The Design and Performance of DASH

Transcription

1 The Design and Performance of DASH Paul Birkmeyer Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS May 14, 2010

2 Copyright 2010, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Acknowledgement The author would like to thank Ron Fearing for his brainstorming, support and guidance, and Alexis Birkmeyer for her mental support and willingness to share her time and Illustrator skills. Thanks to Bob Full for his advice and support. Thanks also to Aaron Hoover and the rest of the Biomimetic Millisystems Lab for their support, Kevin Peterson for his help with the IROS paper, the Center for Integrative Biomechanics in Education and Research for their advice and the use of their force platform, and the CRAB Lab for their collaboration. This work is supported by the NSF Center of Integrated Nanomechanical Systems.

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4 Abstract DASH, the Dynamic Autonomous Sprawled Hexapod, is a small, lightweight, power autonomous robot capable of running at speeds up to 15 body lengths per second. Drawing inspiration from biomechanics, DASH has a sprawled posture and uses an alternating tripod gait to achieve dynamic open-loop horizontal locomotion. The kinematic design which uses only a single drive motor and allows for a high power density is presented. The design is implemented using a scaled Smart Composite Manufacturing (SCM) process. Two different means of turning are presented, one of which is actuated. Actuated turning results from altering the kinematics via body deformation, a method supported by a simple 2-D dynamic model. Evidence is given that DASH runs with a gait that can be characterized using the spring-loaded inverted pendulum (SLIP) model. In-situ power measurements are performed to give cost of transport values both on hard ground and granular media. In addition to having high maximum forward velocities, DASH is also well suited to surviving falls from large heights due to the uniquely compliant nature of its structure.

5 Contents 1 Introduction 1 2 Mechanical Design Design Overview Hip Design Differential Drive Leg Design Body Design and Steering Hexapod Simulation Construction Simulation Results Results Running Performance Turning Results Step Climbing Surviving Falls Discussion of Results and Conclusions 39 i

6 List of Figures 1.1 Photo of DASH Simulation of kinematics in MATLAB model Kinematics illustrations of the hip mechanism SolidWorks model of SCM hip joint SolidWorks model of SCM differential drive mechanism Complete SolidWorks model of DASH Horizontal and compliant leg designs Stiff leg rotation versus four-bar linkage leg rotation Two designs using a four-bar linkage leg Kinematic illustration showing effect of body deformation to produce turns Four-bar ankle to provide normal compliance and turning behavior D hexapod model used in dynamic simulations Simulation results of straight running showing center of mass motion and instantaneous velocity Simulation results of straight running showing relative heading, orientation, and angular velocity Simulation results showing center of mass motion during S-turn maneuver Simulation results showing forward velocity and angular velocity during left turn Simulation results showing relative heading and orientation before and after rightturning input is applied Images of dynamic running in DASH ii

7 4.2 Forward velocity vs motor duty cycle when using parallel leg design Tracked center of mass during a horizontal run SLIP-like behavior of DASH Cost of transport on hard ground and granular media Experimental results of S-turn maneuver in DASH Turning rate vs duty cycle using toe-extension turning method Step-climbing behavior in DASH Images of high-velocity impact iii

8 List of Tables 3.1 Parameters used in simulations of 2-D rigid-body hexapod Select properties of comparable robots iv

9 Chapter 1 Introduction Highly mobile, small robotic platforms offer several advantages over larger mobile robots. Their smaller size allows them to navigate into more confined environments that larger robots would be unable to enter or traverse, such as caves or debris. Small robots are easily transported by either vehicles or humans to be deployed to remote locations as needed. One example includes field workers who need access to otherwise dangerous, inaccessible areas such as collapsed buildings or ones damaged in earthquakes. Small, inexpensive robots are also a key component for rapid installation of ad-hoc networks. As robot size decreases, however, maintaining mobility can become a challenge if climbing means are not available. As objects are larger relative to body dimensions, legs gain advantages over traditional wheels or treads by being able to overcome obstacles greater than hip height. Reduced size also introduces challenges involving controls and power; reduced volume leaves less space for the multiple actuators per leg typically seen on larger robots. Biology offers a wealth of examples of small creatures with remarkable locomotion abilities, such as the cockroach. Studying these animals has revealed lessons that guide the design of legged robots. One such example is preflexes wherein passive mechanical elements, such as tendons, 1

10 contribute stabilizing forces faster than the neurological system can respond [1, 2]. Another passive design element that imparts stability is the sprawled alternating tripod gait. In addition to static stability, this limb arrangement has been shown to lend robustness to both fore-aft and lateral perturbations while running [3], suggesting application in legged robots. The gaits of running animals suggest an additional template useful for stable running. The spring-loaded inverted pendulum (SLIP) model can explain the ground reaction forces during running seen in almost all legged systems with the relative stiffness being nearly uniform across all species [4, 5]. The SLIP model has also been shown to impart dynamic stability to horizontal running. Possessing SLIP-like behavior may benefit the mobility of any legged robotic system. Much work has been done in highly mobile legged robots, some of which incorporate these lessons derived from biology. Examples include Sprawlita [6], isprawl [7], and RHex [8]. These robots incorporate passive compliance to achieve preflex-like behavior and SLIP-like motion. They also all use some form of feed-forward controls, mimicking the central-pattern generator feature from biology [2, 5]. 10 cm Figure 1.1: DASH: a power autonomous hexapedal robot. 2

