Robotic system and method for gaits

Abstract

A system is directed to robotic gaits. The system includes a robotic gait and an actuator. The robotic gait includes an intermediate body including a first end and an opposite second end, an end effector at the first end of the intermediate body and a point of rotation positioned at the second end of the intermediate body, and a moment arm extending transversely from the second end of the intermediate body. The robotic gait also includes two elastic beams each coupled to the point of rotation at an inner end, the intermediate body being free to rotate around the point of rotation, each of the two elastic beams extending from the point of rotation and fixed at an outer end opposite the inner end. The actuator is configured to provide a reciprocating input to the moment arm at a distance on the moment arm from the point of rotation.

Description

PATENT COOPERATION TREATY INTERNATIONAL PATENT APPLICATIONFORROBOTIC SYSTEM AND METHOD FOR GAITSBYDAVOOD FARHADIKATIA BERTOLDIBOBBY VAN THIELROBOTIC SYSTEM AND METHOD FOR GAITSCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to and the benefit of U.S. Provisional Application No. 63 / 675,096, filed on July 24, 2024, which is incorporated herein by reference in its entirety.FIELD OF THE INVENTION

[0002] The present invention relates generally to a robotic system for gaits and, more specifically, to a robotic elastic and compliant system for gaits driven by a single-input actuator to perform closed-loop gaits.BACKGROUND OF THE INVENTION

[0003] To advance existing robotic systems, mechanisms are continuously pushed to become smaller. Compact mechanism require fewer materials and are easier to transport due to their reduced mass and volume. Opting for compliant components in small scale mechanisms creates scalable, monolithic, frictionless systems, that require few assembly steps. However, due to the limited range of motion of compliant components, two actuators are needed to complete a closed loop trajectory. Additional actuators demand control schemes, take up space, are heavy, and increase the total power consumption. Even though underactuated mechanisms are a method to prevent introducing additional actuators, a challenge remains in controlling the entire state of system during its gait. A need exists for an alternative approach to designing underactuated mechanisms that requires a single input actuator and is passively controlled. The present disclosure provides a solution to these and other needs.SUMMARY OF THE INVENTION

[0004] The term embodiment and like terms, e.g., implementation, configuration, aspect, example, and option, are intended to refer broadly to all of the subject matter of this disclosure and the claims below. Statements containing these terms should be understood not to limit the subject matter described herein or to limit the meaning or scope of the claims below. Embodiments of the present disclosure covered herein are defined by the claims below, not this summary. This summary is a high-level overview of various aspects of the disclosure and introduces some of the concepts that are further described in the Detailed Description section below. This summary is not intended to identify key or essential features of the claimed subject matter. This summary is also not intended to be used in isolation to determine the scope of theclaimed subject matter. The subject matter should be understood by reference to appropriate portions of the entire specification of this disclosure, any or all drawings, and each claim.

[0005] According to certain aspects of the present disclosure, a system is directed to robotic gaits. The system includes a robotic gait and an actuator. The robotic gait includes an intermediate body including a first end and an opposite second end, an end effector at the first end of the intermediate body and a point of rotation positioned at the second end of the intermediate body, and a moment arm extending transversely from the second end of the intermediate body. The robotic gait also includes two elastic beams each coupled to the point of rotation at an inner end, the intermediate body being free to rotate around the point of rotation, each of the two elastic beams extending from the point of rotation and fixed at an outer end opposite the inner end. The actuator is configured to provide a reciprocating input to the moment arm at a distance on the moment arm from the point of rotation.

[0006] According to some features of the above aspects, the robotic gait has first and second stable configurations in response to the reciprocating input to the moment arm.

[0007] According to some features of the above aspects, the robotic gait has a first stable configuration when the reciprocating input is in a loading direction, and the robotic gait has a second stable configuration when the reciprocating input is in an unloading direction opposite to the loading direction.

[0008] According to some features of the above aspects, in response to the reciprocating input to the moment arm changing from the loading direction to the unloading direction, the robotic gait switches from the first stable configuration to the second stable configuration at a first bifurcation point, and in response to the reciprocating input to the moment arm changing from the unloading direction to the loading direction, the robotic gait switches from the second stable configuration to the first stable configuration at a second bifurcation point.

[0009] According to some features of the above aspects, in response to the reciprocating input to the moment arm, the end effector follows a closed-loop trajectory.

[0010] According to some features of the above aspects, the closed-loop trajectory of the end effector is in a clockwise sense when the reciprocating input is provided to the moment arm on a first side of the center of rotation, and the closed-loop trajectory of the end effector is in a counter clockwise sense when the reciprocating input is provided to the moment arm on a second side of the center of rotation, the second side opposite the first side.

[0011] According to some features of the above aspects, a path of the closed-loop traj ectory of the end effector is dependent on at least a length of the intermediate body, a displacementof the reciprocating input, and the distance on the moment arm from the point of rotation where the reciprocating input is provided.

[0012] According to certain aspects of the present disclosure, a system is directed to robotic gaits. The system includes a robotic gait and an actuator. The robotic gait includes an intermediate body including a first end and an opposite second end, an end effector at the first end of the intermediate body and a point of rotation positioned at the second end of the intermediate body, and a moment arm extending transversely from the second end of the intermediate body. The robotic gait also includes a first set of two elastic beams, each elastic beam of the first set coupled to the point of rotation at an inner end, the intermediate body being free to rotate around the point of rotation, each elastic beam of the first set extending from the point of rotation and fixed at an outer end opposite the inner end. The robotic gait also includes a second set of two elastic beams, each elastic beam of the second set coupled to the other elastic beam of the second set at a coupling point at an inner end, each elastic beam of the second set fixed at an outer end opposite the inner end, the second set of two elastic beams forming a bistable element. The robotic gait further includes an elastic connector beam coupled between the coupling point of the bistable element and the moment arm at a distance on the moment arm from the point of rotation. The actuator is configured to provide a reciprocating input to the point of rotation of the intermediate body.

[0013] According to some features of the above aspects, the robotic gait has first and second stable configurations in response to the reciprocating input to the point of rotation of the intermediate body.

[0014] According to some features of the above aspects, the robotic gait has a first stable configuration when the reciprocating input is in a loading direction, and the robotic gait has a second stable configuration when the reciprocating input is in an unloading direction opposite to the loading direction.

[0015] According to some features of the above aspects, in response to the reciprocating input provided to the point of rotation of the intermediate body changing from the loading direction to the unloading direction, the robotic gait switches from the first stable configuration to the second stable configuration at a first bifurcation point, and in response to the reciprocating input provided to the point of rotation of the intermediate body changing from the unloading direction to the loading direction, the robotic gait switches from the second stable configuration to the first stable configuration at a second bifurcation point.

[0016] According to some features of the above aspects, in response to the reciprocating input provided to the point of rotation of the intermediate body, the end effector follows a closed-loop trajectory.

[0017] According to some features of the above aspects, the closed-loop trajectory of the end effector is in a clockwise sense when the elastic connector beam is coupled to the moment arm on a first side of the center of rotation, and the closed-loop trajectory of the end effector is in a counter clockwise sense when the elastic connector beam is coupled to the moment arm on a second side of the center of rotation, the second side opposite the first side.

