Anonymous Supplementary Material
This page presents the physical-robot experiments accompanying the manuscript, arranged in the order of the Validation section: NA-FTC on a seven-DOF Franka Emika Panda (Cases 1–5) and on a six-DOF UR5 (Cases 1–2), a hardware comparison with three literature controllers, two ablations, and partial-loss-of-effectiveness trials. In every video, the scheduled joints are locked (or weakened) at preset instants while the controller receives no prior fault information and can learn about the fault only from the real arm's command–response mismatch. The quantity to observe is whether the mission-defined priority channel — roll/pitch attitude in Franka Cases 1–3 and UR5 Case 1, or end-effector x/y/z position in Franka Cases 4–5 and UR5 Case 2 — stays protected while the lower-priority channels are released.
Seven-DOF Panda commanded through its 1 kHz joint-position interface. Each 40 s trial contains three sequential locks; after the second lock only five joints remain, so the six-dimensional pose task is structurally infeasible.
Case 1 · Roll/Pitch Priority
The end-effector tracks a horizontal XY circle with a radius of 0.13 m while carrying a liquid-filled beaker.
Fault sequence: joints 1, 7, and 4 become stuck at 10/20/30 s. NA-FTC estimates the loss of joint effectiveness from measured command-response signals and prioritizes the roll/pitch attitude of the beaker as the available motion directions decrease.
Case 2 · Roll/Pitch Priority
The end-effector tracks a vertical circle with a radius of 0.15 m, stressing velocity reversals and the coupling between yaw and position.
Fault sequence: joints 3, 5, and 6 become stuck at 10/20/30 s. Lower-priority position or yaw tracking may degrade after the faults, while the controller keeps the safety-critical roll/pitch attitude bounded. The largest priority peak of the five cases (0.11 rad) occurs here when joint 5 locks in the middle of a fast velocity reversal.
Case 3 · Roll/Pitch Priority
The robot performs a low-arc round-trip transfer while carrying the liquid-filled beaker.
Fault sequence: joints 2, 4, and 3 become stuck at 12.5/22.5/32.5 s. This is the representative roll/pitch-priority task reused in the comparison, ablation, and partial-loss experiments below. NA-FTC suppresses lower-priority motion when necessary and completes the full multi-fault transfer without triggering the robot's safety protection; the roll/pitch error remains within 0.026 rad while the position is released to decimeter scale.
Case 4 · Position Priority
The tool center point traces a 0.15 m-diameter circle on a virtual wall (vertical YZ plane, tool axis horizontal, one lap per 5 s). Unlike Cases 1–3, this case protects the end-effector x/y/z position rather than roll and pitch, which is the natural choice when a tool tip must stay on its path while the tool attitude may drift.
Fault sequence: joints 7, 5, and 6 become stuck at 10/20/30 s. The first fault in joint 7 has little impact on tool-center-point tracking. After the second fault in joint 5, orientation tracking begins to deviate so that position tracking can be maintained. After the third fault in joint 6 the orientation error continues to increase, while position tracking remains stable: the position error stays within 3.6 mm throughout all three locks.
Case 5 · Position Priority
The tool center point traces a 0.30 m-diameter circle on a virtual floor (horizontal XY plane, tool pointing down, one lap per 10 s), again protecting the x/y/z position channel.
Fault sequence: joints 1, 7, and 5 become stuck at 10/20/30 s. The first fault in joint 1 has little impact on the trajectory. After the second fault in joint 7, the controller sacrifices yaw control to maintain position tracking. After the third fault in joint 5, orientation control is completely relinquished while the position error stays within 3.3 mm. This is the representative position-priority task reused in the comparison and partial-loss experiments below.
To test transfer to a different kinematic structure and number of DOFs, the same controller is deployed on a six-DOF UR5 commanded at 100 Hz. Because a single lock leaves only five actuated joints, the full six-dimensional task is infeasible after the first lock.
UR5 Case 1 · Roll/Pitch Priority
Following the Franka Case 3 trajectory, the end-effector performs the low-arc liquid-transfer motion while a dexterous hand holds a liquid-filled beaker. The trial lasts 30 s.
