Underactuated single wheeled mobile robot mimicking a human ridden unicycle 1

Chapter Introduction 1.1 Overview In recent years, single-wheeled robots and single-wheeled vehicles have gained a lot of attention. Several companies have developed single-wheeled platforms and demonstrated their functionality. They were introduced to the market as either toys or modes of personal transports. Murata Girl (Fig. 1.1), developed by Murata Manufacturing Co., Ltd., was marketed as a toy [6]. Honda Motor Co., Ltd., on the other hand, developed U3-X (Fig. 1.2), a personal transporter [3]. Figure 1.1: Murata Girl Figure 1.2: U3-X Personal Transporter Before reaching this stage in the industrial world, single-wheeled robots have been in existence in the academic research community for at least three decades. In early 1980s, several groups of researchers in Japan made an attempt to construct a single-wheeled robot which, in our knowledge, is the first prototype of this class of robots. As it was reported in an article written in Japanese, their achievements remain unknown and results inaccessible to non-Japanese-speaking research community. In 1987, Arnoldus Schoonwinkel developed a single-wheeled robot with a rotating turntable as part of his Ph.D. research in Stanford University [68]. Schoonwinkel managed to balance that robot longitudinally, i.e., in its forward-backward direction. Nothing was reported about making the robot balanced laterally. However, Schoonwinkel’s work is a significant milestone in this field of research as a wider research community, especially outside Japan, was introduced to single-wheeled robots through that work. Ever since Schoonwinkel’s doctorate thesis was published, interests in single-wheeled-robot research gained momentum. David W. Vos and Andreas H. von Flotow of Massachusetts Institute of Technology investigated exactly the same type of single-wheeled robot in 1990 [78], three years after the publication of Schoonwinkel’s thesis. Research on single-wheeled robots is driven by several factors including (1) the investigation of nonlinear and unstable dynamics [68]; (2) the emulation of human intelligence [71]; (3) the development of advanced mobile surveillance system [80] and (4) the search for better means of transportation. Section 1.2 provides the background of single-wheeled robots including their historical development, current state, classification and our view and evaluation of the trend of single-wheeled robotic research. Subsequently, some preliminaries are presented in Section 1.3 and the motivations of this research are explained in Section 1.4. Finally, an outline of this thesis is given in Section 1.5 for easy reference. 1.2 1.2.1 Background Single-Wheeled Mobile Robots Definition 1.1. [52] A wheeled mobile robot is defined as a robot capable of locomotion on a surface solely through the actuation of wheel assemblies mounted on the robot and in contact with the surface. A wheel assembly is a device which provides or allows relative motion between its mount and a surface on which it is intended to have a single point of rolling contact. Wheeled mobile robots commonly have three or more wheels. With three or more wheels, a wheeled mobile robot maintains its static stability by keeping its center of gravity inside its polygon of support. However, most four-wheeled robots, have limited manoeuvrability and require wheel suspension systems to ensure that the wheels are always in contact with the ground [51]. As for three-wheeled robots, their two drive wheels must rotate at slightly different speeds in order to achieve accurate control of turning [51]. Single-wheeled robots and two-wheeled robots avoid the problems encountered by the multi-wheeled robots at the expense of