Modeling and Control of Casterboard Robot

Transcription

1 9th IFAC Symposium on Nonlinear Control Systems Toulouse, France, September 4-6, 23 FrB2.3 Modeling and Control o Casterboard Robot Kazuki KINUGASA Masato ISHIKAWA Yasuhiro SUGIMOTO Koichi OSUKA Department o Mechanical Engineering, Graduate School o Engineering, Osaka University, Japan, ( ishikawa@mech.eng.osaka-u.ac.jp) Abstract: In this paper, we propose a robotic model o a casterboard, which is a commercial variants o skateboards with twistable ootplates and passive inclined caster wheels. We then derive its mathematical model in the orm o nonlinear state equation; this system is o much interest rom both mechanical and control points o view as a new challenging example o nonholonomic mechanics. Based on the observation on preceding works concerning locomotion control or nonholonomic systems, we propose a locomotion control method with sinusoidal periodic control to realize orwarding and turning locomotion. The proposed idea is examined by simulations and physical experiments using the prototype robot developed by the authors. Moreover, we also examine the inluences on the driving o the robot o parameter in the sinusoidal reerence signals.. INTRODUCTION In this paper, we propose a robotic model o a casterboard, which is a commercial variants o skateboards with twistable ootplates and passive inclined caster wheels (as shown in Fig. ). The casterboard has an peculiar body structure that the board itsel is divided into two parts, the ront and the rear wheels can be steered separately to each other. By appropriate combinatory actions o one s waist and eet, a human rider can drive the casterboard without kicking the ground directory. This seems a quite challenging and ascinating example rom viewpoints o both control theory and robotics. Our purpose is to clariy the principle o casterboard locomotion through control experiment o its prototype robot. As closely related works to the current problem, there have been a series o studies on snakeboard system conducted by Ostrowski et al. (see Ostrowski et al. [995, 997], Ostrowski and Burdick [998], McIsaac and Ostrowski [23] and the reerences therein). The snakeboard is also a variant o skateboards having twistable casters at the both ends, and undulatory propulsion without kicking the ground is possible likewise in the casterboards. Ostrowski et. al. derived a mathematical model o the snakeboard based on nonholonomic mechanics, in the orm o nonlinear state equation composed o (i) geometric equation, (ii) generalized momentum equation and (iii) reduced equation (see Murray et al. [994], Bloch [23]). A notable eature here is the presense o (ii) and coupled inluences between (i) and (ii). The authors also worked on point-to-point eedback control o the snakeboard considering repetitive stability o its generalized momentum (see Kiyasu and Ishikawa [26]). Modeling and control o the casterboard appears to be harder than the snakeboard, in that it has complicated ootplate-caster mechanism and its dynamics is inherently three-dimensional. Shammas and Choset proposed variable inertia snakeboard which is a general ramework including the snakeboard as its special case, in that the inertia matrix depends on the coniguration o the robot, and simulated orwarding locomotion o casterboard by controlling it in speciic conigurations (see Shammas and Choset [26]). However they have not exploited various driving modes o casterboard which appear by changing parameters o inputs, and with experiments. In this paper, we suggest a control approach or casterboard locomotion together with its dynamical model. Slightly rom a dierent viewpoint compared to the previous work, we show that the casterboard can be controled with an approach similar to the aorementioned snakeboard at the cost o some simpliying assumptions. We then model the casterboard as an autonomous