92-629: UKF may refer to: Unscented Kalman filter , a special case of an algorithm to handle measurements containing noise and other inaccuracies UK funky , a genre of electronic dance music from the United Kingdom UKF Music , an electronic music brand based in the United Kingdom United Kingdom First , a small short-lived populist, Eurosceptic British political party Univerzita Konštantína Filozofa ,
184-429: A Markov chain built on linear operators perturbed by errors that may include Gaussian noise . The state of the target system refers to the ground truth (yet hidden) system configuration of interest, which is represented as a vector of real numbers . At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from
276-463: A Kalman Filter is often difficult due to the difficulty of getting a good estimate of the noise covariance matrices Q k and R k . Extensive research has been done to estimate these covariances from data. One practical method of doing this is the autocovariance least-squares (ALS) technique that uses the time-lagged autocovariances of routine operating data to estimate the covariances. The GNU Octave and Matlab code used to calculate
368-425: A base case, or testing for it incorrectly, can cause an infinite loop . For some functions (such as one that computes the series for e = 1/0! + 1/1! + 1/2! + 1/3! + ... ) there is not an obvious base case implied by the input data; for these one may add a parameter (such as the number of terms to be added, in our series example) to provide a 'stopping criterion' that establishes the base case. Such an example
460-523: A depth-first search. Single recursion is often much more efficient than multiple recursion, and can generally be replaced by an iterative computation, running in linear time and requiring constant space. Multiple recursion, by contrast, may require exponential time and space, and is more fundamentally recursive, not being able to be replaced by iteration without an explicit stack. Multiple recursion can sometimes be converted to single recursion (and, if desired, thence to iteration). For example, while computing
552-408: A hybrid merge sort/insertion sort. Recursion and iteration are equally expressive: recursion can be replaced by iteration with an explicit call stack , while iteration can be replaced with tail recursion . Which approach is preferable depends on the problem under consideration and the language used. In imperative programming , iteration is preferred, particularly for simple recursion, as it avoids
644-498: A join-tree or Markov tree . Additional methods include belief filtering which use Bayes or evidential updates to the state equations. A wide variety of Kalman filters exists by now: Kalman's original formulation - now termed the "simple" Kalman filter, the Kalman–Bucy filter , Schmidt's "extended" filter, the information filter , and a variety of "square-root" filters that were developed by Bierman, Thornton, and many others. Perhaps
736-618: A linear interpolation, x = ( 1 − t ) ( a ) + t ( b ) {\displaystyle x=(1-t)(a)+t(b)} for t {\displaystyle t} between [0,1]. In our case: This expression also resembles the alpha beta filter update step. If the model is accurate, and the values for x ^ 0 ∣ 0 {\displaystyle {\hat {\mathbf {x} }}_{0\mid 0}} and P 0 ∣ 0 {\displaystyle \mathbf {P} _{0\mid 0}} accurately reflect
828-509: A natural number), functions such as factorial may also be regarded as structural recursion. Generative recursion is the alternative: Many well-known recursive algorithms generate an entirely new piece of data from the given data and recur on it. HtDP ( How to Design Programs ) refers to this kind as generative recursion. Examples of generative recursion include: gcd , quicksort , binary search , mergesort , Newton's method , fractals , and adaptive integration . This distinction
920-421: A programming technique, it is used most often in the context of lazy programming languages, and can be preferable to recursion when the desired size or precision of a program's output is unknown. In such cases the program requires both a definition for an infinitely large (or infinitely precise) result, and a mechanism for taking a finite portion of that result. The problem of computing the first n prime numbers
1012-462: A set of three or more functions that call each other can be called a set of mutually recursive functions. Recursion is usually done by explicitly calling a function by name. However, recursion can also be done via implicitly calling a function based on the current context, which is particularly useful for anonymous functions , and is known as anonymous recursion . Some authors classify recursion as either "structural" or "generative". The distinction
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#17327901834601104-437: A single equation; however, it is most often conceptualized as two distinct phases: "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from
1196-402: A single expression. A coinductive data definition is one that specifies the operations that may be performed on a piece of data; typically, self-referential coinductive definitions are used for data structures of infinite size. A coinductive definition of infinite streams of strings, given informally, might look like this: This is very similar to an inductive definition of lists of strings;
1288-425: A single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán . Kalman filtering has numerous technological applications. A common application
