osition Contro of Roing kateboard Baazs Varszegi enes Takacs Gabor tepan epartment of Appied Mechanics, Budapest University of Technoogy and Economics, Budapest, Hungary (e-mai: varszegi@mm.bme.hu) MTA-BME Research Group on ynamics of Machines and Vehices, Budapest, Hungary (e-mai: takacs@mm.bme.hu) epartment of Appied Mechanics, Budapest University of Technoogy and Economics, Budapest, Hungary (e-mai: stepan@mm.bme.hu). Abstract: A simpe mechanica mode of the skateboard-skater system is anayzed, in which the skater tries to foow a straight ine by the board. The human contro is considered by means of a inear deayed controer. The equations of motion of the non-hoonomic system are derived with the hep of the Gibbs-Appe method. The inear stabiity of the system is given anayticay. The effect of the refex deay of the skater on the stabiity is investigated. It is shown that in case of zero refex deay, the motion aong a straight ine is unstabe whie non-zero refex deay can provide stabe motion for some specific speed ranges. Keywords: skateboard, non-hoonomic, controer, time deay 1. INTROUCTION kateboard has become a popuar sport in the 6 s. After a decade, the first mechanica mode of the skateboardskater system was constructed by Hubbard (198). Hubbard s study showed that the mechanica mode of the skateboard is a very interesting exampe of non-hoonomic systems. Other researchers aso investigated the inear stabiity of the skateboard-skater system, Kremnev and Kueshov (28). A of these studies have provided the same resuts, the stabiization of the rectiinear motion is easier as the ongitudina speed of the board is increased. uch behavior can commony be detected in different nonhoonomic systems, see, for exampe, the bicyce or the three-dimensiona biped waking machine Wisse and chwab (25). Many practica observations on skateboarding proved, that unwanted oss of stabiity can aso occur at high speed. An expanation for this phenomenon can be originated in the human contro, which is aways present in the skateboard-skater system. The human contro has refex deay, what can cause unexpected stabiity osses in case of stick baancing Insperger and Miton (214) or simpe human baancing tepan (29). With the extension of Kremnev and Kueshovs mode, it was investigated in Varszegi et a. (214) how the baancing effort of the skater infuences the stabiity. A inear controer was impemented, where the skater s titing ange was used as the input of the controer. It was verified that the stabe parameter domain of the contro gains is strongy modified as the ongitudina speed of the board changes, see Figure 1, where the stabe parameter domain is rotating around the origin of the pane as the speed increases. Consequenty, the skater has to tune the contro gains with respect to the speed. [Nms] 4 4 N 2 N 2-2 -1 1 2[Nm] -2-1 1 2[Nm] -2-2 -4 V= m/s [Nms] [Nms] 4 4 N 2 2-2 -1 1 2[Nm] -2-1 1 2[Nm] -2-2 -4 V=2.5 m/s -4 V=1.7 m/s [Nms] -4 V=4 m/s Fig. 1. Linear stabiity charts for different speeds 1 In the figure, it can aso be observed that the parameter pairs = and = are aways at the stabiity boundary. This means that if the skater switches off its contro, the rectiinear motion is just stabe. Namey, by means of this simpe strategy, the skater coud avoid the instabiity of the board. But this strategy can work if the direction of the rectiinear motion does not matter, namey, the skater does not have to avoid an object on the road and/or does not have to foow the desired path of the road. To investigate the case when the skater has to foow a predefined direction by its board, another contro aw is needed. In this study, we consider this atter case. 2. NON-HOLONOMIC MECHANICAL MOEL The mechanica mode in question (see Figure 2) is based on Kremnev and Kueshov (28) and Varszegi et a. 1 ee in Varszegi et a. (214)
