Linearized Carvallo-Whipple Bicycle Model¶
The bicycle is an interesting system in that it can be modeled with multiple
rigid bodies, non-holonomic constraints, and a holonomic constraint. The
linearized equations of motion of the Carvallo-Whipple bicycle model are
presented and benchmarked in [Meijaard2007]. This example will construct the
same linear equations of motion using sympy.physics.mechanics.
>>> import sympy as sm
>>> import sympy.physics.mechanics as me
>>> me.mechanics_printing(pretty_print=False)
Declaration of Coordinates & Speeds¶
The simple definition of \(\mathbf{u} = \dot{\mathbf{q}}\) is used in this model. The generalized speeds are:
yaw frame angular rate \(u_1\),
roll frame angular rate \(u_2\),
rear wheel frame angular rate (spinning motion) \(u_3\),
frame angular rate (pitching motion) \(u_4\),
steering frame angular rate \(u_5\), and
front wheel angular rate (spinning motion) \(u_6\).
Wheel positions are ignorable coordinates, so they are not introduced.
>>> q1, q2, q3, q4, q5 = me.dynamicsymbols('q1 q2 q3 q4 q5')
>>> q1d, q2d, q4d, q5d = me.dynamicsymbols('q1 q2 q4 q5', 1)
>>> u1, u2, u3, u4, u5, u6 = me.dynamicsymbols('u1 u2 u3 u4 u5 u6')
>>> u1d, u2d, u3d, u4d, u5d, u6d = me.dynamicsymbols('u1 u2 u3 u4 u5 u6', 1)
Declaration of System’s Parameters¶
The constant parameters of the model are:
>>> WFrad, WRrad, htangle, forkoffset = sm.symbols('WFrad WRrad htangle forkoffset')
>>> forklength, framelength, forkcg1 = sm.symbols('forklength framelength forkcg1')
>>> forkcg3, framecg1, framecg3, Iwr11 = sm.symbols('forkcg3 framecg1 framecg3 Iwr11')
>>> Iwr22, Iwf11, Iwf22, Iframe11 = sm.symbols('Iwr22 Iwf11 Iwf22 Iframe11')
>>> Iframe22, Iframe33, Iframe31, Ifork11 = sm.symbols('Iframe22 Iframe33 Iframe31 Ifork11')
>>> Ifork22, Ifork33, Ifork31, g = sm.symbols('Ifork22 Ifork33 Ifork31 g')
>>> mframe, mfork, mwf, mwr = sm.symbols('mframe mfork mwf mwr')
Kinematics of the Bicycle¶
Set up reference frames for the system¶
N- inertialY- yawR- rollWR- rear wheel, rotation angle is ignorable coordinate so not orientedFrame- bicycle frameTempFrame- statically rotated frame for easier reference inertia definitionFork- bicycle forkTempFork- statically rotated frame for easier reference inertia definitionWF- front wheel, again posses an ignorable coordinate
>>> N = me.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 = me.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 = me.ReferenceFrame('WF')
Define relevant points for the system¶
WR_cont - rear wheel contact
WR_mc- rear wheel’s center of mass
Steer - frame/fork connection
Frame_mc - frame’s center of mass
Fork_mc - fork’s center of mass
WF_mc - front wheel’s center of mass
WF_cont - front wheel contact point
>>> WR_cont = me.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*(me.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.
u1is yaw rateu2is roll rateu3is rear wheel rateu4is frame pitch rateu5is fork steer rateu6is 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)
- 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 + 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
The kinematic differential equations are as follows. Each entry in this list is equal to zero.
>>> kd = [q1d - u1, q2d - u2, q4d - u4, q5d - u5]
Setup the constraints¶
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 (one fewer direction cosine matrix connecting them). These constraints force the velocity of the front wheel contact point to be zero 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 = [me.dot(WF_cont.vel(N), Y.x),
... me.dot(WF_cont.vel(N), Y.y),
... me.dot(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 dynamical differential equations, but is necessary for the linearization process.
>>> conlist_coord = [me.dot(WF_cont.pos_from(WR_cont), Y.z)]
Inertia and Rigid Bodies¶
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 = (me.inertia(TempFrame, Iframe11, Iframe22, Iframe33, 0, 0,
... Iframe31), Frame_mc)
>>> Fork_I = (me.inertia(TempFork, Ifork11, Ifork22, Ifork33, 0, 0, Ifork31), Fork_mc)
>>> WR_I = (me.inertia(Frame, Iwr11, Iwr22, Iwr11), WR_mc)
>>> WF_I = (me.inertia(Fork, Iwf11, Iwf22, Iwf11), WF_mc)
Declaration of the RigidBody containers.
>>> BodyFrame = me.RigidBody('BodyFrame', Frame_mc, Frame, mframe, Frame_I)
>>> BodyFork = me.RigidBody('BodyFork', Fork_mc, Fork, mfork, Fork_I)
>>> BodyWR = me.RigidBody('BodyWR', WR_mc, WR, mwr, WR_I)
>>> BodyWF = me.RigidBody('BodyWF', WF_mc, WF, mwf, WF_I)
>>> bodies = [BodyFrame, BodyFork, BodyWR, BodyWF]
Gravitational Loads¶
The force list; each body has the appropriate gravitational force applied at its center of mass.
>>> forces = [(Frame_mc, -mframe * g * Y.z),
... (Fork_mc, -mfork * g * Y.z),
... (WF_mc, -mwf * g * Y.z),
... (WR_mc, -mwr * g * Y.z)]
...