11 The Dynamic Autonomous Sprawled Hexapod, or DASH, uses design principles from biology and employs a differential drive using only a single motor. It is capable of power autonomous, robust dynamic locomotion at speeds of approximately 15 body-lengths per second (1.5 m/s). DASH measures 10 cm in length, weighs 16.2 grams, and uses wireless communication for feed-forward commands. It is constructed using a scaled Smart Composite Manufacturing (SCM) process that enables fast build times, scalable designs, and lightweight systems [9]. The SCM process also enables falling survivability through an energy absorbing structure and high surface-to-weight ratio. 3

12 Chapter 2 Mechanical Design The primary challenge in the development of DASH was to create a robot that is capable of efficiently and effectively coupling the output of a single actuator to the legs to create a dynamic, feed-forward locomotion. Though long-term goals include developing the ability to run vertically, initial efforts were focused on developing a robot that could first run effectively on a horizontal surface. Once the drive mechanism and legs are refined from their original designs, they can then be adapted to more complicated terrain as well as climbing. The first step in development is determining the manufacturing process. Previous work has developed the Smart Composite Microstructures (SCM) manufacturing process [10], made by laser micro-machining carbon fiber laminates to form rigid linkages with flexible, polymer joints. This process was later adapted to a prototyping process which uses cardboard rather than carbon fiber laminates as the material for the rigid linkages [9]. This process easily produces lightweight robots with lengths on the order of centimeters and allows for a manufacturing time several orders of magnitude shorter than with the original SCM process. To create a dynamic, high-power density robot, there should be a minimal number of power- 4

13 dense actuators that serve as the main drive actuators. For example, if there is one actuator per tripod, the tripod in swing contributes no work to the system. For the design of DASH, a single DC motor was chosen as the main drive actuator for all six legs. DC motors, compared to other actuators such as piezoelectric actuators and shape memory alloy (SMA) wires, provide longer stroke lengths, higher efficiencies, greater power densities, easier electrical interfacing, and operate in a region of frequencies comparable to those observed in insect legged locomotion when appropriately geared [11]. A sprawled alternating tripod gait was chosen for DASH. This gait is often seen in insects, and it functions well when locomoting on horizontal, inclined, and vertical surfaces [4, 5, 12, 13]. By keeping the center of mass above the support polygon, an alternating tripod is conveniently statically stable. A sprawled stance with an alternating tripod gait has also been shown to impart stability in the presence of fore-aft and lateral perturbations [3]. Alternating tripod gaits can be seen in prior legged robots, including isprawl [7], RHex [8], and Mini-Whegs [14], and several also exhibit sprawled postures [6, 15]. 5

14 2.1 Design Overview #" E3 (mm) "!#"!"!$" "!" #! " E 2 (mm)!#!!!"!!" E 1 (mm) Figure 2.1: Simulated foot trajectory for DASH. The circles are traces of the feet as they move through one stride. The two different colors represent the two different tripods which are 180 degrees out of phase. The coupling of the DC motor output to the legs functions much like the oar of a rowboat: the circular input from the motor drives the end of the oar-like leg in an amplified circular motion through a mechanical coupling. By orienting the output of the motor in the sagittal plane of DASH and coupling it to the oar mechanism, the resulting circular foot trajectories are aligned roughly with the sagittal plane. This creates leg trajectories which have both fore-aft and vertical displacements. To ensure that the hips never contact the ground, the circular orbit is biased slightly downwards. Figure 2.1 shows a MATLAB model of the desired kinematics. The model offers a quick method of varying design parameters and seeing the resultant changes in kinematics in 3-D space. The circular foot trajectories traced by each tripod are marked with different colors. 6

15 2.2 Hip Design In the drive mechanism of DASH, the hip linkages provide both the fore-aft and vertical motion of the legs. These two degrees of freedom are largely decoupled and can be understood separately. Figure 2.2(a) shows a kinematic drawing of the vertical degree of freedom and Figure 2.2(b) shows the fore-aft degree of freedom in the hip. In both figures, the grounded linkages are those mounted to the motor casing, and the linkages adjacent to the input arrows are those mounted to the output of the motor gearbox. In Figure 2.2(a), looking at the front of DASH, the vertical motion of the top beam relative to the grounded beam causes the hips to rotate in opposite directions. This rotation results in vertical motion of the feet, and the anti-phase motion of the opposing hips is required for alternating tripod gaits. Looking from the top of DASH, Figure 2.2(b) shows how fore-aft motion of the middle horizontal beam moves the two opposing four-bar linkages, swinging one leg forward and the other leg backward. The hip mechanism appropriately couples the input beams of Figure 2.2 to the vertical and fore-aft displacement of the final stage of the motor gearbox to create the desired leg trajectory. 7

16 Top FRONT VIEW Bottom (a) Front TOP VIEW Back (b) Figure 2.2: Visualization of the degrees of freedom that enable (a) vertical motion and (b) lateral motion of the legs. In both images, the beams to which the hips are attached are grounded. The realization of the kinematic drawings of Figure 2.2 in a SCM manufacturing process is shown in Figure 2.3. Viewed horizontally from the side of DASH, one can see how a circular input applied to the hip linkages create the circular motion of the feet. The grounded linkages and input linkages from Figure 2.2 are the transparent beam and black beams in Figure 2.3, respectively. The opaque beige linkages are the hip four-bar mechanisms and their attached legs. The output of the motor gearbox is applied through mechanical constraints discussed in Figure 2.4, and the resulting motion of the black input linkages is shown with a green dashed line. Motion of the end of the black linkages along this circular dashed line result in the motion the legs as shown. Two adjacent hips are shown to illustrate that they do, in fact, move 180 degrees out of phase with each other. When these two hip designs are propagated appropriately to the remaining hips, it creates 8

