Analytical model of one flexible link system with nonlinear kinematics

Abstract

The dynamical model of a flexible Euler–Bernoulli link system subject to high-speed motions is developed. Nonlinear kinematics are considered to take into account the foreshortening effect of the link. A continuous model with boundary conditions is derived using Hamilton's principle and spatially discretized dynamical equations are derived using Lagrange's equation based on the expression of the kinetic and potential energies of the flexible link system. The effect of the links foreshortening on the dynamical response is shown for a sinusoidal torque input at the links hub. The simulation results showed that for small torque amplitudes the difference between linear and nonlinear kinematics is small. However, for large amplitudes, this difference is prominent.

Keywords

Flexible link linear kinematics nonlinear kinematics foreshortening effects

1. Introduction

In manufacturing and space applications, the use of lightweight structures in robot manipulators is motivated by their capacity for high-speed maneuvers, their high payload-to-arm-weight ratio, higher mobility, reduced energy consumption, and lower inertia forces for accurate positioning. It is also advantageous to use thin and flexible links in surgical robotics for minimal invasiveness (Tavakoli and Howe, 2009). To ensure satisfactory performances of such systems, their flexibility should be included in modeling and in control design. This flexibility becomes more significant in cases of larger structures and more stringent performance demands.

In modeling flexible links manipulators, the most widely used methods to develop manipulators dynamics are based on energy principles such as Hamilton's principle (Cetinkunt and Yu, 1991; Junkins and Kim, 1993) and Lagrange's equations (Book, 1984; De Luca and Siciliano, 1991). Energy principle-based models of flexible manipulators are spatially continuous of infinite order. To generate spatially discrete models the assumed-mode method (AMM) or the finite-element method (FEM) are generally used. The accuracy of the dynamical model obtained from the analytical formulation is highly dependent on the adopted mode shapes of links deflections and their number. For high-speed maneuvers, it is also dependent on the foreshortening effects of the links, i.e. the axial displacement due to bending deformation of the links, by considering a linear first-order or a nonlinear second-order kinematics.

Static and dynamic analysis of flexible manipulators have been the subject of many investigations. In particular, different comparisons have been devoted to the study of the effectiveness and accuracy of spatially discretized models. Recently, the dynamics of a flexible-hub geometrically nonlinear beam carrying a tip mass was presented in (Emam, 2010). The hub–beam system is assumed to move in plane and the hub is restrained by a vertical translational movement and a rotational spring. The dynamical responses of the system with one, two, and three modes show the effect of the number of modes on the dynamical response. Saad et al. (2006) presented a detailed comparison between assumed-based models and finite-element ones for a single flexible link system for one to eight flexible modes. This comparison shows that clamped beam shape functions for the AMM, and cubic splines finite elements for the FEM describe the shape functions of flexible links best.

A common way to describe a flexible link is by means of the Euler–Bernoulli equation. This model is valid under the assumptions that: (i) the link is slender with uniform geometric and inertial characteristics, (ii) the link is flexible in the lateral directions and stiff with respect to axial forces and to torsion and bending forces due to gravity, and (iii) nonlinear deformation and friction can be neglected. The Euler–Bernoulli beam with linear kinematics is a linear infinite-dimensional model which takes into account only perpendicular deformation with respect to an unstressed reference configuration. For high-speed maneuvers, however, nonlinear kinematics that take into account foreshortening effects of a flexible link are to be considered (Boyer et al., 2002).

Owing to the assumption of small deformation, the Euler–Bernoulli beam model of a single flexible link system is linear in the link deformation, see Cannon and Schmitz (1984), Cetinkunt and Yu (1991), and Jnifene (2007), to name a few. This assumption implies a first-order approximation of the deformation in the final dynamical model. The energy expressions are, however, nonlinear. To consider a nonlinear model, quadratic terms in the deformation are considered, particularly in the mass matrix M(v²), where v is the link's deformation at a point along its axis (Yung and Chen, 1990; Yim et al., 1992; Zaad and Khorasani, 1996; Geniele et al., 1997; Lee, 2005). Coriolis and centrifugal terms are also added such that $\overset{\cdot}{M} - 2 C (v, \overset{\cdot}{v})$ is an antisymmetric matrix, where C is the coriolis and centrifugal matrix terms. In addition, the effects of the links foreshortening are also quadratic in the deformation and are to be considered when using a nonlinear model.

Al-Bedoor and Hamdan (2001) developed a nonlinear model for a flexible link moving in the horizontal plane. They consider the foreshortening effect by considering a fourth-order kinematics. The resulting model is well suited for simulation. For control purposes, however, particularly in robotics, Euler–Bernoulli beams are often used to describe flexible links where links deformation is considered to be small relative to their length. A second-order kinematics is largely sufficient for robotics applications with high-speed maneuvers.

