Asymptotic Stability of Unicycle-Like Robots: The Bessel’s Controller

Email: andresgarcia@frbb.utn.edu.ar Abstract: Asymptotic stability of unicycle-like robots proved to be involved due to Brockett's condition. By using a smooth, time-invariant controller constructed out of Bessel’s functions, in this paper unicycle-like robots are uniformly-exponentially stabilized to the origin. The pure feedback controller obtained provides closed-form trajectories with the possibility of a simple and feasible (hardware) non-linear observer construction from posture angle measurements solely. Two examples are presented: Asymptotic steering of a unicycle to the origin using gyros and a perfect non-linear observer reconstruction states along with conclusions and future work.


Introduction
Modelling mechanical systems can be carried out in two main ways (Angeles and Kecskemethy, 1995):  Dynamic models including forces and torques  Kinematic models excluding forces and torques Both approaches aim to collect a system of Ordinary Differential Equations (ODE's) parameterized in its control inputs (Astolfi et al., 1997;Bloch, 2015).
These ODE systems are mainly non-linear, with a universal modeling given by Sarkar et al. (1994) in the case of kinematic rolling constraints.
It turns out that dynamic models represent the most general approach to account for all possible physical interactions that may occur. However, these models are out of as many ODE's as the mechanical systems' degrees of freedom (Kane and Levinson, 1985;Angeles and Kecskemethy, 1995).
At this point, two main challenges must be dealed with:  A great amount of ODE's  A control law rendering the system asymptotically stable/stable (Kostić et al., 2009;Udwadia and Kalaba, 1994;Skowronski, 2012) For these reasons, many researchers focus the attention on more tractable models, yet keeping the nonlinear richness with fewer amounts of ODE's (Muñoz-Lecanda and Yániz Fernandez, 2008).
This explains the great interest in controlling kinematic models (Siegwart et al., 2011) with the case of mobile robots as a subset of kinematic modeling, mainly, wheel planar kinematic models.
Moreover, it happens that these models can be classified into two universal classes: Holonomic and Nonholonomic (Garcia and Agamennoni (2012) for a universal classification and models).
On the other hand, it is well-known that a nonholonmic robot cannot be stabilized asymptotically with a smooth controller due to Brockett's condition (Brockett, 1983).
For this reason, many different techniques have been proposed to control nonholonomic robots avoiding the use of time-invariant controllers (Zambelli et al. (2015) and the references therein).
However, none of the available techniques considers the stabilization's problem in closed-form. In fact, according to Lizárraga (2004), it is not possible to track some desired trajectories with an equi-continuous control law.
To summarize the literature's drawbacks in controlling nonholonomic robots, either path-following or asymptotic stability:  Planar curves must be parameterized to be followed f(x,y) = 0 (Morro et al. (2011))  Path's curvature must satisfy some specifications: f(x,y)  0 ( Morro et al. (2011))  Very oscillatory and slow motion (see for instance Moon Kim and Tsiotras (2002))  Brockett's condition  Lizárraga's obstructions  Closed-form solution's unavailability for a universal set of models  Closed-form algorithms to track/follow any desired pre-specified trajectory In this paper, generalizing the solution presented in Garcia et al. (2008), a smooth time-invariant controller is presented to steer in closed-loop a unicycle-like robot to the origin uniformly exponentially stable.
The contributions in this paper are as follows:  Closed-form solution to steer unicycle robots to the origin  Continuous feedback controller with guaranteed stability  The proof that a unicycle can be asymptotically stabilized measuring considering modularity  A non-linear observer with angle output measurement (gyros) This paper is organized as follows: Section Rolling constraints presents the modeling of rolling constraints to be considered, trajectories' closed-form solution using a smooth, time-invariant controller, Section Unicycle's asymptotic stability presents the asymptotic stability analysis, Section Practical controller: Only gyros presents a practical algorithm, whereas Section Examples simulates in Matlab. Finally, Section Conclusions depicts some conclusions and future work.

Notations and Definitions
In this short section, some definitions are provided to use all along the paper:

Rotation Matrix
With I the identity matrix.

Matrix Transpose
Given a matrix nn A   , the transpose is denoted by A'.

Derivatives with Respect to Time
Time derivatives are indicated as:

Rolling Constraints: Unicycle-Like Robots
As mentioned previously, mechanical systems that roll without slipping encompass the modeling for many mechanical systems (Bloch, 2015).
Moreover, according to Murray and Shankar (1993) any nonholonomic system can be written in a universal chain form, so unicycle models can be considered as a general nonholonomic dynamics.
In particular, unicycle-like robots represents a universal modeling for a wide variety of wheeled robots (Garcia and Agamennoni, 2012) and Fig. 1: where the control inputs (u1, u2)   1   1 .

