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This article is about the family of vitamers. For vitamin K1 the form usually used as a supplement, see Phytomenadione.
Vitamin K structures. MK-4 and MK-7 are both subtypes of K2.
Vitamin K deficiency, Warfarin overdose
Vitamin K is a group of structurally similar, fat-soluble vitamins the human body requires for complete synthesis of certain proteins that are prerequisites for blood coagulation and which the body also needs for controlling binding of calcium in bones and other tissues. The vitamin K-related modification of the proteins allows them to bind calcium ions, which they cannot do otherwise. Without vitamin K, blood coagulation is seriously impaired, and uncontrolled bleeding occurs. Low levels of vitamin K also weaken bones and promote calcification of arteries and other soft tissues[citation needed].
Chemically, the vitamin K family comprises 2-methyl-1,4-naphthoquinone (3-) derivatives. Vitamin K includes two natural vitamers: vitamin K1 and vitamin K2.[1] Vitamin K2, in turn, consists of a number of related chemical subtypes, with differing lengths of carbon side chains made of isoprenoid groups of atoms.
Vitamin K1, also known as phylloquinone, is made by plants, and is found in highest amounts in green leafy vegetables because it is directly involved in photosynthesis. It may be thought of as the plant form of vitamin K. It is active as a vitamin in animals and performs the classic functions of vitamin K, including its activity in the production of blood-clotting proteins. Animals may also convert it to vitamin K2.
Bacteria in the gut flora can also convert K1 into vitamin K2. In addition, bacteria typically lengthen the isoprenoid side chain of vitamin K2 to produce a range of vitamin K2 forms, most notably the MK-7 to MK-11 homologues of vitamin K2. All forms of K2 other than MK-4 can only be produced by bacteria, which use these forms in anaerobic respiration. The MK-7 and other bacterially derived forms of vitamin K2 exhibit vitamin K activity in animals, but MK-7's extra utility over MK-4, if any, is unclear and is a matter of investigation.
Three synthetic types of vitamin K are known: vitamins K3, K4, and K5. Although the natural K1 and all K2 homologues and synthetic K4 and K5 have proven nontoxic, the synthetic form K3 (menadione) has shown toxicity.[2]
1.2 Cardiovascular health
1.4 Coumarin poisoning
4.1 Conversion of vitamin K1 to vitamin K2
4.2 Vitamin K2
6 Absorption and dietary need
7 Dietary reference intake
10 Biochemistry
10.1 Function in animals
10.2 Gamma-carboxyglutamate proteins
10.3 Methods of assessment
10.4 Function in bacteria
11 Injection in newborns
11.3 Controversy
A review of 2014 concluded that there is positive evidence that monotherapy using MK-4, one of the forms of Vitamin K2, reduces fracture incidence in post-menopausal women with osteoporosis, and suggested further research on the combined use of MK-4 with bisphosphonates. In contrast, an earlier review article of 2013 concluded that there is no good evidence that vitamin K supplementation helps prevent osteoporosis or fractures in postmenopausal women.[3]
A Cochrane systematic review of 2006 suggested that supplementation with Vitamin K1 and with MK4 reduces bone loss; in particular, a strong effect of MK-4 on incident fractures among Japanese patients was emphasized.[4]
A review article of 2016 suggested to consider, as one of several measures for bone health, increasing the intake of foods rich in vitamins K1 and K2.[5]
Cardiovascular health[edit]
Adequate intake of vitamin K is associated with the inhibition of arterial calcification and stiffening,[6] but there have been few interventional studies and no good evidence that vitamin K supplementation is of any benefit in the primary prevention of cardiovascular disease.[7]
One 10-year population study, the Rotterdam Study, did show a clear and significant inverse relationship between the highest intake levels of menaquinone (mainly MK-4 from eggs and meat, and MK-8 and MK-9 from cheese) and cardiovascular disease and all-cause mortality in older men and women.[8]
Vitamin K has been promoted in supplement form with claims it can slow tumor growth; there is however no good medical evidence that supports such claims.[9]
Coumarin poisoning[edit]
Vitamin K is part of the suggested treatment regime for poisoning by rodenticide (coumarin poisoning).[10]
Although allergic reaction from supplementation is possible, no known toxicity is associated with high doses of the phylloquinone (vitamin K1) or menaquinone (vitamin K2) forms of vitamin K, so no tolerable upper intake level (UL) has been set.[11]
Blood clotting (coagulation) studies in humans using 45 mg per day of vitamin K2 (as MK-4)[12] and even up to 135 mg per day (45 mg three times daily) of K2 (as MK-4),[13] showed no increase in blood clot risk. Even doses in rats as high as 250 mg/kg, body weight did not alter the tendency for blood-clot formation to occur.[14]
Unlike the safe natural forms of vitamin K1 and vitamin K2 and their various isomers, a synthetic form of vitamin K, vitamin K3 (menadione), is demonstrably toxic at high levels. The U.S. FDA has banned this form from over-the-counter sale in the United States because large doses have been shown to cause allergic reactions, hemolytic anemia, and cytotoxicity in liver cells.[2]
Phylloquinone (K1)[15][16] or menaquinone (K2) are capable of reversing the anticoagulant activity of the anticoagulant warfarin (tradename Coumadin). Warfarin works by blocking recycling of vitamin K, so that the body and tissues have lower levels of active vitamin K, and thus a deficiency of vitamin K.
Supplemental vitamin K (for which oral dosing is often more active than injectable dosing in human adults) reverses the vitamin K deficiency caused by warfarin, and therefore reduces the intended anticoagulant action of warfarin and related drugs.[17] Sometimes small amounts of vitamin K are given orally to patients taking warfarin so that the action of the drug is more predictable.[17] The proper anticoagulant action of the drug is a function of vitamin K intake and drug dose, and due to differing absorption must be individualized for each patient.[citation needed] The action of warfarin and vitamin K both require two to five days after dosing to have maximum effect, and neither warfarin or vitamin K shows much effect in the first 24 hours after they are given.[18]
The newer anticoagulants dabigatran and rivaroxaban have different mechanisms of action that do not interact with vitamin K, and may be taken with supplemental vitamin K.[19][20]
Vitamin K2 (menaquinone). In menaquinone, the side chain is composed of a varying number of isoprenoid residues. The most common number of these residues is four, since animal enzymes normally produce menaquinone-4 from plant phylloquinone.
A sample of phytomenadione for injection, also called phylloquinone
The three synthetic forms of vitamin K are vitamins K3 (menadione), K4, and K5, which are used in many areas, including the pet food industry (vitamin K3) and to inhibit fungal growth (vitamin K5).[21]
Conversion of vitamin K1 to vitamin K2[edit]
Vitamin K1 (phylloquinone) – both forms of the vitamin contain a functional naphthoquinone ring and an aliphatic side chain. Phylloquinone has a phytyl side chain.
