From 34525add68c7f90c8b4bf75c8c4dc232bfbc2e3e Mon Sep 17 00:00:00 2001 From: Videh Patel Date: Fri, 5 Jan 2024 17:03:37 +0530 Subject: [PATCH] Mathematical description of RS curves --- .../path_planning/reeds_shepp_path/LR_L.png | Bin 0 -> 280028 bytes .../path_planning/reeds_shepp_path/LR_LR.png | Bin 0 -> 381584 bytes .../path_planning/reeds_shepp_path/LSL.png | Bin 0 -> 239825 bytes .../reeds_shepp_path/LSL90xR.png | Bin 0 -> 301712 bytes .../path_planning/reeds_shepp_path/LSR.png | Bin 0 -> 257435 bytes .../reeds_shepp_path/LSR90_L.png | Bin 0 -> 339492 bytes .../reeds_shepp_path/L_R90SL.png | Bin 0 -> 316723 bytes .../reeds_shepp_path/L_R90SL90_R.png | Bin 0 -> 335979 bytes .../reeds_shepp_path/L_R90SR.png | Bin 0 -> 291754 bytes .../path_planning/reeds_shepp_path/L_RL.png | Bin 0 -> 277432 bytes .../path_planning/reeds_shepp_path/L_RL_R.png | Bin 0 -> 323010 bytes .../path_planning/reeds_shepp_path/L_R_L.png | Bin 0 -> 271666 bytes .../reeds_shepp_path_main.rst | 375 ++++++++++++++++++ 13 files changed, 375 insertions(+) create mode 100644 docs/modules/path_planning/reeds_shepp_path/LR_L.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/LR_LR.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/LSL.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/LSL90xR.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/LSR.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/LSR90_L.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/L_R90SL.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/L_R90SL90_R.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/L_R90SR.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/L_RL.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/L_RL_R.png create mode 100644 docs/modules/path_planning/reeds_shepp_path/L_R_L.png diff --git 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z8G1`9-otbc=VLbxY&HD%Pxu~0rF(eFnvPf`Cj6k|>S->XpB|_G7@i&yZz6AdhIHTu zk!j*5ohIVdI{u{6B5%vIi0YZ6~Y`wQEq9FAQpPbx!1QM1F8|nV* zqFn;B1%4aqzoS5Oi6{BrMu7=LVm=hzc*6u_F76$1uWFlDrJQR*C zo7$Fm_CKfv7*8J#5#GQW)HaB~B0xW53oIA?iN#(}va2@{D#Pl_iqj*_t1GCCF*_9n%vegRTpJ7uoMVAH3D@ zfBAF^IS60cr)d!X2IfcT5Q$3pqXj-4(VMW|x5-@-wq@6|6K|buWcOf9UZ}!j#IFS) zI;@>bPd_Ee+WmbyV8v0@DtA~ew;@As09J$Nm(K}%h*uVmh6M~rrq4u?bWd_9{1_(m eaZ~BdH!@-Iy>!yjJ9D?ezti$()` + +Hence, we have: + +- :math:`u, t = polar(x - sin(φ), y + cos(φ) - 1)` +- :math:`v = φ - t` + + +2. **Left-Straight-Right** + +.. image:: LSR.png + +With followng notations: + +- :math:`∠MBD = t1` +- :math:`∠BDF = θ` +- :math:`BC = u1` + +We can deduce the following facts using geometry. + +- D is mid-point of BC and FG. +- :math:`t - v = φ` +- :math:`C(x + sin(φ), y - cos(φ))` +- :math:`A(0, 1)` +- :math:`u1, t1 = polar(vector)` +- :math:`\frac{u1^2}{4} = 1 + \frac{u^2}{4}` +- :math:`BF = 