Practical Position Tracking for 3D Unicycle Model

Lin, Haoyan; Cai, He

doi:https://doi.org/10.1155/2022/5725917

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5725917 | https://doi.org/10.1155/2022/5725917

Practical Position Tracking for 3D Unicycle Model

Haoyan Lin¹and He Cai¹

Academic Editor: Javier Moreno-Valenzuela

Received27 Jul 2022

Revised30 Aug 2022

Accepted14 Sept 2022

Published30 Sept 2022

Abstract

This paper considers practical position tracking for the 3D unicycle model. In contrast to the existing hand position coordinate transformation, a novel modified hand position coordinate transformation is proposed for the 3D unicycle model which can guarantee that the induced velocity transformation matrix is nonsingular for all the possible orientations. By employing the proposed modified hand position coordinate transformation, the nonlinear 3D unicycle position kinematics can be converted into the single integrator form, which enables arbitrary position reference trajectory tracking given any prespecified upper bound for the steady-state tracking error. The control performance is examined by numerical examples.

1. Introduction

The guidance and control problem for robotic system has long been a central topic for both the control and robotic communities [1–3], which leads to fruitful results in the literature, for instance, the intelligent neural network-based control approaches [4–6] and the trajectory tracking for bio-inspired smart robots [7–9], just to name a few. Unicycle is widely adopted as a simplified mathematical model for mobile vehicles, such as the differential drive vehicles [10, 11], aircraft-like vehicles [12, 13], underwater vehicles [14], and so on. The control of the unicycle model, including both stabilization and tracking, is a well-known benchmark problem which has attracted extensive attention from both the control and robotic communities. Though appearing deceptively simple, the control of the unicycle model is indeed rather challenging in that such a system cannot be stabilized by any smooth time-invariant feedback due to its nonholonomic constraint [15].

The vector of generalized coordinate of a unicycle model consists of both position and orientation, and all the vectors of generalized coordinate constitute the configuration space of the unicycle model. Note that the nonholonomic constraint requires that all the admissible vectors of generalized velocity belong to the null space of the constraint matrix, while the motion in the configuration space is not limited. Therefore, the unicycle model is underactuated for the case of configuration tracking, where the configuration reference trajectory must satisfy the nonholonomic constraint. The local and global configuration tracking for 2D unicycle model has so far been well investigated, which leads to various discontinuous and/or time-varying control schemes (see [16–19] and references therein). On the other hand, if only the position tracking of the unicycle model is of interest, then the number of system inputs and outputs will be equal and thus the unicycle model becomes fully actuated. In [20], a hand position coordinate transformation was proposed to solve the stabilization problem for the 2D unicycle model. A salient feature of the hand position coordination transformation is that the induced velocity transformation matrix is nonsingular for all the possible orientations, through which the nonlinear position kinematics of the unicycle model can be converted into the single integrator form, thus allowing arbitrary position reference trajectory tracking given any prespecified upper bound for the steady-state tracking error, called practical position tracking. Since [20], there have been many results based on this hand position coordination transformation, especially on the formation control of multiple unicycle systems [21–24].

The 3D unicycle model is an extension of the 2D unicycle model from planar motion to space motion. Unlike the 2D unicycle model, there have been thus far less results on the motion control of the 3D unicycle model since it is much more complex than that of the 2D unicycle model. In particular, the 3D unicycle model has three Cartesian coordinates and two angle coordinates, while the 2D unicycle model has two Cartesian coordinates and only one angle coordinate. As a result, two extra variables are involved with the nonholonomic constraint for the 3D unicycle model, thus making the kinematic equations for the 3D unicycle model being subject to more complex nonlinearities in terms of the product of sinusoidal functions of the angles. For the 3D unicycle model, the dipolar navigation function-based navigation methodology was invoked in [13] for solving the collision avoidance problem. Later, Wiig et al. [25] studied the same collision avoidance problem by maintaining a constant avoidance angle to the obstacle. Besides, the authors in [12, 26, 27] considered the flocking or formation control problem for a group of 3D unicycle models. In particular, inspired by the work of Pomet et al. [20], a 3D hand position coordinate transformation was proposed by Li et al. [27]. However, the induced velocity transformation matrix of the 3D hand position coordinate transformation would be singular for certain orientations. Therefore, the resulting practical position tracking control shall also suffer from the singularity issues.

To address this issue, in this paper, a novel modified hand position coordinate transformation for the 3D unicycle model is proposed, which, different from the hand position coordinate transformation, can guarantee that the induced velocity transformation matrix is nonsingular for all the possible orientations, and thus it makes it possible to achieve practical position tracking for arbitrary position reference trajectory for the 3D unicycle model.