[0018] According to some features of the above aspects, a path of the closed-loop traj ectory of the end effector is dependent on at least a length of the intermediate body, a displacement of the reciprocating input, and the distance on the moment arm from the point of rotation where the elastic connector beam is coupled.

[0019] According to certain aspects of the present disclosure, a system is directed to robotic gaits. The system includes a single-input actuator. The robotic system further includes a structural mechanism having a bistable element. The structural mechanism is coupled with the single-input actuator and is configured to perform one or more of the robotic gaits in response to actuation via the single-input actuator.

[0020] According to some features of the above aspects, the robotic gaits include one or more of walking, jumping, rowing, and flapping.

[0021] According to some features of the above aspects, the structural mechanism is integrated in a hexapod mechanism.

[0022] According to some features of the above aspects, the hexapod mechanism is symmetric.

[0023] According to some features of the above aspects, the hexapod mechanism is a sixlegged walker.

[0024] According to some features of the above aspects, the six-legged walker has two triplets of walking legs, each of the two triplets providing three contact points with an environment surface, the three contact points preserving stability at all times.

[0025] According to some features of the above aspects, the single-input actuator provides a single degree of freedom (DoF) input with a 180° phase difference between the two triplets.

[0026] According to some features of the above aspects, the structural mechanism is configured to perform a semi-rotation through a single degree of freedom (DoF) input displacement.

[0027] According to some features of the above aspects, the structural mechanism is configured to switch between two stable positions via at least one of a translational motion and a rotational motion.

[0028] According to some features of the above aspects, the rotational motion includes one or more of a clockwise rotation and a counterclockwise rotation performed between two bifurcation points, the two bifurcation points including a first bifurcation point at a first stable position of the two stable positions, the two bifurcation points including a second bifurcation point at a second stable position of the two stable positions.

[0029] The above summary is not intended to represent each embodiment or every aspect of the present disclosure. Rather, the foregoing summary merely provides an example of some of the novel aspects and features set forth herein. The above features and advantages, and other features and advantages of the present disclosure, will be readily apparent from the following detailed description of representative embodiments and modes for carrying out the present invention, when taken in connection with the accompanying drawings and the appended claims. Additional aspects of the disclosure will be apparent to those of ordinary skill in the art in view of the detailed description of various embodiments, which is made with reference to the drawings, a brief description of which is provided below.BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The disclosure, and its advantages and drawings, will be better understood from the following description of representative embodiments together with reference to the accompanying drawings. These drawings depict only representative embodiments and are therefore not to be considered as limitations on the scope of the various embodiments or claims.

[0031] FIG. 1A shows an exemplary 8 degree of freedom input ambulatory micro robot, according to certain aspects of the present disclosure.

[0032] FIG. IB shows a schematic representation of a 5 bar mechanism from a single ambulatory micro robot leg such as the leg circled by the dashed line in FIG. 1 A, according to certain aspects of the present disclosure.

[0033] FIG. 1C shows an ideal, six-legged and single input locomotion mechanism, according to certain aspects of the present disclosure.

[0034] FIG. ID shows a theoretical trajectory with actuator and leg sequencing, according to certain aspects of the present disclosure.

[0035] FIG. 2 A shows a bifurcation diagram of a trajectory traced by an end effector when a unidirectional input along the vertical axis is exerted on the body, according to certain aspects of the present disclosure.

[0036] FIG. 2B shows an overview of the design space of a compliant mechanism where different gait trajectories are indicated by the various closed shapes, located at the corresponding end effector position, according to certain aspects of the present disclosure.

[0037] FIG. 3 A shows a process of applying a switching moment Ms to switch between both stable paths for a given input displacement, according to certain aspects of the present disclosure.

[0038] FIG. 3B shows a graph of switching moment Ms over the input displacementaccording to certain aspects of the present disclosure.

[0039] FIG. 3C shows a process of applying a positive Fs to initiate a clockwise rotation from the neutral position, according to certain aspects of the present disclosure.

[0040] FIG. 3D shows a force-displacement curve of the bistable and switching force over the input displacement, where the cross-hatched areas mark the required bistable force surplus to switch between stable paths, according to certain aspects of the present disclosure.

[0041] FIG. 3E shows a process of applying a negative Fs to switch to the opposite path close to the second bifurcation point, according to certain aspects of the present disclosure.

[0042] FIG. 3F shows the design space of the bistable element, according to certain aspects of the present disclosure.

[0043] FIG. 3G shows a close up of the dimensionalization of a single lumped bistable beam such as the bistable beam circled by the dashed line in FIG. 3F, according to certain aspects of the present disclosure.

[0044] FIG. 3H shows a force-displacement curve of the assembly depicted in FIG. 3C, according to certain aspects of the present disclosure.

[0045] FIG. 4 A shows a 2D presentation of an ANSYS® model with respective constraints and displacement, according to certain aspects of the present disclosure.

[0046] FIG. 4B shows an image of the Instron pull test, including clamping location and actuation point, according to certain aspects of the present disclosure.

[0047] FIG. 4C shows an end effector trajectory of the mechanisms without bistable element in ANSYS®, full leg assembly in ANSYS®, and tested mechanism, according to certain aspects of the present disclosure.

[0048] FIG. 4D shows force-displacement characteristics of the mechanism without bistable element, full leg assembly, and tested mechanism, according to certain aspects of the present disclosure.

[0049] FIG. 5 A shows a picture of joint transmission components, where upwards movement of the middle transmission point results in an equal but opposite motion of the front and back transmission points, according to certain aspects of the present disclosure.

[0050] FIG. 5B shows a picture of the transmission connection point and actuation point of the middle leg, before applying prestress, according to certain aspects of the present disclosure.

[0051] FIG. 5C shows a picture of a triple-leg assembly before applying prestress, according to certain aspects of the present disclosure.

[0052] FIG. 5D shows a picture of the triple-leg assembly of FIG 5C when applying prestress and the transmission in its neutral position, according to certain aspects of the present disclosure.

[0053] FIG. 5E shows a picture of the transmission connection point and actuation point of the middle leg, after applying prestress, according to certain aspects of the present disclosure.

[0054] FIG. 5F shows a picture of the triple-leg assembly of FIG 5C with the front and back leg in its neutral position, according to certain aspects of the present disclosure.

[0055] FIG. 5G shows a picture of the triple-leg assembly ofFIG. 5C, with for each leg the corresponding trajectory trace of the end effector from the test, according to certain aspects of the present disclosure.

[0056] FIG. 5H shows ANSYS® end effector trajectories of a triple-leg assembly compared to the ANSYS® end effector trajectory of the single leg mechanism, according to certain aspects of the present disclosure.

[0057] FIG. 51 shows end effector trajectories of the tested triple-leg assembly compared to the end effector trajectory of the tested single leg mechanism, according to certain aspects of the present disclosure.

[0058] FIG. 5J shows x-displacement of the modelled and tested end effectors over time, alongside the modelled and tested force characteristic over time, according to certain aspects of the present disclosure.

[0059] FIG. 5K shows y-displacement of the modelled and tested end effectors over time, alongside the modelled and tested force characteristic over time, according to certain aspects of the present disclosure.

[0060] FIG. 6A shows a picture of a full prototype assembly, according to certain aspects of the present disclosure.

[0061] FIG. 6B shows a picture of the front view of actuator assembly, according to certain aspects of the present disclosure.

