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Biped walking on a low friction floor

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Biped walking on a low friction floor
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  Biped Walking on a Low Friction Floor Shuuji KAJITA, Kenji KANEKO, Kensuke HARADA, Fumio KANEHIRO, Kiyoshi FUJIWARAand Hirohisa HIRUKAWA National Institute of Advanced Industrial Science and Technology (AIST)Tsukuba Central 2, 1-1-1 Umezono, Tsukuba, Ibaraki, 305-8568 JapanEmail:  { s.kajita,k.kaneko,kensuke.harada,f-kanehiro,k-fujiwara,hiro.hirukawa } @aist.go.jp  Abstract —Biped walking on a low friction floor is analyzedin this paper. For a given walking pattern, we can calculatea necessary friction coefficient which allows the robot to per-form the expected motion. To reduce the maximum necessaryfriction coefficient, a pattern generation based on previewcontrol theory is explained. We also describe a calculationof slip concerned ZMP, which provides a good prediction of falling caused by slips. Finally, we test a walk of 1.35 [km/h]on low friction environment using a humanoid robot HRP-2. The robot could successfully walk over the slippery areawhose friction coefficient is 0.14. I. I NTRODUCTION For a practical use of a humanoid robot, it is required totraverse on  real  environments which are not conditioned tooffer flat and firm support of the feet. A robot for indooruse does not need to handle uneven or deforming ground,but it still needs to handle various level of slipperinesssince a floor can become very slippery with water, oilor powders. Such conditions are dangerous not only fora robot but also for a human. For example, 3,397 Japanesedied from slipping or tripping on flat ground in 2002 [1].In the field of ergonomics and biomechanics, therefore,intensive research works have been performed to analyzeslipping and slip induced falling [2]–[4]. However, it seemslike there still exist controversy on the conditions whichresult falling [5].Back to the field of robotics, a few works treated a bipedlocomotion on slippery environment. Boone and Hodginssimulated a running biped on a floor with low frictionarea [6]. Their srcinal running controller could negotiatefriction coefficients as low as 0.28. By introducing reflexcontrol strategy, their robot could run over surfaces withcoefficients as low as 0.025. Park and Kwon simulateda 12-DOF biped robot walking on slippery surface [7].They designed a controller to enlarge the frictional forceat slipping and it allowed their robot to traverse a surfacewith friction coefficient as low as 0.3.In this paper, we examine a biped walk on slippery floorusing pre-calculated walking pattern. For a given walkingpattern and known friction coefficient, we want to answerthe following basic questions.Q1: Can we predict the occurrence of slip?Q2: Can we predict falling ?We believe answering these questions are valuable not onlyfor robotics but also for our daily life.The rest part of this paper is organized as follows.Section II tries to answer Q1, prediction of slipping. It alsodescribes a walking pattern generation which may reducethe occurrence of slip. Section III answers Q2, predictionof falling. It is shown that a simple measure based on Zero-Moment Point can well predict the occurrence of falling.Section IV describes our experiment of actual humanoidrobot walking on a low friction environment. Section Vsummarizes the results and concludes this paper.II. W ALKING  P ATTERN AND  F RICTION  C OEFFICIENT  A. Foot-Ground Interaction Suppose a robot foot is slipping on a horizontalfloor(Fig.1(a)). The foot is subjected by distributed forcevectors (small black arrows) generated by microscopicinteractions between the sole and the floor surface. Thoseforce vectors can be integrated as the total floor force vector f  , which can be measured by a force sensor embeddedin the foot. Its tangential magnitude  f  t  and the verticalmagnitude  f  z  has a relationship of  f  t  =  µf  z ,  (1)where  µ  is dynamic friction coefficient. f  t f  z slip direction (a) f  µ 1 f  (b) Fig. 1. Forces acting from floor to robot When the foot is not slipping, we have f  t  < µ s f  z ,  (2)where  µ s  is static friction coefficient. For simplicity, wespecify  µ s  =  µ  in this paper. As the result, we can assumethat the force vector  f   always lies inside or on the surfaceof the friction cone(Fig.1(b)).  