""" Full Linearized State-Space Model for the Guadaloop Maglev Pod ============================================================== Combines three dynamic layers into a single LTI system ẋ = Ax + Bu, y = Cx: Layer 1 — Coil RL dynamics (electrical): di/dt = (V·pwm − R·i) / L This is already linear. A first-order lag from PWM command to current. Layer 2 — Electromagnetic force/torque map (from Ansys polynomial): (F, τ) = f(iL, iR, roll, gap) Nonlinear, but the MaglevLinearizer gives us the Jacobian at any operating point, making it locally linear. Layer 3 — Rigid-body mechanics (Newton/Euler): m·z̈ = F_front + F_back − m·g (heave) Iy·θ̈ = L_arm·(F_front − F_back) (pitch from force differential) Ix·φ̈ = τ_front + τ_back (roll from magnetic torque) These are linear once the force/torque are linearized. The key coupling: the pod is rigid, so front and back yoke gaps are NOT independent. They are related to the average gap and pitch angle: gap_front = gap_avg − L_arm · pitch gap_back = gap_avg + L_arm · pitch This means a pitch perturbation changes both yoke gaps, which changes both yoke forces, which feeds back into the heave and pitch dynamics. The electromagnetic Jacobian captures how force/torque respond to these gap changes, creating the destabilizing "magnetic stiffness" that makes maglev inherently open-loop unstable. State vector (10 states): x = [gap_avg, gap_vel, pitch, pitch_rate, roll, roll_rate, i_FL, i_FR, i_BL, i_BR] - gap_avg [m]: average air gap (track-to-yoke distance) - gap_vel [m/s]: d(gap_avg)/dt - pitch [rad]: rotation about Y axis (positive = back hangs lower) - pitch_rate [rad/s] - roll [rad]: rotation about X axis - roll_rate [rad/s] - i_FL..BR [A]: the four coil currents Input vector (4 inputs): u = [pwm_FL, pwm_FR, pwm_BL, pwm_BR] (duty cycles, dimensionless) Output vector (3 outputs): y = [gap_avg, pitch, roll] """ import numpy as np import os from maglev_linearizer import MaglevLinearizer # --------------------------------------------------------------------------- # Physical constants and unit conversions # --------------------------------------------------------------------------- GRAVITY = 9.81 # m/s² DEG2RAD = np.pi / 180.0 RAD2DEG = 180.0 / np.pi # State indices (for readability) GAP, GAPV, PITCH, PITCHV, ROLL, ROLLV, I_FL, I_FR, I_BL, I_BR = range(10) # =================================================================== # StateSpaceResult — the output container # =================================================================== class StateSpaceResult: """ Holds the A, B, C, D matrices of the linearized plant plus operating-point metadata and stability analysis. """ STATE_LABELS = [ 'gap_avg [m]', 'gap_vel [m/s]', 'pitch [rad]', 'pitch_rate [rad/s]', 'roll [rad]', 'roll_rate [rad/s]', 'i_FL [A]', 'i_FR [A]', 'i_BL [A]', 'i_BR [A]', ] INPUT_LABELS = ['pwm_FL', 'pwm_FR', 'pwm_BL', 'pwm_BR'] OUTPUT_LABELS = ['gap_avg [m]', 'pitch [rad]', 'roll [rad]'] def __init__(self, A, B, C, D, operating_point, equilibrium_force_error, plant_front, plant_back): self.A = A self.B = B self.C = C self.D = D self.operating_point = operating_point self.equilibrium_force_error = equilibrium_force_error self.plant_front = plant_front # LinearizedPlant for front yoke self.plant_back = plant_back # LinearizedPlant for back yoke @property def eigenvalues(self): """Eigenvalues of A, sorted by decreasing real part.""" eigs = np.linalg.eigvals(self.A) return eigs[np.argsort(-np.real(eigs))] @property def is_open_loop_stable(self): return bool(np.all(np.real(self.eigenvalues) < 0)) @property def unstable_eigenvalues(self): eigs = self.eigenvalues return eigs[np.real(eigs) > 1e-8] def