17 the desired alternating gait. (a) (b) (c) (d) Figure 2.3: Four different positions of two adjacent hip joints from DASH presented from a lateral view. The black beams are attached to a circular input. The beams move in the dotted paths, causing the motion of the legs. The axes of rotation for the hips lie on opposite sides relative to their respective inputs, creating an anti-phase motion. The image in the top left shows when the input is in the top left position of its circular trajectory and the image in the top right shows when the input is in the top right position the circular trajectory, and likewise for the (c) and (d). The input motion prescribed here would move the robot to the right. 2.3 Differential Drive The hip design of Figure 2.3 requires the motion of the black input beams relative to the motion of the transparent beam. Moreover, this relative motion is designed to be the circular motion of the motor output. We created the differential drive mechanism of Figure 2.4 to ensure that the single output of the drive motor drives each of the six hips with a similar circular input. Zoomed out and 9

18 (a) (b) (c) (d) Figure 2.4: Looking from the side of DASH, linkages connect the top beams with the bottom beams and force them to remain parallel. The linkages form two parallel four-bar mechanisms that share the horizontal beam that runs between the bottom and top beams. The round motor is mounted rigidly to the bottom beams, which are transparent to reveal the motor. The motor output is rigidly connected to all of the dark beams. The parallel constraint means that every point on the top beams move in the same circular path as the motor output, keeping the motion of the legs in each tripod identical. Note that the positions correspond with the position of the input in Figure 2.3. omitting the hip mechanisms and input beams from Figure 2.3, one can see how this mechanism functions. The transparent beam on the bottom is the same as in Figure 2.3, and the vertical black beams of Figure 2.3 are attached to the horizontal black beams of Figure 2.4. The final stage of the drive motor gearbox is shown as the grey circle, and the motor itself is rigidly mounted on the transparent bottom beam. The dark diagonal beams are a mechanical linkage that connects the motor output to the black input beams. The five thin linkages between the top and bottom horizontal linkages in Figure 2.4 enforce the constraint that the top and bottom linkages remain parallel. Original designs did not have the thin horizontal linkage, and while the remaining four thin beams did prevent the top structure from rolling relative to the bottom structure, the constraint did not enforce the desired parallel constraint. The thin horizontal linkage in the middle of Figure 2.4 enforces the parallel constraint by forming 10

19 coupled four-bar linkages. Figure 2.5 shows how all of these linkages combine together to form DASH. Beyond verifying how the SCM design assembles together, this SolidWorks model can be used as a design tool. The effect of adding and removing constraints can be observed in this model, as well as the effects of varying link lengths on the kinematics. The model also allows one to see how new components will integrate into the existing structure. This differential drive mechanism has a floating ground where only the relative motion of the top and bottom structures of the robot create the desired output of the legs. The motor could be mounted to either structure and a similar circular leg motion would be achieved. The motor was mounted to the bottom structure to keep the center of mass (COM) close to the ground to reduce pitching moments during climbing. This differential drive differs from other robots such as isprawl and RHex wherein the entire drive train is grounded to the same structure to which the hips are attached [7, 8]. Figure 2.5: Fully-constrained, functional SolidWorks model of DASH used to verify how SCM linkages assemble together and create the resulting kinematics. The color of beams correspond with colors seen in Figures 2.3 and

20 2.4 Leg Design Several different leg designs have been used for horizontal locomotion in DASH. The most simple leg design is a rigid, nearly straight leg design shown in Figure 2.6(a). It angles down slightly and then has a rigidly constrained flexure which creates a flat underside. A second leg is a compliant leg design seen in Figure 2.6(b). This leg has one flexure joint that is free to rotate when pressed into the ground. The flexure joint serves as a torsional spring as it is deformed from the presence of normal forces, and the equivalent spring constant can be controlled by adjusting the flexure material. To achieve larger forward accelerations and more repeatable behavior during locomotion, polydimethylsiloxane (PDMS) boots were molded to fit over the ends of these leg designs. These press-fit rubber boots increased friction generated with the surface and reduced slipping. (a) (b) Figure 2.6: Two different leg designs used by DASH. (a) is the stiff, horizontal design and (b) is the compliant design. Both of the aforementioned legs are rigidly mounted to the hip, and this causes the legs and feet to rotate through the same angular deflection as that of the hip, as seen in Figure 2.7(a). While these leg designs seem to function well on horizontal ground where ground engagement is not difficult, foot rotation may cause loss of adhesion during climbing due to twisting the foot contact 12

21 relative to the ground. Thus, the leg design of Figure 2.7(b) was created to eliminate the rotation of the feet during stance phase. A foot is mounted to the end of the four-bar leg, and during the stance the foot still moves through an arc but its orientation relative to the robot does not change. The SCM implementation of this design can be seen in Figure 2.8. Front TOP VIEW Foot Back Front Foot (a) TOP VIEW Foot Back Foot (b) Figure 2.7: The foot orientation of the original rigid-legs rotates with the fore-aft rotation of the leg, twisting the foot contact along the ground, as seen in (a). To prevent the rotation of the foot, a parallel four-bar leg linkage was created. This linkage maintains the orientation of the foot with respect to the body during stance, as seen in (b). The vertical arrows indicate direction of actuation. The press-fit PDMS boots created for DASH were designed to increase lateral and fore-aft forces and were not concerned with generating adhesive normal forces. Subsequent foot designs include various efforts to include controlled compliances and claw structures, all of which can be attached to the four-bar leg design. One such design uses a polymer-based spring element. This design, shown in Figure 2.8(a), uses a loop formed from the polymer flexure layer extending from the bottom of the leg. This polymer loop deforms and elastically stores energy during a foot touchdown, adding compliance while adding minimal mass and inertia to the leg. The polymer 13