B. Friction Requirement for a Given Walking Pattern A humanoid robot can be modeled as a set of rigidbodies connected by joints(Fig.2). Let  n  be the numberof joints, so the robot consists of   n  + 1  links. We definethe  0 -th frame on the pelvis as  Σ 0 , whose position and 0-7803-8463-6/04/$20.00 ©2004 IEEE Proceedings of 2004 IEEE/RSJ International Conference onIntelligent Robots and SystemsSeptember 28 - October 2, 2004, Sendai, Japan 3546  orientation are  p 0  and  R  0  with respect to the world frame Σ w . We also define a vector of joint angles  θ  ( n × 1 ). Awalking pattern is determined by a set of time profiles of   p 0 ,  R  0  and  θ . Σ w Σ 0 Fig. 2. A model of humanoid robot From a given walking pattern, we can calculate the totallinear momentum of the robot. P   = n  i =0 m i ˙ c i ,  (3)where  m i  and  c i  are the mass and the center of mass of the  i -th link respectively.By differentiating this linear momentum, we can obtainthe vertical and tangential forces during the walking as f  z  = ˙ P  z  +  Mg,  (4) f  t  =    ˙ P  2 x  + ˙ P  2 y ,  (5)where  M   is the total mass of the robot,  g  is the gravityacceleration.If the forces  f  z  and  f  t  are not obtained from the floor,the robot will not behave as expected. Now, let us define necessary friction coefficient   as µ nec  ≡  f  t /f  z .  (6) µ nec  indicates the minimum friction coefficient to keep therobot on the ground without slip. The non-slip condition(2) can be rewritten as µ nec  < µ.  (7)If this inequality is satisfied for a given walking pattern,we can conclude that the robot will walk without slip. C. Walking Pattern for a Low Friction Environment  In this section, we outline a walking pattern generationbased on preview control theory [8]. With this method, wecan easily modify the necessary friction coefficient of thewalking pattern.Under proper condition a walking dynamics can beapproximated by  p x  =  x −  z c g  ¨ x,  (8)  p y  =  y −  z c g  ¨ y,  (9)where  ( x,y )  represent the horizontal displacement of thewhole robot’s center of mass (CoM),  z c  is the height of the CoM and  (  p x ,p y )  is the Zero-Moment Point (ZMP)[9]. Fig. 3 shows a suggestive model for these equations.The CoM of the robot is represented by a running cartand the leg configuration is represented by a mass lesspedestal table. A walking pattern generation is equivalentto calculate the cart motion so that it yields the prescribedZMP.   ɺɺ x x  c  z  x  p O  τ  = zmp  0 M  Fig. 3. A cart table model Equation 8 can be rewritten as a standard system equa-tion by taking the jerk ( u x  ≡ ... x ) as its input  1 . ddt  x ˙ x ¨ x   =  0 1 00 0 10 0 0  x ˙ x ¨ x  +  001  u x  p x  =   1 0  − z c g  x ˙ x ¨ x  . (10)The system output  p x  follows the reference  p ref x  byapplying the following  preview controller   [10]. u x ( k ) =  − G I k  i =0 e ( i ) − Gx ( k ) − N  L  j =1 G  p (  j )  p ref x  ( k  +  j ) , (11)where  G I   and  G  are the feedback gains and  G  p (  j )  is apreview gain which care about the reference ZMP up to N  L  steps future.  e ( i )  is the tracking error of ZMP and x ( i )  is the state vector, e ( i )  ≡  p x ( i ) −  p ref x  ( i ) , x ( i )  ≡  [  x ( i ) ˙ x ( i ) ¨ x ( i ) ] T  . 1 Hereafter, we only explain the sagittal( x - z ) motion but the lateral ( y - z ) motion can be easily obtained by the same procedure. 3547  TABLE IW ALK PARAMETERS TO EVALUATE SLIPS AND FALLING Height of CoM  z c : 0.814 [m]Sagittal step length: 0.3 [m]Lateral step width: 0.19 [m]Step period: 0.8 [s]Single support duration: 0.7 [s]Double support duration: 0.1 [s]Speed at steady walking: 0.375 [m/s]1.35 [km/h]Number of steps: 4 [steps]Total travel distance: 1.2 [m] The gains of the preview controller are determined tominimize the following performance index. J   = ∞  i = k { Q e e ( i ) 2 + ∆ x T  ( i )  Q 1  0 00  Q 2  00 0  Q 3  ∆ x ( i )+ R ∆ u 2 ( i ) } ,  (12)where  Q e ,Q 1 ,Q 2 ,Q 3  and  R  are non-negative weights, ∆ x ( i )  ≡  x ( i )  − x ( i −  1)  is the incremental state vectorand  ∆ u ( i )  ≡  u ( i ) − u ( i − 1)  is the incremental input.Table I lists the walk parameters used to determine thereference ZMP. In all simulations and experiments of thispaper, we used consistent walking patterns generated fromthe same reference ZMP, since the occurrence of slips andfalls might be highly influenced by the walking speed, thestep length and other conditions.From the same ZMP reference, we can still generatewalking patterns with different characters by adjusting theweight of (12). A walking pattern with smaller necessaryfriction coefficient  µ nec  can be obtained by enlarging theweight  Q 3 , because it penalizes the horizontal accelerationof CoM. We made three walking patterns for  Q 3  =0 . 