EdHRddlmZmZmZddlmZddlmZdgZGddZ dS))eyeMatrixzeros)dynamicsymbols)find_dynamicsymbolsSymbolicSystemc FeZdZdZddddiddddf dZedZedZedZedZ edZ ed Z ed Z ed Z d Zed ZedZdZdZedZedZdS)raB"SymbolicSystem is a class that contains all the information about a system in a symbolic format such as the equations of motions and the bodies and loads in the system. There are three ways that the equations of motion can be described for Symbolic System: [1] Explicit form where the kinematics and dynamics are combined x' = F_1(x, t, r, p) [2] Implicit form where the kinematics and dynamics are combined M_2(x, p) x' = F_2(x, t, r, p) [3] Implicit form where the kinematics and dynamics are separate M_3(q, p) u' = F_3(q, u, t, r, p) q' = G(q, u, t, r, p) where x : states, e.g. [q, u] t : time r : specified (exogenous) inputs p : constants q : generalized coordinates u : generalized speeds F_1 : right hand side of the combined equations in explicit form F_2 : right hand side of the combined equations in implicit form F_3 : right hand side of the dynamical equations in implicit form M_2 : mass matrix of the combined equations in implicit form M_3 : mass matrix of the dynamical equations in implicit form G : right hand side of the kinematical differential equations Parameters ========== coord_states : ordered iterable of functions of time This input will either be a collection of the coordinates or states of the system depending on whether or not the speeds are also given. If speeds are specified this input will be assumed to be the coordinates otherwise this input will be assumed to be the states. right_hand_side : Matrix This variable is the right hand side of the equations of motion in any of the forms. The specific form will be assumed depending on whether a mass matrix or coordinate derivatives are given. speeds : ordered iterable of functions of time, optional This is a collection of the generalized speeds of the system. If given it will be assumed that the first argument (coord_states) will represent the generalized coordinates of the system. mass_matrix : Matrix, optional The matrix of the implicit forms of the equations of motion (forms [2] and [3]). The distinction between the forms is determined by whether or not the coordinate derivatives are passed in. If they are given form [3] will be assumed otherwise form [2] is assumed. coordinate_derivatives : Matrix, optional The right hand side of the kinematical equations in explicit form. If given it will be assumed that the equations of motion are being entered in form [3]. alg_con : Iterable, optional The indexes of the rows in the equations of motion that contain algebraic constraints instead of differential equations. If the equations are input in form [3], it will be assumed the indexes are referencing the mass_matrix/right_hand_side combination and not the coordinate_derivatives. output_eqns : Dictionary, optional Any output equations that are desired to be tracked are stored in a dictionary where the key corresponds to the name given for the specific equation and the value is the equation itself in symbolic form coord_idxs : Iterable, optional If coord_states corresponds to the states rather than the coordinates this variable will tell SymbolicSystem which indexes of the states correspond to generalized coordinates. speed_idxs : Iterable, optional If coord_states corresponds to the states rather than the coordinates this variable will tell SymbolicSystem which indexes of the states correspond to generalized speeds. bodies : iterable of Body/Rigidbody objects, optional Iterable containing the bodies of the system loads : iterable of load instances (described below), optional Iterable containing the loads of the system where forces are given by (point of application, force vector) and torques are given by (reference frame acting upon, torque vector). Ex [(point, force), (ref_frame, torque)] Attributes ========== coordinates : Matrix, shape(n, 1) This is a matrix containing the generalized coordinates of the system speeds : Matrix, shape(m, 1) This is a matrix containing the generalized speeds of the system states : Matrix, shape(o, 1) This is a matrix containing the state variables of the system alg_con : List This list contains the indices of the algebraic constraints in the combined equations of motion. The presence of these constraints requires that a DAE solver be used instead of an ODE solver. If the system is given in form [3] the alg_con variable will be adjusted such that it is a representation of the combined kinematics and dynamics thus make sure it always matches the mass matrix entered. dyn_implicit_mat : Matrix, shape(m, m) This is the M matrix in form [3] of the equations of motion (the mass matrix or generalized inertia matrix of the dynamical equations of motion in implicit form). dyn_implicit_rhs : Matrix, shape(m, 1) This is the F vector in form [3] of the equations of motion (the right hand side of the dynamical equations of motion in implicit form). comb_implicit_mat : Matrix, shape(o, o) This is the M matrix in form [2] of the equations of motion. This matrix contains a block diagonal structure where the top left block (the first rows) represent the matrix in the implicit form of the kinematical equations and the bottom right block (the last rows) represent the matrix in the implicit form of the dynamical equations. comb_implicit_rhs : Matrix, shape(o, 1) This is the F vector in form [2] of the