1380-439: A structure that contains a string and a list of strings. The self-reference in the definition permits the construction of lists of any (finite) number of strings. Another example of inductive definition is the natural numbers (or positive integers ): Similarly recursive definitions are often used to model the structure of expressions and statements in programming languages. Language designers often express grammars in
1472-471: A syntax such as Backus–Naur form ; here is such a grammar, for a simple language of arithmetic expressions with multiplication and addition: This says that an expression is either a number, a product of two expressions, or a sum of two expressions. By recursively referring to expressions in the second and third lines, the grammar permits arbitrarily complicated arithmetic expressions such as (5 * ((3 * 6) + 8)) , with more than one product or sum operation in
1564-469: A system's state to calculate a new state. The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's gain . The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on
1656-522: A tree, which can be linear in the number of function calls, hence significant savings for O ( n ) algorithms; this is illustrated below for a depth-first search. Short-circuiting on a tree corresponds to considering a leaf (non-empty node with no children) as the base case, rather than considering an empty node as the base case. If there is only a single base case, such as in computing the factorial, short-circuiting provides only O (1) savings. Conceptually, short-circuiting can be considered to either have
1748-445: A truck. The truck can be equipped with a GPS unit that provides an estimate of the position within a few meters. The GPS estimate is likely to be noisy; readings 'jump around' rapidly, though remaining within a few meters of the real position. In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel revolutions and
1840-510: A university in Nitra, Slovakia Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title UKF . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=UKF&oldid=985163634 " Category : Disambiguation pages Hidden categories: Short description
1932-620: A worthwhile alternative to the Autocovariance Least-Squares methods. Another approach is the Optimized Kalman Filter ( OKF ), which considers the covariance matrices not as representatives of the noise, but rather, as parameters aimed to achieve the most accurate state estimation. These two views coincide under the KF assumptions, but often contradict each other in real systems. Thus, OKF's state estimation
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#17327901834602024-449: A wrapper function for the case when the overall recursion starts with the base case itself. For example, in the factorial function, properly the base case is 0! = 1, while immediately returning 1 for 1! is a short circuit, and may miss 0; this can be mitigated by a wrapper function. The box shows C code to shortcut factorial cases 0 and 1. Short-circuiting is primarily a concern when many base cases are encountered, such as Null pointers in
2116-411: Is a difficult one and is treated as a problem of control theory using robust control . The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows,
2208-403: Is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works recursively and requires only the last "best guess", rather than the entire history, of
2300-481: Is a strong analogy between the equations of a Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and Ghahramani (1999) and Hamilton (1994), Chapter 13. In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the following framework. This means specifying
2392-454: Is also called mutual recursion , which is a more symmetric term, though this is simply a difference of emphasis, not a different notion. That is, if f calls g and then g calls f, which in turn calls g again, from the point of view of f alone, f is indirectly recursing, while from the point of view of g alone, it is indirectly recursing, while from the point of view of both, f and g are mutually recursing on each other. Similarly
2484-416: Is an important topic in control theory and control systems engineering. Together with the linear-quadratic regulator (LQR), the Kalman filter solves the linear–quadratic–Gaussian control problem (LQG). The Kalman filter, the linear-quadratic regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the most fundamental problems of control theory. In most applications,
2576-522: Is better than the estimate obtained by using only one measurement alone. As such, it is a common sensor fusion and data fusion algorithm. Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of
2668-456: Is different from Wikidata All article disambiguation pages All disambiguation pages Unscented Kalman filter In statistics and control theory , Kalman filtering (also known as linear quadratic estimation ) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on
2760-421: Is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically . Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics . Kalman filtering is also important for robotic motion planning and control, and can be used for trajectory optimization . Kalman filtering also works for modeling
2852-508: Is generally credited with developing the first implementation of a Kalman filter. He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements. It was during a visit by Kálmán to the NASA Ames Research Center that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for
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2944-407: Is generally less efficient , and, for certain problems, algorithmic or compiler-optimization techniques such as tail call optimization may improve computational performance over a naive recursive implementation. A common algorithm design tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as
