diff --git a/docs/Manifest.toml b/docs/Manifest.toml
index 896b5252..d1a3b49b 100644
--- a/docs/Manifest.toml
+++ b/docs/Manifest.toml
@@ -1,6 +1,6 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.8.5"
+julia_version = "1.9.1"
 manifest_format = "2.0"
 project_hash = "9c401d26ed53d16ba6cb9cd45e6a8b3a2e875eec"
 
@@ -19,10 +19,16 @@ uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"
 [[deps.Base64]]
 uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"
 
+[[deps.Calculus]]
+deps = ["LinearAlgebra"]
+git-tree-sha1 = "f641eb0a4f00c343bbc32346e1217b86f3ce9dad"
+uuid = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9"
+version = "0.5.1"
+
 [[deps.CompilerSupportLibraries_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
-version = "1.0.1+0"
+version = "1.0.2+0"
 
 [[deps.Dates]]
 deps = ["Printf"]
@@ -45,19 +51,30 @@ deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
 uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
 version = "1.6.0"
 
+[[deps.DualNumbers]]
+deps = ["Calculus", "NaNMath", "SpecialFunctions"]
+git-tree-sha1 = "5837a837389fccf076445fce071c8ddaea35a566"
+uuid = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"
+version = "0.6.8"
+
 [[deps.FileWatching]]
 uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"
 
 [[deps.IOCapture]]
 deps = ["Logging", "Random"]
-git-tree-sha1 = "f7be53659ab06ddc986428d3a9dcc95f6fa6705a"
+git-tree-sha1 = "d75853a0bdbfb1ac815478bacd89cd27b550ace6"
 uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89"
-version = "0.2.2"
+version = "0.2.3"
 
 [[deps.InteractiveUtils]]
 deps = ["Markdown"]
 uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"
 
+[[deps.IrrationalConstants]]
+git-tree-sha1 = "630b497eafcc20001bba38a4651b327dcfc491d2"
+uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
+version = "0.2.2"
+
 [[deps.JLLWrappers]]
 deps = ["Preferences"]
 git-tree-sha1 = "abc9885a7ca2052a736a600f7fa66209f96506e1"
@@ -105,7 +122,7 @@ uuid = "9c8b4983-aa76-5018-a973-4c85ecc9e179"
 version = "0.9.0"
 
 [[deps.LinearAlgebra]]
-deps = ["Libdl", "libblastrampoline_jll"]
+deps = ["Libdl", "OpenBLAS_jll", "libblastrampoline_jll"]
 uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 
 [[deps.Literate]]
@@ -114,6 +131,22 @@ git-tree-sha1 = "1c4418beaa6664041e0f9b48f0710f57bff2fcbe"
 uuid = "98b081ad-f1c9-55d3-8b20-4c87d4299306"
 version = "2.14.0"
 
+[[deps.LogExpFunctions]]
+deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
+git-tree-sha1 = "c3ce8e7420b3a6e071e0fe4745f5d4300e37b13f"
+uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
+version = "0.3.24"
+
+    [deps.LogExpFunctions.extensions]
+    LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
+    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
+    LogExpFunctionsInverseFunctionsExt = "InverseFunctions"
+
+    [deps.LogExpFunctions.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+    ChangesOfVariables = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0"
+    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"
+
 [[deps.Logging]]
 uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"
 
@@ -130,14 +163,20 @@ uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"
 [[deps.MbedTLS_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"
-version = "2.28.0+0"
+version = "2.28.2+0"
 
 [[deps.Mmap]]
 uuid = "a63ad114-7e13-5084-954f-fe012c677804"
 
 [[deps.MozillaCACerts_jll]]
 uuid = "14a3606d-f60d-562e-9121-12d972cd8159"
-version = "2022.2.1"
+version = "2022.10.11"
+
+[[deps.NaNMath]]
+deps = ["OpenLibm_jll"]
+git-tree-sha1 = "0877504529a3e5c3343c6f8b4c0381e57e4387e4"
+uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
+version = "1.0.2"
 
 [[deps.NetworkOptions]]
 uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"
@@ -146,18 +185,35 @@ version = "1.2.0"
 [[deps.OpenBLAS_jll]]
 deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
 uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
-version = "0.3.20+0"
+version = "0.3.21+4"
+
+[[deps.OpenLibm_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "05823500-19ac-5b8b-9628-191a04bc5112"
+version = "0.8.1+0"
+
+[[deps.OpenSpecFun_jll]]
+deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
+uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
+version = "0.5.5+0"
 
 [[deps.Parsers]]
-deps = ["Dates", "SnoopPrecompile"]
-git-tree-sha1 = "478ac6c952fddd4399e71d4779797c538d0ff2bf"
+deps = ["Dates", "PrecompileTools", "UUIDs"]
+git-tree-sha1 = "a5aef8d4a6e8d81f171b2bd4be5265b01384c74c"
 uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
-version = "2.5.8"
+version = "2.5.10"
 
 [[deps.Pkg]]
-deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
+deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
 uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
-version = "1.8.0"
+version = "1.9.0"
+
+[[deps.PrecompileTools]]
+deps = ["Preferences"]
+git-tree-sha1 = "9673d39decc5feece56ef3940e5dafba15ba0f81"
+uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
+version = "1.1.2"
 
 [[deps.Preferences]]
 deps = ["TOML"]
@@ -170,10 +226,10 @@ deps = ["Unicode"]
 uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 
 [[deps.Quaternions]]
-deps = ["LinearAlgebra", "Random", "RealDot"]
-git-tree-sha1 = "da095158bdc8eaccb7890f9884048555ab771019"
+deps = ["DualNumbers", "LinearAlgebra", "Random"]
+git-tree-sha1 = "0b345302b17b0e694092621915de0e0dc7443a1a"
 uuid = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
-version = "0.7.4"
+version = "0.4.9"
 
 [[deps.REPL]]
 deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
@@ -183,12 +239,6 @@ uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"
 deps = ["SHA", "Serialization"]
 uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
-[[deps.RealDot]]
-deps = ["LinearAlgebra"]
-git-tree-sha1 = "9f0a1b71baaf7650f4fa8a1d168c7fb6ee41f0c9"
-uuid = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
-version = "0.1.0"
-
 [[deps.Reexport]]
 git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
 uuid = "189a3867-3050-52da-a836-e630ba90ab69"
@@ -198,13 +248,13 @@ version = "1.2.2"
 deps = ["DocStringExtensions", "LightXML", "LinearAlgebra", "LoopThrottle", "Random", "Reexport", "Rotations", "SparseArrays", "StaticArrays", "TypeSortedCollections", "UnsafeArrays"]
 path = ".."
 uuid = "366cf18f-59d5-5db9-a4de-86a9f6786172"
-version = "2.3.2"
+version = "2.4.0"
 
 [[deps.Rotations]]
 deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays", "Statistics"]
-git-tree-sha1 = "72a6abdcd088764878b473102df7c09bbc0548de"
+git-tree-sha1 = "405148000e80f70b31e7732ea93288aecb1793fa"
 uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc"
-version = "1.4.0"
+version = "1.2.0"
 
 [[deps.SHA]]
 uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
@@ -213,24 +263,30 @@ version = "0.7.0"
 [[deps.Serialization]]
 uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"
 
-[[deps.SnoopPrecompile]]
-deps = ["Preferences"]
-git-tree-sha1 = "e760a70afdcd461cf01a575947738d359234665c"
-uuid = "66db9d55-30c0-4569-8b51-7e840670fc0c"
-version = "1.0.3"
-
 [[deps.Sockets]]
 uuid = "6462fe0b-24de-5631-8697-dd941f90decc"
 
 [[deps.SparseArrays]]
-deps = ["LinearAlgebra", "Random"]
+deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
 uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 
+[[deps.SpecialFunctions]]
+deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
+git-tree-sha1 = "ef28127915f4229c971eb43f3fc075dd3fe91880"
+uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
+version = "2.2.0"
+
+    [deps.SpecialFunctions.extensions]
+    SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"
+
+    [deps.SpecialFunctions.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+
 [[deps.StaticArrays]]
 deps = ["LinearAlgebra", "Random", "StaticArraysCore", "Statistics"]
-git-tree-sha1 = "c262c8e978048c2b095be1672c9bee55b4619521"
+git-tree-sha1 = "8982b3607a212b070a5e46eea83eb62b4744ae12"
 uuid = "90137ffa-7385-5640-81b9-e52037218182"
-version = "1.5.24"
+version = "1.5.25"
 
 [[deps.StaticArraysCore]]
 git-tree-sha1 = "6b7ba252635a5eff6a0b0664a41ee140a1c9e72a"
@@ -240,16 +296,22 @@ version = "1.4.0"
 [[deps.Statistics]]
 deps = ["LinearAlgebra", "SparseArrays"]
 uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
+version = "1.9.0"
+
+[[deps.SuiteSparse_jll]]
+deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"]
+uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
+version = "5.10.1+6"
 
