SKATEBOARD: A HUMAN CONTROLLED NON-HOLONOMIC SYSTEM

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1 Proceedings of the ASME 205 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 205 August 2-5, 205, Boston, Massachusetts, USA DETC SKATEBOARD: A HUMAN CONTROLLED NON-HOLONOMIC SYSTEM Balazs Varszegi Department of Applied Mechanics Budapest University of Technology and Economics Budapest, Hungary H- varszegi@mm.me.hu Denes Takacs MTA-BME Research Group on Dynamics of Machines and Vehicles Budapest, Hungary, H- takacs@mm.me.hu ABSTRACT A simple mechanical model of the skateoard-skater system is analyzed, in which a linear PD controller with delay is included to mimic the effect of human control. The equations of motion of the non-holonomic system are derived with the help of the Gis-Appell method. The linear staility analysis of rectilinear motion is carried out analytically using the D-sudivision method. It is shown how the control gains have to e varied with respect to the speed of the skateoard in order to stailize the uniform motion. The critical reflex delay of the skater is determined as a function of the speed and the fore-aft location of the skater on the oard. Based on these, an explanation is given for the well-known instaility of the skateoard-skater system at high speed. Gaor Stepan Department of Applied Mechanics Budapest University of Technology and Economics Budapest, Hungary, H- stepan@mm.me.hu ple, [3]) appear even nowadays. The speed dependent staility of non-holonomic systems still have unexplored ehaviors although their dynamics have een investigated for more than 00 years. This is clearly shown y recent pulications, where the lateral staility of the icycle and the three-dimensional iped walking machines are studied (see, for example, [4, 5]). Another interesting challenge is to explain the alancing effort of the person on the skateoard. Reflex delay in the control loop usually plays an important role in dynamical systems, as is shown y papers on the human alancing prolem from iological to engineering points of view [6 8]. In this paper, a mechanical model of the skateoard is constructed in which human control is taken into account. We can say, that the skater wants to control his angle y a torque at his ankle, what is produced y muscles. This is an usual approach in modeling of human control, when we model the human as a root, what is ale to act with forces and torques [6, 8]. INTRODUCTION Skateoard was invented only in 950 s and already enjoys worldwide popularity. The challenge of understanding the motion of the skateoard is due to the kinematic constraints and the complexity of the skater-skateoard system. If the skateoard is tilted around its longitudinal axis then the wheel-pairs also turn around their steering axes ecause of the wheel mounting geometry. This phenomenon makes the mechanical model non-holonomic and the equation of motion can e derived y means of the Gis-Appell method []. The effect of speed on the staility of the skateoard was investigated a not long after the invention of skateoards [2] ut pulications (see, for exam- So we use a very simple linear PD control loop to apply torque at the skaters ankle, and we also consider the reflex delay of human control. The staility analysis of the rectilinear motion is investigated analytically, and a case study is presented to emphasize the physical meaning of the results. Let us note here, other control models can etter represent the higher-frequency ehavior (for example McRuer approach [9]). This phenomenon can e investigated in further researches after this paper. Copyright 205 y ASME

2 m Z ϕ C h C m ϕ dinal direction of the skateoard; ϕ is the deflection angle of the skater from the vertical direction; and finally, β is the deflection angle of the oard (see Fig. ). As it is also shown in Fig., the rotation of the skateoard aout its longitudinal axis is resisted y the torsional spring of stiffness s t, which is originated in the special wheel suspension system of the skateoard. The rotation of the skater around the longitudinal axis of the skateoard is opposed y the internal torque of the PD controller. Herey, the rectilinear motion is considered as the desired motion, which means that zero torque is produced y the controller when the deflection angle of the skater is zero (i.e.: ϕ = 0). Thus, the control torque can e calculated as s t PD S R Z Y ψ X d S v R R S F V a l l FIGURE. THE MECHANICAL MODEL OF THE