We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there. For low temperatures, the disordered region consists of two disjoint ellipses. In the high temperature regime the two ellipses merge into a single simply connected region. At the transition from the low to the high temperature a tacnode appears. The key to our asymptotic study is a recent approach introduced by Duits and Kuijlaars providing a double integral representation for the correlation kernel. One of the factors in the integrand is the Christoffel–Darboux kernel associated to polynomials that satisfy non-Hermitian orthogonality relations with respect to a complex-valued weight on a contour in the complex plane. We compute the asymptotic behavior of these orthogonal polynomials and the Christoffel–Darboux kernel by means of a Riemann–Hilbert analysis. After substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments. N), , ,  and  and consider tilings of the hexagon with the following three types of lozenges  see also Fig. W is a weight function on all possible tilings defined byWe study random lozenge tilings of large regular hexagons. We place the regular hexagon so that it has corners at , , the aufound in  the locafound in . In a mofound in , 66, 67.An important part of the recent literature on random tilings is concerned with proving the universality of the global fluctuations and the emergence of the Gaussian Free Field. For the uniform measure on all possible tilings of the hexagon there are now various techniques in the literature that prove this claim. In  the convMeasures on tilings of the (finite) hexagon that are not uniform are known to be difficult to analyze asymptotically and much less results are known. For instance, in  the authapproach  for studapproach  was to aapproach , 22), thapproach , 29. ThiIt is possible to take the limit of our model in which the vertical sides of the hexagon tend to infinity  is the coordinate for the black dot. From simple geometric considerations, we note that these probabilities add up to 1. Our main result, Theorem Our main result concerns the limiting densities of the lozenges as the size of the hexagon goes to infinity. We introduce the scaled variables egral in . The sadThe rational function For For For each Let us comment on how ked from  that \doFor nt}Qα in  has one  line by . For \do=19 both  and }Ξα from , 2.13)  \documenLet The proof of Proposition liquid region Let For a given N such that  near this part of the boundary of the liquid domain, we haveGiven two points entclass1pt{minimas defines a diffeomorphism from the liquid region In the high temperature regime For each Proposition The boundary equation . Since t because  has a doThe discriminant also vanishes at all points see Fig. \documentAt points on the subarc of the boundary  we have , between we have , and bet we have . This is we have  between A notable difference compared with the low temperature regime is that the liquid region in the high temperature regime is connected. As a result, the frozen region with the two types of tiles  becomes disconnected into two disjoint components.For  the Eq.  has a doWe chose to present Theorem N. Then we have the limits in (Let The proof of this theorem is given in Sect. If  side of  can be cduced in . It is tTheorem  side of .We end this section by commenting on further possible results on the asymptotic behavior of the random tilings.Frozen regions). The complement of the liquid region. At the boundary of the liquid region  we expect Airy behavior. There is a vast amount of literature around this type of universality, and we only refer to [(refer to  for an oTurning points). The turning points are the points where the boundary of the liquid region touches a side of the hexagon. Here we need to distinguish between the turning points that touch the hexagon at a vertical side from the other turning points. In both the low and high temperature regimes (assuming (ight) in . At the ight) in  to appeaCusp points). In the high temperature limit, the boundary of the liquid region has cusp points. Such cusp points have appeared before in the context of random tilings. It is known that the local limit process near such a cusp point is the Pearcey process [. At the critical value . Another interesting feature of random tilings are the fluctuation of the height function. It was found in [2.By , the zerThe leading coefficient of Let These are all simple calculations based on . The ineLet If ked from . Hence \We now give the proof of Proposition for s on the Riemann surface (different sign on different sheets).The proof is similar to the proof of Proposition )1/2,see  and (2.2)1/2,see . As in tfined in , we recoThis shows that w values We finally prove that the line segment For see from  that \dos moves along the circle with radius Suppose N, where we putV is referred to as an external field [S-contours. Later, Rakhmanov [S-contours, and with Martínez-Finkelshtein [S-contours as trajectories of quadratic differentials. See [The orthogonality  does notnity. In  the orthal field . Since tal field  it is knakhmanov  made a selshtein  introducals. See , 59 for nth degree polynomial For al field  has onlyudied in , 56, 57.udied in , and it g-functionV as in  we also need the condition  it it4.5) i we have  and4.8\dWe recall from Sect. We let g-function, and the variational constant With  and 4.1, we defiz in the exterior of the circle (a) If given by  with cougiven by .g-function is defined byThe associated (b) If given by .g-function is defined byThe associated (c) We define the variational constant The definition  is such For the steepest descent analysis of the RH problem, it is convenient to introduce a function z does not intersect (a) If given by , and thez does not intersect (b) If given by , and theThe formulas  and 4.1 define \g and z in the domain of Let fined in  and   \documenbased on  reveals C is a closed contour going around C to infinity, we pick up residue contributions at the poles. It is a straightforward calculation to show thatWe compute  Then by  and the t goes from Note also that the right-hand side of  vanishes}t=θα by . Therefo}t=θα by  is monotC as in  to P on the graph from vertex A to vertex B. If there are no such paths then Consider the directed graph on We assume GV lemma  states tN with l is the number of horizontal edges at an even level. We pick l out of the possible N even levels, and N odd levels, and we see that there are l yieldsTo calculate omparing  and  as z approaches The limits of Let N and The RH problem Y of the RH problem The kernel Also for The formula  is a ref formula  is obtai formula  in a simY is fairly standard by now. It is modelled after the method developed by Deift et al. [The steepest descent analysis of the RH problem t et al.  for orthy follow  includinction in  for orthY can be expressed asg-function from Definition T are conveniently expressed in terms of the function In terms of the function fined in , the jumfined in  for Y cafined in  and ,,\documenfined in  and the T satisfies T has boundary values on T depends on N, even though this is not indicated in the notation. What is important for us, is that T and Note that T(z) and If T(z) and z in the domain If The proposition is a result of the steepest descent analysis that we will perform next for the two regimes separately.Because of  and the If If e domain  for somew and z in the domain (If e domain  for someParts (a) and (b) are immediate from  and PropT along w in the region under consideration in part (c), see for example Fig. Because of  and the Suppose S satisfies the following RH problem. We defineS has boundary values on   R should not be confused with the reproducing kernel R satisfies the following RH problem.We remove the constant jump on R has boundary values on R are exponentially close to the identity matrix as R and T and Since problems , we findrmations  and (5.1rmations  it then In case uch that  holds onrametrix  that we Y(z) denote the solution of the RH problem Suppose mentclasspt{minimaT satisfies the RH problem S as (it is similar to (S satisfies. The first transformation given by  and T samilar to )5.17\docS has boundary values on The global parametrix N and they slightly grow with N, namelyR is defined and analytic inR and N), and we choose them with radii smaller than T and N, and then T and In small disks rmations  and  w w6.2\docFor The equality leading to the third line in  follows equation  for \docThus The two functions are defined and analytic in een from  and the een from –6.5), n, n\documFor each Let We use the explicit inverses for the map given in  and  of Lemma s,We are interested in the level set of s since it is a simple saddle. Four curves emanate from s that are denoted by Typical behaviors of s with It is important for us that three of these curves stay inside To be able to prove this we need information on the behavior of the two functions If If We have the following for z, with double poles at (a) We consider the case To summarize,  with \doz moves over z moves over On the other hand, the calculations  and 4.2 in the pequation  has two ified in .x, where we include the solution Since  is a pol(b) For z, z for which ((c) If z moves along If (a) It is clear that Let z(t), (b) Let be as in . Then wiIf Thus in both cases we find that  is positz moves along the circle The increase along z traverses Suppose Indeed, from the definition  and the Due to Corollary The contours unded by . By the again by  one domaIf Thus cause of  and so iAs To summarize, we have a situation as in Fig. Figures It is now clear that we can find contours as described next. See also Figs. s and s and In the low temperature regime there are closed contours s and s and In the high temperature regime there exist contours Let In the low temperature regime we will also use In the final part of the analysis we are going to deform contours in the double contour integral to the ones from Corollary F as in  as in  and for The following statements hold:H) as in , and7.5\x, y). Hence x, y) which is , and we interchange variables (a) From  we deducintegral  with \doidentity  with \dogiven by  follows.x, y). Then (b) We now apply the change of variables integral  with \doms as in  which wems as in  transforrrive at  with \doH~ as in . FinallyIn the proof of part (b) of Proposition Let The Christoffel–Darboux kernel Part (a) follows since the right-hand side of  satisfiePart (b) follows after inserting  into 5., again wThere are corresponding symmetries for the location of the saddle point.Let From , we have implies .It can be readily verified from  and 2.72.7 that z)1/2 by  and (4.1z)1/2 by  and keep-ηz.From  and  from  has the H is one of the functions from  vary with N such that . Using  and afteally use  and we fally use  with \doSimilarly, but now using –7.7) an an7.7) aThus in order to prove Proposition We start the analysis of the double integral  with a cF is given by  of  we modify the definition . We use ote that  for w inable) of  with \docause of  and the cause of  of T, wex, y) be coordinates inside the hexagon. Then the double contour integral  is a point inside the hexagon, we have inequalities The integrand in the second double integral has no singularities for  thus by \documentf F, see , we haveing with , we see ing with  is thereegral in  vanishesIn the first double integral we deform  term in . The rem term in . \documeWe are now ready for the proof of Proposition We first assume N large enough, we then also have We write N, but in a controlled way. As Then also In particularLet  Then by 7.21\docu view of  it is enBy Corollary initions  and  is suchWe want to estimate the double integral in (N dependent contours as in the proofs above (which can be handled but would obscure the exposition) we assume T whenever w and/or z are on We treat the case see Fig. , left\doegral in  with \doombining , 6.4) w w\documeombining  we find grand in ) is equaT and Because of  and 7.2 we see tT across On the jump  of T acr0. Using  we splits lying inside the disk z on the deformed contour. Fortunately, z on the deformed contour, since The integral with the first term of the right-hand side of  is deforlows. By  and  over s to s and The integral (in the  that by \documentation of  across \s and w close to the branch points, see Proposition What remains are local contributions near the saddles For  that by  we have\The integrand of the double integral in (We approach this case as a limit of egral in  behaves egral in  as \docuidentity  that we w-dependent part isw-integral. Namely the integral with the first term is deformed from For  line of  which tew and z near the branch points T and Then there is exponentially decay on the deformed contours as This completes the proof of Proposition N. The asymptotic behavior of With the coordinates in  (and thent}KN in  as7.31\dis as in  with\doc