 [[deps.TOML]]
 deps = ["Dates"]
 uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
-version = "1.0.0"
+version = "1.0.3"
 
 [[deps.Tar]]
 deps = ["ArgTools", "SHA"]
 uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
-version = "1.10.1"
+version = "1.10.0"
 
 [[deps.Test]]
 deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
@@ -281,12 +343,12 @@ version = "2.10.3+0"
 [[deps.Zlib_jll]]
 deps = ["Libdl"]
 uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
-version = "1.2.12+3"
+version = "1.2.13+0"
 
 [[deps.libblastrampoline_jll]]
-deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
+deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.1.1+0"
+version = "5.8.0+0"
 
 [[deps.nghttp2_jll]]
 deps = ["Artifacts", "Libdl"]
diff --git a/docs/make.jl b/docs/make.jl
index f9d87d27..3669c637 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -11,8 +11,8 @@ let
     exampledir = joinpath(@__DIR__, "..", "examples")
     excludedirs = String[]
     excludefiles = String[]
-    if VERSION < v"1.1.0"
-        push!(excludefiles, "Symbolic double pendulum.jl")
+    if VERSION >= v"1.9.0"
+        push!(excludefiles, "6. Symbolics.jl")
     end
     for subdir in readdir(exampledir)
         subdir in excludedirs && continue
diff --git a/examples/1. Quickstart - double pendulum/Manifest.toml b/examples/1. Quickstart - double pendulum/Manifest.toml
index d6adfac8..aef4b24b 100644
--- a/examples/1. Quickstart - double pendulum/Manifest.toml	
+++ b/examples/1. Quickstart - double pendulum/Manifest.toml	
@@ -1,6 +1,6 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.9.0"
+julia_version = "1.9.1"
 manifest_format = "2.0"
 project_hash = "8f7cc3388be6494a7cb012dfb39416bbbb94df82"
 
@@ -14,6 +14,12 @@ uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"
 [[deps.Base64]]
 uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"
 
+[[deps.Calculus]]
+deps = ["LinearAlgebra"]
+git-tree-sha1 = "f641eb0a4f00c343bbc32346e1217b86f3ce9dad"
+uuid = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9"
+version = "0.5.1"
+
 [[deps.CompilerSupportLibraries_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
@@ -25,15 +31,21 @@ uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"
 
 [[deps.DocStringExtensions]]
 deps = ["LibGit2"]
-git-tree-sha1 = "b19534d1895d702889b219c382a6e18010797f0b"
+git-tree-sha1 = "2fb1e02f2b635d0845df5d7c167fec4dd739b00d"
 uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
-version = "0.8.6"
+version = "0.9.3"
 
 [[deps.Downloads]]
 deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
 uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
 version = "1.6.0"
 
+[[deps.DualNumbers]]
+deps = ["Calculus", "NaNMath", "SpecialFunctions"]
+git-tree-sha1 = "5837a837389fccf076445fce071c8ddaea35a566"
+uuid = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"
+version = "0.6.8"
+
 [[deps.FileWatching]]
 uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"
 
@@ -41,6 +53,11 @@ uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"
 deps = ["Markdown"]
 uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"
 
+[[deps.IrrationalConstants]]
+git-tree-sha1 = "630b497eafcc20001bba38a4651b327dcfc491d2"
+uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
+version = "0.2.2"
+
 [[deps.JLLWrappers]]
 deps = ["Preferences"]
 git-tree-sha1 = "abc9885a7ca2052a736a600f7fa66209f96506e1"
@@ -85,6 +102,22 @@ version = "0.9.0"
 deps = ["Libdl", "OpenBLAS_jll", "libblastrampoline_jll"]
 uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 
+[[deps.LogExpFunctions]]
+deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
+git-tree-sha1 = "c3ce8e7420b3a6e071e0fe4745f5d4300e37b13f"
+uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
+version = "0.3.24"
+
+    [deps.LogExpFunctions.extensions]
+    LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
+    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
+    LogExpFunctionsInverseFunctionsExt = "InverseFunctions"
+
+    [deps.LogExpFunctions.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+    ChangesOfVariables = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0"
+    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"
+
 [[deps.Logging]]
 uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"
 
@@ -107,6 +140,12 @@ version = "2.28.2+0"
 uuid = "14a3606d-f60d-562e-9121-12d972cd8159"
 version = "2022.10.11"
 
+[[deps.NaNMath]]
+deps = ["OpenLibm_jll"]
+git-tree-sha1 = "0877504529a3e5c3343c6f8b4c0381e57e4387e4"
+uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
+version = "1.0.2"
+
 [[deps.NetworkOptions]]
 uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"
 version = "1.2.0"
@@ -116,6 +155,17 @@ deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
 uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
 version = "0.3.21+4"
 
+[[deps.OpenLibm_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "05823500-19ac-5b8b-9628-191a04bc5112"
+version = "0.8.1+0"
+
+[[deps.OpenSpecFun_jll]]
+deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
+uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
+version = "0.5.5+0"
+
 [[deps.Pkg]]
 deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
 uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
@@ -132,10 +182,10 @@ deps = ["Unicode"]
 uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 
 [[deps.Quaternions]]
-deps = ["LinearAlgebra", "Random", "RealDot"]
-git-tree-sha1 = "da095158bdc8eaccb7890f9884048555ab771019"
+deps = ["DualNumbers", "LinearAlgebra", "Random"]
+git-tree-sha1 = "0b345302b17b0e694092621915de0e0dc7443a1a"
 uuid = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
-version = "0.7.4"
+version = "0.4.9"
 
 [[deps.REPL]]
 deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
@@ -145,12 +195,6 @@ uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"
 deps = ["SHA", "Serialization"]
 uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
-[[deps.RealDot]]
-deps = ["LinearAlgebra"]
-git-tree-sha1 = "9f0a1b71baaf7650f4fa8a1d168c7fb6ee41f0c9"
-uuid = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
-version = "0.1.0"
-
 [[deps.Reexport]]
 git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
 uuid = "189a3867-3050-52da-a836-e630ba90ab69"
@@ -158,15 +202,15 @@ version = "1.2.2"
 
 [[deps.RigidBodyDynamics]]
 deps = ["DocStringExtensions", "LightXML", "LinearAlgebra", "LoopThrottle", "Random", "Reexport", "Rotations", "SparseArrays", "StaticArrays", "TypeSortedCollections", "UnsafeArrays"]
-git-tree-sha1 = "067a44aead5b587252e8cf73773cca422c6ae19b"
+git-tree-sha1 = "313ea256ced33aa6039987c9fef57e59aa229589"
 uuid = "366cf18f-59d5-5db9-a4de-86a9f6786172"
-version = "2.3.2"
+version = "2.4.0"
 
 [[deps.Rotations]]
-deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays"]
-git-tree-sha1 = "54ccb4dbab4b1f69beb255a2c0ca5f65a9c82f08"
+deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays", "Statistics"]
+git-tree-sha1 = "405148000e80f70b31e7732ea93288aecb1793fa"
 uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc"
-version = "1.5.1"
+version = "1.2.0"
 
 [[deps.SHA]]
 uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
@@ -182,6 +226,18 @@ uuid = "6462fe0b-24de-5631-8697-dd941f90decc"
 deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
 uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 
+[[deps.SpecialFunctions]]
+deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
+git-tree-sha1 = "ef28127915f4229c971eb43f3fc075dd3fe91880"
+uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
+version = "2.2.0"
+
+    [deps.SpecialFunctions.extensions]
+    SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"
+
+    [deps.SpecialFunctions.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+
 [[deps.StaticArrays]]
 deps = ["LinearAlgebra", "Random", "StaticArraysCore", "Statistics"]
 git-tree-sha1 = "8982b3607a212b070a5e46eea83eb62b4744ae12"
@@ -248,7 +304,7 @@ version = "1.2.13+0"
 [[deps.libblastrampoline_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
 
 [[deps.nghttp2_jll]]
 deps = ["Artifacts", "Libdl"]
diff --git a/examples/2. Closed-loop simulation and visualization/Manifest.toml b/examples/2. Closed-loop simulation and visualization/Manifest.toml
index ef389510..5a662c91 100644
--- a/examples/2. Closed-loop simulation and visualization/Manifest.toml	
+++ b/examples/2. Closed-loop simulation and visualization/Manifest.toml	
@@ -1,6 +1,6 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.9.0"
+julia_version = "1.9.1"
 manifest_format = "2.0"
 project_hash = "3f88f80c58c2ca0eb8b6a454d11120e410982bda"
 
@@ -208,9 +208,9 @@ version = "0.5.0"
 
 [[deps.GPUArraysCore]]
 deps = ["Adapt"]
-git-tree-sha1 = "1cd7f0af1aa58abc02ea1d872953a97359cb87fa"
+git-tree-sha1 = "2d6ca471a6c7b536127afccfa7564b5b39227fe0"
 uuid = "46192b85-c4d5-4398-a991-12ede77f4527"
-version = "0.1.4"
+version = "0.1.5"
 