SKATEBOARD-SKATER SYSTEM THE MECHANICAL MODEL The mechanical model in question (see Fig. ) is ased on [3] and [0], where simpler mechanical models of the skateoard were used ut a human control-loop was also implemented. Here, a more realistic model is constructed, in which the oard and the skater can move separately and a control torque acts at the skater s ankle. The model consists of two massless rods. One of them models the skateoard etween the front (F) and rear (R) points. Although we neglect the mass of the oard its mass moment of inertia around its longitudinal axis is taken into account. The other rod etween the center point (S) of the contact patch etween the skater and the oard and the mass center (C) models the skater having a lumped mass. These two rods are connected y a hinge parallel to the longitudinal axis at the point S. The geometrical parameters are the following: 2h denotes the height of the skater, 2l is the length of the oard, a characterizes the location of the skater on the oard (a > 0 means that the skater stands in front of the center of the oard), the mass point m represents the mass of the skater, J is the mass moment of inertia of the skateoard around its longitudinal axis and g stands for of the gravity acceleration (g =9.8 [m/s 2 ]). In this study, we do not consider the loss of contact etween the wheels and the ground, consequently that the longitudinal axis of the skateoard is always parallel to the ground. One can choose five generalized coordinates to descrie the motion: X and Y are the coordinates of the point S; ψ descries the longitu- v F d S M PD (t) = Pϕ(t τ) + D ϕ(t τ), () where τ refers to the time delay, P and D represent the proportional and the differential control gains, respectively. Regarding the rolling wheels of the skateoard, kinematic constraining equations can e given for the velocities v F, v R of points F and R. The directions of the velocities of these points depend on the deflection angle β of the oard through the socalled steering angle δ S (see Fig. ). This connection can e descried y the expression: sinβ tanκ = tanδ S, (2) where κ is the fixed complementary angle of the so-called rake angle in the skateoard wheel suspension system [3, 0]. Based on this, two scalar kinematic constraining equations can e constructed: ( sinψ + cosψ sinβ tanκ)ẋ+ (cosψ + sinψ sinβ tanκ)ẏ + (l a) ψ = 0, (sinψ + cosψ sinβ tanκ)ẋ+ ( cosψ + sinψ sinβ tanκ)ẏ + (l + a) ψ = 0. We also introduce a third kinematic constraint, namely, the longitudinal speed V of the oard is kept constant: (3) (4) Ẋ cosψ +Ẏ cosψ = V. (5) We do not present details, ut it can e proven that the condition of the linear staility of rectilinear motion of skateoard remains the same even if the constraint Eqn. (5) does not hold, nevertheless, it makes our calculations simpler. Since all of the kinematic constraints (Eqns. (3) (5)) are linear cominations of the generalized velocities they can e written in the following form: A(q) q = A 0, (6) 2 Copyright 205 y ASME

3 where A = q T = [ X Y ψ φ β ], (7) [ ] cosψ sinβ tanκ sinψ sinψ sinβ tanκ + cosψ l a 0 0 cosψ sinβ tanκ + sinψ sinψ sinβ tanκ cosψ l + a 0 0, (8) cosψ sinψ where G refers to the center of the gravity of the ody and a G is the acceleration of G. The angular acceleration and the angular velocity are denoted y α and ω, respectively. The mass of the ody and its matrix of mass moment of inertia are m and J G, respectively. In case of skateoarding, there are two rigid odies: the skater with a lumped mass m at C and the oard with mass moment of inertia J around the hinge axis, while the mass of the oard is neglected compared to the mass of the skater. So the energy of acceleration forms: A T 0 = [ 0 0 V ]. (9) Note, that if the mass moment of inertia J of the oard is zero, then β and ϕ are not independent and β can e expressed as a function of ϕ and ϕ. Thus, the kinematic constraints are not linear functions of the generalized velocities if J = 0, and the Gis-Appell method can not e used. Therefore, we consider the case when J > 0. In order to apply the Gis-Appell equation (see in []), pseudo velocities have to e chosen, y which the kinematic constraints can e eliminated. In our case, two pseudo velocities are required since the difference of the numers of the generalized coordinates and the kinematic constraints is two. An appropriate choice can e the angular velocity components of the skater and the skateoard around the longitudinal axis, respectively: σ (t) := ϕ(t) and σ 2 (t) := β(t). (0) The generalized velocities can e expressed with the help of these two pseudo velocities and with the generalized coordinates: Ẋ V ( cosψ + a l tanκ sinβ sinψ) Ẏ ψ ϕ = V ( sinψ a l tanκ sinβ cosψ) V l tanκ sinβ σ. () β σ 2 The derivation of oth sides of this expression with respect to time leads to the generalized accelerations as functions of the pseudo accelerations, pseudo velocities and generalized coordinates. During the derivation of the equation of motion, the socalled energy of acceleration (A ) is needed. For a rigid ody it can e computed as: A = 2 ma G a G + 2 αt J G α+α T (ω (J G ω))+ 2 ωt (J G ω)ω 2, (2) A = mhv l 2 tanκ cosϕv sinβ(l htanκ sinβ sinϕ) σ + mhav l tanκ cosϕ cosβσ 2 σ + mh2 2 σ J cos 2 (2ψ) σ (3) The expressions that do not contain pseudo accelerations is not computed, ecause the equation of motion can e otained in the form: A σ i = Γ i, (4) where the right hand side is the pseudo force Γ i related to the ith pseudo velocity σ i. It can e determined from the virtual power of the active forces namely: the gravitational force, the torque produced y the controller and the torque produced y the spring: δp = mghsinϕδσ s t βδσ 2 + M PD (t)(δσ 2 δσ ), (5) where notation δ refers to the virtual quantities. Thus the equations of motion are: ( σ = sinϕ g h + V 2 tan 2 κ sin 2 β cosϕ l 2 a h V l tanκ cosβ cosϕσ 2 V 2 hl σ 2 = M PD(t) s t sinhβ, cos 2 (2ψ)J Ẋ = V ( cosψ + a l tanκ sinβ sinψ), Ẏ = V ( sinψ a l tanκ sinβ cosψ), ψ = V l tanκ sinβ, ϕ = σ, β = σ 2. ) h M PD(t)+ tanκ sinβ cosϕ, (6) Note that X, Y and ψ are cyclic coordinates, so only the first two and the last two equations of Eqn. (6) have to e used for our further investigation. LINEAR STABILITY ANALYSIS In this section we are going to investigate the linear staility of the rectilinear motion of a skateoard-skater system. First, we take the linearized equation of motion around the stationary 3 Copyright 205 y ASME

4 solution with respect to small perturations in σ, σ 2, ϕ and β. It can e written as where Ẋ(t) = J X(t) + T X(t τ), (7) 0 av hl tanκ g h V 2 hl tanκ σ (t) J = s t J 0 0 0, X(t) = σ 2 (t) ϕ(t), ψ(t) D 0 P 0 mh 2 mh 2 and T = J D 0 J P (8) The characteristic function D c (λ) can e determined y means of the sustitution of the trial solution X(t) = Ke λt (K C 4 and λ C) into Eqn. (7). It leads to: D c (λ) = V e λτ HJ l ( + e λτ mh 2 tanκe λτ (aλ +V )(Dλ + P)+ ) λ 2 + s t (Dλ J + mhe λτ ( hλ 2 g ) + P ) (9) and the characteristic equation can e written in the form D c (λ) = 0. This equation has infinitely many complex roots. The fixed point in question, i.e. the rectilinear motion, is asymptotically stale if and only if all of the characteristic roots are situated in the left-half of the complex plane. The limit of staility corresponds to the case when the characteristic roots are located at the imaginary axis for some particular system parameters. Two different types of staility oundaries can e distinguished; saddle-node (SN) ifurcation when oth the real and imaginary parts of the characteristic root are zero, and Hopf ifurcation when the characteristic roots are pure complex. In our model, SN ifurcation occurs if D c (0) = 0, namely: gs ( t J H + st mj H 2 + V 2 ) tanκ P = 0. (20) J hl The critical proportional gain P SN can e determined: P SN = mghls t mhv 2 tanκ + s t l. (2) In case of Hopf ifurcation, the critical characteristic exponent can e written as pure imaginary numers (λ H =±iω, TABLE. Hopf PARAMETERS OF THE SKATER-BOARD SYSTEM h [m] m [kg] a [m] τ [s] s t [Nm/rand] l [m] κ [ o ] J [kgm 2 ] D SN FIGURE 2. P SN ω D stale P ω D SN ω stale P STRUCTURE OF STABILITY CHART ω R + ). The characteristic equation D c (iω) can e separated into real and imaginary parts, which procedure is referred to D- sudivision method. The critical control gains can e expressed: where P H = p (ω)p 2 (ω) p 4 (ω), D H = p (ω)p 3 (ω) ω p 4 (ω), (22) p (ω) = 2hlmcosκ ( g + hω 2)( J ω 2 ) s t, p 2 (ω) = l cosκ ( J ω 2 ) s t cos(τω) mhv sinκ(aω sin(τω) +V cos(τω)), p 3 (ω) = l cosκ ( J ω 2 ) ( s t sin(τω) + mhv sinκ(aω cos(τω) V sin(τω)), p 4 (ω) = cos(2κ) l 2 ( s t J ω 2) 2 h 2 m 2 V 2 ( a 2 ω 2 +V 2)) + l 2 ( s t J ω 2) m 2 h 2 V 2 ( a 2 ω 2 +V 2) + 2mhlV 2 sin(2κ) ( s t J ω 2). (23) Let us construct the linear staility chart of the rectilinear motion in the P D parameter plane (see in Fig. 2) for the realistic parameters of Ta.. The staility oundary given y Eqn. (22) starts like in case of the PD controlled inverted pendulum [6], ut it does not spiral outwards continuously. Actually it will cross the origin of the parameter plane at ω = ω, where st ω = (24) J is the natural angular frequency of the skateoard without control (i.e. P = 0 and D = 0 corresponds to the switched off control). As a consequence, a loop of the staility oundary is showing up like for the simplified model in [0]). It can e proved, that only the internal region of this loop can e stale. As a summary, there are three different oundaries of the stale parameter domain: 4 Copyright 205 y ASME

