Blanking and piercing

Blanking and piercing are shearing processes in which a punch and die are used to produce parts from coil or sheet stock. Blanking produces the outside features of the component, while piercing produces internal holes or shapes. The web is created after multiple components have been produced and is considered scrap material. The "slugs" produced by piercing internal features are also considered scrap. The terms "piercing" and "punching" can be used interchangeably.
Die roll and burr formation.
Burrs and die roll are typical features of stamped components. Die roll is created when the material being stamped in compressed before the material begins to shear. Die roll takes the form of a radius around the outside edge of the blank and the pierced holes. After compression, the part shears for about 10% of the part thickness, and then fractures free of the strip or sheet. This fracturing produces a raised, jagged edge which is called a "burr". Burrs are typically removed by tumbling in a secondary process. Burr height can be used as an important indicator of tool wear.
Tooling design guidelines.
The selection criteria of all process parameters are governed by the sheet thickness and by the strength of the work-piece material being pierced.
The punch/die clearance is a crucial parameter, which determines the load or pressure experienced at the cutting edge of the tool, commonly known as point pressure. Excessive point pressure can lead to accelerated wear and ultimately failure. the surface quality of the trimmed edge is severely affected by the clearance, too.
Material specific design guidelines are developed by companies in order to define the minimum acceptable values of hole diameters, bridge sizes, slot dimensions. Similarly, the strip lay-out must be determined (strip width and pitch). The bridge width between the parts and the edge allowance between the part and the edge of the strip also have to be selected.
A simple operation may only need a pancake die. While many dies perform complex procedures simultaneously, a pancake die may only perform one simple procedure with the finished product being removed by hand.
Process variants.
There are various types of blanking and piercing: lancing, perforating, notching, nibbling, shaving, cutoff, and dinking.
Lancing.
Lancing is a piercing operation in which the workpiece is sheared and bent with one strike of the die. A key part of this process is that there is not reduction of material, only a modification in its geometry. This operation is used to make tabs, vents, and louvers.
The cut made in lancing is not a closed cut, like in perforation even though a similar machine is used, but a side is left connected to be bent sharply or in more of a rounded manner.
Lancing can be used to make partial contours and free up material for other operations further down the production line. Along with these reasons, lancing is also used to make tabs (where the material is bent at a 90 degree angle to the material), vents (where the bend is around 45 degrees), and louvers (where the piece is rounded or cupped). Lancing also helps to cut or slight shear of sheet on cylindrical shape.
Normally lancing is done on a mechanical press, lancing requires the use of punches and dies to be used. The different punches and dies determine the shape and angle (or curvature) of the newly made section of the material. The dies and punches are needed to be made of tool steel to withstand the repetitious nature of the procedure.
Perforating.
Perforating is a piercing tooling that involves punching a large number of closely spaced holes.
Notching.
Notching is a piercing operation that removes material from the edge of the workpiece.
Nibbling.
The nibbling process cuts a contour by producing a series of overlapping slits or notches. A nibbler may be employed to do this. This allows for complex shapes to be formed in sheet metal up to 6 mm (0.25 in) thick using simple tools. that is essentially a small punch and die that reciprocates quickly; around 300–900 times per minute. Punches are available in various shape and sizes; oblong and rectangular punches are common because they minimize waste and allow for greater distances between strokes, as compared to a round punch. Nibbling can occur on the exterior or interior of the material, however interior cuts require a hole to insert the tool.
The process is often used on parts that do not have quantities that can justify a dedicated blanking die. The edge smoothness is determined by the shape of the cutting die and the amount the cuts overlap; naturally the more the cuts overlap, the cleaner the edge. For added accuracy and smoothness, most shapes created by nibbling undergo filing or grinding processes after completion.
Shaving.
The shaving process is a finishing operation where a small amount of metal is sheared away from an already blanked part. Its main purpose is to obtain better dimensional accuracy, but secondary purposes include squaring the edge and smoothing the edge. Blanked parts can be shaved to an accuracy of up to 0.025 mm (0.001 in).
Shaving of metals is done in order to remove excess or scrap metal. A straight, smooth edge is provided and therefore shaving is frequently performed on instrument parts, watch and clock parts, and the like. Shaving is accomplished in shaving dies especially designed for the purpose.
Trimming.
The trimming operation is the last operation performed, because it cuts away excess or unwanted irregular features from the walls of drawn sheets.
Fine blanking.
Fine blanking is a specialized form of blanking where there is no fracture zone when shearing. This is achieved by compressing the whole part and then an upper and lower punch extract the blank. This allows the process to hold very tight tolerances, and perhaps eliminate secondary operations.
Materials that can be fine blanked include aluminium, brass, copper, and carbon, alloy, and stainless steels.
Fine blanking presses are similar to other metal stamping presses, but they have a few critical additional parts. A typical compound fine blanking press includes a hardened die punch (male), the hardened blanking die (female), and a guide plate of similar shape/size to the blanking die. The guide plate is the first applied to the material, impinging the material with a sharp protrusion or "stinger" around the perimeter of the die opening. Next, a counter pressure is applied opposite the punch, and finally, the die punch forces the material through the die opening. Since the guide plate holds the material so tightly, and since the counter pressure is applied, the material is cut in a manner more like extrusion than typical punching. Mechanical properties of the cut benefit similarly with a hardened layer at the cut edge of the part. Because the material is so tightly held and controlled in this setup, part flatness remains very true, distortion is nearly eliminated, and edge burr is minimal. Clearances between the die and punch are generally around 1% of the cut material thickness, which typically varies between . Currently parts as thick as can be cut using fine blanking. Tolerances between ± are possible, depending on the base material thickness and tensile strength, and part layout.
With standard compound fine blanking processes, multiple parts can often be completed in a single operation. Parts can be pierced, partially pierced, offset (up to 75°), embossed, or coined, often in a single operation. Some combinations may require progressive fine blanking operations, in which multiple operations are performed at the same pressing station. Due to the higher lifetime, blanking punches are usually covered by PVD protective coatings.
The advantages of fine blanking are:
One of the main advantages of fine blanking is that slots or holes can be placed very near to the edges of the part, or near to each other. Also, fineblanking can produce holes that are much smaller (as compared to material thickness) than can be produced by conventional stamping.
The disadvantages are:


Shaft alignment

Shaft alignment is the process of aligning two or more shafts with each other to within a tolerated margin. The resulting fault if alignment is not achieved within the demanded specifications is shaft misalignment, which may be offset or angular. Faults can lead to premature wear and damage to systems.
Background.
When a driver like an electric motor or a turbine is coupled to a pump, generator, or any other piece of equipment, the shafts of the two pieces must be aligned. Any misalignment increases the stress on the shafts and will almost certainly result in excessive wear and premature breakdown of the equipment. This can be very costly. When the equipment is down, production requiring the equipment may be delayed. Bearings or mechanical seals may be damaged and need to be replaced.
Shaft alignment is the process of aligning two or more shafts with each other to within a tolerated margin. The process is used for machinery before the machinery is put in service.
Technology.
Before shaft alignment can be done, the foundations for the driver and the driven piece must be designed and installed correctly.
Flexible couplings are designed to allow a driver (e.g., electric motor, engine, turbine, hydraulic motor) to be connected to the driven equipment. Flexible couplings use an elastomeric insert to allow a slight degree of misalignment. Flexible couplings can also use shim packs. These couplings are called disc couplings.
Tools used to achieve alignment may be mechanical, optical (e.g., laser shaft alignment), or gyroscope–based. The gyroscopic systems can be operated very time efficiently and can also be used if the shafts have a large distance (e.g., on marine vessels).
Misalignment.
The resulting fault if alignment is not achieved within the demanded specifications is shaft misalignment, which may be offset, angular, or both. Misalignment can cause increased vibration and loads on the machine parts for which they have not been designed (i.e. improper operation).
Types of misalignment.
There are two types of misalignment: offset or parallel misalignment and angular, gap, or face misalignment. With offset misalignment, the center lines of both shafts are parallel but they are offset. With angular misalignment, the shafts are at an angle to each other. Errors of alignment can be caused by parallel misalignment, angular misalignment or a combination of the two.
Offset misalignment can be further divided up in horizontal and vertical misalignment. Horizontal misalignment is misalignment of the shafts in the horizontal plane and vertical misalignment is misalignment of the shafts in the vertical plane:
Similar, angular misalignment can be divided up in horizontal and vertical misalignment:


Automotive hemming

Hemming is a technology used in the automotive industry to join inner and outer closure panels together (hoods, doors, tailgates, etc.). It is the process of bending/folding the flange of the outer panel over the inner one. The accuracy of the operation significantly affects the appearance of the car’s outer surfaces and is therefore a critical factor in the final quality of a finished vehicle.
Hemming processes.
Press hemming.
Hemming presses are widely used in automotive manufacturing for hemming of sheet-metal body components. The process uses traditional hydraulically operated ‘stamping presses’ to hem closure parts, and, being the last forming process in stamping, it largely determines the external quality of such automotive parts as doors, hood and trunk lid.
Table top hemming.
Tabletop hemming machines are utilised for the manufacture of medium to high production volumes, with the ability to achieve cycle times as low as 15 seconds.
Robot (roller hemming).
Robot hemming is utilized for the manufacture of Low to medium production volumes. It uses a standard industrial robot integrated with a roller hemming head to provide a flexible method for forming closures. The flange of the outer panel is bent over the inner panel in progressive steps, by means of a roller-hemming head.
One advantage of this process is that it can use the robot-controlled hemming head to hem several different components within a single cell. Another is that minor changes or fluctuations in panel-hemming conditions can be quickly and cost-effectively accommodated. If equipped with a tool-changing system, the robot could serve a variety of additional functions within the same assembly cell, such as operating dispensing equipment for adhesives and sealants, or carrying out panel manipulations, using a gripper unit.