Fault sequence: joint 3 (elbow) and joint 2 (shoulder lift) become stuck at 12.5/22.5 s. Unlike the Franka Panda, the UR5 no longer has sufficient joints to maintain full six-DOF end-effector control once one joint fails. A noticeable position-tracking error appears after the first fault and increases significantly after the second, while roll and pitch remain highly stable (peaks of 0.053 rad and 0.029 rad at the two locks, mean 2.1 mrad).
UR5 Case 2 · Position Priority
Similar to Franka Case 4, the UR5 tracks a 0.15 m-radius circular end-effector trajectory in the YZ plane (one lap per 5 s) with a superimposed roll oscillation. This task prioritizes stable position tracking along the x, y, and z axes.
Fault sequence: joint 3 (elbow) and joint 5 (wrist 2) become stuck at 10/20 s. The elbow carries most of the vertical authority on the UR5, so a 0.12 m transient (mostly along z) appears in the 1.5 s after the first lock while the estimator gathers command–response evidence; the wrist then undergoes substantial twisting to maintain position tracking, and the position error returns below 0.045 m for the rest of the trial. After the second fault the end-effector orientation deviates noticeably, but the target position continues to be tracked.
Franka Cases 3 and 5 are repeated on the same robot with three kinematic-level controllers from the literature under identical references, fault schedules, sampling, joint and velocity limits, and saturation chain. Each video shows the four runs side by side: SRTP (top left), FT-QP (top right), FT-QP/ZNN (bottom left), and NA-FTC (bottom right).
Comparison · Franka Case 3
SRTP is the singularity-robust task-priority inverse kinematics of Chiaverini, which keeps the healthy Jacobian and receives no fault information. FT-QP is the velocity-level fault-tolerant quadratic program of Li–Zhang, and FT-QP/ZNN is the time-varying fault-tolerant program of Yang solved by an LVI-based zeroing neural network; both quadratic-program methods are given the true failed-joint identity and fault instant. NA-FTC, like SRTP, operates without prior fault information.
Fault sequence: joints 2, 4, and 3 become stuck at 12.5/22.5/32.5 s in all four runs. All four controllers complete the 40 s trial. SRTP continuously allocates through the healthy Jacobian, so part of its command remains assigned to locked joints and its roll/pitch error rises to 0.086 rad after the locks. FT-QP, although told which joint has failed, has no mechanism to give up the lower-priority task and reaches 0.082 rad. FT-QP/ZNN keeps the roll/pitch peak at 0.018 rad but releases the position error to 0.54 m during the single-fault phase. NA-FTC keeps roll/pitch within 0.026 rad with the lowest RMS priority error and completes the transfer.
Comparison · Franka Case 5
The same three literature controllers repeat the position-priority floor-circle task of Franka Case 5.
Fault sequence: joints 1, 7, and 5 become stuck at 10/20/30 s in all four runs. NA-FTC keeps the protected position error within 3.3 mm (RMS 0.6 mm) for the full 40 s. SRTP reaches a 0.040 m maximum position error. FT-QP/ZNN reaches 0.15 m. FT-QP releases the protected position to 0.30 m and its run is terminated by the robot's protective stop at 36.7 s.
Both ablations use the same low-arc trajectory and fault schedule as Franka Case 3.
Ablation A
The complete NA-FTC controller is disabled, leaving only open-loop inverse kinematics.
After the first joint fault, the nominal Jacobian no longer represents the damaged robot and the end-effector drifts away from the desired motion. The trial runs for 40 s, but task tracking is not properly regulated.
Ablation B
Neural-adaptive feedback and RBF neural compensation are enabled, but online joint-efficiency estimation and fault-aware control allocation are disabled.
The controller maintains partial tracking after one fault. After the second fault, only five controllable degrees of freedom remain for the six-dimensional pose task; the motion diverges and the Franka safety mechanism terminates the experiment at 24.94 s. In contrast, the complete NA-FTC architecture in Case 3 finishes the 40 s multi-fault task.
The schedules of Franka Cases 3 and 5 are repeated with the scheduled joints losing 50 % or 80 % of their effectiveness (λ = 0.5 or 0.2) instead of locking, at the same joints and instants. The estimator converges to the injected residual efficiencies in all four trials, and whether the lower-priority channel must be released is decided by the remaining authority rather than by the fault class.
Franka Case 3 · λ = 0.5
Franka Case 3 · λ = 0.2
Franka Case 5 · λ = 0.5
Franka Case 5 · λ = 0.2