from sympy import evalf, symbols, zeros, pi, sin, cos, sqrt, acos, Matrix
from sympy.physics.mechanics import (ReferenceFrame, dynamicsymbols, inertia,
                                     KanesMethod, RigidBody, Point, dot)
from sympy.utilities.pytest import slow, ON_TRAVIS, skip

@slow
def test_bicycle():
    if ON_TRAVIS:
        skip("Too slow for travis.")
    # Code to get equations of motion for a bicycle modeled as in:
    # J.P Meijaard, Jim M Papadopoulos, Andy Ruina and A.L Schwab. Linearized
    # dynamics equations for the balance and steer of a bicycle: a benchmark
    # and review. Proceedings of The Royal Society (2007) 463, 1955-1982
    # doi: 10.1098/rspa.2007.1857

    # Note that this code has been crudely ported from Autolev, which is the
    # reason for some of the unusual naming conventions. It was purposefully as
    # similar as possible in order to aide debugging.

    # Declare Coordinates & Speeds
    # Simple definitions for qdots - qd = u
    # Speeds are: yaw frame ang. rate, roll frame ang. rate, rear wheel frame
    # ang.  rate (spinning motion), frame ang. rate (pitching motion), steering
    # frame ang. rate, and front wheel ang. rate (spinning motion).
    # Wheel positions are ignorable coordinates, so they are not introduced.
    q1, q2, q4, q5 = dynamicsymbols('q1 q2 q4 q5')
    q1d, q2d, q4d, q5d = dynamicsymbols('q1 q2 q4 q5', 1)
    u1, u2, u3, u4, u5, u6 = dynamicsymbols('u1 u2 u3 u4 u5 u6')
    u1d, u2d, u3d, u4d, u5d, u6d = dynamicsymbols('u1 u2 u3 u4 u5 u6', 1)

    # Declare System's Parameters
    WFrad, WRrad, htangle, forkoffset = symbols('WFrad WRrad htangle forkoffset')
    forklength, framelength, forkcg1 = symbols('forklength framelength forkcg1')
    forkcg3, framecg1, framecg3, Iwr11 = symbols('forkcg3 framecg1 framecg3 Iwr11')
    Iwr22, Iwf11, Iwf22, Iframe11 = symbols('Iwr22 Iwf11 Iwf22 Iframe11')
    Iframe22, Iframe33, Iframe31, Ifork11 = symbols('Iframe22 Iframe33 Iframe31 Ifork11')
    Ifork22, Ifork33, Ifork31, g = symbols('Ifork22 Ifork33 Ifork31 g')
    mframe, mfork, mwf, mwr = symbols('mframe mfork mwf mwr')

    # Set up reference frames for the system
    # N - inertial
    # Y - yaw
    # R - roll
    # WR - rear wheel, rotation angle is ignorable coordinate so not oriented
    # Frame - bicycle frame
    # TempFrame - statically rotated frame for easier reference inertia definition
    # Fork - bicycle fork
    # TempFork - statically rotated frame for easier reference inertia definition
    # WF - front wheel, again posses a ignorable coordinate
    N = ReferenceFrame('N')
    Y = N.orientnew('Y', 'Axis', [q1, N.z])
    R = Y.orientnew('R', 'Axis', [q2, Y.x])
    Frame = R.orientnew('Frame', 'Axis', [q4 + htangle, R.y])
    WR = ReferenceFrame('WR')
    TempFrame = Frame.orientnew('TempFrame', 'Axis', [-htangle, Frame.y])
    Fork = Frame.orientnew('Fork', 'Axis', [q5, Frame.x])
    TempFork = Fork.orientnew('TempFork', 'Axis', [-htangle, Fork.y])
    WF = ReferenceFrame('WF')