mobile robot with active weights, and propose a control approach or orwarding and turning locomotion based on sinusoidal reerence signals. The proposed idea is examined by both simulations and experiments. Moreover, we also examine the inluences on the driving o phase dierence and requency o sinusoidal reerence signals by both simulations and experiments. This paper is organized as ollows. Ater brie review o snakeboard research in Section 2, we propose a robotic version o the casterboard and derive its mathematical model in Section 3. Control strategy is proposed and examined numerically and experimentally in Section 4. Inluences o phase dierence in the reerence signals are examined in Section 5. Section 6 concludes the paper. 2. BACKGROUND: SNAKEBOARD ROBOT In this section, we briely review the relevant studies on snakeboard robots mainly conducted by Ostrowski et. al. (see Ostrowski and Burdick [998]). Copyright 23 IFAC 785

2 (a) Casterboard (b) Snakeboard According to these results, given that each wheels is in contact with the ground without slipping nor sliding sideways, orwarding locomotion and turning locomotion o the snakeboard robot are controllable by varying the rotational angle o the rotor Ψ and the steering angle o the wheels Φ periodically with a common requency. Their success motivates us to challenge to other sort o weired dynamic boards. 3. CASTERBOARD ROBOT Fig.. Casterboard and Snakeboard Snakeboard is a commercial variant o skateboard which has been somewhat popular in playground scene since mid- 9 s. It is composed o a main beam and two ootplates with passive wheels at the both ends (ront and rear) as shown in Fig. 2. Each ootplate is connected to the main beam via a rotary joint about the vertical axis, so that the ront wheels and the rear wheels can be steered using the rider s eet relative to the main beam. Basically, maneuvering principle o the snakeboard or a rider is a (skillul) combination o the ollowing actions: to steer the ootplates into the direction opposite to each other to exert yaw-torque around the body center by twisting one s waist In this section, we introduce the dynamics o casterboard and propose its robotic model. 3. Modeling Fig. 4 illustrates a schematic side view o the casterboard (shown in Fig. ). The casterboard consists o a ront board and a rear board joined to each other via rotary joint about the horizontal (roll) axis. Each board is equipped with a passive caster inclined by a certain constant angle. Both boards can be twisted to each other about the roll-axis as the rider presses the boards by his/her eet. The casters also rotate about their incline axes. Thereore, when the rider presses down the boards to make them roll, each wheel is pushed out to the opposite direction, so as to minimize the potential energy as shown in Fig. 5. In short, the caster s yaw angle ψ is indirectly steered when one steers the board s roll angle ϕ. Fig. 2. Maneuvering basics o the snakeboard Ostrowski et al. discussed a mathematical description o the snakeboard dynamics, and proposed an autonomous robot model o the snakeboard, i.e., a snakeboard robot (see Ostrowski et al. [995]). It has a rotor which exerts torque to the snakeboard about the vertical axis passing through the body center, via an active joint Ψ as shown in Fig. 3. The model has been ormulated as a nonholonomic dynamical system which has generalized momentum as a state (see Murray et al. [994], Bloch [23]). Moreover, inputs to the system or propulsion control have been derived based on the analysis o controllability with Lie brackets o vector ields in the state equation o the model. They also simpliied the model by assuming that the ront and the rear ootplates are driven in symmetric way (with a common control parameter Φ), thus reduced the problem to a nonlinear system with two control inputs Φ and Ψ. Fig. 3. Model o the snakeboard robot Fig. 