equations of motion. The top part of the vector represents the right hand side of the implicit form of the kinemaical equations and the bottom of the vector represents the right hand side of the implicit form of the dynamical equations of motion. comb_explicit_rhs : Matrix, shape(o, 1) This vector represents the right hand side of the combined equations of motion in explicit form (form [1] from above). kin_explicit_rhs : Matrix, shape(m, 1) This is the right hand side of the explicit form of the kinematical equations of motion as can be seen in form [3] (the G matrix). output_eqns : Dictionary If output equations were given they are stored in a dictionary where the key corresponds to the name given for the specific equation and the value is the equation itself in symbolic form bodies : Tuple If the bodies in the system were given they are stored in a tuple for future access loads : Tuple If the loads in the system were given they are stored in a tuple for future access. This includes forces and torques where forces are given by (point of application, force vector) and torques are given by (reference frame acted upon, torque vector). Example ======= As a simple example, the dynamics of a simple pendulum will be input into a SymbolicSystem object manually. First some imports will be needed and then symbols will be set up for the length of the pendulum (l), mass at the end of the pendulum (m), and a constant for gravity (g). :: >>> from sympy import Matrix, sin, symbols >>> from sympy.physics.mechanics import dynamicsymbols, SymbolicSystem >>> l, m, g = symbols('l m g') The system will be defined by an angle of theta from the vertical and a generalized speed of omega will be used where omega = theta_dot. :: >>> theta, omega = dynamicsymbols('theta omega') Now the equations of motion are ready to be formed and passed to the SymbolicSystem object. :: >>> kin_explicit_rhs = Matrix([omega]) >>> dyn_implicit_mat = Matrix([l**2 * m]) >>> dyn_implicit_rhs = Matrix([-g * l * m * sin(theta)]) >>> symsystem = SymbolicSystem([theta], dyn_implicit_rhs, [omega], ... dyn_implicit_mat) Notes ===== m : number of generalized speeds n : number of generalized coordinates o : number of states Nc ||mt|_|d|_n"fd|D} t| |_| d|_nofd| D} t| |_nLt|_t||_|j|j|_+|_||_||_d|_d|_ d|_ nW|+d|_d|_d|_||_||_ d|_ n*d|_d|_d|_d|_d|_ ||_ |fd|D}||_ ||_ t| ts| t| } t| ts| t| } | |_| |_dS)z#Initializes a SymbolicSystem objectNc g|] }| Sr .0i coord_statess >lib/python3.11/site-packages/sympy/physics/mechanics/system.py z+SymbolicSystem.__init__..s>>>a,q/>>>c g|] }| Sr r r s rrz+SymbolicSystem.__init__..sDDDA QDDDrc4g|]}|tzSr )len)rrcoordinate_derivativess rrz+SymbolicSystem.__init__..s'HHH1q35666HHHr)r_states _coordinates_speedscol_join_kin_explicit_rhs_dyn_implicit_rhs_dyn_implicit_mat_comb_implicit_rhs_comb_implicit_mat_comb_explicit_rhs_alg_con output_eqns isinstancetuple_bodies_loads)selfrright_hand_sidespeeds mass_matrixralg_conr# coord_idxs speed_idxsbodiesloadscoords speeds_inters ` ` r__init__zSymbolicSystem.__init__s   D!,//DL 3$(!!>>>>:>>>$*6NN! 4# DDDDDDD %l33 &| 4 4D !&>>DL,55dlCCDL " 6%;D "%4D "%0D "&*D #&*D #&*D # #  6%)D "%)D "%)D "&5D #&1D #&*D # #%)D "%)D "%)D "&*D #&*D #&5D #  I#9 IHHHHHHHG &&%(( #V #6]]F%'' !E !%LLE  rc<|jtd|jS)z8Returns the column matrix of the generalized coordinatesNz#The coordinates were not specified.)rAttributeErrorr(s r coordinateszSymbolicSystem.coordinatess'   % !FGG G$ $rc<|jtd|jS)z/Returns the column matrix of generalized speedsNzThe speeds were not specified.)rr5r6s rr*zSymbolicSystem.speedss% <  !ABB B< rc|jS)z0Returns the column matrix of the state variables)rr6s rstateszSymbolicSystem.states!s |rc|jS)zReturns a list with the indices of the rows containing algebraic constraints in the combined form of the equations of motion)r"r6s rr,zSymbolicSystem.alg_con&s }rc<|jtd|jS)zReturns the matrix, M, corresponding to the dynamic equations in implicit form, M x' = F, where the kinematical equations are not includedNzJdyn_implicit_mat is not specified for equations of motion form [1] or [2].)rr5r6s rdyn_implicit_matzSymbolicSystem.dyn_implicit_mat,5  ! * "HII I) )rc<|jtd|jS)zReturns the column matrix, F, corresponding to the dynamic equations in implicit form, M x' = F, where the kinematical equations are not includedNzJdyn_implicit_rhs is not specified for equations of motion form [1] or [2].)rr5r6s rdyn_implicit_rhszSymbolicSystem.dyn_implicit_rhs7r>rc|j|jt|j}t|j}t ||}t ||}t ||}||j}|||_|jStd|jS)zReturns the matrix, M, corresponding to the equations of motion in implicit form (form [2]), M x' = F, where the kinematical equations are includedNzDcomb_implicit_mat is not specified for equations of motion form [1].) r rrrrrrrow_joinrr5)r( num_kin_eqns num_dyn_eqnszeros1zeros2inter1inter2s rcomb_implicit_matz SymbolicSystem.comb_implicit_matBs  " +% F"4#9:: "4#9:: |\::|\::\**33F;;)?@@*0//&*A*A'..$&EFFF* *rc|jE|j/|j}|j}|||_|jSt d|jS)zReturns the column matrix, F, corresponding to the equations of motion in implicit form (form [2]), M x' = F, where the kinematical equations are includedNzGcomb_implicit_mat is not specified for equations of motion in form [1].)rrrrr5)r( kin_inter dyn_inters rcomb_implicit_rhsz SymbolicSystem.comb_implicit_rhsWsp  " +% I 2  2 *3*<*B!$DT$>>>>@%%X%  X XX **X***X*++X+(++X+ &&& ++X+**X*((($   "  X XrN) sympy.core.backendrrrsympy.physics.mechanicsr!sympy.physics.mechanics.functionsr__all__rr rrros1111111111222222AAAAAA  uuuuuuuuuur