22 is cut to create small claw structures to better engaged surfaces such as carpet. During construction, the polymer is plastically deformed to have a flatter bottom to promote more surface contact. A different design replaces the polymer loop spring with a four-bar linkage that serves a similar purpose. This design has the advantage of being more repeatable across build iterations and more easily adapted to new ground engagement mechanisms due to having rigid mounting structures. The flexures in the four-bar mechanism act as torsional springs when deflected through some angular displacement (see Figure 2.10), creating an effective stiffness when the four-bar linkage is deflected. These four-bar linkages could also be actuated for turning, of which a proof-of-concept can be seen later in Section 4.2. Figure 2.8(b) shows an example where simple claw structures are added. (a) (b) Figure 2.8: Two different leg-foot combinations using the parallel four-bar leg design. (a) has a foot made of a polyethylene terephthalate (PET) loop etched with claws to create a compliant foot mechanism. (b) replaces the PET loop with a mechanical four-bar ankle that is compliant in the normal direction and is attached with a rigid claw structure. 14

23 2.5 Body Design and Steering The cardboard beams used to construct DASH are rigid when forces are directed along the face of the beam; however, because the beams are only approximately 900 microns thick, they lack rigidity when subjected to forces directed into the face of the beam. All the beams in DASH are oriented to present the most stiffness in both the fore-aft and vertical axes, since these are the two directions which experience the greatest forces during locomotion. Stiffness in these directions is required for good power transmission to the legs. There are instances, however, when loading occurs in the weak axes of the beams. The lack of off-axis rigidity introduces compliance in the structure as the beams deform under loads. This is in addition to the small amounts of compliance already present due to polymer hinges not perfectly emulating pin joints. Figure 2.9: A model of how the kinematics change when the body is contorted. The midpoint of the fore-aft strokes are biased forward on one side of the body and backwards on the opposite side. This induces a moment on the body and results in turning. During straight-ahead running, the arc swept out by the legs fore-aft motion is centered about a midpoint perpendicular to the body, as in Figure 2.2(b). These leg motions are balanced on both sides of the body and propel the robot straight. To steer, skewing the frame of DASH biases the midpoint of the arcs swept by the legs as shown in Figure 2.9. This shifts the location and direction 15

24 of the ground reaction forces, imparting a moment on the body. A one gram shape-memory alloy (SMA) SmartServo RC-1 from TOKI Corporation, mounted to the rear of DASH, pulls on the front corners of DASH s frame, resulting in a skewed frame and an induced turn. The direction of the turn is dependent upon which corner of DASH is pulled toward the SmartServo. No control input to the SmartServo results in straight-ahead motion. This behavior is explored both in simulation in Chapter 3 and experimentally in Section 4.2. Alternative methods to the body distortion method of turning mentioned above are also being developed. One such method leverages the four-bar ankles on the end of the parallel leg design shown in 2.8(b). By extending either the front-left or front-right foot and applying a force to keep it extended, the robot can be made to turn either right or left, respectively. The extended leg not only touches the ground earlier than the other feet, on average, but also has a higher stiffness than the other feet and creates larger impulses. These effects combine to induce turning behavior in DASH. When no foot is extended, the feet should be balanced and the robot will run straight. Foot Ankle Leg Foot Ankle Leg (a) (b) Figure 2.10: The four-bar ankle linkage is normally horizontal with no loading (a). During locomotion, the four-bar ankle acts as a spring when normal loads deflect the four-bar linkage from its neutral position. To induce turning, the four-bar linkage of either the front-right or front-left leg is locked into the position shown in (b). This induces a left or right turn, respectively. This extension of the four-bar ankle could be done with SMA actuation in the future. 16

25 Chapter 3 Hexapod Simulation A model was created that is capable of emulating the dynamic running of a hexapedal robot in the horizontal plane. The model can simulate straight running as well as the turning method described in Figure 2.9. Though the model is simple, it proves to be fairly effective at predicting the behavior of DASH. It also serves to verify that the modification of the leg trajectories shown in Figure 2.9 actually induces moments on the body and results in turning, even under the assumptions outlined below. 3.1 Construction A reflection-symmetric rigid body of mass m and moment of inertia I was created. The model has massless legs and uses an alternating tripod gait with a 50% duty cycle in which one tripod enters stance as soon as the other tripod exits. The model and design parameters are shown in Figure 3.1. Touchdown angle β, angular sweep during stance φ, stride frequency f, turning input angle, and individual foot friction values can be varied between simulations. All legs are assumed to be 17

26 Θ φ rigid and coupled so that defining the position of one leg defines the position of all of the legs and feet. For later reference, the left tripod is defined as the tripod using legs 1, 3, and 5, and the right tripod is defined as the tripod using legs 2, 4, and 6. The parameters listed in Table 3.1 are used by the simulations shown in this chapter to emulate DASH as configured for turning experiments in Section β δ ν 6 2 d 5 3 rcom 4 Figure 3.1: The 2-D model used in the turning simulation showing various parameters used to define the motion of the rigid body through the plane. Though all six legs are shown, only a single tripod is engaging the surface at any time. Note that the turning input does not effect the total angular deflection of the legs during stance, but does change the touchdown angle β. When turning, all of the legs on one side will have their touchdown position moved either forward or backward, and the legs on the opposite side will have their touchdown position moved backward or forward, respectively, as shown in Figure 2.9. Legs 4, 5, and 6 have their touchdown position moved forward if β is increased while legs 1, 2, and 3 have their touchdown position moved backward. β, as defined in Figure 3.1, is reduced by the 18