4. Side view o the casterboard (a) Top view (b) Front view (c) Actual caster Fig. 5. Relationship between the boards and the caster wheels This leads us to explain the basic maneuvering principle or the casterboard, as shown in Fig. 6. Rolling both boards alternately, exerting orce in the direction which each board is tilted toward (labeled as orce in the igure) by each oot o the rider. This explanation is in rough accordance with the techniques perormed by experienced riders. These actions o the eet generate torque around the vertical axis o the casterboard, with both wheels being steered in the opposite direction rom each other, which result in orward movement o the casterboard. Copyright 23 IFAC 786

3 Fig. 6. Maneuvering basics o the casterboard 3.2 Casterboard Robot Now we are ready to model the casterboard as an autonomous mobile robot (casterboard robot) using active mass drivers instead o a human rider. See Fig. 7 or the proposed robotic model o the casterboard. The robot is supposed to consist o two blocks, the ront and the rear ones. Each block is equipped with a mass on its top side, connected by a rigid beam, where the center o the wheel is located just beneath the bottom end o the rigid beam. The ront and the rear block are connected to each other by an active twistable joint, which we suppose to apply control inputs. (a) Side view Fig. 7. Model o the casterboard robot 3.3 Derivation o the state equation (b) Top view Let us derive the state equation o the casterboard robot. In Fig. 3.2, (x, y) denotes the robot s position, i.e., the coordinates o the centeral joint on X-Y plane. θ is the attitude angle relative to the X-axis. ϕ, ϕ r are roll angles o ront and rear board, respectively. ψ, ψ r are steering angles o ront and rear wheel, respectively. Here we dare to compromise on ollowing assumptions to simpliy the problem. [Assumptions] () Roll orientation o the robot is always kept upright. (2) Roll angle o boards ϕ i (i =, r) is very small, so we obtain the ollowing approximate expressions: sin ϕ i ϕ i, cos ϕ i. () (3) In response to the change o roll angle o the board ϕ i, yaw angle o the wheels ψ i changes instantaneously due to their inclined coniguration (Fig. 7). Namely, ϕ i and ψ i satisy the ollowing holonomic constraint: ψ i (t) = tan (tan α sin ϕ i (t)). (2) Eq. (2) is derived rom the coniguration o each caster when the potential energy o the robot is minimum. These assumptions may seem somewhat restrictive. However, at the cost o this compromise, we can reduce the problem to control problem or an almost planar mobile robot nevertheless the original system is threedimensional. It would be clear in the rest o paper that essence o the casterboard locomotion still remains in this reduced planar model. Now we introduce several physical parameters. m denotes mass o the entire robot, m 2 denotes mass o a weight, l is length rom the center o the robot to one wheel, l 2 is length o the rigid beam (distance between the mass and the board), J denotes moment o inertia o the robot about the vertical axis. α is a constant that indicate incline angle o the caster axis relative to the vertical axis. The Lagrangian o the whole system is given by L = 2 m (ẋ 2 + ẏ 2 ) + 2 J θ 2 [ { + } 2 d 2 m 2 dt (x + l cos θ l 2 ϕ sin θ) { } ] 2 d + dt (y + l sin θ + l 2 ϕ cos θ) + [{ } 2 d 2 m 2 dt (x l cos θ l 2 ϕ r sin θ) { } ] 2 d + dt (y l sin θ + l 2 ϕ r cos θ). (3) In Eq. (3), the irst line denotes kinetic energy o the body except weights, the second and the third lines denote kinetic energy o the ront weight, and the ourth and the ith lines denote kinetic energy o the rear weight. Here, gravitational potential energy is not considered due to Eq. (). Assuming all the wheels do not slide sideways, we obtain the ollowing nonholonomic constraints: { ẋ sin(θ + ψ ) ẏ cos(θ + ψ ) l θ cos ψ = ẋ sin(θ + ψ r ) ẏ cos(θ + ψ r ) + l θ cos ψr =. (4) Thanks