static stability. Two-wheeled configurations, such as the Segway [9] and bicycle, are statically unstable in either longitudinal or lateral direction. Single-wheeled configuration, as found in unicycle, is statically unstable in both longitudinal and lateral directions. Apparently, single-wheeled configuration seems to be at a disadvantage compared to its two-wheeled counterpart. However, at moderate to high speed, disturbance torque arising from uneven ground may make a two-wheeled platform turned over [80]. The distance between the two wheels, while providing static stability, acts as a leverage to make small disturbance force result in large disturbance torque. On the other hand, single-wheeled configuration has minimum disturbance torque because it only has one point contact with the ground. Therefore, disturbance force at the wheel is not magnified. Theoretical and experimental research in the field of wheeled mobile robotics has progressed rapidly. It benefits from the rapid development in related areas such as artificial intelligence, sensing, actuation, control and computing. In particular, for a fully functional practical wheeled mobile robot, a systematic integration of all of these areas is necessary [51]. Below are some important theoretical and experimental results on wheeled mobile robots reported in the literature. 1. Kinematic formulation of wheeled mobile robots is reported in [13], [52] and [61]. Analysis of the internal dynamics of wheeled mobile robots is reported in [83]. 2. Motion control of wheeled mobile robots has been achieved using various techniques such as receding horizon control [32], self-organized fuzzy controller using an evolutionary algorithm [46], fuzzy control with backstepping [35], combined kinemat5 ic/torque control [29], state-feedback linearization [23], sliding-mode control [82], neural sliding-mode control [58], cross-coupling control [28], iterative learning control [42], computed torque control [62], neural control [22], neural control with backstepping [26], virtual-vehicle approach [27], dynamic surface control [72] and model-based adaptive approach [36]. 3. Sensor development and navigation for wheeled mobile robots are reported in [16], [33], [25], [77] and [24]. 4. Actuator, computer and mechanism developments for wheeled mobile robots are reported in [59], [14] and [41]. 1.2.1.1 Monocycle Robot or Unicycle Robot Several terminologies have been used to refer to a mobile robot that uses only one wheel for locomotion. These nomenclatures include single-wheeled robot, one-wheeled robot, mono-wheeled robot, monocycle robot and unicycle robot. While the first three terms are self-explanatory and generally cause no confusion, monocycle robot and unicycle robot must be used with care as they refer to two different groups of single-wheeled platforms [19]. Monocycle refers to a single-wheeled platform having all of its components enclosed inside a wheel, giving it an appearance of a plain wheel from the outside. Unicycle, on the other hand, refers to a single-wheeled platform having an exposed chassis sitting on the wheel’s shaft. An example of monocycle is McLean Monocycle, created by Kerry McLean [5] and shown in Fig. 1.3, while U3-X personal transporter, shown in Fig. 1.2, is a unicycle. Figure 1.3: McLean Monocycle It is important to point out that several authors have used the term unicycle to describe a two-wheeled platform especially two-wheeled inverted pendulum and differential-drive mobile robots. In order to remain consistent, definitions of monocycle and unicycle given in [19] are adopted throughout this thesis. Figure 1.4: The Original Design of A. Schoonwinkel’s Unicycle Robot Figure 1.5: The First Successful Unicycle Robot Developed by Zaiquan