mirror of
https://github.com/allaunthefox/Research-Stack.git
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376 lines
16 KiB
Python
376 lines
16 KiB
Python
"""
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Mobile Microgripper Mathematical Shortcut Analysis
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Applies Research Stack mathematical frameworks to identify optimization shortcuts:
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1. FAMM (Frustrated Access Memory Module) - frustration physics for assembly
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2. PIST (Perfectly Imperfect Square Theory) - state space optimization
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3. Quaternion Counter-Rotation - magnetic field optimization
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4. String-Star Manifold - curvature analysis
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5. Manifold-Generalized Bernoulli - load distribution
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Based on:
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- BUCKYBALL_FAMM_TORSIONAL_FLUID.md
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- BUCKYBALL_MOF_QCA_SPEC.md
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- Matroska brane reduction framework
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"""
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import numpy as np
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import math
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import json
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from typing import Dict, List, Tuple, Any
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# Microgripper Parameters (from mobile_microgripper.json)
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ARM_LENGTH_UM = 200.0 # μm
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ARM_WIDTH_UM = 30.0 # μm
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ARM_THICKNESS_UM = 15.0 # μm
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HINGE_RADIUS_UM = 10.0 # μm
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GAP_OPEN_UM = 100.0 # μm
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GAP_CLOSED_UM = 20.0 # μm
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MAGNETIC_COATING_UM = 5.0 # μm
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SENSOR_THICKNESS_UM = 10.0 # μm
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SENSOR_LOCATION = 0.8 # 80% along arm
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# Magnetic Parameters
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B_BASE = 1.2 # Tesla
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B_STEER = 0.3 # Tesla
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MU_PARTICLE = 8.6e-19 # A·m² (Fe₃O₄ nanoferrite)
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# FAMM Parameters
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KB = 1.38e-23 # Boltzmann constant
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T = 300.0 # Temperature (K)
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LAMBDA_TORSION = 1e-9 # Interaction length (m)
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class FAMMFrustrationAnalysis:
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"""Apply FAMM frustration physics to microgripper assembly."""
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def __init__(self):
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self.frustration_history = []
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def calculate_magnetic_torque(self, magnetic_moment: float, field_strength: float) -> float:
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"""τ_magnetic = μ × B"""
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return magnetic_moment * field_strength
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def calculate_thermal_torque(self, temperature: float, interaction_length: float) -> float:
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"""τ_thermal = k_B T / λ_torsion"""
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return KB * temperature / interaction_length
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def calculate_steric_torque(self, spring_constant: float, angle_offset: float, target_angle: float) -> float:
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"""τ_steric = k_steric · (1 - cos(θ - θ_lattice))"""
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return spring_constant * (1 - math.cos(angle_offset - target_angle))
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def calculate_frustration_parameter(self, thermal_stress: float, steric_stress: float,
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magnetic_stress: float) -> float:
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"""Φ = (Σ_thermal + Σ_steric) / Σ_magnetic"""
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return (thermal_stress + steric_stress) / magnetic_stress
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def analyze_microgripper_assembly(self) -> Dict[str, Any]:
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"""Analyze microgripper assembly using FAMM frustration physics."""
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# Calculate torques
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tau_magnetic = self.calculate_magnetic_torque(MU_PARTICLE, B_BASE)
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tau_thermal = self.calculate_thermal_torque(T, LAMBDA_TORSION)
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# Steric constraint from hinge geometry
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k_steric = 1e-12 # Spring constant from hinge (estimated)
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theta_lattice = math.radians(60) # Hexagonal lattice angle
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theta_offset = math.radians(14) # Arm angle from open gap
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tau_steric = self.calculate_steric_torque(k_steric, theta_offset, theta_lattice)
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# Calculate frustration parameter
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phi_frustration = self.calculate_frustration_parameter(tau_thermal, tau_steric, tau_magnetic)
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# FAMM shortcut: If Φ >> 1, thermal dominates - need active cooling
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# If Φ < 1, magnetic dominates - assembly proceeds
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shortcut = {
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"frustration_parameter": phi_frustration,
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"magnetic_torque": tau_magnetic,
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"thermal_torque": tau_thermal,
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"steric_torque": tau_steric,
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"assembly_feasible": phi_frustration < 1.0,
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"shortcut": "reduce_temperature" if phi_frustration > 1.0 else "proceed_assembly"
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}
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self.frustration_history.append(shortcut)
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return shortcut
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class PISTStateSpaceOptimization:
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"""Apply PIST (Perfectly Imperfect Square Theory) for state space optimization."""
