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from sympy import evalf, symbols, zeros, pi, sin, cos, sqrt, acos, Matrix
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from sympy.physics.mechanics import (ReferenceFrame, dynamicsymbols, inertia,
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KanesMethod, RigidBody, Point, dot)
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from sympy.utilities.pytest import slow, ON_TRAVIS, skip
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skip("Too slow for travis.")
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# Code to get equations of motion for a bicycle modeled as in:
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# J.P Meijaard, Jim M Papadopoulos, Andy Ruina and A.L Schwab. Linearized
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# dynamics equations for the balance and steer of a bicycle: a benchmark
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# and review. Proceedings of The Royal Society (2007) 463, 1955-1982
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# doi: 10.1098/rspa.2007.1857
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# Note that this code has been crudely ported from Autolev, which is the
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# reason for some of the unusual naming conventions. It was purposefully as
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# similar as possible in order to aide debugging.
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# Declare Coordinates & Speeds
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# Simple definitions for qdots - qd = u
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# Speeds are: yaw frame ang. rate, roll frame ang. rate, rear wheel frame
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# ang. rate (spinning motion), frame ang. rate (pitching motion), steering
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# frame ang. rate, and front wheel ang. rate (spinning motion).
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# Wheel positions are ignorable coordinates, so they are not introduced.
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q1, q2, q4, q5 = dynamicsymbols('q1 q2 q4 q5')
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q1d, q2d, q4d, q5d = dynamicsymbols('q1 q2 q4 q5', 1)
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u1, u2, u3, u4, u5, u6 = dynamicsymbols('u1 u2 u3 u4 u5 u6')
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u1d, u2d, u3d, u4d, u5d, u6d = dynamicsymbols('u1 u2 u3 u4 u5 u6', 1)
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# Declare System's Parameters
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WFrad, WRrad, htangle, forkoffset = symbols('WFrad WRrad htangle forkoffset')
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forklength, framelength, forkcg1 = symbols('forklength framelength forkcg1')
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forkcg3, framecg1, framecg3, Iwr11 = symbols('forkcg3 framecg1 framecg3 Iwr11')
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Iwr22, Iwf11, Iwf22, Iframe11 = symbols('Iwr22 Iwf11 Iwf22 Iframe11')
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Iframe22, Iframe33, Iframe31, Ifork11 = symbols('Iframe22 Iframe33 Iframe31 Ifork11')
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Ifork22, Ifork33, Ifork31, g = symbols('Ifork22 Ifork33 Ifork31 g')
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mframe, mfork, mwf, mwr = symbols('mframe mfork mwf mwr')
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# Set up reference frames for the system
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# WR - rear wheel, rotation angle is ignorable coordinate so not oriented
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# Frame - bicycle frame
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# TempFrame - statically rotated frame for easier reference inertia definition
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# TempFork - statically rotated frame for easier reference inertia definition
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# WF - front wheel, again posses a ignorable coordinate
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N = ReferenceFrame('N')
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Y = N.orientnew('Y', 'Axis', [q1, N.z])
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R = Y.orientnew('R', 'Axis', [q2, Y.x])
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Frame = R.orientnew('Frame', 'Axis', [q4 + htangle, R.y])
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WR = ReferenceFrame('WR')
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TempFrame = Frame.orientnew('TempFrame', 'Axis', [-htangle, Frame.y])
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Fork = Frame.orientnew('Fork', 'Axis', [q5, Frame.x])
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TempFork = Fork.orientnew('TempFork', 'Axis', [-htangle, Fork.y])
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WF = ReferenceFrame('WF')
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# Kinematics of the Bicycle First block of code is forming the positions of
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# rear wheel contact -> rear wheel mass center -> frame mass center +
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# frame/fork connection -> fork mass center + front wheel mass center ->
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# front wheel contact point
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WR_cont = Point('WR_cont')
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WR_mc = WR_cont.locatenew('WR_mc', WRrad * R.z)
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Steer = WR_mc.locatenew('Steer', framelength * Frame.z)
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Frame_mc = WR_mc.locatenew('Frame_mc', - framecg1 * Frame.x
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Fork_mc = Steer.locatenew('Fork_mc', - forkcg1 * Fork.x
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WF_mc = Steer.locatenew('WF_mc', forklength * Fork.x + forkoffset * Fork.z)
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WF_cont = WF_mc.locatenew('WF_cont', WFrad * (dot(Fork.y, Y.z) * Fork.y -
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# Set the angular velocity of each frame.
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# Angular accelerations end up being calculated automatically by
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# differentiating the angular velocities when first needed.