Sheng and Kazuo Yamafuji 1.2.1.2 Early Development & Current State The earliest published literature on single-wheeled mobile robots is the article by Ozaka, Kano and Masubuchi [57], published in 1980. Two more articles were published around that time, one [34] by Honma et al. in 1984 and the other [81] by Yamafuji and Inoue in 1986. However, all these articles were in Japanese and, hence, attracted little or no attention from non-Japanese-speaking researchers. The Ph.D. thesis of Schoonwinkel of Stanford University published in 1987 is the first recorded reference on this subject written in English [68]. In his work, Schoonwinkel developed a prototype of unicycle robot with a turntable as its balancing mechanism. This unicycle, shown in Fig. 1.4, was designed to mimic a young person in terms of masses and moments of inertia. This unicycle and its dynamic behaviour were studied by both simulations and experiments. Though simulations of both longitudinal and lateral motions were reported to be successful, experimental results are shown for longitudinal motion only. Since the publication of this research, interest in single-wheeled mobile robots outside Japan gained momentum, as evident from the work published by Vos and Flotow of Massachusetts Institute of Technology who investigated exactly the same unicycle robot in 1990 [78], three years after the publication of Schoonwinkel’s thesis. In 1995, Sheng and Yamafuji of University Electro-Communication made the first claim of successfully balancing a unicycle robot, the design of which follows that of Schoonwinkel with some modifications in the turntable’s shape and the addition of a pair of closed-link mechanisms to imitate human legs [70]. Fig. 1.5 shows the unicycle robot developed by Sheng and Yamafuji. After Sheng’s and Yamafuji’s success, the research on single-wheeled mobile robots spread into several different directions as marked by two important events. The first event is the introduction of Gyrover by Brown and Xu of Carnegie Mellon University in 1996 [39]. As the first monocycle robot in the world, Gyrover paved the way to generate more interest in research on monocycle robots. This is evident by the development of Gyrobots by Cavin et al. of University of Florida in 2000 [20] and by Saleh et al. of National University of Singapore in 2004 [66], Reactobot by Joydeep et al. of Indian Institute of Technology Bombay in 2008 [53], GYROBO by Kim et al. of Chungnam National University in 2007 [44] and Mono-Wheel Robot by Cieslak et al. of AGH University of Science and Technology in 2011 [21]. The second event is the construction of Yamabico ICHIRO, a unicycle robot with a rugby-ball-shaped wheel and a side-leaning head, by Nakajima et al. of University of Tsukuba in 1997 [54]. The introduction of Yamabico ICHIRO paved the way to the exploration in new stabilization mechanisms and the use of unconventional wheels. In the exploration of new mechanisms, lateral pendulum has been studied by Fujimoto and Uchida of Yokohama National University in 2007 [30] and Xu, Mamun and Daud of National University of Singapore in 2011 [79]. In addition, reaction wheel has been explored by Majima, Kasai and Kadohara of University of Tsukuba in 2006 [50], Ruan, Hu and Wang of Beijing Institute of Technology in 2009 [63] and Lee, Han and Lee of Pusan National University in 2011 [38]. In the use of unconventional wheels, Ballbot, a unicycle robot having a spherical wheel, was developed by Lauwers, Kantor and Hollis 10  ˙ = • Along (δ)    • Blong =  −(I [ 21 rw w2 + rw )(σ4 +ρ1 )] ˙ rw ) g δ − (Iw2 +dlong dlong 2r g [− rw (Ic2 +Ip2 +ρ1 )+(σ4 +ρ1 ) rw ] ˙ w δ + dlong dlong  0   0  rw +Iw2 + rw dlong −Ic2 −Ip2 −ρ1 − rw dlong     ˙ α, ¨ =  wlong2  • Wlong (α, γ, δ, ˙ γ, ˙ δ) wlong3 • wlong2 = [− {[−(Iw2 dlong + rw + rw ) rw + (Iw2 + rw )(Ic2 + Ip2 + ρ1 + rw )]αδ¨ + ¨ rw + (σ3 + ρ3 )(Iw2 + rw )]γ δ + [−(Iw2 + rw + rw ) rw + (Ic1 + σ2 + Ip1 + σ3 + 2ρ1 + rw )(Iw2 + • wlong3 = {[(Ic2 dlong ˙ + [−2 ˙δ rw )]α 2 rw + (Ip1 + σ3 + 2ρ3 )(Iw2 + ˙ ˙ δ} rw )]γ + Ip2 + ρ1 )(Iw2 + rw + rw ) − (Ic2 + Ip2 + ρ1 + rw ) rw ]αδ¨ + [(Ic2 + Ip2 + ρ1 ) rw − (σ3 + ρ3 ) rw ]γ δ¨ + [(Ic2 + Ip2 + ρ1 )(Iw2 + rw + rw ) − (Ic1 + ˙ σ2 + Ip1 + σ3 + 2ρ1 + rw ) rw ]α˙ δ˙ + [(Ic2 + Ip2 + ρ1 )2 rw − (Ip1 + σ3 + 2ρ3 ) rw ]γ˙ δ} • dlong = 2 rw − (Iw2 + rw )(Ic2 + Ip2 + ρ1 ). The longitudinal system matrix Along is perturbed by δ˙ and the system is disturbed by Wlong which is called longitudinal coupling disturbance. The lateral dynamics (Eqs. 2.72 and 2.73) can be put into the following state-space form: ˙ lat + Blat τp + Wlat (β, δ, ˙ ω, ˙ δ) ¨ x˙ lat = Alat (δ)x ˙ β, where: • xlat = α γ α˙ γ˙ T 86 (2.94)  0 0 0  ˙ = • Alat (δ)   alat31 alat32 alat41 alat42 • alat31 = rw     rw + ρ3 ) g + ( + )g(Ip1 + lp )]} {[−(σ3 + lp )(Ip1 + rw + ρ3 ) + (σ3 + ρ3 + rw )(Ip1 + lp )]δ˙ + [−(Ip1 + dlat + ρ3 ) g + g(Ip1 {[−(Ip1 dlat • alat41 = + + lp )]} rw + ρ3 )(σ1 + σ2 + σ3 + Ip1 + rw + ρ1 + rw )(σ3 + Ip1 + rw + ρ1 + rw ) g]} • alat42 =  + σ2 + σ3 + rw + ρ1 + rw )(Ip1 + lp ) − (Ip1 + rw + ρ3 )(σ3 + + ρ3 )]δ˙ + [−(Ip1 + • alat32 = rw {[(σ1 dlat 0 rw rw + ρ3 )]δ˙ + [(Ip1 + {[−(σ3 + ρ3 + rw )(Ip1 + rw dlat + ρ1 + rw ) + (Iw1 + Ic1 + rw + ρ3 ) )g + (Iw1 + Ic1 + + ρ3 ) + (σ3 + lp )(Iw1 + Ic1 + Ip1 + rw + ρ1 + rw )]δ˙ + [− g(Ip1 + rw + ρ3 ) + g(Iw1 + Ic1 + Ip1 + rw + ρ1 + rw )]}       • Blat =   −(Ip1 + rw +ρ3 )   dlat (Iw1 +Ic1 +Ip1 + rw +ρ1 +2 rw ) dlat   0   ˙ ˙ ¨ • Wlat (β, δ, ω, ˙ β, δ) =   wlat3 wlat4 • wlat3 = rw {[−(Ip1 + rw dlat     + ρ3 )(Ip1 + ρ3 ) + (Ip1 + lp )(σ4 + rw + ρ1 )]β δ¨ + [−(Ip1 + + ρ3 )(Ip1 + σ3 + 2ρ3 ) + (Ip1 + [−(Ip1 + • wlat4 = rw + ρ3 ) rw + (Iw2 + {[−(Ip1 dlat + rw lp )(Ic1 rw + ρ3 )(σ4 + + rw )(Ip1 rw 87 + σ2 + Ip1 + σ3 + 2ρ1 + rw )]β˙ δ˙ + + ˙ ˙ lp )]δ ω} + ρ1 ) + (Iw1 + Ic1 + Ip1 + rw + ρ1 + rw )(Ip1 + ρ3 )]β δ¨ + [−(Ip1 + rw + ρ3 )(Ic1 + σ2 + Ip1 + σ3 + 2ρ1 + rw ) + (Iw1 + Ic1 + Ip1 + rw + ρ1 + rw )(Ip1 + σ3 + 2ρ3 )]β˙ δ˙ + [−(Ip1 + rw + ρ3 )(Iw2 + rw + rw ) + (Iw1 + Ic1 + Ip1 + • dlat = (Iw1 + Ic1 + Ip1 + rw rw + ρ1 + rw ) rw ]δ˙ ω} ˙ + ρ1 + rw )(Ip1 + lp ) − (Ip1 + rw + ρ )2 The lateral system matrix Alat is perturbed by δ˙ and the system is disturbed by Wlat which is called lateral coupling disturbance. The turning dynamics (Eq. 2.74) can be put into the following state-space form: ¨ γ, x˙ turn = Aturn (β, ω)x ˙ turn + Bturn (ω) ˙ α˙ + Wturn (α, β, γ, α ¨ , β, ˙ ω ¨) (2.95) where: • xturn = δ˙ ˙ rw β ω • Aturn (β, ω) ˙ = − (Iw3 +I c3 +Ip3 ) Iw2 ω˙ • Bturn (ω) ˙ = − (Iw3 +I c3 +Ip3 ) ¨ γ, • Wturn (α, β, γ, α ¨ , β, ˙ ω ¨) = ¨ [(σ4 +ρ1 + rw )β α ¨ −(Ic2 +Ip2 +ρ1 + rw )αβ− (Iw3 +Ic3 +Ip3 ) (σ3 + ρ3 )γ β¨ + (Ip1 + ρ3 )β¨ γ − (Iw2 + rw + ω rw )α¨ − ¨] rw γ ω For the turning dynamics, the system matrix Aturn is a function of β and ω, ˙ and the input matrix is a function of ω. ˙ The system is disturbed by Wturn which is called turning coupling disturbance. If the perturbations in the three state-space representations are ignored, the resulting nominal state-space models can be used for the linear control design. The designed con88 trol laws must be able to withstand the perturbations and therefore, the understanding of these perturbations is valuable. In linear motion, ALP Cycle moves forward on a straight line. At steady state, the forward speed is constant and since there is no turning motion, the turning speed is zero and hence, Eqs. 2.94 and 2.95 degenerates and the perturbations in Eq. 2.93 disappear. In circular motion, ALP Cycle moves following a circular trajectory. At steady state, the