[0062] FIG. 6C shows a picture of the side view of the full prototype assembly, according to certain aspects of the present disclosure.

[0063] FIG. 7A is a picture of the side view of the prototype, including numbering of the traced left three legs and coordinate system, according to certain aspects of the present disclosure.

[0064] FIG. 7B is a traced displacement of the center of the prototype, according to certain aspects of the present disclosure.

[0065] FIG. 7C is a rotation of the center of the prototype over time, including the step numbers, according to certain aspects of the present disclosure.

[0066] FIG. 7D is a rotation difference according to each step, according to certain aspects of the present disclosure.

[0067] FIG. 7E is a graphical representation of the mechanism, including weight distribution on each leg, according to certain aspects of the present disclosure.

[0068] FIG. 7F is a graphical representation of the influence of the normal force on the leg dynamics, according to certain aspects of the present disclosure.

[0069] FIG. 7G is a theoretical closed loop end effector traj ectory for different gravitational loads, where the step size is indicated for all load cases, and the actuation point motion is indicated along the rotation direction, according to certain aspects of the present disclosure.

[0070] FIG. 7H is an average measured closed loop end effector trajectory of each leg with standard deviation alongside the modelled end effector trajectory for the corresponding load case, according to certain aspects of the present disclosure.

[0071] FIG. 71 is a gait pattern of all six legs over six actuation cycles, where the patterns indicate the success of the trajectory during extending and retracting, and the given distances indicate the forward or backward slip between the end effector and the ground, according to certain aspects of the present disclosure.

[0072] FIG. 7J is a graphical representation of the cause for backward slip due to unstable snapping of the leg, according to certain aspects of the present disclosure.

[0073] FIG. 7K is a graphical representation of the cause for forward slip due to oscillation of the body, according to certain aspects of the present disclosure.

[0074] FIG. 8 a schematic view of an exemplary compliant (i.e., elastic and monolithic) robotic gait that generates a closed-loop end-effector trajectory when subjected to a reciprocating actuator input, according to certain aspects of the present disclosure.

[0075] FIG. 9 shows a first stable configuration for the robotic gait of FIG. 8 to achieve a closed-loop end-effector trajectory, according to certain aspects of the present disclosure.

[0076] FIG. 10 shows a second stable configuration for the robotic gait of FIG. 8 to achieve a closed-loop end-effector trajectory, according to certain aspects of the present disclosure.

[0077] FIG. 11 shows a schematic view of another exemplary complaint (i.e., elastic and monolithic) robotic gait that generates a closed-loop end-effector trajectory when subjected to a reciprocating actuator input applied at an intermediate body’s center of rotation, according to certain aspects of the present disclosure.

[0078] FIG. 12 shows the robotic gait of FIG. 8 results in a counter clock-wise rotation for the end-effector when traveling along closed-loop trajectory, according to certain aspects of the present disclosure.

[0079] FIG. 13 shows an exemplary robotic gait similar to the robotic gait of FIG. 8, but for which the input displacement is placed on the opposite side of the center of rotation, which results in a clockwise rotation for the end-effector when traveling along closed-loop trajectory, according to certain aspects of the present disclosure.

[0080] FIG. 14 shows the robotic gait of FIG. 11 results in a clock-wise rotation for the end-effector when traveling along closed-loop trajectory, according to certain aspects of the present disclosure.

[0081] FIG. 15 shows an exemplary robotic gait similar to the robotic gait of FIG. 11, but for which the input displacement is placed on the opposite side of the center of rotation, which results in a counter clockwise rotation for the end-effector when traveling along closed-loop trajectory, according to certain aspects of the present disclosure.

[0082] FIG. 16 shows that the robotic gait of FIG. 8 can achieve different shapes for the end-effector trajectory by changing design parameters and / or changing the location of the endeffector on intermediate body, according to certain aspects of the present disclosure.

[0083] FIG. 17 shows a picture of an exemplary hexapod compliant walker that uses six identical robotic gaits that are interconnected in parallel and all are derived with a single actuator, according to certain aspects of the present disclosure.DETAILED DESCRIPTION

[0084] Various embodiments are described with reference to the attached figures, where like reference numerals are used throughout the figures to designate similar or equivalent elements. The figures are not necessarily drawn to scale and are provided merely to illustrate aspects and features of the present disclosure. Numerous specific details, relationships, and methods are set forth to provide a full understanding of certain aspects and features of the present disclosure, although one having ordinary skill in the relevant art will recognize that these aspects and features can be practiced without one or more of the specific details, with other relationships, or with other methods. In some instances, well-known structures or operations are not shown in detail for illustrative purposes. The various embodiments disclosed herein are not necessarily limited by the illustrated ordering of acts or events, as some acts may occur in different orders and / or concurrently with other acts or events. Furthermore, not all illustrated acts or events are necessarily required to implement certain aspects and features of the present disclosure.

[0085] For purposes of the present detailed description, unless specifically disclaimed, and where appropriate, the singular includes the plural and vice versa. The word “including” means “including without limitation.” Moreover, words of approximation, such as “about,” “almost,” “substantially,” “approximately,” and the like, can be used herein to mean “at,” “near,” “nearly at,” “within 3-5% of,” “within acceptable manufacturing tolerances of,” or any logical combination thereof. Similarly, terms “vertical” or “horizontal” are intended to additionally include “within 3-5% of’ a vertical or horizontal orientation, respectively. Additionally, words of direction, such as “top,” “bottom,” “left,” “right,” “above,” and “below” are intended to relate to the equivalent direction as depicted in a reference illustration; as understood contextually from the object(s) or element(s) being referenced, such as from a commonly used position for the object(s) or element(s); or as otherwise described herein.

[0086] In a Von Mises Truss, one cannot define certainly how (which direction) an intermediate body attached to a center of the truss would deform during loading and unloading. However, as disclosed herein, an elastic and compliant structural mechanism combined with a bistable embodiment shows potential for a variety of gaits, for example including without limitation, locomotion gaits such as walking, jumping, rowing and flapping. In the current disclosure, by carefully designing the angle and placement of the actuator input at the right location, or by adding a bistable control mechanism, we can derive a mechanism where the intermediate body is programmed to take two distinct directions during loading and unloading, thus making a closed loop end-effector trajectory. The attainable complex closed-looptrajectories are presented. A method for designing the bistable control embodiment has been developed. To demonstrate the mechanism performance, a monolithic path generator suitable for walking locomotion was designed. The mechanism has been integrated in a single actuated symmetric hexapod prototype that has successfully demonstrated walking locomotion along a straight trajectory. An underactuated monolithic elastic and compliant structural mechanism is disclosed that is suitable for locomotion purposes and allows for the integration in small scale robotic systems.

[0087] The disclosure herein allows reducing a number of actuators to control complex robotic gaits. According to one example, a single-actuator controls gaits in microscale robotic applications. The illustrations and disclosure provided below further describes novel and inventive aspects of the robotic system.

[0088] The pursuit for smaller robotic systems necessitates fundamental reconsideration of conventional mechanism designs. Scaling mechanisms down creates compact and lightweight systems that reduce power consumption and economize on the used materials. Conventionally, performing a closed loop trajectory is accomplished through implementing rotating gears, axles and actuators. A compliant alternative is more appealing for small mechanisms, since they are scalable, monolithic, frictionless and require fewer assembly steps. On the other hand, compliant mechanisms are restricted by their limited range of motion and cannot complete full rotations.

