from sympy import symbols, Matrix, sin, cos, diff
from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, Point, Particle
from sympy.physics.mechanics import Lagrangian, LagrangesMethod
from sympy.physics.mechanics import mprint
t = symbols('t') # 時間
m1, m2, l, g = symbols('m1 m2 l g') # パラメータ
p, theta = dynamicsymbols('p theta') # 一般化座標
F = dynamicsymbols('F') # 外力
q = Matrix([p, theta]) # 座標ベクトル
qd = q.diff(t) # 座標の時間微分
N = ReferenceFrame('N') # 参照座標系
# 質点1（台車）の質点、位置
P1 = Point('P1')
x1 = p # 質点1の位置
y1 = 0
vx1 = diff(x1, t) # x1.diff(t)?でもOK?
vy1 = diff(y1, t)
P1.set_vel(N, vx1 * N.x + vy1 * N.y)
Pa1 = Particle('Pa1', P1, m1)
Pa1.potential_energy = m1 * g * y1
# 質点2（振子）の質点、位置
P2 = Point('P2')
x2 = p - l * sin(theta)
y2 = cos(theta)
vx2 = diff(x2, t)
def test_Vector_diffs():
q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
q1dd, q2dd, q3dd, q4dd = dynamicsymbols('q1 q2 q3 q4', 2)
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q3, N.z])
B = A.orientnew('B', 'Axis', [q2, A.x])
v1 = q2 * A.x + q3 * N.y
v2 = q3 * B.x + v1
v3 = v1.dt(B)
v4 = v2.dt(B)
assert v1.dt(N) == q2d * A.x + q2 * q3d * A.y + q3d * N.y
assert v1.dt(A) == q2d * A.x + q3 * q3d * N.x + q3d * N.y
assert v1.dt(B) == (q2d * A.x + q3 * q3d * N.x + q3d * N.y - q3 * cos(q3) *
q2d * N.z)
assert v2.dt(N) == (q2d * A.x + (q2 + q3) * q3d * A.y + q3d * B.x + q3d *
N.y)
assert v2.dt(A) == q2d * A.x + q3d * B.x + q3 * q3d * N.x + q3d * N.y
assert v2.dt(B) == (q2d * A.x + q3d * B.x + q3 * q3d * N.x + q3d * N.y -
q3 * cos(q3) * q2d * N.z)
assert v3.dt(N) == (q2dd * A.x + q2d * q3d * A.y + (q3d**2 + q3 * q3dd) *
N.x + q3dd * N.y + (q3 * sin(q3) * q2d * q3d -
cos(q3) * q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v3.dt(A) == (q2dd * A.x + (2 * q3d**2 + q3 * q3dd) * N.x + (q3dd -
q3 * q3d**2) * N.y + (q3 * sin(q3) * q2d * q3d -
cos(q3) * q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v3.dt(B) == (q2dd * A.x - q3 * cos(q3) * q2d**2 * A.y + (2 *
q3d**2 + q3 * q3dd) * N.x + (q3dd - q3 * q3d**2) *
N.y + (2 * q3 * sin(q3) * q2d * q3d - 2 * cos(q3) *
q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v4.dt(N) == (q2dd * A.x + q3d * (q2d + q3d) * A.y + q3dd * B.x +
(q3d**2 + q3 * q3dd) * N.x + q3dd * N.y + (q3 *
sin(q3) * q2d * q3d - cos(q3) * q2d * q3d - q3 *
cos(q3) * q2dd) * N.z)
assert v4.dt(A) == (q2dd * A.x + q3dd * B.x + (2 * q3d**2 + q3 * q3dd) *
N.x + (q3dd - q3 * q3d**2) * N.y + (q3 * sin(q3) *
q2d * q3d - cos(q3) * q2d * q3d - q3 * cos(q3) *
q2dd) * N.z)
assert v4.dt(B) == (q2dd * A.x - q3 * cos(q3) * q2d**2 * A.y + q3dd * B.x +
(2 * q3d**2 + q3 * q3dd) * N.x + (q3dd - q3 * q3d**2) *
N.y + (2 * q3 * sin(q3) * q2d * q3d - 2 * cos(q3) *
q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v3.diff(q1d, N) == 0
assert v3.diff(q2d, N) == A.x - q3 * cos(q3) * N.z
assert v3.diff(q3d, N) == q3 * N.x + N.y
assert v3.diff(q1d, A) == 0
assert v3.diff(q2d, A) == A.x - q3 * cos(q3) * N.z
assert v3.diff(q3d, A) == q3 * N.x + N.y
assert v3.diff(q1d, B) == 0
assert v3.diff(q2d, B) == A.x - q3 * cos(q3) * N.z
assert v3.diff(q3d, B) == q3 * N.x + N.y
assert v4.diff(q1d, N) == 0
assert v4.diff(q2d, N) == A.x - q3 * cos(q3) * N.z
assert v4.diff(q3d, N) == B.x + q3 * N.x + N.y
assert v4.diff(q1d, A) == 0
assert v4.diff(q2d, A) == A.x - q3 * cos(q3) * N.z
assert v4.diff(q3d, A) == B.x + q3 * N.x + N.y
assert v4.diff(q1d, B) == 0
assert v4.diff(q2d, B) == A.x - q3 * cos(q3) * N.z
assert v4.diff(q3d, B) == B.x + q3 * N.x + N.y
# Define generalized coordinates, speeds, and constants:
q0, q1, q2 = dynamicsymbols('q0:3')
q0d, q1d, q2d = dynamicsymbols('q0:3', level=1)
u1, u2, u3 = dynamicsymbols('u1:4')
LA, LB, LP = symbols('LA LB LP')
p1, p2, p3 = symbols('p1:4')
A1, A2, A3 = symbols('A1:4')
B1, B2, B3 = symbols('B1:4')
C1, C2, C3 = symbols('C1:4')
D11, D22, D33, D12, D23, D31 = symbols('D11 D22 D33 D12 D23 D31')
g, mA, mB, mC, mD, t = symbols('g mA mB mC mD t')
TA_star, TB_star, TC_star, TD_star = symbols('TA* TB* TC* TD*')
## --- reference frames ---
E = ReferenceFrame('E')
A = E.orientnew('A', 'Axis', [q0, E.x])
B = A.orientnew('B', 'Axis', [q1, A.y])
C = B.orientnew('C', 'Axis', [0, B.x])
D = C.orientnew('D', 'Axis', [0, C.x])
## --- points and their velocities ---
pO = Point('O')
pA_star = pO.locatenew('A*', LA * A.z)
pP = pO.locatenew('P', LP * A.z)
pB_star = pP.locatenew('B*', LB * B.z)
pC_star = pB_star.locatenew('C*', q2 * B.z)
pD_star = pC_star.locatenew('D*', p1 * B.x + p2 * B.y + p3 * B.z)
pO.set_vel(E, 0) # Point O is fixed in Reference Frame E
# Projection of COM on X-Y plane
x1 = lg_1 * cos(theta_1)
y1 = lg_1 * sin(theta_1)
x2 = l_1 * cos(theta_1) + lg_2 * cos(theta_1 + theta_2)
y2 = l_1 * sin(theta_1) + lg_2 * sin(theta_1 + theta_2)
# Velocity on X-Y plane
vx1 = diff(x1, t)
vy1 = diff(y1, t)
vx2 = diff(x2, t)
vy2 = diff(y2, t)
# Setting Reference Frames
N = ReferenceFrame("N")
# Lumped mass abstraction
P_1 = Point("P_1")
P_2 = Point("P_2")
# Velocity of Point mass in X-Y plane
P_1.set_vel(N, vx1 * N.x + vy1 * N.y)
P_2.set_vel(N, vx2 * N.x + vy2 * N.y)
# Making a particle from point mass
Pa_1 = Particle("P_1", P_1, m_1)
Pa_2 = Particle("P_2", P_2, m_2)
# Potential energy of system
Pa_1.potential_energy = m_1 * g * y1
from __future__ import division
from sympy import cos, pi, solve, symbols, S
from sympy.physics.mechanics import ReferenceFrame, Point
from sympy.physics.mechanics import dynamicsymbols
from util import msprint, partial_velocities, generalized_active_forces
from util import potential_energy
q1 = dynamicsymbols('q1')
q1d = dynamicsymbols('q1', level=1)
u1 = dynamicsymbols('u1')
g, m, r, theta = symbols('g m r θ')
omega, t = symbols('ω t')
# reference frames
M = ReferenceFrame('M')
# Plane containing W rotates about a vertical line through center of W.
N = M.orientnew('N', 'Axis', [omega * t, M.z])
C = N.orientnew('C', 'Axis', [q1, N.x])
R = C # R is fixed relative to C
A = C.orientnew('A', 'Axis', [-theta, R.x])
B = C.orientnew('B', 'Axis', [theta, R.x])
# points, velocities
pO = Point('O') # Point O is at the center of the circular wire
pA = pO.locatenew('A', -r * A.z)
pB = pO.locatenew('B', -r * B.z)
pR_star = pO.locatenew('R*', 1/S(2) * (pA.pos_from(pO) + pB.pos_from(pO)))
pO.set_vel(N, 0)
pO.set_vel(C, 0)
def test_sub_qdot():
