def inertia_torque(rb, frame, kde_map=None, constraint_map=None, uaux=None): # get central inertia # I_S/O = I_S/S* + I_S*/O I = rb.inertia[0] - inertia_of_point_mass( rb.mass, rb.masscenter.pos_from(rb.inertia[1]), rb.frame) alpha = rb.frame.ang_acc_in(frame) omega = rb.frame.ang_vel_in(frame) if not isinstance(alpha, Vector) and alpha == 0: alpha = Vector([]) if not isinstance(omega, Vector) and omega == 0: omega = Vector([]) if uaux is not None: # auxilliary speeds do not change alpha, omega # use doit() to evaluate terms such as # Derivative(0, t) to 0. uaux_zero = dict(zip(uaux, [0] * len(uaux))) alpha = subs(alpha, uaux_zero) omega = subs(omega, uaux_zero) if kde_map is not None: alpha = subs(alpha, kde_map) omega = subs(omega, kde_map) if constraint_map is not None: alpha = subs(alpha, constraint_map) omega = subs(omega, constraint_map) return -dot(alpha, I) - dot(cross(omega, I), omega)
def Matrix_coord(list_vectores): M = [] for vector in list_vectores: coordx_v, coordy_v, coordz_v = dot(N.x, vector), dot(N.y, vector), dot( N.z, vector) M.append([coordx_v, coordy_v, coordz_v]) return Matrix(M)
def inertia_torque(rb, frame, kde_map=None, constraint_map=None, uaux=None): # get central inertia # I_S/O = I_S/S* + I_S*/O I = rb.inertia[0] - inertia_of_point_mass(rb.mass, rb.masscenter.pos_from(rb.inertia[1]), rb.frame) alpha = rb.frame.ang_acc_in(frame) omega = rb.frame.ang_vel_in(frame) if not isinstance(alpha, Vector) and alpha == 0: alpha = Vector([]) if not isinstance(omega, Vector) and omega == 0: omega = Vector([]) if uaux is not None: # auxilliary speeds do not change alpha, omega # use doit() to evaluate terms such as # Derivative(0, t) to 0. uaux_zero = dict(zip(uaux, [0] * len(uaux))) alpha = subs(alpha, uaux_zero) omega = subs(omega, uaux_zero) if kde_map is not None: alpha = subs(alpha, kde_map) omega = subs(omega, kde_map) if constraint_map is not None: alpha = subs(alpha, constraint_map) omega = subs(omega, constraint_map) return -dot(alpha, I) - dot(cross(omega, I), omega)
def main(): bus.orient(sun, "Axis", [30 / 180 * sp.pi, sun.z]) for i in range(0, n): # for each panel, create a new reference frame. panel_ors[i] = Panel(dimensions=[1, 1], connections=[0, "bus", 0, 0], frame=bus.orientnew("panel" + str(i), "Axis", [10 / 180 * sp.pi, bus.z])) upd = 3.6**-1 # hertz t = 1 / upd omega_0 = 0 for panel in panel_ors: panel = panel_ors[panel] c = 0 theta = sp.acos(mech.dot(panel.frame.x, bus.x).evalf()) * panel.frame.z y_axis_theta = [] y_axis_force = [] y_axis_moment_srp = [] y_axis_moment_spr = [] bus_location = 1.5 * sun.x # location in AU referenced on the sun frame. while True: # get_angle(panel.frame, bus) > -0.18: # When the x-axis of of the bus projects on the y-axis of the bus, the panel latches into place. print("current angle is " + str(mech.dot(theta, bus.z).evalf()) + "\n") f = panel.area * calc_srp(bus_location, bus, state=0) angled_force = calc_torque(panel, f) spring = get_spring_moment(panel.frame, bus) alpha = (panel.torque + spring) / panel.MOI d_theta = omega_0 * t + alpha * t * t / 2 theta += d_theta c += 1 omega_0 = omega_0 + alpha * t panel.frame.orient(bus, "Axis", [mech.dot(theta, panel.frame.z), bus.z]) print(str(theta) + ", " + str(omega_0) + ", " + str(c)) y_axis_theta.append(theta.magnitude().evalf()) y_axis_force.append(angled_force) y_axis_moment_spr.append(spring.magnitude().evalf()) y_axis_moment_srp.append(panel.torque.magnitude().evalf()) print("Solution found after " + str(c) + " iterations") #x_axis = list(frange(0, t*c, t)) x_axis = np.linspace(0.0, c * t, c) fig, ax = plt.subplots() # Create a figure and an axes. ax.plot(x_axis, y_axis_theta, label='theta') # Plot some data on the axes. ax.plot(x_axis, y_axis_force, label='force') # Plot more data on the axes... ax.plot(x_axis, y_axis_moment_spr, label='moment_spr') ax.plot(x_axis, y_axis_moment_srp, label='moment_srp') #ax.plot(x_axis, x**3, label='cubic') # ... and some more. ax.set_xlabel('x label') # Add an x-label to the axes. ax.set_ylabel('y label') # Add a y-label to the axes. ax.set_title("Simple Plot") # Add a title to the axes. ax.legend() # Add a legend. plt.show()
def generalized_inertia_forces(partials, bodies, kde_map=None, constraint_map=None, uaux=None): # use the same frame used in calculating partial velocities ulist = partials.ulist frame = partials.frame if uaux is not None: uaux_zero = dict(zip(uaux, [0] * len(uaux))) Fr_star = [0] * len(ulist) for b in bodies: if isinstance(b, RigidBody): p = b.masscenter m = b.mass elif isinstance(b, Particle): p = b.point m = b.mass else: raise TypeError('{0} is not a RigidBody or Particle'.format(b)) # get acceleration of point a = p.acc(frame) if uaux is not None: # auxilliary speeds do not change a a = subs(a, uaux_zero) if kde_map is not None: a = subs(a, kde_map) if constraint_map is not None: a = subs(a, constraint_map) # get T* for RigidBodys if isinstance(b, RigidBody): T_star = _calculate_T_star(b, frame, kde_map, constraint_map, uaux) for i, u in enumerate(ulist): force_term = 0 torque_term = 0 # inertia force term force_term = dot(partials[p][u], -m * a) # add inertia torque term for RigidBodys if isinstance(b, RigidBody): torque_term = dot(partials[b.frame][u], T_star) # auxilliary speeds have no effect on original inertia forces if uaux is not None and u not in uaux: force_term = subs(force_term, uaux_zero) torque_term = subs(torque_term, uaux_zero) Fr_star[i] += force_term + torque_term return Fr_star, ulist
def get_angle(frame1, frame2): angle = sp.asin(mech.dot(frame1.y.express(frame2), frame2.x).evalf()) sector = mech.dot(frame1.x, frame2.x).evalf() sign = sector / abs(sector) if sign == -1.0: angle = sp.pi - angle angle = -angle.evalf() if threshold(angle, -sp.pi): angle = sp.pi return angle # positive when the y axis turns anti-CW from frame2's y-axis.
