Biomechanics
BIOMECHANICAL ANALYSIS
Advanced Framework for Human Motion Analysis
CORE CAPABILITIES
Research-grade biomechanical analysis
with proven mathematical foundations.
The Graffiti biomechanical analysis framework provides comprehensive tools for human motion analysis, featuring advanced algorithms for joint kinematics, dynamics, and sport-specific performance evaluation. Built on proven biomechanical principles, it delivers research-grade analysis capabilities.
PIPELINE ARCHITECTURE
Hierarchical analysis structure
for comprehensive motion evaluation
Scene Analysis
Motion Analysis
Biomechanics
Sport-Specific
Step 1Pose & Tracking
Multi-person detection
Keypoint extraction, temporal smoothing using Savitzky-Golay filtering for accurate motion tracking.
Step 2Kinematic Analysis
Joint angle calculation
Velocity and acceleration computation, movement pattern recognition with quaternion-based calculations.
Step 3Dynamic Analysis
Force estimation
Moment calculation, ground reaction force modeling using Newton-Euler formulation.
Step 4Performance Metrics
Energy analysis
Stability assessment, sport-specific metrics calculation with comprehensive reporting.
MATHEMATICAL FOUNDATIONS
Proven algorithms and formulations
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Joint Position & Tracking
For each joint j in frame t, we define the position vector with confidence:
$$ P_j(t) = \{x_j(t), y_j(t), z_j(t), c_j(t)\} $$
Movement tracking between consecutive frames with displacement and velocity calculations.
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Joint Angle Calculations
Three-dimensional angles computed using quaternion algebra to avoid gimbal lock:
$$ \theta = \arccos\left(\frac{\vec{v_1} \cdot \vec{v_2}}{\|\vec{v_1}\| \|\vec{v_2}\|}\right) $$
Euler angle extraction from quaternions for complete 3D orientation analysis.
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Dynamic Analysis
Force and moment calculations using Newton-Euler formulation:
$$ \vec{F}_i = m_i(\vec{a}_i + \vec{g}) $$
$$ \vec{M}_i = I_i\vec{\alpha}_i + \vec{\omega}_i \times (I_i\vec{\omega}_i) $$
Complete inverse dynamics solution for joint forces and moments throughout motion.
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Energy & Power Analysis
Kinetic and potential energy calculations with instantaneous power:
$$ E_k = \frac{1}{2}m\|\vec{v}\|^2 + \frac{1}{2}I\omega^2 $$
$$ P(t) = \vec{F}(t) \cdot \vec{v}(t) + \vec{M}(t) \cdot \vec{\omega}(t) $$
Work-energy analysis and mechanical power computation throughout movement phases.
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Stability Metrics
Center of mass calculation and dynamic stability assessment:
$$ \text{CoM} = \frac{\sum_{i=1}^n m_i\vec{r_i}}{\sum_{i=1}^n m_i} $$
$$ S_d = 1 - \frac{\sum_{t=1}^{T-1}\|\text{CoM}(t+1) - \text{CoM}(t)\|}{T \cdot d_\text{max}} $$
Multi-segment body model with dynamic stability indices for balance assessment.
IMPLEMENTATION
Advanced algorithms and code examples
class KinematicsAnalyzer:
def __init__(self, filter_window=5, filter_order=3):
self.filter_window = filter_window
self.filter_order = filter_order
def calculate_joint_angles(self, pose_data):
"""Calculate 3D joint angles using quaternion algebra"""
angles = {}
for joint_name, vectors in self._get_joint_vectors(pose_data):
angle = self._quaternion_angle(vectors)
angles[joint_name] = angle
return angles
class DynamicsAnalyzer:
def calculate_ground_reaction_forces(self, kinematics):
"""Estimate GRF using spring-mass model"""
grf_data = []
for frame in kinematics.contact_phases:
vertical_force = self._spring_mass_model(
frame.center_of_mass,
frame.velocity,
frame.acceleration
)
grf_data.append(vertical_force)
return grf_data
class StrideAnalyzer:
def analyze_stride_patterns(self, ankle_trajectory):
"""Detect stride events using ankle kinematics"""
contact_events = []
# Zero-crossing detection for stance/swing
velocity = np.gradient(ankle_trajectory)
zero_crossings = self._find_zero_crossings(velocity)
for crossing in zero_crossings:
if self._validate_contact_event(crossing):
contact_events.append(crossing)
return self._calculate_stride_metrics(contact_events)
class SynchronizationAnalyzer:
def analyze_movement_coordination(self, joint_data):
"""Cross-correlation analysis with DTW alignment"""
coordination_metrics = {}
for joint_pair in self.joint_pairs:
signal1 = joint_data[joint_pair[0]]
signal2 = joint_data[joint_pair[1]]
# Dynamic Time Warping alignment
aligned_signals = self._dtw_align(signal1, signal2)
correlation = self._cross_correlation(aligned_signals)
coordination_metrics[joint_pair] = correlation
return coordination_metrics
APPLICATIONS
Real-world biomechanical analysis
Sprint Biomechanics
Injury Prevention
Performance Optimization
Technique Analysis
"The biomechanical analysis framework provides research-grade accuracy with real-time processing capabilities. Essential for performance analysis."
Dr. Sarah Johnson
Sports Biomechanist, Olympic Training Center
"Quaternion-based joint calculations eliminate gimbal lock issues we've struggled with in other systems. Mathematical rigor is impressive."
Prof. Michael Chen
Biomechanics Research Lab, Stanford University
"The force estimation without force plates opens new possibilities for field-based analysis. Game-changing technology."
Dr. Emma Rodriguez
Lead Researcher, Sports Science Institute