In contrast to the existing results, the main contributions of this paper are as follows:(i)In [12], complex projection operators were designed to achieve local leaderless consensus. In contrast, the control approach proposed in this paper is a simple PD type controller which can achieve global practical position reference tracking.(ii)The control laws in [13, 26] are discontinuous. Moreover, the nonsmooth analysis of Roussos et al. [13] is based on an approximated scheme and the initial angular values of the agents in [26] should be within some specified range, while the control approach proposed in this paper is continuous and valid for any system initial condition, and the result is proven without any approximation.(iii)The underwater vehicles considered in [25] are assumed to be passively stable with respect to the motion of rolling, while in this paper, no assumption is imposed on the motion of the 3D unicycle.(iv)Different from the hand position coordinate transformation proposed in [27] which will be singular at certain angle, the modified hand position coordinate transformation proposed in this paper is singularity free.

2. 2D Unicycle Model Revisited

In this section, we will revisit the practical position tracking for the 2D unicycle model, which is given by (to save space, given an angle , define and )where denotes the Cartesian position of the vehicle center, denotes the orientation, and denote the linear and angular velocity input, respectively, and denotes the vector of generalized coordinate. As in [10, 28, 29], the nonholonomic constraint of (1 1 1) is given byor in matrix form

By (3), the vector of generalized velocity must belong to the null space of the constraint matrix , while the motion in the configuration space is free.

Suppose the position reference trajectory for 2D unicycle model is which is differentiable. Then, the practical position tracking problem for 2D unicycle model is given as follows.

Problem 1. (practical position tracking for 2D unicycle model). Given system (1a)–(1c), define the trajectory tracking error asGiven any , design control input such thatIllustrated by Figure 1, the following hand position coordinate transformation was proposed in [20]:where . Then, it follows thatSince det , the induced velocity transformation matrix is nonsingular for all . Thus, we have the following result.

Theorem 1. By choosing , Problem1is solvable by the following control law:for any .

Proof 1. Submitting (8) into (7) givesLet and . Then by (9), we haveAs a result, since , it follows thati.e.,Note thatTherefore, by (12), it follows that

Remark 1. Through the hand position transformation, the nonlinear position kinematics (1a) and (1b) have been converted into the form of equation (7). By compensating the nonsingular velocity transformation matrix , equation (7) becomes a single integrator, thus allowing arbitrary hand position reference trajectory tracking, which is equivalent to arbitrary position reference trajectory tracking given any prespecified upper bound for the steady-state tracking error, depending on the design parameter , while a trade-off on the selection of the distance parameter lies in that smaller indicates smaller steady-state tracking error, but possibly larger control inputs and .

3. Main Results

In this section, we will present the practical position tracking for 3D unicycle model, which is given bywhere denotes the Cartesian position of the vehicle center, denotes the orientation, denotes the linear velocity control input, denotes the angular velocity control inputs, and denotes the vector of generalized coordinate. Similar to (2), the nonholonomic constraint of (15 15 15 15 15) is given byor in matrix form

By the Lie bracket verification as in [28, 29], it can be verified that the velocity constraint (17) is not integrable.

Suppose the position reference trajectory for 3D unicycle model is given by which is differentiable. Then, the practical position tracking problem for 3D unicycle model is given as follows.

Problem 2. (practical position tracking for 3D unicycle model). Given system (12), define the trajectory tracking error asGiven any , design the control input such thatInspired by Pomet et al. [20], the following 3D hand position coordinate transformation was adopted in [27]:where . Then, it follows thatSince det , the velocity transformation matrix will be singular at .
To address this issue, in this paper, we define a modified hand position coordinate transformation as follows:where . In geometry, the term represents a position offset along the direction of the velocity, while the term represents a position offset along the direction of the projection of the velocity in the XOY plane. A two-step illustration is given in Figure 2.
It follows thatSince det , the velocity transformation matrix will be nonsingular for all as long as . Then, we have the following result.

Theorem 2. By choosing , Problem2is solvable by the following control law:for any .

Proof 2. Submitting (24) into (23) givesAs a result, similar to the proof of the 2D unicycle case, it follows thatWe haveTherefore, by (26), it follows that

Remark 2. Similar to the case of the 2D unicycle model, there is a trade-off on the selection of the distance parameters . In particular, the smaller are, the smaller the steady-state tracking error is, but possibly the larger the magnitude of the control inputs is.

Remark 3. By checking (24) and (25) together, the control law (24) is indeed a PD type controller where can be viewed as a feedback linearization term. In particular, the vectors andare the proportional part and derivative part of the PD type controller, respectively.

4. Numerical Examples

In this section, numerical examples are presented to examine the performance of the proposed control method. Suppose the position reference trajectory is given by

Note that in this case, the orientation reference trajectories can be calculated by

Moreover, suppose and let . In what follows, we examine the control performance for three different cases.