    # Kinematics of the Bicycle First block of code is forming the positions of
    # the relevant points
    # rear wheel contact -> rear wheel mass center -> frame mass center +
    # frame/fork connection -> fork mass center + front wheel mass center ->
    # front wheel contact point
    WR_cont = Point('WR_cont')
    WR_mc = WR_cont.locatenew('WR_mc', WRrad * R.z)
    Steer = WR_mc.locatenew('Steer', framelength * Frame.z)
    Frame_mc = WR_mc.locatenew('Frame_mc', - framecg1 * Frame.x
                                           + framecg3 * Frame.z)
    Fork_mc = Steer.locatenew('Fork_mc', - forkcg1 * Fork.x
                                         + forkcg3 * Fork.z)
    WF_mc = Steer.locatenew('WF_mc', forklength * Fork.x + forkoffset * Fork.z)
    WF_cont = WF_mc.locatenew('WF_cont', WFrad * (dot(Fork.y, Y.z) * Fork.y -
                                                  Y.z).normalize())

    # Set the angular velocity of each frame.
    # Angular accelerations end up being calculated automatically by
    # differentiating the angular velocities when first needed.
    # u1 is yaw rate
    # u2 is roll rate
    # u3 is rear wheel rate
    # u4 is frame pitch rate
    # u5 is fork steer rate
    # u6 is front wheel rate
    Y.set_ang_vel(N, u1 * Y.z)
    R.set_ang_vel(Y, u2 * R.x)
    WR.set_ang_vel(Frame, u3 * Frame.y)
    Frame.set_ang_vel(R, u4 * Frame.y)
    Fork.set_ang_vel(Frame, u5 * Fork.x)
    WF.set_ang_vel(Fork, u6 * Fork.y)

    # Form the velocities of the previously defined points, using the 2 - point
    # theorem (written out by hand here).  Accelerations again are calculated
    # automatically when first needed.
    WR_cont.set_vel(N, 0)
    WR_mc.v2pt_theory(WR_cont, N, WR)
    Steer.v2pt_theory(WR_mc, N, Frame)
    Frame_mc.v2pt_theory(WR_mc, N, Frame)
    Fork_mc.v2pt_theory(Steer, N, Fork)
    WF_mc.v2pt_theory(Steer, N, Fork)
    WF_cont.v2pt_theory(WF_mc, N, WF)

    # Sets the inertias of each body. Uses the inertia frame to construct the
    # inertia dyadics. Wheel inertias are only defined by principle moments of
    # inertia, and are in fact constant in the frame and fork reference frames;
    # it is for this reason that the orientations of the wheels does not need
    # to be defined. The frame and fork inertias are defined in the 'Temp'
    # frames which are fixed to the appropriate body frames; this is to allow
    # easier input of the reference values of the benchmark paper. Note that
    # due to slightly different orientations, the products of inertia need to
    # have their signs flipped; this is done later when entering the numerical
    # value.

    Frame_I = (inertia(TempFrame, Iframe11, Iframe22, Iframe33, 0, 0, Iframe31), Frame_mc)
    Fork_I = (inertia(TempFork, Ifork11, Ifork22, Ifork33, 0, 0, Ifork31), Fork_mc)
    WR_I = (inertia(Frame, Iwr11, Iwr22, Iwr11), WR_mc)
    WF_I = (inertia(Fork, Iwf11, Iwf22, Iwf11), WF_mc)

    # Declaration of the RigidBody containers. ::

    BodyFrame = RigidBody('BodyFrame', Frame_mc, Frame, mframe, Frame_I)
    BodyFork = RigidBody('BodyFork', Fork_mc, Fork, mfork, Fork_I)
    BodyWR = RigidBody('BodyWR', WR_mc, WR, mwr, WR_I)
    BodyWF = RigidBody('BodyWF', WF_mc, WF, mwf, WF_I)


    # The kinematic differential equations; they are defined quite simply. Each
    # entry in this list is equal to zero.
    kd = [q1d - u1, q2d - u2, q4d - u4, q5d - u5]