forward speed, turning speed, lean angle and pendulum angle are all constant. If the lean angle and the forward speed are sufficiently small, the turning speed is small and therefore, its squared value can be ignored. Hence, the perturbations of the longitudinal and lateral system matrices can largely be ignored. At steady state, the longitudinal and turning coupling disturbances disappear, but the lateral coupling disturbance becomes:       ˙ ˙ ¨ Wlat (β, δ, ω, ˙ β, δ) =  −(Ip1 + rw +ρ3 ) rw +(Iw2 + rw + rw )(Ip1 + lp ) . ˙ δ ω ˙   (Iw1 +Ic1 +Ip1 + rw +ρ1 +2 rw )(Ip1 + lp )−(Ip1 + rw +ρ3 )2 −(Ip1 + rw +ρ3 )(Iw2 + rw + rw )+(Iw1 +Ic1 +Ip1 + rw +ρ1 +2 rw ) rw ˙ δ ω˙ (Iw1 +Ic1 +Ip1 + rw +ρ1 +2 rw )(Ip1 + lp )−(Ip1 + rw +ρ3 )2 (2.96) Since the lateral coupling disturbance becomes constant at steady state, an integral action is required for the lateral control law in order to compensate this disturbance. If linear control is used, we can estimate the maximum lateral coupling disturbance by using the previous results. Let us assume that the maximum angle which can be considered small is 30 deg. Hence, the maximum lean angle, |α|max , is: |α|max = 300 0.52 rad = . Kls Kls 89 ˙ max , is: With maximum lean angle, the maximum turning speed, |δ| ˙ max = |δ| 0.52Kturn V Kls where V is the forward speed and is related to the wheel speed by V = rw ω. ˙ Substituting Eq. 2.97 into Eq. 2.96 and expressing the wheel speed as the forward speed, the maximum lateral coupling disturbance is given by:    max(Wlat ) =  −(Ip1 + rw +ρ3 ) rw +(Iw2 + rw + rw )(Ip1 + lp ) 0.52Kturn V  (Iw1 +Ic1 +Ip1 + rw +ρ1 +2 rw )(Ip1 + lp )−(Ip1 + rw +ρ3 )2 Kls rw  −(Ip1 + rw +ρ3 )(Iw2 + rw + rw )+(Iw1 +Ic1 +Ip1 + rw +ρ1 +2 rw ) rw 0.52Kturn V (Iw1 +Ic1 +Ip1 + rw +ρ1 +2 rw )(Ip1 + lp )−(Ip1 + rw +ρ3 )2 Kls rw   .  (2.97) Therefore, the maximum lateral coupling disturbance is related quadratically to the forward speed. For the ALP Cycle prototype which we have constructed, using the parameters in Appendix C, the maximum value of lateral coupling disturbance, max(Wlat ), is: max(Wlat ) = 2.4.7 0 0.26V −0.3V T . (2.98) Closed-Loop Responses We have previously shown that the control of turning motion is possible by the combination of the longitudinal and lateral motions and we have also derived the lateral-statics boundary, lateral-statics constant and turning constant which are important characteristics of ALP Cycle. In order to illustrate the control of turning and the usefulness of these constants, we perform some closed-loop numerical simulations. We require some feedback control because ALP Cycle is inherently unstable. In this subsection, we only use 90 linear quadratic method with unity weighting matrices for the simulation study because we only require stability. In the next chapter, more control laws will be investigated to achieve good performance in addition to stability. 2.4.7.1 Lateral Set-Point Control For ALP Cycle with the parameters given in Appendix C, the lateral-statics boundary, Klsb is 8.08 deg while the lateral-statics constant, Kls , is -8.12. The value of Klsb implies that ALP Cycle can only successfully regulate its lean angle to a set point with maximum magnitude of 8.08 deg. The value of Kls implies that if 30 deg is the maximum angle which can be considered to be small, then according to Eq. 2.83, ALP Cycle can successfully regulate its lean angle only to a set point with the maximum magnitude of: |αref | = 300 = 3.690 . | − 8.12| (2.99) Let us consider three cases of lateral set-point control. Cases I, II and III present the problems of controlling the lean angle, α, to the set points of 30 , 50 and 100 respectively. Linear quadratic integral (LQI) control law with the following performance index is used to stabilize the system. ∞ T (ψlat Qlat ψlat + τpT Rlat τp ) dt Jlat = (2.100) The state vector ψlat is defined as ψlat = α γ α˙ γ˙ t α dt matrices are given as: Qlat = diag(1, 1, 1, 1, 1), Rlat = 1. 