
Control for an Autonomous Bicycle Neil H. Getz Jerrold E. Marsden Department of Electrical Engineering Department of Mathematics University of California University of California Berkeley,CA94720 Berkeley,CA94720 [email protected] [email protected] IEEE International Conference on Rob otics and Automation, 21-27 May 1995, Nagoya m Abstract The control of nonholonomic and underactuated systems with symmetry is il lustrated by the problem ontrol ling a bicycle. We derive a control ler which, of c Component of tire force on ground Steering axis ering and rear-wheel torque, causes a model using ste p parallel to ground plane of a riderless bicycle to recover its balancefrom a near onverge to a time parameterizedpath fal l as wel l as c c in the ground plane. Our construction utilizes new re- Contact Line oth the derivation of equations of motion for sults for b ur nonholonomic systems with symmetry, as wel l as the Constrained directions ontrol of underactuatedrobotic systems. c of wheel travel b 1 Intro duction Figure 1: Side view of the bicycle mo del with =0. Control of the bicycle is a rich problem o ering a the tracking control of nonminimum phase systems. numb er of considerable challenges of current research The pap er pro ceeds as follows: By exploitation of interest in the area of mechanics and rob ot control. the bicycle's constraints and symmetry we rst derive The bicycle is an underactuated system, sub ject to a reduced set of equations of motion for the bicycle. nonholonomic contact constraints asso ciated with the We then review how the bicycle can b e made to fol- rolling constraints on the front and rear wheels. It is low arbitrary roll-angle tra jectories, thus allowing the unstable (except under certain combinations of fork vehicle to recover from near falls and disturbances. A geometry and sp eed) when not controlled. It is also, purely kinematic mo del of the bicycle is then intro- when considered to traverse at ground, a system sub- duced in order to showhow, disregarding the unstable ject to symmetries; its Lagrangian and constraints are roll-angle dynamics, the bicycle may b e made to con- invariant with resp ect to translations and rotations in verge to a desired tra jectory in the plane. Wethen the ground plane. showhowwemay use our knowledge of howtosteer Though a numb er of researchers have studied the the kinematic bicycle to construct a controller that stability of bicycles and motorcycles under a nominal allows a leaning bicycle to track planar tra jectories linear mo del of rider control (See Hand [1] for a sur- without falling. vey), as far as weknow our work presents the rst controller allowing tracking of arbitrary tra jectories while maintaining balance. Control of balance and 2 The Mo del roll-angle tracking for the bicycle mo del we use here has b een addressed byGetz[2]. In addition to extend- The control of a simpli ed bicycle mo del illustrated ing those results to tracking in the plane we also utilize in Figures 1 and 2 will b e considered. The wheels of some new results from Blo ch, et al. [3] on the deriva- the bicycle are considered to have negligible inertial tion of equations of motion for nonholonomic systems moments, mass, radii, and width, and to roll without with symmetries, and from Getz and Hedrick[4]on side or longitudinal-slip. The vehicle is assumed to lo city along the path traveled, and the generalized co- ordinate corresp onding to v is the integral in time of ? elo city along a direction p erp endicular b φ the rear-wheel v h, by virtue of the constraints, α θ to the rear-wheel whic ays zero. v⊥ c is alw It will b e assumed that the bicycle exerts a control force u (see Figure 1) on the ground at the site of y vr r tofcontact b etween the rear-wheel and the Contact Line the p oin ground. The force u will b e considered to act along r the contact line as indicated in Figure 1 and is the gen- eralized force corresp onding to v . A torque generator x r ariable , the gener- Ground Frame is asso ciated with the steering v alized torque b eing u .We consider a vehicle with a rigid or non-existent passenger under automatic con- Figure 2: Top view of the bicycle mo del rolled away trol. from uprightby angle . Bold arrows indicate wheel directions at the ground plane. 