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Verified Task-Space Motion Planning Under Joint-Space Constraints

Reactive task-space planners such as Bug2 operate with fixed Cartesian step sizes and are unaware of the manipulator's joint-angle limits. When the Jacobian is poorly conditioned, even small Cartesian steps can demand joint changes that exceed admissible bounds; clipping the joints to their limits causes tracking drift and can prevent goal reaching entirely. We address this by computing, at each planning step, the largest Cartesian hyperrectangle that is \emph{certifiably reachable} under joint displacement bounds. Using a second-order polynomial approximation of the inverse kinematics and the S-procedure, we formulate a small semidefinite program whose solution yields the certified half-width~$λ^\star$. An equivalent bisection procedure exploiting the quadratic structure solves the certification in sub-millisecond time. Integrating this certificate with Bug2 yields a planner whose step size adapts to local kinematic conditioning. In a statistical evaluation over 94 adversarial scenarios spanning six joint-limit settings, the SOS-verified planner achieves \emph{zero} joint-limit violations with a 100\% goal-reaching rate, whereas a standard Bug2 planner violates joint limits in 6--11\% of steps and fails to reach the goal in up to 18\% of scenarios.

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