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# u3 is rear wheel rate
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# u4 is frame pitch rate
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# u5 is fork steer rate
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# u6 is front wheel rate
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Y.set_ang_vel(N, u1 * Y.z)
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R.set_ang_vel(Y, u2 * R.x)
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WR.set_ang_vel(Frame, u3 * Frame.y)
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Frame.set_ang_vel(R, u4 * Frame.y)
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Fork.set_ang_vel(Frame, u5 * Fork.x)
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WF.set_ang_vel(Fork, u6 * Fork.y)
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# Form the velocities of the previously defined points, using the 2 - point
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# theorem (written out by hand here). Accelerations again are calculated
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# automatically when first needed.
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WR_mc.v2pt_theory(WR_cont, N, WR)
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Steer.v2pt_theory(WR_mc, N, Frame)
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Frame_mc.v2pt_theory(WR_mc, N, Frame)
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Fork_mc.v2pt_theory(Steer, N, Fork)
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WF_mc.v2pt_theory(Steer, N, Fork)
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WF_cont.v2pt_theory(WF_mc, N, WF)
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# Sets the inertias of each body. Uses the inertia frame to construct the
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# inertia dyadics. Wheel inertias are only defined by principle moments of
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# inertia, and are in fact constant in the frame and fork reference frames;
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# it is for this reason that the orientations of the wheels does not need
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# to be defined. The frame and fork inertias are defined in the 'Temp'
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# frames which are fixed to the appropriate body frames; this is to allow
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# easier input of the reference values of the benchmark paper. Note that
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# due to slightly different orientations, the products of inertia need to
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# have their signs flipped; this is done later when entering the numerical
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Frame_I = (inertia(TempFrame, Iframe11, Iframe22, Iframe33, 0, 0, Iframe31), Frame_mc)
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Fork_I = (inertia(TempFork, Ifork11, Ifork22, Ifork33, 0, 0, Ifork31), Fork_mc)
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WR_I = (inertia(Frame, Iwr11, Iwr22, Iwr11), WR_mc)
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WF_I = (inertia(Fork, Iwf11, Iwf22, Iwf11), WF_mc)
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# Declaration of the RigidBody containers. ::
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BodyFrame = RigidBody('BodyFrame', Frame_mc, Frame, mframe, Frame_I)
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BodyFork = RigidBody('BodyFork', Fork_mc, Fork, mfork, Fork_I)
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BodyWR = RigidBody('BodyWR', WR_mc, WR, mwr, WR_I)
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BodyWF = RigidBody('BodyWF', WF_mc, WF, mwf, WF_I)
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# The kinematic differential equations; they are defined quite simply. Each
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# entry in this list is equal to zero.
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kd = [q1d - u1, q2d - u2, q4d - u4, q5d - u5]
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# The nonholonomic constraints are the velocity of the front wheel contact
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# point dotted into the X, Y, and Z directions; the yaw frame is used as it
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# is "closer" to the front wheel (1 less DCM connecting them). These
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# constraints force the velocity of the front wheel contact point to be 0
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# in the inertial frame; the X and Y direction constraints enforce a
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# "no-slip" condition, and the Z direction constraint forces the front
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# wheel contact point to not move away from the ground frame, essentially
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# replicating the holonomic constraint which does not allow the frame pitch
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# to change in an invalid fashion.
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conlist_speed = [WF_cont.vel(N) & Y.x, WF_cont.vel(N) & Y.y, WF_cont.vel(N) & Y.z]
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# The holonomic constraint is that the position from the rear wheel contact
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# point to the front wheel contact point when dotted into the
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# normal-to-ground plane direction must be zero; effectively that the front
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# and rear wheel contact points are always touching the ground plane. This
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# is actually not part of the dynamic equations, but instead is necessary
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# for the lineraization process.
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conlist_coord = [WF_cont.pos_from(WR_cont) & Y.z]
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# The force list; each body has the appropriate gravitational force applied
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# at its mass center.
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FL = [(Frame_mc, -mframe * g * Y.z),
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(Fork_mc, -mfork * g * Y.z),
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(WF_mc, -mwf * g * Y.z),
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(WR_mc, -mwr * g * Y.z)]
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BL = [BodyFrame, BodyFork, BodyWR, BodyWF]
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# The N frame is the inertial frame, coordinates are supplied in the order
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# of independent, dependent coordinates, as are the speeds. The kinematic
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# differential equation are also entered here. Here the dependent speeds
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# are specified, in the same order they were provided in earlier, along
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# with the non-holonomic constraints. The dependent coordinate is also
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# provided, with the holonomic constraint. Again, this is only provided
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# for the linearization process.