Robotics conventions

There are many conventions used in the robotics research field. This article summarises these conventions.
Line representations.
Lines are very important in robotics because:
Non-minimal vector coordinates.
A line formula_1 is completely defined by the ordered set of two vectors:
Each point formula_5 on the line is given a parameter value formula_6 that satisfies:
formula_7. The parameter t is unique once formula_2 and formula_4 are chosen. The representation formula_1 is not minimal, because it uses six parameters for only four degrees of freedom. The following two constraints apply:
Plücker coordinates.
Arthur Cayley and Julius Plücker introduced an alternative representation using two free vectors. This representation was finally named after Plücker.
 The Plücker representation is denoted by formula_15. Both formula_4 and formula_17 are free vectors: formula_4 represents the direction of the line and formula_17 is the moment of formula_4 about the chosen reference origin. formula_21 (formula_17 is independent of which point formula_2 on the line is chosen!)
 The advantage of the Plücker coordinates is that they are homogeneous.
 A line in Plücker coordinates has still four out of six independent parameters, so it is not a minimal representation. The two constraints on the six Plücker coordinates are
Minimal line representation.
A line representation is minimal if it uses four parameters, which is the minimum needed to represent all possible lines in the Euclidean Space (E³).
Denavit–Hartenberg line coordinates.
Jaques Denavit and Richard S. Hartenberg presented the first minimal representation for a line which is now widely used. The common normal between two lines was the main geometric concept that allowed Denavit and Hartenberg to find a minimal representation. Engineers use the Denavit–Hartenberg convention(D–H) to help them describe the positions of links and joints unambiguously. Every link gets its own coordinate system. There are a few rules to consider in choosing the coordinate system:
Once the coordinate frames are determined, inter-link transformations are uniquely described by the following four parameters:
Hayati–Roberts line coordinates.
The Hayati–Roberts line representation, denoted formula_45, is another minimal line representation, with parameters:
This representation is unique for a directed line. The coordinate singularities are different from the DH singularities: it has singularities if the line becomes parallel to either the formula_48 or formula_49 axis of the world frame.
Product of exponentials formula.
The product of exponentials formula represents the kinematics of an open-chain mechanism as the product of exponentials of twists, and may be used to describe a series of revolute, prismatic, and helical joints.


Matsuura Machinery

 is a machine tool manufacturing company based in Fukui, Fukui Prefecture, Japan, in operation since August 1935.
Matsuura Machinery began as a manufacturer and distributor of lathes in 1935. Production of milling machines began in 1957, and the company went public in 1960. Production of automatic-controlled milling machines began in 1961 and numerically controlled milling machines from 1964. Production of automatically controlled drilling machines began in 1972, and vertical machining centers from 1974.
The company began exporting to the United States from 1975. In 1981, Matsuura Machinery began production of high-speed machining centers and twin-spindle vertical machining centers, and horizontal machining centers from 1983. The total number of machining centers shipped surpassed 10,000 units in 1993 and 20,000 in 2015.
The company's machining centers are used in a variety of industries, among them aerospace equipment manufacturers. Machine tools manufactured by Matsuura were used by NASA on the Space Shuttle Discovery's fuel tanks in 1998, making them four tons lighter than before.


Hurwitz polynomial

In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. Such a polynomial must have coefficients that are positive real numbers. The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the imaginary axis (i.e., a Hurwitz stable polynomial).
A polynomial function "P"("s") of a complex variable "s" is said to be Hurwitz if the following conditions are satisfied:
Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the Routh–Hurwitz stability criterion.
Examples.
A simple example of a Hurwitz polynomial is:
The only real solution is −1, because it factors as
In general, all quadratic polynomials with positive coefficients are Hurwitz.
This follows directly from the quadratic formula:
where, if the discriminant "b"2−4"ac" is less than zero, then the polynomial will have two complex-conjugate solutions with real part −"b"/2"a", which is negative for positive "a" and "b".
If the discriminant is equal to zero, there will be two coinciding real solutions at −"b"/2"a". Finally, if the discriminant is greater than zero, there will be two real negative solutions,
because formula_4 for positive "a", "b" and "c".
Properties.
For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive (except for quadratic polynomials, which also imply sufficiency). A necessary and sufficient condition that a polynomial is Hurwitz is that it passes the Routh–Hurwitz stability criterion. A given polynomial can be efficiently tested to be Hurwitz or not by using the Routh continued fraction expansion technique.


Telecommunications network

A telecommunications network is a group of nodes interconnected by telecommunications links that are used to exchange messages between the nodes. The links may use a variety of technologies based on the methodologies of circuit switching, message switching, or packet switching, to pass messages and signals.
Multiple nodes may cooperate to pass the message from an originating node to the destination node, via multiple network hops. For this routing function, each node in the network is assigned a network address for identification and locating it on the network. The collection of addresses in the network is called the address space of the network.
Examples of telecommunications networks include computer networks, the Internet, the public switched telephone network (PSTN), the global Telex network, the aeronautical ACARS network, and the wireless radio networks of cell phone telecommunication providers.
Network structure.
In general, every telecommunications network conceptually consists of three parts, or planes (so-called because they can be thought of as being and often are, separate overlay networks):
Data networks.
Data networks are used extensively throughout the world for communication between individuals and organizations. Data networks can be connected to allow users seamless access to resources that are hosted outside of the particular provider they are connected to. The Internet is the best example of the internetworking of many data networks from different organizations.
Terminals attached to IP networks like the Internet are addressed using IP addresses. Protocols of the Internet protocol suite (TCP/IP) provide the control and routing of messages across the and IP data network. There are many different network structures that IP can be used across to efficiently route messages, for example:
There are three features that differentiate MANs from LANs or WANs:
Data center networks also rely highly on TCP/IP for communication across machines. They connect thousands of servers, are designed to be highly robust, provide low latency and high bandwidth. Data center network topology plays a significant role in determining the level of failure resiliency, ease of incremental expansion, communication bandwidth and latency.
Capacity and speed.
In analogy to the improvements in the speed and capacity of digital computers, provided by advances in semiconductor technology and expressed in the bi-yearly doubling of transistor density, which is described empirically by Moore's law, the capacity and speed of telecommunications networks have followed similar advances, for similar reasons. In telecommunication, this is expressed in Edholm's law, proposed by and named after Phil Edholm in 2004. This empirical law holds that the bandwidth of telecommunication networks doubles every 18 months, which has proven to be true since the 1970s. The trend is evident in the Internet, cellular (mobile), wireless and wired local area networks (LANs), and personal area networks. This development is the consequence of rapid advances in the development of metal-oxide-semiconductor technology.


Teacher forcing

Teacher forcing is an algorithm for training the weights of recurrent neural networks (RNNs). It involves feeding observed sequence values (i.e. ground-truth samples) back into the RNN after each step, thus forcing the RNN to stay close to the ground-truth sequence.
The term "teacher forcing" can be motivated by comparing the RNN to a human student taking a multi-part exam where the answer to each part (for example a mathematical calculation) depends on the answer to the preceding part. In this analogy, rather than grading every answer in the end, with the risk that the student fails every single part even though they only made a mistake in the first one, a teacher records the score for each individual part and then tells the student the correct answer, to be used in the next part.
The use of an external teacher signal is in contrast to real time recurrent learning (RTRL). Teacher signals are known from oscillator networks. The promise is, that teacher forcing helps to reduce the training time.
The term "teacher forcing" was introduced in 1989 by Ronald J. Williams and David Zipser, who reported that the technique was already being "frequently used in dynamical supervised learning tasks" around that time.
A NeurIPS 2016 paper introduced the related method of "professor forcing".


Ideal machine

The term ideal machine refers to a hypothetical mechanical system in which energy and power are not lost or dissipated through friction, deformation, wear, or other inefficiencies. Ideal machines have the theoretical maximum performance, and therefore are used as a baseline for evaluating the performance of real machine systems.
A simple machine, such as a lever, pulley, or gear train, is "ideal" if the power input is equal to the power output of the device, which means there are no losses. In this case, the mechanical efficiency is 100%.
Mechanical efficiency is the performance of the machine compared to its theoretical maximum as performed by an ideal machine. The mechanical efficiency of a simple machine is calculated by dividing the actual power output by the ideal power output. This is usually expressed as a percentage.
Power loss in a real system can occur in many ways, such as through friction, deformation, wear, heat losses, incomplete chemical conversion, magnetic and electrical losses.
Criteria.
A machine consists of a power source and a mechanism for the controlled use of this power. The power source often relies on chemical conversion to generate heat which is then used to generate power. Each stage of the process of power generation has a maximum performance limit which is identified as ideal.
Once the power is generated the mechanism components of the machine direct it toward useful forces and movement. The ideal mechanism does not absorb any power, which means the power input is equal to the power output.
An example is the automobile engine (internal combustion engine) which burns fuel (an exothermic chemical reaction) inside a cylinder and uses the expanding gases to drive a piston. The movement of the piston rotates the crank shaft. The remaining mechanical components such as the transmission, drive shaft, differential, axles and wheels form the power transmission mechanism that directs the power from the engine into friction forces on the road to move the automobile.
"The ideal machine has the maximum energy conversion performance combined with a lossless power transmission mechanism that yields maximum performance."


