Chapter Model and 3D Simulator Development of Gyrobot 4.1 Introduction This chapter presents the development of Gyrobot dynamic model and a 3D simulator in ADAMS with integration of Matlab. The dynamics of the system is first explored through analyzing system kinetic and potential energy. A nonlinear model is derived using constrained Lagrangian approach. The order of obtained dynamic equations is reduced by eliminating the Lagrange multipliers in the dynamic model. Zero generalized force is used to further simplify the model. The nonlinear model is linearized at Gyrobotnospace’s vertical position to develop a linear controller, which will be described in Chapter 5. A 3-Dimensional simulation platform is developed as an aid for designing complex mechatronics system and controller development. ADAMS is used for simulation with animation of the dynamic behavior of the mechanism whose parts are drawn using a 3-D drawing software, e.g., SolidWorks. The overall simulation platform integrates 93 the 3-D simulator in ADAMS with control design software MATLAB. This integration allows the designer to adopt a control-centric approach for designing complex mechanical structure to be used in a mechatronics system. The simulating environment can easily be extended to any complex mechanical system simply by altering SolidWorks drawings. 4.2 Dynamic Model Development and Verification 4.2.1 Coordinate Frame Though the Gyrobot wheel is flying saucer shaped, for the sake of simplicity in formulation of equations, we assume it to be a rigid, homogenous disk that rolls without slipping over a flat horizontal surface. Flywheel, the stabilizing component of the Gyrobot, hangs from the central shaft of the axle of the wheel. There is a vertical offset of the flywheel from the center of the Gyrobot wheel. We designed the internal mechanism, the outer gimbals and the inner gimbals, in a way so that the center of flywheel remains below the center of the wheel. As a result of this design, there is no horizontal offset between the center of Gyrobot and the center of flywheel. According to this description of components, we define the following coordinate frames. 1. The inertial frame Σo {X0 , Y0 , Z0 }: The X-Y plane of this frame is anchored to the flat surface on which the GYROBOT rolls. 2. The wheel coordinate frame ΣW {XW , YW , ZW }: Origin of this frame is located at the center of the main wheel. Its Z-axis represents the axis of rotation of 94 Figure 4.1: Definition of Coordinate Frame and System Variables the wheel. The X-axis passes through the wheel’s point of contact on the flat surface, and the Y-axis in the forward direction of the wheel. 3. The flywheel coordinate frame ΣF {XF , YF , ZF }: Origin of this frame is located at the center of the flywheel. Its Z-axis represents the axis of rotation of the flywheel. The yF -axis is always parallel to the yW -axis. These coordinate frames are illustrated in Figure 4.1. This figure also illustrates different variables used in deriving the dynamic model. Definitions for these variables are listed in Table 4.1. 4.2.2 Kinematic Constraints In this section, we explore the kinematic constraints of the Gyrobotnospace. Let(i, j, k) and (n, m, n) be the unit vectors of the coordinate systems 95 o and W, respectively. Table 4.1: Gyrobot Variable Definition α β γ αa βa γa MW Ml MF m R r IxW , IyW , IzW IxF , IyF , IzF µy , µ p u1 u2 Precession Angle of the wheel measured about the vertical axis(Z0 ) Precession rate (dα/dt) is shown in the figure Lean Angle of the Wheel Leaning rate (dβ/dt) is shown in the figure Spin Angle of the Wheel measured about zW -axis Spinning rate (dβ/dt) is shown in the figure Precession Angle of the flywheel Tilt Angle of the flywheel Spin Angle of the flywheel measured about zF -axis Spinning rate (dγa /dt) is shown in the figure Mass of the wheel Mass of the inner mechanism Mass of the flywheel Total