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Virginia Tech | In this study, we are primarily concerned with how we reach the final voltage and
consider the discontinuity in voltage after reaching the desired value to be of little
concern. Consequences that result from not smoothing the waveforms will be left as a
field for future research.
Finally, we must also consider the initial conditions from which we begin applying input.
Since we wish to investigate performance for a range of slopes b, it makes sense to
normalize the input to a standard reference voltage. If we consider a commanded motion
from the static equilibrium point the appropriate reference voltage would be that needed
to remain stationary on an arbitrary slope as discussed in chapter three.
5.2 Performance Envelope
Figure 5.2 illustrates the results of the performance envelope calculations described
above for T = 5. As T is increased all these input curves tend to coalesce.
Performance Plot for the Biplanar Bicycle
3.5
3
2.5
2
1.5
1
0.5
0
0 5 10 15 20 25 30 35
Slope Angle Beta (degrees)
41
Performance Envelope of the Planar Biplanar Bicycle
)lanoisnemid-non(
egatloV
erutamrA
Lower Bound
Step
1/4 Sin
Whirling Region Ramp
1/2 Cos
1/4 Cycloid
Parabola
Can Accelerate Uphill
Rolls
Figure 5.2: Performance envelopes for various input waveforms.
No
Equilibrium
Exists |
Virginia Tech | There are several interesting conclusions that may be drawn from the performance
envelope study. Perhaps the most important and influential observation from the
perspective of control strategy design is the relative difference in maximum input voltage
between the various waveform types at any given terrain slope b. We saw in chapter
three that every input voltage within the operational envelope results in finite forward
velocity equilibrium. Therefore, the greater the steady state input voltage for a given b,
the faster the vehicle will move at dynamic equilibrium. Here, we see the ramp input
allows the highest voltage increase over a finite time span. This is a particularly
attractive result when considering vehicle control since its implementation in hardware
and software requires minimal effort.
The use of a ramp function as input makes the control signal generation simple to
implement. We may further justify the use of ramp inputs by considering what happens
if the controller is discrete (as may be expected in a real vehicle). We have not only
demonstrated that the ramp is the best for an arbitrary change in voltage, but we have also
shown that the step input falls short of every other input type. If we assume the vehicle
controller is digital, any input waveform will consist of many small step inputs. Though
undesirable, this is for the most part an unavoidable consequence caused by zero-order-
hold digital to analog conversion. To avoid whirling, we would like to minimize the step
change between any two time-intervals. In comparing two arbitrary waveforms, the
minimum step change is associated with the waveform with the smallest instantaneous
first derivative. Therefore, if we optimize any input function between two given points
based on minimizing the peak of the first derivative, the resulting curve is a line (or
ramp) connecting the endpoints. This heuristic argument is the simplest way to
understand why the ramp input allows the largest change in input voltage in the shortest
time.
Because we have already proven the relationship between command voltage and steady-
state wheel velocity, it is a simple step in logic to assert that the ramp input will
ultimately result in a higher maximum speed on any given slope. It must be remembered
42
Performance Envelope of the Planar Biplanar Bicycle |
Virginia Tech | that this result is, in the most general case, dependent on the input time period T. If
interest lies in fast response times, this result is of great significance. However, we still
have not gained any insight concerning the development of a control feedback term based
on control proximity to the bifurcation point. Unfortunately, the Lyapunov exponent and
Fiegenbaum’s number are the only tools available to deal with the locations of
bifurcation points. Although accurate, neither lends itself to the quick prediction of a
system’s first bifurcation. Further, if the slope changes, the nodal points would change as
well. Therefore, the incorporation of a control term based on the proximity of operation
to bifurcation is not a viable option. Further, if the performance envelope for a specific
vehicle is developed numerically, it would be a much easier and probably more robust
measure to simply regress the data, introduce a factor of safety and hard-wire the nodal
locations into a slope-adaptive control algorithm.
Biplanar bicycle performance envelopes, regardless of geometry or non-dimensional
parameters, take on the form presented here. The ramp input prevails in all designs as the
input with the largest operational envelope. However, it must be remembered that each
specific design configuration will result in a numerically different envelope and should be
simulated prior to the development an adaptive linear controller.
43
Performance Envelope of the Planar Biplanar Bicycle |
Virginia Tech | Chapter 6
3-D Dynamics on an Arbitrarily Inclined Plane
By this point we have investigated and learned quite a bit about the dynamics of the
biplanar bicycle. We have demonstrated complex non-linear behavior including
behavioral bifurcation, Lyapunov stability characteristics, and some heuristic control
techniques to avoid the unattractive operational regime of whirling. However, the study
thus far has been confined to the plane and the effects of two wheels being driven off one
reaction mass have not been considered. There are two primary reasons why the full 3-D
model has not been stressed as highly as the planar system. First, the relative importance
of parameters such as viscous damping and non-linear terms are easily discerned in the
planar model. The mathematics in the three-dimensional model, as will soon be
demonstrated, are much more complex and subsequently more difficult to dissect into
informative results. Second, the present application of the vehicle class has been
restricted to low-speed autonomous ground vehicles. The planar model is sufficient to
explain design criteria necessary to physically construct such a vehicle. The only missing
information involves the control of navigation. However, the control on biplanar bicycles
turning at low speed does not deviate substantially from that of more common
differentially driven three and four-wheeled vehicles.
