Carrying heavy loads costs additional energy during walking and leads to fatigue of the user. Conventionally, the load is fixed on the body. Some recent studies showed energy cost reduction when the relative motion of the load with respect to the body was allowed. However, the influences of the load's relative motion on the user are still not fully understood. We employed an optimization-based biped model which can generate human-like walking motion to study the load-carrier interaction. The relative motion can be achieved by a passive mechanism (such as springs) or a powered mechanism (such as actuators), and the relative motion can occur in the vertical or fore-aft directions. The connection between the load and body is added to the biped model in four scenarios (two types × two directions). The optimization results indicate that the stiffness values affect energy cost differently and the same stiffness value in different directions may have opposite effects. Powered relative motion in either direction can potentially reduce energy cost but the vertical relative motion can achieve a higher reduction than fore-aft relative motion. Surprisingly, powered relative motion only performs marginally better than the passive conditions at similar peak interaction force levels. This work provides insights into developing more economical load-carrying methods and the model presented may be applied to the design and control of wearable load-carrying devices.