Controlling jumps through latches in small jumping robots

Small jumping robots can use springs to maximize jump performance, but they are typically not able to control the height of each jump owing to design constraints. This study explores the use of the jumper’s latch, the component that mediates the release of energy stored in the spring, as a tool for controlling jumps. A reduced-order model that considers the dynamics of the actuator pulling the latch and the effect of spring force on the latch is presented. This model is then validated using high speed video and ground reaction force measurements from a 4g jumper. Both the model and experimental results demonstrate that jump performance in small insect-inspired resource-constrained robots can be tuned to a range of outputs using latch mediation, despite starting with a fixed spring potential energy. For a fixed set of input voltages to the latch actuator, the results also show that a jumper with a larger latch radius has greater tunability. However, this greater tunability comes with a trade-off in maximum performance. Finally, we define a new metric, ‘Tunability Range,’ to capture the range of controllable jump behaviors that a jumper with a fixed spring compression can attain given a set of control inputs (i.e. latch actuation voltage) to choose from.


Introduction
Jumping can enable robotic and biological systems alike to traverse obstacles taller than their body lengths.This is particularly important for locomotion on irregular terrain.One way for robotic and biological systems to jump is to use an elastic mechanism, e.g. a spring, that is loaded by an actuator, e.g. a muscle or a motor, and released by a latch [1][2][3][4].Since the jump is powered directly by the spring and not by the actuator, controlling the resultant jump's take-off energy requires controlling the energy stored and subsequently released by the spring.
One way that larger jumping robots tune their jump take-off energy is through tuning the amount of energy stored when the spring is loaded [5,6].In contrast, smaller jumping robots have generally been more limited in their abilities to control their jump take-off energy.In many of these small robots, the spring compression is fixed during the design stage resulting in a jump take-off energy that cannot be tuned without changing the design or adding a secondary mechanism [7].For example, the 7 g EPFL jumper [8] and a later version of Grillo [9] use eccentric cam mechanisms to load and release the springs.Since the cam profile does not vary from jump to jump and the spring compression is fixed, the take-off energy is ultimately constant.The only way to tune the take-off energy between jumps is to change the cam profile by manually replacing the cam or through changing the spring.A design feature in the EPFL jumper allows for manual intervention to change the spring compression between jumps.Similarly, in the older versions of the Grillo robot [10], a magnet is used opposite the spring to block the release of the spring until the spring force is sufficient to overcome the magnetic force.For a given magnet, the spring compression is fixed resulting in fixed take-off energy.
As a further example, the hook that secures the loaded hind legs of the locust-inspired TAUB robot does not allow the spring compression to vary [11].In jumpers such as the MSU jumper [12,13] and the water strider inspired jumping robot [14], the spring loading and release is coupled through a geometric reconfiguration that also results in a consistent jump height due to fixed spring compression.Some small jumping robots have been able to change their jumps without redesign or manual intervention.Most notably, the shape memory alloy (SMA)-based flea-inspired jumpers from [15,16] can change jump performance on the fly.The performance is changed by changing the spring stiffness of their SMA springs by varying the input current; the Joule heating changes the material properties of the SMA spring, thereby changing the total energy stored.The JumpRoACH robot is able to change its stored elastic energy and thus its jump height, by using an active clutch mechanism [17,18].Another novel jumping robot includes the magnetically actuated 3.1 mm jumper that consists of a latex spring attached to a five-bar linkage mechanism driven by a pair of magnetically-actuated gears [19].By using a magnet with a different remnant magnetization, the jumper is able to change its stored elastic energy.It should also be noted that several jumping robots can change their jump heights without changing their take-off energy by controlling the take-off angle rather than controlling the energy at take-off (e.g.[20,21]).
Jumping with springs, and more generally using springs as actuators, is described in the latchmediated spring actuation (LaMSA) framework [22,23].A generic LaMSA system has a spring that is loaded and then released by a latch to power a movement.This latch can be implemented in a variety of ways, including as a physical constraint on the spring or a geometric configuration as described for the robots above.New research on LaMSA systems that incorporate non-ideal springs and latches show that LaMSA systems are capable of a range of outputs [4,[24][25][26][27].While latches have historically been viewed through a binary lens, either present and blocking energy release from a spring, or not present and enabling energy release, previous work shows a more nuanced role for latches, in which they can influence the timing and rate of energy release [4,25].The design and operating parameters of latches affect how stored energy in a spring is converted into jumper kinetic energy.
These new results from [4,25,26] reveal how latches can be a tool for controlling the performance (such as take-off velocity), albeit in a feedforward manner in ultrafast systems.This foundational framework established in previous work assumes that the latch is removed at a constant latch velocity (v L ).This simplifying assumption has been crucial for understanding the fundamental importance of latches.However, in practice, the latch actuator, in the form of a muscle or a motor, introduces a layer of complexity, where the transient dynamics of the actuator are on similar time scales to the recoiling of the spring.Furthermore, in previous work, the latch was applied as a kinematic constraint.This results in the modeled latch being infinitely stiff and unaffected by the spring force pushing on the latch during the unlatching phase.These assumptions were important for understanding the foundational principles of latches, but are not valid for real-world systems, especially robots.This is because latches need to be removed by an actuator, which has its own dynamics, to release the potential energy stored in springs, and the spring force pushes on the latch during the unlatching phase.For the design and control of robots, it is important to understand how the motor-latch dynamics affect latch-mediation, and in turn, the jump output.
In this work, we refine and demonstrate the utility of latches and latch-mediation for tuning the jump performance of jumping robots.The reduced-order model from previous work [4,25] is expanded to incorporate the dynamic interactions of the motorlatch and mass-spring subsystems.This expanded model is then used to probe the following questions: (1) Is it feasible to utilize latch-mediation for tuning jump output of a jumping robot?(2) What is the range of performance outputs that a jumper can achieve given a fixed set of control inputs, i.e. latch actuation?And (3) what are the associated tradeoffs of this approach?The aforementioned model is then validated experimentally using a 4 g jumper.Answering these questions is crucial for designing LaMSA-based jumping robots for control rather than maximum jump performance, thereby incorporating a degree of mechanical intelligence [28] into these small resource-constrained systems for the first time.

