A Convenient Approximation of the Projectile Motion with Quadratic Air Resistance

A convenient approximated analytic solution is proposed for the problem of the motion of a body under a resistive force, acting in the magnitude of the squared velocity of the body. This solution is an explicit function of time, that keeps a good behavior both near the initial state and far from the initial state. To obtain a general analytic solution, we firstly used a reduction principle to be able to manipulate scalar objects, and we analyzed limit behaviors, both near the initial state and far from the initial state. Secondly, we proposed an approximated analytic solution with heuristics based on the built knowledge. Finally, a robust and stable integration scheme is proposed, based on the obtained analytic solution. We compared the scheme with other standard integration schemes.

Design and Experimental Evaluation of an Electrorheological Haptic Module with Embedded Sensing

We present a miniature haptic module based on electrorheological fluid, designed for conveying combined stiffness and vibrotactile sensations at a small scale. Haptic feedback is produced through electrorheological fluid’s controllable resistive force and varies with the actuator’s deformation. To demonstrate the proposed actuator’s feedback in realistic applications, a method for measuring the actuator’s deformation must be implemented for active control. To this end, in this study, we incorporate a sensor design based on a bend-sensitive resistive film to the ER haptic actuator. The combined actuator and sensor module was tested for its ability to simultaneously actuate and sense the actuator’s state under indentation. The results show that the bend sensor can accurately track the actuator’s displacement over its stroke. Thus, the proposed sensor may enable control of the output resistive force according to displacement, which may lead to more informed and engaging combined kinesthetic and tactile feedback.

The metabolic cost of overcoming air resistive forces in distance running

We lack a mechanistic understanding of the relationship between aerodynamic drag forces and metabolic power during running. Further, the energetic and time savings possible from reducing aerodynamic drag (drafting) are still unclear due to the different methods previously assumed for converting from force reductions to metabolic power savings. Here, we quantified how small horizontal impeding forces (equivalent to aerodynamic forces) affect metabolic power and ground reaction forces over a range of velocities in competitive runners. In three sessions, 12 runners completed six 5-minute trials with 5 minutes of recovery in-between. We tested one velocity per session (12, 14 and 16 km/h), at three horizontal impeding force conditions (0, 4 and 8 N). On average, metabolic power increased by 6.13% per 1% body weight of horizontal impeding force but varied considerably between individuals. With greater horizontal impeding force, braking impulses decreased while propulsive impulses increased (p < 0.001). Across running velocities, the changes in braking and propulsive impulses with greater impeding force were correlated (r = -0.97; p < 0.001), but were not related to individual changes in metabolic power. We estimate that at ∼2-hour marathon pace, overcoming air resistive force comprises 8.52% of the gross metabolic power on average.

Modeling of Resistive Forces and Buckling Behavior in Variable Recruitment Fluidic Artificial Muscle Bundles

Fluidic artificial muscles (FAMs), also known as McKibben actuators, are a class of fiber-reinforced soft actuators that can be pneumatically or hydraulically pressurized to produce muscle-like contraction and force generation. When multiple FAMs are bundled together in parallel and selectively pressurized, they can act as a multi-chambered actuator with bioinspired variable recruitment capability. The variable recruitment bundle consists of motor units (MUs)—groups of one of more FAMs—that are independently pressurized depending on the force demand, similar to how groups of muscle fibers are sequentially recruited in biological muscles. As the active FAMs contract, the inactive/low-pressure units are compressed, causing them to buckle outward, which increases the spatial envelope of the actuator. Additionally, a FAM compressed past its individual free strain applies a force that opposes the overall force output of active FAMs. In this paper, we propose a model to quantify this resistive force observed in inactive and low-pressure FAMs and study its implications on the performance of a variable recruitment bundle. The resistive force behavior is divided into post-buckling and post-collapse regions and a piecewise model is devised. An empirically-based correction method is proposed to improve the model to fit experimental data. Analysis of a bundle with resistive effects reveals a phenomenon, unique to variable recruitment bundles, defined as free strain gradient reversal.