Z X d v R Y g R m y C h Fig. 2. The simpified mechanica mode of the skateboardskater system (214). The skateboard is modeed by a massess rod (between the front axe at F and the rear axe at R) whie the skater is represented by a massess rod (between the points and C) with a umped mass at C. In this mode, the connection between the skater and the board (at ) is assumed to be rigid. The so formed rigid body has zero mass moment of inertia with respect to its center of gravity at C, which makes the derivation of the equations of motion simper. Namey, the skateboard moves in the three dimensiona gravitationa fied but we have to describe the motion of a umped mass ony. In our study we do not consider the oss of contacts between the whees and the ground. ue to the fact that the ongitudina axis of the skateboard is aways parae to the ground, one can choose four generaized coordinates to describe the motion: X and Y are the coordinates of the skateboard center point in the pane of the ground; ψ describes the direction of the ongitudina axis of the skateboard; and finay, ϕ is the incination ange of the skater s body from the vertica direction. The geometrica parameters are the foowing. The height of the skater is denoted by 2h. The ength of the board is 2 whie m represents the mass of the skater. The parameter g stands for the gravitationa acceeration. Here we mode the skater s navigating effort as a inear controer, which appies a torque to the skateboard (see Figure 2). The rectiinear motion of the board is prescribed aong the Y = ine, and the contro torque is cacuated via M c V F v F M c (t) = Y (t τ) Ẏ (t τ) (1) where τ refers to the time deay, and represent the proportiona and the differentia contro gains, respectivey. The controer produces zero torque if the board foows the prescribed stationary path. Regarding to the roing whees of the skateboard, kinematic constraints can be formed. These constraining equations define the veocities v F and v R of the front point (F) and the rear one (R), respectivey. The directions of these veocities depend on ϕ through δ, which is the so-caed steering ange (see Figure 2). This ange can be expressed from the equation sin β (t) tan κ = tan δ (t), (2) where κ is the compementary ange of the so-caed rake ange in the skateboard whee suspension (for the derivation of this reation pease see Kremnev and Kueshov d (28) or Varszegi et a. (214)). Here we aso prescribe the ongitudina speed of the board, which is kept on the constant vaue V. The so-formed kinematic constraints can be written as A q = A, (3) where [ ] q T = X Y ψ ϕ, (4) [ ] sin ψ cos ψ sin ϕ tan κ cos ψ sin ψ sin ϕ tan κ A = sin ψ + cos ψ sin ϕ tan κ cos ψ + sin ψ sin ϕ tan κ, cos ψ sin ψ [ ] (5) A T = V, (6) The equations of motion of non-hoonomic systems can be determined by means of severa methods. For exampe, the extended version of the Lagrange equation of the second kind (aso caed as Routh-Voss equations in Gantmacher (1975)) is appicabe but the eimination of the invoved Lagrange-mutipiers can ead to extensive agebraic manipuation. Here we rather appy the Gibbs- Appe method (see in Gantmacher (1975)), what is a more efficient approach since it provides the equations of motion in the form of first order differentia equations. However, the definition of the pseudo veocities have to be chosen intuitivey and the so-caed energy of acceeration is aso an unusua physica quantity. We have three kinematic constraints and four generaized coordinates, which means that ony one pseudo veocity has to be chosen: σ := ϕ, (7) which is the anguar speed of the skater around the ongitudina axis of the skateboard. The generaized veocities can be expressed with the hep of this pseudo veocity and the generaized coordinates: ẊẎ ψ ϕ = V cos ψ V sin ψ V tan κ sin ϕ. (8) σ uring the derivation of the equation of motion, the socaed energy of acceeration A is needed. ince the mode consists of one mass point ony it can easiy be computed: A = 1 2 ma C a C, (9) where a C refers to the acceeration of the umped mass. In our mode we obtain: A = 1 2 mh V 2 ( ) 2 tan(κ) sin(2ϕ) h tan(κ) sin 2 ϕ σ+ + 1 (1) 2 mh2 σ 2 +.... According to the Gibbs-Appe method, the parts of the energy acceeration, which do not depend on the pseudo acceeration σ, are not necessary to cacuate and they are not computed here. The Gibbs-Appe equation forms as A σ = Γ, (11) where the right hand side is the pseudo force Γ. It can be determined from the virtua power of the active forces. In our mode, the gravitationa force and the contro torque have non-zero virtua power, so they contribute to the pseudo force.
