# Using Buckingham’s $\pi$ Theorem for Multi-System Learning Transfer:
a Case-study with 3 Vehicles Sharing a Database

William Therrien, Olivier Lecompte and Alexandre Girard1 1All authors are with
the Department of Mechanical Engineering, Universite de Sherbrooke, Qc, Canada
Contact<EMAIL_ADDRESS>
###### Abstract

Learning schemes for planning and control are limited by the difficulty of
collecting large amounts of experimental data or having to rely on high-
fidelity simulations. This paper explores the potential of a proposed learning
scheme that leverages dimensionless numbers based on Buckingham’s $\pi$
theorem to improve data efficiency and facilitate knowledge sharing between
similar systems. A case study using car-like robots compares traditional and
dimensionless learning models on simulated and experimental data to validate
the benefits of the new dimensionless learning approach. Preliminary results
show that this new dimensionless approach could accelerate the learning rate
and improve the accuracy of the model and should be investigated further.

##  I Introduction

The typical approach to solving motion control problems for autonomous
vehicles and other types of robotic systems is to use a physics-based
kinematic or dynamic model of the system to plan trajectories and establish
feedback laws [1][2][3]. With the rise of efficient machine learning
algorithms, much effort has gone into using learning in planners and
controllers so that autonomous vehicles can learn from experience and improve
over time[4][5][6]. However, major difficulties limit the success of the
learning scheme when it comes to controlling real physical platforms. One of
the major challenges is the difficulty of collecting the large amount of
experimental data required for advanced learning schemes such as deep-learning
[6][7]. A popular solution is to generate data using high-fidelity simulations
instead of experiments. Nevertheless, this approach also has multiple
challenges, such as the difficulty of capturing certain behaviors in a
simulator and transferring the results to the real world [8]. All in all, data
efficiency is a critical bottleneck.

Sharing learned data and policies across multiple systems could help solve the
problem of collecting huge amounts of data. For example, instead of having to
start learning from scratch, a centralized database for a fleet of vehicles
would enable the newly deployed vehicle to use knowledge based on thousands of
hours of motion data collected by hundreds of vehicles. However, this is
difficult to achieve when the studied systems do not share identical
characteristics as their behavior, and therefore the appropriate control
policy, differs. This paper investigates the potential of leveraging
dimentionless number and the Buckingham’s $\pi$ theorem to help learning more
general models. Section II presents some related work and discusses the
original contribution of this paper. Section III defines the problem used as
the case study in this paper. Section IV presents the learning results based
on simulated data and section V presents the learning results based on an
experimental validation with three small scale vehicles shown at Fig. 1 .

(a) Small vehicle

(b) Long vehicle

(c) Large vehicle

Figure 1: Small scale platforms of various dimensions used for experimental
validation.

##  II Background

Multi-task transfer learning has received significant research attention
recently [9][10][11]. This paper focuses on multi-system transfer where the
objective is to share knowledge between similar systems that accomplish the
same task. Specifically for multi-system transfer, an interesting approach is
presented in [12] where a shared database can outperform a robot-specific
training approach that uses 4x-20x more data. In [13], the authors address the
problem of inappropriate transfer, which can result in a decrease in
performance. By assessing how similar two systems are, it is possible to know
with which systems the transfer learning would be most beneficial. The closest
related work is probably [14], where the authors rather tackle the problem
directly by guiding the policies network with a representation vector built
from information regarding the hardware. This approach is similar to the one
presented in this work, but instead of two vectors, one containing the states
and the other containing the characteristics of the hardware, a single vector
is built based on these features with the intention of reducing the dimensions
of the system to zero by using the Buckingham’s $\pi$ theorem.

Dimensional analysis [15], i.e. the Buckingham’s $\pi$ theorem, is a useful
tool to simplify input/output relationships that involved physical quantities.
It is used extensively in the field of fluid mechanics to minimize the number
of experiments required to characterize a behaviour. Also, there is a renew
interest in using dimensional analysis in the context of learning [16] [17]
[18]. However, dimensional analysis it still very seldomly explored in the
field of motion control and robotics, to our knowledge only a few projects
used that concept. In the case of an in-lane collision avoidance system, [19]
created dimensionless indices, based on a vehicle’s relative longitudinal
velocity, road surface conditions and terminal lateral distance, which were
used in the selection of the most efficient maneuver and timing for an
intervention. The use of these indices was shown to be computationally
efficient. The authors of [20] proposed an improved method for kinematic
calibration based on dimensionless error mapping matrices (EMMs). Simulation
results show that the residual pose errors with the proposed dimensionless
EMMs were lower than with the conventional EMM in various units. Finally, [21]
discuss the notion that similar dynamic systems must have an equivalent
optimal policy when expressed in dimensionless form.

