A Dynamically Stable Single-Wheeled Mobile Robot With .

Fig. 4.Ballbot inverse mouse-ball drive mechanismFig. 5.ball to produce motion. The initial ball was a 200 mm diameterhydroformed steel shell covered with a 3.2 mm thick urethaneouter layer. We have fabricated balls with urethane formulationsof several different durometers. The ball is actuated by a pair of12.7 mm diameter smooth stainless steel rollers placed orthogonally at the sphere’s equator. These rollers are linked throughtiming belts to two high torque DC servomotors. Opposite thedrive rollers are two spring-loaded passive idler rollers thatapply force at the ball’s equator to maintain contact betweenthe drive rollers and the ball. This arrangement representsa compromise since some slippage is always present. Forexample, if one roller is being driven, the orthogonal roller mustbe slipping. This simultaneously demands both a high-frictionand low-friction material for the ball. On the other hand, it isalways desirable to have high friction between the ball and thefloor. The drive works well but the initial ball eventually woreout. A second ball design with a lighter 190.5 mm diameterspun aluminum shell and 12.7 mm of urethane has unobservablewear, presumably due to the lower shear stresses in the thickerurethane layer. The entire drive mechanism is attached to thebody with a large diameter thrust bearing, allowing a thirdactuator (currently not installed) to re-orient the body in yaw.Finally, the entire Ballbot body rests on top of the ball on threecommercial low friction, omni-directional ball transfer devices.IV. S IMPLIFIED BALLBOT M ODELFor the purposes of developing a stabilizing controller, weintroduce and derive equations of motion for a simplified modelof Ballbot. In this model, the Ballbot ball wheel is a rigidsphere, the body is rigid, and the control inputs are torquesapplied between the ball and the body. There is no slip betweenthe wheel and the floor. Further, we assume that the motion inthe median sagital plane and median coronal plane is decoupledand that the equations of motion in these two planes areidentical. As a result, we can design a controller for the full3D system by designing independent controllers for the twoseparate and identical planar systems.It is worth making special note of the modeling assumptionsthat are made regarding friction. Friction between the wheeland the floor and between the wheel and the body is modeledas pure viscous damping. Forces due to static friction andnonlinear dynamic friction are neglected. The inclusion of aviscous term in the friction model makes sense: there arePlanar simplified Ballbot model used for controller design.hysteresis losses associated with with the compression andrelaxation of the urethane layer that occurs at the ball-floorand ball-roller contact points, and these losses can reasonablybe assumed to be velocity dependent. However the exclusionof the effects of static and nonlinear dynamic friction is notas easily justified, and we have determined experimentallythat these effects are in fact significant (see Section VII).Still we choose to neglect these terms because they wouldintroduce discontinuities and extreme nonlinearities that wouldrender the resulting Ballbot model unusable for linear controllersynthesis. As described in Section V, the controller presentedhere employs an inner PI loop to mitigate the effect of thesemodeling omissions.Fig. 5 is a diagram depicting the planar model. The Lagrangian formulation is used to derive the nonlinear equationsof motion for the simplified model (see, e.g., [3]). The first stepis to compute the kinetic energy Kb of the ball:mb (rb θ̇)2Ib θ̇2 ,22where Ib , mb , and rb are, respectively, the moment of inertia,mass, and radius of the ball. The potential energy of the ballis Vb 0. The kinetic energy KB and potential energy VB ofthe body are mB 2 2KB rb θ̇ 2rb ℓ(θ̇2 θ̇ φ̇) cos(θ φ) ℓ2 (θ̇ φ̇)22IB (θ̇ φ̇)2 ,2VB mB gℓ cos(φ θ),Kb where IB is the moment of inertia of the body about the centerof the ball, ℓ is the distance between the center of the ball andthe center of mass of the body, mB is the mass of the body, andg is the acceleration due to gravity. The total kinetic energy isK Kb KB and the total potential energy is V Vb VB .Define the system configuration vector q [ θ φ ]T . TheLagrangian L is a function of q and q̇ and is defined to beL(q, q̇) K V.Let τ be the the component of the torque applied betweenthe ball and the body in the direction normal to the plane. Tomodel the viscous friction terms, define the vector µ θ̇D(q̇) θ ,µφ φ̇