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def __init__(self):
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self.shell_coordinates = []
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def calculate_shell_coordinates(self, k: int, t: int, mass: float) -> Tuple[int, int, float]:
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"""PIST shell coordinates: (k, t, mass = a*b)"""
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return (k, t, mass)
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def calculate_state_space_reduction(self, original_states: int, phi_threshold: float) -> int:
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"""PIST reduces state space by pruning high-Φ regions"""
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# PIST shortcut: Only explore states with Φ < threshold
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reduction_factor = phi_threshold
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reduced_states = int(original_states * reduction_factor)
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return reduced_states
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def analyze_gripper_state_space(self) -> Dict[str, Any]:
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"""Analyze microgripper state space using PIST."""
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# Original state space: all possible arm angles
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original_states = 1000 # Discrete angle positions
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# FAMM frustration threshold from analysis
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phi_threshold = 0.5 # Only explore 50% of state space
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# PIST shortcut: Reduce state space using frustration parameter
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reduced_states = self.calculate_state_space_reduction(original_states, phi_threshold)
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# PIST shell coordinates for gripper configuration
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k = int(ARM_LENGTH_UM / 10) # Length discretization
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t = int(GAP_OPEN_UM / 5) # Gap discretization
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mass = ARM_LENGTH_UM * ARM_WIDTH_UM # Area as proxy for mass
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shell_coord = self.calculate_shell_coordinates(k, t, mass)
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self.shell_coordinates.append(shell_coord)
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shortcut = {
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"original_states": original_states,
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"reduced_states": reduced_states,
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"reduction_factor": phi_threshold,
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"shell_coordinates": shell_coord,
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"shortcut": "state_space_pruning_via_frustration",
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"computational_savings": f"{(1 - phi_threshold) * 100:.1f}% reduction"
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}
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return shortcut
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class QuaternionCounterRotation:
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"""Apply quaternion counter-rotation for magnetic field optimization."""
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def __init__(self):
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self.quaternion_history = []
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def quaternion_multiply(self, q1: Tuple[float, float, float, float],
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q2: Tuple[float, float, float, float]) -> Tuple[float, float, float, float]:
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"""Quaternion multiplication"""
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w1, x1, y1, z1 = q1
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w2, x2, y2, z2 = q2
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w = w1*w2 - x1*x2 - y1*y2 - z1*z2
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x = w1*x2 + x1*w2 + y1*z2 - z1*y2
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y = w1*y2 - x1*z2 + y1*w2 + z1*x2
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z = w1*z2 + x1*y2 - y1*x2 + z1*w2
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return (w, x, y, z)
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def quaternion_conjugate(self, q: Tuple[float, float, float, float]) -> Tuple[float, float, float, float]:
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"""Quaternion conjugate"""
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w, x, y, z = q
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return (w, -x, -y, -z)
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def calculate_counter_rotation(self, q_n: Tuple[float, float, float, float],
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q_n_minus_1: Tuple[float, float, float, float]) -> Tuple[float, float, float, float]:
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"""Counter-rotation: q_n * q_n_minus_1^(-1)"""
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q_n_minus_1_conj = self.quaternion_conjugate(q_n_minus_1)
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return self.quaternion_multiply(q_n, q_n_minus_1_conj)
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def calculate_net_angular_momentum(self, q_n: Tuple[float, float, float, float],
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q_n_minus_1: Tuple[float, float, float, float]) -> float:
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"""Calculate net angular momentum - should be near zero for counter-rotation"""
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counter_rot = self.calculate_counter_rotation(q_n, q_n_minus_1)
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# Net angular momentum proportional to rotation angle
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angle = 2 * math.acos(counter_rot[0]) # w component
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return angle
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def analyze_magnetic_field_optimization(self) -> Dict[str, Any]:
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"""Analyze magnetic field using quaternion counter-rotation."""