# This test solves exercises 8.12, 8.17 from Kane 1985 and defines
# some velocities in terms of q, qdot.
## --- Declare symbols ---
q1, q2, q3 = dynamicsymbols('q1:4')
q1d, q2d, q3d = dynamicsymbols('q1:4', level=1)
u1, u2, u3 = dynamicsymbols('u1:4')
u_prime, R, M, g, e, f, theta = symbols('u\' R, M, g, e, f, theta')
a, b, mA, mB, IA, J, K, t = symbols('a b mA mB IA J K t')
IA22, IA23, IA33 = symbols('IA22 IA23 IA33')
Q1, Q2, Q3 = symbols('Q1 Q2 Q3')
# --- Reference Frames ---
F = ReferenceFrame('F')
P = F.orientnew('P', 'axis', [-theta, F.y])
A = P.orientnew('A', 'axis', [q1, P.x])
A.set_ang_vel(F, u1*A.x + u3*A.z)
# define frames for wheels
B = A.orientnew('B', 'axis', [q2, A.z])
C = A.orientnew('C', 'axis', [q3, A.z])
## --- define points D, S*, Q on frame A and their velocities ---
pD = Point('D')
pD.set_vel(A, 0)
# u3 will not change v_D_F since wheels are still assumed to roll w/o slip
pD.set_vel(F, u2 * A.y)
pS_star = pD.locatenew('S*', e*A.y)
pQ = pD.locatenew('Q', f*A.y - R*A.x)
# masscenters of bodies A, B, C
pA_star = pD.locatenew('A*', a*A.y)
pB_star = pD.locatenew('B*', b*A.z)
pC_star = pD.locatenew('C*', -b*A.z)
for p in [pS_star, pQ, pA_star, pB_star, pC_star]:
p.v2pt_theory(pD, F, A)
# points of B, C touching the plane P
pB_hat = pB_star.locatenew('B^', -R*A.x)
pC_hat = pC_star.locatenew('C^', -R*A.x)
pB_hat.v2pt_theory(pB_star, F, B)
pC_hat.v2pt_theory(pC_star, F, C)
# --- relate qdot, u ---
# the velocities of B^, C^ are zero since B, C are assumed to roll w/o slip
kde = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]]
kde += [u1 - q1d]
kde_map = solve(kde, [q1d, q2d, q3d])
for k, v in list(kde_map.items()):
kde_map[k.diff(t)] = v.diff(t)
# inertias of bodies A, B, C
# IA22, IA23, IA33 are not specified in the problem statement, but are
# necessary to define an inertia object. Although the values of
# IA22, IA23, IA33 are not known in terms of the variables given in the
# problem statement, they do not appear in the general inertia terms.
inertia_A = inertia(A, IA, IA22, IA33, 0, IA23, 0)
inertia_B = inertia(B, K, K, J)
inertia_C = inertia(C, K, K, J)
# define the rigid bodies A, B, C
rbA = RigidBody('rbA', pA_star, A, mA, (inertia_A, pA_star))
rbB = RigidBody('rbB', pB_star, B, mB, (inertia_B, pB_star))
rbC = RigidBody('rbC', pC_star, C, mB, (inertia_C, pC_star))
## --- use kanes method ---
km = KanesMethod(F, [q1, q2, q3], [u1, u2], kd_eqs=kde, u_auxiliary=[u3])
forces = [(pS_star, -M*g*F.x), (pQ, Q1*A.x + Q2*A.y + Q3*A.z)]
bodies = [rbA, rbB, rbC]
# Q2 = -u_prime * u2 * Q1 / sqrt(u2**2 + f**2 * u1**2)
# -u_prime * R * u2 / sqrt(u2**2 + f**2 * u1**2) = R / Q1 * Q2
fr_expected = Matrix([
f*Q3 + M*g*e*sin(theta)*cos(q1),
Q2 + M*g*sin(theta)*sin(q1),
e*M*g*cos(theta) - Q1*f - Q2*R])
#Q1 * (f - u_prime * R * u2 / sqrt(u2**2 + f**2 * u1**2)))])
fr_star_expected = Matrix([
-(IA + 2*J*b**2/R**2 + 2*K +
mA*a**2 + 2*mB*b**2) * u1.diff(t) - mA*a*u1*u2,
-(mA + 2*mB +2*J/R**2) * u2.diff(t) + mA*a*u1**2,
0])
fr, fr_star = km.kanes_equations(forces, bodies)
assert (fr.expand() == fr_expected.expand())
assert (trigsimp(fr_star).expand() == fr_star_expected.expand())
def test_linearize_pendulum_kane_nonminimal():
# Create generalized coordinates and speeds for this non-minimal realization
# q1, q2 = N.x and N.y coordinates of pendulum
# u1, u2 = N.x and N.y velocities of pendulum
q1, q2 = dynamicsymbols('q1:3')
q1d, q2d = dynamicsymbols('q1:3', level=1)
u1, u2 = dynamicsymbols('u1:3')
u1d, u2d = dynamicsymbols('u1:3', level=1)
L, m, t = symbols('L, m, t')
g = 9.8
# Compose world frame
N = ReferenceFrame('N')
pN = Point('N*')
pN.set_vel(N, 0)
# A.x is along the pendulum
theta1 = atan(q2 / q1)
A = N.orientnew('A', 'axis', [theta1, N.z])
# Locate the pendulum mass
P = pN.locatenew('P1', q1 * N.x + q2 * N.y)
pP = Particle('pP', P, m)
# Calculate the kinematic differential equations
kde = Matrix([q1d - u1, q2d - u2])
dq_dict = solve(kde, [q1d, q2d])
# Set velocity of point P
P.set_vel(N, P.pos_from(pN).dt(N).subs(dq_dict))
# Configuration constraint is length of pendulum
f_c = Matrix([P.pos_from(pN).magnitude() - L])
# Velocity constraint is that the velocity in the A.x direction is
# always zero (the pendulum is never getting longer).
f_v = Matrix([P.vel(N).express(A).dot(A.x)])
f_v.simplify()
# Acceleration constraints is the time derivative of the velocity constraint
f_a = f_v.diff(t)
f_a.simplify()
# Input the force resultant at P
R = m * g * N.x
# Derive the equations of motion using the KanesMethod class.
KM = KanesMethod(N,
q_ind=[q2],
u_ind=[u2],
q_dependent=[q1],
u_dependent=[u1],
configuration_constraints=f_c,
velocity_constraints=f_v,
acceleration_constraints=f_a,
kd_eqs=kde)
(fr, frstar) = KM.kanes_equations([(P, R)], [pP])
# Set the operating point to be straight down, and non-moving
q_op = {q1: L, q2: 0}
u_op = {u1: 0, u2: 0}
ud_op = {u1d: 0, u2d: 0}
A, B, inp_vec = KM.linearize(op_point=[q_op, u_op, ud_op],
A_and_B=True,
new_method=True,
simplify=True)
assert A == Matrix([[0, 1], [-9.8 / L, 0]])
assert B == Matrix([])
def test_rolling_disc():
# Rolling Disc Example
# Here the rolling disc is formed from the contact point up, removing the
# need to introduce generalized speeds. Only 3 configuration and three
# speed variables are need to describe this system, along with the disc's
# mass and radius, and the local gravity (note that mass will drop out).
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3')
q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1)
r, m, g = symbols('r m g')
# The kinematics are formed by a series of simple rotations. Each simple
# rotation creates a new frame, and the next rotation is defined by the new
# frame's basis vectors. This example uses a 3-1-2 series of rotations, or
# Z, X, Y series of rotations. Angular velocity for this is defined using
# the second frame's basis (the lean frame).
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
R.set_ang_acc(N,
R.ang_vel_in(N).dt(R) + (R.ang_vel_in(N) ^ R.ang_vel_in(N)))
# This is the translational kinematics. We create a point with no velocity
# in N; this is the contact point between the disc and ground. Next we form
# the position vector from the contact point to the disc's center of mass.
# Finally we form the velocity and acceleration of the disc.
C = Point('C')
C.set_vel(N, 0)
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
Dmc.a2pt_theory(C, N, R)
# This is a simple way to form the inertia dyadic.
I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
# Kinematic differential equations; how the generalized coordinate time
# derivatives relate to generalized speeds.
kd = [q1d - u3 / cos(q2), q2d - u1, q3d - u2 + u3 * tan(q2)]
# Creation of the force list; it is the gravitational force at the mass
# center of the disc. Then we create the disc by assigning a Point to the
# center of mass attribute, a ReferenceFrame to the frame attribute, and mass
# and inertia. Then we form the body list.
ForceList = [(Dmc, -m * g * Y.z)]
BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
BodyList = [BodyD]
