Finite Time Stabilization of Simple Mechanical Systems using

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To appear in the International Journal of Control Vol. 00, No. 00, Month 20XX, 1–12

Finite Time Stabilization of Simple Mechanical Systems using Continuous Feedback Amit K. Sanyal∗ and Jan Bohn Mechanical and Aerospace Engineering New Mexico State University Las Cruces, NM 88011 Tel: 575-646-2580 e-mails: {asanyal, janb1980}@ nmsu.edu

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(Received 00 Month 20XX; accepted 00 Month 20XX)

Stabilization of simple mechanical systems in finite time with continuous state feedback is considered here. The dynamics is represented by generalized (local) coordinates. A general methodology to construct control Lyapunov functions that are H¨ older continuous and that can be used to show finite-time stability of the feedback controlled system, is presented. This construction also gives the feedback control law, and results in the feedback system being H¨ older continuous as well. Unlike Lipschitz continuous feedback control systems, the feedback control scheme given here converges to the desired equilibrium in finite time. Moreover, unlike discontinuous and hybrid control schemes, the feedback control law does not lead to chattering in the presence of measurement noise, does not excite unmodeled high-frequency dynamics, and can be implemented with actuators that can only deliver continuous control inputs. The advantages of continuous finite-time stabilization over continuous asymptotic stabilization of mechanical systems, has been described in some prior research on finite-time stabilization of the double integrator. The finite-time stabilization scheme given here generalizes this prior research to multiple degree-of-freedom mechanical systems. A numerical comparison is carried out through numerical simulations on two example systems that are representative of a broad class of simple mechanical systems.

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Introduction

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1.

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Finite-time stability of nonlinear systems has the advantage of providing a guarantee on the time it takes for the system to converge to a desired state, besides being more robust to bounded temporary and persistent disturbances than asymptotic stability (Bhat & Bernstein, 1998, 2000; Dorato, 2006; Haddad, Nersesov, & Du, 2009). In addition, low-level persistent disturbances are better rejected by a finite-time stable system in comparison to an asymptotically stable system, because the ultimate bound on the state is of higher order than the bound on the disturbance (Bhat & Bernstein, 2000). Continuous finite-time stability has been dealt with in (Bhat & Bernstein, 1998, 2000; Haddad et al., 2009) for example, while discontinuous sliding mode control schemes that also provide finite-time stability have been extensively researched in the past (Ding & Li, 2009; Lagrouche, Plestan, & Glumineau, 2004; Levant, 2001; Yu, Ma, & Yang, 2007; Zhu, Xia, & Fu, 2011). The main advantages of continuous feedback stabilization in finite time are: (1) faster convergence than asymptotic or exponential stabilization for a given transient control effort; (2) can be implemented with actuators that cannot provide discontinuous inputs; (3) does not suffer from chattering in the presence of measurement noise; (4) does not require analysis of solutions of discontinuous systems in the sense of Filippov (Lagrouche et al., 2004; Levant, 2001; Yu et al., 2007); and (5) does not excite unmodeled high frequency dynamics in contrast to discontinuous