5 τ a [s] a = 0. [m] stailizale for only skater s front position a = - 0. [m] stailizale for any skater s position V [m/s] FIGURE 3. THE EFFECT OF THE SPEED ON THE ULTIMATE REFLEX TIME the SN for static loss of staility and a low-frequency and a highfrequency Hopf ifurcation. One can verify, that the aove descried closed loop of Hopf oundaries rotates counterclockwise around the origin as the reflex delay τ increases. EFFECTS OF THE LONGITUDINAL SPEED AND THE REFLEX DELAY When the effects of a human controller are neglected it can e shown that the higher the speed, the easier it is to stailize the skateoard [2, 3]. However, a more complex picture arises when the human control is taken into consideration. This was recognized in [0] using a simpler mechanical model that also exhiits rotating closed loop staility oundaries in the P D parameter plane. Here we determine the necessary and sufficient conditions to stailize the skateoard with respect to the skater s reaction time. A condition can e developed y examining the starting point of the D-curve. If the curve starts to the left hand side at ω = 0 then stale domain can not exist. This leads to the ultimate critical time delay τ cr,u of the controller. If the delay is larger than this value, then the investigated equilirium is for sure. Fig. 3 shows the variation of the ultimate time delay versus the longitudinal speed for two different cases. The continuous line corresponds to a = 0.m, i.e. the skater stands ahead of the center of the skateoard, while the dashed line corresponds to a = 0.m, i.e. the skater stands ehind the center of the oard. As it can e oserved in the figure, the maximal time delay, y which the system can e stailized, is larger for any speed when the skater stands in front of the center point of the oard. However, in case of smaller reflex delay, the stailization of the rectilinear motion is possile even when the skater stands ehind the center of the oard. This can also e confirmed y practical experiences. Previous models without control-loop do not explain such ehaviors of the skateoard. For example, it is proved in [2, 3] that the stailizale stailizale stailizale stailizale FIGURE 4. BILITY a = [m] a = 0.05 [m] V=2 [m/s] V=3 [m/s] V=5 [m/s] V=7 [m/s] THE EFFECT OF THE REFLEX TIME ON THE STA- rectilinear motion cannot e stale for any speed in case a < 0. Figure 3 also represents that the skater s position is more relevant at low speeds ut oth curves of the ultimate critical delays tend to the same value as the speed tends to infinity. Independently from the skater s position, the critical time delay is also the same at zero speed. The curves of Fig. 3 are characterized y where τ V 0 ut,max = 2 ω 2 s and ω s = τ V ut,max = 2 ω 2 s τ [s] ω 2, (25) g h. (26) The ultimate reflex time at zero speed (τut,max) V 0 is identical with the critical time delay of the human alancing prolem [6]. Based on the formulas, it is easy to show that τut,max V 0 > τut,max V ut the difference is very small for realistic parameters. For example, the parameters of Ta. leads to 0.04% relative difference due to the fact that the natural angular frequency ω is very high. The ultimate critical time (see in Fig. 3) is close, ut higher, to the human reflex delay for hand [7]. As it was mentioned, former studies [3] proved that the system is for a < 0. Here we show that this statement can e reconsidered in some special cases, moreover, standing ehind the center of the oard can e even advantageous. In order to do this, the effect of the reflex time has to e investigated in more details. In Fig. 4, the existence of the stale P D parameter domain is illustrated versus the reflex delay. The figures are constructed for different longitudinal speeds. The continuous and the dashed lines elong to a = 0.05m and a = 0.05m, respectively. It can e oserved that the condition τ < τ ut,max is not sufficient from point of view of the staility. There are several reflex time ranges, where the system is not stailizale for any control gains, which is due to the rotation of the Hopf loop around the origin 5 Copyright 205 y ASME

