A bicycle

The bicycle is an interesting system in that it has multiple rigid bodies, non-holonomic constraints, and a holonomic constraint. The linearized equations of motion are presented in [Meijaard2007]. This example will go through construction of the equations of motion in mechanics.

>>> from sympy import *
>>> from sympy.physics.mechanics import *
>>> print('Calculation of Linearized Bicycle \"A\" Matrix, '
...       'with States: Roll, Steer, Roll Rate, Steer Rate')
Calculation of Linearized Bicycle "A" Matrix, with States: Roll, Steer, Roll Rate, Steer Rate

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 aid initial porting & debugging. We set Vector.simp to False (in case it has been set True elsewhere), since it slows down the computations:

>>> Vector.simp = False
>>> mechanics_printing()

Declaration of Coordinates & Speeds: A simple definition for qdots, qd = u,is used in this code. 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, q3, q4, q5 = dynamicsymbols('q1 q2 q3 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)

Declaration of System’s Parameters: The below symbols should be fairly self-explanatory.

>>> 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’s center of mass -> frame’s center of mass + frame/fork connection -> fork’s center of mass + front wheel’s center of mass -> 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 points, using the 2-point theorem. Accelerations again are calculated automatically when first needed.

>>> WR_cont.set_vel(N, 0)
>>> WR_mc.v2pt_theory(WR_cont, N, WR)
WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y
>>> Steer.v2pt_theory(WR_mc, N, Frame)
WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y
>>> Frame_mc.v2pt_theory(WR_mc, N, Frame)
WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framecg3*(u1*sin(q2) + u4)*Frame.x + (-framecg1*(u1*cos(htangle + q4)*cos(q2) + u2*sin(htangle + q4)) - framecg3*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4)))*Frame.y + framecg1*(u1*sin(q2) + u4)*Frame.z
>>> Fork_mc.v2pt_theory(Steer, N, Fork)
WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + forkcg3*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.x + (-forkcg1*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkcg3*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y + forkcg1*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.z
>>> WF_mc.v2pt_theory(Steer, N, Fork)
WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + forkoffset*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.x + (forklength*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkoffset*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y - forklength*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.z
>>> WF_cont.v2pt_theory(WF_mc, N, WF)
WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + (-WFrad*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1) + forkoffset*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5)))*Fork.x + (forklength*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkoffset*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y + (WFrad*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5)/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1) - forklength*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5)))*Fork.z - WFrad*((-sin(q2)*sin(q5)*cos(htangle + q4) + cos(q2)*cos(q5))*u6 + u4*cos(q2) + u5*sin(htangle + q4)*sin(q2))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1)*Y.x + WFrad*(u2 + u5*cos(htangle + q4) + u6*sin(htangle + q4)*sin(q5))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1)*Y.y

Sets the inertias of each body. Uses the inertia frame to construct the inertia dyadics. Wheel inertias are only defined by principal 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)

>>> print('Before Forming the List of Nonholonomic Constraints.')
Before Forming the List of Nonholonomic Constraints.

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 linearization process.

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

The force list; each body has the appropriate gravitational force applied at its center of mass.

>>> 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. The kinematic differential equations are also entered here. Here the independent speeds are specified, followed by the dependent speeds, along with the non-holonomic constraints. The dependent coordinate is also provided, with the holonomic constraint. Again, this is only comes into play in the linearization process, but is necessary for the linearization to correctly work.

>>> KM = KanesMethod(N, q_ind=[q1, q2, q3],
...           q_dependent=[q4], configuration_constraints=conlist_coord,
...           u_ind=[u2, u3, u5],
...           u_dependent=[u1, u4, u6], velocity_constraints=conlist_speed,
...           kd_eqs=kd)
>>> print('Before Forming Generalized Active and Inertia Forces, Fr and Fr*')
Before Forming Generalized Active and Inertia Forces, Fr and Fr*
>>> (fr, frstar) = KM.kanes_equations(FL, BL)
>>> print('Base Equations of Motion Computed')
Base Equations of Motion Computed

This is the start of entering in the numerical values from the benchmark paper to validate the eigenvalues of the linearized equations from this model to the reference eigenvalues. 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 = Symbol('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}
>>> kdd = KM.kindiffdict()
>>> print('Before Linearization of the \"Forcing\" Term')
Before Linearization of the "Forcing" Term

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 centripetal 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. (Note that below this is commented out, as it takes a few minutes to run, which is not good when performing the doctests)

>>> # forcing_lin = KM.linearize()[0].subs(sub_dict)

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._k_kqdot).row_join(zeros(4, 6)).col_join(
...           (zeros(6, 4)).row_join(KM.mass_matrix))
>>> print('Before Substitution of Numerical Values')
Before Substitution of Numerical Values

I think this is pretty self explanatory. It takes a really long time though. I’ve experimented with using evalf with substitution, this failed due to maximum recursion depth being exceeded; I also tried lambdifying this, and it is also not successful. (again commented out due to speed)

>>> # MM_full = MM_full.subs(val_dict)
>>> # forcing_lin = forcing_lin.subs(val_dict)
>>> # print('Before .evalf() call')

>>> # MM_full = MM_full.evalf()
>>> # forcing_lin = forcing_lin.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. (this is all commented out due to being dependent on the above code, which is also commented out):

>>> # Amat = MM_full.inv() * forcing_lin
>>> # A = Amat.extract([1,2,4,6],[1,2,3,5])
>>> # print(A)
>>> # print('v = 1')
>>> # print(A.subs(v, 1).eigenvals())
>>> # print('v = 2')
>>> # print(A.subs(v, 2).eigenvals())
>>> # print('v = 3')
>>> # print(A.subs(v, 3).eigenvals())
>>> # print('v = 4')
>>> # print(A.subs(v, 4).eigenvals())
>>> # print('v = 5')
>>> # print(A.subs(v, 5).eigenvals())

Upon running the above code yourself, enabling the commented out lines, compare the computed eigenvalues to those is the referenced paper. This concludes the bicycle example.

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