Margin classifier

In machine learning, a margin classifier is a classifier which is able to give an associated distance from the decision boundary for each example. For instance, if a linear classifier (e.g. perceptron or linear discriminant analysis) is used, the distance (typically euclidean distance, though others may be used) of an example from the separating hyperplane is the margin of that example.
The notion of margin is important in several machine learning classification algorithms, as it can be used to bound the generalization error of the classifier. These bounds are frequently shown using the VC dimension. Of particular prominence is the generalization error bound on boosting algorithms and support vector machines.
Support vector machine definition of margin.
See support vector machines and maximum-margin hyperplane for details.
Margin for boosting algorithms.
The margin for an iterative boosting algorithm given a set of examples with two classes can be defined as follows. The classifier is given an example pair formula_1 where formula_2 is a domain space and formula_3 is the label of the example. The iterative boosting algorithm then selects a classifier formula_4 at each iteration formula_5 where formula_6 is a space of possible classifiers that predict real values. This hypothesis is then weighted by formula_7 as selected by the boosting algorithm. At iteration formula_8, the margin of an example formula_9 can thus be defined as
By this definition, the margin is positive if the example is labeled correctly and negative if the example is labeled incorrectly.
This definition may be modified and is not the only way to define margin for boosting algorithms. However, there are reasons why this definition may be appealing.
Examples of margin-based algorithms.
Many classifiers can give an associated margin for each example. However, only some classifiers utilize information of the margin while learning from a data set.
Many boosting algorithms rely on the notion of a margin to give weights to examples. If a convex loss is utilized (as in AdaBoost, LogitBoost, and all members of the AnyBoost family of algorithms) then an example with higher margin will receive less (or equal) weight than an example with lower margin. This leads the boosting algorithm to focus weight on low margin examples. In nonconvex algorithms (e.g. BrownBoost), the margin still dictates the weighting of an example, though the weighting is non-monotone with respect to margin. There exists boosting algorithms that probably maximize the minimum margin (e.g. see ).
Support vector machines probably maximize the margin of the separating hyperplane. Support vector machines that are trained using noisy data (there exists no perfect separation of the data in the given space) maximize the soft margin. More discussion of this can be found in the support vector machine article.
The voted-perceptron algorithm is a margin maximizing algorithm based on an iterative application of the classic perceptron algorithm.
Generalization error bounds.
One theoretical motivation behind margin classifiers is that their generalization error may be bound by parameters of the algorithm and a margin term. An example of such a bound is for the AdaBoost algorithm. Let formula_11 be a set of formula_12 examples sampled independently at random from a distribution formula_13. Assume the VC-dimension of the underlying base classifier is formula_14 and formula_15. Then with probability formula_16 we have the bound
for all formula_18.


Invariant extended Kalman filter

The invariant extended Kalman filter (IEKF) (not to be confused with the iterated extended Kalman filter) was first introduced as a version of the extended Kalman filter (EKF) for nonlinear systems possessing symmetries (or "invariances"), then generalized and recast as an adaptation to Lie groups of the linear Kalman filtering theory. Instead of using a linear correction term based on a linear output error, the IEKF uses a geometrically adapted correction term based on an invariant output error; in the same way the gain matrix is not updated from a linear state error, but from an invariant state error. The main benefit is that the gain and covariance equations have reduced dependence on the estimated value of the state. In some cases they converge to constant values on a much bigger set of trajectories than is the case for the EKF, which results in a better convergence of the estimation.
Filter derivation.
Discrete-time framework.
Consider a system whose state is encoded at time step formula_1 by an element formula_2 of a Lie group formula_3 and dynamics has the following shape:
where formula_5 is a group automorphism of formula_3, formula_7 is the group operation and formula_8 an element of formula_3. The system is supposed to be observed through a measurement formula_10 having the following shape:
where formula_12 belongs to a vector space formula_13 endowed with a left action of the elements of formula_3 denoted again by formula_7 (which cannot create confusion with the group operation as the second member of the operation is an element of formula_13, not formula_3). Alternatively, the same theory applies to a measurement defined by a right action:
Filter equations.
The invariant extended Kalman filter is an observer formula_19 defined by the following equations if the measurement function is a left action:
where formula_22 is the exponential map of formula_3 and formula_24 is a gain matrix to be tuned through a Riccati equation.
If measurement function is a right action then the update state is defined as:
Continuous-time framework.
The discrete-time framework above was first introduced for continuous-time dynamics of the shape:
where the vector field formula_27 verifies at any time formula_28 the relation:
where the identity element of the group is denoted by formula_30 and is used the short-hand notation formula_31 (resp. formula_32) for the left translation formula_33 (resp. the right translation formula_34) where formula_35 denots the tangent space to formula_3 at formula_37. It leads to more involved computations than the discrete-time framework, but properties are similar.
Main properties.
The main benefit of invariant extended Kalman filtering is the behavior of the invariant error variable, whose definition depends on the type of measurement. For left actions we define a left-invariant error variable as:
while for right actions we define a right-invariant error variable as:
Indeed, replacing formula_2, formula_43, formula_44 by their values we obtain for left actions, after some algebra:
and for right actions:
We see the estimated value of the state is not involved in the equation followed by the error variable, a property of linear Kalman filtering the classical extended Kalman filter does not share, but the similarity with the linear case actually goes much further. Let formula_49 be a linear version of the error variable defined by the identity:
Then, with formula_52 defined by the Taylor expansion formula_53 we actually have:
In other words, there are no higher-order terms: the dynamics is linear for the error variable formula_55. This result and error dynamics independence are at the core of theoretical properties and practical performance of IEKF.
Relation to symmetry-preserving observers.
Most physical systems possess natural symmetries (or invariance), i.e. there exist transformations (e.g. rotations, translations, scalings) that leave the system unchanged. From mathematical and engineering viewpoint, it makes sense that a filter well-designed for the considered system should preserve the same invariance properties. The idea for the IEKF is a modification of the EKF equations to take advantage of the symmetries of the system.
Definition.
Consider the system
where formula_58 are independent white Gaussian noises.
Consider formula_59 a Lie group with identity formula_60, and
(local) transformation groups formula_61 (formula_62) such that formula_63. The previous system with noise is said to be "invariant" if it is left unchanged by the action the transformations groups formula_61; that is, if
Filter equations and main result.
Since it is a symmetry-preserving filter, the general form of an IEKF reads
where
To analyze the error convergence, an invariant state error formula_73 is defined, which is different from the standard output error formula_74, since the standard output error usually does not preserve the symmetries of the system.
Given the considered system and associated transformation group, there exists a constructive method to determine formula_75, based on the moving frame method.
Similarly to the EKF, the gain matrix formula_72 is determined from the equations
where the matrices formula_79 depend here only on the known invariant vector formula_71, rather than on formula_81 as in the standard EKF. This much simpler dependence and its consequences are the main interests of the IEKF. Indeed, the matrices formula_79 are then constant on a much bigger set of trajectories (so-called "permanent trajectories") than equilibrium points as it is the case for the EKF. Near such trajectories, we are back to the "true", i.e. linear, Kalman filter where convergence is guaranteed. Informally, this means the IEKF converges in general at least around any slowly varying permanent trajectory, rather than just around any slowly varying equilibrium point for the EKF.
Application examples.
Attitude and heading reference systems.
Invariant extended Kalman filters are for instance used in attitude and heading reference systems. In such systems the orientation, velocity and/or position of a moving rigid body,
e.g. an aircraft, are estimated from different embedded sensors, such as inertial sensors, magnetometers, GPS or sonars. The use of an IEKF naturally leads to consider the quaternion error formula_83, which is often used as an "ad hoc" trick to preserve the constraints of the quaternion group. The benefits of the IEKF compared to the EKF are experimentally shown for a large set of trajectories.
Inertial navigation.
A major application of the Invariant extended Kalman filter is inertial navigation, which fits the framework after embedding of the state (consisting of attitude matrix formula_84, velocity vector formula_85 and position vector formula_37) into the Lie group formula_87 defined by the group operation:
Simultaneous localization and mapping.
The problem of simultaneous localization and mapping also fits the framework of invariant extended Kalman filtering after embedding of the state (consisting of attitude matrix formula_84, position vector formula_37 and a sequence of static feature points formula_91) into the Lie group formula_92 (or formula_93 for planar systems) defined by the group operation:
The main benefit of the Invariant extended Kalman filter in this case is solving the problem of false observability.


Automated fueling

Automated fueling or robotic fueling involves the use of automation to remove human labor from the fueling process. The fueling is performed by a robotic arm, which opens the car's flap, unscrews the cap, picks up the fuel nozzle and inserts it into the tank opening. It requires the contours and dimensions of the fuel cap to be present in the database.


