Design And Analysis Of A Mechanical Hopping Mechanism Suited For Exploring
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Design and Analysis of a Mechanical Hopping MechanismSuited for Exploring Low-gravity EnvironmentsHimangshu KalitaUniversity of Arizona1130 N Mountain AveTucson, AZ 85721hkalita@email.arizona.eduTroy M. JamesonUniversity of Arizona1130 N Mountain AveTucson, AZ 85721tjameson@email.arizona.eduGeorge StancuUniversity of Arizona1130 N Mountain AveTucson, AZ 85721georgestancu@email.arizona.eduJekan ThangavelauthamUniversity of Arizona1130 N Mountain AveTucson, AZ 85721jekan@email.arizona.eduAbstract—Exploration of extreme environments, includingcaves, canyons and cliffs on low-gravity surfaces such as theMoon, Mars and asteroid surfaces can provide insight into thegeological history of the solar system, origins of life, andprospects for future habitation and resource exploitation.Although current methods of exploration utilizing wheeledground rovers have excellent performance on relatively flat,benign, even terrains, they are unsuitable for exploring theseextreme environments due to their inability to travers ruggedenvironments as their obstacle traversing capabilities aretypically limited to wheel diameter, and reduced traction onlow-gravity environments. So, developing small, cost-effectiverobots that can utilize unconventional mode of mobility throughballistic hopping can overcome these limitations. Our past workhas proposed using a spherical robot (SphereX) that achievesballistic hopping mobility through the use of a miniaturizedpropulsion system and 3-axis reaction wheel system. In thispaper, we present the design and control analysis of amechanical hopping mechanism that can be used for SphereX.The mechanism is comprised of two mechanical systems toproduce its ability to maneuver terrain and achieve mobilitythrough ballistic hopping. On the robot’s interior, it consists ofan electric gearmotor attached to a set of gears, a spring, and arubber foot. These components make up the hoppingmechanism used to hop the robot by applying a force along thelongitudinal axis of the spring between the rubber foot and theground. However, the robot needs to be oriented in a desiredorientation in order to achieve ballistic hopping and intercept adesired target. This is achieved through a secondary mechanicalsystem that consists of three linear actuators each connected tolevers which are mounted to the exterior of the robot’s shell. Thelever and the linear actuator system are used to orient the robotin a desired orientation so that when the hopping mechanism isdeployed it will be launched in a ballistic trajectory to intercepta desired target. Although the spring based hopping mechanismprovides a constant force, but the lever and linear actuatorbased system is used to orient the robot at different angles toproduce range in mobility. The robot also consists of electronicsand sensors equivalent to current smartphones, an array ofguidance, navigation and control sensors, lithium-ion batterybased power system and a volume for science payload.978-1-7821-2734-7/20/ 31.00 2020 IEEE1TABLE OF CONTENTS1. INTRODUCTION . 12. BACKGROUND AND MOTIVATION . 23. SYSTEM DESIGN . 24. MODELING OF THE HOPPING PROCESS . 45. DESIGN ANALYSIS . 55. RESULTS AND DISCUSSIONS . 76. CONCLUSION . 8REFERENCES . 9BIOGRAPHY . 91. INTRODUCTIONThe recent trend towards small and frequent space missionsto the Moon, Mars, other planetary moons, asteroids, andcomets has sparked new interest on the development ofmobility platforms with dedicated scientific instruments.However, the best method to achieve mobility on planetarybodies is still the subject of discussion. So far, mobilewheeled robots have become integral for surface explorationof the Moon, Mars and other planetary bodies. These rovershave proven their merit, but they are large, in the order ofseveral hundred kilograms and house state-of-the-art sciencelaboratories. Moreover, they can drive over obstacles that area fraction of the vehicle’s body length and uses a significantnumber of actuators and complex suspension linkages. Assuch exploration of extreme and rugged environmentsremains out of reach for current planetary rovers.The last decade has seen a revolution in the miniaturizationof satellites for Low Earth Orbit (LEO) applications. Theadvent of small satellites and micro-satellites has changed thecost models associated with space operations and has also ledto rapid advancement in lightweight structural materials,miniaturization of electronics, sensors and actuators. Withthese recent developments, it is now possible to develop