    # The nonholonomic constraints are the velocity of the front wheel contact
    # point dotted into the X, Y, and Z directions; the yaw frame is used as it
    # is "closer" to the front wheel (1 less DCM connecting them). These
    # constraints force the velocity of the front wheel contact point to be 0
    # in the inertial frame; the X and Y direction constraints enforce a
    # "no-slip" condition, and the Z direction constraint forces the front
    # wheel contact point to not move away from the ground frame, essentially
    # replicating the holonomic constraint which does not allow the frame pitch
    # to change in an invalid fashion.

    conlist_speed = [WF_cont.vel(N) & Y.x, WF_cont.vel(N) & Y.y, WF_cont.vel(N) & Y.z]

    # The holonomic constraint is that the position from the rear wheel contact
    # point to the front wheel contact point when dotted into the
    # normal-to-ground plane direction must be zero; effectively that the front
    # and rear wheel contact points are always touching the ground plane. This
    # is actually not part of the dynamic equations, but instead is necessary
    # for the lineraization process.

    conlist_coord = [WF_cont.pos_from(WR_cont) & Y.z]

    # The force list; each body has the appropriate gravitational force applied
    # at its mass center.
    FL = [(Frame_mc, -mframe * g * Y.z),
        (Fork_mc, -mfork * g * Y.z),
        (WF_mc, -mwf * g * Y.z),
        (WR_mc, -mwr * g * Y.z)]
    BL = [BodyFrame, BodyFork, BodyWR, BodyWF]


    # The N frame is the inertial frame, coordinates are supplied in the order
    # of independent, dependent coordinates, as are the speeds. The kinematic
    # differential equation are also entered here.  Here the dependent speeds
    # are specified, in the same order they were provided in earlier, along
    # with the non-holonomic constraints.  The dependent coordinate is also
    # provided, with the holonomic constraint.  Again, this is only provided
    # for the linearization process.

    KM = KanesMethod(N, q_ind=[q1, q2, q5],
            q_dependent=[q4], configuration_constraints=conlist_coord,
            u_ind=[u2, u3, u5],
            u_dependent=[u1, u4, u6], velocity_constraints=conlist_speed,
            kd_eqs=kd)
    (fr, frstar) = KM.kanes_equations(FL, BL)

    # This is the start of entering in the numerical values from the benchmark
    # paper to validate the eigen values of the linearized equations from this
    # model to the reference eigen values. Look at the aforementioned paper for
    # more information. Some of these are intermediate values, used to
    # transform values from the paper into the coordinate systems used in this
    # model.
    PaperRadRear                    =  0.3
    PaperRadFront                   =  0.35
    HTA                             =  evalf.N(pi / 2 - pi / 10)
    TrailPaper                      =  0.08
    rake                            =  evalf.N(-(TrailPaper*sin(HTA)-(PaperRadFront*cos(HTA))))
    PaperWb                         =  1.02
    PaperFrameCgX                   =  0.3
    PaperFrameCgZ                   =  0.9
    PaperForkCgX                    =  0.9
    PaperForkCgZ                    =  0.7
    FrameLength                     =  evalf.N(PaperWb*sin(HTA)-(rake-(PaperRadFront-PaperRadRear)*cos(HTA)))
    FrameCGNorm                     =  evalf.N((PaperFrameCgZ - PaperRadRear-(PaperFrameCgX/sin(HTA))*cos(HTA))*sin(HTA))
    FrameCGPar                      =  evalf.N((PaperFrameCgX / sin(HTA) + (PaperFrameCgZ - PaperRadRear - PaperFrameCgX / sin(HTA) * cos(HTA)) * cos(HTA)))
    tempa                           =  evalf.N((PaperForkCgZ - PaperRadFront))
    tempb                           =  evalf.N((PaperWb-PaperForkCgX))
    tempc                           =  evalf.N(sqrt(tempa**2+tempb**2))
    PaperForkL                      =  evalf.N((PaperWb*cos(HTA)-(PaperRadFront-PaperRadRear)*sin(HTA)))
    ForkCGNorm                      =  evalf.N(rake+(tempc * sin(pi/2-HTA-acos(tempa/tempc))))
    ForkCGPar                       =  evalf.N(tempc * cos((pi/2-HTA)-acos(tempa/tempc))-PaperForkL)