44
3-D Dynamics on an Arbitrarily Inclined Plane |
Virginia Tech | During the 3-D vehicle’s linear traversal of an inclined plane, its governing dynamics are
the same as those developed for the planar model. Only during turns does it perform
differently. Without any design constraints on navigation we are left to an arbitrary
inspection of the 3-D dynamics. It is for this reason that we only derive the information
necessary to calculate the equations of motion and provide neither solutions nor
simulations to the resulting system. That work will be left to future researchers who need
specific results for specific applications.
6.1 Kinematic Model
Figure 6.1 presents the idealized kinematic diagram of the three-dimensional biplanar
bicycle. The side view is identical to the planar model of chapter three with the
exception of an additional wheel and associated angular coordinate.
n3
n2
n1 Top View
2 d
qq l P L
P L aa 2 d a
x(t), y(t)
R
x(t),
P
R Q y(t)
n2
qq r
P
ff R Q, Reaction mass
modeled as a
point mass.
n1
Figure 6.1: Kinematic Diagram of the 3-D Bicycle
It becomes immediately evident that the complexity of the system definitions has
increased substantially. Like the planar model, we first define the positions of all body
masses in terms of the Newtonian fixed reference frame (denoted by nˆ). However,
because the complex number notation used previously can only be implemented with a
45
3-D Dynamics on an Arbitrarily Inclined Plane |
Virginia Tech | However, Eq. (6.4) is complicated by the fact that we wish to examine the vehicle
dynamics while traversing an arbitrarily inclined plane. The gravitational field term of
Eq. (6.4) needs to be defined in a more rigorous manner. Equations (6.1) and (6.2)
suggest the Newtonian fixed frame is coincident with the plane on which the vehicle is in
contact. This convention in the kinematic definition has been enforced by design. It is
easier to redefine the gravitational field for different planes than it is to redefine the
position vectors.
Consider a plane that is rotated using 2-1 Newtonian angles b and b . It can be shown
x y
that we may arbitrarily orient a plane in Newtonian space using only two rotations. The
Newtonian cosine direction matrix associated with transforming a general directional
reference from the gravity-coincident system to the plane-fixed Newtonian reference is
derived as
Ø gˆ ø Ø cosb sinb sinb - cosb sinb ø Ø nˆ ø
Œ 1œ Œ x y x y xœ Œ 1œ
Œ gˆ œ = Œ 0 cosb sinb œ Œ nˆ œ (6.5)
2 y y 2
Œ º gˆ œ ß Œ º sinb - sinb cosb cosb cosb œ ß Œ º nˆ œ ß
3 x y x y x 3
In the case of a gravitational field, we only wish to know the rotational components
operating on the original k direction. Decomposing Eq. (6.5) and applying it to the
known gravitation field results in an expression for the local gravitational field in terms
of our Newtonian fixed reference coordinates. Since we are concerned with the vehicle
on a plane, we can realign the Newtonian reference frame of figure 6.1 such that n and
1
n lie within the plane of motion. By doing so, the gravitational field as viewed from the
2
Newtonian reference becomes
v [ ]
G = - g sinb nˆ - sinb cosb nˆ +cosb cosb nˆ (6.6)
x 1 y x 2 y x 3
Substituting this result back into Eq. (6.4) completes the energy definitions in the three-
dimensional biplanar bicycle system.
47
3-D Dynamics on an Arbitrarily Inclined Plane |
Virginia Tech | 6.3 Generalized Forces
Before equations of motion can be developed, we must consider any external generalized
forces acting on the system. Like the planar model, we neglect the effects of
aerodynamic body forces and focus only on the forces generated by the DC drive motors.
Although the input torque generated by the motors has been derived in previous chapters,
the model is repeated here for convenience. The torque developed by each motor is
defined using the motor torque constant and the armature current.
t = K i (6.7)
t a
The armature current is modeled using both the armature resistance and the electrical
back EMF constant. In this case, the rotational coordinate in q is for the right and left
wheels. One torque equation must be developed for each wheel.
V K ( )
i = a - B q&+f& (6.8)
a R R
a a
Combining equation (6.7) and (6.8) we develop the final torque equation.
K K K ( ) ( )
t = T V - T B q&+f& = K V - K q&+f& (6.9)
R a R 1 a 2
a a
At this point the derivation differs in form from that in chapter three. Consider the
general definition for calculating generalized forces acting on j generalized coordinates
Qj
=(cid:229)
Fi (cid:215)
¶ rA
+ MA (cid:215)
¶ w
(6.10)
¶ qj ¶ q& j
i
where r are vectors locating points at which i forces are applied and M are moments
A A
acting on the bodies rotating at w. From this, we deduce that the force affecting the
48
3-D Dynamics on an Arbitrarily Inclined Plane |
Virginia Tech | reaction mass is no longer a single torque, but rather a linear combination of the torque
produced by both drive motors. Implementing Eq. (6.10) and defining motor torques
with Eq. (6.9) we calculate the generalized forces acting on each of the generalized
coordinates. The resulting forces are
( )
Q q = K1Va - K2 q& R +f&
R R
( )
Q q = K1Va - K2 q& L +f& (6.11)
(L L)
( )
Q f = K1Va +Va - K2 q& R +q& L +2f&
R L
6.4 Dynamic Model
In order to complete the dynamic model, we have to define further kinematic constraints
to relate the vehicles spatial position to the motion of the generalized variables. First,
there exists a constraint that relates the angular position a of the vehicle and its total time
derivatives to the angular positions q of the wheels and their total time derivatives.
