The quadriceps are particularly susceptible to fatigue during repetitive lifting, lowering, and carrying (LLC), affecting worker performance, posture, and ultimately lower-back injury risk. Although robotic exoskeletons have been developed and optimized for specific use cases like lifting-lowering, their controllers lack the versatility or customizability to target critical muscles across many fatiguing tasks. Here, we present a task-adaptive knee exoskeleton controller that automatically modulates virtual springs, dampers, and gravity and inertia compensation to assist squatting, level walking, and ramp and stairs ascent/descent. Unlike end-to-end neural networks, the controller is composed of predictable, bounded components with interpretable parameters that are amenable to data-driven optimization for biomimetic assistance and subsequent application-specific tuning, for example, maximizing quadriceps assistance over multiterrain LLC. When deployed on a backdrivable knee exoskeleton, the assistance torques holistically reduced quadriceps effort across multiterrain LLC tasks (significantly except for level walking) in 10 human users without user-specific calibration. The exoskeleton also significantly improved fatigue-induced deficits in time-based performance and posture during repetitive lifting-lowering. Last, the system facilitated seamless task transitions and garnered a high effectiveness rating postfatigue over a multiterrain circuit. These findings indicate that this versatile control framework can target critical muscles across multiple tasks, specifically mitigating quadriceps fatigue and its deleterious effects.