mass of the Gyrobot Radius of the Gyrobot wheel Radius of the flywheel Moment of inertia of the wheel about axes Moment of inertia of the flywheel about axes Friction coefficient in yaw and pitch direction Torque input to drive motor Torque input to tilt motor 96 The transformation between these two coordinate frames is given by,      i − sin α cos β − cos α − sin αsinβ l       = j cos α cos β − sin α cos αsinβ m   sin β cos β n k (4.1) Let vA and ωB denote the velocity of the center of the wheel and its angular velocity with respect to the coordinate frame O. Then, ωB = −αsinβ ˙ l + β˙ m + (γ˙ + α˙ cos β)n (4.2) As the wheel rolls without slipping on a flat surface, the velocity of the contact point must be at any instant, i.e., vC = where vC is the velocity of the contact point. The velocity of the rolling wheel is vA = ωB × rAC + vC (4.3) Here rAC = −Rl is the vector from the contact point to the center of wheel. Combining Equations 4.1, 4.2 and 4.3, we get, vA = X˙ i + Y˙ j + Z˙ k (4.4) where, X˙ = R(γ˙ cos α + α˙ cos α cos β − β˙ sin α sin β) Y˙ = R(γ˙ sin α + α˙ sin α cos β + β˙ cos α sin β) Z˙ = Rβ˙ cos β 97 (4.5) The third one in the arrays of Equation 4.5 is integrable, i.e., Z = R sin β . Therefore the motion of the Gyrobot can be represented by six independent variables X, Y, α, β, γ, and βa . 4.2.3 Equations of Motion We derive the equations of motion by calculating the Lagrangian, where T and P are the kinetic energy and potential energy of the system, respectively. To calculate the energies, the system is divided into two parts: (1) the wheel, and (2) the spinning flywheel. Energy of the Wheel The kinetic energy of the wheel is given by, 1 TW = mW [X˙ + Y˙ + Z˙ ] + [IxW ωx2 + IyW ωy2 + IzW ωz2 ] 2 (4.6) Substituting Equations 4.2 - 4.5 into Equation 4.6, following equation is obtained. TW = mW [X˙ + Y˙ + (Rβ˙ cos β) ] (4.7) ˙ 2] + [IxW (α˙ sin β)2 + IyW β˙ + IzW (α˙ cos β + γ) Potential energy of the wheel is, PW = mW gR sin β Here, g is the acceleration due to gravity. 98 (4.8) Energy of the Internal Mechanism and Flywheel Contrary to the design used by researchers of the Carnegie Melon University, the center of the mass of the flywheel coincides with that of the inner mechanism in our design. In CMU design, the flywheel is attached to the end of the second link (length l2 ) of a 2-link arm. Since this length (l2 ) is very small compared to the length (l1 ) of the first link, i.e., the distance between wheel’s center and the center of the inner mechanism, it was assumed in their derivation that l2 ∼ = 0. Let xF , yF , zF be the coordinate of the center of mass of the internal mechanism and flywheel with respect to the coordinate frame O. Then,     X − l1 sin β cos β xF  yF  = Y + l1 cos β cos β  zF Z − l1 sin β (4.9) The translational kinetic energy of the flywheel and the inner mechanism is, TFt = (mI + mF )[x˙ 2F + y˙ F2 + z˙F2 ] (4.10) We can obtain an expression for this energy by differentiating Equation 4.9 and substituting it in Equation 4.10. In our design, the internal mechanism does not swing. This is one of the differences between our design and the design followed by CMU researchers. In CMU’s gyrover, the internal mechanism is swung forward or backward by the drive motor in an attempt to change the center of gravity of the structure. The wheel then rotates to bring the CG underneath the center of the wheel causing forward (backward) mo99 tion in the wheel. In our design, on the other hand, the drive motor is fixed to the internal structure but the axle is passed through a pair of ball-bearings attached to the platform. The drive motor directly drives the axle. While deriving the kinetic energy of the internal mechanism and flywheel, the CMU researchers assumed very slow swing of the pendulum-structured internal mechanism and, therefore, insignificant influence of internal mechanism on the rotational kinetic energy. In