# Finally we form the equations of motion, using the same steps we did
# before. Specify inertial frame, supply generalized speeds, supply
# kinematic differential equation dictionary, compute Fr from the force
# list and Fr* from the body list, compute the mass matrix and forcing
# terms, then solve for the u dots (time derivatives of the generalized
# speeds).
KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd)
KM.kanes_equations(ForceList, BodyList)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
kdd = KM.kindiffdict()
rhs = rhs.subs(kdd)
assert rhs.expand() == Matrix([
(10 * u2 * u3 * r - 5 * u3**2 * r * tan(q2) + 4 * g * sin(q2)) /
(5 * r), -2 * u1 * u3 / 3, u1 * (-2 * u2 + u3 * tan(q2))
]).expand()
particles = []
links = []
points = []
com = []
frames = []
sym_pars = [*m_links, *d_links, *m_point, b_cart, b_joint, g, k, l0]
num_pars = [*par.m_links, *par.d_links, *par.m_point, par.b_cart, par.b_joint,
par.g, par.k, par.l0]
subs_dict = dict(zip(sym_pars, num_pars))
# ----------------------------------------------------------------------------
# Reference frames
# ----------------------------------------------------------------------------
N = ReferenceFrame("N") # Inertial reference frame
(origin := Point("O")).set_vel(N, 0) # Set velocity to zero
pend_frame = N.orientnew("pend", "axis", (q[1], N.z))
# pend_frame.set_ang_vel(N, dq[1]*N.z)
# Link 1: upper left link
link1_frame = pend_frame.orientnew("link_1", "axis", (q[2], pend_frame.z))
link1_frame.set_ang_vel(pend_frame, dq[2]*pend_frame.z)
# Link 2: upper right link
link2_frame = pend_frame.orientnew("link_2", "axis", (-q[2], pend_frame.z))
link2_frame.set_ang_vel(pend_frame, -dq[2]*pend_frame.z)
# Compute angle between upper and lower links
beta = q[2] + asin(d_links[0]/d_links[1]*sin(q[2]))
def test_operator_match():
"""Test that the output of dot, cross, outer functions match
operator behavior.
"""
A = ReferenceFrame('A')
v = A.x + A.y
d = v | v
zerov = Vector(0)
zerod = Dyadic(0)
# dot products
assert d & d == dot(d, d)
assert d & zerod == dot(d, zerod)
assert zerod & d == dot(zerod, d)
assert d & v == dot(d, v)
assert v & d == dot(v, d)
assert d & zerov == dot(d, zerov)
assert zerov & d == dot(zerov, d)
raises(TypeError, lambda: dot(d, S(0)))
raises(TypeError, lambda: dot(S(0), d))
raises(TypeError, lambda: dot(d, 0))
raises(TypeError, lambda: dot(0, d))
assert v & v == dot(v, v)
assert v & zerov == dot(v, zerov)
assert zerov & v == dot(zerov, v)
raises(TypeError, lambda: dot(v, S(0)))
raises(TypeError, lambda: dot(S(0), v))
raises(TypeError, lambda: dot(v, 0))
raises(TypeError, lambda: dot(0, v))
# cross products
raises(TypeError, lambda: cross(d, d))
raises(TypeError, lambda: cross(d, zerod))
raises(TypeError, lambda: cross(zerod, d))
assert d ^ v == cross(d, v)
assert v ^ d == cross(v, d)
assert d ^ zerov == cross(d, zerov)
assert zerov ^ d == cross(zerov, d)
assert zerov ^ d == cross(zerov, d)
raises(TypeError, lambda: cross(d, S(0)))
raises(TypeError, lambda: cross(S(0), d))
raises(TypeError, lambda: cross(d, 0))
raises(TypeError, lambda: cross(0, d))
assert v ^ v == cross(v, v)
assert v ^ zerov == cross(v, zerov)
assert zerov ^ v == cross(zerov, v)
raises(TypeError, lambda: cross(v, S(0)))
raises(TypeError, lambda: cross(S(0), v))
raises(TypeError, lambda: cross(v, 0))
raises(TypeError, lambda: cross(0, v))
# outer products
raises(TypeError, lambda: outer(d, d))
raises(TypeError, lambda: outer(d, zerod))
raises(TypeError, lambda: outer(zerod, d))
raises(TypeError, lambda: outer(d, v))
raises(TypeError, lambda: outer(v, d))
raises(TypeError, lambda: outer(d, zerov))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(d, S(0)))
raises(TypeError, lambda: outer(S(0), d))
raises(TypeError, lambda: outer(d, 0))
raises(TypeError, lambda: outer(0, d))
assert v | v == outer(v, v)
assert v | zerov == outer(v, zerov)
assert zerov | v == outer(zerov, v)
raises(TypeError, lambda: outer(v, S(0)))
raises(TypeError, lambda: outer(S(0), v))
raises(TypeError, lambda: outer(v, 0))
raises(TypeError, lambda: outer(0, v))
from pydy.viz.scene import Scene
# import dill
# import pickle
import cloudpickle
import os
import inspect
#this file is used to generate a serialized function that can be used to estimate
# next states given current states
#ASSUMES GRAVITY SUFFICIENTLY COMPENSATED BY ROBOT
init_vprinting(use_latex='mathjax', pretty_print=True)
#Kinematics ------------------------------
#init reference frames, assume base attached to floor
inertial_frame = ReferenceFrame('I')
j0_frame = ReferenceFrame('J0')
j1_frame = ReferenceFrame('J1')
j2_frame = ReferenceFrame('J2')
#declare dynamic symbols for the three joints
theta0, theta1, theta2 = dynamicsymbols('theta0, theta1, theta2')
#orient frames
j0_frame.orient(inertial_frame, 'Axis', (theta0, inertial_frame.y))
j1_frame.orient(j0_frame, 'Axis', (theta1, j0_frame.z))
j2_frame.orient(j1_frame, 'Axis', (theta2, j1_frame.z))
#TODO figure out better name for joint points
#init joints
joint0 = Point('j0')
q1, q2, q3, q4, q5, q6, q7 = q = dynamicsymbols('q1:8')
u1, u2, u3, u4, u5, u6, u7 = u = dynamicsymbols('q1:8', level=1)
M, J, I11, I22, m, r, b = symbols('M J I11 I22 m r b',
real=True,
positive=True)
omega, t = symbols('ω t')
theta = 30 * pi / 180 # radians
b = r * (1 + sin(theta)) / (cos(theta) - sin(theta))
# Note: using b as found in Ex3.10. Pure rolling between spheres and race R
# is likely a typo and should be between spheres and cone C.
# define reference frames
R = ReferenceFrame('R') # fixed race rf, let R.z point upwards
A = R.orientnew('A', 'axis', [q7, R.z]) # rf that rotates with S* about R.z
# B.x, B.z are parallel with face of cone, B.y is perpendicular
B = A.orientnew('B', 'axis', [-theta, A.x])
S = ReferenceFrame('S')
S.set_ang_vel(A, u1 * A.x + u2 * A.y + u3 * A.z)
C = ReferenceFrame('C')
C.set_ang_vel(A, u4 * B.x + u5 * B.y + u6 * B.z)
#C.set_ang_vel(A, u4*A.x + u5*A.y + u6*A.z)
# define points
pO = Point('O')
pS_star = pO.locatenew('S*', b * A.y)
pS_hat = pS_star.locatenew('S^', -r * B.y) # S^ touches the cone
pS1 = pS_star.locatenew('S1',
-r * A.z) # S1 touches horizontal wall of the race
def test_replace_qdots_in_force():
# Test PR 16700 "Replaces qdots with us in force-list in kanes.py"
# The new functionality allows one to specify forces in qdots which will
# automatically be replaced with u:s which are defined by the kde supplied
# to KanesMethod. The test case is the double pendulum with interacting
# forces in the example of chapter 4.7 "CONTRIBUTING INTERACTION FORCES"
# in Ref. [1]. Reference list at end test function.
q1, q2 = dynamicsymbols('q1, q2')
qd1, qd2 = dynamicsymbols('q1, q2', level=1)
u1, u2 = dynamicsymbols('u1, u2')
l, m = symbols('l, m')
N = ReferenceFrame('N') # Inertial frame
A = N.orientnew('A', 'Axis', (q1, N.z)) # Rod A frame
B = A.orientnew('B', 'Axis', (q2, N.z)) # Rod B frame
O = Point('O') # Origo
O.set_vel(N, 0)
P = O.locatenew('P', (l * A.x)) # Point @ end of rod A
P.v2pt_theory(O, N, A)
Q = P.locatenew('Q', (l * B.x)) # Point @ end of rod B
Q.v2pt_theory(P, N, B)
Ap = Particle('Ap', P, m)
Bp = Particle('Bp', Q, m)
# The forces are specified below. sigma is the torsional spring stiffness
# and delta is the viscous damping coefficient acting between the two
# bodies. Here, we specify the viscous damper as function of qdots prior
# forming the kde. In more complex systems it not might be obvious which
# kde is most efficient, why it is convenient to specify viscous forces in
# qdots independently of the kde.
sig, delta = symbols('sigma, delta')
Ta = (sig * q2 + delta * qd2) * N.z
forces = [(A, Ta), (B, -Ta)]
# Try different kdes.
kde1 = [u1 - qd1, u2 - qd2]
kde2 = [u1 - qd1, u2 - (qd1 + qd2)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1)
fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2)
fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
# Check EOM for KM2:
# Mass and force matrix from p.6 in Ref. [2] with added forces from
# example of chapter 4.7 in [1] and without gravity.
forcing_matrix_expected = Matrix(
[[m * l**2 * sin(q2) * u2**2 + sig * q2 + delta * (u2 - u1)],
[m * l**2 * sin(q2) * -u1**2 - sig * q2 - delta * (u2 - u1)]])
mass_matrix_expected = Matrix([[2 * m * l**2, m * l**2 * cos(q2)],
[m * l**2 * cos(q2), m * l**2]])
assert (KM2.mass_matrix.expand() == mass_matrix_expected.expand())
assert (KM2.forcing.expand() == forcing_matrix_expected.expand())