i
These relationships can be expressed as
( )
Rq - q
a = R L
2d
( )
Rq& - q&
a& = R L (6.12)
2d
( )
Rq&& - q&&
a&&= R L
2d
Equation (6.12) is commonly used in the process of vehicular ground navigation by
means of dead reckoning. For example, the differential mechanism at the heart of the
notorious South pointing chariot is inherently based on the same concepts. However, it is
important to note that Eq. (6.12) is derived and proved assuming the vehicle operates in
accordance with conditions of no slip. Even though we have shown in chapter four that
49
3-D Dynamics on an Arbitrarily Inclined Plane |
Virginia Tech | this assumption is robust for real vehicles, it remains a kinematic constraint that must be
dealt with in any mathematical simulation.
Unlike the planar case in which the no-slip condition can be enforced by equating wheel
rotation to linear distance, the same constraint in spatial coordinates creates a non-
holonomic constraint between the wheels and the rigid rolling surface. We must
therefore ensure proper contact forces under each wheel’s no-slip condition so that
angular velocity of a wheel remains proportional to its linear velocity. We define the
velocities of the wheels to be
d v ( ) ( )
V = P = x&+a&d cosa nˆ + y&+a&dsina nˆ
R dt R 1 2
(6.13)
d v ( ) ( )
V = P = x&- a&dcosa nˆ + y&- a&dsina nˆ
L dt R 1 2
Because Eq. (6.12) already enforces constant distance between the wheel centers (it
assumes a non-extensible axle) we need only employ the no slip condition for a single
wheel. Examining the similarities between the resulting constraint equations for each
wheel can mathematically demonstrate this idea.
( ) ( )
( ) ( )
V = x&+a&dcosa nˆ + y&+a&dsina nˆ = Rq& cosa nˆ + Rq& sina nˆ
R 1 2 ( R ) 1 ( R ) 2 (6.14)
( ) ( )
V = x&- a&dcosa nˆ + y&- a&dsina nˆ = Rq& cosa nˆ + Rq& sina nˆ
L 1 2 L 1 L 2
We can now separate one of the relationships in Eq. (6.14) into two scalar equations. This
yields the two constraint equations with which we enforce the no-slip condition. Using
the vector equation for the right wheel only we find the resulting constraints to be
x&+a&dcosa = Rq& cosa
R (6.15)
y&+a&dsina = Rq& sina
R
Because the constraint equations in Eq. (6.15) have first order terms, it is necessary when
applying the Lagrange multipliers that the full variational result is used. Therefore, the
50
3-D Dynamics on an Arbitrarily Inclined Plane |
Virginia Tech | full extended Lagrange equation with Raleigh dissipation is used. C denotes one of n
i c
constraint equations and l is the multiplier associated with the constraint. In this case,
i
four Lagrange multipliers are required to fully constrain the system to the non-holonomic
no-slip condition.
(cid:229)nc ŒØ d (cid:231)(cid:230) ¶L (cid:247)(cid:246)
-
¶L
+
¶D
- l
¶C
i -
d (cid:231)(cid:230)
l
¶C
i
(cid:247)(cid:246) Ͽ
=Q (6.16)
i=1Œº dt (cid:231) Ł ¶q&
j
(cid:247) ł ¶q
j
¶q&
j
i ¶q
j
dt (cid:231) Ł i ¶q&
j
(cid:247) ł œß
We now have a system of seven coupled differential equations and seven unknown
accelerations. However, the two constraint equations (6.15) are functions of the states
only. This reduces our set to five equations and seven unknowns. Therefore, before the
system may be solved, the total time derivatives of the constraint equations must be
taken, remembering that a is a function of q. We now have sufficient equations to solve
i
the system. Further, the unknowns in the system are now
q&& q&& f&& &x& &y& l& l& (6.17)
R L 1 2
It should be noted that this system is considerably more complicated than the planar
model developed in chapter three. To date, the mathematics involved in solving this
system are too complicated for most symbolic and numerical solution software packages.
Further, the potential payoff in the solution of this system is relatively low and resides
primarily in navigation and control of high-speed vehicles. For these reasons, the
solution of the three-dimensional system is not completed and left as a topic of future
research. The Mathematica code and resulting coupled equations of motion are presented
in the Appendix as a resource during future work.
