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Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control
Alexandria Engineering Journal (2017) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control Yasir Awais Butt Department of Electronic Engineering, Capital University of Science and Technology, Islamabad, Pakistan Received 13 January 2017; revised 7 May 2017; accepted 10 May 2017
KEYWORDS Stabilization; Nonholonomic systems; Switching control; Integral sliding mode control
Abstract The problem of stabilization of a class of nonholonomic systems that can be transformed into chained form is considered. Since nonholonomic systems cannot be stabilized using smooth and time invariant control, a switching algorithm comprising three distinct steps is employed to stabilize the system. The proposed algorithm allows the system to be reduced into simpler subsystems and affords employing different controllers in each step. Using this approach, desired performance and robustness properties of the feedback control system can be guaranteed. The effectiveness of the proposed algorithm is established by applying it on a unicycle type system moving on hyperbolic plane. In order to make the controller robust, integral sliding mode control is introduced that guarantees robust stabilization. Simulation results confirm the mathematical developments. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction A system whose velocity constraints do not integrate into position constraints is called a nonholonomic system [1,2]. In differential geometric terms, a nonholonomic system is a double ðM; DÞ where M is the n-dimensional configuration manifold of the system and D is its velocity distribution with dimension m < n i.e., the distribution D is a sub-bundle of the system’s tangent bundle D TM [2,3]. The definition stated above has nontrivial physical implications. Restriction of velocities to a sub-bundle of the tangent bundle implies that for any point q1 2 M, the system is restricted to move only along certain directions i.e., the given velocity vector fields. However, it E-mail address: [email protected] Peer review under responsibility of Faculty of Engineering, Alexandria University.
is a well-known fact that nonholonomic systems satisfy the Hormander condition or the bracket generating condition and therefore by Rashevsky-Chow’s theorem they are completely controllable [4]. The bracket generating condition ensures that in addition to available velocity vector fields, the system has a velocity vector field that is orthogonal to the given vector fields which is why a nonholonomic system is completely controllable. Consequently, q1 2 M can be joined with any other point q2 2 M using a combination of available velocities. Nonholonomic systems are popular because they are naturally encountered in robotics e.g., parking of a car problem [5,6], Reeds Shepp car [7], car with trailer system [8], UAVs [9] and in other diverse fields such as neurogeometry of vision [10], economics [11] etc. Due to applied nature, there is tremendous interest in stabilization and tracking of nonholonomic systems. Roger Brockett proved that nonholonomic systems cannot be stabilized by
http://dx.doi.org/10.1016/j.aej.2017.05.017 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
2 continuous and time invariant controls [12]. Intuitively, every state of nonholonomic systems can be steered to origin by a combination of some control laws, but, such control laws cannot be joined in a continuous manner to yield a continuous globally stabilizing feedback control law [13]. This makes stabilization of nonholonomic systems rather interesting and somewhat challenging research problem. Number of approaches have been proposed to solve the stabilization problem including continuous and time varying control, discontinuous time invariant control and a combination of these two. Dynamic feedback stabilization technique was proposed in [14–17]. Stabilization via discontinuous control that guaranteed exponential stabilization was proposed in [18–21]. It was shown that through suitable coordinate transformation the system can be rendered discontinuous and in these coordinates suitable control law can be designed that ensures exponential rate of convergence [18]. Switching control based on continuous dynamics and discrete logic that switched controllers in various regions of the state space has also been employed effectively to solve the stabilization problem [22,23]. Most of the proposed time varying control techniques suffer from either of the two problems i.e., slow convergence rate and oscillatory behavior [14–16] or lack of robustness against disturbances or parametric variations [17]. The limitations of robustness and finite time convergence can be mitigated using another discontinuous control technique i.e., sliding mode control. Sliding modes are the main modes of variable structure systems in which the system is forced to follow a sliding surface s ¼ fðxÞ; x 2 M; s M, where x is the system state and M is the state manifold [24]. The control