Underactuated single wheeled mobile robot mimicking a human ridden unicycle 3

Chapter Control Design, Simulation & Implementation In Chapter 2, we have derived the ALP Cycle’s dynamic model and its pseudolinearised version. We have also identified several unique characteristics of ALP Cycle in relation to lateral stability, turning motion, path curvature and coupling disturbances. Issues related to the dynamic control of the ALP Cycle are discussed in this chapter. Specifically, we define two control tasks namely (1) posture balancing and (2) manoeuvring, and propose suitable control schemes for fulfilling these tasks. Control of ALP Cycle is challenging and tricky because of many reasons. Firstly, ALP Cycle is inherently unstable. Secondly, ALP Cycle is underactuated in which two control inputs are responsible for five degrees of freedom, so each degree of freedom can not be independently controlled. Thirdly, ALP Cycle possesses a non-minimum-phase behaviour, so the achievable control performance is theoretically limited. Fourthly, ALP Cycle’s dynamics is highly nonlinear, so linear control can only be guaranteed to work in the small vicinity of its upright equilibrium posture. Lastly, ALP Cycle can not move 186 in the direction parallel to its wheel axis due to its nonholonomy and, therefore, motion control presents a real challenge. This chapter is organised into five sections. Section 6.1 describes the problem of posture balancing and presents a control scheme based on separate-regulations concept and H∞ control design method to achieve robust posture balancing. Simulation results are presented to show the effectiveness of the proposed control scheme. In Section 6.2, the problem of manoeuvring is described and two control schemes are proposed to achieve stable manoeuvring. The first control scheme is based on direct set-point controls deploying linear quadratic integral (LQI) control and gain-scheduling technique. The second control scheme is based on cascade set-point controls with LQI control and gain-scheduling technique. Simulation results are presented to show effectiveness of these control schemes and to compare their performances. Experimental result with the constructed prototype is presented in Section 6.3 to demonstrate the working principle of ALP Cycle and show its practical issues. 6.1 6.1.1 Posture Balancing Description of Control Task ALP Cycle must maintain the stability of its posture and prevent itself from falling down at all times by active controls of the wheel torque and the pendulum torque. The upright equilibrium posture of ALP Cycle is described as: α = β = γ = 00 , 187 α˙ = β˙ = γ˙ = 00 /s, where α, β and γ are lateral lean angle of the unicycle, longitudinal angular deviation of the chassis and pendulum from the vertical position and lateral angular deviation of the pendulum from the chassis respectively. The task of posture balancing is stated formally as followed. Definition 6.1. Given that ALP Cycle is released from some static initial posture which is not upright, i.e. one or more of the angles α, β and γ are non-zero and derivatives of these angles are zero, design a control system such that it can bring ALP Cycle back to the vicinity of its upright equilibrium posture and maintain it there for all future times despite the presence of external disturbances. The practical significance of posture-balancing task is in the start-up state of ALP Cycle prototype when the system is turned on and released by the operator’s hands. Due to the imperfection in human handling, the robot is inevitably released from some static non-upright initial posture. After the release, the applied control algorithms must be able to bring ALP Cycle to the vicinity of its upright equilibrium posture such that the posture balance is then achieved. In the real world, disturbances are inevitable. They may come from uneven ground surface, wind etc Hence, the applied control algorithms must also be able to make ALP Cycle withstand the disturbances which are likely to occur, so ALP Cycle’s upright equilibrium posture can be robustly maintained. It must be noted that the posture-balancing task disregards turning angle δ, wheel angle ω and their derivatives δ˙ and ω. ˙ This means that as long as the upright equilibrium 188 posture can be achieved, the ALP Cycle’s position, orientation, wheel speed and turning speed are not of concern. 