[0089] Completing a full rotation is a method to accomplish cyclic patterns of motion characteristics, such as walking, running, or jumping, otherwise referred to as gaits. The closed loop paths created by an end effector during the gaits, are interpreted as gait trajectories. Most commonly, gait trajectories are mentioned as the path performed by a walking gait. Additionally, the terminology is adopted in, for instance, aerial flapping, swimming, or jumping gaits.

[0090] In compliant mechanisms, completing a gait trajectory requires two actuators, one for each degree of freedom (DoF). Aiming for smaller scale mechanisms whilst introducing additional actuators increases the actuator mass, volume, and power consumption ratio. Furthermore, a complex control architecture is needed to synchronize the motion of both actuators. Underactuated mechanisms formulate an opportunity to prevent implementing abundant actuators and avoid additional control schemes. Under actuation is a classification that applies to all mechanisms in which the number of output DoF exceeds the number of actuators.

[0091] A pivotal challenge for underactuated systems is controlling the complete state of the system over its entire gait. In existing underactuated mechanisms, the actuation deficit is predominantly compensated for by extensive and complex control. Alternatively, temporary interaction with the environment introduces additional constraints, during which the mechanism has an equal number of DoF and actuators. During the remaining gait, mechanism components are uncontrollable. However, passively controlling the abundant DoF will prevent uncontrollable components and reduces the need for a complex control architecture.

[0092] A Von Mises Truss inspired mechanism performs a semi-rotation through a single DoF input displacement. Inherently, in a Von Mises Truss no preference exists for either of both rotation directions, resulting in bifurcation points at both stable positions. A system that can produce closed loop complex trajectories created by an elastic and compliant structural mechanism is disclosed. Furthermore, a design strategy has been formulated that regulates the movement of the structural mechanism at the bifurcation points to take a desired path that results in a closed-loop path through implementing a bistable element. As such, the mechanism is single actuated and passively controlled. To demonstrate the mechanism’s functioning, a sixlegged walker prototype was designed, modelled and tested.

[0093] Conventionally, every DoF of an end effector is matched by an equal amount of actuators. For instance, FIG. 1A and FIG. IB shows the Harvard ambulatory microrobot (HAMR) with eight independently actuated DoF. When scaling down mechanisms, the weight, volume, and energy contribution of each actuator increases compared to the total design. Through decreasing the number of required actuators, under actuation potentially permits even lower scale mechanisms.

[0094] To demonstrate the locomotion capabilities, a prototype performing a walking gait was selected. By implementing six legs into a walker prototype, three contact points with the environment are preserved at all times to ensure stability. Ideally, a single DoF input results in two triplets of walking gaits, with a 180° phase difference between both triplets. This is depicted in FIG. 1C and FIG. ID. Fundamentally, this configuration allows for the lowest actuator weight, volume and energy ratio in low speed applications.

[0095] An elastic and compliant structural mechanism was implemented to create the walking gait. The elastic and compliant structural mechanism has two stable positions between which is switched through a combination of a translational and rotational motion of the body, as depicted in FIG. 2A. The double output motion from a unidirectional input displacement makes this mechanism underactuated. Inherently, there is no preference between the clockwise or counterclockwise rotation of the body, hence bifurcation points are present at both stablepositions. Tracing any point on the elastic and compliant structural mechanism body creates a path, and this point is henceforth referred to as the end effector. Whilst performing a clockwise or counterclockwise rotation between both bifurcation points, the end effector creates two stable paths. What makes the primary and secondary path stable is the energy barrier between both, created by higher order bending modes of the compliant beams. In addition to both stable trajectories formed by the clockwise and counterclockwise rotation, an unstable path exists for pure translation of the end effector.

[0096] Joining both stable paths together creates a closed gait trajectory. Comparable to a Coupler curve, FIG. 2B shows that generation of different gait trajectories is possible through tracing the end effector of the elastic and compliant structural mechanism. The gait trajectories with a flat side are potentially suitable for walking applications, where the eight-shaped gait trajectories could be used for flapping gaits, or the vertical oval for rowing gaits. Furthermore, gait trajectories with symmetric properties are observable between shapes opposite the horizontal and vertical axis. The design space of a compliant, underactuated gait generator consists of five independent variables, L,A, and , as displayed in FIG. 2B. The range of the design parameters over which the gait trajectories are explored is indicated in Table 1.Parameter Dimensions UnitL [20, 30, 40, 50] mm w [10, 15, 20, 25] mm< / > [10, 15, 20, 25] h [10, 20, 30, 40, 50, 60] mm [0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330]Table 1. For each parameter indicated in FIG. 2B, the range over which the results are explored is given.

[0097] All gait trajectories are extracted from the Mechanical APDL software, or in short, ANSYS®. The displacement and rotation data from both semi-rotations are merged into a single closed gait trajectory.

[0098] The design from FIG. 2B determines the kinematics of the end effector, henceforth referred to as the kinematic mechanism. The previously presented closed gait trajectories are combined from joining both stable paths, whilst both bifurcation points are still present. To eliminate the bifurcation points, a moment is exerted on the body to switch from the primary stable path to the secondary stable path and vice versa. This is shown in FIG. 3 A. To establisha threshold for the minimum moment to switch between stable paths, at each input position uina switching moment Msis required, which is depicted FIG. 3B.

[0099] Creating this moment is an external force Fsout of line of the actuation force. When a downward motion is initiated by uin, an upward force is needed for Fsto cause clockwise rotation of the end effector, as shown in FIG. 3C. FIG. 3D shows the switching threshold Fs, deduced from Msand the arm. When the bistable force Fbi is higher than the switching threshold, the end effector switches path. The required bistable force surplus is marked by cross-hatched regions in FIG. 3D. Upon committing to the clockwise rotation, the end effector shall remain stable in this path until approaching the opposite bifurcation point. As FIG. 3E shows, at the second bifurcation point, the switching force Fsdirects downward to create a clockwise moment around the actuation point. Due to the opposing direction of the switching force, the force threshold in FIG. 3D is inverted for uin> 10 mm. The magnitude of the switching force Fsexceeds the switching threshold to overcome the energy barrier between both stable paths. When sufficiently large, the end effector switches from one to the next stable path. Then, the input motion is inverted and returned to its initial position to complete the rotation. In the instance that the bistable force Fbi exceeds the force threshold, the end effector body snaps to its opposing stable path. This instance is indicated as switching point in FIG. 3D.

[0100] Near both bifurcation points, the direction of switching force Fsopposes each other. Translating this kinetic behavior to a mechanism design is best done through a bistable element. Adding a bistable element to the kinematic mechanism will introduce preference for one of both stable paths at each bifurcation point. The design concept is depicted in FIG. 3F. The springs of this bistable element have stepwise variable thickness over its length, as shown in FIG. 3G, to avoid applying preload whilst ensuring bistable properties. With this design, the leg is compliant and monolithic, eliminating complex assembly steps and allowing for scaling. The complete input force-displacement characteristics of such a mechanism, including the switching points, is depicted in FIG. 3H.