 [[deps.GeoInterface]]
 deps = ["Extents"]
@@ -244,9 +244,9 @@ version = "1.3.14+0"
 
 [[deps.HTTP]]
 deps = ["Base64", "CodecZlib", "ConcurrentUtilities", "Dates", "Logging", "LoggingExtras", "MbedTLS", "NetworkOptions", "OpenSSL", "Random", "SimpleBufferStream", "Sockets", "URIs", "UUIDs"]
-git-tree-sha1 = "ba9eca9f8bdb787c6f3cf52cb4a404c0e349a0d1"
+git-tree-sha1 = "5e77dbf117412d4f164a464d610ee6050cc75272"
 uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
-version = "1.9.5"
+version = "1.9.6"
 
 [[deps.HarfBuzz_jll]]
 deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
@@ -271,9 +271,9 @@ uuid = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59"
 version = "0.14.7"
 
 [[deps.IterTools]]
-git-tree-sha1 = "fa6287a4469f5e048d763df38279ee729fbd44e5"
+git-tree-sha1 = "4ced6667f9974fc5c5943fa5e2ef1ca43ea9e450"
 uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
-version = "1.4.0"
+version = "1.8.0"
 
 [[deps.IteratorInterfaceExtensions]]
 git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
@@ -489,10 +489,10 @@ uuid = "4d8831e6-92b7-49fb-bdf8-b643e874388c"
 version = "1.4.1"
 
 [[deps.OpenSSL_jll]]
-deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
-git-tree-sha1 = "9ff31d101d987eb9d66bd8b176ac7c277beccd09"
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "1aa4b74f80b01c6bc2b89992b861b5f210e665b5"
 uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
-version = "1.1.20+0"
+version = "1.1.21+0"
 
 [[deps.Opus_jll]]
 deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
@@ -541,9 +541,9 @@ version = "1.9.0"
 
 [[deps.PrecompileTools]]
 deps = ["Preferences"]
-git-tree-sha1 = "259e206946c293698122f63e2b513a7c99a244e8"
+git-tree-sha1 = "9673d39decc5feece56ef3940e5dafba15ba0f81"
 uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
-version = "1.1.1"
+version = "1.1.2"
 
 [[deps.Preferences]]
 deps = ["TOML"]
@@ -571,9 +571,9 @@ uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
 [[deps.Ratios]]
 deps = ["Requires"]
-git-tree-sha1 = "6d7bb727e76147ba18eed998700998e17b8e4911"
+git-tree-sha1 = "1342a47bf3260ee108163042310d26f2be5ec90b"
 uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439"
-version = "0.4.4"
+version = "0.4.5"
 weakdeps = ["FixedPointNumbers"]
 
     [deps.Ratios.extensions]
@@ -719,9 +719,9 @@ version = "1.0.4"
 
 [[deps.WebIO]]
 deps = ["AssetRegistry", "Base64", "Distributed", "FunctionalCollections", "JSON", "Logging", "Observables", "Pkg", "Random", "Requires", "Sockets", "UUIDs", "WebSockets", "Widgets"]
-git-tree-sha1 = "976d0738247f155d0dcd77607edea644f069e1e9"
+git-tree-sha1 = "0eef0765186f7452e52236fa42ca8c9b3c11c6e3"
 uuid = "0f1e0344-ec1d-5b48-a673-e5cf874b6c29"
-version = "0.8.20"
+version = "0.8.21"
 
 [[deps.WebSockets]]
 deps = ["Base64", "Dates", "HTTP", "Logging", "Sockets"]
@@ -821,7 +821,7 @@ version = "0.15.1+0"
 [[deps.libblastrampoline_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
 
 [[deps.libfdk_aac_jll]]
 deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
diff --git a/examples/3. Four-bar linkage/Manifest.toml b/examples/3. Four-bar linkage/Manifest.toml
index a44d1e14..38262666 100644
--- a/examples/3. Four-bar linkage/Manifest.toml	
+++ b/examples/3. Four-bar linkage/Manifest.toml	
@@ -1,6 +1,6 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.9.0"
+julia_version = "1.9.1"
 manifest_format = "2.0"
 project_hash = "7a77bdb04ffe215b26280152602ca7aaf8d3a3e0"
 
@@ -208,9 +208,9 @@ version = "0.5.0"
 
 [[deps.GPUArraysCore]]
 deps = ["Adapt"]
-git-tree-sha1 = "1cd7f0af1aa58abc02ea1d872953a97359cb87fa"
+git-tree-sha1 = "2d6ca471a6c7b536127afccfa7564b5b39227fe0"
 uuid = "46192b85-c4d5-4398-a991-12ede77f4527"
-version = "0.1.4"
+version = "0.1.5"
 
 [[deps.GeoInterface]]
 deps = ["Extents"]
@@ -244,9 +244,9 @@ version = "1.3.14+0"
 
 [[deps.HTTP]]
 deps = ["Base64", "CodecZlib", "ConcurrentUtilities", "Dates", "Logging", "LoggingExtras", "MbedTLS", "NetworkOptions", "OpenSSL", "Random", "SimpleBufferStream", "Sockets", "URIs", "UUIDs"]
-git-tree-sha1 = "ba9eca9f8bdb787c6f3cf52cb4a404c0e349a0d1"
+git-tree-sha1 = "5e77dbf117412d4f164a464d610ee6050cc75272"
 uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
-version = "1.9.5"
+version = "1.9.6"
 
 [[deps.HarfBuzz_jll]]
 deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
@@ -271,9 +271,9 @@ uuid = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59"
 version = "0.14.7"
 
 [[deps.IterTools]]
-git-tree-sha1 = "fa6287a4469f5e048d763df38279ee729fbd44e5"
+git-tree-sha1 = "4ced6667f9974fc5c5943fa5e2ef1ca43ea9e450"
 uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
-version = "1.4.0"
+version = "1.8.0"
 
 [[deps.IteratorInterfaceExtensions]]
 git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
@@ -489,10 +489,10 @@ uuid = "4d8831e6-92b7-49fb-bdf8-b643e874388c"
 version = "1.4.1"
 
 [[deps.OpenSSL_jll]]
-deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
-git-tree-sha1 = "9ff31d101d987eb9d66bd8b176ac7c277beccd09"
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "1aa4b74f80b01c6bc2b89992b861b5f210e665b5"
 uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
-version = "1.1.20+0"
+version = "1.1.21+0"
 
 [[deps.Opus_jll]]
 deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
@@ -541,9 +541,9 @@ version = "1.9.0"
 
 [[deps.PrecompileTools]]
 deps = ["Preferences"]
-git-tree-sha1 = "259e206946c293698122f63e2b513a7c99a244e8"
+git-tree-sha1 = "9673d39decc5feece56ef3940e5dafba15ba0f81"
 uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
-version = "1.1.1"
+version = "1.1.2"
 
 [[deps.Preferences]]
 deps = ["TOML"]
@@ -571,9 +571,9 @@ uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
 [[deps.Ratios]]
 deps = ["Requires"]
-git-tree-sha1 = "6d7bb727e76147ba18eed998700998e17b8e4911"
+git-tree-sha1 = "1342a47bf3260ee108163042310d26f2be5ec90b"
 uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439"
-version = "0.4.4"
+version = "0.4.5"
 weakdeps = ["FixedPointNumbers"]
 
     [deps.Ratios.extensions]
@@ -719,9 +719,9 @@ version = "1.0.4"
 
 [[deps.WebIO]]
 deps = ["AssetRegistry", "Base64", "Distributed", "FunctionalCollections", "JSON", "Logging", "Observables", "Pkg", "Random", "Requires", "Sockets", "UUIDs", "WebSockets", "Widgets"]
-git-tree-sha1 = "976d0738247f155d0dcd77607edea644f069e1e9"
+git-tree-sha1 = "0eef0765186f7452e52236fa42ca8c9b3c11c6e3"
 uuid = "0f1e0344-ec1d-5b48-a673-e5cf874b6c29"
-version = "0.8.20"
+version = "0.8.21"
 
 [[deps.WebSockets]]
 deps = ["Base64", "Dates", "HTTP", "Logging", "Sockets"]
@@ -821,7 +821,7 @@ version = "0.15.1+0"
 [[deps.libblastrampoline_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
 
 [[deps.libfdk_aac_jll]]
 deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
diff --git a/examples/4. Jacobian IK and Control/Manifest.toml b/examples/4. Jacobian IK and Control/Manifest.toml
index a2e3c0c5..b142223d 100644
--- a/examples/4. Jacobian IK and Control/Manifest.toml	
+++ b/examples/4. Jacobian IK and Control/Manifest.toml	
@@ -1,8 +1,8 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.9.0"
+julia_version = "1.9.1"
 manifest_format = "2.0"
-project_hash = "41bf50131458289df41c0f0f4e1f057c1ad2f29d"
+project_hash = "57912b674860b02adbb24c85dcfc1ca99ed81fed"
 