input that forces the system to stick to the surface is usually given as KsignðsÞ where K is the design parameter that determines rate of convergence to the sliding surface. Once on the surface, system is guaranteed to stabilize to origin due to the properties of the sliding surface. Sliding mode control (SMC) is inherently robust and can reject parametric variations as well as matched disturbances [24]. Sliding mode control due to its inherent robustness properties has been extensively used for stabilization of nonholonomic systems see e.g., [25–27] and references contained therein. Sliding mode is characterized by two distinct phases i.e., a reaching phase when the system trajectories are forced toward the sliding surface and the sliding phase when the system slides along the surface. An inherent limitation of the conventional sliding mode control is that robustness properties are guaranteed in the sliding phase only. It has been shown that the disturbances can render the system uncontrollable in the reaching phase. In order to overcome this limitation, integral sliding mode control (ISMC) was proposed that eliminates the reaching phase thus making the system robust throughout [28]. However, it is not straightforward to design SMC or ISMC for driftless kinematic nonholonomic systems. The difficulty stems from the fact that once the reaching phase is over and the states have reached the sliding surface, the control input vanishes and there are no dynamics/drift in the system that can guarantee system stabilization to the origin. In this paper a robust switching controller based on discrete switching logic and ISMC is proposed for stabilization of a class of nonholonomic systems. The class of systems under consideration is two input kinematic nonholonomic systems in chained form for two reasons i.e.,
Y.A. Butt For two input nonholonomic systems, the chained, power and Caplygin form and hence the proposed control technique are equivalent. Nonholonomic systems up to order n ¼ 4 can always be transformed into chained form [29]. Using switching decisions based on system state and ISMC, robustness and finite time stabilization are guaranteed. The proposed controller is simulated for stabilization of a unicycle moving on hyperbolic plane. Simulations confirm the mathematical claims. Rest of the paper is organized as follows. Section 2 states the detailed description of the problem under investigation. Section 3 presents the controller design technique and application of proposed control technique to a unicycle type nonholonomic system moving on hyperbolic plane. Simulation results are presented in Sections 4 and 5 concludes the paper. 2. Problem statement Consider a two input nonholonomic systems: z_ ¼ g1 ðzÞv1 þ g2 ðzÞv2 ;
z 2 Rn ;
ð2:1Þ
where gi ðzÞ are the velocity vector fields and vi ; i ¼ 1; 2 are the control inputs. System (2.1) in chained form is given as follows: z_1 ¼ v1 ; z_2 ¼ v2 ; z_3 ¼ z2 v1 ;
ð2:2Þ
z_4 ¼ z3 v1 ; .. . z_n ¼ zn1 v1 : Given a two input nonholonomic system (2.2) with zð0Þ ¼ z0 , the stabilization problem is to design vi ; i ¼ 1; 2 such that origin becomes globally asymptotically stabilizing equilibrium point. 3. Controller design 3.1. Controller design algorithm Following properties of the system in chained form (2.2) aid development of control algorithm: The system (2.2) can be divided into two subsystems. The equation z_ 1 ¼ v1 forms the first subsystem and the rest of the equations in (2.2) form the second subsystem. First subsystem can be stabilized independently of the second subsystem. If v1 is kept constant, the second subsystem becomes n 1 dimensional linear system in the controllable canonical form that is fully controllable through v2 . Hence, it can be stabilized independently of the first subsystem. Based on these properties, following switching control scheme is proposed:
Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
Robust stabilization of a class of nonholonomic systems
3
1. Step 1 – Stabilize first subsystem using v1 while keeping v2 ¼ 0.In order to achieve robust stabilization of state z1 , it is proposed to use conventional SMC. Define a sliding surface s ¼ z1 and a Lyapunov function V ¼ 12 s2 . Now, V_ ¼ ss_ ¼ sz_1 ¼ sv1 :
ð3:1Þ
Let us define v1 ¼ k1 s k2 signðsÞ. Consequently, V_ ¼ k1 s2 k2 j s j< 0; that guarantees strong reachability. Note that for stabilization of single state z1 , any linear control technique could be employed but that does not provide robustness against parametric variations or matched disturbance. Consider for example the subsystem 1 with bounded matched disturbance j c1 j6 C1 : z_1 ¼ v1 þ c1 : Using the controller v1 ¼ k1 s k2 signðsÞ; (3.1) can be rewritten as follows:
ð3:2Þ k1 ; k2 > 0,
V_ ¼ z1 ðk1 z1 k2 signðz1 Þ þ c1 Þ; ¼ k1 z21 þ z1 ðk2 signðz1 Þ þ c1 Þ; which is negative definite for C1 6 k2 . Hence, k2 can be used for disturbance rejection and k1 for desired speed of response. 2. Step 2 – Keeping v1 ¼ a where a – 0 is a constant, stabilize the second subsystem by suitably designing v2 using any control technique. Note that since a does not play any significant role in system performance, it can be set to a ¼ 1. Setting v1 ¼ a renders the n 1 dimensional second subsystem linear. Resulting second subsystem is given as follows: z_2 z_3 z_4 .. . z_n
