654 lines
27 KiB
Python
654 lines
27 KiB
Python
|
"""
|
|||
|
|
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)")
|