 [[deps.Adapt]]
 deps = ["LinearAlgebra", "Requires"]
@@ -121,11 +121,19 @@ git-tree-sha1 = "96d823b94ba8d187a6d8f0826e731195a74b90e9"
 uuid = "f0e56b4a-5159-44fe-b623-3e5288b988bb"
 version = "2.2.0"
 
-[[deps.Conda]]
-deps = ["Downloads", "JSON", "VersionParsing"]
-git-tree-sha1 = "e32a90da027ca45d84678b826fffd3110bb3fc90"
-uuid = "8f4d0f93-b110-5947-807f-2305c1781a2d"
-version = "1.8.0"
+[[deps.ConstructionBase]]
+deps = ["LinearAlgebra"]
+git-tree-sha1 = "738fec4d684a9a6ee9598a8bfee305b26831f28c"
+uuid = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
+version = "1.5.2"
+
+    [deps.ConstructionBase.extensions]
+    ConstructionBaseIntervalSetsExt = "IntervalSets"
+    ConstructionBaseStaticArraysExt = "StaticArrays"
+
+    [deps.ConstructionBase.weakdeps]
+    IntervalSets = "8197267c-284f-5f27-9208-e0e47529a953"
+    StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [[deps.Contour]]
 git-tree-sha1 = "d05d9e7b7aedff4e5b51a029dced05cfb6125781"
@@ -273,21 +281,21 @@ version = "3.3.8+0"
 
 [[deps.GPUArraysCore]]
 deps = ["Adapt"]
-git-tree-sha1 = "1cd7f0af1aa58abc02ea1d872953a97359cb87fa"
+git-tree-sha1 = "2d6ca471a6c7b536127afccfa7564b5b39227fe0"
 uuid = "46192b85-c4d5-4398-a991-12ede77f4527"
-version = "0.1.4"
+version = "0.1.5"
 
 [[deps.GR]]
 deps = ["Artifacts", "Base64", "DelimitedFiles", "Downloads", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Preferences", "Printf", "Random", "Serialization", "Sockets", "TOML", "Tar", "Test", "UUIDs", "p7zip_jll"]
-git-tree-sha1 = "d014972cd6f5afb1f8cd7adf000b7a966d62c304"
+git-tree-sha1 = "8b8a2fd4536ece6e554168c21860b6820a8a83db"
 uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
-version = "0.72.5"
+version = "0.72.7"
 
 [[deps.GR_jll]]
 deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
-git-tree-sha1 = "f670f269909a9114df1380cc0fcaa316fff655fb"
+git-tree-sha1 = "19fad9cd9ae44847fe842558a744748084a722d1"
 uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
-version = "0.72.5+0"
+version = "0.72.7+0"
 
 [[deps.GeoInterface]]
 deps = ["Extents"]
@@ -332,9 +340,9 @@ version = "1.0.2"
 
 [[deps.HTTP]]
 deps = ["Base64", "CodecZlib", "ConcurrentUtilities", "Dates", "Logging", "LoggingExtras", "MbedTLS", "NetworkOptions", "OpenSSL", "Random", "SimpleBufferStream", "Sockets", "URIs", "UUIDs"]
-git-tree-sha1 = "ba9eca9f8bdb787c6f3cf52cb4a404c0e349a0d1"
+git-tree-sha1 = "5e77dbf117412d4f164a464d610ee6050cc75272"
 uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
-version = "1.9.5"
+version = "1.9.6"
 
 [[deps.HarfBuzz_jll]]
 deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
@@ -364,9 +372,9 @@ uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
 version = "0.2.2"
 
 [[deps.IterTools]]
-git-tree-sha1 = "fa6287a4469f5e048d763df38279ee729fbd44e5"
+git-tree-sha1 = "4ced6667f9974fc5c5943fa5e2ef1ca43ea9e450"
 uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
-version = "1.4.0"
+version = "1.8.0"
 
 [[deps.IteratorInterfaceExtensions]]
 git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
@@ -528,9 +536,9 @@ uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 
 [[deps.LogExpFunctions]]
 deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
-git-tree-sha1 = "0a1b7c2863e44523180fdb3146534e265a91870b"
+git-tree-sha1 = "c3ce8e7420b3a6e071e0fe4745f5d4300e37b13f"
 uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
-version = "0.3.23"
+version = "0.3.24"
 
     [deps.LogExpFunctions.extensions]
     LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
@@ -664,10 +672,10 @@ uuid = "4d8831e6-92b7-49fb-bdf8-b643e874388c"
 version = "1.4.1"
 
 [[deps.OpenSSL_jll]]
-deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
-git-tree-sha1 = "9ff31d101d987eb9d66bd8b176ac7c277beccd09"
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "1aa4b74f80b01c6bc2b89992b861b5f210e665b5"
 uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
-version = "1.1.20+0"
+version = "1.1.21+0"
 
 [[deps.OpenSpecFun_jll]]
 deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
@@ -738,10 +746,10 @@ uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
 version = "1.3.5"
 
 [[deps.Plots]]
-deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "JLFzf", "JSON", "LaTeXStrings", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "Pkg", "PlotThemes", "PlotUtils", "PrecompileTools", "Preferences", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "RelocatableFolders", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs", "UnicodeFun", "Unzip"]
-git-tree-sha1 = "d03ef538114b38f89d66776f2d8fdc0280f90621"
+deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "JLFzf", "JSON", "LaTeXStrings", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "Pkg", "PlotThemes", "PlotUtils", "PrecompileTools", "Preferences", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "RelocatableFolders", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs", "UnicodeFun", "UnitfulLatexify", "Unzip"]
+git-tree-sha1 = "ceb1ec8d4fbeb02f8817004837d924583707951b"
 uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
-version = "1.38.12"
+version = "1.38.15"
 
     [deps.Plots.extensions]
     FileIOExt = "FileIO"
@@ -759,9 +767,9 @@ version = "1.38.12"
 
 [[deps.PrecompileTools]]
 deps = ["Preferences"]
-git-tree-sha1 = "259e206946c293698122f63e2b513a7c99a244e8"
+git-tree-sha1 = "9673d39decc5feece56ef3940e5dafba15ba0f81"
 uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
-version = "1.1.1"
+version = "1.1.2"
 
 [[deps.Preferences]]
 deps = ["TOML"]
@@ -773,18 +781,6 @@ version = "1.4.0"
 deps = ["Unicode"]
 uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 
-[[deps.PyCall]]
-deps = ["Conda", "Dates", "Libdl", "LinearAlgebra", "MacroTools", "Serialization", "VersionParsing"]
-git-tree-sha1 = "62f417f6ad727987c755549e9cd88c46578da562"
-uuid = "438e738f-606a-5dbb-bf0a-cddfbfd45ab0"
-version = "1.95.1"
-
-[[deps.PyPlot]]
-deps = ["Colors", "LaTeXStrings", "PyCall", "Sockets", "Test", "VersionParsing"]
-git-tree-sha1 = "92e7ca803b579b8b817f004e74b205a706d9a974"
-uuid = "d330b81b-6aea-500a-939a-2ce795aea3ee"
-version = "2.11.1"
-
 [[deps.Qt5Base_jll]]
 deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
 git-tree-sha1 = "0c03844e2231e12fda4d0086fd7cbe4098ee8dc5"
@@ -807,9 +803,9 @@ uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
 [[deps.Ratios]]
 deps = ["Requires"]
-git-tree-sha1 = "6d7bb727e76147ba18eed998700998e17b8e4911"
+git-tree-sha1 = "1342a47bf3260ee108163042310d26f2be5ec90b"
 uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439"
-version = "0.4.4"
+version = "0.4.5"
 weakdeps = ["FixedPointNumbers"]
 
     [deps.Ratios.extensions]
@@ -937,9 +933,9 @@ version = "1.6.0"
 
 [[deps.StatsBase]]
 deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
-git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916"
+git-tree-sha1 = "75ebe04c5bed70b91614d684259b661c9e6274a4"
 uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
-version = "0.33.21"
+version = "0.34.0"
 
 [[deps.StructArrays]]
 deps = ["Adapt", "DataAPI", "GPUArraysCore", "StaticArraysCore", "Tables"]
@@ -1018,6 +1014,24 @@ git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
 uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
 version = "0.4.1"
 
+[[deps.Unitful]]
+deps = ["ConstructionBase", "Dates", "LinearAlgebra", "Random"]
+git-tree-sha1 = "ba4aa36b2d5c98d6ed1f149da916b3ba46527b2b"
+uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
+version = "1.14.0"
+
+    [deps.Unitful.extensions]
+    InverseFunctionsUnitfulExt = "InverseFunctions"
+
+    [deps.Unitful.weakdeps]
+    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"
+
+[[deps.UnitfulLatexify]]
+deps = ["LaTeXStrings", "Latexify", "Unitful"]
+git-tree-sha1 = "e2d817cc500e960fdbafcf988ac8436ba3208bfd"
+uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
+version = "1.6.3"
+
 [[deps.UnsafeArrays]]
 git-tree-sha1 = "3350f94f6caa02f324a23645bf524fc9334c7488"
 uuid = "c4a57d5a-5b31-53a6-b365-19f8c011fbd6"
@@ -1028,11 +1042,6 @@ git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
 uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
 version = "0.2.0"
 