91 T and the weighting The final control law is expressed by the following equation.   α − αref   γ − γref     τp = −Klat  α˙      γ˙ t (α − αref ) dt Klat = (2.101) −230.47 −28.67 −51.39 −9.33 Numerical simulation is performed in MATLAB with the designed LQI control law and the explicit Runge-Kutta (4, 5) formula, the Dormand-Prince pair, is used to solve the involved ordinary differential equations. The full nonlinear model, represented by Eqs. 2.38 - 2.42, is used in the simulations. The simulation results are shown in Figs. 2.9 - 2.11 for cases I, II and III respectively. Figure 2.9: Simulation Result of Lean-Angle Regulation with LQI and αref of 30 92 Figure 2.10: Simulation Result of Lean-Angle Regulation with LQI and αref of 50 Figure 2.11: Simulation Result of Lean-Angle Regulation with LQI and αref of 100 For the three cases, αref ’s are compared with Klsb , calculated using Eq. 2.82, and the magnitudes of γref ’s, calculated using Eq. 2.83, are checked as shown below. • Case I: αref = 30 ≤ Klsb = 8.080 93 |γref | = | − 8.12 × 30 | = 24.360 ≤ 300 • Case II: αref = 50 ≤ Klsb = 8.080 |γref | = |Kls × αref | = | − 8.12 × 50 | = 40.60 ≥ 300 • Case III: αref = 100 ≥ Klsb = 8.080 |γref | = |Kls × αref | = | − 8.12 × 100 | = 81.20 ≥ 300 . (2.102) In Fig. 2.9, good set-point control is achieved. In Fig. 2.10, set-point control is still successful, but there is an obvious offset between γ and γref . In Fig. 2.11, the system is unstable. These findings can be explained by referring to the values of αref and |γref | for each case. In case I, αref is less than Klsb , so the set-point control is feasible. The computed γref is less than 30 deg and, therefore, it can be considered small, so linear control is expected to be successful. In case II, αref is less than Klsb , so the setpoint control is also still feasible. However, the computed γref is greater than 30 deg and, therefore, it can not generally be considered small, so the success of linear control is questionable. In case III, αref is greater than Klsb , so the set-point control is not feasible and, therefore, the instability of the system is completely expected and there is no need to further evaluate γref . In this case study, we have shown how the lateral-statics boundary, Klsb , and the lateral-statics constant, Kls , can be used to predict and properly set references for lean 94 angle and pendulum angle in the case of lateral set-point control. 2.4.7.2 Turning-Motion Control The generation and control of the turning motion are illustrated with a simple case study. This study will make use of the lateral-statics constant, Kls , and the turning constant, Kturn , developed previously. Let us suppose that we want ALP Cycle to move forward with a constant speed of m/s and then, at t = 10s, ALP Cycle must turn at a rate of deg/s for 40 s while maintaining its forward speed at m/s. This simple scenario consists of linear and circular motions. At any moment, the robot’s forward speed V is the resultant of the robot’s speeds in x direction x˙ and y direction y, ˙ as expressed by: V = x˙ + y˙ . Due to the kinematic constraints Eqs. 2.2 and 2.3, the forward speed is related to the wheel speed ω˙ by: V = rw ω. ˙ Using the equation above, the wheel-speed reference during the linear-motion phase is computed as: ω˙ ref = Vref m/s = = 6.67 rad/s = 381.970 /s rw 0.15 m and, since there is no turning motion involved, the references for the lean and pendulum angles are both set to zero: αref = 00 , γref = 00 . 