Q. This value was determined for a layered material obtained as the result of explosive welding of three alloys: AA2519, Ti6Al4V and AA1050, and separately for each material. In both cases tests were conducted for two temperatures—the ambient temperature (293 K), and cryogenic temperature (77 K). A model for initial assessment of the KQ value of AA2519–AA1050–Ti6Al4V (Al–Ti) layered material has also been presented. The proposed model has been developed so as to enable the determination of the curve course of load–COD for Al–Ti layered material using nominal stresses defined on the basis of a real load–COD course, obtained for the base materials, for both temperature conditions.The article presents the results of the author’s tests involving the determination of the maximal experimental value of the stress intensity factor K Layered materials are being more and more frequently used in the construction of engineering structures. This is due to at least two factors. The first is associated with a decrease in the object mass without losing the mechanical properties of the structure’s top layer. Naturally, the mechanical properties of the internal layer material differ from the external ones though they are characterized by lower density. The proposed solution can be applied in objects that are exposed to an aggressive environment ,4,5,6,7.One of the methods for joining materials with significantly different mechanical properties is explosive welding. Unlike low energy methods, explosive welding provides the possibility of bonding materials which cannot be connected by standard methods. These composites include, e.g., stainless steel and copper , stainleAn explosive welded layered material made from three alloys with significantly different mechanical properties is an example of such a material. These materials are AA2519, AA1050 and Ti6Al4V. The mechanical characteristics of these alloys used as base materials are known and widely described in the literature ,16,17,18This study covers two aspects. The first includes selected characteristics of the layered material and base materials, the research equipment and selected mechanical properties of the tested materials. A special focus is put on quantitatively characterizing the fracture toughness of the tested materials.Q value for a layered material made of base materials whose KQ value is known.Each of the base materials used to produce Al–Ti layered material is characterized by individual features describing its fracture toughness. After explosive welding a completely new material with different mechanical characteristics is created. The question has arisen whether it is possible to estimate the value of this characteristic for the layer material, resulting from explosive welding of these materials, on the basis of known characteristics determined for base materials. The second aspect addressed in this paper is an attempt to answer this question. This part of the article is a presentation and verification of an analytic model which was assumed to be a tool allowing an initial assessment of the KQ value for the Al–Ti layered material, with the one obtained by means of an analytic model.The main goal of this study is to compare an experimentally obtained crack opening diagram, and thus the maximal experimental KQP—load determined on the basis of the experimental curve, B—sample thickness, W—nominal sample width and a—crack length.The value of the maximal experimental stress intensity factor has already been determined in tests for both ambient and cryogenic conditions. The described characteristics were determined according to the equation:Details of those tests can be found in Boroński et al. . HoweverThe article shown contains an attempt to describe the strength of the layer connecting the base materials, already after the explosive welding. The main problem encountered during this attempt to describe the strength of this layer is the delamination occurring between the materials.However, this article uses previous work to propose an analytical model, based on the information contained therein, which would allow to determine the stress intensity factor for Al–Ti material on the basis of the same amount but determined for base materials. It should be noted that the presented model strongly depends on the geometry of the sample.The base materials used for manufacture of the Al–Ti composite include two aluminum alloys. One of them is AA2519 alloy. This is a new material which has not been explicitly described in the literature. One of its main advantages is its outstanding resistance to adverse weather and environmental conditions such as saline sea water. Furthermore, the alloy is also resistant to stress-related corrosion. It is referred to as hard to weld, which complicates some technological production and montage processes.The characteristics listed above make the material applicable in military vehicles. It is used for covering external layers of military amphibious vehicles, reducing their mass and providing them with corrosion and ballistic resistance. The material gained popularity as soon as its plastic treatment technology had been developed, and then it stopped being used only in the form of sheet metals and plates ,26,27. STi6Al4V titanium alloy , is a coA series of experiments involving the determination of the static properties of the main base materials, that is, the AA2519 and Ti6Al4V alloys, were conducted prior to fracture mechanics tests. The results and a detailed description of the tests including cryogenic conditions can be found in Boroński et al. k) ranged from 1850 m·s−1 to 2000 m·s−1, with the approach angle equal to 15°, whereas the impact zone shift speed (Vc) for the analyzed material was about 500 m·s−1. A schematic presentation of this process is given in The main research object is an explosive welded material which consists of the three alloys: AA2519–AA1050–Ti6Al4V. According to the developed technology, the welding of these materials involved parallel plating of AA2519 aluminum and Ti6Al4V titanium layers, whereas the lightweight alloy was a layer to be applied. As a result of this, the titanium had no direct contact with the explosion. Prior to the welding process, the aluminum layer was rolled onto a 0.2 mm thick layer of AA1050 alloy. This procedure was in order to create a buffer layer between the materials. The process of welding was carried out in such a way that the speed of the flame front during detonation  surface, whose size is given in pixels with dimensions x and y, sum S is calculated, which can be described as:α, β—initial and final value of x coordinate index of the correlated images (β − α = w), γ, ζ—initial and final value of y coordinate index of the correlated images (ζ − γ = h), nI—intensity of the image at the point with ix, jy, coordinates recorded for the loading n stage and mI—intensity of the image at the point with ix, jy, coordinates recorded in the loading m stage.From a fragment of the analyzed image, being the environment of a selected point of the P specimens with the dimension ratio W/B = 4  for bothThe use of such a low temperature to conduct the tests was dictated by the potential application of the tested material. Aviation structures and particularly the aerospace ones for which the analyzed material was developed, are exposed to the impact of low temperatures. Due to the fact that it is difficult to anticipate the impact of differences in the thermal expansion of the base materials on the Al–Ti material strength, it was decided to conduct experiments under cryogenic conditions.In contrast to the determination of static properties, tests of the fracture toughness in the linear- elastic range consisted of three stages. The first of which involved generating a fatigue crack whose length fitted in the normative range . Due to −1. The process of tension was carried on until the loss of coherence between the stretched parts of the specimen. Since during the second stage of the experiment only the load and crack opening displacement were measured, it was possible to carry out the experiments under both ambient and cryogenic conditions. This was done for all three materials.The second involving carrying out a monotonic tensile test of the CT specimen with the above-discussed fatigue crack. In the second stage of the tests the speed of the machine piston stroke was 0.02 mm·sThe third stage involved determining the geometric features of a crack generated in the tested specimens. Commercially available programs were used for the analysis of the image to enable the determination of the basic planimetric values. Consistently with the basic assumptions of fracture mechanics, cracks can never be equal. Thus, following the experiment, it is necessary to measure the crack length of the CT specimens. In this study this measurement was performed using optical methods.Diagrams of the curves for load-crack opening determined during tests of the maximal experimental value of the stress intensity factor for heat-treated AA2519 aluminum alloy obtained from the experimental tests are presented in The diagrams obtained as a result of experiments with the use of Ti6Al4V titanium alloy were similar to those obtained for the AA2519 aluminum alloys. Thus, the diagrams for the analyzed Ti6Al4V titanium alloy are presented in The maximal experimental value of the stress intensity factor and the values characteristic of the described materials were also determined for the Al–Ti material. As in the case of the base materials, the tests