From d24da261a5aa5b339f65309e38126dd3f4fa1d48 Mon Sep 17 00:00:00 2001 From: Tmoroz0077 Date: Wed, 27 Dec 2017 18:30:06 +0300 Subject: [PATCH] issue #54 added better picture and supplemented the model description --- doc/chap_examples.rst | 83 +++++++++++++++------ doc/pic/chapter06_section05_inpendulum.png | Bin 36563 -> 154870 bytes 2 files changed, 61 insertions(+), 22 deletions(-) diff --git a/doc/chap_examples.rst b/doc/chap_examples.rst index 6a4ffd24..6e9e9731 100644 --- a/doc/chap_examples.rst +++ b/doc/chap_examples.rst @@ -531,7 +531,57 @@ Inverted Pendulum On a Cart Inverted pendulum -The mechanical system presented in :num:`figure #invpendfig` is described by the following system of equations: +Consider a mechanical system consisting of a cart (body 1) and a rod (body 2) attached to the cart with a joint. We assume that movement of the bodies is plane. The system represents an inverted pendulum with unstable equilibrium at :math:`\theta = \frac{\pi}{2}.` There is a control :math:`u(t)` that presents a force applied to the cart along the axis OX. + +Therefore we define + +- :math:`\theta` - angle between axis OX and the rod; + +- :math:`m_1` and :math:`m_2` - weights of the bodies 1 and 2 respectively; + +- :math:`L` - length of the rod; + +- :math:`k_1, k_2` - the coefficient of dynamic viscosity for the bodies 1 and 2 respectively; + +- :math:`J_c` - inertia moment of the second body relative to its center of mass; + +- :math:`u` - the force applied to cart and we assume :math:`|u| \leqslant \alpha`. + + +Firstly we consider the system without any control. Writing down expressions for kinetic and potential energies of the bodies 1 and 2 we obtain: + +.. math:: + :label: invpend_energy + + \begin{aligned} + T_1 &= \frac{m_1v^2}{2}, \\ + T_2 &= \frac{J_c\omega^2}{2}, \\ + \Pi_1 &= 0, \\ + \Pi_2 &= m_2g\frac{L}{2}\sin(\theta). + \end{aligned} + +Here :math:`T_1` and :math:`T_2` stand for kinetic energy of cart and rod respectively, :math:`\Pi_1` and :math:`\Pi_2` stand for potential energy, :math:`v` is a speed of a cart and :math:`\omega` is an angular velocity of the rod. Further we will replace :math:`v` with :math:`\dot x`, :math:`\omega` with :math:`\dot \theta`. + +As there is viscosity, we write down generalized forces: + +.. math:: + :label: invpend_forces + + \begin{aligned} + Q_1 &= k_1\dot x, \\ + Q_2 &= k_2L\dot \theta. + \end{aligned} + +Writing down the Langrange equations in case of potential and nonpotential forces we obtain: + +.. math:: + :label: invpend_lagrange + + \frac{d}{dt}(\frac{\partial T}{\partial \dot q_i}) - \frac{\partial T}{\partial q_i} + \frac{\partial \Pi}{\partial q_i} = Q_i. + +Here the Lagrange coordinates :math:`q_1 = x, q_2 = \theta`. + +Applying previously obtained values to the equations and