95 In the circular-motion phase, the wheel-speed reference is not changed. Since turning motion is involved, the proper references for lean and pendulum angles need to be calculated using Kls and Kturn . αref = δ˙ref 60 /s = = −4.050 Kturn Vref −1.48 × 1m/s γref = Kls × αref = −8.12 × (−4.050 ) = 32.890 Numerical simulation is performed in MATLAB with the previously designed LQI control law and the explicit Runge-Kutta (4, 5) formula, the Dormand-Prince pair, is used to solve the involved ordinary differential equations. The full nonlinear model, represented by Eqs. 2.38 - 2.42, is used in the simulations. The simulation result is shown in Figs. 2.12 - 2.19. Figure 2.12: Trajectory of ALP Cycle on X-Y Plane 96 Figure 2.13: Forward Speed of ALP Cycle Figure 2.14: Longitudinal State of ALP Cycle 97 Figure 2.15: Lateral State of ALP Cycle Figure 2.16: Turning Angle of ALP Cycle 98 Figure 2.17: Coupling Disturbance for Longitudinal Dynamics (Wlong ) Figure 2.18: Coupling Disturbance for Lateral Dynamics (Wlat ) 99 Figure 2.19: Coupling Disturbance for Turning Dynamics (Wturn ) Figs. 2.17 and 2.19 shows that during the transient period, the coupling disturbance for longitudinal dynamics increases and peaks up, but it slowly goes down to zero as the circling motion becomes steady. Fig. 2.18 shows that the coupling disturbance for lateral dynamics increases, peaks up and slowly approaches constant values. The results are in agreement with the previous analysis. 2.5 Conclusions In this chapter, detailed work on dynamic modelling and its characteristics evaluation have been presented. We have derived the dynamic model of ALP Cycle by EulerLagrange formulation. The dynamic model has been validated by comparison with the dynamic models of single-wheeled inverted pendulum and Acrobot. Unlike the other works in the literature, we did not perform simplification or elimination on the dynamic model before or during the modelling process, so this dynamic model represents the 100 complete dynamics of ALP Cycle. Due to the large scale and high complexity of the dynamic model, it is simplified by pseudolinearisation with respect to the state representing the equilibrium upright posture of ALP Cycle. With the dynamic model and its pseudolinearised version, we have investigated the equilibrium characteristics of ALP Cycle and their dependence on the robot’s physical parameters. We also have investigated the process of balancing the upright posture and the generation of turning motion by the combination of the longitudinal and lateral motions. All of these investigations are very useful for the control design and prototype improvement in the future. In Chapter 6, the models and results presented in this chapter will be used to design the control laws for posture balancing and manoeuvring of ALP Cycle. 101 [...]... Schoonwinkel’s unicycle and Gyrover This may be the reason that despite the low interest in experimental research of single- wheeled mobile robots, this field has not encountered a natural death yet 1. 2.3 Classification & Comparative Analysis of Single- Wheeled Mobile Robots A lot of designs and mechanisms for single- wheeled mobile robots have been proposed in the literature These various single- wheeled mobile robots... which is stabilised and controlled by a lateral pendulum Its conceptual design is shown in Fig 1. 18 Figure 1. 