3036-476: Is important in proving termination of a function. In actual implementation, rather than a pure recursive function (single check for base case, otherwise recursive step), a number of modifications may be made, for purposes of clarity or efficiency. These include: On the basis of elegance, wrapper functions are generally approved, while short-circuiting the base case is frowned upon, particularly in academia. Hybrid algorithms are often used for efficiency, to reduce
3128-564: Is independent of time. The initial state, and the noise vectors at each step { x 0 , w 1 , … , w k , v 1 , … , v k } {\displaystyle \{\mathbf {x} _{0},\mathbf {w} _{1},\dots ,\mathbf {w} _{k},\mathbf {v} _{1},\dots ,\mathbf {v} _{k}\}} are all assumed to be mutually independent . Many real-time dynamic systems do not exactly conform to this model. In fact, unmodeled dynamics can seriously degrade
3220-415: Is more naturally treated by corecursion , where successive terms in the output are the partial sums; this can be converted to a recursion by using the indexing parameter to say "compute the n th term ( n th partial sum)". Many computer programs must process or generate an arbitrarily large quantity of data . Recursion is a technique for representing data whose exact size is unknown to the programmer :
3312-473: Is more robust to modeling inaccuracies. It follows from theory that the Kalman filter provides an optimal state estimation in cases where a) the model matches the real system perfectly, b) the entering noise is "white" (uncorrelated), and c) the covariances of the noise are known exactly. Correlated noise can also be treated using Kalman filters. Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in
3404-582: Is named for Hungarian émigré Rudolf E. Kálmán , although Thorvald Nicolai Thiele and Peter Swerling developed a similar algorithm earlier. Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory, causing it to be known sometimes as Kalman–Bucy filtering. Kalman was inspired to derive the Kalman filter by applying state variables to the Wiener filtering problem . Stanley F. Schmidt
3496-611: Is one of the central ideas of computer science. The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions. Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure ) do not define any looping constructs but rely solely on recursion to repeatedly call code. It
3588-462: Is one that can be solved with a corecursive program (e.g. here ). Recursion that contains only a single self-reference is known as single recursion , while recursion that contains multiple self-references is known as multiple recursion . Standard examples of single recursion include list traversal, such as in a linear search, or computing the factorial function, while standard examples of multiple recursion include tree traversal , such as in
3680-494: Is proved in computability theory that these recursive-only languages are Turing complete ; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as while and for . Repeatedly calling a function from within itself may cause the call stack to have a size equal to the sum of the input sizes of all involved calls. It follows that, for problems that can be solved easily by iteration, recursion
3772-464: Is related to where a recursive procedure gets the data that it works on, and how it processes that data: [Functions that consume structured data] typically decompose their arguments into their immediate structural components and then process those components. If one of the immediate components belongs to the same class of data as the input, the function is recursive. For that reason, we refer to these functions as (STRUCTURALLY) RECURSIVE FUNCTIONS. Thus,
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3864-596: The Apollo program resulting in its incorporation in the Apollo navigation computer . This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed by the Soviet mathematician Ruslan Stratonovich . In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before
3956-435: The central nervous system 's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback , the use of Kalman filters provides a realistic model for making estimates of the current state of a motor system and issuing updated commands. The algorithm works via a two-phase process: a prediction phase and an update phase. In the prediction phase, the Kalman filter produces estimates of
4048-481: The divide-and-conquer method ; when combined with a lookup table that stores the results of previously solved sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be referred to as dynamic programming or memoization . A recursive function definition has one or more base cases , meaning input(s) for which the function produces a result trivially (without recurring), and one or more recursive cases , meaning input(s) for which
4140-599: The Fibonacci sequence naively entails multiple iteration, as each value requires two previous values, it can be computed by single recursion by passing two successive values as parameters. This is more naturally framed as corecursion, building up from the initial values, while tracking two successive values at each step – see corecursion: examples . A more sophisticated example involves using a threaded binary tree , which allows iterative tree traversal, rather than multiple recursion. Most basic examples of recursion, and most of
4232-421: The angle of the steering wheel. This is a technique known as dead reckoning . Typically, the dead reckoning will provide a very smooth estimate of the truck's position, but it will drift over time as small errors accumulate. For this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update. In the prediction phase, the truck's old position will be modified according to