small, lightweight and low-cost platforms to tackle some ofthe hardest challenges in planetary exploration. In our pastworks we had presented an architecture for a spherical robotcalled SphereX with several kilograms in mass and severalliters in volume that can perform exploration in low-gravityenvironments through hopping and rolling [1-4]. Weproposed the use of a miniaturized propulsion system and a3-axis reaction wheel system to obtain hopping and rollingmobility. In this paper, we extended our work and present amechanical hopping mechanism that fits inside the SphereXrobot and suited for hopping in lo-gravity environments.goal of launching fewer robots, that are better equipped withscience grade instruments.3. SYSTEM DESIGNSphereX is a small, low-cost, modular spherical robot that isdesigned for exploring extreme environments on low-gravityenvironments like the Moon, Mars, icy moons and asteroidsas shown in Figure 1. It consists of a mobility system toperform optimal exploration of these target environments. Italso consists of space-grade electronics like computer boardfor command and data handling, power board for powermanagement and radio transceiver for communicating amongmultiple robots. Moreover, it also consists of a power systemfor power generation/storage, multiple UHF/S-band antennasand accommodates payloads in the rest of the volume. A largerover or lander may carry several of these SphereX robotsthat can be tactically deployed to explore and access ruggedenvironments inaccessible by it.2. BACKGROUND AND MOTIVATIONSmall spherical robots have been widely proposed in the past.Their spherical shape enables them to roll on loose, eventerrain. Examples include spherical robots developed at Univ.of Sherbrooke [5], Kickbot [6] developed at MIT, Cyclops[7] at Carnegie Mellon University and inflatable ball robotsdeveloped at North Carolina State University [8] andUniversity of Toronto [9]. Typically, these spherical robotsuse a pair of direct drive motors in a holonomic configuration.Others such as the Cyclops and the inflatables pivot a heavymass, thus moving center of gravity that results in rolling.Other mobility techniques including use of spinningflywheels attached to a two-link manipulator on the Gyrover[10] or 3-axis reaction wheels to spin and summersault aswith the Hedgehog developed by Stanford and NASA JPL[11]. Hedgehog’s use of reaction wheels enables it toovercome rugged terrain by simply creeping over the obstacleno matter how steep or uneven. However, it’s unclear if agyro-based system can overcome both steep and largeobstacles. In reality, even a gyro-based system is bound toslip on steep surfaces, but under low gravity environmentssuch as asteroids, they may be able to reach meters in height.Our past work has proposed the use of a miniaturizedpropulsion system and a 3-axis reaction wheel system toachieve controlled ballistic hopping for mobility. Althoughpropulsive ballistic hopping is the most optimal mode ofmobility for long-range exploration in low-gravityenvironments, we are also interested in mechanical hoppingmechanisms for short-range explorations [16]. In this paper,we present the design and control analysis of a mechanicalhopping mechanism that can be used for SphereX along withdetails of other subsystems.An alternative to rolling and creeping is hopping. A typicalapproach to hopping is to use a hopping spring mechanism toovercome large obstacles [12]. One is the Micro-hopper forMars exploration developed by the Canadian Space Agency[13]. The Micro-hopper has a regular tetrahedron geometrythat enables it to land in any orientation at the end of a jump.The hopping mechanism is based on a novel cylindricalscissor