to_scipy(self): """Convert to scipy.signal.StateSpace for frequency-domain analysis.""" from scipy.signal import StateSpace return StateSpace(self.A, self.B, self.C, self.D) def print_A_structure(self): """Print the A matrix with row/column labels for physical insight.""" labels_short = ['gap', 'ġap', 'θ', 'θ̇', 'φ', 'φ̇', 'iFL', 'iFR', 'iBL', 'iBR'] print("\nA matrix (non-zero entries):") print("-" * 65) for i in range(10): for j in range(10): if abs(self.A[i, j]) > 1e-10: print(f" A[{labels_short[i]:>3}, {labels_short[j]:>3}] " f"= {self.A[i,j]:+12.4f}") print("-" * 65) def print_B_structure(self): """Print the B matrix with labels.""" labels_short = ['gap', 'ġap', 'θ', 'θ̇', 'φ', 'φ̇', 'iFL', 'iFR', 'iBL', 'iBR'] u_labels = ['uFL', 'uFR', 'uBL', 'uBR'] print("\nB matrix (non-zero entries):") print("-" * 50) for i in range(10): for j in range(4): if abs(self.B[i, j]) > 1e-10: print(f" B[{labels_short[i]:>3}, {u_labels[j]:>3}] " f"= {self.B[i,j]:+12.4f}") print("-" * 50) def __repr__(self): op = self.operating_point eigs = self.eigenvalues at_eq = abs(self.equilibrium_force_error) < 0.5 eq_str = ('AT EQUILIBRIUM' if at_eq else f'NOT AT EQUILIBRIUM — {self.equilibrium_force_error:+.2f} N residual') lines = [ "=" * 70, "LINEARIZED MAGLEV STATE-SPACE (ẋ = Ax + Bu, y = Cx)", "=" * 70, f"Operating point:", f" gap = {op['gap_height']:.2f} mm, " f"currL = {op['currL']:.2f} A, " f"currR = {op['currR']:.2f} A, " f"roll = {op['roll']:.1f}°, " f"pitch = {op['pitch']:.1f}°", f" F_front = {self.plant_front.f0:.3f} N, " f"F_back = {self.plant_back.f0:.3f} N, " f"F_total = {self.plant_front.f0 + self.plant_back.f0:.3f} N, " f"Weight = {op['mass'] * GRAVITY:.3f} N", f" >> {eq_str}", "", f"System: {self.A.shape[0]} states × " f"{self.B.shape[1]} inputs × " f"{self.C.shape[0]} outputs", f"Open-loop stable: {self.is_open_loop_stable}", "", "Eigenvalues of A:", ] # Group complex conjugate pairs printed = set() for i, ev in enumerate(eigs): if i in printed: continue re_part = np.real(ev) im_part = np.imag(ev) stability = "UNSTABLE" if re_part > 1e-8 else "stable" if abs(im_part) < 1e-6: lines.append( f" λ = {re_part:+12.4f} " f" τ = {abs(1/re_part)*1000 if abs(re_part) > 1e-8 else float('inf'):.2f} ms" f" ({stability})" ) else: # Find conjugate pair for j in range(i + 1, len(eigs)): if j not in printed and abs(eigs[j] - np.conj(ev)) < 1e-6: printed.add(j) break omega_n = abs(ev) lines.append( f" λ = {re_part:+12.4f} ± {abs(im_part):.4f}j" f" ω_n = {omega_n:.1f} rad/s" f" ({stability})" ) lines.extend(["", "=" * 70]) return '\n'.join(lines) # =================================================================== # MaglevStateSpace — the builder # =================================================================== class MaglevStateSpace: """ Assembles the full 10-state linearized state-space from the electromagnetic Jacobian + rigid body dynamics + coil dynamics. Physical parameters come from the URDF (pod.xml) and MagLevCoil. """ def __init__(self, linearizer, mass=5.8, I_roll=0.0192942414, # Ixx from pod.xml [kg·m²] I_pitch=0.130582305, # Iyy from pod.xml [kg·m²] coil_R=1.1, # from MagLevCoil in lev_pod_env.py coil_L=0.0025, # 2.5 mH V_supply=12.0, # supply voltage [V] L_arm=0.1259): # front/back yoke X-offset [m] self.linearizer = linearizer self.mass = mass self.I_roll = I_roll self.I_pitch = I_pitch self.coil_R = coil_R self.coil_L = coil_L self.V_supply = V_supply self.L_arm = L_arm @staticmethod def _convert_jacobian_to_si(jac): """ Convert a linearizer Jacobian from mixed units to pure SI. The linearizer returns: Row 0: Force [N] per [A, A, deg, mm] Row 1: Torque [mN·m] per [A, A, deg, mm] We