4324-412: The auxiliary function can be nested inside the wrapper function and use a shared scope. In the absence of nested functions, auxiliary functions are instead a separate function, if possible private (as they are not called directly), and information is shared with the wrapper function by using pass-by-reference . Short-circuiting the base case, also known as arm's-length recursion , consists of checking
4416-460: The base case before making a recursive call – i.e., checking if the next call will be the base case, instead of calling and then checking for the base case. Short-circuiting is particularly done for efficiency reasons, to avoid the overhead of a function call that immediately returns. Note that since the base case has already been checked for (immediately before the recursive step), it does not need to be checked for separately, but one does need to use
4508-436: The base case check before the recursive step. Alternatively, these can be considered a different form of base case and recursive step, respectively. Note that this requires a wrapper function to handle the case when the tree itself is empty (root node is Null). In the case of a perfect binary tree of height h, there are 2 −1 nodes and 2 Null pointers as children (2 for each of the 2 leaves), so short-circuiting cuts
4600-428: The clear separation of base case and recursive step in standard recursion, it is often considered poor style, particularly in academia. A basic example of short-circuiting is given in depth-first search (DFS) of a binary tree; see binary trees section for standard recursive discussion. The standard recursive algorithm for a DFS is: In short-circuiting, this is instead: In terms of the standard steps, this moves
4692-418: The controls on the system if they are known. Then, another linear operator mixed with more noise generates the measurable outputs (i.e., observation) from the true ("hidden") state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the difference that the hidden state variables have values in a continuous space as opposed to a discrete state space as for the hidden Markov model. There
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#17327901834604784-403: The covariances are set, it is useful to evaluate the performance of the filter; i.e., whether it is possible to improve the state estimation quality. If the Kalman filter works optimally, the innovation sequence (the output prediction error) is a white noise, therefore the whiteness property of the innovations measures filter performance. Several different methods can be used for this purpose. If
4876-402: The current state variables , including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a weighted average , with more weight given to estimates with greater certainty. The algorithm is recursive . It can operate in real time , using only the present input measurements and
4968-417: The current timestep. In the update phase, the innovation (the pre-fit residual), i.e. the difference between the current a priori prediction and the current observation information, is multiplied by the optimal Kalman gain and combined with the previous state estimate to refine the state estimate. This improved estimate based on the current observation is termed the a posteriori state estimate. Typically,
5060-404: The defining characteristic of a structurally recursive function is that the argument to each recursive call is the content of a field of the original input. Structural recursion includes nearly all tree traversals, including XML processing, binary tree creation and search, etc. By considering the algebraic structure of the natural numbers (that is, a natural number is either zero or the successor of
5152-417: The difference is that this definition specifies how to access the contents of the data structure—namely, via the accessor functions head and tail —and what those contents may be, whereas the inductive definition specifies how to create the structure and what it may be created from. Corecursion is related to coinduction, and can be used to compute particular instances of (possibly) infinite objects. As
5244-449: The distribution of the initial state values, then the following invariants are preserved: where E [ ξ ] {\displaystyle \operatorname {E} [\xi ]} is the expected value of ξ {\displaystyle \xi } . That is, all estimates have a mean error of zero. Also: so covariance matrices accurately reflect the covariance of estimates. Practical implementation of
5336-493: The dynamic systems will be linear." Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense , although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian. Extensions and generalizations of
5428-552: The examples presented here, demonstrate direct recursion , in which a function calls itself. Indirect recursion occurs when a function is called not by itself but by another function that it called (either directly or indirectly). For example, if f calls f, that is direct recursion, but if f calls g which calls f, then that is indirect recursion of f. Chains of three or more functions are possible; for example, function 1 calls function 2, function 2 calls function 3, and function 3 calls function 1 again. Indirect recursion
5520-431: The filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability (it diverges). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of distinguishing between measurement noise and unmodeled dynamics
5612-512: The guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station . Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its state ) that