    # Here is the final assembly of the numerical values. The symbol 'v' is the
    # forward speed of the bicycle (a concept which only makes sense in the
    # upright, static equilibrium case?). These are in a dictionary which will
    # later be substituted in. Again the sign on the *product* of inertia
    # values is flipped here, due to different orientations of coordinate
    # systems.
    v = symbols('v')
    val_dict = {WFrad: PaperRadFront,
                WRrad: PaperRadRear,
                htangle: HTA,
                forkoffset: rake,
                forklength: PaperForkL,
                framelength: FrameLength,
                forkcg1: ForkCGPar,
                forkcg3: ForkCGNorm,
                framecg1: FrameCGNorm,
                framecg3: FrameCGPar,
                Iwr11: 0.0603,
                Iwr22: 0.12,
                Iwf11: 0.1405,
                Iwf22: 0.28,
                Ifork11: 0.05892,
                Ifork22: 0.06,
                Ifork33: 0.00708,
                Ifork31: 0.00756,
                Iframe11: 9.2,
                Iframe22: 11,
                Iframe33: 2.8,
                Iframe31: -2.4,
                mfork: 4,
                mframe: 85,
                mwf: 3,
                mwr: 2,
                g: 9.81,
                q1: 0,
                q2: 0,
                q4: 0,
                q5: 0,
                u1: 0,
                u2: 0,
                u3: v / PaperRadRear,
                u4: 0,
                u5: 0,
                u6: v / PaperRadFront}

    # Linearizes the forcing vector; the equations are set up as MM udot =
    # forcing, where MM is the mass matrix, udot is the vector representing the
    # time derivatives of the generalized speeds, and forcing is a vector which
    # contains both external forcing terms and internal forcing terms, such as
    # centripital or coriolis forces.  This actually returns a matrix with as
    # many rows as *total* coordinates and speeds, but only as many columns as
    # independent coordinates and speeds.

    forcing_lin = KM.linearize()[0]

    # As mentioned above, the size of the linearized forcing terms is expanded
    # to include both q's and u's, so the mass matrix must have this done as
    # well.  This will likely be changed to be part of the linearized process,
    # for future reference.
    MM_full = KM.mass_matrix_full

    MM_full_s = MM_full.subs(val_dict)
    forcing_lin_s = forcing_lin.subs(KM.kindiffdict()).subs(val_dict)


    MM_full_s = MM_full_s.evalf()
    forcing_lin_s = forcing_lin_s.evalf()

    # Finally, we construct an "A" matrix for the form xdot = A x (x being the
    # state vector, although in this case, the sizes are a little off). The
    # following line extracts only the minimum entries required for eigenvalue
    # analysis, which correspond to rows and columns for lean, steer, lean
    # rate, and steer rate.
    Amat = MM_full_s.inv() * forcing_lin_s
    A = Amat.extract([1, 2, 4, 6], [1, 2, 3, 5])

    # Precomputed for comparison
    Res = Matrix([[               0,                                           0,                  1.0,                    0],
                  [               0,                                           0,                    0,                  1.0],
                  [9.48977444677355, -0.891197738059089*v**2 - 0.571523173729245, -0.105522449805691*v, -0.330515398992311*v],
                  [11.7194768719633,   -1.97171508499972*v**2 + 30.9087533932407,   3.67680523332152*v,  -3.08486552743311*v]])


    # Actual eigenvalue comparison
    eps = 1.e-12
    for i in range(6):
        error = Res.subs(v, i) - A.subs(v, i)
        assert all(abs(x) < eps for x in error)