were conducted for two temperature conditions, that is, ambient and cryogenic temperatures. A number of simplifications related to the quantities that refer to the mechanical properties of the AA2519–AA1050–Ti6Al4V composite were used for the description of the layered material. The above-mentioned characteristics were analyzed and adopted for a description of the individual materials, not the composites. When analyzing the results, it must be remembered that in the considered case the tests covered not only two materials placed near each other but also a “structure” which, apart from materials with significantly differing mechanical properties, contains a layer where the materials interact with each other.Q was assumed to be a comparative quantity, regardless of whether it meets the conditions that qualify it to be a critical value. The diagrams so achieved for the considered Al–Ti are presented in This was one of the reasons why the maximal experimental value of the stress intensity factor KIt should be noted that the details of determining individual mechanical properties can be found in Boroński et al. .The possibility of determining the mechanical properties of a layered material by using the information about the mechanical properties of the base materialsis advantageous. In other words, on the basis of the experimental results determined for the base materials, there is a possibility of analytical determination of selected properties for the layered material.To enable such a solution, it is necessary to develop an analytical model based on e.g., based on a simplified analysis of the nominal stresses of the CT specimens. However, due to the uncertainty of the analytically determined results, it is necessary to an scientific experiments in the field of solid mechanics.To develop the described analytical model, certain assumptions were made. The primary assumption was that the force loading the Al–Ti layered material is a sum of the partial forces loading the Al and Ti layers:In order to remain independent of the differences associated with different lengths of the fatigue crack, the input values were assumed to refer to real cross-sections of CT specimens, hence further analysis based on the nominal stresses was performed.gM—bending moment, xW—axial index of the cross-section fracture toughness and F—uncracked surface area.According to the scheme shown in The value of the maximal nominal normal stresses in the direction of CT specimen loading can be calculated including the specimen geometric dimensions and its crack length, according to the following Dependence:P, the diagram curves were determined for nominal stresses Smax as a function of the crack opening COD. The measured crack lengths were used for calculations of the considered nominal stresses. Averaged values of stress diagrams for both base materials are shown in Using Dependence (8) and the experimental results for In order to calculate the loads in the layers of Al and Ti, the values of the nominal stresses for the same COD values were read from S-COD diagrams determined for both base materials.AlP and TiP As regards the experimental value of the stress intensity factor, when determined for Al–Ti layered material, it was not lower than for AA2519 aluminum alloy, for which this value was the lowest.(2)Cryogenic conditions caused a slight drop in the brittle fracture toughness of the Al–Ti layered material and Ti6Al4V titanium alloy. AA2519 aluminum alloy in the linear–elastic range did not react to the considered conditions.(3)The results obtained from the proposed analytic model differ from those obtained experimentally, while the differences are lower for the analysis results obtained for cryogenic conditions.(4)When comparing the results of experimental and analytic analyses, it can be said that the test results obtained using the proposed analytic model are overestimated in almost every case.(5)The main disadvantage of the proposed analytic model is that it neglects the middle layer of AA1050 aluminum alloy situated between the base materials.(6)The inconsistency of the experimental results with the analytic ones may be caused by the joining zone between the welded materials. This is due to a reaction that occurs in the interlayer zone which is difficult to describe. An attempt to provide a description of the interaction can set a new direction of research to be continued in order to get to know the exact mechanical characteristics of layered materials.(7)Despite the differences between the results obtained experimentally and analytically, the proposed model can find application in the initial assessment of the fracture toughness of layered materials built on the basis of AA2519 aluminum alloy and Ti6Al4V titanium alloy.QK—stress-intensity factorkV—speed of the flame front during detonationcV—speed of joining the base materialsW—nominal sample widthB, eB—sample thicknessH*—distance between the horizontal axis of symmetry of the sample and the axis of the clamping slotsa—crack length0b—uncracked specimen fragment in the crack planeα, β, γ, ζ—values of the coordinates of indexed imagesP, maxP, QP—loadgM—bending momentxW—axial index of the cross-section fracture toughnessF—uncracked surface areal—the distance between the force application point (pin axis) and the back of the sample


Hybrid exoskeletons are a recent development which combine Functional Electrical Stimulation with actuators to improve both the mental and physical rehabilitation of stroke patients. Hybrid exoskeletons have been shown capable of reducing the weight of the actuator and improving movement precision compared to Functional Electrical Stimulation alone. However little attention has been given towards the ability of hybrid exoskeletons to reduce and manage Functional Electrical Stimulation induced fatigue or towards adapting to user ability. This work details the construction and testing of a novel assist-as-need upper-extremity hybrid exoskeleton which uses model-based Functional Electrical Stimulation control to delay Functional Electrical Stimulation induced muscle fatigue. The hybrid control is compared with Functional Electrical Stimulation only control on a healthy subject.The hybrid system produced 24° less average angle error and 13.2° less Root Mean Square Error, than Functional Electrical Stimulation on its own and showed a reduction in Functional Electrical Stimulation induced fatigue.As far as the authors are aware, this is the study which provides evidence of the advantages of hybrid exoskeletons compared to use of Functional Electrical Stimulation on its own with regards to the delay of Functional Electrical Stimulation induced muscle fatigue. Stroke is the second largest cause of disability worldwide after dementia . TemporaFES is the application of high frequency electrical pulses to the nerves or directly to the muscle belly in order to elicit contractions in the muscle. FES devices are typically lightweight and FES is well suited to reducing muscle atrophy in patients with no or extremely limited movement. The trade off to this is that precise control of FES is extremely difficult and controlling specific, repetitive, and functional movement is not easily accomplished. Furthermore, extended use of FES is limited by the introduction of muscle fatigue caused by the unnatural motor unit recruitment order . The forCombining the use of FES and an electromechanical actuator within an exoskeleton can potentially overcome the limitations of each individual system. Despite the potential advantages of hybrid exoskeletons, so far only limited studies have been done on their effectiveness. A recent review was conducted into upper-extremity hybrid exoskeletons  which hiThis work describes the design and testing of an assist-as-need upper-extremity hybrid exoskeleton which uses model-based control of FES with a focus on reducing FES-induced muscle fatigue. The control system is described in Section “It is highly desirable in stroke rehabilitation robotics that a robot or exoskeleton be capable of performing assistance-as-needed. This way the patient is encouraged to make the effort to achieve movement rather than learning to rely on the robot to perform the movement –9. ApproThe assistance provided from the FES and motor should be the minimum which the patient requires to perform the movement at a given time.The FES should perform the bulk of the movement which the patient is physically unable to. This ensures that most of the movement performed requires effort form the patient’s muscles and thus improves muscular strength.To implement the concept of assist-as-need there are two important features which are desired:In the system proposed here the angle of the arm can be affected and controlled by three different inputs; volitional movement from the subject, FES-induced movement, and rotation of the motor. Any one of these on their own could potentially produce the desired angle. However to achieve the two defined desires, there is a necessary hierarchy of control.The control system may in general be clearly divided into at least two control systems, each related to a different output variable which can be considered independently Fig. . There iSection “Previous work  investigWhere:θ is the elbow angle in degreesv is the voltage in voltspw is the pulse-width in microseconds (full pulse length = positive + negative portions)f is the frequency in Hertzk is the overall gaing= 14, voltage gainvg = 0.15, pulse-width gainpwg = 0.22, frequency gainfThe user is instructed to relax their arm so that the palm faces the user  with fingers pointed down. Once the user is relaxed the system is switched on.At the beginning of setup the motor rotates the arm to 90°. Five measurements each are taken of the angle and torque. These readings are averaged and used to calculate the weight of the arm under the assumption that the arm and exoskeleton are a point mass at distance 0.13 m from the elbow. The motor then lowers the arm back to 0°.The voltage threshold test is conducted. Stimulation is applied at a frequency of 30.5 Hz, and pulse-width of 200 μs. Voltage steps are applied in increments of 0.5 V starting at 10 V. Each step is applied for a duration of 3 s and the peak arm response is recorded in degrees. When a step results in a peak arm angle of 20° the voltage threshold test is complete and the input voltage is recorded and defined as the threshold voltage. In between each step if a 20° angle has not been achieved then then stimulation is turned off for a duration of 3 s before the next step is applied. This short rest is to prevent the arm getting used to the stimulation which would affect the voltage threshold (more stimulation would be required to achieve a given angle).The model performed well for different subjects and only the threshold voltage and the overall gain needed to be found for the subject. As described in Section “The entire system takes less than 6 min for a complete set up including attachment of the exoskeleton and electrodes. Once the setup has completed, the control system runs on the right arm in response to a desired arm angle based on the position of the left arm. Control of this system is described