¼ v2 ; ¼ z2 ; ¼ z3 ;
ð3:3Þ
¼ zn1 ;
which is a linear system in controllable canonical form. System (3.3) can be trivially stabilized using pole placement with v2 ¼ l2 z2 l3 z3 ln zn , where li > 0; i ¼ 2; 3; . . . ; n are the state feedback gains that can be designed to achieve exponential stabilization, but, not robustness. It is therefore proposed to stabilize this subsystem with ISMC using gains computed for pole placement. In order to use ISMC, input v2 is redefined as v2 ¼ v20 þ v21 where v20 ¼ l2 z2 l3 z3 ln zn is the nominal controller obtained from pole placement. Let the sliding surface for conventional SMC be given as s0 ¼ c2 z2 þ c3 z3 þ þ zn , then the sliding surface for ISMC is given as follows: s ¼ s0 þ r;
ð3:4Þ
where r is the integral term that is used to eliminate reaching phase in ISMC. In order to eliminate reaching phase it must be true that sð0Þ ¼ 0 ) rð0Þ ¼ s0 ð0Þ [28]. Dynamics of the auxiliary variable r are defined by setting s_ ¼ 0:
s_ ¼ s_0 þ r_ ¼ 0; s_ ¼ z2 ðc2 l2 Þ þ z3 ðc3 l3 Þ þ þ zn ðcn ln Þ þ r_ ¼ 0; r_ ¼ z2 ðc2 l2 Þ z3 ðc3 l3 Þ zn ðcn ln Þ;
rð0Þ ¼ s0 ð0Þ:
ð3:5Þ
This ensures that sliding mode is established from t ¼ 0. During sliding, system dynamics are governed by the nominal controller v20 that grantees exponential stabilization. In order to ensure robustness against parametric variations, matched disturbance and to enforce sliding mode along the manifold, discontinuous component of the input is defined as follows: v21 ¼ MsignðsÞ: Hence, overall controller for stabilization of states zi ; i ¼ 2; 3; . . . ; n is given as follows: v2 ¼ l2 z2 l3 z3 ln zn MsignðsÞ:
ð3:6Þ
3. Step 3 – Keeping v1 constant in step 2 causes z1 to diverge from origin. To overcome this, repeat step 1. Since zi ¼ 0; i ¼ 2; 3; . . . ; n in step 2, repeating step 1 does not diverge these states.
3.2. Controller design for unicycle system In order to establish effectiveness of the control technique, the switching controller is applied to the mathematical model of a unicycle which is a benchmark nonholonomic system. Unicycle motion is primarily considered on Euclidean plane which is a flat surface with zero curvature. It is however an established fact that real world surfaces are seldom flat and are generally described by a combination of spherical and hyperbolic surfaces. Due to practical significance, motion of a unicycle is considered on a hyperbolic plane or a pseudo Euclidean plane i.e., a saddle-like surface that has constant negative curvature j ¼ 1 . Optimal control via geometric control techniques of this system was considered in [30–32]. However, geometric control techniques presented in these references are hard to implement for real world applications and therefore stabilization of this system is considered with the control technique presented in Section 3. A unicycle moving on hyperbolic plane is described as follows: 2 3 2 3 2 3 x_ cosh z 0 6 7 6 7 6 7 ð3:7Þ 4 y_ 5 ¼ 4 sinh z 5u1 þ 4 0 5u2 ; z_ 0 1 where ðx; yÞ is the point of contact of unicycle with the hyperbolic plane, z is its orientation and u1 ; u2 are the translational and rotational velocities respectively. The configuration manifold M along with the motions of the hyperbolic plane form a Lie group called special hyperbolic group SH(2). 3.2.1. Transformation to chained form v1 Let u1 ¼ cosh ; u2 ¼ v2 cosh2 z and redefine the state variables as z z1 ¼ x; z2 ¼ tanh z and z3 ¼ y. System (3.7) is now given as follows:
Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
4
Y.A. Butt
2 3 1 0 z_1 6 7 6 7 6 7 4 z_2 5 ¼ 4 0 5v1 þ 4 1 5v2 : z2 z_3 0 2
3
2
3
ð3:8Þ
It is trivial to see that stabilization of state variables zi ; i ¼ 1; 2; 3 is equivalent to stabilization of ðx; y; zÞ. Therefore, in the remainder text stabilization of states zi is investigated. 3.2.2. Controller design 1. Step 1 – Set v2 ¼ 0 and v1 ¼ k 1 z1 k 2 signðz1 Þ to stabilize z1 . For simulation purposes gains are given as k 1 ¼ k 2 ¼ 5. 2. Step 2 – Set v1 ¼ 1 so that the resulting subsystem 2 is given as follows: z_2 0 0 z2 1 ð3:9Þ ¼ þ v2 : z_3 1 0 0 z3
Figure 1
System states z1 ; z2 ; z3 ; r.
For simulation purposes the poles of (3.9) are placed at 1 for which the system is critically damped that ensures fastest speed of response without overshoot. State feedback gains for this choice of poles are ½ l2 l3 ¼ ½ 2 1 . Hence the nominal controller is given as v20 ¼ 2z2 z3 . In order to design ISMC choose the sliding surface s0 ¼ z2 þ z3 and therefore sliding surface with auxiliary integral term r is given as follows: s ¼ s0 þ r ¼ z2 þ z3 þ r; _ ) s_ ¼ z_2 þ z_3 þ r_ ¼ v20 þ z2 þ r: From (3.5): r_ ¼ z2 z3 ;
rð0Þ ¼ s0 ð0Þ ¼ z2 ð0Þ z3 ð0Þ:
Discontinuous input that ensures that sliding surface s is attractive is given as v21 ¼ MsignðsÞ. For simulations use M ¼ 5. Hence, v2 ¼ 2z2 z3 5signðsÞ:
Figure 2
Sliding surface s ¼ z2 þ z3 þ r ¼ 0.