[0101] The design parameters of the kinematic mechanism with bistable element, as depicted in FIG. 2B and FIG. 3F respectively, are extracted with nonlinear analysis in ANSYS®. For large values of Lunk the bistable elements has little effect on the traj ectory of the end effector. As such, both components are treated as parallel systems and a parametrization search is performed for both components independently.

[0102] Parameter selection is primarily restricted by design and manufacturing constraints. The selected production method and material is 3-dimensional printed PLA with the BambuLab XI Carbon. The yield strength used for PLA is 3. 12 MPa. A 0.2 mm stainless steel nozzle is fitted to the printer, allowing for a 0.25 mm line width and 0. 1 mm layer thickness. A minimal spring width of 0.5 mm is adopted to prevent layer defects during printing. Other than that, the design is constrained by the actuator specifications and dimensions. The compact and lightweight Actuonix PQ12-R linear servo is selected. It has a maximum stroke of 20 mm with a maximum force of 18 N.

[0103] Respecting the presented manufacturing and design constraints, a selection is made for the parameters of the kinematic mechanism and bistable element. The values for / , tiumped and tsprare set according to fabrication constraints, where the value for Lumped is set arbitrarily. The design parameters are selected according to two objectives. First, the bistable element must have a positive Fbi for a uinclose to 0 mm and a negative Fbi for a uinapproaching 20 mm. Secondly, the switching point must be close to both ends of the actuator’s stroke. The instance that Fbi exceeds Fs the end effector will snap to the opposing stable path. Therefore, selecting parameters with switching points close to the bifurcation points will create a larger rotation with shorter unstable trajectories. All design parameters with their corresponding value are given in Table 2.Parameter DimensionsL 50 mm w 10 mm15° h 40 mm p 0° arm 10 mmLunk 15 mm t 0.5 mm tiumped 0.75 mm tspr1 mmLbt 60 mmWbi 5 mm6 11 mmLspr5 mm^lumped 20 mmTable 2. Parameter overview of the final design.

[0104] Subsequently, the full leg assembly is modelled and tested, as shown in FIG. 4A and FIG. 4B respectively. In the model, all components are constructed from BEAM188 elements, suitable for slender beams. For both kinematic mechanism springs, an asymmetric arc is used to help the model converge. A separate load step for extending and retracting the actuation point in the y-direction is applied through displacement control. Displacement in x- and z-direction are kept constrained while allowing for free rotation. A maximum and minimum of 2000 and 300 substeps per load step is implemented respectively, whilst allowing for automatic time stepping. A maximum of 200 iterations per substep is allowed. The Young’s modulus of the PLA compliant beams with identical thickness is measured at 2.3 GPa and is set in the model accordingly. A trace of the end effector displacement alongside the force-displacement characteristics is provided in FIG. 4C and FIG. 4D.

[0105] A tensile and compression test is performed on the Instron® 5900 Series. An SLA printed actuation bracket is attached to a 50 N load cell, which is in turn mounted on the Instron®. Through the corresponding Bluehill® software, a stroke of 19 mm with an actuation speed of 40 mm / min is set. The PLA leg is camped into an Instron vice. A single leg mechanism is tested over nine actuation cycles. An actuation cycle consists of extending and retracting the actuation point once. Video tracking provides the gait trajectories, of which the average presented in FIG. 4C. The average measured force-displacement characteristic is shown in FIG. 4D.

[0106] The test shows that the leg design completes a full walking gait trajectory with a single input actuator. The end effector trajectory at each switching point are indicated with dotted lines. This path is unstable, which is represented in a rapid snapping motion. During walking applications, when the end effector is in contact with the ground, a rapid snapping motion has the potential side effect that the leg slips on the surface and prevents body motion. On the other hand, a snap through motion offers opportunities in gait applications such as jumping. As such, the mechanism design allows for continuous jumping gaits through a single input actuation.

[0107] Sequencing three legs to the same actuator generates a triplet of legs for a symmetric hexapod walker prototype. Each triplet includes a front leg and a rear leg of one side and a middle leg of the other side of the hexapod walker. Due to the flat nature of the leg mechanism, a symmetric hexapod configuration is selected for the prototype. To maintain a minimum of three contact points with the ground, the two triplets of legs are connected in phase, with a 180°phase difference between them. A 180° phase difference is achieved through applying a prestress between the legs during assembly.

[0108] The three legs of a single side are connected to each other with the transmission chain depicted in FIG. 5 A. Displacing the middle input point results in an equal but opposite displacement of the front and back connection points. The symmetric design creates a purely vertical motion, which is required for the input of each leg component. The front and back legs are connected in line with dovetails. Then, the transmission chain is fixed to both legs, vertically aligning both connection points of the transmission chain with the respective leg actuation points as shown in FIG. 5B. Through stacking the middle leg in the center of the transmission chain, the actuation point of the middle leg aligns vertically with the middle connection point, resulting in the assembly showed in FIG. 5C. All in plane connections are made with dovetail connections, where all out of plane connections are made with pin connections. The prestressed mechanism with horizontal transmission will result in FIG. 5D. Prestress is applied by displacing the connection point of the transmission to the actuation point of the leg and fitting it over the pin, as shown in FIG. 5E. The equilibrium position of the prestressed mechanism is displayed in FIG. 5F.

[0109] The assembly shown in FIG. 5F is modelled in ANSYS®. The three legs are connected to each other with MPC184 frictionless and dampingless rotational joints. Prestress is applied in two steps. First, the three leg constraints are displaced half of the total input stroke downwards, whilst keeping the actuation point fixed. Then the actuation point is displaced half of the total input stroke upwards, attaining the configuration depicted in FIG. 5F. Due to the high degree of nonlinearity, the large number of elements, and simultaneous unstable snapping, a minimum and maximum of 800 and 4000 substeps per load steps are allowed, respectively. Additionally, the length of the stroke is decreased to 8.6 mm in both directions from the center. This is enough to take into account the snapping dynamics, yet, does compute the gait trajectory close to the bifurcation points.

[0110] The triple-leg assembly is tested on a Physik Instrumente single DoF displacement stage with 50 N load cell. The middle transmission connection point and the actuation point of the middle leg are attached to an actuation pin. FIG. 5G shows the end effector trajectories established over a 19 mm actuation stroke. Furthermore, the trajectories of the actuation points of the legs are shown. The proposed assembly successfully completes a full end effector rotation with an 180° phase difference between the legs, from a unidirectional input actuation. The theoretical and tested end effector trajectories are compared in FIG. 5H and FIG. 51,respectively. The x- and y-displacement over time of the ANSYS® and tested gait trajectories are indicated in FIG. 5J and FIG. 5K, respectively.[OHl] As FIG. 5H shows, the modelled trajectories of all three legs are identical to the modelled gait trajectory of the single leg. Thus, the transmission chain does not influence the shape of the end effector trajectory. FIG. 5 J shows that the modelled snapping moment of the right and left leg are simultaneous and earlier than the snapping moment of the middle legs. This is because the snapping moment is not symmetrically distributed around the middle of the actuator stroke. When the middle leg is in its upward stroke, the left and right leg are in its downward stroke. Therefore, there is a discrepancy between the timing of the snap. FIG. 5J also shows that the snapping moments align with little peaks in the force-displacement characteristic.