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#17327901834605704-499: The internal state is much larger (has more degrees of freedom ) than the few "observable" parameters which are measured. However, by combining a series of measurements, the Kalman filter can estimate the entire internal state. For the Dempster–Shafer theory , each state equation or observation is considered a special case of a linear belief function and the Kalman filtering is a special case of combining linear belief functions on
5796-622: The matrices, for each time-step k {\displaystyle k} , following: As seen below, it is common in many applications that the matrices F {\displaystyle \mathbf {F} } , H {\displaystyle \mathbf {H} } , Q {\displaystyle \mathbf {Q} } , R {\displaystyle \mathbf {R} } , and B {\displaystyle \mathbf {B} } are constant across time, in which case their k {\displaystyle k} index may be dropped. The Kalman filter model assumes
5888-501: The method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems . The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion , and distributed sensor networks to develop distributed or consensus Kalman filtering. The filtering method
5980-425: The most commonly used type of very simple Kalman filter is the phase-locked loop , which is now ubiquitous in radios, especially frequency modulation (FM) radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment. Kalman filtering is based on linear dynamic systems discretized in the time domain. They are modeled on
6072-411: The most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness. When performing the actual calculations for the filter (as discussed below),
6164-603: The noise covariance matrices using the ALS technique is available online using the GNU General Public License . Field Kalman Filter (FKF), a Bayesian algorithm, which allows simultaneous estimation of the state, parameters and noise covariance has been proposed. The FKF algorithm has a recursive formulation, good observed convergence, and relatively low complexity, thus suggesting that the FKF algorithm may possibly be
6256-701: The noise has no explicit knowledge of time. At time k {\displaystyle k} an observation (or measurement) z k {\displaystyle \mathbf {z} _{k}} of the true state x k {\displaystyle \mathbf {x} _{k}} is made according to where Analogously to the situation for w k {\displaystyle \mathbf {w} _{k}} , one may write v ∙ {\displaystyle \mathbf {v} _{\bullet }} instead of v k {\displaystyle \mathbf {v} _{k}} if R {\displaystyle \mathbf {R} }
6348-402: The noise terms are distributed in a non-Gaussian manner, methods for assessing performance of the filter estimate, which use probability inequalities or large-sample theory, are known in the literature. Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of
6440-489: The notation x ^ n ∣ m {\displaystyle {\hat {\mathbf {x} }}_{n\mid m}} represents the estimate of x {\displaystyle \mathbf {x} } at time n given observations up to and including at time m ≤ n . The state of the filter is represented by two variables: The algorithm structure of the Kalman filter resembles that of Alpha beta filter . The Kalman filter can be written as
6532-505: The number of function calls in half in the worst case. In C, the standard recursive algorithm may be implemented as: The short-circuited algorithm may be implemented as: Note the use of short-circuit evaluation of the Boolean && (AND) operators, so that the recursive call is made only if the node is valid (non-Null). Note that while the first term in the AND is a pointer to a node,
6624-618: The overhead of recursion in small cases, and arm's-length recursion is a special case of this. A wrapper function is a function that is directly called but does not recurse itself, instead calling a separate auxiliary function which actually does the recursion. Wrapper functions can be used to validate parameters (so the recursive function can skip these), perform initialization (allocate memory, initialize variables), particularly for auxiliary variables such as "level of recursion" or partial computations for memoization , and handle exceptions and errors. In languages that support nested functions ,
6716-556: The overhead of repeated function calls and returns. For this reason efficient implementations of recursive algorithms often start with the recursive algorithm, but then switch to a different algorithm when the input becomes small. An important example is merge sort , which is often implemented by switching to the non-recursive insertion sort when the data is sufficiently small, as in the tiled merge sort . Hybrid recursive algorithms can often be further refined, as in Timsort , derived from
6808-417: The physical laws of motion (the dynamic or "state transition" model). Not only will a new position estimate be calculated, but also a new covariance will be calculated as well. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain about the position estimate at low speeds. Next, in
6900-444: The position estimate back toward the real position but not disturb it to the point of becoming noisy and rapidly jumping. The Kalman filter is an efficient recursive filter estimating the internal state of a linear dynamic system from a series of noisy measurements. It is used in a wide range of engineering and econometric applications from radar and computer vision to estimation of structural macroeconomic models, and
6992-423: The program recurs (calls itself). For example, the factorial function can be defined recursively by the equations 0! = 1 and, for all n > 0 , n ! = n ( n − 1)! . Neither equation by itself constitutes a complete definition; the first is the base case, and the second is the recursive case. Because the base case breaks the chain of recursion, it is sometimes also called the "terminating case". The job of