3. Step 3 – Repeat step 1 to stabilize z1 .
4. Simulation results and discussion The intuitive reasoning behind the proposed algorithm is more comprehensible in terms of the unicycle system. In step 1, x is stabilized that brings the unicycle to y-axis. In step 2, y; z are stabilized, thus moving the unicycle to x-axis, but, in doing so the unicycle moves away causing x – 0 and y; z ¼ 0. Physically, the unicycle is somewhere on the x-axis with orientation of z ¼ 0. In step 3, the control forces the unicycle to move along x-axis and stabilizes it to the origin. Simulation results for stabilization of unicycle system (3.8) are presented in Figs. 1–5. From Fig. 1, it is seen that all states are stabilized to origin in finite time. Switching of controls takes place at times t ¼ T1 and t ¼ T2 . The auxiliary state variable r is visible only for t 2 ½T1 ; T2 . The sliding surface for ISMC of step 2 is given in Fig. 2. It is evident that there is no reaching phase, however, high frequency switching called chattering is present that nearly vanishes when the states converge to origin. The xy plot is presented in Fig. 3 which confirms our intuitive reasoning of the proposed algorithm. Step 1 stabilizes state x ¼ z1 to y-axis. In step 2, state y ¼ z3 and tanh z ¼ z2 is stabilized to the
Figure 3
xy plot.
x-axis. Finally, in step 3, the unicycle moves on the x-axis to stabilize state x to origin. The controller effort for both control inputs v1 and v2 is show in Fig. 4. Magnitude of control inputs v1 ; v2 in steps 1–3 and chattering due to sliding control can be
Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
Robust stabilization of a class of nonholonomic systems
Figure 4
Figure 5
5
Controller effort.
System states z1 ; z2 ; z3 ; r and controller effort with input disturbance.
clearly seen in both plots. It is also evident that the control input is finite. Effect of input disturbance is shown in Fig. 5. A unit step disturbance is introduced in the state z1 during step 3 at t ¼ 13 s. This causes a positive magnitude shift in the state z1 . However, the disturbance is compensated by the controller and the state z1 converges to origin in finite time. Effect of disturbance is also visible in the control effort plot. Since, the disturbance is introduced in state z1 during step 3, only v1 adjusts to respond to the disturbance whereas v2 remains unaffected. 5. Conclusion The proposed algorithm along with SMC/ISMC ensures exponential stabilization of nonholonomic systems in chained form. ISMC eliminates reaching phase ensuring robust stabilization, nonetheless it suffers from signature chattering problem of SMC. One of the ways to eliminate chattering while still ensuring robustness is using integral higher order sliding mode control with the same logical switching control technique. Other trivial way is to replace the sign term with the saturation of inverse hyperbolic function which can reduce the chattering problem.
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Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
6 [9] P. Panyakeow, M. Mesbahi, Decentralized deconfliction algorithms for unicycle UAVs, in: American Control Conference (ACC), 2010, pp. 794–799. [10] A.P. Mashtakov, A.A. Ardentov, Yu.L. Sachkov, Parallel algorithm and software for image inpainting via subRiemannian minimizers on the group of rototranslations, Numer. Math. Theor. Meth. Appl. 6 (2013) 95–115. [11] C. Udriste, M. Ferrara, Nonholonomic geometry of economic systems, in: 4th European Computing Conference, 2010. [12] R.W. Brockett, Differential geometric control theory, Asymptotic Stability and feedbak stabilization, Birkhauser, Boston, 1983. [13] Daniel Liberzon, Switching in Systems and Control, Systems not stabilizable by continuous feedback, Birkhauser, Boston, 2003. [14] R. Murray, S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Autom. Control 38 (5) (1993) 700– 716. [15] N. Khaneja, R. Brockett, Dynamic feedback stabilization of NH systems, in: 38th Conference on Decision and Control, Pheonix, Arizona, USA, 1999. [16] C. Samson, Control of chained systems: application to path following and time-varying point-stabilization of mobile robots, IEEE Trarns. Autom. Control 40 (1995) 64–77. [17] Yu-Ping Tian, Shihua Li, Smooth exponential stabilization of nonholonomic systems via time-varying feedback, in: 39th IEEE CDC, 2000. [18] A. Astolfi, Discontinuous control of nonholonomic systems, Lett. Syst. Control 27 (1) (1996) 37–45. [19] A. Astolfi, Asymptotic stabilization of nonholonomic systems with discontinuous control (Ph.D. thesis), Swiss federal institute of technology, Zurich, 1996. [20] N. Marchand, M. Alamir, Discontinuous exponential stabilization of chained form systems, Automatica (2003). [21] A. Astolfi, Exponential stabilization of nonholonomic systems via discontinuous control, in: Symposium on Nonlinear Control System Design, Lake Tahoe, USA, 1995, pp. 741–746.