Model
The reduced-order model from previous studies assumes that a linear spring is compressed a fixed distance and a latch with radius R is removed with a constant latch velocity, v L [4,25].In the current work, we have adapted this model to represent a more realistic latch actuator by relaxing the constraint of constant latch velocity and including the dynamics of the latch motor.The influence of force applied by the mass-spring pair onto the latch is also considered.Furthermore, the effect of damping in the mass-spring pair is also considered. Figure 1(A) illustrates the reduced-order model of the LaMSA system, with the latch characterized by latch radius R and latch-motor actuation characterized by voltage V S (in contrast to a constant latch-release velocity used in previous work).The mass-spring pair is characterized by mass m and spring stiffness k as in previous models.Figure 1(B) illustrates a close-up view of the interface between the latch and the mass-spring pair.The output parameter of interest is the jumper's takeoff velocity v to .
Figure 1(C) shows different phases of the LaMSA jumper.In the latched phase, the latch is engaged on the mass-spring pair with a fixed potential energy stored in the spring.When a latch-motor actuation voltage is applied, the motor pulls the latch which begins the unlatching phase.During this phase, the spring recoil is constrained by the latch.The translational velocity of the latch ẋ is influenced not only by the motor dynamics, but also by the force applied by the spring onto the latch, which further accelerates the latch.When the latch is completely removed, i.e. unlatched, the jumper's spring now recoils in an unconstrained manner with the initial conditions for this phase (unconstrained spring-actuation) being set by the end of the previous phase (unlatching phase).When the jumper's spring reaches its equilibrium length, the jumper takes off and enters the ballistic phase.
The displacement (y), velocity (ẏ) and acceleration (ÿ) of the mass-spring pair during the unlatching process are given by equations ( 1)-( 3).Here, x, ẋ, and ẍ refer to the translational displacement, velocity and acceleration of the latch, respectively, Given the rotary nature of the latch in the jumper used in this study (jumper design discussed in section 2.2), the rotational quantities of the latching subsystem are linked to the translational quantities of the latch as shown in equation ( 4).Here, γ denotes the gear ratio, D latch represents the moment-arm distance measured from the latch's axis of rotation, and ω denotes the rotational velocity of the latch motor, Equations ( 5) and ( 6) represent the dynamics of the latch motor considering the force from the mass-spring pair, the jumper's inherent damping, and a resistive torque term.J, k t , k b , R e , L e , and I represent the net moment of inertia, torque constant, back-EMF constant, winding resistance, winding inductance and current of the latch motor respectively.∆y denotes the initial spring compression in the jumper's spring.Damping ratio ζ represents the jumper's inherent damping [29].θ denotes the angle of contact between the latch and the mass-spring pair as shown in figure 1(B) and is described by equation ( 7), In the jumper modeled here, the latch motor attempts to pull the latch out of the way of the massspring pair.ϕ M represents the angle of rotation if the rotation was purely due to the applied motor torque.At the same time, the mass-spring pair applies a force in the same direction as that of the latch motor, thereby also contributing force to push the latch out of the way.ϕ MS represents the angle of rotation from the combination of both the motor torque and the torque reflected from the mass-spring pair.ϕ M and ϕ MS can be obtained by solving equations ( 8) and ( 9) respectively, Given that the torque applied by the mass-spring pair on the latch is in the same direction as that of the motor rotation, a resistance term, κ, is included to capture the general resistance that a geared motor will experience when rotating.An empirical value of 0.3 is used for κ based on the jumper's performance for an R4(4 mm) latch operated at 6 V.
The above equations were solved to obtain the jumper position, velocity and acceleration (y, ẏ, ÿ) during the unlatching phase.Once unlatched, the jumper enters unconstrained spring-actuation and y, ẏ, ÿ are obtained by solving equation (10).Equation (11) was solved to obtain the unlatching time t L , which is the transition point between the unlatching and unconstrained spring-actuation phases,