Here in this paper, the novelty is to leverage dimensional analysis to improve
data efficiency for learning motion models of similar vehicles. The
contributions are: 1) A new way to describe a system using an Augmented
Buckingham’s $\pi$ theorem based model is presented. 2) A demonstration of how
that dimensionless model can be a way to generalize learning in the context of
motion control by being able to share knowledge with different systems is
made. 3) A case study with robotic vehicles is used to show through
simulations and experiments that this dimensionless learning scheme is
advantageous compared to baseline approaches.

##  III Problem definition for the case-study

The main goal of this paper is to validate whether a learning scheme can
benefit from a dimensionless mapping of its inputs and outputs. These benefits
could take the form of an increase in learning speed, a reduction in mean
absolute error (MAE) and the possibility of sharing knowledge between similar
systems. To do so, different optimized distributed gradient boosting (XGBoost)
models have been trained with simulated and experimental data and compared
using their respective MAE. Two dimensionless learning schemes have been
created and compared with a baseline traditional dimensional model. The first
one is based on applying the Buckingham’s $\pi$ theorem to the input/output
relationship that is modeled. The second, called the Augmented Buckingham’s
$\pi$ theorem based model, is also based on the Buckingham’s $\pi$ theorem but
with additionnal dimensionless inputs (hand crafted features based on expert
knowledge of the system’s physics).

Inspired by the challenges of learning good control policies for emergency
braking maneuvers on various ground conditions (snow, ice, etc.), such as
explored in [22], the presented case study is the prediction of the final
relative position of a vehicle after a sudden emergency maneuver, based on
initial conditions, control inputs (braking and steering), and environmental
parameters. The learned model could be then used in a control pipeline to
select the best maneuver in an emergency braking situation for instance. To
simplify the case study a few assumptions are made. Firstly, the control
inputs are constant throughout a maneuver, i.e. the steering angle $\delta$
and braking action (constant deceleration rate $a$ imposed on both rear
wheels) are decided once at the start of the maneuver and remain constant
until the vehicle comes to a complete stop. Also, only the final position of
the vehicle is predicted, not the complete trajectory. Furthermore, the
environment is always a horizontal surface and only 2D planar motion is taken
into account. The simulation only consider a kinematic motion model, while for
the experimental data the dynamics of the system is taken into account,
neglecting any roll or pitch effects and considering the complex soil-tire
relationship simply as a constant coefficient of friction $\mu$. Table I shows
the main variables involved in this case study.

TABLE I: Variables used throughout this paper Variables |  Descriptions |  Units [Dimensions]
---|---|---
State variables
$X$ |  Position of the vehicle in X-axis of world frame |  m [L]
$Y$ |  Position of the vehicle in Y-axis of world frame |  m [L]
$\theta$ |  Yaw of the vehicle in world frame |  rad
Environment related variables
$\mu$ |  Friction coefficient wheels/road |  -
$v$ |  Longitudinal velocity of the vehicle |  m/s [LT-1]
$g$ |  Gravitational acceleration |  m/s2 [LT-2]
maneuvers related variables
$a$ |  Deceleration of the wheel |  m/s2 [LT-2]
$\delta$ |  Steering angle of front wheels |  rad
Vehicles related variables
$N_{f}$ |  Normal force on front wheels |  N [MLT-2]
$N_{r}$ |  Normal force on rear wheels |  N [MLT-2]
$l$ |  Length between vehicle’s axles |  m [L]

Three small scale car-like vehicles of various dimension are used for both
validation processes (simulation and experimentation). Vehicle 1, shown in
figure 1a, is a modified 1:10 scale platform based on a Traxxas SLASH. The
vehicle 2, shown in figure 1b, is also a modified 1:10 scale platform based on
a Traxxas SLASH with the chassis modified to have a longer wheelbase. Finally,
vehicle 3, shown in figure 1c, is a modified 1:5 scale platform based on a
Traxxas X-MAXX. From now on, they will be referred to as small vehicle, long
vehicle and large vehicle respectively. Their specifications are summarized in
table II. Since the simulation only takes kinematics into account, only the
wheelbase length l of these vehicles is used to describe them in that
particular section.