our design, the internal structure does not move at all and therefore has no influence on the rotational kinetic energy of the flywheel. The angular velocity of the flywheel with respect to the coordinate frame O is ωF = ωF x i + ωF y j + ωF z k (4.11) where, ωF x = α˙ cos (β + βa ) + γ˙ cos β ωF y = β˙ + β˙a (4.12) ωF z = −α˙ sin (β − βa ) + γ˙ sin βa + γ˙a The kinetic energy of the flywheel is TFr = [(ωF x )2 IxF + (ωF y )2 IyF + (ωF z )2 IzF ] (4.13) The flywheel is assumed to be a uniform disk of radius r, and the moments of inertia are IxF = IyF = 41 mF r2 and IzF = 12 mF r2 . 100 The potential energy of the internal mechanism and flywheel is PF = (mI + mF )(R sin β − l1 sin β) (4.14) Lagrangian of the System Taking different energies into consideration, we now define the Lagrangian as, L = (TW + TFt + TFr ) − (PW + PF ) (4.15) Substituting the expressions for different energies derived in the previous two sub-sections, we can determine L. For a system with m generalized coordinates, qj = (q1 , q2 , ., qm ) and n degrees of freedom, there will be (m − n) constraint conditions that must explicitly be satisfied. For such a system, the set of Lagrangian equations is given by, d dt ∂L ∂ q˙j − ∂L ∂qj m−n λs A˙ sj , j ∈ 1, 2, ., m = (4.16) s=1 Equation 4.16 is evaluated for the energy equations derived in previous two subsections to obtain the dynamic equation of the entire system. M (q)¨ q + N (q, q) ˙ = AT (q)λ + B(q)u (4.17) Here M (q) ∈ R6×6 and N ∈ R6×1 are the inertia matrix and nonlinear terms, respectively and λ = [λ1 , λ2 ]T is the Lagrange multiplier. 101 4.2.4 Dynamic Model of GYROBOT Taking derivatives, both in time and space, of the Lagrangian of Equation 4.15, a set of equations that includes Asj and λs is obtained. The coefficients Asj can be evaluated by comparing these equations with the constraint Equation 4.5. The Lagrange multiplier can be eliminated by partitioning the matrix A in accordance to the method suggested in [13]. As the wheel is disk-like shape, it’s practical to assume the inertias of the wheel are: IxW = IyW = 12 mW R2 and IzW = mW R2 . We also assume that l1 = 0, i.e., the center of the flywheel is collocated with the center of the wheel. The flywheel spins at a constant rate, i.e., γ¨a = . The normal form of the dynamics of Gyrobot becomes, M (q)¨ q = F (q, q) ˙ + Bu (4.18) X˙ = R(γ˙ cos α + α˙ cos α cos β − β˙ sin α sin β) (4.19) = R(γ˙ sin α + α˙ sin α cos β + β˙ cos α sin β) (4.20) Y˙ Z˙ = Rβ˙ cos β (4.21) Here, q = [α β γ βa ]T excludes the Cartesian coordinates of the center through a system reduction process. The procedure of such reduction is explained in subsection 4.2.5 and 4.2.6. Motion of the center of the wheel is expressed by Equations 4.19 to 102 If xv and ωv are set as the reference velocities for the Gyrobot which is controlled to follow these references, then both will trace the same path provided the initial poses of the two are same. However, such assumption on the initial condition is not realistic as the initial configuration of Gyrobot may not at all coincide with the trajectory it is expected to follow. We use the error between configurations of the real robot (Gyrobot) and the virtual robot to generate a feedback component of the reference such that the error converges asymptotically to zero. Convergence of actual robot’s path to that of the virtual robot can be guaranteed by ensuring stability of the feedback loop. Assuming that vv and ωv required to keep the virtual robot on the desired path are known, we use them as the feedforward references for tangential velocity vR and ωR angular velocity R of Gyrobot. We also take into consideration the kinematic equations of the Gyrobot, x˙ R = vR cos θR (5.12) y˙ R = vR sin θR (5.13) θ˙R = ωR (5.14) The difficulty in navigation of robot under these constraints lies in the fact that there are three degrees of freedom but only two control variables