Stabilization of exact nonlinear Timoshenko Beams in Space by Boundary Feedback

Boundary feedback controllers are designed to stabilize Timoshenko beams with large translational and rotational motions in space under external disturbances. The exact nonlinear partial di ﬀ erential equations governing motion of the beams are derived and used in the control design. The designed controllers guarantee globally practically asymptotically (and locally practically exponentially) stability of the beam motions at the reference state. The control design, well-posedness and stability analysis are based on various relationships between the earth-ﬁxed and body-ﬁxed coordinates, Sobolev embeddings, and a Lyapunov-type theorem developed to study well-posedness and stability for a class of evolution systems in Hilbert space. Simulation results are included to illustrate the e ﬀ ectiveness of the proposed control design.


Introduction
Timoshenko beams are widely used in practical structures such as poles, bars, columns, and robot arms.In modeling and control of Timoshenko beams, shear deformation and rotational bending effects need to be considered as opposed to neglection of shearing in Bernoulli beams.These effects result in nonlinear couplings between translational and rotational motions of the beams.Usually, the beam's motions are already stable but exhibit an unacceptably slow rate of decay.It is therefore of practical interest to introduce boundary controls that increase the margin of stability.Control of Timoshenko beams is different from that of (Bernoulli) slender beams due to the fact that slender beams are usually supported by tension, which providing structural stiffness, in addition to boundary control forces and moments.
There are excellent works on boundary control on small (vibration) motions of Timoshenko beams, where the classical Timoshenko beam model [1] was used, (e.g., [2][3][4][5][6][7][8][9][10] based on Lyapunov's direct method or [11,12] based on the backstepping method [13]).Since the model in [1] is obtained by linearizing exact nonlinear partial differential equations (PDEs) governing motions of shear beams, the results of the above works are only valid in the neighborhood of the origin, see also Remark 2.1.In practice, Timoshenko beams work in a wide range of operations, under which the beams deform with a large magnitude of both translational and rotational motions.It is therefore necessary to consider Timoshenko beams using exact nonlinear PDEs in three-dimensional space.Recently, an exact nonlinear model of Timoshenko beams, of which motions are restricted in two-dimensional space, and their boundary control design are addressed in [14].Related works include (e.g., [15][16][17][18][19][20][21][22][23][24][25][26]) on boundary control of Bernoulli-type beams with small motions; (e.g., [27][28][29][30][31]) on boundary control of slender beams (i.e., the shear magnitude is smaller than that of the spatial gradient of the Email address: duc@curtin.edu.au(K.D. Do) Preprint submitted to Elsevier February 5, 2018 transverse displacements) with large motions.Modelling, boundary control, and stability analysis of Timoshenko beams governed by exact nonlinear PDEs in space have not been considered.The above discussion motivates the writing of this paper on modelling, boundary control design, and stability analysis of Timoshenko beams in space under external loads.One end of the beam is connected to an actuation system while the other end carries a payload.The new contributions of this paper are highlighted as follows: • First, exact nonlinear PDEs governing motions of the beam are derived using deformation theory and Newton's law.The unit quaternion is also used for attitude representation of the beams to resolve singularities caused by Euler angles.
• Second, boundary feedback controllers are designed for global practical asymptotical (and local practical exponential) stabilization of the beams based on the Lyapunov direct method.In the control design, a new Lyapunov functional, various Young's and Hölder's inequalities and Sobolev embedding, a nontrivial combination of Earth-fixed and body-fixed coordinates, and cross vector products are used.Moreover, a special intention is made during the control design so that the development can be extended to spinning beams in future.
• Third, well-posedness and stability analysis of the variational solution of the closed-loop system are rigorously studied based on a Lyapunov-type theorem developed for study of well-posedness and stability analysis for a class of nonlinear evolution systems in Hilbert space.The variational solution (instead of the classical solution) is considered because initial conditions and boundary conditions, which include the boundary controls, are allowed to be general (smooth or nonsmooth), see [32] for discussion of the variational solution in depth.This allowance makes the variational solution more practical than the classical solution, which requires very specific (such as smooth) initial conditions and specific (such as natural/compatible) boundary conditions [32].
Notations.The symbols ∧ and ∨ denote the infimum and supremum operators, respectively.These operators are also applied to more than two arguments.The symbol col denotes the column operator.The symbol × denotes the vector cross product operator.