in sections “When the desired support is less than 100% the motor speed is set using proportional-differential (PD) control based on the support error. When the desired support is 100% the motor speed is set using proportional-derivative (PD) control based on the angle error for errors larger than 5°. If the angle error is negative and greater than 20° then the motor will lower the arm at the maximum speed. This last state is to ensure that the motor does not impede movement of the user. Due to the setup of the pulley and cable system the motor cannot physically impede movement of the user when raising the arm but could potentially impede movement during lowering.Control of FES is performed using the model described by Eq. The control of the desired amount of support is performed based on the angle error over time. In general if the error over time is consistently positive then desired support should increase. If, on the other hand, the error is consistently negative then support should decrease. In general the desired support should not respond too quickly to errors in the angle and it should ignore any large short term errors. Thus rather than use the average error over time a median filter of length 50 is used. A measurement of the angle error is taken every 0.5 s.If the median error over the last 25 s is within 5° then no changes are made to the desired amount of support. If the median error is positive and larger than 5° the desired support is increased. If the median error is negative and larger than 5° the desired support is decreased. The previous median error is also used to calculate how much the desired support should be changed. If there has been a change in both the desired support and the median error then those values are used to calculate the new support using Eq. Where:S is the desired % supportt is the current timeEvery 0.5 s Eq. Ten tests were conducted on Exoskeleton using a healthy 27-year old female subject with different initial values for the overall gain and desired assistance. Selected plots of the test results are displayed in Sections “All tests, except Test 5, were conducted with the user providing no volitional input from their right arm. Test 5 involved the user moving both arms volitionally together in a mirroring pattern. Test 6 used FES only with no assistance form the motor. Test 9 and 10 used only the motor and no FES. Only Test 10 did not perform assist-as-need. The tests are listed in the order they were conducted and only short rests (a few minutes) were taken between each test. All tests were conducted on the same day. A discussion is given for the tests and results in Section “The first figure in each test subsection displays the desired angle  and measured angle  during the test. In cases where there is a second figure this shows the change in the desired support and the change in the gain during the test in response to the assist-as-need control scheme.The results for the tests are given in Table This is likely why at initial glance the volitional test (Test 5) and the motor only without assist-as-need test (Test 10) appear to have larger average and median errors than the hybrid tests. It is expected that these two tests should produce the smallest errors. That said, from the plots for these two tests Figs.  and 7 itOverall, the first four Hybrid Control Tests (Tests 1–4) performed similarly to that of the motor only control (Test 10) with similar sized error measurements across the board. There was some variation in performance among the Hybrid control tests depending on the initial test parameters (k and desired assistance), variations in test time, and variations in fatigue, however no noticeable trend was observed and the differences were not large. The difference in RMSE between the best and worst of the first four hybrid control tests was 7.44 degrees.There are three tests which stand out as having large errors. These are Test 6, 8 and 9. Test 6 is the FES only control test and given that the difficulties with performing large movements with FES and FES-induced fatigue are well known problems for FES, it is not surprising that Test 6 has the worst performance. The results for Test 8 and 9 are less expected and will be discussed later on in this section.thresh) and the general decrease in the overall gain (k). It is for this reason that the 6 min FES only tests (Test 6) was conducted prior to the 6 min long Hybrid test (Test 7). If the hybrid control is able to reduce the impact of FES-induced fatigue then it is expected that the performance of Test 7 should be better than that of Test 6, however it is also expected that a good performance would be harder to achieve in the presence of more fatigue. Thus given that Test 7 has better performance than Test 6 despite being performed after Test 6 there is much stronger support for the argument that the hybrid control does indeed reduce the impact of FES-induced Fatigue. This is further backed up by the smaller final overall gain (k) for Test 6.Due to all of these tests being conducted on the same day and one after another it is expected that the arm will be more fatigued for the later tests. This is backed up by the general increase seen for the voltage threshold  is the most commonly used measurement of performance for prosthesis and exoskeleton control systems  however It is important to note that the tracking tests for the FES/Robot hand were performed on the index finger which comparatively has a smaller range of motion as compared to the elbow joint so is it is difficult to directly compare to this work. Furthermore, the tests performed in this work involved a comparison between a healthy subject performing volitional movement and the same subject putting no effort in at all with FES and with the hybrid system, whereas the tests conducted on the FES/Robot Hand compared Stroke patients performing volitional movement with and without the help of the different exoskeleton systems , 15. As One other exoskeleton described in , the WeaA key novel contribution of the current work is to test whether the hybrid exoskeleton is able to reduce the level of FES-induced fatigue as this is something which has not been tested by the hybrid exoskeletons described in . As has Given that all of these tests were conducted on the same individual and same day it is expected that the final value for the overall gain would be of roughly similar magnitude, with some variation due to fatigue as described above. Thus it is promising to see that despite the large differences in the initial value of the overall gain, the final values are all a similar magnitude to one another, with one exception. The volitional test (Test 5) did not cause variations in the overall gain. This can be attributed to a lack of errors during the test which would normally cause the software to apply the FES. The value of the overall gain can only be updated if the FES has been applied a sufficient number of times. Given that Test 5 involved the subject moving both arm simultaneously there was very little error and thus very few reasons for the FES to be applied. This is not an issue from a control perspective as if the error were to increase then the FES would be applied and the overall gain would be calculated. Given that the user is very capable it is not a problem that the overall gain has yet to be calculated and from a patient monitoring perspective one can still observe that the FES input parameter values are small, desired assistance has decreased and remains sitting at 0% Fig. , and yetThe assist-as-need of the motor is not quite as smooth as that of the overall gain which can be seen by comparing Test 9 and Test 10. It is expected that the desired assistance will fluctuate somewhat given that short rests can improve the effectiveness of the FES parameters, however the rate at which the desired assistance varies during these tests is faster and larger than is desirable. It’s possible that the rest angle of the right arm being greater than 0° contributes to this as well as the slow lowering speed of the motor. However, what is more likely is that the assumption made in Section “So far the hybrid exoskeleton described in this work has only been tested on one healthy subject. This is something which is a common issue with regards to current exoskeleton research in general. Furthermore, very few exoskeletons, and even less hybrid exoskeletons have been tested on stroke patients, let alone on large numbers of them. Cost is one of the main barriers to widespread testing of exoskeleton devices. The cost to construct the exoskeleton described in this work is very low (a few hundred NZD) which may help to lessen this barrier in future.Overall the hybrid control and assist-as-need control methods perform well in comparison with complete volitional movement and non-hybrid control. In particular the hybrid system shows an improved performance with regards to FES-induced fatigue compared with using FES only demonstrated by larger change in overall gain (k) and a larger average and RMSE error for the FES only control. As far as the authors are aware this is the first upper-extremity hybrid exoskeleton which uses model-based FES control to perform assist-as-need.The design of a voltage controlled functional electrical stimulator has been described in other works  and is tprodctodc.com - Item ID #090438, set to 13.5 V for this system) and relay circuitry for added safety. The relay section of the circuit was constructed by one of the lab technicians. A 5 V regulator (L78s05cv) was used to step the 13.5 V down for the Arduino and motor. For testing described this work, a desktop DC power supply was used in place of a 3 V regulator for the input to the FES circuit as a slightly more consistent supply could be achieved. A 3 V regulator could be used instead for portability. Ethical approval for testing of this device was granted by the University of Canterbury Human Ethics Committee. Section “For simplicity and portability the Rhino Motion Controls High Torque Servo Motor (RMCS-2251) has been selected as the actuator for this exoskeleton This motor is more than capable of providing all of the torque requirements for movement of the elbow joint . A smallThe powered exoskeleton was arbitrarily selected to be designed for the right-arm and a second non-powered smaller exoskeleton was designed for the left arm to be used as the control input. Construction of the both exoskeletons was based around Actobotics components sourced from Sparkfun . The powA swivel hub allows free movement of the elbow joint for both exoskeletons. A pulley is affixed to one side of the swivel hub on the powered exoskeleton. A metal cable is wound around this pulley and runs up the inside of a protective plastic tube. At the other end of the plastic tube the cable is wound around a second pulley which is affixed to the shaft of the motor situated on the shoulder of the user.To attach the motor to the user a soft shoulder brace is worn. The exoskeleton is manually lined up with the user’s arm and the motor is placed gently on the shoulder. Velcro straps are used to hold the motor and exoskeleton in place. The FES sleeve is placed on the user’s arm and