Case 1. Effect of initial configuration.
In this case, we show the control performance of the proposed control method subject to different initial configurations. Let , . Three initial configurations are considered:(1)IC1: .(2)IC2: .(3)IC3: .The control performance is shown in Figure 3. It can be seen that successful practical tracking has been achieved for all these three initial configurations. Moreover, both the transient and steady-state control inputs are of the same magnitude.

(a)

(b)

(c)

(d)

Case 2. Effect of distance parameters.
In this case, we show the control performance of the proposed control method subject to different distance parameters. Let . Three sets of distance parameters are considered:(1)DP1: , .(2)DP2: , .(3)DP3: , .The control performance is shown in Figure 4. It can be seen that smaller distance parameters lead to smaller steady-state tracking errors, yet resulting in larger control magnitude.

(a)

(b)

(c)

(d)

Case 3. Effect of noisy orientation measurements.
In this case, we show the control performance of the proposed control method subject to noisy orientation measurements. Let and be the measured orientations, which are supposed to be and with being the stochastic measurement noises. Suppose . Let , , and . The control performance is shown in Figure 5. It can be seen that noisy orientation measurements do not affect much the control performance.

(a)

(b)

(c)

(d)

Case 4. Comparison with the classic hand position coordinate transformation.
The modified hand position coordinate transformation proposed in this paper is global without singularity issues, while the classic hand position coordinate transformation proposed in [27] is not. In this case, we will show this fact by numerical simulations. To this end, we modify the position reference trajectory to beSuppose . Let , , and . The initial configuration is given byThe simulation results for both methods are shown in Figure 6, where M-HPCT is short for modified hand position coordinate transformation proposed in this paper and HPCT is short for hand position coordinate transformation proposed in [27]. It can be seen that the though the control input for the M-HPCT has a large transient at sec, it is still bounded. However, all the three control inputs for the HPCT method tend to go unbounded since in this case, , which is singular for the HPCT method.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

5. Conclusion

This paper proposed a novel modified hand position coordinate transformation for the 3D unicycle model. Since the induced velocity transformation matrix is nonsingular for all the possible orientations, the system kinematics can be converted into the single integrator form, which makes it simple to achieve practical position tracking for the 3D unicycle model, while a drawback of the current method might be that the resulting control input may incur large transient for some special types of position reference trajectories, such as Case 4 in the numerical example. In the future, it would be interesting to overcome this issue possibly by integrating the modified hand position coordinate transformation with path planning techniques.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was funded in part by the National Natural Science Foundation of China under grant no. 62173149 and in part by the Guangdong Natural Science Foundation under grant no. 2022A1515011262.