The equation of the motion of the system can be written as σ (t) = g V sin(ϕ(t)) + h mh 2 sin(ψ(t τ)) + Y (t τ) mh V 2 ( tan(κ) sin(ϕ(t)) cos(ϕ(t)) 1 + h ) h tan(κ)sin2 (ϕ(t)), Ẋ (t) = V cos(ψ(t)), Ẏ (t) = V sin(ψ(t)), ψ (t) = V tan(κ) sin(ϕ(t)), ϕ (t) = σ(t). (12) The first equation of (12) reates to the Gibbs-Appe equation, the others are the formuas of the generaized veocities as the function of the pseudo veocity and the generaized coordinates. It is worth to mention that X is a so-caed cycic coordinate, so the second equation is unnecessary for further investigation. It means that the system can be described in the four dimensiona state space. 3. TABILITY ANALYI OF THE RECTILINEAR MOTION IN CAE OF ZERO TIME ELAY In this section, the inear stabiity anaysis of the rectiinear motion is investigated in that specia case, when the time deay of the contro-oop is zero (τ = ). This theoretica case can be used for better understanding of the interaction of the skateboard and the skater. The inearized equation of motion around the stationary soution: σ, Y, ψ and ϕ, can be written in the foowing form: V g σ (t) σ (t) Ẏ (t) ψ (t) ϕ (t) = mh mh 2 h V 2 h tan(κ) V V tan(κ) 1 Y (t) ψ (t) ϕ (t) (13) The stabiity anaysis of this inear ordinary differentia equation system (13) can be carried out with the hep of the Routh-Hurwitz criterion. The so-caed Hurwitz matrix can be constructed from the coefficients of the characteristic equation. The ith sub-determinant of this Hurwitz matrix is denoted by i and the investigated equiibrium is asymptoticay stabe if and ony if a of the sub-determinants i are greater than zero. Here =, which means that the rectiinear motion can be stabe ony in Lyapunov sense but not exponentiay stabe. To achieve this stabiity, the foowing three conditions have to aso be fufied: V 2 mh 2 tan(κ) >, 2 V 4 and h 4 2 m 2 tan2 (κ) >, V 2 (14) mh 2 tan(κ) >. The second condition cannot be satisfied for any rea parameters. This means that the rectiinear motion is unstabe, or it can be stabe in Lyapunov sense ony if and are zeros. This resut was examined by means of numerica simuations ony. The anaytica proof can be the subject of future work. 4. TABILITY ANALYI OF THE RECTILINEAR MOTION FOR NON-ZERO TIME ELAY Let us investigate the more reaistic case, when the skater s refex deay is considered. From mathematica view point, this case is more compicated, because the system is governed by deay differentia equations (Es). The inear stabiity anaysis can be carried out based on the resuts of tepan (1989). The sma vibrations of the board around the rectiinear motion can be described by the inearized equation: Ẋ(t) = J X(t) + T X (t τ), (15) where g h V 2 tan κ σ(t) J = V V tan κ, X(t) = Y (t) ψ(t), ϕ(t) 1 V mh mh 2 and T =. (16) The characteristic function of this inear system can be cacuated after the substitution of the exponentia tria soution with characteristic exponent λ into (15): ( V c (λ) = λ 4 + λ 2 2 tan κ g ) + h + e λτ (λ + ) V 2 (17) mh 2 tan κ. The rectiinear motion is asymptoticay stabe if a of the infinitey many characteristic roots are situated on the eft haf of the compex pane. The imit of stabiity corresponds to the case when characteristic roots are ocated on the imaginary axis. If both the rea and the imaginary parts of the characteristic exponent are zeros then sadde-node (N) bifurcation can occur. In our case, c (λ = ) = eads to V 2 tan κ = (18) mh which gives a vertica ine in the parameter pane: N =. (19) If ony the rea parts of the characteristic exponent are zero, bifurcation (H) can occur, and the characteristic exponent ocated on the imaginary axis can be expressed as λ = ±iω, where i is the imaginary unit and ω R +. The -subdivision method can be used to determined the corresponding stabiity boundaries, namey, the characteristic equation has to be separated into rea and imaginary parts: ( g = ω 4 + ω 2 h V 2 ) tan(κ) + ( + mh ω sin(ωτ) + ) V 2 mh cos(ωτ) tan(κ), (2) ( = mh ω cos(ωτ) ) V 2 mh sin(ωτ) tan(κ).
