DESIGN AND CONTROL OF AUTONOMOUS MOBILE ROBOTS WITH IMPROVED DYNAMIC STABILITY 2

Energy of the Internal Mechanism and FlywheelContrary to the design used by researchers of the Carnegie Melon University, thecenter of the mass of the flywheel coincides with that of the

simu-is derived using constrained Lagrangian approach The order of obtained dynamicequations is reduced by eliminating the Lagrange multipliers in the dynamic model.Zero generalized force is used to further simplify the model The nonlinear model islinearized at Gyrobotnospace’s vertical position to develop a linear controller, whichwill be described in Chapter 5.

A 3-Dimensional simulation platform is developed as an aid for designing complexmechatronics system and controller development ADAMS is used for simulation withanimation 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

the 3-D simulator in ADAMS with control design software MATLAB This tion allows the designer to adopt a control-centric approach for designing complexmechanical structure to be used in a mechatronics system The simulating environ-ment can easily be extended to any complex mechanical system simply by alteringSolidWorks drawings.

Though the Gyrobot wheel is flying saucer shaped, for the sake of simplicity in mulation of equations, we assume it to be a rigid, homogenous disk that rolls withoutslipping over a flat horizontal surface Flywheel, the stabilizing component of theGyrobot, hangs from the central shaft of the axle of the wheel There is a verticaloffset of the flywheel from the center of the Gyrobot wheel We designed the internalmechanism, the outer gimbals and the inner gimbals, in a way so that the center offlywheel remains below the center of the wheel As a result of this design, there is nohorizontal offset between the center of Gyrobot and the center of flywheel According

for-to this description of components, we define the following 3 coordinate frames

the flat surface on which the GYROBOT rolls

at the center of the main wheel Its Z-axis represents the axis of rotation of

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 flatsurface, and the Y-axis in the forward direction of the wheel

at the center of the flywheel Its Z-axis represents the axis of rotation of the

These coordinate frames are illustrated in Figure 4.1 This figure also illustratesdifferent variables used in deriving the dynamic model Definitions for these variablesare listed in Table 4.1

In this section, we explore the kinematic constraints of the Gyrobotnospace Let(~i,~j, ~k)

Table 4.1: Gyrobot Variable Definition

Precession rate (dα/dt) is shown in the figure

Leaning rate (dβ/dt) is shown in the figure

Spinning rate (dβ/dt) is shown in the figure

The transformation between these two coordinate frames is given by,

− sin α cos β − cos α − sin αsinβ

~n





As the wheel rolls without slipping on a flat surface, the velocity of the contact

point The velocity of the rolling wheel is

Combining Equations 4.1, 4.2 and 4.3, we get,

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

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 theenergies, the system is divided into two parts: (1) the wheel, and (2) the spinningflywheel

Energy of the Wheel

The kinetic energy of the wheel is given by,

Energy of the Internal Mechanism and Flywheel

Contrary to the design used by researchers of the Carnegie Melon University, thecenter of the mass of the flywheel coincides with that of the inner mechanism in ourdesign In CMU design, the flywheel is attached to the end of the second link (length

of the first link, i.e., the distance between wheel’s center and the center of the inner

differ-an attempt to chdiffer-ange the center of gravity of the structure The wheel then rotates

to bring the CG underneath the center of the wheel causing forward (backward)

mo-tion in the wheel In our design, on the other hand, the drive motor is fixed to theinternal structure but the axle is passed through a pair of ball-bearings attached tothe platform The drive motor directly drives the axle.

While deriving the kinetic energy of the internal mechanism and flywheel, theCMU researchers assumed very slow swing of the pendulum-structured internal mech-anism and, therefore, insignificant influence of internal mechanism on the rotationalkinetic energy In our design, the internal structure does not move at all and thereforehas no influence on the rotational kinetic energy of the flywheel The angular velocity

The potential energy of the internal mechanism and flywheel is

Lagrangian of the System

Taking different energies into consideration, we now define the Lagrangian as,

L = (T W + T t

F + T r

Substituting the expressions for different energies derived in the previous two

sub-sections, we can determine L.