Fig. 6.Structure of stabilizing linear feedback controller.where µθ and µφ are the viscous damping coefficients thatmodel ball–ground and ball–body friction, respectively. Usingthis notation, the Euler-Lagrange equations of motion for thesimplified Ballbot model are d L L0 D(q̇).τdt q̇ qAfter computing the derivatives in the Euler-Lagrange equations and rearranging terms, the equations of motion can beexpressed as 0M (q)q̈ C(q, q̇) G(q) D(q̇) .(1)τmust be overcome to achieve velocity tracking, thus reducingthe effect of the unmodeled static and dynamic friction. Theintegral term adds an extra state to the system. Define theaugmented state vector xa [ xT x5 ]T . The closed loopequations of motion of the inner loop can then be written as" #f x, kp (ωd θ̇) ki (x5 θ) fa (xa , ωd ).ẋa ωdThe outer loop is designed by linearizing the inner loopequations of motion and applying LQR (see e.g., [1]). Notethat the simplified Ballbot system is at equilibrium wheneversin(θ φ) 0 and φ̇ θ̇ 0. The objective is to designa controller that will balance Ballbot with the body straightup and hold it in a fixed position θ 0, which is equivalentto stabilizing the equilibrium point at xa 0. We begin bylinearizing the equations of motion about this point:ẋa fa faxa ωd . xa xa 0,ωd 0 ωd xa 0,ωd 0 {z}{z}ABThe pair (A, B) is controllable, from which we can infer theabsence of any nonholonomic constraints and the existence ofa smooth stabilizing controller.The mass matrix M (q) isNow LQR can be used to find a linear state feedback con Γ1 2mB rb ℓ cos(θ φ) Γ2 mB rb ℓ cos(θ φ)troller that stabilizes the system about xa 0 and minimizes,M (q) Γ2 mB rb ℓ cos(θ φ)Γ2the cost functionZwhereJ (xa (t)T Qxa (t) Rωd (t)2 )dt.Γ1 Ib IB mb rb2 mB rb2 mB ℓ2 ,We choose the structure of Q to beΓ2 mB ℓ2 IB . γb γB γB000The vector of Coriolis and centrifugal forces is γbγB000 0 γḃ γḂ γḂ 0 Q 0 mB rb ℓ sin(θ φ)(θ̇ φ̇)2 ,C(q, q̇) 00γḂγḂ 0 00000 γ5and the vector of gravitational forces iswhere γb , γB , γḃ , γḂ , and γ5 can be loosely thought of as mB gℓ sin(θ φ)controlling the relative convergence rates of the ball angle,G(q) . mB gℓ sin(θ φ)body angle, ball angular velocity, body angular velocity, andTo put these equations into standard nonlinear state space x5 , respectively. In practice, these parameters were hand tunedform, define the state vector to be x [ q T q̇ T ]T and define based on simulation results. For a given choice of Q and R,Matlab’s LQR command can be used to compute the associatedthe input u τ . This together with Eq. 1 yieldsgain matrix K, which defines the stabilizing feedback control q̇law ωd Kxa . f (x, u).0ẋ When implementing the controller on the actual robot, weM (q) 1 C(q, q̇) G(q) D(q̇)uwere forced to deviate slightly from the controller presentedabove. We found that there is a practical limit on the magnitudeV. S TABILIZING F EEDBACK C ONTROLLERof the gain k4 that multiplies φ̇. Exceeding this limit induces anThe linear controller used to stabilize Ballbot has two loops: oscillation not present in the simplified Ballbot model. The Kan inner loop that feeds ball velocity θ̇ back into a PI controller, matrix generated by the LQR algorithm gives a k4 that exceedsand an outer loop linear quadratic regulator (LQR) that uses the practical limit, so we manually adjusted k4 to an allowablefull state feedback. This architecture is shown in Fig. 6. The level. We hypothesize that this oscillation is due to flexibilityproportional gain kp and integral gain ki in the PI controller in the body frame and the mechanics of the soft urethane layerare chosen and experimentally tuned so that the actual ball that couples the drive roller to the ball. This may also be avelocity θ̇ tracks the desired ball velocity ωd . This inner loop consequence of the decision to neglect static and nonlinearautomatically compensates for the various frictional torques that dynamic friction in the simplified model.