def generalized_inertia_forces(partials, bodies, kde_map=None, constraint_map=None, uaux=None): # use the same frame used in calculating partial velocities ulist = partials.ulist frame = partials.frame if uaux is not None: uaux_zero = dict(zip(uaux, [0] * len(uaux))) Fr_star = [0] * len(ulist) for b in bodies: if isinstance(b, RigidBody): p = b.masscenter m = b.mass elif isinstance(b, Particle): p = b.point m = b.mass else: raise TypeError('{0} is not a RigidBody or Particle'.format(b)) # get acceleration of point a = p.acc(frame) if uaux is not None: # auxilliary speeds do not change a a = subs(a, uaux_zero) if kde_map is not None: a = subs(a, kde_map) if constraint_map is not None: a = subs(a, constraint_map) # get T* for RigidBodys if isinstance(b, RigidBody): T_star = _calculate_T_star(b, frame, kde_map, constraint_map, uaux) for i, u in enumerate(ulist): force_term = 0 torque_term = 0 # inertia force term force_term = dot(partials[p][u], -m*a) # add inertia torque term for RigidBodys if isinstance(b, RigidBody): torque_term = dot(partials[b.frame][u], T_star) # auxilliary speeds have no effect on original inertia forces if uaux is not None and u not in uaux: force_term = subs(force_term, uaux_zero) torque_term = subs(torque_term, uaux_zero) Fr_star[i] += force_term + torque_term return Fr_star, ulist
def test_dot(): assert dot(A.x, A.x) == 1 assert dot(A.x, A.y) == 0 assert dot(A.x, A.z) == 0 assert dot(A.y, A.x) == 0 assert dot(A.y, A.y) == 1 assert dot(A.y, A.z) == 0 assert dot(A.z, A.x) == 0 assert dot(A.z, A.y) == 0 assert dot(A.z, A.z) == 1
def angle_between_vectors(a, b): """Return the minimum angle between two vectors. The angle returned for vectors a and -a is 0. """ angle = (acos(dot(a, b) / (a.magnitude() * b.magnitude())) * 180 / pi).evalf() return min(angle, 180 - angle)
def angle_between_vectors(a, b): """Return the minimum angle between two vectors. The angle returned for vectors a and -a is 0. """ angle = (acos(dot(a, b)/(a.magnitude() * b.magnitude())) * 180 / pi).evalf() return min(angle, 180 - angle)
def calc_srp(bus_location, bus): alpha = sp.acos(mech.dot(bus_location.normalize(), bus.x)).evalf() pressure_ref = 4.56E-6 # n/m^2 at 1 AU Rs = bus_location.magnitude() pressure = pressure_ref * (1 / Rs)**2 * sp.cos(alpha) pressure = pressure * bus.y return pressure
def test_aux(): # Same as above, except we have 2 auxiliary speeds for the ground contact # point, which is known to be zero. In one case, we go through then # substitute the aux. speeds in at the end (they are zero, as well as their # derivative), in the other case, we use the built-in auxiliary speed part # of KanesMethod. The equations from each should be the same. 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) u4, u5, f1, f2 = dynamicsymbols('u4, u5, f1, f2') u4d, u5d = dynamicsymbols('u4, u5', 1) r, m, g = symbols('r m g') 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) C = Point('C') C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x)) Dmc = C.locatenew('Dmc', r * L.z) Dmc.v2pt_theory(C, N, R) Dmc.a2pt_theory(C, N, R) I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2) kd = [dot(R.ang_vel_in(N) - w_R_N_qd, uv) for uv in L] ForceList = [(Dmc, -m * g * Y.z), (C, f1 * L.x + f2 * (Y.z ^ L.x))] BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc)) BodyList = [BodyD] KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3, u4, u5], kd_eqs=kd) with warnings.catch_warnings(): warnings.filterwarnings("ignore", category=SymPyDeprecationWarning) (fr, frstar) = KM.kanes_equations(ForceList, BodyList) fr = fr.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar = frstar.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) KM2 = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd, u_auxiliary=[u4, u5]) with warnings.catch_warnings(): warnings.filterwarnings("ignore", category=SymPyDeprecationWarning) (fr2, frstar2) = KM2.kanes_equations(ForceList, BodyList) fr2 = fr2.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar2 = frstar2.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar.simplify() frstar2.simplify() assert (fr - fr2).expand() == Matrix([0, 0, 0, 0, 0]) assert (frstar - frstar2).expand() == Matrix([0, 0, 0, 0, 0])
def main(): for i in range(0, n): # for each panel, create a new reference frame. panel_ors[i] = Panel( dimensions=[1,1], connections=[0,"bus",0, 0], frame = bus.orientnew(str(i), "Axis", [80/180 * sp.pi, bus.z]) ) upd = 100 # hertz t= 1/upd omega_0 = 0 for panel in panel_ors: panel = panel_ors[panel] iter = 0 theta = sp.acos(mech.dot(panel.frame.x, bus.x).evalf()) * panel.frame.z y_axis_theta = [] y_axis_force = [] while mech.dot(theta, bus.z).evalf() > 0: # When the x-axis of of the bus projects on the y-axis of the bus, the panel latches into place. print("current angle is "+str(mech.dot(theta, bus.z).evalf())+"\n") angled_force = calc_torque(panel, f) alpha = panel.torque/ panel.MOI d_theta = omega_0 * t + alpha*t*t/2 theta += d_theta iter += 1 omega_0 = omega_0 + alpha*t panel.frame.orient(bus, "Axis", [mech.dot(theta, panel.frame.z), bus.z]) print(str(theta)+", "+str(omega_0)+", "+str(iter)) y_axis_theta.append(theta.magnitude().evalf()) y_axis_force.append(sp.Abs(angled_force)) print("Solution found after "+str(iter)+" iterations") #x_axis = list(frange(0, t*iter, t)) x_axis = np.linspace(0.0, iter*t, iter) fig, ax = plt.subplots() # Create a figure and an axes. ax.plot(x_axis, y_axis_theta, label='theta') # Plot some data on the axes. ax.plot(x_axis, y_axis_force, label='force') # Plot more data on the axes... #ax.plot(x_axis, x**3, label='cubic') # ... and some more. ax.set_xlabel('x label') # Add an x-label to the axes. ax.set_ylabel('y label') # Add a y-label to the axes. ax.set_title("Simple Plot") # Add a title to the axes. ax.legend() # Add a legend. plt.show()
def calc_torque(panel, force, bus): force_z = mech.dot(force, bus.y) * sp.cos(get_angle(panel.frame, bus)) if force_z > 0: force_z = force_z * ( 1 + panel.state ) # if the light hits on the front of the panel, use state to determine a coefficient of reflection. #force_z = mech.dot(force, panel.frame.y).evalf() torque_zz = -panel.dimensions[0] * 3**(1 / 2) / 6 * force_z panel.torque = 0 * panel.frame.y + 0 * panel.frame.x + torque_zz * panel.frame.z return force_z
