ó ¡¼™\c@sãddlmZmZmZmZmZmZmZmZ m Z ddl m Z m Z mZmZddlmZmZddlmZmZddlmZddgZdefd„ƒYZdefd „ƒYZd „Zd S( iÿÿÿÿ( tdifftexpandtsintcostsympifyteyetsymbolstImmutableMatrixt MatrixBase(ttrigsimptsolvetSymboltDummy(t string_typestrange(tVectort _check_vector(t translatet CoordinateSymtReferenceFramecBsAeZdZd„Zed„ƒZd„Zd„Zd„ZRS(s¦ A coordinate symbol/base scalar associated wrt a Reference Frame. Ideally, users should not instantiate this class. Instances of this class must only be accessed through the corresponding frame as 'frame[index]'. CoordinateSyms having the same frame and index parameters are equal (even though they may be instantiated separately). Parameters ========== name : string The display name of the CoordinateSym frame : ReferenceFrame The reference frame this base scalar belongs to index : 0, 1 or 2 The index of the dimension denoted by this coordinate variable Examples ======== >>> from sympy.physics.vector import ReferenceFrame, CoordinateSym >>> A = ReferenceFrame('A') >>> A[1] A_y >>> type(A[0]) >>> a_y = CoordinateSym('a_y', A, 1) >>> a_y == A[1] True cCs~i}tt|ƒj||ƒtt|ƒj|||}t|ƒ|tddƒkrktdƒ‚n||f|_|S(NiisInvalid index specified(tsuperRt _sanitizet__xnew__t _check_frameRt ValueErrort_id(tclstnametframetindext assumptionstobj((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt__new__1s cCs |jdS(Ni(R(tself((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyR=scCs,t|tƒr(|j|jkr(tSntS(N(t isinstanceRRtTruetFalse(R!tother((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt__eq__AscCs ||k S(N((R!R%((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt__ne__IscCs*t|jdjƒ|jdfƒjƒS(Nii(ttupleRt__hash__(R!((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyR)Ls( t__name__t __module__t__doc__R tpropertyRR&R'R)(((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyR s $   cBsæeZdZdZdddd„Zd„Zd„Zd„ZeZ d„Z d„Z d„Z d „Z d „Zd „Zd d „Zd dddd„Zd„Zd„Zed„ƒZed„ƒZed„ƒZd„ZRS(s÷A reference frame in classical mechanics. ReferenceFrame is a class used to represent a reference frame in classical mechanics. It has a standard basis of three unit vectors in the frame's x, y, and z directions. It also can have a rotation relative to a parent frame; this rotation is defined by a direction cosine matrix relating this frame's basis vectors to the parent frame's basis vectors. It can also have an angular velocity vector, defined in another frame. icCslt|tƒstdƒ‚n|d k r‚t|ttfƒsNtdƒ‚nt|ƒdkrotdƒ‚nx,|D]$}t|tƒsvtdƒ‚qvqvW|d|dd|d|d d|d|d dg|_|j ƒd |d|j ƒd |d |j ƒd |d g|_ d |j ƒ|dfd |j ƒ|d fd |j ƒ|d fg|_ ||_ n|d |d|dg|_|j ƒd|j ƒd|j ƒdg|_ d|j ƒd|j ƒd|j ƒg|_ dddg|_ |d k ržt|ttfƒsBtdƒ‚nt|ƒdkrctdƒ‚nx,|D]$}t|tƒsjtdƒ‚qjqjW||_ n||_ i|_i|_i|_i|_i|_|j|j|jg|_d|_ttd ddgƒ|fgƒ|_ttdd dgƒ|fgƒ|_ttddd gƒ|fgƒ|_|d k rðt|ttfƒstdƒ‚nt|ƒdkr¾tdƒ‚nxJ|D]$}t|tƒsÅtdƒ‚qÅqÅWn|d|d|dg}t|d|dƒt|d |d ƒt|d |d ƒf|_tjd 7_tj|_d S(!s0ReferenceFrame initialization method. A ReferenceFrame has a set of orthonormal basis vectors, along with orientations relative to other ReferenceFrames and angular velocities relative to other ReferenceFrames. Parameters ========== indices : list (of strings) If custom indices are desired for console, pretty, and LaTeX printing, supply three as a list. The basis vectors can then be accessed with the get_item method. latexs : list (of strings) If custom names are desired for LaTeX printing of each basis vector, supply the names here in a list. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, vlatex >>> N = ReferenceFrame('N') >>> N.x N.x >>> O = ReferenceFrame('O', indices=('1', '2', '3')) >>> O.x O['1'] >>> O['1'] O['1'] >>> P = ReferenceFrame('P', latexs=('A1', 'A2', 'A3')) >>> vlatex(P.x) 'A1' sNeed to supply a valid namesSupply the indices as a listisSupply 3 indicessIndices must be stringss['is']iiu_s\mathbf{\hat{%s}_{%s}}s.xs.ys.zu_xu_yu_zs\mathbf{\hat{%s}_x}s\mathbf{\hat{%s}_y}s\mathbf{\hat{%s}_z}txtytzsLatex entries must be stringss)Supply the variable names as a list/tuplesSupply 3 variable namessVariable names must be stringst_xt_yt_zN(R"R t TypeErrortNoneR(tlisttlenRtstr_vecstlowert pretty_vecst latex_vecstindicesRt _var_dictt _dcm_dictt _dcm_cachet _ang_vel_dictt _ang_acc_dictt_dlistt_curRtMatrixR1R2R3RtvarlistRt_countR(R!RR<tlatexst variablesti((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt__init___s€$                   '''  cCs“t|tƒs5|dkr&|j|Stdƒ‚n|jd|krO|jS|jd|kri|jS|jd|krƒ|jStdƒ‚dS(sà Returns basis vector for the provided index, if the index is a string. If the index is a number, returns the coordinate variable correspon- -ding to that index. isInvalid index providediiisNot a defined indexN(R"R RERR<R.R/R0(R!tind((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt __getitem__Ïs  cCst|j|j|jgƒS(N(titerR.R/R0(R!((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt__iter__äscCs|jS(sReturns the name of the frame. (R(R!((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt__str__çsc CsG|gg}gg}x¦||kr½|}xŒt|ƒD]~\}}|dj|jƒ}xXt|ƒD]J\}} |j| ƒsh|| g} |j| ƒs²|j| ƒq²qhqhWq8WqWx:t|ƒD],\}}|d|krË|j|ƒqËqËW|jdtƒt|ƒdkr%|dStd|j d|j ƒ‚dS(s3Creates a list from self to other using _dcm_dict. iÿÿÿÿtkeyis!No Connecting Path found between s and N( t enumerateRBtkeyst __contains__tappendtremovetsortR7RR( R!R%tnumtoutlisttoldlistRItvttemplistti2tv2tlittletemplist((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt _dict_listís&   c Csµddlm}|j|ƒ}|j|jƒ}||j}tt|dƒdtƒ}tt|dƒdtƒ}tt|dƒdtƒ}t t |||gƒ|fgƒ S(s4Angular velocity from time differentiating the DCM. iÿÿÿÿ(tdynamicsymbolsit recursiveii( tsympy.physics.vector.functionsR`tdcmRt_ttTR RR#RRD( R!t otherframeR`tdcm2difftdiffedt angvelmattw1tw2tw3((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt _w_diff_dcms cCsÕt|ƒ|tjf|jkr6|j|tjfS|j|ƒt|jƒ}i}x\t|ƒD]N\}}tjržt||ddƒ||j|>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols >>> A = ReferenceFrame('A') >>> q = dynamicsymbols('q') >>> B = A.orientnew('B', 'Axis', [q, A.z]) >>> A.variable_map(B) {A_x: B_x*cos(q(t)) - B_y*sin(q(t)), A_y: B_x*sin(q(t)) + B_y*cos(q(t)), A_z: B_z} tmethodtfuN( RRtsimpR=RcRDRERQR (R!Rft vars_matrixtmappingRIR.((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt variable_maps  $cCs>t|ƒ||jkr$|j|S|j|ƒj|ƒSdS(sReturns the angular acceleration Vector of the ReferenceFrame. Effectively returns the Vector: ^N alpha ^B which represent the angular acceleration of B in N, where B is self, and N is otherframe. Parameters ========== otherframe : ReferenceFrame The ReferenceFrame which the angular acceleration is returned in. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, Vector >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> V = 10 * N.x >>> A.set_ang_acc(N, V) >>> A.ang_acc_in(N) 10*N.x N(RRAt ang_vel_intdt(R!Rf((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt ang_acc_in:s  cCsjt|ƒ|j|dƒ}tdƒ}x;tt|ƒdƒD]#}|||j||d7}q?W|S(s‘Returns the angular velocity Vector of the ReferenceFrame. Effectively returns the Vector: ^N omega ^B which represent the angular velocity of B in N, where B is self, and N is otherframe. Parameters ========== otherframe : ReferenceFrame The ReferenceFrame which the angular velocity is returned in. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, Vector >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> V = 10 * N.x >>> A.set_ang_vel(N, V) >>> A.ang_vel_in(N) 10*N.x ii(RR_RRR7R@(R!RftflisttoutvecRI((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyRt[s   !cCs¡t|ƒ||jkr$|j|S|j|dƒ}tdƒ}x;tt|ƒdƒD]#}|||j||d}qYW||j|<|j|j|<|S(sÝThe direction cosine matrix between frames. This gives the DCM between this frame and the otherframe. The format is N.xyz = N.dcm(B) * B.xyz A SymPy Matrix is returned. Parameters ========== otherframe : ReferenceFrame The otherframe which the DCM is generated to. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, Vector >>> from sympy import symbols >>> q1 = symbols('q1') >>> N = ReferenceFrame('N') >>> A = N.orientnew('A', 'Axis', [q1, N.x]) >>> N.dcm(A) Matrix([ [1, 0, 0], [0, cos(q1), -sin(q1)], [0, sin(q1), cos(q1)]]) iii(RR?R_RRR7R>Re(R!RfRwtoutdcmRI((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyRc}s   ! tc) CsÍddlm}t|ƒ|dkrGt|tƒs’tdƒ‚q’nKt|ƒ}x<t|ƒD].\}}t|tƒs`t |ƒ||>> from sympy.physics.vector import ReferenceFrame, Vector >>> from sympy import symbols, eye, ImmutableMatrix >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') >>> N = ReferenceFrame('N') >>> B = ReferenceFrame('B') Now we have a choice of how to implement the orientation. First is Body. Body orientation takes this reference frame through three successive simple rotations. Acceptable rotation orders are of length 3, expressed in XYZ or 123, and cannot have a rotation about about an axis twice in a row. >>> B.orient(N, 'Body', [q1, q2, q3], 123) >>> B.orient(N, 'Body', [q1, q2, 0], 'ZXZ') >>> B.orient(N, 'Body', [0, 0, 0], 'XYX') Next is Space. Space is like Body, but the rotations are applied in the opposite order. >>> B.orient(N, 'Space', [q1, q2, q3], '312') Next is Quaternion. This orients the new ReferenceFrame with Quaternions, defined as a finite rotation about lambda, a unit vector, by some amount theta. This orientation is described by four parameters: q0 = cos(theta/2) q1 = lambda_x sin(theta/2) q2 = lambda_y sin(theta/2) q3 = lambda_z sin(theta/2) Quaternion does not take in a rotation order. >>> B.orient(N, 'Quaternion', [q0, q1, q2, q3]) Next is Axis. This is a rotation about an arbitrary, non-time-varying axis by some angle. The axis is supplied as a Vector. This is how simple rotations are defined. >>> B.orient(N, 'Axis', [q1, N.x + 2 * N.y]) Last is DCM (Direction Cosine Matrix). This is a rotation matrix given manually. >>> B.orient(N, 'DCM', eye(3)) >>> B.orient(N, 'DCM', ImmutableMatrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]])) iÿÿÿÿ(R`tDCMs+Amounts must be a sympy Matrix type object.cSsý|dkrStdddgdt|ƒt|ƒ gdt|ƒt|ƒggƒS|dkr¦tt|ƒdt|ƒgdddgt|ƒ dt|ƒggƒS|dkrùtt|ƒt|ƒ dgt|ƒt|ƒdgdddggƒSdS(s(DCM for simple axis 1,2,or 3 rotations. iiiiN(RDRR(taxistangle((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt_rotýs     t123t231t312t132t213t321t121t131t212t232t313t323RztXYZxyzt123123s*The supplied order is not an approved typetAXISs(Axis orientation takes no rotation orderis'Amounts are a list or tuple of length 2iisAxis cannot be time-varyingit QUATERNIONs.Quaternion orientation takes no rotation orderis'Amounts are a list or tuple of length 4tBODYs*Body orientation takes 3 values & 3 orderstSPACEs+Space orientation takes 3 values & 3 orderss#That is not an implemented rotation(tCoercionFailed(tkinematic_equationss u1, u2, u3RN( RR€RR‚RƒR„R…R†R‡RˆR‰RŠRz(1RbR`RR"RR4R6RQRRRtstrtupperR(R7RRuRtexpresst normalizetargsRReRRDRtinttNotImplementedErrorR?RRR>RBtupdateRdRRmtsympy.polys.polyerrorsR‘R’RR RR R.R/R0tAssertionErrorR@R=()R!tparenttrot_typetamountst rot_orderR`RIRZR~tapproved_orderst parent_orienttthetaR|tq0tq1tq2tq3ta1ta2ta3tframest dcm_dict_delt dcm_cache_delRtttq0dtq1dtq2dtq3dRjRkRltwvectthetadR‘R’tu1tu2tu3R[ttd((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pytorient¨sÜF        %   ‚   %.2/; ,8 ,8         &&&'    'c Cs;|j|d|d|d|ƒ}|j||||ƒ|S(sÀCreates a new ReferenceFrame oriented with respect to this Frame. See ReferenceFrame.orient() for acceptable rotation types, amounts, and orders. Parent is going to be self. Parameters ========== newname : str The name for the new ReferenceFrame rot_type : str The type of orientation matrix that is being created. amounts : list OR value The quantities that the orientation matrix will be defined by. rot_order : str If applicable, the order of a series of rotations. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, Vector >>> from sympy import symbols >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3') >>> N = ReferenceFrame('N') Now we have a choice of how to implement the orientation. First is Body. Body orientation takes this reference frame through three successive simple rotations. Acceptable rotation orders are of length 3, expressed in XYZ or 123, and cannot have a rotation about about an axis twice in a row. >>> A = N.orientnew('A', 'Body', [q1, q2, q3], '123') >>> A = N.orientnew('A', 'Body', [q1, q2, 0], 'ZXZ') >>> A = N.orientnew('A', 'Body', [0, 0, 0], 'XYX') Next is Space. Space is like Body, but the rotations are applied in the opposite order. >>> A = N.orientnew('A', 'Space', [q1, q2, q3], '312') Next is Quaternion. This orients the new ReferenceFrame with Quaternions, defined as a finite rotation about lambda, a unit vector, by some amount theta. This orientation is described by four parameters: q0 = cos(theta/2) q1 = lambda_x sin(theta/2) q2 = lambda_y sin(theta/2) q3 = lambda_z sin(theta/2) Quaternion does not take in a rotation order. >>> A = N.orientnew('A', 'Quaternion', [q0, q1, q2, q3]) Last is Axis. This is a rotation about an arbitrary, non-time-varying axis by some angle. The axis is supplied as a Vector. This is how simple rotations are defined. >>> A = N.orientnew('A', 'Axis', [q1, N.x]) RHR<RG(t __class__R¹( R!tnewnameRžRŸR RHR<RGtnewframe((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt orientnew}s>cCsd|dkrtdƒ}nt|ƒ}t|ƒ|jji||6ƒ|jji| |6ƒdS(s5Define the angular acceleration Vector in a ReferenceFrame. Defines the angular acceleration of this ReferenceFrame, in another. Angular acceleration can be defined with respect to multiple different ReferenceFrames. Care must be taken to not create loops which are inconsistent. Parameters ========== otherframe : ReferenceFrame A ReferenceFrame to define the angular acceleration in value : Vector The Vector representing angular acceleration Examples ======== >>> from sympy.physics.vector import ReferenceFrame, Vector >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> V = 10 * N.x >>> A.set_ang_acc(N, V) >>> A.ang_acc_in(N) 10*N.x iN(RRRRARš(R!Rftvalue((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt set_ang_accÀs    cCsd|dkrtdƒ}nt|ƒ}t|ƒ|jji||6ƒ|jji| |6ƒdS(s!Define the angular velocity vector in a ReferenceFrame. Defines the angular velocity of this ReferenceFrame, in another. Angular velocity can be defined with respect to multiple different ReferenceFrames. Care must be taken to not create loops which are inconsistent. Parameters ========== otherframe : ReferenceFrame A ReferenceFrame to define the angular velocity in value : Vector The Vector representing angular velocity Examples ======== >>> from sympy.physics.vector import ReferenceFrame, Vector >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> V = 10 * N.x >>> A.set_ang_vel(N, V) >>> A.ang_vel_in(N) 10*N.x iN(RRRR@Rš(R!RfR¾((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyt set_ang_veläs    cCs|jS(s=The basis Vector for the ReferenceFrame, in the x direction. (R1(R!((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyR.scCs|jS(s=The basis Vector for the ReferenceFrame, in the y direction. (R2(R!((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyR/ scCs|jS(s=The basis Vector for the ReferenceFrame, in the z direction. (R3(R!((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyR0scGs\g|D]'}|j|ƒj||dtƒ^q}t|ƒdkrN|dSt|ƒSdS(s¬Returns the partial angular velocities of this frame in the given frame with respect to one or more provided generalized speeds. Parameters ========== frame : ReferenceFrame The frame with which the angular velocity is defined in. gen_speeds : functions of time The generalized speeds. Returns ======= partial_velocities : tuple of Vector The partial angular velocity vectors corresponding to the provided generalized speeds. Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> u1, u2 = dynamicsymbols('u1, u2') >>> A.set_ang_vel(N, u1 * A.x + u2 * N.y) >>> A.partial_velocity(N, u1) A.x >>> A.partial_velocity(N, u1, u2) (A.x, N.y) t var_in_dcmiiN(RtRR$R7R((R!Rt gen_speedstspeedtpartials((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pytpartial_velocitys 1N(R*R+R,RFR5RJRLRNROt__repr__R_RmRsRvRtRcR¹R½R¿RÀR-R.R/R0RÅ(((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyRPs, p     * ! " + ÕB $ $cCs;ddlm}t|tƒs7||tdƒƒ‚ndS(Ni(tVectorTypeErrortA(tvectorRÇR"R(R%RÇ((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyR@sN(tsympy.core.backendRRRRRRRRRDRtsympyR R R R tsympy.core.compatibilityR Rtsympy.physics.vector.vectorRRtsympy.utilities.miscRt__all__RtobjectRR(((s9lib/python2.7/site-packages/sympy/physics/vector/frame.pyts@" Eÿÿò