need: Row 0: Force [N] per [A, A, rad, m] Row 1: Torque [N·m] per [A, A, rad, m] Conversion factors: col 0,1 (current): ×1 for force, ×(1/1000) for torque col 2 (roll): ×(180/π) for force, ×(180/π)/1000 for torque col 3 (gap): ×1000 for force, ×(1000/1000)=×1 for torque """ si = np.zeros((2, 4)) # Force row — already in N si[0, 0] = jac[0, 0] # N/A → N/A si[0, 1] = jac[0, 1] # N/A → N/A si[0, 2] = jac[0, 2] * RAD2DEG # N/deg → N/rad si[0, 3] = jac[0, 3] * 1000.0 # N/mm → N/m # Torque row — from mN·m to N·m si[1, 0] = jac[1, 0] / 1000.0 # mN·m/A → N·m/A si[1, 1] = jac[1, 1] / 1000.0 # mN·m/A → N·m/A si[1, 2] = jac[1, 2] / 1000.0 * RAD2DEG # mN·m/deg → N·m/rad si[1, 3] = jac[1, 3] # mN·m/mm → N·m/m (factors cancel) return si def build(self, gap_height, currL, currR, roll=0.0, pitch=0.0): """ Build the A, B, C, D matrices at a given operating point. Parameters ---------- gap_height : float Average gap [mm] currL : float Equilibrium left coil current [A] (same front & back) currR : float Equilibrium right coil current [A] roll : float Equilibrium roll angle [deg], default 0 pitch : float Equilibrium pitch angle [deg], default 0 Non-zero pitch means front/back gaps differ. Returns ------- StateSpaceResult """ m = self.mass Ix = self.I_roll Iy = self.I_pitch R = self.coil_R Lc = self.coil_L V = self.V_supply La = self.L_arm # ------------------------------------------------------------------ # Step 1: Compute individual yoke gaps from average gap + pitch # # The pod is rigid. If it pitches, the front and back yoke ends # are at different distances from the track: # gap_front = gap_avg − L_arm · sin(pitch) ≈ gap_avg − L_arm · pitch # gap_back = gap_avg + L_arm · sin(pitch) ≈ gap_avg + L_arm · pitch # # Sign convention (from lev_pod_env.py lines 230-232): # positive pitch = back gap > front gap (back hangs lower) # ------------------------------------------------------------------ pitch_rad = pitch * DEG2RAD # L_arm [m] * sin(pitch) [rad] → meters; convert to mm for linearizer gap_front_mm = gap_height - La * np.sin(pitch_rad) * 1000.0 gap_back_mm = gap_height + La * np.sin(pitch_rad) * 1000.0 # ------------------------------------------------------------------ # Step 2: Linearize each yoke independently # # Each U-yoke has its own (iL, iR) pair and sees its own gap. # Both yokes see the same roll angle (the pod is rigid). # The linearizer returns the Jacobian in mixed units. # ------------------------------------------------------------------ plant_f = self.linearizer.linearize(currL, currR, roll, gap_front_mm) plant_b = self.linearizer.linearize(currL, currR, roll, gap_back_mm) # ------------------------------------------------------------------ # Step 3: Convert Jacobians to SI # # After this, all gains are in [N or N·m] per [A, A, rad, m]. # Columns: [currL, currR, roll, gap_height] # ------------------------------------------------------------------ Jf = self._convert_jacobian_to_si(plant_f.jacobian) Jb = self._convert_jacobian_to_si(plant_b.jacobian) # Unpack for clarity — subscript _f = front yoke, _b = back yoke # Force gains kFiL_f, kFiR_f, kFr_f, kFg_f = Jf[0] kFiL_b, kFiR_b, kFr_b, kFg_b = Jb[0] # Torque gains kTiL_f, kTiR_f, kTr_f, kTg_f = Jf[1] kTiL_b, kTiR_b, kTr_b, kTg_b = Jb[1] # ------------------------------------------------------------------ # Step 4: Assemble the A matrix (10 × 10) # # The A matrix encodes three kinds of coupling: # # (a) Kinematic identities: gap_vel = d(gap)/dt, etc. # These are always 1.0 on the super-diagonal of the # position/velocity pairs. # # (b) Electromagnetic coupling through current states: # Coil currents produce forces/torques. The