6.1.2 Posture Classification Assuming that there is no lateral motion (α = γ = 00 and α˙ = γ˙ = 00 /s), ALP Cycle can only have two types of configurations as shown in Fig. 6.1. In both cases, neither the ALP Cycle moves sideways from the upright position nor the pendulum swings sideways. In the first case, the chassis and pendulum stand together vertically upright (longitudinally balanced) while, in the second case, the two deviate together from the upright position (longitudinally unbalanced). Figure 6.1: (a) Longitudinal Balanced Configuration and (b) Longitudinal Unbalanced Configuration In this case, the pendulum and the chassis can be considered as a single body, named pendulum-chassis body. Longitudinal unbalanced configuration does not pose any serious control difficulty. This is because only one body needs to be balanced, i.e. the pendulum-chassis body, and there is no restriction whatsoever on the motion of the 189 wheel. Therefore, wheel torque can be applied in any way necessary to balance the pendulum-chassis body without much concern on its effect on the wheel. This case is effectively the well-known problem of controlling an inverted pendulum. Assuming that there is no longitudinal motion (β = ω = 00 and β˙ = ω˙ = 00 /s), six types of configurations, shown in Figs. 6.2 and 6.3, are possible. The ALP Cycle can be laterally balanced with wheel, chassis and pendulum in vertically upright position as shown in Fig. 6.2(a). It is also possible to keep the ALP Cycle balanced with the wheel and chassis tilted, but counterbalanced by the pendulum tilting in the opposite direction, as shown in Fig. 6.2(b). Both of these are laterally balanced posture. Different combinations of wheel, chassis and pendulum angles may result in laterally unbalanced posture as illustrated in Fig. 6.3. Figure 6.2: (a) Upright Lateral Balanced Configuration and (b) Non-Upright Lateral Balanced Configuration 190 Figure 6.3: Lateral Unbalanced Configurations of (a) Type I, (b) Type II, (c) Type III and (d) Type IV In all these cases, the wheel and the chassis can be considered as one body, named wheel-chassis body. Lateral balancing is more difficult than longitudinal balancing because of three reasons. Firstly, lateral balancing involves two bodies which are the pendulum and the wheel-chassis body. Secondly, the motions of both the pendulum and and wheel-chassis body are physically restricted to ±1800 and ±900 respectively, i.e. the pendulum can not swing over passing the wheel-chassis body and the wheel-chassis body can not swing over passing the flat ground. Lastly, the pendulum torque acts on the pendulum and the wheel-chassis body in the opposite directions as an action torque and 191 a reaction torque respectively. These reasons are behind the fact that the four types of lateral unbalanced configurations in Fig. 6.3 have varying degrees of control difficulty. Type III configuration is the easiest to be balanced while type IV configuration is the most difficult. In type III configuration, the pendulum and the wheel-chassis body lean in the opposite directions, so if the pendulum actuator exerts a torque on the pendulum to bring it closer to γ = 00 , the reaction torque, acting on the wheel-chassis body in the opposite direction, brings the wheel-chassis body closer to α = 00 . In type IV configuration, the pendulum and wheel-chassis body lean in the same directions, so if the pendulum motor exerts a torque on the pendulum to bring it closer to γ = 00 , the reaction torque, acting on the wheel-chassis body in the opposite direction, brings the wheel-chassis body farther from α = 00 and therefore, the whole system becomes unbalanced. It is not immediately clear which one of the other two lateral unbalanced configurations is more difficult, but through numerical simulations, it is found that type I configuration is more difficult than type II configuration. The four unbalanced configuration types not have clearly defined boundaries and their relative difficulties depend heavily on the angle magnitudes. In the general case of 3-D posture balancing, the number of possible configurations is the permutation of the longitudinal configurations and lateral configurations. There are in total two types of balanced configurations and ten types of unbalanced configurations. The balanced configurations are shown below. 192 • Type 1: β = 00 , α = 00 , γ = 00 • Type 2: β = 00 , α = 00 , γ = f (α) The ten types of unbalanced configurations are named and expressed as followed. • Type A: β = 00 , α = 00 , γ = 00 • Type B: β = 00 , α = 00 , γ = f (α) • Type C: β = 00 , α = 00 , γ = 00 • Type D: β = 00 , α = 00 , γ = 00 • Type E: β = 00 , α = 00 , γ = 00 , sgn(α) = sgn(γ) • Type F: β = 00 , α = 00 , γ = 00 , sgn(α) = sgn(γ) • Type G: β = 00 , α = 00 , γ = 00 • Type H: β = 00 , α = 00 , γ = 00 • Type I: β = 00 , α = 00 , γ = 00 , sgn(α) = sgn(γ) • Type J: β = 00 , α = 00 , γ = 00 , sgn(α) = sgn(γ) Similar to the previous case, all of these configurations not have clearly defined boundaries and angle magnitudes are significant factors influencing the relative difficulties of these configurations. Rigorous mathematical analysis is needed for deeper understanding of this varying control difficulty, but it is beyond the scope of the current research. 