Linearized model

lin_sym.py

@@ -0,0 +1,130 @@
+from nonlin_sym import *
+
+import sympy as sm
+from sympy.solvers import linear_eq_to_matrix
+from sympy.physics.mechanics.functions import center_of_mass
+
+# A_all*x = B_all
+# [Mp  0,  0, -Mf]*[up ] = [F     ]
+# [0, Cf,  0,  Cz] [fyp]   [-Fy   ]
+# [0,  0, Df,  Dz] [mzp]   [-Mz   ]
+# [Ap, 0,  0,  Af] [fz ]   [-B_aux]
+# TODO : Maybe add these rows for completeness
+# [I,  0,  0,  0 ] [qp ] = [0     ]
+
+# f(x', x, r) = f(xeq', xeq, req) + Jfx(xeq', xeq, req)*([x, r] - [xeq, req])
+# f(x', x, r) = Mp*x' + Mx*x + Mr*r
+# x' = A*x + B*r, where A = Mp
+
+total_mass = mc + md + me + mf
+mass_center = center_of_mass(dn, rear_frame, rear_wheel, front_frame,
+                             front_wheel)
+
+# These are approximations of the mass center that will be exact when the
+# equilibrium points are subsituted in.
+# total_mass*g*x = Ffz*w
+# TODO : Check if these shoudl be positive or negative.
+w = fn.pos_from(dn).dot(A.x)
+Ffz_eq = total_mass*g*mass_center.dot(A.x)/w
+# total_mass*g*(w-x) = Frz*w
+Frz_eq = total_mass*g*(mass_center - w*A.x).dot(A.x)/w
+
+q5eq, q7eq, u6eq, u8eq = sm.symbols('q5eq, q7eq, u6eq, u8eq')
+
+Fryp, Ffyp, Mrzp, Mfzp = sm.symbols('Fryp, Ffyp, Mrzp, Mfzp')
+u1p, u2p, u3p, u4p, u5p, u6p, u7p, u8p = sm.symbols('u1p, u2p, u3p, u4p, u5p, u6p, u7p, u8p')
+
+xp = sm.Matrix([
+    u1p,
+    u2p,
+    u3p,
+    u4p,
+    u5p,
+    u6p,
+    u7p,
+    u8p,
+    Fryp,
+    Ffyp,
+    Mrzp,
+    Mfzp,
+    Frz,
+    Ffz,
+])
+
+x = sm.Matrix([
+    u1,
+    u2,
+    u3,
+    u4,
+    u5,
+    u6,
+    u7,
+    u8,
+    Fry,
+    Ffy,
+    Mrz,
+    Mfz,
+])
+
+r = sm.Matrix([
+    T4,
+    T6,
+    T7,
+    Fkp,
+])
+
+v0 = {
+    # dx/dt
+    u1p: 0,
+    u2p: 0,
+    u3p: 0,
+    u4p: 0,
+    u5p: 0,
+    u6p: 0,
+    u7p: 0,
+    u8p: 0,
+    Fryp: 0,
+    Ffyp: 0,
+    Mrzp: 0,
+    Mfzp: 0,
+    Frz: Frz_eq,
+    Ffz: Ffz_eq,
+    # x
+    u1: 0,
+    u2: 0,
+    u3: 0,
+    u4: 0,
+    u5: 0,
+    u6: u6eq,
+    u7: 0,
+    u8: u8eq,
+    q1: 0,
+    q2: 0,
+    q3: 0,
+    q4: 0,
+    q5: q5eq,
+    q6: 0,
+    q7: q7eq,  # needed due to divide-by-zero issues
+    q8: 0,
+    # r
+    T4: 0,
+    T6: 0,
+    T7: 0,
+    Fkp: 0,
+}
+
+v = sm.Matrix(list(v0.keys()))
+v_val = sm.Matrix(list(v0.values()))
+v_zerod = {k: 0 for k, _ in v0.items()}
+
+f = A_all*xp - b_all
+f_lin = f.xreplace(v0) + f.jacobian(v).xreplace(v0)*(v - v_val)
+
+Mp_lin = linear_eq_to_matrix(f_lin, xp)[0]
+Mx_lin = linear_eq_to_matrix(f_lin, x)[0]
+Mr_lin = linear_eq_to_matrix(f_lin, r)[0]
+remainder = f_lin.xreplace(v_zerod)
+
+eval_lin = sm.lambdify([qs, us, fs, rs, ps, eqs],
+                       [Mp_lin, Mx_lin, Mr_lin],
+                       cse=True)