Y.A. Butt [22] A.S. Morse, J.P. Hespanha, Stabilization of nonholonomic integrators via logic-bases switching, Automatica 35 (3) (1999) 385–393. [23] J.P. Hespanha, D. Liberzon, A.S. Morse, Logic-based switching control of a nonholonomic system with parametric modeling uncertainty, Syst. Control Lett. 38 (1999) 167–177. [24] V.I. Utkin, J. Guldner, J. Shi, Sliding Mode Control in Electromechanical Systems, second ed., CRC Press, 2009. [25] A.M. Bloch, S. Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes, Syst. Control Lett. 29 (2) (1996) 91–99. [26] J. Min Yang, J.-H. wan Kim, SMC for trajectory tracking of nonholonomic wheeled mobile robots, IEEE Trans. Robot. Autom. (1999). [27] T. Floquet, J.-P. Barbot, W. Perruquetti, HOSM stabilization for a class of nonholonomic perturbed systems, Automatica 39 (6) (2003) 1077–1083. [28] J. Shi, V. Utkin, Integral sliding mode in systems operating under uncertainly conditions, in: IEEE CDC, 1996, pp. 4591– 4596. [29] Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, 1994. [30] Y.A. Butt, Yu. L. Sachkov, A.I. Bhatti, Extremal trajectories and Maxwell strata in sub-Riemannian problem on group of motions of pseudo-Euclidean plane, JDCS 20 (3) (2014) 341– 364. [31] Y.A. Butt, Yu. L. Sachkov, A.I. Bhatti, Maxwell strata and conjugate points in the sub-Riemannian problem on the Lie group SH(2), J. Dyn. Control Syst. 22 (4) (2016) 747–770, Springer. [32] Yasir Awais Butt, Yuri L. Sachkov, Aamer Iqbal Bhatti, Cut locus and optimal synthesis in sub-riemannian problem on the lie group sh(2), J. Dyn. Control Syst. 23 (1) (2017) 155–195.
Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control Yasir Awais Butt Department of Electronic Engineering, Capital University of Science and Technology, Islamabad, Pakistan Received 13 January 2017; revised 7 May 2017; accepted 10 May 2017
KEYWORDS Stabilization; Nonholonomic systems; Switching control; Integral sliding mode control
Abstract The problem of stabilization of a class of nonholonomic systems that can be transformed into chained form is considered. Since nonholonomic systems cannot be stabilized using smooth and time invariant control, a switching algorithm comprising three distinct steps is employed to stabilize the system. The proposed algorithm allows the system to be reduced into simpler subsystems and affords employing different controllers in each step. Using this approach, desired performance and robustness properties of the feedback control system can be guaranteed. The effectiveness of the proposed algorithm is established by applying it on a unicycle type system moving on hyperbolic plane. In order to make the controller robust, integral sliding mode control is introduced that guarantees robust stabilization. Simulation results confirm the mathematical developments. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction A system whose velocity constraints do not integrate into position constraints is called a nonholonomic system [1,2]. In differential geometric terms, a nonholonomic system is a double ðM; DÞ where M is the n-dimensional configuration manifold of the system and D is its velocity distribution with dimension m < n i.e., the distribution D is a sub-bundle of the system’s tangent bundle D TM [2,3]. The definition stated above has nontrivial physical implications. Restriction of velocities to a sub-bundle of the tangent bundle implies that for any point q1 2 M, the system is restricted to move only along certain directions i.e., the given velocity vector fields. However, it E-mail address: [email protected] Peer review under responsibility of Faculty of Engineering, Alexandria University.