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

4.2.4 Dynamic Model of GYROBOT

Taking derivatives, both in time and space, of the Lagrangian of Equation 4.15,

be evaluated by comparing these equations with the constraint Equation 4.5 TheLagrange multiplier can be eliminated by partitioning the matrix A in accordance tothe method suggested in [13]

As the wheel is disk-like shape, it’s practical to assume the inertias of the wheel

center of the flywheel is collocated with the center of the wheel The flywheel spins at

system reduction process The procedure of such reduction is explained in subsection4.2.5 and 4.2.6 Motion of the center of the wheel is expressed by Equations 4.19 to

4.21 The inertia matrix and other vectors and matrices used in Equation 4.18 are,

4.2.5 Zero Generalized Force

In the derivation of full-scale dynamic model, one challenge comes from how to deal

speed, it contributes largely to the kinetic energy When the lagrangian formulation

constant while the robot is running so that the variable should not exist in the motionformula

To deal with this problem, the following technique should be followed:

1 Iterms that contains the flywheel velocity in the the system kinematic energy

T1

term

d dt

Applying T s with respect to generalized variables in the form of lagrangian mulate

−2I xf ω s ( ˙β + ˙β a )S ββ a 2I xf ω s ˙αS ββ a

As stated in the previous section, the terms in kinetic energy containing the flywheel

Sym-bols is applied to generate the nonlinear terms Thus the dynamic model (Equation

eqn:gyrover.dyn.model) can be written in

M(q)¨ q + N(q, ˙q) = M(q)¨ q + K ˙q + L ˙q + ( ∂U

∂q)

where K is the matrix derived from Christoffel Symbols [50] and U is the potential

energy The nonholonomic constraints remains unchanged as shown in Equation 4.19

4 Considering C T (q)A T (q) = 0 and Equation 4.29, 4.30, Equation 4.31 becomes

However, for this simulation, this variation is not caused by the reference velocitygenerator, rather it is changed directly Corresponding results illustrating tilt angle

by the Gyrobot are shown in Figure 4.2 - Figure 4.4 The wheel leans in the tion opposite to the tilt of the flywheel, and leaning the wheel makes it steer in thedirection of the lean

In order to simply the design of Gyrobot controller, a PD controller is first developedand will be discussed in section 5.2.3 and 5.2.4 To design a linear controller, a linear

Figure 4.2: Viariation in Lean Angle (β) in response to (δ β,ref)

Figure 4.3: Wheel Leans in the direction opposite to the tilt of flywheel

Figure 4.4: Trace of Path Traversed by the Gyrobotmodel is derived in this section Below are some assumptions made for linear modeldevelopment:

• The speed of the flywheel is very large compared to the rate at which the wheel

RPM whereas the drive motor operates at a maximum speed of 340 RPM With

this assumption, we can consider terms like ˙β ˙α or ˙β ˙γ negligible compared to

terms like ˙γ a ˙α or ˙ γ a ˙β

• The tilt angle of the flywheel and therefore the lean angle of the wheel are

• The term ˙α is negligible The precession rate of the wheel depends on the lean

angle of the wheel We do not force the wheel to lean by a large angle in order

to avoid falling So the rate of precession is assumed small.

• The wheel operates in a fairly upright position, i.e., β = 90 o ± δ β

• The nominal rolling speed of the wheel is Ω o , i.e., ˙γ = Ω0± Ω

With these assumptions, the dynamic model of the GYROBOT can be rewrittenas,

(I xF + I xW)¨α = 2(I xWΩ0+ I xF ˙γ a ) ˙δ β + 2I xF ˙γ a u β a − µ s ˙α (4.33)

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 dedicated closedloop is available to control the rotation speed of the wheel and thus the longitudinalmotion of the wheel It should be noted that the velocity of the longitudinal motion

is v = RΩ.

linear model Then the equations 4.33 and 4.34 are reorganized as,

design-to any complex mechanical system simply by altering SolidWorks drawings.

