The Design of an Arm Suspension Mechanism foran Underactuated Single Legged Hopping Robot.

Christopher Schmidt-Wetekam, Thomas Bewley

Abstract—The geometry, kinematics, and mechanical devel-opment of a symmetric and adjustable arm suspension mecha-nism for an actively stabilized single-legged hopping robot arepresented. A novel two-armed vehicle design is presented as amultimodal alternative to existing monopedal platforms, and isenabled by the development of this arm suspension mechanism.The mechanism is shown to behave essentially as a digressive-rate torsional spring placed between two arms, in series witha weaker progressive-rate torsional spring tied to a leg/centralbody. This makes it well-suited to recovering substantial energyfrom hopping motion, whilst presenting negligible resistance toantisymmetric motions used to maintain side-to-side stability.Furthermore, by storing energy purely in tension, the problem ofbuckling encountered with compression springs is avoided, and,the effective torsional spring constant may be varied by adjustingthe spring pretension. Finally, the mechanism is self-latchingnear maximum deflection, which facilitates the rapid releaseof gradually accumulated spring energy for highly dynamicmaneuvers with low input power.

I. INTRODUCTION

Robots vastly outperform their human counterparts in repet-itive, precise tasks such as manufacturing, yet most lack themobility to compete in even simple tasks such as climbingstairs. This difficulty arises chiefly from the requirements ofhigh power-density, and complex, real-time path-planning forcontrolling a humanoid robot with a high number of degrees offreedom. Therefore, mechanically simple robots that operatein unstable modes in order to enhance agility have gainedresearch interest. A classic example is the monopedal hoppingrobot, or “self-balancing” pogo-stick, the most well-known ofwhich was originally demonstrated by Marc Raibert at MITin the 1980’s [6]. Raibert’s hopper was hydraulically actuatedusing offboard power, and was able to control its orientationand forward speed by a combination of foot placement andtorque applied between the leg and main body of the robot.Other notable monopedal designs incorporating on-board bat-tery power have been created since [4], [2], however, none ofthese designs are capable of self-uprighting if they fall over.

In the spring of 2003, a natural evolution of this designwas proposed: a reaction wheel stabilized monopod (fig. 1).The robot’s two main drive wheels roll efficiently over smoothterrain, yet serve also to upright the robot, and, in coordinationwith an orthogonal set of wheels, stabilize the robot duringflight, by exchanging angular momentum between the legand the wheels. Several working prototypes based upon thisconcept have been developed; most notably, “iHop v.2” (fig

The authors are affiliated with the Coordinated Robotics Lab, UC SanDiego, La Jolla, CA 92093-0411 USA; e-mail: cmschmid@ucsd.edu, be-wley@ucsd.edu

(i)

(iv)

(iii)(ii)

Fig. 1. Cartoons illustrating the multimodal dynamic robot concept. As seenat the right, the proposed vehicle can roll around in a horizontal roving mode(i), transition (ii) to a vertical roving mode (iii), and, when necessary, hopover obstacles (iv).

Fig. 2. Three-wheeled hopping robot, “iHop v.2” prototype. During hoppingmaneuvers, a reaction wheel centrally-housed within the main body stabilizesabout the roll axis, and the two drive wheels stabilize about the pitch axis. Adual four bar hopping mechanism creates linear leg motion with a torque/forcerelationship favorable for DC motor actuation, and stores hopping energyduring roving by latching into a fully-tensioned state.

2) which incorporated a dual four-bar hopping mechanism,and an “H-Pattern” reaction wheel array (RWA), consisting ofa large central reaction wheel mounted orthogonally betweentwo drive wheels [3].

II. MOTIVATION: CHALLENGES WITH REACTION WHEELSTABILIZATION

A. Background

The term “reaction wheel” refers to a driven flywheel thatis used to regulate the attitude of any attached bodies aboutthe wheel’s axis of rotation. They are commonly installed inan x,y,z array on small satellites, however have also been used

Fig. 3. Maximum recovery angle and associated gear reduction from zero-velocity initial conditions for a single-axis inverted pendulum stabilized viaa DC motor driven reaction wheel. Three main parameters are varied: bodyrotational inertia, J1, reaction wheel rotational inertia, J2, and distance fromground pivot point to center of mass, L.

to stabilize terrestrial platforms [7]. In contrast to gyroscopes,which spin at a fixed velocity, reaction wheels use feedbackmeasurements to actively transfer angular momentum impartedby external disturbances from the body to the flywheel. Thereaction wheel speed, in turn, is regulated by transferringangular momentum back into the surrounding environment(for the inverted pendulum, this is achieved by overshootingthe unstable equilibrium). Reaction wheels are nominally keptnearly stationary, since, for a fixed power input, the appliedtorque, acting equally and oppositely upon both the bodyand the reaction wheel, is generally inversely proportional toreaction wheel velocity.

The angular momentum capacity is maximized for a highspeed (low-torque) flywheel with large rotational inertia, butthis is not necessarily the optimal configuration for invertedpendulum stabilization, since the reaction torque must exceedthe gravity induced moment. As shown in fig. 3, a high-torque/low-speed flywheel with a large moment of inertia,J2, maximizes the angle from which the system is ableto recover, starting from zero velocity initial condition. Ingeneral, increasing the reaction wheel rotational inertia oftenincreases the overall mass, and raises the center of mass,however, these properties are assumed to be decoupled in thisanalysis. Increased body inertia, J1 only slightly diminishesthe maximum recovery angle, however, also decreases robust-ness to steady-state estimation errors, by virtue of the factthat large external disturbances will cause smaller changes inorientation, which may be obscured by sensor noise.

B. Reaction Wheel Arrays

The performance of each reaction wheel diminishes asadditional orthogonal wheels necessary for controlling otheraxes are added. Configuring the wheels concentrically createsthe tightest packing, and minimizes the extent to which each

+r

2r

r/A

θ

m1 J1

ρ

Fig. 4. An H-pattern RWA consisting of a centrally mounted roll axis (θ)wheel of uniform density, ρ, radius, r, and thickness, r/A, surrounded bytwo orthogonal drive wheels with identical mass properties. The wheels aremounted 2r above the foot in order to create consistent ground clearance tothe outer wheels for nonzero θ.

Fig. 5. Maximum roll axis recovery angle for a generic H-pattern RWA, asa function of: wheel radius (dashed), and wheel thickness/aspect ratio (solid).The applied torque, τ = kt

R(u− ktω), where ω is the reaction wheel angular

velocity, and u is a step voltage input.

reaction wheel is burdened by the minor-axis moment ofinertia of orthogonal wheels, however is not suitable for aroving operation in which least two parallel drive wheelscontrol both speed and direction.

The H-pattern RWA used in the iHop v.2 prototype is thesimplest fixed configuration suitable for multimodal operation,but is fundamentally limited in terms of performance bythe central reaction wheel. Approximating all three reactionwheels as solid, uniform density discs, and assuming that theground clearance to the wheels is equal to the wheel radius,results in a unique wheel radius for which the maximumrecovery angle is achieved (fig. 5). This outcome is ratherintuitive in light of the fact that a large center wheel increasesand elevates the center of mass, and also increases the minoraxis rotational inertia of the orthogonal wheels. Testing onthe iHop v.2 prototype confirms the weakness of the roll axisreaction wheel, particularly as bias in the angular estimatedevelops during hopping maneuvers.

Fig. 6A. Canting reaction wheel configuration. The wheels are pivoted intoan orthogonal configuration during legged maneuvers in order to stabilizemotions about the pitch and roll axes. A nonorthogonal canting angle may bepreferred in order to project a greater portion of reaction wheel torque abouteither the pitch or roll axes.

Fig. 6B. Pivoting reaction wheel configuration. The roll axis is stabilized byapplying torque against a nominally horizontal “balance beam” mechanism,ballasted by the drive wheels on each end. The balance beam employs aparallelogram linkage, in order to maintain parallel alignment of the wheelsrelative to the body.

Fig. 6C. Independently pivoting reaction wheel configuration. Two maindrive wheels mounted at the ends of independently actuated arms producehopping and/or stabilization motion via a combination of symmetric andantisymmetric rotation. Parallelogram linkages within each arm preserve therelative orientation of the wheels, so as to properly align the wheels for roving,and to minimize coupling of the pitch/roll/yaw axis dynamics.

III. ARM SUSPENSION DESIGN FOR UNDERACTUATEDSTABILIZATION OF HOPPING

Three alternative designs have been developed in order toovercome the performance limitations of the H-pattern array:1. the canted reaction wheel pair (fig. 6A), 2. the pivotingreaction wheel pair (fig. 6B), and 3. the independently pivotingreaction wheel pair (fig. 6C).

The canted design transitions between parallel and orthog-onal wheel orientations as necessary for rolling or balancing,significantly reducing overall weight by eliminating the centerwheel, and reducing the overall rotational inertia by bring-ing the drive wheels closer together. Transitioning betweenwheeled and hopping modes with the canted array, however,requires delicate coordination of leg retraction and wheelcanting in order to avoid misalignment of the wheels whilerolling, without sacrificing control about the roll axis after thewheels leave the ground.

The pivoting design avoids this vulnerability by maintainingparallel alignment of the reaction wheels, and, instead, utilizesthem as ballast mounted at the ends of a finite rotation beammechanism. In order to compensate for the finite rotation, thebeam assembly is shaped such that its center of mass liesbelow the pivot, causing the overall center of mass to shiftas the beam rotates, which enhances steady-state disturbancerejection.

The independently pivoting design, consisting of two inde-pendently actuated arms with wheels mounted at the ends,functions like the pivoting pair when the arms are forcedantisymmetrically, however also produces hopping motionwhen actuated symmetrically (fig. 6C). This arrangement isunderactuated in the sense that, during hopping, two controlinputs (left arm, right arm) simultaneously control the attitude,and x-y position of the central body, as well as the attitudeof each arm. The fact that both hopping and stabilization aretriggered by the same set of actuators enables resources tobe shared between these two functions. The remainder of thisdocument focuses on the design illustrated in fig. 6C.

A. Independently Pivoting Reaction Wheel Pair Concept

Several unique design challenges arise in a configurationin which the arms simultaneously produce hopping motionand provide stabilization. Most importantly, the arms must be“antisymmetrically compliant”, so as not to stifle stabilizingmotions, yet be “symmetrically stiff”, in order to passivelysupport the heavy drive wheels and store energy, when de-flected, for hopping. While simply placing a torsion springacross the arms would fulfill these requirements, additionaldesign constraints necessitate a more involved solution.

Each arm is actuated by torque applied at one of theoutward joints by a high-speed / low-torque motor, therefore,a digressive symmetric stiffness (decreasing with increasingdeflection) is desirable in order to provide high support atsmall deflections, without overwhelming the motors at largedeflections. A digressive stiffness also increases hopping effi-ciency, by emulating a “soft” mechanical stopper that arreststhe arms near horizontal during takeoff.

In order to store energy for large jumps, the arms shouldlatch into a fully-tensioned state at an angle which is less than90 degrees from horizontal. A latching action is preferred,as this eliminates the need for added actuators (clutches,ratches), which are succeptible to jamming under high loads.The latching angle requirement prevents collision betweencoplanar mechanism links, and simplifies the vehicle dynamicsby enabling the arm axes of rotation to be collocated.

In order to enable “end-over-end” hopping, the spring resis-tance must be symmetric about the horizontal axis; the abilityto adjust the spring stiffness without altering this symmetry isalso desired.

B. Arm Suspension Mechanism Layout

The design constraints for this mechanism differ signif-icantly from those encountered for existing legged robotsemploying leg mechanisms [5]. The parallelogram linkagecomprising the “frame” of each arm is based upon suspensiondesigns used in high-performance vehicles [1], which serves tomaintain parallel alignment of the reaction wheels with respectto each other, and to the rest of the robot, throughout the rangeof motion. This is necessary for efficient wheeled locomotion,as well as to decouple the pitch and roll axis dynamics byensuring that the effect of the reaction wheel torque on thebody is not a function of arm position or velocity.

Iteratively working within this tightly constrained designspace, an entirely new suspension mechanism consisting pri-marily of two extension springs tensioned by levers actuatedwithin parallelogram mechanisms has been developed. Asshown in fig. 7, each arm has an associated lever and extensionspring that attaches to the opposite arm. Each spring leverattaches to a prismatic joint that pivots with respect to itsattachment to the parallelogram mechanism. The symmetry ofthe mechanism about horizontal enables the robot to operateon either end of the leg, which is most useful for end-over-endmaneuvers, such as hopping over tall obstacles by orientingthe leg tangent to its parabolic flight trajectory.

C. Suspension Mechanism Kinematics

The motion of each arm suspension mechanism planeinvolves the interaction of at least five members, with twodegrees of freedom, αL and αR (fig. 8). Note that length of Ahas no effect on the kinematics, however it must be sufficientlysized to prevent collision between links L and C, when theyare coplanar.

Defining the origin as the midpoint of the central armsupport, the coordinates of first the spring attachment, SL1,are given by

SL1 =

[L cosαR −B − C cosφRL sinαR − C sinφR

](1)

where

φR = − tan−1

((L−D) sinαR

(L−D) cosαR +B

)(2)

PivotingPrismaticJoint

Bottom PlaneTop PlaneLeftSpringLever

RightSpringLever

RightExtensionSpring

LeftExtensionSpring

Fig. 7. The suspension mechanism consists of two symmetric halves thatare mounted in separate planes to prevent collisions.

L

A

B

C

B2

D

αRαL

φR

SL2

SL1SL

Fig. 8. Geometry of bottom linkage plane. The left spring length SL variesas a function of the left and right arm angles, αL and αR.

As shown in fig. 9, when B = 0, the “spring lever” (C) angle,φR = αR, however, as B is increased, the motion of the springlever w.r.t. the arm frame becomes increasingly non-parallel.

Furthermore, as shown in fig. 10, the instantaneous centerof rotation of the spring lever is invariant throughout the nom-inal range of motion, for both symmetric and antisymmetricrotation. The invariant coordinate is given by

F =

[BD

L−D, 0

](3)

Therefore, a kinematically equivalent mechanism can be con-structed by rigidly attaching the spring fulcrum to the centralleg (fig. 11). As B and D are lengthened, F moves away fromthe origin, and the spring lever motion becomes less parallel.Recall that strictly parallel motion occurs for B = 0, which isagreement with the fact that F is at the origin for B = 0. Thelatter linkage configuration may be more desirable as it usesfour fewer revolute joints, and exerts lighter radial loads onthe spring lever support joints, however, it is more challengingto package around other hardware.

Fig. 9. The relationship between arm angle, α, and lever angle, φ becomesincreasingly nonlinear as the spring lever attachment is offset inwards fromthe arm frame.

Fig. 10. Lever (solid), fulcrum (triangle), and spring (thick solid) positionsthroughout the nominal range of symmetric (top) and antisymmetric (bottom)motion.

L

B

D

BDL-D

L

B

B

B

Fig. 11. Two mechanism realizations with equivalent kinematics.

D. Suspension Mechanism Spring Action

The left spring vector, SL = SL2 − SL1, is given by:

SL =

[B + B2 + C cosφR − L (cosαR − cosαL)

C sinφR − L (sinαR − sinαL)

](4)

for the case when B = B2 and αL = −αR, (symmetricrotation) the left spring length is given by:

‖ SL ‖2 = 4L2 + 4B2 + C2 − 8LB cosα

− 4LC cos (α− φR) + 4BC cosφR (5)

for the case when B = B2 and αL = αR (antisymmetricrotation) the spring length is given by:

‖ SL ‖2 = 4L2 cosα2 + 4B2 + C2 − 8LB cosα

− 4LC cosφR cosα+ 4BC cosφR (6)

Therefore, no spring deflection occurs when B = 0 and C =L. The minimum spring length,

SL0 = 2L− 2B − C

occurs at αL = αR = 0 (arms outstretched and orthogonal toleg).

The combined deflection of the left and right springs issymmetric about the axes of symmetric (αL = −αR) andantisymmetric (αL = αR) rotation, with a greater sensitivityto symmetric rotation (figure 12) .

As the springs are pretensioned, both the symmetric andantisymmetric stiffness increase near horizontal (figs. 13, 14).The effective spring constant associated with symmetric rota-tion is digressive; i.e. it tends to decrease for large deflection,creating a more constant resistive torque throughout the rangeof motion, as is suitable for a motor geared for high-speedhopping motion. By comparison, the effective antisymmetric

Symmetric rotation

Antisymmetric rotation

Fig. 12. Combined (left and right) spring lengths as a function of left armangle (αL), and right arm angle (αR). B

L= B2

L= 0.125, C

L= 1.094,

DL

= 0.438

Fig. 13. Total displacement, torsional resistance, and stiffness for symmetric(hopping) motion. Self-latching behavior occurs as the torsional resistancecrosses zero, at α = +/−650. Added spring pretension significantly stiffenssymmetric motion about the rest position.

spring constant is much weaker and, for low pretension, isprogressive; i.e. increases with deflection, as is suitable foraccommodating small antisymmetric corrections necessary foractive stabilization, while behaving in some sense as a softlimit near the full extent of angular displacement.

E. Suspension Mechanism Latching Properties

For symmetric rotation, the applied torque to both armschanges sign as the linkage travels past αmax = +/ − 650

(fig. 13), causing the arms to latch in tension, provided thatthe arms are constrained to travel at most slightly past αmax.The latching angle cannot be solved for directly, however, asshown in 15, is primarily a function of the spring lever length,C, and ranges between 400 to 700.

Fig. 14. Total displacement, torsional resistance, and stiffness for antisym-metric (balancing) motion. The resistance profile becomes more linear asspring pretension is added, resembling that of a conventional torsional spring.

IV. WORKING PROTOTYPE

A working prototype employing the linkage design hasbeen fabricated (fig. 16), and has already demonstrated theability to execute feedback-stabilized monopedal locomotion.A continuous hopping height of 5 cm is achieved by bang-bang control input at 9.6 Volts (35% PWM duty cycle). Thedetails regarding the control design are beyond the scope ofthis paper, however videos of the robot are available at ????.

Curved arms are used in the physical prototype, in orderto permit coplanar placement of members in each half ofthe mechanism, creating a load path free of any cantileveredsupports. Furthermore, in order to avoid linear bearings, theconstructed prototype employs a Watts straight-line linkagein place of the prismatic joint. Elastomer springs designedfor spear guns are incorporated into the design, and providehigher energy density and greater strain capacity, as comparedto helical metal springs. The prototype is fabricated from acombination of 1/16” thick aluminum sheet, and 3/4” squarealuminum tubing, with 1/4” I.D. ball bearings used in the jointsattaching the arms to the central leg, and 1/8” I.D. bronzesleeve bearings are used elsewhere. The vehicle weighs 3.5kg, including batteries and electronics.

V. CONCLUSION

A novel suspension mechanism suitable for supporting thearms of a multimodal hopping robot has been developedand successfully demonstrated in a working prototype. Atwo-armed design enables stabilization and hopping functionsto share the same resources; i.e. depending on the controlinput, actuation can be fully devoted to either task or somecombination thereof. The arm suspension mechanism engagestwo extension springs, producing large resistance to symmetricrotations and small resistance to antisymmetric rotations. Asthe extension springs are pretensioned, the symmetric stiffnessbecomes increasingly digressive, which is complementary to

75 80 85 90 95 100 105 110 115 120 12535

40

45

50

55

60

65

70

75

% of Nominal Length

Lock

ing

Ang

le [d

eg]

Mechanism Locking Angle for Varying Link Lengths B/L = 0.125, C/L = 1.094, D/L = 0.438

D

CB

Fig. 15. The arm mechanism latching angle is a function of the lengths B, C,and D. It is most sensitive to length changes of the spring bar, C, particularlyfor small B and D.

Fig. 16. “iHop v.3” prototype featuring independently actuated armssuspended via two spring-coupled linkages mirrored about the vertical axis.

hopping motion actuated via DC gearmotor torque. Further-more, the mechanism possesses two unique unstable equilibria,resulting in a latching behavior as the springs reach theirmaximum tension. Geometric symmetry of the mechanismenables bidirectional arm operation necessary for attempting“end-over-end” hopping and climbing maneuvers. Future workwill focus on implementing such advanced maneuvers on thephysical prototype, as well as investigating alternative sym-metric actuation policies, in order to raise hopping efficiencyby decreasing stalled operation, and by adding hopping energyduring flight.

ACKNOWLEDGMENT

The authors would like to thank the Los Alamos NationalLabs, the Von Liebig Institute, the San Diego Foundation,and Texas Instruments for financial/hardware donations to thisproject.

REFERENCES

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[3] C. Schmidt-Wetekam, D. Zhang, R. Hughes, & T. Bewley (2007) Design,optimization, and control of a new class of reconfigurable hopping rovers.48th IEEE CDC, 5150 - 5155.

[4] H. Rad, P. Gregorio and M. Buehler, “Design, Modeling and Control ofa Hopping Robot,” Proceedings of IROS, Yokohama, Japan (Jul. 1993)pp. 1778-1785

[5] K. V. Papantoniou, “Electromechanical Design for an Electrically Pow-ered, Actively Balanced One Leg Planar Robot,” Proceedings of IROS,Workshop on Intelligence for Mechanical Systems, Osaka, Japan (Nov.1991) pp. 1553- 1560

[6] M. H. Raibert, H. B. Brown and M. Chepponis, “Experiments in balancewith a 3D one-legged hopping machine,” Int. J. Robot. Res. 3(2), 75-92(1984).

[7] http://www.murataboy.com/en/index.html[8] N. Cherouvim and E. Papadopoulos, “Single actuator control analysis of

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