193 6.1.3 Proposed Control Scheme In order to achieve robust posture balancing, the control structure, shown in Fig. 6.4 and called Longitudinal-Lateral Control Structure (LLCS), is proposed. LLCS is based on the idea of separate regulations of the longitudinal mode and lateral mode. The longitudinal controller is solely responsible for the ALP Cycle’s longitudinal stability. Similarly, the lateral controller is only responsible for the ALP Cycle’s lateral stability. The two controllers work as totally separate entities, but the longitudinal control system must have faster convergence than the lateral control system. This is because as β increases, the effectiveness of the pendulum torque in influencing α and γ is reduced. Figure 6.4: Block Diagram of Longitudinal-Lateral Control Structure (LLCS) for Posture Balancing of ALP Cycle The longitudinal controller and lateral controller in LLCS can be designed with any 194 suitable control-design method. In this research, H∞ control method is proposed for designing both controllers. The longitudinal dynamics and the lateral dynamics of ALP Cycle are highly coupled. This dynamic coupling can be regarded as internal disturbance. Therefore, the longitudinal controller and the lateral controller must be robust enough to withstand this disturbance. There are a number of available robust MIMO control design methods such as H∞ control, H2 control and sliding-mode control. Sliding-mode control has good robustness property, but it has the drawback of having chattering phenomenon. The use of robust H∞ control method is proposed because it is considered that minimising the effect of maximum disturbance instead of its average value as in H2 control is more relevant to this case. There are a number of available H∞ control methods in the literature [49], [85], [12]. The method by Liu, Chen and Lin [49] is adopted for the designs of both longitudinal controller and lateral controller. In the designs of both controllers, the control-design parameters are tuned to optimise the system robustness. Longitudinal controller is designed based on the linearised state-space model in Eq. 2.93. By using the parameters in Appendix C and assuming zero δ˙ and Wlong , Eq. 2.93 is written as followed. x˙ long = Along xlong + Blong τw where: • xlong = β β˙ ω˙ T 195 (6.1) Figure 6.19: Wheel Torque and Pendulum Torque Based on Fig. 6.17, it can be observed that the performance of the forward-speedreference following is identical for both control schemes. This is fully understood as thrust controller and longitudinal controller are exactly the same. In the turning-speedreference following, both control schemes show similar performances in terms of convergence speed and overshoot, but LLTCS has apparent oscillations with pretty large amplitudes. Therefore, in term of performance, TSCS is better than LLTCS. In Fig. 6.18, the ALP Cycle’s posture during the path following can be observed. For both control schemes, the longitudinal lean angles β and lateral lean angles α remained within 20 and 50 from the upright equilibrium posture. However, as the lateral-statics constant Kls is much greater than 1, the pendulum angles γ in both cases, reached 220 up to 150 , which, although can be considered as small, may be an issue in practical implementation. Fig. 6.19, the input control torques to the wheel and pendulum can be observed. For both control schemes, the wheel torques τw and the pendulum torques τp are reasonable and not excessive. Therefore, both schemes are implementable. Although the designed control schemes should have been implemented on the real prototype, it is unfortunate that experiments could not be performed due to limitations on time and resources. 6.3 Experiments Three types of experiments (1) 1-D longitudinal posture balancing, (2) 1-D lateral posture balancing and (3) 2-D posture balancing have been conducted with the constructed prototype to demonstrate the stabilising concept of ALP Cycle. During each experiment, ALP Cycle is released from the upright equilibrium posture and the implemented control scheme has to maintain this upright posture as long as possible. In the experiments, the prototype is placed on a carpet with approximate size of 40 cm × 60 cm. 6.3.1 1-D Longitudinal Posture Balancing In the first experiment, a training wheel is attached to the prototype’s left side in order to ensure lateral stability. Only the wheel actuator is needed and, hence, the pendulum actuator is turned off. The prototype is released from the upright equilibrium posture and it has to balance itself around the upright posture. The experimental result is 221 shown in Figs. 6.20 and 6.21 while Fig. 6.22 shows the snapshot of ALP Cycle during the posture-balancing process. Figure 6.20: Longitudinal Lean Angle and Wheel Angle during 1-D Longitudinal Posture Balancing 222 Figure 6.21: Torque Generated by Wheel Actuator during 1-D Longitudinal Posture Balancing Figure 6.22: Snapshot of ALP Cycle during 1-D Longitudinal Posture Balancing 223 As shown in the figures, the constructed prototype is able to balance itself in the longitudinal direction. Although the result is only shown for approximately 105 s, in fact, ALP Cycle remains balanced in prolonged period of time. The longitudinal balance can consistently be achieved upon experiment repetition. 