or C} and input {1 pr 0}, whereby the dimension of time is not significant nor explicitly indicated. For instance, state A will change to state B only when the input will be 1; when the input will become 1, however, is not known.
Combinatorial or sequential logic does not address many knowledge intensive and real-time processes where temporal reasoning plays an important role. The logic that extends the traditional logic and predicate calculus to include the notion of time is called temporal logic. However, combinatorial, sequential, and temporal logics are crisp and the parameter and variable values true/false, exactly 1 or 0.
¢ Fuzzy logics
Fuzzy logics may be considered a generalized combinatorial or sequential logic; however, the passage of time is not necessarily of the essence. In fuzzy control an if¦.then, else approach is also followed, where again the passage of time is not of the essence.
As with combinatorial and sequential processes, there are real-time fuzzy processes where temporal reasoning is important. However, existing fuzzy control approaches are not result related, they are algorithmically oversimple, and they do not reflect real-time evaluation of the control objectives. To overcome this difficulty, different approaches have been proposed but again,, time is not explicit in these approaches.
How Does Fuzzy Logic Work
In order to illustrate some basic concepts in Fuzzy Logic, consider a simplified example of a thermostat controlling a heater fan .The room temperature detected through a sensor is input to a controller which outputs a control force to adjust the heater fan speed.
A conventional thermostat works like an on-off switch (Figure 2). If we set it at 78oF then the heater is activated only when the temperature falls below 75oF . When it reaches 81oF the heater is turned off. As a result the desired room temperature is either too warm or too hot.
A fuzzy thermostat works in shades of gray where the temperature is treated as a series of overlapping ranges. For example, 78oF is 60% warm and 20% hot. The controller is programmed with simple if-then rules that tell the heater fan how fast to run. As a result, when the temperature changes the fan speed will continuously adjust to keep the temperature at the desired level.
Our first step in designing such a fuzzy controller is to characterize the range of values for the input and output variables of the controller. Then we assign labels such as cool for the temperature and high for the fan speed, and we write a set of simple English-like rules to control the system. Inside the controller all temperature regulating actions will be based on how the current room temperature falls into these ranges and the rules describing the system behavior. The controller's output will vary continuously to adjust the fan speed.
The temperature controller described above can be defined in four simple rules:
IF temperature IS cold THEN fan_speed IS high
IF temperature IS cool THEN fan_speed IS medium
IF temperature IS warm THEN fan_speed IS low
IF temperature IS hot THEN fan_speed IS zero
Here the linguistic variables cool, warm, high, etc. are labels which refer to the set of overlapping values shown in figure 2. These triangular shaped values are called membership functions.
A fuzzy controller works similar to a conventional system: it accepts an input value, performs some calculations, and generates an output value. This process is called the Fuzzy Inference Process and works in three steps illustrated in Figure 3: (a) Fuzzification where a crisp input is translated into a fuzzy value, (b) Rule Evaluation, where the fuzzy output truth values are computed, and © Defuzzification where the fuzzy output is translated to a crisp value.
During the fuzzification step the crisp temperature value of 78oF is input and translated into fuzzy truth values. For this example, 78oF is fuzzified into warm with truth value 0.6 (or 60%) and hot with truth value 0.2 (or 20%).
During the rule evaluation step the entire set of rules is evaluated and some rules may fire up. For 78oF only the last two of the four rules will fire. Specifically, using rule three the fan_speed will be low with degree of truth 0.6. Similarly, using rule four the fan_speed will be zero with degree of truth 0.2.
During the defuzzification step the 60% low and 20% zero labels are combined using a calculation method called the Center of Gravity (COG) in order to produce the crisp output value of 13.5 RPM for the fan speed.
Why Use Fuzzy Logic
¢ An Alternative Design Methodology Which Is Simpler, And Faster
o Fuzzy Logic reduces the design development cycle
o Fuzzy Logic simplifies design complexity
o Fuzzy Logic improves time to market
¢ A Better Alternative Solution To Non-Linear Control
o Fuzzy Logic improves control performance
o Fuzzy Logic simplifies implementation
o Fuzzy Logic reduces hardware costs
Fuzzy Logic is a paradigm for an alternative design methodology which can be applied in developing both linear and non-linear systems for embedded control. By using fuzzy logic, designers can realize lower development costs, superior features, and better end product performance. Furthermore, products can be brought to market faster and more cost-effectively.
An Alternative Design Methodology Which Is Simpler, And Faster
In order to appreciate why a fuzzy based design methodology is very attractive in embedded control applications let us examine a typical design flow. Figure 4 illustrates a sequence of design steps required to develop a controller using a conventional and a Fuzzy approach.