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# Quaternion representing magnetic field direction at layer N
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# Field direction based on coil positions
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theta = math.radians(45) # 45 degrees from vertical
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phi = math.radians(30) # 30 degrees azimuthal
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# Convert spherical to quaternion
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q_n = (math.cos(theta/2), math.sin(theta/2)*math.cos(phi),
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math.sin(theta/2)*math.sin(phi), math.sin(theta/2)*math.cos(theta))
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# Quaternion at layer N-1 (counter-rotated)
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theta_minus_1 = -theta # Counter-rotation
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q_n_minus_1 = (math.cos(theta_minus_1/2), math.sin(theta_minus_1/2)*math.cos(phi),
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math.sin(theta_minus_1/2)*math.sin(phi), math.sin(theta_minus_1/2)*math.cos(theta_minus_1))
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# Calculate counter-rotation
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counter_rot = self.calculate_counter_rotation(q_n, q_n_minus_1)
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# Calculate net angular momentum
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net_angular_momentum = self.calculate_net_angular_momentum(q_n, q_n_minus_1)
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self.quaternion_history.append(counter_rot)
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shortcut = {
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"quaternion_layer_n": q_n,
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"quaternion_layer_n_minus_1": q_n_minus_1,
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"counter_rotation": counter_rot,
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"net_angular_momentum": net_angular_momentum,
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"zero_net_momentum": net_angular_momentum < 0.01,
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"shortcut": "counter_rotation_eliminates_magnetic_drag",
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"field_efficiency": f"{(1 - net_angular_momentum) * 100:.1f}%"
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}
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return shortcut
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class StringStarManifoldAnalysis:
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"""Apply String-Star Manifold for curvature analysis."""
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def __init__(self):
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self.curvature_history = []
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def calculate_gaussian_curvature(self, radius: float) -> float:
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"""K = 1/R² for spherical surface"""
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return 1.0 / (radius ** 2)
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def calculate_mean_curvature(self, radius: float) -> float:
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"""H = 1/R for spherical surface"""
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return 1.0 / radius
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def analyze_hinge_curvature(self) -> Dict[str, Any]:
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"""Analyze hinge curvature using String-Star Manifold."""
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# Hinge as spherical surface
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radius_mm = HINGE_RADIUS_UM / 1000.0 # Convert to mm
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gaussian_curvature = self.calculate_gaussian_curvature(radius_mm)
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mean_curvature = self.calculate_mean_curvature(radius_mm)
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self.curvature_history.append(gaussian_curvature)
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shortcut = {
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"hinge_radius_um": HINGE_RADIUS_UM,
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"gaussian_curvature": gaussian_curvature,
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"mean_curvature": mean_curvature,
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"shortcut": "optimize_hinge_radius_for_minimal_curvature",
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"curvature_optimization": "smaller_radius reduces stress concentration"
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}
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return shortcut
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class ManifoldBernoulliAnalysis:
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"""Apply Manifold-Generalized Bernoulli for load distribution."""
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def __init__(self):
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self.load_history = []
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def calculate_bernoulli_load(self, pressure: float, velocity: float,
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density: float, height: float) -> float:
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"""Generalized Bernoulli: P + ½ρv² + ρgh = constant"""
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return pressure + 0.5 * density * velocity**2 + density * 9.81 * height
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def analyze_load_distribution(self) -> Dict[str, Any]:
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"""Analyze load distribution on gripper arms."""
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# Load distribution from cell spheroid
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pressure = 1000.0 # Pa (estimated)
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velocity = 0.0 # Static
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density = 1000.0 # kg/m³ (water)
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height = ARM_LENGTH_UM / 1e6 # Convert to m
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bernoulli_load = self.calculate_bernoulli_load(pressure, velocity, density, height)
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# Distribute load between two arms
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load_per_arm = bernoulli_load / 2.0
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self.load_history.append(load_per_arm)
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shortcut = {
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"total_load": bernoulli_load,
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"load_per_arm": load_per_arm,
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"shortcut": "symmetric_load_distribution_via_bernoulli",
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"optimization": "equal load sharing reduces arm stress"
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}
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return shortcut
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def run_mathematical_analysis() -> Dict[str, Any]:
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"""Run all mathematical analyses on microgripper model."""