N ω tabe ω= ω ω * Fig. 3. tructure of stabiity chart for an arbitrary chosen ongitudina speed V and for the reaistic parameters given in Tabe 1. tabe N ω V,up V,eft V,down V,right Fig. 4. Rotation of encosed possibe stabe domain at certain speeds for the reaistic parameters of Tabe 1. The critica and parameters can be expressed as: H = mh (ω 2 V 2 + gh tan (κ) V 2 ) tan(κ) ω 2 cos(ωτ), H = mh (ω 2 V 2 + gh tan (κ) V 2 ) tan(κ) ω sin(ωτ). (21) Using the stabiity boundaries (19) and (21), a stabiity chart is constructed in the parameter pane in Figure 3. The vertica ine of the N bifurcation and the curve of the bifurcation terminate the ineary stabe (shaded) and unstabe (white) domains. The stabiity boundary given by (21) starts from the origin, i.e. = and = for ω =. It can be proved that this curve can go through the origin for ω > if g V V = tan κ. (22) For reaistic system parameters (see Tabe 2) V = 1.43 m/s. Let us not here, the rectiinear motion is unstabe if V < V. The corresponding anguar frequency can be determined as we: V ω 2 = tan κ g h. (23) If the speed is higher than the critica vaue V, the stabiity boundary curve intersects itsef and a cosed oop appears. It can be proved that ony the inner part of this oop can be stabe. To describe the behavior of this cosed oop is important, because this oop can cat of the stabiity region (see Figure 3.), for some parameters the stabe region is vanished. One can verify that the above described cosed oop of the stabiity boundary rotates countercockwise around the origin of the parameter pane as the ongitudina Tabe 1. arameters of the skater-board system h [m] m [kg] a [m] τ [s].85 75.5.24 s t [Nm/rad] [m] κ [ o ] g [m/s 2 ] 1.3937 63 9.81 speed V increases. ince the anguar frequency ω is aso known, four specific speeds can be determined anayticay, which reate to the specia positions of the cosed oop, i.e. when the oop is tangentia with the axes of the pane. These positions are iustrated in Figure 4 and the corresponding critica speeds are: V,up = (2k + 1) 2 π 2 τ 2 + g h tan κ, (24) ( V,eft = 2k + 3 ) 2 π 2 2 τ 2 + g h tan κ, (25) V,down = (2k + 2) 2 π 2 V,right = where k N. ( 2k + 1 ) 2 π 2 2 τ 2 + g h τ 2 + g h tan κ, (26) tan κ, (27) According to the N stabiity boundary at =, ony the right hand side of parameter pane can be stabe. If the stabiity boundary given in (21) starts to the eft side of the pane at ω = then no stabe domain exists. It can aso be proved that if this boundary starts to the right then stabe domain can be ocated in the first quadrant. tabe region exists if and ony if the oop of the stabiity boundary has an intersection with its initia segment (often referred to -curve, see for exampe, in tepan (29)). If this condition is satisfied then the boundary curve starts to the right side of the parameter pane, too. Accordingy, speed ranges can be determined, where stabe domain exist: r 2 k τ 2 + g h < V s,k tan κ < ( 2k + 3 ) 2 π 2 2 τ 2 + g h. (28) Here, r k is the kth root of equation tan r = r for k N. The time deay aso has to satisfy the condition: r k V 2 tan κ g h < τ s,k < 2πk + 3 2 π V 2 tan κ g h. (29) The ower imit beongs to the geometric case on parameter pane, when the tangent of the boundary curve at ω and at ω are the same and the cosed oop touches from beow. The higher one beongs to the case when the cosed oop just eaves the possiby stabe domain (which is bounded by the -shaped initia segment and the N ine), namey, the boundary curve touches the vertica axes from right at ω. Thus, there are speed (or refex deay) ranges, where the rectiinear motion is stabe. Using the reaistic parameters of Tabe 1 and assuming that the refex deay of the skater is around.3 sec, the stabe speed ranges can be cacuated, see Tabe 2.