Using the conventional approach our first step is to understand the physical system and its control requirements. Based on this understanding, our second step is to develop a model which includes the plant, sensors and actuators. The third step is to use linear control theory in order to determine a simplified version of the controller, such as the parameters of a PID controller. The fourth step is to develop an algorithm for the simplified controller. The last step is to simulate the design including the effects of non-linearity, noise, and parameter variations. If the performance is not satisfactory we need to modify our system modeling, re-design the controller, re-write the algorithm and re-try.
With Fuzzy Logic the first step is to understand and characterize the system behavior by using our knowledge and experience. The second step is to directly design the control algorithm using fuzzy rules, which describe the principles of the controller's regulation in terms of the relationship between its inputs and outputs. The last step is to simulate and debug the design. If the performance is not satisfactory we only need to modify some fuzzy rules and re-try.
Although the two design methodologies are similar, the fuzzy-based methodology substantially simplifies the design loop. This results in some significant benefits, such as reduced development time, simpler design and faster time to market:
Fuzzy Logic reduces the design development cycle
With a fuzzy logic design methodology some time consuming steps are eliminated. Moreover, during the debugging and tuning cycle you can change your system by simply modifying rules, instead of redesigning the controller. In addition, since fuzzy is rule based, you do not need to be an expert in a high or low level language which helps you focus more on your application instead of programming. As a result, Fuzzy Logic substantially reduces the overall development cycle.
Fuzzy Logic simplifies design complexity
Fuzzy logic lets you describe complex systems using your knowledge and experience in simple English-like rules. It does not require any system modeling or complex math equations governing the relationship between inputs and outputs. Fuzzy rules are very easy to learn and use, even by non-experts. It typically takes only a few rules to describe systems that may require several of lines of conventional software. As a result, Fuzzy Logic significantly simplifies design complexity.
Fuzzy Logic improves time to market
Commercial applications in embedded control require a significant development effort a majority of which is spent on the software portion of the project. Development time is a function of design complexity, and the number of iterations required in a debugging and tuning cycle. As we explained above, a fuzzy based design methodology addresses both issues very effectively. Moreover, due to its simplicity the description of a fuzzy controller not only is transportable across design teams, but also provides a superior media to preserve, maintain, and upgrade intellectual property. As a result, Fuzzy Logic can dramatically improve time to market.
A Better Alternative Solution To Non-Linear Control
Most real life physical systems are actually non-linear systems. Conventional design approaches use different approximation methods to handle non-linearity. Some typical choices are, linear, piecewise linear, and lookup table approximations to trade off factors of complexity, cost, and system performance.
A linear approximation technique is relatively simple, however it tends to limit control performance and may be costly to implement in certain applications. A piecewise linear technique works better, although it is tedious to implement because it often requires the design of several linear controllers. A lookup table technique may help improve control performance, but it is difficult to debug and tune. Furthermore in complex systems where multiple inputs exist, a lookup table may be impractical or very costly to implement due to its large memory requirements.
Fuzzy logic provides an alternative solution to non-linear control because it is closer to the real world. Non-linearity is handled by rules, membership functions, and the inference process which results in improved performance, simpler implementation, and reduced design costs:
Fuzzy Logic improves control performance
In many applications Fuzzy Logic can result in better control performance than linear, piecewise linear, or lookup table techniques. For instance, a typical problem associated with traditional techniques is trading-off the controller's response time versus overshoot. For the simple one-input temperature controller example this is illustrated in Figure 5:
The first linear approximation for the desired curve generates a slow output response with no overshoot, which implies that the room would be too cold for a while. The second linear approximation results in faster response with an overshoot and subsequent fluctuations, which implies that the temperature will be uncomfortable for a period of time.
With fuzzy logic we can use rules and membership functions to approximate any continuous function to any degree of precision. Figure 6 illustrates how we can approximate the desired control curve for our temperature controller using four points (or four rules). We can also add more rules to increase the accuracy of the approximation (similar to a Fourier transform), which yields an improved control performance. Rules are much simpler to implement and much easier to debug and tune than piecewise linear or lookup table techniques.
IF temperature IS cold THEN force IS high
IF temperature IS cool THEN force IS medium
IF temperayure IS warm THEN force IS low
IF temperature IS hot THEN force IS zero