6.3.2 1-D Lateral Posture Balancing In the second experiment, longitudinal stability is maintained by turning off the wheel actuator and sandwiching the wheel between two styrofoam pieces in order to prevent it from moving. ALP Cycle is released from the upright equilibrium posture and, then, it has to maintain its lateral balance as long as possible. The experimental result is shown in Figs. 6.23 and 6.24 while Fig. 6.25 shows the snapshot of ALP Cycle during the posture-balancing process. 224 Figure 6.23: Lateral Lean Angle and Pendulum Angle during 1-D Lateral Posture Balancing 225 Figure 6.24: Torque Generated by Pendulum Actuator during 1-D Lateral Posture Balancing Figure 6.25: Snapshot of ALP Cycle during 1-D Lateral Posture Balancing 226 As seen from the plots, lateral posture balancing shows jerkier motion than longitudinal posture balancing. In fact, the repeatability of this experiment is not good. This may be contributed to hardware imperfection in addition to the inherent control difficulty of ALP Cycle’s lateral mode. 6.3.3 2-D Posture Balancing In the last experiment, ALP Cycle has to balance itself in both longitudinal direction and lateral direction. The experimental result is shown in Figs. 6.26 and 6.27 while Fig. 6.28 shows the snapshot of ALP Cycle during the posture-balancing process. Figure 6.26: Longitudinal Lean Angle, Lateral Lean Angle and Pendulum Angle during 2-D Posture Balancing 227 Figure 6.27: Torque Generated by Wheel Actuator and Pendulum Actuator during 2-D Posture Balancing Figure 6.28: Snapshot of ALP Cycle during 2-D Posture Balancing 228 As seen from the figures, the prototype is able to balance itself in two dimensions for a short time. However, it is apparent that the robot motion is very jerky. Its repeatability is not good either. Therefore, although this result shows that, at least, 2-D posture balancing can successfully be achieved by the use of lateral pendulum, it also suggests that lateral pendulum poses much greater difficulty than other mechanisms and, hence, it is not suitable for practical applications. 6.4 Conclusions In this chapter, ALP Cycle’s control tasks of posture balancing and manoeuvring have been defined, investigated and solved. In posture balancing, various postures of ALP Cycle have been classified and qualitative explanation has been given to explain the varying control difficulties of the postures. A control scheme based on separate-regulations concept has been proposed and H∞ control design method has been adopted to design both the longitudinal controller and the lateral controller. The control performances have been verified by numerical simulations. For manoeuvring, by making use of the ALP Cycle’s characteristics from the previous chapter, two control schemes have been proposed. One is based on the separateregulations concept and integration of lateral model and turning model. Another one is based on a cascade structure where the output of the turning controller becomes the reference input for the lateral controller. LQI control design method and gain-scheduling technique have been combined to design the controllers for both control schemes. The 229 control performances have been verified by numerical simulations for a simple manoeuvring case. Experiments have been performed with the constructed prototype. The results have shown that the ALP Cycle can successfully balance itself at its upright posture. This result is significant in that, to the best of our knowledge, we are the first group which have experimentally shown successful posture balancing of lateral-pendulum unicycle robot. 230 Chapter Conclusions This research is aimed at the comprehensive study of lateral-pendulum unicycle robot, which represents a rarely explored niche of single-wheeled mobile robots. A working prototype has been successfully developed and it provides a solid platform for experimental investigation of this robot. To provide a framework for theoretical study, a complete dynamic model of the robot has been derived and validated. Control tasks of posture balancing and manoeuvring have been identified, investigated and solved by the proposed control schemes. Experiments with the constructed prototype have been conducted through which the working principle and concept of this robot, together with its practical challenges, have been demonstrated. In the rest of this chapter, the contributions of this research are presented and useful recommendations for further research are given. 7.1 Summary of Contributions This research contributes to the study of lateral-pendulum unicycle robot, a specific type of single-wheeled mobile robot, in the following six aspects. 