vR and ωR . Moreover, other constraints specific to the Gyrobot must be taken care of. For example, not only we control the configuration [xR yR θR ]T of robot but also we must keep its lean 151 angle within stable range. Let us consider the error vector e = [e1 e2 e3 ]T that defines the difference between the configurations of the virtual robot and the Gyrobot as illustrated in Figure 5.11. e1 = (xv − xR ) cos θR + (yv − yR ) sin θR (5.15) e2 = −(xv − xR ) sin θR + (yv − yR ) cos θR (5.16) e3 = θv − θR (5.17) The errors e1 and e2 are shown in Figure 5.11. The reference velocities vR and ωR are to be generated such that the errors in Equations 5.15 to 5.17 approach zero asymptotically. The reference velocities are generated by combining feedback components and feedforward components (vR , ωR ). Before we derive the expressions for feedback components, let’s examine the error dynamics by differentiating Equations 5.15 to 5.17. It can be easily shown that, e˙ = vv cos e3 − vR + ωR e2 (5.18) e˙ = vv sin e3 − ωR e1 (5.19) e˙ = ωv − ωR (5.20) The velocities of the virtual robot, vv and ωv , are pre-computed such that the virtual robot follow the desired path. vR and ωR are calculated as functions of error such that the error dynamics (Equation 5.18 to 5.20) is stabilized. If the velocity vv is decomposed along the XR -axis and YR -axis, then the compo152 nent vv cose3 acts to increase the error e1 while the component vv sine3 acts to increase e2 . In order to reduce e1 , we choose the tangential velocity vR as, vR = vv cos e3 + h1 e1 (5.21) The angular velocity ωR affects the orientation error e3 directly. Gyrobotmust be steered towards the direction of the virtual robot in order to reduce the error e2 . While affecting the error e2 by changing angular velocity, the direction of motion of the virtual robot must be taken into consideration. Accordingly, the angular velocity of the robot is chosen as, ωR = ωv + sgn(vv )h2 e2 + h3 e3 (5.22) Figure 5.12: Schematic of the Reference Velocity Generator The schematic of Figure 5.12 illustrates the operation of the reference velocity 153 generator. The error dynamics is converted of Equation 5.18 to 5.20 to e˙ = −h1 e1 + ωv e2 + sgn(vv )h2 e22 + h3 e2 e3 (5.23) e˙ = vv sin e3 − ωv e1 − sgn(vv )h2 e1 e2 − h3 e1 e3 (5.24) e˙ = −sgn(vv )h2 e2 − h3 e3 (5.25) The choice of feedback gain h = [h1 h2 h3 ] has direct consequence on the stability of the error dynamics (Equations 5.23 to 5.25). A method for finding time-varying gains is proposed by Klancar et al [72]. In the following paragraphs, we derive, using a Lyapunov function, constraints on the gains such that the stability is guaranteed. 5.3.2 Stability Analysis Consider a candidate Lyapunov function, 1 V (t, e1 , e2 , e3 ) = e21 + e22 + e23 2 (5.26) This function V is positive-definite, decresent, and radially unbounded. Taking the time derivative of the Lyapunov function, V˙ (t, e1 , e2 , e3 ) = e1 e˙ + e2 e˙ + e2 e˙ (5.27) Substituting Equations 5.23 to 5.25 to Equation 5.27, V˙ = −h1 e21 + ωv e1 e2 + sgn(vv )h2 e1 e22 + h3 e1 e2 e3 + vv e2 sin e3 −ωv e1 e2 − sgn(vv )h2 e1 e22 − h3 e1 e2 e3 − sgn(vv )h2 e2 e3 − h3 e23 154 (5.28) Reorganize Equation 5.28, we have V˙ = −h1 e21 + vv e2 sin e3 − sgn(vv )h2 e2 e3 − h3 e23 (5.29) Assuming positive gains, the terms (−h1 e21 ) and (−h3 e23 ) are always negative. So the time derivative of Lyapunov function is guaranteed to be negative if vv e2 sin e3 − sgn(vv )h2 e2 e3 ≤ (5.30) h2 e2 e3 ≥ |vv |e2 sin e3 , if vv > (5.31) h2 e2 e3 ≤ |vv |e2 sin e3 , if vv < (5.32) This implies that which can be transformed to, h2 ≥ |vv | sin e3 , if vv > e3 (5.33) h2 ≤ |vv | sin e3 , if vv < e3 (5.34) From the above result, it can be concluded that the stability of the error dynamics is guaranteed if we choose h1 ≥ 0, h2 = |vv | sin e3 and h3 ≥ 0. 