Brain.js

Brain.js is a JavaScript library used for neural networking, which is released as free and open-source software under the MIT License. It can be used in both the browser and Node.js backends.
Brain.js is most commonly used as a simple introduction to neural networking, as it hides complex mathematics and has a familiar modern JavaScript syntax. It is maintained by members of the Brain.js organization and open-source contributors.
Examples.
Creating a feedforward neural network with backpropagation:
const net = new brain.NeuralNetwork;
net.train([
 { input: [0, 0], output: [0] },
 { input: [0, 1], output: [1] },
 { input: [1, 0], output: [1] },
 { input: [1, 1], output: [0] },
console.log(net.run([1, 0]));
Creating a recurrent neural network:
const net = new brain.recurrent.RNN;
net.train([
 { input: [0, 0], output: [0] },
 { input: [0, 1], output: [1] },
 { input: [1, 0], output: [1] },
 { input: [1, 1], output: [0] },
let output = net.run([0, 0]); // [0]
output = net.run([0, 1]); // [1]
output = net.run([1, 0]); // [1]
output = net.run([1, 1]); // [0]
Train the neural network on RGB color contrast:
const net = new brain.NeuralNetwork;
net.train([{
 input: {
 r: 0.03,
 g: 0.7,
 b: 0.5
 output: {
 black: 1
 },
 input: {
 r: 0.16,
 g: 0.09,
 b: 0.2
 output: {
 white: 1
 },
 input: {
 r: 0.5,
 g: 0.5,
 b: 1.0
 output: {
 white: 1
const output = net.run({
 r: 1,
 g: 0.4,
 b: 0
console.log(output)


Control-Lyapunov function

In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function formula_1 to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is "(Lyapunov) stable" or (more restrictively) "asymptotically stable". Lyapunov stability means that if the system starts in a state formula_2 in some domain "D", then the state will remain in "D" for all time. For "asymptotic stability", the state is also required to converge to formula_3. A control-Lyapunov function is used to test whether a system is "asymptotically stabilizable", that is whether for any state "x" there exists a control formula_4 such that the system can be brought to the zero state asymptotically by applying the control "u".
The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D. Sontag in the 1980s and 1990s.
Definition.
Consider an autonomous dynamical system with inputs
where formula_5 is the state vector and formula_6 is the control vector. Suppose our goal is to drive the system to an equilibrium formula_7 from every initial state in some domain formula_8. Without loss of generality, suppose the equilibrium is at formula_9 (for an equilibrium formula_10, it can be translated to the origin by a change of variables).
Definition. A control-Lyapunov function (CLF) is a function formula_11 that is continuously differentiable, positive-definite (that is, formula_1 is positive for all formula_13 except at formula_14 where it is zero), and such that for all formula_15 there exists formula_16 such that
where formula_18 denotes the inner product of formula_19.
The last condition is the key condition; in words it says that for each state "x" we can find a control "u" that will reduce the "energy" "V". Intuitively, if in each state we can always find a way to reduce the energy, we should eventually be able to bring the energy asymptotically to zero, that is to bring the system to a stop. This is made rigorous by Artstein's theorem.
Some results apply only to control-affine systems—i.e., control systems in the following form:
where formula_20 and formula_21 for formula_22.
Theorems.
E. D. Sontag showed that for a given control system, there exists a continuous CLF if and only if the origin is asymptotic stabilizable. It was later shown by Francis H. Clarke that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback.
Artstein proved that the dynamical system  has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback "u"("x").
Constructing the Stabilizing Input.
It is often difficult to find a control-Lyapunov function for a given system, but if one is found, then the feedback stabilization problem simplifies considerably. For the control affine system , "Sontag's formula" (or "Sontag's universal formula") gives the feedback law formula_23 directly in terms of the derivatives of the CLF. In the special case of a single input system formula_24, Sontag's formula is written as
where formula_26 and formula_27 are the Lie derivatives of formula_28 along formula_29 and formula_30, respectively.
For the general nonlinear system , the input formula_31 can be found by solving a static non-linear programming problem
for each state "x".
Example.
Here is a characteristic example of applying a Lyapunov candidate function to a control problem.
Consider the non-linear system, which is a mass-spring-damper system with spring hardening and position dependent mass described by
Now given the desired state, formula_34, and actual state, formula_35, with error, formula_36, define a function formula_37 as
A Control-Lyapunov candidate is then
which is positive definite for all formula_40, formula_41.
Now taking the time derivative of formula_28
The goal is to get the time derivative to be
which is globally exponentially stable if formula_28 is globally positive definite (which it is).
Hence we want the rightmost bracket of formula_47,
to fulfill the requirement
which upon substitution of the dynamics, formula_50, gives
Solving for formula_31 yields the control law
with formula_54 and formula_55, both greater than zero, as tunable parameters
This control law will guarantee global exponential stability since upon substitution into the time derivative yields, as expected
which is a linear first order differential equation which has solution
And hence the error and error rate, remembering that formula_58, exponentially decay to zero.
If you wish to tune a particular response from this, it is necessary to substitute back into the solution we derived for formula_28 and solve for formula_60. This is left as an exercise for the reader but the first few steps at the solution are:
which can then be solved using any linear differential equation methods.


Finsler's lemma

Finsler's lemma is a mathematical result named after Paul Finsler. It states equivalent ways to express the positive definiteness of a quadratic form "Q" constrained by a linear form "L".
Since it is equivalent to another lemmas used in optimization and control theory, such as Yakubovich's S-lemma, Finsler's lemma has been given many proofs and has been widely used, particularly in results related to robust optimization and linear matrix inequalities.
Statement of Finsler's lemma.
Let , and . The following statements are equivalent:
Variants.
In the particular case that "L" is positive semi-definite, it is possible to decompose it as . The following statements, which are also referred as Finsler's lemma in the literature, are equivalent:
There is also a variant of Finsler's lemma for quadratic matrix inequalities, known as matrix Finsler's lemma, which states that the following statements are equivalent for symmetric matrices "Q" and "L" belonging to R("l"+"k")x("l"+"k"):
under the assumption that
formula_9 and formula_10
satisfy the following assumptions:
Generalizations.
Projection lemma.
The following statement, known as Projection Lemma (or also as Elimination Lemma), is common on the literature of linear matrix inequalities:
This can be seen as a generalization of one of Finsler's lemma variants with the inclusion of an extra matrix and an extra constraint. Furthermore, there exists a version of the projection lemma that utilizes non-strict inequalities.
Robust version.
Finsler's lemma also generalizes for matrices "Q" and "B" depending on a parameter "s" within a set "S". In this case, it is natural to ask if the same variable μ (respectively "X") can satisfy formula_13 for all formula_14 (respectively, formula_15). If "Q" and "B" depends continuously on the parameter "s", and "S" is compact, then this is true. If "S" is not compact, but "Q" and "B" are still continuous matrix-valued functions, then μ and "X" can be guaranteed to be at least continuous functions.
Applications.
Data-driven control.
The matrix variant of Finsler lemma has been applied to the data-driven control of Lur'e systems and in a data-driven robust linear matrix inequality-based model predictive control scheme.
S-Variable approach to robust control of linear dynamical systems.
Finsler's lemma can be used to give novel linear matrix inequality (LMI) characterizations to stability and control problems. The set of LMIs stemmed from this procedure yields less conservative results when applied to control problems where the system matrices has dependence on a parameter, such as robust control problems and control of linear-parameter varying systems. This approach has recently been called as S-variable approach and the LMIs stemming from this approach are known as SV-LMIs (also known as dilated LMIs).
Sufficient condition for universal stabilizability of non-linear systems.
A nonlinear system has the universal stabilizability property if every forward-complete solution of a system can be globally stabilized. By the use of Finsler's lemma, it is possible to derive a sufficient condition for universal stabilizability in terms of a differential linear matrix inequality.


Kalman decomposition

In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces.
Definition.
Consider the continuous-time LTI control system
or the discrete-time LTI control system
The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:
where formula_9 is the coordinate transformation matrix defined as
and whose submatrices are
It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then formula_18, making the other matrices zero dimension.
Consequences.
By using results from controllability and observability, it can be shown that the transformed system formula_19 has matrices in the following form:
This leads to the conclusion that
Variants.
A Kalman decomposition also exists for linear dynamical quantum systems. Unlike classical dynamical systems, the coordinate transformation used in this variant requires to be in a specific class of transformations due to the physical laws of quantum mechanics.


U-JIN Tech Corp.