Cai et al. (2005) developed a nonlinear model for a single flexible link system taking into account foreshortening effects. A second-order kinematics was considered for the large motion at high speed. A spatially discretized model was also developed using Hermite cubic finite elements. However, the frequencies of the resultant model do not converge monotonically from above when the number of elements is increased (Meirovitch and Silverberg, 1983)

The objective of this work is to develop a nonlinear dynamical model of a one flexible link system. The systems movement is not restricted to the horizontal plane, but considered in the vertical one. The horizontal plane dynamic becomes a special case of the vertical one. An analytical continuous dynamical model is developed using Hamilton's principle. A spatially discrete dynamical model is given using Lagrange's equations with the AMM. In contrast to Cai et al. (2005), the resultant frequencies in the AMM model are known to be convergent from above when the number of modes is increased. The rotating beam eigenfunctions are also developed for the linearized system. The developed nonlinear dynamical model is particularly useful for nonlinear control purposes. The deformations are assumed to be small while the speed of the motion is considered to be high. We consider the foreshortening effect of the link by considering second-order nonlinear kinematics. The effect of the foreshortening is shown by simulating the response of the developed dynamics to various sinusoidal inputs.

Section 2 describes the features of the flexible link system under consideration and presents its continuous model. The eigenvalue problem (EVP) and the discretized model are presented in Sections 3 and 4, respectively. Section 5 deals with the analytical aspects of the shape functions. The effect of the link foreshortening is shown by simulation in Section 6. A conclusion is presented in Section 7.

2. The continuous model

Figure 1 shows the one link flexible system. It consists of a motor, a flexible beam, and a payload. The motor has an angular position θ(t), a viscous friction b_m, an inertia I_m, and develops a torque T_m. The beam length is l, its linear density ρ, its rigidity EI_z, and its internal damping coefficient κ_e. The payload has a mass m_p, an inertia I_p, and a center of mass r_p. We assume that the system parameters are known. The gravity acts along the x-axis of frame ( ${\vec{ı}}_{0}$ , ${\vec{j}}_{0}$ ). The link deformation is v(x, t). For simplification, x and t are omitted in the following. The flexible link is modeled as a Euler–Bernoulli beam. We take into account the foreshortening effect of the link by assuming a second-order kinematics. In the following, v(0,t) = v₀, v(l,t) = v_l, $\overset{\cdot}{v} = \partial v (x, t) / \partial t$ , v′ = ∂v(x, t)/∂x, ${\overset{\cdot}{v}}_{l} = \partial v (x, t) / \partial t |_{x = l}$ , and $v'_{l} = \partial v (x, t) / \partial x |_{x = l}$ .

Figure 1.

A flexible rotating beam.

The continuous model consists of one ordinary differential equation (ODE) for the motor's motion, and a partial differential equation (PDE) with four associated boundary conditions (BCs) for the flexible link.

2.1. Kinematics

The position of reference frame ℛ₁ relative to the inertial reference system ℛ₀ is

0 P_{1} = [000]^{T}

. The rotation matrix of ℛ₁ relative to ℛ₀ is

0 R_{1} = [\cos θ - \sin θ 0 \sin θ \cos θ 0001]

Let dm be an element of the link at distance x from the base along the link's neutral axis,

{\vec{ı}}_{1}

. The position of dm relative to ℛ₁ is

^{1} P_{dm} = [x - \underset{O_{2} (v)}{\underset{︸}{1/2 \int_{0}^{x} {v'}^{2} (s, t) d s}} v (x, t) 0]

The

₂(v) term is a second-order term in v. This term is to be neglected if first-order kinematics was considered. The rotation matrix of ℛ_dm relative to ℛ₁ is (Piedbœuf, 1998):

1 R_{dm} = [1 - 1/2 {v'}^{2} - v' 0 v' 1 - 1/2 {v'}^{2} 0001]

Note that ¹R_dm verifies the equation

1 R_{dm} 1 R_{dm}^{T} = I

, where I is the identity matrix.

The position vector of the payload center of mass relative to ℛ₁ is:

^{1} P_{c} = 1 P_{2} + 1 R_{2} 2 P_{c} = [l + r_{p} - 1/2 \int_{0}^{l} {v'}^{2} d x - 1/2 r_{p} {v'}_{l}^{2} v_{l} + r_{p} v'_{l} 0]

The ¹R₂ matrix and ¹P₂ vector are obtained from ¹R_dm and ¹P_dm, respectively, by substituting x by l, and ²P_c = [r_p 0 0]^T. If the payload is concentrated at the link's extremity, then r_p = 0. The rotation matrix of the payload center of mass relative to ℛ₁ is

^{1} R_{c} = 1 R_{2} 2 R_{c} = [1 - 1/2 {v'}_{l}^{2} - v'_{l} 0 v'_{l} 1 - 1/2 {v'}_{l}^{2} 0001]

where ²R_c is the identity matrix.

The angular velocity of ℛ₁ relative to ℛ₀ is

ω_{1} = \overset{\cdot}{θ} {\vec{k}}_{1}

where

{\vec{k}}_{1}

is the motor's axis of motion and is orthonormal to (

{\vec{}}_{1}, {\vec{}}_{1}

) axes.