5.3.3 Dynamic Controller The virtual robot approach is used to generate the reference velocities vR and ωR of the Gyrobot. The linear velocity vR is controlled by changing the rolling speed of the Gyrobot whereas the precession rate ωR is controlled by manipulating the lean 155 angle of the Gyrobot. However, the lean angle β of the Gyrobot can not be controlled directly. Instead, we control the tilt angle βa of the flywheel. Details of the dynamic controller is illustrated in section 5.2.3 and 5.2.4. The complete diagram of control scheme using virtual robot approach is shown in Figure 5.13. Figure 5.13: Block Diagram of the Virtual Robot Controller 5.3.4 Simulation Result Straight Line Tracking For this simulation, the reference path is described by the equation of a straight line y = mx + c. In order to have the virtual robot follow such straight-line path, its angular speed ωv is set to 0. The linear speed of the virtual robot vv is set to the nominal speed of the Gyrobot moving in its vertical posture. Since the nominal speed of the drive motor is 240 RPM, the linear speed of the virtual robot is set to 2π(240/60)R, which is equal to 4.27m/s. Assuming the initial position of the robot at (0, 0) of the X-Y plane, we set the initial position and orientation of the virtual robot at (0, c) and tan−1 (m), respectively. Simulation result is shown in Figures 5.14 156 and 5.15. 30 25 Gyrobot path Reference Path Y (m) 20 15 10 −5 10 15 X (m) 20 25 30 35 Figure 5.14: Straight Line Tracking Trajectory by Virtual Robot Approach Circle Tracking A circular path of radius with center at (X0 ,Y0 ) is described by the equation ((x − X0 )2 + (y − Y0 )2 = ρ2 ). We set the tangential speed vv of the virtual robot to the nominal speed of the Gyrobot which is 4.27 m/s. Then the angular speed v should be vv /ρ, i.e., 4.27/ρ rad/s. Setting the initial configuration of the virtual robot is an important issue. It is observed that the path of the Gyrobotconverges to that of the virtual robot starting from any arbitrary initial state. However, initial condition affects the time it takes to converge. Methods for selecting an optimal initial configuration will be published in a sequel of this work. Result for following a 157 50 60 Wheel Lean Angle Flywheel Tilt Angle 40 40 30 20 Angel(Deg) Angel(Deg) 20 10 −10 −20 −20 −40 −30 −40 −60 10 Time (t) 10 Time (t) Figure 5.15: Wheel Lean Angle (deg) and Flywheel Tilt Angle (deg) circular path is shown in Figures 5.16 and 5.17. Sinusoidal Path Tracking For the virtual robot following a sinusoidal path, its angular velocity should be changed as function of time. The following set of equations can be used to generate a sinusoidal path, ϕπ cos ϕt x˙ v = vv cos θv , y˙ v = vv sin θv , θ˙v = (5.35) The parameter ϕ can be adjusted to change the length of one full cycle of the sinusoidal path. Since the angular velocity of the Gyrobot is controlled by controlling its lean angle, there exists an upper limit to the angular velocity that can be achieved. Tilting the Gyrobot beyond the limit causes it to fall on its side. Let the maximum allowable angular velocity of the Gyrobot be ωR,max ,max, which depends on the physical parameters of the robot as well as on the lean angle. Then, according to Equation 5.35, the maximum angular velocity is ϕ = 2ωR,max /π . Simulation results are shown 158 Gyrobot path Reference Path 35 30 Y (m) 25 20 15 10 10 20 X (m) 30 40 Figure 5.16: Circle Tracking Trajectory by Virtual Robot Approach 60 80 Wheel Lean Angle Flywheel Tilt Angle 60 40 40 Angel(Deg) Angel(Deg) 20 20 −20 −20 −40 −60 −40 10 15 Time (t) 20 25 −60 30 10 15 Time (t) 20 25 30 Figure 5.17: Wheel Lean Angle (deg) and Flywheel Tilt Angle (deg) by Virtual Robot Approach 159 in Figure 5.18 and 5.19. Gyrobot path Reference Path 70 60 50 Y (m) 40 30 20 10 −10 −20 20 40 60 80 100 120 X (m) Figure 5.18: Sinusoidal Tracking Trajectory by Virtual Robot Approach 5.4 Conclusion In this chapter, two tracking control algorithms have been proposed