Mathematical model
A beam in space as shown in Fig. 1 is considered in this paper.One end of the beam is connected to a payload via a fixed joint while the other end is connected to actuators, which provide boundary control forces and moments.The (three) boundary control forces are provided by three linear actuators while the (three) boundary control moments are provided by three rotary actuators.These actuators, which can be electric or hydraulic motors, are installed on a mechanism such that they are completely decouple.An example of a mechanism that decouples all six motions is a 3D guide tube.We assume that plane sections are rigid; and the beam material is (nonlinear) elastic, homogeneous and isotropic.In what follows, equations of motion are briefly derived, see [30] for details.

Kinematics
The reference configuration B 0 of the beam in space is described by the position of the base straight line C 0 parameterized by its arclength coordinate s and the fixed basis (b 0 1 , b 0 2 , b 0 3 ), where (b 0 1 , b 0 2 ) are collinear with the principal axes of inertia of the cross section S 0 (s) through the base point N 0 , see Fig. 1a.The triple (e 1 , e 2 , e 3 ) is paralleled to (b 0 1 , b 0 2 , b 0 3 ).Thus, C 0 is described by the position vector r 0 (s), which in the fixed basis is expressed as r 0 (s) = 0e 1 + 0e 2 + se 3 .We denote by Γ the beam length in its reference state.
The actual configuration B of the curved beam is described by the actual position C(s, t) of the base curve and the actual configuration S (s, t) of cross sections through the base point N.The base curve is described by the position vector r(s, t) while the material cross section is described by the unit vectors {b 1 (s, t), b 2 (s, t), b 3 (s, t)} with b 3 being aligned with r s (s, t) and b The deformation from B 0 to B is achieved by means of the vector r(s, t) expressed in the local basis, i.e., r(s, t) = r 1 (s, t)b 1 + r 2 (s, t)b 2 + r 3 (s, t)b 3 , and the orthogonal tensor R 1 (θ(s, t)) describing the incremental rigid rotation suffered by S 0 (s ) is given in components by: where θ := col(θ 1 , θ 2 , θ 3 ); c θ i := cos(θ i ) and s θ i := sin(θ i ).This gives b ks = R 1s b 0 k , where R 1s = µ × b k with µ being the axial vector of R 1s R T 1 .The generalized strains (i.e., the stretch ε and the shear strains η 1 and η 2 ) are expressed by the stretch vector ν = η 1 b 1 + η 2 b 2 + (1 + ε)b 3 in its local basis: ν = r s .Thus, we have From ( 1) and (2) we have where t θ 2 := tan(θ 2 ).