Using the antisymmetric matrix to represent the velocity of an element dm of the link:

S = 1 {\overset{\cdot}{R}}_{dm} 1 R_{dm}^{- 1} = [0 - \overset{\cdot}{v}' 0 \overset{\cdot}{v}' 00000] = [0 - ω_{z} ω_{y} ω_{z} 0 - ω_{x} - ω_{y} ω_{x} 0]

the angular velocity of ℛ_dm relative to ℛ₁ is then given by

^{1} Ω_{dm} = [ω_{x} ω_{y} ω_{z}] = [00 \overset{\cdot}{v}']

and relative to ℛ₀, it is given by

ω_{dm} = ω_{1} + 0 R_{1} 1 Ω_{dm} = [00 \overset{\cdot}{θ} + \overset{\cdot}{v}']

The linear velocity of the origin of ℛ_dm relative to ℛ₀ is

v_{dm} = 0 R_{1} 1 V_{dm} + ω_{1} \times (0 R_{1} 1 P_{dm})

When expressed in ℛ₁, it becomes

1 v dm = 0 R_{1}^{T} v_{dm} = d 1 P_{dm} d t + ω_{1} \times 1 P_{dm} = [- \overset{\cdot}{θ} v \overset{\cdot}{v} + x \overset{\cdot}{θ} 0] + \underset{O_{2} (v)}{\underset{︸}{[- \int_{0}^{x} v' \overset{\cdot}{v}' d s - 1/2 \overset{\cdot}{θ} \int_{0}^{x} {v'}^{2} d s 0]}}

The payload angular velocity is obtained from ω_dm by replacing x by l:

ω_{c} = [00 \overset{\cdot}{θ} + \overset{\cdot}{v}'_{l}]^{T}

The absolute velocity of the payload, written in ℛ₁, is

^{1} v_{c} = d d t (1 P_{c}) + ω_{1} \times 1 P_{c} = [- \overset{\cdot}{θ} (v_{l} + r_{p} v'_{l}) - \int_{0}^{l} v' \overset{\cdot}{v}' d x - r_{p} v'_{l} \overset{\cdot}{v}'_{l} (l + r_{p}) \overset{\cdot}{θ} + {\overset{\cdot}{v}}_{l} + r_{p} \overset{\cdot}{v}'_{l} - 1/2 \overset{\cdot}{θ} \int_{0}^{l} {v'}^{2} d x - 1/2 r_{p} \overset{\cdot}{θ} {v'}_{l}^{2} 0]

The gravity vector is represented in ℛ₀ by

0 g = [g 00]^{T}

where g is the gravitational acceleration. In ℛ₁, it is given by

^{1} g = 1 R_{0} 0 g = 0 R_{1}^{T} 0 g = [g \cos θ - g \sin θ 0]

2.2. Kinetic energy

The kinetic energy of the given system is given by

T = T_{B} + \int_{0}^{l} T_{dm} + T_{p}

(1)

where T_B is the kinetic energy of the base,

T_{B} = 1/2 I_{m} {\overset{\cdot}{θ}}^{2}

T_dm is the kinetic energy of an element of the link,

T_{dm} = 1/2 ρ v_{dm}^{T} v_{dm} d x = 1/2 ρ 1 v_{dm}^{T} 1 v_{dm} d x

and T_p is the kinetic energy of the payload,

T_{p} = 1/2 I_{p} ω_{c}^{2} + 1/2 m_{p} v_{c}^{T} v_{c} = 1/2 I_{p} ω_{c}^{2} + 1/2 m_{p} 1 v_{c}^{T} 1 v_{c}

Linearizing the kinetic energy to the second order in v gives

T = 1/2 (I_{m} + 1/3 ρ l^{3} + I_{p} + m_{p} (l + r_{p})^{2}) {\overset{\cdot}{θ}}^{2}

+ \overset{\cdot}{θ} (ρ \int_{0}^{l} x \overset{\cdot}{v} d x + m_{p} (l + r_{p}) {\overset{\cdot}{v}}_{l} + (I_{p} + m_{p} r_{p} (l + r_{p})) \overset{\cdot}{v}'_{l})

+ 1/2 ρ \int_{0}^{l} {\overset{\cdot}{v}}^{2} d x + 1/2 m_{p} {\overset{\cdot}{v}}_{l}^{2} + 1/2 (I_{p} + m_{p} r_{p}^{2}) {\overset{\cdot}{v'}}^{2}_{l} + m_{p} r_{p} {\overset{\cdot}{v}}_{l} \overset{\cdot}{v}'_{l}

+ 1/2 {\overset{\cdot}{θ}}^{2} (ρ \int_{0}^{l} v^{2} d x + m_{p} v_{l}^{2} + 2 m_{p} r_{p} v_{l} v'_{l})

- 1/2 {\overset{\cdot}{θ}}^{2} (ρ \int_{0}^{l} x \int_{0}^{x} {v'}^{2} d s d x + m_{p} (l + r_{p}) \int_{0}^{l} {v'}^{2} d x + m_{p} r_{p} l {v'}_{l}^{2})

(2)