mechanism enabled by a Shape Memory Alloy(SMA) actuator. However, the design allows only one jumpper day on Mars. Another technique for hopping developedby Plante and Dubowsky at MIT utilize Polymer ActuatorMembranes (PAM) to load a spring. The system is only 18grams and can enable hopping of Microbots with a mass of100 g up to 1 m [14],[15]. Microbots are cm-scale sphericalrobots equipped with power and communication systems, amobility system that enables it to hop, roll and bounce and anarray of miniaturized sensors such as imagers, spectrometers,and chemical analysis sensors developed at MIT. They areintended to explore caves, lava tubes, canyons and cliffs.Ideally, many hundreds of these robots would be deployedenabling large-scale in-situ exploration.Figure 1. SphereX system architectureHopping MechanismSphereX is the direct descendant of the Microbot platform.SphereX has the same goals as the Microbots, but with the2The mechanical hopping mechanism consists of an electricgear motor attached to a pinion and rack gear system, a springand a foot as shown in Figure 2 (Left and Middle). The designof the pinion gear is shown in Figure 2(Right), where theteeth subtended by the angle Φ are removed. As such, themechanism has two phases: a) Compression phase, and b)Release phase. When the pinion gear rotates in a clockwisedirection, for a rotation of 2𝜋𝜋 Φ, the rack travels in theupward direction compressing the spring which correspondto the compression phase. Next, as soon as the pinion gearrotates an angle of 2𝜋𝜋 Φ, it unlocks from the rack and the
spring is released to its original length which corresponds tothe release phase. Considering 𝐷𝐷𝑝𝑝 as the pitch diameter of thepinion, the displacement of the rack during the compressionphase is 𝑥𝑥𝑟𝑟 𝜋𝜋𝐷𝐷𝑝𝑝 (2𝜋𝜋 Φ)/2𝜋𝜋. As such, the spring iscompressed by a distance 𝑥𝑥 𝑥𝑥𝑟𝑟 and the restoring forcegenerated by the spring is equal to 𝐹𝐹 𝑘𝑘 𝑥𝑥, where 𝑘𝑘 is thespring constant. During the release phase, when the spring isreleased, the foot hits the ground resulting in the action of anormal force acting on the robot.Figure 2. (Left and Middle) Two different views of the hopping mechanism consisting a gear motor, rack and piniongear system, a spring and a foot, (Right) Design of the pinion gear with the teeth subtended by the angle 𝚽𝚽 removed.Steering MechanismAlthough the mechanism discussed above results in theaction of a normal force on the robot, a steering mechanismis still needed to orient the robot at a desired angle so that itcan perform ballistic hops. The steering mechanism consistsof three linear actuators, each connected to levers which aremounted to the exterior of the robot’s chassis (shell) as shownin Figure 3.Figure 4. (Left) Position of the lever when the linearactuator is actuated to its full stroke length, (Middle)Position of the lever when the linear actuator is retracted,(Right) Possible orientations of the robot with respect tothe vertical axis 𝒆𝒆𝟑𝟑 defined by the cone with angle 𝚯𝚯.Figure 5. The hopping mechanism and the steeringmechanism assembled together inside the lower half ofSphereX.Figure 3. Steering mechanism consisting of three linearactuators, each connected to levers mounted on the robotshell.By actuating each lever independently, it is possible toposition the robot onto the ground and then orient the robotat a desired angle before releasing the spring of the hoppingmechanism causing the robot to perform ballistic hops.3The lever is designed such that it’s radius of curvature isequal to the radius of the robot. The orientation of the leverwith respect to the axis of actuation of the linear actuator isdefined by an angle Ψ, and is a function of the stroke of thelinear actuator as shown in Figure 4. As such when the linearactuator is actuated to its full stroke length, Ψ 0, and whenthe linear actuator is retracted, Ψ Ψ𝑚𝑚𝑚𝑚𝑚𝑚 . Hence, byactuating each linear actuator independently, the robot can be
