from sympy.core.function import expand from sympy.core.symbol import symbols from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.matrices.dense import Matrix from sympy.simplify.trigsimp import trigsimp from sympy.physics.mechanics import ( PinJoint, JointsMethod, RigidBody, Particle, Body, KanesMethod, PrismaticJoint, LagrangesMethod, inertia) from sympy.physics.vector import dynamicsymbols, ReferenceFrame from sympy.testing.pytest import raises, warns_deprecated_sympy from sympy import zeros from sympy.utilities.lambdify import lambdify from sympy.solvers.solvers import solve t = dynamicsymbols._t # type: ignore def test_jointsmethod(): with warns_deprecated_sympy(): P = Body('P') C = Body('C') Pin = PinJoint('P1', P, C) C_ixx, g = symbols('C_ixx g') q, u = dynamicsymbols('q_P1, u_P1') P.apply_force(g*P.y) with warns_deprecated_sympy(): method = JointsMethod(P, Pin) assert method.frame == P.frame assert method.bodies == [C, P] assert method.loads == [(P.masscenter, g*P.frame.y)] assert method.q == Matrix([q]) assert method.u == Matrix([u]) assert method.kdes == Matrix([u - q.diff()]) soln = method.form_eoms() assert soln == Matrix([[-C_ixx*u.diff()]]) assert method.forcing_full == Matrix([[u], [0]]) assert method.mass_matrix_full == Matrix([[1, 0], [0, C_ixx]]) assert isinstance(method.method, KanesMethod) def test_rigid_body_particle_compatibility(): l, m, g = symbols('l m g') C = RigidBody('C') b = Particle('b', mass=m) b_frame = ReferenceFrame('b_frame') q, u = dynamicsymbols('q u') P = PinJoint('P', C, b, coordinates=q, speeds=u, child_interframe=b_frame, child_point=-l * b_frame.x, joint_axis=C.z) with warns_deprecated_sympy(): method = JointsMethod(C, P) method.loads.append((b.masscenter, m * g * C.x)) method.form_eoms() rhs = method.rhs() assert rhs[1] == -g*sin(q)/l def test_jointmethod_duplicate_coordinates_speeds(): with warns_deprecated_sympy(): P = Body('P') C = Body('C') T = Body('T') q, u = dynamicsymbols('q u') P1 = PinJoint('P1', P, C, q) P2 = PrismaticJoint('P2', C, T, q) with warns_deprecated_sympy(): raises(ValueError, lambda: JointsMethod(P, P1, P2)) P1 = PinJoint('P1', P, C, speeds=u) P2 = PrismaticJoint('P2', C, T, speeds=u) with warns_deprecated_sympy(): raises(ValueError, lambda: JointsMethod(P, P1, P2)) P1 = PinJoint('P1', P, C, q, u) P2 = PrismaticJoint('P2', C, T, q, u) with warns_deprecated_sympy(): raises(ValueError, lambda: JointsMethod(P, P1, P2)) def test_complete_simple_double_pendulum(): q1, q2 = dynamicsymbols('q1 q2') u1, u2 = dynamicsymbols('u1 u2') m, l, g = symbols('m l g') with warns_deprecated_sympy(): C = Body('C') # ceiling PartP = Body('P', mass=m) PartR = Body('R', mass=m) J1 = PinJoint('J1', C, PartP, speeds=u1, coordinates=q1, child_point=-l*PartP.x, joint_axis=C.z) J2 = PinJoint('J2', PartP, PartR, speeds=u2, coordinates=q2, child_point=-l*PartR.x, joint_axis=PartP.z) PartP.apply_force(m*g*C.x) PartR.apply_force(m*g*C.x) with warns_deprecated_sympy(): method = JointsMethod(C, J1, J2) method.form_eoms() assert expand(method.mass_matrix_full) == Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2*l**2*m*cos(q2) + 3*l**2*m, l**2*m*cos(q2) + l**2*m], [0, 0, l**2*m*cos(q2) + l**2*m, l**2*m]]) assert trigsimp(method.forcing_full) == trigsimp(Matrix([[u1], [u2], [-g*l*m*(sin(q1 + q2) + sin(q1)) - g*l*m*sin(q1) + l**2*m*(2*u1 + u2)*u2*sin(q2)], [-g*l*m*sin(q1 + q2) - l**2*m*u1**2*sin(q2)]])) def test_two_dof_joints(): q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2') m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2') with warns_deprecated_sympy(): W = Body('W') B1 = Body('B1', mass=m) B2 = Body('B2', mass=m) J1 = PrismaticJoint('J1', W, B1, coordinates=q1, speeds=u1) J2 = PrismaticJoint('J2', B1, B2, coordinates=q2, speeds=u2) W.apply_force(k1*q1*W.x, reaction_body=B1) W.apply_force(c1*u1*W.x, reaction_body=B1) B1.apply_force(k2*q2*W.x, reaction_body=B2) B1.apply_force(c2*u2*W.x, reaction_body=B2) with warns_deprecated_sympy(): method = JointsMethod(W, J1, J2) method.form_eoms() MM = method.mass_matrix forcing = method.forcing rhs = MM.LUsolve(forcing) assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m) assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 * c2 * u2) / m) def test_simple_pedulum(): l, m, g = symbols('l m g') with