2.3. Potential energy

The potential energy of the given system is given by

V = \int_{0}^{l} V_{dm} + V_{p}

(3)

where V_dm is the potential energy of the link's element dm,

V_{dm} = 1/2 {EI}_{z} {v ″}^{2} d x - ρ 1 g^{T} 1 P_{dm} d x = 1/2 {EI}_{z} {v ″}^{2} d x - ρ g x \cos θ d x + ρ g v \sin θ d x + 1/2 ρ g \cos θ \int_{0}^{x} {v'}^{2} d s d x

and V_p is the potential energy of the payload:

V_{p} = - m_{p} 1 g^{T} 1 P_{c} = - m_{p} g (l + r_{p}) \cos θ + m_{p} g (v_{l} + r_{p} v'_{l}) \sin θ + 1/2 m_{p} g \cos θ (\int_{0}^{l} {v'}^{2} d x + r_{p} {v'}_{l}^{2})

2.4. Rayleigh dissipation function

Friction is due to the motor's viscous friction and the link's internal damping. The Rayleigh dissipation function is then given by

R = 1/2 b_{m} {\overset{\cdot}{θ}}^{2} + 1/2 κ_{e} {EI}_{z} \int_{0}^{l} \overset{\cdot}{v}'' 2 d x

(4)

2.5. Dynamics

To develop the dynamics of the given system, we apply Hamilton's principle using the developed kinetic and potential energies. The variational of these expressions are simplified using integration by parts. First we ignore damping forces. The only nonconservative applied force is the motor torque T_m. Two dynamical equations are then obtained. The first is associated with the motor's angle θ, and the second is associated with the deformation of the link v. Moreover, four BCs are associated with the dynamical equation of the deformation. These conditions describe the way in which the arm is attached to the base (geometrical BCs) and to the payload (natural BCs).

To complete these dynamics, we then add the viscous friction at the base and the link's internal damping. To take into account the friction at the base, the motor torque T_m is replaced by $T_{m} - b_{m} \overset{\cdot}{θ}$ . The internal damping of the link is considered using the standard linear model of Voigt–Kelvin (Nashif et al., 1985). This model gives the following relation between the constraint (σ) and the strain (ε):

σ = E (ε + k_{e} d ε d t)

In the deformation equation and its associated BCs, Young's modulus E is replaced by

E_{k} = E (1 + k_{e} D)

where D is the partial derivatives operator, i.e. Dv = ∂v/∂t. Taking motor and internal damping into account, the dynamical equations are then given as follows.

Dynamical equation of the motor's angle θ

T_{m} = (I_{m} + 1/3 ρ l^{3} + I_{p} + m_{p} (l + r_{p})^{2}) \overset{··}{θ} + b_{m} \overset{\cdot}{θ} + \int_{0}^{l} ρ x \overset{··}{v} d x + m_{p} (l + r_{p}) {\overset{··}{v}}_{l} + (I_{p} + m_{p} r_{p} (l + r_{p})) \overset{··}{v}'_{l} + (1/2 ρ g l^{2} + m_{p} g (l + r_{p})) \sin θ

(5)

+ (\int_{0}^{l} ρ g v d x + m_{p} g (v_{l} + r_{p} v'_{l})) \cos θ + \int_{0}^{l} \partial \partial t (ρ \overset{\cdot}{θ} v^{2}) d x - 1/2 \int_{0}^{l} ρ (l^{2} - x^{2}) \partial \partial t (\overset{\cdot}{θ} {v'}^{2}) d x + m_{p} \partial \partial t (\overset{\cdot}{θ} v_{l}^{2}) - m_{p} (l + r_{p}) \int_{0}^{l} \partial \partial t (\overset{\cdot}{θ} {v'}^{2}) d x + 2 m_{p} r_{p} \partial \partial t (\overset{\cdot}{θ} v_{l} v'_{l}) - m_{p} l r_{p} \partial \partial t (\overset{\cdot}{θ} {v'}_{l}^{2})

Dynamical equation of the link's deformation v

0 = - ρ (\overset{··}{v} + x \overset{··}{θ}) - {EI}_{z} v ″^{″} - k_{e} {EI}_{z} \overset{\cdot}{v} ″^{″} + g \cos θ (ρ (- v' + (l - x) v ″) + m_{p} v ″) - ρ g \sin θ + {\overset{\cdot}{θ}}^{2} (ρ v + m_{p} (l + r_{p}) v ″ + 1/2 ρ (- 2 x v' + (l^{2} - x^{2}) v ″))

(6)

Boundary conditions. The BCs are compatible with the geometric configuration of the given system. The link is clamped at the joint, i.e. v(0,t) = v′(0,t) = 0. The geometrical BCs are then given by

at x = 0 : v_{0} = 0 and v'_{0} = 0

(7)

at x = l, the natural BCs are

E I_{z} v_{l} ″^{'} = m_{p} (l \overset{··}{θ} + {\overset{··}{v}}_{l}) + m_{p} r_{p} (\overset{··}{θ} + \overset{··}{v}'_{l}) - m_{p} {\overset{\cdot}{θ}}^{2} (v_{l} + r_{p} v'_{l}) + m_{p} (l + r_{p}) {\overset{\cdot}{θ}}^{2} v'_{l} + m_{p} g v'_{l} \cos θ + m_{p} g \sin θ