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∗ Corresponding

author

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feedback (Bhat & Bernstein, 1998). This makes continuous finite-time stabilizing control schemes appealing for applications requiring guaranteed convergence and robustness properties, while being implementable with actuators that can only provide continuous control inputs. In particular, continuous finite-time stable control schemes are effective in applications where there are bounded disturbance inputs due to unmodeled dynamics (Sanyal, Bohn, & Bloch, 2013). This work deals with design of finite-time stabilizing control schemes for simple mechanical systems represented in generalized (local) coordinates. While the work in (Bhat & Bernstein, 2000) dealt with continuous finite-time control schemes and ref. (Bhat & Bernstein, 1998) dealt with double integrator dynamics, there has been no systematic treatment of continuous finite-time stabilizing control for mechanical systems with multiple degrees of freedom in the prior literature. In this work, a Lyapunov framework for finite time stabilization of fully actuated mechanical systems that results in H¨ older continuous feedback dynamics, is obtained. The Lyapunov function is constructed as a quadratic form of a vector-valued function that is linear in the generalized velocities and H¨ older continuous in the generalized coordinates. This vector function is constructed such that when its value is the zero vector, the generalized coordinate vector converges to the desired equilibrium in finite time. The stabilizing control law for generalized forces, then ensures that this vector converges to the zero vector in finite time, and therefore the system states (generalized coordinates and velocities) converge to the desired equilibrium in a finite time duration. This article is organized as follows. Section 2 gives the dynamics model of simple mechanical systems as defined in (Bullo & Lewis, 2005), along with two practical examples: an inverted pendulum on a cart and a two-link planar manipulator. Section 3 states and proves the main result on finite time stabilization of a simple mechanical system using a H¨older continuous Lyapunov function. Section 4 presents a set of numerical simulation results for the fully actuated inverted pendulum on a cart in section 4.A and the two link planar manipulator in section 4.B. These results illustrate the stability and robustness of this finite time continuous feedback stabilization scheme in comparison to an asymptotically stabilizing continuous feedback scheme. Finally, section 5 provides a summary of the main contributions of this article and planned future work.

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Simple Mechanical Systems Represented by Generalized Coordinates

Consider a simple mechanical system with n degrees of freedom and whose configuration is described by a vector of generalized coordinates q ∈ Q ⊂ Rn where Q is the configuration space. The Lagrangian of such a system is given by (Bullo & Lewis, 2005)

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1 L(q, q) ˙ = q˙T M (q)q˙ − U (q), 2

(1)

where T (q, q) ˙ = 12 q˙T M (q)q˙ is the kinetic energy of the system, q˙ ∈ Rn is the vector of generalized velocities, M (q) ∈ Rn×n is the positive definite inertia matrix, and U (q) is the energy due to a conservative potential. The equation of motion of these systems can be derived using Lagrange’s equation which gives the dynamics of a simple mechanical system as M (q)¨ q + C(q, q) ˙ q˙ + G(q) = F (q, q), ˙

(2)

d where C(q, q) ˙ = dt M (q) ∈ Rn×n is the time derivative of the inertia matrix, G(q) = ∂V∂q(q) ∈ Rn is the vector of conservative generalized forces, and F (q, q) ˙ ∈ Rn is the vector of non-conservative generalized forces acting on the system, which could include control inputs (forces and moments). This vector of generalized forces can be expressed as

F (q, q) ˙ = Fc (q, q) ˙ + FN C (q, q), ˙ http://mc.manuscriptcentral.com/tcon

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where Fc (q, q) ˙ ∈ Rn is the vector of control inputs and FN C (q, q) ˙ ∈ Rn is the vector of external nonconservative generalized forces (like friction) acting on the system. If the system is fully actuated, then all components of Fc (q, q) ˙ will be non-zero. One can convert the kinematics and dynamics to the state space form: q˙ = v, v˙ = M −1 (q){−C(q, v)v − G(q) + F (q, v)} = f (q, v).

(4)

This state space form is used for constructing control schemes for stabilizing the simple mechanical system. The remainder of this section describes two simple mechanical systems that are representative of this large class of systems that occur in several practical applications. Two illustrative examples of simple mechanical systems are considered next.

2.1

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Inverted Pendulum on a Cart

The first example system considered is an inverted pendulum on a cart, a schematic of which is shown in Fig. 1. This system has two degrees of freedom. The generalized coordinates are chosen as the horizontal displacement of the cart, x, and the rotational displacement of the pendulum from the upward vertical, θ; thus, q = [x θ]T . The kinetic and potential energy for the inverted pendulum on a cart are obtained in terms of these generalized coordinates and their generalized velocities as

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1 1 1 T = mc x˙ 2 + mp (x˙ 2 + 2dx˙ cos θ θ˙ + d2 θ˙ 2 ) + I θ˙ 2 , 2 2 2 U = mp gd cos θ,

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(5) (6)

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where mc and mp are the mass of the cart and the pendulum respectively. The pendulum inertia about its center of mass is I, and its center of mass is a distance d from the pendulum pivot; g is the acceleration due to uniform gravity. It is assumed that the only nonconservative generalized

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Figure 1. Inverted Pendulum on a Cart

forces are the control inputs Fx = fc (t),

(7)

Fθ = τc (t).