warns_deprecated_sympy(): C = Body('C') b = Body('b', mass=m) q = dynamicsymbols('q') P = PinJoint('P', C, b, speeds=q.diff(t), coordinates=q, child_point=-l * b.x, joint_axis=C.z) b.potential_energy = - m * g * l * cos(q) with warns_deprecated_sympy(): method = JointsMethod(C, P) method.form_eoms(LagrangesMethod) rhs = method.rhs() assert rhs[1] == -g*sin(q)/l def test_chaos_pendulum(): #https://www.pydy.org/examples/chaos_pendulum.html mA, mB, lA, lB, IAxx, IBxx, IByy, IBzz, g = symbols('mA, mB, lA, lB, IAxx, IBxx, IByy, IBzz, g') theta, phi, omega, alpha = dynamicsymbols('theta phi omega alpha') A = ReferenceFrame('A') B = ReferenceFrame('B') with warns_deprecated_sympy(): rod = Body('rod', mass=mA, frame=A, central_inertia=inertia(A, IAxx, IAxx, 0)) plate = Body('plate', mass=mB, frame=B, central_inertia=inertia(B, IBxx, IByy, IBzz)) C = Body('C') J1 = PinJoint('J1', C, rod, coordinates=theta, speeds=omega, child_point=-lA * rod.z, joint_axis=C.y) J2 = PinJoint('J2', rod, plate, coordinates=phi, speeds=alpha, parent_point=(lB - lA) * rod.z, joint_axis=rod.z) rod.apply_force(mA*g*C.z) plate.apply_force(mB*g*C.z) with warns_deprecated_sympy(): method = JointsMethod(C, J1, J2) method.form_eoms() MM = method.mass_matrix forcing = method.forcing rhs = MM.LUsolve(forcing) xd = (-2 * IBxx * alpha * omega * sin(phi) * cos(phi) + 2 * IByy * alpha * omega * sin(phi) * cos(phi) - g * lA * mA * sin(theta) - g * lB * mB * sin(theta)) / (IAxx + IBxx * sin(phi)**2 + IByy * cos(phi)**2 + lA**2 * mA + lB**2 * mB) assert (rhs[0] - xd).simplify() == 0 xd = (IBxx - IByy) * omega**2 * sin(phi) * cos(phi) / IBzz assert (rhs[1] - xd).simplify() == 0 def test_four_bar_linkage_with_manual_constraints(): q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1:4, u1:4') l1, l2, l3, l4, rho = symbols('l1:5, rho') N = ReferenceFrame('N') inertias = [inertia(N, 0, 0, rho * l ** 3 / 12) for l in (l1, l2, l3, l4)] with warns_deprecated_sympy(): link1 = Body('Link1', frame=N, mass=rho * l1, central_inertia=inertias[0]) link2 = Body('Link2', mass=rho * l2, central_inertia=inertias[1]) link3 = Body('Link3', mass=rho * l3, central_inertia=inertias[2]) link4 = Body('Link4', mass=rho * l4, central_inertia=inertias[3]) joint1 = PinJoint( 'J1', link1, link2, coordinates=q1, speeds=u1, joint_axis=link1.z, parent_point=l1 / 2 * link1.x, child_point=-l2 / 2 * link2.x) joint2 = PinJoint( 'J2', link2, link3, coordinates=q2, speeds=u2, joint_axis=link2.z, parent_point=l2 / 2 * link2.x, child_point=-l3 / 2 * link3.x) joint3 = PinJoint( 'J3', link3, link4, coordinates=q3, speeds=u3, joint_axis=link3.z, parent_point=l3 / 2 * link3.x, child_point=-l4 / 2 * link4.x) loop = link4.masscenter.pos_from(link1.masscenter) \ + l1 / 2 * link1.x + l4 / 2 * link4.x fh = Matrix([loop.dot(link1.x), loop.dot(link1.y)]) with warns_deprecated_sympy(): method = JointsMethod(link1, joint1, joint2, joint3) t = dynamicsymbols._t qdots = solve(method.kdes, [q1.diff(t), q2.diff(t), q3.diff(t)]) fhd = fh.diff(t).subs(qdots) kane = KanesMethod(method.frame, q_ind=[q1], u_ind=[u1], q_dependent=[q2, q3], u_dependent=[u2, u3], kd_eqs=method.kdes, configuration_constraints=fh, velocity_constraints=fhd, forcelist=method.loads, bodies=method.bodies) fr, frs = kane.kanes_equations() assert fr == zeros(1) # Numerically check the mass- and forcing-matrix p = Matrix([l1, l2, l3, l4, rho]) q = Matrix([q1, q2, q3]) u = Matrix([u1, u2, u3]) eval_m = lambdify((q, p), kane.mass_matrix) eval_f = lambdify((q, u, p), kane.forcing) eval_fhd = lambdify((q, u, p), fhd) p_vals = [0.13, 0.24, 0.21, 0.34, 997] q_vals = [2.1, 0.6655470375077588, 2.527408138024188] # Satisfies fh u_vals = [0.2, -0.17963733938852067, 0.1309060540601612] # Satisfies fhd mass_check = Matrix([[3.452709815256506e+01, 7.003948798374735e+00, -4.939690970641498e+00], [-2.203792703880936e-14, 2.071702479957077e-01, 2.842917573033711e-01], [-1.300000000000123e-01, -8.836934896046506e-03, 1.864891330060847e-01]]) forcing_check = Matrix([[-0.031211821321648], [-0.00066022608181], [0.001813559741243]]) eps = 1e-10 assert all(abs(x) < eps for x in eval_fhd(q_vals, u_vals, p_vals)) assert all(abs(x) < eps for x in (Matrix(eval_m(q_vals, p_vals)) - mass_check)) assert all(abs(x) < eps for x in (Matrix(eval_f(q_vals, u_vals, p_vals)) - forcing_check))