to the assumptions mentioned above, the generalized coordinates o the system is composed o just q := [w, ϕ, ϕ r ] T, where w := [x, y, θ] T indicates the o the entire robot. First, the Lagrangian equation o motion with respect to ϕ, ϕ r is described as ollows. m 2 l 2 ϕ 2 m 2 l 2 (ẍ sin θ ÿ cos θ l θ) ( ) 2 m 2l l 2 cos 2θ l2ϕ 2 θ 2 = τ m 2 l 2 ϕ 2 r m 2 l 2 (ẍ sin θ ÿ cos θ + l θ) ( ) + 2 m 2l l 2 cos 2θ l2ϕ 2 r θ 2 = τ r Here τ, τ r are external torques applied as the actuator inputs. Following the standard manner o computed torque method, it is not diicult to consider u := [ ϕ, ϕ r ] T as a virtual control input. (5) On the other hand, combining Eq. (2) and Eq. (4) gives us a kinematic constraint linear in ẇ Copyright 23 IFAC 787

4 D(q)ẇ =, D(q) := [ ] sin(θ + ψ (ϕ )) cos(θ + ψ (ϕ )) l cos ψ (ϕ ) sin(θ + ψ r (ϕ r )) cos(θ + ψ r (ϕ r )) l cos ψ r (ϕ r ) (6) Thus the Lagrangian equation o motion with respect to the vector w under the constraint (6) is derived as in the ollowing orm, with the Lagrange multiplier λ = [λ, λ 2 ] T to indicate constraint orces: Mẅ + W (q) E(q)u D(q) T λ = (7) where M(q), W (q) and E(q) are given by [ ] m + 2m 2 M 3 M(q) = m + 2m 2 M 23 M 3 M 23 M 33 M 3 = m 2 l 2 (ϕ + ϕ r ) cos θ M 23 = m 2 l 2 (ϕ + ϕ r ) sin θ M 33 = J + m 2 {2l 2 + l l 2 (ϕ ϕ r ) cos 2θ + l2(ϕ ϕ 2 r)} ] W (q) = [ W W 2 W 3 W = m 2 l 2 {(ϕ + ϕ r ) sin θ θ 2( ϕ + ϕ r ) cos θ} θ + cẋ W 2 = m 2 l 2 {(ϕ + ϕ r ) cos θ θ 2( ϕ + ϕ r ) sin θ} θ + cẏ W 3 = m 2 l 2 [l {( ϕ ϕ r ) cos 2θ (ϕ ϕ r ) sin 2θ θ} + 2l 2 (ϕ ϕ + ϕ r ϕr )] θ + 2cl 2 θ [ ] m2 l 2 sin θ m 2 l 2 sin θ E(q) = m 2 l 2 cos θ m 2 l 2 cos θ m 2 l l 2 m 2 l l 2 Here, we assume that both wheels are subjected to viscosity resistance rom ground along their moving direction. Above c is proportionality constant in the viscosity resistance. Solving Eq. (7) using Eq. (6), λ is given by λ = (DM D T ) ( Ḋẇ + DM W DM Eu). (8) Eliminating λ rom Eq. (7) using Eq. (8), we have: ẅ + QḊẇ + P M W P M Eu = (9) where Q(q) and P (q) are given by Q(q) = M D T (DM D T ) P (q) = I QD Here, I is an unit matrix. Thereore the state equation o the casterboard robot is described as ollows. w ẇ ẇ d ϕ QḊẇ P M W P M E ϕ dt ϕ r = + u () ϕ ϕ r ϕ r Note again that D, W, E, Q and P are all matrix-valued nonlinear unctions depend on the state variables. 4. LOCOMOTION CONTROL In this section, we propose a control approach or the casterboard robot, whose validity is examined both by numerical simulations and physical experiments. 4. Control approach Now we intend to control the roll angle ϕ, ϕ r o the casterboard robot to track respectively the sinusoidal reerence angle ϕ re, ϕre r given as ollows, so that the steering angle o each wheel and torque exerted on the body changes periodically with a common requency: ϕ re (t) = A sin ωt + B () ϕ re r (t) = ϕ re (t) (2) where A is the amplitude, B is the average o oscillation and ω denotes the angular requency. B determines the direction (or turning radius) o locomotion : B = corresponds to orwarding locomotion, while B corresponds to turning locomotion where B > or counter-clockwise rotation and B < or clockwise rotation. As or the tracking control o ϕ, ϕ r to the reerence angle ϕ re, ϕre r, we simply adopt the ollowing eedback PD-type control law: [ KP (ϕ re re ϕ u = ) + K D ( ϕ ϕ ] re ) + ϕ K P (ϕ re re r ϕ r ) + K D ( ϕ r ϕ re (3) r ) + ϕ r where ϕ re, ϕre r are the reerence angles, K P is the proportional gain and K D is the derivative gain. 4.2 Simulation In this subsection, the proposed approach is examined by several numerical simulations based on the state equation (). Here, the values o the parameters A, B and T in the reerence angles ϕ re, ϕre r Forwarding locomotion : were given as ollows. A = π [rad], B = [rad], ω = 6.4 [rad/s] 9 Turning locomotion : A = π 8 [rad], B = π [rad], ω =.4 [rad/s] 8 The value o each parameter in the state equation () was given as