(8)

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Under this assumption, the dynamics model is given by equations (3)-(4) with:

" ! " mc + mp mp d cos θ 0 −mp dθ˙ sin θ M (q) = , C(q, v) = mp d cos θ I + mp d2 0 0 ! " ! " ! " 0 fc (t) 0 G(q) = , Fc = , FN C (q, v) = . −mp gd sin θ τc (t) 0 !

2.2

Two-Link Planar Manipulator

The second example considered is a two link planar manipulator, a schematic of which is shown in Fig. 2. The model of a two-link planar nonlinear robotic system with the assumption of lumped masses at the joints can be found in (Lin & Wang, 2010). This model is presented and used here.

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Figure 2. Two Link Planar Manipulator

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The generalized coordinates are the rotation angles of the arms q = [θ1 θ2 ]T . The dynamics model is of the form given by equations (3)-(4), with:

! " (m1 + m2 )d21 + m2 d22 + 2m2 d1 d2 cos θ2 m2 (d22 + d1 d2 cos θ2 ) M (q) = , m2 (d22 + d1 d2 cos θ2 ) m2 d22 ! " ! " ! " 21 τ1 (t) 0 ˙ C(q, v) = −m2 d1 d2 θ2 sin θ2 , Fc = , FN C = , 10 τ2 (t) 0 ! " (m1 + m2 )d1 cos θ1 + m2 d2 cos(θ1 + θ2 ) G(q) = g . m2 d2 cos(θ1 + θ2 )

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This model also assumes that the only non-conservative generalized forces acting on the system are the control input torques. 3.

Finite-Time Stabilization of Simple Mechanical Systems

A constructive method to obtain finite-time stabilization schemes for simple mechanical systems is provided here. This method proceeds in two successive steps. In the first step, a continuous vectorvalued function of the states (generalized coordinates and generalized velocities) is constructed with special properties that ensure that when this function converges to zero, the states converge to zero as well. In the second step, a H¨older continuous Lyapunov function that is a positive definite function of the vector-valued function created in the first step, is constructed to prove the finite-time stability of the mechanical system. This two step process is similar to an integrator backstepping approach applied to the mechanical system (Khalil, 2002; Krstic, Kanellakopoulos, & Kokotovic, 1995). The following two subsections describe these two steps in this constructive method.

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3.1

Finite-time kinematic stabilization of mechanical system

In the first step of the finite-time stabilization process, a vector-valued function is constructed that ensures the generalized coordinate vector converges to the desired equilibrium in finite-time when this vector converges to zero. Without loss of generality, we assume that the desired equilibrium to be stabilized is the origin, i.e., (q, v) = (0, 0). The following statement gives the properties of such a function. Proposition 1: Let l(q, v) : Q × Rn → Rn be a function that has the following properties: • l(q, v) is linear in v = q, ˙ • l(q, v) is H¨ older continuous in q, • l(q, v) ≡ 0 ⇒ q(t) = 0 for t ≥ T > 0. These properties ensure that the states (q, v) ∈ Q× Rn converge to zero in finite-time when l(q, v) = 0. For example, for the mechanical system (4) , one can construct l(q, v) as

r Fo l(q, v) = v +

kq )q)1−α

where

1 < α < 1 and k > 0. 2

(9)

This leads to the following result for l(q, v) as defined above.

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Lemma 1: If l(q, v) is defined by (9) where v = q, ˙ then l(q, v) satisfies the properties listed in Proposition 1. Proof. Consider the candidate Lyapunov function

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1 V (q) = q T q. 2

(10)

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The time derivative of this Lyapunov function when evaluated on the submanifold defined by l(q, v) = 0 where l(q, v) is defined by (9), is given by qTq V˙ = q T v = −k )q)1−α

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Noting that 2V = q T q = )q)2 , this time derivative of the Lyapunov function is

Given that

1 2

# $ 1+α V˙ = −k)q)1+α = −k 2V 2 .

(11)

< α < 1, one obtains 1+α 3 < β < 1 where β = . 4 2

Therefore, the time derivative of V can be expressed as V˙ = −κV β where κ = 2β k.