51
3-D Dynamics on an Arbitrarily Inclined Plane |
Virginia Tech | Chapter 7
Orientation-Regulated Platforms for Use in Biplanar Bicycles
The dynamics of the biplanar bicycle have been explored in depth during previous
chapters. We have seen both planar and three-dimensional models and have simulated
and attempted basic control of the planar system. We have also spent extensive energy in
understanding the operational envelops of these vehicles, and have a firm grasp on their
salient operational characteristics. However, we have done little beyond heuristic
thought experimentation in the consideration of application and usage of this new vehicle
class. Some ideas, such as planetary exploration, landmine clearance, and railroad
inspection are striking in their potential. Unfortunately, the design challenges associated
with implementing this type of vehicle do not cease with the design of the vehicle
geometry. Many secondary design considerations must be accounted for prior to the
fielding any such ground vehicle. This chapter demonstrates an example of one of these
problems.
7.1 The Pendulation Problem
Kinematically speaking, the biplanar bicycle contains no grounded or Newtonian-fixed
link within its body-relative system. This can also be restated to say that at no time
during operation can we assume to know the direction of the local gravitational field with
52
Orientation-Regulated Platforms for use in Biplanar Bicycles |
Virginia Tech | respect to vehicle components. In fact, calculating the gravitational direction is made
more difficult, especially during transient motion, because we do not know the local
terrain slope. the lack of a Newtonian ground prevents our design from having a linkage
to maintain directional information. The result of all this is the potentially detrimental
pendulation of any and all devices being carried by the vehicle chassis.
One application in which this problem becomes evident is in the implementation of
computer vision which is needed in most autonomous robotic applications. Regardless of
what kind of camera is being used, its pendulation about the axle predicates the
implementation of rigorous mathematics to compute navigation and obstacle avoidance
routines. It is possible to nearly counter-balance a camera spar with a passive weight, but
the pendulation will continue unless the counter-balance is perfect. Although an
attractive prospect, this design fails with the introduction of any exogenous system inputs
such as aerodynamic drag or viscous damping in the mount bearing. Any generalized
external moment will, during a finite duration of application, force the counterbalanced
system into an orientation other than that which is desired. It becomes evident that in
order to isolate any peripheral or excitation-sensitive equipment from the vehicle
dynamics, a controlled stable platform must be designed. The rest of this chapter
develops such a platform and uses the camera spar used in computer vision as a working
example. However, the control techniques developed herein are applicable for any
platforms.
7.2 Possible Control Techniques
As stated before, one control technique is simple mass counterbalance sufficient enough
to maintain the camera’s position above the axle. Although the resulting pendulum is
stable, the accelerations of the axle result in base-excited oscillations. Further,
adjustment of the CG location does not eliminate the problem; in fact the result is a new
design trade off. Consider, for modest disturbances, the frequency of the induced
nonlinear oscillation is given by (Nayfeh and Mook, 1979)
53
Orientation-Regulated Platforms for use in Biplanar Bicycles |
Virginia Tech | g (cid:230) 1 (cid:246) ( )
w = (cid:231) 1- a2(cid:247) +O a3 (7.1)
l Ł 16 ł
CG
and so a reduction in the distance between the CG and the axle decreases the period of
the motion resulting in a faster return to the unperturbed orientation. However, a
decrease in l results in a reduction in the restoring moment as can be seen by
CG
differentiation of the potential energy with respect to the angular coordinate (Meirovitch,
1970):
¶ V
M = =mgl sinq (7.2)
restoring ¶ q cg
Such reductions culminate with zero restoring moment as the mass moves closer to the
pivot point even as the frequency of motion approaches infinity (the result is marginal
stability; perturbations result in unrecoverable deviations). Having seen the inadequacy
of the counter-balance, the next logical step in system control would be to implement a
simple open-loop control strategy in which employ an equal and opposite rotation with
respect to the wheels. This would certainly result in a fixed camera orientation.
However, this approach relies completely on dead reckoning and may therefore fall short
as a robust control strategy. The only alternative remaining is the implementation of an
active control technique.
In considering any active control, one must reconsider the lack of fixed ground from
which to react any control efforts. Two mass-based control alternatives exist. Used in
space applications for similar reasons, thrusters and reaction masses are employed to
impart forces on structures. Although either would prove sufficient, the reaction mass is
more appropriate to ground-based vehicle architecture. Here, a secondary ballast is
employed in a double pendulum arrangement. Actively controlling the angle between the
bottom pendulum relative to the upper pendulum with a simple DC servomotor provides
the inertial and gravitational resistance necessary to generate the control torque.
54
Orientation-Regulated Platforms for use in Biplanar Bicycles |
Virginia Tech | v v
C =O+ jl ejq(t) (7.6)
c
With the system geometry defined, we can now define the system kinetic and potential
energy. The resulting energy equations are
[ ]
1 v & v & v & v & v & v &
T = m C(cid:215) C + m M (cid:215) M +m R(cid:215) R (7.7)
2 c m r
[ v v v ]
V = g m C(cid:215) j+m M (cid:215) j+m R(cid:215) j (7.8)
c m r
Again, we wish to maintain a level of physical relevance and add a Rayleigh dissipation
function to account for what is modeled as linear viscous damping in the joint bearing.
The primary advantage of this is to ensure all system poles do not lie on the imaginary
axis. This is desirable since losses in any real system prohibit marginal dynamic stability.
The desired function is
1
R= cq&2 (7.9)
2
The equations of motion are then found by the application of the extended Lagrange
equations
d (cid:231)(cid:230) ¶ T (cid:247)(cid:246)
-
¶ T
+
¶ V
+
¶ R
=Q (7.10)
(cid:231) (cid:247)
dt Ł ¶ q& ł ¶ q ¶ q ¶ q& j
j j j j
( ) ( )
where q= q,f and Q = 0,t where t is the controller motor torque. In the servo-
dynamic model the back EMF and armature resistance are considered while the motor
inductance is neglected. The equation that governs the torque output verses the applied
voltage is given by
56
Orientation-Regulated Platforms for use in Biplanar Bicycles |
Virginia Tech | m =1+L2 (M +M )
1 m m r
k = L (M +M )- 1 (7.14)
1 m m r
m = L2M
2 r r
m = L L M
3 m r r
k = L M
2 r r
and overdots now indicate differentiation with respect to non-dimensional time.
7.4 Controller Design
In the design of an active feedback controller, we take advantage of the relatively
generous region of near-linear behavior of pendulums for moderate angular deflections.
Because the temporal character of the disturbance excitation is not known a priori, we
attack the problem as a simple regulation of the equilibrium. By designing a full-state-
feedback regulator for the system, the closed-loop robustness to external forcing is
improved by the increase in effective linear damping.
Here, we seek to design a fixed gain, linear, state feedback controller using the
techniques of optimal control. The basis for the control-law design is the linearization of
the plant via a power series expansion in f and q about the trivial equilibrium point.
Next, a Linear Quadratic Regulator (LQR) controller is designed to minimize the
(quadratic) cost functional based on the linear approximation to the plant.
Upon linearization the equations are transformed to state-space form to allow the tools of
modern controls be applied. The form of the state space equation is
x& = Ax+bv (7.15)
[ ]
The state vector x is established as q f q& f& T . The state matrix A is developed
based on the mass, damping, and stiffness matrices associated with the linearization:
58
Orientation-Regulated Platforms for use in Biplanar Bicycles |
Virginia Tech | 1
PA- PbbTP+ATP+Q =0 (7.21)
r
where
1
gT =- bTP
r
P may be determined using Potter’s algorithm (Meirovitch, 1989).
7.5 Numerical Simulations
Applying the above control law, the closed-loop system was simulated via numerical
simulation. The values for the non-dimensional parameters used in the simulation were
K = 2 M = 3 M = 4
m m r
L = 1/6 L = 1/6 z = 0.2 w = 3.27 rad/s
m r c
The weighting factors used in the LQR cost functional
Q=diag[10000 100 0 0] and r =1
(note that Q is positive semi-definite) results in the state feedback gain vector
K = [92.06 6.84 –49.14 –6.96].
The closed-loop system robustness is demonstrated by considering initial condition
responses and horizontally and vertically forced responses. For example, Fig. 7.2a
demonstrates the system’s fast decay rate when subjected to an initial displacement of
10o. The associated control effort is shown in Fig. 7.2b as f(t).
Figure 7.3a demonstrates system response to base disturbances of &y&=0.1sint and
&x&=0.1cost. After a short transient, the system stabilizes in a 2o sinusoidal oscillation.
60
Orientation-Regulated Platforms for use in Biplanar Bicycles |
Virginia Tech | m
c
Im
y
Re
P (x,y), M, I
l
c
Q, m
q
Cyclic l
l
m
m
m l
r
m
r
g
b
f
Figure 8.1: Coupled Vehicle-Camera Kinematic Definitions
No changes have been made from the vehicle nomenclature presented in chapter three.
However, some new variables had to be added to the camera system from chapter seven.
The angles associated with camera and reaction mass motion have been renamed to avoid
redundancy. Further, the system will ultimately be non-dimensionalized with respect to
the wheel radius r and mass M. Finally, reference for wheel rotation angle q is simply
suggested in figure 8.1 but is assumed to lie along the imaginary axis at time t = 0. We
may do so without loss of generality because of the cyclic nature of the variable as
discussed in chapter three.
In superimposing the systems, we may write the system kinetic and potential energies as
the linear combination of those developed earlier. Likewise, the Rayleigh dissipation
functions may also be combined. All these are repeated and combined here with the
modified coordinate nomenclature.
