Source code for sympy.physics.vector.functions

from __future__ import print_function, division

from sympy import (sympify, diff, sin, cos, Matrix, Symbol, integrate,
                   trigsimp, Function, symbols)
from sympy.core.basic import S
from sympy.core.compatibility import reduce
from .vector import Vector, _check_vector
from .frame import CoordinateSym, _check_frame
from .dyadic import Dyadic
from .printing import vprint, vsprint, vpprint, vlatex, init_vprinting

__all__ = ['cross', 'dot', 'express', 'time_derivative', 'outer',
           'kinematic_equations', 'get_motion_params', 'partial_velocity',
           'dynamicsymbols', 'vprint', 'vsprint', 'vpprint', 'vlatex',
           'init_vprinting']


[docs]def cross(vec1, vec2): """Cross product convenience wrapper for Vector.cross(): \n""" if not isinstance(vec1, (Vector, Dyadic)): raise TypeError('Cross product is between two vectors') return vec1 ^ vec2
cross.__doc__ += Vector.cross.__doc__
[docs]def dot(vec1, vec2): """Dot product convenience wrapper for Vector.dot(): \n""" if not isinstance(vec1, (Vector, Dyadic)): raise TypeError('Dot product is between two vectors') return vec1 & vec2
dot.__doc__ += Vector.dot.__doc__
[docs]def express(expr, frame, frame2=None, variables=False): """ Global function for 'express' functionality. Re-expresses a Vector, scalar(sympyfiable) or Dyadic in given frame. Refer to the local methods of Vector and Dyadic for details. If 'variables' is True, then the coordinate variables (CoordinateSym instances) of other frames present in the vector/scalar field or dyadic expression are also substituted in terms of the base scalars of this frame. Parameters ========== expr : Vector/Dyadic/scalar(sympyfiable) The expression to re-express in ReferenceFrame 'frame' frame: ReferenceFrame The reference frame to express expr in frame2 : ReferenceFrame The other frame required for re-expression(only for Dyadic expr) variables : boolean Specifies whether to substitute the coordinate variables present in expr, in terms of those of frame Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> B = N.orientnew('B', 'Axis', [q, N.z]) >>> d = outer(N.x, N.x) >>> from sympy.physics.vector import express >>> express(d, B, N) cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x) >>> express(B.x, N) cos(q)*N.x + sin(q)*N.y >>> express(N[0], B, variables=True) B_x*cos(q(t)) - B_y*sin(q(t)) """ _check_frame(frame) if expr == 0: return expr if isinstance(expr, Vector): #Given expr is a Vector if variables: #If variables attribute is True, substitute #the coordinate variables in the Vector frame_list = [x[-1] for x in expr.args] subs_dict = {} for f in frame_list: subs_dict.update(f.variable_map(frame)) expr = expr.subs(subs_dict) #Re-express in this frame outvec = Vector([]) for i, v in enumerate(expr.args): if v[1] != frame: temp = frame.dcm(v[1]) * v[0] if Vector.simp: temp = temp.applyfunc(lambda x: trigsimp(x, method='fu')) outvec += Vector([(temp, frame)]) else: outvec += Vector([v]) return outvec if isinstance(expr, Dyadic): if frame2 is None: frame2 = frame _check_frame(frame2) ol = Dyadic(0) for i, v in enumerate(expr.args): ol += express(v[0], frame, variables=variables) * \ (express(v[1], frame, variables=variables) | express(v[2], frame2, variables=variables)) return ol else: if variables: #Given expr is a scalar field frame_set = set([]) expr = sympify(expr) #Subsitute all the coordinate variables for x in expr.atoms(): if isinstance(x, CoordinateSym)and x.frame != frame: frame_set.add(x.frame) subs_dict = {} for f in frame_set: subs_dict.update(f.variable_map(frame)) return expr.subs(subs_dict) return expr