Rules are not like a lookup table because the fuzzy arithmetic interpolates the shape of the non-linear function. The combined memory required for the labels and fuzzy inference is substantially less than a lookup table, especially for multiple input systems. As a result, processing speed can be improved as well.
Another example of robust control that can be achieved with Fuzzy Logic is the classical problem of the inverted pendulum. A conventional controller for the pendulum depends on system parameters such as length, weight, and mass. If the parameters change, then we need to re-design our controller. With fuzzy control this is not necessary because a fuzzy system is robust. Aptronix has demonstrated an actual device where we can vary the weight or length of the pendulum and the system is still stable using the original set of rules.
By using a more natural rule-based approach which is closer to the real world, Fuzzy control can offer a superior performance and a better trade-off between system robustness and sensitivity, which results into handling non-linear control better than traditional methods.
Fuzzy Logic simplifies implementation
The one input temperature controller presented so far has helped us illustrate some fundamental concepts, however real life control is much more complex in nature. Most control applications have multiple inputs and require modeling and tuning of a large number of parameters which makes implementation very tedious and time consuming. Fuzzy rules can help you simplify implementation by combining multiple inputs into single if-then statements while still handling non-linearity.
Fuzzy Logic reduces hardware costs
Using a lookup table the two-input temperature controller requires 64Kb of memory, while the fuzzy approach is accomplished with less than 0.5Kb of memory for labels and object code combined. This difference in memory savings implies a cheaper hardware implementation. In addition, conventional techniques in most real life applications require complex mathematical analysis and modeling, floating point algorithms, and complex branching. This typically yields a substantial size of object code which requires a high end DSP chip to run. Fuzzy Logic enables you to use a simple rule based approach which offers significant cost savings, both in memory and processor class.
Fuzzy Logic - a powerful new technology
As we all know, Japanese products have higher standards in comparision with other countries. Because they use fuzzy logic technique in most of the products. The Japanese have a famous automatically operated train in Sendai that moves so smoothly you can hardly tell it's travelling.
A load of clothes into fuzzy washer and press start, and the machine begins to churn , automatically choosing the best cycle.
Place chili, potatoes, or etc in a fuzzy microwave and push single button, and it cooks for the right time at the proper temperature.
Fuzzy Logic was being made over cars: cushioning their ride, enhancing safety, and cutting gas consumption by certain percentage. There's a prototype of a prototype of a fuzzy helicopter that hovers automatically, without adjustments by the pilot. It schedules elevators and traffic lights, and prevents tunnel cave-ins at constructions at constructions sites. If one were to visit Japan, one could see nation living slightly further in the future.
Fuzzy logic also has its applications in the fields of information retrieval systems, a navigation system for automatic cars. A predicative fuzzy logic controller for automatic operation pf trains, laboratory water level controllers, controllers for robot arc-welders, feature definition controllers for robot vision, graphics controllers for automated police sketchers, and many more.
Fuzzy logic is a powerful problem-solving methodology with a myriad of applications in embedded control and information processing. Fuzzy provides a remarkably simple way to draw definite conclusions from vague, ambiguous or imprecise information. In a sense, fuzzy logic resembles human decision making with its ability to work from approximate data and find precise solutions.
Unlike classical logic which requires a deep understanding of a system, exact equations, and precise numeric values, Fuzzy logic incorporates an alternative way of thinking, which allows modeling complex systems using a higher level of abstraction originating from our knowledge and experience. Fuzzy Logic allows expressing this knowledge with subjective concepts such as very hot, bright red, and a long time which are mapped into exact numeric ranges.
Fuzzy Logic has been gaining increasing acceptance during the past few years. There are over two thousand commercially available products using Fuzzy Logic, ranging from washing machines to high speed trains. Nearly every application can potentially realize some of the benefits of Fuzzy Logic, such as performance, simplicity, lower cost, and productivity.
Fuzzy Logic has been found to be very suitable for embedded control applications. Several manufacturers in the automotive industry are using fuzzy technology to improve quality and reduce development time. In aerospace, fuzzy enables very complex real time problems to be tackled using a simple approach. In consumer electronics, fuzzy improves time to market and helps reduce costs. In manufacturing, fuzzy is proven to be invaluable in increasing equipment efficiency and diagnosing malfunctions.
Daniel Mcneil and Paul Freiberger " Fuzzy Logic".
Fuzzy sets and fuzzy logic (Theory and applications) by George J. Klir/ Bo Yuan.
Understanding neural networks and fuzzy logic by Stamatios V. Kartalopoulos.
http ://
Fuzzy Fundamentals by E.Cox (IEEE Spectrum, October 1992,pp.-58-61).

