- Hybrid Parallel/Serial Kinematics
- Symbolic Jacobian Matrix
- Singularity Boundary Analysis
- Positive Cable Tension Optimization
ROBOTICS RESEARCH / MATHEMATICAL FRAMEWORK
Mathematical Framework for Singularities in Hybrid Cable Robots
Cable-driven parallel robots offer high accelerations and large workspaces, but cables can only pull, never push. In hybrid serial-cable architectures, determining controllable workspaces requires rigorous singularity analysis to detect where the Jacobian matrix loses rank and where cable tensions become non-positive.
Interactive Singularity Analysis Notebook (SymPy & NumPy)
A complete symbolic & numerical computational research notebook deriving closed vector loop kinematics, analytical Jacobian rank deficiency matrices, and quadratic programming for positive cable tensions across planar task space.
Closed-Loop Kinematics
Formulates coupled loop-closure kinematic constraints linking base cable winches, serial revolute coordinates, and planar platform pose [x, y, θ].
Symbolic Differential Mapping
Computes the analytical Jacobian matrix J(q) via SymPy, isolating actuation structure and cable unit direction vectors.
Singularity Boundary Locus
Maps forward, inverse, and architectural singularity manifolds where det(J) = 0 or singular values σ_min → 0.
Positive Cable Tension QP
Applies null-space projection and quadratic programming to verify that cable tensions remain strictly non-negative (τ_i > 0) at all non-singular points.
hybrid_cable_singularity_analysis.ipynbInteractive Kinematics & Singularity Solver# Symbolic Kinematics for Hybrid Planar Cable Robot:
# Closed loop: p_platform + R(theta) @ r_i - a_i = l_i * u_i
# Analytical Hybrid Jacobian J(q) = [ J_cables(q) | J_serial(q) ]
import sympy as sp
import numpy as np
# Evaluate determinant for singularity loci (det(J @ J^T) -> 0):
det_J = sp.simplify(J_hybrid.det())
singularity_surface = sp.solve(det_J, (x, y, theta))
# Quadratic Programming for strictly positive cable tensions (tau_i > 0):
# min 0.5 * tau^T @ H @ tau + f^T @ tau
# s.t. J_cables^T @ tau = -W_ext, tau_min <= tau <= tau_max
res_tension = solve_tension_distribution(J_cables, external_wrench, tau_min=5.0)Loop-closure kinematics
Establishes geometric relationships between base anchor points, actuated serial revolute joints, cable extension lengths, and the mobile platform's planar pose [x, y, θ].
Analytical Jacobian matrix derivation
Derives the differential mapping relating joint rates and cable spool velocities to platform task-space twists, separating structural serial constraints from cable tension directions.
Singularity & rank deficiency identification
Identifies forward, inverse, and architectural singularities where determinant vanishing leads to uncontrollable degrees of freedom or internal cable slackness.
Euler-Lagrange dynamic modeling
Models multi-body mass matrices, centrifugal/Coriolis effects, and gravitational torques, coupled with quadratic programming for tension allocation in real time.
Mathematical Specifications & Framework
- Platform Motion
- Planar 3-DOF workspace (surge, heave, and yaw angle θ)
- Actuation Topology
- Hybrid system: serial revolute joints + antagonistic cable winches
- Kinematics
- Non-linear closed vector loops solved analytically & symbolically
- Jacobian Analysis
- Matrix rank evaluation via symbolic algebra and singular value decomposition
- Environment
- Python, Jupyter Notebook, Google Colab, and LaTeX documentation
- Publication
- Based on peer-reviewed research (ajeassp.2022.197.208)
Research & Control Engineering Impact
Provides closed-form equations and executable Python/Jupyter verification models that ensure safe, singularity-free trajectory planning in robotics and rehabilitation exoskeleton systems.