Jumper
To test the hypothesis that latches can be used to control the release of stored spring potential energy in small jumpers with fixed spring compression, a 4 g jumper was designed and fabricated.Figure 2(A) shows a schematic of the LaMSA jumper with its key elements highlighted: the spring that is manually loaded to store energy, the latch to mediate the release of this stored spring potential energy, and a motor to remove the latch The jumper consists of a main body and latch that were 3D printed with ABS using an FDM printer (Raise 3D Pro2 Plus).The body includes a cylindrical casing and shaft hole to mount the motor and latch respectively.A 2 mm diameter carbon fiber rod provides support and acts as a central shaft for the spring.The jumper was designed to allow a fixed spring compression of 5.4 mm.The spring (Lee Spring LC061A05M) was manually loaded before each jump and held in place by a latch blocking the release of this stored potential energy.A small 1.2 g DC motor (26:1 sub-micro plastic planetary gearmotor, Pololu 2357) rated at 6 V was fit inside the main body.The motor is connected to the latch via a gear train (gear ratio of γ =68/40) consisting of two gears.A gear with a 14.4 mm diameter is connected to the motor while the latch connects to a gear with a diameter of 18 mm.The gears were 3D printed with a polyjet printer (Objet24) using VeroWhitePlus material.The feet of the jumper were made from 1 mm carbon fiber rods held in place by 3D printed ABS joints.
The latch was made of ABS and 3D printed using an FDM printer, similar to the jumper's body.The latch consists of a cylindrical shaft connected to a solid block (figure 2(D)).The shaft fits inside the main body of the jumper and connects to the motor through a gear train.The latches were printed with the desired radius of curvature incorporated into the solid block (highlighted in green in figure 2(B)).
The physical latches used in this study are shown in figure 2(D).Actuating the motor turns the latch through the gear train resulting in latch removal and the subsequent release of stored potential energy.For a given latch radius R, changing the motor voltage (6-12 V) results in the latch turning and releasing at different velocities (i.e.varying v L ).The assembled jumper weighs 4 g and is 30 mm tall in its uncompressed state.Table 1 lists the jumper parameters used in this work.Table 2 lists the latch motor parameters used in the simulations.