The MK-4 form of vitamin K2 is produced by conversion of vitamin K1 in the testes, pancreas, and arterial walls.[22] While major questions still surround the biochemical pathway for this transformation, the conversion is not dependent on gut bacteria, as it occurs in germ-free rats[23][24] and in parenterally-administered K1 in rats.[25][26] In fact, tissues that accumulate high amounts of MK-4 have a remarkable capacity to convert up to 90% of the available K1 into MK-4.[27][28] There is evidence that the conversion proceeds by removal of the phytyl tail of K1 to produce menadione as an intermediate, which is then condensed with an activated geranylgeranyl moiety (see also prenylation) to produce vitamin K2 in the MK-4 (menatetrione) form.[29]
Vitamin K2[edit]
Main article: Vitamin K2
Vitamin K2 (menaquinone) includes several subtypes. The two subtypes most studied are menaquinone-4 (menatetrenone, MK-4) and menaquinone-7 (MK-7).
Vitamin K1, the precursor of most vitamin K in nature, is a stereoisomer of phylloquinone, an important chemical in green plants, where it functions as an electron acceptor in photosystem I during photosynthesis. For this reason, vitamin K1 is found in large quantities in the photosynthetic tissues of plants (green leaves, and dark green leafy vegetables such as romaine lettuce, kale and spinach), but it occurs in far smaller quantities in other plant tissues (roots, fruits, etc.). Iceberg lettuce contains relatively little. The function of phylloquinone in plants appears to have no resemblance to its later metabolic and biochemical function (as "vitamin K") in animals, where it performs a completely different biochemical reaction.
Vitamin K (in animals) is involved in the carboxylation of certain glutamate residues in proteins to form gamma-carboxyglutamate (Gla) residues. The modified residues are often (but not always) situated within specific protein domains called Gla domains. Gla residues are usually involved in binding calcium, and are essential for the biological activity of all known Gla proteins.[30]
At this time[update], 17 human proteins with Gla domains have been discovered, and they play key roles in the regulation of three physiological processes:
Blood coagulation: prothrombin (factor II), factors VII, IX, and X, and proteins C, S, and Z[31]
Bone metabolism: osteocalcin, also called bone Gla protein (BGP), matrix Gla protein (MGP),[32] periostin,[33] and the recently discovered Gla-rich protein (GRP).[34][35]
Vascular biology: growth arrest-specific protein 6 (Gas6)[36]
Unknown function: proline-rich γ-carboxyglutamyl proteins (PRGPs) 1 and 2, and transmembrane γ-carboxy glutamyl proteins (TMGs) 3 and 4.[37]
Like other lipid-soluble vitamins (A, D and E), vitamin K is stored in the fatty tissue of the human body.
Absorption and dietary need[edit]
Previous theory held that dietary deficiency is extremely rare unless the small intestine was heavily damaged, resulting in malabsorption of the molecule. Another at-risk group for deficiency were those subject to decreased production of K2 by normal intestinal microbiota, as seen in broad spectrum antibiotic use.[38] Taking broad-spectrum antibiotics can reduce vitamin K production in the gut by nearly 74% in people compared with those not taking these antibiotics.[39] Diets low in vitamin K also decrease the body's vitamin K concentration.[40] Those with chronic kidney disease are at risk for vitamin K deficiency, as well as vitamin D deficiency, and particularly those with the apoE4 genotype.[41] Additionally, in the elderly there is a reduction in vitamin K2 production.[42]
The National Academy of Medicine (NAM) updated an estimate of what constitutes an adequate intake (AI) for vitamin K in 2001. The NAM does not distinguish between K1 and K2 – both are counted as vitamin K. At that time there was not sufficient evidence to set the more rigorous estimated average requirement (EAR) or recommended dietary allowance (RDA) given for most of the essential vitamins and minerals. The current daily AIs for vitamin K for adult women and men are 90 μg and 120 μg respectively. The AI for pregnancy and lactation is 90 μg. For infants up to 12 months the AI is 2–2.5 μg, and for children aged 1 to 18 years the AI increases with age from 30 to 75 μg. As for safety, the FNB also sets tolerable upper intake levels (known as ULs) for vitamins and minerals when evidence is sufficient. In the case of vitamin K no UL is set, as evidence for adverse effects is not sufficient. Collectively EARs, RDAs, AIs and ULs are referred to as dietary reference intakes.[43] The European Food Safety Authority reviewed the same safety question and did not set an UL.[44]
For U.S. food and dietary supplement labeling purposes, the amount in a serving is expressed as a percentage of daily value (%DV). For vitamin K labeling purposes the daily value was 80 μg, but as of May 2016 it has been revised upwards to 120 μg. A table of the pre-change adult daily values is provided at reference daily intake. Food and supplement companies have until 28 July 2018 to comply with the change.
See also: Vitamin K2 § Dietary sources
K1 (μg)[45]
Kale, cooked
Collards, cooked
Collards, raw
Swiss chard, cooked
Swiss chard, raw
Turnip greens, raw
Romaine lettuce, raw
Table from "Important information to know when you are taking: Warfarin (Coumadin) and Vitamin K", Clinical Center, National Institutes of Health Drug Nutrient Interaction Task Force.[46]
Vitamin K1 is found chiefly in leafy green vegetables such as dandelion greens (which contain 778.4 μg per 100 g, or 741% of the recommended daily amount), spinach, swiss chard, lettuce and Brassica vegetables (such as cabbage, kale, cauliflower, broccoli, and brussels sprouts) and often the absorption is greater when accompanied by fats such as butter or oils; some fruits, such as avocados, kiwifruit and grapes, are also high in vitamin K. By way of reference, two tablespoons of parsley contains 153% of the recommended daily amount of vitamin K.[47] Some vegetable oils, notably soybean oil, contain vitamin K, but at levels that would require relatively large calorie consumption to meet the USDA-recommended levels.[48] colonic bacteria synthesize a significant portion of humans' vitamin K needs; newborns often receive a vitamin K shot at birth to tide them over until their colons become colonized at five to seven days of age from the consumption of breast milk.
The tight binding of vitamin K1 to thylakoid membranes in chloroplasts makes it less bioavailable. For example, cooked spinach has a 5% bioavailability of phylloquinone, however, fat added to it increases bioavailability to 13% due to the increased solubility of vitamin K in fat.[49]
Main article: Vitamin K deficiency
Average diets are usually not lacking in vitamin K, and primary deficiency is rare in healthy adults. Newborn infants are at an increased risk of deficiency. Other populations with an increased prevalence of vitamin K deficiency include those who suffer from liver damage or disease (e.g. alcoholics), cystic fibrosis, or inflammatory bowel diseases, or have recently had abdominal surgeries. Secondary vitamin K deficiency can occur in people with bulimia, those on stringent diets, and those taking anticoagulants. Other drugs associated with vitamin K deficiency include salicylates, barbiturates, and cefamandole, although the mechanisms are still unknown. Vitamin K1 deficiency can result in coagulopathy, a bleeding disorder.[50]Symptoms of K1 deficiency include anemia, bruising, nosebleeds and bleeding of the gums in both sexes, and heavy menstrual bleeding in women.