[docs]def time_derivative(expr, frame, order=1): """ Calculate the time derivative of a vector/scalar field function or dyadic expression in given frame. References ========== http://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames Parameters ========== expr : Vector/Dyadic/sympifyable The expression whose time derivative is to be calculated frame : ReferenceFrame The reference frame to calculate the time derivative in order : integer The order of the derivative to be calculated Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols >>> from sympy import Symbol >>> q1 = Symbol('q1') >>> u1 = dynamicsymbols('u1') >>> N = ReferenceFrame('N') >>> A = N.orientnew('A', 'Axis', [q1, N.x]) >>> v = u1 * N.x >>> A.set_ang_vel(N, 10*A.x) >>> from sympy.physics.vector import time_derivative >>> time_derivative(v, N) u1'*N.x >>> time_derivative(u1*A[0], N) N_x*Derivative(u1(t), t) >>> B = N.orientnew('B', 'Axis', [u1, N.z]) >>> from sympy.physics.vector import outer >>> d = outer(N.x, N.x) >>> time_derivative(d, B) - u1'*(N.y|N.x) - u1'*(N.x|N.y) """ t = dynamicsymbols._t _check_frame(frame) if order == 0: return expr if order % 1 != 0 or order < 0: raise ValueError("Unsupported value of order entered") if isinstance(expr, Vector): outvec = Vector(0) for i, v in enumerate(expr.args): if v[1] == frame: outvec += Vector([(express(v[0], frame, variables=True).diff(t), frame)]) else: outvec += time_derivative(Vector([v]), v[1]) + \ (v[1].ang_vel_in(frame) ^ Vector([v])) return time_derivative(outvec, frame, order - 1) if isinstance(expr, Dyadic): ol = Dyadic(0) for i, v in enumerate(expr.args): ol += (v[0].diff(t) * (v[1] | v[2])) ol += (v[0] * (time_derivative(v[1], frame) | v[2])) ol += (v[0] * (v[1] | time_derivative(v[2], frame))) return time_derivative(ol, frame, order - 1) else: return diff(express(expr, frame, variables=True), t, order)
[docs]def outer(vec1, vec2): """Outer product convenience wrapper for Vector.outer():\n""" if not isinstance(vec1, Vector): raise TypeError('Outer product is between two Vectors') return vec1 | vec2
outer.__doc__ += Vector.outer.__doc__
[docs]def kinematic_equations(speeds, coords, rot_type, rot_order=''): """Gives equations relating the qdot's to u's for a rotation type. Supply rotation type and order as in orient. Speeds are assumed to be body-fixed; if we are defining the orientation of B in A using by rot_type, the angular velocity of B in A is assumed to be in the form: speed[0]*B.x + speed[1]*B.y + speed[2]*B.z Parameters ========== speeds : list of length 3 The body fixed angular velocity measure numbers. coords : list of length 3 or 4 The coordinates used to define the orientation of the two frames. rot_type : str The type of rotation used to create the equations. Body, Space, or Quaternion only rot_order : str If applicable, the order of a series of rotations. Examples ======== >>> from sympy.physics.vector import dynamicsymbols >>> from sympy.physics.vector import kinematic_equations, vprint >>> u1, u2, u3 = dynamicsymbols('u1 u2 u3') >>> q1, q2, q3 = dynamicsymbols('q1 q2 q3') >>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'), ... order=None) [-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3'] """ # Code below is checking and sanitizing input approved_orders = ('123', '231', '312', '132', '213', '321', '121', '131', '212', '232', '313', '323', '1', '2', '3', '') rot_order = str(rot_order).upper() # Now we need to make sure XYZ = 123 rot_type = rot_type.upper() rot_order = [i.replace('X', '1') for i in rot_order] rot_order = [i.replace('Y', '2') for i in rot_order] rot_order = [i.replace('Z', '3') for i in rot_order] rot_order = ''.join(rot_order) if not isinstance(speeds, (list, tuple)): raise TypeError('Need to supply speeds in a list') if len(speeds) != 3: raise TypeError('Need to supply 3 body-fixed speeds') if not isinstance(coords, (list, tuple)): raise