Experimental setup
To characterize jump performance, the jumper was filmed using two high-speed cameras.One highspeed camera (Photron FASTCAM Nova S12) was equipped with a macro lens (Zeiss Milvus 2/100 M) and recorded the jumps from the front view of the jumper with a pixel resolution of 1024 × 1024.A second high-speed camera (Photron Mini AX200) equipped with a Nikon 35 mm lens was used to record the rotation of the latch gear from the top with a pixel resolution of 1024 × 672.Both cameras were synchronized and recorded at 10 000 fps.The jumper jumps from a 3D printed ABS jump platform that is attached to a Futek LSB205 load cell to measure the ground reaction forces during the course of the jump.This load cell has a rated bandwidth of 25 kHz and the data collected are synchronized with the jump data by synchronizing the measured peak force with the jumper's tracked acceleration peak.
Prior to each jump, the jumper was loaded manually, compressing the spring by a fixed distance of 5.4 mm.The jumper was supplied with power via a 32 gauge wire tether connected to a power supply (Sorensen XPF60-20DP) with a set voltage to drive the latch-motor.The jumper was filmed jumping over four trials at motor voltages (V S ) ranging from 6 V to 12 V, in 2 V increments for each latch radius.Three different latch radii were tested: 1 mm, 2 mm, and 4 mm.The jumper's jump trajectory as well as the latch trajectory were then analyzed using tracking software (Image Systems, TEMA).

Model simulation
To understand how the jumper's take-off energy varies across different latch radii and latch actuation voltages, the jumps were simulated in MATLAB using the ode45 solver.The simulation begins from the start of the unlatching phase and ends when the jumper takes-off.The jumper's take-off point is when the spring reaches its equilibrium length.Figure 3(A) shows the simulated take-off velocities at different latch-motor actuation voltages (6-30 V) for three different latch radii: R = 1 mm (R1), R = 2 mm (R2), and R = 4 mm (R4).A wide voltage range of 6-30 V is chosen for the simulations to observe the output behavior over a large range of input parameters.Increasing the latch-motor voltage V S for a given latch radius R results in a faster latch-release and a smaller unlatching time t L , thereby leading to a faster takeoff.This increase in performance output is greater for a larger R, that is, a larger latch radius results in more performance variation in the jumper's take-off characteristics.
Figure 3(B) depicts the jumper's normalized performance where the jumper's energy at take-off is normalized to the maximum energy at take-off across all simulated latch configurations (i.e.latch radius and actuation voltage).The performance range indicated for each latch radius denotes the tunability range.Tunability range is a metric that we define in this work to refer to the range of kinetic energy outputs attainable with a specific latch radius R, given a fixed set of latch-motor actuation inputs V S .Tunability range is defined by equation (12), where E VS to refers to the set of a jumper's kinetic energy outputs at take-off for a specific latch radius over the set of latch-motor actuation inputs (i.e.V S ∈ [6-30 V] in this case), and E max to refers to the jumper's maximum kinetic energy at take-off across all latches, A larger latch radius results in a higher tunability range given a fixed set of actuation inputs (V S ).For example, as shown in figure 3(B), a latch radius of R = 4 mm results in a tunability range of 36.8% considering an actuation voltage set of V S ∈ [6-30 V].In contrast to the example of the R1 latch, it is clear that this tunability results in a trade-off in the maximum energy output.Therefore, jumpers can be designed to enhance control of jump performance using latch geometry with the latch-motor voltage V S acting as the control input.Jumpers with larger latch geometries result in greater tunability, but at the expense of maximum performance.
Figures 4(A), (C) and (E) depict how the jumper's velocity changes over time for different actuation voltages V S across 1 mm (R1), 2 mm (R2), and 4 mm (R4) latch radii respectively.Figures 4(B), (D) and (F) depict how the latch velocity changes over time for different actuation voltages V S across the three different latch designs.Studying latch velocities is important as it helps better explain the unlatching process and underscores how differently a latch with its own dynamics behaves when compared to the constant latch velocity assumption from previous work.The resulting variation in turn helps explain the variation in the jumper's velocities.The black squares indicate the point at which the unlatching phase ends (i.e. the latch is unlatched) which dictates the initial conditions for the next phase: unconstrained springactuation.The green circles indicate the point at which take-off occurs.
When latch actuator dynamics are added to the model, the latch velocities increase over time.As the latch motor begins to unlatch, the spring begins to recoil and applies a force back onto the latch, further accelerating the latch out of the way of the spring.This process can be considered as positive mechanical feedback.It can be seen that higher V S results in faster latch velocities which in turn dictate the rate of spring recoil during the unlatching phase.For smaller latch radii such as the R1 latch, the change in actuation input results in little change in the jumper's take-off velocity and the change in latch velocities is also small.A larger latch radius (R4) results in a larger range of take-off velocities for a given set of actuation inputs.Latch velocities also vary considerably across this range of input voltages.This is expected given that larger latch geometries increase the range of tunable output behaviors as previously seen in [25].This is reflected in the tunability range shown in figure 3(B).These results demonstrate that the unlatching process can be used to control the timing and magnitude of the energy released from a spring, despite starting with a fixed spring potential energy.