electrode gel (Spectra 360) is applied prior to attachment of the exoskeleton. Correct placement of the electrodes are also checked and any adjustments made prior to exoskeleton attachment. Figure The exoskeleton can be made shorter or longer for the shoulder to elbow section by unscrewing the upper metal rod and moving the screw up or down a hole. The entire attachment of the FES electrodes and exoskeleton to the user takes only 1–2 min and can be performed by the user themselves without the need for movement in the right arm. Despite the motor only weighing 180 g (and making up the bulk of the weight of the exoskeleton) the structure was still found to be heavy enough to cause mild discomfort during prolonged wearing (45 min) for a healthy subject. Future designs should consider methods to shift the weight of the motor to the centre of the back of the user and away from the shoulder.To measure the angle of the arm, the shaft of a potentiometer was attached to the pulley at the elbow joint using a set screw hub (Actobotics) and the body of the potentiometer was soldered to a small Vero board and affixed to a rod. The rod is attached to the upper portion of the exoskeleton as shown in Fig. . Thus byTo measure the force applied to and by the exoskeleton arm, a 10 kg straight bar load cell is used to connect the elbow section of the exoskeleton to the wrist section Fig. . An HX71The interaction forces between the user’s forearm and the exoskeleton all occur at the wrist strap point (x), which is located 0.2 m from the elbow pivot joint. The support forces from the motor are applied very close to the elbow pivot joint. The total torque applied at the exoskeleton elbow is the due to the torque produced from the interaction forces between the user and exoskeleton, and the support torque. The load sensor is located 0.13 m from the elbow pivot joint and measures the perpendicular force to the exoskeleton at this point. Thus the total torque at the exoskeleton elbow joint is the product of the force measured by the load sensor and the distance to the load sensor (0.13 m) as described by Eqs. Where:τtot is the total torque around the exoskeleton elbow joint (anticlockwise).τsupport is torque produced by the motor and cable system (anticlockwise).τuser is the torque produced by the user, includes volitional movement and effects from gravity (anticlockwise).x is the distance from the elbow joint to the wrist strap (0.2 m).Fmuscle is the force produced by volitional movement from the user measured in Newtons (perpendicular to the arm and upwards).m is the combined mass of the arm and exoskeleton in kilograms (at the wrist strap).g is acceleration due to gravity (9.8 ms-2) (perpendicular to Earth and downwards)θ is the elbow angle in degrees. This is the angle which the arm makes measured from 0° (when the arm hangs straight and perpendicular to Earth) in an anticlockwise direction.τ is (90 – θ) for arm angles below 90° and (θ – 90) for arm angles above 90°.θL is the distance from the elbow joint to the load sensor (0.13 m).Fmeas is the force measured by the load sensor in Newtons (perpendicular to the arm and upwards).M is the gradientc is the offset.R is the output of the load sensor amplifier (volts).Thus the measured total torque is given by Eq. The exoskeleton arm was rotated using the motor and cable system and measurements of the load sensor were taken at 14 different angles. This was repeated with three different weights attached to the end of the exoskeleton arm: 100 g, 200 g, and 500 g. Using the measurements and expected torque produced by each weight the gradient and offset in Eq. Using these values, the expected torque produced by the mass of the exoskeleton can be calculated for a given angle using Eq. In order to calculate the percent of support which the exoskeleton is providing, knowledge of the arm weight of the subject is required. During setup the system rotates the exoskeleton arm to 90° while the subject relaxes their arm. Several measurements are taken by the software and the results are averaged. From these measurements the arm mass of the subject can be calculated and the 100% support torque point is defined. Any torque measurements between this point and the set point mean that the exoskeleton is providing a certain percent support. For example half way between these two values would be 50% support.


Estimating parameters for distributed hydrological models is a challenging and long studied task. Parameter transfer functions, which define model parameters as functions of geophysical properties of a catchment, might improve the calibration procedure, increase process realism, and can enable prediction in ungauged areas. We present the function space optimization (FSO), a symbolic regression method for estimating parameter transfer functions for distributed hydrological models. FSO is based on the idea of transferring the search for mathematical expressions into a continuous vector space that can be used for optimization. This is accomplished by using a text generating neural network with a variational autoencoder architecture that can learn to compress the information of mathematical functions. To evaluate the performance of FSO, we conducted a case study using a parsimonious hydrological model and synthetic discharge data. The case study consisted of two FSO applications: single‐criteria FSO, where only discharge was used for optimization, and multicriteria FSO, where additional spatiotemporal observations of model states were used for transfer function estimation. The results show that FSO is able to estimate transfer functions correctly or approximate them sufficiently. We observed a reduced fit of the parameter density functions resulting from the inferred transfer functions for less sensitive model parameters. For those it was sufficient to estimate functions resulting in parameter distributions with approximately the same mean parameter values as the real transfer functions. The results of the multicriteria FSO showed that using multiple spatiotemporal observations for optimization increased the quality of estimation considerably. This study introduces a method to infer transfer functions for the multiscale parameter regionalization approachWe demonstrate its ability in a case study using synthetic runoff dataWe show that multicriteria optimization can improve the estimation of parameter transfer functions The modeled processes include the simulation of spatially distributed land surface fluxes . Hence, the main restriction of MPR today is the selection of suitable parameter transfer functions . These limitations will be explained in detail in the methods part of this publication and are a main motivation for this work.To overcome these limitations, the proposed method is based on the interpretation of mathematical functions as text, where each symbol of a function is seen as a “word.” Recent developments in Natural Language Processing (NLP) resulted in powerful Artificial Intelligence (AI) architectures which are able to translate  as a method for estimating parameter transfer functions for distributed hydrological models and applies the FSO method in a case study. The case study uses synthetic runoff data and consists of three tests. In the first test only runoff data are used for estimating transfer functions. In the second and third tests we investigate how the use of additional spatial‐temporal information in a multicriteria optimization can improve the transfer function estimation.22.1O and M, for the spatial scales of observations and the model, respectively. Note that O<M is a necessary condition for MPR. Let θO∈ℝn be the model parameters on the spatial scale of observation, defined bytransfer function. It uses a set of k numerical parameters XO to θO. Here, XO∈ℝn×sp contains the spatial predictors. While θO and XO are spatially distributed, Let the two spatial scales used in the MPR approach be denoted by θM∈ℝ, can thus be defined asModel parameters on the spatial scale of the model, aggregation function which upscales the values of θO to θM. Theoretically, any kind of aggregation function is possible. Samaniego et al.  finding the correct transfer function and global parameters and (b) finding the correct aggregation function. We here apply the arithmetic mean aggregation and focus mainly on the estimation procedure of transfer functions and global parameters. For (b), one can either define an aggregation function using knowledge from previous investigations about the parameter, trying different aggregation functions, or use the generalized mean as an additional parameter to optimize. It is important to note that the estimated transfer functions are valid for the chosen aggregation function, and it cannot generally be assumed that they perform equally well with other types of aggregation.With this we can define the aim of any transfer function estimation procedure:where Considering these definitions, assumptions, and restrictions, we developed a method to infer parameter transfer functions and their global numerical parameters simultaneously from data.2.2The main difficulty of inferring This leads to the main idea of FSO: define a numerical space which defines distance between functions by (1) semantic closeness and (2) closeness in the resulting parameter distributions. Property (1) specifically includes the interpretation of functions as text, in which function symbols  are interpreted as words. Property (2) is necessary since physical catchment properties are often highly correlated. Hence, the functions “Property (2) implies the a priori choice of global parameters By transferring the problem in a numerical space with the above‐mentioned properties, any continuous global optimization method would be applicable. The steps for creating such a space and its use in the estimation of a transfer function will be described in the following sections.2.2.1In the first step of creating a search space, it is necessary to define a realm of possible transfer functions. A context‐free grammar (CFG) Knuth,  is used a in the vector representation in Figure In the simple example in Figure Klotz et al.  used theLooking at the characteristics of These two properties influence the optimization of These issues were one of the main motivations for developing FSO. The main advantage of using a CFG for FSO is the possibility of sampling functions, while preserving the defined function properties. Thereby, we use it to create a (very large) realm of possible function for the transfer function search.2.2.2latent representation or latent space of the input information. After training an autoencoder to correctly encode and decode the information of a set of strings, the decoder can be used to generate strings from the latent space.One type of generative models often used in NLP is autoencoders  architecture  the cross‐entropy loss resulting from the reconstruction of functions strings, (2) the mean square error of the parameter distribution reconstruction, and (3) the Kullback‐Leibler divergence  meaning. This allows for global scale application.2.2.4The full workflow of FSO is shown in Figure The optimization phase of the FSO is a fully automatic procedure