U-JIN Tech Corp. is a South Korean manufacturer of friction welding machines and automated manufacturing cells.
History.
U-JIN Tech Corp. was founded in February 2009. It stablished its own R&D center within the Korea Industrial Technology Association  in 2010. The R&D center has the objective to develop new products.
U-Jin has initially developed and manufactured hydraulic friction welding machines, and it built Korea's first CNC friction welding machine in 2012.
In 2015 the company was recognized as "Contributor for Development of Excellent Capital Goods" by the Minister of Trade, Industry, and Energy. In November 2016 it received the European CE Certificate and started exporting machines to Europe. On Trade Day in December 2016, it received the "10 Million Dollar Export Tower Award".
Friction welding machines.
CNC technology is used by U-JIN Tech Corp both for automatic material transport and in cases where high accuracy is required. Due to the position measuring devices known from CNC milling machines, the length tolerance of the components can be maintained more accurately than with conventional hydraulic machines. It is even possible, to bring the spindle to a standstill in a given position so that the two eyes of a drive shaft can be positioned at an angle to each other.
The two spindles of U-JIN's computer numerical controlled double-head friction welding machines are driven by servo motors that allow the angular position of their motor shaft to be controlled, as well as the speed of rotation and acceleration, since they are equipped with position sensors. If the spindles are controlled in the same way as CNC-controlled servo motors, angular accuracies of ±0.5° can be achieved, e.g. at both ends of a cardan shaft.
Friction welded products.
As friction welding operates below the melting point of the materials, even dissimilar material joints can be produced with high tensile strength. In many cases, the tensile strength of the bimetallic joint is higher than that of the softer base material.
U-JIN's friction welding machines are used industrially for a wide variety of products:


Machine learning control

Machine learning control (MLC) is a subfield of machine learning, intelligent control and control theory
which solves optimal control problems with methods of machine learning.
Key applications are complex nonlinear systems
for which linear control theory methods are not applicable.
Types of problems and tasks.
Four types of problems are commonly encountered.
MLC comprises, for instance, neural network control,
genetic algorithm based control,
genetic programming control,
reinforcement learning control,
and has methodological overlaps with other data-driven control,
like artificial intelligence and robot control.
Applications.
MLC has been successfully applied
to many nonlinear control problems,
exploring unknown and often unexpected actuation mechanisms.
Example applications include
As for all general nonlinear methods,
MLC comes with no guaranteed convergence,
optimality or robustness for a range of operating conditions.


Maria Elena Valcher

Maria Elena Valcher is an Italian control theorist, and a professor at the Department of Information Engineering at the University of Padova.
Valcher was the president for IEEE Control Systems Society in 2015. She was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2012, "for contributions to positive systems theory and the behavioral approach to system analysis and control". She is also a Fellow of the International Federation of Automatic Control.


Algorithmic complexity attack

An algorithmic complexity attack (ACA) is a form of attack in which the system is attacked by an exhaustion resource to take advantage of worst-case performance.
Algorithmic complexity.
Algorithmic complexity is the rate in which an algorithm performs. Although there are multiple ways to solve a computational problem, the best and most effective way in doing so matters. For real programs, factors such as the hardware, networking, programming language, and performance constraints play into the time a program takes to output the desired result.


Routh–Hurwitz stability criterion

In the control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants (formula_1) than computing them directly. A polynomial satisfying the Routh–Hurwitz criterion is called a Hurwitz polynomial.
The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions ept of the system that are stable (bounded). Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly. For discrete systems, the corresponding stability test can be handled by the Schur–Cohn criterion, the Jury test and the Bistritz test. With the advent of computers, the criterion has become less widely used, as an alternative is to solve the polynomial numerically, obtaining approximations to the roots directly.
The Routh test can be derived through the use of the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices. Hurwitz derived his conditions differently.
Using Euclid's algorithm.
The criterion is related to Routh–Hurwitz theorem. From the statement of that theorem, we have formula_2 where:
By the fundamental theorem of algebra, each polynomial of degree "n" must have "n" roots in the complex plane (i.e., for an "ƒ" with no roots on the imaginary line, "p" + "q" = "n"). Thus, we have the condition that "ƒ" is a (Hurwitz) stable polynomial if and only if "p" − "q" = "n" (the proof is given below). Using the Routh–Hurwitz theorem, we can replace the condition on "p" and "q" by a condition on the generalized Sturm chain, which will give in turn a condition on the coefficients of "ƒ".
Using matrices.
Let "f"("z") be a complex polynomial. The process is as follows:
Example.
Notice that we had to suppose "b" different from zero in the first division. The generalized Sturm chain is in this case formula_25. Putting formula_26, the sign of formula_27 is the opposite sign of "a" and the sign of "by" is the sign of "b". When we put formula_28, the sign of the first element of the chain is again the opposite sign of "a" and the sign of "by" is the opposite sign of "b". Finally, -"c" has always the opposite sign of "c".
Suppose now that "f" is Hurwitz-stable. This means that formula_29 (the degree of "f"). By the properties of the function "w", this is the same as formula_30 and formula_31. Thus, "a", "b" and "c" must have the same sign. We have thus found the necessary condition of stability for polynomials of degree 2.
Higher-order example.
A tabular method can be used to determine the stability when the roots of a higher order characteristic polynomial are difficult to obtain. For an "n"th-degree polynomial
the table has "n" + 1 rows and the following structure:
where the elements formula_50 and formula_51 can be computed as follows:
When completed, the number of sign changes in the first column will be the number of non-negative roots.
In the first column, there are two sign changes (0.75 → −3, and −3 → 3), thus there are two non-negative roots where the system is unstable.
The characteristic equation of a servo system is given by:
for stability, all the elements in the first column of the Routh array must be positive. So the conditions that must be satisfied for stability of the given system as follows:
formula_55
We see that if
formula_56
then
formula_57
Is satisfied.
We have the following table :
there are two sign changes. The system is unstable, since it has two right-half-plane poles and two left-half-plane poles. The system cannot have jω poles since a row of zeros did not appear in the Routh table.
Sometimes the presence of poles on the imaginary axis creates a situation of marginal stability. In that case the coefficients of the "Routh array" in a whole row become zero and thus further solution of the polynomial for finding changes in sign is not possible. Then another approach comes into play. The row of polynomial which is just above the row containing the zeroes is called the "auxiliary polynomial".
We have the following table:
In such a case the auxiliary polynomial is formula_60 which is again equal to zero. The next step is to differentiate the above equation which yields the polynomial formula_61. The coefficients of the row containing zero now become
"8" and "24". The process of Routh array is proceeded using these values which yield two points on the imaginary axis. These two points on the imaginary axis are the prime cause of marginal stability.


Optomechatronics

In engineering, optomechatronics is a field that investigates the integration of optical components and technology into mechatronic systems. The optical components in these systems are used as sensors to measure mechanical quantities such as surface structure and orientation. Optical sensors are used in a feedback loop as part of control systems for mechatronic devices. Optomechatronics has applications in areas such as adaptive optics, vehicular automation, optofluidics, optical tweezers and thin-film technology.


Stochastic Petri net

Stochastic Petri nets are a form of Petri net where the transitions fire after a probabilistic delay determined by a random variable.
Definition.
A "stochastic Petri net" is a five-tuple "SPN" = ("P", "T", "F", "M"0, "Λ") where:
Correspondence to Markov process.
The reachability graph of stochastic Petri nets can be mapped directly to a Markov process. It satisfies the Markov property, since its states depend only on the current marking.
Each state in the reachability graph is mapped to a state in the Markov process, and the firing of a transition with firing rate λ corresponds to a Markov state transition with probability λ.


Context tree weighting

The context tree weighting method (CTW) is a lossless compression and prediction algorithm by . The CTW algorithm is among the very few such algorithms that offer both theoretical guarantees and good practical performance (see, e.g. ).
The CTW algorithm is an “ensemble method”, mixing the predictions of many underlying variable order Markov models, where each such model is constructed using zero-order conditional probability "estimators".


Oracle Clusterware

Oracle Clusterware is the cross-platform cluster software required to run the Real Application Clusters (RAC) option for Oracle Database. It provides the basic clustering services at the operating-system level that enable Oracle Database software to run in clustering mode. In earlier versions of Oracle (release 9i and earlier), RAC required a vendor-supplied clusterware like Sun Cluster or Veritas Cluster Server (except when running on Linux or on Microsoft Windows).
Oracle Clusterware Components.
Oracle Clusterware is the software which enables the nodes to communicate with each other, allowing them to form the cluster of nodes which behaves as a single logical server. Oracle Clusterware is run by Cluster Ready Services (CRS) consisting of two key components: Oracle Cluster Registry (OCR), which records and maintains the cluster and node membership information; voting disk, which polls for consistent heartbeat information from all the nodes when the cluster is running, and acts as a tiebreaker during communication failures.
The CRS service has four components, each handling a variety of functions: Cluster Ready Services daemon (CRSd), Oracle Cluster Synchronization Service Daemon (OCSSd), Event Volume Manager Daemon (EVMd), and Oracle Process Clusterware Daemon (OPROCd). Failure or death of the CRS daemon can cause node failure, which triggers automatic reboots of the nodes to avoid the corruption of data (due to the possible failure of communication between the nodes), also known as fencing. The CRS daemon runs as "root" (super user) on UNIX platforms and runs as a service on Windows platforms.
CRSd.
The following functions are provided by the Oracle Cluster Ready Services daemon (CRSd):
OCSSd.
Oracle Cluster Synchronization Services daemon (OCSSd) provides basic ‘group services’ support. Group Services is a distributed group membership system that allows the applications to coordinate activities to achieve a common result. As such, it provides synchronization services between nodes, access to the node membership information, as well as enabling basic cluster services, including cluster group services and cluster locking. It can also run without integration with vendor clusterware. Failure of OCSSd causes the machine to reboot to avoid a split-brain situation. This is also required in a single instance configuration if Automatic Storage Management (ASM) is used. ASM was a new feature in Oracle 10g. OCSSd runs as the "oracle" user.
The following functions are provided by the Oracle Cluster Synchronization Services daemon (OCSSd):
EVMd.
The third component in OCS is the Event Volume Management Logger daemon (EVMd). EVMd spawns a permanent child process called "evmlogger" and generates events. The EVMd child process ‘evmlogger’ spawns new children processes on demand and scans the callout directory to invoke callouts. It will restart automatically on failures and death of the EVMd process does not halt the instance. EVMd runs as the "oracle" user.
OPROCd.
OPROCd provides the server fencing solution for the Oracle Clusterware. It is the process monitor for Oracle Clusterware and it uses the hang check timer or watchdog timer (depending on the implementation) for the cluster integrity. OPROCd is locked in the memory and runs as a real time process. This sleeps for a fixed time and runs as the "root" user. Failure of the OPROCd process causes the node to restart. OPROCd is so important that even it is being monitored by a process called OCLSOMON and causes a cluster node to reboot if OPROCd is hung.