TABLE II: Specifications of all vehicles Vehicle | $l$ | $N_{f}$ | $N_{r}$
---|---|---|---
| (m) | (N) | (N)
Small | 0.345 | 37.77 | 28.84
Long | 0.853 | 22.74 | 52.89
Large | 0.475 | 71.12 | 71.12

##  IV Learning with simulated data

###  IV-A Simulator presentation

This section presents an analysis in which the proposed learning approach is
evaluated using data generated in a simplified simulated environment. The
simulation was based on the kinematic bicycle model frequently used in the
literature. The non-linear equations for the bicycle model are shown at 1.

$\begin{bmatrix}\dot{X}\\\ \dot{Y}\\\ \dot{\theta}\\\ \dot{v}\\\
\end{bmatrix}=\begin{bmatrix}vcos(\theta)\\\ vsin(\theta)\\\
\frac{vtan(\delta)}{l}\\\ a\\\ \end{bmatrix}$ (1)

For all three vehicles presented at figure 1, 5 500 simulations were conducted
with the initial values and maneuvers shown at table III

TABLE III: Values for the simulations for all 3 vehicles $v_{i}$ | $a$ | $\delta$
---|---|---
(m/s) | (m/s2) | (rad)
From 0.1 | From -0.1 g | From 0.0000
to 5.0 | to -1.0 g | to 0.7854
by 0.1 | by -0.1 g | by 0.0785
50 values | 10 values | 11 values

All simulations start with the vehicle at position [X,Y] = (0,0), orientation
$\theta$ = 0 rad and initial speed $v_{i}$. The maneuver starts right away
with deceleration $a$ and steering angle $\delta$. The simulation stops when
the vehicle reach zero velocity and the final position and orientation are
noted.

###  IV-B Learning model types for the simulation

####  IV-B1 Traditional dimensionalized learning model

The nonlinear equations describing the kinematic bicycle model used for the
simulation presented at 1 can be written in the form of 2.

$\begin{bmatrix}X,Y,\theta\end{bmatrix}=f(v_{i},a,\delta,l)$ (2)

Thus, the dimensionalized parameters for the initial velocity $v_{i}$, the
decelaration a, the steering angle $\delta$ and the length of the wheelbase l
are used as inputs to the XGBoost algorithm to predict the final pose
[X,Y,$\theta$] of the vehicle. This traditional dimensionalized learning
scheme is illustrated at figure 2

Figure 2: Baseline learning scheme for simulated data

####  IV-B2 Buckingham’s $\pi$ theorem based model

According to Buckingham’s $\pi$ theorem, with N dimensionalized variables and
P independent dimensions, we obtain m = N \- P dimensionless numbers. Here, N
= 7 [$X$, $Y$, $\theta$, $v_{i}$, $a$, $\delta$, $l$] and P = 2 [L,T]. We thus
have five dimensionless numbers. By using the distance between the two axles
$l$ [M] and the initial velocity of the vehicle $v_{i}$ [MT-1] as repeated
variables, we can create the five dimensionless numbers. The inputs and ouputs
of the XGBoost algorithm used to predict the outcome of a maneuver are shown
on figure 3.

Figure 3: Buckingham’s $\pi$ theorem based model for simulated data

####  IV-B3 Augmented Buckingham’s $\pi$ theorem based model

Based on the mathematical shape of the nonlinear equation for $\dot{\theta}$
presented at equation 1, we obtain the arbitrary dimensionless number
$\pi_{6}$ derived at 3. The inputs and ouputs of the Augmented Buckingham’s
$\pi$ theorem based model used to predict the outcome of a maneuver are shown
on figure 4.