References

S. G. Tzafestas, “Mobile robot control and navigation: a global overview,” Journal of Intelligent and Robotic Systems, vol. 91, no. 1, pp. 35–58, 2018.
View at: Publisher Site | Google Scholar
H. Y. Zhang, W. M. Lin, and A. X. Chen, “Path planning for the mobile robot: a review,” Symmetry, vol. 10, no. 10, 450 pages, 2018.
View at: Publisher Site | Google Scholar
A. Billard and D. Kragic, “Trends and challenges in robot manipulation,” Science, vol. 364, 6446 pages, 2019.
View at: Publisher Site | Google Scholar
C. Yang, Y. Jiang, J. Na, Z. Li, L. Cheng, and C.-Y. Su, “Finite-time convergence adaptive fuzzy control for dual-arm robot with unknown kinematics and dynamics,” IEEE Transactions on Fuzzy Systems, vol. 27, no. 3, pp. 574–588, 2019.
View at: Publisher Site | Google Scholar
L. Kong, W. He, W. Yang, Q. Li, and O. Kaynak, “Fuzzy approximation-based finite-time control for a robot with actuator saturation under time-varying constraints of work space,” IEEE Transactions on Cybernetics, vol. 51, no. 10, pp. 4873–4884, 2021.
View at: Publisher Site | Google Scholar
C. Liu, G. Wen, Z. Zhao, and R. Sedaghati, “Neural-network-based sliding-mode control of an uncertain robot using dynamic model approximated switching gain,” IEEE Transactions on Cybernetics, vol. 51, no. 5, pp. 2339–2346, 2021.
View at: Publisher Site | Google Scholar
W. He, X. Tang, T. Wang, and Z. Liu, “Trajectory tracking control for a three-dimensional flexible wing,” IEEE Transactions on Control Systems Technology, vol. 30, no. 5, pp. 2243–2250, 2022.
View at: Publisher Site | Google Scholar
H. Huang, W. He, J. Wang, L. Zhang, and Q. Fu, “An all servo-driven bird-like flapping-wing aerial robot capable of autonomous flight,” IEEE, 2022.
View at: Publisher Site | Google Scholar
A. A. Transeth, R. I. Leine, C. Glocker, K. Y. Pettersen, and P. LiljebÄck, “Snake robot obstacle-aided locomotion: modeling, simulations, and experiments,” IEEE Transactions on Robotics, vol. 24, no. 1, pp. 88–104, 2008.
View at: Publisher Site | Google Scholar
Z. Qu, Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles, Springer, Berlin, 2009.
W. Ren and R. W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Applications, Springer, Berlin, 2008.
N. Moshtagh and A. Jadbabaie, “Distributed geodesic control laws for flocking of nonholonomic agents,” IEEE Transactions on Automatic Control, vol. 52, no. 4, pp. 681–686, 2007.
View at: Publisher Site | Google Scholar
G. Roussos, D. V. Dimarogonas, and K. J. Kyriakopoulos, “3D navigation and collision avoidance for nonholonomic aircraft-like vehicles,” International Journal of Adaptive Control and Signal Processing, vol. 24, no. 10, pp. 900–920, 2010.
View at: Publisher Site | Google Scholar
P. Millán, L. Orihuela, I. Jurado, and F. R. Rubio, “Formation control of autonomous underwater vehicles subject to communication delays,” IEEE Transactions on Control Systems Technology, vol. 22, no. 2, pp. 770–777, 2014.
View at: Publisher Site | Google Scholar
R. P. Brockett, “Asymptotic stability and feedback stabilization,” Differential Geometric Control Theory, Birkäuser, Basel, Switzerland, 1983.
View at: Google Scholar
T. Lee, K. Song, C. Lee, and C. Teng, “Tracking control of unicycle-modeled mobile robots using a saturation feedback controller,” IEEE Transactions on Control Systems Technology, vol. 9, no. 2, pp. 305–318, 2001.
View at: Publisher Site | Google Scholar
J. Huang, C. Wen, W. Wang, and Z. P. Jiang, “Adaptive output feedback tracking control of a nonholonomic mobile robot,” Automatica, vol. 50, no. 3, pp. 821–831, 2014.
View at: Publisher Site | Google Scholar
Y. H. Choi and D. Kim, “Distance-based formation control with goal assignment for global asymptotic stability of multi-robot systems,” IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 2020–2027, 2021.
View at: Publisher Site | Google Scholar
P. Zhang, T. Liu, and Z. Jiang, “Tracking control of unicycle mobile robots with event-triggered and self-triggered fedback,” IEEE Transactions on Automatic Control, 2022.
View at: Publisher Site | Google Scholar
J.-B. Pomet, B. Thuilot, G. Bastin, and G. Campion, “A hybrid strategy for the feedback stabilization of nonholonomic mobile robots,” in Proceedings of the IEEE Int. Conf. Robot. Autom, pp. 129–134, Nice, France, May 1992.
View at: Publisher Site | Google Scholar
W. Ren and E. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” International Journal of Robust and Nonlinear Control, vol. 17, no. 10-11, pp. 1002–1033, 2007.
View at: Publisher Site | Google Scholar
S. Liu, L. Xie, and F. L. Lewis, “Synchronization of multi-agent systems with delayed control input information from neighbors,” Automatica, vol. 47, no. 10, pp. 2152–2164, 2011.
View at: Publisher Site | Google Scholar
Y. Su and J. Huang, “Cooperative output regulation with application to multi-agent consensus under switching network,” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 42, no. 3, pp. 864–875, 2012.
View at: Publisher Site | Google Scholar
X. Cai and M. D. Queiroz, “Adaptive rigidity-based formation control for multi-robotic vehicles with dynamics,” IEEE Transactions on Control Systems Technology, vol. 23, no. 1, pp. 389–396, 2015.
View at: Publisher Site | Google Scholar
M. S. Wiig, K. Y. Pettersen, and T. R. Krogstad, “A 3D reactive collision avoidance algorithm for underactuated underwater vehicles,” Journal of Field Robotics, vol. 37, no. 6, pp. 1094–1122, 2020.
View at: Publisher Site | Google Scholar
V. Nazarzehi and A. V. Savkin, “Decentralized three dimensional formation building algorithms for a team of nonholonomic mobile agents,” International Journal of Control, Automation and Systems, vol. 17, no. 5, pp. 1283–1292, 2019.
View at: Publisher Site | Google Scholar
X. Li, C. Wen, and C. Chen, “Adaptive formation control of networked robotic systems with bearing-only measurements,” IEEE Transactions on Cybernetics, vol. 51, no. 1, pp. 199–209, 2021.
View at: Publisher Site | Google Scholar
A. M. Bloch and B. Brogliato, Nonholonomic Mechanics and Control, Springer, Berlin, 2000.
G. Klancar, A. Zdesar, S. Blazic, and I. Skrjanc, Wheeled Mobile Robotics: From Fundamentals towards Autonomous Systems, Butterworth-Heinemann, Oxford, UK, 2017.

Copyright

Copyright © 2022 Haoyan Lin and He Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies