OtjZCdgZddlmZmZmZddlmZddlmZddl m Z ddl m Z m Z ddlmZddlmZGd dZd Zd S) Linearizer)MatrixeyezerosDummy)flatten)dynamicsymbols)msubs_parse_linear_solver) namedtuple)Iterablec@eZdZdZ d dZdZdZdZdZd d Z dS) raIThis object holds the general model form for a dynamic system. This model is used for computing the linearized form of the system, while properly dealing with constraints leading to dependent coordinates and speeds. The notation and method is described in [1]_. Attributes ========== f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a : Matrix Matrices holding the general system form. q, u, r : Matrix Matrices holding the generalized coordinates, speeds, and input vectors. q_i, u_i : Matrix Matrices of the independent generalized coordinates and speeds. q_d, u_d : Matrix Matrices of the dependent generalized coordinates and speeds. perm_mat : Matrix Permutation matrix such that [q_ind, u_ind]^T = perm_mat*[q, u]^T References ========== .. [1] D. L. Peterson, G. Gede, and M. Hubbard, "Symbolic linearization of equations of motion of constrained multibody systems," Multibody Syst Dyn, vol. 33, no. 2, pp. 143-161, Feb. 2015, doi: 10.1007/s11044-014-9436-5. NLUct||_t||_t||_t||_t||_t||_t||_t||_ t||_ t| |_ t| |_ d}|| |_ || |_|| |_|||_|||_|||_|j t(j|_|j t(j|_t1|j|j tfd|jD|_t7|j}t7|j }t7|j }t7|j }t7|j}t7|j}t9dgd}||||||||_d|_d|_d|_ d|_!d|_"d|_#d|_$d|_%d|_&d|_'d|_(dS)a Parameters ========== f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a : array_like System of equations holding the general system form. Supply empty array or Matrix if the parameter does not exist. q : array_like The generalized coordinates. u : array_like The generalized speeds q_i, u_i : array_like, optional The independent generalized coordinates and speeds. q_d, u_d : array_like, optional The dependent generalized coordinates and speeds. r : array_like, optional The input variables. lams : array_like, optional The lagrange multipliers linear_solver : str, callable Method used to solve the several symbolic linear systems of the form ``A*x=b`` in the linearization process. If a string is supplied, it should be a valid method that can be used with the :meth:`sympy.matrices.matrixbase.MatrixBase.solve`. If a callable is supplied, it should have the format ``x = f(A, b)``, where it solves the equations and returns the solution. The default is ``'LU'`` which corresponds to SymPy's ``A.LUsolve(b)``. ``LUsolve()`` is fast to compute but will often result in divide-by-zero and thus ``nan`` results. c@|rt|n tS)N)r)xs _/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/physics/mechanics/linearize.pyz%Linearizer.__init__..^sa!=VXXc8g|]}|vr|n tSr).0vardup_varss r z'Linearizer.__init__..ms7,,,3c&9&9ssuww,,,rdims)lmnoskNF))r linear_solverrf_0f_1f_2f_3f_4f_cf_vf_aquq_iq_du_iu_drlamsdiffr _t_qd_udset intersection_qd_duplenr _dims_Pq_Pqi_Pqd_Pu_Pui_Pud_C_0_C_1_C_2perm_mat _setup_done)selfr%r&r'r(r)r*r+r,r-r.r/r0r1r2r3r4r$ none_handlerrrr r!r"r#rrs @r__init__zLinearizer.__init__,sgF2-@@#;;#;;#;;#;;#;;#;;#;;#;;== <$$<$$<$$<$$a L&& 6;;~0116;;~011tx==--df55,,,,"&(,,,--  MM MM KK KK KK  NN&"@"@"@AAT!Q1a++          rc|||d|_dS)NT)_form_permutation_matrices_form_block_matrices_form_coefficient_matricesrH)rIs r_setupzLinearizer._setupsH ''))) !!### '')))rc |j\}}}}}}|dkrt|jt|j|jg|_|dkr3|jddd| f|_|jdd| df|_n|j|_t|_|dkrt|j t|j |j g|_ |dkr3|j ddd| f|_ |j dd| df|_n|j |_ t|_t|jt||z||z g}tt|||z |j t|||z g}|r'|r|||_dS||_dS||_dS)z(Form the permutation matrices Pq and Pu.rN)r=permutation_matrixr-rr/r0r>r?r@r.r1r2rArBrCrrow_joinrG) rIrrr r!r"r#P_col1P_col2s rrMz%Linearizer._form_permutation_matricess :1aAq 66)$&&$(DH9M2N2NOODH1uu HQQQ!V,  HQQQV,  H "HH 66)$&&$(DH9M2N2NOODH1uu HQQQ!V,  HQQQV,  H "HH E!a%Q$7$7899q!a%$)U1a!e__EFF  # ' & 7 7 & "DMMMrc|j\}}}}}}|dkrc|j|j}t ||j|||jz|zz |jz|_nt ||_|dkr|j |j }||j z} |dkrD|j |j} |j || | z|_ nt|||_ t ||j || |zz |jz|_dSt|||_ t ||_dS)z0Form the coefficient matrices C_0, C_1, and C_2.rN)r=r*jacobianr-rr@r$r?rDr+r.rCrErrBrF) rIrrr r!r"r# f_c_jac_q f_v_jac_utemp f_v_jac_qs rrOz%Linearizer._form_coefficient_matricessn :1aAq q55))$&11IQ$)++Idi,?,577#778< BDIIADI q55))$&11Ity(DAvv H--df55 !YJ););D))L)LL !!QKK Q$)++D)<<#==>BiHDIIIa DIADIIIrc|j\}}}}}}|dkrR|j|j|_|j|jz|j |_n&t|_t|_|dkrP|dkrJ|j |j |_ |j |j |_ n&t|_ t|_ |dkrf||z |zdkrZ|j |j |_|j|j z|jz|j |_n&t|_t|_|dkrP|dkrJ|j |j|_|j |j |_n&t|_t|_|dkr^||z |zdkrR|j|j|_|j|j z|j |_n&t|_t|_|dkr,|dkr&|j|j |_nt|_|dkr1||z |zdkr%|j|j|_nt|_|dkr3||z |zdkr'|j |j |_dSt|_dS)z2Form the block matrices for composing M, A, and B.rN)r=r%rWr7_M_qqr&r-_A_qqrr,r;_M_uqc_A_uqcr(_M_uqdr'r)_A_uqdr8_M_uucr._A_uuc_M_uud_A_uud_A_qur4_M_uldr3_B_u)rIrrr r!r"r#s rrNzLinearizer._form_block_matricess :1aAq 66**4844DJ8dh.88@@@DJJDJDJ 66a1ff(++DL99DK8,,TV444DKK ((DK ((DK 66a!eai1nn(++DL99DK Htx/$(:DDTVLLLDKK ((DK ((DK 66a1ff(++DH55DK8,,TV444DKK ((DK ((DK 66a!eai1nn(++DH55DK Htx/99$&AAADKK ((DK ((DK 66a1ff(++DF333DJJDJ 66a!eai1nn(++DI66DKK ((DK 66a!eai1nn**46222DIIIDIIIrFcF|js|t|tr|}n4t|tri}|D]}||ni}|j\}}}} } } |j} |j} |j }|j }|j }|j }|j }|j}|j}|j}|j}|j}|j}|j}|j}|j}| dkr t/t1|| ||g}| dkr"t/t1||z| |g}|dkrft/| | |g}| dkr/| dkr)|||}n=| dkr||}n!|}n| dkr||}n|}t5||} |dkr|}!| dkr|!||zz }!|!|z}!|dkr|}"| dkr|"||zz }"|"|z}"nt/}"| |z | zdkr|}#| dkr|#||zz }#|#|z}#nt/}#t/|!|"|#g}nt/}| dkrg|dkr||z}$nt/}$|dkr||z}%nt/}%| |z | zdkr||z}&nt/}&t/|$|%|&g}nt/}|r|r||}'n|}'n|}'t5|'|}(| dkrC| |z | zdkr7t1||z| |})t5|)|}*nt/}*|ry|jj|| |(z}+|*r$|jj|| |*z},n|*},|r(|+|,|+|,fS|r<| |(|*| |(|*fS)a!Linearize the system about the operating point. Note that q_op, u_op, qd_op, ud_op must satisfy the equations of motion. These may be either symbolic or numeric. Parameters ========== op_point : dict or iterable of dicts, optional Dictionary or iterable of dictionaries containing the operating point conditions for all or a subset of the generalized coordinates, generalized speeds, and time derivatives of the generalized speeds. These will be substituted into the linearized system before the linearization is complete. Leave set to ``None`` if you want the operating point to be an arbitrary set of symbols. Note that any reduction in symbols (whether substituted for numbers or expressions with a common parameter) will result in faster runtime. A_and_B : bool, optional If A_and_B=False (default), (M, A, B) is returned and of A_and_B=True, (A, B) is returned. See below. simplify : bool, optional Determines if returned values are simplified before return. For large expressions this may be time consuming. Default is False. Returns ======= M, A, B : Matrices, ``A_and_B=False`` Matrices from the implicit form: ``[M]*[q', u']^T = [A]*[q_ind, u_ind]^T + [B]*r`` A, B : Matrices, ``A_and_B=True`` Matrices from the explicit form: ``[q_ind', u_ind']^T = [A]*[q_ind, u_ind]^T + [B]*r`` Notes ===== Note that the process of solving with A_and_B=True is computationally intensive if there are many symbolic parameters. For this reason, it may be more desirable to use the default A_and_B=False, returning M, A, and B. More values may then be substituted in to these matrices later on. The state space form can then be found as A = P.T*M.LUsolve(A), B = P.T*M.LUsolve(B), where P = Linearizer.perm_mat. r) rHrP isinstancedictrupdater=r]r_rarcrerhr^r`rbrgrdrfrirDrErFrrrSr col_joinrGTr$simplify)-rIop_pointA_and_Brp op_point_dictoprrr r!r"r#M_qqM_uqcM_uqdM_uucM_uudM_uldA_qqA_uqcA_uqdA_quA_uucA_uudB_uC_0C_1C_2col2col3col1MM_eqr1c1r2c1r3c1r1c2r2c2r3c2AmatAmat_eqBmatBmat_eqA_contB_conts- r linearizezLinearizer.linearizes\  KKMMM h % % $MM ( + + M ) )$$R(((( )M :1aAqz     z  z  iiii 665A;;u566D 665Q??E233D 664.//DAvv!q&&MM$''0066aMM$'' !VV d##AAAQ && 66DAvv$#:DAvv66US[)Dczxx1uqyA~~66US[)Dczxx4t,--DD88D 66AvvczxxAvvs{xx1uqyA~~s{xx4t,--DD88D   }}T**Dm,, 66a!eai1nnQ??++C00DD-00GGhhG  *]_t'9'9$'H'HHF !4+=+=dG+L+LL! "!!!!!!6> ! #   """  """') )r)NNNNNNr)NFF) __name__ __module__ __qualname____doc__rKrPrMrOrNrrrrrr s<JN<@#Y!Y!Y!Y!v   ###B@0!0!0!dx*x*x*x*x*x*rctttfstt|ttfst|}t t |krt dfd|D}t t}t|D] \}}d|||f< |S)akCompute the permutation matrix to change order of orig_vec into order of per_vec. Parameters ========== orig_vec : array_like Symbols in original ordering. per_vec : array_like Symbols in new ordering. Returns ======= p_matrix : Matrix Permutation matrix such that orig_vec == (p_matrix * per_vec). zKorig_vec and per_vec must be the same length, and contain the same symbols.c:g|]}|Sr)index)riorig_vecs rrz&permutation_matrix..s%333aq!!333r) rklisttupler r9 ValueErrorrr< enumerate)rper_vecind_listp_matrixrjs` rrRrRs$ hu . .%8$$ ge} - -#'"" 8}}G $$9:: :33337333HS]]##H(##1A OrN)__all__sympyrrrsympy.core.symbolrsympy.utilities.iterablesr sympy.physics.vectorr !sympy.physics.mechanics.functionsr r collectionsr collections.abcrrrRrrrrs .$$$$$$$$$$######------//////IIIIIIII""""""$$$$$$m*m*m*m*m*m*m*m*` r