3position (m)ball position (m)0.250.10.050.20.100 0.1 0.0505101520time (s)25303540 0.2 0.4 0.200.20.4position (m)Fig. 7.Position step response for a point-to-point move.Fig. 10.Plot of the ball path while attempting to move in a square.0.6simulationexperimentalfinish0.30.20.250y (meters)body angle (deg)0.4 0.2 0.40.20.150.1 0.60.05 0.80510152025303540startactualdesired0time (s)00.050.10.150.20.250.30.35x (meters)Fig. 8.Body angle during point-to-point move.Fig. 11.VI. I NITIAL R ESULTSA number of tests were conducted to characterize physicalsystem performance, and to make comparisons with simulation.During operation on a hard tiled floor, it was found thatthe machine was able to balance robustly, strongly resistingattempts to tip it over when a person applied torques to thebody. However, it was not able to simultaneously balance andstation keep. When operated on a carpeted surface, Ballbot wasable to do both, presumably due to the extra damping affordedby the carpet material.In the test run shown in Figs. 7 and 8, Ballbot was commanded to move from a starting position in a straight line to agoal position. There is an initial retrograde ball motion causingthe body to lean toward the goal position, followed by a reversemotion to stop at the goal. Differences between simulation and3020y position (mm)100 20 30 40 40 20020x position (mm)Fig. 9.experiment are probably due to unmodeled frictional and springforces. The divergence when station keeping is at most about40 mm in position, and 0.5 in tilt.To see the typical motion jitter experienced during operation,one may plot the paths taken as the ball moves around on thecarpeted floor. Fig. 9 shows data taken from a 99 s run whereBallbot was released slightly out of balance, which was rapidlycorrected by ball motion, followed by station keeping withina roughly circular region of about 40 mm diameter. Fig. 10shows Ballbot’s attempt to track a square trajectory.The straight line path plotted in Fig. 11 was generated bycommanding Ballbot to move at a constant velocity for a periodof 40 seconds, demonstrating that we can specify trajectories interms of both desired position and desired velocity. This motionis much slower that the motion depicted in the previous straightline plot (Figs. 7 and 8) and in the square plot (Fig. 10). Thesefast motions exhibit fairly straight trajectories because Ballbotis essentially “falling” toward its goal, then recovering when itreaches the goal. In contrast, the motion depicted in Fig. 11 isslow, steady, and tightly controlled over the entire path.VII. F UTURE W ORK 10 50 60Plot of the ball path for straight line move with trajectory control.Plot of the ball path during balancing and station keepingThe LQR controller presented here is sufficient to balanceBallbot and drive it along rudimentary trajectories on carpetedsurfaces. These early capabilities fall short of the robust balancing and agile mobility that will be required in order for Ballbotto operate effectively in human environments.The overly simplified friction model is likely to be a majorreason for the poor performance of the resulting controller.Experimentally, we have determined that the effects of static

ball velocity (m/s)ball velocity (m/s)10.500actuating the idler rollers so that the ball is driven from bothsides with equal and opposite tangential forces.test datanew modelold modelVIII. D ISCUSSION510torque applied (Nm, forward direction)15510torque applied (Nm, reverse direction)150 0.5 10Fig. 12.Plot of ball velocity vs. applied torque. The plot comparesexperimental results to values predicted by the pure viscous model used inthis paper and the nonlinear model that will be used in future work.Our results are preliminary and there is much that remainsto refine Ballbot’s model and control. Nevertheless, it wouldappear that Ballbot and its progeny might well represent thevanguard of a new type of wheeled mobile robot capable ofagile, omni-directional motion. Such robots, combined with theresearch community’s ongoing work in perception, navigation,and cognition, could yield truly capable intelligent mobilerobots for use in physical contact with people.AcknowledgmentThis work was supported in part by NSF grant IIS-0308067.The authors also thank Eric Schearer and Anish Mampetta fortheir indispensable assistance in running Ballbot experiments.R EFERENCESand nonlinear dynamic friction on Ballbot behavior are significant. For example, the break-away torque necessary toovercome static friction is nearly half of the maximum torquethat can be applied