TypeError('Need to supply coordinates in a list') if rot_type.lower() in ['body', 'space']: if rot_order not in approved_orders: raise ValueError('Not an acceptable rotation order') if len(coords) != 3: raise ValueError('Need 3 coordinates for body or space') # Actual hard-coded kinematic differential equations q1, q2, q3 = coords q1d, q2d, q3d = [diff(i, dynamicsymbols._t) for i in coords] w1, w2, w3 = speeds s1, s2, s3 = [sin(q1), sin(q2), sin(q3)] c1, c2, c3 = [cos(q1), cos(q2), cos(q3)] if rot_type.lower() == 'body': if rot_order == '123': return [q1d - (w1 * c3 - w2 * s3) / c2, q2d - w1 * s3 - w2 * c3, q3d - (-w1 * c3 + w2 * s3) * s2 / c2 - w3] if rot_order == '231': return [q1d - (w2 * c3 - w3 * s3) / c2, q2d - w2 * s3 - w3 * c3, q3d - w1 - (- w2 * c3 + w3 * s3) * s2 / c2] if rot_order == '312': return [q1d - (-w1 * s3 + w3 * c3) / c2, q2d - w1 * c3 - w3 * s3, q3d - (w1 * s3 - w3 * c3) * s2 / c2 - w2] if rot_order == '132': return [q1d - (w1 * c3 + w3 * s3) / c2, q2d + w1 * s3 - w3 * c3, q3d - (w1 * c3 + w3 * s3) * s2 / c2 - w2] if rot_order == '213': return [q1d - (w1 * s3 + w2 * c3) / c2, q2d - w1 * c3 + w2 * s3, q3d - (w1 * s3 + w2 * c3) * s2 / c2 - w3] if rot_order == '321': return [q1d - (w2 * s3 + w3 * c3) / c2, q2d - w2 * c3 + w3 * s3, q3d - w1 - (w2 * s3 + w3 * c3) * s2 / c2] if rot_order == '121': return [q1d - (w2 * s3 + w3 * c3) / s2, q2d - w2 * c3 + w3 * s3, q3d - w1 + (w2 * s3 + w3 * c3) * c2 / s2] if rot_order == '131': return [q1d - (-w2 * c3 + w3 * s3) / s2, q2d - w2 * s3 - w3 * c3, q3d - w1 - (w2 * c3 - w3 * s3) * c2 / s2] if rot_order == '212': return [q1d - (w1 * s3 - w3 * c3) / s2, q2d - w1 * c3 - w3 * s3, q3d - (-w1 * s3 + w3 * c3) * c2 / s2 - w2] if rot_order == '232': return [q1d - (w1 * c3 + w3 * s3) / s2, q2d + w1 * s3 - w3 * c3, q3d + (w1 * c3 + w3 * s3) * c2 / s2 - w2] if rot_order == '313': return [q1d - (w1 * s3 + w2 * c3) / s2, q2d - w1 * c3 + w2 * s3, q3d + (w1 * s3 + w2 * c3) * c2 / s2 - w3] if rot_order == '323': return [q1d - (-w1 * c3 + w2 * s3) / s2, q2d - w1 * s3 - w2 * c3, q3d - (w1 * c3 - w2 * s3) * c2 / s2 - w3] if rot_type.lower() == 'space': if rot_order == '123': return [q1d - w1 - (w2 * s1 + w3 * c1) * s2 / c2, q2d - w2 * c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / c2] if rot_order == '231': return [q1d - (w1 * c1 + w3 * s1) * s2 / c2 - w2, q2d + w1 * s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / c2] if rot_order == '312': return [q1d - (w1 * s1 + w2 * c1) * s2 / c2 - w3, q2d - w1 * c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / c2] if rot_order == '132': return [q1d - w1 - (-w2 * c1 + w3 * s1) * s2 / c2, q2d - w2 * s1 - w3 * c1, q3d - (w2 * c1 - w3 * s1) / c2] if rot_order == '213': return [q1d - (w1 * s1 - w3 * c1) * s2 / c2 - w2, q2d - w1 * c1 - w3 * s1, q3d - (-w1 * s1 + w3 * c1) / c2] if rot_order == '321': return [q1d - (-w1 * c1 + w2 * s1) * s2 / c2 - w3, q2d - w1 * s1 - w2 * c1, q3d - (w1 * c1 - w2 * s1) / c2] if rot_order == '121': return [q1d - w1 + (w2 * s1 + w3 * c1) * c2 / s2, q2d - w2 * c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / s2] if rot_order == '131': return [q1d - w1 - (w2 * c1 - w3 * s1) * c2 / s2, q2d - w2 * s1 - w3 * c1, q3d - (-w2 * c1 + w3 * s1) / s2] if rot_order == '212': return [q1d - (-w1 * s1 + w3 * c1) * c2 / s2 - w2, q2d - w1 * c1 - w3 * s1, q3d - (w1 * s1 - w3 * c1) / s2] if rot_order == '232': return [q1d + (w1 * c1 + w3 * s1) * c2 / s2 - w2, q2d + w1 * s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / s2] if rot_order == '313': return [q1d + (w1 * s1 + w2 * c1) * c2 / s2 - w3, q2d - w1 * c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / s2] if rot_order == '323': return [q1d - (w1 * c1 - w2 * s1) * c2 / s2 - w3, q2d - w1 * s1 - w2 * c1, q3d - (-w1 * c1 + w2 * s1) / s2] elif rot_type.lower() == 'quaternion': if rot_order != '': raise ValueError('Cannot have rotation order for quaternion') if len(coords) != 4: raise ValueError('Need 4 coordinates for quaternion') # Actual hard-coded kinematic differential equations e0, e1, e2, e3 = coords w = Matrix(speeds + [0]) E = Matrix([[e0, -e3, e2, e1], [e3, e0, -e1, e2], [-e2, e1, e0, e3], [-e1, -e2, -e3, e0]]) edots = Matrix([diff(i, dynamicsymbols._t) for i in [e1, e2, e3, e0]]) return list(edots.T - 0.5 * w.T * E.T) else: raise ValueError('Not an approved rotation type for this function')
[docs]def get_motion_params(frame, **kwargs): """ Returns the three motion parameters - (acceleration, velocity, and position) as vectorial functions of time in the given frame. If a higher order differential function is provided, the lower order functions are used as boundary conditions. For example, given the acceleration, the velocity and position parameters are taken as boundary conditions. The values of time at which the boundary conditions are specified are taken from timevalue1(for position boundary condition) and timevalue2(for velocity boundary condition). If any of the boundary conditions are not provided, they are taken to be zero by default (zero vectors, in case of vectorial inputs). If the boundary conditions are also functions of time, they are converted to constants by substituting the time values in the dynamicsymbols._t time Symbol. This function can also be used for calculating rotational motion parameters. Have a look at the Parameters and Examples for more clarity. Parameters ========== frame : ReferenceFrame The frame to express the motion parameters in acceleration : Vector Acceleration of the object/frame as a function of time velocity : Vector Velocity as function of time or as boundary condition of velocity at time = timevalue1 position : Vector Velocity as function of time or as boundary condition of velocity at time = timevalue1 timevalue1 : sympyfiable Value of time for position boundary condition timevalue2 : sympyfiable Value of time for velocity boundary condition Examples ======== >>> from sympy.physics.vector import ReferenceFrame, get_motion_params, dynamicsymbols >>> from sympy import symbols >>> R = ReferenceFrame('R') >>> v1, v2, v3 = dynamicsymbols('v1 v2 v3') >>> v = v1*R.x + v2*R.y + v3*R.z >>> get_motion_params(R, position = v) (v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z) >>> a, b, c = symbols('a b c') >>> v = a*R.x + b*R.y + c*R.z >>> get_motion_params(R, velocity = v) (0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z) >>> parameters = get_motion_params(R, acceleration = v) >>> parameters[1] a*t*R.x + b*t*R.y + c*t*R.z >>> parameters[2] a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z """ ##Helper functions def _process_vector_differential(vectdiff, condition, \ variable, ordinate, frame): """ Helper function for get_motion methods. Finds derivative of vectdiff wrt variable, and its integral using the specified boundary condition at value of variable = ordinate. Returns a tuple of - (derivative, function and integral) wrt vectdiff """ #Make sure boundary condition is independent of 'variable' if condition != 0: condition = express(condition, frame, variables=True) #Special case of vectdiff == 0 if vectdiff == Vector(0): return (0, 0, condition) #Express vectdiff completely in condition's frame to give vectdiff1 vectdiff1 = express(vectdiff, frame) #Find derivative of vectdiff vectdiff2 = time_derivative(vectdiff, frame) #Integrate and use boundary condition vectdiff0 = Vector(0) lims = (variable, ordinate, variable) for dim in frame: function1 = vectdiff1.dot(dim) abscissa = dim.dot(condition).subs({variable : ordinate}) # Indefinite integral of 