7084-438: The programmer can specify this data with a self-referential definition. There are two types of self-referential definitions: inductive and coinductive definitions. An inductively defined recursive data definition is one that specifies how to construct instances of the data. For example, linked lists can be defined inductively (here, using Haskell syntax): The code above specifies a list of strings to be either empty, or
7176-420: The recursive cases can be seen as breaking down complex inputs into simpler ones. In a properly designed recursive function, with each recursive call, the input problem must be simplified in such a way that eventually the base case must be reached. (Functions that are not intended to terminate under normal circumstances—for example, some system and server processes —are an exception to this.) Neglecting to write
7268-400: The same base case and recursive step, checking the base case only before the recursion, or it can be considered to have a different base case (one step removed from standard base case) and a more complex recursive step, namely "check valid then recurse", as in considering leaf nodes rather than Null nodes as base cases in a tree. Because short-circuiting has a more complicated flow, compared with
7360-519: The second term is a Boolean, so the overall expression evaluates to a Boolean. This is a common idiom in recursive short-circuiting. This is in addition to the short-circuit evaluation of the Boolean || (OR) operator, to only check the right child if the left child fails. In fact, the entire control flow of these functions can be replaced with a single Boolean expression in a return statement, but legibility suffers at no benefit to efficiency. Recursive algorithms are often inefficient for small data, due to
7452-400: The section above. More generally, if the model assumptions do not match the real system perfectly, then optimal state estimation is not necessarily obtained by setting Q k and R k to the covariances of the noise. Instead, in that case, the parameters Q k and R k may be set to explicitly optimize the state estimation, e.g., using standard supervised learning . After
7544-488: The state calculated previously and its uncertainty matrix; no additional past information is required. Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán : "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean;
7636-412: The state estimate and covariances are coded into matrices because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances. As an example application, consider the problem of determining the precise location of
7728-401: The state of the system as an average of the system's predicted state and of the new measurement using a weighted average . The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the covariance , a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average
7820-561: The summer of 1961, when Kalman met with Stratonovich during a conference in Moscow. This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. The CPU was built from ICs [...]. Clock speed was under 100 kHz [...]. The fact that the MIT engineers were able to pack such good software (one of
7912-439: The truck are described by the linear state space Recursion (computer science) In computer science , recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion
8004-439: The truck every Δ t seconds, but these measurements are imprecise; we want to maintain a model of the truck's position and velocity . We show here how we derive the model from which we create our Kalman filter. Since F , H , R , Q {\displaystyle \mathbf {F} ,\mathbf {H} ,\mathbf {R} ,\mathbf {Q} } are constant, their time indices are dropped. The position and velocity of
8096-535: The true state at time k {\displaystyle k} is evolved from the state at k − 1 {\displaystyle k-1} according to where If Q {\displaystyle \mathbf {Q} } is independent of time, one may, following Roweis and Ghahramani ( op. cit. ), write w ∙ {\displaystyle \mathbf {w} _{\bullet }} instead of w k {\displaystyle \mathbf {w} _{k}} to emphasize that
8188-508: The two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction procedures performed. Likewise, if multiple independent observations are available at the same time, multiple update procedures may be performed (typically with different observation matrices H k ). The formula for
8280-464: The update phase, a measurement of the truck's position is taken from the GPS unit. Along with this measurement comes some amount of uncertainty, and its covariance relative to that of the prediction from the previous phase determines how much the new measurement will affect the updated prediction. Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull
8372-536: The updated ( a posteriori ) estimate covariance above is valid for the optimal K k gain that minimizes the residual error, in which form it is most widely used in applications. Proof of the formulae is found in the derivations section, where the formula valid for any K k is also shown. A more intuitive way to express the updated state estimate ( x ^ k ∣ k {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}} ) is: This expression reminds us of
8464-416: The very first applications of the Kalman filter) into such a tiny computer is truly remarkable. Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarines , and in the guidance and navigation systems of cruise missiles such as the U.S. Navy's Tomahawk missile and the U.S. Air Force 's Air Launched Cruise Missile . They are also used in
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