-[[deps.VersionParsing]]
-git-tree-sha1 = "58d6e80b4ee071f5efd07fda82cb9fbe17200868"
-uuid = "81def892-9a0e-5fdd-b105-ffc91e053289"
-version = "1.3.0"
-
 [[deps.Wayland_jll]]
 deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
 git-tree-sha1 = "ed8d92d9774b077c53e1da50fd81a36af3744c1c"
@@ -1047,9 +1056,9 @@ version = "1.25.0+0"
 
 [[deps.WebIO]]
 deps = ["AssetRegistry", "Base64", "Distributed", "FunctionalCollections", "JSON", "Logging", "Observables", "Pkg", "Random", "Requires", "Sockets", "UUIDs", "WebSockets", "Widgets"]
-git-tree-sha1 = "976d0738247f155d0dcd77607edea644f069e1e9"
+git-tree-sha1 = "0eef0765186f7452e52236fa42ca8c9b3c11c6e3"
 uuid = "0f1e0344-ec1d-5b48-a673-e5cf874b6c29"
-version = "0.8.20"
+version = "0.8.21"
 
 [[deps.WebSockets]]
 deps = ["Base64", "Dates", "HTTP", "Logging", "Sockets"]
@@ -1239,7 +1248,7 @@ version = "0.15.1+0"
 [[deps.libblastrampoline_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
 
 [[deps.libfdk_aac_jll]]
 deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
diff --git a/examples/5. Derivatives and gradients using ForwardDiff/Manifest.toml b/examples/5. Derivatives and gradients using ForwardDiff/Manifest.toml
index bfec1210..84b28b08 100644
--- a/examples/5. Derivatives and gradients using ForwardDiff/Manifest.toml	
+++ b/examples/5. Derivatives and gradients using ForwardDiff/Manifest.toml	
@@ -1,6 +1,6 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.9.0"
+julia_version = "1.9.1"
 manifest_format = "2.0"
 project_hash = "510b48287c09a8847f0a555bda4a640614bff15e"
 
@@ -20,6 +20,12 @@ git-tree-sha1 = "d9a9701b899b30332bbcb3e1679c41cce81fb0e8"
 uuid = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
 version = "1.3.2"
 
+[[deps.Calculus]]
+deps = ["LinearAlgebra"]
+git-tree-sha1 = "f641eb0a4f00c343bbc32346e1217b86f3ce9dad"
+uuid = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9"
+version = "0.5.1"
+
 [[deps.CommonSubexpressions]]
 deps = ["MacroTools", "Test"]
 git-tree-sha1 = "7b8a93dba8af7e3b42fecabf646260105ac373f7"
@@ -43,21 +49,27 @@ version = "1.1.0"
 
 [[deps.DiffRules]]
 deps = ["IrrationalConstants", "LogExpFunctions", "NaNMath", "Random", "SpecialFunctions"]
-git-tree-sha1 = "a4ad7ef19d2cdc2eff57abbbe68032b1cd0bd8f8"
+git-tree-sha1 = "23163d55f885173722d1e4cf0f6110cdbaf7e272"
 uuid = "b552c78f-8df3-52c6-915a-8e097449b14b"
-version = "1.13.0"
+version = "1.15.1"
 
 [[deps.DocStringExtensions]]
 deps = ["LibGit2"]
-git-tree-sha1 = "b19534d1895d702889b219c382a6e18010797f0b"
+git-tree-sha1 = "2fb1e02f2b635d0845df5d7c167fec4dd739b00d"
 uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
-version = "0.8.6"
+version = "0.9.3"
 
 [[deps.Downloads]]
 deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
 uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
 version = "1.6.0"
 
+[[deps.DualNumbers]]
+deps = ["Calculus", "NaNMath", "SpecialFunctions"]
+git-tree-sha1 = "5837a837389fccf076445fce071c8ddaea35a566"
+uuid = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"
+version = "0.6.8"
+
 [[deps.FileWatching]]
 uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"
 
@@ -132,9 +144,9 @@ uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 
 [[deps.LogExpFunctions]]
 deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
-git-tree-sha1 = "0a1b7c2863e44523180fdb3146534e265a91870b"
+git-tree-sha1 = "c3ce8e7420b3a6e071e0fe4745f5d4300e37b13f"
 uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
-version = "0.3.23"
+version = "0.3.24"
 
     [deps.LogExpFunctions.extensions]
     LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
@@ -216,9 +228,9 @@ version = "1.9.0"
 
 [[deps.PrecompileTools]]
 deps = ["Preferences"]
-git-tree-sha1 = "259e206946c293698122f63e2b513a7c99a244e8"
+git-tree-sha1 = "9673d39decc5feece56ef3940e5dafba15ba0f81"
 uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
-version = "1.1.1"
+version = "1.1.2"
 
 [[deps.Preferences]]
 deps = ["TOML"]
@@ -235,10 +247,10 @@ deps = ["Printf"]
 uuid = "9abbd945-dff8-562f-b5e8-e1ebf5ef1b79"
 
 [[deps.Quaternions]]
-deps = ["LinearAlgebra", "Random", "RealDot"]
-git-tree-sha1 = "da095158bdc8eaccb7890f9884048555ab771019"
+deps = ["DualNumbers", "LinearAlgebra", "Random"]
+git-tree-sha1 = "0b345302b17b0e694092621915de0e0dc7443a1a"
 uuid = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
-version = "0.7.4"
+version = "0.4.9"
 
 [[deps.REPL]]
 deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
@@ -248,12 +260,6 @@ uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"
 deps = ["SHA", "Serialization"]
 uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
-[[deps.RealDot]]
-deps = ["LinearAlgebra"]
-git-tree-sha1 = "9f0a1b71baaf7650f4fa8a1d168c7fb6ee41f0c9"
-uuid = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
-version = "0.1.0"
-
 [[deps.Reexport]]
 git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
 uuid = "189a3867-3050-52da-a836-e630ba90ab69"
@@ -261,15 +267,15 @@ version = "1.2.2"
 
 [[deps.RigidBodyDynamics]]
 deps = ["DocStringExtensions", "LightXML", "LinearAlgebra", "LoopThrottle", "Random", "Reexport", "Rotations", "SparseArrays", "StaticArrays", "TypeSortedCollections", "UnsafeArrays"]
-git-tree-sha1 = "067a44aead5b587252e8cf73773cca422c6ae19b"
+git-tree-sha1 = "313ea256ced33aa6039987c9fef57e59aa229589"
 uuid = "366cf18f-59d5-5db9-a4de-86a9f6786172"
-version = "2.3.2"
+version = "2.4.0"
 
 [[deps.Rotations]]
-deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays"]
-git-tree-sha1 = "54ccb4dbab4b1f69beb255a2c0ca5f65a9c82f08"
+deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays", "Statistics"]
+git-tree-sha1 = "405148000e80f70b31e7732ea93288aecb1793fa"
 uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc"
-version = "1.5.1"
+version = "1.2.0"
 
 [[deps.SHA]]
 uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
@@ -363,7 +369,7 @@ version = "1.2.13+0"
 [[deps.libblastrampoline_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
 
 [[deps.nghttp2_jll]]
 deps = ["Artifacts", "Libdl"]
diff --git a/examples/6. Symbolics/6. Symbolics.jl b/examples/6. Symbolics/6. Symbolics.jl
index a304190a..227d0c54 100644
--- a/examples/6. Symbolics/6. Symbolics.jl	
+++ b/examples/6. Symbolics/6. Symbolics.jl	
@@ -64,15 +64,17 @@ x = MechanismState(double_pendulum);
 
 # Set the joint configuration vector of the MechanismState to a new vector of symbolic variables
 q = configuration(x)
-for i in eachindex(q)
-    q[i] = symbols("q_$i", real=true)
-end
+# for i in eachindex(q)
+#     q[i] = symbols("q_$i")
+# end
+q = @variables q[1:2]
 
 # Set the joint velocity vector of the MechanismState to a new vector of symbolic variables
 v = velocity(x)
-for i in eachindex(v)
-    v[i] = symbols("v_$i", real=true)
-end
+# for i in eachindex(v)
+#     v[i] = symbols("v_$i", real=true)
+# end
+v = @variables v[1:2]
 
 
 # ## Compute dynamical quantities in symbolic form
diff --git a/examples/6. Symbolics/Manifest.toml b/examples/6. Symbolics/Manifest.toml
index 4562be10..d93fbdd4 100644
--- a/examples/6. Symbolics/Manifest.toml	
+++ b/examples/6. Symbolics/Manifest.toml	
@@ -1,14 +1,19 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.9.0"
+julia_version = "1.9.1"
 manifest_format = "2.0"
 project_hash = "b85fe0e9ec615134f7a540386dbd29224a001a21"
 