18: Conceptual Design of ALP Cycle The unicycle robot was initially named Pendulum-Balanced Autonomous Unicycle (PBAU), but its name was later changed to Automatic Lateral-Pendulum Unicycle (ALP 32 Cycle) to reflect the fact that the pendulum functions not only as a balancer, but also as a steering... can be classified based on four criteria namely wheel shapes, platform types, driving mechanisms and steering mechanisms Based on the wheel shapes, single- wheeled mobile robots can be divided into (1) conventional -single- wheeled robots and (2) innovative -single- wheeled robots Following the classification of single- wheeled vehicles based on Cardini’s article [19 ], based on the platform types, single- wheeled. .. will remain commercially non-competitive 3 Low entertainment value: The resemblance of humanoid robots to human and some biomimetic robots to animals easily gains a lot of attraction and popularity especially in conferences and exhibitions Compared to those robots, single- wheeled robots may not easily attract the crowd’s attention, although external modification can certainly be done to improve it, as in... monocycle, a unicycle is a single- wheeled platform with an exterior chassis sitting on the wheel’s shaft in an inverted position The chassis arrangement 18 makes unicycle statically unstable in both longitudinal and lateral directions Unicycle is more common than monocycle as a personal transport and an entertainment equipment in circus The adoption of unicycle type of platform in a robotic system was first... have any clear advantage over the electric motor and, given the potential complexity of its implementation, it has not been implemented in unicycle robots so far 21 Figure 1. 12: Gyrover, The First Monocycle Robot 1. 2.3.4 Steering Mechanisms Steering mechanism is the means by which a single- wheeled mobile robot maintains its balance and steer itself to follow a designated path on Cartesian x-y plane Many... Wheel, an advanced version of omni-wheel made by Honda Motor Co., Ltd [3] 3 Spherical wheel: As its name suggests, spherical wheel has the shape of a sphere It is holonomic and its implementation has been reported in Volvot, a monocycle robot made by Ishikawa, Kitayoshi and Sugie [37], and Ballbot, a unicycle robot made by Lauwers, Kantor and Hollis [47] Figure 1. 7: Yamabico ICHIRO, Its Rugby-Ball-Shaped... the case of Murata Girl 4 High academic value: Being a good candidate for complex and challenging systems to be controlled, a single- wheeled mobile robot carries enormous academic value 14 Should a prototype of single- wheeled mobile robot be made to work successfully with a certain controller, it would be of much value for academic research and give some recognition to the pioneers, as in the case of... Monocycle as a vehicle was much explored in late 18 00s and early 19 00s [19 ] However, due to its control difficulty especially at low speeds, it has not been adapted as a daily vehicle Despite the fact, a fully working monocycle vehicle has been developed by McLean, a freelance machinist, since 19 70 [5] A single- wheeled mobile robot with monocycle type of platform was first created by Brown and Xu in 19 96 Different... monocycle and unicycle platforms can be seen in McLean Monocycle and U3-X shown in Figs 1. 3 and 1. 2 respectively 19 1. 2.3.3 Driving Mechanisms Single- wheeled mobile platforms can also be classified according to the means by which driving thrust is generated Most, if not all, of single- wheeled mobile robots are relatively small in size and, therefore, electric motors are suitable as the actuator of choice . in Section 1. 5 for easy reference. 1. 2 Background 1. 2 .1 Single- Wheeled Mobile Robots Definition 1. 1. [52] A wheeled mobile robot is defined as a robot capable of locomotion on a surface solely. a fully functional practical wheeled mobile robot, a systematic integration of all of these areas is necessary [ 51] . Below are some important theoretical and experimental results on wheeled mobile. and Xu, Mamun and Daud of National University of Singapore in 2 011 [79]. In addition, reaction wheel has been explored by Majima, Kasai and Kadohara of University of Tsukuba in 2006 [50], Ruan, Hu