27 Mass m 16g Moment of inertia I kg m 2 Leg length m µ d of legs 2 and µ d of legs 1, 3, 4, and π Touchdown angle β 12 Sweep during stance φ 5.4 π 12 Center of mass offset d m Turning input angular offset π 10 Table 3.1: Parameters used in simulations of 2-D rigid-body hexapod. turning input angular offset to turn right, and increased by the turning offset to turn left. Ground reaction forces are determined using a Coulomb friction model. Due to the planar assumptions, the lack of roll and pitch of the rigid body dictates that the normal forces generated by each foot are equivalent. When also considering that the opposing legs move in opposite lateral directions, it emerges that the system is dominated by dynamic friction with coefficient of friction µ d. A time-varying normal force, similar to those seen in experimental trials and modeled by SLIP (see Figure 4.4), was dictated at each foot during every stance. Friction forces are generated at the end of each leg and are directed along the relative motion between the foot and ground. The friction reaction forces from the engaged tripod sum to impart accelerations and moments to the body. The center of mass position and velocity, the orientation and heading, as well as the stride phase of the system are tracked. Model simulations are done using the Runge-Kutta integrator ode45 in MATLAB. Two similar sets of equations of motion are created, one describing the motion during each of the two tripod stances. Integration is performed through the duration of a single stance, and the event functionality of ode45 signals the end of stance and the transition between tripods. Between integrations of the sequential tripods, final state vectors become the initial state vectors of the subsequent integration. 19

28 3.2 Simulation Results Initial trials were conducted without an applied turning input to serve as a control. The system is operated at a stride frequency of 17 Hz, which is near the upper limit of operating frequencies of DASH. Results can be seen in Figures 3.2 and 3.3. The system accelerates quickly and levels off at a top-speed of approximately 1.6 m/s. During running, the system decelerates at the beginning of stance, accelerates through the middle of stance, and decelerates again near the end of the stroke. These fore-aft accelerations have a frequency of half of the stride frequency. The orientation of the body oscillates at the stride frequency, as does the relative heading of the system. The abrupt acceleration in heading and angular velocity seen in Figure 3.2 occurs at the middle of the stance phase. More precisely, it occurs when the legs are positioned perpendicular to the body. It is at this point when the horizontal component of the net reaction forces change direction. Prior to this point, the net reaction forces push the body away from the side with two feet in stance (pushes left when the right tripod is in stance and right when the left tripod is in stance); after this time, the net reaction forces pull the body towards the side with two feet in stance. The behavior seen during straight-ahead running matches expectations and results from more complicated hexapedal models. The deceleration and accelerations match the behavior seen in many legged animals, the SLIP model, and also more complicated hexapedal models [5, 16]. Likewise, the orientation oscillations are similar to those exhibited in legged animals, the lateral leg spring (LLS) model, as well as previous models [5, 16, 17]. The angular acceleration Θ also closely resembles the results from [16] with the decelerations near the change of tripods. The abrupt jerk of the orientation at mid-stance in Figure 3.3(c) is likely due to the rigidity of the legs; the results in [16] have smaller jerk at mid-stance but benefit from having compliance modeled in the legs. One may note that the orientation of the body is not directed vertically in Figure 3.2(a). This is 20

29 Tracked Center of Mass Velocity vs Time Distance (m) Distance (m) (a) Velocity (meters per second) Time (s) (b) Velocity vs Time Velocity (meters per second) Time (s) (c) Figure 3.2: Results of the simulated system running straight without turning inputs operating at the fastest stride frequencies seen in DASH. (a) shows the trajectory of the system across 2-D Euclidean space. Though initially the system was oriented straight-ahead, the system had some transients at the beginning that turned the system slightly. The arrow indicates the motion of the system as time progresses. (b) shows the acceleration from a stop to a steady-state velocity. Once at a steady-state velocity, the forward velocity oscillates slightly (c). The dashed line in (c) indicates the steady-state mean forward velocity. Red circles indicate the state when a transition between tripods occurs. 21

30 because there are some transients that occur during startup. The robot has no initial velocity, and during the acceleration to steady-state velocity, the body experiences unbalanced moments created from the different tripods at different points in the acceleration Relative Heading vs Time 0.36 Orientation vs Time (radians per second) 0 Orientation (radians) Time (s) (a) Time (s) (b) Angular Velocity vs Time Angular Velocity (radians per second) Time (s) (c) Figure 3.3: The rigid body exhibits slight changes in orientation and heading during straight running. (a) shows how the heading changes during running, with the greatest heading acceleration occurring when the legs reach the middle of stance. The orientation oscillates with the same frequency as the stride frequency (b). The angular velocity is seen in (c). Dashed lines indicate the mean values at steady-state. The average relative heading is 0 radians, and the average angular velocity is 0 radians per second. Red circles indicate the state of the system when a transition between tripods occurs. Results of a simulation of a single run with multiple turning inputs applied can be seen in Figures 3.4, 3.5, and 3.6. The simulation consists of a period of running straight, followed by a 22

31 period of applying a left turning input, then another period of running straight, and then a period of applying a right turning input. This creates a S-turn maneuver. The system was operated at 10Hz, near the experimentally-determined best stride frequency for turning in DASH of 11.5 Hz. Figure 3.4 tracks the center of mass as the simulation evolves over time, and the regions where the turning inputs are applied is apparent. Tracked Center of Mass 5 4 Distance (m) Distance (m) Figure 3.4: Simulation results of running at 10Hz stride frequency with turning input of π/10 radians and higher friction mid-legs. The simulation has four distinct regions of operation: the first with no turning input; the second with a left turning input; the third with no turning input again; and the final with a right turning input. The COM is plotted over 2-D Euclidean space. Arrows indicate the direction of motion and are adjacent to the sections during which no turning input is applied. When a turning input is applied, the instantaneous forward velocity of the system changes from a smooth, SLIP-like motion with equivalent behaviors for both tripod stances to a behavior with decidedly different behaviors between the two tripod stances. During a left turn in Figure 3.5(a), the body has a larger acceleration during the left tripod stance phase and a smaller acceleration during the right tripod stance phase. During a right turn, the larger acceleration occurs during the right tripod stance phase and the smaller acceleration occurs during the left tripod stance phase. The angular velocity exhibits similar behavioral differences between the two tripod stances, as seen in Figure 3.5(b). In this figure, the hexapod is turning left and has an average negative angular 23