+[[deps.ADTypes]]
+git-tree-sha1 = "dcfdf328328f2645531c4ddebf841228aef74130"
+uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
+version = "0.1.3"
+
 [[deps.AbstractAlgebra]]
-deps = ["GroupsCore", "InteractiveUtils", "LinearAlgebra", "MacroTools", "Random", "RandomExtensions", "SparseArrays", "Test"]
-git-tree-sha1 = "3ee5c58774f4487a5bf2bb05e39d91ff5022b4cc"
+deps = ["GroupsCore", "InteractiveUtils", "LinearAlgebra", "MacroTools", "Preferences", "Random", "RandomExtensions", "SparseArrays", "Test"]
+git-tree-sha1 = "1bd8a536c949eb3de9b58042d57790ded6b70fa6"
 uuid = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
-version = "0.29.4"
+version = "0.30.9"
 
 [[deps.AbstractTrees]]
 git-tree-sha1 = "faa260e4cb5aba097a73fab382dd4b5819d8ec8c"
@@ -31,9 +36,9 @@ version = "1.1.1"
 
 [[deps.ArrayInterface]]
 deps = ["Adapt", "LinearAlgebra", "Requires", "SparseArrays", "SuiteSparse"]
-git-tree-sha1 = "917286faa2abb288796e75b88ca67edc016f3219"
+git-tree-sha1 = "d3f758863a47ceef2248d136657cb9c033603641"
 uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9"
-version = "7.4.5"
+version = "7.4.8"
 
     [deps.ArrayInterface.extensions]
     ArrayInterfaceBandedMatricesExt = "BandedMatrices"
@@ -51,18 +56,6 @@ version = "7.4.5"
     StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
     Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"
 
-[[deps.ArrayInterfaceCore]]
-deps = ["LinearAlgebra", "SnoopPrecompile", "SparseArrays", "SuiteSparse"]
-git-tree-sha1 = "e5f08b5689b1aad068e01751889f2f615c7db36d"
-uuid = "30b0a656-2188-435a-8636-2ec0e6a096e2"
-version = "0.1.29"
-
-[[deps.ArrayInterfaceStaticArraysCore]]
-deps = ["ArrayInterfaceCore", "LinearAlgebra", "StaticArraysCore"]
-git-tree-sha1 = "01a9f8e6cfc2bfdd01d333f70b8014a04893103c"
-uuid = "dd5226c6-a4d4-4bc7-8575-46859f9c95b9"
-version = "0.1.4"
-
 [[deps.Artifacts]]
 uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"
 
@@ -92,9 +85,9 @@ uuid = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
 version = "1.0.2"
 
 [[deps.CommonSolve]]
-git-tree-sha1 = "9441451ee712d1aec22edad62db1a9af3dc8d852"
+git-tree-sha1 = "0eee5eb66b1cf62cd6ad1b460238e60e4b09400c"
 uuid = "38540f10-b2f7-11e9-35d8-d573e4eb0ff2"
-version = "0.2.3"
+version = "0.2.4"
 
 [[deps.CommonSubexpressions]]
 deps = ["MacroTools", "Test"]
@@ -161,9 +154,9 @@ version = "1.1.0"
 
 [[deps.DiffRules]]
 deps = ["IrrationalConstants", "LogExpFunctions", "NaNMath", "Random", "SpecialFunctions"]
-git-tree-sha1 = "a4ad7ef19d2cdc2eff57abbbe68032b1cd0bd8f8"
+git-tree-sha1 = "23163d55f885173722d1e4cf0f6110cdbaf7e272"
 uuid = "b552c78f-8df3-52c6-915a-8e097449b14b"
-version = "1.13.0"
+version = "1.15.1"
 
 [[deps.Distributed]]
 deps = ["Random", "Serialization", "Sockets"]
@@ -171,9 +164,9 @@ uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b"
 
 [[deps.Distributions]]
 deps = ["FillArrays", "LinearAlgebra", "PDMats", "Printf", "QuadGK", "Random", "SparseArrays", "SpecialFunctions", "Statistics", "StatsAPI", "StatsBase", "StatsFuns", "Test"]
-git-tree-sha1 = "5eeb2bd01e5065090ad591a205d8cad432ae6cb6"
+git-tree-sha1 = "c72970914c8a21b36bbc244e9df0ed1834a0360b"
 uuid = "31c24e10-a181-5473-b8eb-7969acd0382f"
-version = "0.25.93"
+version = "0.25.95"
 
     [deps.Distributions.extensions]
     DistributionsChainRulesCoreExt = "ChainRulesCore"
@@ -185,9 +178,9 @@ version = "0.25.93"
 
 [[deps.DocStringExtensions]]
 deps = ["LibGit2"]
-git-tree-sha1 = "b19534d1895d702889b219c382a6e18010797f0b"
+git-tree-sha1 = "2fb1e02f2b635d0845df5d7c167fec4dd739b00d"
 uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
-version = "0.8.6"
+version = "0.9.3"
 
 [[deps.DomainSets]]
 deps = ["CompositeTypes", "IntervalSets", "LinearAlgebra", "Random", "StaticArrays", "Statistics"]
@@ -227,9 +220,9 @@ uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"
 
 [[deps.FillArrays]]
 deps = ["LinearAlgebra", "Random", "SparseArrays", "Statistics"]
-git-tree-sha1 = "7072f1e3e5a8be51d525d64f63d3ec1287ff2790"
+git-tree-sha1 = "589d3d3bff204bdd80ecc53293896b4f39175723"
 uuid = "1a297f60-69ca-5386-bcde-b61e274b549b"
-version = "0.13.11"
+version = "1.1.1"
 
 [[deps.Formatting]]
 deps = ["Printf"]
@@ -264,15 +257,15 @@ uuid = "9fa8497b-333b-5362-9e8d-4d0656e87820"
 
 [[deps.GPUArraysCore]]
 deps = ["Adapt"]
-git-tree-sha1 = "1cd7f0af1aa58abc02ea1d872953a97359cb87fa"
+git-tree-sha1 = "2d6ca471a6c7b536127afccfa7564b5b39227fe0"
 uuid = "46192b85-c4d5-4398-a991-12ede77f4527"
-version = "0.1.4"
+version = "0.1.5"
 
 [[deps.Groebner]]
 deps = ["AbstractAlgebra", "Combinatorics", "Logging", "MultivariatePolynomials", "Primes", "Random", "SnoopPrecompile"]
-git-tree-sha1 = "b6c3e9e1eb8dcc6fd9bc68fe08dcc7ab22710de6"
+git-tree-sha1 = "c8b55b624a83f60bcd6574cc999ad148d0a47dd6"
 uuid = "0b43b601-686d-58a3-8a1c-6623616c7cd4"
-version = "0.3.4"
+version = "0.3.6"
 
 [[deps.GroupsCore]]
 deps = ["Markdown", "Random"]
@@ -292,9 +285,9 @@ uuid = "615f187c-cbe4-4ef1-ba3b-2fcf58d6d173"
 version = "0.1.1"
 
 [[deps.IntegerMathUtils]]
-git-tree-sha1 = "f366daebdfb079fd1fe4e3d560f99a0c892e15bc"
+git-tree-sha1 = "70f65ced5129d36dbf200b07c51ea8a955294660"
 uuid = "18e54dd8-cb9d-406c-a71d-865a43cbb235"
-version = "0.1.0"
+version = "0.1.1"
 
 [[deps.InteractiveUtils]]
 deps = ["Markdown"]
@@ -352,6 +345,12 @@ version = "0.16.0"
     DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
     SymEngine = "123dc426-2d89-5057-bbad-38513e3affd8"
 
+[[deps.Lazy]]
+deps = ["MacroTools"]
+git-tree-sha1 = "1370f8202dac30758f3c345f9909b97f53d87d3f"
+uuid = "50d2b5c4-7a5e-59d5-8109-a42b560f39c0"
+version = "0.15.1"
+
 [[deps.LibCURL]]
 deps = ["LibCURL_jll", "MozillaCACerts_jll"]
 uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"
@@ -392,9 +391,9 @@ uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 
 [[deps.LogExpFunctions]]
 deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
-git-tree-sha1 = "0a1b7c2863e44523180fdb3146534e265a91870b"
+git-tree-sha1 = "c3ce8e7420b3a6e071e0fe4745f5d4300e37b13f"
 uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
-version = "0.3.23"
+version = "0.3.24"
 
     [deps.LogExpFunctions.extensions]
     LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
@@ -508,9 +507,9 @@ version = "0.4.12"
 
 [[deps.PrecompileTools]]
 deps = ["Preferences"]
-git-tree-sha1 = "259e206946c293698122f63e2b513a7c99a244e8"
+git-tree-sha1 = "9673d39decc5feece56ef3940e5dafba15ba0f81"
 uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
-version = "1.1.1"
+version = "1.1.2"
 
 [[deps.Preferences]]
 deps = ["TOML"]
@@ -535,10 +534,10 @@ uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
 version = "2.8.2"
 