6 D [Nms] stale 0 stale V=.2 [m/s] V=.5 [m/s] V=2.25 [m/s] control gains, which can enlarge the effects of the dead-zones of the human control. Clearly, the skater cannot apply close to zero control torque and cannot even sense very small tilting angles of the oard. In human alancing models, the existences of dead-zones are also suspected as the reason of micro-chaotic and transient-chaotic virations around linearly equilirium with large surviving times [8]. V= 3.5 [m/s] V=7 [m/s] P [Nm] FIGURE 5. STABLE PARAMETER DOMAIN FOR DIFFERENT LONGITUDINAL SPEEDS ACKNOWLEDGMENT This research was partly supported y the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences and y the Hungarian National Science Foundation under grant no. OTKA PD with increasing τ. However, the locations of these ranges depend on the standing positions, namely, in some cases a > 0, in other cases a < 0 can lead to stale rectilinear motion. Nevertheless, if the speed tends to infinity, the applicale time delay ranges do not depend on the skater s position anymore. Having a time delay elow the sharp critical value is not enough to ensure stale rectilinear motion, the choice of adequate control gains (P and D) is also important. The variation of the stale domains in the P D parameter plane can e seen in Fig. 5 for τ = 0.24s and for different speeds. The stale domain tends to the origin (P = 0, D = 0) as the speed increases. As a consequence, the skater has to tune the control parameters more and more precisely in order to catch the very narrower and narrower stale parameter domain at increasing high speeds. Nevertheless, the switching off of the control (i.e., P = 0 and D = 0) could e a solution ut only at infinite (very high) speed. CONCLUSION A mechanical model of the skateoard-skater system was constructed, in which the effect of the human alancing was taken into account y a linear delayed PD controller. The reflex time of the skater was also considered. The staility of the rectilinear motion of the skateoard was analyzed and staility charts were composed with special attention to the effects of the reflex time and the longitudinal speed. Time delays were determined for realistic parameters, y which the skateoarding can e performed. The effect of the skater s position on the oard was also investigated. It was verified, that the variation of the skater s position can qualitatively influence the staility of the rectilinear motion. The presented staility charts can also explain the loss of staility at high speed. The stale parameter domain of the control gains reduces and its location tends towards the origin as the speed increases. As a consequence, the skater must decrease the REFERENCES [] Gantmacher, F., 975. Lectures in Analytical Mechanics. MIR Pulisher, Moscow, Russia. [2] Huard, M., 980. Human control of the skateoard. Journal of Biomechanics, 3, pp [3] Kremnev, A. V., and Kuleshov, A. S., Dynamics and simulation of the simplest model of skateoard. In Proceedings of Sixth EUROMECH Nonlinear Dynamics Conference. [4] Kooijman, J. D. G., Meijaard, J. P., Papadopoulos, J. M., Ruina, A., and Schwa, A. L., 20. A icycle can e self-stale without gyroscopic or caster effects. Science, 332, pp [5] Wisse, M., and A, S., Skateoard, icycles, and three-dimensional iped walking machines: Velocitydependent staility y means of lean-to-yaw coupling. The International Journal of Rootics Research, 24(6), pp [6] Stepan, G., Delay effects in the human sensory system during alancing. Philosophical Transactions of The Royal Society, 367, pp [7] Milton, J., Solodkin, A., Hlustik, P., and Small, S. L., The mind of expert motor performance is cool and focused. NeuroImage, 35, pp [8] Insperger, T., and Milton, J., 204. Sensory uncertainty and stick alancing at the fingertip. Biological Cyernetics, 08, pp [9] McRuer, D. T., Weir, D. H., Jex, H. R., Magdeleno, R. E., and Allen, E. W., 975. Measurement of driver-vehicle multiloop response properties with a single disturance input. IEEE Transactions on systems, Man, and Cyernetics, SMC-5, pp [0] Varszegi, B., Takacs, D., Stepan, G., and Hogan, S. J., 204. Balancing of the skateoard with reflex delay. In Proceedings of Eighth EUROMECH Nonlinear Dynamics Conference, ENOC Copyright 205 y ASME