Nonlinear Equations of Motion¶
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.
>>> kane = me.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,
... constraint_solver='CRAMER')
>>> fr, frstar = kane.kanes_equations(bodies, loads=forces)
Linearized Equations of Motion¶
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 = sm.evalf.N(sm.pi/2 - sm.pi/10)
>>> TrailPaper = 0.08
>>> rake = sm.evalf.N(-(TrailPaper*sm.sin(HTA) - (PaperRadFront*sm.cos(HTA))))
>>> PaperWb = 1.02
>>> PaperFrameCgX = 0.3
>>> PaperFrameCgZ = 0.9
>>> PaperForkCgX = 0.9
>>> PaperForkCgZ = 0.7
>>> FrameLength = sm.evalf.N(PaperWb*sm.sin(HTA) - (rake -
... (PaperRadFront - PaperRadRear)*sm.cos(HTA)))
>>> FrameCGNorm = sm.evalf.N((PaperFrameCgZ - PaperRadRear -
... (PaperFrameCgX/sm.sin(HTA))*sm.cos(HTA))*sm.sin(HTA))
>>> FrameCGPar = sm.evalf.N((PaperFrameCgX / sm.sin(HTA) +
... (PaperFrameCgZ - PaperRadRear -
... PaperFrameCgX / sm.sin(HTA)*sm.cos(HTA))*sm.cos(HTA)))
>>> tempa = sm.evalf.N((PaperForkCgZ - PaperRadFront))
>>> tempb = sm.evalf.N((PaperWb-PaperForkCgX))
>>> tempc = sm.evalf.N(sm.sqrt(tempa**2 + tempb**2))
>>> PaperForkL = sm.evalf.N((PaperWb*sm.cos(HTA) -
... (PaperRadFront - PaperRadRear)*sm.sin(HTA)))
>>> ForkCGNorm = sm.evalf.N(rake + (tempc*sm.sin(sm.pi/2 -
... HTA - sm.acos(tempa/tempc))))
>>> ForkCGPar = sm.evalf.N(tempc*sm.cos((sm.pi/2 - HTA) -
... sm.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 = sm.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,
... }
...
Linearize the equations of motion about the equilibrium point:
>>> eq_point = {
... u1d: 0,
... u2d: 0,
... u3d: 0,
... u4d: 0,
... u5d: 0,
... u6d: 0,
... q1: 0,
... q2: 0,
... q4: 0,
... q5: 0,
... u1: 0,
... u2: 0,
... u3: v/PaperRadRear,
... u4: 0,
... u5: 0,
... u6: v/PaperRadFront,
... }
...
>>> Amat, _, _ = kane.linearize(A_and_B=True, op_point=eq_point, linear_solver='CRAMER')
>>> Amat = me.msubs(Amat, val_dict)
Calculate the Eigenvalues¶
Finally, we construct an “A” matrix for the form \(\dot{\mathbf{x}} = \mathbf{A} \mathbf{x}\) (\(\mathbf{x}\) 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.
>>> A = Amat.extract([1, 2, 3, 5], [1, 2, 3, 5])
>>> A
Matrix([
[ 0, 0, 1, 0],
[ 0, 0, 0, 1],
[9.48977444677355, -0.891197738059089*v**2 - 0.571523173729245, -0.105522449805691*v, -0.330515398992311*v],
[11.7194768719633, 30.9087533932407 - 1.97171508499972*v**2, 3.67680523332152*v, -3.08486552743311*v]])
>>> print('v = 1')
v = 1
>>> print(A.subs(v, 1).eigenvals())
{-3.13423125066578 - 1.05503732448615e-65*I: 1, 3.52696170990069 - 0.807740275199311*I: 1, 3.52696170990069 + 0.807740275199311*I: 1, -7.11008014637441: 1}
>>> print('v = 2')
v = 2
>>> print(A.subs(v, 2).eigenvals())
{2.68234517512745 - 1.68066296590676*I: 1, 2.68234517512745 + 1.68066296590676*I: 1, -3.07158645641514: 1, -8.67387984831737: 1}
>>> print('v = 3')
v = 3
>>> print(A.subs(v, 3).eigenvals())
{1.70675605663973 - 2.31582447384324*I: 1, 1.70675605663973 + 2.31582447384324*I: 1, -2.63366137253665: 1, -10.3510146724592: 1}
>>> print('v = 4')
v = 4
>>> print(A.subs(v, 4).eigenvals())
{0.413253315211239 - 3.07910818603205*I: 1, 0.413253315211239 + 3.07910818603205*I: 1, -1.42944427361326 + 1.65070329233125e-64*I: 1, -12.1586142657644: 1}
>>> print('v = 5')
v = 5
>>> print(A.subs(v, 5).eigenvals())
{-0.775341882195845 - 4.46486771378823*I: 1, -0.322866429004087 + 3.32140410564766e-64*I: 1, -0.775341882195845 + 4.46486771378823*I: 1, -14.0783896927982: 1}
The eigenvalues shown above match those in Table 2 on pg. 1971 of [Meijaard2007]. This concludes the bicycle example.
References¶
Meijaard, J. P., Papadopoulos, J. M., Ruina, A., & Schwab, A. L. (2007). Linearized dynamics equations for the balance and steer of a bicycle: A benchmark and review. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2084), 1955–1982. https://doi.org/10.1098/rspa.2007.1857