65
Coupled Vehicle-Camera Dynamics and Control |
Virginia Tech | T = 1 ί mQv & (cid:215) Qv & + 1 MPv & (cid:215) Pv & +m Cv & (cid:215) Cv & +m Mv & (cid:215) Mv & +m Rv & (cid:215) Rv & Ͽ + 1 Iq&2 (8.1)
2º 2 c m r ß 2
(v ) (v ) (v ) ( v ) (v )
V =mg Q(cid:215) j +Mg P(cid:215) j +m g C(cid:215) j +m g M (cid:215) j +m g R(cid:215) j (8.2)
c m r
( )
R= 1 Cq&+f& 2 +1 Cy&2 (8.3)
2 2
With T, V, and R defined we may now solve the extended Lagrange equation for the
dynamic response. This is represented as
(cid:230) (cid:246)
d (cid:231) ¶T (cid:247) - ¶T + ¶V + ¶R =Q (8.3)
(cid:231) (cid:247)
dt Ł ¶q& ł ¶q ¶q ¶q& j
j j j j
( ) ( )
where q= q,f,y,g and Q = t,t,0,t . To maintain consistency between the models
j
presented in chapters three and seven, the motor torques t are defined with identical
motor constants. The DC servomotor model developed earlier remains the same and the
resulting expression for motor torque on both drive and control motors is shown to be
K K K ( )
t = T Vˆ - B T q&+f& (8.4)
R a R
a a
As we have done in all past models, the system is simplified by introducing non-
dimensional variables and parameters. Because the vehicle is of primary concern we will
maintain the wheel radius and mass as the characteristic length and mass. By doing so,
the parameters and variables defined in chapter seven are rendered invalid, as they are
now in terms of a vehicle dimension. Also note the addition of two new parameters to
describe the camera spar length and camera mass. We introduce the non-dimensional
parameters
66
Coupled Vehicle-Camera Dynamics and Control |
Virginia Tech | l l l l
L= L = m L = p L = r
r m r p r r r
m m m m
a = M = m M = r M = c
M +m m M r M c M (8.5)
C K K I
z = r2w + R ar2( MB +T m) w m = ( M +m) r2
and variables
K Vˆ K vˆ
U = T a , v = T and t=tˆw (8.6)
( )
R r2 M +mw2 R l2m w2
a a c c c
where w = g/r . Once again we take advantage of the relatively generous region of
near-linear behavior of pendulums for moderate angular deflections. As before, the
temporal character of the disturbance excitation is not known a priori. However, we
designed the previous full-state feedback regulator without any regard to the disturbance
source. Therefore, it would be a logical assumption that the regulator design should
remain unchanged by the addition of the vehicle. Mathematically, we see a difference in
the system dynamic matrices caused by the change in source description. We no longer
describe the base excitation with a arbitrary x and y motion but rather incorporate the
motion at point P based on the vehicle coordinates q and f. Using the same process of
power expansion and LQR design presented in chapter seven we linearize the camera
system about the trivial solution. In doing so we may see the new mass, stiffness, and
damping matrices to be
Ø L2M +L2 M +L2 M +L2M +2L L M L L M +L2M ø
M =Œ P C M M M R R R M R R M R R R Rœ
º L L M +L2M L2M ß
M R R R R R R
Ø - L M +L M +L M +L M L M ø
K = Œ P C M M M R R R R R œ (8.7)
º L M L M ß
R R R R
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Coupled Vehicle-Camera Dynamics and Control |
Virginia Tech | Using the same relative link lengths, masses, and control gains as found in chapter seven,
we see the camera spar is controlled to a maximum angular deviation of approximately 4
degrees. When compared to the uncontrolled deviation of more than 60 degrees, we
determine the controller is performing as expected. Additionally, we see the expected
linear damping effects of the feedback.
Of course, it is also helpful to determine the control effort involved in accomplishing
these results. Figure 8.3 presents the absolute angle of the controller reaction mass.
Controlled and Uncontrolled Absolute Camera Reaction Mass Angle g
80
60
40
20
0
-20
-40
-60
0 10 20 30 40 50
Time (w )
n
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Coupled Vehicle-Camera Dynamics and Control
seerged
ni
g
elgnA
fo
edutingaM
Uncontrolled
Controlled
Figure 8.3: Control Effort in Reaction Mass Degrees for a Vehicular Step Input
Recall the reaction mass angle is measured relative to the projected camera spar. Here,
the angle is presented as an absolute measurement from the negative imaginary axis to
help the reader visualize the system in action. It is interesting to note that the maximum
angular displacement of the controlled system is not substatially greater than that of the
uncontrolled counterpart. |
Virginia Tech | Again, the results of the controller performance simulations are in no way surprising
since the original regulator was designed to reject base-excitation disturbances regardless
of input waveform. However, what may not be intuitive in any way are the effects in
vehicle performance characteristics. One would probably conclude from intuition that
the vehicle will respond substantially differently as a result of mounting the controlled
camera system to the axle. However, as is true with many aspects of this new vehicle
class, intuition would prove incorrect in this case.
8.3 Coupled System Stability
If we re-examine stability characteristics as we did in chapter three, we see the static and
dynamic equilibrium conditions remain identical in nature; they simple have added mass
terms. First, consider again the equilibrium solution in which the vehicle remains static
on an arbitrarily inclined plane. The resulting conditions for static equilibrium are
( ) ( ) ( )
1+M +M +M sin b =U and La sinf =U (8.8)
C M R o o o
This result is analogous to the result found in Eq. (3.14). In fact, the only difference
between the two is the addition of the camera system masses. The unity term in the mass
is the non-dimensional representation of the wheel mass and existed implicitly in the
conditions of Eq. (3.14). The relationship correlating the pendulum angle f to the control
voltage U remains unchanged. Buried in Eq. (8.8) is an expression for the maximum
o
slope on which the biplanar bicycle may rest statically. By using the same logic as we
did in chapter three in which we denied the existence of complex angles in f and b we
see the maximum slope to be bounded above and below by
(cid:230) (cid:246) (cid:230) (cid:246)
La La
- sin- 1 (cid:231)(cid:231) (cid:247)(cid:247) £ b £ sin- 1 (cid:231)(cid:231) (cid:247)(cid:247) (8.9)
Ł 1+M +M +M ł Ł 1+M +M +M ł
C M R C M R
Since the static equilibrium is based solely on a torque balance between the body mass
and reaction mass, the result of Eq. (8.9) seems a logical result. It is important to note,
70
Coupled Vehicle-Camera Dynamics and Control |
Virginia Tech | however, that the control of f by the propulsive drive-motor has no effect on the two new
generalized coordinates y and g. In steady state, the camera-system center of mass
necessarily has to lie on the same vertical axis as the vehicle axle. Regardless of the
controlled angle g, the system will pendulate to its natural equilibrium. This can be
shown mathematically in the static equilibrium solutions of the controller’s generalized
coordinates.