Bessel's Functions Closed-form Solutions
Following the ideas in Garcia et al. (2008), a lemma can be proved.

Lemma 1
Given the dynamics in Equation 1 driven by the controller: For any arbitrary N  , with a <0. The robot's trajectories are given by: With Ji the Bessel's functions of first kind and Ci arbitrary constants depending on the initial conditions.

Unicycle's Asymptotic Stability
Equation 1 defines a dynamics that it is endowed with uniform exponential stability by using Lemma 1 (Rugh (1995) for a definition on stability).

Theorem 1
The controller in Lemma 1 possess uniform exponentially stability to the origin.

Proof
The proof is in the Appendix. Clearly, it is a closed-loop and time-invariant controller, except for (0) = 0, (rendering the controller identically zero).
However, the posture angle is a modular quantity: This modularity property avoids Brockett's condition collision. On the other hand, in a paper by Aicardi et al. (1995) a closed-loop controller using Lyapunov techniques was presented, however that controller becomes singular at x(0) = 0, y(0) = 0, whereas the controller in this paper is well defined for the whole  3 .

Regular Embedded Sub-Manifold: Trajectories' First Integral
Notwithstanding that the closed-form solution obtained represents the complete system's time evolution; a geometrical point of view is of interest in what follows.
Considering the closed-form trajectories in Lemma 1 with N = 2: is nonsingular in the view of the Bourget's hypothesis proved in 1929 by Siegel (Watson, 1966).

Non-Linear Observer
Equation 3 can be utilized to derive a non-linear observer measuring only the angle posture (Luenberger (1966) and Isidori (1995) for linear and non-linear observers):

Practical Controller: Only Gyros
Once that uniform exponential stability has been proved in Theorem 1, a practical algorithm can be described to control in closed-loop a unicycle robot:  Determine the constant vector C from Equation 3 given the initial conditions  Use the controller in Lemma 1with N=2 and single gyro measurements: Notice that only the initial condition must be provided to initiate the algorithm, endowing the controller with a very strong property for the well-known SLAM problem (Lavalle, 2006).
Only SLAM must be made at t = 0 as opposed to the available literature where SLAM or time-tracking has to be performed on-line.

Examples
Using Matlab, Lemma 1 is implemented with both objectives: steering to the origin asymptotically stable and non-linear observer reconstruction.

Non-Linear Observer Verification
Equation 4 provides an interesting verification to numerically reconstruct robot's states. Then, Fig. 3 is obtained.

Discussion
Unicycle robot's asymptotic stability is not possible using a time-invariant feedback controller due to Brockett's condition.
In this paper, the modularity of the angular posture along with a novel pure feedback and time-invariant controller, allows asymptotic stability for unicycle robots.
It should be clear that modularity was not studied previously in the literature, excluding also closed form solutions addressed in this paper.
It turns out that, besides the fact of providing the important concept of a modular controller, SLAM and non-linear observers con be constructed in hardware using posture angle solely.

Conclusion
Nonholonomic trajectories of a unicycle-like robot is solved in closed-form using a smooth, closed-loop and time-invariant controller.
Uniform exponential stability was proved, even for this case (unicycle robot) where Brockett's condition is satisfied on the basis of modularity. In fact, postural angle's modularity was the cornerstone to avoid obstructions using smooth, closed-loop, time-invariant control laws.
Besides the wide variety of available literature, trajectories' closed-form knowledge allowing an explicit non-linear observer derivation to completely reconstruct states using only a single gyro sensor, makes a salient property of this paper.
Possible future work encompasses:  Numerical algorithm to drive a set of multiple robots with additional constraints (formation of robots)  Non-linear observer's numerical analysis robustness  Real-time applications using on-board gyros and microprocessors  Satellite control using gyros, applying the universal transformation in Murray and Shankar (1993)  Optimal control

Acknowledgement
The author would like to acknowledge María de los Angeles, María de los Angeles and Alicia for their constant support.

Funding Information
This work is supported by Universidad Tecnológica Nacional-Facultad Regional Bahía Blanca under the project 5122TC.

Ethics
This article is original and contains unpublished material. The corresponding author confirms that all of the other authors have read and approved the manuscript and no ethical issues involved.