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KM = KanesMethod(N, q_ind=[q1, q2, q5],
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q_dependent=[q4], configuration_constraints=conlist_coord,
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u_dependent=[u1, u4, u6], velocity_constraints=conlist_speed,
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(fr, frstar) = KM.kanes_equations(FL, BL)
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# This is the start of entering in the numerical values from the benchmark
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# paper to validate the eigen values of the linearized equations from this
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# model to the reference eigen values. Look at the aforementioned paper for
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# more information. Some of these are intermediate values, used to
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# transform values from the paper into the coordinate systems used in this
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HTA = evalf.N(pi / 2 - pi / 10)
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rake = evalf.N(-(TrailPaper*sin(HTA)-(PaperRadFront*cos(HTA))))
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FrameLength = evalf.N(PaperWb*sin(HTA)-(rake-(PaperRadFront-PaperRadRear)*cos(HTA)))
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FrameCGNorm = evalf.N((PaperFrameCgZ - PaperRadRear-(PaperFrameCgX/sin(HTA))*cos(HTA))*sin(HTA))
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FrameCGPar = evalf.N((PaperFrameCgX / sin(HTA) + (PaperFrameCgZ - PaperRadRear - PaperFrameCgX / sin(HTA) * cos(HTA)) * cos(HTA)))
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tempa = evalf.N((PaperForkCgZ - PaperRadFront))
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tempb = evalf.N((PaperWb-PaperForkCgX))
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tempc = evalf.N(sqrt(tempa**2+tempb**2))
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PaperForkL = evalf.N((PaperWb*cos(HTA)-(PaperRadFront-PaperRadRear)*sin(HTA)))
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ForkCGNorm = evalf.N(rake+(tempc * sin(pi/2-HTA-acos(tempa/tempc))))
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ForkCGPar = evalf.N(tempc * cos((pi/2-HTA)-acos(tempa/tempc))-PaperForkL)
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# Here is the final assembly of the numerical values. The symbol 'v' is the
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# forward speed of the bicycle (a concept which only makes sense in the
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# upright, static equilibrium case?). These are in a dictionary which will
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# later be substituted in. Again the sign on the *product* of inertia
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# values is flipped here, due to different orientations of coordinate
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val_dict = {WFrad: PaperRadFront,
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forklength: PaperForkL,
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framelength: FrameLength,
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framecg1: FrameCGNorm,
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framecg3: FrameCGPar,
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u3: v / PaperRadRear,
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u6: v / PaperRadFront}
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# Linearizes the forcing vector; the equations are set up as MM udot =
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# forcing, where MM is the mass matrix, udot is the vector representing the
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# time derivatives of the generalized speeds, and forcing is a vector which
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# contains both external forcing terms and internal forcing terms, such as
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# centripital or coriolis forces. This actually returns a matrix with as
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# many rows as *total* coordinates and speeds, but only as many columns as
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# independent coordinates and speeds.
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forcing_lin = KM.linearize()[0]
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# As mentioned above, the size of the linearized forcing terms is expanded
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# to include both q's and u's, so the mass matrix must have this done as
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# well. This will likely be changed to be part of the linearized process,
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# for future reference.
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MM_full = KM.mass_matrix_full
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MM_full_s = MM_full.subs(val_dict)
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forcing_lin_s = forcing_lin.subs(KM.kindiffdict()).subs(val_dict)
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MM_full_s = MM_full_s.evalf()
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forcing_lin_s = forcing_lin_s.evalf()
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# Finally, we construct an "A" matrix for the form xdot = A x (x being the
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# state vector, although in this case, the sizes are a little off). The
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# following line extracts only the minimum entries required for eigenvalue
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# analysis, which correspond to rows and columns for lean, steer, lean
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# rate, and steer rate.
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Amat = MM_full_s.inv() * forcing_lin_s
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A = Amat.extract([1, 2, 4, 6], [1, 2, 3, 5])
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# Precomputed for comparison
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Res = Matrix([[ 0, 0, 1.0, 0],
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[9.48977444677355, -0.891197738059089*v**2 - 0.571523173729245, -0.105522449805691*v, -0.330515398992311*v],
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[11.7194768719633, -1.97171508499972*v**2 + 30.9087533932407, 3.67680523332152*v, -3.08486552743311*v]])
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# Actual eigenvalue comparison
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error = Res.subs(v, i) - A.subs(v, i)
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assert all(abs(x) < eps for x in error)
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lib/py/sympy/physics/mechanics/tests/test_kane3.py