[0112] The gait trajectories of the tested counterparts are displayed in FIG. 51. All gait trajectories are aligned according to the end effector position in its neutral state. The shape of all gait trajectories are similar, however, they are displaced from the center. Since the left leg actuation point is displaced 1.7 mm upward in its neural state, the end effector is forced to enter is preferred path and move to the left. Therefore, the gait trajectory is displaced to the right. The actuation point displacement is caused by the stiffness of the transmission system, since it creates a force in the y-direction. Hence, the y- displacement of the end effector compared to the model, shown in the left leg segment of FIG. 5K. The initial displacement of the right leg actuation point, however, is negligible, aligning the right leg and middle leg trajectory in FIG. 51. The discrepancy between the actuation point displacement of the right and left leg is potentially caused by a reduced stiffness of the left leg mechanism, likely due to imperfections in the mechanism. The changed end effector trajectory of the left leg argues in favor of this presumption.

[0113] Similar to FIG. 51, in FIG. 5J the initial x-displacement is aligned according to the initial end effector position of the legs. The theoretical results, however, indicate an initial displacement from its center position. Thus, the tested gait trajectories are displaced relative to the theoretical gait trajectories.

[0114] Comparing the x- and y-displacement components of the tested and modelled end effectors, FIG. 5J and FIG. 5K show that the instance the tested right and middle leg switch from one stable path to the other aligns with their theoretical counterpart. The left leg, however, has a switching point later than indicated by the theoretical x-displacement. This is due to the deformed gait trajectory of the left leg compared to the middle and right leg. Manufacturing defects or deformations whilst applying prestress are feasible explanations for the displacedswitching instances. Alternatively, friction between rotation points could also yield a delay of the switching points.

[0115] To create a prototype, two triple-leg assemblies are connected, as shown in FIG. 6A. Both sides are joined with four frames, constraining displacement and rotation in all directions. Between both sides, an actuator assembly is fitted. Through this mechanism, the actuator is connected to the actuation point of both sides, as displayed in FIG. 6B. Implementing the same principle as for the transmission chain, a rotating rigid bar creates a 180° difference between the phase of the left and right leg assemblies. Both sides of this mechanism have male pin connectors which fits the female connection points of the middle transmission and middle leg. To prevent the pins from slipping out of their assigned female connection points, compliant linear guides are attached. The compliant beams are lumped for higher buckling stiffness. FIG. 6C shows the side view of the walker when all legs make contact with the ground. All prototype properties are listed in Table 3.Parameter DimensionsLength 305 mmWidth 128 mmHeight 128 mmMass 128 gNumber of Legs 6Number of Actuators 1Table 3. Overview of the dimensions and properties of the prototype.

[0116] The walker from FIG. 6C was tested on a flat surface over 25 cm. An Arduino Uno is connected to the actuator with a 3 seconds delay between extending and retracting. The trajectories of leg 1, 2 and 3, as shown in FIG. 7A, are measured with the same point tracking code as with the previous two tests. Two color markers are placed on top of the prototype. A second camera measures the x-displacement and y-displacement of the prototype from the top. The coordinate system provided in FIG. 7A is adopted. In this context, the left leg in FIG. 5G is a front leg in FIGs. 7A, 7E, and 7H, and the right leg in FIG. 5G is a rear leg in FIGs. 7A, 7E, and 7H.

[0117] The prototype successfully completes the 25 cm distance with a single DoF actuator. FIG. 7B shows the path taken by the center of the prototype during the test. Over the test, the center of the prototype has a y-displacement of 24 mm. This is due to a gradual bodyrotation towards 0.21 rad, as shown in FIG. 7C. FIG. 7D shows the rotation data separated for each step individually. All even steps result in a positive rotation of the body, where the odd steps cause a negative rotation. This pattern is due to an uneven weight distribution between the legs. FIG. 7E shows that the front and back leg each carry 23% of the prototype weight, where the middle leg supports 54% of the weight. This is assuming the center of mass of the mechanism is at the center of the prototype body.

[0118] FIG. 7F shows how different loads influence the steps size of a leg, and consequently cause an alternating pattern in the body rotation. Upon touching the ground, a normal force is introduced on the tip of the end effector. A moment is exerted around the actuation point of the leg, proportional to the normal force on the end effector. This moment causes deformation opposite the direction of rotation. The actuation point is extended further, and the beam springs incrementally counteract the inverted rotation through their elasticity. Consequently, the end effector switches to its second stable path and the prototype body is displaced forwards. In the second stable path, the prototype body leans into the rotational stiffness of the compliant beams until the opposing legs take over and the leg loses contact with the ground. FIG. 7G shows that the end effect trajectories change when a normal force is introduced on the leg. These gait trajectories are established through ANSYS®, upon implementing a vertical force of 23% and 54% of the prototype weight on the end effectors for input displacement exceeding 9.5 mm. The theoretical step size is determined according to the instance that the load is shared by all six legs simultaneously. As the load in the model is increased linearly, this is exactly halfway through the load step. The theoretical change in step size is indicated for the different external loads.

[0119] The measured motion of the end effector with respect to the stationary frame of the camera is built up from the closed loop gait trajectory of the end effector with respect to the prototype body and the motion of the body with respect to the stationary frame, as indicated by Equation 1. From the motion of the end effectors with respect to the frame and the motion of the body with respect to the frame, the closed loop gait trajectories of the legs are extracted. The average closed loop gait trajectories from all six actuation cycles are displayed in FIG. 7H. The standard deviations of the average gait trajectories are indicated with error bars. The end effector trajectories are displayed along their respective loaded modelled end effector trajectories.

[0121] From FIG. 7H it is concluded that the size of the average gait trajectories is comparable to their loaded modelled counterparts. Furthermore, the average gait trajectories show that when interacting with the ground, the mechanism deforms into its loaded deformed state. Where the left and middle leg follow a relatively round gait trajectory, the unloaded half of the right gait trajectory shows a much larger deviation from the modelled gait trajectory. This is partially due to the right leg not completing its full rotation during two of the six actuation cycles, as indicated by the large error bars. However, the raw data shows that the successful rotations also have a deviation from the loaded model. This is potentially due to imperfections in the mechanism sustained during printing.

[0122] During the dominant part of the actuation cycle, the leg is at rest at the top or bottom of the closed loop. Hence, the large sample density there. The relatively larger standard deviation there is due to oscillation of the prototype body. As the grounded end effectors snap from one stable path to the other, the prototype body is accelerated. Due to the rotational stiffness around the actuation point and the low damping of the leg mechanisms, decelerating the body inertia causes oscillations. This is too observable in FIG. 7B and FIG. 7C.

[0123] Apart from the back and forth rotation of the body due to the uneven gravity distribution on the legs, an asymmetric rotation is observable. This has two root causes. Firstly, GIG. 71 shows in the gait pattern that not all steps completed the full trajectory. In the first actuation cycle the left back leg failed to snap to its second stable path and in step twelve the left back leg only partially completed its trajectory. Accordingly, FIG. 7C shows an increase in body rotation during those steps. In step nine, there is a delay in switching to its second stable path of all legs, possibly due to a voltage drop in the actuator. This did not yield additional rotation since it was uniform for all legs. Secondly, slip between the tip of the legs and the surface causes a decrease in step size and subsequent rotation of the body. The middle leg shows a relatively higher amount of slip than the back and front leg, as quantified in FIG. 7H. This slip is accompanied by a relatively larger rotation around the bodies center. Slip data from the right set of legs is not available. To further investigate the effects of slip, tracing the position of the right set of legs is required.