231 • Development of Complete Dynamic Model: The approaches used by other researchers for modelling the dynamics of lateral-pendulum unicycle robot assume that the robot’s longitudinal dynamics and lateral dynamics are completely separate. Therefore, the model of turning dynamics must be derived separately and the dynamic couplings resulting from the interactions among the longitudinal dynamics, lateral dynamics and turning dynamics are not represented. Consequently, numerical simulations with these simplified models produce inaccurate and deceptively good results which, in turn, undermine the real control difficulty of the robot. In addition, with such models, it is not possible to study the behaviours of dynamic couplings and investigate their effects on system performance. In this research, the complete dynamic model has been derived and it serves as an important framework in which more accurate simulation study and theoretical analyses can be performed. • Analyses of Posture Balancing and Manoeuvring: There is an apparent tendency to approach the problems of posture balancing and manoeuvring too heavily from control point of view and less from dynamics perspective. With such approach, there is an implicit assumption that these problems are solvable and it is not fully known under what circumstances balancing the robot’s posture and controlling its manoeuvre become impossible or extremely difficult. In this research, four formulae have been derived from the complete dynamic model which help in the design of control systems for posture balancing and manoeuvring. Besides, 232 through analyses of these formulae, the robot’s limitations, i.e. the maximum lateral lean angle in set-point control and the minimum radius of curvature in manoeuvring, and how its dynamics can be improved by structural optimisation become known. • Design and Verification of Control Scheme for Posture Balancing: A control scheme has been proposed for posture balancing which is based on the concept of separate regulations and H∞ control design method. The effectiveness of this scheme has been verified by extensive numerical simulations. Compared to other simulation results of the same robot, the simulation study in this research has been performed using the complete dynamic model and, as an outcome, the results are more reliable. • Design and Verification of Control Schemes for Manoeuvring: Two control schemes have been proposed for manoeuvring. One comprises two controllers and the other one comprises three controllers, two of which are in cascade form. LQI control design method is combined with gain-scheduling technique to design the controllers in both schemes. The performances of these schemes have been verified and compared through numerical simulations. • Construction of a Successfully Working Prototype: A working prototype of lateral-pendulum unicycle robot has been developed from total scratch. During the design and construction processes, great attention has been paid to the robot’s 233 structural design, the design and selection of actuation system, sensing system and power system, and the developments of the computer system’s hardware and software. This prototype serves as a useful platform for further experimental research in this topic, especially in National University of Singapore. • Practical Verification: Although simulation study of this robot has been reported in the literature, no success in implementation has ever been reported. In this research, with our fully working prototype, we have demonstrated the robot’s posture-balancing capability in practice to a certain extent. This can serve as a stepping stone towards better and more consistent performance through improvement in the prototype and control scheme. 7.2 Recommendations for Future Research The research work presented in this thesis forms a solid foundation from which the following extensions are recommended for further research. • Mechanical Improvement of Prototype: The current prototype applies directdrive concept with the drawback that counterweights are necessary in order to symmetrically position the robot’s centre of gravity. Mechanical power transmissions, in particular the combination of timing belts and pulleys or chains and sprockets, can be experimented in order to enable the electric motors to be placed at the centre of the robot’s chassis and, hence, the counterweights can be eliminated and the robot’s total mass can be reduced. 234 • Addition of Wireless-Communication Capability: In the current prototype, communication between the robot’s computer system and the personal computer used for programming and data acquisition is carried out through wired communication. In this way, the robot must always stay nearby the personal computer during posture-balancing experiment while it is totally not convenient to perform manoeuvring and path-following experiments. Therefore, wireless module can be integrated into the robot system, so that the robot’s motion is not limited by the length of communication wire and execution of manoeuvring and path-following experiments can be more conveniently performed. • Experimental Verifications of Manoeuvring Control Schemes: The manoeuvring control schemes presented in the current research have been verified only through numerical simulations. Mathematically intensive stability analyses or experimental verifications need to be done for these control schemes to provide stronger claims of their effectiveness and expose their drawbacks. However, the scale of the robot dynamics is very large, so it is questionable if it is even possible to conduct complete mathematical stability analyses. Therefore, experiments are seen as better and viable options for verifying the designed control schemes. 