# Check fr1 with reference fr_expected from [1] with u:s instead of qdots.
fr1_expected = Matrix([0, -(sig * q2 + delta * u2)])
assert fr1.expand() == fr1_expected.expand()
# Check fr2
fr2_expected = Matrix(
[sig * q2 + delta * (u2 - u1), -sig * q2 - delta * (u2 - u1)])
assert fr2.expand() == fr2_expected.expand()
# Specifying forces in u:s should stay the same:
Ta = (sig * q2 + delta * u2) * N.z
forces = [(A, Ta), (B, -Ta)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1)
fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
assert fr1.expand() == fr1_expected.expand()
Ta = (sig * q2 + delta * (u2 - u1)) * N.z
forces = [(A, Ta), (B, -Ta)]
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2)
fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
assert fr2.expand() == fr2_expected.expand()
# Test if we have a qubic qdot force:
Ta = (sig * q2 + delta * qd2**3) * N.z
forces = [(A, Ta), (B, -Ta)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1)
fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
fr1_cubic_expected = Matrix([0, -(sig * q2 + delta * u2**3)])
assert fr1.expand() == fr1_cubic_expected.expand()
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2)
fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
fr2_cubic_expected = Matrix(
[sig * q2 + delta * (u2 - u1)**3, -sig * q2 - delta * (u2 - u1)**3])
assert fr2.expand() == fr2_cubic_expected.expand()
a, a_star, b, b_star, c_star = symbols('a a⁺ b b⁺ c⁺') # distances
#a, a_star, b, b_star, c_star = symbols('a (a*) b (b*) (c*)') # distances
# torque constants
alpha1, alpha2, alpha3 = symbols('α1:4') # exerted on A
beta1, beta2, beta3 = symbols('β1:4') # exerted on C
gamma1, gamma2, gamma3 = symbols('γ1:4') # exerted on A by B
# force constants
P1, P2, P3 = symbols('P1:4') # exerted on A*
Q1, Q2, Q3 = symbols('Q1:4') # exerted on C*
R1, R2, R3 = symbols('R1:4') # exerted on A at P by B
# inertia variables
A1, A3, B1, B3 = symbols('A1 A3 B1 B3')
C1, C2, C3 = symbols('C1:4')
# reference frames
N = ReferenceFrame('N') # also equal to reference frame of cylinder D
A = N.orientnew('A', 'Axis', [q1, N.y]) # counter-weighted crank
B = N.orientnew('B', 'Axis', [q2, -N.y]) # connected rod
C = N.orientnew('C', 'Axis', [0, N.x]) # piston
# points, velocities
pO = Point('O')
pO.set_vel(N, 0)
pA_star = pO.locatenew('A*', -a_star * A.z)
pA_star.v2pt_theory(pO, N, A)
pP = pO.locatenew('P', a * A.z)
pP.v2pt_theory(pO, N, A)
pB_star = pP.locatenew('B*', (b - b_star) * B.z)
def test_rolling_disc():
# Rolling Disc Example
# Here the rolling disc is formed from the contact point up, removing the
# need to introduce generalized speeds. Only 3 configuration and three
# speed variables are need to describe this system, along with the disc's
# mass and radius, and the local gravity (note that mass will drop out).
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3')
q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1)
r, m, g = symbols('r m g')
# The kinematics are formed by a series of simple rotations. Each simple
# rotation creates a new frame, and the next rotation is defined by the new
# frame's basis vectors. This example uses a 3-1-2 series of rotations, or
# Z, X, Y series of rotations. Angular velocity for this is defined using
# the second frame's basis (the lean frame).
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
w_R_N_qd = R.ang_vel_in(N)
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
# This is the translational kinematics. We create a point with no velocity
# in N; this is the contact point between the disc and ground. Next we form
# the position vector from the contact point to the disc's center of mass.
# Finally we form the velocity and acceleration of the disc.
C = Point('C')
C.set_vel(N, 0)
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
# This is a simple way to form the inertia dyadic.
I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
# Kinematic differential equations; how the generalized coordinate time
# derivatives relate to generalized speeds.
kd = [dot(R.ang_vel_in(N) - w_R_N_qd, uv) for uv in L]
# Creation of the force list; it is the gravitational force at the mass
# center of the disc. Then we create the disc by assigning a Point to the
# center of mass attribute, a ReferenceFrame to the frame attribute, and mass
# and inertia. Then we form the body list.
ForceList = [(Dmc, -m * g * Y.z)]
BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
BodyList = [BodyD]
# Finally we form the equations of motion, using the same steps we did
# before. Specify inertial frame, supply generalized speeds, supply
# kinematic differential equation dictionary, compute Fr from the force
# list and Fr* from the body list, compute the mass matrix and forcing
# terms, then solve for the u dots (time derivatives of the generalized
# speeds).
KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
KM.kanes_equations(ForceList, BodyList)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
kdd = KM.kindiffdict()
rhs = rhs.subs(kdd)
rhs.simplify()
assert rhs.expand() == Matrix([
(6 * u2 * u3 * r - u3**2 * r * tan(q2) + 4 * g * sin(q2)) / (5 * r),
-2 * u1 * u3 / 3, u1 * (-2 * u2 + u3 * tan(q2))
]).expand()
assert simplify(KM.rhs() -
KM.mass_matrix_full.LUsolve(KM.forcing_full)) == zeros(
6, 1)
# This code tests our output vs. benchmark values. When r=g=m=1, the
# critical speed (where all eigenvalues of the linearized equations are 0)
# is 1 / sqrt(3) for the upright case.
A = KM.linearize(A_and_B=True)[0]
A_upright = A.subs({
r: 1,
g: 1,
m: 1
}).subs({
q1: 0,
q2: 0,
q3: 0,
u1: 0,
u3: 0
})
import sympy
assert sympy.sympify(A_upright.subs({u2: 1 / sqrt(3)})).eigenvals() == {
S(0): 6
}
Vector.simp = False # Prevent the use of trigsimp and simplify
t, g = symbols('t g') # Time and gravitational constant
a, b, c = symbols('a b c') # semi diameters of ellipsoid
d, e, f = symbols('d e f') # mass center location parameters
# Mass and Inertia scalars
m, Ixx, Iyy, Izz, Ixy, Iyz, Ixz = symbols('m Ixx Iyy Izz Ixy Iyz Ixz')
q = dynamicsymbols('q:3') # Generalized coordinates
qd = [qi.diff(t) for qi in q] # Generalized coordinate time derivatives
wx, wy, wz = symbols('wx wy wz')
rx, ry, rz = symbols('rx ry rz') # Coordinates, in R frame, from O to P
Fx, Fy, Fz = symbols('Fx Fy Fz')
mu_x, mu_y, mu_z = symbols('mu_x mu_y mu_z')
N = ReferenceFrame('N') # Inertial Reference Frame
Y = N.orientnew('Y', 'Axis', [q[0], N.z]) # Yaw Frame
L = Y.orientnew('L', 'Axis', [q[1], Y.x]) # Lean Frame
R = L.orientnew('R', 'Axis', [q[2], L.y]) # Rattleback body fixed frame
print R.dcm(Y)
I = inertia(R, Ixx, Iyy, Izz, Ixy, Iyz, Ixz) # Inertia dyadic
print I.express(Y)
stop
# Rattleback ground contact point
P = Point('P')
# Rattleback ellipsoid center location, see:
# "Realistic mathematical modeling of the rattleback", Kane, Thomas R. and
# David A. Levinson, 1982, International Journal of Non-Linear Mechanics
def test_non_central_inertia():
# This tests that the calculation of Fr* does not depend the point
# about which the inertia of a rigid body is defined. This test solves
# exercises 8.12, 8.17 from Kane 1985.
# Declare symbols
q1, q2, q3 = dynamicsymbols('q1:4')
q1d, q2d, q3d = dynamicsymbols('q1:4', level=1)
u1, u2, u3, u4, u5 = dynamicsymbols('u1:6')
u_prime, R, M, g, e, f, theta = symbols('u\' R, M, g, e, f, theta')
a, b, mA, mB, IA, J, K, t = symbols('a b mA mB IA J K t')
Q1, Q2, Q3 = symbols('Q1, Q2 Q3')
IA22, IA23, IA33 = symbols('IA22 IA23 IA33')
# Reference Frames
F = ReferenceFrame('F')
P = F.orientnew('P', 'axis', [-theta, F.y])
A = P.orientnew('A', 'axis', [q1, P.x])
A.set_ang_vel(F, u1*A.x + u3*A.z)
# define frames for wheels
B = A.orientnew('B', 'axis', [q2, A.z])
C = A.orientnew('C', 'axis', [q3, A.z])
B.set_ang_vel(A, u4 * A.z)
C.set_ang_vel(A, u5 * A.z)
# define points D, S*, Q on frame A and their velocities
pD = Point('D')
pD.set_vel(A, 0)