is a well-known fact that nonholonomic systems satisfy the Hormander condition or the bracket generating condition and therefore by Rashevsky-Chow’s theorem they are completely controllable [4]. The bracket generating condition ensures that in addition to available velocity vector fields, the system has a velocity vector field that is orthogonal to the given vector fields which is why a nonholonomic system is completely controllable. Consequently, q1 2 M can be joined with any other point q2 2 M using a combination of available velocities. Nonholonomic systems are popular because they are naturally encountered in robotics e.g., parking of a car problem [5,6], Reeds Shepp car [7], car with trailer system [8], UAVs [9] and in other diverse fields such as neurogeometry of vision [10], economics [11] etc. Due to applied nature, there is tremendous interest in stabilization and tracking of nonholonomic systems. Roger Brockett proved that nonholonomic systems cannot be stabilized by
http://dx.doi.org/10.1016/j.aej.2017.05.017 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
2 continuous and time invariant controls [12]. Intuitively, every state of nonholonomic systems can be steered to origin by a combination of some control laws, but, such control laws cannot be joined in a continuous manner to yield a continuous globally stabilizing feedback control law [13]. This makes stabilization of nonholonomic systems rather interesting and somewhat challenging research problem. Number of approaches have been proposed to solve the stabilization problem including continuous and time varying control, discontinuous time invariant control and a combination of these two. Dynamic feedback stabilization technique was proposed in [14–17]. Stabilization via discontinuous control that guaranteed exponential stabilization was proposed in [18–21]. It was shown that through suitable coordinate transformation the system can be rendered discontinuous and in these coordinates suitable control law can be designed that ensures exponential rate of convergence [18]. Switching control based on continuous dynamics and discrete logic that switched controllers in various regions of the state space has also been employed effectively to solve the stabilization problem [22,23]. Most of the proposed time varying control techniques suffer from either of the two problems i.e., slow convergence rate and oscillatory behavior [14–16] or lack of robustness against disturbances or parametric variations [17]. The limitations of robustness and finite time convergence can be mitigated using another discontinuous control technique i.e., sliding mode control. Sliding modes are the main modes of variable structure systems in which the system is forced to follow a sliding surface s ¼ fðxÞ; x 2 M; s M, where x is the system state and M is the state manifold [24]. The control input that forces the system to stick to the surface is usually given as KsignðsÞ where K is the design parameter that determines rate of convergence to the sliding surface. Once on the surface, system is guaranteed to stabilize to origin due to the properties of the sliding surface. Sliding mode control (SMC) is inherently robust and can reject parametric variations as well as matched disturbances [24]. Sliding mode control due to its inherent robustness properties has been extensively used for stabilization of nonholonomic systems see e.g., [25–27] and references contained therein. Sliding mode is characterized by two distinct phases i.e., a reaching phase when the system trajectories are forced toward the sliding surface and the sliding phase when the system slides along the surface. An inherent limitation of the conventional sliding mode control is that robustness properties are guaranteed in the sliding phase only. It has been shown that the disturbances can render the system uncontrollable in the reaching phase. In order to overcome this limitation, integral sliding mode control (ISMC) was proposed that eliminates the reaching phase thus making the system robust throughout [28]. However, it is not straightforward to design SMC or ISMC for driftless kinematic nonholonomic systems. The difficulty stems from the fact that once the reaching phase is over and the states have reached the sliding surface, the control input vanishes and there are no dynamics/drift in the system that can guarantee system stabilization to the origin. In this paper a robust switching controller based on discrete switching logic and ISMC is proposed for stabilization of a class of nonholonomic systems. The class of systems under consideration is two input kinematic nonholonomic systems in chained form for two reasons i.e.,
Y.A. Butt For two input nonholonomic systems, the chained, power and Caplygin form and hence the proposed control technique are equivalent. Nonholonomic systems up to order n ¼ 4 can always be transformed into chained form [29]. Using switching decisions based on system state and ISMC, robustness and finite time stabilization are guaranteed. The proposed controller is simulated for stabilization of a unicycle moving on hyperbolic plane. Simulations confirm the mathematical claims. Rest of the paper is organized as follows. Section 2 states the detailed description of the problem under investigation. Section 3 presents the controller design technique and application of proposed control technique to a unicycle type nonholonomic system moving on hyperbolic plane. Simulation results are presented in Sections 4 and 5 concludes the paper. 2. Problem statement Consider a two input nonholonomic systems: z_ ¼ g1 ðzÞv1 þ g2 ðzÞv2 ;
z 2 Rn ;
ð2:1Þ
where gi ðzÞ are the velocity vector fields and vi ; i ¼ 1; 2 are the control inputs. System (2.1) in chained form is given as follows: z_1 ¼ v1 ; z_2 ¼ v2 ; z_3 ¼ z2 v1 ;
ð2:2Þ