Experimental validation
Figure 5(A) depicts the measured jumper take-off velocities at different latch actuation voltages for R1, R2, and R4 latches.For the experiments, the latch motor actuation voltage V S is varied between 6-12 V.This range was chosen given the motor nominal voltage of 6 V; increasing the voltage beyond   12 V increased the risk of burning out the motor.The experimental velocities, including take-off velocities, are obtained from the tracking software TEMA (Image Systems AB) which uses a fourth-order phaseless Butterworth filter with default settings of 3 and 9 frames as the minimum and maximum filter lengths respectively.Take-off velocity corresponds to the velocity during the frame at which the jumper takes off from the ground.Simulated model results are also included in figure 5(A) depicted by dashed lines.As expected from the behavior observed in the model, R1 and R2 latches result in smaller variations in take-off velocities, whereas R4 results in a larger range of take-off velocities.The differences in experimentally observed take-off velocities and simulations can be due to sources of energy dissipation such as sliding friction and other mechanistic losses that occur during the spring recoil phase.Figure 5(B) shows the jumper's normalized performance.Brown patches indicate the experimentally observed tunability range for the 4 g LaMSA jumper, whereas the green patches indicate the model results for the 6-12 V range.
As expected from the previous modeling results, a larger latch radius results in a greater tunability range for a given set of actuation inputs, V S ∈ [6-12 V].The R4 latch results in an experimental tunability range of 35.1% while R1 and R2 latches result in 11.4% and 13.4% experimental tunability range respectively.
As expected, the R4 latch results in a greater range of energy outputs, while trading off maximum performance/energy output.In contrast, the R1 and R2 latches are closer to the maximum energy output while having a smaller tunability range.This is also observed in the jump heights attained by the jumper for each latch as shown in figure 5(C)-(E).With the R4 latch, the jumper attains about 2.6x its body height when actuated at 6 V and 4.5x its body height when actuated at 12 V.Thus, despite starting with a fixed spring potential energy, a latch can be used as a tool for open-loop or feedforward control of energy output in LaMSA-inspired synthetic jumpers where the energy output obtained at take-off is determined by the latch geometry and actuation.Furthermore, it shows that despite the ultrafast nature of LaMSA systems, a designer can choose to design for control, but at the cost of maximum performance for this particular latch design.
It is also notable that the experimentally observed tunability range is greater than what is predicted by the model (indicated by the green patch in figure 5(B)).This is explained by a closer examination of latch velocity and ground force measurements.Figures 6(A)-(C) depict the experimentally measured jumper velocity, latch velocity, and ground force measurements for the jumper with an R1 latch at different actuation voltages.Given the small latch geometry, there is little variation in the jumper velocities across different actuation voltages.When unlatching begins (at t = 0), the force applied by the spring on the latch, begins to act in addition to the motor torque which accelerates the latch.Given that this force from the spring pushes the latch in the same direction that motor torque pulls the latch, the force from the spring tries to accelerate the latch faster than the motor can pull.This dynamic interaction results in the latch accelerating out of the way and then slowing down.At higher V S , the pull of the latch motor is higher resulting in faster velocities over time.Figures 6(D)-(F) show the measured jumper velocity, latch velocity, and ground force measurements for the jumper with R2 latch at different actuation voltages.Similarly, figures 6(G)-(I) depict the same for the R4 latch at different actuation voltages.The latch velocity measurements as well as the ground reaction force measurements show that the latch release process is not as simple as modeled in figure 4 likely because of this spring/latch interaction and deceleration of the latch.The ground reaction forces show that these interactions can lead to multiple force peaks, especially for higher-radius latches.