linearized # gains (∂F/∂i, ∂T/∂i) appear in the acceleration rows. # This is the path from current states to mechanical # acceleration — the "plant gain" that PID acts through. # # (c) Electromagnetic coupling through mechanical states: # Gap and roll perturbations change the force/torque. # This creates feedback loops: # # - ∂F/∂gap < 0 → gap perturbation changes force in a # direction that AMPLIFIES the gap error → UNSTABLE # (magnetic stiffness is "negative spring") # # - ∂T/∂roll → roll perturbation changes torque; # sign determines whether roll is self-correcting or not # # - Pitch couples through gap_front/gap_back dependence # on pitch angle, creating pitch instability too # ------------------------------------------------------------------ A = np.zeros((10, 10)) # (a) Kinematic identities A[GAP, GAPV] = 1.0 A[PITCH, PITCHV] = 1.0 A[ROLL, ROLLV] = 1.0 # ------------------------------------------------------------------ # HEAVE: m · Δgap̈ = −(ΔF_front + ΔF_back) # # The negative sign is because force is upward (+Z) but gap # is measured downward (gap shrinks when pod moves up). # At equilibrium F₀ = mg; perturbation ΔF pushes pod up → gap shrinks. # # Expanding ΔF using the rigid-body gap coupling: # ΔF_front = kFg_f·(Δgap − La·Δpitch) + kFr_f·Δroll + kFiL_f·ΔiFL + kFiR_f·ΔiFR # ΔF_back = kFg_b·(Δgap + La·Δpitch) + kFr_b·Δroll + kFiL_b·ΔiBL + kFiR_b·ΔiBR # ------------------------------------------------------------------ # Gap → gap acceleration (magnetic stiffness, UNSTABLE) A[GAPV, GAP] = -(kFg_f + kFg_b) / m # Pitch → gap acceleration (cross-coupling through differential gap) A[GAPV, PITCH] = -(-kFg_f + kFg_b) * La / m # Roll → gap acceleration A[GAPV, ROLL] = -(kFr_f + kFr_b) / m # Current → gap acceleration (the control path!) A[GAPV, I_FL] = -kFiL_f / m A[GAPV, I_FR] = -kFiR_f / m A[GAPV, I_BL] = -kFiL_b / m A[GAPV, I_BR] = -kFiR_b / m # ------------------------------------------------------------------ # PITCH: Iy · Δpitcḧ = La · (ΔF_front − ΔF_back) # # Pitch torque comes from DIFFERENTIAL FORCE, not from the # electromagnetic torque (which acts on roll). This is because # the front yoke is at x = +La and the back at x = −La: # τ_pitch = F_front·La − F_back·La = La·(F_front − F_back) # # At symmetric equilibrium, F_front = F_back → zero pitch torque. ✓ # A pitch perturbation breaks this symmetry through the gap coupling. # ------------------------------------------------------------------ # Gap → pitch acceleration (zero at symmetric equilibrium) A[PITCHV, GAP] = La * (kFg_f - kFg_b) / Iy # Pitch → pitch acceleration (pitch instability — UNSTABLE) # = −La²·(kFg_f + kFg_b)/Iy. Since kFg < 0 → positive → unstable. A[PITCHV, PITCH] = -La**2 * (kFg_f + kFg_b) / Iy # Roll → pitch acceleration A[PITCHV, ROLL] = La * (kFr_f - kFr_b) / Iy # Current → pitch acceleration A[PITCHV, I_FL] = La * kFiL_f / Iy A[PITCHV, I_FR] = La * kFiR_f / Iy A[PITCHV, I_BL] = -La * kFiL_b / Iy A[PITCHV, I_BR] = -La * kFiR_b / Iy # ------------------------------------------------------------------ # ROLL: Ix · Δroll̈ = Δτ_front + Δτ_back # # Unlike pitch (driven by force differential), roll is driven by # the electromagnetic TORQUE directly. In the Ansys model, torque # is the moment about the X axis produced by the asymmetric flux # in the left vs right legs of each U-yoke. # # The torque Jacobian entries determine stability: # - ∂T/∂roll: if this causes torque that amplifies roll → unstable # - ∂T/∂iL, ∂T/∂iR: how current asymmetry controls roll # ------------------------------------------------------------------ # Gap → roll acceleration A[ROLLV, GAP] = (kTg_f + kTg_b) / Ix # Pitch → roll acceleration (cross-coupling) A[ROLLV, PITCH] = (-kTg_f + kTg_b) * La / Ix # Roll → roll acceleration (roll stiffness) A[ROLLV, ROLL] = (kTr_f + kTr_b) / Ix # Current → roll acceleration A[ROLLV, I_FL] = kTiL_f / Ix A[ROLLV, I_FR] = kTiR_f / Ix A[ROLLV, I_BL] = kTiL_b / Ix A[ROLLV, I_BR] = kTiR_b / Ix # ------------------------------------------------------------------ # COIL DYNAMICS: L·di/dt = V·pwm − R·i # # Rearranged: di/dt = −(R/L)·i + (V/L)·pwm # # This is a simple first-order lag with: # - Time constant τ_coil = L/R = 2.5ms/1.1 = 2.27 ms # - Eigenvalue = −R/L = −440 (very fast, well-damped) # # The coil dynamics act as a low-pass filter between the PWM # command and the actual current. For PID frequencies below # ~100 Hz, this lag is small but not negligible. # ------------------------------------------------------------------ for k in range(I_FL, I_BR + 1): A[k, k] = -R / Lc # ------------------------------------------------------------------ # Step 5: B matrix (10 × 4) # # Only the coil states respond directly to the PWM inputs. # The mechanical states are affected INDIRECTLY: pwm → current # → force/torque → acceleration. This indirect path shows up # as the product A_mech_curr × B_curr_pwm in the transfer function. # # B[coil_k, pwm_k] = V_supply / L_coil # ------------------------------------------------------------------ B = np.zeros((10, 4)) for k in range(4): B[I_FL + k, k] = V / Lc # ------------------------------------------------------------------ # Step 6: C matrix (3 × 10) # # Default outputs are the three controlled DOFs: # gap_avg, pitch, roll # These are directly the position states. # ------------------------------------------------------------------ C = np.zeros((3, 10)) C[0, GAP] = 1.0 # gap_avg C[1, PITCH] = 1.0 # pitch C[2, ROLL] = 1.0 # roll # D = 0 (no direct feedthrough from PWM to position) D = np.zeros((3, 4)) # ------------------------------------------------------------------ # Step 7: Equilibrium check # # At a valid operating point, the total magnetic force should # equal the pod weight. A large residual means the linearization # is valid mathematically but not physically meaningful (the pod # wouldn't hover at this point without acceleration). # ------------------------------------------------------------------ F_total = plant_f.f0 + plant_b.f0 weight = m * GRAVITY eq_error = F_total - weight return StateSpaceResult( A=A, B=B, C=C, D=D, operating_point={ 'gap_height': gap_height, 'currL': currL, 'currR': currR, 'roll': roll, 'pitch': pitch, 'mass': m, }, equilibrium_force_error=eq_error, plant_front=plant_f, plant_back=plant_b, ) def find_equilibrium_current(self, gap_height, roll=0.0, tol=0.01): """ Find the symmetric current (currL = currR = I) that makes total force = weight at the given gap height. Uses bisection over the current range. The search assumes negative currents produce attractive (upward) force, which matches the Ansys model convention. Parameters ---------- gap_height : float Target gap [mm] roll : float Roll angle [deg], default 0 tol : float Force tolerance [N] Returns ------- float : equilibrium current [A] """ target_per_yoke = self.mass * GRAVITY / 2.0 def force_residual(curr): f, _ = self.linearizer.evaluate(curr, curr, roll, gap_height) return f - target_per_yoke # Bisection search over negative current range # (More negative = stronger attraction) a, b = -20.0, 0.0 fa, fb = force_residual(a), force_residual(b) if fa * fb > 0: # Try positive range too a, b = 0.0, 20.0 fa, fb = force_residual(a), force_residual(b) if fa * fb > 0: raise ValueError( f"Cannot find equilibrium current at gap={gap_height}mm. " f"Force at I=−20A: {target_per_yoke + force_residual(-20):.1f}N, " f"at I=0: {target_per_yoke + force_residual(0):.1f}N, " f"at I=+20A: {target_per_yoke + force_residual(20):.1f}N, " f"target per yoke: {target_per_yoke:.1f}N" ) for _ in range(100): mid = (a + b) / 2.0 fmid = force_residual(mid) if abs(fmid) < tol: return mid if fa * fmid < 0: b = mid else: a, fa = mid, fmid return (a + b) / 2.0 # ====================================================================== # Demo # ====================================================================== if __name__ == '__main__': model_path = os.path.join(os.path.dirname(__file__), 'maglev_model.pkl') lin = MaglevLinearizer(model_path) ss = MaglevStateSpace(lin) # ------------------------------------------------------------------ # Find the equilibrium current at the target gap # ------------------------------------------------------------------ TARGET_GAP_MM = 16.491741 # from lev_pod_env.py print("=" * 70) print("FINDING EQUILIBRIUM CURRENT") print("=" * 70) I_eq = ss.find_equilibrium_current(TARGET_GAP_MM) F_eq, T_eq = lin.evaluate(I_eq, I_eq, 0.0, TARGET_GAP_MM) print(f"Target gap: {TARGET_GAP_MM:.3f} mm") print(f"Pod weight: {ss.mass * GRAVITY:.3f} N ({ss.mass} kg)") print(f"Required per yoke: {ss.mass * GRAVITY / 2:.3f} N") print(f"Equilibrium current: {I_eq:.4f} A (symmetric, currL = currR)") print(f"Force per yoke at equilibrium: {F_eq:.3f} N") print(f"Equilibrium PWM duty cycle: {I_eq * ss.coil_R / ss.V_supply:.4f}") print() # ------------------------------------------------------------------ # Build the state-space at equilibrium # ------------------------------------------------------------------ result = ss.build( gap_height=TARGET_GAP_MM, currL=I_eq, currR=I_eq, roll=0.0, pitch=0.0, ) print(result) print() # ------------------------------------------------------------------ # Show the coupling structure # ------------------------------------------------------------------ result.print_A_structure() result.print_B_structure() # ------------------------------------------------------------------ # Physical interpretation of key eigenvalues # ------------------------------------------------------------------ eigs = result.eigenvalues unstable = result.unstable_eigenvalues print(f"\nUnstable modes: {len(unstable)}") for ev in unstable: # Time to double = ln(2) / real_part t_double = np.log(2) / np.real(ev) * 1000 # ms print(f" λ = {np.real(ev):+.4f} → amplitude doubles in {t_double:.1f} ms") print() print("The PID loop must have bandwidth FASTER than these unstable modes") print("to stabilize the plant.") # ------------------------------------------------------------------ # Gain schedule: how eigenvalues change with gap # ------------------------------------------------------------------ print("\n" + "=" * 70) print("GAIN SCHEDULE: unstable eigenvalues vs gap height") print("=" * 70) gaps = [8, 10, 12, 14, TARGET_GAP_MM, 18, 20, 25] header = f"{'Gap [mm]':>10} {'I_eq [A]':>10} {'λ_heave':>12} {'t_dbl [ms]':>12} {'λ_pitch':>12} {'t_dbl [ms]':>12}" print(header) print("-" * len(header)) for g in gaps: try: I = ss.find_equilibrium_current(g) r = ss.build(g, I, I, 0.0, 0.0) ue = r.unstable_eigenvalues real_ue = sorted(np.real(ue), reverse=True) # Typically: largest = heave, second = pitch lam_h = real_ue[0] if len(real_ue) > 0 else 0.0 lam_p = real_ue[1] if len(real_ue) > 1 else 0.0 t_h = np.log(2) / lam_h * 1000 if lam_h > 0 else float('inf') t_p = np.log(2) / lam_p * 1000 if lam_p > 0 else float('inf') print(f"{g:10.2f} {I:10.4f} {lam_h:+12.4f} {t_h:12.1f} " f"{lam_p:+12.4f} {t_p:12.1f}") except ValueError as e: print(f"{g:10.2f} (no equilibrium found)")