(8)

- E I_{z} v ″_{l} = (I_{p} + m_{p} r_{p}^{2}) (\overset{··}{θ} + \overset{··}{v}'_{l}) + m_{p} r_{p} (l \overset{··}{θ} + {\overset{··}{v}}_{l}) - m_{p} r_{p} (v_{l} + r_{p} v'_{l}) {\overset{\cdot}{θ}}^{2} + m_{p} r_{p} (l + r_{p}) v'_{l} {\overset{\cdot}{θ}}^{2} + m_{p} r_{p} g \cos θ v'_{l} + m_{p} r_{p} g \sin θ

(9)

3. Eigenvalue problem

The EVP consists of solving for the deformation v the dynamics of the system represented by the ODE (5) and the PDE with the associated BCs (6)–(9). Let us consider the case of a homogeneous problem where the motor's torque is null, i.e. T_m = 0. We consider a solution for v separable in space and time. The deformation is then given by

v (x, t) = φ (x) q_{f} (t)

(10)

where φ(x) represents the link's configuration and is only dependent on the spatial variable x, and q_f(t) indicates the nature of the movement carried out by the configuration and depends only on time t. By substituting (10) in the dynamical equations and the associated BCs, the PDE associated with v is transformed into an ODE in q_f(t). To have a stable solution, a harmonic movement is selected for q_f(t) so that

{\overset{··}{q}}_{f} (t) = - ω^{2} q_{f} (t)

where ω represents the natural frequency of the system.

The EVP thus amounts finding one ω and a nontrivial solution φ(x) that verifies homogeneous discretized equations together with the associated BCs. The corresponding ω are the characteristic values or the eigenvalues and the φ(x) are the eigenfunctions. EVP generally generates the solution of a characteristic equation having an infinite countable number of solutions w_r. For each eigenvalue w_r corresponds an eigenfunction φ_r(x). In the general case where dynamics is represented by nonlinear equations and with nonuniform parameters, a solution of the EVP is practically not possible. Approximate methods are used to solve this kind of problem (Mirovitch, 1967).

4. Spatial discretization

The order of the solution is infinite. In order to analyze and control this system, a spatially discrete model of finite order is suitable. A finite number of terms is then retained in the discretization of the deformation which is rewritten in the form:

v (x, t) = \sum_{i = 1}^{ν} φ_{i} (x) q_{f_{i}} (t) = φ^{T} (x) q_{f} (t)

(11)

where ν is the number of retained modes, φ = [φ₁, φ₂,…, φ_ν]^T and q_f = [q_{f
₁}, q_{f
₂},…, q_{f
_ν}]^T. The discretization of the kinetic energy (2) gives

T = 1/2 I_{t} {\overset{\cdot}{θ}}^{2} + \overset{\cdot}{θ} M_{rf}^{T} {\overset{\cdot}{q}}_{f} + 1/2 {\overset{\cdot}{q}}_{f}^{T} M_{ff} {\overset{\cdot}{q}}_{f} + 1/2 {\overset{\cdot}{θ}}^{2} q_{f}^{T} C_{rr} q_{f}

(12)

where I_t is the total inertia at the base given by

I_{t} = I_{m} + 1/3 ρ l^{3} + I_{p} + m_{p} (l + r_{p})^{2}

M_{rf} = \int_{0}^{l} ρ x φ d x + m_{p} (l + r_{p}) φ_{l} + (I_{p} + m_{p} r_{p} (l + r_{p})) φ'_{l} M_{ff} = ρ \int_{0}^{l} φ φ^{T} d x + m_{p} φ_{l} φ_{l}^{T} + (I_{p} + m_{p} r_{p}^{2}) φ'_{l} {φ'}_{l}^{T} + m_{p} r_{p} (φ'_{l} φ_{l}^{T} + φ_{l} {φ'}_{l}^{T})

C_{rr} = ρ \int_{0}^{l} φ φ^{T} d x + m_{p} φ_{l} φ_{l}^{T} + m_{p} r_{p} (φ_{l} {φ'}_{l}^{T} + φ'_{l} φ_{l}^{T}) - ρ \int_{0}^{l} x \int_{0}^{x} φ' {φ'}^{T} d s d x - m_{p} (l + r_{p}) \int_{0}^{l} φ' {φ'}^{T} d x - m_{p} r_{p} l φ'_{l} {φ'}_{l}^{T}

In a matrix form,

T = 1/2 [\overset{\cdot}{θ} {\overset{\cdot}{q_{f}}}^{T}] [I_{t} + q_{f}^{T} C_{rr} q_{f} M_{rf}^{T} M_{rf} M_{ff}] [\overset{\cdot}{θ} \overset{\cdot}{q_{f}}] = 1/2 {\overset{\cdot}{q}}^{T} M (q) \overset{\cdot}{q}

(13)

In the mass matrix M(q), the second-order element in v, i.e. $q_{f}^{T} C_{rr} q_{f}$ , will be neglected because we assumed that the deformation is small.

The discretization of the potential energy (3) gives

V = 1/2 q_{f}^{T} K_{ff} q_{f} - G_{rr} \cos θ + \sin θ G_{rf}^{T} q_{f} + 1/2 \cos θ q_{f}^{T} G_{ff} q_{f}

(14)

where

K_{ff} = {EI}_{z} \int_{0}^{l} φ ″ {φ ″}^{T} d x G_{rr} = 1/2 ρ g l^{2} + m_{p} g (l + r_{p}) G_{rf} = ρ g \int_{0}^{l} φ d x + m_{p} g (φ_{l} + r_{p} φ'_{l}) G_{ff} = ρ g \int_{0}^{l} (l - x) φ' {φ'}^{T} d x + m_{p} g \int_{0}^{l} φ' {φ'}^{T} d x + m_{p} g r_{p} φ'_{l} {φ'}_{l}^{T}

The discretization of the Rayleigh dissipation function (4) gives

R = 1/2 b_{m} \overset{\cdot}{θ} + 1/2 {\overset{\cdot}{q}}_{f}^{T} (κ_{e} \int_{0}^{l} {EI}_{z} φ ″ {φ ″}^{T} d x) {\overset{\cdot}{q}}_{f} = 1/2 [\overset{\cdot}{θ} {\overset{\cdot}{q_{f}}}^{T}] [b_{m} 00 B_{ff}] [\overset{\cdot}{θ} \overset{\cdot}{q_{f}}] = 1/2 {\overset{\cdot}{q}}^{T} B \overset{\cdot}{q}

(15)

where B_ff = κ_e K_ff.

We then introduce these expressions into the Lagrangian ℒ = T − V and apply Lagrange equations:

d d t (\partial L \partial {\overset{\cdot}{q}}_{j}) - \partial L \partial q_{j} + \partial R \partial {\overset{\cdot}{q}}_{j} = Q_{j}, j = 1, 2, \dots, n

(16)

where q_j are the generalized coordinates, Q_j are the generalized forces, j = 1, 2,…, n and n = ν + 1 is the total number of generalized coordinates. In Lagrange equations, q_f(t) is the generalized coordinate vector associated with the flexibility. Let q_r be the generalized coordinate associated with the base movement (rigid coordinate), i.e. q_r(t) = θ(t). The generalized coordinate vector is then given by

q (t) = [q_{r} (t), q_{f}^{T} (t)]^{T}

The spatially discrete dynamical model is then given by

T_{m} = I_{t} {\overset{··}{q}}_{r} + {\overset{··}{q}}_{r} q_{f}^{T} C_{rr} q_{f} + M_{rf}^{T} {\overset{··}{q}}_{f} + b_{m} {\overset{\cdot}{q}}_{r} + G_{rr} \sin q_{r} + \cos q_{r} G_{rf} q_{f} + 2 \overset{\cdot}{q_{r}} q_{f}^{T} C_{rr} {\overset{\cdot}{q}}_{f}}

(17)

0 = M_{rf} {\overset{··}{q}}_{r} + M_{ff} {\overset{··}{q}}_{f} + B_{ff} {\overset{\cdot}{q}}_{f} - {\overset{\cdot}{q}}_{r}^{2} C_{rr} q_{f} + K_{ff} q_{f} + \cos q_{r} G_{ff} q_{f} + \sin q_{r} G_{rf}}

(18)

Written in a matrix form, Equations (17)–(18) become

[I_{t} + q_{f}^{T} C_{rr} q_{f} M_{rf}^{T} M_{rf} M_{ff}] [{\overset{··}{q}}_{r} {\overset{··}{q}}_{f}] + [2 {\overset{\cdot}{q}}_{r} q_{f}^{T} C_{rr} {\overset{\cdot}{q}}_{f} - {\overset{\cdot}{q}}_{r}^{2} C_{rr} q_{f}] + [b_{m} 00 B_{ff}] [{\overset{\cdot}{q}}_{r} {\overset{\cdot}{q}}_{f}] + [000 K_{ff}] [q_{r} q_{f}] + [G_{rr} G_{rf}^{T} G_{rf} G_{ff}] [\sin q_{r} q_{f} \cos q_{r}] = [10] T_{m}

(19)

or also in the compact form

M (q) \overset{··}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + B \overset{\cdot}{q} + K q + G (q) = L T_{m}

(20)

where M, B, and K are the mass, damping, and rigidity matrices, respectively, and

C (q, \overset{\cdot}{q}) \overset{\cdot}{q}

, and G(q) are the Coriolis and centrifugal, and gravity force vectors. Here M, B and K_ff are positive definite symmetric matrices.

Remark 1.

The mass matrix M and the Coriolis and centrifugal force vector $C (q, \overset{\cdot}{q}) \overset{\cdot}{q}$ verify the following:

\overset{\cdot}{M} (q) - 2 C (q, \overset{\cdot}{q}) = S

(21)

where S is an antisymmetric matrix.