adding the control to the first equation, we get: .. math:: :label: invpend1 @@ -541,21 +591,11 @@ The mechanical system presented in :num:`figure #invpendfig` is described by the .. math:: :label: invpend2 - (J_c + \frac{m_2L^2}{4})\ddot{\theta}+\frac{m_2gL\cos(\theta)}{2} + k_2\dot{\theta}L = 0 . - -Here :math:`k_1, k_2` are the coefficient of dynamic viscosity for the bodies 1 and 2, with masses :math:`m_1` and :math:`m_2` respectively, - -:math:`J_c` is inertia moment of the second body relatively to its center of mass, + (J_c + \frac{m_2L^2}{4})\ddot{\theta}+\frac{m_2gL\cos(\theta)}{2} + k_2\dot{\theta}L = 0 . -:math:`u` is the force applied to cart and we shall assume -:math:`|u| \leqslant \alpha`. - -The initial conditions can -be taken as :math:`x(0)=0`, :math:`x_2(0)=\pi/2 - \varepsilon`, where :math:`\varepsilon=10^{-3}`. After linerarization in the neighbourhood of :math:`\frac{\pi}{2}` we have :math:`\cos(\theta) \approx \frac{\pi}{2} - \theta`. -Defining :math:`x_1 = x, x_2=\dot{x}_1, x_3 = \theta` and :math:`x_4=\dot{x}_3`, we can rewrite -:eq:`invpend1`-:eq:`invpend2` as a linear system in standard form: +Defining :math:`x_1 = x, x_2=\dot{x}_1, x_3 = \theta` and :math:`x_4=\dot{x}_3`, we can rewrite :eq:`invpend1`-:eq:`invpend2` as a linear system in standard form: .. math:: :label: invpendls @@ -573,18 +613,17 @@ Defining :math:`x_1 = x, x_2=\dot{x}_1, x_3 = \theta` and :math:`x_4=\dot{x}_3`, x_1 \\ x_2 \\ x_3 \\ - x_4 \end{array}\right] + \left[\begin{array}{cc} - 0 \\ - 0 \\ - 0 \\ - -\frac{m_2Lg\pi}{4J_c+m_2L^2} \end{array}\right] + \left[\begin{array}{cc} + x_4 \end{array}\right] + \left[\begin{array}{cc} 0 \\ \frac{1}{m_1} \\ 0 \\ - 0 \end{array}\right]u. - -Now we can compute the reach set of system :eq:`invpend1`-:eq:`invpend2` for given time by computing the reach set of the linear system :eq:`invpendls` -and taking its projection onto :math:`(x_1, x_3)` subspace. + 0 \end{array}\right]u + \left[\begin{array}{cc} + 0 \\ + 0 \\ + 0 \\ + -\frac{m_2Lg\pi}{4J_c+m_2L^2} \end{array}\right]. + +Consider some moment of time :math:`t_1` and final position :math:`x_1(t_1) = x_1, x_2(t_1) = 0, x_3(t_1) = \frac{\pi}{2}, x_4(t_1) = 0`. It is required to calculate the backward reachability sets (tube) for the linearized system :eq:`invpendls` emanating from the given final position and project it onto :math:`(x_1, x_3)` subspace. It is also required to identify whether it’s possible to reach the final position from a given initial position :math:`x_1(t_1) = x_1, x_2(t_1) = v_1, x_3(t_1) = \theta_1, x_4(t_1) = \omega_1` using some admissible control function. .. raw:: html diff --git a/doc/pic/chapter06_section05_inpendulum.png b/doc/pic/chapter06_section05_inpendulum.png index b7a88d2a0f34690c56fe5f5c66b088906e0fd4e8..be76ac7123934772dcb55e54ef24b6aff7e77ae8 100644 GIT binary patch literal 154870 zcmeFacOVve`#*k?