[0124] One of the causes of the body rotation is incomplete rotation of the legs. Body rotation due to incomplete gait trajectories offers an opportunity for future research. Controlling the full and semi-rotation of legs will create prototypes that can make turns. Adjusting the force-displacement characteristics of the bistable element, either actively or through influences of the environment, will cause legs to perform semi-rotations instead of full rotations. Further research is required to implement this into the prototype.

[0125] Referring to FIG. 8, a schematic view of an exemplary embodiment of a compliant (i.e., elastic and monolithic) robotic gait 100 is shown. The robotic gait 100 generates closed- loop end-effector trajectory when subjected to a reciprocating actuator input 150, i.e., back and forth displacement. The robotic gait 100 comprises of two angled elastic beams 110 and 120 Each of the two beams 110 and 120 is grounded on one side. The second end of each beam 110, 120 is clamped to an intermediate body 130. For an applied displacement from a reciprocating actuator input 150, the end-effector 160 follows a closed-loop trajectory 170. In this embodiment, the applied displacement of the reciprocating actuator input 150 must be distanced from the center of rotation 140 of the intermediate body 130.

[0126] FIGs. 9 and 10 show the conditions for the robotic gait 100 to achieve a closed-loop end-effector trajectory. The robotic gait 100 must at least have two stable configurations within the applied range of displacement of the reciprocating actuator input 150. A first stable configuration as shown in FIG. 9 is within the loading displacement of the reciprocating actuator input 150, referred to as the loading phase. A second stable configuration as shown in FIG. 10 is within unloading displacement of the reciprocating actuator input 150, referred to as the unloading phase. During each of the loading phase and the unloading phase, the endeffector 160 follows distinct trajectories, thus resulting in a closed-loop end-effector trajectory 170 for the robotic gait 100.

[0127] FIG. 11 shows schematic view of a complaint (i.e., elastic and monolithic) robotic gait 400 that generates a closed-loop end-effector trajectory when subjected to a reciprocating actuator input 150 that is applied at the center of rotation 140 of the intermediate body 130. In this embodiment, an additional set of beams 410, 420, 430 are added to the robotic gait 100 to achieve a closed-loop trajectory 170 when the reciprocating actuator input 150 is applied at the center of rotation 140 of the intermediate body 130. The two elastic angled beams 410 and 420 form a bistable element. The elastic beam 430 allows for relative rotation between the set of angled beams 110, 120 and the set of angled beams 410, 420 throughout the range of motion.

[0128] FIG. 12 shows that the robotic gait 100 results in a counter clock-wise rotation for the end-effector 160 when traveling along the closed-loop trajectory 170. This is because when the input displacement of the reciprocating actuator input 150 is applied it generates a clockwise moment to the intermediate body 130 around its center of rotation 140 during the loading phase. The input displacement of the reciprocating actuator input 150 generates a counter clockwise moment to the intermediate body 130 around its center of rotation 140 during the unloading phase.

[0129] Referring to FIG. 13, a schematic view of an exemplary embodiment of a compliant (i.e., elastic and monolithic) robotic gait 500 is shown. The robotic gait 500 has the input displacement of the reciprocating actuator input 150 placed on the opposite side of center of rotation 140 as compared to the robotic gait 100 in FIG. 12. This results in a clockwise rotation for the end-effector 160 when traveling along the closed-loop trajectory 170. The clockwise rotation occurs because when the input displacement of the reciprocating actuator input 150 is applied it generates a counter clockwise moment to the intermediate body 130 around its center of rotation 140 during the loading phase. The input displacement of the reciprocating actuator input 150 generates a clockwise moment to the intermediate body 130 around its center of rotation 140 during the unloading phase.

[0130] FIG. 14 shows that the robotic gait 400 generates a clockwise rotation for the endeffector 160 when traveling along the closed-loop trajectory 170. The elastic bistable beams 410 and 420 generate an internal reaction force transferred by elastic beam 430 to the intermediate body 130 throughout the movement. During the loading phase, beams 410 and 420 are at their first stable configuration, and through beam 430, this results in an internal reaction force along the positive y direction on the intermediate body 130. This forces the intermediate body 130 to rotate counter clock-wise around the center of rotation 140 and forces the end-effector 160 to move clockwise along the trajectory 170.

[0131] During the unloading phase, beams 410 and 420 are at their second stable configuration, and through beam 430, this results in an internal reaction force along the negative y direction on the intermediate body 130. This forces the intermediate body 130 to rotate clock-wise around the center of rotation 140 and forces end-effector 160 to move clockwise along the trajectory 170. Referring to FIG. 15, a schematic view of an exemplary embodiment of a compliant (i.e., elastic and monolithic) robotic gait 600 is shown. Robotic gait 600 is configured to achieve counter clockwise motion for the end-effector 160 when traveling along the trajectory 170. By connecting the second set of bistable beams 410 and 420 through beam 430 to the left side of the center of rotation 140 (instead of to the right side of the center of rotation 140 as shown in FIG. 14), a counter clockwise motion for end-effector 160 can be achieved when traveling along the trajectory 170.

[0132] Referring to FIG. 16, the robotic gait 100 can achieve different shapes for the trajectory 170 of the end-effector 160 as shown by the closed-loop dashed patterns. The different shapes are achieved, for example, by changing design parameters and / or changing the location of the end-effector 160 on the intermediate body 130.

[0133] Disclosed herein is a robotic device that includes two or more of robotic gaits 100, 400, 500, 600 that together accomplish a robotic locomotion or manipulation task. FIG. 17 shows an example of a hexapod compliant walker that uses six identical robotic gaits 400 that are interconnected in parallel and all are derived with a single actuator 700.

[0134] An underactuated passive locomotion mechanism is disclosed that completes a complex trajectory. Through parametrization of the elastic and compliant structural mechanism a variety of complex gait trajectories are attained, possibly suitable for walking, flapping, rowing, jumping, or swimming gaits. Introducing a bistable element creates a passive preference for any of the two stable path and eliminates the bifurcation points. The resulting mechanism is modelled and tested, showing comparable results. The suggested mechanism is monolithic, allowing for scaling and eliminating the need for assembly.

[0135] A phase difference between legs is attained by introducing a transmission chain. Through applying prestress in the mechanism, a 180° phase difference is created. An assembly of three legs is modelled and tested. The test results show comparable behavior to the modelled counterpart. For two out of three legs, the switching point is identical to the theoretical model. Discrepancies are possibly due to manufacturing defects or deformations whilst applying prestress.

[0136] An actuator assembly is disclosed to connect two triple-leg assemblies together into a single prototype. The prototype successfully completes a 25 cm distance over a flat surface with a single actuator, demonstrating the potential of the underactuated locomotion mechanism. A parasitic displacement in the y-direction is observed as a result of failure to complete the full rotation and slip. The body rotation due to the failure to complete the gait trajectory, marks an opportunity for possible steering mechanisms.