that uses the text generating VAE which was trained in the assembly phase. It searches for the optimal point in the function space  to minimize the loss function. In each iteration a new function is generated from the function space, which is used to produce a parameter field. This new parameter field is used in the hydrological model and results in a loss function output. After a previously defined number of iterations, the function with minimum loss is chosen as the estimation for the parameter transfer function.In general, any continuous optimization algorithm can be applied in the optimization phase. We experimented with three commonly used algorithms: Genetic Algorithm Taylor, , DynamicFSO can optimize multiple parameters at the same time. This can be done by optimizing multiple function spaces. Since each function is represented as a six‐dimensional continuous vector in function space, optimizing two transfer functions results in a 12‐dimensional continuous optimization problem.3To test whether FSO is able to sufficiently approximate transfer functions, we conducted a test in a virtual reality setting. This case study applies FSO on a parsimonious distributed model using synthetic runoff data.3.1The case study was performed using hydrological, meteorological, climate, and geographic data from the Mur catchment , slope (slope), height above nearest drainage (hand), percentage of clay (clay), percentage of sand (sand), soil depth (bdim), the enhanced vegetation index (evi), and a noise layer (noise). Topographic properties  were calculated from a digital elevation model obtained from Rechenraum  was obtained from SoilGrids  that consists of values sampled from a uniform distribution over the interval noise layer was created for further testing FSO by providing irrelevant information as a possible predictor.The 250 m gridded geophysical properties used in this case study are height above sea level . Very high correlation coefficients could be observed for clay/sand (−0.95), clay/elevation (−0.83), and elevation/bdim (−0.82), and the lowest was observed for evi/hand (−0.35).All geophysical properties are strongly correlated with an overall mean absolute correlation coefficient of 0.55 , X2 catchment water exchange coefficient (mm/day), X3 1 day maximal capacity of the routing reservoir (mm), and X4 unit hydrograph time base (days). A more detailed description of the GR4J model is given by Perrin et al.   result in a large production store. Their resulting parameter fields are shown in Figures We chose the true underlying transfer functions that use one or two different spatial predictors. The transfer functions for the three parameters are chosen to reflect the possible physical interpretation of the parameters, for example, large values of The CFG for the case study is chosen to be complex, to create a large search space. It included multiple recursive nonterminals, exp/log functions, power functions, reciprocal functions, and linear combinations. Additionally, to include the search for the global parameters n Figure . The thrFrom this CFG we sampled 5 million unique functions. We used 80% as the training set and 20% as the validation set for the VAE. The minimum validation loss was reached after 132 epochs of training the VAE with a batch size of 1,000.After selecting the true transfer functions, the resulting parameter fields are used to produce a synthetic discharge time series using d‐GR4J on a 2 km grid. Also, the internal states of the storage S and R for each grid and each time step are calculated. An overview of the case study model setup is shown in Figure 3.4floss. Therefore, two different tests were conducted using:a single‐objective criterion, with loss only dependent on discharge, anda multivariate objective criterion, with loss dependent on discharge and state “observations.” For both tests, the optimization procedure was repeated five times to evaluate the variations in the retrieval process, while a maximum number of 3,000 iterations for the DDS global optimization algorithm was chosen.The performance of FSO is strongly dependent on the choice of the loss function We applied the multicriteria FSO tests (Tests 2 and 3) using S, R, and both simultaneously. For brevity, we here present only the optimization using the time series of state S. The results for the additional tests can be found in the supporting information Figures –S9.3.4.1NSE=∑it=1TQmt−Qot2∑it=1TQmt−Qo¯2  has a standard deviation of 0.337, while evi*1.5 has a standard deviation of 0.094.The comparison between true Figure A comparison of the estimated parameter fields and the true parameter fields is shown in the supporting information Figure .4.4Figure For brevity, we will show the detailed results of one run only, which describes the general behavior of all runs. Figure The comparison between true Figure A comparison of maps of the estimated parameter fields and the true parameter fields is shown in the supporting information Figure .4.5Figure For brevity, we show the detailed results of only one run. Due to the increased variance of the results, we chose the best training run (Run 2). Figure The comparison between true Figure 5In this study, we present a method to automatically estimate parameter transfer functions for distributed hydrological models. Defining parameters as functions of the geophysical properties of a basin results in an increased physical interpretability of the model parameters, seamless parameter fields, and the possibility of prediction in ungauged basins. Our approach is based on the compression of functions from a CFG into a searchable continuous space (function space), which subsequently can be used for continuous optimization.To demonstrate the predictive ability of FSO, we conducted a case study using synthetic data to avoid any influence of potential sources of errors, such as measurements errors and model assumptions, on the estimation procedure. The underlying true transfer functions were defined a priori and used for generating synthetic parameter fields that, in combination with the rainfall‐runoff model d‐GR4J, result in synthetic runoff and storage data.We demonstrated that the developed function space has the desired properties of being “searchable” and that our chosen model parameters are sensitive. FSO is then tested in a case study using synthetic parameter fields and corresponding synthetic runoff and/or storage data.The case study consists of two tests. First, we apply a single‐criteria calibration, optimizing the transfer functions only on runoff data in the calibration procedure. Second, this is then extended to additionally include spatially distributed time series of storage data in a multicriteria optimization. For both tests we could find transfer functions that produce a nearly perfect discharge prediction with an NSE of 0.999 in “ungauged” basins.The results of the single‐criteria optimization show that FSO can find transfer functions that result in a perfect match for runoff and that FSO results are stable and multiple runs vary only insignificantly in their resulting mean basin NSE.The results of both multicriteria optimization showed an increasing performance when estimating the parameter (X1), which is associated with the storage observations that are added to the loss function. Looking at the results of the different optimization runs, it is noteworthy that having the additional term in the loss function increases the difficulty of the optimization problem  large space of possible functions for FSO and therefore have a high probability of including the true or a sufficiently approximating function in it.Knowing the restrictions due to equifinality and assumptions related to FSO, we could show that the multicriteria tests increased its predictive capability for the state related parameter. FSO was able to estimate transfer functions which included the correct spatial predictors and had either the exact same or a very similar parameter field on the model scale compared to the true one. Therefore, similar to other studies that showed an increased model performance by using multicriteria parameter estimation, we could demonstrate an improvement in the search for a transfer function. The only disadvantage resulting from using multicriteria optimization is the increased complexity of the optimization task, which potentially increases the number of iterations needed.In this study we could show that it is generally possible to estimate true underlying parameter fields with mathematical equations. However, it is important to keep in mind that the physical realism of the resulting functions does not solely rely on the here proposed method but also to a main part on the hydrological model definition and available data. This is especially relevant for large‐scale complex process‐based hydrological models, and it can be assumed that optimization restrains in the form of multicriteria optimization are necessary to find physical meaningfull relationships. Furthermore, FSO will only be able to find transferable functions when applied on a wide range of catchments. Consequently, the case where a given relationship is by chance only apparent in a certain catchment will not give too much weight to an unrelated function. It could still be the case that a given function is only applicable in a certain climatic region; however, this would still not result in a physical unrelated transfer function. We hope that this study can be a methodological starting point to further enhance data‐driven methods for process‐based hydrology.6So far, the FSO performance has been purely evaluated in a synthetic setting, where all transfer functions, resulting parameter fields, as well as discharge and internal states are precisely known for any location and time step. Such a setting allows to test the principle functioning of the method and to explore potentials and limitations. In future work, the FSO method as presented here will be applied to more complex hydrological models using also real runoff data and spatial observations of relevant hydrological variables. This will provide further insight in the predictive capabilities of the FSO and the difficulties of estimating transfer functions in a real‐world setting. The application of the FSO to more physically based hydrological models will explore the potential of the FSO to identify and extract well‐known transfer function concepts as have been developed in the area of soil physics. The extension of the FSO application to larger regions will allow testing the transferability of derived model parameter transfer functions to “unkown” regions. The larger spectrum of geophysical properties will enable a more robust estimation of transfer functions and reduce the uncertainties in derived parameter fields.However, an application of the FSO to larger regions with different climate conditions and land surface characteristics will potentially require more complex process descriptions, with more transfer functions to be estimated. In this context, it will be interesting to evaluate what type of additional spatially distributed data  to benchmark our results.Supporting Information Figure S1Click here for additional data file.