32 velocity of approximately radians per second, or 16 degrees per second for a stride frequency of 10Hz. When the right tripod is in stance during a left turn, the body actually begins to turn right with an angular velocity as high as 0.4 radians per second. It is during the left stance that the body has a negative angular velocity for nearly the entire duration of stance. The pattern is reversed during right turns, when the right tripod induces the largest right angular velocity. Figure 3.6 shows how the heading and orientation change when a right-turn turning input is applied. Prior to t =7s, the system is running straight with no turning input. At t =7s, the rightturn input is applied, and the relative heading δ of the robot quickly changes. After several strides, it settles to a steady-state oscillation centered around 0.33 radians, implying that the orientation is always lagging behind the instantaneous velocity. The quick change in heading explains why the center of mass appears to abruptly change directions at the transition between turning and straight running in Figure 3.4. The orientation, shown in Figure 3.6(b), continues to oscillate after the turning input is applied, but now begins to steadily increase as the body turns. The cockroach Blaberus discoidalis does have these oscillations and changes in heading and orientation, but the magnitudes are significantly larger and less uniform between strides [18]. 24

33 Velocity (meters per second) Velocity vs Time Time (s) Angular Velocity (radians per second) Angular Velocity vs Time Time (s) (a) (b) Figure 3.5: Simulation results of running at 10Hz stride frequency with turning input of π/10 radians and higher friction mid-legs. These results track the velocity (a) and angular velocity (b) of the orientation while a left turning input is applied. The average velocities are marked with dashed lines. Red circles indicate the state when a transition between tripods occurs. Relative Heading vs Time Orientation vs Time (radians) Orientation (radians) Time (s) Time (s) (a) (b) Figure 3.6: Simulation results of running at 10Hz stride frequency with turning input of π/10 radians and higher friction mid-legs. These figures track the changes to the heading (a) and the orientation (b) before and while a turning input is applied. Prior to t =7s, there is no turning input; After t =7s, there is a right-turning input applied. Red circles indicate the state when a transition between tripods occurs. 25

34 It should be noted that turning could be induced in the model using means other than those used to generate the results above. For example, increasing the friction on only one foot creates an imbalance and turns the robot away from the side with the foot with increased friction, as one would expect. Moving the high friction feet from feet 2 and 4 to 1 and 6 still produces turns in the same direction when the turning input is applied; the high friction on the mid-legs was used only to compare directly to experimental results from Section 4.2. Reducing the frequency below 10Hz increases the turning rate but also results in more hard accelerations and jerky motions. 26

35 Chapter 4 Results DASH is capable of running at 15 body-lengths per second (1.5 meters per second). This speed was observed on a DASH with no steering actuator mounted, but the addition of this actuator appears to have little effect on the top speed. Turning is achieved using a single actuator to deform the body, though an alternative turning method has been demonstrated as a proof of concept. In addition to running and turning on smooth terrain, DASH has been shown to be able to climb over relatively large obstacles and survive falls from large heights [19]. The entire system weighs only 16.2 grams with battery and electronics. Its body is 10 cm long and 5 cm wide. Including legs, it measures approximately 10 cm wide and 5 cm tall. The electronics package includes a microcontroller, motor driver, and Bluetooth communication module for wireless operation [20]. Running at full power with no steering control, its 3.7V 50mAh lithium polymer battery provides approximately 40 minutes of battery life. Larger batteries, such as a 2.6 gram 90mAH lithium polymer battery, can easily replace the 1.8 gram 50 mah battery to augment the battery life with minimal effect on the performance metrics described below. 27

36 4.1 Running Performance Figure 4.1 shows DASH running at 11 body-lengths per second. From the sequence of images, the alternating tripod gait produced by the structure can be seen. In general, there are aerial phases between the touchdown of the alternating tripods with aerial phases taking up to 30 percent of a full cycle [19]. (a) t = 0 ms (b) t = 16.7 ms (c) t = 33.3 ms (d) t = 50 ms (e) t = 66.7 ms (f) t = 83.3 ms Figure 4.1: DASH exhibits dynamic horizontal locomotion. (a) depicts the beginning of one tripod stance and (b) shows the end of the tripod stance. (c) captures the middle of the second tripod stance. (e) shows the middle of the first tripod stance. (d) and (f) both show aerial phases. The ends of the legs are painted white for better visibility. Performance is highly dependent upon on stride frequency, leg geometry, and foot design. The rigid legs of Figure 2.6(a) are most effective for running on smooth terrain, and with these legs DASH was able to reach speeds of 1.5 m/s [19]. There is an observed, roughly-linear relationship between stride frequency and forward velocity [21]. The leg design of Figure 2.8(b) was able to reach speeds near 1.5 m/s, and the relationship between the forward velocity and duty cycle applied to the drive motor is shown in Figure 4.2. Note that this plot shows the relationship between velocity and duty cycle, not the relationship between velocity and stride frequency. The duty cycle 28