3 Equations of Motion have a xed steering-axis that is p erp endicular to the Wecho ose a b o dy-frame for the bicycle centered at at ground when the bicycle is upright. For simplic- the rear-wheel ground contact, with one axis p oint- itywe concern ourselves with a p oint mass bicycle. ing forward along the line of intersection of the rear The rigid frame of the bicycle will b e assumed to b e wheel plane with the ground, another axis orthogonal symmetric ab out a plane containing the rear wheel. to the rst and in the ground plane, and an axis nor- Consider a ground- xed inertial reference frame mal to the ground, p ointing in the direction opp osite with x and y axes in the ground plane and z -axis p er- to gravity (see Figure 2). The b o dy frame is a natural p endicular to the ground plane in the direction opp o- frame in which to write the Lagrangian of the bicy- site to gravity.Theintersection of the vehicle's plane cle for a numb er of reasons. In particular the rolling of symmetry with the ground plane forms a contact- constraints takeonavery simple form. The general- line. The contact-line is rotated ab out the z -direction ized velo cities of the bicycle are contained in the par- T T _ bya yaw-angle, . The contact-line is considered di- titioned co ordinatesr _ =[_ ;v ; _ ] ands _ =[; v ] . r ? rected, with its p ositive direction from the rear to the In these velo city co ordinates the nonholonomic con- front of the vehicle. The yaw-angle is zero when the straints asso ciated with the front and rear wheels, as- contact-line is in the x direction. The angle that the sumed to roll without slipping, are expressed very sim- bicycle's plane of symmetry makes with the ground is ply by_s + A(r;s)_r =0 or the rol l-angle, 2 (=2;=2). Front and rear-wheel 2 3 _ contacts are constrained to havevelo cities parallel to _ 0 0 4 5 + v =0 (1) the lines of intersection of their resp ective wheel planes r v 0 0 0 ? _ and the ground-plane, but free to turn ab out an axis through the wheel/ground contact and parallel to the The mapping represented bymatrixA(r;s)isan z -axis. Ehresmann connection [3], connecting the base velo ci- Let 2 (=2;=2) b e the steering-angle between tiesr _ to the ber velo citiess _ . Due to symmetries of the the front-wheel-plane/ground-plane intersection and Lagrangian with resp ect to translations and rotations the contact-line as shown in Figure 2. With we in the plane, A(r;s) is a function only of r . asso ciate a moment of inertia J .For simplicitywe Let s := sin( )and c := cos ( ): The Lagrangian will parameterize the steering angle by := tan(=b). for the bicycle is The comp onent of the velo city of the rear- 2 wheel/ground contact along the contact line is v . b_ 1 r J L = mg pc + 2 2 2 1+b The velo city of the rear contact p erp endicular to this 1 2 2 2 _ _ (v + ps + ) +(v p _ c + c ) +(p s_ ) line and in the ground plane is v . The angle of the r ? ? 2 contact-line with resp ect to the x-axis of the ground- (2) xed inertial frame is . where m is the mass of the bicycle, considered for Note that the generalized co ordinate corresp onding simplicitytobea point mass, and J is the moment to v is the the integral in time of the rear-wheel ve- of inertia asso ciated with the steering action. r 2 .Thevariable is thus decoupled. For convenience Incorp orating the constraints into the Lagrangian later wewillcho ose _ as a control and call it w .The we obtain the constrainedLagrangian for the bicycle equations of motion then take on the simpler form 2 2 b J _ L = (g mpc )+ 2 2 2 c 2(1+b ) 1 2 2 2 2 2 _ = w +m((v + p s v ) + p s _ + (c v pc _ ) ) r r r 2 u (3) r ~ ~ ~ M = F + B (7) v_ w Of course the equations of motion for the constrained r Lagrangian are not Lagrange's equations. The correct where formulation of the equations of motion based up on the 2 constrained Lagrangian are derived in [3] and shown p cpc ~ M = (8) 2 2 2 2 to b e equivalent to d'Alemb ert's equations for con- cpc 1+ (c + p s ) +2p s strained systems. They are 2 gps +(1+ p s )pc v r ~ F = 2 (1 + p s )2pc v _ cp s _ @L @L @L d @L r c c c a b + A = B r_ (4) b cpc v 0 dt @ r_ @r @s @ s_ r ~ B = (9) 2 (c + ps (1 + p s ))v 1=m r b where B denote the co ordinates of the curvature of the connection A(r;s), ~ Note that the rst column of B has v as a factor r ! con rming the intuitive notion that if v = 0 then the r b b b b @A @A @A @A b a a steering action can have no a ect on either or v .
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