# u3 will not change v_D_F since wheels are still assumed to roll without slip.
pD.set_vel(F, u2 * A.y)
pS_star = pD.locatenew('S*', e*A.y)
pQ = pD.locatenew('Q', f*A.y - R*A.x)
for p in [pS_star, pQ]:
p.v2pt_theory(pD, F, A)
# masscenters of bodies A, B, C
pA_star = pD.locatenew('A*', a*A.y)
pB_star = pD.locatenew('B*', b*A.z)
pC_star = pD.locatenew('C*', -b*A.z)
for p in [pA_star, pB_star, pC_star]:
p.v2pt_theory(pD, F, A)
# points of B, C touching the plane P
pB_hat = pB_star.locatenew('B^', -R*A.x)
pC_hat = pC_star.locatenew('C^', -R*A.x)
pB_hat.v2pt_theory(pB_star, F, B)
pC_hat.v2pt_theory(pC_star, F, C)
# the velocities of B^, C^ are zero since B, C are assumed to roll without slip
kde = [q1d - u1, q2d - u4, q3d - u5]
vc = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]]
# inertias of bodies A, B, C
# IA22, IA23, IA33 are not specified in the problem statement, but are
# necessary to define an inertia object. Although the values of
# IA22, IA23, IA33 are not known in terms of the variables given in the
# problem statement, they do not appear in the general inertia terms.
inertia_A = inertia(A, IA, IA22, IA33, 0, IA23, 0)
inertia_B = inertia(B, K, K, J)
inertia_C = inertia(C, K, K, J)
# define the rigid bodies A, B, C
rbA = RigidBody('rbA', pA_star, A, mA, (inertia_A, pA_star))
rbB = RigidBody('rbB', pB_star, B, mB, (inertia_B, pB_star))
rbC = RigidBody('rbC', pC_star, C, mB, (inertia_C, pC_star))
km = KanesMethod(F, q_ind=[q1, q2, q3], u_ind=[u1, u2], kd_eqs=kde,
u_dependent=[u4, u5], velocity_constraints=vc,
u_auxiliary=[u3])
forces = [(pS_star, -M*g*F.x), (pQ, Q1*A.x + Q2*A.y + Q3*A.z)]
bodies = [rbA, rbB, rbC]
fr, fr_star = km.kanes_equations(forces, bodies)
vc_map = solve(vc, [u4, u5])
# KanesMethod returns the negative of Fr, Fr* as defined in Kane1985.
fr_star_expected = Matrix([
-(IA + 2*J*b**2/R**2 + 2*K +
mA*a**2 + 2*mB*b**2) * u1.diff(t) - mA*a*u1*u2,
-(mA + 2*mB +2*J/R**2) * u2.diff(t) + mA*a*u1**2,
0])
assert (trigsimp(fr_star.subs(vc_map).subs(u3, 0)).doit().expand() ==
fr_star_expected.expand())
# define inertias of rigid bodies A, B, C about point D
# I_S/O = I_S/S* + I_S*/O
bodies2 = []
for rb, I_star in zip([rbA, rbB, rbC], [inertia_A, inertia_B, inertia_C]):
I = I_star + inertia_of_point_mass(rb.mass,
rb.masscenter.pos_from(pD),
rb.frame)
bodies2.append(RigidBody('', rb.masscenter, rb.frame, rb.mass,
(I, pD)))
fr2, fr_star2 = km.kanes_equations(forces, bodies2)
assert (trigsimp(fr_star2.subs(vc_map).subs(u3, 0)).doit().expand() ==
fr_star_expected.expand())
from sympy import symbols, simplify, latex
from sympy.physics.mechanics import (inertia, inertia_of_point_mass, mprint,
mlatex, Point, ReferenceFrame)
Ixx, Iyy, Izz, Ixz, I, J = symbols("I_xx I_yy I_zz I_xz I J")
mA, mB, lx, lz = symbols("m_A m_B l_x l_z")
A = ReferenceFrame("A")
IA_AO = inertia(A, Ixx, Iyy, Izz, 0, 0, Ixz)
IB_BO = inertia(A, I, J, I)
BO = Point('BO')
AO = BO.locatenew('AO', lx * A.x + lz * A.z)
CO = BO.locatenew('CO', mA / (mA + mB) * AO.pos_from(BO))
print(mlatex(CO.pos_from(BO)))
IC_CO = IA_AO + IB_BO +\
inertia_of_point_mass(mA, AO.pos_from(CO), A) +\
inertia_of_point_mass(mB, BO.pos_from(CO), A)
mprint((A.x & IC_CO & A.x).expand())
mprint((A.y & IC_CO & A.y).expand())
mprint((A.z & IC_CO & A.z).expand())
mprint((A.x & IC_CO & A.z).expand())
def test_aux_dep():
# This test is about rolling disc dynamics, comparing the results found
# with KanesMethod to those found when deriving the equations "manually"
# with SymPy.
# The terms Fr, Fr*, and Fr*_steady are all compared between the two
# methods. Here, Fr*_steady refers to the generalized inertia forces for an
# equilibrium configuration.
# Note: comparing to the test of test_rolling_disc() in test_kane.py, this
# test also tests auxiliary speeds and configuration and motion constraints
#, seen in the generalized dependent coordinates q[3], and depend speeds
# u[3], u[4] and u[5].
# First, mannual derivation of Fr, Fr_star, Fr_star_steady.
# Symbols for time and constant parameters.
# Symbols for contact forces: Fx, Fy, Fz.
t, r, m, g, I, J = symbols('t r m g I J')
Fx, Fy, Fz = symbols('Fx Fy Fz')
# Configuration variables and their time derivatives:
# q[0] -- yaw
# q[1] -- lean
# q[2] -- spin
# q[3] -- dot(-r*B.z, A.z) -- distance from ground plane to disc center in
# A.z direction
# Generalized speeds and their time derivatives:
# u[0] -- disc angular velocity component, disc fixed x direction
# u[1] -- disc angular velocity component, disc fixed y direction
# u[2] -- disc angular velocity component, disc fixed z direction
# u[3] -- disc velocity component, A.x direction
# u[4] -- disc velocity component, A.y direction
# u[5] -- disc velocity component, A.z direction
# Auxiliary generalized speeds:
# ua[0] -- contact point auxiliary generalized speed, A.x direction
# ua[1] -- contact point auxiliary generalized speed, A.y direction
# ua[2] -- contact point auxiliary generalized speed, A.z direction
q = dynamicsymbols('q:4')
qd = [qi.diff(t) for qi in q]
u = dynamicsymbols('u:6')
ud = [ui.diff(t) for ui in u]
ud_zero = dict(zip(ud, [0.]*len(ud)))
ua = dynamicsymbols('ua:3')
ua_zero = dict(zip(ua, [0.]*len(ua)))
# Reference frames:
# Yaw intermediate frame: A.
# Lean intermediate frame: B.
# Disc fixed frame: C.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q[0], N.z])
B = A.orientnew('B', 'Axis', [q[1], A.x])
C = B.orientnew('C', 'Axis', [q[2], B.y])
# Angular velocity and angular acceleration of disc fixed frame
# u[0], u[1] and u[2] are generalized independent speeds.
C.set_ang_vel(N, u[0]*B.x + u[1]*B.y + u[2]*B.z)
C.set_ang_acc(N, C.ang_vel_in(N).diff(t, B)
+ cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Velocity and acceleration of points:
# Disc-ground contact point: P.
# Center of disc: O, defined from point P with depend coordinate: q[3]
# u[3], u[4] and u[5] are generalized dependent speeds.
P = Point('P')
P.set_vel(N, ua[0]*A.x + ua[1]*A.y + ua[2]*A.z)
O = P.locatenew('O', q[3]*A.z + r*sin(q[1])*A.y)
O.set_vel(N, u[3]*A.x + u[4]*A.y + u[5]*A.z)
O.set_acc(N, O.vel(N).diff(t, A) + cross(A.ang_vel_in(N), O.vel(N)))
# Kinematic differential equations:
# Two equalities: one is w_c_n_qd = C.ang_vel_in(N) in three coordinates
# directions of B, for qd0, qd1 and qd2.
# the other is v_o_n_qd = O.vel(N) in A.z direction for qd3.
# Then, solve for dq/dt's in terms of u's: qd_kd.
w_c_n_qd = qd[0]*A.z + qd[1]*B.x + qd[2]*B.y
v_o_n_qd = O.pos_from(P).diff(t, A) + cross(A.ang_vel_in(N), O.pos_from(P))
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_o_n_qd - O.vel(N), A.z)])
qd_kd = solve(kindiffs, qd)
# Values of generalized speeds during a steady turn for later substitution
# into the Fr_star_steady.
steady_conditions = solve(kindiffs.subs({qd[1] : 0, qd[3] : 0}), u)
steady_conditions.update({qd[1] : 0, qd[3] : 0})
# Partial angular velocities and velocities.
partial_w_C = [C.ang_vel_in(N).diff(ui, N) for ui in u + ua]
partial_v_O = [O.vel(N).diff(ui, N) for ui in u + ua]
partial_v_P = [P.vel(N).diff(ui, N) for ui in u + ua]