by the motor. Additionally, the frictionaleffects are asymmetric; Ballbot moves in the reverse directionmuch more easily than it moves in the forward direction. Wehave developed a Ballbot actuation model that includes static,Coulomb, and viscous friction as well as the observed asymmetry. This improved model matches experimentally observedBallbot motion significantly better than the pure viscous frictionmodel used in this paper (see Fig. 12). In future work, we willuse this model to guide the design of a more robust nonlinearbalancing controller. Specifically, we will investigate the use ofsliding mode control, which has been proven to be effective forsystems with high static friction [5].In addition to designing new controllers, we will make anumber of mechanical modifications that will expand Ballbot’scapabilities. One such modification is the installation of theactuator to control the yaw of the Ballbot body. This actuatorcontrols the relative yaw angle between the body and the inversemouse-ball actuator at its base, and it relies on static frictionbetween the ball and the floor in order to control the absoluteyaw of the body. For the aluminum ball with a half-inch thickcoating of 70A-72A durometer urethane, we have measured theyaw-axis break away torque between the ball and the floor tobe approximately 4.25 Nm on tile floor and 3.55 Nm on carpet.The area of contact between this ball and the floor is a circlewith a diameter of 30mm. These values should be sufficient toprevent the ball from slipping during yaw maneuvers.We will also consider a redesign of the inverse mouse-ballactuator. The observed asymmetry in the mouse-ball actuationis due to force imbalance that results from driving the ball froma single roller on the side. When the actuator moves forward,the ball is pushed up into the body, increasing the frictionbetween the ball and body. When the actuator moves backward,the ball is pulled down, away from the body, decreasing friction.We will investigate designs that eliminate this asymmetry by[1] B.D.O. Anderson and J.B. Moore. Optimal Contol: Linear QuadraticMethods. Prentice Hall, 1990.[2] G. Baltus, D. Fox, F. Gemperle, J. Goetz, T. Hirsch, D. Magaritis,M. Montemerlo, J. Pineau, N. Roy, J. Schulte, and S. Thrun. Towardspersonal service robots for the elderly. In Proc. Workshop on InteractiveRobotics and Entertainment (WIRE), Pittsburgh, PA, 2000.[3] A.M. Bloch. Nonholonomic Mechanics and Control. Springer, 2003.[4] A. Diaz-Calderon and A. Kelly. On-line stability margin and attitudeestimation for dynamic articulating mobile robots. Int’l. J. of RoboticsResearch, 2005. (to be published).[5] C. Edwards and S.K. Spurgeon. Sliding Mode Control: Theory andApplications. Taylor and Francis Ltd, 1998.[6] Y.-S. Ha and S. Yuta. Trajectory tracking control for navigation of selfcontained mobile inverse pendulum. In Proc. IEEE/RSJ Int’l. Conf. onIntelligent Robots and Systems, pages 1875–1882, 1994.[7] O. Khatib, K. Yokoi, K. Chang, D. Ruspini, R. Holmberg, and A. Casal.Vehicle/arm coordination and multiple mobile manipulator decentralizedcooperation. In Proc. of the IEEE/RSJ Int’l Conf. on Intelligent Robotsand Systems, pages 546–553, Osaka, 1996.[8] T. Lauwers, G. Kantor, and R. Hollis. One is enough! In Proc. Int’l.Symp. for Robotics Research, San Francisco, October 12-15 2005. Int’l.Foundation for Robotics Research.[9] R. Nakajima, T. Tsubouchi, S. Yuta, and E. Koyanagi. A developmentof a new mechanism of an autonomous unicycle. In Proc. IEEE/RSJInt’l. Conf. on Intelligent Robots and Systems, pages 906–12, Grenoble,France, September 7-11 1997.[10] H. G. Nguyen, J. Morrell, K. Mullens, A. Burmeister, S. Miles, N. Farrington, K. Thomas, and D. Gage. Segway robotic mobility platform. InSPIE Proc. 5609: Mobile Robots XVII, Philadelphia, PA, October 2004.[11] R. Simmons, J. Fernandez, R. Goodwin, S. Koenig, and JosephO’Sullivan. Xavier: An autonomous mobile robot on the web. Roboticsand Automation Magazine, 1999.[12] S. Thrun, M. Bennewitz, W. Burgard, A.B. Cremers, Frank Dellaert,Dieter Fox, D. Haehnel, Chuck Rosenberg, Nicholas Roy, JamiesonSchulte, and D. Schulz. MINERVA: A second generation mobile tourguide robot. In Proc. of the IEEE Int’l Conf. on Robotics and Automation(ICRA’99), 1999.

A two-wheeled robot with inverse pendulum control devel-oped in Japan was demonstrated in 1994 [6]. The two-wheeled design eliminated the need for a third castoring wheel. The same group introduced a one-wheel balancing robot [9]. The wheel is a prolate ellipsoid like a rugby ball and is driv