shown in Table. Here, or simplicity, let J = m l 2 3 by putting the body a bar o uniorm crosssection. The values o K P and K D were given so that ϕ i ollows ϕ re i suiciently. The other values in Table were given so as to be the same as in the prototype shown in the ollowing section. Incidentally, the initial was w() = [,, ] T. Table. Physical parameters o the casterboard robot l length rom the center o the body to a.95 [m] wheel l 2 length o the inverted pendulum.5 [m] m mass o the body.29 [kg] m 2 mass o a weight.29 [kg] J inertia o the body about the vertical axis.39 [kg m 2 ] 7 α incline angle o the caster axis 8 π [rad] c proportionality constant in viscosity resistance.8 [N s/m] between wheels and ground First, the simulation result o orwarding locomotion is shown in Fig. 8. As shown in the igures, the casterboard robot moves orward winding its way. Copyright 23 IFAC 788

5 Y [m] (b) Robot s Fig. 8. Simulation o orwarding locomotion Second, the simulation result o turning locomotion is shown in Fig. 9. As shown in the igures, the casterboard robot turns. Here, the turning radius is almost constant. Moreover, it was also conirmed that the model o the Y [m] Fig. 9. Simulation o turning locomotion (b) Robot s In addition, we attach a support caster consisting o two ball casters to the bottom o the central block, through a parallel link mechanism so that the casters are kept horizontal. As a result, the central block does not roll irrespective o its vertical motion. Thereore the roll angle o each board ϕ i is given on the basis o the central block by each servomotor. The proposed approach is examined by several experiments. We adopted the same reerence signals to the servomotors ϕ re (t), ϕre r (t) as in simulations mentioned above. Fig. shows the experimental result o orwarding locomotion. The casterboard robot surely moves orward with a certain amount o winding, likewise in the simulation results. Y [m] (b) Robot s Fig.. Experiment o orwarding locomotion Second, the experimental result o turning locomotion is shown in Fig. 2. The casterboard robot also exhibits casterboard robot was valid rom the above results. 4.3 Experiment We developed a prototype o microcomputer-controlled casterboard robot as shown in Fig.. This prototype has Y [m] (b) Robot s Fig. 2. Experiment o turning locomotion counter-clockwise turning with almost constant radius, also likewise in the simulation results. 5. INFLUENCE OF PHASE DIFFERENCE Fig.. Overview o the casterboard robot the ront and the rear boards (ootplates), each o which is equipped with a weight and a caster. It also has the central block put between the both boards so as to control rolling o each board. The central block has two servomotor at the ront and the rear side, connected to the ront and the rear boards, respectively. In the last section, we have been considering to drive roll angles o the both boards in anti-phase manner. Now we examine how the phase dierence between ϕ re and ϕ re r aects on the resulting locomotion. 5. Simulation First, we examine the inluence o phase dierence o ront and rear boards by numerical simulations based on the Copyright 23 IFAC 789

6 state equation (). The reerence signals ϕ re given in the ollowing orms: { ϕ re (t) = A sin ωt ϕ re (t) = A sin(ωt β), ϕre r were (4) where A is the common amplitude, ω is the common angular requency and β is the phase dierence between the two signals. Now we ix A = π/9 [rad], ω = 6.4 [rad/s], set the initial as w() = [,, ] T, and see what happens with various phase gaps β =, π/2, π [rad] (a) β = [rad].5.5 Fig. 3. Robot s (simulation) (b) β = π/2 [rad] Fig. 3 shows the simulation results. Note we omit the case β = π [rad] or it corresponds to the anti-phase drive mentioned in the last section (see Fig. 8). The robot moves almost orward at any phase dierence β. Although the robot turns a little rightward at β = π/2 [rad], it travels