Osteoporosis[51][52] and coronary heart disease[53][54] are strongly associated with lower levels of K2 (menaquinone). Vitamin K2 (as menaquinones MK-4 through MK-10) intake level is inversely related to severe aortic calcification and all-cause mortality.[8]
Function in animals[edit]
Mechanism of action of vitamin K1.
The function of vitamin K2 in the animal cell is to add a carboxylic acid functional group to a glutamate (Glu) amino acid residue in a protein, to form a gamma-carboxyglutamate (Gla) residue. This is a somewhat uncommon posttranslational modification of the protein, which is then known as a "Gla protein". The presence of two −COOH (carboxylic acid) groups on the same carbon in the gamma-carboxyglutamate residue allows it to chelate calcium ions. The binding of calcium ions in this way very often triggers the function or binding of Gla-protein enzymes, such as the so-called vitamin K-dependent clotting factors discussed below.
Within the cell, vitamin K undergoes electron reduction to a reduced form called vitamin K hydroquinone, catalyzed by the enzyme vitamin K epoxide reductase (VKOR).[55] Another enzyme then oxidizes vitamin K hydroquinone to allow carboxylation of Glu to Gla; this enzyme is called gamma-glutamyl carboxylase[56][57] or the vitamin K-dependent carboxylase. The carboxylation reaction only proceeds if the carboxylase enzyme is able to oxidize vitamin K hydroquinone to vitamin K epoxide at the same time. The carboxylation and epoxidation reactions are said to be coupled. Vitamin K epoxide is then reconverted to vitamin K by VKOR. The reduction and subsequent reoxidation of vitamin K coupled with carboxylation of Glu is called the vitamin K cycle.[58] Humans are rarely deficient in vitamin K1 because, in part, vitamin K1 is continuously recycled in cells.[59]
Warfarin and other 4-hydroxycoumarins block the action of VKOR.[60] This results in decreased concentrations of vitamin K and vitamin K hydroquinone in tissues, such that the carboxylation reaction catalyzed by the glutamyl carboxylase is inefficient. This results in the production of clotting factors with inadequate Gla. Without Gla on the amino termini of these factors, they no longer bind stably to the blood vessel endothelium and cannot activate clotting to allow formation of a clot during tissue injury. As it is impossible to predict what dose of warfarin will give the desired degree of clotting suppression, warfarin treatment must be carefully monitored to avoid overdose.
Gamma-carboxyglutamate proteins[edit]
Main article: Gla domain
The following human Gla-containing proteins ("Gla proteins") have been characterized to the level of primary structure: blood coagulation factors II (prothrombin), VII, IX, and X, anticoagulant proteins C and S, and the factor X-targeting protein Z. The bone Gla protein osteocalcin, the calcification-inhibiting matrix Gla protein (MGP), the cell growth regulating growth arrest specific gene 6 protein (Gas6), and the four transmembrane Gla proteins (TMGPs), the function of which is at present unknown. Gas6 can function as a growth factor to activate the Axl receptor tyrosine kinase and stimulate cell proliferation or prevent apoptosis in some cells. In all cases in which their function was known, the presence of the Gla residues in these proteins turned out to be essential for functional activity.
Gla proteins are known to occur in a wide variety of vertebrates: mammals, birds, reptiles, and fish. The venom of a number of Australian snakes acts by activating the human blood-clotting system. In some cases, activation is accomplished by snake Gla-containing enzymes that bind to the endothelium of human blood vessels and catalyze the conversion of procoagulant clotting factors into activated ones, leading to unwanted and potentially deadly clotting.
Another interesting class of invertebrate Gla-containing proteins is synthesized by the fish-hunting snail Conus geographus.[61] These snails produce a venom containing hundreds of neuroactive peptides, or conotoxins, which is sufficiently toxic to kill an adult human. Several of the conotoxins contain two to five Gla residues.[62]
Methods of assessment[edit]
Vitamin K status can be assessed by:
The prothrombin time (PT) test measures the time required for blood to clot. A blood sample is mixed with citric acid and put in a fibrometer; delayed clot formation indicates a deficiency. This test is insensitive to mild deficiency, as the values do not change until the concentration of prothrombin in the blood has declined by at least 50%.[63]
Undercarboxylated prothrombin (PIVKA-II); in a study of 53 newborns, found "PT (prothrombin time) is a less sensitive marker than PIVKA II",[64] and as indicated above, PT is unable to detect subclinical deficiencies that can be detected with PIVKA-II testing.
Plasma phylloquinone was found to be positively correlated with phylloquinone intake in elderly British women, but not men,[65] but an article by Schurgers et al. reported no correlation between FFQ[further explanation needed] and plasma phylloquinone.[66]
Urinary γ-carboxyglutamic acid responds to changes in dietary vitamin K intake. Several days are required before any change can be observed. In a study by Booth et al., increases of phylloquinone intakes from 100 μg to between 377 and 417 μg for five days did not induce a significant change. Response may be age-specific.[67]
Undercarboxylated osteocalcin (UcOc) levels have been inversely correlated with stores of vitamin K[68] and bone strength in developing rat tibiae. Another study following 78 post-menopausal Korean women found a supplement regimen of vitamins K and D, and calcium, but not a regimen of vitamin D and calcium, was inversely correlated with reduced UcOc levels.[69]
Function in bacteria[edit]
Many bacteria, such as Escherichia coli found in the large intestine, can synthesize vitamin K2 (menaquinone-7 or MK-7, up to MK-11),[70] but not vitamin K1 (phylloquinone). In these bacteria, menaquinone transfers two electrons between two different small molecules, during oxygen-independent metabolic energy production processes (anaerobic respiration).[71] For example, a small molecule with an excess of electrons (also called an electron donor) such as lactate, formate, or NADH, with the help of an enzyme, passes two electrons to menaquinone. The menaquinone, with the help of another enzyme, then transfers these two electrons to a suitable oxidant, such fumarate or nitrate (also called an electron acceptor). Adding two electrons to fumarate or nitrate converts the molecule to succinate or nitrite plus water, respectively.
Some of these reactions generate a cellular energy source, ATP, in a manner similar to eukaryotic cell aerobic respiration, except the final electron acceptor is not molecular oxygen, but fumarate or nitrate. In aerobic respiration, the final oxidant is molecular oxygen (O2), which accepts four electrons from an electron donor such as NADH to be converted to water. E. coli, as facultative anaerobes, can carry out both aerobic respiration and menaquinone-mediated anaerobic respiration.
Injection in newborns[edit]