Discussion
The major contribution of this study is in defining the utility of latches in small jumping robots, where the transient dynamics of latch removal actuators play a role in mediating the release of a fixed, stored spring potential energy.For a fixed spring compression, latches can be used to mediate the release of this stored spring potential energy to vary the energy output, jump take-off velocity, and ultimately jump height.This is achieved through a combination of latch geometry and latch removal without adding additional mechanisms or adjusting the initial energy stored in a spring.While varied methods exist to control jump trajectories and heights like changing takeoff angle [20,21] or adding winged structures to control landings [30,31], latches provide an additional pathway for control in small resource-constrained jumping robots as we build towards autonomous jumping robots.
Similar to the findings in [25], our modeling and experimental results indicate that higher latch-motor voltages, corresponding to higher latch removal velocities, result in faster take-off velocities.For a fixed set of latch motor voltages, this range of take-off velocities is larger for larger latch radii, showing that resource-constrained robots need to consider design features, like latch radii, when designing for control.However, this work also relaxes some of the simplifying assumptions made previously, and the relationship between latch velocity and jumper take-off velocity is more nuanced that previously described.This results from considering the real-world limitations of actuators that are feasible for use in small, jumping robots.Many off-the-shelf actuators at smaller scales, such as the 0.6 g Pololu DC motor (26:1 gear ratio) used in this study, come with their own actuation bandwidth, input limitations, and performance limitations.As a result, our results show that a tradeoff exists between our newly defined Tunability Range metric and maximum jump height.This is in contrast to the predicted ability to achieve both maximum jump height and tunability in previous work [25].Our results instead show that if the maximum energy output is desired, then using a latch with a sharper radius is preferred.However, if the goal is to design the jumper to attain a desirable range of tunable outputs, a larger latch radius is advisable.
Understanding the actuator limitations given a fixed set of inputs can be used in the design process through our novel metric of tunability range.We show that these ultrafast synthetic mechanisms can be designed for control through latch geometry rather than for maximum performance.These synthetic systems can be designed with larger latch geometries that enable a large span of tunability range given a set of actuation inputs, thus shifting the design goal from absolute performance to designing these systems for control through latch-mediation.However, the effectiveness of this approach has practical limitations too.Given the limits of actuation bandwidth, latch geometry cannot be increased beyond a certain measure for a given spring compression.In other words, this strategy is effective when having a comparable size between the latch radius and the spring, otherwise the tunability range of the jumper will be very small for a fixed range of actuation voltage inputs.
Another prominent feature of the results is the time scale involved.As shown in figure 6, jumps occur in less than 5 ms and the latch motor used does not reach its steady-state behavior over this time period.The unlatching phase occurs during the transient period of the latch motor.This contrasts with previous work that assumed a constant latch velocity.However, despite this, latch-mediation through geometry and actuation can still be used to tune the release of stored spring potential energy by influencing the motor's transient dynamics.
There are also differences between the model and experiments.The model assumes a mass-spring-damper system without friction.The physical jumper consists of losses due to various sliding components.Furthermore, the model assumes that unlatching begins at time t = 0 when the latch radius comes into play.However, in the physical jumper, the latch accelerates horizontally before reaching the latch curvature where unlatching begins.This results in faster latch velocities in the physical jumper than in the model.The model can be further improved by modeling the friction acting between the 3D printed sliding surfaces.In addition, the process of latch removal can be modeled to incorporate a small distance over which the latch initially slides before its latch radius comes into play, which may prove interesting for potentially removing some of the transient actuator dynamics.
While this work used latches with a fixed radius and varied the rates of removal through actuation motor inputs, it is also possible to engineer jumpers in which the latch's geometry is changed while the removal is actuated at a fixed input voltage.This is an exciting avenue for future research, particularly given the prevalence of smart materials, such as SMAs in small, jumping robots.For example, SMA-based latches could be thermally activated to change the latch curvature or even vary latch stiffness.Furthermore, although this study used contact latches (physical blocks that rely on contact), the physical implementation of latches can vary.The contact latch in this work simply acts as a constraint force on the spring recoil, and the combination of geometry and latch removal results in different rates at which this constraint force decays to zero.How the results in this work map to other types of latches, including fluidic [32], geometric [15,16,[33][34][35], electrostatic [36,37], and magnetic mechanisms, remains an open question in the study of ultrafast systems.