37 applied to the drive motor does not have a linear relationship to stride frequency due to mechanical stiffness in the differential drive mechanism. The compliant leg in Figure 2.6(b) was not able to achieve as high velocities as these other legs. The simulation results of Figure 3.2 seem to suggest that the maximum forward velocity is just above 1.6 m/s, which is only slightly higher than DASH manages to achieve at the same stride frequency. Indeed, foot designs haven t been shown to increase the maximum forward velocity above 1.6 m/s, though they can significantly impede performance if not designed well. Foot design does have a dramatic impact on the acceleration of the system, however. Figure 3.2 shows the simulated system accelerates to full speed in roughly 11 complete strides. However, using a similar stride frequency and configuration of friction on the feet of DASH, it takes nearly 30 strides to reach 1.5 m/s. Using either the flexure-loop foot with claws or the cardboard claws on the four-bar ankle on a rough surface such as carpet or rigid foam, DASH was able to accelerate to 1.5 m/s within four strides. The simulation also shows a transient effect during startup that results in a small deflection in orientation during acceleration, which is seen in DASH during acceleration on smooth surfaces with very high probability. 1.6 Forward Velocity vs Time Velocity (meters per second) Duty Cycle (%) Figure 4.2: Forward velocity of DASH with the parallel leg design and four-bar ankle on tile floor. There is a monotonic relationship of velocity with applied motor duty cycle. 29

38 Figure 4.3: Center of mass tracked over nearly one half second. The center of mass follows an approximate sinusoidal trajectory. The grid pattern marks 5 cm increments. The dashed line shows the resting position of the center of mass. When DASH runs on hard ground, the center of mass follows a roughly sinusoidal trajectory, which is indicative of the SLIP model. Fig. 4.3 shows the trajectory during another example of horizontal running. The center of mass is tracked throughout the run, and it follows a stable sinusoidal motion. This trajectory of the center of mass is similar to those seen in running animals [4, 5]. The SLIP model of ground reaction forces is shown in Fig. 4.4(a). In the model, when a system makes initial contact with the ground, it decelerates as it stores energy in the system. As the normal force peaks, however, the fore-aft force becomes positive as the energy is returned to the system and it accelerates forward. Analysis of the SLIP model shows that these ground reaction forces lend dynamic stability when running on a level surface [5]. The ground reaction forces of a cockroach are shown in Figure 4.4(b) [22]. The cockroach data show a cockroach running in an alternating tripod gait where the animal never enters an aerial phase, causing the normal ground reaction force to never reach zero. DASH was tested on a force platform to see if it exhibits similar ground reaction forces to those of natural systems, including the cockroach. A representative sample of filtered data showing DASH running on a horizontal force platform is in Fig. 4.4(c). Each large increase in normal force corresponds with a single tripod making contact with the ground. Just 30

39 as in the SLIP model and with the cockroach, there is a deceleration upon initial contact followed by an acceleration. The phase relationship between the fore-aft and normal forces are the same as both the model and what is observed in the cockroaches. The shapes of the force curves are also very similar, with some slight differences between the measured forces and the sinusoidal forces of the SLIP model. The fore-aft accelerations were also predicted in the simulation (Figure 3.2), though the simulation had steeper accelerations than decelerations while the experimental results were generally more balanced. (a) (b) (c) Figure 4.4: Dynamic locomotion in the horizontal plane has been shown to be similar across many legged systems as modeled by SLIP. In the figures, red solid lines indicate fore-aft forces and blue dotted lines indicate normal forces into the ground. Negative fore-aft forces represent decelerating forces and positive fore-aft forces represent accelerating forces. Positive normal forces indicate forces pushing down on the surface. The SLIP model is shown in (a), the ground reaction forces of a cockroach are shown in (b), and the ground reaction forces of DASH are shown in (c). Results in (b) are borrowed from [22] with permission. 31

40 DASH has also been shown to be able to run on loose granular media [21]. In early experiments done jointly with Li and Hoover, DASH has shown only partial performance loss when running through loosely-packed media (velocity decreased to approximately 5 body-lengths per second). The performance loss is much less than was been observed in earlier trials with Sandbot (velocity went down to 0.1 body-lengths per second [23]). The sprawled posture and low mass of DASH keep the body above the surface of the granular media, and the robot appears to operate in a type of swimming mode. Figure 4.5(a) shows the velocity of DASH running on a hard surface (drywall), and on a granular media (loosely-packed poppy seeds) [21]. One may note that the hard surface performance wasn t as fast as in other hard ground trials. This was due to using motor-driver electronics that enabled real-time power measurements but were unable to provide the instantaneous current levels required for the fastest forward velocities. The real-time power measurements taken from the trials in Figure 4.5 enabled accurate cost of transport calculations for DASH. The mechanical power is determined from P mechanical = τω, where τ = K t I is the motor torque, and ω =2πV EMF /K e is the motor angular velocity. The motor torque constant was previously computed empirically to be K t = N/A. The motor back EMF constant was also found to be K e =0.15 V/Hz. The current to the motor can be computed by I =(V ref V EMF )/R, where V ref is the battery voltage and R is the motor resistance. The metabolic power is simply the electrically power consumed by the motor, or P metabolic = I 2 R. The battery voltage was measured before and after each run to aid in the calculation of the current. The cost of transport is then given by COT = P/mv, where m is the mass of DASH, and v is the forward velocity, and P is either P mechanical or P metabolic. The forward velocity and stride frequency is obtained from slow-motion video and the back EMF V EMF is measured from the onboard electronics. 32