0 , 0 . 5 , 1 . 0 , and the necessary friction coefficients areshown in Fig. 4 and Table II. As expected, by enlarging Q 3 , we got a walking pattern with smaller  µ nec  which ismore suitable for lower friction environment. 0123456700.020.040.060.080.10.120.140.160.180.2      µ   n  e  c    [ −   ]  time [s]Q 3 =0.0Q 3 =0.5Q 3 =1.0 Fig. 4. Necessary friction for walking patterns Fig. 5 shows the walking pattern of   Q 3  = 0 . 0 . This isa setting of our usual walking pattern generation, wherethe ZMP tracks the reference with good accuracy. Fig. 6 TABLE IIW ALKING PATTERN AND NECESSARY FRICTION COEFFICIENTS Walking pattern maximum  µ nec Q 3  = 0 . 0  0.196 Q 3  = 0 . 5  0.146 Q 3  = 1 . 0  0.131Other parameters are  Q e  = 1 . 0 ,Q 1  =  Q 2  = 0 . 0 ,R  = 1 . 0 × 10 − 6 is the walking pattern of   Q 3  = 1 . 0 , which realizes smallnecessary friction coefficient. As observed in the graph,the ZMP does not accurately track the reference with thissetting. Nevertheless, this is also a valid walking pattern.These patterns of CoM and reference ZMP were trans-lated into the actual walking pattern of   (  p 0 , R  0 , θ )  by usinginverse kinematics. 0123456700.20.40.60.811.2   z  m  p  x   [  m   ] zmp ref xzmp01234567−0.1−0.0500.050.10.15   z  m  p  y   [  m   ] time [s]zmp ref yzmp Fig. 5. Walking pattern of   Q 3  = 0 0123456700.20.40.60.811.2   z  m  p  x   [  m   ] zmp ref xzmp01234567−0.1−0.0500.050.10.15   z  m  p  y   [  m   ] time [s]zmp ref yzmp Fig. 6. Walking pattern of   Q 3  = 1 . 0  D. Trajectory Error Caused by Slip The trajectory errors caused by slips were evaluatedby simulation. We used OpenHRP, which is a dynamicsimulator developed in the Humanoid Robotics Project(HRP) [11]. Simulated robot model is HRP-2, a 30 DOFhumanoid robot which was also developed in HRP. Fig.7 illustrates the simulation result at  µ  = 0 . 08  usingthe walking pattern of   Q 3  = 0 . The reference pelvistrajectory  p ref  0  and the footholds are plotted by dotted 3548  lines. Since slip occurred in the simulation, the simulatedpelvis trajectory  p 0  (bold line) and the foot placements(broken lines) did not follow the reference. 00.20.40.60.811.2−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5 x [m]    y   [  m   ] Reference Achievedµ = 0.08 Fig. 7. Simulation result of walking on a low friction floor  µ  = 0 . 08 ,walking pattern of   Q 3  = 0 To evaluate the amount of slip, we defined the followingindex. SlipIndex  ≡    |  p 0 ( t ) −  p ref  0  ( t ) | dt  (13)Fig. 8 shows  SlipIndex  of simulated walk using threewalking patterns and friction coefficients between 0.08 and0.2. 0.080.10.120.140.160.180.200.050.10.150.20.250.30.35 friction coefficient µ [−]     S   l   i  p   I  n   d  e  x   [  m   *  s   ] Q 3 =0Q 3 =0.5Q 3 =1.0 Fig. 8. Amount of slip on various friction coefficients Although it was expected that  Q 3  = 1 . 0  gives thesmallest  SlipIndex  by its small  µ nec , it was not clear. Q 3  = 1 . 0  gave even the largest  SlipIndex  at  µ  = 0 . 16  whiletheoretically slip will not occur at this friction coefficient.From this simulation, we could not confirm that a walkingpattern having smaller  µ nec  gives smaller amount of slip.We need further analysis on this subject as well as thecheck of the accuracy of OpenHRP simulator.III. S LIP  I NDUCED  F ALLING  A. Simulated Falling When a robot walks on a floor of very low friction, itmay fall by slipping. Fig. 9 shows an example of the falling 33.05[s]34.40[s] 34.65[s]35.15[s] Fig. 9. Simulated slip induced falling. Walking pattern  Q 3  = 0 ,µ  =0 . 05 . Arrows indicate the slip direction of the support foot. process. We used a walking pattern of   Q 3  = 0  and set thefloor friction coefficient as  0 . 05 .The robot started slip toward the left at the third step(top left) and inclined because of the fast slip speed of the left foot (top right). That body inclination resultedthe unexpected touchdown of the right foot, then a slipto the opposite direction started (bottom left). The slipof the right foot quickly evolved, and finally, the robotlost balance at all (bottom right). Note that the fallingprocess was not limited to this, but many variations wereobserved depending on the walking pattern and the frictioncoefficient.In our simulation, all walking patterns of Table II fellwhen  µ  ≤  0 . 