Kinetic
Balancing linear and angular momentum on a beam element, see Fig. 1b, gives the equations of motion: where m 0 is the beam mass per unit length; J 0 is the mass moment matrix of inertia; n and m denote the contact force and moment vectors; and (see Fig. 1a) The nonconservative force and moment vectors f 1 and f 2 are given in the body-fixed frame as where D i j , (i, j) = 1, 2 are diagonal and positive definite matrices; a ⊗ a := diag(a 2 1 , a 2 2 , a 2 3 ) with a = col(a 1 , a 2 , a 3 ); and f 10 (t) and f 20 (t) are external disturbances bounded in L 2 -norm, and v is the linear velocity vector with coordinates in the body-fixed frame, i.e., When θ 2 = ± π 2 , there are singularities in (3).Thus, we use the unit quaternion vector q = col(q 1 , q 2 , q 3 , q 4 ) for attitude representation with ∥q∥ 2 = 1 relating to (θ 1 , θ 2 , θ 3 ) via the sequence θ 1 → θ 2 → θ 3 as follows [33]: The rotational matrix R 1 is given in terms of q as follows: where q := col(q 2 , q 3 , q 4 ) and the matrix S(x) is defined as S(x)y = x × y for all (x, y) ∈ R 3 .Let us also define the matrix: [ − qT With ( 3), ( 7), (9), and (10), we can write (4) as the following system of PDEs: The contact force and moment vectors n and m are given by: where Q 1 and Q 2 are the shear forces; N is the axial force; M 1 and M 2 are the bending moments; and T is the twisting moment.Using the third-order Maclaurin series expansion of nonlinear stress-strain relations in [34] results in the constitutive equations: where i = 1, 2; E is the Young modulus; G is the shear modulus; A, Ā1 , Ā2 are cross section and shear areas; I k , k = 1, 2, 3 are principal mass moments of inertia about b k .
Finally, referring to Fig. 1 the boundary conditions are given by At s = 0 : where At s = 0 : and m p and J P are the mass and matrix of inertia moment of the payload; M H and J H are mass and inertia moment matrices of the actuator systems; D 0 i j and D Γ i j are positive definite and diagonal damping matrices; ϕ 1B = col(ϕ 11B , ϕ 12B , ϕ 13B ) and ϕ 2B = col(ϕ 21B , ϕ 22B , ϕ 23B ) are force and moment boundary control input vectors; and f 0 i0 and f Γ i0 are external force and moment vectors acting on the payload and actuators.Note that f B0 10 and f BΓ 10 are coordinated in the earth-fixed frame while f B0 20 and f BΓ 20 are coordinated in the body-fixed frame.Remark 2.1.Linearization of (11), (3), and (13) will result in a system modelling the classical Timoshenko beams, which were considered in existing works (e.g., [2-9, 11, 12]).The linearized system excludes couplings of the longitudinal motion with transverse and rotational motions.Moreover, recent work [14] restricts motions of the beam to two dimensional space.From (11) and (3), it is clearly seen that all the translational and rotational motions of the beam are nonlinearly coupled in all dimensions via the matrices R 1 (q) and K(q), and the term r s × n, see (11) and (3).Therefore, the existing works ([2-9, 11, 12, 14]) were not able to address the aforementioned couplings.This paper directly considers the exact nonlinear model comprising (11) and (3).Thus, the aforementioned couplings are considered, and both large and small amplitude translational and rotational motions are controlled.It will be seen in the sequel that due to complex nonlinear couplings between all the motions of the beam in all dimensions, the control design is much more involved than the one for two-dimensional space in [14].
2) The external loads are bounded in appropriate norms, i..e, there exist nonnegative constants f M i0 , f B0M i0 , and f BΓM i0 such that Control Objective 3.1.Under Assumption 3.1, design the boundary control vectors ϕ iB , i = 1, 2 such that the beam system consisting of ( 11)-( 15) is globally practically asymptotically (and locally practically exponentially) stable at the origin in the sense that where c is a positive constant depending on the initial data when the data are large, and is independent of the initial data when the data are small, and the constant c 0 is a nonnegative constant.The function E(t) is given by where γ, γ 1 , and γ 2 are positive constants (to be chosen later), and ϑ(s, t) where K 1B is a positive definite matrix.
Remark 3.1.It is clear that E(t) is a positive definite and radially unbounded functional of velocities, stretch, shear strain, bending and torsional curvatures.While this functional penalizes (translational and rotational) displacement and velocity motions of the actuated end, it is only necessary to include translational and rotational velocities at the payload end.Convergence of the translational and rotational displacements at any point of the beam including the payload end are ensured by convergence of E(t) via Sobolev embedding, see the last five inequalities in the lemma below.By doing so, difficulties in the control design will be relaxed.
Several useful equalities and inequalities, which will be used in the control design and stability analysis, are given in the following lemma.
Lemma 3.1.For all t ≥ t 0 ≥ 0 and s ∈ [0, Γ], we have: where we have dropped the argument (s, t) for clarity.The inequality 8) is of its own interest because the right hand-side depends on (q 2 2 (Γ, t) + q 2 3 (Γ, t)) not on q 1 (Γ, t) and q 4 (Γ, t).This makes the boundary control design in this paper applicable to spinning beams such as drillers because the right hand-side of the inequality 8) does not depend on θ 3 (Γ, t), which is inferred from (8), i.e., q 2  2 + q 2 ) for all s ∈ [0, Γ] and t ∈ [t 0 , ∞).Proof.Proof of the first 5 equalities is given in the proof of Lemma 3.1 in [30] while proof of the sixth and seventh inequalities is given in the proof of Lemma 2.1 in [14].We here provide the proof of the last three inequalities.From (5) and the first equation of (3), we have where ϑ is defined in ( 20) and e 3 := col(0, 0, 1).By taking norm-2 both sides of ( 22) and applying Young's inequality together with the expression of R 1 (q) in ( 9), we have Using the simplified Poincaré inequality, see proof of Lemma 2.1 in [14] and integrating both sides of (23 where we have use q s = K(q)µ ⇒ ∥q s ∥ 2 ≤ 6∥µ∥ 2 with K(q) given in (10), which completes proof of the inequality 8).Proof of the inequality 9) is similar to that of the inequality 7) with the use of The inequality 10) is proved by applying the simplified Poincaré inequality to ∥ |q 1 − 1 qT | T ∥ 2 and noting that q s = K(q)µ and K T (q)K(q) = 1 4 .6

Well-posedness and Stability of Nonlinear Evolution Systems
This section presents results on well-posedness (existence, uniqueness, and continuous dependence on initial conditions) and stability of nonlinear evolution systems to be used in control design and stability analysis of the beam system.