41 v (cm/s) Hard ground Loosely packed COT mechanical (J/kg*m) Hard ground Loosely packed f (Hz) (a) f (Hz) (b) Hard ground Loosely packed COT metabolic (J/kg*m) f (Hz) (c) Figure 4.5: Performance metrics of DASH on hard ground and loosely-packed granular media. Hard ground results are indicated by red squares and granular media results are indicated by blue circles. (a) shows the velocity of DASH as a function of stride frequency in both media, and the results show a roughly linear relationship between stride frequency and velocity. (b) shows the mechanical cost of transport as a function of stride frequency and (c) shows the metabolic cost of transport. On both media, the mechanical cost of transport tends to increase with the stride frequency, though the highest velocity data points on hard ground and granular media show a dip in the cost of transport. The hard ground metabolic cost of transport of DASH also seems to increase with stride frequency but shows a decrease at the highest stride frequency tested. The decrease 33

42 in the hard ground metabolic cost of transport above a stride frequency of 15Hz is large and may represent the system operating at a resonant frequency or other preferred mechanical operating frequency. The metabolic cost of transport values computed on hard ground have a trend similar to the mechanical cost of transport values, but the metabolic cost of transport values on granular media vary dramatically. Because the metabolic cost of transport depends on the square of the battery voltage and the battery voltage cannot be measured instantaneously during a run but instead before and after a run, there is some uncertainty about the accuracy of these values. Future electronics boards may allow for direct measurement of battery voltage during a run, which may observe decreases in voltage during loaded conditions and thus lend more credibility to the metabolic cost of transport values. The metabolic cost of transport of DASH on hard ground determined in [21] is close to the earlier values estimated in [19], which used the total electrical power of the system including not only the motor electrical power but also wireless communications power. 4.2 Turning Results DASH demonstrates that turning can be achieved with relatively simple modifications of kinematics. The top speed of DASH is negligibly affected by the turning actuator, but turning is most successful when operating at a 20 percent duty cycle applied to the main DC motor. Figure 4.6 shows a turning maneuver of DASH, first turning left and then immediately turning right. DASH turns approximately 50 degrees per second to left and 55 degrees per second to the right when applying a maximum turning input and a 20 percent duty cycle to the main drive, which in this case corresponds with stride frequency of 11.5Hz. 34

43 Figure 4.6: Controlled S-turn maneuver using a single turning actuator. The simulation from Figure 3.4 underestimated the ability of the turning input to induce turns in DASH. The simulation, which was run at a 10Hz stride frequency as opposed to the 11.5Hz stride frequency from the experiment in Figure 4.6, suggested a turning rate of approximately 16 degrees. Increasing the friction in the simulation did increase the turning rate, but the friction values used were fairly close to the conditions on the floor used in Figure 4.6. This is turning is significantly slower than in Blaberus discoidalis, which can have average turning rates of a couple hundred degrees per second [18]. The extended-toe turning method described in Section 2.5 and Figure 2.10 also resulted in turning behavior. In fact, it was able to produce turning rates higher than using the method used in Figure 4.6. This method was only a proof-of-concept and did not rely on actuators, instead mechanically locking a single toe downward. If the toe extension were actuated rather than locked mechanically, it would likely have less downward displacement and reduced stiffness and thus have a lower turning rate. The turning rate in Figure 4.7 is taken from right-turn trials. When no toe was locked downward, the robot ran straight with the forward velocities of Figure 4.2. At lower duty cycles without a turning input, there was slight turning, but this was likely due to construction error. There was no discernible turning above a 40% duty cycle when there was no applied turning 35

44 input. When applying the turning input, DASH was exhibiting a sort of skipping motion at the lowest duty cycles. Though these achieved the highest turning rates, the skipping behavior was unlike the smooth dynamics seen when turning at higher duty cycles or when performing highspeed straight running. Increasing the duty cycle with a turning input applied resulted in decreased turning rates, eventually getting to nearly straight running despite having a toe rigidly extended. All measurements were determined from video captured by overhead slow-motion video cameras. 120 Angular Velocity vs Duty Cycle (degree/s) Duty Cycle (%) Figure 4.7: Turning performance of DASH with the parallel leg design and four-bar ankle on tile floor. Turning is achieved by extending a single front toe and the turning rate is dependent upon duty cycle applied to the drive motor. These data are from observed right turns only. 4.3 Step Climbing DASH is able to mount obstacles greater than its own body height using only feed-forward controls [19]. The compliance throughout the structure can compensate for the differences in foot position when running on uneven terrain, and power is still successfully delivered to the feet. Though lacking control of individual foot placement, DASH is eventually able clear the 5.5 cm obstacle in Figure 4.8. This is near the upper limit of step heights DASH can surmount. 36

45 Figure 4.8: Sequential images of DASH climbing over a stack of acrylic. The obstacle is approximately 5.5 cm tall. 4.4 Surviving Falls Several tests were carried out to show the capability of DASH to survive falls from large heights. After multiple drops of 7.5 meters, 12 meters, and 28 meters (75, 120 and 280 body lengths) on to concrete, the robot remained operational, with no damage [19]. From 28 meters, the velocity at impact is approximately 10.3 m/s. This has been verified to be the terminal velocity using wind tunnel experiments, indicating it can survive falls from any height. Despite this high impact velocity, the impact energy is only J due to the lightweight construction. As can be seen from Fig. 4.9, the body contorts dramatically upon impact, in the process absorbing a significant amount of the impact energy. In these images, the body was falling at approximately 6.5 meters per second just prior to impact. 37

46 (a) t = 0 ms (b) t = 10 ms (c) t = 16.7 ms (d) t = 23.3 ms Figure 4.9: Sequential images of DASH impacting the ground at a velocity of approximately 6.5 m/s. (b) and (c) show the great contortion that the body undergoes during a high velocity impact. By (d), the body has almost fully recovered its original shape. The compliance of the structure enables it to withstand high impact velocities and survive without damage. 38