 [[deps.Quaternions]]
-deps = ["LinearAlgebra", "Random", "RealDot"]
-git-tree-sha1 = "da095158bdc8eaccb7890f9884048555ab771019"
+deps = ["DualNumbers", "LinearAlgebra", "Random"]
+git-tree-sha1 = "0b345302b17b0e694092621915de0e0dc7443a1a"
 uuid = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
-version = "0.7.4"
+version = "0.4.9"
 
 [[deps.REPL]]
 deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
@@ -554,12 +553,6 @@ git-tree-sha1 = "062986376ce6d394b23d5d90f01d81426113a3c9"
 uuid = "fb686558-2515-59ef-acaa-46db3789a887"
 version = "0.4.3"
 
-[[deps.RealDot]]
-deps = ["LinearAlgebra"]
-git-tree-sha1 = "9f0a1b71baaf7650f4fa8a1d168c7fb6ee41f0c9"
-uuid = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
-version = "0.1.0"
-
 [[deps.RecipesBase]]
 deps = ["PrecompileTools"]
 git-tree-sha1 = "5c3d09cc4f31f5fc6af001c250bf1278733100ff"
@@ -567,10 +560,20 @@ uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
 version = "1.3.4"
 
 [[deps.RecursiveArrayTools]]
-deps = ["Adapt", "ArrayInterfaceCore", "ArrayInterfaceStaticArraysCore", "ChainRulesCore", "DocStringExtensions", "FillArrays", "GPUArraysCore", "IteratorInterfaceExtensions", "LinearAlgebra", "RecipesBase", "StaticArraysCore", "Statistics", "Tables", "ZygoteRules"]
-git-tree-sha1 = "a5ce741acddc02f0d4fc6505463ca89697d7fb23"
+deps = ["Adapt", "ArrayInterface", "DocStringExtensions", "GPUArraysCore", "IteratorInterfaceExtensions", "LinearAlgebra", "RecipesBase", "Requires", "StaticArraysCore", "Statistics", "SymbolicIndexingInterface", "Tables"]
+git-tree-sha1 = "02ef02926f30d53b94be443bfaea010c47f6b556"
 uuid = "731186ca-8d62-57ce-b412-fbd966d074cd"
-version = "2.32.3"
+version = "2.38.5"
+
+    [deps.RecursiveArrayTools.extensions]
+    RecursiveArrayToolsMeasurementsExt = "Measurements"
+    RecursiveArrayToolsTrackerExt = "Tracker"
+    RecursiveArrayToolsZygoteExt = "Zygote"
+
+    [deps.RecursiveArrayTools.weakdeps]
+    Measurements = "eff96d63-e80a-5855-80a2-b1b0885c5ab7"
+    Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"
+    Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 
 [[deps.Reexport]]
 git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
@@ -585,9 +588,9 @@ version = "1.3.0"
 
 [[deps.RigidBodyDynamics]]
 deps = ["DocStringExtensions", "LightXML", "LinearAlgebra", "LoopThrottle", "Random", "Reexport", "Rotations", "SparseArrays", "StaticArrays", "TypeSortedCollections", "UnsafeArrays"]
-git-tree-sha1 = "067a44aead5b587252e8cf73773cca422c6ae19b"
+git-tree-sha1 = "313ea256ced33aa6039987c9fef57e59aa229589"
 uuid = "366cf18f-59d5-5db9-a4de-86a9f6786172"
-version = "2.3.2"
+version = "2.4.0"
 
 [[deps.Rmath]]
 deps = ["Random", "Rmath_jll"]
@@ -602,26 +605,32 @@ uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f"
 version = "0.4.0+0"
 
 [[deps.Rotations]]
-deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays"]
-git-tree-sha1 = "54ccb4dbab4b1f69beb255a2c0ca5f65a9c82f08"
+deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays", "Statistics"]
+git-tree-sha1 = "405148000e80f70b31e7732ea93288aecb1793fa"
 uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc"
-version = "1.5.1"
+version = "1.2.0"
 
 [[deps.RuntimeGeneratedFunctions]]
 deps = ["ExprTools", "SHA", "Serialization"]
-git-tree-sha1 = "d7d9ebe28062161c1e314ed643097b0c6fe657d9"
+git-tree-sha1 = "0b9b18d6236e9ab2b092defaacdffd929d572642"
 uuid = "7e49a35a-f44a-4d26-94aa-eba1b4ca6b47"
-version = "0.5.7"
+version = "0.5.9"
 
 [[deps.SHA]]
 uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
 version = "0.7.0"
 
 [[deps.SciMLBase]]
-deps = ["ArrayInterfaceCore", "CommonSolve", "ConstructionBase", "Distributed", "DocStringExtensions", "EnumX", "FunctionWrappersWrappers", "IteratorInterfaceExtensions", "LinearAlgebra", "Logging", "Markdown", "Preferences", "RecipesBase", "RecursiveArrayTools", "RuntimeGeneratedFunctions", "StaticArraysCore", "Statistics", "Tables"]
-git-tree-sha1 = "fe89a8113ea445bcff9ee570077830674babb534"
+deps = ["ADTypes", "ArrayInterface", "CommonSolve", "ConstructionBase", "Distributed", "DocStringExtensions", "EnumX", "FunctionWrappersWrappers", "IteratorInterfaceExtensions", "LinearAlgebra", "Logging", "Markdown", "PrecompileTools", "Preferences", "RecipesBase", "RecursiveArrayTools", "Reexport", "RuntimeGeneratedFunctions", "SciMLOperators", "StaticArraysCore", "Statistics", "SymbolicIndexingInterface", "Tables", "TruncatedStacktraces"]
+git-tree-sha1 = "ddf57a23d610d30d0c4bd01132f4646a2a77d171"
 uuid = "0bca4576-84f4-4d90-8ffe-ffa030f20462"
-version = "1.81.0"
+version = "1.92.5"
+
+[[deps.SciMLOperators]]
+deps = ["ArrayInterface", "DocStringExtensions", "Lazy", "LinearAlgebra", "Setfield", "SparseArrays", "StaticArraysCore", "Tricks"]
+git-tree-sha1 = "8b3d4f7f3d80b58b0a87b56d22cf1138e0a99aa4"
+uuid = "c0aeaf25-5076-4817-a8d5-81caf7dfa961"
+version = "0.2.11"
 
 [[deps.Serialization]]
 uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"
@@ -712,6 +721,12 @@ deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"]
 uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
 version = "5.10.1+6"
 
+[[deps.SymbolicIndexingInterface]]
+deps = ["DocStringExtensions"]
+git-tree-sha1 = "f8ab052bfcbdb9b48fad2c80c873aa0d0344dfe5"
+uuid = "2efcf032-c050-4f8e-a9bb-153293bab1f5"
+version = "0.2.2"
+
 [[deps.SymbolicUtils]]
 deps = ["AbstractTrees", "Bijections", "ChainRulesCore", "Combinatorics", "ConstructionBase", "DataStructures", "DocStringExtensions", "DynamicPolynomials", "IfElse", "LabelledArrays", "LinearAlgebra", "MultivariatePolynomials", "NaNMath", "Setfield", "SparseArrays", "SpecialFunctions", "StaticArrays", "TimerOutputs", "Unityper"]
 git-tree-sha1 = "5cb1f963f82e7b81305102dd69472fcd3e0e1483"
@@ -720,9 +735,9 @@ version = "1.0.5"
 
 [[deps.Symbolics]]
 deps = ["ArrayInterface", "ConstructionBase", "DataStructures", "DiffRules", "Distributions", "DocStringExtensions", "DomainSets", "Groebner", "IfElse", "LaTeXStrings", "LambertW", "Latexify", "Libdl", "LinearAlgebra", "MacroTools", "Markdown", "NaNMath", "RecipesBase", "Reexport", "Requires", "RuntimeGeneratedFunctions", "SciMLBase", "Setfield", "SparseArrays", "SpecialFunctions", "StaticArrays", "SymbolicUtils", "TreeViews"]
-git-tree-sha1 = "e23ec62c083ca8f15a4b7174331b3b8d1c511e47"
+git-tree-sha1 = "c4e5688a9a5a2f4088cd1a779b0742be31cbe9aa"
 uuid = "0c5d862f-8b57-4792-8d23-62f2024744c7"
-version = "5.3.1"
+version = "5.5.0"
 
 [[deps.TOML]]
 deps = ["Dates"]
@@ -762,6 +777,17 @@ git-tree-sha1 = "8d0d7a3fe2f30d6a7f833a5f19f7c7a5b396eae6"
 uuid = "a2a6695c-b41b-5b7d-aed9-dbfdeacea5d7"
 version = "0.3.0"
 
