A Mathematical Implementation of a Global Human Walking Model with Real-Time Kinematic Personification by Boulic, Thalmann and Thalmann.

Transcription

1 A Mathematical Implementation of a Global Human Walking Model with Real-Time Kinematic Pesonification by Boulic, Thalmann and Thalmann. Mashall Badley National Cente fo Physical Acoustics Univesity of Mississippi June 29 Southeasten Louisiana Univesity Hammond, Louisiana Abstact: The human motion algoithms fom the pape "A Global Human Walking Model with Real-Time Kinematic Pesonification by Boulic, Thalmann and Thalmann have been implemented in the Mathematica-pogamming language. The algoithms have been modified and extended to enable the motion of a 3-D stick figue. The velocity of 19 body pats (points on the stick figue) ae computed as a function of time in the human walk cycle. Model pedictions ae visually consistent with actual human walking and Dopple acoustic sona measuements made at the National Cente fo Physical Acoustics. Discussion. The Boulic-Thalmann (BT) model (efeence 1) is based upon expeimental data. It is an empiical athe than dynamical model. It is based upon the motion of an aveage human being. It was developed to povide computationally efficient tools fo aiding compute animation. Pocedues ae discussed in efeence 1 fo adding pesonification effects to pedictions of human motion, but we do not conside these moe complicated extensions in this bief pape. In the BT model, the body is epesented as a stick figue. The components of motion which ae addessed ae: 1) pelvic oll, pitch and yaw, 2) flexion of the knee, and ankle, 3) yaw (o tosion) of the toso independent of the lowe body ( and legs), 4) flexion of the shoulde and elbow, and 5) thee component tanslation of the body as a whole. The coodinate system used in the BT model is shown in figue 1. The cente of the system is the spine cente. The z-axis is in the vetical diection and the x-axis points in the diection the individual is walking. The lowe body is fee to otate in thee diections (oll, pitch and yaw) about the spine cente. The toso is fee to otate about the spine cente independently of the lowe body. Flexion occus at the, knee and ankle fo the ight and left legs. Flexion occus at the shoulde and elbow fo the ight and left ams. Left and ight side leg and am motions ae symmetical. Additionally, the body tanslates about spine cente. In total thee ae twelve degees of feedom. spine cente z y z x H thigh H spine Boulic, Thalmann and Thalmann (199) Figue 1.Coodinate system fo body motion used in the BT model.

2 Cycle = Two contacts of left heel with floo Duation of Cycle left foot ight foot HS TO HS TO HS TO D ds D ds D b D ds =Duation of dole suppot D b =Duation of balance Figue 2. Definition of key tems in the walk cycle. One cycle of walking is illustated in figue 2. The cycle consists of two steps. It begins with left foot heel stike (HS), continues though ight foot HS, left foot toe off (TO) and ends with the second HS of the left foot. In the BT model, the velocity of walking is nondimensionalized in tems of thigh height. If V denotes the velocity of walking in m/s, then V is V = RV, H thigh whee RV is the elative velocity in units of thigh height pe sec. Boulic and Thalmann conside elative velocities in the ange RV A bisk walk of about 4 mph coesponds to a elative velocity of about depending upon thigh height of the individual. In ode to actually make pedictions fo a stick-figue body like the one shown in figue 1, we must have actual body dimensions. The autho of this pape has a thigh height of one mete. His body dimensions (in inches) ae shown in the accompanying table. These table values ae used in the pedictions to follow. Each measuement in the table is a tiple of the fom (x,y,z) with efeence to the coodinate system in figue 1. Right Left Body (,-7,-7) (,7,-7) spine cente (,,) knee (,-7,25.37) knee (,7,25.37) thoat bottom (,,11.63) ankle (,-7,-46.37) ankle (,7,-46.37) thoat top (,,15.63) toe (8,-7,-46.37) toe (8,7,-46.37) head cente (,,2.67) shoulde (,-8.5,11.63) shoulde (,8.5,11.63) head top (,,25.67) elbow (,-8.5,-.37) elbow (,8.5,-.37) finges (,-8.5,-17.37) finges (,8.5,17.37) Table 1. Body dimensions in inches fo an individual with a one mete thigh height.

3 In ode to compute the position of the citical body pats shown in table 1, we will make use of the fact that the motion of a igid body can be descibed by applying an appopiate set of otations and tanslations. A set of paametes fo doing this is descibed in efeence 1. We will fist conside motion of the lowe body pats. This equies a fou-step pocess. Fist, we conside the effect of flexion on the position of the knee, ankle and toe. If denotes the angle of the at some instant of time, then the knee, ankle and toe in elation to the will be at the vecto locations ]( knee,, ) +, ankle,1 ]( ankle,, ) +, [ ]( ) + toe,1 y toe,,, whee M y [ ] is the otation matix about the y-axis that descibes the flexion: cos sin M ] = 1 sin cos Second, if knee denotes the angle of the knee at some instant, the ankle and toe in elation to the knee will be at the vecto locations ankle,2 toe,2 knee ]( [ ]( y knee ankle,1 toe,1 ) + ) + whee M y [ knee ] is the otation matix about the y-axis that descibes the knee flexion. Thid, if ankle denotes the angle of the ankle at some instant, the toe in elation to the ankle will be at the vecto location ]( + toe, 3 ankle toe,2 ankle,2 ) ankle,2 whee M y [ ankle ] is the otation matix about the y-axis that descibes the ankle flexion. Finally thee dimensional otation of the about the spine cente though oll, pitch and yaw angles and tanslation of the will cause the, knee, ankle and toe to move to the final locations xyz[ φ,, ψ ], + knee xyz[ φ,, ψ ] + ankle xyz[ φ,, ψ ] ankle,2 + [ φ,, ψ ] + toe xyz toe,3 whee cosψ M xyz [ φ,, ψ ] = sinψ sinψ cosψ cos 1 sin 1 sin 1 cos cosφ sinφ sinφ cosφ