The blood clotting factors of newborn babies are roughly 30–60% that of adult values; this may be due to the reduced synthesis of precursor proteins and the sterility of their guts. Human milk contains 1–4 μg/L of vitamin K1, while formula-derived milk can contain up to 100 μg/L in supplemented formulas. Vitamin K2 concentrations in human milk appear to be much lower than those of vitamin K1. Occurrence of vitamin K deficiency bleeding in the first week of the infant's life is estimated at 0.25–1.7%, with a prevalence of 2–10 cases per 100,000 births.[72] Premature babies have even lower levels of the vitamin, so they are at a higher risk from this deficiency.
Bleeding in infants due to vitamin K deficiency can be severe, leading to hospitalization, blood transfusions, brain damage, and death. Supplementation can prevent most cases of vitamin K deficiency bleeding in the newborn. Intramuscular administration is more effective in preventing late vitamin K deficiency bleeding than oral administration.[73][74]
As a result of the occurrences of vitamin K deficiency bleeding, the Committee on Nutrition of the American Academy of Pediatrics has recommended 0.5–1 mg of vitamin K1 be administered to all newborns shortly after birth.[74]
In the UK vitamin K supplementation is recommended for all newborns within the first 24 hours.[75] This is usually given as a single intramuscular injection of 1 mg shortly after birth but as a second-line option can be given by three oral doses over the first month.[76]
Controversy arose in the early 1990s regarding this practice, when two studies suggested a relationship between parenteral administration of vitamin K and childhood cancer,[77] however, poor methods and small sample sizes led to the discrediting of these studies, and a review of the evidence published in 2000 by Ross and Davies found no link between the two.[78] Doctors reported emerging concerns in 2013,[79] after treating children for serious bleeding problems. They cited lack-of newborn vitamin K administration, as the reason that the problems occurred, and recommended that breastfed babies could have an increased risk unless they receive a preventative dose.
In the early 1930s, Danish scientist Henrik Dam investigated the role of cholesterol by feeding chickens a cholesterol-depleted diet.[80] He initially replicated experiments reported by scientists at the Ontario Agricultural College (OAC).[81] McFarlane, Graham and Richardson, working on the chick feed program at OAC, had used chloroform to remove all fat from chick chow. They noticed that chicks fed only fat-depleted chow developed hemorrhages and started bleeding from tag sites.[82] Dam found that these defects could not be restored by adding purified cholesterol to the diet. It appeared that – together with the cholesterol – a second compound had been extracted from the food, and this compound was called the coagulation vitamin. The new vitamin received the letter K because the initial discoveries were reported in a German journal, in which it was designated as Koagulationsvitamin. Edward Adelbert Doisy of Saint Louis University did much of the research that led to the discovery of the structure and chemical nature of vitamin K.[83] Dam and Doisy shared the 1943 Nobel Prize for medicine for their work on vitamin K (K1 and K2) published in 1939. Several laboratories synthesized the compound(s) in 1939.[84]
For several decades, the vitamin K-deficient chick model was the only method of quantifying vitamin K in various foods: the chicks were made vitamin K-deficient and subsequently fed with known amounts of vitamin K-containing food. The extent to which blood coagulation was restored by the diet was taken as a measure for its vitamin K content. Three groups of physicians independently found this: Biochemical Institute, University of Copenhagen (Dam and Johannes Glavind), University of Iowa Department of Pathology (Emory Warner, Kenneth Brinkhous, and Harry Pratt Smith), and the Mayo Clinic (Hugh Butt, Albert Snell, and Arnold Osterberg).[85]
The first published report of successful treatment with vitamin K of life-threatening hemorrhage in a jaundiced patient with prothrombin deficiency was made in 1938 by Smith, Warner, and Brinkhous.[86]
The precise function of vitamin K was not discovered until 1974, when three laboratories (Stenflo et al.,[87] Nelsestuen et al.,[88] and Magnusson et al.[89]) isolated the vitamin K-dependent coagulation factor prothrombin (factor II) from cows that received a high dose of a vitamin K antagonist, warfarin. It was shown that, while warfarin-treated cows had a form of prothrombin that contained 10 glutamate (Glu) amino acid residues near the amino terminus of this protein, the normal (untreated) cows contained 10 unusual residues that were chemically identified as γ-carboxyglutamate (Gla). The extra carboxyl group in Gla made clear that vitamin K plays a role in a carboxylation reaction during which Glu is converted into Gla.
The biochemistry of how vitamin K is used to convert Glu to Gla has been elucidated over the past thirty years in academic laboratories throughout the world.
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BMJ (Clinical Research Edition). 316 (7126): 189–193. doi:10.1136/bmj.316.7126.189. PMC 2665412. PMID 9468683. ^ McMillan, D. D. (1997). "Routine administration of vitamin K to newborns". Paediatric Child Health. 2 (6): 429–431. ^ "Newborns get rare disorder after parents refused shots". Having four cases since February just at Vanderbilt was a little bit concerning to me ^ Dam, C. P. H. (1935). "The Antihaemorrhagic Vitamin of the Chick: Occurrence And Chemical Nature". Nature. 135 (3417): 652–653. doi:10.1038/135652b0. ^ Dam, C. P. H. (1941). "The discovery of vitamin K, its biological functions and therapeutical application" (PDF). Nobel Prize Laureate Lecture. ^ McAlister, V. C. (2006). "Control of coagulation: a gift of Canadian agriculture" (PDF). Clinical and Investigative Medicine. 29 (6): 373–377. ^ MacCorquodale, D. W.; Binkley, S. B.; Thayer, S. A.; Doisy, E. A. (1939). "On the constitution of Vitamin K1". 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Identification of gamma-carboxyglutamic acid as a component of prothrombin" (PDF). Journal of Biological Chemistry. 249 (19): 6347–6350. PMID 4214105. ^ Magnusson, S.; Sottrup-Jensen, L.; Petersen, T. E.; Morris, H. R.; Dell, A. (Aug 1974). "Primary structure of the vitamin K-dependent part of prothrombin". FEBS Letters. 44 (2): 189–193. doi:10.1016/0014-5793(74)80723-4. PMID 4472513. Bibliography[edit]
Rhéaume-Bleue, Kate (2012). Vitamin K2 and the Calcium Paradox. John Wiley & Sons, Canada. ISBN 1-118-06572-7. External links[edit]
"Vitamin K: Another Reason to Eat Your Greens". v
TPP / ThDP (B1)
FMN, FAD (B2)
NAD+, NADH, NADP+, NADPH (B3)
Coenzyme A (B5)
PLP / P5P (B6)
THFA / H4FA, DHFA / H2FA, MTHF (B9)
AdoCbl, MeCbl (B12)
Phylloquinone (K1), Menaquinone (K2)
non-vitamins
Coenzyme B
Heme / Haem (A, B, C, O)
Molybdopterin/Molybdenum cofactor
THMPT / H4MPT
Fe2+, Fe3+
vitamins: see vitamins
Antihemorrhagics (B02)
(coagulation)
Phytomenadione (K1)
Menadione (K3)
intrinsic: IX/Nonacog alfa
VIII/Moroctocog alfa/Turoctocog alfa
extrinsic: VII/Eptacog alfa
common: X
II/Thrombin
I/Fibrinogen
XIII/Catridecacog
combinations: Prothrombin complex concentrate (II, VII, IX, X, protein C and S)
Carbazochrome
thrombopoietin receptor agonist (Romiplostim
Eltrombopag)
Tetragalacturonic acid hydroxymethylester
Epinephrine/Adrenalone
amino acids (Aminocaproic acid
Aminomethylbenzoic acid)
serpins (Aprotinin
Alfa1 antitrypsin
Camostat).