[0137] An underactuated locomotion mechanism is disclosed, and its potential is demonstrated through a fully functional walking prototype. Future research is recommended, however. As the results from the triple-leg tests and prototype have shown, the performance between leg mechanisms differ. Different end effector trajectories are achieved between legs, and not all mechanisms perform their gait trajectory robustly. Further research is required to identify the effect of imperfections on the mechanism. Other than that, manufacturing methods that allow for materials with higher yield strengths will open up the full array of possible end effector trajectories and will facilitate scaling the mechanism down. When implementing different materials, can also be opted for a manufacturing method less sensitive to imperfections. Furthermore, investigating the dynamic effects of the prototype will increase performance metrics such as speed and stability. Lastly, semi-rotation of the locomotionmechanisms resulted in rotation of the prototype body, which potentially allows for a steering mechanism. Expanding on the current design and changing the bistable properties during operation will introduce controllable steering.MATERIALS AND METHODS

[0138] All modelled results are extracted from ANSYS® Mechanical APDL. The design process consists of five distinct steps. First, all possible end effector trajectories are gathered from a simple Von Mises Truss mechanism. In the second and third step, the switching threshold is established and the corresponding bistable element dimensions are selected. Step four offers the closed loop end effector trajectory from the combined kinematic mechanism with bistable embodiment. Lastly, step five assembles three mechanisms with transmission chain.

[0139] During the design process, a design strategy is explored that would allow for bidirectional rotation. Through displacing the mechanism boundary conditions, the bistable properties of the embodiment change, resulting in changing rotation preferences. Attaining robust results, however, proved to be time consuming and the development of this mechanism was terminated prematurely.

[0140] The models from step four and five are validated with physical tests, for which physical mechanisms are produced. The 3 -dimensional printed PLA mechanisms are manufactured with a Bambu Lab XI Carbon printer, fitted with a 0.2 mm stainless steel nozzle. In step five a transmission chain is introduced. In addition to the triple-leg assembly test, a symmetric hexapod prototype is tested.

[0141] Although the disclosed embodiments have been illustrated and described with respect to one or more implementations, equivalent alterations and modifications will occur or be known to others skilled in the art upon the reading and understanding of this specification and the annexed drawings. In addition, while a particular feature of the invention may have been disclosed with respect to only one of several implementations, such feature may be combined with one or more other features of the other implementations as may be desired and advantageous for any given or particular application.

[0142] While various embodiments of the present disclosure have been described above, it should be understood that they have been presented by way of example only, and not limitation. Numerous changes to the disclosed embodiments can be made in accordance with the disclosure herein, without departing from the spirit or scope of the disclosure. Thus, the breadth and scope of the present disclosure should not be limited by any of the above described

Claims

embodiments. Rather, the scope of the disclosure should be defined in accordance with the following claims and their equivalents.CLAIMSWhat is claimed is:

1. A system for robotic gaits, the system comprising: a robotic gait, comprising: an intermediate body including a first end and an opposite second end, an end effector at the first end of the intermediate body and a point of rotation positioned at the second end of the intermediate body, and a moment arm extending transversely from the second end of the intermediate body; two elastic beams each coupled to the point of rotation at an inner end, the intermediate body being free to rotate around the point of rotation, each of the two elastic beams extending from the point of rotation and fixed at an outer end opposite the inner end; and an actuator configured to provide a reciprocating input to the moment arm at a distance on the moment arm from the point of rotation.

2. The system of claim 1, wherein the robotic gait has first and second stable configurations in response to the reciprocating input to the moment arm.

3. The system of claim 2, wherein the robotic gait has a first stable configuration when the reciprocating input is in a loading direction, and the robotic gait has a second stable configuration when the reciprocating input is in an unloading direction opposite to the loading direction.

4. The system of claim 3, wherein in response to the reciprocating input to the moment arm changing from the loading direction to the unloading direction, the robotic gait switches from the first stable configuration to the second stable configuration at a first bifurcation point, and in response to the reciprocating input to the moment arm changing from the unloading direction to the loading direction, the robotic gait switches from the second stable configuration to the first stable configuration at a second bifurcation point.

5. The system of claim 1, wherein in response to the reciprocating input to the moment arm, the end effector follows a closed-loop trajectory.

6. The system of claim 5, wherein the closed-loop trajectory of the end effector is in a clockwise sense when the reciprocating input is provided to the moment arm on a first side of the center of rotation, and the closed-loop trajectory of the end effector is in a counter clockwise sense when the reciprocating input is provided to the moment arm on a second side of the center of rotation, the second side opposite the first side.

7. The system of claim 5, wherein a path of the closed-loop trajectory of the end effector is dependent on at least a length of the intermediate body, a displacement of the reciprocating input, and the distance on the moment arm from the point of rotation where the reciprocating input is provided.

8. A system for robotic gaits, the system comprising: a robotic gait, comprising: an intermediate body including a first end and an opposite second end, an end effector at the first end of the intermediate body and a point of rotation positioned at the second end of the intermediate body, and a moment arm extending transversely from the second end of the intermediate body; a first set of two elastic beams, each elastic beam of the first set coupled to the point of rotation at an inner end, the intermediate body being free to rotate around the point of rotation, each elastic beam of the first set extending from the point of rotation and fixed at an outer end opposite the inner end; a second set of two elastic beams, each elastic beam of the second set coupled to the other elastic beam of the second set at a coupling point at an inner end, each elastic beam of the second set fixed at an outer end opposite the inner end, the second set of two elastic beams forming a bistable element; an elastic connector beam coupled between the coupling point of the bistable element and the moment arm at a distance on the moment arm from the point of rotation; and an actuator configured to provide a reciprocating input to the point of rotation of the intermediate body.

9. The system of claim 8, wherein the robotic gait has first and second stable configurations in response to the reciprocating input to the point of rotation of the intermediate body.

10. The system of claim 9, wherein the robotic gait has a first stable configuration when the reciprocating input is in a loading direction, and the robotic gait has a second stable configuration when the reciprocating input is in an unloading direction opposite to the loading direction.

11. The system of claim 10, wherein in response to the reciprocating input provided to the point of rotation of the intermediate body changing from the loading direction to the unloading direction, the robotic gait switches from the first stable configuration to the second stable configuration at a first bifurcation point, and in response to the reciprocating input provided to the point of rotation of the intermediate body changing from the unloading direction to the loading direction, the robotic gait switches from the second stable configuration to the first stable configuration at a second bifurcation point.

12. The system of claim 8, wherein in response to the reciprocating input provided to the point of rotation of the intermediate body, the end effector follows a closed-loop trajectory.

13. The system of claim 12, wherein the closed-loop trajectory of the end effector is in a clockwise sense when the elastic connector beam is coupled to the moment arm on a first side of the center of rotation, and the closed-loop trajectory of the end effector is in a counter clockwise sense when the elastic connector beam is coupled to the moment arm on a second side of the center of rotation, the second side opposite the first side.

14. The system of claim 12, wherein a path of the closed-loop trajectory of the end effector is dependent on at least a length of the intermediate body, a displacement of the reciprocating input, and the distance on the moment arm from the point of rotation where the elastic connector beam is coupled.

15. A system for robotic gaits, the system comprising: a single-input actuator; and a structural mechanism having a bistable element and being coupled with the singleinput actuator, the structural mechanism performing one or more of the robotic gaits in response to actuation via the single-input actuator.

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