z_4 ¼ z3 v1 ; .. . z_n ¼ zn1 v1 : Given a two input nonholonomic system (2.2) with zð0Þ ¼ z0 , the stabilization problem is to design vi ; i ¼ 1; 2 such that origin becomes globally asymptotically stabilizing equilibrium point. 3. Controller design 3.1. Controller design algorithm Following properties of the system in chained form (2.2) aid development of control algorithm: The system (2.2) can be divided into two subsystems. The equation z_ 1 ¼ v1 forms the first subsystem and the rest of the equations in (2.2) form the second subsystem. First subsystem can be stabilized independently of the second subsystem. If v1 is kept constant, the second subsystem becomes n 1 dimensional linear system in the controllable canonical form that is fully controllable through v2 . Hence, it can be stabilized independently of the first subsystem. Based on these properties, following switching control scheme is proposed:
Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
Robust stabilization of a class of nonholonomic systems
3
1. Step 1 – Stabilize first subsystem using v1 while keeping v2 ¼ 0.In order to achieve robust stabilization of state z1 , it is proposed to use conventional SMC. Define a sliding surface s ¼ z1 and a Lyapunov function V ¼ 12 s2 . Now, V_ ¼ ss_ ¼ sz_1 ¼ sv1 :
ð3:1Þ
Let us define v1 ¼ k1 s k2 signðsÞ. Consequently, V_ ¼ k1 s2 k2 j s j< 0; that guarantees strong reachability. Note that for stabilization of single state z1 , any linear control technique could be employed but that does not provide robustness against parametric variations or matched disturbance. Consider for example the subsystem 1 with bounded matched disturbance j c1 j6 C1 : z_1 ¼ v1 þ c1 : Using the controller v1 ¼ k1 s k2 signðsÞ; (3.1) can be rewritten as follows:
ð3:2Þ k1 ; k2 > 0,
V_ ¼ z1 ðk1 z1 k2 signðz1 Þ þ c1 Þ; ¼ k1 z21 þ z1 ðk2 signðz1 Þ þ c1 Þ; which is negative definite for C1 6 k2 . Hence, k2 can be used for disturbance rejection and k1 for desired speed of response. 2. Step 2 – Keeping v1 ¼ a where a – 0 is a constant, stabilize the second subsystem by suitably designing v2 using any control technique. Note that since a does not play any significant role in system performance, it can be set to a ¼ 1. Setting v1 ¼ a renders the n 1 dimensional second subsystem linear. Resulting second subsystem is given as follows: z_2 z_3 z_4 .. . z_n
¼ v2 ; ¼ z2 ; ¼ z3 ;
ð3:3Þ
¼ zn1 ;
which is a linear system in controllable canonical form. System (3.3) can be trivially stabilized using pole placement with v2 ¼ l2 z2 l3 z3 ln zn , where li > 0; i ¼ 2; 3; . . . ; n are the state feedback gains that can be designed to achieve exponential stabilization, but, not robustness. It is therefore proposed to stabilize this subsystem with ISMC using gains computed for pole placement. In order to use ISMC, input v2 is redefined as v2 ¼ v20 þ v21 where v20 ¼ l2 z2 l3 z3 ln zn is the nominal controller obtained from pole placement. Let the sliding surface for conventional SMC be given as s0 ¼ c2 z2 þ c3 z3 þ þ zn , then the sliding surface for ISMC is given as follows: s ¼ s0 þ r;
ð3:4Þ
where r is the integral term that is used to eliminate reaching phase in ISMC. In order to eliminate reaching phase it must be true that sð0Þ ¼ 0 ) rð0Þ ¼ s0 ð0Þ [28]. Dynamics of the auxiliary variable r are defined by setting s_ ¼ 0:
s_ ¼ s_0 þ r_ ¼ 0; s_ ¼ z2 ðc2 l2 Þ þ z3 ðc3 l3 Þ þ þ zn ðcn ln Þ þ r_ ¼ 0; r_ ¼ z2 ðc2 l2 Þ z3 ðc3 l3 Þ zn ðcn ln Þ;
rð0Þ ¼ s0 ð0Þ:
ð3:5Þ
This ensures that sliding mode is established from t ¼ 0. During sliding, system dynamics are governed by the nominal controller v20 that grantees exponential stabilization. In order to ensure robustness against parametric variations, matched disturbance and to enforce sliding mode along the manifold, discontinuous component of the input is defined as follows: v21 ¼ MsignðsÞ: Hence, overall controller for stabilization of states zi ; i ¼ 2; 3; . . . ; n is given as follows: v2 ¼ l2 z2 l3 z3 ln zn MsignðsÞ:
ð3:6Þ
3. Step 3 – Keeping v1 constant in step 2 causes z1 to diverge from origin. To overcome this, repeat step 1. Since zi ¼ 0; i ¼ 2; 3; . . . ; n in step 2, repeating step 1 does not diverge these states.
3.2. Controller design for unicycle system In order to establish effectiveness of the control technique, the switching controller is applied to the mathematical model of a unicycle which is a benchmark nonholonomic system. Unicycle motion is primarily considered on Euclidean plane which is a flat surface with zero curvature. It is however an established fact that real world surfaces are seldom flat and are generally described by a combination of spherical and hyperbolic surfaces. Due to practical significance, motion of a unicycle is considered on a hyperbolic plane or a pseudo Euclidean plane i.e., a saddle-like surface that has constant negative curvature j ¼ 1 . Optimal control via geometric control techniques of this system was considered in [30–32]. However, geometric control techniques presented in these references are hard to implement for real world applications and therefore stabilization of this system is considered with the control technique presented in Section 3. A unicycle moving on hyperbolic plane is described as follows: 2 3 2 3 2 3 x_ cosh z 0 6 7 6 7 6 7 ð3:7Þ 4 y_ 5 ¼ 4 sinh z 5u1 þ 4 0 5u2 ; z_ 0 1 where ðx; yÞ is the point of contact of unicycle with the hyperbolic plane, z is its orientation and u1 ; u2 are the translational and rotational velocities respectively. The configuration manifold M along with the motions of the hyperbolic plane form a Lie group called special hyperbolic group SH(2). 3.2.1. Transformation to chained form v1 Let u1 ¼ cosh ; u2 ¼ v2 cosh2 z and redefine the state variables as z z1 ¼ x; z2 ¼ tanh z and z3 ¼ y. System (3.7) is now given as follows:
Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
4
Y.A. Butt
2 3 1 0 z_1 6 7 6 7 6 7 4 z_2 5 ¼ 4 0 5v1 þ 4 1 5v2 : z2 z_3 0 2
3
2
3
ð3:8Þ
It is trivial to see that stabilization of state variables zi ; i ¼ 1; 2; 3 is equivalent to stabilization of ðx; y; zÞ. Therefore, in the remainder text stabilization of states zi is investigated. 3.2.2. Controller design 1. Step 1 – Set v2 ¼ 0 and v1 ¼ k 1 z1 k 2 signðz1 Þ to stabilize z1 . For simulation purposes gains are given as k 1 ¼ k 2 ¼ 5. 2. Step 2 – Set v1 ¼ 1 so that the resulting subsystem 2 is given as follows: z_2 0 0 z2 1 ð3:9Þ ¼ þ v2 : z_3 1 0 0 z3
Figure 1
System states z1 ; z2 ; z3 ; r.