Answer: 90 μg for women and 120 μg for men.


Input: What are the benefits of using binary variables in the SLAS formulation?

Context:
Paper Info

Title: SLAS: Speed and Lane Advisory System for Highway Navigation
Publish Date: Unkown
Author List: Faizan Tariq, David Isele, John Baras, Sangjae Bae

Figure

Fig. 1.Motivational Example.With a slow moving vehicle ahead, the ego vehicle (in blue) may decide to either change lane to the fast moving lane (left) to minimize travel time or adjust its speed without changing lanes to preserve safety but it would be unwise for it to switch to the slow moving lane (right) as that would not benefit travel time or safety.
Fig. 3. Simulation Setup.Scenario Runner sets up the scenario for the CARLA Simulator, which then communicates with the SLAS and the Planning and Control ROS (Robot Operating System) nodes through the ROS bridge node.
Fig. 4. Testing scenario with three lanes: lane 0 (left), lane 1 (center) and lane 2 (right).The expected motion of the ego vehicle, over the course of the simulation, is shown with numbered frames.The right most lane (lane 3) is reserved for merging traffic so it is not utilized in our simulation.
Fig. 5. Left: Travel time comparison.Center: Lane choice (lateral position) comparison.The center lines of lanes 0 (left), 1 (center) and 2 (right) have fixed lateral displacements of 0m, 3.5m and 7m respectively.Right: Headway comparison.With no leading vehicle, the headway is restricted by the visibility range of 50m.

abstract

This paper proposes a hierarchical autonomous vehicle navigation architecture, composed of a high-level speed and lane advisory system (SLAS) coupled with low-level trajectory generation and trajectory following modules. Specifically, we target a multi-lane highway driving scenario where an autonomous ego vehicle navigates in traffic.
We propose a novel receding horizon mixed-integer optimization based method for SLAS with the objective to minimize travel time while accounting for passenger comfort. We further incorporate various modifications in the proposed approach to improve the overall computational efficiency and achieve real-time performance.
We demonstrate the efficacy of the proposed approach in contrast to the existing methods, when applied in conjunction with state-of-the-art trajectory generation and trajectory following frameworks, in a CARLA simulation environment.

INTRODUCTION

Lane changing is considered to be one of the most risky driving behaviors since it is highly contingent upon multimodal trajectory predictions of neighboring vehicles and requires timely decision making . It is further influenced by a number of uncertainty factors such as road conditions, measurement accuracy, and a long tail of behavioral uncertainty of on-road agents.
However, if executed efficiently, lane changing coupled with speed adjustment can yield significant improvement in minimizing overall travel time while ensuring passenger comfort . To elaborate further, consider the scenario presented in Fig. . Based on the predicted motion (shown in a lighter shade) of the neighboring vehicles (shown in orange), the ego vehicle (shown in blue) may decide to either change lane left in an attempt to minimize its travel time or slow down in the current lane to maintain safety.
However, it would be imprudent for the ego vehicle to risk changing lane right and consequently get stuck behind a slow moving vehicle even though there is presently a greater headway. This simple scenario highlights the importance of foresight and long planning-horizon in strategic decision making for autonomous vehicles.
Existing methods like MOBIL give us the ability to change lanes but behave greedily (prioritizing immediate rewards) oftentimes, which can lead to sub-optimal performance. It was shown in that the lane changing performance can be improved with an A inspired approach, but the formulation was limited to constant speed.
Such an approach is unable to assess the benefits of speed adjustment 1 University of Maryland, College Park, MD, USA. Email: {mftariq,baras}@umd.edu. 2 Honda Research Institute, San Jose, CA, USA. Email: {disele,sbae}@honda-ri.com. Research supported by Honda Research Institute, USA.  in minimizing overall travel time.
As will become apparent in Section IV, it may be necessary at times to sacrifice on shortterm benefits to gain long-term performance improvements. In such a scenario, an approach with speed adjustment coupled with long planning horizon has the foresight to deliver significantly better results. Moreover, the inclusion of speed adjustment in the decision making process inhibits the risk of incurring trajectory infeasibility as the environment conditions may prevent the ego vehicle from traveling at a constant reference speed and the low-level planner may be unable to handle such a discrepancy.
Therefore, in this work, we propose a low complexity receding horizon optimization based approach that outputs the lane change maneuvers coupled with speed adjustments for long planning horizons (> 15s) while guaranteeing safety. The long horizon strategic decision making gives ego vehicle the ability to proactively anticipate and handle challenging driving situations.
Literature review: In the literature, speed and lane changing decisions are generally considered from a motion planner's perspective , which allows for a simultaneous determination of target lanes and waypoints to perform the maneuver. The motion planning methods present in the literature can broadly be categorized into sampling-based, learning-based and optimization-based approaches.
In regards to the sampling-based approaches, single-query methods, in particular the different variants of RRT, are preferred over multi-query methods, like roadmap-based methods, due to the faster execution time and their ability to incorporate non-holonomic constraints . Even though these methods are able to incorporate safety guarantees by sampling feasible trajectories from a reachable safe set , the overall driving experience is often rather uncomfortable due to the concatenation of individual trajectories.
Moreover, the asymptotic optimality guarantees availed by arXiv:2303.00861v1 [cs.RO] 1 Mar 2023 these methods do not help with real-world implementation in complex driving scenarios since they tend to have high sample complexity . In terms of the learning-based methods, the preferred approach seems to be the variations of Reinforcement Learning techniques applied in a simulated environment , , , , .
These approaches, although seeming to work well in simulation, have concerns regarding real-world implementation due to the large amount of training data that they require, the exploration of unsafe behaviors during training, and a general inability to handle edge cases. They mainly utilize neural networks as function approximators which yields low computational complexity but also results in a lack of explainability and safety guarantees.
Lastly, the optimization-based approaches, especially the derivatives of optimal control methods, are abundant in the literature. In contrast to the potential-field based approaches that yield decent collision avoidance performance but are unable to accommodate vehicle dynamics, the optimal control methods , especially the derivatives of Model Predictive Control (MPC) approach , , , yield excellent collision avoidance performance while accommodating vehicle dynamics.
However, this performance comes at a cost of high computational complexity, arising mainly from the non-convex collision avoidance and the non-linear dynamics constraints. This, in turn, restricts the planning horizon to merely a few seconds. The key requirements for the algorithmic design of an autonomous vehicle include real-time operation, safety guarantees, optimality with respect to some metric(s), and accounting for the behavior variability of on-road agents.
Considering these requirements, we propose an optimization-based behavioral planning framework that enables autonomous vehicle maneuvering on multilane highways. While having the benefits of optimizationbased approaches, our method achieves a low computational complexity by employing a binary representation of the decoupled lane indicator dynamics in lieu of lateral dynamics, and utilizing algorithmic modifications to aid numerical computations.
Specifically, our method provides: • optimality with respect to travel time and comfort; • safety and feasibility guarantees; • real-time applicability for a long planning horizon; and • modularity in design, which enables the integration of external trajectory prediction modules. The proposed method fills in the research gap by meeting all the key algorithmic requirements while simultaneously gaining the foresight to make strategic decisions that yield long-term performance benefits, as verified in Section IV.
In this section, we present the algorithmic pipeline and formalize the road, observation and vehicle dynamics models that will be utilized in the subsequent sections.

Algorithmic Pipeline

Fig. illustrates the algorithmic pipeline of the proposed navigation architecture, in reference to the various existing . Algorithmic pipeline of the proposed navigation architecture. The raw sensory input data is processed by the Perception, and Simultaneous Localization and Mapping (SLAM) modules to place the autonomous vehicle relative to the various environmental entities in a unified frame of reference.
This information is then passed on to the navigation stack, composed of the behavioral planning, motion planning, and vehicle control modules. The output of the navigation module is passed down further in terms of actuation commands (brake, throttle and steering) to the actuators. algorithmic modules deployed on an autonomous vehicle.
The taxonomy of the various components of the navigation stack (highlighted by the dotted rectangle) is borrowed from . This pipeline essentially improves the pipeline introduced in by adding a speed advisory system. Our main focus is the development of the behavior planning module, highlighted as SLAS in Fig. . SLAS outputs the target lane and reference speed which are utilized by the motion planning module to generate a reference trajectory for the ego vehicle.
The vehicle controllers compute the throttle and steering commands to track the trajectory accordingly. For the motion planning module, we adopt the Neural Networks integrated Model Predictive Control (NNMPC) due to its ability to accommodate the behaviors of neighboring vehicles in the trajectory generation process.
In our approach, we assume that the perception (of other vehicles) and the localization (of ego vehicle) are known without any uncertainty, for simplicity, but the modular architecture avails us the ability to integrate any perception or SLAM module in the overall framework. Throughout the manuscript, Z will denote the set of integers and R the set of real numbers.
For some a, c ∈ Z and a < c, we will write For some e, g ∈ R and e < g, we will write

Road Model

The physical road structure is modeled as a continuous multi-lane highway with negligible curvature and unidirectional traffic flow. The lanes on the highway are clearly demarcated and at any given time k, the number of available lanes for the vehicles to travel on is denoted by N l (k) while the road speed limit is denoted by V l .
Therefore, the set of lanes available for traveling at a given time instant k is denoted by . We work with the Frenet coordinate system where the distance along the road is denoted by the longitudinal displacement (s) and the distance perpendicular to the road is defined by the lateral displacement (d).
Each lane is assigned a lane indicator variable l. The leftmost lane, with respect to the direction of traffic flow, is assigned a value of l = 0 while each subsequent lane is assigned an increasing integer value for l, as depicted in Fig. .

Vehicle Model

Since we aim to have real-time computations for a long planning horizon (> 15s), we model the vehicle dynamics with a linearized decoupled dynamical system. For the highway driving scenario, where the road curvature is typically small, it is reasonable to assume a decoupling between the lateral and the longitudinal dynamics , especially for the behavior planning layer.
Therefore, we utilize a linear constant acceleration model for the longitudinal dynamics and abstract out the lateral dynamics with a lane indicator variable. For the lane change dynamics, we use a moving average filter coupled with a rounding function to model the time required by the ego vehicle to change lanes.
This is compactly represented as: where s 0 (k), v 0 (k), l 0 (k) and L(k) denote the ego vehicle's longitudinal displacement, speed, lane indicator and target lane, respectively, at time instant k; the subscript i indexes the vehicles on the road with 0 being reserved for the ego vehicle; T s denotes the discretization time step; and N corresponds to the number of time steps required to change lane.
The state (x 0 (k)) and control input (u 0 (k)) to the system at time instant k are defined as: where V m denotes the maximum speed of the ego vehicle.

Observation Model

For practical considerations, we restrict the ego vehicle's visibility range to the sensory perception limit, denoted by R v . Then, the set of vehicles in ego vehicle's visibility range at time instant k, represented by O(k), is defined as: where s i (k) corresponds to the longitudinal displacement of the observed vehicle.
Remark 1: For the multi-lane highway driving scenario, occlusion does not play a prominent role so we do not account for it in the existing formulation. However, the proposed framework can easily accommodate occlusion and measurement uncertainties since the receding horizon approach bases its decision on the most up-to-date information available at any given time, as demonstrated in .
In this section, we describe the prediction model to generate the predicted future trajectories of observed vehicles and present a discussion on the proposed receding horizon optimization-based behavioral planning module.

Trajectory Prediction

Reliable behavior and trajectory prediction of other traffic participants is crucial for safe maneuvering of autonomous vehicles. The algorithm proposed in Section III-B is able to incorporate any generic prediction module available in the literature as long as it can provide a deterministic predicted future trajectory for a given vehicle.
In this work, we formulate a low-complexity prediction model that highlights the flexibility and efficiency of our proposed approach. For an observed vehicle i ∈ O(k), the future speed profile is predicted using a piece-wise linear function while the lane profile is assumed to stay constant for the duration of the prediction horizon.
At a given time step k, the estimated acceleration (ā k i ) and the estimated speed (v k i ) parameters are obtained through linear regression with mean-squared error on the past o k i > 1 speed observations. Based on the estimated parameters, we predict the future speed and longitudinal displacement as follows:
Here, H a corresponds to the acceleration horizon while vk i (j) and ŝk i (j) respectively represent the predicted speed and longitudinal displacement for vehicle i, j time steps into the future starting from the current time instant k. Remark 2: Due to the modular nature of the proposed framework, the behavior planning module detailed in Section III-B can work with advanced maneuver-based (e.g.
Markov Chain ) and interaction-based (e.g. Social Generative Adversarial Networks ) trajectory prediction modules, allowing for interactive maneuvering behaviors.

Speed and Lane Advisory System

The goal of our behavior planning module, Speed and Lane Advisory System or in short, SLAS, is to determine a sequence of speed and lane change commands that would enable the ego vehicle to maximize its speed, thus minimizing the travel time, while accounting for driver comfort and abiding by its dynamical, actuator, and safety limits.
The output of this module is a relatively smooth speed and lane change profile which is then passed on to a motion planner. It is necessary to incorporate the dynamical and actuator limits in the behavioral planning module so as not to provide the motion planner with goals that are not reachable, and jeopardize the safety of the overall system as a result.
In the subsequent discussion, we provide a formulation of the optimization problem for SLAS; highlight the modifications necessary to improve the computational complexity; and, present safety and feasibility analysis. 1) Optimization Problem with Integer Constraints: SLAS is posed as an optimization problem, with the objective to maximize speed while minimizing frequent lane changes and abrupt changes in speed.
The output of SLAS, at time instant k, is the control input u 0 (k + 1), as defined in . The optimization problem is formulated as follows: Objective Function: In the formulation above, the optimization variables are the ego vehicle's speed (v k (j)) and target lane (L k (j)), j step into the future, starting from time instant k.
Here, H corresponds to the planning horizon. The scalarization parameters γ 1 , γ 2 and γ 3 in the objective function account for a relative tradeoff between maximizing speed, minimizing lane changes and minimizing abrupt changes in speed respectively. Increasing γ 1 yields a more aggressive behavior with the priority placed on maximizing speed while γ 2 and γ 3 combine to place an emphasis on maximizing passenger comfort by reducing lane and speed changes respectively.
Dynamical Constraints: These constraints are put in place to ensure the dynamical feasibility of the solution. The constraints ( ), and serve to initialize the longitudinal displacement, speed and target lane respectively for the optimizer, based on the values observed at time instant k. The constraints and ( ) bound the ego vehicle's speed by the speed limit and the acceleration limits of the vehicle respectively.
The ego vehicle's speed is then used to calculate the projected longitudinal displacement in . The target lane values at any planning step (j) are restricted to the set of reachable values by ( ), ( ) and . Here, restricts the target lane to the set of available lanes (L(k)), ensures that the lane change, if needed, is made to the adjacent lane only and (17) models the time steps (N ) required for a lane change.
The flooring function can easily be transformed into a couple of linear constraints by the introduction of an auxiliary integer variable, as shown in the Appendix. Finally, l k (j) is merely the internal representation of the lane the ego vehicle is projected to travel on at planning step j. Safety Constraint: The safety constraint ensures that the ego vehicle maintains a minimum safe distance (L s i (j)) to the nearest vehicle i, in its projected lane of travel (l k (j)), at planning instant j.
We borrow the definition of this safe distance from , where the authors provide a formalization, based on the clause from Vienna Convention on Road Traffic that states that "A vehicle [...] shall keep at a sufficient distance [...] to avoid collision if the vehicle in front should suddenly slow down or stop."
Furthermore, the absolute value constraint can be decomposed into linear constraints by the application of big-M method and the introduction of an auxiliary variable, as shown in the Appendix. Remark 3: The proposed formulation can accommodate arbitrary number of lanes at any given time instant k. This means that if at any given time, the number of available lanes for traveling either increases or decreases, the proposed formulation will still continue to hold.
This is an important consideration since many a times on highways, some lanes are blocked due to various unanticipated situations such as road accidents, roadwork, narrowing of road etc. 2) Computational Complexity Reduction: This section details the optimization problem reformulation with binary variables, optimization warm start technique and lazy constraint implementation, all of which combine to improve the computational complexity of our SLAS module.
Binary Variables: The proposed formulation in Section III-B has relatively high computation complexity (computation time of ∼ 2s in the worst case scenario -slow moving traffic blocking all the lanes) due to the integer decision variables yielding a mixed-integer optimization problem . To circumvent the computational overload, we reformulate the problem with binary variables that replace the integer variables, as follows:
where the Lk (i, j) represents the modified target lane variable, indexed by the lane (i) as well as the planning step (j) and Lk (a, b) = 1 represents the choice of lane a ∈ L as the target lane at planning step b ∈ Z . Then, some of the constraints from the SLAS formulation in Section III-B are modified as follows:
Here, initializes the target lane, (21) restricts the target lane at any planning step to the set of available lanes, restricts the lane change between consecutive planning steps to the adjacent lanes, and ( ) represents the augmented safety constraint. The implication ( =⇒ ) in ( ) can easily be transformed into a linear constraint (see Appendix).
The augmented minimum safety distance ( Ls i (j)) incorporates the time required to execute the lane change maneuver (N ) from ( ) into the following unified safety constraint: where L l is the width of the lanes (see Fig. ), δ(k) is the signed lateral deviation of the ego vehicle from the previous target lane's boundary at time step k, and γ d (δ k (j)) is the dynamic cost of deviation from the previous target lane (L(k − 1)).
Moreover, in the cost function , we take L k (0) = L(k − 1). These costs are introduced to prevent the swerving (canceling of lane switch before completion) behavior, unless absolutely necessary (for safety purposes). Remark 4: Since the ego vehicle is considered to have changed lane once it crosses a lane boundary, the deviation δ k (j) is considered from the lane boundary instead of the center of the target lane to maintain the continuity of γ d (δ k (j)) with respect to the lateral displacement of the ego vehicle.
Specifically, δ k (j) > 0 if the ego vehicle has crossed the previous target lane boundary and 0 otherwise. This is an important consideration since a discontinuity in γ d (δ k (j)), upon completion of lane change, may lead to infeasibility. Remark 5: The swerving behavior is suppressed but not completely eliminated with a hard constraint since such a behavior is necessary at times to react to the environment's unpredictability.
This reactive strategy, which is a distinctive feature of our approach, avails the algorithm the ability to proactively 'change its mind' in case something unanticipated happens in the environment that can jeopardize safety. Optimization Warm Start: To aid the optimizer in finding an initially feasible solution, we provide the solution from the previous time step as a reference.
Formally, This doesn't imply that the solution from time step k − 1 will hold exactly at time step k, owing to the unmodeled disturbances, but providing this reference aids the optimizer in finding an initially feasible solution in the vicinity of the reference solution. This observation is rooted in the premise that the solution for the long planning horizon is not expected to change significantly between time steps, given the sampling time is not too large, and the predicted behavior of on-road agents does not alter significantly.
It is also worth pointing out that the priority here is quickly finding a feasible solution that obeys the safety constraints and actuator limits, and recursively improving it rather than excessively iterating to reach at a global optimum. In our experiments, it was observed that a suboptimal solution was qualitatively not significantly different from the optimal one.
Therefore, we utilize the cutting planes method for optimization , which first looks for a feasible solution, using our provided reference, and then recursively updates it until either the globally minimal solution is found or the time limit is reached. Lazy Constraints: To further enhance the computational efficiency, we introduce a lazy implementation of the lane changing constraints .
It was observed in our experiments that a feasible solution without the lane changing constraints ( ) can be found several order of magnitude (∼ 10×) quicker than if we include these constraints so we decided to have a lazy implementation for them. With a lazy implementation , the solver finds a set of feasible solutions without the inclusion of these constraints and then determines the feasibility of those solutions from the reduced problem with respect to the lazy constraints.
3) Feasibility: By an argument similar to the one presented in , it is a relatively straightforward proof for recursive feasibility of the problem, i.e. the optimization problem will continue to stay feasible, if initially feasible, with the trivial solution being matching the speed of the leading vehicle and not changing lanes.
In this section, we detail our experimental setup, demonstrate the performance of SLAS, and report a qualitative as well as a quantitative comparative analysis. The baselines in our comparative analysis are set to: Extended-Astar (EA ) , MOBIL , and no lane-change model (No-Change).

Experimental Setup

The implementation setup, depicted in Fig. , is composed of the CARLA simulator (Version 0.9.11) , SLAS module (Section III-B), and the planner and controller module . To solve the optimization problem for SLAS, we use Gurobi Optimizer (Version 9.1.1) . The simulations are performed on a computer equipped with an Intel Xeon(R) CPU E5-2643 v4 @ 3.40GHz × 12 and NVIDIA Titan XP, running Ubuntu 20.04 LTS.
On average, the time required for each optimization step is ∼ 0.096s, while the maximum time limit for the optimizer is set to 0.2s, indicating the strong potential for real-time applicability.

Case Study

Figure illustrates the test case scenario for our comparative analysis. The scenario is composed of a highway segment with four lanes and the rightmost lane reserved for merging vehicles. The ego vehicle is initialized to follow a slow moving vehicle in lane 1 and has even slower moving traffic to its right in lane 2. Thus, the only option for it, in order to minimize travel time, is to switch left to lane 0 with faster moving traffic and greater headway.
Once it moves to lane 0, and overtakes the slow moving vehicle in lane 1, it has two options: either to keep traveling in lane 0 without making any lane change decisions until getting close to the lead vehicle or proactively exploiting the gap in lane 2 to switch to lane 3 in anticipation of traffic buildup in lanes 1 and 2. A strategic decision maker with foresight will choose to take the later option and make the decision proactively for a greater overall benefit.
The evaluation metrics for the comparative analysis include: travel time, lateral displacement, headway and distance to the closest vehicle. As for the simulation parameters, the simulation step size is set to 0.05s (simulation frequency of 20Hz); the velocities of vehicles in lanes 0, 1 and 2 are set to 8, 5 and 2 m/s respectively while the speed limit V l is set to 15m/s; the length of the highway patch is set to 350m while the width between the lanes is set to 3.5m; and the sensor visibility range is set to R v = 50m.
The parameters for SLAS are set as follows: T s = 0.4s, H = 40, N = 3, A min = −5m/s 2 , A max = 3.5m/s 2 , γ 1 = 1, γ 2 = 0.1 and γ 3 = 0.01. The values of these parameters can be tuned to yield an aggressive or defensive behavior of the algorithm. 1) Travel Time: The left plot in Fig. depicts the travel time as a function of longitudinal displacement for the four algorithms.
As seen in the plot, our method (SLAS) maintains a lower overall travel time as compared to the other methods. Quantitatively speaking, SLAS outperforms EA , MOBIL and No-change methods by 12.72%, 23.52% and 54.34% respectively in terms of the time required to complete the simulation scenario. This shows that our method's foresight compensates for its apparent conservativeness arising from the need to preserve passenger comfort.
2) Lateral Displacement: To identify the differences in lane changing behaviors between the four approaches, the relationship between lateral and longitudinal displacements over the course of the simulation is highlighted in the center plot of Fig. . In the plot, the lateral displacement of 0 corresponds to the center of lane 0 while the center of each following lane is 3.5m away.
Comparing the performance of the four algorithms, we see SLAS and EA showing relatively similar performances, resulting from proactive decision making. In contrast, since MOBIL only assesses the advantage of switching to the adjacent lanes, it is unable to see the benefit of proactively switching to lane 2. This explains why EA and SLAS start outperforming MOBIL in terms of travel time (left plot) at around the 130 [m] mark for longitudinal displacement.
As for a direct comparison between SLAS and EA , the benefits of having speed advisory system become apparent in this center plot. Due to speed control, SLAS is able to constantly maintain a greater headway (right plot) without having to brake significantly upon getting too close to the lead vehicle. This results in a smooth lateral displacement profile which allows the vehicle to change lanes with minimal jerk (quantitative analysis to follow in Section IV-C) and deliver better overall timing performance (left plot).
3) Headway: The right plot in Fig. shows the headway maintained by the ego vehicle over the course of the simulation. In accordance with our prior discussion, MOBIL cruises behind the front vehicle, maintaining a relatively low headway until a sufficient space in the adjacent lane is found to perform the lane-change maneuver.
On the other hand, EA and SLAS show a comparable headway trajectory, however, SLAS maintains a greater headway throughout and achieves the maximum headway prior to EA . Quantitatively, SLAS maintains on average 9.43%, 36.57% and 113.17% more headway than the EA , MOBIL and No-change approaches respectively.
This strong performance by SLAS can be attributed to its incorporation of safety guarantees coupled with its consideration for passenger comfort. 4) Distance to closest vehicle: Finally, we compare the distance that ego vehicle maintains from the closest vehicle throughout the simulation. On average, SLAS maintains 9.28%, 32.01%, and 22.84% more distance in comparison to EA , MOBIL and No-change approaches respectively.
These numbers are a testament to the strength of our approach resulting from consideration of long planning horizon coupled with speed control.

Monte Carlo Simulations

To demonstrate the long-term performance of the three approaches (SLAS, EA and MOBIL), we run a series of Monte Carlo simulations on scenarios with randomized initial positions (within a range of 8m) and velocities (within    ranges of 8, 5 and 2 m/s assigned to each of the three lanes randomly) of traffic participants.
The result from 50 simulations is presented in Table . In this table, the columns represent the different evaluation metrics, the rows identify the three algorithms, and the values highlighted in green represent the best result with respect to each evaluation metric. The evaluation metrics, going from left to right in the table, are completion time (s), brake (R [−1,0] ), brake jerk (R [−1,0] ), throttle (R [0,1] ), throttle jerk (R [0,1] ), angular acceleration ( • /s 2 ) and angular jerk ( • /s 3 ).
Apart from completion time, the remaining metrics, based on the commands passed to the vehicular actuators (Fig. ), are used to model passenger comfort. In terms of average performance, SLAS greatly outperforms the other methods when it comes to passenger comfort since it explicitly accounts for comfort in the formulation.
However, it does so at a cost of slightly reduced performance in regards to travel time, when compared to EA , since SLAS tries to strike a balance between minimizing travel time and maximizing passenger comfort. SLAS also secures the lowest standard deviation, for each of the evaluation metrics, when compared to the other methods, which points to the consistency in its long-term performance.

CONCLUSION

We propose a novel behavior planning module for the multi-lane highway maneuvering scenario, that outputs strategic target lane and reference speed commands, and incorporate it with a state-of-the-art motion planning and control framework. We formulate the approach as a receding horizon mixed integer optimization with the goal to minimize travel time while accounting for passenger comfort for a long planning horizon.
In order to reduce the computational overload, we reformulate the problem by replacing integer variables with binary ones and further incorporate various modifications to aid numerical computations. We also carry out a detailed comparative analysis to demonstrate the performance of our approach on the CARLA simulator.
Our future work includes incorporating various delays and uncertainty measures in the perception, localization and prediction modules to evaluate the robustness properties of our approach.

Flooring Constraint

For y ∈ Z and x ∈ R, the constraint y = x can be represented by the following linear constraints: y ≤ x, y + 1 ≥ x + where > 0 accounts for the feasibility tolerance. where M 0 (big-M) and > 0 accounts for numerical errors (chosen to be 0.1 in our implementations).

Absolute Value Constraint

For ∆s, L s ∈ R, the constraint |∆s| − L s ≥ 0 can be represented as: This can further be generalized, as done in our implementation, to have different forward and rear safety margins as: ∆s ≥ L f s ∨ ∆s ≤ −L r s where L f s and L r s are forward and rear safety margins respectively. This can be represented with the following linear constraints:
s ≥ 0 where M 0 (big-M) and c ∈ {0, 1} is responsible for making a choice between the two constraints.

Answer: Reduced computational complexity.