%BbA3O4&kLrDStQ(+*{Cl09=1(OpC#%8JZN8py~fGb0&kD0^fl zo9y+w-nY*AoaZ_B{XE})e*ZoGIgaDJ@AtT_*Y&zy*Y$q8rLKB<2Ng5bAAkI@TQ^f|g`bGuj2VJ|Y;x5&E&oSG!@gnoUe{Vn$4y5?S<>9mUdYtK(aciF z!`=yg>yJOAJtW~?ZcdWI!Z;jG2zNxt(Zx#mu!Mw!@F5Xl z5fMT74MA5=2RBm>K?hf^jY0k#=Y*xJxr?=vo3*0@I~v#2%<;0D3@0c0pufI0=ILgA z@&7)_!S(mDV1dHuzX%@|Iwbtp*zmtfqn}Euxma7mC!_IY4@(n%^Iv}U`*Wm)(TD$S zG8-!;{1lcdOC>G**S5)0DVOu={_zL)kFzI^Yk6!M?c8K%sO>2}+1Ojo|G1vMajVT5 zHRk5FP1hN@H*ej3L@!CmJ^IqI>_YO~v?Tlwt%K*1DL*~^LC$a{#fIOe?b02aAac78 zDKS5HY!)i`{*0CUP}uc7LVowICvV+8`y+AYSm_(<(V4jZ)``w#se6$lxKiB2qR)_S z|HihS8sTGNVh+j37_N9uzTH+aI%+3=<=3StD+dRm`i~z2%&YIn2;pzuyqRg; z_@wva;D-kaE7o>)+lLZ;eOlgJwjEB*Xy|)b-Q$Tz?Cl_@(>pHTeGIR&gLaYr(6qDc zjBLuv%6^3tC4~zUcezfh>Ud{93Ch@IU~FtGpw&`x)oY=ybfxaomoFLuW;z3csojB}Ij9NC7Dh(K@??d8@(*I&``*2KC%;mx zciL_p6GXqZHnA@IDmQn_82j_fDfL0H7r7k3FON0V2oMG*LT&NPOo3ASf6dPN-@SVm zgHwH)OWS_j&E5T`jMtoBQ&W@PgOWwGiC6{-KJ};9PafJrEUcoVWPd>BjPuiXc)i~M z*Lm6W;zgb#XB|4`YeYv)~cFMOdWe7J5Fdrsi)XXSI)NpM(pyf$IlR zQxlV<=lbGtKfhEOGIXtu;{y0I3OFmxZx9xf#*>6K6=fOvn=Sp@iQ8gy4l4hcn4k#? z3bOUEvJW_@^>l3Zh!-p)E&3Xn|Li??Hj+!qA#-(g?fLoQrBI&hdri~wDQaeF2D6cX zf=BX(hMiNXtE$KbB$(8b!e(f${UW)XaHha2ie@7(`-1T|Bk}?oPrtV0+3O<G8c6(GAeo-#}D=Qi5xS*o}QB7pFMjvnSm?j4-O4YY-+XZ zDw24Wmv>XU_~-emhkhw$evMbc<%|Rvg(~I!#n-_xq$<5tai8U_tgH&A-tASWui4mL zDthm&|7ptu3wgYMD6mmA8@Q^78;*?zjCz^{zFHO&Ays=n^rkMB)VAq7DmyK=^tm^KGE20)=8npPWBx63->2(@X|k}%7v3RPTw7hP zz;E7G4qs>0r1Gu=o}|k1_dkw|W04Mwqf|8dFe#VL+=^dZT!hgT6clVdn#~pZGB!mU zy6bv$FL!A-N>S{SxpH)6MsD4;VED-c)i>%Kc`pM6*}4`RaI%awj%XjkJU4BjiZM2$ zB#Cys{1~?P#@XIOQ_~GE!1|Gv54s ze(2DlFe;gz#ksk;N#XTrVbS6jm6UWdoM=bl)Uo(-O5GxoVA`+yJxbq!a>Aops=T+i zH+GMnzJA4QvoTv9!06z7)6cztfwzqs8yjs^O_%4!dK#oAwm%SaG@9`$CPV`prSJ7^ zu^Fu-H;4Qn|70@F)gO;<(=qSeTQQd3%?e(cVZgv!`7Q9|p6-QrV&wsHD?RFb^i8j@ z;HNzU!oHsOV`6k0m&uV`V#g?3jwjDG)d!ORfHG^^z{1Uy7s%f|BFhr*VOyVKW4oGM z|7gBwNG8T{NS)_oU?E>+)9${$zJZdI-AvTdvg?Z<1Jd>LGls8TM}sE0laD5;OW9gJ z`OEPC^_%anDC<9b@V9+gJom~_n3`#+Y@2N6V6Pv<%*ob|KF7TKx!5RYXJ>Uy#0U1p z#>U<*JX85FBsaLVwRQ5QSkYTmD%<09O5hI6V6W=t)ED3U>-c}gy6;+BdK)O24BgHZ zOq49E&T7PX`gTb4++bVz9y@LA;IPwpxvd!hV<@&OI^cWDAP1xU%1G(jNGz3@*@J-Q z;_=HpKR!S5G`YtG$M3=m=7m!dG9_+a?+eT1Rx0GuJeOAm9>go_2k1m#+G*`tv-j5> zO^wqB@cV2%Gx)*Mn4j@LQ7^+w^~dUY_pvb6*E@jhZMytN_LC4+QUxY&sd;l&vi13H zadYI{f@9JkRM%zkG zf0t7Nstco5a(m~-JM~oy?w=<&(+_T8H5WWdHF2f6P+mam5DFVfa}fW(t})mC>%jjP zQPK`#@@o>~;tWK2)OgLaPQ7MTs-m>@OiJLb3$K~AxJ{4n9_Zp#k^C7}!lXITq+jiR zB2QvS)%>#DW~043_lY~b5!TOXUjA`fj``H7Q?&`}ymE^*qF38XN2!OT4>2C-@L3)+ zg@sw36&NFXc=D{S!F8CJdM-rQ=kC_Z!6d>&Tx=PN1l!~m-f9+}x!EOi-MoorKo;;BkICPk>7w%{;#@k{fNEy;Oc835q{XGls7T0$P z3JKk8;bHfoD~+2%_FA+$<|xdoReyluYjbmen(tC$GJK`IsJ#B*)~#FFW?K+t#m-U! zcPkM+%!Rg!NFa4mBm=&&!_m8MZ$(E(#{olLCC9Z-0kZV(s;lc1pXKBnS!tA8)zr

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