to drive the Gyrobotto follow various type of trajectories, namely straight line path, circular path and sinusoidal path. In the first approach, three key equations that dominate the motion characteristics are identified. These equations are used to generate the reference signals for the designed linear controller which is applied to the single-wheeled robot. Simulation results show it works well. A feedforward controller is designed by utilizing the kinematics of virtual robot, which is virtually moving along the desired trajectory. The controller generates the 160 60 80 Wheel Lean Angle Flywheel Tilt Angle 60 40 40 20 Angel(Deg) Angel(Deg) 20 −20 −20 −40 −40 −60 −60 10 20 30 40 −80 50 Time (t) 10 20 30 40 50 Time (t) Figure 5.19: Wheel Lean Angle (deg) and Flywheel Tilt Angle (deg) by Virtual Robot Approach reference signals, i,e, the lean angle and wheel speed for the dynamic controller. The effectiveness of this controller design is verified by the simulation of three types of trajectories, namely a straight line path, a circular path and a sinusoidal path. Since the Gyrobotcan only execute continuous-curvature path, simulations done in this chapter are fundamental to more complex trajectories. It is also demonstrated that the time to complete tracking a circle is implicitly related to the available tilt torque to flywheel. This finding will help refine the design requirement for building the actual robot. 161 Chapter Conclusions This thesis has discussed the design, implementation and control of a gyroscopicallystabilized, single wheel robot, including its working principles, dynamics and control systems. Three prototypes have been built and tested with automatic sensing and control capabilities. A virtual 3D simulator has been setup for mechatronic design and controller simulation, in collaboration with Matlab. Although the implementation of Gyrobot is still subject to mechanical limitations, the work presented in this thesis places a solid foundation for future experimentation and research on Gyrobot as well as other robots with improved dynamic stability. This chapter summarizes the contribution that has been presented in this thesis from a broad point of view. Recommendation for future work is presented as well. 6.1 Contribution The contribution of this thesis can be identified in the following aspects: • Implementation of Gyrobot: Hardware and Software 162 The procedure of designing and building the gyroscopically-stabilized singlewheel robot, named Gyrobot, is presented thoroughly in this thesis, which is important from the design approach point of view. Mechatronic approach is employed in the robot development, which combines both the mechanical and electrical aspects of the system. Three prototypes of the Gyrobot have been build with design improvements for each new realization. The test results verifies the workability and feasibility of the concept and design approach. Automatic control is enabled with onboard computing and sensing subsystems integrated with a customized software system complied in a real-time manner, as presented in Chapter 3. • Dynamic model and Virtual Simulator The dynamic model of the gyrobot is derived, with nonholonomic constraints and zero generalized force identified. Simulation results show that the model exhibits the intended behavior of Gyrobot. The nonlinar model is linearized for controller development presented in Chapter 5. The adoption of 3D virtual simulation system serves as an important supplement in terms of mechanical design for a better realization for improved stability performance. A co-simulation system, in which connection with Matlab is established, is also developed and tested. This simulation environment provides 163 a visual-friendly approach for controller design and verification. • Control Systems From control perspective, this thesis presents two control designs for the