The arrival time problem for the free particle in one dimension may be formulated as the problem of determining a joint probability for the particle being found on opposite sides of the x-axis at two different times. We explore this problem using a two-time quasi-probability linear in the projection operators, a natural counterpart of the corresponding classical problem. We show that it can be measured either indirectly, by measuring its moments in different experiments, or directly, in a single experiment using a pair of sequential measurements in which the first measurement is weak (or more generally, ambiguous). We argue that when positive, it corresponds to a measurementindependent arrival time probability. For small time intervals it coincides approximately with the time-averaged current, in agreement with semiclassical expectations. The quasi-probability can be negative and we exhibit a number of situations in which this is the case. We interpret these situations as the presence of "quantumness", in which the arrival time probability is not properly defined in a measurement-independent manner. Backflow states, in which the current flows in the direction opposite to the momentum, are shown to provide an interesting class of examples such situations. We also show that the quasi-probability is closely linked to a set of two-time Leggett-Garg inequalities, which test for macroscopic realism.represents the amplitude for the history in which the particle lies in x < 0 at times t 1 , t 2 , · · · t n−1 and is in x > 0 at t n , where P + (t) denotes the projection operator in the Heisenberg picture. For sufficiently close spacing of the times, this object is then a plausible candidate for the amplitude for the particle to make a left-right crossing of the origin, for the first time, during the time interval [t n−1 , t n ]. The probability for the crossing is then the norm of this state.Note that in an expression of the form Eq.(1.1), one would expect the Zeno effect [11] to come into play for sufficiently frequent measurements. This is indeed the case -it becomes significant when the time interval between projectors is smaller than /E, where E is the energy scale of the incoming packet [12,13].

Locomotion is a key aspect associated with ecologically relevant tasks for many organisms, therefore, survival often depends on their ability to perform well at these tasks. Despite this significance, we have little idea how different performance tasks are weighted when increased performance in one task comes at the cost of decreased performance in another. Additionally, the ability for natural systems to become optimized to perform a specific task can be limited by structural, historic or functional constraints. Climbing lizards provide a good example of these constraints as climbing ability likely requires the optimization of tasks which may conflict with one another such as increasing speed, avoiding falls and reducing the cost of transport (COT). Understanding how modifications to the lizard bauplan can influence these tasks may allow us to understand the relative weighting of different performance objectives among species. Here, we reconstruct multiple performance landscapes of climbing locomotion using a 10 d.f. robot based upon the lizard bauplan, including an actuated spine, shoulders and feet, the latter which interlock with the surface via claws. This design allows us to independently vary speed, foot angles and range of motion (ROM), while simultaneously collecting data on climbed distance, stability and efficiency. We first demonstrate a trade-off between speed and stability, with high speeds resulting in decreased stability and low speeds an increased COT. By varying foot orientation of fore- and hindfeet independently, we found geckos converge on a narrow optimum of foot angles (fore 20°, hind 100°) for both speed and stability, but avoid a secondary wider optimum (fore −20°, hind −50°) highlighting a possible constraint. Modifying the spine and limb ROM revealed a gradient in performance. Evolutionary modifications in movement among extant species over time appear to follow this gradient towards areas which promote speed and efficiency.

Deep learning-based computer vision methods are transforming animal behavioural research. Transfer learning has enabled work in non-model species, but still requires hand-annotation of example footage, and is only performant in well-defined conditions. To overcome these limitations, we created replicAnt, a configurable pipeline implemented in Unreal Engine 5 and Python, designed to generate large and variable training datasets on consumer-grade hardware instead. replicAnt places 3D animal models into complex, procedurally generated environments, from which automatically annotated images can be exported. We demonstrate that synthetic data generated with replicAnt can significantly reduce the hand-annotation required to achieve benchmark performance in common applications such as animal detection, tracking, pose-estimation, and semantic segmentation; and that it increases the subject-specificity and domain-invariance of the trained networks, so conferring robustness. In some applications, replicAnt may even remove the need for hand-annotation altogether. It thus represents a significant step towards porting deep learning-based computer vision tools to the field.

Robotic systems for complex tasks, such as search and rescue or exploration, are limited for wheeled designs, thus the study of legged locomotion for robotic applications has become increasingly important. To successfully navigate in regions with rough terrain, a robot must not only be able to negotiate obstacles, but also climb steep inclines. Following the principles of biomimetics, we developed a modular bio-inspired climbing robot, named X4, which mimics the lizard’s bauplan including an actuated spine, shoulders, and feet which interlock with the surface via claws. We included the ability to modify gait and hardware parameters and simultaneously collect data with the robot’s sensors on climbed distance, slip occurrence and efficiency. We first explored the speed-stability trade-off and its interaction with limb swing phase dynamics, finding a sigmoidal pattern of limb movement resulted in the greatest distance travelled. By modifying foot orientation, we found two optima for both speed and stability, suggesting multiple stable configurations. We varied spine and limb range of motion, again showing two possible optimum configurations, and finally varied the centre of pro- and retraction on climbing performance, showing an advantage for protracted limbs during the stride. We then stacked optimal regions of performance and show that combining optimal dynamic patterns with either foot angles or ROM configurations have the greatest performance, but further optima stacking resulted in a decrease in performance, suggesting complex interactions between kinematic parameters. The search of optimal parameter configurations might not only be beneficial to improve robotic in-field operations but may also further the study of the locomotive evolution of climbing of animals, like lizards or insects.

The life and death of an organism often depends on its ability to perform well at some ecologically relevant task. Yet despite this significance we have little idea how well species are optimised for competing locomotor tasks. Most scientists generally accept that the ability for natural systems to become optimised for a specific task is limited by structural, historic or functional constraints. Climbing lizards provide a good example of constraint where climbing ability requires the optimization of conflicting tasks such as speed, stability, or efficiency. Here we reconstruct multiple performance landscapes of climbing locomotion using a 10-DOF robot based upon the lizard bauplan, including an actuated spine, shoulders, and feet, the latter which interlock with the surface via claws. This design allows us to independently vary speed, foot angles, and range of motion, while simultaneously collecting data on climbed distance, stability and efficiency. We first demonstrate a trade-off between speed and stability with high speeds resulting in decreased stability and low speeds an increased cost of transport. By varying foot orientation of fore and hindfeet independently, we found geckos converge on a narrow optimum for both speed and stability, but avoid a secondary wider optimum highlighting a possible constraint. Modifying the spine and limb range of movement revealed a gradient in performance. Evolutionary modifications in movement among extant species appear to follow this gradient towards areas which promote speed and efficiency. This approach can give us a better understanding about locomotor optimization, and provide inspiration for industrial and search-and-rescue robots.Significance StatementClimbing requires the optimization of conflicting tasks such as speed, stability, or efficiency, but understanding the relative importance of these competing performance traits is difficult.We used a highly modular bio-inspired climbing robot to reconstruct performance landscapes for climbing lizards. We then compared the performance of extant species onto these and show strong congruence with lizard phenotypes and robotic optima.Using this method we can show why certain phenotypes are not present among extant species, illustrating why these would be potentially mal-adaptive.These principles may be useful to compare with relative rates of evolution along differing evolutionary histories. It also highlights the importance of biological inspiration towards the optimization of industrial climbing robots, which like lizards, must negotiate complex environments.