Taylor-free exact-trigonometric dynamic modeling and PSO-tuned adaptive super-twisting control of a two-section spatial continuum robot: a simulation study

Cable-driven continuum robots are difficult to model and control because of their strongly coupled spatial deformation and configuration-dependent tendon actuation. This simulation study develops a control-oriented dynamic model of a two-section spatial continuum robot in which the piecewise-constant-curvature geometry is evaluated with exact sine and cosine functions instead of Taylor polynomials; the Jacobians and inertia derivatives of this exact geometry are computed by central finite differences, whose relative error was verified to be below 1 0 − 9 against complex-step derivatives. The generalized coordinates q = [ θ 1 , ϕ 1 , θ 2 , ϕ 2 ] T let both bending angles and bending-plane orientations evolve dynamically. A particle-swarm-optimization-tuned adaptive super-twisting sliding-mode controller (PSO-ASTSMC) is proposed, and a strict Lyapunov function establishes finite-time convergence of its sliding variables for perturbations whose rate satisfies a computable linear-matrix-inequality bound. In simulation, PSO-ASTSMC ranked first among six controllers (PID, computed-torque PD, SMC, fast terminal SMC, and fixed-gain STSMC) on the two tuning trajectories (joint/Cartesian RMSE 1.628 ° /14.328 mm and 1.918 ° /13.631 mm) and on two trajectories not used for tuning, reducing the Cartesian RMSE by 27%–51% at comparable RMS effort; twenty independent PSO runs reproduced this result (coefficient of variation ≤ 1.5 % ). Compared with a second-order Taylor-expanded model, the exact geometry removed about 20 mm of open-loop prediction error at no computational cost. A robustness analysis showed that, with gains tuned on the nominal model, all six controllers fail when the stiffness or damping of the robot differs from the model by about 10%, because the generalized inertia of the robot is small. After a common uncertainty-aware retuning, PSO-ASTSMC was the only controller without failures over 40 random plants with ± 15 % stiffness and damping errors (median post-transient tip error 0.59 mm), at the cost of a 500-Hz control loop and a control signal that is sensitive to measurement noise. Disturbance tests with section-wise forces, tendon follower forces, and repeated pulses showed complete tip recovery when sufficient time was available. All results are numerical: the reported margins are simulated and require experimental confirmation.