gyrobot, which is a highly nonlinear and nonholonomic system with coupled dynamics. Three equations that dominates the Gyrobot motion are identified and their relationship is analyzed. It contributes to a better understanding of the dynamics of this gyro-stabilized single wheel robot. The virtual robot approach includes a reference velocity generator which considers the kinematic constraints. Test results show it works for various types of trajectories, such as the straight, circle and sinusoidal path that is basic for more complex trajectories. An interesting property of Gyrobot has been revealed in terms of orientation performance, with respect to the tilt torque. Time for completing a circular path with fixed radius is found to be implicitly related to the tilt torque. This finding will eventually influence the mechanical and electrical design for future prototypes. 6.2 Recommendation for Future Work This thesis has presented the research works Gyrobot - a gyroscopically-stabilized single wheeled robot. Further research topics in this field are suggested as follows. 164 The stability of Gyrobot depends on the angular momentum that is determined by the flywheel spin velocity and mass. Due to the limitation of current mechanical structure, the flywheel spin velocity is capped at 7000 rpm. Increasing the spin speed may be required for better stability performance, especially for application on bumpy surface. A vacuum chamber may be an good option to host the flywheel for reducing air friction. Vibration reduction measures have been developed in this thesis that have shown satisfactory result. However, due to the breakdown of the controller board, it is not able to put into a field test. A new controller board is required to continue examining the effectiveness of new system. Other measures might be required if the flywheel runs faster than current velocity. Due to the mechanical structure, the flywheel tilt angle is bounded at ±27 degrees. The removal of such limitation may largely widen the usage of Gyrobot. Other hardware improvements include, but not limited to: a sealed shell with sharp rigid edge and a more efficient heat dissipation system for on-board power supply. Wireless communication might also be necessary for future field test. A ground station with real-time display of robot status would be useful for long-distance testing. The path following controller developed in this thesis worked well with straight, circular and sinusoidal trajectories. This technique can theoretically work for more complex trajectories that are combinations of the above-mentioned three type of trajectories. Further research can investigate developing a nonlinear dynamic controller 165 to work with the velocity generator presented in this thesis to better cater for the nonlinearity of the system. The structure and components of Gyrobot have been proven to be workable. Further research may consider a miniature realization for collaborative application that requires to cover a large area with a number of small robotic systems. It poses new challenges to both the robot implementation and controller design. 166 [...]...4 .21 The inertia matrix and other vectors and matrices used in Equation 4.18 are,   M11 0 M13 0   0 IxF   0 IxF + IxW + mR2 M =  (4 .22 )  0 2IxW + mR2 0  M13 0 IxF 0 IxF     F1 0 0     u1 F  0 0 F =  2 , B =  ,u = u2 F3  1 0 F4 0 1 where, M11 = IxF + IxW + IxW cos2 β + mR2 cos2 β + IxF cos2 (β + βa ) M13 = 2IxW cos β + mR2 cos β F1 = (IxW + mR2 ) sin (2 )αβ + IxF sin 2( β... sin 2( β + βa )αβa + 2IxW sin β β γ + 2IxF sin (β + βa )β γ˙a ˙ ˙ ˙ +2IxF sin (β + βa )βa γ˙a − µs α ˙ F2 = −gmR cos β − (IxW + mR2 ) cos β sin β 2 − IxF cos (β + βa )sin(β + βa ) 2 ˙ ˙ −(2IxW + mR2 ) sin β αγ − 2IxF sin (β + βa )αγa ˙˙ ˙˙ F3 = 2( IxW + mR2 ) sin β αβ ˙ ˙ F4 = −IxF cos (β + βa ) sin (β + βa ) 2 − 2IxF sin (β + βa )αγ˙a ˙ ˙ 4 .2. 