Space notations
Let H be a separable Hilbert space identified with its dual H * by the Riesz isomorphism.Let V be a real reflexible Banach space such that V ⊂ H continuously and densely, and V * be the dual of V. From the definitions of H and V, we have that the embedding V ⊂ H ≡ H * ⊂ V * is continuous and dense.We denote by ∥.∥ H , ∥.∥ V , and ∥.∥ V * the norms in H, V, and V * , respectively; by ⟨ ., .
the duality product between V and V * ; and by ⟨ ., .⟩ H the inner product in H.The duality product between V and V * has the following property [35]:

Evolution systems
Let us consider the nonlinear evolution system on the space H: where X is assumed to be in H for almost every (a.e.) t ∈ [t 0 , ∞) and The following definition is a deterministic version of the stochastic one in [36].
for each t ∈ [t 0 , T ].If T is replaced by ∞, then X(t), t ≥ t 0 , is said to be a global variational solution of (26).
The following definition is an extended version of the one in finite dimensional space in [37] to infinite dimensional space.Definition 4.2.Let α be a class K ∞ -function.The variational solution of ( 26) is said to be , where c is a positive constant depending on the initial condition; 3. globally practically K ∞ -exponentially stable if it is globally stable and , where c is a positive constant independent of the initial condition, and c 0 is positive constant.
If α(∥X(t 0 )∥ H ) ≡ ∥X(t 0 )∥ H , then "K ∞ " is dropped in the above definition.Moreover, if c > 0 depends on the initial conditions, then "exponentially" is replaced by asymptotically.

Well-posedness and stability of evolution systems
We assume that F : H × [t 0 , ∞) → V * is measurable and satisfies the following continuity and local monotonicity and growth conditions.Assumption 4.1. 1

3) [Local growth]
There exists a constant δ such that: a.e.(X, t) ∈ V × [t 0 , ∞), and that the generator LU := dU dt given by with U t (X, t) and U X (X, t) being the (Fréchet) derivatives of U(X, t) with respect to t and X, respectively.
1) [well-posedness] If the generator LU(X, t) satisfies where c is a nonnegative constant.Then the system ( 26) is globally well-posed in terms of the variational solution for each X 0 ∈ H.
2) [stability] If the generator LU(X, t) satisfies where c 3 is a positive constant.If c 0 = 0, ϱ 1 = 0, and ϱ 2 = 0, the equilibrium X ≡ 0 is globally K ∞ -exponentially stable.If any of c 0 , ϱ 1 , and ϱ 2 is a positive constant, the equilibrium H with c 1 and c 2 being positive constants, then "K ∞ " is dropped in the above statement.Moreover, if c 3 > 0 depends on the initial conditions, then "exponentially" is replaced by asymptotically.
Proof.See [38].The above theorem covers most common types of stability of evolution systems.For example, dX(t) dt = −X(t) is globally exponentially stable while dX(t) dt = −X 3 (t) and are globally asymptotically stable with a note that is also locally exponentially stable.