def test_aux(): # Same as above, except we have 2 auxiliary speeds for the ground contact # point, which is known to be zero. In one case, we go through then # substitute the aux. speeds in at the end (they are zero, as well as their # derivative), in the other case, we use the built-in auxiliary speed part # of KanesMethod. The equations from each should be the same. 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) u4, u5, f1, f2 = dynamicsymbols('u4, u5, f1, f2') u4d, u5d = dynamicsymbols('u4, u5', 1) r, m, g = symbols('r m g') 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) C = Point('C') C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x)) Dmc = C.locatenew('Dmc', r * L.z) Dmc.v2pt_theory(C, N, R) Dmc.a2pt_theory(C, N, R) I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2) kd = [dot(R.ang_vel_in(N) - w_R_N_qd, uv) for uv in L] ForceList = [(Dmc, - m * g * Y.z), (C, f1 * L.x + f2 * (Y.z ^ L.x))] BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc)) BodyList = [BodyD] KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3, u4, u5], kd_eqs=kd) with warnings.catch_warnings(): warnings.filterwarnings("ignore", category=SymPyDeprecationWarning) (fr, frstar) = KM.kanes_equations(ForceList, BodyList) fr = fr.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar = frstar.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) KM2 = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd, u_auxiliary=[u4, u5]) with warnings.catch_warnings(): warnings.filterwarnings("ignore", category=SymPyDeprecationWarning) (fr2, frstar2) = KM2.kanes_equations(ForceList, BodyList) fr2 = fr2.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar2 = frstar2.subs({u4d: 0, u5d: 0}).subs({u4: 0, u5: 0}) frstar.simplify() frstar2.simplify() assert (fr - fr2).expand() == Matrix([0, 0, 0, 0, 0]) assert (frstar - frstar2).expand() == Matrix([0, 0, 0, 0, 0])
def inertia_coefficient_contribution(body, partials, r, s): """Returns the contribution of a rigid body (or particle) to the inertia coefficient m_rs of a system. 'body' is an instance of a RigidBody or Particle. 'partials' is an instance of a PartialVelocity. 'r' is the first generalized speed. 's' is the second generlized speed. """ if isinstance(body, Particle): m_rs = body.mass * dot(partials[body.point][r], partials[body.point][s]) elif isinstance(body, RigidBody): m_rs = body.mass * dot(partials[body.masscenter][r], partials[body.masscenter][s]) m_rs += dot(dot(partials[body.frame][r], body.central_inertia), partials[body.frame][s]) else: raise TypeError(('{0} is not a RigidBody or Particle.').format(body)) return m_rs
def get_angle(frame1, frame2): angle = sp.acos(mech.dot(frame1.y, frame2.y)).evalf() try: projection = frame1.y.express(frame2).args[0][0][0] // abs( frame1.y.express(frame2).args[0][0][0] ) # If the y axis of the panel has a non-zero projection on the x-axis of the bus, it is at an angle. It falls -ve if CCW, it falls +ve if CW. except ZeroDivisionError: projection = 1 # if the angle of the panel is perfectly 0, assume CW direction. if projection > 0: angle = -angle return angle
def Volumen_CP(a1, a2, a3): """ Parameters ---------- a1 : 1 vector de la base VP a2 : 2 vector de la base VP a3 : 3 vector de la base VP Returns ------- volumen de 1 CP """ return dot(a1, cross(a2, a3))
def generalized_active_forces(partials, forces, uaux=None): # use the same frame used in calculating partial velocities ulist = partials.ulist if uaux is not None: uaux_zero = dict(zip(uaux, [0] * len(uaux))) Fr = [0] * len(ulist) for pf in forces: p = pf[0] f = pf[1] for i, u in enumerate(ulist): if partials[p][u] != 0 and f != 0: r = dot(partials[p][u], f) # if more than 2 args, 3rd is an integral function, where the # input is the integrand if len(pf) > 2: r = pf[2](r) # auxilliary speeds have no effect on original active forces if uaux is not None and u not in uaux: r = subs(r, uaux_zero) Fr[i] += r return Fr, ulist
def generalized_active_forces(partials, forces, uaux=None): # use the same frame used in calculating partial velocities ulist = partials.ulist if uaux is not None: uaux_zero = dict(zip(uaux, [0] * len(uaux))) Fr = [0] * len(ulist) for pf in forces: p = pf[0] # first arg is point/rf f = pf[1] # second arg is force/torque for i, u in enumerate(ulist): if partials[p][u] != 0 and f != 0: r = dot(partials[p][u], f) # if more than 2 args, 3rd is an integral function, where the # input is the integrand if len(pf) > 2: r = pf[2](r) # auxilliary speeds have no effect on original active forces if uaux is not None and u not in uaux: r = subs(r, uaux_zero) Fr[i] += r return Fr, ulist
R = L.orientnew('R', 'Axis', [q[2], L.y]) # Rattleback body fixed frame I = inertia(R, Ixx, Iyy, Izz, Ixy, Iyz, Ixz) # Inertia dyadic # Angular velocity using u's as body fixed measure numbers of angular velocity R.set_ang_vel(N, u[0]*R.x + u[1]*R.y + u[2]*R.z) # Rattleback ground contact point P = Point('P') P.set_vel(N, ua[0]*Y.x + ua[1]*Y.y + ua[2]*Y.z) # Rattleback paraboloid -- parameterize coordinates of contact point by the # roll and pitch angles, and the geometry # TODO: FIXME!!! # f(x,y,z) = a*x**2 + b*y**2 + z - c mu = [dot(rk, Y.z) for rk in R] rx = mu[0]/mu[2]/2/a ry = mu[1]/mu[2]/2/b rz = 1 - (mu[0]**2/a + mu[1]**2/b)*(1/2/mu[2])**2 # Locate origin of parabaloid coordinate system relative to contact point O = P.locatenew('O', -rx*R.x - ry*R.y - rz*R.z) O.set_vel(N, P.vel(N) + cross(R.ang_vel_in(N), O.pos_from(P))) # Mass center position and velocity RO = O.locatenew('RO', d*R.x + e*R.y) RO.set_vel(N, P.vel(N) + cross(R.ang_vel_in(N), RO.pos_from(P))) qd_rhs = [(u[2]*cos(q[2]) - u[0]*sin(q[2]))/cos(q[1]), u[0]*cos(q[2]) + u[2]*sin(q[2]), u[1] + tan(q[1])*(u[0]*sin(q[2]) - u[2]*cos(q[2])),