+[[deps.Tricks]]
+git-tree-sha1 = "aadb748be58b492045b4f56166b5188aa63ce549"
+uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
+version = "0.1.7"
+
+[[deps.TruncatedStacktraces]]
+deps = ["InteractiveUtils", "MacroTools", "Preferences"]
+git-tree-sha1 = "7bc1632a4eafbe9bd94cf1a784a9a4eb5e040a91"
+uuid = "781d530d-4396-4725-bb49-402e4bee1e77"
+version = "1.3.0"
+
 [[deps.TypeSortedCollections]]
 git-tree-sha1 = "d539b357e7695d80c75f5b066cec5d8c45886ab2"
 uuid = "94a5cd58-49a0-5741-bd07-fa4f4be8babf"
@@ -796,16 +822,10 @@ deps = ["Libdl"]
 uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
 version = "1.2.13+0"
 
-[[deps.ZygoteRules]]
-deps = ["ChainRulesCore", "MacroTools"]
-git-tree-sha1 = "977aed5d006b840e2e40c0b48984f7463109046d"
-uuid = "700de1a5-db45-46bc-99cf-38207098b444"
-version = "0.2.3"
-
 [[deps.libblastrampoline_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
 
 [[deps.nghttp2_jll]]
 deps = ["Artifacts", "Libdl"]
diff --git a/examples/7. Rigorous error bounds using IntervalArithmetic/Manifest.toml b/examples/7. Rigorous error bounds using IntervalArithmetic/Manifest.toml
index 4b3a07f6..9b094486 100644
--- a/examples/7. Rigorous error bounds using IntervalArithmetic/Manifest.toml	
+++ b/examples/7. Rigorous error bounds using IntervalArithmetic/Manifest.toml	
@@ -1,6 +1,6 @@
 # This file is machine-generated - editing it directly is not advised
 
-julia_version = "1.9.0"
+julia_version = "1.9.1"
 manifest_format = "2.0"
 project_hash = "3899da068c88cead675439c424121794dbc2c0d1"
 
@@ -26,6 +26,12 @@ git-tree-sha1 = "e329286945d0cfc04456972ea732551869af1cfc"
 uuid = "4e9b3aee-d8a1-5a3d-ad8b-7d824db253f0"
 version = "1.0.1+0"
 
+[[deps.Calculus]]
+deps = ["LinearAlgebra"]
+git-tree-sha1 = "f641eb0a4f00c343bbc32346e1217b86f3ce9dad"
+uuid = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9"
+version = "0.5.1"
+
 [[deps.CompilerSupportLibraries_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
@@ -37,15 +43,21 @@ uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"
 
 [[deps.DocStringExtensions]]
 deps = ["LibGit2"]
-git-tree-sha1 = "b19534d1895d702889b219c382a6e18010797f0b"
+git-tree-sha1 = "2fb1e02f2b635d0845df5d7c167fec4dd739b00d"
 uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
-version = "0.8.6"
+version = "0.9.3"
 
 [[deps.Downloads]]
 deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
 uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
 version = "1.6.0"
 
+[[deps.DualNumbers]]
+deps = ["Calculus", "NaNMath", "SpecialFunctions"]
+git-tree-sha1 = "5837a837389fccf076445fce071c8ddaea35a566"
+uuid = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"
+version = "0.6.8"
+
 [[deps.ErrorfreeArithmetic]]
 git-tree-sha1 = "d6863c556f1142a061532e79f611aa46be201686"
 uuid = "90fa49ef-747e-5e6f-a989-263ba693cf1a"
@@ -70,6 +82,11 @@ git-tree-sha1 = "c1c88395d09366dae431556bcb598ad08fa1392b"
 uuid = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
 version = "0.20.8"
 
+[[deps.IrrationalConstants]]
+git-tree-sha1 = "630b497eafcc20001bba38a4651b327dcfc491d2"
+uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
+version = "0.2.2"
+
 [[deps.JLLWrappers]]
 deps = ["Preferences"]
 git-tree-sha1 = "abc9885a7ca2052a736a600f7fa66209f96506e1"
@@ -114,6 +131,22 @@ version = "0.9.0"
 deps = ["Libdl", "OpenBLAS_jll", "libblastrampoline_jll"]
 uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 
+[[deps.LogExpFunctions]]
+deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
+git-tree-sha1 = "c3ce8e7420b3a6e071e0fe4745f5d4300e37b13f"
+uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
+version = "0.3.24"
+
+    [deps.LogExpFunctions.extensions]
+    LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
+    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
+    LogExpFunctionsInverseFunctionsExt = "InverseFunctions"
+
+    [deps.LogExpFunctions.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+    ChangesOfVariables = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0"
+    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"
+
 [[deps.Logging]]
 uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"
 
@@ -136,6 +169,12 @@ version = "2.28.2+0"
 uuid = "14a3606d-f60d-562e-9121-12d972cd8159"
 version = "2022.10.11"
 
+[[deps.NaNMath]]
+deps = ["OpenLibm_jll"]
+git-tree-sha1 = "0877504529a3e5c3343c6f8b4c0381e57e4387e4"
+uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
+version = "1.0.2"
+
 [[deps.NetworkOptions]]
 uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"
 version = "1.2.0"
@@ -145,6 +184,17 @@ deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
 uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
 version = "0.3.21+4"
 
+[[deps.OpenLibm_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "05823500-19ac-5b8b-9628-191a04bc5112"
+version = "0.8.1+0"
+
+[[deps.OpenSpecFun_jll]]
+deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
+uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
+version = "0.5.5+0"
+
 [[deps.Pkg]]
 deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
 uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
@@ -152,9 +202,9 @@ version = "1.9.0"
 
 [[deps.PrecompileTools]]
 deps = ["Preferences"]
-git-tree-sha1 = "259e206946c293698122f63e2b513a7c99a244e8"
+git-tree-sha1 = "9673d39decc5feece56ef3940e5dafba15ba0f81"
 uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
-version = "1.1.1"
+version = "1.1.2"
 
 [[deps.Preferences]]
 deps = ["TOML"]
@@ -167,10 +217,10 @@ deps = ["Unicode"]
 uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 
 [[deps.Quaternions]]
-deps = ["LinearAlgebra", "Random", "RealDot"]
-git-tree-sha1 = "da095158bdc8eaccb7890f9884048555ab771019"
+deps = ["DualNumbers", "LinearAlgebra", "Random"]
+git-tree-sha1 = "0b345302b17b0e694092621915de0e0dc7443a1a"
 uuid = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
-version = "0.7.4"
+version = "0.4.9"
 
 [[deps.REPL]]
 deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
@@ -180,12 +230,6 @@ uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"
 deps = ["SHA", "Serialization"]
 uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
-[[deps.RealDot]]
-deps = ["LinearAlgebra"]
-git-tree-sha1 = "9f0a1b71baaf7650f4fa8a1d168c7fb6ee41f0c9"
-uuid = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
-version = "0.1.0"
-
 [[deps.RecipesBase]]
 deps = ["PrecompileTools"]
 git-tree-sha1 = "5c3d09cc4f31f5fc6af001c250bf1278733100ff"
@@ -199,15 +243,15 @@ version = "1.2.2"
 
 [[deps.RigidBodyDynamics]]
 deps = ["DocStringExtensions", "LightXML", "LinearAlgebra", "LoopThrottle", "Random", "Reexport", "Rotations", "SparseArrays", "StaticArrays", "TypeSortedCollections", "UnsafeArrays"]
-git-tree-sha1 = "067a44aead5b587252e8cf73773cca422c6ae19b"
+git-tree-sha1 = "313ea256ced33aa6039987c9fef57e59aa229589"
 uuid = "366cf18f-59d5-5db9-a4de-86a9f6786172"
-version = "2.3.2"
+version = "2.4.0"
 
 [[deps.Rotations]]
-deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays"]
-git-tree-sha1 = "54ccb4dbab4b1f69beb255a2c0ca5f65a9c82f08"
+deps = ["LinearAlgebra", "Quaternions", "Random", "StaticArrays", "Statistics"]
+git-tree-sha1 = "405148000e80f70b31e7732ea93288aecb1793fa"
 uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc"
-version = "1.5.1"
+version = "1.2.0"
 
 [[deps.RoundingEmulator]]
 git-tree-sha1 = "40b9edad2e5287e05bd413a38f61a8ff55b9557b"
@@ -233,6 +277,18 @@ uuid = "6462fe0b-24de-5631-8697-dd941f90decc"
 deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
 uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 
+[[deps.SpecialFunctions]]
+deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
+git-tree-sha1 = "ef28127915f4229c971eb43f3fc075dd3fe91880"
+uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
+version = "2.2.0"
+
+    [deps.SpecialFunctions.extensions]
+    SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"
+
+    [deps.SpecialFunctions.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+
 [[deps.StaticArrays]]
 deps = ["LinearAlgebra", "Random", "StaticArraysCore", "Statistics"]
 git-tree-sha1 = "8982b3607a212b070a5e46eea83eb62b4744ae12"
@@ -299,7 +355,7 @@ version = "1.2.13+0"
 [[deps.libblastrampoline_jll]]
 deps = ["Artifacts", "Libdl"]
 uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
 
 [[deps.nghttp2_jll]]
 deps = ["Artifacts", "Libdl"]