5 Zero Generalized Force In the derivation of full-scale dynamic. .. IxW )¨ = 2( IxW Ω0 + IxF γa )δβ + 2IxF γa uβa − µs α α ˙ ˙ ˙ ˙ (4.33) ¨ (IxW + mR2 )δβ = gmRδβ − (2IxW + mR2 )Ω0 α − 2IxF αγa ˙ ˙˙ (4.34) ˙ (2IxW + mR2 )Ω = −µg Ω + u1 (4.35) ˙ The roll and yaw dynamics of the Gyrobot, i.e., β and α, are determined by the ˙ Equations 4.33 and 4.34, whereas the rotational speed of the wheel (Ω) determined by Equation 4.35 is independent of the roll and yaw dynamics A... 4.3   a21     a  23  a  32  a  33     b 3 gmR = IxW +mR2 2 )Ω ˙ o = − (2IxW +mR+mR+2IxF γa 2 IxW o +IxF ˙ = 2( IxW Ω+IxW γa ) IxF = − IxF−µsxW +I 2I ˙ = IxFxF γa +IxW (4.39) Gyrobot 3D Simulator In this section, a 3-Dimensional simulation platform is developed as an aid for designing complex mechatronics system It uses ADAMS for simulation with animation of the dynamic behavior of the mechanism... is available to control the rotation speed of the wheel and thus the longitudinal motion of the wheel It should be noted that the velocity of the longitudinal motion is v = RΩ ˙ A new set of state variables,x1 = δβ , x2 = δβ , x3 = α is defined for reforming the ˙ linear model Then the equations 4.33 and 4.34 are reorganized as, x1 = x2 ˙ (4.36) x2 = a21 x1 + a23 x3 ˙ (4.37) x3 = a 32 x2 + a33 x3 + b3... angle of the flywheel βa , change in the lean angle of the wheel δβ , and the path traversed by the Gyrobot are shown in Figure 4 .2 - Figure 4.4 The wheel leans in the direction opposite to the tilt of the flywheel, and leaning the wheel makes it steer in the direction of the lean 4 .2. 8 Linear Model of Gyrobot In order to simply the design of Gyrobot controller, a PD controller is first developed and will... loop control is essential In this section, we present simulation results for closed loop control with controller realized in ADAMS, as shown in Figure 4.13 and Figure 4.14 Figure 4.13: Control template to create a PD controller 124 Figure 4.14: PD Controller Setting in ADAMS Design of the controller used here is described in Section 5.1 Controller used for this simulation is Equation 5.7, which is designed... inter- 127 mediate control block and Adams cmd file The Simulink intermediate control block created in shown in the Figure 4.18 With such block available, the entire control scheme in Simulink is shown in Figure 4.19 with PD controller implemented Figure 4.18: Intermediate Simulink Block and Expanded Simulink Block Figure 4.19: PD Controller with ADAMS Simulink Block Closed loop responses with controller... each software is used for the task it is appropriate for - ADAMS for 3D dynamic simulation of mechanical structure and MATLAB for design of controller Creating ADAMS/Matlab Interface To enable the co-simulation between Matlab and ADAMS, a Simulink block is created in ADAMS This process is called Control Plant Creation and shown in Figure 4.17 Figure 4.17: Load and Export ADAMS Control Plant When the Control. .. - 4 .21 To eliminate the Lagrange multipliers so that a minimum set if differential equations is obtained, the matrix partitioning method is applied [13] as follows: 1 Partition the matrix A(q) into A1 and A2 , and let C(q) = −A−1 A2 1 I4×4 (4 .28 ) 2 Consider the following relationship q = C(q)q2 ˙ ˙ (4 .29 ) where q1 = [X, Y ]T and q2 = [α, β, γ, βa ]T Differentiaing Equation 4 .29 yields, ˙ q = C(q) 2 + . the wheel, and (2) the spinning flywheel. Energy of the Wheel The kinetic energy of the wheel is given by, T W = 1 2 m W [ ˙ X 2 + ˙ Y 2 + ˙ Z 2 ] + 1 2 [I xW ω 2 x + I yW ω 2 y + I zW ω 2 z ] (4.6) Substituting. I xW cos 2 β + mR 2 cos 2 β + I xF cos 2 (β + β a ) M 13 = 2I xW cos β + mR 2 cos β F 1 = (I xW + mR 2 ) sin (2 ) ˙α ˙ β + I xF sin 2( β + β a ) ˙α ˙ β +I xF sin 2( β + β a ) ˙α ˙ β a + 2I xW sin. the center of the mass of the flywheel coincides with that of the inner mechanism in our design. In CMU design, the flywheel is attached to the end of the second link (length l 2 ) of a 2- link arm.