Contents :
¢ Introduction
¢ Boolean Vs Fuzzy
¢ Fuzzy Sets
¢ Fuzzy Set Operations
¢ Time Dependent Fuzzy Logic
¢ How does fuzzy logic works
¢ Why use fuzzy logic
¢ Applications
¢ Conclusion
¢ References
Post: #2
Presented by:ANSHU ANAND


Fuzzy logic has rapidly become one of the most successful of today's technologies for developing sophisticated control systems. The reason for which is very simple. Fuzzy logic addresses such applications perfectly as it resembles human decision making with an ability to generate precise solutions from certain or approximate information. It fills an important gap in engineering design methods left vacant by purely mathematical approaches (e.g. linear control design), and purely logic-based approaches (e.g. expert systems) in system design.

While other approaches require accurate equations to model real-world behaviors, fuzzy design can accommodate the ambiguities of real-world human language and logic. It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models.


The first applications of fuzzy theory were primaly industrial, such as process control for cement kilns. However, as the technology was further embraced, fuzzy logic was used in more useful applications. In 1987, the first fuzzy logic-controlled subway was opened in Sendai in northern Japan. Here, fuzzy-logic controllers make subway journeys more comfortable with smooth braking and acceleration. Best of all, all the driver has to do is push the start button! Fuzzy logic was also put to work in elevators to reduce waiting time. Since then, the applications of Fuzzy Logic technology have virtually exploded, affecting things we use everyday.
Take for example, the fuzzy washing machine . A load of clothes in it and press start, and the machine begins to churn, automatically choosing the best cycle. The fuzzy microwave, Place chili, potatoes, or etc in a fuzzy microwave and push single button, and it cooks for the right time at the proper temperature. The fuzzy car, manuvers itself by following simple verbal instructions from its driver. It can even stop itself when there is an obstacle immedeately ahead using sensors. But, practically the most exciting thing about it, is the simplicity involved in operating it.
Before illustrating the mechanisms which make fuzzy logic machines work, it is important to realize what fuzzy logic actually is. Fuzzy logic is a superset of conventional(Boolean) logic that has been extended to handle the concept of partial truth- truth values between "completely true" and "completely false". As its name suggests, it is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.
The essential characteristics of fuzzy logic as founded by Zader Lotfi are as follows.
In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning.
In fuzzy logic everything is a matter of degree.
Any logical system can be fuzzified
In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables
Inference is viewed as a process of propagation of elastic constraints.
The third statement hence, define Boolean logic as a subset of Fuzzy logic.

Human beings make descisions based on rules. Although, we may not be aware of it, all the descisions we make are all based on computer like if-then statements. If the weather is fine, then we may decide to go out. If the forecast says the weather will be bad today, but fine tommorow, then we make a descision not to go today, and postpone it till tommorow. Rules associate ideas and relate one event to another.
Fuzzy machines, which always tend to mimic the behaviour of man, work the same way. However, the descision and the means of choosing that descision are replaced by fuzzy sets and the rules are replaced by fuzzy rules. Fuzzy rules also operate using a series of if-then statements. For instance, if X then A, if y then b, where A and B are all sets of X and Y. Fuzzy rules define fuzzy patches, which is the key idea in fuzzy logic.

A machine is made smarter using a concept designed by Bart Kosko called the Fuzzy Approximation Theorem (FAT). The FAT theorem generally states a finite number of patches can cover a curve as seen in the figure below. If the patches are large, then the rules are sloppy. If the patches are small then the rules are fine.

In a fuzzy system this simply means that all our rules can be seen as patches and the
input and output of the machine can be associated together using these patches.
Graphically, if the rule patches shrink, our fuzzy subset triangle gets narrower. Simple
enough? Yes, because even novices can build control systems that beat the best math
models of control theory. Naturally, it is math-free system.
Fuzzy control, which directly uses fuzzy rules is the most important application in fuzzy theory. Using a procedure originated by Ebrahim Mamdani in the late 70s, three steps are taken to create a fuzzy controlled machine:
Fuzzification(Using membership functions to graphically describe a situation)
Rule evaluation(Application of fuzzy rules)
Defuzzification(Obtaining the crisp or actual results)

As a simple example on how fuzzy controls are constructed, consider the following classic situation: the inverted pendulum. Here, the problem is to balance a pole on a mobile platform that can move in only two directions, to the left or to the right. The angle between the platform and the pendulum and the angular velocity of this angle are chosen as the inputs of the system. The speed of the platform hence, is chosen as the corresponding output.
Step 1
First of all, the different levels of output (high speed, low speed etc.) of the platform is defined by specifying the membership functions for the fuzzy_sets. The graph of the function is shown below
Similary, the different angles between the platform and the pendulum and...

the angular velocities of specific angles are also defined

Note: For simplicity, it is assumed that all membership functions are spreaded equally. Hence, this explains why no actual scale is included in the graphs.
Step 2
The next step is to define the fuzzy rules. The fuzzy rules are mearly a series of if-then statements as mentioned above. These statements are usually derived by an expert to achieve optimum results. Some examples of these rules are:
i) If angle is zero and angular velocity is zero then speed is also zero. ii) If angle is zero and angular velocity is low then the speed shall be low.
The full set of rules is summarised in the table below. The dashes are for conditions, which have no rules ascociated with them. This is to simplify the situation.
Post: #3
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