$\dot{\theta}=\frac{vtan(\delta)}{l}\hskip 2.84544pt;\hskip
14.22636pt\pi_{6}=\frac{{v^{2}}tan(\delta)}{al}$ (3)

Figure 4: Augmented Buckingham’s $\pi$ theorem based model for simulated data

###  IV-C Results

The results for the three model types will be examined from two perspectives.
On the one hand, we will study the MAEs of self-predictions, cross-predictions
and shared-predictions for each model type. On the other hand, we will examine
the effect of the size of the training set on the convergence of the MAEs for
self-predictions, or the so-called the learning rate of the models.

####  IV-C1 Mean absolute error

The MAE is one of the most widely used metrics for regression algorithms since
it has the same units as the model’s dependent variables which makes the
results easier to understand. Equation 4 shows how MAE is calculated, where N
is the number of tests used, $y_{i}$ are the actual result values and
$\hat{y}_{i}$ are the predicted values. Since the output values for both
dimensionless models are $\pi_{1}$ = $\frac{X}{l}$, $\pi_{2}$ = $\frac{Y}{l}$
and $\pi_{3}$ = $\theta$, $\pi_{1}$ and $\pi_{2}$ are multiplied by the
wheelbase $l$ of the tested vehicle before computing the MAE so that all
resultats share the same units.

$MAE=\frac{1}{N}\sum_{i=1}^{N}|(y_{i}-\hat{y}_{i})|$ (4)

80$\%$ of the simulated data were used to train a model of each type for the
three vehicles. The remaining 20$\%$ is used to test the models. Table IV
contains the results of the traditional dimensionalized model, the results of
the model based on Buckingham’s $\pi$ theorem are presented in table V and
table VI shows the results of the Augmented Buckingham model. For all three
tables, the light gray cells correspond to the results of what we call self-
predictions, i.e. predictions that use a vehicle’s test set in its own trained
model. The white cells correspond to the results of a vehicle’s test set in
another vehicle’s trained model, known as cross-predictions. Finally, the dark
gray cells are the results of shared predictions. For shared predictions, the
test set of a particular vehicle is ran in a model that has been trained with
data from all three vehicles.

TABLE IV: MAEs for the traditional dimensionalized model (X and Y are in meters and $\theta$ in rad) Models | Data Vehicle 1 | Data Vehicle 2 | Data Vehicle 3
---|---|---|---
| (Small) | (Long) | (Large)
Model | X: 0.0494 | X: 0.3703 | X: 0.1601
Vehicle 1 | Y: 0.0466 | Y: 0.3071 | Y: 0.1459
(Small) |  $\theta$: 0.0587 |  $\theta$: 0.9851 |  $\theta$: 0.4502
Model | X: 0.3653 | X: 0.0316 | X: 0.2622
Vehicle 2 | Y: 0.3032 | Y: 0.0257 | Y: 0.2061
(Long) |  $\theta$: 0.9919 |  $\theta$: 0.0238 |  $\theta$: 0.5369
Model | X: 0.1667 | X: 0.2679 | X: 0.0414
Vehicle 3 | Y: 0.1440 | Y: 0.2125 | Y: 0.0362
(Large) |  $\theta$: 0.4599 |  $\theta$: 0.5324 |  $\theta$: 0.0422
Model | X: 0.0467 | X: 0.0433 | X: 0.0514
MERGED | Y: 0.0465 | Y: 0.0368 | Y: 0.0498
(All 3) |  $\theta$: 0.0457 |  $\theta$: 0.0302 |  $\theta$: 0.0431
TABLE V: MAEs for the Buckingham’s $\pi$ theorem based model (X and Y are in meters and $\theta$ in rad) Models | Data Vehicle 1 | Data Vehicle 2 | Data Vehicle 3
---|---|---|---
| (Small) | (Long) | (Large)
Model | X: 0.0204 | X: 0.0144 | X: 0.0125
Vehicle 1 | Y: 0.0261 | Y: 0.0102 | Y: 0.0099
(Small) |  $\theta$: 0.0331 |  $\theta$: 0.0109 |  $\theta$: 0.0159
Model | X: 0.0354 | X: 0.0145 | X: 0.0303
Vehicle 2 | Y: 0.0394 | Y: 0.0176 | Y: 0.0262
(Long) |  $\theta$: 0.2007 |  $\theta$: 0.0127 |  $\theta$: 0.0917
Model | X: 0.0153 | X: 0.0103 | X: 0.0190
Vehicle 3 | Y: 0.0162 | Y: 0.0095 | Y: 0.0182
(Large) |  $\theta$: 0.0666 |  $\theta$: 0.0097 |  $\theta$: 0.0241
Model | X: 0.0080 | X: 0.0087 | X: 0.0083
MERGED | Y: 0.0086 | Y: 0.0087 | Y: 0.0083
(All 3) |  $\theta$: 0.0143 |  $\theta$: 0.0080 |  $\theta$: 0.0115
TABLE VI: MAEs for the Augmented Buckingham’s $\pi$ theorem based model (X and Y are in meters and $\theta$ in rad) Models | Data Vehicle 1 | Data Vehicle 2 | Data Vehicle 3
---|---|---|---
| (Small) | (Long) | (Large)
Model | X: 0.0126 | X: 0.0117 | X: 0.0104
Vehicle 1 | Y: 0.0121 | Y: 0.0090 | Y: 0.0077
(Small) |  $\theta$: 0.0131 |  $\theta$: 0.0043 |  $\theta$: 0.0063
Model | X: 0.0271 | X: 0.0103 | X: 0.0219
Vehicle 2 | Y: 0.0286 | Y: 0.0134 | Y: 0.0222
(Long) |  $\theta$: 0.1407 |  $\theta$: 0.0052 |  $\theta$: 0.0591
Model | X: 0.0130 | X: 0.0090 | X: 0.0117
Vehicle 3 | Y: 0.0133 | Y: 0.0091 | Y: 0.0129
(Large) |  $\theta$: 0.0389 |  $\theta$: 0.0038 |  $\theta$: 0.0098
Model | X: 0.0045 | X: 0.0055 | X: 0.0046
MERGED | Y: 0.0050 | Y: 0.0061 | Y: 0.0051
(All 3) |  $\theta$: 0.0060 |  $\theta$: 0.0028 |  $\theta$: 0.0035