Let

C (q, \overset{\cdot}{q}) = (q_{f}^{T} C_{rr} {\overset{\cdot}{q}}_{f} {\overset{\cdot}{q}}_{r} q_{f}^{T} C_{rr} - {\overset{\cdot}{q}}_{r} C_{rr} q_{f} 0)

(22)

Then,

x^{T} (\overset{\cdot}{M} (q) - 2 C (q, \overset{\cdot}{q})) x = 0, \forall x \in ℝ^{(ν + 1)}

where ν is the flexible modes number, and ℝ is the set of real numbers.

By neglecting the second-order elements (relative to v) in (20), i.e. in the mass matrix and the gravity element G_ff, equation (20) becomes

M \overset{··}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + B \overset{\cdot}{q} + K q + G (q) = L T_{m}

(23)

Remark 2.

Nonlinear models are considered in the literature by considering the nonlinear element in the mass matrix, which is a second-order element in v. However, as shown in the dynamical equations (20), Coriolis nonlinear terms are also to be considered when a second-order term in v is to be considered. Actually, the amplitude of the nonlinear Coriolis elements is greater than the nonlinear mass element. The latter is small due to the hypothesis of small deformation. However, the Coriolis nonlinear elements are not eliminated by this hypothesis, they are the result of the multiplications of v and $\overset{\cdot}{v}$ . v is small, but $\overset{\cdot}{v}$ and $\overset{\cdot}{θ}$ are not assumed to be small.

5. Description of admissible functions

The analytical solution of the eigenfunctions of the system shown in Figure 1 is given by (Saad et al., 2006):

φ_{i} (x) = A_{i} (μ_{3} λ_{i}^{3} (\cos β_{i} x - \cosh β_{i} x) + (1 + c_{i}) \sin β_{i} x + (1 - c_{i}) \sinh β_{i} x - 2 β_{i} x)

(24)

Here, $β_{i} = λ_{i} l$ and λ_i is the solution of the following characteristic equation :

(μ_{1} μ_{2} μ_{3} + μ_{2} μ_{3} μ_{5} - μ_{3} μ_{4}^{2}) λ^{7} (1 - C Ch) - μ_{3} (μ_{1} + μ_{5}) λ^{6} (S Ch + C Sh) - 2 μ_{3} μ_{4} λ^{5} S Sh + (μ_{2} μ_{5} + μ_{1} μ_{2} + μ_{2} μ_{3} - μ_{4}^{2}) λ^{4} (C Sh - S Ch) + μ_{3} λ^{3} (1 + C Ch) + 2 (μ_{1} + μ_{5}) λ^{3} C Ch + 2 μ_{4} λ^{2} (C Sh + S Ch) + 2 μ_{2} λ S Sh + S Ch - C Sh = 0

(25)

where

μ_{1} = I_{p} ρ l^{3}, μ_{2} = m_{p} ρ l, μ_{3} = I_{m} ρ l^{3}, μ_{4} = m_{p} r_{p} ρ l^{2}, μ_{5} = m_{p} r_{p}^{2} ρ l^{3},

C = \cos λ, S = \sin λ, Ch = \cosh λ, Sh = \sinh λ

The admissible shape functions are generally the eigenfunctions of a simpler but related problem. The eigenfunctions of a rotating beam with a payload concentrated at its end (bp) are obtained from (24) and (25) by replacing the payload center of mass r_p by zero. The eigenfunctions of a clamped beam with a concentrated payload (cp) are deduced from the bp eigenfunctions by taking the motor inertia I_m as infinity. The eigenfunctions of a clamped–free beam (cf) are deduced from the cp ones by taking the payload inertia I_p and mass m_p as zero. The first four eigenfunctions of a rotating beam are shown in Figure 2.

Figure 2.

First four eigenfunctions of a rotating beam.

6. Simulations

To show the foreshortening effects on the system's response, a sinusoidal torque input is applied to the system shown in Figure 1. For the sake of comparison with published results, the system is considered moving in the horizontal plane, i.e. gravity is eliminated (g = 0). The system's parameters are (Emam, 2010): the hub rotary inertia is J = 0.30 kg m², the beam length is l = 1.8 m, the cross-sectional area is A = 2.5 × 10⁻⁴ m², the area moment of inertia is I = 1.3021 × 10⁻¹⁰ m⁴, the mass density is ρ = 2.766 × 10³ kg/m³, the modulus of elasticity is E = 6.90 × 10¹⁰ N/m², and the tip mass is m_p = 0.085 kg. As in (Emam, 2010), the selected input torque is

τ = A_{0} \sin 2 π T t, 0 \leq t \leq T and τ = 0 for t > T

where 1/T is its frequency and A₀ is its amplitude. To investigate the foreshortening effect on the tip deflection response, the input torque amplitude A₀ is varied from 1 to 6 N m, for a fixed frequency, T = 2 s. Investigating the responses, we note that for small amplitude the effect of the foreshortening is almost not visible. However, for large amplitude, the foreshortening effect is prominent. Figures 3 –5 compare the responses of the tip deformation with two flexible modes. The difference between linear and nonlinear kinematics responses is 1.63% for A₀ = 2 N m, 6.09% for A₀ = 4 N m, and 12.40% for A₀ = 6 N m, respectively.

Figure 3.

Tip deflection with linear and nonlinear kinematics, A₀ = 2 N m.

Figure 4.