1` [Radius Of Curvature] +- :math:`FD = \frac{u}{2}` +- :math:`θ = arctan(\frac{BF}{FD})` +- :math:`t1 + θ = t` + +Hence, we have: + +- :math:`u1, t1 = polar(x + sin(φ), y - cos(φ) - 1)` +- :math:`u = \sqrt{u1^2 - 4}` +- :math:`θ = arctan(\frac{2}{u})` +- :math:`t = t1 + θ` +- :math:`v = t - φ` + +3. **LeftxRightxLeft** + +.. image:: L_R_L.png + +With followng notations: + +- :math:`∠CBD = ∠CDB = A` [BCD is an isoceles triangle] +- :math:`∠DBK = θ` +- :math:`BD = u1` + +We can deduce the following facts using geometry. + +- :math:`t + u + v = φ` +- :math:`D(x - sin(φ), y + cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`A = arccos(\frac{BD/2}{CD})` +- :math:`u = (π - 2*A)` +- :math:`∠ABK = \frac{π}{2}` +- :math:`∠KBD = θ` +- :math:`t = ∠ABK + ∠KBD + ∠DBC` + +Hence, we have: + +- :math:`u1, θ = polar(x - sin(φ), y + cos(φ) - 1)` +- :math:`A = arccos(\frac{u1/2}{2})` +- :math:`t = \frac{π}{2} + θ + A` +- :math:`u = (π - 2*A)` +- :math:`v = (φ - t - u)` + +4. **LeftxRight-Left** + +.. image:: L_RL.png + +With followng notations: + +- :math:`∠CBD = ∠CDB = A` [BCD is an isoceles triangle] +- :math:`∠DBK = θ` +- :math:`BD = u1` + +We can deduce the following facts using geometry. + +- :math:`t + u - v = φ` +- :math:`D(x - sin(φ), y + cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`A = arccos(\frac{BD/2}{CD})` +- :math:`u = (π - 2*A)` +- :math:`∠ABK = \frac{π}{2}` +- :math:`∠KBD = θ` +- :math:`t = ∠ABK + ∠KBD + ∠DBC` + +Hence, we have: + +- :math:`u1, θ = polar(x - sin(φ), y + cos(φ) - 1)` +- :math:`A = arccos(\frac{u1/2}{2})` +- :math:`t = \frac{π}{2} + θ + A` +- :math:`u = (π - 2*A)` +- :math:`v = (-φ + t + u)` + +5. **Left-RightxLeft** + +.. image:: LR_L.png + +With followng notations: + +- :math:`∠CBD = ∠CDB = A` [BCD is an isoceles triangle] +- :math:`∠DBK = θ` +- :math:`BD = u1` + +We can deduce the following facts using geometry. + +- :math:`t - u - v = φ` +- :math:`D(x - sin(φ), y + cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`BC = CD = 2` [2 * radius of curvature] +- :math:`cos(2π - u) = \frac{BC^2 + CD^2 - BD^2}{2 * BC * CD}` [Cosine Rule] +- :math:`\frac{sin(A)}{BC} = \frac{sin(u)}{u1}` [Sine Rule] +- :math:`∠ABK = \frac{π}{2}` +- :math:`∠KBD = θ` +- :math:`t = ∠ABK + ∠KBD - ∠DBC` + +Hence, we have: + +- :math:`u1, θ = polar(x - sin(φ), y + cos(φ) - 1)` +- :math:`u = arccos(1 - \frac{u1^2}{8})` +- :math:`A = arcsin(\frac{sin(u)}{u1}*2)` +- :math:`t = \frac{π}{2} + θ - A` +- :math:`v = (t - u - φ)` + +6. **Left-RightxLeft-Right** + +.. image:: LR_LR.png + +With followng notations: + +- :math:`∠CLG = ∠BCL = ∠CBG = ∠LGB = A = u` [BGCL is an isoceles trapezium] +- :math:`∠KBG = θ` +- :math:`BG = u1` + +We can deduce the following facts using geometry. + +- :math:`t - 2u + v = φ` +- :math:`G(x + sin(φ), y - cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`BC = CL = LG = 2` [2 * radius of curvature] +- :math:`CG^2 = CL^2 + LG^2 - 2*CL*LG*cos(A)` [Cosine rule in LGC] +- :math:`CG^2 = CL^2 + LG^2 - 2*CL*LG*cos(A)` [Cosine rule in LGC] +- From the previous two equations: :math:`A = arccos(\frac{u1 + 2}{4})` +- :math:`∠ABK = \frac{π}{2}` +- :math:`t = ∠ABK + ∠KBG + ∠GBC` + +Hence, we have: + +- :math:`u1, θ = polar(x + sin(φ), y - cos(φ) - 1)` +- :math:`u = arccos(\frac{u1 + 2}{4})` +- :math:`t = \frac{π}{2} + θ + u` +- :math:`v = (φ - t + 2u)` + +7. **LeftxRight-LeftxRight** + +.. image:: L_RL_R.png + +With followng notations: + +- :math:`∠GBC = A` [BGCL is an isoceles trapezium] +- :math:`∠KBG = θ` +- :math:`BG = u1` + +We can deduce the following facts using geometry. + +- :math:`t - v = φ` +- :math:`G(x + sin(φ), y - cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`BC = CL = LG = 2` [2 * radius of curvature] +- :math:`CD = 1` [radius of curvature] +- D is midpoint of BG +- :math:`BD = \frac{u1}{2}` +- :math:`cos(u) = \frac{BC^2 + CD^2 - BD^2}{2*BC*CD}` [Cosine rule in BCD] +- :math:`sin(A) = CD*\frac{sin(u)}{BD}` [Sine rule in BCD] +- :math:`∠ABK = \frac{π}{2}` +- :math:`t = ∠ABK + ∠KBG + ∠GBC` + +Hence, we have: + +- :math:`u1, θ = polar(x + sin(φ), y - cos(φ) - 1)` +- :math:`u = arccos(\frac{20 - u1^2}{16})` +- :math:`A = arcsin(2*\frac{sin(u)}{u1})` +- :math:`t = \frac{π}{2} + θ + A` +- :math:`v = (t - φ)` + + +8. **LeftxRight90-Straight-Left** + +.. image:: L_R90SL.png + +With followng notations: + +- :math:`∠FBM = A` [BGCL is an isoceles trapezium] +- :math:`∠KBF = θ` +- :math:`BF = u1` + +We can deduce the following facts using geometry. + +- :math:`t + \frac{π}{2} - v = φ` +- :math:`F(x - sin(φ), y + cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`BM = CB = 2` [2 * radius of curvature] +- :math:`MD = CD = 1` [CGDM is a rectangle] +- :math:`MC = GD = u` [CGDM is a rectangle] +- :math:`MF = MD + DF = 2` +- :math:`BM = \sqrt{BF^2 - MF^2}` [Pythagoras theorem on BFM] +- :math:`tan(A) = \frac{MF}{BM}` +- :math:`u = MC = BM - CB` +- :math:`t = ∠ABK + ∠KBF + ∠FBC` + +Hence, we have: + +- :math:`u1, θ = polar(x - sin(φ), y + cos(φ) - 1)` +- :math:`u = arccos(\sqrt{u1^2 - 4} - 2)` +- :math:`A = arctan(\frac{2}{\sqrt{u1^2 - 4}})` +- :math:`t = \frac{π}{2} + θ + A` +- :math:`v = (t - φ + \frac{π}{2})` + + +9. **Left-Straight-Right90xLeft** + +.. image:: LSR90_L.png + +With followng notations: + +- :math:`∠MBH = A` [BGCL is an isoceles trapezium] +- :math:`∠KBH = θ` +- :math:`BH = u1` + +We can deduce the following facts using geometry. + +- :math:`t - \frac{π}{2} - v = φ` +- :math:`H(x - sin(φ), y + cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`GH = 2` [2 * radius of curvature] +- :math:`CM = DG = 1` [CGDM is a