(12)

Equation (12) guarantees that V = 21 )q)2 converges to zero in the finite time duration T =

V01−β , κ(1 − β)

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where V0 = V (q(t0 )) is the finite initial value of the Lyapunov function at time t0 . Therefore q(t) and hence v(t) = q(t) ˙ converge to zero in this finite time duration. The following subsection shows the second step of the constructive method to obtain a finite-time stabilization scheme for mechanical systems.

3.2

Finite-time stabilization of mechanical systems with generalized input forces

In the second step of this finite-time stabilization method for a simple mechanical system with multiple degrees of freedom, like (4), a vector of generalized input forces is obtained that stabilizes (q, v) = (0, 0) in finite time. With this goal in mind, a Lyapunov function that is positive definite in a vector-valued function l(q, v) that satisfies the properties listed in Proposition 1, is constructed. As an example, the Lyapunov function V(q, v) =

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1 l(q, v)T l(q, v), 2

(14)

can be used to obtain the finite-time stabilization scheme. The following statement gives the main result on finite-time stabilization of simple mechanical systems with multiple degrees of freedom. Theorem 1: Consider the simple multiple degree-of-freedom mechanical system (4) with modified feedback control given by

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f (q, v) = −γ

v qq T l + k(α − 1) v, −k )q)1−α )q)3−α (lT l)1−α

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(15)

where γ > 0, k, α, and l = l(q, v) are given by equation (9). Then the feedback system given by (4), (9) and (15) is finite-time stable.

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Proof. The time derivative of l(q, v), where l(q, v) is defined by equation (9), is ˙ v) = v˙ + l(q,

kv + k(1 − α))q)α−3 qq T v. )q)1−α

(16)

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One can evaluate the time derivative of V(q, v) as defined by (14), along trajectories of (4) with l(q, v) defined by (9), as follows ˙ v), ˙ v) = l(q, v)T l(q, V(q, = l(q, v)T {f (q, v) + k

v qq T + k(1 − α) v}, )q)1−α )q)3−α

Now if f (q, v) is as given by equation (15), then V˙ = −γ(lT l)α = −γV α , which implies that V converges to zero in finite time as α < 1. This in turn guarantees that l(q, v) converges to zero in the finite time duration T1 =

1 V01−α where V0 = l0T l0 , l0 = l(q(t0 ), v(t0 )). γ(1 − α) 2 http://mc.manuscriptcentral.com/tcon

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When l(q, v) ≡ 0, we have v=

−kq )q)1−α

(18)

Substituting for v from (18), the second and third terms in (15) converge to zero if )q)1−(2−2α) = )q)2α−1 coverges to zero, i.e., if 1 2α − 1 > 0 or α > . 2 Therefore, to ensure that the control f (q, v) remains bounded and the feedback system converges to l(q, v) = 0 in finite time, α has to be in the interval 12 < α < 1. Thereafter, according to Lemma 1 and equation (17), the states (q, v) are guaranteed to converge to zero in the finite time duration T + T1 .

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In order to stabilize the mechanical system given in section II to (q, v) = (0, 0) in finite time using continuous state feedback and assuming it to be fully actuated, we substitute the expression (15) into the dynamics of the system (4) to obtain the control law

% qq T l v & Fc = M k(α − 1) v − γ − k + Cv + G + FN C . )q)3−α (lT l)1−α )q)1−α

(19)

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Remark: Note that the H¨ older continuous feedback system given by (4) and (15) can be continuously transformed to a Lipschitz continuous mechanical system in two ways. The first way to do this would be to take the limit α = 1 in (15), or alternately, the control law (19). The second method would be to transform the feedback system into the form

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q˙ = v,

v˙ = W (q, v)f (q, v)

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(20)

where W (q, v) : R2n → R is non-negative, continuous, with W (q, v) = 0 iff f (q, v) = 0. For f (q, v) given by (15), a Lipschitz continuous system is given by W (q, v) = )l(q, v))2 )q)2−α . The resulting Lipschitz continuous systems obtained both ways have the same equilibria and domain of attraction as the H¨older continuous system (4), only the slopes of trajectories in the velocity phase space are different. For example, if system (20) is globally/locally asymptotically stable, then system (4) is globally/locally finite-time stable.