Figure 1 .
Figure 1.(A) Reduced-order LaMSA model with the latch characterized by latch radius (R) and latch-motor actuation voltage (VS).(B) Close-up view of the interface between the latch and mass-spring pair.Here, the constraint of constant latch-velocity is relaxed, by allowing the latch retraction to be influenced by the feedback from the mass-spring pair.(C) Different LaMSA phases.

Figure 2 .
Figure 2. (A) Schematic illustration of the jumper depicting the spring, latch and latch motor.(B) CAD illustration of the jumper.(C) Physical realization of the jumper loaded with energy sitting on top of the jump platform.A force sensor measures the ground reaction force during the course of the jump.(D) Sample latches with different radius of curvature (R1 : 1 mm, R2 : 2 mm, R4 : 4 mm).

Figure 4 .
Figure 4. (A) Simulated velocity trajectories of the jumper for different latch-motor actuation voltages VS for R1 latch.The black squares indicate the point at which the unlatching phase ends, while the green circles indicate the point of take-off (i.e.end of unconstrained spring-actuation phase).(B) Latch velocities for different latch actuation inputs VS for the R1 latch.(C) Jumper's velocity trajectories for R2 latch.(D) Latch velocity trajectories for the R2 latch.(E) Jumper's velocity trajectories for the R4 latch.(F) Latch velocity trajectories for the R4 latch.

Figure 5 .
Figure 5. (A) velocities recorded for the 4 g LaMSA jumper at different VS ∈ [6-12 V] for three different latch radii.Dashed lines indicate simulated take-off velocities while circular markers indicate experimental results.Error bars indicate standard deviation over four jump trials.(B) Experimentally measured tunability range for the jumper for three different latch radii compared against the simulated tunability range (VS ∈ [6-12 V]).(C)-(E) Jump height trajectories for the jumper with R1, R2 and R4 latches respectively.Solid lines indicate mean height trajectories while shading indicates the standard deviation in trajectory across four trials.

Figure 6 .
Figure 6.(A), (B) Jumper and latch velocity trajectories for the 4 g LaMSA jumper with an R1 latch at different latch-motor actuation voltages.(C) Ground force measurements during the course of the jump for the jumper with an R1 latch.(D), (E) Jumper and latch velocity trajectories for the 4 g LaMSA jumper with an R2 latch at different latch-motor actuation voltages.(F) Ground force measurement during the course of the jump for the jumper with an R2 latch.(G), (H) Jumper and latch velocity trajectories for the 4 g LaMSA jumper with an R4 latch at different latch-motor actuation voltages.(I) Ground force measurements during the course of the jump for the jumper with an R4 latch.Solid lines indicate mean trajectories while shading indicates the standard deviation in trajectory across four trials.

Table 1 .
List of jumper parameters.

Table 2 .
Model parameters for the latch motor used in the simulations.
through a gear train.Figures2(B) and (C) show a CAD illustration of the jumper and its physical realization.