The validated model was then used to estimate the diagnostic threshold of iFR. The model determined a quadratic relationship between cFFR and the ciFR. The iFR diagnostic threshold was determined to be 0.8910 from a receiver operating characteristic curve that is in the range of 0.89 to 0.9 that is normally reported in clinical studies.In this work, we estimate the diagnostic threshold of the instantaneous wave‐free ratio (iFR) through the use of a one‐dimensional haemodynamic framework. To this end, we first compared the computed fractional flow reserve (cFFR) predicted from a 1D computational framework with invasive clinical measurements. The framework shows excellent promise and utilises minimal patient data from a cohort of 52 patients with a total of 66 stenoses. The diagnostic accuracy of the cFFR model was 75.76 In this work we estimate the diagnostic threshold of iFR through the use of a one‐dimentional haemodynamic model. The 1D model is first validated through a comparison with invasive fractional flow reserve measurements, for which a diagnostic accuracy of 75.76% was achieved. The one‐dimentional model was then utilised to compare cFFR and ciFR on patient‐specific arterial geometries that were extracted via CCTA. The diagnostic threshold of iFR was determined to be 0.891 which is in the normal range reported from invasive clinical studies. The prevalence of CHD creates a significant economic burden because of the direct health care costs, reduced productivity, and informal public aftercare caused by CHD and other cardiovascular diseases (CVD).% of all Americans have a CVD.Coronary heart disease (CHD) is the leading cause of mortality worldwide and has been attributed to approximately 15.5% of patients experience adverse reactions to adenosine, with 8% having such a severe reaction that the adenosine infusion had to be discontinued during the procedure.CHD is caused by a narrowing (stenosis) of a coronary artery and is often from the process of atherosclerosis. The current gold standard of care for assessing the functional significance of a stenotic lesion is fractional flow reserve (FFR).Alternative diagnostic tools for CHD are continually being developed. One such technique is diastolic FFR (dFFR),Another promising alternative to FFR is the instantaneous wave‐free ratio (iFR).Non‐invasive methods for estimating FFR are based on reconstructed and segmented coronary computed tomography angiography (CCTA) images. Techniques from computational fluid dynamics are then applied to the patient geometry to determine the FFR while avoiding the need for any invasive cardiac catheterisation. The technique is often called computed FFR (cFFR) and relies on the estimation of several important modelling components, which includes the patients' cardiovascular response to the hyperaemic drug infusion, which influences the elasticity of arteries, and boundary conditions of the model. Different computational methodologies have been proposed, which include computationally expensive three‐dimensional models,The purpose of this work is to compare clinical FFR measurements with cFFR estimated via a one‐dimensional haemodynamic model in order to validate the cFFR strategy and to determine the diagnostic cut‐off point of iFR through the application of computational models on patient‐specific coronary arterial networks that have been extracted from non‐invasive CCTA. By using the same extracted patient‐specific coronary network geometry, the boundary conditions of the 1D model can be easily adjusted to change from hyperaemic conditions to resting physiological conditions, which essentially adapts the cFFR model into a ciFR model. The correlation between ciFR and cFFR and the clinically accepted FFR diagnostic threshold of 0.8 will then be used in order to estimate the diagnostic threshold for the iFR procedure. An advantage of using this computational framework is that variability of surgical techniques and of the patients' physiological conditions can be eliminated, which allows the iFR diagnostic threshold to be estimated in a more controlled environment. The patient data were collected retrospectively, and no patient specific measurements were used to aid the model parameter estimation.The paper is organised into the following parts: in Section 22.1%) considered in this work are in the intermediate FFR range of 0.7<FFR<0.9, with 68.2% of the total lesions having a negative FFR value and 31.8% having a positive FFR value. A stenosis location of the left anterior descending (LAD) artery was most prevalent (60.6%). Patients who had undergone previous coronary surgical interventions were included only if the intervention was performed within vessels that were not of interest for this study. Patients with serial stenoses and CCTA data images that contained motion artefacts and significant levels of calcification were also included in this study. Table In this study, an anonymised retrospective data set of 52 patients with a total of 66 lesions was collected. The exact clinical FFR values were not available for a subset of 10 of these patients (15 stenosis). Only whether the FFR was positive or negative for these patients was available. An overview of the locations and characteristics of these stenotic lesions can be seen in Table 2.2The segmentation of CCTA images, centreline and vessel geometry extraction, were all performed in the image segmentation software VMTKLab, .2.2.1Ca is the vessel compliance, P is the mean hydrostatic pressure in a cross section, and Q is the volumetric flow rate; and the momentum equation: ρ=1.05g/cm3 is the density of blood, A is the cross‐sectional area of the vessel, μ=0.04Poise is the blood viscosity, and the magnitude of the viscous coefficient of 22 is from Smith et al∂A∂P from the following viscoelastic constitutive law,c0 is calculated from the vessel diameter using the empirical formulaD0 is the reference diameter, and the fitting parameters are k1=20 g/s2/cm, k2=−22.5 cm−1, and k3=86.5 g/s2/cm. The reference pressure P0 is set to equal the diastolic pressure, and the collapsing pressure Pcollapse=−10 mmHg. The wall viscous coefficient ΓD is the vessel lumen diameter.The modelling methodology implemented in this work is described in CarsonThe system of equations are solved using a subdomain collocation scheme2.2.2The inlet and outlet boundary conditions for the model are predicted by using a two‐tiered parameter estimation technique. The first tier utilises a general, nonpatient‐specific, closed‐loop 1D‐0D network to estimate the volumetric inflow rate for the coronary arteries, while the second tier uses the inflow rate determined from the first tier as the coronary inflow boundary condition and then estimates the vascular bed resistances for the patient‐specific coronary arterial network that was extracted from the CCTA image data.2.2.2.1The first‐tier of the parameter estimation utilising the closed‐loop cardiovascular model with an initial and adaptive parameter estimation technique that are described in the previous studies.%, this causes the mean flow rate in the coronary arteries for the hyperaemic condition to be 3.5 times larger than in resting conditions.An additional change from the resting condition to the hyperaemic condition is performed by reducing the total coronary vascular bed resistance by 782.2.2.2i represents the branch (left or right) of the coronary artery network, and Qi is the defined inflow into the coronary branch from the first tier. The resistance of each branch is then distributed to each terminal vessel using the proximal Murray's law with a power of 2.27.The inlet boundary condition for all simulations is a defined flow rate (one for hyperaemic conditions and one for resting conditions) that was generated by the closed‐loop cardiovascular model described in the first parameter estimation tier. The second tier uses the patient‐specific coronary geometry that was extracted via segmentation. The outlet boundary condition is a lumped‐parameter model that includes external pressure from the left and right ventricles2.2.2.3Pp) to a stenosis  and the pressure distal (Pd) to a stenosis. The ratio of Pd/Pp is used for all cases. The main differences between these two methods are as follows: conventional FFR is measured under maximal hyperaemic conditions, which requires a drug such as adenosine to be administered and is the mean of the pressure ratio Pd/Pp over one cardiac cycle ; iFR is performed under resting conditions and is the mean of the pressure ratio Pd/Pp during the wave‐free period, which is shown in Figure th into diastole to the end of the cardiac cycle.Both conventional FFR and iFR are measurements that involve the ratio of the pressure proximal  to determine the optimum threshold of a methodology. Figure 4%, which is in the region seen from other studies; however, the current methodology did not have patient information along with the CCTA data, and hence, a population average blood pressure, heart rate, and cardiac output from published studies%, a specificity of 77.78%, a positive predictive value of 60% and a negative predictive value of 85.37%. Furthermore, the mean difference between cFFR and invasive FFR was −0.015314, which is lower in magnitude than the mean difference shown in the three‐dimensional methodology performed for different patient data in the DISCOVER‐FLOW%) of invasive cFFR measurements were in the range 0.7< FFR <0.9. Overall, the results indicate that the reduced‐order methodology presented here provides a high level of diagnostic accuracy for cFFR.The performance of the reduced‐order cFFR methodology in comparison with the invasive FFR measurements is very satisfactory at this early stage of development and shows a similar level of accuracy to several three‐dimensional cFFR methodologies.The framework presented does need to be tested on a larger cohort in fully blinded conditions and ideally using prospective data. In the current cohort, there are several patients that would have been excluded in other studies because of poor image quality4.1The use of iFR in combination with or instead of FFR has received increased attention in recent years. However, many studies use different threshold values of iFR to determine whether a patient needs further treatment.Another interesting aspect to consider is whether the cardiovascular system is a pressure‐driven or flow‐driven system, which depends on whether the heart is a flow generator or pressure generator.Many of the fundamental assumptions needed for FFR have since been proven incorrect. For example, the pressure and flow are assumed to be directly proportional (have a linear relationship) when the resistance is constant and minimalMany studies compare the diagnostic performance of iFR to FFR and consider the latter index to be the “perfect measure.” This generally means that iFR looks worse than FFR, but this is a rather unfair comparison. Further studies have indicated that iFR and FFR have a similar number of negative cardiovascular events after 1 yearFrom a non‐invasive standpoint for the determination of iFR and FFR, iFR has more advantages over FFR as it