4.

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Numerical Simulation Results

The finite time control law given by equation (19) of Theorem 1 is numerically simulated for the two example systems described in section 2. The inverted pendulum on a cart is simulated first, followed by the two-link planar manipulator. 4.1

Inverted Pendulum on a Cart

For this simulation, the parameter values are mc = 5 kg and mp = 1 kg with d = 0.5 m, which gives I = 0.0833 kgm2 . The scalar control gains are k = 1, γ = 1.5, and α = 21/27. The initial states are set to x0 = 1 m, θ0 = 1.5 rad, x˙ 0 = 0 m/s, and θ˙0 = 0 rad/s. Friction forces and torques are assumed to be unknown for the control law design; however, a small Coulomb friction force on the cart wheels and a small viscous friction torque on the pendulum pivot are provided as unknown disturbance inputs to the system. The vector of disturbance inputs is therefore given by Fu = http://mc.manuscriptcentral.com/tcon

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˙ T , where we set the the Coulomb friction force fu = −0.005 sgn(x) [fu cθ] ˙ N where sgn(·) denotes the signum function, and the viscous friction coefficient c = 0.1 Nm/(rad/s). The continuous finite time feedback stabilization (FTS) scheme is compared with a continuous asymptotic feedback stabilization (AS) scheme that is obtained by setting α = 1 in the control law (19), while k and γ remain the same. We run both simulations for 10 seconds using the same initial conditions. In 1

−0.1

x˙ (m/s)

x (m)

0

FTS AS

0.8 0.6 0.4 0.2 0

0

2

4

6

t (s)

−0.2 −0.3

−0.5

8

FTS AS

−0.4 0

Figure 3. Displacement of the Cart

2

4

t (s)

6

8

10

Figure 4. Velocity of the Cart

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Fig. 3, the horizontal displacement of the cart is seen to converge to zero in about 5.9 seconds using the FTS scheme, whereas it takes the AS scheme about 12.6 seconds to converge to a small value of x (such that |x| < 0.0006 m). Fig. 4 shows time plot of the velocity of the cart over this time period. The time plot In Fig. 5 for the FTS scheme shows that the rotational displacement

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1.5

θ˙ (rad/s)

1

θ (rad)

0

FTS AS

−0.2

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0.5

−0.4 FTS AS

−0.6

0

0

2

4

t (s)

6

8

10

Figure 5. Rotational Displacement of the Pendulum

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2

4

t (s)

6

8

10

Figure 6. Rate of Change of the Rotational Displacement

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of the pendulum converges to zero after about 6.0 sec, while for the AS scheme it converges after about 12.3 sec to a small value such that |θ| < 0.0013 rad. In Fig. 6, the angular velocity of the pendulum is plotted using these two control schemes. In Fig. 7 and Fig. 8, the time plots of the cart 0.1

0.1 FTS AS

0.08

0.08

θ (rad)

0.06

x (m)

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0.04 0.02 0

FTS AS

0.06 0.04 0.02 0

3

4

5

6

7

t (s)

8

9

10

3

Figure 7. Enlarged plot of Displacement of the Cart

4

5

6

7

t (s)

8

9

10

Figure 8. Enlarged plot of the Rotational Displacement

displacement and pendulum rotational displacement are shown. The FTS scheme gives a quicker convergence than the AS scheme. In Fig. 9 and Fig. 10, the control force and control torque are depicted. These figures show that the transient control inputs for both schemes are very similar. Fig. 11 and Fig. 12 show the control efforts of the FTS and the AS scheme, where the control efforts are defined as the integrals of the absolute values of the control force and control torque acting on the system. The FTS scheme is seen to use less control effort while stabilizing the system

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2 0

τc (t) (Nm)

0

fc (t) (N)

−2 −4 −6 FTS AS

−8 −10

0

2

4

t (s)

6

8

| τc (t) | dt (Nms)

4

2 FTS AS

2

4

t (s)

6

8

FTS AS 0

2

4

t (s)

6

8

10

10

5 FTS AS 0

'

0

−4

Figure 10. Control Torque acting on the Pendulum

6

'

| fc (t) | dt (Ns)

Figure 9. Control Force acting on the Cart

0

−2

−6

10

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10

0

Figure 11. Integral of the Control Force

2

4

t (s)

6

8

10

Figure 12. Integral of the Control Torque

faster than the AS scheme. Furthermore it is seen that the FTS scheme is more robust than the AS scheme to the unknown disturbance inputs due to friction.