(cid:230) (cid:246)
y =sin- 1(cid:231)
vˆ(t)
(cid:247) (8.10)
(cid:231) Ł L M - L M - L M (cid:247) ł
p c m m m r
(cid:230) vˆ(t) (cid:246)
g(t)=sin- 1 (cid:231)(cid:231) (cid:247)(cid:247) - y (8.11)
Ł L M ł
r r
Once again, it is easily seen that Eqs.(8.10) and (8.11) represent nothing more than a
torque balance on the stabilization components. Further, these relationships do not
change for the steady-state vehicle velocity equilibrium case.
We can deduce from the stability analysis that the vehicle acts very much like the original
uncoupled system when in its steady-state configurations. To determine if any transient
differences exist, we turn to numerical simulations. It stands to reason that if we wish to
compare the coupled and uncoupled system responses, we should examine both
simultaneously under the same input conditions. To do so we apply a step input to the
vehicle drive motor in both cases and then apply the controlled camera system to the
coupled simulation. The coordinates of interest are those associated with the vehicle
performance, q and f. Figure 8.4 presents the wheel rotation angle q for both the
uncoupled and controlled-coupled dynamic configurations.
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Coupled Vehicle-Camera Dynamics and Control |
Virginia Tech | Coupled and Uncoupled Vehicle Reaction Mass Angle f
80
60
40
20
0
-20
-40
0 10 20 30 40 50
Time (w )
n
73
Coupled Vehicle-Camera Dynamics and Control
seerged
ni
f
muludneP
fo
edutingaM
Uncoupled
Coupled
Figure 8.5: Comparison of Uncoupled and Controlled-Coupled Reaction Mass Angle
Indeed, we see the reaction mass performance to be near identical to that of the
uncoupled case. More interestingly, figures 8.4 and 8.5 are representative of the system
while experiencing input voltages near the critical whirling input. Therfore, what we see
here is the near worst-case comparison.
8.4 Results and Further Considerations
Surprisingly, the results of this study suggest that adding a platform stabilization system
with a reaction mass as large as 15% that of the vehicle mass does not affect the vehicle
performace curves in a significant way. Therefore, the control strategies and performace
envelopes developed in previous chapters will still apply in the case of the coupled
system design. However, it would also be prudent to consider other design issues
associated with coupling these two systems. For example, in order to stabilize a platform
in the manner presented here, we necessarily must add additional mass to the overall
vehicular system. In applications such as planetary exploration and autonomous ground
vehicles where weight-saving is a major driving force, this additional reaction mass may
be more detrimental in the end. Several methods of countering this problem are |
Virginia Tech | presented here as food for thought. The development of dynamics and control for these
cases is left as an exercise for future research.
If the primary concern with platform stabilization is weight addition, it would only seen
reasonable to use the already existing vehicle reaction mass as the stabilization mass. For
example, figure 8.6 presents a concept whereby the platform is actuated against the
reaction mass via a four-bar linkage.
Camera Arm
Axle and
Propulsive Moment
Control Arm
Drive Arm
Control Moment
Reaction Mass
Figure 8.6: Four-Bar Camera Stabilization Concept
The link lengths of the driving four-bar must be designed according to Grashoff’s law
such that the pendulum link can rotate fully with respect to the camera arm. This way,
camera-ground impingement may be avoided during whirling. However, using linkages
of this type present issues associated with the kinematics of the drive. Singular positions
in the four-bar motion will require instantaneous changes in drive direction in the event
of reaction mass whirling. The forces and responses times resulting from such a singular
point may prove detrimental to an otherwise robust control strategy.
Another similar drive mechanism is presented in figure 8.7 in which motion is
transmitted though linear rather than rotary actuation.
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Coupled Vehicle-Camera Dynamics and Control |
Virginia Tech | Camera Arm
Axle and
Propulsive Moment
Drive Arm
Control Arm
Reaction Mass
Figure 8.7: Linear Actuated Platform Control Concept
There are obvious limitations to this concept. The control arm must be pinned to the
camera spar with an offset. The case of static equilibrium on no grade dictates the
colinearity of the camera and drive arms. This would in turn require the control arm to
assume a zero-length position. Further, though must be given to what may occur when
the reaction mass swings in the negative direction. In spite of these obvious drawbacks,
this concept should not be entirely dismissed for low-speed applications in which the
vehicle reaction mass angle f is expected to remain relatively small.