Control and indicating equipment

Control and indicating equipment is equipment for receiving, processing, controlling, indicating and initiating the onward transmission of information as used in fire alarm systems. The fire detection and fire alarm system subcommittee of ISO/TC 21, Equipment for Fire Protection and Fire Fighting, had oversight for development of five standards covering detectors, control and indicating equipment. ISO 7240-2:3003 specifies requirements, test methods and performance criteria for control and indicating equipment (c.i.e.) for use in fire detection and fire alarm systems installed in buildings.
By country.
The Australian CSIRO under the Active Fire Protection Equipment Scheme sets the technical specifications and standards for lead acid batteries in Control and Indicating Equipment.
The United Kingdom Loss Prevention Council sets requirements for the design and testing of control and indicating equipment (CIE) for use with intruder alarm and hold-up alarm systems. It covers two grades of CIE (Grade A and Grade B).
Shuichi Murao was granted a United States patent on May 29, 2001 for a fire alarm system for use in control and indicating equipment.


Shear forming

Shear forming, also referred as shear spinning, is similar to metal spinning. In shear spinning the area of the final piece is approximately equal to that of the flat sheet metal blank. The wall thickness is maintained by controlling the gap between the roller and the mandrel. In shear forming a reduction of the wall thickness occurs.
Before the 1950s, spinning was performed on a simple turning lathe. When new technologies were introduced to the field of metal spinning and powered dedicated spinning machines were available, shear forming started its development in Sweden.
Schematics.
Figure 2 shows the schematics of a shear forming process.
1. A sheet metal blank is placed between the mandrel and the chuck of the spinning machine. The mandrel has the interior shape of the desired final component.
2. A roller makes the sheet metal wrap the mandrel so that it takes its shape.
As can be seen, s1 which is the initial wall thickness of the workpiece is reduced to s0.
Workpiece and roller tool profiles.
In shear forming, the starting workpiece can have circular or rectangular cross sections. On the other hand, the profile shape of the final component can be concave, convex or a combination of these two.
A shear-forming machine looks very much like a conventional spinning machine, except that it has to be much more robust to withstand the higher forces necessary to perform the shearing operation.
The design of the roller must be considered carefully, because it affects the shape of the component, the wall thickness, and dimensional accuracy. The smaller the tool nose radius, the higher the stresses and poorest thickness uniformity achieved.
Spinnability.
Spinnability, sometimes referred as shear spinnability, can be defined as the ability of a metal to undergo shear spinning deformation without exceeding its tensile strength and tearing. Published work on spinnability is available from the authors Kegg and Kalpakcioglu.
Kegg predicted that for materials with a tensile reduction of 80%, the limiting spinning reduction will be equal or greater than 80%.
Kalpakciouglu concluded that for metals with a true fracture strain of 0.5 or greater, there is a maximum limit for the shear forming reduction. For materials with a true strain below 0.5, the spinnability depends on the ductility of the material.
Highly spinnable materials include ductile materials like aluminum and certain steel alloys.
Importance of shear forming operations in manufacturing.
Shear forming and conventional spinning are being used less than other manufacturing processes such as deep drawing and ironing. Being able to achieve almost any shape, thin sectioned parts, makes spinning a versatile process used widely in the production of lightweight items. Other advantages of shear spinning include the good mechanical properties of the final item and a very good surface finish.
Typical components produced by mechanically powered spinning machines include rocket nose cones, gas turbine engine and dish aerials.
Flow forming.
Flow Forming is an incremental metal-forming technique in which a disk or tube of metal is formed over a mandrel by one or more rollers using tremendous pressure. The roller deforms the workpiece, forcing it against the mandrel, both axially lengthening and radially thinning it. Since the pressure exerted by the roller is highly localized and the material is incrementally formed, often there is a net savings in energy in forming over drawing or ironing processes. However, these savings are often not realized because of the inherent difficulties in predicting the resulting deformation for a given roller path. Flow forming subjects the workpiece to a great deal of friction and deformation. These two factors may heat the workpiece to several hundred degrees if proper cooling fluid is not utilized.
Flow forming is often used to manufacture automobile wheels and can be used to draw a wheel to net width from a machined blank.
During flow forming, the workpiece is cold worked, changing its mechanical properties, so its strength becomes similar to that of forged metal.
References.
 3. https://www.pmfind.com/benefits/flowforming-process-benefits-process


Ultrasonic machining

Ultrasonic machining is a subtractive manufacturing process that removes material from the surface of a part through high frequency, low amplitude vibrations of a tool against the material surface in the presence of fine abrasive particles. The tool travels vertically or orthogonal to the surface of the part at amplitudes of 0.05 to 0.125 mm (0.002 to 0.005 in.). The fine abrasive grains are mixed with water to form a slurry that is distributed across the part and the tip of the tool. Typical grain sizes of the abrasive material range from 100 to 1000, where smaller grains (higher grain number) produce smoother surface finishes.
Ultrasonic vibration machining is typically used on brittle materials as well as materials with a high hardness due to the microcracking mechanics.
Process.
An ultrasonically vibrating machine consists of two major components, an electroacoustic transducer and a sonotrode, attached to an electronic control unit with a cable. The abrasive grains in the slurry now act as a free cutting tool as they strike the workpiece thousands of times per second. An electronic oscillator in the control unit produces an alternating current oscillating at a high frequency, usually between 18 and 40 kHz in the ultrasonic range. The transducer converts the oscillating current to a mechanical vibration. Two types of transducers have been used in ultrasonic machining; either piezoelectric or magnetostrictive:
The transducer vibrates the sonotrode at low amplitudes and high frequencies. The sonotrode is usually made of low carbon steel. A constant stream of abrasive slurry flows between the sonotrode and work piece. This flow of slurry allows debris to flow away from the work cutting area. The slurry usually consists of abrasive boron carbide, aluminum oxide or silicon carbide particles in a suspension of water (20 to 60% by volume). The sonotrode removes material from the work piece by abrasion where it contacts it, so the result of machining is to cut a perfect negative of the sonotrode's profile into the work piece. Ultrasonic vibration machining allows extremely complex and non-uniform shapes to be cut into the workpiece with extremely high precision.
Machining time depends on the workpiece's strength, hardness, porosity and fracture toughness; the slurry's material and particle size; and the amplitude of the sonotrode's vibration. The surface finish of materials after machining depends heavily on hardness and strength, with softer and weaker materials exhibiting smoother surface finishes. The inclusion of microcrack and microcavity features on the materials surface depend highly on the crystallographic orientation of the work piece's grains and the materials fracture toughness.
Mechanics.
Ultrasonic vibration machining physically operates by the mechanism of microchipping or erosion on the work piece's surface. Since the abrasive slurry is kept in motion by high frequency, low amplitude vibrations, the impact forces of the slurry are significant, causing high contact stresses. These high contact stresses are achieved by the small contact area between the slurry's particles and the work piece's surface. Brittle materials fail by cracking mechanics and these high stresses are sufficient to cause micro-scale chips to be removed from its surface. The material as a whole does not fail due to the extremely localized stress regions. The average force imparted by a particle of the slurry impacting the work piece's surface and rebounding can be characterized by the following equation:
Where "m" is the mass of the particle, "v" is the velocity of the particle when striking the surface and "to" is the contact time, which can be approximated according to the following equation:
Where "r" is the radius of the particle, "co" is the elastic wave velocity of the work piece, "E" is the work pieces Young's Modulus and "ρ" is the materials density.
Types.
Rotary ultrasonic vibration machining.
In rotary ultrasonic vibration machining (RUM), the vertically oscillating tool is able to revolve about the vertical center line of the tool. Instead of using an abrasive slurry to remove material, the surface of the tool is impregnated with diamonds that grind down the surface of the part. Rotary ultrasonic machines are specialized in machining advanced ceramics and alloys such as glass, quartz, structural ceramics, Ti-alloys, alumina, and silicon carbide. Rotary ultrasonic machines are used to produce deep holes with a high level of precision.
Rotary ultrasonic vibration machining is a relatively new manufacturing process that is still being extensively researched. Currently, researchers are trying to adapt this process to the micro level and to allow the machine to operate similar to a milling machine.
Chemical-assisted ultrasonic vibration machining.
In chemical-assisted ultrasonic machining (CUSM), a chemically reactive abrasive fluid is used to ensure greater machining of glass and ceramic materials. Using an acidic solution, such as hydrofluoric acid, machining characteristics such as material removal rate and surface quality can be improved greatly compared to traditional ultrasonic machining. While time spent machining and surface roughness decrease with CUSM, the entrance profile diameter is slightly larger than normal due to the additional chemical reactivity of the new slurry choice. In order to limit the extent of this enlargement, the acid content of the slurry must be carefully selected as to ensure user safety and a quality product.
Applications.
Since ultrasonic vibration machining does not use subtractive methods that may alter the physical properties of a workpiece, such as thermal, chemical, or electrical processes, it has many useful applications for materials that are more brittle and sensitive than traditional machining metals. Materials that are commonly machined using ultrasonic methods include ceramics, carbides, glass, precious stones and hardened steels. These materials are used in optical and electrical applications where more precise machining methods are required to ensure dimensional accuracy and quality performance of hard and brittle materials. Ultrasonic machining is precise enough to be used in the creation of microelectromechanical system components such as micro-structured glass wafers.
In addition to small-scale components, ultrasonic vibration machining is used for structural components because of the required precision and surface quality provided by the method. The process can safely and effectively create shapes out of high-quality single crystal materials that are often necessary but difficult to generate during normal crystal growth. As advanced ceramics become a greater part of the structural engineering realm, ultrasonic machining will continue to provide precise and effective methods of ensuring proper physical dimensions while maintaining crystallographic properties.
Advantages.
Ultrasonic vibration machining is a unique non-traditional manufacturing process because it can produce parts with high precision that are made of hard and brittle materials which are often difficult to machine. Additionally, ultrasonic machining is capable of manufacturing fragile materials such as glass and non-conductive metals that can not be machined by alternative methods such as electrical discharge machining and electrochemical machining. Ultrasonic machining is able to produce high-tolerance parts because there is no distortion of the worked material. The absence of distortion is due to no heat generation from the sonotrode against the work piece and is beneficial because the physical properties of the part will remain uniform throughout. Furthermore, no burrs are created in the process, thus fewer operations are required to produce a finished part.
Disadvantages.
Because ultrasonic vibration machining is driven by microchipping or erosion mechanisms, the material removal rate of metals can be slow and the sonotrode tip can wear down quickly from the constant impact of abrasive particles on the tool. Moreover, drilling deep holes in parts can prove difficult as the abrasive slurry will not effectively reach the bottom of the hole. Note, rotary ultrasonic machining is efficient at drilling deep holes in ceramics because the absence of a slurry cutting fluid and the cutting tool is coated in harder diamond abrasives. In addition, ultrasonic vibration machining can only be used on materials with a hardness value of at least 45 HRC.