# Configuration constraint: f_c, the projection of radius r in A.z direction
# is q[3].
# Velocity constraints: f_v, for u3, u4 and u5.
# Acceleration constraints: f_a.
f_c = Matrix([dot(-r*B.z, A.z) - q[3]])
f_v = Matrix([dot(O.vel(N) - (P.vel(N) + cross(C.ang_vel_in(N),
O.pos_from(P))), ai).expand() for ai in A])
v_o_n = cross(C.ang_vel_in(N), O.pos_from(P))
a_o_n = v_o_n.diff(t, A) + cross(A.ang_vel_in(N), v_o_n)
f_a = Matrix([dot(O.acc(N) - a_o_n, ai) for ai in A])
# Solve for constraint equations in the form of
# u_dependent = A_rs * [u_i; u_aux].
# First, obtain constraint coefficient matrix: M_v * [u; ua] = 0;
# Second, taking u[0], u[1], u[2] as independent,
# taking u[3], u[4], u[5] as dependent,
# rearranging the matrix of M_v to be A_rs for u_dependent.
# Third, u_aux ==0 for u_dep, and resulting dictionary of u_dep_dict.
M_v = zeros(3, 9)
for i in range(3):
for j, ui in enumerate(u + ua):
M_v[i, j] = f_v[i].diff(ui)
M_v_i = M_v[:, :3]
M_v_d = M_v[:, 3:6]
M_v_aux = M_v[:, 6:]
M_v_i_aux = M_v_i.row_join(M_v_aux)
A_rs = - M_v_d.inv() * M_v_i_aux
u_dep = A_rs[:, :3] * Matrix(u[:3])
u_dep_dict = dict(zip(u[3:], u_dep))
# Active forces: F_O acting on point O; F_P acting on point P.
# Generalized active forces (unconstrained): Fr_u = F_point * pv_point.
F_O = m*g*A.z
F_P = Fx * A.x + Fy * A.y + Fz * A.z
Fr_u = Matrix([dot(F_O, pv_o) + dot(F_P, pv_p) for pv_o, pv_p in
zip(partial_v_O, partial_v_P)])
# Inertia force: R_star_O.
# Inertia of disc: I_C_O, where J is a inertia component about principal axis.
# Inertia torque: T_star_C.
# Generalized inertia forces (unconstrained): Fr_star_u.
R_star_O = -m*O.acc(N)
I_C_O = inertia(B, I, J, I)
T_star_C = -(dot(I_C_O, C.ang_acc_in(N)) \
+ cross(C.ang_vel_in(N), dot(I_C_O, C.ang_vel_in(N))))
Fr_star_u = Matrix([dot(R_star_O, pv) + dot(T_star_C, pav) for pv, pav in
zip(partial_v_O, partial_w_C)])
# Form nonholonomic Fr: Fr_c, and nonholonomic Fr_star: Fr_star_c.
# Also, nonholonomic Fr_star in steady turning condition: Fr_star_steady.
Fr_c = Fr_u[:3, :].col_join(Fr_u[6:, :]) + A_rs.T * Fr_u[3:6, :]
Fr_star_c = Fr_star_u[:3, :].col_join(Fr_star_u[6:, :])\
+ A_rs.T * Fr_star_u[3:6, :]
Fr_star_steady = Fr_star_c.subs(ud_zero).subs(u_dep_dict)\
.subs(steady_conditions).subs({q[3]: -r*cos(q[1])}).expand()
# Second, using KaneMethod in mechanics for fr, frstar and frstar_steady.
# Rigid Bodies: disc, with inertia I_C_O.
iner_tuple = (I_C_O, O)
disc = RigidBody('disc', O, C, m, iner_tuple)
bodyList = [disc]
# Generalized forces: Gravity: F_o; Auxiliary forces: F_p.
F_o = (O, F_O)
F_p = (P, F_P)
forceList = [F_o, F_p]
# KanesMethod.
kane = KanesMethod(
N, q_ind= q[:3], u_ind= u[:3], kd_eqs=kindiffs,
q_dependent=q[3:], configuration_constraints = f_c,
u_dependent=u[3:], velocity_constraints= f_v,
u_auxiliary=ua
)
# fr, frstar, frstar_steady and kdd(kinematic differential equations).
(fr, frstar)= kane.kanes_equations(forceList, bodyList)
frstar_steady = frstar.subs(ud_zero).subs(u_dep_dict).subs(steady_conditions)\
.subs({q[3]: -r*cos(q[1])}).expand()
kdd = kane.kindiffdict()
assert Matrix(Fr_c).expand() == fr.expand()
assert Matrix(Fr_star_c.subs(kdd)).expand() == frstar.expand()
assert (simplify(Matrix(Fr_star_steady).expand()) ==
simplify(frstar_steady.expand()))
LagrangesMethod)
from sympy.physics.mechanics.functions import inertia
init_vprinting()
# Variáveis Simbólicas
THETA_1, THETA_2 = dynamicsymbols('theta_1 theta_2')
DTHETA_1, DTHETA_2 = dynamicsymbols('theta_1 theta_2', 1)
TAU_1, TAU_2 = symbols('tau_1 tau_2')
L_1, L_2 = symbols('l_1 l_2', positive=True)
R_1, R_2 = symbols('r_1 r_2', positive=True)
M_1, M_2, G = symbols('m_1 m_2 g')
I_1_ZZ, I_2_ZZ = symbols('I_{1zz}, I_{2zz}')
# Referenciais
# Referencial Inercial
B0 = ReferenceFrame('B0')
# Referencial móvel: theta_1 em relação a B0.z
B1 = ReferenceFrame('B1')
B1.orient(B0, 'Axis', [THETA_1, B0.z])
# Referencial móvel: theta_2 em relação a B1.z
B2 = ReferenceFrame('B2')
B2.orient(B1, 'Axis', [THETA_2, B1.z])
# Pontos e Centros de Massa
O = Point('O')
O.set_vel(B0, 0)
A = Point('A')
A.set_pos(O, L_1 * B1.x)
A.v2pt_theory(O, B0, B1)
CM_1 = Point('CM_1')
CM_1.set_pos(O, R_1 * B1.x)
def test_linearize_rolling_disc_kane():
# Symbols for time and constant parameters
t, r, m, g, v = symbols('t r m g v')
# Configuration variables and their time derivatives
q1, q2, q3, q4, q5, q6 = q = dynamicsymbols('q1:7')
q1d, q2d, q3d, q4d, q5d, q6d = qd = [qi.diff(t) for qi in q]
# Generalized speeds and their time derivatives
u = dynamicsymbols('u:6')
u1, u2, u3, u4, u5, u6 = u = dynamicsymbols('u1:7')
u1d, u2d, u3d, u4d, u5d, u6d = [ui.diff(t) for ui in u]
# Reference frames
N = ReferenceFrame('N') # Inertial frame
NO = Point('NO') # Inertial origin
A = N.orientnew('A', 'Axis', [q1, N.z]) # Yaw intermediate frame
B = A.orientnew('B', 'Axis', [q2, A.x]) # Lean intermediate frame
C = B.orientnew('C', 'Axis', [q3, B.y]) # Disc fixed frame
CO = NO.locatenew('CO', q4 * N.x + q5 * N.y + q6 * N.z) # Disc center
# Disc angular velocity in N expressed using time derivatives of coordinates
w_c_n_qd = C.ang_vel_in(N)
w_b_n_qd = B.ang_vel_in(N)
# Inertial angular velocity and angular acceleration of disc fixed frame
C.set_ang_vel(N, u1 * B.x + u2 * B.y + u3 * B.z)
# Disc center velocity in N expressed using time derivatives of coordinates
v_co_n_qd = CO.pos_from(NO).dt(N)
# Disc center velocity in N expressed using generalized speeds
CO.set_vel(N, u4 * C.x + u5 * C.y + u6 * C.z)
# Disc Ground Contact Point
P = CO.locatenew('P', r * B.z)
P.v2pt_theory(CO, N, C)
# Configuration constraint
f_c = Matrix([q6 - dot(CO.pos_from(P), N.z)])
# Velocity level constraints
f_v = Matrix([dot(P.vel(N), uv) for uv in C])
# Kinematic differential equations
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_co_n_qd - CO.vel(N), uv) for uv in N])
qdots = solve(kindiffs, qd)
# Set angular velocity of remaining frames
B.set_ang_vel(N, w_b_n_qd.subs(qdots))
C.set_ang_acc(
N,
C.ang_vel_in(N).dt(B) + cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Active forces
F_CO = m * g * A.z
# Create inertia dyadic of disc C about point CO
I = (m * r**2) / 4
J = (m * r**2) / 2
I_C_CO = inertia(C, I, J, I)
Disc = RigidBody('Disc', CO, C, m, (I_C_CO, CO))
BL = [Disc]
FL = [(CO, F_CO)]
KM = KanesMethod(N, [q1, q2, q3, q4, q5], [u1, u2, u3],
kd_eqs=kindiffs,
q_dependent=[q6],
configuration_constraints=f_c,
u_dependent=[u4, u5, u6],
velocity_constraints=f_v)
(fr, fr_star) = KM.kanes_equations(FL, BL)
# Test generalized form equations
linearizer = KM.to_linearizer()
assert linearizer.f_c == f_c
assert linearizer.f_v == f_v
assert linearizer.f_a == f_v.diff(t)
sol = solve(linearizer.f_0 + linearizer.f_1, qd)
for qi in qd:
assert sol[qi] == qdots[qi]
assert simplify(linearizer.f_2 + linearizer.f_3 - fr - fr_star) == Matrix(
[0, 0, 0])
# Perform the linearization
# Precomputed operating point
q_op = {q6: -r * cos(q2)}
u_op = {
u1: 0,
u2: sin(q2) * q1d + q3d,