Tip deflection with linear and nonlinear kinematics, A₀ = 4 N m.

Figure 5.

Tip deflection with linear and nonlinear kinematics, A₀ = 6 N m.

With linear kinematics, the tip deformation is underestimated. These results are similar quantitatively and qualitatively to those shown in Emam (2010). For a one flexible link, the results show that the foreshortening effect is similar to the effect of a nonlinear dynamics when the hub is restrained by a vertical translational and a rotational spring. The number of the flexible modes has little effect on the results. With one flexible mode and A₀ = 6 N m, the tip deformation is 12.13% lower with linear kinematics than with a nonlinear kinematics, that is comparable to the 12.40% when two modes where considered. For additional modes, the difference between the responses is insignificant. This shows that the lower modes are dominant.

In addition, to investigate the foreshortening effect on the dynamics when the speed is increasing, a constant torque of 4 N m is applied at the links base. Figures 6 and 7 show the speed and the tip deformation, respectively, for linear and nonlinear kinematics. The results show that with linear kinematics the tip deformation is overestimated. The nonlinear model shows that, for high speed, the deformations are smaller and hence the flexible link is stiffer.

Figure 6.

Motion speed, A₀ = 4 N m.

Figure 7.

Tip deflection, A₀ = 4 N m.

7. Conclusion

In this work, the mathematical model of a flexible link system using Lagrange equations has been presented by taking into account foreshortening effect of the link. This effect was considered using a nonlinear kinematics, that is the elongation of the link along its longitudinal axis. We also showed the effect of the foreshortening on the dynamical response for a sinusoidal torque input at the links hub. The simulations results showed that for small amplitudes the difference between linear and nonlinear kinematics are small. However, for large amplitudes, this difference is more significant and cannot be ignored while controlling flexible links systems.

Footnotes

Funding

This work was supported in part by the NSERC, Canada (grant number 341901-07).

References

AL-Bedoor

Hamdan

(2001) Geometrically non-linear dynamic model of a rotating flexible arm. Journal of Sound and Vibration 240: 59–72.

Book

(1984) Recursive Lagragian dynamics of flexible manipulator arms. The International Journal of Robotics Research 3(3): 87–101.

Boyer

Glandais

Khalil

(2002) Flexible multibody dynamics based on a non-linear Euler–Bernoulli kinematics. International Journal for Numerical Methods in Engineering 54: 27–59.

Cai

Hong

Yang

(2005) Dynamic analysis of a flexible hub–beam system with tip mass. Mechanics Research Communications 32: 173–190.

De Luca

Siciliano

(1991) Closed-form dynamic model of planar multi-link lightweight robots. IEEE Transactions on Systems, Man, and Cybernetics 21: 826–839.

Cannon

Schmitz

(1984) Initial experiments of the end-point control of a flexible one-link robot. The International Journal of Robotics Research 3(3): 62–75.

Cetinkunt

(1991) Closed-loop behavior of a feedback-controlled flexible arm: a comparative study. The International Journal of Robotics Research 10: 263–275.

Emam

(2010) Dynamics of a flexible-hub geometrically nonlinear beam with a tip mass. Journal of Vibration and Control 16: 1989–2000.

Geniele

Patel

Khorasani

(1997) End-point control of a flexible-link manipulator: theory and experiments. IEEE Transactions on Control Systems Technology 5: 556–570.

10.

Jnifene

(2007) Active vibration control of flexible structures using delayed position feedback. System and Control Letters 56: 215–222.

11.

Junkins

Kim

(1993) Introduction to Dynamics and Control of Flexible Structure, Washington, DC: AIAA Education Series Publishers.

12.

Lee

(2005) New dynamic modeling of flexible link robots. Journal of Dynamic Systems Measurement and Control 127: 307–309.

13.

Meirovitch

(1967) Analytical Methods in Vibrations, New York: MacMillan.

14.

Meirovitch

Silverberg

(1983) Two bracketing theorems characterizing the eigensolution for theh-version of the finite element method. International Journal for Numerical Methods in Engineering 19: 1691–1704.

15.

Nashif

Jones

DIG

Henderson

(1985) Vibration Damping, New York: John Wiley & Sons.

16.

Piedbœuf

(1998) Recursive modeling of serial flexible manipulators. Journal of the Astronautical Sciences 47: 1–24.

17.

Saad

Piedbœuf

Akhrif

Saydy

(2006) Modal analysis of assumed-mode models of a flexible slewing beam. International Journal of Modelling, Identifcation and Control 1: 325–337.

18.

Tavakoli

Howe

(2009) Haptic effects of surgical teleoperator flexibility. The International Journal of Robotics Research 28: 1289–1302.

19.

Yim

Zuang

Singh

(1992) Experimental dual-mode control of a flexible robotic arm. Robotica 10: 135–145.

20.

Yung

Chen

(1990) Sliding-mode controller design of a single-link flexible manipulator under gravity. International Journal of Control 52: 101–117.

21.

Zaad

Khorasani

(1996) Control of non minimum phase singularly perturbed systems with application to flexible-link manipulators. International Journal of Control 63: 679–701.