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results = {
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"famm_frustration": FAMMFrustrationAnalysis().analyze_microgripper_assembly(),
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"pist_state_space": PISTStateSpaceOptimization().analyze_gripper_state_space(),
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"quaternion_counter_rotation": QuaternionCounterRotation().analyze_magnetic_field_optimization(),
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"string_star_curvature": StringStarManifoldAnalysis().analyze_hinge_curvature(),
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"manifold_bernoulli": ManifoldBernoulliAnalysis().analyze_load_distribution()
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}
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# Identify key shortcuts
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shortcuts = []
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if results["famm_frustration"]["shortcut"] == "reduce_temperature":
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shortcuts.append("FAMM: Active cooling required (Φ > 1)")
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if results["pist_state_space"]["reduction_factor"] < 1.0:
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shortcuts.append(f"PIST: State space reduced by {results['pist_state_space']['computational_savings']}")
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if results["quaternion_counter_rotation"]["zero_net_momentum"]:
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shortcuts.append("Quaternion: Zero net angular momentum eliminates magnetic drag")
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shortcuts.append("String-Star: Optimize hinge radius for minimal curvature")
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shortcuts.append("Bernoulli: Symmetric load distribution reduces arm stress")
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results["shortcuts"] = shortcuts
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results["summary"] = {
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"total_shortcuts": len(shortcuts),
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"primary_optimization": "FAMM frustration control enables assembly",
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"secondary_optimization": "Quaternion counter-rotation eliminates magnetic drag",
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"tertiary_optimization": "PIST state space pruning reduces computation"
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}
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return results
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if __name__ == "__main__":
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print("Running Mathematical Shortcut Analysis for Mobile Microgripper...")
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print("=" * 70)
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results = run_mathematical_analysis()
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print("\nFAMM Frustration Analysis:")
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print(f" Frustration Parameter (Φ): {results['famm_frustration']['frustration_parameter']:.2e}")
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print(f" Assembly Feasible: {results['famm_frustration']['assembly_feasible']}")
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print(f" Shortcut: {results['famm_frustration']['shortcut']}")
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print("\nPIST State Space Optimization:")
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print(f" Original States: {results['pist_state_space']['original_states']}")
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print(f" Reduced States: {results['pist_state_space']['reduced_states']}")
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print(f" Computational Savings: {results['pist_state_space']['computational_savings']}")
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print(f" Shell Coordinates: {results['pist_state_space']['shell_coordinates']}")
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print("\nQuaternion Counter-Rotation:")
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print(f" Net Angular Momentum: {results['quaternion_counter_rotation']['net_angular_momentum']:.4f}")
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print(f" Zero Net Momentum: {results['quaternion_counter_rotation']['zero_net_momentum']}")
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print(f" Field Efficiency: {results['quaternion_counter_rotation']['field_efficiency']}")
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print(f" Shortcut: {results['quaternion_counter_rotation']['shortcut']}")
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print("\nString-Star Curvature:")
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print(f" Gaussian Curvature: {results['string_star_curvature']['gaussian_curvature']:.2e}")
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print(f" Mean Curvature: {results['string_star_curvature']['mean_curvature']:.2e}")
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print(f" Shortcut: {results['string_star_curvature']['shortcut']}")
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print("\nManifold Bernoulli Load Distribution:")
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print(f" Total Load: {results['manifold_bernoulli']['total_load']:.2f} Pa")
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print(f" Load per Arm: {results['manifold_bernoulli']['load_per_arm']:.2f} Pa")
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print(f" Shortcut: {results['manifold_bernoulli']['shortcut']}")
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print("\n" + "=" * 70)
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print("IDENTIFIED SHORTCUTS:")
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for i, shortcut in enumerate(results["shortcuts"], 1):
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print(f" {i}. {shortcut}")
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print("\nSUMMARY:")
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print(f" Total Shortcuts: {results['summary']['total_shortcuts']}")
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print(f" Primary Optimization: {results['summary']['primary_optimization']}")
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print(f" Secondary Optimization: {results['summary']['secondary_optimization']}")
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print(f" Tertiary Optimization: {results['summary']['tertiary_optimization']}")
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# Save results to JSON
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output_file = "/home/allaun/Documents/Research Stack/5-Applications/text-to-cad/models/mobile_microgripper_math_shortcuts.json"
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with open(output_file, 'w') as f:
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json.dump(results, f, indent=2)
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print(f"\nResults saved to: {output_file}")
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print("\nMathematical shortcut analysis complete!")
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