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))