Grinding (abrasive cutting)

Grinding is a type of abrasive machining process which uses a grinding wheel as cutting tool.
A wide variety of machines are used for grinding, best classified as portable or stationary:
Milling practice is a large and diverse area of manufacturing and toolmaking. It can produce very fine finishes and very accurate dimensions; yet in mass production contexts, it can also rough out large volumes of metal quite rapidly. It is usually better suited to the machining of very hard materials than is "regular" machining (that is, cutting larger chips with cutting tools such as tool bits or milling cutters), and until recent decades it was the only practical way to machine such materials as hardened steels. Compared to "regular" machining, it is usually better suited to taking very shallow cuts, such as reducing a shaft's diameter by half a thousandth of an inch or 12.7 μm.
Grinding is a subset of cutting, as grinding is a true metal-cutting process. Each grain of abrasive functions as a microscopic single-point cutting edge (although of high negative rake angle), and shears a tiny chip that is analogous to what would conventionally be called a "cut" chip (turning, milling, drilling, tapping, etc.) . However, among people who work in the machining fields, the term "cutting" is often understood to refer to the macroscopic cutting operations, and "grinding" is often mentally categorized as a "separate" process. This is why the terms are usually used separately in shop-floor practice.
Lapping and sanding are subsets of grinding.
Processes.
Selecting which of the following grinding operations to be used is determined by the size, shape, features and the desired production rate.
Creep-feed grinding.
Creep-feed grinding (CFG) was a grinding process which was invented in Germany in the late 1950s by Edmund and Gerhard Lang. Normal grinding is used primarily to finish surfaces. But CFG is used for high rates of material removal, competing with milling and turning as a manufacturing process choice. CFG has grinding depth up to 6 mm (0.236 inches) and workpiece speed is low. Surfaces with a softer-grade resin bond are used to keep workpiece temperature low and an improved surface finish up to 1.6 μm Rmax.
CFG can take 117 s to remove of material. Precision grinding would take more than 200 s to do the same. CFG has the disadvantage of a wheel that is constantly degrading, requires high spindle power , and is limited in the length of part it can machine.
To address the problem of wheel sharpness, continuous-dress creep-feed grinding (CDCF) was developed in 1970s. The wheel is dressed constantly during machining in CDCF process and keeps the wheel in a state of specified sharpness. It takes only 17 s to remove of material, a huge gain in productivity. 38 hp (28 kW) spindle power is required, with a low to conventional spindle speeds. The limit on part length was erased.
High-efficiency deep grinding (HEDG) is another type of grinding. This process uses plated superabrasive wheels. These wheels never need dressing and last longer than other wheels. This reduces capital equipment investment costs. HEDG can be used on long part lengths and removes material at a rate of in 83 s. HEDG requires high spindle power and high spindle speeds.
Peel grinding, patented under the name of Quickpoint in 1985 by Erwin Junker Maschinenfabrik, GmbH in Nordrach, Germany, uses a thin superabrasive grinding disk oriented almost parallel to a cylindrical workpiece and operates somewhat like a lathe turning tool.
Ultra-high speed grinding (UHSG) can run at speeds higher than 40,000 fpm (200 m/s), taking 41 s to remove of material, but is still in the research and development (R&D) stage. It also requires high spindle power and high spindle speeds.
Cylindrical grinding.
Cylindrical grinding (also called center-type grinding) is used to grind the cylindrical surfaces and shoulders of the workpiece. The workpiece is mounted on centers and rotated by a device known as a lathe dog or center driver. The abrasive wheel and the workpiece are rotated by separate motors and at different speeds. The table can be adjusted to produce tapers. The wheel head can be swiveled. The five types of cylindrical grinding are: outside diameter (OD) grinding, inside diameter (ID) grinding, plunge grinding, creep feed grinding, and centerless grinding.
A cylindrical grinder has a grinding (abrasive) wheel, two centers that hold the workpiece, and a chuck, grinding dog, or other mechanism to drive the work. Most cylindrical grinding machines include a swivel to allow the forming of tapered pieces. The wheel and workpiece move parallel to one another in both the radial and longitudinal directions. The abrasive wheel can have many shapes. Standard disk-shaped wheels can be used to create a tapered or straight workpiece geometry, while formed wheels are used to create a shaped workpiece. The process using a formed wheel creates less vibration than using a regular disk-shaped wheel.
Tolerances for cylindrical grinding are held within ± for diameter and ± for roundness. Precision work can reach tolerances as high as ± for diameter and ± for roundness. Surface finishes can range from to , with typical finishes ranging from .
Surface grinding.
"Surface grinding" uses a rotating abrasive wheel to remove material, creating a flat surface. The tolerances that are normally achieved with grinding are ± for grinding a flat material and ± for a parallel surface.
The surface grinder is composed of an abrasive wheel, a workholding device known as a chuck, either electromagnetic or vacuum, and a reciprocating table.
Grinding is commonly used on cast iron and various types of steel. These materials lend themselves to grinding because they can be held by the magnetic chuck commonly used on grinding machines and do not melt into the cutting wheel, clogging it and preventing it from cutting. Materials that are less commonly ground are aluminum, stainless steel, brass, and plastics. These all tend to clog the cutting wheel more than steel and cast iron, but with special techniques it is possible to grind them.
Others.
Centerless grinding is when the workpiece is supported by a blade instead of by centers or chucks. Two wheels are used. The larger one is used to grind the surface of the workpiece and the smaller wheel is used to regulate the axial movement of the workpiece. Types of centerless grinding include through-feed grinding, in-feed/plunge grinding, and internal centerless grinding.
Electrochemical grinding is a type of grinding in which a positively charged workpiece in a conductive fluid is eroded by a negatively charged grinding wheel. The pieces from the workpiece are dissolved into the conductive fluid.
Electrolytic in-process dressing (ELID) grinding is one of the most accurate grinding methods. In this ultra precision grinding technology the grinding wheel is dressed electrochemically and in-process to maintain the accuracy of the grinding. An ELID cell consists of a metal bonded grinding wheel, a cathode electrode, a pulsed DC power supply and electrolyte. The wheel is connected to the positive terminal of the DC power supply through a carbon brush whereas the electrode is connected to the negative pole of the power supply. Usually alkaline liquids are used as both electrolytes and coolant for grinding. A nozzle is used to inject the electrolyte into the gap between wheel and electrode. The gap is usually maintained to be approximately 0.1mm to 0.3 mm. During the grinding operation one side of the wheel takes part in the grinding operation whereas the other side of the wheel is being dressed by electrochemical reaction. The dissolution of the metallic bond material is caused by the dressing which in turns results continuous protrusion of new sharp grits.
 is a specialized type of cylindrical grinding where the grinding wheel has the exact shape of the final product. The grinding wheel does not traverse the workpiece.
Internal grinding is used to grind the internal diameter of the workpiece. Tapered holes can be ground with the use of internal grinders that can swivel on the horizontal.
Pre-grinding - When a new tool has been built and has been heat-treated, it is pre-ground before welding or hardfacing commences. This usually involves grinding the outside diameter (OD) slightly higher than the finish grind OD to ensure the correct finish size.
Grinding wheel.
A grinding wheel is an expendable wheel used for various grinding and abrasive machining operations. It is generally made from a matrix of coarse abrasive particles pressed and bonded together to form a solid, circular shape, various profiles and cross sections are available depending on the intended usage for the wheel. Grinding wheels may also be made from a solid steel or aluminium disc with particles bonded to the surface.
Lubrication.
The use of fluids in a grinding process is often necessary to cool and lubricate the wheel and workpiece as well as remove the chips produced in the grinding process. The most common grinding fluids are water-soluble chemical fluids, water-soluble oils, synthetic oils, and petroleum-based oils. It is imperative that the fluid be applied directly to the cutting area to prevent the fluid being blown away from the piece due to rapid rotation of the wheel.
The workpiece.
Workholding methods.
The workpiece is manually clamped to a lathe dog, powered by the faceplate, that holds the piece in between two centers and rotates the piece. The piece and the grinding wheel rotate in opposite directions and small bits of the piece are removed as it passes along the grinding wheel. In some instances special drive centers may be used to allow the edges to be ground. The workholding method affects the production time as it changes set up times.
Workpiece materials.
Typical workpiece materials include aluminum, brass, plastics, cast iron, mild steel, and stainless steel. Aluminum, brass and plastics can have poor to fair machinability characteristics for cylindrical grinding. Cast Iron and mild steel have very good characteristics for cylindrical grinding. Stainless steel is very difficult to grind due to its toughness and ability to work harden, but can be worked with the right grade of grinding wheels.
Workpiece geometry.
The final shape of a workpiece is the mirror image of the grinding wheel, with cylindrical wheels creating cylindrical pieces and formed wheels creating formed pieces. Typical sizes on workpieces range from 0.75 in to 20 in (18 mm to 1 m) and 0.80 in to 75 in (2 cm to 4 m) in length, although pieces from 0.25 in to 60 in (6 mm to 1.5 m) in diameter and 0.30 in to 100 in (8 mm to 2.5 m) in length can be ground. Resulting shapes can be straight cylinders, straight-edged conical shapes, or even crankshafts for engines that experience relatively low torque.
Effects on workpiece materials.
Chemical property changes include an increased susceptibility to corrosion because of high surface stress.
Mechanical properties will change due to stresses put on the part during finishing. High grinding temperatures may cause a thin martensitic layer to form on the part, which will lead to reduced material strength from microcracks.
Physical property changes include the possible loss of magnetic properties on ferromagnetic materials.