05 . Therefore, the necessary friction coeffi-cients  µ nec  gives too conservative figure(more than twice)to predict the risk of slip induced falling.  B. Slip and ZMP calculation Now, we reconsider the cart table model of Fig. 3.For simplicity, we treat the two dimensional case(sagittalmotion) in this subsection.Suppose a reference cart motion was given as  x ref  ( t ) .According to the ZMP equation (8) the corresponding ZMPwill be  p x  =  x ref  −  z c g  ¨ x ref  . However, this is valid as long as the table does not slip.Indeed, the possible acceleration of the cart is bounded by 3549  the friction between the table and the floor. f  x  =  M  ¨ x ≤  µMg ¨ x ≤  µg As illustrated in Fig.10, if the cart attempts to acceleratewith  ¨ x ref  > µg , the table starts slipping with an accel-eration of   µg  −  ¨ x ref  which compensates the excessiveacceleration. As the result, the cart can only accelerate  µg in the world frame. In this situation, the ZMP is given bythe following equation.  p x  =  x ref  − µz c  (14)   τ  = zmp  0 µ = ɺɺ x g O µ = − ɺɺ ɺɺ ref tbl x g x x  p x  µ 1 M  Fig. 10. Cart-table on a low friction floor: The friction coefficientbetween the table and the floor is  µ . C. Slip concerned ZMP Taking into account of the above consideration, let uscalculate the ZMP of a humanoid robot of Fig.2 for a givenfriction coefficient.The horizontal floor reaction force  f  t  at slipping is f  t  =  µf  z  =  µµ nec    ˙ P  2 x  + ˙ P  2 y ,  (15)where  µ nec  is the necessary friction coefficient,  P   is thelinear momentum calculated from the walking pattern.To treat slip and non-slip condition, we introduce aparameter  γ   ∈ [0 , 1]  as γ   =  µ/µ nec  (if   µ nec  > µ ), 1  (else). (16) 0  ≤  γ <  1  at slipping, and  γ   = 1  at non-slipping. Using γ  , we can calculate a slip concerned ZMP as follows.  p x  =  c x −  γ   ˙ P  x c z  + ˙ L y ˙ P  z  +  Mg,  (17)  p y  =  c y  −  γ   ˙ P  y c z  −  ˙ L x ˙ P  z  +  Mg,  (18)where  [ c x ,c y ,c z ]  is the position of the total CoM and  L is the angular momentum around the CoM. In this paper,we refer the ZMP calculated by these equations as  slipconcerned ZMP 2 . 2 Rigorously speaking, we should say that Eq.(17) and (18) representthe  true  ZMP. Fig. 11 illustrates this slip concerned ZMP (slip-ZMP inshort) at  µ  = 0 . 05  and the srcinal ZMP. While the srcinalZMP trajectory (thin solid line) always runs in the middleof the support area (boundaries are shown by dashedlines), the slip-ZMP (bold line) grazes the boundaries. Thisindicates that the walking with low friction is less stable.   0123456700.20.40.60.811.2   z  m  p  x   [  m   ] zmpslip zmpsup. area01234567−0.15−0.1−0.0500.050.10.15   z  m  p  y   [  m   ] time [s]zmpslip zmpsup. area Fig. 11. ZMP, Slip concerned ZMP  ( µ  = 0 . 05)  and support area.Walking pattern of   Q 3  = 0 As a quantitative measure, we calculated the StabilityIndex  which is defined by the minimumdistance between the slip-ZMP and the edges of thesupport polygon (convex hull).  StabilityIndex  is definedto be positive when the slip-ZMP is inside of the supportpolygon and to be negative when the slip-ZMP is outof the polygon. Fig. 12 shows the  StabilityIndex corresponding to Fig.11. 01234567−0.0500.050.10.15     S   t  a   b   i   l   i   t  y   I  n   d  e  x   [  m   ]  time [s] Fig. 12.  StabilityIndex  of walking pattern of   Q 3  = 0  with  ( µ  =0 . 05) For a given walking pattern and the friction coefficient,the minimum value of   StabilityIndex  indicates the risk of falling. Fig. 13 visualizes  min ( StabilityIndex )  for thefriction coefficients between 0.01 and 0.15. The plots thatgo less than zero at  µ ≤  0 . 05  explain the simulation resultsthat all walking pattern fell at  µ ≤  0 . 05 .IV. W ALK  E XPERIMENT ON A  L OW  F RICTION  F LOOR In this section we describe preliminary walking experi-ments on low friction floors using a humanoid robot HRP-2 [12]. The soles of HRP-2 are covered by a rubber-likematerial, which offer the friction coefficient  µ >  1 . 0  onthe lab floor. We tested two settings of low friction. The 3550
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