oriented by an angle Θ 𝑓𝑓(Ψ1 , Ψ2 , Ψ3 ), with respect to thevertical axis 𝑒𝑒3 defined by a conical section as shown inFigure 4(Right). Figure 5 shows both the hopping andsteering mechanism assembled together inside the lower halfof SphereX.PayloadThe robot will have a payload capacity of mass 500 g, volume0.5 liter and power less than 5 W. The payload will beaccommodated on the top half of the robot and can includestereo cameras, LiDARs or bio-detection instruments.Command & Data Handling4. MODELING OF THE HOPPING PROCESSThe main computer selected for the robot is RinconResearch’s AstroSDR which is a complete RF payload:software-defined radio (SDR), FPGA signal processor, ARMprocessor, and data storage. The single board computercontains the Dual-core ARM Cortex A9 with NEONprocessor that can operate at up to 733 MHz and a XilinxZynq 7045 FPGA. It also contains 512 Mbyte DDR3 RAMmemory and 2 GByte Flash for radiation-tolerant OS storageand an option for 64 GByte eMMC flash storage. The tuningrange for the receiver and transmitter is 70 MHz to 6 GHzwith a maximum bandwidth of 56 MHz. It also has 30 pins1.8 V GPIO and 24 pins 3.3 V GPIO interfaces. Thedimension of the board is 90 x 90 mm, weighs only 95 g,consumes 5.5 W power under nominal conditions and has anoperating temperature range of -40 C to 85 C.A simplified model is developed for the hopping process asshown in Figure 6. The first step is to compress the spring tostore energy 𝐸𝐸0 𝑘𝑘 𝑥𝑥 2 /2 (where, 𝑘𝑘 is the spring constantand 𝑥𝑥 is the displacement of the spring), the next step is toorient the robot at a desired angle and the last step is to releasethe stored energy causing the robot to hop. During the laststep, the body of the robot first accelerates upward due to thespring force, while the lower part remains stationary. Oncethe body moves to a specific height, a perfect inelasticcollision happens between the body and the foot if the springconstant is large. After the collision, both parts move with thesame velocity, which is the robot’s take-off velocity 𝑣𝑣0 .PowerPower required for the operation of the robot will be donethrough NanoPower BP4 battery pack. The battery packconsists of four 18650 series lithium-ion cells resulting in acapacity of 38.5 Wh. The weight of the battery pack is 258 gwith dimensions 94 x 84 x 23 mm. Power management willbe done through the GomSpace NanoPower P31u board thatis configurable with the battery pack. It features amicrocontroller that provides maximum power-pointtracking (MPPT) capability, measures and logs voltages,currents and temperatures of the system. With an I2Cinterface, it is possible to read out measurements, control theon/off-state of 3.3 V and 5 V busses, switch on/off the MPPTand to set/read various parameters. The incoming power fromthe batteries is used to feed two buck-converters supplying a3.3 V @ 5 A and a 5 V @ 4 A output bus. It also contains sixindividually controllable output switches with over-currentshut-down and latch-up protection, each separatelyconfigurable to either 3.3 V or 5.0 V output. The dimensionof the board is 89.3 x 92.9 x 15.3 mm, weighs only 100 g andconsumes 0.165 W power under nominal conditions.Figure 6. Simplified model of exchange of energy for thehopping process.Let the mass of the body be 𝑚𝑚𝑏𝑏 and that of the foot be 𝑚𝑚𝑓𝑓 . Inthe ideal case, all the energy 𝐸𝐸0 stored in the spring isconverted to the kinetic energy of the body. Therefore, thespeed of the body before the inelastic collision is 𝑣𝑣𝑏𝑏 2𝐸𝐸0 /𝑚𝑚𝑏𝑏 . By the conservation of momentum, 𝑚𝑚𝑏𝑏 𝑣𝑣𝑏𝑏 (𝑚𝑚𝑏𝑏 𝑚𝑚𝑓𝑓 )𝑣𝑣0 , thus the take-off velocity is calculatedaccording to Equation (1).𝑣𝑣0 𝑚𝑚𝑏𝑏 2𝑚𝑚𝑏𝑏 𝐸𝐸0𝑣𝑣 𝑚𝑚𝑏𝑏 𝑚𝑚𝑓𝑓 𝑏𝑏 𝑚𝑚𝑏𝑏 𝑚𝑚𝑓𝑓(1)The kinetic energy at take-off is expressed as Equation (2).1𝑚𝑚𝑏𝑏1 𝑚𝑚𝑏𝑏 𝑚𝑚𝑓𝑓 𝑣𝑣02 𝐸𝐸0 𝐸𝐸𝑚𝑚𝑏𝑏 𝑚𝑚𝑓𝑓2𝑟𝑟 1 0(2)1𝑥𝑥(𝑡𝑡) 𝑣𝑣0 𝑡𝑡 cos 𝜃𝜃 , 𝑧𝑧(𝑡𝑡) 𝑣𝑣0 𝑡𝑡 sin 𝜃𝜃 𝑔𝑔𝑡𝑡 22(3)𝐸𝐸 CommunicationWith multiple robots deployed to perform cooperativeexploration, efficient inter-robot communication is a keyfactor. The robots need to transmit information withsynchronization among them and a communicationsynchronization protocol will be used, where each robottransmit one at a time and wait for its time to transmit again[4]. To accommodate this protocol, multiple UHF/S bandantennas will be used as an array along the circumference ofthe robot such that the directivity of the array is maximized[4].where, 𝑟𝑟 𝑚𝑚𝑓𝑓 𝑚𝑚𝑏𝑏 is the mass ratio between the foot and thebody. The robot after leaving the ground with the take ofvelocity 𝑣𝑣0 will be subjected to gravitational force and airresistance. If the air resistance is negligible, then the robotperforms a projectile motion. With a coordinate frame whoseorigin is at the take-off point, x-axis along the horizontaldirection and z-axis along the vertical direction, the robot’strajectory is given by the Equation (3).4