Table VII summarizes the three previous tables (IV, V and VI). For self-
predictions, we can see that the Buckingham’s $\pi$ theorem based model and
the Augmented Buckingham model are respectively 1.93 times and 3.60 times more
precise than the traditional model. These numbers climb to 11.76 times and
15.80 times for cross-predictions. Finaly, the Buckingham’s $\pi$ theorem
based model and the Augmented Buckingham model are respectively 4.80 times and
9.17 times more precise than the traditional model in shared-predictions.

TABLE VII: MAEs comparison for all models trained with simulated data (X and Y are in meters and $\theta$ in rad) Prediction | Traditional | Buckingham’s | Augmented
---|---|---|---
types | dimensionalized |  $\pi$ theorem | Buckingham’s
| model | based model |  $\pi$ model
SELF | X: 0.0408 | X: 0.0180 | X: 0.0115
Predictions | Y: 0.0362 | Y: 0.0206 | Y: 0.0128
|  $\theta$: 0.0416 |  $\theta$: 0.0233 |  $\theta$: 0.0094
CROSS | X: 0.2654 | X: 0.0197 | X: 0.0155
Predictions | Y: 0.2198 | Y: 0.0186 | Y: 0.0128
|  $\theta$: 0.6594 |  $\theta$: 0.0659 |  $\theta$: 0.0422
SHARED | X: 0.0471 | X: 0.0083 | X: 0.0049
Predictions | Y: 0.0444 | Y: 0.0085 | Y: 0.0054
|  $\theta$: 0.0397 |  $\theta$: 0.0113 |  $\theta$: 0.0041

####  IV-C2 Learning rate

The amount of data required to obtain an operational predictive function for
self-predictions for all model types is then investigated. Figure 5 shows the
MAE convergence for the final longitudinal position $X$, final lateral
position $Y$ and final yaw $\theta$ of the three model types with the large
vehicle’s self-predictions. For all outputs, the Augmented Buckingham model
converges the fastest, followed by the Buckingham’s $\pi$ theorem based model.
More data would be required to achieve full convergence of the final $X$ and
$Y$ positions for the traditional dimensionalized model. Only the data for the
large vehicle is presented to avoid any redundancy since all curves were
alike.

Figure 5: MAE convergence according to training size for the simulated data of
the large vehicle

##  V Learning with experimental data

###  V-A Tests used for generating the data

For all three vehicles, 540 experimental tests were carried out with the
initial values and maneuvers shown in table VIII. All tests start with the
vehicle in position (0,0), orientation 0 rad, initial speed $v_{i}$ and
ground/tire friction coefficient $\mu$. The maneuver begins immediately with
deceleration $a$ and steering angle $\delta$. The test stops when the vehicle
has come to a complete stop, and the final position and orientation are
recorded using a VICON triangulation camera system.

TABLE VIII: Values for the experimental tests for all 3 vehicles $\mu$ | $v_{i}$ | $a$ | $\delta$
---|---|---|---
| (m/s) | g (9.81 m/s2) | (rad)
0.2 | From 1.0 | From 0.1 | 0.0000
0.4 | to 3.5 | to 1.0 | 0.3927
0.9 | by 0.5 | by 0.1 | 0.7854
3 values | 6 values | 10 values | 3 values

###  V-B Learning model types for the experimental validation

####  V-B1 Traditional dimensionnalized learning model

With the assumptions made in section III and based on table I, the equation 5
can be written for the experimental end position [$X$,$Y$,$\theta$]. The
baseline learning scheme for the experimental validation is illustrated at
figure 6.

$\begin{bmatrix}X,Y,\theta\end{bmatrix}=f(\mu,v_{i},g,a,\delta,N_{f},N_{r},l)$
(5)

Figure 6: Baseline learning scheme for experimental data

####  V-B2 Buckingham $\pi$ theorem based model

According to Buckingham $\pi$ theorem, N = 11 [$\mu$, $v_{i}$, $g$, $a$,
$\delta$, $N_{f}$, $N_{r}$ , $l$, $X$, $Y$, $\theta$] and P = 3 [M,L,T]. We
thus have eight dimensionless numbers. By using the distance between the two
axles $l$ [M], the initial velocity of the vehicle $v_{i}$ [MT-1] and the
normal force on front wheels $N_{f}$ [MLT-2] as repeated variables, we get the
eight dimensionless numbers. The inputs and ouputs of the XGBoost algorithm
are used to predict the outcome of a maneuver are shown on figure 7.

Figure 7: Buckingham’s $\pi$ theorem based model for experimental data

####  V-B3 Augmented Buckingham $\pi$ theorem based model for experimental
data

By simple manipulations of the same physical variables used in previous
models, the limits in longitudinal force $\pi_{9}$ and lateral force
$\pi_{10}$ before sliding can be introduced using equations 6 and 7. An
infinite number of corrective dimensionless numbers could be created depending
on the system under study to minimize the output error of the Augmented
Buckingham model. However, these two numbers have been arbitrarily chosen for
their high dynamic significance.

$\pi_{9}=\frac{F_{limit}}{F_{x_{maneuver}}}=\frac{N_{r}\mu}{ma}=\frac{N_{r}\mu
g}{(N_{f}+N_{r})a}$ (6)

$\pi_{10}=\frac{F_{limit}}{F_{y_{maneuver}}}=\frac{N_{r}\mu}{\frac{mv^{2}}{R}}=\frac{g\mu
R}{v^{2}}=\frac{g\mu l}{v^{2}\tan\delta}$ (7)

Figure 8: Augmented Buckingham’s $\pi$ theorem based model for experimental
data

###  V-C Results

####  V-C1 Mean Absolute Error

To avoid any redudancy, only the summarizing table IX for self, cross and
shared-predictions is presented for the experimental data. For self-
predictions, we can see that the Buckingham’s $\pi$ theorem based model and
the Augmented Buckingham model are respectively 1.11 times and 1.14 times more
precise than the traditional model on average. These numbers drop to 1.05
times and 1.07 times more precise for cross-predictions. Finaly, the
Buckingham’s $\pi$ theorem based model and the Augmented Buckingham model are
respectively 1.11 times and 1.28 times more precise than the traditional model
in shared-predictions.

TABLE IX: MAE comparison for all models trained with experimental data (X and Y are in meters and $\theta$ in rad) Prediction | Traditional | Buckingham’s | Augmented
---|---|---|---
types | dimensionalized |  $\pi$ theorem | Buckingham’s
| model | based model |  $\pi$ model
SELF | X: 0.1229 | X: 0.1228 | X: 0.1197
Predictions | Y: 0.0488 | Y: 0.0321 | Y: 0.0335
|  $\theta$: 0.0482 |  $\theta$: 0.0596 |  $\theta$: 0.0554
CROSS | X: 0.3027 | X: 0.3752 | X: 0.3775
Predictions | Y: 0.2198 | Y: 0.0186 | Y: 0.0128
|  $\theta$: 0.2007 |  $\theta$: 0.1494 |  $\theta$: 0.1454
SHARED | X: 0.0539 | X: 0.0507 | X: 0.0400
Predictions | Y: 0.0130 | Y: 0.0105 | Y: 0.0089
|  $\theta$: 0.0193 |  $\theta$: 0.0187 |  $\theta$: 0.0187

####  V-C2 Learning rate

Figure 9 shows the MAE convergence for the final longitudinal position $X$,
final lateral position $Y$ and final yaw $\theta$ of the three model types
with the large vehicle’s experimental self-predictions. For all outputs, all
three models follow a similar learning curve. More training data (limited here
by the number of experimental test conducted) would be required to fully
converge since all curves keep on improving at a relatively high rate.

Figure 9: MAE convergence according to training size for the experimental data
of the large vehicle

##  VI Conclusion

To conclude, this paper presents a new dimensionless learning scheme based on
Bucking’s $\pi$ theorem. The results with data based on kinematic simulations
show that the proposed dimensionless models are much more accurate when using
a shared database, and have faster learning rates than a traditional
dimensional model. The experimental validation also shown some advantage with
the proposed dimensionless scheme over a baseline approach in a complex
dynamical context, but the gain was less significant ($\approx$10% compared to
10x for the kinematic behavior in simulation), probably due to a discrepancy
between the assumptions made (i.e. when selecting the list of input variables
for the dimensional analysis) and the high complexity of the tire dynamics
involved in the experiments. All in all, the early results presented in this
case-study show that leveraging Bucking’s $\pi$ theorem can minimize the
experimental data required to train a learning algorithm modelling a physical
phenomena. Hence, this novel approach should be investigated further to better
understand its full potential and limitation.

## References

  * [1] N. H. Amer, H. Zamzuri, K. Hudha, and Z. A. Kadir, “Modelling and control strategies in path tracking control for autonomous ground vehicles: A review of state of the art and challenges,” vol. 86, no. 2, pp. 225–254. [Online]. Available: http://link.springer.com/10.1007/s10846-016-0442-0
  * [2] B. Paden, M. Čáp, S. Z. Yong, D. Yershov, and E. Frazzoli, “A Survey of Motion Planning and Control Techniques for Self-Driving Urban Vehicles,” _IEEE Transactions on Intelligent Vehicles_ , vol. 1, no. 1, pp. 33–55, Mar. 2016.
  * [3] C. Katrakazas, M. Quddus, W.-H. Chen, and L. Deka, “Real-time motion planning methods for autonomous on-road driving: State-of-the-art and future research directions,” vol. 60, pp. 416–442. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0968090X15003447
  * [4] S. Aradi, “Survey of deep reinforcement learning for motion planning of autonomous vehicles,” vol. 23, no. 2, pp. 740–759, conference Name: IEEE Transactions on Intelligent Transportation Systems.
  * [5] R. H. Crites and A. G. Barto, “Improving elevator performance using reinforcement learning,” p. 7.
  * [6] X. Di and R. Shi, “A survey on autonomous vehicle control in the era of mixed-autonomy: From physics-based to AI-guided driving policy learning,” vol. 125, p. 103008. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0968090X21000401
  * [7] B. M. Lake, T. D. Ullman, J. B. Tenenbaum, and S. J. Gershman, “Building machines that learn and think like people,” vol. 40, publisher: Cambridge University Press. [Online]. Available: http://www.cambridge.org/core/journals/behavioral-and-brain-sciences/article/building-machines-that-learn-and-think-like-people/A9535B1D745A0377E16C590E14B94993
  * [8] Y. Liu, H. Xu, D. Liu, and L. Wang, “A digital twin-based sim-to-real transfer for deep reinforcement learning-enabled industrial robot grasping,” vol. 78, p. 102365. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0736584522000539
  * [9] S. Dey, S. Boughorbel, and A. F. Schilling, “Learning a shared model for motorized prosthetic joints to predict ankle-joint motion.” [Online]. Available: http://arxiv.org/abs/2111.07419
  * [10] O. M. Andrychowicz, B. Baker, M. Chociej, R. Józefowicz, B. McGrew, J. Pachocki, A. Petron, M. Plappert, G. Powell, A. Ray, J. Schneider, S. Sidor, J. Tobin, P. Welinder, L. Weng, and W. Zaremba, “Learning dexterous in-hand manipulation,” vol. 39, no. 1, pp. 3–20, publisher: SAGE Publications Ltd STM. [Online]. Available: https://doi.org/10.1177/0278364919887447
  * [11] A. Nagabandi, I. Clavera, S. Liu, R. S. Fearing, P. Abbeel, S. Levine, and C. Finn, “Learning to adapt in dynamic, real-world environments through meta-reinforcement learning.” [Online]. Available: http://arxiv.org/abs/1803.11347
  * [12] S. Dasari, F. Ebert, S. Tian, S. Nair, B. Bucher, K. Schmeckpeper, S. Singh, S. Levine, and C. Finn, “RoboNet: Large-scale multi-robot learning.” [Online]. Available: http://arxiv.org/abs/1910.11215
  * [13] M. J. Sorocky, S. Zhou, and A. P. Schoellig, “Experience selection using dynamics similarity for efficient multi-source transfer learning between robots.” [Online]. Available: http://arxiv.org/abs/2003.13150
  * [14] T. Chen, A. Murali, and A. Gupta, “Hardware conditioned policies for multi-robot transfer learning,” in _Advances in Neural Information Processing Systems_ , vol. 31. Curran Associates, Inc. [Online]. Available: https://proceedings.neurips.cc/paper/2018/hash/b8cfbf77a3d250a4523ba67a65a7d031-Abstract.html
  * [15] M. E. Buckingham, “On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” _Physical Review_ , Oct. 1914, publisher: American Physical Society (APS). [Online]. Available: https://www.scienceopen.com/document?vid=805fe995-1849-413a-b228-3fe616732290
  * [16] J. Bakarji, J. Callaham, S. L. Brunton, and J. N. Kutz, “Dimensionally consistent learning with Buckingham Pi,” _Nature Computational Science_ , vol. 2, no. 12, pp. 834–844, Dec. 2022. [Online]. Available: https://www.nature.com/articles/s43588-022-00355-5
  * [17] K. Fukami and K. Taira, “Robust machine learning of turbulence through generalized Buckingham Pi-inspired pre-processing of training data,” p. A31.004, Jan. 2021, conference Name: APS Division of Fluid Dynamics Meeting Abstracts ADS Bibcode: 2021APS..DFDA31004F. [Online]. Available: https://ui.adsabs.harvard.edu/abs/2021APS..DFDA31004F
  * [18] X. Xie, A. Samaei, J. Guo, W. K. Liu, and Z. Gan, “Data-driven discovery of dimensionless numbers and governing laws from scarce measurements,” _Nature Communications_ , vol. 13, no. 1, p. 7562, Dec. 2022, number: 1 Publisher: Nature Publishing Group. [Online]. Available: https://www.nature.com/articles/s41467-022-35084-w
  * [19] A. S. P. Singh and N. Osamu, “Nondimensionalized indices for collision avoidance based on optimal control theory,” in _36th FISITA World Automotive Congress, 2016, September 26, 2016 - September 30, 2016_ , ser. FISITA 2016 World Automotive Congress - Proceedings. FISITA.
  * [20] X. Luo, F. Xie, X.-J. Liu, and Z. Xie, “Kinematic calibration of a 5-axis parallel machining robot based on dimensionless error mapping matrix,” vol. 70, p. 102115. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0736584521000016
  * [21] A. Girard, “Dimensionless policies based on the buckingham $\pi$ theorem: Is it a good way to generalize numerical results?” 2023.
  * [22] O. Lecompte, W. Therrien, and A. Girard, “Experimental investigation of a maneuver selection algorithm for vehicles in low adhesion conditions,” vol. 7, no. 3, pp. 407–412, conference Name: IEEE Transactions on Intelligent Vehicles.