Laser guided and stabilized arc welding

Laser guided and stabilized welding (LGS-welding) is a process in which a laser beam irradiates an electrical heated plasma arc to set a path of increased conductivity. Therefore, the arc's energy can be spatial directed and the plasma burns more stable. The process must be distinguished from laser-hybrid welding, since only low power laser energy of a couple hundred Watts is used and the laser does not contribute significantly to the welding process in terms of energy input.
Operation.
The principle of laser enhanced welding is based on the interaction between the electrical arc and laser radiation. Due to the optogalvanic effect (OGE) a channel of higher conductivity in the plasma is established along the path of the laser. Therefore, a movement of the laser beam results in a movement of the electrical arc. This effect is limited to a range of some millimeters, but shows the influence of the radiation to the plasma. A raise of welding speed of over 100% is described by using a diode laser with a wavelength of 811 nm without a significant loss in penetration depth. Furthermore, this technique is used in cladding. Depending on the welded material argon or argon with CO2 is used as shielding gas. The laser source must be tuned to emit at a wavelength of 811 nm and is focused into the plasma.
Laser guided and stabilized GMA-Welding.
The process is used for welding thin metal sheets up to about 2 mm when welding in overlap or butt joint. LGS-GMA-welding is most advantageous when welding fillet welds. The guidance effect of the laser radiation forces the arc into the fillet. Therefore, a steady seam can be reached. Furthermore, the stabilization of the plasma enables the GMA-process to weld thin sheets without burning holes in the material.
Equipment and setup.
The setup requires the GMA welding head tilted at 60° to the work piece surface. In order to realize a maximum overlap between the electric arc and the laser beam in the process area, the laser is installed upright to the workpiece and focused in the electrical arc. Standard welding equipment can be used for the process. The laser source is described above.
Laser guided and stabilized double head TIG-welding.
In laser guided and stabilized double head TIG-welding the laser forces two arcs together. The goal of this technique is to increase the welding speed of TIG-welding without compromising the quality.
Equipment and setup.
For this process two TIG-sources are needed and the laser described above. The TIG-torches are set up with the laser beam perpendicular in the middle. All welding modes of the two torches are possible (DC/DC, AC/AC, AC/DC).
Laser guided and stabiliszed GMA-Cladding.
In LGS-GMA-cladding the stabilization effect is used enable the GMA-process to work with low energy. This is needed to reduce the penetration depth and therefore the dilution of base and deposition material. The combination of GMA-welding and a diode laser leads to a cheap and energy efficient process.
Equipment and setup.
The setup for the LGS-GMA-cladding is almost alike the one for LGS-GMA-welding beside that the GMA-source needs to have a "Cold-MIG" process. This means, that the welding current is controlled my microcontrollers and produced by power electronics. That way not only the current peaks can be controlled, but also the slopes.


Manfred Morari

Manfred Morari (born 1951) is a world-leading control theorist who has made pioneering contributions to the theory and applications of Model Predictive Control, Internal Model Control (IMC) and Hybrid Systems. His book on Robust Process Control is considered to be definitive text on the subject. He is currently Peter and Susanne Armstrong Faculty Fellow at the University of Pennsylvania. He received his Ph.D. in Chemical Engineering from the University of Minnesota in 1977. Dr. Morari held positions at the University of Wisconsin, Madison from 1977–1983, the California Institute of Technology from 1983-1991, and the Swiss Federal Institute of Technology in Zurich ETH Zurich. He is considered as pioneer in field of Model Predictive Control, Control of Hybrid Systems, Internal Model Control (IMC), and robust control.
In recognition of his research contributions he received numerous awards, among them the Donald P. Eckman Award and the John R. Ragazzini Award of the Automatic Control Council, the Allan P. Colburn Award and the Professional Progress Award of the AIChE, the Curtis W. McGraw Research Award of the ASEE, Doctor Honoris Causa from Babes-Bolyai University, Fellow of IEEE and IFAC, and the IEEE Control Systems Field Award. He was also elected a member of the US National Academy of Engineering in 1993 for analysis of the effects of design on process operability and the development of techniques for robust process control. Manfred Morari has held appointments with Exxon and ICI plc and serves on the technical advisory boards of several major corporations. He received in 2005 the IEEE Control Systems Award, and in 2011 the Richard E. Bellman Control Heritage Award.


Tournament selection

Tournament selection is a method of selecting an individual from a population of individuals in a genetic algorithm. Tournament selection involves running several "tournaments" among a few individuals (or "chromosomes") chosen at random from the population. The winner of each tournament (the one with the best fitness) is selected for crossover. "Selection pressure", a probabilistic measure of a chromosome's likelihood of participation in the tournament based on the participant selection pool size, is easily adjusted by changing the tournament size. The reason is that if the tournament size is larger, weak individuals have a smaller chance to be selected, because, if a weak individual is selected to be in a tournament, there is a higher probability that a stronger individual is also in that tournament.
The tournament selection method may be described in pseudo code:
 choose k (the tournament size) individuals from the population at random
 choose the best individual from the tournament with probability p
 choose the second best individual with probability p*(1-p)
 choose the third best individual with probability p*((1-p)^2)
 and so on
Deterministic tournament selection selects the best individual (when "p" = 1) in any tournament. A 1-way tournament ("k" = 1) selection is equivalent to random selection. There are two variants of the selection: "with" and "without" replacement. The variant without replacement guarantees that when selecting "N" individuals from a population of "N" elements, each individual participates in exactly "k" tournaments. An algorithm is proposed in. Note that depending on the number of elements selected, selection "without" replacement does "not" guarantee that no individual is selected more than once. It just guarantees that each individual has an equal chance of participating in the same number of tournaments.
In comparison with the (stochastic) fitness proportionate selection method, tournament selection is often implemented in practice due to its lack of stochastic noise.
Tournament selection has several benefits over alternative selection methods for genetic algorithms (for example, fitness proportionate selection and reward-based selection): it is efficient to code, works on parallel architectures and allows the selection pressure to be easily adjusted. Tournament selection has also been shown to be independent of the scaling of the genetic algorithm fitness function (or 'objective function') in some classifier systems.