## 2 Materials and Methods  
### Data Sources and Preprocessing  
This study leveraged data from a previously published experimental protocol evaluating a semi-rigid bilateral hip exoskeleton designed to assist hip extension during treadmill walking [4]. Ten healthy adults participated (4 males, 6 females; mean age: 27.6 \(\pm\) 5.9 years; body mass: 65.3 \(\pm\) 13.1 kg; height: 1.66 \(\pm\) 0.08 m). Participants walked at a fixed speed of 1.25 m/s on a split-belt instrumented treadmill (Bertec, Columbus, OH, USA). Kinematic data were recorded at 120 Hz using a Vicon motion capture system (Vicon Motion Systems Inc., Oxford, UK), and kinetic data were collected at 1000 Hz via integrated force plates. The hip exoskeleton was actuated by an off-board motor connected through a series-elastic interface and controlled in real-time using a Simulink-based system (SpeedGoat, MathWorks, Natick, MA, USA). A temporal force-tracking controller applied extension torque profiles to the hip joint, with torque onset fixed at 90% of the stride cycle (just before heel strike) and peak torque occurring at 17% (early stance). The duration of torque application varied systematically across conditions by adjusting the end-timing of assistance, ranging from 21% to 49% of the stride cycle. Twelve experimental conditions were tested, comprising ten powered assistance trials, a passive condition (PowerOff, where participants wore the exoskeleton without actuation), and an unassisted condition (NoExo, where the exoskeleton was not worn). The powered conditions combined two torque magnitudes--low (\(\sim\)0.06 Nm/kg) and high (\(\sim\)0.12 Nm/kg)--with five end-timings, labeled as LowEarly, LowEarlier, LowMid, LowLater, LowLatest, HighEarly, HighEarlier, HighMid, HighLater, and HighLatest. For example, LowEarly delivered low-magnitude torque with an end-timing at approximately 21% of the stride cycle, while HighLatest provided high-magnitude torque ending near 49%. The NoExo and PowerOff conditions served as controls, isolating the effects of natural walking and the exoskeleton's passive mass (5.77 kg). Gait data were segmented into strides based on heel strike events detected from ground reaction forces and time-normalized to 100 points representing the whole gait cycle (0-100%). Center of mass (COM) velocity trajectories were computed in the anteroposterior (Vx), vertical (Vy), and mediolateral (Vz) directions using kinematic data. Velocities were converted from meters per second to millimeters per second to enhance numerical resolution for variability analyses, consistent with gait tube visualization requirements [4]. For each condition, average COM velocity trajectories were calculated across valid strides per participant. Trials with fewer than five clean strides were excluded to ensure reliable variability estimates. Clean strides were defined as those free of marker loss, tracking artifacts, or irregular timing patterns that deviated more than \(\pm\)2 standard deviations from the subject's average stride time [4]. Strides with inconsistent COM trajectories or discontinuities in velocity profiles were excluded based on visual inspection. This ensured that each included stride reflected typical, uninterrupted walking behavior under the given condition. This preprocessing enabled a systematic investigation of how assistance magnitude and timing variations influenced gait mechanics across all twelve conditions. Participants were included if they were between 18 and 40 years of age, had no known neuromuscular or musculoskeletal impairments, and were capable of walking unaided for prolonged periods. Exclusion criteria included any recent lower limb injury, neurological disorder, cardiovascular condition, or prior use of wearable robotic devices that could interfere with gait mechanics. These criteria ensured the safety of participants and the consistency of gait measurements under exoskeleton conditions, as detailed in the original data collection protocol [1].