Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems Maui, Hawaii, USA, Oct. 29 - Nov. 03, 2001 The D Linear Inverted Pendulum Mo de A simple mo deling for a bip ed walking pattern generation Shuuji Ka jita Fumio Kanehiro Kenji Kaneko Kazuhito Yok oiand Hirohisa Hirukawa National Institute of Advanced Industrial Science and Technology AIST Umezono Tsukuba Ibaraki Japan Email ska jitaaistgojp Abstract The mo deling the ThreeDimensional Linear Inv erted P endulum Mo de DLIPM is derived from a general threedimensional inv erted p endulum whose motion is F or D walking c ontr ol of a bipedrobot we analyze constrained to mov e along an arbitrarily dened plane the dynamics of a thr e edimensional inverted p endu It allows a separate controller design for the sagit lum in which motion is constr aine d to move along an tal xz and the lateral yz motion and simplies arbitr arily dened plane This analysis leads us a sim awalking pattern generation a great deal ple line ar dynamics the Thre eDimensional Linear In verte d Pendulum Mode DLIPM Geometric nature of trajectories under the DLIPM and a method for Derivation of D Linear Inverted walking p attern gener ation ar e discusse d A simula tion r esult of a walking contr ol using a dof bip e d Pendulum Mo de robot model is also shown Motion equation of a D in verted p endulum Intro duction When a bip ed rob ot is supp orting its b o dy on one leg its dominant dynamics can be represented by a Research on h umanoid rob ots and bip ed lo como single inv erted p endulum which connects the supp ort tion is currently one of the most exciting topics in the ing fo ot and the center of mass of the whole rob ot eld of rob otics and there exist many ongoing pro jects Figure depicts suc h an inverted p endulum consist Although some of those w orks hav e already demon ing of a p oint mass and a massless telescopic leg The strated very reliable dynamic bip ed walking we p osition of the point mass p x y z is uniquely believe it is still imp ortant to understand the theoret sp ecied by a set of state variables q r r p ical background of bip ed lo comotion A lot of researc hes dedicated to the bip ed walk x rS ing pattern generation can b e classied into t w o cate p gories The rst group uses precise knowledge of dy y rS r namic parameter of a rob ot eg mass lo cation of cen z rD ter of mass and inertia of each link to prepare walking q patterns Therefore it mainly relies on the accuracy S S S sin S sin D p r r r p p of the mo del data Let f b e the actuator torque and force asso r p Contrary the second group uses limited knowledge ciated with the state variables r With these r p of dynamics eg lo cation of total center of mass total inputs the equation of motion of the D inverted p en angular momentum etc Since the con troller knows dulum in Cartesian co ordinates is given as follows little ab out the system structure this approachmuch relies on a feedbackcontrol In this pap er w e tak e a standp oint of the second x r T A A A approach and intro duce a new mo deling which rep y m J p resents a rob ot dynamics with limited parameters z f mg 0-7803-6612-3/01/$10.00 2001 IEEE 239 D Linear In v ertedPendulum Mo de Although the moving pattern of the p endulum has v ast p ossibilities wewant to select a class of motion that would b e suitable for walking For this reason we apply constraints to limit the motion of the p endulum The rst constraint limits the motion in a plane with given normal vector k k and z intersection z x y c z k x k y z x y c Forarobotwalking on a rugged terrain the normal v ector should match the slop e of the ground and the zintersection should b e the exp ected average distance of the center of the rob ots mass from the ground For further calculation we prepare the second derivatives of Figure D Pendulum z k x k y x y Substituting these constraints into equations and where m is the mass of the p endulum and g is gravit y w e obtain the dynamics of the p endulum under acceleration The structure of the Jacobian J is the constraints F rom straightforward calculations we get rC S p p p A g k rC S J r r y y xy xy u q r rC S D rC S D D z z mz r r p p c c c k g x xy xy u x p z z mz c c c C cos C cos r r p p where u u are new virtual inputs which are intro r p T o erase the inv ersed Jacobian that app ears in duced to comp ensate input nonlinearity T let us multiply the matrix J from the left C r u r r rC rC S D x r r r D A A rC rC S D y m p p p C p u p p S S D z p r D rC S r r r In the case of the walkingona atplane wecan A A rC S D mg p p p set the horizontal constraint plane k k x y f D and