where, 𝜃𝜃 is the take-off angle, and g is the gravitationalconstant. Based on the trajectory, the hopping height ℎ anddistance 𝑑𝑑 can be obtained as Equation (4) and (5).𝐸𝐸0 sin2 𝜃𝜃𝑣𝑣02sin2 𝜃𝜃 ℎ (1 𝑟𝑟)𝑚𝑚𝑚𝑚2𝑔𝑔𝑑𝑑 2𝐸𝐸0 sin 2𝜃𝜃𝑣𝑣02sin 2𝜃𝜃 (1 𝑟𝑟)𝑚𝑚𝑚𝑚𝑔𝑔𝑘𝑘 4𝐺𝐺𝑑𝑑𝑤𝑤8𝑛𝑛𝑎𝑎 𝐷𝐷𝑐𝑐3(6)where, 𝐺𝐺 is the material shear modulus, 𝑛𝑛𝑎𝑎 is the number ofactive coils, 𝐷𝐷𝑐𝑐 is the mean coil diameter and 𝑑𝑑𝑤𝑤 is the wirediameter of the spring. The number of active coils iscalculated as 𝑛𝑛𝑎𝑎 𝑛𝑛𝑡𝑡 𝑛𝑛 , where 𝑛𝑛𝑡𝑡 is the total number ofcoils and 𝑛𝑛 is the number of end coils. The maximum shearstress 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚,𝑠𝑠 occurs on the inner face of the spring coil and isexpressed as Equation (7).(4)(5)where, 𝑚𝑚 𝑚𝑚𝑏𝑏 𝑚𝑚𝑓𝑓 is the total mass of the robot. Fromthese equations, it can be seen that in order to maximize thehopping height and distance, the mass ratio 𝑟𝑟 and the totalmass 𝑚𝑚 should be minimized, while the stored energy 𝐸𝐸0should be maximized. Considering 𝑟𝑟 0, since the mass ofthe body is significantly more than the mass of the foot, ourgoal is to minimize the total mass 𝑚𝑚 so that the hopingdistance 𝑑𝑑 is � 𝑤𝑤(7)where, 𝑊𝑊 is the Wahl correction factor which accounts forshear stress resulting from spring curvature and is expressedas Equation (8).𝑊𝑊 5. DESIGN ANALYSIS4𝐶𝐶 1 0.615 𝐶𝐶4𝐶𝐶 4(8)where, 𝐶𝐶 𝐷𝐷𝑐𝑐 𝑑𝑑𝑤𝑤 is the spring index. Considering 𝐿𝐿𝑠𝑠 to bethe uncompressed length of the spring, the maximumdisplacement possible is 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 𝐿𝐿𝑠𝑠 𝑛𝑛𝑡𝑡 𝑑𝑑𝑤𝑤 . Finally, the2𝜌𝜌/4,mass of the spring can be calculated as 𝑚𝑚𝑠𝑠 𝜋𝜋 2 𝐷𝐷𝑐𝑐 𝑛𝑛𝑡𝑡 𝑑𝑑𝑤𝑤where 𝜌𝜌 is the density of the material used. Figure 7 showsthe spring constant 𝑘𝑘, mass 𝑚𝑚𝑠𝑠 and maximum shear stress𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚,𝑠𝑠 for the spring as a function wire diameter and meancoil diameter for a load of 𝐹𝐹 500N, 𝑛𝑛𝑡𝑡 10, and 𝑛𝑛 2.The material used for the design of the spring is Music Wire– ASTM A228 – Spring Wire with a density of 𝜌𝜌 7861 kg/m3 .As discussed above, the mass of the robot has to beminimized in order to maximize its hopping distance.Considering the mass of all the other subsystems except thehopping mechanism to be fixed, we analyzed eachcomponent of the hopping mechanism to find the optimaldesign.Spring DesignFor the design of the spring for SphereX, Hooke’s law isassumed to hold 𝐹𝐹 𝑘𝑘 𝑥𝑥, where 𝐹𝐹 is the restoring forceexerted by the spring and 𝑥𝑥 is the displacement of the spring.The spring constant 𝑘𝑘 can be expressed as a function of thematerial properties of the spring as Equation (6).Figure 7. (Left) Spring constant 𝒌𝒌 in N/m, (Middle) Mass of the spring 𝒎𝒎𝒔𝒔 in kg, and (Right) Maximum shear stress onthe spring 𝝉𝝉𝒎𝒎𝒎𝒎𝒎𝒎,𝒔𝒔 in N/m2 as a function of wire diameter 𝒅𝒅𝒘𝒘 and mean coil diameter 𝑫𝑫𝒄𝒄 .Gear DesignThe gear system consists of a rack and a pinion. During thecompression of the spring, both the rack and the pinionexperiences a load equal to 𝐹𝐹 𝑘𝑘 𝑥𝑥. The classical methodof estimating the bending stresses in a gear tooth is the Lewisequation. It models a gear tooth taking the full load at its tipas a simple cantilever beam. The maximum bending stress𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚,𝑔𝑔 developed is given by Equation (9).𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚,𝑔𝑔 5𝐹𝐹𝑡𝑡 𝑃𝑃𝑤𝑤𝑤𝑤(9)where, 𝐹𝐹𝑡𝑡 is the tangential tooth load, 𝑃𝑃 is the diametral pitch,𝑤𝑤 is the face width of tooth, and 𝑌𝑌 is the Lewis form factor.The Lewis form factor 𝑌𝑌 is a function of the number of teeth,pressure angle, and involute depth of the gear. The diametralpitch is calculated as 𝑃𝑃 𝑁𝑁/𝐷𝐷𝑝𝑝 , where, 𝑁𝑁 is the number ofteeth and 𝐷𝐷𝑝𝑝 is the pitch diameter of the pinion. The rack isdesigned to have the same face width and pitch as the pinion.