It is important to note that the solutions presented here are not intended to cover all
possible design solutions to the platform stabilization problem. In fact, we assume many
application-specific solutions exist. What has been presented here are the ideas
considered from a very general perspective and the solution to one of the simplest
configurations conceived to date. However, since the biplanar bicycle is novel in design
and application, the designer must remain open to innovative kinematic and control
configurations when trying to meet a specific mission.
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Coupled Vehicle-Camera Dynamics and Control |
Virginia Tech | Chapter 9
Conclusions and Recommendations for Future Work
This work has outlined a foundation of mathematics, analytical methods, and design
strategies necessary to complete the robust and reliable design for any biplanar bicycle
application. Although the mathematics and simulations presented herein can prove useful
in their extension to mission-specific vehicles, they are really intended to provide a solid
background in the vehicle class from which the designer may cultivate an intuitive
understanding of generalized performance and control characteristics. Understanding the
nature of what has been done here is essential in any future development of the biplanar
bicycle. Having said this, we can now consider some of the natural spin-offs of this
research that must be considered in any future development efforts.
9.1 Future Work
The dynamics for the three-dimensional vehicle have been presented but not solved. At
this time, the symbolic representations of the resulting equations of motion are beyond
the computing power available. However, the solutions to the dynamics, symbolic or
numerical, should be investigated in terms of ground navigation. Until then, dead
reckoning seems the logical algorithmic choice for the autonomous ground vehicle
applications currently under consideration. Although dead reckoning is a widely
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Recommendations for Future Work and Conclusions |
Virginia Tech | accepted method of navigation, we may find the integration of global positioning to be
better suited to some vehicle applications. In the event GPS is desired, navigation no
longer stands as the major design challenge. Instead, a deeper understanding of the three-
dimensional dynamics will be needed in the planning and control of specific trajectories
over known terrain. Careful navigational control schemes may prove beneficial as a
means of reducing control effort, minimizing energy dissipation, governing travel time,
and controlling obstacle avoidance. Therefore, it stands to reason that as the technology
involved in the building and implementation of the biplanar bicycle increases, our ability
to analytically model and predict system behavior will become more important. In fact,
the three-dimensional dynamic model may one day supercede the planar system
presented here as the backbone of vehicular design methodology.
Another focus of future work is experimental validation of the biplanar bicycle’s ability
to traverse discontinuous terrain. We have suggested the biplanar bicycle may provide
distinct advantages in stair climbing. To date we have only verified this concept through
calculations and prototype testing. A more rigorous investigation may be prudent if
application warrants this capability. As a natural extension of the validation process the
generation of performance envelopes based on the relative size of terrain discontinuities
would be prudent. This kind of information would be useful not only during the design
phase but may also be integrated into the navigational obstacle avoidance algorithms. No
everything must be avoided, some things can be climbed and conquered.
In general, we must continue to find and prove the application worthiness of the biplanar
bicycle vehicle class. We have learned enough to believe the class provides advantages
over classical ground vehicle designs. However, it will never be accepted as a viable
solution until we can prove its performance and applicability using prototypes.
Therefore, it would be very useful if future researchers choose reasonable applications
such as planetary exploration, landmine clearance, and railway inspection and build
vehicles capable of completing the tasks as well or better than traditional designs. To this
end, the development of an autonomous railway inspection vehicle is suggested as a
primary target for vehicle application. Because rail systems naturally constrain the
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Recommendations for Future Work and Conclusions |
Virginia Tech | vehicle in its directional navigation, the planar models presented in this work can be very
easily extended to this application. Few issues like non-holonomic constraints and added
kinematics would arise in this design. Therefore, we suggest this field as the first likely
source of application.
Another very interesting field of work surrounds to continuation of control algorithms for
the stabilized platform concept. For example, the four-bar drive linkage concept
presented in chapter eight requires the development of a non-learning adaptive control
scheme based in system linearization around the constant forward velocity equilibrium
points. Using such techniques is the only way the linkage approach would work
throughout the vehicle’s performance envelope. Further, the non-linear control of the
vehicle itself could potentially be improved by implementing the same type of adaptive
control strategy. By doing so, the analytical performance envelop information regarding
dynamic bifurcation points may be included in the control system design as a way of
making the system considerably more robust and reliable with respect to vehicle
whirling.
Finally, work must be completed on the physical design of the vehicle. Many ideas
concerning the physical realization of a useful vehicle have been considered during this
research. Moving all body mass components into the wheel rims would be a method of
increasing ground clearance. Adding hemispherical hubs to the drive wheels would
allow the vehicle to self-right itself if dropped from aircraft or spacecraft. This idea can
be take one step further by designing an extendable axle so the entire vehicle can be
deployed as a sphere. Another type of performance improvement involves using more
than one point of contact during the traversal of discontinuous terrain as a kinematic
ground. By doing so, other internal linkages could be implemented to enhance
performance in stair or rock climbing applications.
9.2 Conclusions
The research presented in this work has convinced us that the biplanar bicycle is a viable
option in the design of autonomous ground vehicles. We have seen the dynamic
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Recommendations for Future Work and Conclusions |
Subsets and Splits