u3: cos(q2) * q1d,
u4: -r * (sin(q2) * q1d + q3d) * cos(q3),
u5: 0,
u6: -r * (sin(q2) * q1d + q3d) * sin(q3)
}
qd_op = {
q2d: 0,
q4d: -r * (sin(q2) * q1d + q3d) * cos(q1),
q5d: -r * (sin(q2) * q1d + q3d) * sin(q1),
q6d: 0
}
ud_op = {
u1d:
4 * g * sin(q2) / (5 * r) + sin(2 * q2) * q1d**2 / 2 +
6 * cos(q2) * q1d * q3d / 5,
u2d:
0,
u3d:
0,
u4d:
r * (sin(q2) * sin(q3) * q1d * q3d + sin(q3) * q3d**2),
u5d:
r * (4 * g * sin(q2) /
(5 * r) + sin(2 * q2) * q1d**2 / 2 + 6 * cos(q2) * q1d * q3d / 5),
u6d:
-r * (sin(q2) * cos(q3) * q1d * q3d + cos(q3) * q3d**2)
}
A, B = linearizer.linearize(op_point=[q_op, u_op, qd_op, ud_op],
A_and_B=True,
simplify=True)
upright_nominal = {q1d: 0, q2: 0, m: 1, r: 1, g: 1}
# Precomputed solution
A_sol = Matrix([[0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[sin(q1) * q3d, 0, 0, 0, 0, -sin(q1), -cos(q1), 0],
[-cos(q1) * q3d, 0, 0, 0, 0,
cos(q1), -sin(q1), 0],
[0, 4 / 5, 0, 0, 0, 0, 0, 6 * q3d / 5],
[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2 * q3d, 0, 0]])
B_sol = Matrix([])
# Check that linearization is correct
assert A.subs(upright_nominal) == A_sol
assert B.subs(upright_nominal) == B_sol
# Check eigenvalues at critical speed are all zero:
assert A.subs(upright_nominal).subs(q3d, 1 / sqrt(3)).eigenvals() == {0: 8}
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)
with warns_deprecated_sympy():
(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 = msubs(MM_full, val_dict)
forcing_lin_s = msubs(forcing_lin, KM.kindiffdict(), 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)
from __future__ import print_function, division
from sympy import symbols, simplify
from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, Point
from sympy.physics.vector import init_vprinting
init_vprinting(use_latex='mathjax', pretty_print=False)
inertial_frame = ReferenceFrame('I')
lower_leg_frame = ReferenceFrame('L')
theta1, theta2, theta3 = dynamicsymbols('theta1, theta2, theta3')
lower_leg_frame.orient(inertial_frame, 'Axis', (theta1, inertial_frame.z))
lower_leg_frame.dcm(inertial_frame)
omega1, omega2, omega3 = dynamicsymbols('omega1, omega2, omega3')
lower_leg_frame.set_ang_vel(inertial_frame, omega1 * inertial_frame.z)
print(lower_leg_frame.ang_vel_in(inertial_frame))
def __init__(self):
self.coords = dynamicsymbols('q:3')
self.speeds = dynamicsymbols('p:3')
self.dynamic = list(self.coords) + list(self.speeds)
self.states = ([radians(45) for x in self.coords] +
[radians(30) for x in self.speeds])
self.inertial_ref_frame = ReferenceFrame('I')
self.A = self.inertial_ref_frame.orientnew('A', 'space', self.coords,
'XYZ')
self.B = self.A.orientnew('B', 'space', self.speeds, 'XYZ')
self.camera_ref_frame = \
self.inertial_ref_frame.orientnew('C', 'space', (pi / 2, 0, 0),
'XYZ')
self.origin = Point('O')
self.P1 = self.origin.locatenew(
'P1', 10 * self.inertial_ref_frame.x +
10 * self.inertial_ref_frame.y + 10 * self.inertial_ref_frame.z)
self.P2 = self.P1.locatenew(
'P2', 10 * self.inertial_ref_frame.x +
10 * self.inertial_ref_frame.y + 10 * self.inertial_ref_frame.z)
self.point_list1 = [[2, 3, 1], [4, 6, 2], [5, 3, 1], [5, 3, 6]]
self.point_list2 = [[3, 1, 4], [3, 8, 2], [2, 1, 6], [2, 1, 1]]
self.shape1 = Cylinder(1.0, 1.0)
self.shape2 = Cylinder(1.0, 1.0)
self.Ixx, self.Iyy, self.Izz = symbols('Ixx, Iyy, Izz')
self.mass = symbols('m')
self.parameters = [self.Ixx, self.Iyy, self.Izz, self.mass]
self.param_vals = [2.0, 3.0, 4.0, 5.0]
self.inertia = inertia(self.A, self.Ixx, self.Iyy, self.Izz)
self.rigid_body = RigidBody('rigid_body1', self.P1, self.A, self.mass,
(self.inertia, self.P1))
self.global_frame1 = VisualizationFrame('global_frame1', self.A,
self.P1, self.shape1)
self.global_frame2 = VisualizationFrame('global_frame2', self.B,
self.P2, self.shape2)
self.scene1 = Scene(self.inertial_ref_frame,
self.origin,
(self.global_frame1, self.global_frame2),
name='scene')
self.particle = Particle('particle1', self.P1, self.mass)
q = self.coords
c = cos
s = sin
# Here is the dragon ..
self.transformation_matrix = [
[c(q[1]) * c(q[2]),
s(q[2]) * c(q[1]), -s(q[1]), 0],
[
s(q[0]) * s(q[1]) * c(q[2]) - s(q[2]) * c(q[0]),
s(q[0]) * s(q[1]) * s(q[2]) + c(q[0]) * c(q[2]),
s(q[0]) * c(q[1]), 0
],
[
s(q[0]) * s(q[2]) + s(q[1]) * c(q[0]) * c(q[2]),
-s(q[0]) * c(q[2]) + s(q[1]) * s(q[2]) * c(q[0]),
c(q[0]) * c(q[1]), 0
], [10, 10, 10, 1]
]
from sympy import cos, Matrix, sin, symbols, pi
from sympy.abc import x, y, z
from sympy.physics.mechanics import Vector, ReferenceFrame, dot, dynamicsymbols
Vector.simp = True
A = ReferenceFrame('A')
def test_dyadic():
d1 = A.x | A.x
d2 = A.y | A.y
d3 = A.x | A.y
assert d1 * 0 == 0
assert d1 != 0
assert d1 * 2 == 2 * A.x | A.x
assert d1 / 2. == 0.5 * d1
assert d1 & (0 * d1) == 0
assert d1 & d2 == 0
assert d1 & A.x == A.x
assert d1 ^ A.x == 0
assert d1 ^ A.y == A.x | A.z
assert d1 ^ A.z == - A.x | A.y
assert d2 ^ A.x == - A.y | A.z
assert A.x ^ d1 == 0
assert A.y ^ d1 == - A.z | A.x
assert A.z ^ d1 == A.y | A.x
assert A.x & d1 == A.x
assert A.y & d1 == 0
assert A.y & d2 == A.y
assert d1 & d3 == A.x | A.y
assert d3 & d1 == 0
"""
from sympy import symbols
from sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame, Point,
Particle, KanesMethod)
# from sympy.printing.pycode import NumPyPrinter, pycode
n = 1
q = dynamicsymbols('q:' + str(n + 1)) # Generalized coordinates
u = dynamicsymbols('u:' + str(n + 1)) # Generalized speeds
f = dynamicsymbols('f') # Force applied to the cart
m = symbols('m:' + str(n + 1)) # Mass of each bob
length = symbols('l:' + str(n)) # Length of each link
g, t = symbols('g t') # Gravity and time
ref_frame = ReferenceFrame('I') # Inertial reference frame
origin = Point('O') # Origin point
origin.set_vel(ref_frame, 0) # Origin's velocity is zero
P0 = Point('P0') # Hinge point of top link
P0.set_pos(origin, q[0] * ref_frame.x) # Set the position of P0
P0.set_vel(ref_frame, u[0] * ref_frame.x) # Set the velocity of P0
Pa0 = Particle('Pa0', P0, m[0]) # Define a particle at P0
# List to hold the n + 1 frames
frames = [ref_frame]
points = [P0] # List to hold the n + 1 points
particles = [Pa0] # List to hold the n + 1 particles
# List to hold the n + 1 applied forces, including the input force, f
forces = [
def test_ang_vel():
q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q1, N.z])
B = A.orientnew('B', 'Axis', [q2, A.x])
C = B.orientnew('C', 'Axis', [q3, B.y])
D = N.orientnew('D', 'Axis', [q4, N.y])
u1, u2, u3 = dynamicsymbols('u1 u2 u3')
assert A.ang_vel_in(N) == (q1d)*A.z
assert B.ang_vel_in(N) == (q2d)*B.x + (q1d)*A.z
assert C.ang_vel_in(N) == (q3d)*C.y + (q2d)*B.x + (q1d)*A.z
A2 = N.orientnew('A2', 'Axis', [q4, N.y])
assert N.ang_vel_in(N) == 0
assert N.ang_vel_in(A) == -q1d*N.z
assert N.ang_vel_in(B) == -q1d*A.z - q2d*B.x
assert N.ang_vel_in(C) == -q1d*A.z - q2d*B.x - q3d*B.y
assert N.ang_vel_in(A2) == -q4d*N.y
assert A.ang_vel_in(N) == q1d*N.z
assert A.ang_vel_in(A) == 0