displacement of the rack is 𝑥𝑥𝑟𝑟 𝜋𝜋𝐷𝐷𝑝𝑝 (2𝜋𝜋 Φ)/2𝜋𝜋. Thedesign of the angle Φ has to be such that the maximumdisplacement of the rack is less than or equal to the maximumdisplacement of the spring 𝑥𝑥𝑟𝑟 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 . Figure 8 shows themaximum bending stress 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚,𝑔𝑔 , mass of the rack 𝑚𝑚𝑟𝑟 andmass of the pinion 𝑚𝑚𝑝𝑝 as a function of pitch diameter 𝐷𝐷𝑝𝑝 andface width 𝑤𝑤 for a load of 𝐹𝐹 500N, and 𝑁𝑁 20. Thematerial used for the design of the rack and pinion is 1045Carbon Steel with a density of 𝜌𝜌 7870 kg/m3 .The minimum length of the rack 𝐿𝐿𝑟𝑟 required is equal to sumof the length of the spring and the diameter of the pinion,𝐿𝐿𝑟𝑟 𝐿𝐿𝑠𝑠 𝐷𝐷𝑝𝑝 . Considering the rack to have a height of pitch𝐻𝐻, the mass of the rack and the pinion is then approximatedas 𝑚𝑚𝑟𝑟 𝐻𝐻𝐻𝐻𝐿𝐿𝑟𝑟 𝜌𝜌 and 𝑚𝑚𝑝𝑝 𝜋𝜋𝐷𝐷𝑝𝑝2 𝑤𝑤𝑤𝑤/4, where 𝜌𝜌 is the densityof the material used.Moreover, as discussed in Section 3, if the angle subtendedby the removed teeth of the pinion is Φ, the maximumFigure 8. (Left) Maximum bending stress 𝝈𝝈𝒎𝒎𝒎𝒎𝒎𝒎,𝒈𝒈 in N/m2 on the gear tooth, (Middle) Mass of the rack 𝒎𝒎𝒓𝒓 in kg, and(Right) Mass of the pinion 𝒎𝒎𝒑𝒑 in kg as a function of the pitch diameter 𝑫𝑫𝒑𝒑 and face width 𝒘𝒘.where, 𝜈𝜈 is the Poisson’s ratio of the material used. Thus, themass of the foot can be approximated as 𝑚𝑚𝑓𝑓 𝜋𝜋𝑟𝑟𝑓𝑓2 𝑡𝑡𝑓𝑓 𝜌𝜌, where𝜌𝜌 is the density of the material used. Figure 9 shows themaximum stress 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚,𝑓𝑓 , and mass of the foot 𝑚𝑚𝑓𝑓 as a functionof foot radius 𝑟𝑟𝑓𝑓 and foot thickness 𝑡𝑡𝑓𝑓 for a load of 𝐹𝐹 500N.The material used for the design of the foot is 2024-T4Aluminum with a Poisson’s ratio 𝜈𝜈 0.32 and density of𝜌𝜌 2780 kg/m3 .Foot DesignWhen the spring is released and the foot impacts the ground,it will experience a normal force equal to 𝐹𝐹 𝑘𝑘 𝑥𝑥. The footis designed as a circular disk of radius 𝑟𝑟𝑓𝑓 and thickness 𝑡𝑡𝑓𝑓 .Assuming the load acting on it during its impact with theground as a concentrated load at its center, the maximumstress occurs at the center on the lower surface and can beexpressed as Equation (10)𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚,𝑓𝑓 𝑟𝑟𝑓𝑓𝐹𝐹(1 𝜈𝜈) 0.485 ln 0.52 𝑡𝑡𝑓𝑓𝑡𝑡𝑓𝑓2(10)Figure 9. (Left) Maximum stress 𝝈𝝈𝒎𝒎𝒎𝒎𝒎𝒎,𝒇𝒇 in N/m2, and (Right) Mass of the foot 𝒎𝒎𝒇𝒇 in kg as a function of foot radius 𝒓𝒓𝒇𝒇and foot thickness 𝒕𝒕𝒇𝒇 .Motor SelectionDesign OptimizationWith the gear system experiencing a maximum force of 𝐹𝐹,when the spring is fully compressed, the torque required torotate the pinion gear is 𝔪𝔪 𝐹𝐹𝐷𝐷𝑝𝑝 /2. Thus, a motor iscarefully selected such that it can provide a maximum torquegreater than 𝔪𝔪 and sustain a maximum load greater than 𝐹𝐹.The goal of the optimization process is to minimize the massof hopping mechanism (spring, gear system and foot) for agiven mass of the other subsystems (avionics, power,communication, shell) such that the robot can hop amaximum distance 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚 , which is user defined. The designvariables for the optimization process are 𝕩𝕩 6