Simulation is important for designing any mechatronics system that involves complexstructure and operations Traditionally, most simulations are computer based andrequire mathematical modeling Such model is used for describing system dynamicsand for finding its dynamic response The models are used to predict system behaviorfor a given set of known parameters and initial conditions

ADAMS, which was developed for 3-D motion simulation, is integrated withMATLAB/SIMULINK, popular software among control community, to create thecontrol-centric design platform The motion simulator uses three dimensional me-chanical drawings of different sub-systems to be combined into a complex mechatron-ics system This eliminates the need for deriving complex, nonlinear mathematicalmodels for the dynamic behavior of the system The simulator has been effectivelyused for the analysis of dynamic behavior of the Gyrobot in both open loop conditionand with closed loop controller.

ADAMS (Automatic Dynamic Analysis of Mechanical Systems) started as a eral purpose program that can analyze systems undergoing large non-linear displace-ments while under the effect of non-linear force and motion input The methodologywas developed by Nicolae Orlandea [52] It can be categorized as a general purposenumeric code utilizing a non-minimal set of co-ordinates to develop the equations ofmotion This motion simulation software analyzes complex behavior of mechanicalassemblies and thus enables users to test virtual prototypes and optimize designs with-out having to build numerous physical prototypes It uses stiff integrators to solvethese equations and sparse matrix algebra to solve the linear algebraic equations in itsinnermost computation loop [53] Since its inception, significant development invest-ments resulted in sophisticated virtual prototyping tools for a wide range of industrialapplications Use of virtual prototyping for design and development of a humanoidrobot has been reported in a recently published article [54] A tire modeling is build

gen-in ADAMS [55] as a test case for vehicle dynamics tire models to provide a wide range

of test models

ADAMS allows users to import geometry from CAD systems or to build a solidmodel of the system from scratch In this thesis, the first approach is adopted.Different parts of the gyrobot have been created using SolidWorks These componentsare then transferred to ADAMS where final model is elaborated

The integrated simulator platform will simplify the phases of design of a tronics system Moreover, it can be used by students and researchers alike to have

mecha-an in-depth understmecha-anding of mecha-any complex mechmecha-anism Integration of ADAMS withMATLAB enables control engineers to complete the design cycle without buildingany prototype Simulation results can also be used to fine tune mechanical design.Results presented in this section include (i) open loop simulation of gyrobot usingADAMS, and (ii) simulation of the closed loop system The integrated system can beused for the control-centric design of any mechatronic system simply by replacing the3D drawings of the mechanical structure of gyrobot with that of any complex system

Gyrobot Model in Solidworks

Principle of operation of the gyrobot can be explained using the schematic drawingshown in Figure 4.5 The flywheel is suspended from the axle of the gyrobot using agimbal assembly that enables tilting of the spinning flywheel The flywheel is attached

to the inner gimbal and is spun by the spin motor The tilt motor, which is attached

to the outer gimbal, can tilt the inner gimbal plus fast spinning flywheel to eitherside The gimbal and flywheel structure is suspended from a platform and the axle ofgyrobot is passed through this platform The outer shell of gyrobot is rigidly joined

to the axle Running the drive motor makes the wheel roll; rolling speed can becontrolled by controlling RPM of drive motor

Figure 4.5: Internal mechanism of gyrobot

Gyrobot model drawn in SolidWorks is shown in Figure 4.6 Size and shape

of each part are created as par size and shape of the actual component To reducethe complexity of model in ADAMS, only the mechanical parts are included in theSolidWorks drawing; these are the outer wheel, the inner gimbal, the outer gimbal,the flywheel with its supporting structure This structure is sufficient to reflect thephysical properties of gyrobot, which will be examined later Mass, density andfrictions are defined in ADAMS/View

Figure 4.6: Gyrobot 3D Drawing in SolidworksGyrobot Model in ADAMS

The ADAMS model of Gyrobot is created by importing the 3D drawings in Solidworks

as parasolids file, as shown in Figure 4.7 Material, mass, density and friction arethen defined Mass and inertial are automatically calculated Table 4.2 lists maindynamic characteristics of the robot components for the proposed ADAMS model.Gyrobot model in ADAMS environment takes several aspects into account, such asgravity, contact constraints, friction, inertial properties and reference markers, whichenable good approximation of real robot behavior Each joint of the mechanism isdefined with a particular motion For example, the driving joint is specified as arevolute joint so the rotation associated with this joint specifies the rolling motion

of the wheel Table 4.3 show some of the main joints and motions that have beendefined for Gyrobot model in ADAMS Friction force is one of the key components

for successfully implementation.