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Two-Link Planar Manipulator System

4.2

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For this simulation, the parameter values chosen are the link masses m1 = 3 kg and m2 = 10 kg, the link lengths d1 = 1 m and d2 = 1 m; it is assumed that there is no friction at the joints. These parameters are as defined in Fig. 2. The scalar control gains are k = 0.6, γ = 1, and α = 21/29. We chose the initial values (θ1 )0 = 4.5 rad, (θ2 )0 = 3 rad, (θ˙1 )0 = 0 rad/s, and (θ˙2 )0 = 0 rad/s. The objective is to stabilize the two link planar manipulator to the equilibrium configuration [θ1 θ2 ]T = [π/2 0]T . This simulation also compares the finite time stabilization (FTS) control law of (19) with an asymptotically stabilizing (AS) control law obtained by setting α = 1 in this control law, while the parameters k and γ remain the same. Both control laws are simulated for 15 sec with the same initial conditions. In Fig. 13, time plots of the rotation angle θ1 of the first link are

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θ˙1 (rad/s)

FTS AS

4

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3.5 3 2.5 2

−0.2 −0.4 −0.6

FTS AS

−0.8

1.5 0

5

t (s)

10

15

0

Figure 13. Link 1 Rotation Angle

5

t (s)

10

15

Figure 14. Angular Velocity of Link 1

shown. With the FTS scheme, θ1 converges to π2 radians within 9.9 s, while it takes 14.6 s for θ1 to approach within 0.01 radians of π2 using the AS scheme. Fig. 14 shows the time plots of the angular velocity θ˙1 of the first link with the FTS and AS control schemes. In Fig. 15, time plots of the rotation angle of the second link are shown. With the FTS scheme, θ2 converges to zero in 9.9 s, while it takes 14.6 s for θ2 to approach less than 0.01 radians in absolute value with the AS scheme. Fig. 16 shows the time plots of θ˙2 of the angular velocity of the second link using these two http://mc.manuscriptcentral.com/tcon

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3 0

θ˙2 (rad/s)

θ2 (rad)

FTS AS 2

1

−0.2 −0.4 −0.6 FTS AS

−0.8

0 0

5

t (s)

10

15

0

Figure 15. Rotation Angle of Link 2

5

t (s)

10

15

Figure 16. Angular Velocity of Link 2

control schemes. In Fig. 17 and Fig. 18, time plots of θ1 and θ2 between 5 to 15 s, using the FTS 1.8

0.25 FTS AS

1.7 1.65 1.6 1.55

5

10

0.15 0.1 0.05 0

15

5

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t (s)

FTS AS

0.2

θ2 (rad)

θ1 (rad)

1.75

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Figure 17. Plot of rotation angle θ1 between 5 to 15 s

10

15

t (s)

Figure 18. Plot of rotation angle θ2 between 5 to 15 s

and AS schemes, are plotted. Fig. 19 and Fig. 20 depict the control torques applied to the two links

er τ2 (t) (Nm)

τ1 (t) (Nm)

0 −50 −100 −150

FTS AS 0

5

t (s)

10

FTS AS

50 0

−50

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−200

100

Re −100

15

Figure 19. Control Torque applied on Joint 1

0

5

t (s)

10

15

Figure 20. Control Torque applied Joint 2

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| τ2 (t) | dt (Nms)

600 400 200 0

FTS AS 0

5

t (s)

10

400 300 200 100 0

'

| τ1 (t) | dt (Nms)

using the FTS and AS control schemes. Fig. 21 and Fig. 22 depict the control efforts, defined as the

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Figure 21. Integral of Control Torque applied on Joint 1

FTS AS 0

5

t (s)

10

15

Figure 22. Integral of Control Torque applied Joint 2

L1 norms of the control torques, for the FTS and AS control laws stabilizing the two-link planar manipulator in a vertical orientation. As is clearly seen in these figures, the AS scheme requires larger control effort than the FTS scheme over the time duration of this stabilization.