𝐷𝐷𝑐𝑐 , 𝑑𝑑𝑤𝑤 , 𝐿𝐿𝑠𝑠 , 𝑤𝑤, 𝐷𝐷𝑝𝑝 , Φ, 𝑟𝑟𝑓𝑓 , 𝑡𝑡𝑓𝑓 . We have considered the totalnumber of coils of the spring and the number of teeth of thegear as constant. Also, considering the mass of all the othersubsystems except the hopping mechanism to be 𝑚𝑚𝑜𝑜 , the totalmass of the robot is 𝑚𝑚 𝑚𝑚𝑜𝑜 𝑚𝑚𝑠𝑠 𝑚𝑚𝑟𝑟 𝑚𝑚𝑝𝑝 𝑚𝑚𝑓𝑓 andmass of the body 𝑚𝑚𝑏𝑏 𝑚𝑚 𝑚𝑚𝑓𝑓 . Five constraints are addedto the optimization problem. The first three constraints aresuch that the stresses developed in the spring, gear and footare less than 50% of the yield strength of the materials used.The fourth constraint is such that the displacement of the rackis less than or equal to the maximum possible displacementof the spring. And finally, the fifth constraint is such that thehopping distance 𝑑𝑑 of the robot at 𝜃𝜃 45 is greater than orequal to the user defined maximum distance 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚 . Theproblem can be mathematically formulated as Equation (11).min 𝑓𝑓(𝕩𝕩) 𝑚𝑚𝑠𝑠 𝑚𝑚𝑟𝑟 𝑚𝑚𝑝𝑝 𝑚𝑚𝑓𝑓𝑔𝑔1 (𝕩𝕩) 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚,𝑠𝑠 0.5𝜎𝜎𝑡𝑡,𝑠𝑠 𝑔𝑔2 (𝕩𝕩) 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚,𝑔𝑔 ��𝑠𝑠𝑒𝑒𝑒𝑒𝑒𝑒 𝑡𝑡𝑡𝑡 𝑔𝑔3 (𝕩𝕩) 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚,𝑓𝑓 0.5𝜎𝜎𝑡𝑡,𝑓𝑓 𝑔𝑔4 (𝕩𝕩) 𝑥𝑥𝑟𝑟 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 𝑔𝑔5 (𝕩𝕩) 𝑑𝑑 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚With the mathematical model of each element of the hoppingmechanism developed and the optimization model defined,we investigated the design of the hopping mechanism forexploration in different environments. Exploration of anenvironment with SphereX through hopping will be done byemploying a Hop Map Stop Science Process cycle,with each cycle taking 5 minutes. During the Hop Mapphase, the robot hops from its current position to a desiredposition while performing mapping of the environment. Afterthe hop is completed, the robot stops and performs scienceoperations, processes the collected data and then hops to thenext location. Table 1 shows the mass and powerconsumption of each component of SphereX except thehopping mechanism. The power consumption for thecomputer & transceiver, power management board, antennasand payload are assumed to be constant throughout themission. The steering mechanism consumes power only toorient the hop to execute a hop and is calculated to consumean average of 2W per hop. The power consumed by the motorto operate the hopping mechanism will depend on the designsolution of the hopping mechanism found through theoptimization process and will be discussed later.Computer Transceiver955.5Battery anism2502 (avg)Motor300 Connectors Mounting gth(MPa)SpringMusic Wire –ASTM A2287861200Gear1045 ation on the surface of the MoonThe first target environment for which we analyzed thedesign of the hopping mechanism is on the surface of theMoon with gravity 𝑔𝑔 1.62 m/s 2 . The design optimizationis performed such that the robot can hop at least a maximumdistance of 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚 5𝑚𝑚 at an angle of 𝜃𝜃 45 . Form Table1, the mass of all the other subsystems except the hoppingmechanism is 𝑚𝑚𝑜𝑜 2.2 kg. The bounds for each design(𝑏𝑏)(𝑏𝑏)variable are 𝐷𝐷𝑐𝑐 [10 50] mm, 𝑑𝑑𝑤𝑤 [2 6] mm,(𝑏𝑏)(𝑏𝑏)𝑤𝑤 (𝑏𝑏) [5 15] mm,𝐷𝐷𝑝𝑝 𝐿𝐿𝑠𝑠 [10 80] mm,[10 30] mm, Φ(𝑏𝑏) [10 80] , 𝑟𝑟𝑓𝑓(𝑏𝑏) [10 50] mm,(𝑏𝑏)and 𝑡𝑡𝑓𝑓 [1 10] mm. The problem is modeled as anonlinear optimization problem (NLP) and we used thesequential quadratic programming (SQP) method to solve it.At each iteration of SQP, the gradient is calculated using thefinite difference method and the hessian is calculated usingthe Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.Figure 10 shows the variation of the objective function value,constraint violation and the first-order optimality of theoptimization process. It can be seen that the objectivefunction value is stationary, and the constraint violation andthe first-order optimality approached zero.Table 1. Mass and power consumption of each componentof SphereX except the hopping mechanismPowerConsumption (W)2Table 2. Material properties used for the design of thespring, gear system and foot of the hopping mechanism5. RESULTS AND DISCUSSIONSMass (g)120Table 2 shows the material properties (density and yieldstrength) used for the design of the spring, gear system andfoot of the hopping mechanism.(11)ComponentAntennas7