we obtain Using the rst ro w of this equation and multiplying g y u y r DC weget r z mz c c g D x u x rS mg mrDy rS z p r r r z mz C c c r In the case of the walking on a slop e or stairs By substituting kinematic relationship of equations and w e get a go o dlo oking equation that de where k k weneed another constraint From x y scrib es the dynamics along the yaxis xy wesee D mg y mz y y z u x u y xy xy r r p C mz r A similar pro cedure for the second row of yields Therefore wehav e the same dynamics of and the equation for the dynamics along the xaxis in the case of an inclined constraint plane byintro duc ing the following new constraint ab out the inputs D mg x mz x xz p C p u x u y r p 240 A unit mass is driven by a force vector of magnitude Equations and are indep endent linear f that is prop ortional to the distance r betw een the equations The only parameter which go verns those mass and the origin The force magnitude can b e dis dynamics is z ie the z intersection of the constraint c tributed into x and y elementasfollows plane and the inclination of the plane never aects the horizontal motion Note that the original dynamics w ere nonlinear and we derived linear dynamics with y g f f y out using any approximation y r z c Let us call this the ThreeDimensional Linear In g x x f f verted Pendulum Mo de DLIPM The rst author x r z c and Tani intro duced a tw odimensional version of this dynamics mo de in and Hara Yokokaw a and Equation reminds us the celestial mechanics Sadao extended it to three dimensions in the case of under the gravityeld In this case the force magni zero input torque in tude is k G f G r Nature of the D Linear Inv erted where k is a parameter determined from the gravity G Pendulum Mo de constant and the mass of the gra vit y source Both in the equations and the force vectors are In this section w e examine nature of tra jectories parallel to the p osition vectors from the origin to the under the DLIPM with zero input torques u r mass This results wellknown Keplers second lawof u p planetary motion areal velo cit y conservation g Let us see howtheforceofgravit y can b e separated y y z into x and y direction c g x x xk k x G G z c f Gx r r x y With a given initial condition these equations deter k yk y G G mine a tra jectory in D space Figure sho ws two f Gy r r x y examples Since x aects f and y aects f wemust always Gy Gx wo equations as a set Contrarilythe x 1 treat these t and y element of the DLIPM can be treated inde 0.8 p endently at all times This gives us great advan tages alking patterns as we 0.6 in analyzing and in designing w z will see in the following sections 0.4 0.2 Geometry of the tra jectory 0 0.5 0.5 0 0 -0.5 -0.5 y x Y Y' D Linear Inverted Pendulum Mode Figure (v , v ) x y (x, y) Similarity and dierence with gravity X' eld θ X e can regard that equations and repre W O sen t a force eld for a unit mass g r f Figure DLIPM proje ctedontoXYplane z c 241 following condition must be satised Since B Figure shows a DLIPM tra jectory whichispro rarely happ ens we do not consider eq and jected on to XY plane Motions along Y and X are go v erned by the equations and resp ectively gz xy x y c By integrating each equation we obtain a time invari an t parameter named the orbital energy Using this equation wecan calculate the geometric g shap e of the DLIPM By substituting eq and y y E y z c g x x E x gz x y x y c z c E gz x E gz y x c y c In Figure it is sho wn another co ordinate frame X Y which rotates from the original frame XY The nal form is a simple quadratic equation Since the DLIPM is a dynamics under the central g g force eld as discussed in the last section the new x y z E z E c x c y frame X Y also gives a prop er representation of the DLIPM The new orbital energy is calculated as Since E andE eq forms a hyp erb olic x y g curve cx sy cx sy E x Hyp erb olic curves also app ear in Kepler motion z c and one of its examples is a swing by ight of the g sx cy sx cy E y V o y ager spacecraft approached Jupiter in z c It seems in teresting that w e obtained same shap e of where c cos s sin By simple calculations we tra jectory from a totally dierent p otential eld can verify that the total energy of the system do es not c hange by the wa y of the co ordinate setting D walking pattern generation E E E E constant x y x y When the Y axis corresp onds to the axis of sym Outline metry like in Figure E and E b ecome maximum x y and minimum resp ectively Therefore w e can cal Figure sho ws an example of a w alking pattern culate the axis of symmetry b y solving the following based on the DLIPM In this