235 [...]... Cycle may be required to approach a certain location by manoeuvring to follow a particular path During the course of manoeuvre, it is important to always maintain the longitudinal stability and lateral stability, so that ALP Cycle will not fall down halfway Manoeuvring is a challenging control task because ALP Cycle is inherently unstable, underactuated, non-minimum phase and nonholonomic 204 6.2.2 Proposed... fact is attributed to two reasons Firstly, the initial value of β is sufficiently small in that it does not give visible effect on the lateral balancing Secondly, the convergence of β is faster than those of α and γ If β has greater initial value or converges very slowly, the stability 2 03 and performance of the system are greatly a ected 6.2 6.2.1 Manoeuvring Description of Control Task As a mobile robot, ... posture, so the handling 197 imperfection must not cause the angles to be very far from zero and 30 is considered to be a reasonable magnitude Unbalanced configuration of type B is not included in the numerical simulations because the lateral-statics constant Kls is -8.12, so for lateral lean angle α = 30 , the pendulum angle γ = −24 .36 0 , which is not possible to happen in reality At time t = 2.5 s,... Each fixed steering gain matrix Ks was computed with the following formula n −1 T Ks = Rs Bs Ps The matrix Ps is obtained by solving the following algebraic Riccati equation −1 T AT Ps + Ps As + Qs − Ps Bs Rs Bs Ps = 0 s After all of the computations, the following eight fixed steering gain matrices are obtained as followed 210 1 • Ks (1.5) = 32 1 .36 − 43. 33 −71.96 − 13. 80 −28. 93 −2.24 2 • Ks (3. 0) = 32 8. 23. .. neglected as well The wheel speed ω, on the other hand, ˙ is moderate and, therefore, ω is significant After the simplification, it is apparent that ˙ Eq 6.7 is a linear parameter-varying system with the varying parameter ω ˙ For designing steering controller based on linear parameter-varying model, gain-scheduling technique is adapted with the scheduling variable ω The steering model is first frozen ˙ at several... design a control system which can enable ALP Cycle’s forward speed and turning speed to follow their respective references with sufficiently small error while maintaining the posture’s longitudinal stability and lateral stability at all times The significance of this control task is that after the upright equilibrium posture has been achieved upon the system’s start-up stage, ALP Cycle may be required to approach... Torques in Posture Balancing from Unbalanced Posture of Type D 199 Figure 6.8: Angles and Torques in Posture Balancing from Unbalanced Posture of Type E Figure 6.9: Angles and Torques in Posture Balancing from Unbalanced Posture of Type F 200 Figure 6.10: Angles and Torques in Posture Balancing from Unbalanced Posture of Type G Figure 6.11: Angles and Torques in Posture Balancing from Unbalanced Posture... references for lateral lean angle αref and pendulum angle γref are computed using the following formulae, which are essentially Eqs 2. 83 and 2.88 in Section 2.4 of Chapter 211 2 αref = ˙ δref Kturn Vref γref = Kls αref 6.2.2.2 (6.12) (6. 13) Longitudinal-Lateral-Turning Control Structure with Gain-Scheduled Linear Quadratic Integral Control Laws Similar to TSCS, LLTCS is also a state-feedback control structure,... peaks of α are still relatively small, but the peaks of γ are very large in magnitude In both cases, linear controllers are, in principle, no longer reliable because the system state has strayed too far from its operating point These results demonstrate that, although H∞ controllers are still successful due to their robustness, nonlinear control methods may provide better alternatives, but nonlinear... the current research In addition, these results have shown that, indeed, all four configurations have varying control difficulties despite having the same magnitudes of initial values In Figs 6.10 - 6. 13, it can be observed that despite the non-zero β, the proposed control system can still manage to stabilise the robot s posture In all four cases, the trajectories of α and γ are similar to those in Figs . always maintain the longitudinal stability and lateral stability, so that ALP Cycle will not fall down halfway. Manoeuvring is a challenging control task because ALP Cycle is inherently unstable,. illustrated in Fig. 6 .3. Figure 6.2: (a) Upright Lateral Balanced Configuration and (b) Non-Upright Lateral Balanced Configuration 190 Figure 6 .3: Lateral Unbalanced Configurations of (a) Type I, (b) Type. Configuration and (b) Longitudinal Unbalanced Configuration In this case, the pendulum and the chassis can be considered as a single body, named pendulum-chassis body. Longitudinal unbalanced configuration