almost straightorward or all choices o β. Thus, it is the traveling speed in the x-direction that varies depending on β. The astest is the case β =, say in-phase drive mode, and the speed tends to be slower as the β increases. Now let us turn to ocus on history o the attitude angle θ. θ(t) is kept almost constant in the case o β = (note that θ = holds in Eq. (6) or this case). On the other hand, θ starts to oscillate as β gets larger, in particular with larger amplitude A. This partially explains the loss o energy in the orwarding locomotion, as well as the change o traveling speed as depending on β. 5.2 Experiment We perormed physical experiments under the same reereces and the conditions as the simulation above; compare Fig. 4 or the cases β =, π/2 and Fig. or the case β = π. As shown in these igures, the robot moves almost straightorward in all cases, while the relationship between phase dierence and traveling speed shows similar tendency as in the simulation results. 6. CONCLUSION In this paper, we proposed a mathematical model o the casterboard as an autonomous mobile robot equipped with a pair o mass-beam mechanisms, in the orm o nonlinear state equation. We then proposed a control approach or orwarding and turning locomotion that drives the massbeam actuators so as to track periodic reerences. Then (a) β = [rad] Fig. 4. Robot s (experiment) (b) β = π/2 [rad] we developed an experimental system o the proposed casterboard robot and realized the orwarding and turning locomotion, to show that the experimental results are well consistent with simulation results. Moreover, we examined the inluence on the driving o phase dierence o periodic reerences. In low-requency range, moving velocity o the robot got higher as the phase dierence got smaller, and increased exponentially as the requency got higher by both simulations and experiments. In the experiments, it was conirmed that a peak o moving velocity appeared in a requency range, although the peak did not appear in the simulations. Moreover, in the experiments, it was also conirmed that a peak o moving velocity appeared in two requency ranges, respectively in in-phase drive. As or physical experiments, we are keen to realize locomotion without the support casters by actively balancing the robot s roll angle. We also neglected the dynamics o ψ i s behavior in response to ϕ i by assuming their relationship is static. The mathematical model should be reined in the uture by taking this dynamics into account. Acknowledgement This research is partially supported by Aihara Project, the FIRST program rom JSPS, initiated by CSTP. REFERENCES A.M. Bloch. Nonholonomic mechanics and control. Interdisciplinary Applied Mathematics (IAM), 24, 23. Y. Kiyasu and M. Ishikawa. Feedback control o nonholonomic mobile robot concerning momentum dynamics: a snakeboard example. The 3rd IFAC worksop on Lagrangian and Hamiltonian Methods in Nonlinear Control (LHMNLC 6), 26. K.A. McIsaac and J.P. Ostrowski. A ramework or steering dynamic robotic locomotion systems. The International Journal o Robotics Research, pages 22:83 97, 23. R. M. Murray, S. S. Sastry, and Z. Li. A Mathematical Introduction to Robotic Manipulation. CRC Press, 994. J. Ostrowski and J. Burdick. he geometric mechanics o undulatory robotic locomotion. The International Journal o Robotics Research, pages 7(7):683 7, 998. J. Ostrowski, J. Burdick, A.D. Lewis, and R.M. Murray. The mechanics o undulatory locomotion: The mixed kinematic and dynamic case. International Conerence o IEEE Robotics and Automation 95, pages , 995. J. Ostrowski, J.D. Desai, and V. Kumar. Optimal gait selection or nonholonomic locomotion systems. International Conerence o IEEE Robotics and Automation97, pages 2 25, 997. E. A. Shammas and H. Choset. Generalized Motion Planning or Underactuated Mechanical Systems. PhD thesis, Carnegie Mellon University, 26. Copyright 23 IFAC 79