Control design
To design the boundary control vectors ϕ iB , i = 1, 2, we consider the following Lyapunov functional candidate: where the functionals U 0 , U 1 , and U 2 are chosen as follows: where γ, γ 1 , γ 2 , k 2B are positive constants and K 1B with λ m (K 1B ) ≥ 1 is a positive definite matrix to be chosen later, and The constant 0 ≤ ϱ ≤ 1 (to be specified later) is included in U 1 to handle both large and small bending stiffness relatively to the shear stiffness.
Remark 5.1.The choice of the Lyapunov function candidate U in (37) with U 0 , U 1 , and U 2 being given in (38) is elaborated as follows.The function U 0 is the sum of kinetic and potential energies of the beam, where the potential energy is referred to as the energy due to elastic deformation.The function U 1 is motivated by the backstepping method [40], for example with the first equation of (3) and (20) it can be shown that dr 1  dt = γm 0 ϑs (with a note that Dr = R 1 (q)ϑ + R 1 (q)Dr 0 , see ( 3)) is globally exponentially stable at the origin by an appropriate boundary control.The function U B puts appropriate weights on translational displacements and velocities of the beam at the boundaries.The translational and rotational displacements at the left-end are not included in U B while fairly complicated motions at the right-end are included in U B to avoid difficulties in the control design and stability analysis, see also Remark 3.1.Finally, the term γ 1 We now find the bounds of U.
x 4 for all x ∈ R, we can bound U 0 as: where λ m (•) and λ M (•) denote the minimum and maximum eigenvalues of •, respectively.Using Dr = R 1 (q)ϑ + R 1 (q)Dr 0 , (7), and Young's inequality, we can find the bound of |U 1 | as follows where ϱ 0i , i = 1, 2, 3 are positive constants to be determined.The bounds of U 2 can be calculated as Using ( 39), ( 40), (41), and , we can bound U as follows: where E is defined in (19), and the constants c 1 , c 2 , and ϱ 0 are given by The constants γ, ϱ 0i , i = 1, 2, 3, and ϱ are chosen such that where c ⋄ 1 is a strictly positive constant.This is always possible by choosing a small γ for given ϱ 0i , i = 1, 2, 3, and ϱ.Thus, U is a proper functional of E.
We now calculate the infinite generator LU.It is obvious from (37) that where LU 0 , LU 1 , and LU B are detailed in what follows.
Calculation of LU 0 : Differentiating U 0 given in (38) along the solutions of (34), and using integration by parts results in: (46) To further calculate LU 0 , we note the following facts: where we have used equalities 1) and 4) in Lemma 3.1, see (21), and integration by parts to obtain the last two equations.Substituting (47) into (46) results in Calculation of LU 1 : This is the most complicated and difficult task in the control design.Thus, detailed calculation of LU 1 is given.We use the fact that ⟨ ϑ, vs 7) and then differentiate U 1 given in (38) along the solutions of (34) to obtain: where We now calculate LU i j , (i, j) = 1, 2. We can calculate LU 11 as follows: where ϵ 11 is a positive constant to be chosen, and we have used Dr 0 = col(0, 0, 1) and the second equation of (34).Using the third equation of (34) and integration by parts, we can calculate LU 12 as follows: To further calculate LU 12 , we observe the following facts by using ( 12) and ( 13), and Young's inequality: ) where ϵ 12 , ϵ 13 , ε12 , and ε13 are positive constants to be chosen.Substituting (53) into (52) gives
2) The constants (k i j , k N i j ), (i, j) = 1, 2: If the bending stiffness is strictly larger than the shear stiffness multiplied by square of the beam's length, i.e., ).In this case, under no external disturbances, the closed-loop system is globally asymptotically stable at the origin as seen by substituting f i0 = 0, f B0 i0 = 0, f BΓ i0 = 0, and ϱ = 1 into (43 ) and ( 73).This is because the beam is easier to be sheared than to be bent.However, if the condition (75) does not hold, the constant ϱ needs to be a small positive constant.In this case, the closed loop system is globally practically asymptotically stable at the origin even under no external disturbances.
3) The constants (k i jB , k N i jB ), i = 1, 2, 3, j = 1, 2, ): Interpretation of how to ensure these constants strictly positive can be carried out similarly to the constants (k i j , k N i j ), (i, j) = 1, 2.
We now verify all the conditions of Theorem 4.1.The continuity condition in Assumption 4.1 holds due to continuity of F(X, t).By using with the use of the local inner product in LH defined as above and integration by parts similarly to the calculation of LU in Section 5.2, it is readily shown that the local monotonicity condition (28) and local growth condition (29) hold.From Eqs. ( 42) and ( 76) with E defined in Eq. ( 19), the conditions ( 30) and ( 33) hold with , where c 1 and c 2 are defined in Eq. ( 43).Proof of Theorem 5.1 is therefore completed.This section illustrates the effectiveness of the proposed boundary controller on a steel beam with a cross section of an I-shape as shown in Fig. 2 and a

Conclusions
The exact nonlinear model describing nonlinear couple motions of Timoshenko beams under external loads in space was derived and used for boundary control design.The developed boundary control laws stabilize both large and small amplitudes of translational and rotational motions of the beams and their gradients, and fully consider the coupling between all the motions.These control laws ensure that the closed-loop system is globally practically asymptotically stable at the origin.Analysis of both well-posedness and stability was carried out based on a Lyapunov-type theorem, which was developed to study well-posedness and stability for a class of evolution systems.Future work is to consider spinning Timoshenko beams such as spindles and drillers in space.This will require a further development of the control design proposed in this paper to address an angular velocity tracking objective.
Forces and moments acting on a beam element.

Figure 1 :
Figure 1: Beam deformation geometry and loading diagram on a beam element.

rBΓ 1 ∆
BΓ is used in U B instead of γ 1 rBΓ 1 to overcome impossibility in calculating the upper-bound of the generator LU B .

Figure 2 :
Figure 2: Dimensions of the beam's cross section.

Figure 3 :
Figure 3: Simulation results with gradient feedbacks θ 3 (s, t)) are plotted in Sub-figs.3b.A, 3b.B and 3b.C, while the controls ϕ 1B and ϕ 2B are plotted in Sub-fig.3a.D and Sub-fig.3b.D.It is seen that the displacements and rotations oscillate with quite large magnitudes due to the external loads but are bounded due to the gradient feedbacks, Translational displacements and boundary control ϕ 1B (ϕ 11B : black, ϕ 12B : blue, ϕ 13B : red).