does not require an estimation of how the patient will react to the administered hyperaemic inducing drug. Many cardiac and haemodynamic parameters can be measured by non‐invasive means, including cardiac output estimations, brachial artery blood pressures, and heart rate. These parameters can be directly utilised by any non‐invasive iFR model predictions rather than attempting to predict the effects of a hyperaemic condition, which are variable between patients and are required for FFR estimations. In addition, in the majority, the patient CCTA scans are performed at resting conditions. This is particularly important as the inducing of hyperaemic conditions through the use of a drug, such as adenosine, was observed to increase coronary vessel diameters by up to 15%4.2The main limitations of this study are that the size of the cohort is relatively small with 52 patients and a total of 66 stenoses. Furthermore, we assume that there are no pressure losses at vessel junctions and do not have patient data such as age, blood pressures, heart rate, or gender, which may play a role in FFR prediction. We do not have any invasive iFR measurements; only FFR measurements and the patient CCTA images were available. As the data are retrospective, we do not know the exact location of the pressure measurement taken during invasive FFR, as only the general location is normally recorded in the clinic. However, the main target of this study was to determine the diagnostic threshold of iFR in a more controlled environment, and although the cFFR performance of the model is very good, it was not the focus of this paper, and thus, the additional patient data are not of importance here.It has been observed that vessels with coronary artery disease5The diagnostic performance of the cFFR with the invasive FFR measurement is very promising. The methodology not only showed a lower correlation between cFFR and FFR than in other studies but also showed the lowest magnitude of mean difference and one of the lowest standard deviations between the computed and measured FFR when compared with other studies. The lower correlation coefficient may be due to the lower patient numbers considered in this cohort and also on the nonselective nature in this study with regard to image quality such as the presence of significant motion artefacts, blooming artefacts, and high levels of calcification that can have a large impact on the accuracy of the segmentation process. The methodology must now be performed on a significantly larger cohort in a fully blinded fashion and ideally prospectively in order to improve the confidence in the 1D modelling methodology for both FFR and iFR predictions.A comparison between cFFR and computed iFR was also performed by utilising the patient CCTA data. The model predicted an iFR diagnostic cut‐off point of 0.891 with the correlation and polynomial of best fit between cFFR and ciFR being quadratic in nature. Further studies involving comparisons between FFR and iFR must be performed in order to determine whether FFR or iFR is the more reliable measure. There is a significant advantage of iFR, as it does not require a hyperaemic drug infusion which can cause negative side affects in patients and would also be less expensive. When considering a computational methodology for prediction purposes, ciFR is also more attractive as a diagnosis index when compared with cFFR, as the patients non‐invasive measurements can be utilised directly without the need to predict hyperaemic conditions.


Everyone is talking about coronavirus from the last couple of months due to its exponential spread throughout the globe. Lives have become paralyzed, and as many as 180 countries have been so far affected with 928,287 (14 September 2020) deaths within a couple of months. Ironically, 29,185,779 are still active cases. Having seen such a drastic situation, a relatively simple epidemiological SIR model with Caputo derivative is suggested unlike more sophisticated models being proposed nowadays in the current literature. The major aim of the present research study is to look for possibilities and extents to which the SIR model fits the real data for the cases chosen from 1 April to 15 March 2020, Pakistan. To further analyze qualitative behavior of the Caputo SIR model, uniqueness conditions under the Banach contraction principle are discussed and stability analysis with basic reproduction number is investigated using Ulam–Hyers and its generalized version. The best parameters have been obtained via the nonlinear least-squares curve fitting technique. The infectious compartment of the Caputo SIR model fits the real data better than the classical version of the SIR model (Brauer et al. in Mathematical Models in Epidemiology  Literature includes mathematical models for pathways for the transmission of infectious ailments. These models play an important role in quantifying and assessing the efficient control and preventive measures of infectious ailments –4. It haIn December 2019, there was an emergence of a novel ailment in China which was later declared as pandemic named COVID-19 by the World Health Organization . It is wOut of the seven known human COVID-19, four are the known human influenza pathogens. SARS-CoV, MERS-CoV, and 2019-nCoV are responsible for severe respiratory diseases . While COver the last few decades, many scientists have shown that the fractional models can more accurately explain natural phenomena than the differential equations of the integer order. Because of this advantage, the fractional calculus has taken on the importance and popularity of modeling realistic cases, especially those with memory effects –23. Furtβ) between the infectives and the susceptibles. Similarly, the number of infectives will increase due to contact between people and decrease because of people either recovering or dying as a result of the disease. Finally, the removed category includes people that no longer can catch the disease either because they have recovered or died, and this is going to increase at the constant rate (recovery rate =γ) depending on how many infectives there are. Next, we need some initial data before we can solve the system of differential equations. Therefore,Although there have been various complex models proposed recently for transmission dynamics of COVID-19 as discussed in Sect. quations  that areν in the power of biological parameters β and γ:Having been inspired by a plethora of research works carried out in fractional mathematical epidemiology  to relativelycompact subsets). Additionally, ∃ a constantsuch thatand eachwhich is an equivalence of .  which isP. Moreover, take Take into account tion for  represenon rule,  possesseNow, we want show the existence of solutions of  by meansTakeas a bounded, convex, and closed subset of a Banach spaceConsideras operators in obedience of the following: whenis continuous and compact;is a contraction mapping.Thenso thatSurmise thatis continuous and satisfies conditionAdditionally, take into account thatThen (ii)A function atisfies  if and oSurmise thatholds for  of mentclass2pt{minimSurmise thatis continuous ∀ and assumptionholds withThen  of model (http://covid.gov.pk/stats/pakistan). Daily cases of the disease are taken from 1 April to 15 May 2020 when the present research study was being conceived and prepared.A disease model like the one being investigated in the present research study is widely accepted and validated if its simulations for infectious compartment agree well with those of available real cases of the disease. In order to do so, one needs to obtain values of the parameters of the model in a best possible way so that they cause the model to be in good agreement with the available data. This brings into attention a few existing methods for getting best values of such parameters. The Bayesian technique, probability plotting, maximum likelihood estimation, and least-squares method are among the methods that exist for parameter estimation. In this study, we have used the least-squares method to compute the parameters (of model . Since tn points (data pairs) m adjustable parameters are kept in the vector p. The goal is to notice the parameters for the model that fit the data in a best possible way. Such a fit of a model to a data point is measured by its residual, defined as the difference between the real available value of the dependent variable and the value predicted by the model:ν which is 8.561591e-01. Hence, the basic reproduction number is obtained as In the least-squares approach, the objective function requires to be minimized. The objective is based upon adjusting the parameters for a model function to fit a data set in a best possible way. A simple data set consists of mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimIn this portion, the concept of sensitivity analysis is employed to discover the robust significance of the generic parameters that are present in the basic reproduction number  follows :23\docuor model  under coβ, while that of negative relations is γ. A negative relationship indicates that an increase in this parameter’s value would help to reduce the brutality of the disease. While a positive relationship indicates that an increase in the values of that parameter would have a substantial effect on the frequency of the spread of the ailment. Figure The numerical values showing the relative significance of the entclass1pt{minimaentclass1pt{minimaβ is responsible for the spread of the virus  plot of Fig. This is the place where we get deep insights into the model’s dynamical behavior. The present section offers the numerical simulations of Caputo model  while usν on the dynamical behavior of Caputo model (ν increases, whereas Fig. ν, and this makes sense because the susceptible population, after being infected, is shifted to I compartment thereby leading to an increase therein. It is observed in Fig. ν gets large. Finally, we have also shown the dynamics of the basic reproduction number β and the recovery rate γ in Fig. β brings the reproduction number near to 1, which is clearly an alarming situation for policy makers to devise an effective approach to prevent We have also investigated the effects of fractional order to model  for all ν without violating dimensional consistency between parameters and the operator itself. The proposed Caputo SIR model is used to comprehend the behavior of the devastating disease of COVID-19 that has recently emerged in the entire community of humankind. The model has shown promising results for it is proved to have a unique solution on the basis of the Banach contraction principle. Using Ulam–Hyers and its generalized version, the Caputo model is shown to be stable. Fitted parameters of the model, using real incidence cases of the virus from 1 April to 15 May 2020, are obtained under the least-squares approach wherein the fractional order ν is also optimized to be β and γ relating with Using Caputo differential operator famous to have nonlocal nature, which is perfectly suitable to investigate transmission dynamics of a disease, we have fractionalized the SIR epidemic model having order