'function1' wrt 'variable', using # the given initial condition (ordinate, abscissa). vectdiff0 += (integrate(function1, lims) + abscissa) * dim #Return tuple return (vectdiff2, vectdiff, vectdiff0) ##Function body _check_frame(frame) #Decide mode of operation based on user's input if 'acceleration' in kwargs: mode = 2 elif 'velocity' in kwargs: mode = 1 else: mode = 0 #All the possible parameters in kwargs #Not all are required for every case #If not specified, set to default values(may or may not be used in #calculations) conditions = ['acceleration', 'velocity', 'position', 'timevalue', 'timevalue1', 'timevalue2'] for i, x in enumerate(conditions): if x not in kwargs: if i < 3: kwargs[x] = Vector(0) else: kwargs[x] = S(0) elif i < 3: _check_vector(kwargs[x]) else: kwargs[x] = sympify(kwargs[x]) if mode == 2: vel = _process_vector_differential(kwargs['acceleration'], kwargs['velocity'], dynamicsymbols._t, kwargs['timevalue2'], frame)[2] pos = _process_vector_differential(vel, kwargs['position'], dynamicsymbols._t, kwargs['timevalue1'], frame)[2] return (kwargs['acceleration'], vel, pos) elif mode == 1: return _process_vector_differential(kwargs['velocity'], kwargs['position'], dynamicsymbols._t, kwargs['timevalue1'], frame) else: vel = time_derivative(kwargs['position'], frame) acc = time_derivative(vel, frame) return (acc, vel, kwargs['position'])
[docs]def partial_velocity(vel_list, u_list, frame): """Returns a list of partial velocities. For a list of velocity or angular velocity vectors the partial derivatives with respect to the supplied generalized speeds are computed, in the specified ReferenceFrame. The output is a list of lists. The outer list has a number of elements equal to the number of supplied velocity vectors. The inner lists are, for each velocity vector, the partial derivatives of that velocity vector with respect to the generalized speeds supplied. Parameters ========== vel_list : list List of velocities of Point's and angular velocities of ReferenceFrame's u_list : list List of independent generalized speeds. frame : ReferenceFrame The ReferenceFrame the partial derivatives are going to be taken in. Examples ======== >>> from sympy.physics.vector import Point, ReferenceFrame >>> from sympy.physics.vector import dynamicsymbols >>> from sympy.physics.vector import partial_velocity >>> u = dynamicsymbols('u') >>> N = ReferenceFrame('N') >>> P = Point('P') >>> P.set_vel(N, u * N.x) >>> vel_list = [P.vel(N)] >>> u_list = [u] >>> partial_velocity(vel_list, u_list, N) [[N.x]] """ if not hasattr(vel_list, '__iter__'): raise TypeError('Provide velocities in an iterable') if not hasattr(u_list, '__iter__'): raise TypeError('Provide speeds in an iterable') list_of_pvlists = [] for i in vel_list: pvlist = [] for j in u_list: vel = i.diff(j, frame) pvlist += [vel] list_of_pvlists += [pvlist] return list_of_pvlists
def dynamicsymbols(names, level=0): """Uses symbols and Function for functions of time. Creates a SymPy UndefinedFunction, which is then initialized as a function of a variable, the default being Symbol('t'). Parameters ========== names : str Names of the dynamic symbols you want to create; works the same way as inputs to symbols level : int Level of differentiation of the returned function; d/dt once of t, twice of t, etc. Examples ======== >>> from sympy.physics.vector import dynamicsymbols >>> from sympy import diff, Symbol >>> q1 = dynamicsymbols('q1') >>> q1 q1(t) >>> diff(q1, Symbol('t')) Derivative(q1(t), t) """ esses = symbols(names, cls=Function) t = dynamicsymbols._t if hasattr(esses, '__iter__'): esses = [reduce(diff, [t] * level, e(t)) for e in esses] return esses else: return reduce(diff, [t] * level, esses(t)) dynamicsymbols._t = Symbol('t') dynamicsymbols._str = '\''