Where, 𝑖𝑖0 is the no load current, 𝑘𝑘 𝑇𝑇 is the torque constant,and 𝑉𝑉 is the operating voltage. Using the values of theselected motor and averaging out the power consumption ofthe motor during the hoping phase into the exploration cycletime of 5 minutes, we found 𝑃𝑃ℎ 2.3 W/cycle. Thus, withthe selected battery pack, the robot will be able to operate for2.4 hours performing 30 hops and exploring 150 meters onthe surface of the Moon. Using this optimal design of thehopping mechanism designed for the Moon, we investigatedhow the hopping distance varies with gravity. Figure 11shows the variation of the hopping distance of the optimaldesign found for the Moon as a function of gravity with thered squares showing the hopping distance on the surface ofCallisto, Europa, Ganymede, Moon, Io, and Mars. Although,we showed an optimal design for operation on the surface ofthe Moon, this method of desig
has proposed using a spherical robot (SphereX) that achieves ballistic hopping mobility through the use of a miniaturized propulsion system and 3-axis reaction wheel system. In this paper, we present the design and control analysis of a mechanical hopping mechanism that can be used for SphereX. The mechanism is comprised of two mechanical .
Collectively make tawbah to Allāh S so that you may acquire falāḥ [of this world and the Hereafter]. (24:31) The one who repents also becomes the beloved of Allāh S, Âَْ Èِﺑاﻮَّﺘﻟاَّﺐُّ ßُِ çﻪَّٰﻠﻟانَّاِ Verily, Allāh S loves those who are most repenting. (2:22
akuntansi musyarakah (sak no 106) Ayat tentang Musyarakah (Q.S. 39; 29) لًََّز ãَ åِاَ óِ îَخظَْ ó Þَْ ë Þٍجُزَِ ß ا äًَّ àَط لًَّجُرَ íَ åَ îظُِ Ûاَش
Present ICE Analysis in Environmental Document 54 Scoping Activities 55 ICE Analysis Analysis 56 ICE Analysis Conclusions 57 . Presenting the ICE Analysis 59 The ICE Analysis Presentation (Other Information) 60 Typical ICE Analysis Outline 61 ICE Analysis for Categorical Exclusions (CE) 62 STAGE III: Mitigation ICE Analysis Mitigation 47 .
Research Design: Financial Performance Analysis In this study, financial performance analysis will be used. The analysis is based on three types of analysis methods which are horizontal analysis, trend analysis and ratio analysis. All data analysis is based on the items on the financial statement. A financial statement is a written record
Design and Analysis Software v2 file used as a template to contain primary analysis and secondary analysis settings. This option can be used for the analysis of legacy EDS or SDS files when a specific set of primary and secondary analysis settings are needed. The primary and secondary analysis settings will be used when the analysis (-a) option .
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Qualitative analysis, quantitative analysis, non-financial indicator analysis, financial indicator analysis, internal performance analysis, external performance analysis, project-orientated analysis, organization-orientated analysis 8 [36] Area-based Knowledge measurement in products and processes,
Module 7: Fundamental Analysis (NCFM Certification) 1. Introduction of Fundamental Analysis What is Fundamental & Technical Analysis? Difference between technical & fundamental analysis Features & benefits of Fundamental analysis 2. Top-Down Approach in Fundamental Analysis Economic Analysis Industry Analysis Company analysis 3.