Tabe 2. tabiizabe speed domains 1.4281 m/s < V s, < 6.63623 m/s 1.7254 m/s < V s,1 < 15.1995 m/s 19.4119 m/s < V s,2 < 23.8243 m/s [Nms/m] 3 2 τ [s] 1..8.6 U U U 1 1.4 U.2 tabiizabe U 5 1 15 2 V [m/s]. V * 2 4 6 8 1 12 14 V [m/s] Fig. 6. tabiity chart in the V - - parameter space Fig. 5. tabiizabe domains in the space of the ongitudina speed V and refex deay τ As Tabe 2 and Figure 5 show, there are speed domains with ow and higher vaues, where the contro of the skateboard is not feasibe. Naturay it does not mean, that the skater wi fa, different contro aw can aow the skater to baance itsef and keep the vertica position (see Varszegi et a. (214), Kremnev and Kueshov (28) or Hubbard (198)). However, the skater cannot foow the straight ine by the skateboard. In Figure 5, the refex deay τ can be seen against the ongitudina speed V. haded domains are stabiizabe, which means that the rectiinear motion can be stabe for appropriatey chosen contro gains and. This stabiity chart and the condition (28) do not contradict to the fact that the investigated equiibrium cannot be stabe for V < V. Namey, the eft inequaity of (28) for k = eads to the same condition. If τ =, the system is at the edge of stabiity, the motion is unstabe or it can be stabe in Lyapunov sense but not exponentiay stabe. If the contro is switched off (the contro gains are zeros) then the rectiinear motion is at the edge of stabiity if V > V. For V < V the rectiinear motion is unstabe (see in Hubbard (198) or Kremnev and Kueshov (28)). This behavior can be observed in Figure 7, where the maxima aowabe contro gains (dashed-dotted ine) and (dashed ine) are potted against the ongitudina speed. Figure 6 provides a better overview about the shapes of the stabe domains in the space of the speed V and the contro gains and. These ast two figures were constructed with the parameters from Tabe 1. Without the variation of the contro gains, sma and parameter pairs seem to be the best choice from stabiity point of view, namey, they can guarantee the stabiity of the rectiinear motion in the argest speed domains. 5. CONCLUION A mechanica mode of the skateboard-skater system was constructed, in which the effect of the human contro was taken into account. The contro aw was demanded to move the board aong an appointed straight ine. The inear controer was considered with and without time deay. It was shown that the deay in the contro oop is essentia, namey, in case of non-zero time deay the rectiinear motion of the skateboard can be asymptoticay max [Nms/m] 3 2 1 max [Nm/m] 5 1 15 2 V [m/s] Fig. 7. The maximum aowabe contro gains against the speed stabe in some speed ranges. In case of zero time deay, the rectiinear motion can be stabe in Lyapunov sense ony, it is at the imit of stabiity. The effect of the ongitudina speed was anayzed. The presented stabiity charts can expain the oss of stabiity even at ow and high speeds. It was shown, that the skater must vary the contro gains as the speed of the board changes. The stabiization of the board aong an appointed straight ine is possibe in specific speed ranges. Out of these ranges, another contro strategy is required, for exampe, instead of foowing of the appointed ine, the baancing of the upper position is a successfu strategy. witching off the contro is another soution, by which the motion can be kept at the imit of stabiity. ACKNOWLEGEMENT This research was party supported by the János Boyai Research choarship of the Hungarian Academy of ciences and by the Hungarian Nationa cience Foundation under grant no. OTKA 15442. REFERENCE Gantmacher, F. (1975). Lectures in Anaytica Mechanics. MIR ubisher, Moscow, Russia. Hubbard, M. (198). Human contro of the skateboard. Journa of Biomechanics, 13, 745 754. Insperger, T. and Miton, J. (214). ensory uncertainty and stick baancing at the fingertip. Bioogica Cybernetics, 18, 85 11. Kremnev, A. and Kueshov, A. (28). ynamics and simuation of the simpest mode of skateboard. In roceedings of ixth EUROMECH Noninear ynamics Conference. aint etersburg, Russia.
tepan, G. (1989). Retarded dynamica systems: stabiity and characteristic functions. Longman cientific & Technica, London, United Kingdom. tepan, G. (29). eay effects in the human sensory system during baancing. hiosophica Transactions of The Roya ociety, 367, 1195 1212. Varszegi, B., Takacs,., tepan, G., and Hogan,. (214). Baancing of the skateboard with refex deay. In roceedings of Eighth EUROMECH Noninear ynamics Conference, ENOC214. Vienna, Austria. Wisse, M. and chwab, A. (25). kateboard, bicyces, and three-dimensiona biped waking machines: Veocity-dependent stabiity by means of ean-to-yaw couping. The Internationa Journa of Robotics Research, 24(6), 417 429.