4 Motion of the shoulde, elbow and finges ae dealt with in a simila fashion. Flexion of the shoulde will cause the elbow and finges to move with espect to the shoulde in the following way. elbow,1 shoulde ]( elbow, shoulde, ) + shoulde, [ ]( ) + finge,1 y shoulde Flexion of the elbow will poduce the following movement of the finges elative to the elbow: ]( + Rotation of the toso about the spine cente and tanslation of the body as a whole will cause the shouldes, elbow and finges to move to the vecto locations shoulde z[ ψ ] shoulde, + elbow z[ ψ ] elbow,1 + [ ψ ] + whee otation of the toso about the spine cente is descibed by the matix Figue 3 shows pedictions of the 9 angles and 3 displacements used in the peceding equations. The elative velocity is RV = 1. These gaphs wee computed using the empiical equations developed by Boulic and Thalmann (efeence 1). In each plot in figue 3, the hoizontal axis is elative time. A step with the left and ight foot equies a elative time inteval of length one. The BT model makes pedictions in tems of elative time. In ode to compute body pat velocities we must opeate in tems of absolute time. To this end we must conside two dimensional quantities. The most impotant spatial chaacteistic of a walk cycle is the elative cycle length defined by RL c = RV. The fundamental tempoal chaacteistic of a walk cycle is the duation Dc = RLc / RV. The elative time in a walk cycle is τ = t / Dc whee t is absolute time and τ is elative time. If ( τ, RV ) denotes the position of a body pat at elative time τ, then the velocity of this body pat can be shown with the aid of the chain ule to be RV v( τ, RV ) = H thigh ( τ, RV ). RL τ c finge, shoulde, finge, 2 elbow finge,1 elbow,1) elbow,1 finge z finge,2 cosψ sinψ M = z[ ψ ] sinψ cosψ 1 shoulde, This last equation has been used to compute the gaphs of body pat velocities shown in figue 4. The deivative has been computed using numeical diffeentiation. The uppe potion of the figue shows the velocities of the left, knee, ankle and toe (tips). The middle potion of the figue shows the velocities of the left shoulde, elbow and finge (tips). The lowe potion of the figue shows the velocities fo each body pat in table 1. Left-side and ight-side am and leg motion ae one-half cycle out of phase. All computations in figue 4 ae fo a elative velocity RV = 1 and unit thigh height.

5 Figue 5 shows a compaison of measued mico-dopple data and pedictions of body pat velocities appopiate fo the expeimental geomety. The measued data wee ecoded at the National Cente fo Physical Acoustics (NCPA). A continuous-wave Dopple sona opeating at a caie fequency of 4 khz was used to make the measuements. Data wee ecoded using 24 bit technology at a sample ate of 96 khz. The ecoded data shown in the figue ae fo a time peiod that is 1.7 sec long duing which an individual with a height of 67 inches was walking away fom the sona. The colo plot on the left hand side of figue 5 is a shot-time spectogam. The dynamic ange in the plot is 2 db. An FFT of size 8192 with a Hamming window has been used to eveal the individual Dopple components in the data. Fequency has been conveted to velocity of body pat by applying the Dopple fomula f = 2v / λ whee f is the fequency of the appopiate FFT bin elative to 4 khz, λ is acoustic wavelength (nominally.8275 cm) and v is velocity. The bight ed line in the measued spectogam at a velocity of m/s is the caie fequency. The ight hand side of figue 5 shows model pedictions of body pat velocities using the pocedues descibed in this pape. The thigh height of the individual has been estimated to be.93 m. His walking velocity has been estimated fom the measued data. The visual ageement between the measued and modeled esults is quite good. The measuements show velocities of about -4 m/s coesponding to simila model pedictions fo velocities of the ankle-toe complex. These ae velocity components in the measued data that ae consistent with the velocities of the toso and. It is quite stiking that so much of the chaacte in the data can be pedicted with a model with only 12 degees of feedom. Conclusions. The BT model poduces motions that ae visually coect. This can be seen by looking at movie loops of the motions of stick figue models computed using the equations in this pape. Pedictions of body pat velocities ae in qualitative ageement with measued Dopple-sona data. Refeences 1. A Global Human Walking Model with Real-Time Kinematic Pesonification, by Ronan Boulic, Nadia Magnenat-Thalmann and Daniel Thalmann, Visual Compute, Vol. 6, pp ,199.

6 Figue 3. Angles and displacements fo the BT fo eleative velocity RV = 1.

7 Figue 4. Hoizontal velocity of the left, knee, anle and toes (uppe potion), hoizontal velocity of the left shoulde, elbow and finge tips (middle) and all body pats (lowe).

8 Figue 5. Compaison of measued and modeled Dopple data.