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Conclusion

A general methodology to obtain continuous, finite-time stabilizing control schemes for fullyactuated mechanical systems is provided. This methodology is based on constructing a vectorvalued function that is linear in the generalized velocities and H¨older continuous in the generalized coordinates, with properties that ensure that when this function converges to zero, the states converge to the desired equilibrium. Thereafter, a Lyapunov function that is positive definite in this vector-valued function is constructed. This Lyapunov function is then used to obtain a continuous feedback stabilization scheme that gives the vector of generalized control forces required to stabilize the desired equilibrium in finite time. Numerical simulations on two classical example systems, an inverted pendulum on a cart and a planar two-link manipulator, demonstrate the finite-time stability of this control scheme. Comparisons with an asymptotic stabilization scheme for both examples show that the finite-time stabilization scheme has superior convergence rate while requiring lower control effort. In the case of the inverted pendulum on a cart system, these simulation results also show the finite-time stabilization scheme to be more robust to unmodeled friction than the asymptotic stabilization scheme. Future work will generalize and apply such finite-time stabilizing control schemes to rigid body and multi-body systems, which evolve on non-contractible state spaces.

Acknowledgments

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This research was partly supported by NASA grant NNX11AQ35A and NSF grant CMMI 1131643.

References

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Bhat, S. P., & Bernstein, D. S. (1998). Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Transactions on Automatic Control , 43 (5), 678-682. Bhat, S. P., & Bernstein, D. S. (2000). Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 38 (3), 751-766. Bullo, F., & Lewis, A. D. (2005). Geometric control of mechanical systems (No. 49). Springer Verlag. Ding, S., & Li, S. (2009). Stabilization of the attitude of a rigid spacecraft with external disturbances using finite-time control techniques. Aerospace Science and Technology, 13 , 256-265. Dorato, P. (2006). An overview of finite-time stability. In Current trends in nonlinear systems and control: Foundations and applications (p. 185-194). Haddad, W. M., Nersesov, S. G., & Du, L. (2009). Finite-time stability for time-varying nonlinear dynamical systems. In Advances in nonlinear analysis: Theory, methods and applications (p. 139-150). Cambridge, UK. (S. Sivasundaram, J. V. Devi, Z. Drici, and F. McRae, Eds., Cambridge Scientific Publishers) Khalil, H. K. (2002). Nonlinear systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. Krstic, M., Kanellakopoulos, I., & Kokotovic, P. V. (1995). Nonlinear and adaptive control design. New York: John Wiley. Lagrouche, S., Plestan, F., & Glumineau, A. (2004, September). Higher order sliding mode control based on optimal LQ control and integral sliding mode. In IFAC symposium on nonlinear control systems (Vol. 2, p. 1993-1998). Stuttgart, Germany. Levant, A. (2001). Universal single-input single-output (SISO) sliding-mode controllers with finite-time convergence. IEEE Transactions on Automatic Control , 46 (9), 1447-1451. Lin, S. B., & Wang, S.-G. (2010). Robust control design for two-link nonlinear robotic system, advances in robot manipulators. Ernest Hall. Sanyal, A. K., Bohn, J., & Bloch, A. M. (2013, December). Almost global finite time stabilization of rigid body attitude dynamics. In IEEE conf. on decision and control. Florence, Italy. Yu, S., Ma, Z., & Yang, X. (2007). Nonsmooth finite-time control of uncertain second-order nonlinear systems. Journal of Control Theory and Applications, 5 (2), 171-176.

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Zhu, Z., Xia, Y., & Fu, M. (2011). Attitude stabilization of rigid spacecraft with finite-time convergence. International Journal of Robust and Nonlinear Control, 21 (6), 686-702.

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