{"id":"https://openalex.org/W7222961447","doi":"https://doi.org/10.3390/math14203769","title":"Direct Neural Dynamic Programming Adaptive Control of Knee and Ankle Joints in a Simulated Prosthetic Leg Model","display_name":"Direct Neural Dynamic Programming Adaptive Control of Knee and Ankle Joints in a Simulated Prosthetic Leg Model","publication_year":2026,"publication_date":"2026-10-17","ids":{"openalex":"https://openalex.org/W7222961447","doi":"https://doi.org/10.3390/math14203769"},"language":"en","primary_location":{"id":"doi:10.3390/math14203769","doi":"https://doi.org/10.3390/math14203769","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14203769","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/math14203769","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045617522","display_name":"Jordan K. Kralev","orcid":"https://orcid.org/0000-0002-5416-2115","observed_orcids":["https://orcid.org/0000-0002-5416-2115"]},"institutions":[{"id":"https://openalex.org/I31151848","display_name":"Technical University of Sofia","ror":"https://ror.org/052prhs50","country_code":"BG","type":"education","lineage":["https://openalex.org/I31151848"]}],"countries":["BG"],"is_corresponding":true,"raw_author_name":"Jordan K. Kralev","raw_affiliation_strings":["Department of Systems and Control, Technical University of Sofia, 1700 Sofia, Bulgaria"],"raw_orcid":"https://orcid.org/0000-0002-5416-2115","affiliations":[{"raw_affiliation_string":"Department of Systems and Control, Technical University of Sofia, 1700 Sofia, Bulgaria","institution_ids":["https://openalex.org/I31151848"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027705027","display_name":"Tsonyo N. Slavov","orcid":"https://orcid.org/0000-0002-4180-4721","observed_orcids":["https://orcid.org/0000-0002-4180-4721"]},"institutions":[{"id":"https://openalex.org/I31151848","display_name":"Technical University of Sofia","ror":"https://ror.org/052prhs50","country_code":"BG","type":"education","lineage":["https://openalex.org/I31151848"]}],"countries":["BG"],"is_corresponding":false,"raw_author_name":"Tsonyo N. Slavov","raw_affiliation_strings":["Department of Systems and Control, Technical University of Sofia, 1700 Sofia, Bulgaria"],"raw_orcid":"https://orcid.org/0000-0002-4180-4721","affiliations":[{"raw_affiliation_string":"Department of Systems and Control, Technical University of Sofia, 1700 Sofia, Bulgaria","institution_ids":["https://openalex.org/I31151848"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007081128","display_name":"\u0410. \u0413. \u041c\u0430\u0440\u043a\u043e\u0432\u0441\u043a\u0438","orcid":"https://orcid.org/0009-0004-9725-3406","observed_orcids":["https://orcid.org/0009-0004-9725-3406"]},"institutions":[{"id":"https://openalex.org/I31151848","display_name":"Technical University of Sofia","ror":"https://ror.org/052prhs50","country_code":"BG","type":"education","lineage":["https://openalex.org/I31151848"]}],"countries":["BG"],"is_corresponding":false,"raw_author_name":"Asparuh Markovski","raw_affiliation_strings":["Department of Systems and Control, Technical University of Sofia, 1700 Sofia, Bulgaria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Systems and Control, Technical University of Sofia, 1700 Sofia, Bulgaria","institution_ids":["https://openalex.org/I31151848"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065805730","display_name":"Vladislav Slavov","orcid":"https://orcid.org/0000-0001-8736-9764","observed_orcids":["https://orcid.org/0000-0001-8736-9764"]},"institutions":[{"id":"https://openalex.org/I31151848","display_name":"Technical University of Sofia","ror":"https://ror.org/052prhs50","country_code":"BG","type":"education","lineage":["https://openalex.org/I31151848"]}],"countries":["BG"],"is_corresponding":false,"raw_author_name":"Vladislav Slavov","raw_affiliation_strings":["Department of Electrical Measurement, Technical University of Sofia, 1700 Sofia, Bulgaria"],"raw_orcid":"https://orcid.org/0000-0001-8736-9764","affiliations":[{"raw_affiliation_string":"Department of Electrical Measurement, Technical University of Sofia, 1700 Sofia, Bulgaria","institution_ids":["https://openalex.org/I31151848"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5045617522"],"corresponding_institution_ids":["https://openalex.org/I31151848"],"apc_list":{"value":2600,"currency":"CHF","value_usd":3137},"apc_paid":{"value":2600,"currency":"CHF","value_usd":3137},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.71513676,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"20","first_page":"3769","last_page":"3769"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12794","display_name":"Adaptive Dynamic Programming Control","score":0.7470999956130981,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12794","display_name":"Adaptive Dynamic Programming Control","score":0.7470999956130981,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11023","display_name":"Prosthetics and Rehabilitation Robotics","score":0.1979999989271164,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10879","display_name":"Robotic Locomotion and Control","score":0.020400000736117363,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/adaptive-control","display_name":"adaptive control","score":0.7570000290870667},{"id":"https://openalex.org/keywords/knee-joint-control","display_name":"knee joint control","score":0.7940000295639038},{"id":"https://openalex.org/keywords/ankle-joint-control","display_name":"ankle joint control","score":1.0},{"id":"https://openalex.org/keywords/model-free-optimal-control","display_name":"model-free optimal control","score":0.7870000004768372},{"id":"https://openalex.org/keywords/amputees","display_name":"amputees","score":0.7689999938011169},{"id":"https://openalex.org/keywords/real-time-control","display_name":"real-time control","score":0.5289999842643738}],"concepts":[{"id":"https://openalex.org/C177606310","wikidata":"https://www.wikidata.org/wiki/Q5674297","display_name":"Adaptability","level":2,"score":0.682200014591217},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6464999914169312},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6126999855041504},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.6025000214576721},{"id":"https://openalex.org/C151800584","wikidata":"https://www.wikidata.org/wiki/Q2370000","display_name":"Gait","level":2,"score":0.5814999938011169},{"id":"https://openalex.org/C107464732","wikidata":"https://www.wikidata.org/wiki/Q235781","display_name":"Adaptive control","level":3,"score":0.49320000410079956},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.42899999022483826},{"id":"https://openalex.org/C133731056","wikidata":"https://www.wikidata.org/wiki/Q4917288","display_name":"Control engineering","level":1,"score":0.3982999920845032},{"id":"https://openalex.org/C2778640784","wikidata":"https://www.wikidata.org/wiki/Q168002","display_name":"Ankle","level":2,"score":0.38760000467300415},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3785000145435333},{"id":"https://openalex.org/C2780791683","wikidata":"https://www.wikidata.org/wiki/Q846785","display_name":"Action (physics)","level":2,"score":0.37599998712539673},{"id":"https://openalex.org/C37404715","wikidata":"https://www.wikidata.org/wiki/Q380679","display_name":"Dynamic programming","level":2,"score":0.374099999666214},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.36340001225471497},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.35749998688697815},{"id":"https://openalex.org/C82327864","wikidata":"https://www.wikidata.org/wiki/Q835100","display_name":"Intelligent control","level":2,"score":0.33889999985694885},{"id":"https://openalex.org/C2775936607","wikidata":"https://www.wikidata.org/wiki/Q466845","display_name":"Tracking (education)","level":2,"score":0.3375000059604645},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.3294999897480011},{"id":"https://openalex.org/C2780598303","wikidata":"https://www.wikidata.org/wiki/Q65921492","display_name":"Flexibility (engineering)","level":2,"score":0.32420000433921814},{"id":"https://openalex.org/C17500928","wikidata":"https://www.wikidata.org/wiki/Q959968","display_name":"Control system","level":2,"score":0.3066999912261963},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.2655999958515167},{"id":"https://openalex.org/C139807058","wikidata":"https://www.wikidata.org/wiki/Q352374","display_name":"Adaptation (eye)","level":2,"score":0.2581000030040741}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/math14203769","doi":"https://doi.org/10.3390/math14203769","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14203769","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:bfd3eb99811b46efbcdd26b4e1e735ef","doi":null,"is_oa":false,"landing_page_url":"https://doaj.org/article/bfd3eb99811b46efbcdd26b4e1e735ef","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics, Vol 14, Iss 20, p 3769 (2026)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/math14203769","doi":"https://doi.org/10.3390/math14203769","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14203769","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W173093232","https://openalex.org/W1986119156","https://openalex.org/W2023618517","https://openalex.org/W2145339207","https://openalex.org/W2247195028","https://openalex.org/W2788472882","https://openalex.org/W2901188190","https://openalex.org/W2901258812","https://openalex.org/W3082851711","https://openalex.org/W3115597439","https://openalex.org/W3127275112","https://openalex.org/W3158715160","https://openalex.org/W4285280483","https://openalex.org/W4362576326","https://openalex.org/W4393144910","https://openalex.org/W4400124095","https://openalex.org/W4402624667","https://openalex.org/W4404092142","https://openalex.org/W4404897708","https://openalex.org/W4407212638","https://openalex.org/W4409494326","https://openalex.org/W7108206041"],"related_works":["https://openalex.org/W2231476566","https://openalex.org/W4396585382","https://openalex.org/W2256274362","https://openalex.org/W3141406181","https://openalex.org/W2490084223","https://openalex.org/W1977142928","https://openalex.org/W2032464944","https://openalex.org/W4402353557","https://openalex.org/W2371683604","https://openalex.org/W4413269585"],"abstract_inverted_index":{"Effective":[0],"control":[1,41,63,83],"of":[2,15,82],"prosthetic":[3,44],"legs":[4],"is":[5],"essential":[6],"for":[7,17,43,64],"restoring":[8],"natural":[9],"mobility":[10],"and":[11,33,51,67,96],"enhancing":[12],"the":[13,39,57,62,65,72],"quality":[14],"life":[16],"amputees.":[18],"This":[19],"paper":[20],"presents":[21],"a":[22],"Direct":[23],"Neural":[24],"Dynamic":[25],"Programming":[26],"(DNDP)":[27],"algorithm":[28],"that":[29],"leverages":[30],"coupled":[31],"action":[32,59],"critic":[34],"neural":[35],"networks":[36,60],"to":[37,53,70,93],"solve":[38],"optimal":[40],"problem":[42],"gait":[45,55,73],"trajectory":[46],"tracking.":[47],"By":[48],"continuously":[49],"learning":[50],"adapting":[52],"individual":[54],"patterns,":[56],"DNDP":[58],"generate":[61],"knee":[66],"ankle":[68],"joints":[69],"reduce":[71],"tracking":[74],"error.":[75],"The":[76],"proposed":[77],"approach":[78],"enables":[79],"real-time":[80],"optimization":[81],"strategies":[84],"without":[85],"relying":[86],"on":[87],"precise":[88],"system":[89],"models,":[90],"offering":[91],"adaptability":[92],"subject-specific":[94],"dynamics":[95],"varying":[97],"conditions.":[98]},"counts_by_year":[],"updated_date":"2026-10-29T09:56:32.521200","created_date":"2026-10-20T00:00:00"}