Figure 4.7: Gyrobot Model in ADAMS

Table 4.2: Main Inertial Data for Gyrobot Modules

in agreement with [56] Friction coefficients for other joints are listed in Table 4.4

Table 4.3: Some of Joints and Motions defined for Gyrobot ADAMS model

Dummy Joints

for Lean Angle

Acquisition

Table 4.4: Friction Coefficient Values for Various Joints

With proper setting of internal joints friction forces, the virtual model is able to reflectreal robot behavior more accurately

ADAMS/Solver does all the calculations required to simulate the motion Itallows different types of integrator for a particular problem Below is the list for keysettings required in Gyrobot simulation with briefing Detailed explanation on eachintegrator can be found in ADAMS/Solver documents

• Integrator, formulation and corrector

• Simulation frequency and internal frequency

• End time and Steps Vs Duration and Step size

• Optimal step size

For integrator setting, GSTIFF is the recommended integrator for most of themechanical systems We use default integrator, formulation and corrector Onemay use other integrator settings, based on the simulation type (for models otherthan Gyrobotnospace) with clear understanding of integrator setting documents onADAMS/Solver.

One has to specify simulation frequency, internal frequency and step size ulation frequency is the frequency of updating the graphic display, whereas internalfrequency is closely related to the system under study It represents the speed atwhich component states are changed Internal frequency is very important parameter

Sim-in simulation settSim-ing and other solver settSim-ings must be set Sim-in accordance with theinternal frequency

The flywheel is the fastest moving object in the whole structure If flywheel

frequency is Y degrees per second (calculated based on its speed) and the desired

Y × H max = θ Step

For example, when the flywheel speed is set to 7000 rpm or 420000 revolutions per

flywheel rpm the simulation requires

Figure 4.8 illustrates the complete simulation setting procedure for nospace The overall gyrobot virtual model development in ADAMS is shown inFigure 4.9 Simulations will be provided in the sections below for model verificationand controller design.

Gyrobot-Figure 4.8: Simulation Setting procedures

4.3.3 Model Verification

Stability Test

Law of conservation of momentum and gyroscopic precession govern the operation ofthe Gyrobotnospace Large angular momentum of the fast-spinning, heavy flywheelaids to overcome the external disturbance such as tilting torque caused by gravity.According to conservation of angular momentum, the higher the spin wheel velocity,the longer the robot should be able to stand by itself

We verify the effect of flywheel speed on the balancing the wheel by simulating

Figure 4.9: Gyrobot 3D Simulator Development Diagram

the motion with different flywheel speeds while keeping external forces, e.g., gravityunchanged Simulations are carried out with seven different flywheel speeds between

1000 rpm and 7000 rpm, the highest speed of the flywheel For each flywheel speed,simulation is done for two conditions - (a) keeping Gyrobot stationary and (b) making

it roll at a speed of 30 rpm Results are summarized in Table II It is clearly evidentthat time elapsed before the fall of Gyrobot increases with increasing flywheel speed,which conforms to the conservation of angular momentum When the wheel rolls,its angular momentum adds to the angular momentum of the flywheel aiding thebalancing effort It is observed that for identical flywheel speed, time elapsed beforethe fall is longer when Gyrobot rolls

Table 4.5: Results of Stabilization Test

Rolling Speed=0 Rolling Speed=30rpm

(Figure 4.1) is related to the precession rate according to the Equation 5.6

During the tilt test, Gyrobot is assumed standing on a flat surface with theflywheel spinning at 7000 rpm At 0.5 second, the flywheel is tilted from to 23degrees in 0.5 seconds causing an average tilt velocity of 46 deg/s The results of thissimulation are shown in Figure 4.10 The plots show that while the flywheel is tilted

by 23 degrees, the entire Gyrobot is tilted by -57 deg This conforms to the simulationand experimental results published in [28]and [18] Leaning velocity and precessionrate of the Gyrobot are also plotted in the same figure The test is repeated by tiltingthe flywheel from 0 to -20 degrees in 0.5 s Again the results shown in Figure 4.11are in conformity with the concept of gyroscopic precession

Initial conditions for the simulation are:

• Precession angle, α = 00

• Wheel lean angle, β = 900

• Flywheel tilt angle, β a= 00

• Flywheel Speed, ˙γ a = 7000rpm

• Faceting Tolerance = 0, (surface bumpiness)

Figure 4.10: Tilt Test I (23deg) from upper

left clockwise: tilt angle, lean angle, sion rate, lean velocity

preces-Tilt Test II

The initial condition for this simulation is same as in Tilt Test 1 However, in thistest the flywheel tilts to the opposite direction and the rolling speed of the gyrobot

is set to 30 rpm Results are illustrated in Figure 4.12

It is also observed that if the flywheel tilts angle changes in positive direction, thewheel lean angle rotates in the other direction This is in accordance with simulationresult shown in Figure 4.3 and Tilt Test I After 1s the tilt angle remains unchanged,

Figure 4.11: Tilt Test I (-20deg) from

up-per left clockwise: tilt angle, lean angle, precession rate, lean velocity

Figure 4.12: Tilt Test II (23deg&30rpm)

from upper left clockwise: tilt angle, lean angle, precession rate, lean velocity

the lean angle will converge to a steady-state value in accordance with Equation5.6 The Gyrobot moves in circles as it single-wheeled shape enables it turns to thedirection it leans From the above results, it can be concluded that the 3D virtualsimulator for Gyrobot is successfully build.

Development of this design environment is motivated by the need for a control-centricapproach for mechatronics design Incorporating the capability of simulating underclosed loop control is essential In this section, we present simulation results for closedloop control with controller realized in ADAMS, as shown in Figure 4.13 and Figure4.14

Figure 4.13: Control template to create a PD controller

Figure 4.14: PD Controller Setting in ADAMSDesign of the controller used here is described in Section 5.1 Controller usedfor this simulation is Equation 5.7, which is designed using a linear model of thegyrobot, obtained by linearizing system dynamics at its vertical position Gains of

Initial conditions for this simulation are set as:

• Precession angle, α = 00

• Wheel lean angle, β = 900

• Flywheel tilt angle, β a= 00

• Flywheel Speed, ˙γ a = 7000rpm

• Faceting Tolerance = 0, (surface bumpiness)

Simulation results with a PD controller implemented in ADAMS are shown inFigure 4.15 and 4.16

Figure 4.15: PD Controller in ADAMS δ β,ref = 100

(from clockwise from upper left: lean angle, tilt

veloc-ity, wheel 3D view, tilt angle)

(from clockwise from upper left: lean angle, tilt velocity, robot jectory, tilt angle)

tra-4.3.5 Controller in Matlab

As ADAMS is meant for 3D simulation of dynamics, it has very limited capabilityfor realizing a controller Except for the build-in PID controller template, othertype of controllers can only be implemented using script written in Fortran, which isnot popular among control community It would greatly increase the virtual modelusability if the controller can be implemented in Matlab, which is the most popular

in control design Such interface would ensure that each software is used for the task

it is appropriate for - ADAMS for 3D dynamic simulation of mechanical structureand 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 Plant is created, two files are created namely Simulink

inter-mediate control block and Adams cmd file The Simulink interinter-mediate control blockcreated in shown in the Figure 4.18 With such block available, the entire controlscheme 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 realized in SIMULINK are shown in Figure4.20 These results are identical to those obtained with controller realized in ADAMSshown in Figure 4.15 Integrating ADAMS with MATLAB enable the researcher

exploit the better aspects of both It’s no longer a restriction by the limitations ofADAMS in implementing controllers and experiment with more complex controllerstructures including fuzzy logic control, sliding mode control etc can be done.

(from clockwise from upper left: lean angle, tilt ity, precession angle, tilt angle)

This 3-D virtual system not only facilitate controller design and verification, it alsoprovide the fast prototyping method for mechatronic design before real implementa-tion For Gyrobotnospace, the stability test result shown in Figure 4.10 exhibits largeoscillations in lean angle To improve this system stability, increasing the flywheelspeed and loading a heavier flywheel both contribute to larger angular momentum,which will bring in more stability It’s verified in Figure 4.21 and Figure 4.22.However, due to the limitation of the hardware, either increasing the flywheel