For simulation purposes the poles of (3.9) are placed at 1 for which the system is critically damped that ensures fastest speed of response without overshoot. State feedback gains for this choice of poles are ½ l2 l3 ¼ ½ 2 1 . Hence the nominal controller is given as v20 ¼ 2z2 z3 . In order to design ISMC choose the sliding surface s0 ¼ z2 þ z3 and therefore sliding surface with auxiliary integral term r is given as follows: s ¼ s0 þ r ¼ z2 þ z3 þ r; _ ) s_ ¼ z_2 þ z_3 þ r_ ¼ v20 þ z2 þ r: From (3.5): r_ ¼ z2 z3 ;
rð0Þ ¼ s0 ð0Þ ¼ z2 ð0Þ z3 ð0Þ:
Discontinuous input that ensures that sliding surface s is attractive is given as v21 ¼ MsignðsÞ. For simulations use M ¼ 5. Hence, v2 ¼ 2z2 z3 5signðsÞ:
Figure 2
Sliding surface s ¼ z2 þ z3 þ r ¼ 0.
3. Step 3 – Repeat step 1 to stabilize z1 .
4. Simulation results and discussion The intuitive reasoning behind the proposed algorithm is more comprehensible in terms of the unicycle system. In step 1, x is stabilized that brings the unicycle to y-axis. In step 2, y; z are stabilized, thus moving the unicycle to x-axis, but, in doing so the unicycle moves away causing x – 0 and y; z ¼ 0. Physically, the unicycle is somewhere on the x-axis with orientation of z ¼ 0. In step 3, the control forces the unicycle to move along x-axis and stabilizes it to the origin. Simulation results for stabilization of unicycle system (3.8) are presented in Figs. 1–5. From Fig. 1, it is seen that all states are stabilized to origin in finite time. Switching of controls takes place at times t ¼ T1 and t ¼ T2 . The auxiliary state variable r is visible only for t 2 ½T1 ; T2 . The sliding surface for ISMC of step 2 is given in Fig. 2. It is evident that there is no reaching phase, however, high frequency switching called chattering is present that nearly vanishes when the states converge to origin. The xy plot is presented in Fig. 3 which confirms our intuitive reasoning of the proposed algorithm. Step 1 stabilizes state x ¼ z1 to y-axis. In step 2, state y ¼ z3 and tanh z ¼ z2 is stabilized to the
Figure 3
xy plot.
x-axis. Finally, in step 3, the unicycle moves on the x-axis to stabilize state x to origin. The controller effort for both control inputs v1 and v2 is show in Fig. 4. Magnitude of control inputs v1 ; v2 in steps 1–3 and chattering due to sliding control can be
Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017
Robust stabilization of a class of nonholonomic systems
Figure 4
Figure 5
5
Controller effort.
System states z1 ; z2 ; z3 ; r and controller effort with input disturbance.
clearly seen in both plots. It is also evident that the control input is finite. Effect of input disturbance is shown in Fig. 5. A unit step disturbance is introduced in the state z1 during step 3 at t ¼ 13 s. This causes a positive magnitude shift in the state z1 . However, the disturbance is compensated by the controller and the state z1 converges to origin in finite time. Effect of disturbance is also visible in the control effort plot. Since, the disturbance is introduced in state z1 during step 3, only v1 adjusts to respond to the disturbance whereas v2 remains unaffected. 5. Conclusion The proposed algorithm along with SMC/ISMC ensures exponential stabilization of nonholonomic systems in chained form. ISMC eliminates reaching phase ensuring robust stabilization, nonetheless it suffers from signature chattering problem of SMC. One of the ways to eliminate chattering while still ensuring robustness is using integral higher order sliding mode control with the same logical switching control technique. Other trivial way is to replace the sign term with the saturation of inverse hyperbolic function which can reduce the chattering problem.
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Please cite this article in press as: Y.A. Butt, Robust stabilization of a class of nonholonomic systems using logical switching and integral sliding mode control, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.05.017