pC_star = pD.locatenew('C*', +b * A.z) for p in [pA_star, pB_star, pC_star]: p.set_vel(A, 0) 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.set_vel(B, 0) pC_hat.set_vel(C, 0) pB_hat.v2pt_theory(pB_star, F, B) pC_hat.v2pt_theory(pC_star, F, C) # kinematic differential equations and velocity constraints kde = [ u1 - dot(A.ang_vel_in(F), A.x), u2 - dot(pD.vel(F), A.y), u3 - q3d, u4 - q4d, u5 - q5d ] kde_map = solve(kde, [q1d, q2d, q3d, q4d, q5d]) vc = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]] + [dot(pD.vel(F), A.z)] vc_map = solve(subs(vc, kde_map), [u3, u4, u5]) forces = [(pS_star, -M * g * F.x), (pQ, Q1 * A.x)] # no friction at point Q torques = [(A, -TB * A.z), (A, -TC * A.z), (B, TB * A.z), (C, TC * A.z)] partials = partial_velocities(zip(*forces + torques)[0], [u1, u2], F, kde_map, vc_map, express_frame=A) Fr, _ = generalized_active_forces(partials, forces + torques)
def inertia_matrix(dyadic, rf): """Return the inertia matrix of a given dyadic for a specified reference frame. """ return Matrix([[dot(dot(dyadic, i), j) for j in rf] for i in rf])
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) with warns_deprecated_sympy(): (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, Rational(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_Vector(): assert A.x != A.y assert A.y != A.z assert A.z != A.x v1 = x*A.x + y*A.y + z*A.z v2 = x**2*A.x + y**2*A.y + z**2*A.z v3 = v1 + v2 v4 = v1 - v2 assert isinstance(v1, Vector) assert dot(v1, A.x) == x assert dot(v1, A.y) == y assert dot(v1, A.z) == z assert isinstance(v2, Vector) assert dot(v2, A.x) == x**2 assert dot(v2, A.y) == y**2 assert dot(v2, A.z) == z**2 assert isinstance(v3, Vector) # We probably shouldn't be using simplify in dot... assert dot(v3, A.x) == x**2 + x assert dot(v3, A.y) == y**2 + y assert dot(v3, A.z) == z**2 + z assert isinstance(v4, Vector) # We probably shouldn't be using simplify in dot... assert dot(v4, A.x) == x - x**2 assert dot(v4, A.y) == y - y**2 assert dot(v4, A.z) == z - z**2
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 warns_deprecated_sympy(): 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.Zero: 6 }
I = inertia(R, Ixx, Iyy, Izz, Ixy, Iyz, Ixz) # Inertia dyadic print(I.express(Y)) # 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 #mu = [dot(rk, Y.z) for rk in R] #eps = sqrt((a*mu[0])**2 + (b*mu[1])**2 + (c*mu[2])**2) O = P.locatenew('O', -rx*R.x - rx*R.y - rx*R.z) RO = O.locatenew('RO', d*R.x + e*R.y + f*R.z) w_r_n = wx*R.x + wy*R.y + wz*R.z omega_dict = {wx: dot(qd[0]*Y.z, R.x), wy: dot(qd[0]*Y.z, R.y), wz: dot(qd[0]*Y.z, R.z)} v_ro_n = cross(w_r_n, RO.pos_from(P)) a_ro_n = cross(w_r_n, v_ro_n) mu_dict = {mu_x: dot(R.x, Y.z), mu_y: dot(R.y, Y.z), mu_z: dot(R.z, Y.z)} #F_RO = m*g*Y.z + Fx*Y.x + Fy*Y.y + Fz*Y.z #F_RO = Fx*R.x + Fy*R.y + Fz*R.z + m*g*Y.z F_RO = Fx*R.x + Fy*R.y + Fz*R.z + m*g*(mu_x*R.x + mu_y*R.y + mu_z*R.z) newton_eqn = F_RO - m*a_ro_n force_scalars = solve([dot(newton_eqn, uv).expand() for uv in R], [Fx, Fy, Fz]) #print("v_ro_n =", v_ro_n) #print("a_ro_n =", a_ro_n) #print("Force scalars =", force_scalars)
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.set_vel(A, 0) 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.set_vel(B, 0) pC_hat.set_vel(C, 0) pB_hat.v2pt_theory(pB_star, F, B) pC_hat.v2pt_theory(pC_star, F, C) # kinematic differential equations and velocity constraints kde = [u1 - dot(A.ang_vel_in(F), A.x), u2 - dot(pD.vel(F), A.y), u3 - q3d, u4 - q4d, u5 - q5d] kde_map = solve(kde, [q1d, q2d, q3d, q4d, q5d]) vc = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]] + [dot(pD.vel(F), A.z)] vc_map = solve(subs(vc, kde_map), [u3, u4, u5]) forces = [(pS_star, -M*g*F.x), (pQ, Q1*A.x)] # no friction at point Q torques = [(A, -TB*A.z), (A, -TC*A.z), (B, TB*A.z), (C, TC*A.z)] partials = partial_velocities(zip(*forces + torques)[0], [u1, u2], F, kde_map, vc_map, express_frame=A) Fr, _ = generalized_active_forces(partials, forces + torques) q = [q1, q2, q3, q4, q5]
def test_Vector(): assert A.x != A.y assert A.y != A.z assert A.z != A.x v1 = x * A.x + y * A.y + z * A.z v2 = x ** 2 * A.x + y ** 2 * A.y + z ** 2 * A.z v3 = v1 + v2 v4 = v1 - v2 assert isinstance(v1, Vector) assert dot(v1, A.x) == x assert dot(v1, A.y) == y assert dot(v1, A.z) == z assert isinstance(v2, Vector) assert dot(v2, A.x) == x ** 2 assert dot(v2, A.y) == y ** 2 assert dot(v2, A.z) == z ** 2 assert isinstance(v3, Vector) # We probably shouldn't be using simplify in dot... assert dot(v3, A.x) == x ** 2 + x assert dot(v3, A.y) == y ** 2 + y assert dot(v3, A.z) == z ** 2 + z assert isinstance(v4, Vector) # We probably shouldn't be using simplify in dot... assert dot(v4, A.x) == x - x ** 2 assert dot(v4, A.y) == y - y ** 2 assert dot(v4, A.z) == z - z ** 2 assert v1.to_matrix(A) == Matrix([[x], [y], [z]]) q = symbols("q") B = A.orientnew("B", "Axis", (q, A.x)) assert v1.to_matrix(B) == Matrix([[x], [y * cos(q) + z * sin(q)], [-y * sin(q) + z * cos(q)]])
pP = pO.locatenew('P', q1 * A.x + q2 * A.y + q3 * A.z) pP.set_vel(A, pP.pos_from(pO).dt(A)) # kinematic differential equations kde_map = dict(zip(map(lambda x: x.diff(), q), u)) # forces forces = [(pP, -beta * pP.vel(A))] torques = [(B, -alpha * B.ang_vel_in(A))] partials_c = partial_velocities(zip(*forces + torques)[0], u, A, kde_map) Fr_c, _ = generalized_active_forces(partials_c, forces + torques) dissipation_function = function_from_partials(map( lambda x: 0 if x == 0 else -x.subs(kde_map), Fr_c), u, zero_constants=True) from sympy import simplify, trigsimp dissipation_function = trigsimp(dissipation_function) #print('ℱ = {0}'.format(msprint(dissipation_function))) omega2 = trigsimp(dot(B.ang_vel_in(A), B.ang_vel_in(A)).subs(kde_map)) v2 = trigsimp(dot(pP.vel(A), pP.vel(A)).subs(kde_map)) sym_map = dict(zip([omega2, v2], map(lambda x: x**2, symbols('ω v')))) #print('ω**2 = {0}'.format(msprint(omega2))) #print('v**2 = {0}'.format(msprint(v2))) print('ℱ = {0}'.format(msprint(dissipation_function.subs(sym_map)))) dissipation_function_expected = (alpha * omega2 + beta * v2) / 2 assert expand(dissipation_function - dissipation_function_expected) == 0
# reference frames A = ReferenceFrame('A') B = A.orientnew('B', 'body', [q1, q2, q3], 'xyz') # points B*, O pB_star = Point('B*') pB_star.set_vel(A, 0) # rigidbody B I_B_Bs = inertia(B, I11, I22, I33) rbB = RigidBody('rbB', pB_star, B, m, (I_B_Bs, pB_star)) # kinetic energy K = rbB.kinetic_energy(A) # velocity of point B* is zero print('K_ω = {0}'.format(msprint(K))) print('\nSince I11, I22, I33 are the central principal moments of inertia') print('let I_min = I11, I_max = I33') I_min = I11 I_max = I33 H = rbB.angular_momentum(pB_star, A) K_min = dot(H, H) / I_max / 2 K_max = dot(H, H) / I_min / 2 print('K_ω_min = {0}'.format(msprint(K_min))) print('K_ω_max = {0}'.format(msprint(K_max))) print('\nI11/I33, I22/I33 =< 1, since I33 >= I11, I22, so K_ω_min <= K_ω') print('Similarly, I22/I11, I33/I11 >= 1, ' 'since I11 <= I22, I33, so K_ω_max >= K_ω')
pB_star.v2pt_theory(pP, N, B) # P' is the same point as P but on rigidbody B instead of rigidbody A. pP_prime = pB_star.locatenew('P\'', -(b - b_star)*B.z) pP_prime.v2pt_theory(pB_star, N, B) # Define point Q where B and C intersect. pQ = pP.locatenew('Q', b*B.z) pQ.v2pt_theory(pP, N, B) # Define the distance between points Q, C* as c. pC_star = pQ.locatenew('C*', c_star*C.z) pC_star.v2pt_theory(pQ, N, C) # configuration constraint for q2. cc = [dot(pC_star.pos_from(pO), N.x)] cc_map = solve(cc, q2)[1] # kinematic differential equations kde = [u1 - q1d, u2 - dot(B.ang_vel_in(N), N.y)] kde_map = solve(kde, [q1d, q2d]) # velocity constraints vc = subs([u3 - dot(pC_star.vel(N), N.z), cc[0].diff(t)], kde_map) vc_map = solve(vc, [u2, u3]) # verify motion constraint equation match text u2_expected = -a*cos(q1)/(b*cos(q2))*u1 u3_expected = -a/cos(q2)*(sin(q1)*cos(q2) + cos(q1)*sin(q2))*u1 assert trigsimp(vc_map[u2] - u2_expected) == 0 assert trigsimp(vc_map[u3] - u3_expected) == 0
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) (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 = MM_full.subs(val_dict) forcing_lin_s = forcing_lin.subs(KM.kindiffdict()).subs(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)
V = 2 * R * (pi * r)**2 rho = m / V A = ReferenceFrame('A') # Torus fixed, x-y in symmetry plane B = A.orientnew('B', 'Axis', [phi, A.z]) # Intermediate frame C = B.orientnew('C', 'Axis', [-theta, B.y]) # Intermediate frame # Position vector from torus center to arbitrary point of torus # R : torus major radius # s : distance >= 0 from center of torus cross section to point in torus p = R * B.x + s * C.x # Determinant of the Jacobian of the mapping from a, b, c to x, y, z # See Wikipedia for a lucid explanation of why we must comput this Jacobian: # http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Further_examples J = Matrix([dot(p, uv) for uv in A]).transpose().jacobian([phi, theta, s]) dv = J.det().trigsimp() # Need to ensure this is positive print("dx*dy*dz = {0}*dphi*dtheta*ds".format(dv)) # We want to compute the inertia scalars of the torus relative to it's mass # center, for the following six unit vector pairs unit_vector_pairs = [(A.x, A.x), (A.y, A.y), (A.z, A.z), (A.x, A.y), (A.y, A.z), (A.x, A.z)] # Calculate the six unique inertia scalars using equation 3.3.9 of Kane & # Levinson, 1985. inertia_scalars = [] for n_a, n_b in unit_vector_pairs: # Integrand of Equation 3.3.9 integrand = rho * dot(cross(p, n_a), cross(p, n_b)) * dv
m = symbols('m') m_val = 12 N = ReferenceFrame('N') pO = Point('O') pBs = pO.locatenew('B*', -3 * N.x + 2 * N.y - 4 * N.z) I_B_O = inertia(N, 260 * m / m_val, 325 * m / m_val, 169 * m / m_val, 72 * m / m_val, 96 * m / m_val, -144 * m / m_val) print("I_B_rel_O = {0}".format(I_B_O)) I_Bs_O = inertia_of_point_mass(m, pBs.pos_from(pO), N) print("\nI_B*_rel_O = {0}".format(I_Bs_O)) I_B_Bs = I_B_O - I_Bs_O print("\nI_B_rel_B* = {0}".format(I_B_Bs)) pQ = pO.locatenew('Q', -4 * N.z) I_Bs_Q = inertia_of_point_mass(m, pBs.pos_from(pQ), N) print("\nI_B*_rel_Q = {0}".format(I_Bs_Q)) I_B_Q = I_B_Bs + I_Bs_Q print("\nI_B_rel_Q = {0}".format(I_B_Q)) # n_a is a vector parallel to line PQ n_a = S(3) / 5 * N.x - S(4) / 5 * N.z I_a_a_B_Q = dot(dot(n_a, I_B_Q), n_a) print("\nn_a = {0}".format(n_a)) print("\nI_a_a_B_Q = {0} = {1}".format(I_a_a_B_Q, I_a_a_B_Q.subs(m, m_val)))
pP1.v1pt_theory(pO, A, B) pDs.set_vel(E, 0) pDs.v2pt_theory(pP1, B, E) pDs.v2pt_theory(pP1, A, E) # X*B.z, (Y*E.y + Z*E.z) are forces the panes of glass # exert on P1, D* respectively R1 = X*B.z + C*E.x - m1*g*B.y R2 = Y*E.y + Z*E.z - C*E.x - m2*g*B.y resultants = [R1, R2] points = [pP1, pDs] forces = [(pP1, R1), (pDs, R2)] system = [Particle('P1', pP1, m1), Particle('P2', pDs, m2)] # kinematic differential equations kde = [u1 - dot(pP1.vel(A), E.x), u2 - dot(pP1.vel(A), E.y), u3 - q3d] kde_map = solve(kde, qd) # include second derivatives in kde map for k, v in kde_map.items(): kde_map[k.diff(t)] = v.diff(t) # use nonholonomic partial velocities to find the nonholonomic # generalized active forces vc = [dot(pDs.vel(B), E.y).subs(kde_map)] vc_map = solve(vc, [u3]) partials = partial_velocities(points, [u1, u2], A, kde_map, vc_map) Fr, _ = generalized_active_forces(partials, forces) Fr_star, _ = generalized_inertia_forces(partials, system, kde_map, vc_map) # dynamical equations dyn_eq = [x + y for x, y in zip(Fr, Fr_star)]
from sympy import acos, pi, symbols from sympy.physics.mechanics import ReferenceFrame, Point from sympy.physics.mechanics import cross, dot # vectors A, B have equal magnitude 10 N alpha = 10 # [N] N = ReferenceFrame('N') pO = Point('O') pS = pO.locatenew('S', 10*N.x + 5*N.z) pR = pO.locatenew('R', 10*N.x + 12*N.y) pQ = pO.locatenew('Q', 12*N.y + 10*N.z) pP = pO.locatenew('P', 4*N.x + 7*N.z) A = alpha * pQ.pos_from(pP).normalize() B = alpha * pS.pos_from(pR).normalize() R = A + B M = cross(pP.pos_from(pO), A) + cross(pR.pos_from(pO), B) Ms = dot(R, M) * R / dot(R, R) ps = cross(R, M) / dot(R, R) # Replace set S (vectors A, B) with a wrench consisting of the resultant of S, # placed on the central axis of S and a couple whose torque is equal to S's # moment of minimum magnitude. r = R C = Ms print("|C| = {0}, {1}".format(C.magnitude(), C.magnitude().n())) # Since the bound vector r is placed on the central axis of S, |p*| gives the # distance from point O to r. print("|p*| = {0}, {1}".format(ps.magnitude(), ps.magnitude().n()))
def main(): theta_zero = 30 / 180 * sp.pi #10/180 * sp.pi bus = sun.orientnew("bus", "Axis", [angle_to_sun, sun.z]) order = [bus] # MANUAL GENERATION FOR TWO PANELS. ''' panel_ors["0"] = Panel( connections="bus", dimensions=[0.7,0.7], frame = bus.orientnew("0", "Axis", [theta_zero, bus.z]), state = 0 ) panel_ors["1"] = Panel( connections="0", dimensions=[0.7,0.7], frame = panel_ors[0].frame.orientnew("1", "Axis", [theta_zero, panel_ors["0"].frame.z]), state = 0 ) ''' #PANEL GENERATOR FOR LOOP. for i in range(1, n + 1): # for each panel, create a new reference frame. panel_ors[i] = Panel(dimensions=[0.7, 0.7], frame=order[i - 1].orientnew( "panel" + str(i), "Axis", [theta_zero, bus.z]), state=1) order.append(panel_ors[i].frame) upd = 3.6**-1 # hertz t = 1 / upd omega_0 = 0 #return panel_ors for panel in panel_ors: panel = panel_ors[panel] c = 0 alpha = 0 * panel.frame.z theta = sp.acos(mech.dot(panel.frame.x, bus.x).evalf()) * panel.frame.z y_axis_theta = [] y_axis_force = [] y_axis_moment_srp = [] y_axis_moment_spr = [] y_axis_moment_dam = [] y_axis_speed = [] bus_location = 1.5 * sun.x # location in AU referenced on the sun frame. omega_0 = 0 * panel.frame.z spring_trigger = 0 while True: f = panel.area * (1 + panel.state) * calc_srp(bus_location, bus) angled_force = calc_torque(panel, f, bus) #spring = get_spring_moment(panel.frame, bus) ''' if get_angle(panel.frame, bus) < 0: if mech.dot(omega_0, panel.frame.z) < 0: # once it reaches a desired velocity ccw, turn off. spring = -panel.torque * 6 else: spring = 0 *panel.frame.z else: spring = 0 * panel.frame.z ''' spring = get_spring_moment(panel.frame, bus) damper = get_joint_damper_monent(omega_0, panel.frame) #c = 4.29E-6 # spring constant. N*M/rad ''' ref_spring_theta = 0 # rads if get_angle(panel.frame, bus) < 0 and spring_trigger == 0: # start breaking time_to_stop = 300 # seconds. spring_coeff = - alpha.magnitude()* panel.MOI / time_to_stop spring = -spring_coeff * (theta - ref_spring_theta) spring_trigger = 1 elif get_angle(panel.frame, bus) > 0: spring_trigger = 0 spring = 0 * panel.frame.z ''' alpha = (panel.torque + spring + damper) / panel.MOI d_theta = omega_0 * t + alpha * t * t / 2 theta += d_theta c += 1 omega_0 = omega_0 + alpha * t panel.frame.orient(bus, "Axis", [mech.dot(theta, panel.frame.z), bus.z]) print(str(theta) + ", " + str(omega_0) + ", " + str(c)) y_axis_theta.append(get_angle(panel.frame, bus)) y_axis_force.append(angled_force) y_axis_moment_spr.append(mech.dot(spring, bus.z).evalf()) y_axis_moment_srp.append(mech.dot(panel.torque, bus.z).evalf()) y_axis_moment_dam.append(mech.dot(damper, bus.z).evalf()) y_axis_speed.append(mech.dot(omega_0, bus.z).evalf()) if c > 2000: print("Iteration limit reached, escaping...") break if c > 1 and threshold(np.mean( np.abs(np.diff(y_axis_theta[-convergence_mean:]))), 0, thresh=0.0001): print("Solution found after " + str(c) + " iterations") print("Force: " + "%.3g" % (2 * n * y_axis_force[-1]) + "N") break if abs(get_angle(panel.frame, bus)) > sp.pi.evalf() / 2: print("Exception ocurred: Panel rotated more than 90 degrees.") print("Force: " + "%.3g" % (2 * n * y_axis_force[-1]) + "N") break x_axis = np.linspace(0.0, c * t, c) figtheta, axtheta = plt.subplots() # Create a figure and an axes. axtheta.plot(x_axis, y_axis_theta) axtheta.plot([0, c * t], [y_axis_theta[-1], y_axis_theta[-1]], "r--") axtheta.set_xlabel('Time (s)') # Add an x-label to the axes. axtheta.set_ylabel('Panel angle (rads)') # Add a y-label to the axes. axtheta.set_title( "Angle plot against time, initial theta =" + str(round(theta_zero.evalf(), 2))) # Add a title to the axes. axtheta.grid(True) #figtheta.savefig("theta_plot.fig") figtheta.savefig("theta_plot.png") #axtheta.grid(axis="both", linestyle='-', linewidth=2) figforce, axforce = plt.subplots() axforce.plot(x_axis, y_axis_force) axforce.plot([0, c * t], [y_axis_force[-1], y_axis_force[-1]], "r--") axforce.set_xlabel("Time (s)") axforce.set_ylabel("SRP Force (N)") axforce.set_title("Force plot against time, initial theta =" + str(round(theta_zero.evalf(), 2))) axforce.grid(True) #axforce.savefig("force_plot.fig") figforce.savefig("force_plot.png") #axforce.grid(axis="both", linestyle='-', linewidth=2) figmoment, axmoment = plt.subplots() axmoment.plot(x_axis, y_axis_moment_spr, label="Spring moment") axmoment.plot(x_axis, y_axis_moment_srp, label="SRP moment") axmoment.plot(x_axis, y_axis_moment_dam, label="Damper moment") axmoment.plot([0, c * t], [ np.mean(y_axis_moment_srp[-convergence_mean:]), np.mean(y_axis_moment_srp[-convergence_mean:]) ], "r--") axmoment.set_xlabel("Time (s)") axmoment.set_ylabel("Moment (N*m)") axmoment.set_title("Moment plot against time, initial theta =" + str(round(theta_zero.evalf(), 2))) axmoment.legend() axmoment.grid(True) #axmoment.savefig("moment_plot.fig") figmoment.savefig("moment_plot.png") #axmoment.grid(axis="both", linestyle='-', linewidth=2) # figspeed, axspeed = plt.subplots() axspeed.plot(x_axis, y_axis_speed) axspeed.plot([0, c * t], [0, 0], "r--") axspeed.set_xlabel("Time (s)") axspeed.set_ylabel("Angular velocity (Rads/s)") axspeed.set_title( "Angular velocity plot against time, initial theta =" + str(round(theta_zero.evalf(), 2))) axspeed.grid(True) #axspeed.savefig("speed_plot.fig") figspeed.savefig("speed_plot.png") #axspeed.grid(axis="both", linestyle='-', linewidth=2) #plt.show() return panel_ors, bus
# 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. pC_star = pC_hat.locatenew("C*", R * B.y) pC_star.set_vel(C, 0) pC_star.set_vel(B, 0) # calculate velocities in A pC_star.v2pt_theory(pR, A, B) pC_hat.v2pt_theory(pC_star, A, C) # kinematic differential equations kde = [x - y for x, y in zip([dot(C.ang_vel_in(A), basis) for basis in B] + qd[3:], u)] kde_map = solve(kde, qd) # include second derivatives in kde map for k, v in kde_map.items(): kde_map[k.diff(t)] = v.diff(t) vc = map(lambda x: dot(pC_hat.vel(A), x), [A.x, A.y]) vc_map = solve(subs(vc, kde_map), [u4, u5]) # define disc rigidbody I_C = inertia(C, m * R ** 2 / 4, m * R ** 2 / 4, m * R ** 2 / 2) rbC = RigidBody("rbC", pC_star, C, m, (I_C, pC_star)) # forces R_C_hat = Px * A.x + Py * A.y + Pz * A.z R_C_star = -m * g * A.z
# No Input test case initial_altitude=10 numSteps=10 timeStep=1 param1=[1]*len(param) pos1=[0]*len(state) vel1=[0]*len(velocity) pos1[1]=initial_altitude input1=[0]*4 NoInput=QuadcopterMotion(param1,vel1+pos1) for i in range(numSteps): NoInput.move(input1,timeStep) #print NoInput.state from sympy import simplify from sympy.physics.mechanics import dot gC=dot(NewtonFrame.y,RobotFrame.x),dot(NewtonFrame.y,RobotFrame.y),dot(NewtonFrame.y,RobotFrame.z) Frame=[RobotFrame.x,RobotFrame.y,RobotFrame.z] alpha=Symbol('alpha') #print([simplify(dot(RobotFrame.ang_vel_in(NewtonFrame),vec).subs(symdict)) for vec in Frame]) #print([simplify(dot(NewtonFrame.y,vec).subs(symdict)) for vec in Frame]) #print([simplify(dot(cos(alpha)*NewtonFrame.x-sin(alpha)*NewtonFrame.y,vec).subs(symdict)) for vec in Frame]) try: from quadcopterODE import quadcopterODE except ImportError: from warnings import warn warn("quadcopterODE not written yet; run QuadcopterMechanics.py") from scipy.integrate import ode class QuadcopterMotion: def __init__(self,param,init_state): self.param=param self.ODE=ode(quadcopterODE)#.set_integrator('dopri5')
def test_Vector(): assert A.x != A.y assert A.y != A.z assert A.z != A.x v1 = x*A.x + y*A.y + z*A.z v2 = x**2*A.x + y**2*A.y + z**2*A.z v3 = v1 + v2 v4 = v1 - v2 assert isinstance(v1, Vector) assert dot(v1, A.x) == x assert dot(v1, A.y) == y assert dot(v1, A.z) == z assert isinstance(v2, Vector) assert dot(v2, A.x) == x**2 assert dot(v2, A.y) == y**2 assert dot(v2, A.z) == z**2 assert isinstance(v3, Vector) # We probably shouldn't be using simplify in dot... assert dot(v3, A.x) == (x*(x + 1)).expand() assert dot(v3, A.y) == (y*(y + 1)).expand() assert dot(v3, A.z) == (z*(z + 1)).expand() assert isinstance(v4, Vector) # We probably shouldn't be using simplify in dot... assert dot(v4, A.x) == (x*(1 - x)).expand() assert dot(v4, A.y) == (y*(1 - y)).expand() assert dot(v4, A.z) == (z*(1 - z)).expand()
pC_star = pD.locatenew('C*', -b*A.z) for p in [pA_star, pB_star, pC_star]: p.set_vel(A, 0) 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.set_vel(B, 0) pC_hat.set_vel(C, 0) 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 = [dot(p.vel(F), b) for b in A for p in [pB_hat, pC_hat]] kde = [dot(p.vel(F), A.y) for p in [pB_hat, pC_hat]] kde_map = solve(kde, [q2d, q3d]) # need to add q2'', q3'' terms manually since subs does not replace # Derivative(q(t), t, t) with Derivative(Derivative(q(t), t)) for k, v in 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)
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_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) (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. with warnings.catch_warnings(): warnings.filterwarnings("ignore", category=SymPyDeprecationWarning) 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 = MM_full.subs(val_dict) forcing_lin_s = forcing_lin.subs(KM.kindiffdict()).subs(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 xrange(6): error = Res.subs(v, i) - A.subs(v, i) assert all(abs(x) < eps for x in error)
b1, b2, b3 = symbols('b1 b2 b3', real=True) p1, p2, p3 = symbols('p1 p2 p3', real=True) N = ReferenceFrame('N') pO = Point('O') pS = pO.locatenew('S', 10*N.x + 5*N.z) pR = pO.locatenew('R', 10*N.x + 12*N.y) pQ = pO.locatenew('Q', 12*N.y + 10*N.z) pP = pO.locatenew('P', 4*N.x + 7*N.z) A = alpha * pQ.pos_from(pP).normalize() B = alpha * pS.pos_from(pR).normalize() R = A + B M = cross(pP.pos_from(pO), A) + cross(pR.pos_from(pO), B) Ms = dot(R, M) * R / dot(R, R) ps = cross(R, M) / dot(R, R) A_prime = beta * pP.pos_from(pO).normalize() B_prime = b1*N.x + b2*N.y + b3*N.z pB_prime = pO.locatenew("B'", p1*N.x + p2*N.y + p3*N.z) M_prime = cross(pB_prime.pos_from(pO), B_prime) eqs = [dot(R - A_prime - B_prime, n) for n in N] eqs += [dot(M - M_prime, n) for n in N] # choose pB_prime to be perpendicular to B_prime # then pB_prime.magnitude() gives the distance d from O # to the line of action of B' eqs.append(dot(pB_prime.pos_from(pO), B_prime))
# C* is the point at the center of disk C. pC_star = pC_hat.locatenew('C*', R*B.y) pC_star.set_vel(C, 0) pC_star.set_vel(B, 0) # calculate velocities in A pC_star.v2pt_theory(pR, A, B) pC_hat.v2pt_theory(pC_star, A, C) # kinematic differential equations #kde = [dot(C.ang_vel_in(A), x) - y for x, y in zip(B, u[:3])] #kde += [x - y for x, y in zip(qd[3:], u[3:])] #kde_map = solve(kde, qd) kde = [x - y for x, y in zip(u, qd)] kde_map = solve(kde, qd) vc = map(lambda x: dot(pC_hat.vel(A), x), [A.x, A.y]) vc_map = solve(subs(vc, kde_map), [u4, u5]) # define disc rigidbody IC = inertia(C, m*R**2/4, m*R**2/4, m*R**2/2) rbC = RigidBody('rbC', pC_star, C, m, (IC, pC_star)) rbC.set_potential_energy(m*g*dot(pC_star.pos_from(pR), A.z)) # potential energy V = rbC.potential_energy print('V = {0}'.format(msprint(V))) # kinetic energy K = trigsimp(rbC.kinetic_energy(A).subs(kde_map).subs(vc_map)) print('K = {0}'.format(msprint(K)))
pP1.v1pt_theory(pO, A, B) pDs.set_vel(E, 0) pDs.v2pt_theory(pP1, B, E) pDs.v2pt_theory(pP1, A, E) # X*B.z, (Y*E.y + Z*E.z) are forces the panes of glass # exert on P1, D* respectively R1 = X * B.z + C * E.x - m1 * g * B.y R2 = Y * E.y + Z * E.z - C * E.x - m2 * g * B.y resultants = [R1, R2] points = [pP1, pDs] forces = [(pP1, R1), (pDs, R2)] system = [Particle('P1', pP1, m1), Particle('P2', pDs, m2)] # kinematic differential equations kde = [u1 - dot(pP1.vel(A), E.x), u2 - dot(pP1.vel(A), E.y), u3 - q3d] kde_map = solve(kde, qd) # include second derivatives in kde map for k, v in kde_map.items(): kde_map[k.diff(t)] = v.diff(t) # use nonholonomic partial velocities to find the nonholonomic # generalized active forces vc = [dot(pDs.vel(B), E.y).subs(kde_map)] vc_map = solve(vc, [u3]) partials = partial_velocities(points, [u1, u2], A, kde_map, vc_map) Fr, _ = generalized_active_forces(partials, forces) Fr_star, _ = generalized_inertia_forces(partials, system, kde_map, vc_map) # dynamical equations dyn_eq = [x + y for x, y in zip(Fr, Fr_star)]
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 = {udi : 0 for udi in ud} ud_zero = dict(zip(ud, [0.]*len(ud))) ua = dynamicsymbols('ua:3') #ua_zero = {uai : 0 for uai in ua} 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)) #u_dep_dict = {udi : u_depi[0] for udi, u_depi in zip(u[3:], u_dep.tolist())} # 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() # Test # First try Fr_c == fr; # Second try Fr_star_c == frstar; # Third try Fr_star_steady == frstar_steady. # Both signs are checked in case the equations were found with an inverse # sign. assert ((Matrix(Fr_c).expand() == fr.expand()) or (Matrix(Fr_c).expand() == (-fr).expand())) assert ((Matrix(Fr_star_c).expand() == frstar.expand()) or (Matrix(Fr_star_c).expand() == (-frstar).expand())) assert ((Matrix(Fr_star_steady).expand() == frstar_steady.expand()) or (Matrix(Fr_star_steady).expand() == (-frstar_steady).expand()))
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_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) 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() # 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, new_method=True)[0] A_upright = A.subs({r: 1, g: 1, m: 1}).subs({q1: 0, q2: 0, q3: 0, u1: 0, u3: 0}) assert A_upright.subs(u2, 1 / sqrt(3)).eigenvals() == {S(0): 6}
#!/usr/bin/env python # -*- coding: utf-8 -*- """Exercise 7.2 from Kane 1985.""" from __future__ import division from sympy import solve, symbols from sympy.physics.mechanics import ReferenceFrame, Point from sympy.physics.mechanics import cross, dot c1, c2, c3 = symbols('c1 c2 c3', real=True) alpha = 10 # [N] N = ReferenceFrame('N') pO = Point('O') pS = pO.locatenew('S', 10*N.x + 5*N.z) # [m] pR = pO.locatenew('R', 10*N.x + 12*N.y) # [m] pQ = pO.locatenew('Q', 12*N.y + 10*N.z) # [m] pP = pO.locatenew('P', 4*N.x + 7*N.z) # [m] A = alpha * pQ.pos_from(pP).normalize() B = alpha * pS.pos_from(pR).normalize() C_ = c1*N.x + c2*N.y + c3*N.z eqs = [dot(A + B + C_, b) for b in N] soln = solve(eqs, [c1, c2, c3]) C = sum(soln[ci] * b for ci, b in zip(sorted(soln, cmp=lambda x, y: x.compare(y)), N)) print("C = {0}\nA + B + C = {1}".format(C, A + B + C)) M = cross(pP.pos_from(pO), A) + cross(pR.pos_from(pO), B) print("|M| = {0} N-m".format(M.magnitude().n()))
# 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 velocities in A pCs.v2pt_theory(pR, A, B) pC_hat.v2pt_theory(pCs, A, C) print("velocities of points R, C^, C* in rf A:") print("v_R_A = {0}\nv_C^_A = {1}\nv_C*_A = {2}".format( pR.vel(A), pC_hat.vel(A), pCs.vel(A))) ## --- Expressions for generalized speeds u1, u2, u3, u4, u5 --- u_expr = map(lambda x: dot(C.ang_vel_in(A), x), B) u_expr += qd[3:] kde = [u_i - u_ex for u_i, u_ex in zip(u, u_expr)] kde_map = solve(kde, qd) print("using the following kinematic eqs:\n{0}".format(msprint(kde))) ## --- Define forces on each point in 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)] ## --- Calculate generalized active forces --- partials = partial_velocities([pC_hat, pCs], u, A, kde_map) F, _ = generalized_active_forces(partials, forces) print("Generalized active forces:") for i, f in enumerate(F, 1):
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())
x, y = _me.dynamicsymbols('x y') x_d, y_d = _me.dynamicsymbols('x_ y_', 1) e = _sm.cos(x)+_sm.sin(x)+_sm.tan(x)+_sm.cosh(x)+_sm.sinh(x)+_sm.tanh(x)+_sm.acos(x)+_sm.asin(x)+_sm.atan(x)+_sm.log(x)+_sm.exp(x)+_sm.sqrt(x)+_sm.factorial(x)+_sm.ceiling(x)+_sm.floor(x)+_sm.sign(x) e = (x)**2+_sm.log(x, 10) a = _sm.Abs(-1*1)+int(1.5)+round(1.9) e1 = 2*x+3*y e2 = x+y am = _sm.Matrix([e1.expand().coeff(x), e1.expand().coeff(y), e2.expand().coeff(x), e2.expand().coeff(y)]).reshape(2, 2) b = (e1).expand().coeff(x) c = (e2).expand().coeff(y) d1 = (e1).collect(x).coeff(x,0) d2 = (e1).collect(x).coeff(x,1) fm = _sm.Matrix([i.collect(x)for i in _sm.Matrix([e1,e2]).reshape(1, 2)]).reshape((_sm.Matrix([e1,e2]).reshape(1, 2)).shape[0], (_sm.Matrix([e1,e2]).reshape(1, 2)).shape[1]) f = (e1).collect(y) g = (e1).subs({x:2*x}) gm = _sm.Matrix([i.subs({x:3}) for i in _sm.Matrix([e1,e2]).reshape(2, 1)]).reshape((_sm.Matrix([e1,e2]).reshape(2, 1)).shape[0], (_sm.Matrix([e1,e2]).reshape(2, 1)).shape[1]) frame_a = _me.ReferenceFrame('a') frame_b = _me.ReferenceFrame('b') theta = _me.dynamicsymbols('theta') frame_b.orient(frame_a, 'Axis', [theta, frame_a.z]) v1 = 2*frame_a.x-3*frame_a.y+frame_a.z v2 = frame_b.x+frame_b.y+frame_b.z a = _me.dot(v1, v2) bm = _sm.Matrix([_me.dot(v1, v2),_me.dot(v1, 2*v2)]).reshape(2, 1) c = _me.cross(v1, v2) d = 2*v1.magnitude()+3*v1.magnitude() dyadic = _me.outer(3*frame_a.x, frame_a.x)+_me.outer(frame_a.y, frame_a.y)+_me.outer(2*frame_a.z, frame_a.z) am = (dyadic).to_matrix(frame_b) m = _sm.Matrix([1,2,3]).reshape(3, 1) v = m[0]*frame_a.x +m[1]*frame_a.y +m[2]*frame_a.z
def calc_torque(panel, force, bus): force_z = mech.dot(force, bus.y) * sp.cos(get_angle(panel.frame, bus)) #force_z = mech.dot(force, panel.frame.y).evalf() torque_zz = -panel.dimensions[1] / 2 * force_z panel.torque = 0 * panel.frame.y + 0 * panel.frame.x + torque_zz * panel.frame.z return force_z
B.set_ang_vel(A, u2 * A["1"]) C.set_ang_vel(B, u3 * B["2"]) E.set_ang_vel(C, u4 * C["3"]) D.set_ang_vel(C, u5 * C["2"]) F.set_ang_vel(E, u6 * E["2"]) print("ready for special unit vectors; and points and velocities") ###################### # special unit vectors ###################### # pitch back for the unit vector g_3 along front wheel radius; g_3 = (mec.express(A["3"], E) - mec.dot(E["2"], A["3"]) * E["2"]).normalize() # another way: g_3 = E['2'].cross(A['3']).cross(E['2']).normalize() # roll back for longitudinal and lateral unit vector of front wheel long_v = mec.cross(E["2"], A["3"]).normalize() lateral_v = mec.cross(A["3"], long_v).normalize() ######################### # points and velocities ######################### ####################rear wheel contact point, dn dn = mec.Point("dn")
# points and velocities pO = Point('O') pO.set_vel(A, 0) pO.set_vel(B, 0) pP1 = pO.locatenew('P1', q1 * B.x + q2 * B.y) pDs = pP1.locatenew('D*', L * E.x) pP1.set_vel(E, 0) pP1.set_vel(B, pP1.pos_from(pO).dt(B)) pP1.v1pt_theory(pO, A, B) pDs.set_vel(E, 0) pDs.v2pt_theory(pP1, B, E) pDs.v2pt_theory(pP1, A, E) # define generalized speeds and constraints kde = [u1 - dot(pP1.vel(A), E.x), u2 - dot(pP1.vel(A), E.y), u3 - q3d] kde_map = solve(kde, qd) # include second derivatives in kde map for k, v in kde_map.items(): kde_map[k.diff(t)] = v.diff(t) vc = [dot(pDs.vel(B), E.y)] vc_map = solve(subs(vc, kde_map), [u3]) # define system of particles system = [Particle('P1', pP1, m1), Particle('P2', pDs, m2)] # calculate kinetic energy, generalized inertia forces K = sum(map(lambda x: x.kinetic_energy(A), system)) Fr_tilde_star = generalized_inertia_forces_K(K, q, [u1, u2], kde_map, vc_map)