rectangle] +- :math:`CD = MG = u` [CGDM is a rectangle] +- :math:`BM = BC + CM = 2` +- :math:`MH = \sqrt{BH^2 - BM^2}` [Pythagoras theorem on BHM] +- :math:`tan(A) = \frac{HM}{BM}` +- :math:`u = MC = BM - CB` +- :math:`t = ∠ABK + ∠KBH - ∠HBC` + +Hence, we have: + +- :math:`u1, θ = polar(x - sin(φ), y + cos(φ) - 1)` +- :math:`u = arccos(\sqrt{u1^2 - 4} - 2)` +- :math:`A = arctan(\frac{2}{\sqrt{u1^2 - 4}})` +- :math:`t = \frac{π}{2} + θ - A` +- :math:`v = (t - φ - \frac{π}{2})` + + +10. **LeftxRight90-Straight-Right** + +.. image:: L_R90SR.png + +With followng notations: + +- :math:`∠KBG = θ` +- :math:`BG = u1` + +We can deduce the following facts using geometry. + +- :math:`t - \frac{π}{2} - v = φ` +- :math:`G(x + sin(φ), y - cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`BD = 2` [2 * radius of curvature] +- :math:`DG = EF = u` [DGFE is a rectangle] +- :math:`DG = BG - BD = 2` +- :math:`∠ABK = \frac{π}{2}` +- :math:`t = ∠ABK + ∠KBG` + +Hence, we have: + +- :math:`u1, θ = polar(x + sin(φ), y - cos(φ) - 1)` +- :math:`u = u1 - 2` +- :math:`t = \frac{π}{2} + θ` +- :math:`v = (t - φ - \frac{π}{2})` + + +11. **Left-Straight-Left90xRight** + +.. image:: LSL90xR.png + +With followng notations: + +- :math:`∠KBH = θ` +- :math:`BH = u1` + +We can deduce the following facts using geometry. + +- :math:`t + \frac{π}{2} + v = φ` +- :math:`H(x + sin(φ), y - cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`GH = 2` [2 * radius of curvature] +- :math:`DC = BG = u` [DGBC is a rectangle] +- :math:`BG = BH - GH` +- :math:`∠ABC= ∠KBH` + +Hence, we have: + +- :math:`u1, θ = polar(x + sin(φ), y - cos(φ) - 1)` +- :math:`u = u1 - 2` +- :math:`t = θ` +- :math:`v = (φ - t - \frac{π}{2})` + + +12. **LeftxRight90-Straight-Left90xRight** + +.. image:: L_R90SL90_R.png + +With followng notations: + +- :math:`∠KBH = θ` +- :math:`∠HBM = A` +- :math:`BH = u1` + +We can deduce the following facts using geometry. + +- :math:`t - v = φ` +- :math:`H(x + sin(φ), y - cos(φ))` +- :math:`B(0, 1)` +- :math:`u1, θ = polar(vector)` +- :math:`GF = ED = 1` [radius of curvature] +- :math:`BD = GH = 2` [2 * radius of curvature] +- :math:`FN = GH = 2` [ENMD is a rectangle] +- :math:`NH = GF = 1` [FNHG is a rectangle] +- :math:`MN = ED = 1` [ENMD is a rectangle] +- :math:`DO = EF = u` [DOFE is a rectangle] +- :math:`MH = MN + NH = 2` +- :math:`BM = \sqrt{BH^2 - MH^2}` [Pythagoras theorem on BHM] +- :math:`DO = BM - BD - OM` +- :math:`tan(A) = \frac{MH}{BM}` +- :math:`∠ABC = ∠ABK + ∠KBH + ∠HBM` + +Hence, we have: + +- :math:`u1, θ = polar(x + sin(φ), y - cos(φ) - 1)` +- :math:`u = /sqrt{u1^2 - 4} - 4` +- :math:`A = arctan(\frac{2}{u1^2 - 4})` +- :math:`t = \frac{π}{2} + θ + A` +- :math:`v = (t - φ)` + + Ref: - `15.3.2 Reeds-Shepp

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