assert A.ang_vel_in(B) == - q2d*B.x
assert A.ang_vel_in(C) == - q2d*B.x - q3d*B.y
assert A.ang_vel_in(A2) == q1d*N.z - q4d*N.y
assert B.ang_vel_in(N) == q1d*A.z + q2d*A.x
assert B.ang_vel_in(A) == q2d*A.x
assert B.ang_vel_in(B) == 0
assert B.ang_vel_in(C) == -q3d*B.y
assert B.ang_vel_in(A2) == q1d*A.z + q2d*A.x - q4d*N.y
assert C.ang_vel_in(N) == q1d*A.z + q2d*A.x + q3d*B.y
assert C.ang_vel_in(A) == q2d*A.x + q3d*C.y
assert C.ang_vel_in(B) == q3d*B.y
assert C.ang_vel_in(C) == 0
assert C.ang_vel_in(A2) == q1d*A.z + q2d*A.x + q3d*B.y - q4d*N.y
assert A2.ang_vel_in(N) == q4d*A2.y
assert A2.ang_vel_in(A) == q4d*A2.y - q1d*N.z
assert A2.ang_vel_in(B) == q4d*N.y - q1d*A.z - q2d*A.x
assert A2.ang_vel_in(C) == q4d*N.y - q1d*A.z - q2d*A.x - q3d*B.y
assert A2.ang_vel_in(A2) == 0
C.set_ang_vel(N, u1*C.x + u2*C.y + u3*C.z)
assert C.ang_vel_in(N) == (u1)*C.x + (u2)*C.y + (u3)*C.z
assert N.ang_vel_in(C) == (-u1)*C.x + (-u2)*C.y + (-u3)*C.z
assert C.ang_vel_in(D) == (u1)*C.x + (u2)*C.y + (u3)*C.z + (-q4d)*D.y
assert D.ang_vel_in(C) == (-u1)*C.x + (-u2)*C.y + (-u3)*C.z + (q4d)*D.y
q0 = dynamicsymbols('q0')
q0d = dynamicsymbols('q0', 1)
E = N.orientnew('E', 'Quaternion', (q0, q1, q2, q3))
assert E.ang_vel_in(N) == (
2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1) * E.x +
2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2) * E.y +
2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3) * E.z)
F = N.orientnew('F', 'Body', (q1, q2, q3), '313')
assert F.ang_vel_in(N) == ((sin(q2)*sin(q3)*q1d + cos(q3)*q2d)*F.x +
(sin(q2)*cos(q3)*q1d - sin(q3)*q2d)*F.y + (cos(q2)*q1d + q3d)*F.z)
G = N.orientnew('G', 'Axis', (q1, N.x + N.y))
assert G.ang_vel_in(N) == q1d * (N.x + N.y).normalize()
assert N.ang_vel_in(G) == -q1d * (N.x + N.y).normalize()
def test_sub_qdot2():
# This test solves exercises 8.3 from Kane 1985 and defines
# all velocities in terms of q, qdot. We check that the generalized active
# forces are correctly computed if u terms are only defined in the
# kinematic differential equations.
#
# This functionality was added in PR 8948. Without qdot/u substitution, the
# KanesMethod constructor will fail during the constraint initialization as
# the B matrix will be poorly formed and inversion of the dependent part
# will fail.
g, m, Px, Py, Pz, R, t = symbols('g m Px Py Pz R t')
q = dynamicsymbols('q:5')
qd = dynamicsymbols('q:5', level=1)
u = dynamicsymbols('u:5')
## Define inertial, intermediate, and rigid body reference frames
A = ReferenceFrame('A')
B_prime = A.orientnew('B_prime', 'Axis', [q[0], A.z])
B = B_prime.orientnew('B', 'Axis', [pi / 2 - q[1], B_prime.x])
C = B.orientnew('C', 'Axis', [q[2], B.z])
## Define points of interest and their velocities
pO = Point('O')
pO.set_vel(A, 0)
# R is the point in plane H that comes into contact with disk C.
pR = pO.locatenew('R', q[3] * A.x + q[4] * A.y)
pR.set_vel(A, pR.pos_from(pO).diff(t, A))
pR.set_vel(B, 0)
# C^ is the point in disk C that comes into contact with plane H.
pC_hat = pR.locatenew('C^', 0)
pC_hat.set_vel(C, 0)
# C* is the point at the center of disk C.
pCs = pC_hat.locatenew('C*', R * B.y)
pCs.set_vel(C, 0)
pCs.set_vel(B, 0)
# calculate velocites of points C* and C^ in frame A
pCs.v2pt_theory(pR, A, B) # points C* and R are fixed in frame B
pC_hat.v2pt_theory(pCs, A, C) # points C* and C^ are fixed in frame C
## Define forces on each point of the system
R_C_hat = Px * A.x + Py * A.y + Pz * A.z
R_Cs = -m * g * A.z
forces = [(pC_hat, R_C_hat), (pCs, R_Cs)]
## Define kinematic differential equations
# let ui = omega_C_A & bi (i = 1, 2, 3)
# u4 = qd4, u5 = qd5
u_expr = [C.ang_vel_in(A) & uv for uv in B]
u_expr += qd[3:]
kde = [ui - e for ui, e in zip(u, u_expr)]
km1 = KanesMethod(A, q, u, kde)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
fr1, _ = km1.kanes_equations(forces, [])
## Calculate generalized active forces if we impose the condition that the
# disk C is rolling without slipping
u_indep = u[:3]
u_dep = list(set(u) - set(u_indep))
vc = [pC_hat.vel(A) & uv for uv in [A.x, A.y]]
km2 = KanesMethod(A,
q,
u_indep,
kde,
u_dependent=u_dep,
velocity_constraints=vc)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
fr2, _ = km2.kanes_equations(forces, [])
fr1_expected = Matrix([
-R * g * m * sin(q[1]),
-R * (Px * cos(q[0]) + Py * sin(q[0])) * tan(q[1]),
R * (Px * cos(q[0]) + Py * sin(q[0])), Px, Py
])
fr2_expected = Matrix([-R * g * m * sin(q[1]), 0, 0])
assert (trigsimp(fr1.expand()) == trigsimp(fr1_expected.expand()))
assert (trigsimp(fr2.expand()) == trigsimp(fr2_expected.expand()))
from __future__ import division
from sympy import diff, solve, simplify, symbols
from sympy.physics.mechanics import ReferenceFrame, Point
from sympy.physics.mechanics import dynamicsymbols
from util import partial_velocities, generalized_active_forces
## --- Declare symbols ---
q1, q2 = dynamicsymbols('q1 q2')
q1d, q2d = dynamicsymbols('q1 q2', level=1)
u1, u2 = dynamicsymbols('u1 u2')
b, g, m, L, t = symbols('b g m L t')
E, I = symbols('E I')
# --- ReferenceFrames ---
N = ReferenceFrame('N')
B = N.orientnew('B', 'Axis',
[-(q2 - q1) / (2 * b), N.y]) # small angle approx.
# --- Define Points and set their velocities ---
pO = Point('O') # Point O is where B* would be with zero displacement.
pO.set_vel(N, 0)
# small angle approx.
pB_star = pO.locatenew('B*', -(q1 + q2) / 2 * N.x)
pP1 = pO.locatenew('P1', -q1 * N.x - b * N.z)
pP2 = pO.locatenew('P2', -q2 * N.x + b * N.z)
for p in [pB_star, pP1, pP2]:
p.set_vel(N, p.pos_from(pO).diff(t, N))
## --- Define kinematic differential equations/pseudo-generalized speeds ---
from pydy.viz.visualization_frame import VisualizationFrame
from pydy.viz.scene import Scene
from utils import controllable
from scipy.linalg import solve_continuous_are
import control
import sympy
from sympy import symbols, simplify, trigsimp
from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, Point, inertia, RigidBody, KanesMethod
from sympy.physics.vector import init_vprinting, vlatex
#from matplotlib.pyplot import plot, legend, xlabel, ylabel, rcParams
init_vprinting(use_latex='mathjax', pretty_print=True)
#Kinematics ---------------------------------------------------------------------------
#Init reference frames, assume foot is fixed to floor
inertial_frame = ReferenceFrame('I')
lower_leg_frame = ReferenceFrame('L')
upper_leg_frame = ReferenceFrame('U')
torso_frame = ReferenceFrame('T')
#declare dynamic symbols for three joints- dynamic symbols allow derivative notation
theta1, theta2, theta3 = dynamicsymbols('theta1, theta2, theta3')
#Reference frames - rotation only
#orient lower leg frame with respect to inertial (base) frame
lower_leg_frame.orient(inertial_frame, 'Axis', (theta1, inertial_frame.z))
#command below displays direction cosine matrix (DCM)
#pretty_print(lower_leg_frame.dcm(inertial_frame))
#orient upper leg frame with respect to lower leg frame
upper_leg_frame.orient(lower_leg_frame, 'Axis',(theta2,lower_leg_frame.z))
